diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzhpwp" "b/data_all_eng_slimpj/shuffled/split2/finalzzhpwp" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzhpwp" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction and Motivation}\n\nThe discovery of neutrino mass and lepton mixing provides key evidence for \nnew physics beyond the Standard Model (SM) \\cite{King:2015aea,King:2003jb,Altarelli:2010gt,King:2013eh,King:2014nza}.\nThe seesaw mechanism \\cite{Minkowski:1977sc,GellMann:1980vs,Yanagida:1979as,Mohapatra:1979ia,Schechter:1980gr} is an attractive possibilty to account \nfor the origin of neutrino mass and lepton mixing in terms of right-handed neutrinos with large Majorana masses.\n$SO(10)$ Grand Unified Theories (GUTs) \\cite{Fritzsch:1974nn} predict such \nright-handed neutrinos which appear along with SM matter fields in a single $\\mathbf{16}$ multiplet. When the $SO(10)$ gauge group is broken to that of the SM, neutrino mass is an inevitable \nconsequence. In order to satisfy the constraint of gauge coupling unification, we shall here\nassume low energy supersymmetry (SUSY) \\cite{Martin:1997ns}. However to also account for gravity, one needs to \ngo beyond gauge theories, and here we shall focus on an $M$ theory version of string theory \\cite{Witten:1995ex,Horava:1995qa}.\n\nRecently we showed how $SO(10)$ SUSY GUTs could emerge from \n$M$ Theory compactified on a $G_2$-manifold \\cite{Acharya:2015oea}.\nIn this framework, discrete symmetry and Wilson lines \\cite{Witten:2001bf} were used to prevent proton decay while maintaining gauge unification. In contrast to the $SU(5)$ version \\cite{Acharya:2008zi,Acharya:2011te}, the Wilson line symmetry breaking mechanism \nin $SO(10)$ requires additional matter at the TeV scale, with the quantum numbers of an extra \n${\\bf 16}_X$ plus $\\overline{\\bf 16}_X$ \\cite{Acharya:2015oea}. In addition,\nthere were a number of unresolved issues in this approach, notably the mechanism for breaking the extra gauged $U(1)_X$ which accompanies the SM gauge group after the Wilson line symmetry breaking mechanism in $SO(10)$. This gauge group is the usual one in the maximal $SO(10)$ subgroup $SU(5)\\times U(1)_X$ \\footnote{The $U(1)_X$ is also commonly called $U(1)_{\\chi}$ in the literature.},\nwhere $SU(5)$ embeds the SM gauge group.\nThe key point is that, since Abelian Wilson line symmetry breaking preserves the rank of the gauge group,\nthe $U(1)_X$ gauge group needs to be broken by some other mechanism in the low energy effective field theory. Since right-handed\nMajorana neutrino masses can only arise once the $U(1)_X$ is broken, the origin of neutrino mass\nis therefore linked to this symmetry breaking.\n\nIn this paper we address the problem of $U(1)_X$ breaking and neutrino masses arising from\nthe $SO(10)$ $M$ theory, following the construction in \\cite{Acharya:2015oea}, although our approach to solving these problems may be more general than the specific example studied.\nTo break the $U(1)_X$ gauge symmetry, we employ a (generalised) Kolda-Martin mechanism \n\\cite{Kolda:1995iw},\nwhere higher order operators can break the symmetry, inducing vacuum expectation values (VEVs)\nin the scalar right-handed neutrino components of both the \nmatter ${\\bf 16}$ and the extra ${\\bf 16}_X$, as well as their\nconjugate partners. The subsequent induced R-parity violation \\cite{Barbier:2004ez} provides additional sources\nof neutrino mass, in addition to that arising from the seesaw mechanism\n\\cite{Minkowski:1977sc,GellMann:1980vs,Yanagida:1979as,Mohapatra:1979ia,Schechter:1980gr}. The resulting $11\\times 11$ neutrino mass\nmatrix is analysed for one neutrino family (nominally the third family) and it is shown how a\nphenomenologically acceptable neutrino mass can emerge. We defer any discussion of flavour mixing \nto a possible future study of flavour from $M$ theory. Here we only show that symmetry breaking and\nviable neutrino masses can arise within the framework of $M$ theory $SO(10)$, which is a highly non-trivial\nresult, given the constrained nature of $M$ theory constructions.\n\n\nIt is worth remarking that there are other alternative ways that have been proposed to study neutrino masses\nin string theory, which are complementary to the approach followed here.\nFor example, it is possible to obtain large Majorana mass terms from instanton effects \\cite{Acharya:2006ia,Blumenhagen:2006xt,Ibanez:2006da,Cvetic:2007ku,Buchmuller:2007zd}, large volume compactification \\cite{Conlon:2007zza}, or orbifold compactfications of the heterotic string \\cite{Buchmuller:2007zd}. However the origin of Majorana mass terms in $SO(10)$ has been non-trivial to realise from the string theory point of view. In GUTs all matter fields are unified in $\\mathbf{16}$ multiplets whereas Higgs fields and triplet scalars are unified in $\\mathbf{10}$. Since string theory does not predict \nlight particles in \nrepresentations larger than the adjoint, the traditional renormalisable terms involving $\\mathbf{126}, \\overline{\\mathbf{126}}, \\mathbf{210}$, e.g., $W \\sim \\mathbf{126}\\:\\mathbf{16}\\:\\mathbf{16}$, are not possible. The dominant higher order operators are quartic ones such as \n$W = \\overline{\\mathbf{16}}\\:\\overline{\\mathbf{16}}\\:\\mathbf{16}\\:\\mathbf{16}$. \nAssuming that the supersymmetric partner of the right handed neutrino singlet gets a VEV, the Majorana mass is given by $M \\sim \\frac{\\langle \\widetilde{N} \\rangle^2}{M_{PL}}$. However, the required values of neutrino mass imply $M > 10^{14}$ GeV, which gives $\\langle \\widetilde{N}\\rangle \\sim \\sqrt{M m_{Pl}} \\sim 10^{16}$ GeV.\nThe implementation of the seesaw mechanism \\cite{Minkowski:1977sc,GellMann:1980vs,Yanagida:1979as,Mohapatra:1979ia,Schechter:1980gr} in other corners of string compactification\nhas also been discussed \\cite{Faraggi:1990it,Faraggi:1993zh,Coriano:2003ui,Ghilencea:2002da}.\n\n\nThe layout of the remainder of the paper is as follows.\nIn section \\ref{sec:review}, we will review the $SO(10)$ construction from $M$ Theory on $G_2$-manifolds, expanding the discussion in \\cite{Acharya:2015oea}. In section \\ref{sec:sym_breaking_section}, the mechanism for $U(1)_X$ breaking will be given. The neutrino mass matrix will be analysed in section \\ref{neutrino_section}, and the numerical results presented in section \\ref{sec:numerical}. Finally we conclude in section \\ref{sec:conclusion}.\n\n\n\n\n\\section{SO(10) SUSY GUTS from $M$ Theory on $G_2$-manifolds} \\label{sec:review}\n\n\n\n\n$M$ Theory compactified on a $G_2$-manifold leads to a 4 dimensional theory with $\\mathcal{N}=1$ SUSY, where gauge fields and chiral fermions are supported by different types of singularities in the compactified space \\cite{Acharya:2001gy,Acharya:2004qe}. Yang-Mills fields are supported on three dimensional subspaces of the extra dimensions, along which there is an orbifold singularity, while chiral fermions will be further localised on conical singularities localised on these three dimensional spaces and interact with the gauge fields.\n\nOne of the key features of $M$ Theory compactified on $G_2$-manifolds without fluxes is that it provides a framework for generating hierarchies of mass scales. To understand the reason behind this notice that in $M$ Theory, the moduli fields, $s_i$, are paired with the axions, $a_i$, in order to form a complex scalar component of a superfield $\\Phi_i$\n\\begin{equation}\n\\Phi_i = s_i + i a_i + \\mbox{fermionic terms} \\ .\n\\end{equation}\nIn the absence of fluxes, the axions enjoy an approximate shift-symmetry, which is remnant of the higher dimensional gauge symmetry, $a_i \\to a_i + c_i $\nwhere $c_i$ is an arbitrary constant. This Peccei-Quinn symmetry, in conjunction with holomorphicity of the superpotential, severely constrains the superpotential for the moduli. As such, terms which are polynomial in the moduli and matter fields are forbidden at tree-level in superpotential, appearing only in the K\\\"ahler potential.\n\nIn general \nnon-perturbative effects such as instantons\nbreak the above shift symmetry, and generate a non-perturbative\nsuperpotential involving moduli and matter. Interactions will be generated by membrane instantons, whose actions are given by exponentials of the moduli. As the moduli stabilise and acquire VEVs, these exponentials will turn out to be small, and the VEV of the hidden sector superpotential naturally leading to a generation of hierarchical masses at the GUT scale \\cite{Acharya:2007rc}.\nThese ideas were used to construct the $G_2$-MSSM \\cite{Acharya:2008zi,Acharya:2011te}, an $SU(5)$ SUSY GUT from $M$ Theory on a $G_2$ manifold with the MSSM spectrum. Here, we discuss an extention of the program to the $SO(10)$ GUT group \\cite{Acharya:2015oea}, while referring to previous work on $G_2$ compactifications and consequent predictions for the parameters \\cite{Acharya:2008hi,Acharya:2012tw}.\n\n\n\nIn the remainder of this section,\nwe focus on the $SO(10)$ SUSY GUT from $M$ Theory on $G_2$ manifolds which we proposed in \\cite{Acharya:2015oea}.\nThe breaking patterns of an abelian Wilson line are the same as the ones of an adjoint Higgs. The simplest case of a surviving group that is the most resembling to the SM is\n\\begin{equation}\nSO(10) \\to SU(3)_c \\times SU(2)_L \\times U(1)_Y \\times U(1)_X \\ ,\n\\end{equation}\nunder which the branching rules of the GUT irreps read\n\\begin{align}\n{\\bf 10} : \\ &H_u=({\\bf 1},{\\bf 2})_{\\left(\\frac{1}{2}, 2\\right)} \\oplus H_d=({\\bf 1},{\\bf 2})_{\\left(-\\frac{1}{2},-2\\right)} \\oplus D=({\\bf 3},{\\bf 1})_{\\left(-\\frac{1}{3},2\\right)}\\oplus \\overline{D}=({\\bf \\overline{3}},{\\bf 1})_{\\left(\\frac{1}{3},-2\\right)}\\ , \\\\\n{\\bf 16} : \\ &L=( {\\bf 1},{\\bf 2})_{\\left(-\\frac{1}{2},3\\right)}\\oplus e^c=({\\bf 1},{\\bf 1})_{(1,-1)} \\oplus N=({\\bf 1},{\\bf 1})_{(0,-5)}\\oplus u^c = ({\\bf \\overline{3}},{\\bf 1})_{\\left(-\\frac{2}{3},-1\\right)}\\oplus \\nonumber \\\\\n&\\oplus d^c =({\\bf \\overline{3}},{\\bf 1})_{\\left(\\frac{1}{3},3\\right)}\\oplus Q= ({\\bf 3},{\\bf 2})_{\\left(\\frac{1}{6},-1\\right)} \\ ,\n\\end{align}\nand the subscripts are the charges under $U(1)_Y \\times U(1)_X$, which are normalised as\n$Q_Y = \\sqrt{\\frac{5}{3}}Q_1 , \\ Q_X = \\sqrt{40}\\tilde Q_{X} $,\nwhere $Q_1$, $\\tilde Q_X$ are $SO(10)$ generators.\n\nThe Wilson line can be conveniently represented as \n\\begin{equation}\n\\mathcal W = \\exp\\left[\\frac{i 2 \\pi}{N}\\left(a Q_Y + b Q_X\\right)\\right]=\\sum^\\infty_{m=0} \\frac{1}{m!} \\left(\\frac{i 2\\pi}{N}\\right)^m\\left(a Q_Y + b Q_X\\right)^m \\ ,\n\\end{equation}\nwhere the coefficients $a$, $b$ are constrained by the requirement that $\\mathcal{W}^N = 1$ and specify the parametrisation of the Wilson line. Under the linear transformation\n\\begin{align}\n\\frac{1}{2}a+2 b \\to \\alpha \\ , \\\\\n\\frac{1}{3}a-2b \\to \\beta\\ ,\n\\end{align}\nits action on the fundamental irrep then reads\n\\begin{equation}\\label{eq:W10}\n\\mathcal{W} 10 = \\eta^\\alpha H_u \\oplus \\eta^{-\\alpha} H_d \\oplus \\eta^{-\\beta} D \\oplus \\eta^{\\beta} \\overline{D} \\ ,\n\\end{equation}\nwhere $\\eta$ is the $N$th root of unity.\n\nLikewise the Wilson line matrix acts on the 16 irrep as\n\\begin{equation}\\label{eq:W16}\n\\mathcal{W}16=\\eta^{-\\frac{3}{2}\\beta} L \\oplus \\eta^{ \\alpha +\\frac{3}{2}\\beta} e^c \\oplus \\eta^{-\\alpha +\n \\frac{3}{2}\\beta} N \\oplus \\eta^{-\\alpha - \\frac{1}{2}\\beta} u^c \\oplus \\eta^{ \\alpha - \\frac{1}{2}\\beta} d^c \\oplus \\eta^{\\frac{1}{2}\\beta} Q \\ ,\n\\end{equation}\nwhich could be simplified a bit further by replacing $\\beta\\to 2\\beta$ without loss of generality, in order for the parameters to read as integers.\n\nThe effective discrete charges -- of different states on a chiral supermultiplet that absorbs Wilson line phases -- will be the overall charge of the discrete symmetry (common to all states belonging to the same GUT irrep) in addition to the Wilson line phases (different for each state inside the GUT irrep).\n\nHaving all the ingredients required to employ Witten's discrete symmetry proposal, we would like to have a consistent implementation of a well-motivated doublet-triplet splitting mechanism as it was done for $SU(5)$. Unfortunately the customary approach to the problem does not seem to work with $SO(10)$, as shown in \\cite{Acharya:2015oea}. To understand this first notice that Witten's splitting mechanism can only work in order to split couplings between distinct GUT irreps. This is understood as $\\mathcal W$ has the form of a gauge transformation of the surviving group and so it will never be able to split self bilinear couplings of a GUT irrep. For example, if one takes a $\\textbf{10}$ with Wilson line phases to contain the MSSM Higgses, we can see from \\cref{eq:W10} that both mass terms for the Higgses and coloured triplets are trivially allowed. We could consider that in order to split the Higgses, $H_u$ and $H_d$, from the coloured triplets -- $D$, $\\overline D$ -- we would need to add another ${\\bf 10}$, but it was shown that this cannot be achieved and so we are ultimately left with light coloured triplets.\n\nIn order to allow for light $D$, $\\overline{D}$ we need to guarantee that they are sufficiently decoupled from matter to prevent proton-decay. To accomplish this, we can use the discrete symmetry to forbid certain couplings,\nnamely to {\\it decouple $D$ and $\\overline{D}$ from matter}.\nSuch couplings arise from the $SO(10)$ invariant operator $\\mathbf{10}\\ \\mathbf{16}\\ \\mathbf{16}$, with $\\mathbf{16}$ denoting the three $SO(10)$ multiplets, each containing a SM family plus right handed neutrino $N$. If $\\mathbf{16}$ transforms as $\\eta^{\\kappa} \\mathbf{16}$, the couplings and charge constraints are\n\\begin{align}\nH_u \\mathbf{16} \\mathbf{16}\\; : & \\;2\\kappa + \\alpha + \\omega = 0 \\;\\mbox{mod}\\; N \\\\\nH_d \\mathbf{16} \\mathbf{16}\\; : & \\;2\\kappa - \\alpha + \\omega = 0 \\;\\mbox{mod}\\; N \\\\\nD \\mathbf{16} \\mathbf{16}\\; : & \\;2\\kappa - \\beta + \\omega \\neq 0 \\;\\mbox{mod}\\; N \\\\\n\\overline{D} \\mathbf{16} \\mathbf{16} \\; : & \\;2\\kappa + \\beta + \\omega \\neq 0 \\;\\mbox{mod}\\; N,\n\\end{align}\nwhere we allow for up-type quark Yukawa couplings together with couplings to the right-handed neutrinos,\n\\begin{equation}\ny_u^{ij}H_u^w \\mathbf{16}_i \\mathbf{16}_j \\equiv y_u^{ij}H_u^w (Q_iu_j^c+L_iN_j +i\\leftrightarrow j),\n\\label{yu}\n\\end{equation}\nand similarly for down-type quarks and charged leptons.\n\nThe couplings forbidden at a renormalizable tree-level by the discrete symmetry are generically regenerated from K\\\"ahler interactions through the Giudice-Masiero mechanism \\cite{Giudice:1988yz}. While this provides the Higgsinos a TeV scale $\\mu$-term mass, it also originates effective trilinear couplings with an $\\mathcal{O}(10^{-15})$ coefficient. As these are generic, we need to systematically study their physical implications at low energies, such as proton-decay, R-parity violation, and flavour mixing.\n\nFor proton decay, effective superpotential will be generate by the following K\\\"ahler potential\n\\begin{equation}\nK \\supset \\frac{s}{m_{Pl}^2} \\overline{D} d^c u^c + \\frac{s}{m_{Pl}^2} D e^c u^c + \\frac{s}{m_{Pl}^2} D Q Q + \\frac{s}{m_{Pl}^2} \\overline{D} Q L + \\frac{s}{m_{Pl}^2} D N d^c + \\mbox{h.c.} \\ ,\n\\end{equation}\nwhere we assume $\\mathcal{O}(1)$ coefficients. As the moduli acquire non-vanishing VEVs, these become\n\\begin{eqnarray}\nW_{eff} & \\supset \\lambda D Q Q + \\lambda D e^c u^c + \\lambda D N d^c\\label{eq:proton} + \\nonumber \\\\\n& +\\lambda \\overline{D} d^c u^c+\\lambda \\overline{D} Q L,\n\\end{eqnarray}\nwhere we considering all couplings to be similar and taking one family for illustrative purposes. Notice that contrary to $SU(5)$ case, there is no extra contribution from rotation of $L$ and $H_u$ as the bilinear term $\\kappa L H_u$ is not allowed by gauge invariance.\n\nWe estimate the scalar triplet mediated proton decay rate to be\n\\begin{equation}\n\\Gamma_p \\simeq \\frac{\\left| \\lambda^2 \\right|^2}{16 \\pi^2}\\frac{m_p^5}{m_D^4} \\simeq \\left( 10^{42} \\;\\mbox{yrs}\\right)^{-1} ,\n\\end{equation}\nwhere we took the mass of the colour triplets to be $m_D \\simeq 10^3$ GeV.\n\nAnother limit for triplet scalar comes from the cosmological constraints on its decay. As we have seen from proton-decay operators, triplet scalars can decay into quarks. If they start to decay during the Big Bang Nucleosynthesis (BBN) then nucleons could be disassociated, spoiling the predictions for light element \nabundances. We can estimate another limit on the triplet scalar mass by calculating its lifetime as it decay through the processes $D \\rightarrow e^c u^c, Q Q, Q L, d^c u^c$, and we get\n\\begin{equation}\n\\Gamma \\simeq \\lambda^2 m_D \\simeq ( 0.1 \\;\\mbox{sec})^{-1} ,\n\\end{equation}\nwhich is approximately consistent with BBN constraint. They will also give interesting collider signatures due to their long-lived nature.\n\\subsection{The vector-like family splitting}\n\nBecause the presence of a light vector-like pair coloured triplets spoils unification, we need a workaround that will preserve unification while keeping the presented doublet-triplet problem solution. We achieve this by considering the presence of extra matter that would form a complete GUT irrep with the coloured triplets, and hence restore unification. Unification constraints requires heavy states with equivalent SM gauge numbers, say $d^c_X$ and $\\overline{d^c}_X$, that have to be subtracted from the spectrum. This can be achieved by adding a vector-like family pair, ${\\bf 16}_X \\overline {\\bf 16}_X$, and splitting its mass terms using Wilson line phases. \n\nFurthermore, as the Wilson line breaking pattern is rank-preserving, we still need to break the extra abelian gauge factor $U(1)_X$. This can be achieved if a scalar component of the right-handed conjugated neutrino pair of an extra vector-like family ${\\bf 16}_X$, $\\overline{\\bf{16}}_X$ acquires VEVs. On top of this, this VEV can generate a Majorana mass for the matter right-handed conjugated neutrinos, providing a crucial ingredient for a type I see-saw mechanism.\n\nIn order to preserve gauge coupling unification, we notice that the down-type quarks -- $d^c_X$, $\\overline{d^c}_X$ -- have the same SM quantum numbers as the coloured triplet pair -- $D$, $\\overline{D}$ -- coming from the ${\\bf 10}$. We take ${\\bf 16}_X$ to be localised along a Wilson line, and find that it\ntransforms under the discrete symmetry as\n\\begin{equation}\n{\\bf 16}_X \\to \\eta^x \\left( \\eta^{-3\\gamma} L \\oplus \\eta^{ 3\\gamma+\\delta} e^c \\oplus \\eta^{3 \\gamma - \\delta} N \\oplus \\eta^{-\\gamma-\\delta} u^c \\oplus \\ \\eta^{-\\gamma +\\delta} d^c \\oplus \\eta^{\\gamma} Q \\right) .\n\\end{equation}\nOn the other hand, we let ${\\bf {\\overline{16}}}_X$ transform without Wilson line phases, ${\\bf {\\overline{16}}}_X \\to \\eta^{\\overline x}\\, {\\bf {\\overline{16}}}_X$, and the condition for the mass term that will split the vector-like family is\n\\begin{equation}\n\\overline{d^c}_X d^c_X : x - \\gamma + \\delta + \\overline{x} = 0 \\mod N ,\\\n\\end{equation}\nwhilst forbidding all the other self couplings that would arise from ${\\bf 16}_X {\\bf {\\overline{16}}}_X$.\nThe $d^c_X$, $\\overline{d^c}_X$ quarks will then be naturally endowed a GUT scale mass through membrane instantons, provided that the singularities supporting ${\\bf 16}_X$, $\\overline{\\bf{16}}_X$ are close enough to each other in the compactified space. The remaining states of ${\\bf 16}_X$, $\\overline{\\bf{16}}_X$ will have a $\\mu$ term of order TeV through the Giudice-Masiero mechanism.\nThe coloured triplets -- $D$, $\\overline{D}$ -- and the light components of ${\\bf 16}_X$, $\\overline{\\bf{16}}_X$ will effectively account for a full vector-like family. The light spectrum is then the one of MSSM in addition to this vector-like family, which in turn preserves unification, with a larger unification coupling at the GUT scale.\n\n\n\n\n\n\n\n\n\\subsection{R-parity violation\\label{sec:RPV}}\n\nDespite the existence of an effective matter parity symmetry inside $SO(10)$, the presence of a vector-like family will lead to R-parity violating (RPV) interactions though the VEV of the $N_X$, $\\overline{N}_X$ components in the presence of moduli generated interactions. Furthermore, as we will see in detail in Section \\ref{sec:sym_breaking_section}, the scalar component of the matter conjugate right-handed neutrino, $N$, will also acquire a VEV. These VEVs break $SO(10)$ and will inevitably generate RPV.\nThese interactions will mediate proton-decay, enable the lightest supersymmetric particle (LSP) to decay, and generate extra contributions to neutrino masses. In our framework RPV is generic, not only arising from allowed superpotential terms but as well from K\\\"ahler interactions involving moduli fields.\n\nThe interactions that break R-parity can either be trilinear or bilinear (B-RPV), and have different origins in our framework. The first contribution we can find comes from the tree-level renormalizable superpotential allowed by the discrete symmetry. Since we will encounter $\\langle N \\rangle \\neq 0$, this means that even in a minimal setup, there will be an R-RPV contribution from matter Dirac mass coupling\n\\begin{equation}\nW \\supset y_\\nu N H_u L \\ ,\n\\end{equation}\nreading\n\\begin{equation}\\label{eq:BRPVfromDirac}\nW \\supset y_\\nu \\langle N \\rangle H_u L \\ .\n\\end{equation}\n\nNext we turn our attention to the K\\\"ahler potential, where interactions otherwise forbidden by the discrete symmetry might arise if there is a modulus with required charge. In such case, there is another contribution arising from the non-vanishing VEVs of $N_X$, $N$ $\\overline N_X$ in conjugation with moduli VEVs. To see this, notice that in the K\\\"ahler potential there are generically interactions of the form\n\\begin{equation}\nK \\supset \\frac{1}{m_{Pl}}N H_u L+\\frac{s}{m_{Pl}^2} \\overline N_X H_u L+\\frac{s}{m_{Pl}^2} \\overline N_X^\\dagger H_u L +\\mbox{ h.c.}\\ ,\n\\end{equation}\nwhere while the first term exists in zeroth order in moduli (otherwise there would be no neutrino Dirac mass in the superpotential), the last two are otherwise forbidden by the discrete symmetry, and $s$ denotes a generic modulus for each coupling. These terms will generate contributions to B-RPV as $N_X$, $N$ $\\overline N_X$, $s$ acquire VEVs.\n\nThere are two types of contribution arising from the terms above. The first is generates through the Giudice-Masiero mechanism. As the moduli acquire VEVs, new holomorphic couplings will appear in the superpotential\n\\begin{equation}\nW_{eff, 1} = \\frac{m_{3\/2}}{m_{Pl}} \\langle N \\rangle H_u L +0.1 \\frac{m_{3\/2}}{m_{Pl}} \\langle N_X \\rangle H_u L + 0.1 \\frac{m_{3\/2}}{m_{Pl}} \\langle \\overline N_X^\\dagger \\rangle H_u L \\ ,\n\\end{equation}\nwhere $m_{3\/2} \\simeq \\mathcal{O}(10^{4})$ GeV, and since $s\/m_{Pl} \\simeq 0.1$ in $M$ Theory.\nNotice that in principle we would also have a term in the K\\\"ahler potential involving $N$, but this can be found to be subleading in comparison to the term arising from the Dirac mass Eq. \\eqref{eq:BRPVfromDirac}.\n\nThe second contribution arises if the F-terms of the fields $N_X$, $N$, $\\overline N_X$ are non-vanishing. In this case, we expect the appearance of the contributions\n\\begin{equation}\nW_{eff, 2} = \\frac{\\langle F_N \\rangle}{m_{Pl}} H_u L+0.1\\frac{ \\langle F_{N_X} \\rangle}{m_{Pl}} H_u L+0.1 \\frac{ \\langle F_{\\overline{N}^\\dagger_X}\\rangle}{m_{Pl}} H_u L \\ ,\n\\end{equation}\nand its magnitude will depend on how much F-breaking provoked by our symmetry breaking mechanism. Here we are considering that the case where $\\overline N_X^\\dagger H_u L$ cannot exist in the K\\\"ahler potential in zeroth order in a modulus field.\n\nPutting all together, the B-RPV interactions account to the B-RPV paramter\n\n\\begin{equation}\nW \\supset \\kappa H_u L\n\\end{equation}\nwith \n\\begin{equation}\n\\kappa = \\left(y_\\nu+\\frac{m_{3\/2}}{m_{Pl}}\\right) \\langle N \\rangle + 0.1 \\frac{m_{3\/2}}{m_{Pl}} \\langle N_X \\rangle + 0.1 \\frac{m_{3\/2}}{m_{Pl}} \\langle \\overline N^\\dagger_X \\rangle +\\frac{\\langle F_N \\rangle}{m_{Pl}} + 0.1 \\frac{\\langle F_{N_X}\\rangle}{m_{Pl}} + 0.1 \\frac{ \\langle F_{\\overline{N}^\\dagger_X}\\rangle}{m_{Pl}} \\ ,\n\\end{equation}\nand the relative strength of each contribution is model detail dependent, namely on neutrino Yukawa textures, symmetry breaking details, and F-flatness deviation.\n\n\nIn a similar manner, trilinear RPV couplings will be generated when $N$, $N_X$, $\\overline N_X$, $s$ acquire VEVs.\nIn order to systematically study this, we notice that the trilinear RPV couplings come from the term\n\\begin{equation}\n\\mathbf{16}\\ \\mathbf{16}\\ \\mathbf{16}\\ \\mathbf{16} , \\\n\\mathbf{16}_X \\mathbf{16}\\ \\mathbf{16}\\ \\mathbf{16} , \\\n\\mathbf{\\overline{16}}_X^\\dagger \\mathbf{16}\\ \\mathbf{16}\\ \\mathbf{16}\n\\end{equation}\nas the scalar component of $N_X$, $N$ acquires non-vanishing VEVs. Notice that the last term lives in the K\\\"ahler potential.\nThese are made forbidden at tree-level using the discrete symmetry of the compactified space. However, just like the $\\mu$ terms and the B-RPV terms shown above, these terms will in general be present in the K\\\"ahler potential and will effectively be generated as the moduli acquire VEVs.\nThis happens again through the Giudice-Masier mechanism and we will find\n\\begin{equation}\n\\mathcal{O} \\left(\\frac{m_{3\/2}}{m_{Pl}^2} (\\langle N \\rangle + \\langle N_X \\rangle+\\langle \\overline{N}_X^\\dagger \\rangle)\\right) \\{L L e^c,\\ LQ d^c,\\ u^c d^c d^c\\} ,\n\\end{equation}\nwhere $m_{3\/2}\/m_{Pl}\\simeq \\mathcal{O}(10^{-14})$. The apparent suppression of trilinear RPV is understood as these terms can only be generated by non-renormalizable terms in an $SO(10)$ context.\n\nSimilarly to the B-RPV case, there will be further contributions if the F-terms of $N_X$, $N$, $\\overline N_X$ are non-vanishing. Namely we find\n\\begin{equation}\n\\mathcal{O} \\left(\\frac{\\langle F_N \\rangle + \\langle F_{N_X} \\rangle + \\langle F_{\\overline{N}_X^\\dagger}\\rangle}{m_{Pl}^2}\\right) \\{L L e^c,\\ LQ d^c,\\ u^c d^c d^c\\} ,\n\\end{equation}\nand again we expect these to be sub-leading even if the F-terms are not vanishing.\n\nWe see then that the values of all RPV coupling are strictly related to the details of the breaking mechanism employed to break the extra $U(1)_X$. This will be studied in great detail in Section \\ref{sec:sym_breaking_section}. Furthermore, the bilinear B-RPV term generates a contribution to the physical neutrino masses \\cite{Banks:1995by,Barbier:2004ez}. The complete picture of neutrino masses, including B-RPV operators, will be discussed in Section \\ref{neutrino_section}.\n\n\nWe can study now some direct effects of RPV in the dynamics of our class of models. Under the assumption that $\\kappa \\ll \\mu$, performing a small rotation, of\n$\\mathcal{O}(\\kappa\/\\mu)$, in $(H_d,L)$ space, the last term can be absorbed $\\mu H_dH_u$.\nAs a consequence, the first two terms will be enhanced by the Yukawa couplings $y_eH_dLe^c$, etc., leading to\n\\begin{equation}\nW \\supset y_e \\frac{\\kappa}{\\mu} L L e^c + y_d \\frac{\\kappa}{\\mu} LQ d^c + \\lambda \\frac{v}{m_{Pl}} u^c d^c d^c , \\label{eq:RPVrot}\n\\end{equation}\nand we have dropped the $\\mathcal{O}(1\/m_{Pl})$ contributions to the first two terms since now the Yukawa rotated contributions are much larger.\nAlso, we kept the last term with the parametrization $v$ describing all contributions. These will be very small, for example in the case the VEVs are high-scale, $\\langle N_X \\rangle \\simeq 10^{16}$ GeV, the trilinear RPV coupling strength is of $\\mathcal{O}(10^{-16})$. A direct consequence of this result is that proton decay will be slow, even when the $\\Delta L = 1$ terms are enhanced.\n\nWhile the proton is relatively stable, the enhanced terms will provide a decay channel for the LSP, which is now unstable.\nIn the limit that we can take the final states to be massless, and considering that the LSP is a neutralino mainly composed of neutral gauginos, the LSP lifetime through the decay $\\tilde \\chi^0 \\to d^cQL$ can be estimated from a tree-level diagram involving a virtual $\\tilde{d}^c$ with mass $m_0$,\\footnote{See, for example, the diagrams in \\cite{Martin:1997ns}.} \n\\begin{equation}\n\\tau_{LSP} \\simeq \\left(3.9 \\times 10^{-15} \\right)\\left(\\frac{\\mu}{g_w y_d \\kappa}\\right)^2\\left( \\frac{m_0}{10 \\ \\mbox{TeV}}\\right)^4 \\left( \\frac{100 \\ \\mbox{GeV}}{m_{LSP}}\\right)^5 \\sec,\n\\end{equation}\nwhere $g_w$ is a weak gauge coupling. The LSP lifetime is bounded to be either $\\tau_{LSP} \\lesssim 1$ sec or $\\tau_{LSP} \\gtrsim 10^{25}$ sec \\cite{Banks:1995by,Dreiner:1997uz}, from Big Bang Nucleosythesis (BBN) and indirect Dark Matter (DM) experiments, respectively. If we take $m_{LSP} \\simeq 100\\mbox{ GeV}$, $m_0 \\simeq 10 \\mbox{ TeV}$, $y_d=y_b \\simeq 10^{-2}$, $g_w \\simeq 0.1$, we find that the VEV $v_X$ is constrained to be either\n\\begin{align}\n\\kappa & \\gtrsim 6 \\times 10^{-2} \\mbox{ GeV} \\label{eq:kLSPBBN} \\\\\n\\ {\\rm or} \\ \\kappa &\\lesssim 2 \\times 10^{-14} \\mbox{ GeV} ,\n\\end{align}\nfor a short- and long-lived LSP, respectively. In the above estimate we used the fact that the decay involving the bottom Yukawa is the largest contribution to the decay width.\n\nWe can use the above result to infer some parametric dependence on the scale of the $U(1)_X$ breaking. If we have the leading contribution to the B-RPV coupling to be $\\kappa \\simeq \\langle N_X \\rangle \\lambda \\Rightarrow \\langle N_X \\rangle \\gtrsim 10^{12}$ GeV. In this case, the LSP is too short lived to be a good DM candidate, but decays quickly enough to not spoil BBN predictions. On the other-hand, a low-scale VEV is bound to be $\\langle N_X \\rangle \\lesssim 1$ GeV in order to allow for a long-lived LSP. This would imply the abelian gauge boson associated with extra $U(1)_X$ to be light, $m_{Z^\\prime} < \\mathcal{O}(1)$ GeV. This last scenario is completely excluded from experimental searches.\n\n\nThe lack of a good DM candidate in the visible sector indicates us that DM is realised elsewhere. For instance, it has been recently suggested that in the context of String\/$M$ Theory, the generic occurrence of hidden sectors could account for the required DM mechanics \\cite{Acharya:2016fge}.\n\n\n\n\n\n\\subsection{The see-saw mechanism}\n\nThe relevance of the bounds on the rank-breaking VEV is only fully understood when studying the details of symmetry breaking mechanism and neutrino masses. For example, if we start with an $SO(10)$ invariant theory the Yukawas are unified for each family leading to at least one very heavy Dirac neutrino mass, $m_\\nu^D$. However, if the right-handed conjugated neutrino has a heavy Majorana mass, then the physical left-handed neutrino mass will be small through a type I see-saw mechanism. In order to accomplish this, one has to allow the following terms in the superpotential\n\\begin{equation}\nW \\supset y_{\\nu} H_u L N + M N N ,\n\\end{equation}\nwhere $y_\\nu$ are the neutrino Yukawas, $L$ the matter lepton doublets, $N$ the right-handed conjugated neutrino, and $M$ its Majorana mass, which we take $M \\gg m_\\nu^D = y_\\nu \\langle H_u \\rangle$. With the above ingredients, a mostly left-handed light neutrino will have a physical mass\n\\begin{equation}\nm^\\nu_{phy} \\simeq - \\frac{(m_\\nu^D)^2}{M} .\n\\end{equation}\nOne of the most appealing features of $SO(10)$ models is that each family is in a $\\bf{16}$ which includes a natural candidate for the right-handed conjugated neutrino, the $N$. In order to employ a type I see-saw mechanism, we need to generate a Majorana mass term for the matter right-handed conjugated neutrino through the operator $W\\supset \\overline{\\bf 16}_X\\overline{\\bf 16}_X {\\bf 16}\\: {\\bf 16}$ \\footnote{Given that in $M$ Theory one does not account for irreps larger than the adjoint, this is the lowest order term that can generate a right-handed neutrino Majorana mass.} leading to the operator\n\\begin{equation}\n\\frac{1}{m_{Pl}} \\overline{N}_X \\overline{N}_X N N\\ ,\n\\end{equation}\nfrom which the Majorana mass for the (CP conjugated) right-handed neutrino field $N$\nis emerges as\n\\begin{equation}\nM \\simeq \\frac{\\langle \\overline{N}_X \\rangle^2}{m_{Pl}} \\ .\n\\end{equation}\nWe can now relate the bounds on the value of the D-flat VEVs\n$\\langle \\overline{N}_X \\rangle = \\langle N_X \\rangle$ from both RPV and the \nrequirement of a realistic see-saw mechanism. Since the physical neutrino mass in type I see-saw mechanism is given by\n\\begin{equation}\nm_\\nu^{phy} \\simeq \\frac{(m_\\nu ^D)^2}{M} \\ ,\n\\end{equation}\nassuming $m_\\nu ^D \\simeq \\mathcal{O}(100\\mbox{ GeV})$, and knowing that the upper bound on the neutrino masses $m_\\nu^{phy} \\lesssim 0.1$ eV, one finds\n\\begin{equation}\nM \\gtrsim 10^{14} \\mbox{ GeV} \\Rightarrow\n\\langle {N}_X \\rangle \\gtrsim 10^{16} \\mbox{ GeV}.\n\\end{equation}\nThe above argument suggests that we need to break the $U(1)_X$ close to the GUT scale. Since the Wilson line breaking mechanism is rank-preserving, we need to look for an alternative solution. Although the neutral fermion mass matrix will be considerably more intricate, obscuring the relations and hierarchies amongst different contributions to the neutrino masses, the above estimate motivates\nthe need for a high-scale $U(1)_X$ breaking mechanism.\n\n\\subsection{Effective light families}\nFor a simple SUSY $SO(10)$ model where each family is unified into a single irrep with universal soft masses, it is well known that\nelectroweak symmetry is difficult to break \\cite{Hall:1993gn,Rattazzi:1994bm,Murayama:1995fn,Baer:1999mc,Auto:2003ys,Baer:2009ie}. Since the two Higgs soft masses are unified at GUT scale and have similar beta function due to Yukawa unification, either both masses are positive at electroweak scale and symmetry is not broken or both masses are negative and the potential becomes unbounded from below. Another aspect of Yukawa unification problem lies in the fact that low energy spectrum of quarks and leptons requires some degree of tuning in parameter space when their RG runnings are considered.\n\nThe EWSB and Yukawa textures issues are naturally solved if each family is not contained in one single complete ${\\bf 16}$, but is instead formed of states from different Ultra Violet (UV) complete ${\\bf 16}$s. In order to implement this in our framework, first we assume the existence of multiple ${\\bf 16}$ with independent and different Wilson Line phases, alongside the existence of multiple ${\\overline{\\bf 16}}$. Second, we employ Witten's proposal to turn on some vector-like masses such that three effective light ${\\bf 16}$ survive. Since in $M$ Theory the strength of the Yukawa couplings is given by membrane instantons, and are therefore related to distances between the singularities supporting the respective superfields, by constructing effective families from different UV ${\\bf 16}$s one can obtain different Yukawa couplings within each family.\n\nSuch solution can be achieved if one considers $M$ complete ${\\bf \\overline{16}}_j$ and $M+3$ complete ${\\bf 16}_i$ UV irreps. Allowing for masses between different states of these UV irreps to appear, one has schematically the mass terms in the superpotential\n\\begin{equation}\n\\mathbf{16}_i \\mu_{ij} \\overline{\\mathbf{16}}_j,\n\\end{equation}\nbut since $i=1,...,M$ while $j=1,...,M+3$ the mass matrix $\\mu_{ji}$ can only have at most rank $M$ and hence there will be three linear combinations composing three ${\\bf 16}$ that will remain massless. If these masses are truly $SO(10)$ invariant, i.e.\n\\begin{equation}\n\\mathbf{16}_i \\mu_{ij} \\overline{\\mathbf{16}}_j = \\mu_{ij} \\left( Q_i \\overline Q_j + L_i \\overline L_j + \\ldots\\right),\n\\end{equation}\neach effective light family will be $SO(10)$ invariant. Consequently each family will retain unified Yukawa textures, and so this does not solve our problem of splitting the Yukawa couplings within each family.\n\nHowever, Witten's proposal endows our framework with a GUT breaking discrete symmetry which can be employed to ensure that the superpotential mass matrices between the UV states\n\\begin{equation}\n\\mu^Q_{ij} Q_i \\overline Q_j +\\mu^L_{ij} L_i \\overline L_j + \\ldots,\n\\end{equation}\nare not the same, leading to different diagonalisations of $Q$, $L$, etc which in turn break the Yukawa $SO(10)$ invariance. In order to accomplish that, take for example that the ${\\bf 16}_i$ absorb distinct and independent Wilson line phases, while $\\overline{\\bf 16}_j$ do not, i.e. the UV irreps will transform under the discrete symmetry as\n\\begin{align}\n{\\bf 16}_i &\\to\\eta^{m_i}\\left(\\eta^{-3\\gamma_i} L_i \\oplus \\eta^{ 3\\gamma_i+\\delta_i} e_i^c \\oplus \\eta^{3 \\gamma_i - \\delta_i} N_i \\oplus \\eta^{-\\gamma_i-\\delta_i} u_i^c \\oplus \\eta^{-\\gamma_i +\\delta_i} d_i^c \\oplus \\eta^{\\gamma_i} Q_i\\right) \\label{eq:16icharges}\\\\\n\\overline{\\bf 16}_j &\\to \\eta^{\\overline m_i}\\overline{\\bf 16}_j \\label{eq:16bjcharges},\n\\end{align}\nand look for solutions for the discrete charges where different states have different mass matrices. Since explicit examples can only be given by solving extensive modular linear systems, which are computationally prohibitive, a fully working example with three light-families is not provided.\n\n\n\n\n\\section{$U(1)_X$ Breaking scenarios and mechanisms} \\label{sec:sym_breaking_section}\n\n\n\nIn this section we are interested in implementing a symmetry breaking mechanism for the extra $U(1)_X$ in which the breaking VEV is stabilised at high values, more or less close to the GUT scale. In order to do so, we will look into the D-flat direction of the potential that breaks the extra $U(1)_X$. It was shown \\cite{Drees:1986vd,Kolda:1995iw} that in the D-flat direction, non-renormalisable operators can provide such scenario. In its simplest inception, the Kolda-Martin\nmechanism \\cite{Kolda:1995iw} relies on a vector-like pair which lowest order term allowed in the superpotential is non-renormalizable\n\\begin{equation}\n W = \\frac{c}{m_{Pl}} (\\Phi \\bar \\Phi)^2\n\\end{equation}\nand alongside the soft-term Lagrangian\n\\begin{equation}\n -\\mc{L}_{soft} = m_\\Phi^2 |\\Phi|^2 + m_{\\bar\\Phi}^2 |\\bar{\\Phi}|^2,\n\\end{equation}\nit is immediate to find that along the D-flat direction the potential has a non-trivial minimum which fixes the VEVs at a high scale\n\\begin{equation}\n\\Phi^2 = \\sqrt{-\\frac{(m_\\Phi^2+m_{\\bar\\Phi}^2)m_{Pl}^2}{12 c}} ,\n\\end{equation}\nwhere if we take $m \\simeq 10^4$ the VEVs are estimated at $\\Phi \\simeq 10^{11}\\mbox{ GeV}$.\n\nThere are some caveats to this mechanism as presented above. First, there is significant F-breaking as $\\langle F \\rangle \\simeq \\mc{O}(10^{15})\\mbox{ GeV}$. While this is not a problem if the vector-like family does not share gauge interactions with ordinary matter, in our case non-vanishing F-terms will originate undesirable interactions, c.f. Section \\ref{sec:RPV}. We shall therefore focus on F-flat solutions.\n\nSecond, the mechanism is not complete in the absence of the full soft-terms Lagrangian, which has to include\n\\begin{equation}\n-\\mc{L}_{soft} \\supset C \\frac{1}{m_{Pl}} \\Phi^2 \\bar{\\Phi} ^2 + \\mbox{ h.c.} .\n\\end{equation}\nAs we estimate $C\\simeq\\mc{O}(m_{3\/2})$ at the GUT scale from the SUGRA \\cite{Brignole:1997dp}, at the VEV scale this term is competing with the non-renormalisable terms in the potential arising from the superpotential, and therefore cannot be ignored.\n\nFinally the model presented differs from ours as $\\mu$-terms are generically generated by moduli VEVs even if they are disallowed by the discrete symmetry of the compactified space.\n\nIn order to proceed, we turn to a more complete version of the mechanism. To do so, we include the $\\mu$-term\n\\begin{equation}\nW = \\mu \\Phi \\bar{\\Phi}+\\frac{c}{m_{Pl}} (\\Phi \\bar \\Phi)^2\n\\end{equation}\nand the more complete soft Lagrangian,\n\\begin{equation}\n -\\mc{L}_{soft} = m_\\Phi^2 |\\Phi|^2 + m_{\\bar\\Phi}^2 |\\bar{\\Phi}|^2 - (B\\mu \\Phi \\bar \\Phi + \\mbox{ h.c.})+ \\frac{C}{m_{Pl}} \\Phi^2 \\bar{\\Phi} ^2 + \\mbox{ h.c.} .\n\\end{equation}\nDue to the presence of the $\\mu$-term, the F-term\n\\begin{equation}\nF_{\\Phi} = \\mu \\bar \\Phi + \\frac{2 c }{m_{Pl}} \\Phi \\bar \\Phi ^2\n\\end{equation}\ncan be set to zero for two different field configurations\n\\begin{equation}\n F_\\Phi =0 \\Rightarrow \\left\\{\n \\begin{array}{l}\n \\bar \\Phi =0 \\\\\n \\Phi \\bar \\Phi = - \\frac{\\mu m_{Pl}}{2 c}\n \\end{array}\n \\right.\n\\end{equation}\nand the non-trivial VEV can be estimated. Taking $\\mu \\simeq \\mc{O}(10^3)$ GeV, this leads to $| \\Phi | = 10^{10.5}$ GeV. This looks very similar to the original Kolda-Martin case, with the exception being that the F-term can vanish, and the parametric dependence on the VEV is now on $\\mu$ instead of a soft-mass. In general there might be a non-SUSY preserving vacuum elsewhere in field space, but we will work under the assumption that the SUSY vacua discovered with this approach are at least stable enough to host phenomenologically viable models.\n\nWe wish to assess if we can minimise the potential in this SUSY-preserving field configuration. For that, we need to check if the above field configuration will also extremise the soft-term Lagrangian. To see this we take\n\\begin{equation}\n- \\partial_\\Phi \\mc{L}_{soft} = m^2_\\Phi \\Phi^* - B\\mu \\bar \\Phi + \\frac{2 C}{m_{Pl}} \\Phi \\bar \\Phi^2 =0\n\\end{equation}\nand, in the limit the VEVs are real, we find a trivial and a non-trivial solutions\n\\begin{equation}\n- \\partial_\\Phi \\mc{L}_{soft} = 0 \\Rightarrow \\left\\{\n\\begin{array}{l}\n\\Phi =0 \\\\\n\\Phi ^2 = - \\frac{(m_\\Phi^2-B\\mu) m_{Pl}}{2 C}\n\\end{array}\n\\right.\n\\end{equation}\nand the second one seems very similar to the non-trivial configuration derived through the F-term. In fact, both conditions can be met. To see this, we re-parametrise the soft-terms by factoring out their dimensionful dependence on $m_{3\/2}$\n\\begin{align}\n\tB\\mu &= m_{3\/2} \\mu b \\\\\n\tC & = m_{3\/2} \\tilde c \\\\\n\tm_\\Phi & = m_{3\/2} a ,\n\\end{align}\nwhere $a$, $b$, $\\tilde c$ are dimensionless, and from SUGRA formulae they are $\\mc{O}(1)$ at the GUT scale. Of course they will evolve with the scale through RGE evolution, so they need not to be always of the same order. The condition that both the F-flatness and soft-term stabilisation are jointly achieved boils down to be a relation between parameters\n\\begin{equation}\n\\frac{\\tilde c}{c} = \\frac{2 a m_{\\Phi} -\\mu b}{\\mu} ,\n\\end{equation}\nwhich is generically valid.\n\nIn order for the above non-trivial VEV be a minimum, we need the trivial VEV solution to account for a maximum. This is to say that the mass matrix for the system $(\\Phi, \\bar \\Phi^*)$ evaluated at the origin has a negative eigen-value. In our case this accounts for allowing its determinant to be negative\n\\begin{equation}\n(|\\mu|^2+m^2_\\Phi)(|\\mu|^2+m^2_{\\bar \\Phi})-B\\mu^2<0.\n\\label{eq:CondTrivialMaximum}\n\\end{equation}\n\nWe notice as well that the above discussion can be immediately extended for the case that the lowest order non-renormalisable term allowed by the discrete symmetry\n\\begin{equation}\nW \\supset \\frac{c}{m_{Pl}^{2n-3}}(\\Phi \\bar \\Phi)^n \\Rightarrow \\Phi \\simeq (\\mu m_{Pl}^{2n-3})^{\\frac{1}{2n-2}}\n\\end{equation}\nhappens for $n \\geq 2$, and not only for $n=2$. Even so, the presented implementation of the Kolda-Martin mechanism only accounts for a vector-like pair of superfields, while in our case the system breaking the extra $U(1)_X$ is composed of $N$, $N_X$, $\\overline N_X$ states.\n\nTherefore, we want to find similar solutions starting with the superpotential\n\\begin{equation}\nW = \\mu^N_{Xm} N \\overline N_X + \\mu^{N}_X N_X \\overline N_X + \\frac{c_{2,2}}{m_{Pl}} (N \\overline N_X)^2 + \\frac{c_{n,k}}{m_{Pl}^{2n-3}} (N_X \\overline N_X )^{n-k} (N \\overline N_X)^k\n\\end{equation}\nwhere $n\\geq 2$ and $k \\mu^L_{Xm}\n\t\\end{equation}\n\tas required to split $\\nu$ from $\\nu_X$, as discussed in Section \\ref{sec:Mhierarchies}.\n\t\n\tThe above considerations indicate us that the mechanism to split $\\nu$ from $\\nu_X$ will only work for large values of $b_{11}$. This can be seen in Figures \\ref{fig:(2,0)-Scatter-(alpha,b11)} and \\ref{fig:(2,1)-Scatter-(alpha,b11)}, where a slight agglomeration of points around $(\\alpha,b_{11})\\simeq (1,1)$ can be identified.\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\subfigure[$(2,0)$ case\\label{fig:(2,0)-Scatter-(alpha,b11)}]{\\includegraphics[width=0.45\\linewidth]{2,0-Scatter-alpha,b11.pdf}}\n\t\t\\subfigure[$(2,0)$ case\\label{fig:(2,1)-Scatter-(alpha,b11)}]{\\includegraphics[width=0.45\\linewidth]{2,1-Scatter-alpha,b11.pdf}}\t\t\n\t\t\\caption{Scatter plots showing the amplitude $\\alpha$ of the left-handed doublet state\n\t\t$\\nu$ in the lightest mass eigenstate\n\t\t$\\nu_{light}$ as $b_{11}$ varies for the $(2,0)$ and $(2,1)$ cases.\n\t\tThe points are fairly evenly distributed with a slight clustering near the desired value of $\\alpha \\approx 1$ for $b_{11}\\approx 1$.}\n\t\\end{figure}\n\t\t\n\t On the other hand, we find that the $\\kappa_m$ parameter is mostly bounded to be smaller than $1$ GeV, as is shown in Figures \\ref{fig:(2,0)-Scatter-(alpha,kappam)} and \\ref{fig:(2,1)-Scatter-(alpha,kappam)}. Although such small values of $\\kappa_m$ are welcome, the fact that there is no clear preference for $\\kappa_m \\gtrsim 10^{-2}$ GeV suggests this class of models is challenged by BBN constraints, c.f. Eq. \\eqref{eq:kLSPBBN}.\n\t \n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\subfigure[$(2,0)$ case\\label{fig:(2,0)-Scatter-(alpha,kappam)}]{\\includegraphics[width=0.45\\linewidth]{2,0-Scatter-alpha,kappam.pdf}}\n\t\t\\subfigure[$(2,1)$ case\\label{fig:(2,1)-Scatter-(alpha,kappam)}]{\\includegraphics[width=0.45\\linewidth]{2,1-Scatter-alpha,kappam.pdf}}\n\t\t\\caption{Scatter plots showing the amplitude $\\alpha$ of the left-handed doublet state\n\t\t$\\nu$ in the lightest mass eigenstate\n\t\t$\\nu_{light}$ as $\\kappa_m$ varies for the $(2,0)$ and $(2,1)$ cases.\n\t\tThe points are fairly evenly distributed with a slight clustering near the desired value of $\\alpha \\approx 1$. The horizontal dashed line represents the bound on the LSP lifetime, c.f. Eq. \\eqref{eq:kLSPBBN}.}\n\t\\end{figure}\n\t\n\n\t\n\t\\vspace{10mm}\n\t\\underline{\\bf $(3,0)$ case}\n\t\\vspace{5mm}\n\n\tFor the $(3,0)$ Kolda-Martin realisation, we found much promising results. Since the $N_X$, $\\overline N_X$ VEVs are expected to be around $\\mathcal{O}(10^{14.25})$ GeV, if we allow them to be in the range\n\t\\begin{equation}\n\tN_X,\\ \\overline N_X \\in [10^{13.25},10^{15.25}] \\mbox{ GeV}\n\t\\end{equation}\n\twe find\n\t\\begin{equation}\n\t\\mu_{11} \\in b_{11} [10^{8.25},10^{12.25}]\\mbox{ GeV}\n\t\\end{equation}\n\twhich implies that it is natural to achieve\n\t\\begin{equation}\n\t\\mu_{11} \\gg \\mu^L_{Xm}\n\t\\end{equation}\n\tand consequently $\\nu$ will decouple easily from the other $\\nu$-type states.\n\t\n\tThe above expectations are confirmed by the numerical results, and the lightest state will be mostly composed of $\\nu$ even for values of $b_{11}$ below $\\mathcal{O}(1)$. This behaviour can be seen in Figure \\ref{fig:(3,0)-Scatter-(alpha,b11)}.\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\subfigure[Scatter of $(\\alpha,b_{11})$ plane\\label{fig:(3,0)-Scatter-(alpha,b11)}]{\\includegraphics[width=0.45\\linewidth]{3,0-Scatter-alpha,b11.pdf}}\n\t\t\\subfigure[Scatter of $(\\alpha,\\kappa_m)$ plane\\label{fig:(3,0)-Scatter-(alpha,kappam)}]{\\includegraphics[width=0.45\\linewidth]{3,0-Scatter-alpha,kappam.pdf}}\n\t\t\\caption{Scatter plots showing the amplitude $\\alpha$ of the left-handed doublet state\n\t\t$\\nu$ in the lightest mass eigenstate\n\t\t$\\nu_{light}$ as $\\kappa_m$ varies for the $(3,0)$ case.\n\t\tThe points are fairly evenly distributed except for a significant clustering near the desired value of $\\alpha \\approx 1$ for larger values of $b_{11}$. The horizontal dashed line represents the bound on the LSP lifetime, c.f. Eq. \\eqref{eq:kLSPBBN}. The right panel shows that nearly all the points\n\t\tsatisfy $\\kappa_m \\gtrsim 10^{-2}$ GeV.}\n\t\\end{figure}\n\t\n\tInterestingly, in the $(\\alpha,\\kappa_m)$ plane, shown in Figure \\ref{fig:(3,0)-Scatter-(alpha,kappam)} we can see again that the mass matrix prefers $\\kappa_m < 1$ GeV in order to reproduce a mostly-$\\nu$ lightest state. This is a nice result which ensures that whenever we have good physical neutrinos, we also find sufficiently suppressed B-RPV.\n\tFurthermore, all the good points also suggest $\\kappa_m \\gtrsim 10^{-2}$ GeV, \n\tsatisfying the requirement for successful BBN physics, c.f. Eq. \\eqref{eq:kLSPBBN}.\n\t\n\t\\subsection{Matter Neutrino Yukawas and B-RPV couplings}\n\n\t\n\tFrom the above analysis we learned that for the $(2,0)$, $(2,1)$ and $(3,0)$ cases we expect a non-suppressed $b_{11}$ to enhance the component of $\\nu$ in the lightest state. As such, we will now consider this coupling to be of order 1 and re-run the analysis for these cases, with the goal being to assess what typical values $\\kappa_m$ and $y_\\nu$ should take for a successful implementation of the proposed Kolda-Martin mechanism.\n\n\t\\vspace{10mm}\n\t\\underline{\\bf $(2,0)$ and $(2,1)$ cases}\n\t\\vspace{5mm}\t\n\t\n\tIn Figures \\ref{fig:(2,0)-Histogram-Ynu} and \\ref{fig:(2,1)-Histogram-Ynu} we see that the preferred points are those with $y_\\nu \\lesssim 10^{-10}$. This suggests that for theses cases, the see-saw mechanism does not take a great role in explaining the light neutrino masses.\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\subfigure[$(2,0)$ case\\label{fig:(2,0)-Histogram-Ynu}]{\\includegraphics[width=0.45\\linewidth]{2,0-Histogram-Ynu.pdf}}\n\t\t\\subfigure[$(2,1)$ case\\label{fig:(2,1)-Histogram-Ynu}]{\\includegraphics[width=0.45\\linewidth]{2,1-Histogram-Ynu.pdf}}\n\t\t\\caption{Histograms for the values of $y_\\nu$ for the $(2,0)$ and $(2,1)$ cases with unsuppressed $b_{11}$}\n\t\\end{figure}\n\t\n\tIn Figures \\ref{fig:(2,0)-Histogram-Kappam} and \\ref{fig:(2,1)-Histogram-Kappam} we see that for these cases, the B-RPV parameter $\\kappa_m$ is naturally very small. This result is easy to understand, considering the main contribution to $\\kappa_m$ to be\n\t\\[\n\t\\kappa_m \\simeq y_\\nu v_m,\n\t\\]\n\tand given the range of values that we are allowing the VEVs to take, $\\kappa_m$ is expected to be small. Unfortunately, all points returning good neutrino physics also return $\\kappa_m > 10^{-2}$ GeV, which means that these classes of models spoil BBN, c.f. \\eqref{eq:kLSPBBN}.\n\tAlthough not shown here one can also find that $\\kappa_X$, $\\kappa_{\\overline X}$ parameters, which mix $L_X$, $\\overline L_X$ with $H_u$, $H_d$ respectively, are also constrained to be smaller than 1 $GeV$.\n\t\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\subfigure[Histogram for values of $\\kappa_m$\\label{fig:(2,0)-Histogram-Kappam}]{\\includegraphics[width=0.45\\linewidth]{2,0-Histogram-Kappam.pdf}}\n\t\t\\subfigure[Histogram for values of $\\kappa_m$\\label{fig:(2,1)-Histogram-Kappam}]{\\includegraphics[width=0.45\\linewidth]{2,1-Histogram-Kappam.pdf}}\n\t\t\\caption{Histograms for the values of $y_\\nu$ and $\\kappa_m$ for the $(2,1)$ case with unsuppressed $b_{11}$. The vertical dashed line represents the bound on the LSP lifetime, c.f. Eq. \\eqref{eq:kLSPBBN}.}\n\t\\end{figure}\n\t\n\t\n\t\\vspace{10mm}\n\t\\underline{\\bf $(3,0)$ case}\n\t\\vspace{5mm}\n\t\n\tFor this realisation of the Kolda-Martin mechanism, the results are slightly different but in line with our expectations. In Figure \\ref{fig:(3,0)-Histogram-Ynu} we can see that the matter Yukawa coupling is allowed to take values larger than in the previous case. This indicates that the see-saw mechanism is having an effect on reducing the contribution of the matter neutrino Dirac mass to the lightest eigenstate.\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\subfigure[Histogram for values of $y_\\nu$\\label{fig:(3,0)-Histogram-Ynu}]{\\includegraphics[width=0.45\\linewidth]{3,0-Histogram-Ynu.pdf}}\n\t\t\\subfigure[Histogram for values of $\\kappa_m$. The vertical dashed line represents the bound on the LSP lifetime, c.f. Eq. \\eqref{eq:kLSPBBN}.\\label{fig:(3,0)-Histogram-Kappam}]{\\includegraphics[width=0.45\\linewidth]{3,0-Histogram-Kappam}.pdf}\n\t\t\\caption{Histograms for the values of $y_\\nu$ and $\\kappa_m$ for the $(3,0)$ case with unsuppressed $b_{11}$}\n\t\\end{figure}\n\t\n\t\n\t\n\tIn Figure \\ref{fig:(3,0)-Histogram-Kappam} we see that $\\kappa_m$ is bound to be smaller than $1$ GeV. The fact that $\\kappa_m$ takes larger values for $(3,0)$ case than for the $n=2$ cases is easily understandable. The main contributions to $\\kappa_m$ are\n\t\\begin{equation}\n\t\\kappa_m \\simeq y_\\nu N + \\lambda N_X\n\t\\end{equation}\n\twhere the VEVs are expected as in Table \\ref{tab:KMSUSYVacua}. These contributions are in general greater than those in $n=2$ cases, but they are still bounded to be smaller than $1$ GeV. This is fortunate, as $\\kappa_m \\gtrsim 10^{-2}$ GeV and hence this class of models retain the successful predictions of BBN, c.f. \\eqref{eq:kLSPBBN}.\nAs before, although not shown here also finds that $\\kappa_X$, $\\kappa_{\\overline X}$ parameter are also constrained to be smaller than 1 $GeV$.\n\t\n\n\\section{Conclusions and Discussion} \\label{sec:conclusion}\n\nIn this paper we have studied the origin of neutrino mass from $SO(10)$ SUSY GUTs arising from\n$M$ Theory compactified on a $G_2$-manifold. We have seen that this problem is linked to \nthe problem of $U(1)_X$ gauge symmetry breaking, which appears \nin the $SU(5)\\times U(1)_X$ subgroup of $SO(10)$, and remains unbroken by the Abelian Wilson\nline breaking mechanism. In order to break the $U(1)_X$ gauge symmetry, we considered a \n(generalised) Kolda-Martin mechanism.\nOur results show that it is possible to break the $U(1)_X$ gauge symmetry\nwithout further SUSY breaking while achieving high-scale VEVs that play a crucial role in \nachieving the desired value of neutrino mass.\n\nThe subsequent induced R-parity violation provides an additional source\nof neutrino mass, in addition to that arising from the seesaw mechanism from non-renormalisable terms. The resulting $11\\times 11$ neutrino mass \nmatrix was analysed for one neutrino family and it was shown how a\nphenomenologically acceptable neutrino mass can emerge. This happens easily for the $(n,k)=(3,0)$ case of the Kolda-Martin mechanism we developed. For this class of models, not only is the neutrino masses phenomenologically viable, but also the physical light neutrino eigenstate is almost entirely composed of the left-handed (weakly charged) state $\\nu$ in the same doublet as the electron $(\\nu, e)$, as desired.\nFurthermore, our analysis showed that the B-RPV parameters, which play an important role in neutrino masses and low-energy dynamics, are in the required range, being smaller than $1$ GeV. Finally, we notice that contrary to the $n=2$ cases, the $n=3$ type of Kolda-Martin mechanism immediately preserves the successful predictions of BBN by allowing the LSP to decay quickly in early universe.\n\nIn conclusion, we have shown that $SO(10)$ SUSY GUTs from $M$ Theory on $G_2$ manifolds \nprovides a phenomenologically viable framework, in which the rank can be broken in the effective theory\nbelow the compactification scale, leading to acceptable values of neutrino mass,\narising from a combination of the seesaw mechanism\nand induced R-parity breaking contributions. \nIn principle the mechanism presented here could be extended to three neutrino families and eventually \ncould be incorporated into a complete theory of flavour, based on $M$ Theory $SO(10)$,\nhowever such questions are beyond the scope of the present paper. \n\n\n\n\n\n\\section*{Acknowledgements}\nThe work of BSA is supported by UK STFC via the research grant ST\/J002798\/1.\nSFK acknowledges support from the STFC Consolidated grant ST\/L000296\/1 and the\nEuropean Union Horizon 2020 research and innovation programme under the Marie \nSklodowska-Curie grant agreements InvisiblesPlus RISE No. 690575 and \nElusives ITN No. 674896. MCR acknowledges support from the FCT under\nthe grant SFRH\/BD\/84234\/2012. CP is supported by the KCL NMS graduate school and ICTP Trieste.\nThe work of KB is supported by a KCL GTA studentship.\n\\bibliographystyle{JHEP}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{S:intro}\n\nIn this paper, we give new results on the $GL(2, \\bR)$ action on moduli spaces of translation surfaces with marked points, and applications such as the following. \n\n\n\\bold{The finite blocking problem.} We say that two, not necessarily distinct, points $x_1,x_2$ on a rational polygon are finitely blocked if there is a finite set $B$ of points such that every billiard trajectory from $x_1$ to $x_2$ passes through a point of $B$. We call a polygon Gaussian if it can be tiled by $(\\frac\\pi4, \\frac\\pi4, \\frac\\pi2)$ triangles in such a way that triangles that share (part of) an edge are related via reflection. Similarly we call it Eisenstein if it can be tiled by $(\\frac\\pi6, \\frac\\pi3, \\frac\\pi2)$ triangles. We adopt the convention that all polygons are required to have connected boundary.\n\n\\begin{thm}\\label{T:poly}\nLet $P$ be a rational polygon. \n\\begin{enumerate}\n\\item If $P$ is Gaussian or Eisenstein, any two points are finitely blocked. \n\\item If $P$ is not Gaussian or Eisenstein and all its angles are multiples of $\\pi\/2$, then possibly infinitely many pairs of points are finitely blocked, but each point is finitely blocked from only finitely many other points. \n\\item Otherwise, only finitely many pairs of points are finitely blocked in $P$. \n\\end{enumerate}\n\\end{thm}\nThe main content is the third statement; the first two are included for completeness and are closely related to previous results (see for example Theorems 1 and 2 of Leli\\`evre, Monteil, Weiss~\\cite{LMW}). We prove Theorem \\ref{T:poly} by showing that every translation surface that is not a branched cover of a torus has a finite set that, together with covering maps to half-translation surfaces, accounts for all finite blocking (see Theorems \\ref{T:cor} and \\ref{T:blockingsets}). Theorem~\\ref{T:poly} builds upon and recovers results of authors such as Gutkin, Hubert, Leli\\`evre, Monteil, Schmidt, Schmoll, Troubetzkoy, Weiss, which we will discuss shortly. Our results also apply to the illumination problem, which is the special case of the finite blocking problem when the blocking set is required to be empty.\n\n\\bold{Affine invariant submanifolds.} Given a partition of $2g-2$ as a sum of positive integers $2g-2=\\sum_{i=1}^s k_i$, define the stratum $\\cH(k_1, \\ldots, k_s)$ to be the orbifold of all translation surfaces $(X,\\omega)$ of genus $g$ where $X$ has genus $g$ and $\\omega$ has zeros of order $k_1, \\ldots, k_i$. A result of Eskin-Mirzakhani-Mohammadi \\cite{EM, EMM} gives that any closed $GL(2, \\bR)$ invariant subset of a stratum is an affine invariant submanifold, which is by definition a properly immersed manifold whose image is locally described by real homogeneous linear equations in period coordinates. \n\n\\bold{Marked points on translation surfaces.} Let $\\cM$ be an affine invariant submanifold of a stratum $\\cH$ of translation surfaces. Let $\\cH^{*n}$ denote the set of surfaces in $\\cH$ with $n$ distinct marked points, none of which coincide with each other or with zeros of the Abelian differential. Let $\\pi: \\cH^{*n}\\to \\cH$ be the map that forgets the marked points. \n\nDefine an $n$-point marking over $\\cM$ to be an affine invariant submanifold $\\cN$ of $\\cH^{*n}$ such that $\\pi(\\cN)$ is equal to a dense subset of $\\cM$ (equivalently, $\\pi(\\cN)$ is equal to $\\cM$ minus a finite, possibly empty, union of smaller dimensional affine invariant submanifolds). Define an $\\cM$-periodic point to be a $1$-point marking over $\\cM$ of the same dimension as $\\cM$. \n\n\\begin{thm}[Eskin-Filip-Wright]\\label{T:periodic}\nAn affine invariant submanifold has infinitely many periodic points if and only if it consists entirely of branched covers of tori.\n\\end{thm}\n\n Section \\ref{SS:EFW} explains why Theorem \\ref{T:periodic} is a special case of results in \\cite{EFW}. \n\n We say that an $n$-point marking $\\cN$ over $\\cM$ is \\emph{reducible} if there is a $k$-point marking $\\cN'$ and a $(n-k)$-point marking $\\cN''$ over $\\cM$ such that $\\cN$ is a component of \n $$\\{(X,\\omega, S'\\cup S''): (X, \\omega, S')\\in \\cN', (X, \\omega, S'')\\in \\cN'', |S_1\\cup S_2|=n\\}.$$\n\n\n\n\n\n\n\n\n\n\n\n\n\n We call a point marking irreducible if it is not reducible. The study of point markings immediately reduces to the irreducible case. \n \n\nThe following result states that when $\\cM$ does not consist entirely of torus covers the only non-obvious ways to mark points over $\\cM$ are to mark $\\cM$-periodic points. This result arose during conversations with Ronen Mukamel. \n\n\\begin{thm}\\label{T:main}\nLet $\\cM$ be an affine invariant submanifold that does not consist entirely of branched covers of tori. Any irreducible $n$-point marking $\\cN$ over $\\cM$ with $n>1$ arises from a half-translation surface covering construction: for any $(X, \\omega, S)\\in \\cN$, there is a map from $(X,\\omega)$ to a half-translation surface that takes $S$ to a single point. \n\\end{thm} \n\nTheorem \\ref{T:main} implies a stronger statement, which we allude to with the terminology ``covering construction\". See Remark \\ref{R:main}.\n In the case that $\\cM$ does consist entirely of branched covers of tori, point markings are easily described, and there are infinitely many $n$-point markings for all $n\\geq 1$.\n \n\n\\bold{Strata of quadratic differentials.} For any connected component of a stratum $\\cQ$ of quadratic differentials, one can form the affine invariant submanifold $\\tilde{\\cQ}$ consisting of all Abelian differentials which arise as double covers (also called square roots) of quadratic differentials in $\\cQ$. Each $(X,\\omega)\\in \\tilde{\\cQ}$ has a natural involution $J$, so that $(X\/J, \\omega^2)\\in \\cQ$. (Since $(X,\\omega)$ might, in unusual cases, have more than one involution, the data of $J$ should be included in a point of $\\tilde\\cQ$, but rather than write $(X,\\omega, J)\\in \\tilde\\cQ$ we suppress this from the notation.)\n\nIf $\\cQ$ consists entirely of hyperelliptic surfaces we say that $\\cQ$ is hyperelliptic. In this case the hyperelliptic involution on $X\/J$ lifts to a hyperelliptic involution on $X$.\n\n\\begin{thm}\\label{T:Q}\nSuppose $\\cQ$ has higher rank. \nIf $\\cQ$ is not hyperelliptic, the only $\\tilde{\\cQ}$-periodic points are fixed points of the involution $J$. If $\\cQ$ is hyperelliptic, the only $\\tilde{\\cQ}$-periodic points are fixed points for $J$ and fixed points for the hyperelliptic involution. \n\\end{thm}\n\n\\bold{Context.} Leli\\`evre, Monteil, and Weiss recently showed that the only translation surfaces in which every pair of points are finitely blocked are covers of tori \\cite{LMW}; we recover and strengthen this in Corollary~\\ref{C:LMW}. Generalizing work of Hubert, Schmoll, and Troubetzkoy \\cite{HST} in the case of lattice surfaces, they also showed that for any point $p$ on any translation surface, the set of points not illuminated by $p$ is finite. For more background on the finite blocking problem and the closely related illumination problem, see \\cite{Mont1, Mont2, HST, LMW}. \n\n\nIn the case of closed $GL(2, \\bR)$ orbits, Theorem \\ref{T:periodic} is due to Gutkin, Hubert, and Schmidt \\cite{GHS} and was established independently by M\\\"oller using algebro-geometric methods \\cite{M2}. M\\\"oller also showed that for non-arithmetic closed orbits in genus 2 the only periodic points are Weierstrass points, using McMullen classification of such closed orbits \\cite{McM:spin, Mc4}. \n\nThe proof of Theorem \\ref{T:main} builds upon and was inspired by an argument of Hubert, Schmoll, and Troubetzkoy \\cite[Theorem 5]{HST}.\n\nLanneau classified connected components of strata of quadratic differentials \\cite{Lconn}, see also \\cite{CM} for a correction. \n\nApisa classified periodic points over connected components of strata of Abelian differentials: they exist only for hyperelliptic connected components, in which case they must be Weierstrass points \\cite{Apisa}. \n\nThere is an unexpected periodic point over the golden eigenform locus in genus 2, see forthcoming work of Eskin-McMullen-Mukamel-Wright and \\cite{KM2}. \n\n\\begin{prob}\nCompute the periodic points over the (Prym) eigenform loci and the Veech-Ward-Bouw-M\\\"oller Teichm\\\"uller curves, as well as the new orbit closures in the forthcoming work of Eskin-McMullen-Mukamel-Wright and \\cite{MMW}. \n\\end{prob}\n\nWhen studying translation surfaces without marked points it is often helpful to consider degenerations which may have marked points \\cite{MirWri}, and indeed Theorems \\ref{T:main} and \\ref{T:Q} have already been used to study $GL(2, \\bR)$ orbit closures of unmarked translation surfaces \\cite{MirWri2}.\n\n\\bold{Organization.} Section \\ref{S:Tmain} proves Theorem \\ref{T:main}. Section \\ref{S:FB} gives our applications to the finite blocking problem, including Theorem \\ref{T:poly}. The remaining sections, which are independent of Section \\ref{S:FB}, prove Theorem \\ref{T:Q}. Section \\ref{S:background} gives the required background, and Section \\ref{S:Q-overview} gives a proof, conditional on two results established in the remaining two sections. The approach is by induction: Section \\ref{S:FindCyl} produces an appropriate cylinder to degenerate, and Section \\ref{S:BaseCase} provides the base case. \n \n\n\\bold{Acknowledgements.} The authors thank Corentin Boissy, Elise Goujard, Erwan Lanneau, Maryam Mirzakhani, Barak Weiss, and Anton Zorich for helpful conversations, and Ronen Mukamel for significant contributions to this paper regarding Theorem \\ref{T:main}. This research was partially conducted during the period AW served as a Clay Research Fellow. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1144082. PA gratefully acknowledges their support.\n\n\\section{Proof of Theorem \\ref{T:main}}\\label{S:Tmain}\n\n\\begin{defn}\nLet $\\cN$ be an irreducible 2-point marking over $\\cM$. Consider any $(X,\\omega, \\{p_1,p_2\\})\\in \\cN$, and let $\\gamma_1$ be a path from a zero of $\\omega$ to $p_1$, and let $\\gamma_2$ be a path from a zero of $\\omega$ to $p_2$. Let $\\Sigma$ denote the set of zeros of $\\omega$ (so $p_1,p_2\\notin \\Sigma$). A linear equation $\\int_{\\gamma_1} \\omega = a\\int_{\\gamma_2} \\omega + \\int_\\gamma \\omega$ must hold for some relative homology class $\\gamma\\in H_1(X, \\Sigma,\\bR)$ and nonzero real number $a$. We define the slope of $\\cN$ to be $a$ or $1\/a$, whichever is larger in absolute value. \n\\end{defn}\n\nThe slope describes the speed at which one marked point moves when the other marked point is moved at unit speed (and the underlying surface $(X,\\omega)$ without marked points is fixed). Note that if the role of $p_1$ and $p_2$ are interchanged, $1\/a$ will play the role of $a$. The slope depends only on $\\cN$ and not on any of the choices made in the definition.\n\n\\begin{rem}\nLet $\\cN$ be an irreducible $n$-point marking over $\\cM$ of dimension $\\dim \\cM+1$. Then if $(X, \\omega, S) \\in \\cN$ and $p_1,p_2 \\in S$, one can define the slope of $p_1$ with respect to $p_2$ in the same way. If $(X, \\omega, S)$ is generic, this will be equal to the slope of the irreducible 2-point marking given by the orbit closure of $(X, \\omega, \\{p_1,p_2\\})$. \n\\end{rem}\n\n\\begin{ex}\nIf $\\cM$ is the hyperelliptic locus in some stratum, and $\\cN$ is given by a pair of points that are exchanged under the hyperelliptic involution, then the slope is -1. \n\\end{ex}\n\n\\begin{ex}\nSuppose that $\\cM$ is an affine invariant submanifold of translation surfaces of genus $g$, such that the generic translation surface in $\\cM$ has a unique map to a translation surface of genus $hn$, that there is an irreducible $n'$-point marking $\\cN'$ over $\\cM$ of dimension $\\dim \\cN=\\dim \\cM +1$, such that for all $(Y, \\eta, P')$ in $ \\cN'$ minus a union of smaller affine invariant submanifolds there is a set $P \\subset P'$ so that $(Y, \\eta, P)\\in \\cN$. \n\nIn other words, in $\\cN$ there are $n$ marked points whose position locally determine each other given the unmarked translation surface, and $\\cN'$ extends these $n$ points to a larger collection of points such that each locally determines all the others. \n\nWe claim there is an upper bound for how large $n'$ may be. Let $\\Sigma'$ be the set of zeros of $\\omega$ and all points $s$ such that $(X,\\omega, s)$ is contained in a $\\cM$-periodic point. Let $T$ be the sum of the cone angles at points of $\\Sigma'$, divided by $\\pi$. \n\n Fix $(X,\\omega, S')\\in \\cN'$, and move the $n'$-marked points around while remaining in the fiber of $\\cN'$ over $(X,\\omega)$. Move one of the marked points along a horizontal seperatrix until it hits a point of $\\Sigma'$. By Lemma \\ref{L:boundaryperiodic}, all marked points must then lie at points of $\\Sigma'$. \n\nBy Theorem \\ref{T:Slope1}, the slope for any pair of these points is 1 or -1, so each marked point must have travelled along a different directed horizontal line segment towards a point of $\\Sigma'$. There are exactly $T$ such directed horizontal line segments. The claim is proved. \n\nNow assume $n'$ as above was maximal. Then two points not in $\\Sigma'$ being simultaneously marked is in fact an equivalence relation. Indeed, if two sets of $n'$-points partially overlapped, then their union would contradict the maximality of $n'$. This uses that $(X,\\omega, S')$ has dense orbit. \n\nThe quotient of $(X,\\omega)\\setminus \\Sigma'$ by this equivalence relation gives a map to a punctured surface with an atlas of charts to $\\bC$ whose transition functions are translation and translations composed with multiplication by $-1$. This map extends continuously to a map $f$ from $(X,\\omega)$ to the metric completion of the punctured surface, and this $f$ is the desired map. \n\\end{proof}\n\n\\begin{lem}\\label{L:dimplus1}\nSuppose that $\\cM$ has only finitely many $\\cM$-periodic points. Let $\\cN$ be an irreducible $n$-point marking with $n>2$. Then $\\dim \\cN=\\dim \\cM+1$. \n\\end{lem}\n\n\\begin{proof}\nSuppose the lemma is not true. Consider a counterexample with $\\dim \\cN- \\dim \\cM>1$ minimal, and in the smallest genus that admits a counterexample for this minimal value of $\\dim \\cN- \\dim\\cM$.\n\nPick $(X,\\omega, S) \\in \\cN$ such that the $GL(2,\\bR)$ orbit of $(X,\\omega, S)$ is dense in $\\cN$. Let $\\Sigma'$ be defined as above; since $\\cN$ is irreducible we have that $S$ is disjoint from $\\Sigma'$. \n\nFor each $p\\in S$, let $\\gamma_p$ be a path from a zero of $\\omega$ to $p$. Since $S$ is irreducible, and since $\\dim \\cM +n-1 \\geq\\dim \\cN\\geq \\dim \\cM+2$, we may pick three points $p_1,p_2,p_3 \\in S$ so that exactly one equation of the form $ \\sum a_i \\gamma_{p_i}=\\gamma$ for $\\gamma \\in H^1(X, \\Sigma, \\bR)$ holds, up to adding one of the defining equations of $\\cM$ to $\\gamma$. \n\nIn other words, the three points $p_i$ move with two (complex) degrees of freedom in the fiber of $\\cN$ over $(X,\\omega)$. Hence we may move $p_1$ to $\\Sigma'$ without moving the other two points, $p_2$ and $p_3$, to $\\Sigma'$. This gives a point marking $\\mathcal{N}'$ over $\\cM$ with fewer marked points and with $\\dim \\cN'=\\dim \\cN-1$. We do not know if $\\cN'$ is irreducible, but it can be expressed as a union of irreducibles, one of which, call it $\\cN''$, contains $(X, \\omega, S')$ for some $S'$ containing $p_2$ and $p_3$. (The two points $p_1$ and $p_2$ must be in the same irreducible point marking because there is an equation relating $\\gamma_{p_1}$ to $\\gamma_{p_2}$.)\n\nBecause we chose the counterexample $\\cN$ with $\\dim \\cN- \\dim \\cM>1$ as small as possible, we get that $\\dim \\cN''= \\dim \\cM+1$. Without loss of generality, assume $a_2$ and $a_3$ have the same sign. This forces the slope of the 2-point marking given by $\\{p_2, p_3\\}$ to be 1. Now, the proof of Lemma \\ref{L:producecover} shows that $(X,\\omega)$ covers a smaller genus translation surface. (Lemma \\ref{L:producecover} itself says it covers a quadratic differential, but one can easily see that in the slope 1 case there is also a map to a translation surface of positive degree. Since $\\cM$ has only finitely many periodic points, the genus is greater than 1. A positive degree map from a surface of genus greater than 1 must decrease the genus.) \n\nNow, we may push $S$ forward under this map, and construct in this way a counterexample in smaller genus. Note that since $\\cN$ is irreducible, we may assume that for generic $(X,\\omega, S)\\in \\cN$ no two points of $S$ will map to the same point on the lower genus translation surface. Thus we have contradicted the fact that we have chosen $\\cN$ in minimal genus. \n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{T:main}]\nBy Theorem \\ref{T:periodic}, there are only finitely many $\\cM$-periodic points. By Lemma \\ref{L:dimplus1}, we have $\\dim \\cN=\\dim \\cM+1$. Hence Lemma \\ref{L:producecover} gives the result. \n\\end{proof}\n\n\n\n\n\n\n\\section{The finite blocking problem}\\label{S:FB}\n\nIn the first subsection we explain the implications of Theorem \\ref{T:main} for finitely blocked points; in the next two we give applications; and in the final subsection we study possible finite blocking sets. The final three subsections can be read independently of each other but all rely on the first. \n\n\\subsection{Consequences of Theorem \\ref{T:main}}\nThroughout this subsection let $(X, \\omega)$ be a translation surface. Given two not necessarily distinct points $x_1$ and $x_2$ on $(X, \\omega)$ the finite blocking problem asks whether all straight line paths between $x_1$ and $x_2$ may be blocked by a finite collection of points $B$. If this is possible then we say that $x_1$ and $x_2$ are blocked by $B$. The following lemma provides an example of this phenomenon. \n\n\\begin{lem}\\label{L:involution-blocking}\nSuppose that $(X, \\omega)$ has an involution $j$ so that $j^* \\omega = - \\omega$. For any point $p$ that is not a zero and is not fixed by $j$, $p$ and $j(p)$ are finitely blocked by the fixed points of $j$.\n\\end{lem}\n\\begin{proof}\nLet $\\ell$ be a line segment in $(X,\\omega)$ joining $p$ to $j(p)$. Since $j^*\\omega=-\\omega$, we get that $j$ maps $\\ell$ to itself, and hence contains a fixed point in its interior.\n\\end{proof}\n\n Leli\\`evre, Monteil, and Weiss showed that if $(X, \\omega)$ is a translation cover of a torus then any two points are finitely blocked (and, conversely, that this property characterizes torus covers) \\cite[Theorem 1]{LMW}. \n \n \\begin{ass}\n Suppose throughout this section that $(X, \\omega)$ is not a translation cover of a torus.\n \\end{ass}\n\nRecall the result of M\\\"oller that states that there is a unique map $\\pi_{X_{\\operatorname{min}}}:(X,\\omega)\\to \\Xmin$ to a translation surface of minimal genus, and any map from $(X,\\omega)$ to a translation surface is a factor of this map \\cite[Theorem 2.6]{M2}. This can be extended to quadratic differentials as follows. \n\n\\begin{lem}\\label{L:minquad}\nThere is a quadratic differential $(Q_{\\operatorname{min}}, q_{\\operatorname{min}})$ with a degree 1 or 2 map $\\Xmin\\to (Q_{\\operatorname{min}}, q_{\\operatorname{min}})$ such that any map from $(X,\\omega)$ to a quadratic differential is a factor of the composite map $\\pi_{Q_{\\operatorname{min}}}:(X,\\omega)\\to (Q_{\\operatorname{min}}, q_{\\operatorname{min}})$.\n\\end{lem}\n\n\\begin{proof}\nIf $(X,\\omega)$ does not admit any maps to strictly half translation surfaces, we may set $(Q_{\\operatorname{min}}, q_{\\operatorname{min}})=\\Xmin$. \n\nSo suppose there is a map $h: (X,\\omega)\\to (Q', q')$, where $(Q',q')$ is not the square of an Abelian differential. \nRecall that any map from a translation surface to a quadratic differential lifts to a map from the translation surface to the square root of the quadratic differential. Let $(X',\\omega') \\to (Q',q')$ be the square root of $(Q',q')$, and let $J$ be the involution on $(X',\\omega')$ so $(Q',q')=(X',\\omega')\/J$. By the defining property of $\\Xmin$, there exists a map $\\pi: (X',\\omega')\\to \\Xmin$ through which the map $\\pi\\circ J : (X',\\omega')\\to (X_{\\operatorname{min}}, -\\omega_{\\operatorname{min}})$ factors. Hence $X_{\\operatorname{min}}$ must have a self-map $j$ negating $\\omega_{\\operatorname{min}}$ and satisfying $\\pi = j \\circ \\pi \\circ J$. Since $X_{\\operatorname{min}}$ has genus greater than 1, $j$ must be an involution. \n\n\nSince $\\Xmin$ does not cover a smaller genus translation surface, it has at most one involution negating $\\omega_{\\operatorname{min}}$. Hence the lemma is true with $(Q_{\\operatorname{min}}, q_{\\operatorname{min}})=\\Xmin\/j$. \n\\end{proof}\n\nWe define a point in a point marking to be free if it can be moved freely, independently of the unmarked surface and the position of the other points in the point marking.\n\n \\begin{rem}\\label{R:main}\nGiven $\\cM$, let $\\cM_{\\operatorname{min}}$ denote the set of all $(Q_{\\operatorname{min}}, q_{\\operatorname{min}})$ arising from all $(X, \\omega)\\in \\cM$. There is a natural point marking $\\cM_{\\operatorname{min}}^{\\mathrm{br}}$ over $\\cM_{\\operatorname{min}}$ which consists of all $(Q_\\mathrm{min}, q_{\\mathrm{min}}, B)$ such that there is a cover $(X, \\omega)\\to (Q_\\mathrm{min}, q_{\\mathrm{min}})$ branched over $B$, with $(X,\\omega)\\in \\cM$ and $B$ of maximal size. \nIf $\\cN$ is an irreducible $n$-point marking over $\\cM$ with $n>1$, then for each $(X, \\omega, S)\\in \\cN$, Theorem \\ref{T:main} gives that $S$ maps to a single point $p\\in (Q_\\mathrm{min}, q_{\\mathrm{min}})$. We can then consider the point marking $\\cN_\\mathrm{min}$ which consists of all $(Q_\\mathrm{min}, q_{\\mathrm{min}}, B\\cup\\{p\\})$ that arise in this way. By Theorem \\ref{T:main}, the point marking $\\cN_\\mathrm{min}$ over $\\cM_\\mathrm{min}$ consists of a number of $\\cN_\\mathrm{min}$-periodic points together with a number of free marked points. \n\nNote that in general $S$ could be a proper subset of the fiber of $p$, and that $p$ must be free; it cannot be an $\\cN_\\mathrm{min}$-periodic point.\n \\end{rem} \n\n\n\\begin{thm}\\label{T:cor}\nIf $x_1$ and $x_2$ are finitely blocked on $(X,\\omega)$, then either they are both $\\cM$-periodic points or zeros, where $\\cM$ is the orbit closure of $(X,\\omega)$, or $\\pi_{Q_{\\operatorname{min}}} (x_1)=\\pi_{Q_{\\operatorname{min}}} (x_2)$. \n\\end{thm}\n\nTo prove this theorem, which is the main result of this subsection, we first require two lemmas. Given two points $x_1$ and $x_2$ that are finitely blocked by a collection of points $B$, let $\\mathcal{M}_{x_1,x_2,B}$ be the $\\mathrm{GL}_2(\\mathbb R)$ orbit closure of $(X, \\omega; p, q; B)$ in $\\cH^{*n+2}$ where $n$ is the size of $B$. We would like to permit the points $x_1$ and $x_2$ to coincide and to be zeros. We will use the same notation, but if this happens the orbit closure will be taken in $\\cH^{*n+1}$ after forgetting one of the redundant points are deleting a zero (if there are two zeros then delete both and take an orbit closure in $\\cH^{*n}$). Finally, in order to refer to specific zeros, we will work on a finite cover of $\\mathcal{M}$ where the zeros are labelled. We will suppress these details in the sequel.\n\n\\begin{lemma}\\label{L:OB}\nIf $(X', \\omega'; x_1', x_2'; B')$ belongs to $\\mathcal{M}_{x_1,x_2,B}$ then $x_1'$ and $x_2'$ are blocked by $B'$.\n\\end{lemma}\n\n\\begin{proof}\nThe locus of $(X', \\omega'; x_1', x_2'; B')$ such that there is a straight line segment from $x_1'$ to $x_2'$ not intersecting $B'$ or the zeros of $\\omega'$ is open and $\\mathrm{GL}_2(\\mathbb R)$ invariant. \n\\end{proof}\n\nWe define a blocking set to be minimal if no proper subset also blocks the two points. \n\n\\begin{lemma}\\label{L:MB}\nNeither $x_1$ nor $x_2$ is free in $\\mathcal{M}_{x_1,x_2,B}$. If $B$ is minimal, then locally in $\\mathcal{M}_{x_1,x_2,B}$ the position of the points in $B$ are determined by the unmarked surface and $x_1,x_2$. \n\\end{lemma}\n\n\\begin{proof}\nIf $x_1$ is free, we can move it into a small ball around $x_2$ that doesn't contain any points of $B$, and find a straight line segment from $x_1$ to $x_2$ not intersecting $B$. \n\nIf some points in $B$ could be moved without changing the underlying unmarked surface or the position of $x_1,x_2$, we could move at least one of these points off the countable collection of line segments from $x_1$ to $x_2$ to obtain a smaller finite blocking set. \n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{T:cor}]\nLet $B$ be a minimal finite blocking set. \n\nFirst we suppose that $x_1$ is $\\cM$-periodic or a zero and show that so is $x_2$. Suppose otherwise. Let $U$ be a flat disk around $x_1$ on $(X, \\omega)$ on which $\\pi_{Q_{\\operatorname{min}}} (z)=z^n$ in a local coordinate $z$ centered at $x_1$. Remaining in $\\mathcal{M}_{x_1,x_2;B}$ we may move $x_2$ into $U$ without moving $x_1$. Assume $x_2$ is closer to $x_1$ than any other periodic point or zero. Now, there must be a point $b$ of $B$ also contained in $U$, blocking the straight line from $x_1$ to $x_2$ in $U$. \n\nLet $B'$ be the non-empty set of non-periodic points in $B$. Lemma~\\ref{L:MB} gives that $B'\\cup \\{x_2\\}$ is an irreducible point marking, so we see by Theorem \\ref{T:main} that $B'\\cup \\{x_2\\}$ maps to a single point under $\\pi_{Q_{\\operatorname{min}}} $. In particular, $b$ and $x_2$ must map to the same point under $\\pi_{Q_{\\operatorname{min}}} $. This is a contradiction, since $b$ is closer to $x_1$ than $x_2$. \n\nNext suppose neither $x_1$ nor $x_2$ is periodic. By the previous lemma, neither is free in $\\mathcal{M}_{x_1, x_2, B}$, so by Theorem \\ref{T:main} they must map to the same point under $\\pi_{Q_{\\operatorname{min}}} $. \n\\end{proof}\n\n\\begin{cor}\\label{C:LMW}\nA non-singular point on a translation surface that is not a torus-cover is only finitely blocked from finitely many other points.\n\\end{cor}\n\\begin{proof}\nLet $p$ be a non-singular point on a translation surface that is not a torus cover. If $p$ is periodic then it is only finitely blocked from other periodic points, of which there are finitely many by Theorem~\\ref{T:periodic}. If $p$ is not periodic then it is only finitely blocked from other points in $\\pi_{Q_{\\operatorname{min}}}^{-1} \\left( \\pi_{Q_{\\operatorname{min}}}(p) \\right)$, of which there are only finitely many. \n\\end{proof}\n\n\n\n\\subsection{$k$-differentials, $k>2$.}\nThroughout this section we will suppose that $(S, \\theta)$ is a Riemann surface $S$ with a $k$-differential $\\theta$, $k>2$, and $\\theta$ is not a power of a lower order differential. Let $(X, \\omega)$ be the canonical unfolding of $(S, \\theta)$ to an Abelian differential, which comes with a map $\\pi_S:(X, \\omega) \\to (S,\\theta)$. In this section we will prove the following:\n\n\\begin{thm}\\label{T:FBK}\nIf $(X, \\omega)$ is not a translation covering of a torus then there are only finitely many pairs of finitely blocked points on $(S, \\theta)$.\n\\end{thm}\n\n\\begin{proof}\n $X$ has a rotational self-symmetry $T$ of order $k$ with $(S,\\theta)=(X,\\omega)\/\\langle T\\rangle$. As in the proof of Lemma \\ref{L:minquad}, we see that it descends to an automorphism $t$ of $X_{\\operatorname{min}}$, with $\\pi_{X_{\\operatorname{min}}} \\circ T= t\\circ \\pi_{X_{\\operatorname{min}}}$. \n\nIf $s_1$ and $s_2$ are finitely blocked on $(S, \\theta)$, then the set $\\pi_S^{-1}(s_1)$ is finitely blocked from the set $\\pi^{-1}(s_2)$, which means there is a finite set $B$ such that every straight line segment from one set to the other intersects $B$. Equivalently, every point of one set is finitely blocked from every point in the second set.\n\nSuppose that $s_1$ or $s_2$ is such that $\\pi_S^{-1}(s_1)$ and $\\pi_S^{-1}(s_2)$ do not contain any $\\cM$-periodic points and do not map to any of the finitely many points in $\\Xmin$ that are fixed by a non-trivial power of $t$. (As usual, $\\cM$ is the orbit closure of $(X,\\omega)$.)\n\nWe will show that $s_1$ and $s_2$ are not finitely blocked. Suppose in order to find a contradiction that they are. Consider a point of $\\pi_S^{-1}(s_1)$ and a point of $\\pi_S^{-1}(s_2)$. Since these two points are finitely blocked, we see that they map to the same point in $(Q_{\\operatorname{min}}, q_{\\operatorname{min}})$. Hence $\\pi_{Q_{\\operatorname{min}}} $ maps $\\pi_S^{-1}(s_1)$ to a single point of $(Q_{\\operatorname{min}}, q_{\\operatorname{min}})$. But $\\pi_S^{-1}(s_1)$ is a $T$ orbit, so its image on $\\Xmin$ must be a $t$ orbit of size at most two, which is a contradiction since $k>2$. \n\\end{proof}\n\nRecall that our convention is that polygons are assumed to have connected boundary.\n\n\\begin{prop}\\label{P:toruscover}\nA rational polygon unfolds to the cover of a torus if and only if the polygon is Gaussian or Eisenstein. \n\\end{prop}\n\n\\begin{proof}\nLet $P$ be a rational polygon and suppose its unfolding $(X,\\omega)$ is a torus cover; we will show $P$ is Gaussian or Eisenstein. (The other direction is easy.) \n\nLet $k$ be the least common denominator of the angles divided by $\\pi$, so $X$ admits an order $k$ symmetry $T$ with $T^*(\\omega)=\\xi\\omega$, where $\\xi$ is a primitive $k$-th root of unity.\n\nBy assumption, $\\omega$ lies in a two dimensional subspace of $H^1(X, \\bC)$ defined over $\\bQ$, spanned by $\\omega$ and its complex conjugate. Since $T^*(\\omega)=\\xi\\omega$, this subspace is invariant under $T^*$. Hence $T^*$ restricted to this rational subspace must have all Galois conjugates of $\\xi$ as eigenvalues. Hence, the degree of $\\bQ(\\xi)$ as a field extension of $\\bQ$ is at most 2, and we conclude that $k\\in \\{2,3,4, 6\\}$.\n\nFor each zero of $\\omega$ there is some non-trivial power of $T$ that fixes it. Hence if $p:H^1(X,\\Sigma, \\bC)\\to H^1(X,\\bC)$ is the usual map from cohomology relative to the set $\\Sigma$ of zeros of $\\omega$ to absolute cohomology, we get that $T$ acting on $\\ker(p)$ does not have any any primitive $k$-th roots of unity as eigenvalues. (Here we count preimages of corners of $P$ of angle $\\frac{\\pi}{k}$ as zeros of order zero and include them in $\\Sigma$.) Hence the dimension of the $\\xi$-eigenspace of $T$ is the same in absolute and relative cohomology. \n\nThe sum of primitive eigenspaces of $T$ in relative and absolute cohomology are both defined over $\\mathbb Q$. Since $p$ induces a $\\mathbb Q$-linear isomorphism between them it follows that the relative periods of $\\omega$ are rational linear combinations of the absolute periods of $\\omega$. \n\n\nSince $k\\in \\{2, 3,4, 6\\}$ the periods span a lattice in $\\bC$, and after rotating and scaling the relative periods lie in $\\bQ[\\xi]$. Assuming $k \\in \\{ 2, 4 \\}$ gives that $P$ is Gaussian and assuming $k\\in \\{3,6\\}$ gives that $P$ is Eisenstein. \n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{T:poly}]\nIf $P$ is Gaussian or Eisenstein, then it unfolds to a torus cover, where it is known that any two points are finitely blocked. Hence any two sets of points are finitely blocked (just take the union of the blocking sets). \n\nSuppose now that $P$ is not Gaussian or Eisenstein. By Proposition~\\ref{P:toruscover}, $P$ does not unfold to a torus cover and so there are only finitely many periodic points on the unfolding and any point is finitely blocked from only finitely many others by Theorem~\\ref{T:cor}. If some angle is not an integer multiple of $\\frac{\\pi}{2}$ then the pillowcase double of $P$ is a $k$-differential for $k > 2$ and so Theorem~\\ref{T:FBK} implies that there are only finitely many pairs of finitely blocked points on $P$.\n\\end{proof}\n\n\\subsection{Prime triangles.}\nTheorem \\ref{T:main} can sometimes be applied without knowing the orbit closure of a translation surface, since both flat and algebro-geometric methods exist to restrict the number of periodic points on a translation surface without knowing the orbit closure. Here is one example. \n\n\\begin{thm}\\label{T:prime}\nConsider a triangle with angles $\\frac{a}{\\ell}\\pi, \\frac{b}{\\ell}\\pi, \\frac{c}{\\ell}\\pi$ with $\\ell>3$ prime and $\\{a, b, c\\} \\ne \\{1, 2, 4\\}$. If two points are finitely blocked they are both vertices and the minimal blocking set is the collection of vertices $V$ in the non-isosceles case and $V \\cup \\{m\\}$ where $m$ is the midpoint of the line joining the two vertices of equal angle in the isosceles case.\n\\end{thm}\n\\begin{rem}\nThe authors have verified that the result still holds for the $(1,2,4)$ triangle, but have chosen to omit the proof\n\\end{rem}\n\\begin{rem}\nWe permit billiard paths to run along the edge of the table.\n\\end{rem}\n\n\n If $\\ell=3$, the triangle unfolds to a torus and any two points are finitely blocked. We require the following deep result due to Tzermias~\\cite[Theorem 1.1]{Tzermias} in the nonhyperelliptic case and Grant-Shaulis~\\cite[Theorem 1.1]{Grant-Shaulis} in the hyperelliptic case and which builds on work of Coleman~\\cite{Coleman-etale} and Coleman-Tamagawa-Tzermias~\\cite{CTT-Fermat}.\n\n\\begin{thm}\\label{T:prime2}\nLet $(X,\\omega)$ be the unfolding of a triangle with angles $\\frac{a}{\\ell}\\pi, \\frac{b}{\\ell}\\pi, \\frac{c}{\\ell}\\pi$ with $\\ell>5$ prime and so that $(a, b, c) \\ne (1,2,4)$. The only points of $(X,\\omega)$ whose difference from a branch point is torsion are branch points and, when $(X,\\omega)$ is hyperelliptic, Weierstrass points. \n\\end{thm}\n\n\\begin{proof}[Proof of Theorem \\ref{T:prime}.]\nBy work of Filip, the difference between any two points of $(X,\\omega)$ must be torsion in the Jacobian \\cite{Fi2}. (In general, Filip allows for more complicated twisted torsion relations, but to have non-trivial twisting one must consider relations between at least 3 points. In general, Filip also allows for the difference to be merely torsion in a factor of the Jacobian, but since $\\ell$ is prime the relevant factor is in fact the whole Jacobian.)\n\nWhen $\\ell$ is prime, $(X,\\omega)$ does not cover any smaller genus translation surface. So $(X,\\omega)=\\Xmin$, and either it is hyperelliptic or it is also equal to $(Q_{\\operatorname{min}}, q_{\\operatorname{min}})$. By Theorem~\\ref{T:cor} the only pairs of finitely blocked points are points that unfold to periodic points. \n\nSuppose first that $(X, \\omega)$ is not hyperelliptic. When $\\ell > 7$, Theorem~\\ref{T:prime2} implies that the only points that unfold to periodic points are vertices of the triangle. By Lemma~\\ref{L:MB}, a minimal blocking set of two finitely blocked periodic points consists of only periodic points. \n\nNow suppose $(X,\\omega)$ is hyperelliptic. This happens if and only if the triangle is isosceles (see for example \\cite[Section 4]{Coleman-etale}). When $\\ell > 5$, Theorem~\\ref{T:prime2} states the only points that unfold to periodic points (aside from the vertices) are points that unfold to Weierstrass points. The only such point on an isosceles triangle is the midpoint $m$ of the edge between the two vertices of equal angle. \n\nBy Lemma~\\ref{L:MB}, a minimal blocking set of two finitely blocked periodic points consists of only periodic points. Therefore, $m$ is not finitely blocked from any of the vertices of the triangle. The trajectories shown in Figure~\\ref{F:Fagnano} shows that $m$ is not blocked from itself. Therefore, in the hyperelliptic case we have also shown that the only finitely blocked points are pairs of vertices. \n\\begin{figure}[h!]\n \\begin{subfigure}[b]{0.4\\textwidth}\n \\centering\n \\resizebox{.8\\linewidth}{!}{\\begin{tikzpicture} \t\t\n \t\t\\draw (0,0) -- (2, 2) -- (4, 0) -- (0,0);\n \t \\draw[dashed] (2,0) -- (1,1) -- (3,1) -- (2,0);\n\t\\node at (.4, .2) {$\\theta$}; \\node at (3.6, .2) {$\\theta$}; \\node at (1.4, 1.2) {$\\theta$}; \\node at (2.6, 1.2) {$\\theta$};\n\t\t\\draw[black, fill] (2, 0) circle[radius = 1.6pt]; \\node at (2, -.2) {$m$};\n \\end{tikzpicture} }\n \\label{SF:blah1}\n \\caption{ $\\theta > \\pi\/4$ }\n \\end{subfigure}\n \\qquad\n \\begin{subfigure}[b]{0.4\\textwidth}\n \\centering\n \\resizebox{.8\\linewidth}{!}{\\begin{tikzpicture}\n \\draw (0,0) -- (2, 2) -- (4, 0) -- (0,0);\n \\draw[dashed] (1,1) -- (2,0) -- (3,1);\n\t\\node at (.4, .2) {$\\theta$}; \\node at (3.6, .2) {$\\theta$}; \\node at (1.6, .2) {$\\theta$}; \\node at (2.4, .2) {$\\theta$};\n\t\t\\draw[black, fill] (2, 0) circle[radius = 1.6pt]; \\node at (2, -.2) {$m$};\n \\end{tikzpicture} }\n \\label{SF:blah2}\n \\caption{ $\\theta = \\pi\/4$ }\n \\end{subfigure}\n \\qquad\n \\begin{subfigure}[b]{0.4\\textwidth}\n \\centering\n \\resizebox{.8\\linewidth}{!}{\\begin{tikzpicture}\n \\draw (0,0) -- (2, 2) -- (4, 0) -- (0,0);\n \\draw[dashed] (1,0) -- (1,1) -- (2,0) -- (3,1) -- (3,0);\n\t\\node at (.4, .2) {$\\theta$}; \\node at (3.6, .2) {$\\theta$}; \\node at (1.2, .4) {$2\\theta$}; \\node at (2.8, .4) {$2\\theta$};\n\t\t\\draw[black, fill] (2, 0) circle[radius = 1.6pt]; \\node at (2, -.2) {$m$};\n \\end{tikzpicture} }\n \\label{SF:blah2}\n \\caption{ $\\theta < \\pi\/4$ }\n \\end{subfigure}\n\\caption{The degenerate Fagnano trajectory in an isosceles triangle}\n\\label{F:Fagnano}\n\\end{figure}\n\nFor the two excluded primes $\\ell = 5$ and $\\ell = 7$ we have the following special arguments. Both triangles with $\\ell = 5$ unfold to Teichm\\\"uller curves in genus two and hence M\\\"oller~\\cite{M2} implies that the only periodic points are zeros and Weierstrass points. For $\\ell = 7$ the only triangle that is not isosceles is the $(1,2,4)$ triangle and Theorem~\\ref{T:prime2} classifies the periodic points on the unfolding of the isosceles triangles. Since we only must consider isosceles triangles in these cases, the previous analysis applies and we are done.\n\\begin{comment}\nThe unfolding of the $(1,2,4)$ triangle can be described as follows. For simplicity let $\\alpha = \\frac{2\\pi}{7}$ and let $r = \\frac{\\sin(\\alpha)}{\\sin(2\\alpha)}$. Label the seven solutions of $z^7 = 1$ and the seven solutions $\\frac{z^{14} - r^{14}}{z^7 - r^7}$. Connect each seventh root of unity to the two labelled points closest to it on the circle of radius $r$. There is a unique identification of the sides to make the resulting $14$-gon a translation surface. This translation surface is the unfolding of the $(1,2,4)$ triangle. The translation surface is vertically periodic. A picture that shows how the four cylinders are glued together (but which is otherwise an inaccurate picture) is shown below\n\n\\begin{figure}[H]\n \\centering\n\t\\begin{tikzpicture}\n \t\t\\draw (0,0) -- (0,1) -- (-1, 1) -- (-1, 2) -- (0,2) -- (0, 3) -- (1,3) -- (1, 4) -- (2, 4) -- (2, 0) -- (0,0);\n\t\t\\draw[dotted] (1,0) -- (1,1) -- (2,1) -- (0,1) -- (0,2);\n\t\t\\draw[dotted] (0,2) -- (2,2) -- (2,3);\n\t\t\\draw[dotted] (1,0) -- (1,3) -- (2, 3);\n\t\t\\draw[black, fill] (0, 1) circle[radius = 1.6pt];\n\t\t\\node at (-.5, .75) {$1$}; \\node at (.5, .-.25) {$2$}; \\node at (1.5, -.25) {$3$}; \n\t\t\\node at (-.5, 2.25) {$3$}; \\node at (.5, 3.25) {$2$}; \\node at (1.5, 4.25) {$1$}; \n\t\t\\node at (1, .5) {$A$}; \\node at (.5, 1.5) {$B$}; \\node at (1, 2.5) {$D$}; \\node at (1.5, 3.5) {$C$};\n\t\\end{tikzpicture}\n\\end{figure}\n\nWe compute the lengths of the cylinders to be \n\\[ \\ell_C = 2r \\sin \\alpha \\quad \\ell_A = 2 \\sin(2 \\alpha) \\quad \\ell_B = \\ell_A + \\ell_C \\]\nand the heights of \n\\[ h_C = \\cos(\\alpha) + r \\cos(2 \\alpha) \\qquad h_A = \\cos(2 \\alpha) + r \\cos(\\alpha) \\]\nThe ratio of modulus of cylinder $A$ to cylinder $C$ is\n\\[ \\frac{m_A}{m_C} = \\frac{\\cos(2 \\alpha) + r \\cos(\\alpha)}{\\cos(\\alpha) + r \\cos(2 \\alpha)} \\cdot \\frac{2r \\sin \\alpha}{2 \\sin(2 \\alpha)} = \\frac{\\sin(\\alpha) \\tan(\\alpha) + \\frac{1}{2} \\sin(\\alpha) }{ \\cos(\\alpha) \\sin(2 \\alpha) + \\sin(\\alpha) \\cos(2 \\alpha)} \\]\nSimplifying,\n\\[ \\frac{m_A}{m_C} = \\frac{\\sin(\\alpha)}{\\sin(3\\alpha)} \\left( \\cot(2\\alpha) + \\frac{1}{2} \\right) \\]\nGalois theory implies that this number is irrational. Therefore the $(1,2,4)$ triangle does not unfold to a Teichm\\\"uller curve. By Aulicino-Nguyen~\\cite{AN} either the $(1,2,4)$ triangle unfolds to a surface that has a rank one orbit or it unfolds to a surface with a dense orbit. If it unfolds to a surface with a dense orbit, then Apisa~\\cite{Apisa} will imply that the only periodic points are zeroes. Suppose to a contradiction that the unfolding is a rank one rel one orbit closure. The relative deformation is given by $\\gamma_A + \\gamma_C - \\gamma_B$. Therefore, the rational relation that the moduli of the cylinders satisfies implies that there are rational numbers $q_\\cdot$ so that\n\\[ q_A \\frac{h_A + t}{\\ell_A} + q_B \\frac{h_B - t}{\\ell_B} + q_C \\frac{h_C + t}{\\ell_C} + q_D \\frac{h_D }{\\ell_D} = 0 \\]\nfor all $t$. Therefore,\n\\[ q_A \\frac{1}{\\ell_A} - q_B \\frac{1}{\\ell_A + \\ell_C} + q_C \\frac{1}{\\ell_C} = 0 \\]\nMultiplying through by $\\ell_A + \\ell_C$ yields that\n\\[ q_A x + q_C x^{-1} = c \\]\nwhere $x = \\frac{\\ell_C}{\\ell_A} = r^2$ and $c$ is some rational constant. We see that since $x$ is not rational and since the rational relation is not trivial that $q_A$ and $q_C$ are both nonzero. Therefore, we see that $x$ satisfies\n\\[ q_A x^2 - cx + q_C = 0 \\]\nHowever, $\\mathbb Q(x)$ is a degree three extension of $\\mathbb Q$ and not a quadratic extension, so we have a contradiction. \n\\end{comment}\n\\end{proof}\n\n\n\n\\subsection{Description of blocking sets.}\nAssume that $(X,\\omega)$ is not a torus cover. \n\n\\begin{thm}\\label{T:blockingsets}\nLet $x_1$ and $x_2$ be finitely blocked on $(X,\\omega)$, and let $B$ be any minimal blocking set. If $x_1$ and $x_2$ are periodic points, then so are all points in $B$. Otherwise, one of the following holds:\n\\begin{enumerate} \n\\item If $\\pi_{X_{\\operatorname{min}}}(x_1)=\\pi_{X_{\\operatorname{min}}}(x_2)$, then $B$ does not contain any periodic points and $\\pi_{X_{\\operatorname{min}}}$ maps $\\{x_1,x_2\\}\\cup B$ to a single point. \n\\item If $\\pi_{X_{\\operatorname{min}}}(x_1)\\neq \\pi_{X_{\\operatorname{min}}}(x_2)$ but $\\pi_{Q_{\\operatorname{min}}} (x_1)=\\pi_{Q_{\\operatorname{min}}} (x_2)$, and if $B'$ is the set of non-periodic points in $B$, then $\\pi_{Q_{\\operatorname{min}}} $ maps $\\{x_1,x_2\\}\\cup B'$ to a single point. \n\\end{enumerate}\n\\end{thm}\n\n\\begin{rem}\nIn the case of two periodic points and the case of two points that are identified under $\\pi_{X_{\\operatorname{min}}}$, the converse - i.e. that any such points $x_1,x_2$ must be finitely blocked - is false. However, in the third case, the converse follows from Lemma~\\ref{L:involution-blocking}, which shows that $x_1$ and $x_2$ are finitely blocked by the preimages under $\\pi_{X_{\\operatorname{min}}}$ of fixed points of the involution on $X_{min}$. In particular, a minimal blocking set is contained in the set of periodic points in this case.\n\\end{rem} \n\n\n\\begin{proof}[Proof of Theorem \\ref{T:blockingsets}]\nPart 1 follows from Lemma \\ref{L:MB}. So assume neither $x_1$ nor $x_2$ is periodic. \n\nThen Lemma \\ref{L:MB} and Theorem \\ref{T:main} gives that $\\pi_{Q_{\\operatorname{min}}} (x_1)=\\pi_{Q_{\\operatorname{min}}} (x_2)$ and the blocking sets consists of periodic points and the $\\pi_{Q_{\\operatorname{min}}} $ fiber of $x_1$. This proves part 3. \n\nSo assume $\\pi_{X_{\\operatorname{min}}}(x_1)=\\pi_{X_{\\operatorname{min}}}(x_2)$. Any line segment from $x_1$ to $x_2$ maps under $\\pi_{X_{\\operatorname{min}}}$ to a periodic line on $\\Xmin$. Moving $x_1$ slightly, we can assume that $\\pi_{X_{\\operatorname{min}}}(x_1)$ is not on the central core curve of any cylinder. Hence any image of $\\pi_{X_{\\operatorname{min}}}(x_1)$ under the involution is not on one of these periodic lines through $\\pi_{X_{\\operatorname{min}}}(x_1)$, so we may assume that $B'$ maps to $\\pi_{X_{\\operatorname{min}}}(x_1)$. (By Theorem \\ref{T:main}, every point of $B'$ maps to either $\\pi_{X_{\\operatorname{min}}}(x_1)$ or its image under the involution (if there is an involution)). \n\nSimilarly, moving $\\pi_{X_{\\operatorname{min}}}(x_1)$ slightly we can assume it does not lie on any of the countably many periodic lines through periodic points on $\\Xmin$, and so we get that $B$ contains no periodic points. \n\\end{proof}\n\n\n\n\n\n\\section{Background}\\label{S:background}\n\nHere we recall some background that will be used in the rest of the paper. \n\n\\subsection{Affine invariant submanifolds.}\nGiven an affine invariant submanifold $\\mathcal{M}$ and a point $(X, \\omega)$ in $\\mathcal{M}$ the tangent space\\footnote{Formally, an affine invariant submanifold is a properly immersed submanifold in the stratum, and the image of this immersion may have self-crossings. At such a self-crossing, the tangent space depends on not just the surface $(X, \\omega)$ in the stratum but also a point in the abstract manifold $\\cM$. See \\cite{LNW} for more details. For notational simplicity we will use notation adapted to the case when the image of $\\cM$ has no self-crossings and hence $\\cM$ can be identified with its image in the stratum.} $T_{(X, \\omega)} \\mathcal{M}$ is naturally identified with a subspace of $H^1(X, \\Sigma; \\bC)$ where $\\Sigma$ is the zero set of $\\omega$. Let $p: H^1(X, \\Sigma; \\bC) \\to H^1(X; \\bC)$ be the natural map from relative to absolute cohomology. The rank of $\\cM$ is defined as $\\rank(\\cM)=\\frac12 \\dim_\\bC p(T_{(X,\\omega)} \\cM)$ for any $(X,\\omega)\\in \\cM$. This is an integer by work of Avila-Eskin-M\\\"oller \\cite{AEM}. \n\nThe affine field of definition ${\\mathbf{k}}(\\cM)$ of $\\cM$ is the smallest subfield of $\\bR$ such that $\\cM$ can locally be defined by linear equations in period coordinates with coefficients in this field \\cite{Wfield}. It is an algebraic extension of $\\bQ$ of degree at most $\\deg({\\mathbf{k}}(\\cM))\\leq g$, where $g$ is the genus. \n\nWe will use the matrices \n$$u_t=\\left(\\begin{array}{cc} 1&t\\\\0&1\\end{array}\\right), \\quad \\quad a_t=\\left(\\begin{array}{cc}1&0\\\\0& e^t\\end{array}\\right), \\quad \\quad\nr_t=\\left(\\begin{array}{cc} \\cos(t)&-\\sin(t)\\\\\\sin(t)&\\cos(t)\\end{array}\\right).$$\n\nWe will refer to a cylinder on a translation surface $(X,\\omega)$ together with a choice of orientation of its core curve as an oriented cylinder. Given a collection of parallel oriented cylinders, we will say they are consistently oriented if the holonomies of $\\omega$ along the oriented core curves are positive multiples of each other. \n\nGiven an oriented cylinder $C$ on a translation surface $(X,\\omega)$, we define $u_t^C(X,\\omega)$ and $a_t^C(X,\\omega)$ to be the result of the following process. Rotate $(X,\\omega)$ so that $C$ becomes horizontal and the orientation is in the positive real direction, apply $u_t$ or $a_t$ respectively to just $C$ and not to the rest of the surface, and then apply the inverse rotation. Given a collection $\\cC=\\{C_1, \\ldots, C_k\\}$ of parallel consistently oriented cylinders, define $u_t^{\\cC}(X,\\omega)=u_t^{C_1} \\circ \\cdots \\circ u_t^{C_k}(X,\\omega)$ and $a_t^\\cC(X,\\omega)=a_t^{C_1} \\circ \\cdots \\circ a_t^{C_k}(X,\\omega)$. We refer to $u_t^{\\cC}$ as the cylinder shear and $a_t^\\cC$ as the cylinder stretch. Typically, either a choice of orientation for the cylinders will be clear, or else either choice will be equally good. \n\nLet $\\cM$ be an affine invariant submanifold. We say that two cylinders $C_1, C_2$ on a surface $(X, \\omega)\\in \\cM$ are $\\cM$-parallel if they are parallel and remain parallel on nearby\\footnote{If $\\cM$ has self-crossings, then one considers only deformations arising from a neighbourhood in the abstract manifold $\\cM$.} surfaces in $\\cM$. These definitions were introduced in \\cite{Wcyl}, where the following is shown. \n\n\\begin{thm}[Cylinder Deformation Theorem]\\label{T:CDT}\nIf $\\cC$ is an equivalence class of $\\cM$-parallel cylinders on $(X,\\omega)\\in \\cM$, then $u_t^{\\cC}\\circ a_s^\\cC(X,\\omega)\\in \\cM$ for all $s,t\\in \\bR$. \n\\end{thm}\n\nIf $\\cC$ is as above and contains a saddle connection perpendicular to the core curves, we define the ``collapse\" of $\\cC$ to be the limit of $a_s^\\cC(X,\\omega)$ as $s\\to -\\infty$. The condition that $\\cC$ contains a perpendicular saddle connection connection is equivalent to the surface degenerating as $t\\to -\\infty$, and here we take the limit in the partial compactification described in \\cite{MirWri}. If there is a unique $t_0$ (up to Dehn twists) so that $u_{t_0}^{\\cC}(X,\\omega)$ contains a saddle connection in $\\cC$ perpendicular to the core curves, for example if $\\cC$ is a single simple cylinder, then we define the cylinder collapse to be the limit of $a_s^\\cC u_{t_0}^{\\cC}(X,\\omega)$ as $s\\to -\\infty$. (Recall a simple cylinder is one for which each boundary is a single saddle connection.) \n\nCylinder deformations apply equally well to translations surfaces $(X, \\omega, S)$ with marked points $S$. If we write $$(X', \\omega', S')=\\lim_{s\\to -\\infty} a_s^\\cC(X,\\omega, S),$$ then the set $S'$ may have a different size than $S$. In particular, $S'$ maybe be non-empty even when $S$ is empty \\cite{MirWri}. In general, $(X', \\omega', S')$ might also have multiple components, however in all instances in this paper it will have only a single component. \n\n\n\\subsection{Finiteness of periodic points.}\\label{SS:EFW}\nWe now explain why Theorem \\ref{T:periodic} is a special case of results in \\cite{EFW}. All theorems in \\cite{EFW} apply to affine invariant submanifolds in $\\cH^{*n}$, as well as those in strata without marked points. If $\\cN$ is a $\\cM$-periodic point, it is in particular an affine invariant submanifold of $\\cM^{*1}$ (the preimage of $\\cM$ in $\\cH^{*1}$). All of $\\cM, \\cM^{*1},$ and $\\cN$ have the same rank. By assumption, we have that $\\cM^{*1}$ is has either higher rank or degree of affine field of definition greater than 1 (or both), since otherwise $\\cM$ would consist of torus covers. By \\cite[Theorem 1.5]{EFW}, such an affine invariant submanifold cannot properly contain infinitely many affine invariant submanifolds of the same rank. \n\n\n\\subsection{Strata of quadratic differentials.}\n\n\\begin{lemma}\\label{L:q-rank}\nLet $\\cQ(\\kappa)$ where $\\kappa = (k_1, \\hdots, k_n)$ be a stratum of quadratic differentials. Let $m_{odd}$ be the number of odd numbers in $\\kappa$ and $m_{even}$ the number of even numbers. Let $g$ be the genus of the Riemann surfaces on which the quadratic differentials lie. The rank and rel of the component is then\n\\[ \\mathrm{rk}(\\cQ) = g + \\frac{m_{odd}}{2} - 1 \\qquad \\text{and} \\qquad \\mathrm{rel}(\\cQ) = m_{even}. \\]\n\\end{lemma}\n\nWe define the rel to be the dimension minus twice the rank.\n\n\\begin{proof}[Proof sketch.]\nThe rank is the difference of the genera of surfaces in $\\cQ$ and their double covers, which can be computed using Riemann-Hurwitz formula. See \\cite[Section 2.1]{KZ} for the formula for $\\dim \\cQ$. \n\\end{proof}\n\n\\begin{cor}\\label{C:q-rank-one}\nThe only rank one strata of strictly quadratic differentials are $\\cQ(-1^4)$, $\\cQ(2,-1^2)$, and $\\cQ(2^2)$. \n\\end{cor}\n\n\n\n\n\n\\subsection{Hat homologous saddle connections.}\n\nTwo saddle connections or cylinders on $(Q, q)\\in \\cQ$ are called hat homologous if they are parallel and remain so on all nearby surfaces in $\\cQ$. \n Configurations of hat homologous saddle connections were classified in \\cite{MZ}; in particular, we mention that two hat homologous cylinders must have have ratio of lengths in $\\{\\frac12, 1, 2\\}$. \n\nWe say that a quadratic differential is generic in a given direction if any two saddle connections in that direction are hat homologous. A cylinder in a half-translation surface is an isometric map of $\\bR\/(c\\bZ) \\times (0,h)$ into the surface. This always extends to a continuous map of $\\bR\/(c\\bZ) \\times [0,h]$ into the surface. The two boundary components of the cylinder are the images of $\\bR\/(c\\bZ) \\times \\{0\\}$ and $\\bR\/(c\\bZ) \\times \\{h\\}$. The multiplicity of a saddle connection on the component of the boundary corresponding to $\\bR\/(c\\bZ) \\times \\{0\\}$ is the number of preimages of a point in this saddle connection in $\\bR\/(c\\bZ) \\times \\{0\\}$, and similarly for $\\bR\/(c\\bZ) \\times \\{h\\}$. This multiplicity is always 1 or 2. Define a simple cylinder to be one that has one saddle connection, with multiplicity one, in each of its boundaries.\n\nWe recall the following consequences of \\cite[Theorems 1 and 2]{MZ}. \n\n\\begin{prop}\\label{P:MZ}\nSuppose that $C$ is a cylinder in a generic direction on a quadratic differential. Then each boundary component of $C$ consists of either \n\\begin{enumerate}\n\\item one saddle connection with multiplicity one, \n\\item one saddle connection with multiplicity two, or\n\\item two saddle connections, each with multiplicity one. \n\\end{enumerate}\nIn the last case, removing the two saddle connections disconnects the surface, and the component not containing $C$ has trivial linear holonomy. \n\n\\begin{figure}[h!]\n\\includegraphics[width=.3\\linewidth]{mult2.pdf}\n\\caption{The case of a multiplicity two saddle connection in Proposition \\ref{P:MZ}.}\n\\label{F:MZ}\n\\end{figure}\n\nFurthermore if $C$ shares a boundary saddle connection with another cylinder $C'$, then possibly after switching $C$ and $C'$ we have that $C'$ is simple and does not share a boundary saddle connection with any other cylinder, and $C$ has two saddle connections in the given boundary component as in case (3) above. \n\\begin{figure}[h!]\n\\includegraphics[width=.9\\linewidth]{3hat.pdf}\n\\caption{The left and right images indicate the two possible configurations of $C$ and $C'$ in Proposition \\ref{P:MZ}. The middle images reminds us that there may also be another cylinder adjacent to $C$. }\n\\label{F:3hat}\n\\end{figure}\n \\end{prop}\n\n\n\\subsection{Primitivity}\n\n\\begin{lemma}\\label{L:R-H}\nThe generic element of a component of a stratum of Abelian or quadratic differentials admits a non-bijective half-translation cover to another translation or half-translation surface if and only if the component is hyperelliptic, in which case the hyperelliptic involution yields the only such map, or the stratum consists of genus 1 translation surfaces. \n\\end{lemma} \n\nSince we are not aware of a proof in the lemma in the literature, we sketch one way that the lemma can be verified. The proof also shows that the only genus $0$ stratum where every surface has an involution is $\\cQ(-1^4)$, and that in every hyperelliptic stratum other than $\\cQ(-1^4)$ and $\\cH(\\emptyset)$, the hyperelliptic involution is unique on a generic surface (in genus zero or one a surface can have several hyperelliptic involutions).\n\n\\begin{proof}\nThe generic element has a saddle connection not parallel to any other saddle connection. (For example, one can consider a saddle connection contained in a cylinder given by Lemma \\ref{L:firstcyl}, and perform a generic twist in that cylinder.) Using the fact that each saddle connection lifts to a union of saddle connections, we see that if the generic element is a cover, it is by a degree 2 map. \n\nThe lemma then follows from a Riemann-Hurwitz argument, using that if the generic element of one stratum covers an element of another, the dimension of the first stratum must be at least as large as the second, as in the determination of which strata have a hyperelliptic component \\cite{KZ, LanneauHyp}. \n\\end{proof}\n\n\n\n\n\n\n\\section{Proof of Theorem~\\ref{T:Q}}\\label{S:Q-overview}\n\nThroughout the rest of this paper, $\\cQ, \\cQ'$, etc., will denote connected components of strata of quadratic differentials. Recall that point markings over strata of Abelian differentials are classified in \\cite{Apisa}, so we make the following standing assumption for the remainder of the paper. \n\\begin{ass}\\label{A}\nAll strata of quadratic differentials considered will not consist of squares of Abelian differentials.\n\\end{ass}\n\nWe begin by noting that we have already classified irreducible $n$-point markings with $n>1$. \n\n\\begin{cor}\\label{C:nothypbig}\nLet $\\cQ$ be higher rank. \nIf $\\cQ$ is not hyperelliptic, then there are no irreducible $n$-point markings with $n>1$. If $\\cQ$ is hyperelliptic, the only such point markings occur when $n=2$ and the two points are interchanged by the involution. \n\\end{cor}\n\n\\begin{proof}\nOne can rephrase Lemma \\ref{L:R-H} as saying that if $(Q,q)$ is a generic element of a non-hyperelliptic $\\cQ$, then $(Q_{\\operatorname{min}}, q_{\\operatorname{min}})=(Q,q)$, and otherwise $(Q_{\\operatorname{min}}, q_{\\operatorname{min}})$ is the quotient by the hyperelliptic involution. \n\\end{proof}\n\nThe following lemma will provide the inductive step for our arguments. A simple cylinder is one whose boundary consists of two saddle connections, each with multiplicity one. An envelope is a cylinder that has one boundary that consists of two saddle connections of multiplicity two. An envelope is simple if it also has one boundary that only contains a multiplicity one saddle connection. \n\nIn regards to Assumption \\ref{A}, we remark that degenerating a simple cylinder or a simple envelope on a quadratic differential with non-trivial holonomy will not create a quadratic differential with trivial holonomy. (In contrast, degenerating a non-simple envelope may create a quadratic differential with trivial linear holonomy.)\n\n\\begin{prop}\\label{P:induct}\nSuppose that $(Q, q)\\in \\cQ$ has a simple cylinder $C$, and that degenerating $C$ gives $(Q', q', S)$, where $(Q', q')\\in \\cQ'$ and $\\cQ'$ does not have any periodic points. Then $\\cQ$ does not have any periodic points. \n\nThe same conclusion holds if $(Q,q)$ has a disjoint pair of simple envelopes, and degenerating either one similarly gives a $\\cQ'$ without periodic points. \n\\end{prop}\n\nThe difficulty of the proof is that if we naively collapse $C$, it may be that as $C$ decreases in size, the periodic point converges to a zero or pole, so that on the limit there is no periodic point. Notice that the assumptions imply that $\\cQ$ is not rank one.\n\n\\begin{proof}\nWe will handle both cases simultaneously. In the first case, $C$ is the given simple cylinder, and in the second case, we let $C$ be either envelope. \nSuppose to a contradiction that $p$ is a $\\cQ$-periodic point on $(Q, q)$. Let $\\cC$ be the equivalence class of cylinders $\\cQ$-parallel to $C$, and suppose without loss of generality that $\\cC$ consists of horizontal cylinders and that the horizontal direction is generic. Our analysis will place increasingly strong constraints on $p$, until eventually we reach a contradiction. \n\nSuppose without loss of generality, after shearing the surface, that $C$ contains a vertical saddle connection. Recall we have defined the collapse of $C$ to be the limit of $a_s^C(Q,q)$ as $s\\to -\\infty$. Here it will be important that this limit can be thought of as limit of the path $a_{\\log(t)}^C(Q,q)$, which is linear in period coordinates as $t$ ranges from $e$ to $0$. We refer to this whole path as the collapse path of $C$. \n\nThe proof will proceed in three parts. First, we will show that it suffices to show that we can move the marked point out of $\\cC$. Second, we will show that if we can't collapse $C$ without $p$ merging with a singularity then the position of $p$ is entirely controlled by $C$, in a precise sense specified at the beginning of step 2. Finally, we will show that if the position of $p$ is entirely controlled by $C$ then $p$ can be moved out of $\\cC$.\n\n\n\\noindent \\textbf{Step 1: $p$ must belong to a cylinder in $\\cC$.} \n\nSuppose first that the periodic point lies outside of $C$ and that it remains constant in the complement of $C$ when cylinder deformations are performed to $C$. Note that this is always the case when $p$ lies outside of $\\cC$ by \\cite[Lemma 4.6]{MirWri}. Then we may collapse $C$ and pass to a boundary surface where $p$ is a periodic point that is not a zero or pole. This contradicts Mirzakhani-Wright~\\cite{MirWri}, which implies that $p$ becomes a periodic point on the boundary translation surface.\n\nSuppose now that $p$ lies on the boundary of $C$ for all time as $C$ collapses. Then we proceed as follows. If $C$ is a simple cylinder, then we proceed with the collapse and arrive at the same contradiction as before. If $C$ is an envelope, then it may not be possible to collapse $C$ without causing $p$ to coincide with a singularity on the boundary, see Figure~\\ref{F:Shear1} (bottom).\n\n\\begin{figure}[h!]\n\\includegraphics[width=.8\\linewidth]{Shear.pdf}\n\\caption{Top: A full Dehn twist in a simple cylinder. Bottom: A half Dehn twist in an envelope. The bad position on the boundary is marked with an x. }\n\\label{F:Shear1}\n\\end{figure}\n\n\\noindent In this case, we relabel the cylinders so that the second envelope is labelled $C$. \n\n\\noindent \\textbf{Step 2: $C$ determines the position of $p$.}\n\nWe may now suppose that $p$ is contained in the interior of a cylinder $D\\in \\cC$, and either $D=C$ or the position of $p$ in the complement of $C$ is not constant along the collapse path. We will show that we may assume that there is a saddle connection joining $p$ to a zero on the boundary of $D$ whose holonomy is a fixed real multiple of a cross curve of $C$. We will also show $C\\neq D$. Thus, this step could be more completely described as ``$D\\neq C$, and $C$ determines the position of $p$ in $D$.\"\n\nShear $(Q, q)$ so that $p$ does not lie on a vertical separatrix (this could easily cause $C$ to no longer contain a vertical saddle connection). \nRecall the collapse path is defined to start at $t=e$ and end at $t=0$. Because we have assumed $p$ does not lie on a vertical seperatrix, $p$ does not hit a singularity of the metric before $t=0$. We may partition the interval $(0,e)$ into closed subintervals according to which cylinder in $\\cC$ (including the boundary of the cylinder) $p$ is in at a given time $t\\in (0,e)$. (The subintervals overlap at their endpoints, and by convention we require adjacent intervals to correspond to distinct cylinders in $\\cC$.) \n\nLet $\\gamma$ be a path from a singularity to the periodic point. Let $f(t)$ denote the imaginary part of the period of $\\gamma$ at time $t$ along the collapse path. (The function $f$ is only well defined up to replacing it with $-f$). If $h$ is the height of the shortest cylinder in $\\cC - \\{ C \\}$, then $p$ passes through at most $\\frac{|f(0) - f(e)|}{h}$ cylinders in $\\cC - \\{ C \\}$ along the collapse path. This shows that the partition described in the previous paragraph is finite. \n\nLet $D$ be the last cylinder the marked point visits before the collapse is completed at time $0$. By replacing $(Q,q)$ with an appropriate point on the collapse path, we may assume that in fact $p$ started in $D$ and remains in $D$ along the collapse path. Now shear $C$ so that it again contains a vertical saddle connection $v_1$. Let $(Q, q)$ now denote this new translation surface. Note that the assumption that $p$ remains in $D$ is still valid. \n \nSuppose first that $C=D$. If $p$ lies on a vertical separatrix contained in $C$ then we may shear the surface to perform a full Dehn twist (in the case that $C$ is simple) or half a Dehn twist (in the case that $C$ is an envelope) to ensure that $C$ still contains a vertical saddle connection and that $p$ does not lie on a vertical separatrix contained in $C$. See Figure~\\ref{F:Shear1} and, for a non-example, Figure~\\ref{F:Shear2}. Collapsing $C$ now gives a half translation surface where $p$ is not a singularity of the metric, but is a periodic point, which is a contradiction. \n\n\\begin{figure}[h!]\n\\includegraphics[width=.8\\linewidth]{BadShear.pdf}\n\\caption{Cylinders with four hat homologous boundary saddle connections must be avoided, since if $p$ is the midpoint of a vertical saddle connection, a Dehn twist cannot fix this problem. Another reason to avoid degenerating such cylinders is that the double cover may become disconnected (which can also happen for non-simple envelopes).}\n\\label{F:Shear2}\n\\end{figure}\n \nNext we suppose that $C\\neq D$. Suppose that there is a small perturbation of $(Q, q)$, so that $v_1$ remains vertical and $C$ remains horizontal, and such that after replacing $(Q,q)$ with this deformation $p$ does not collide with a singularity at the conclusion of the collapse path. If this occurs, then, as above, $p$ becomes a periodic point on the boundary, which has no periodic points by hypothesis. Therefore, we may suppose that after any small perturbation as above, $p$ collides with a singularity at the conclusion of the collapse path. In particular, this means that there is a saddle connection $v_2$ in $D$ joining a singularity on the boundary of $D$ to $p$, and a positive real constant $c$, so that the orbit closure of $(Q, q; p)$ is cut out by the equation $v_2 = c v_1$. \n\n\\noindent \\textbf{Step 3: $p$ may be moved out of $\\cC$.}\n\nThe idea of the proof will now be to vary $v_1$ to move the periodic point $p$ outside of $\\cC$ (which will be a contradiction). Since $v_2 = c v_1$ we see that by shearing $C$ to the left or right while fixing the complement of $C$ we may move the marked point to the left or right. Similarly, increasing the height of $C$ while fixing the rest of the surface causes $p$ to move up or down. We will use these two operations to force $p$ to move through the complement of $C$ (and we will be careful not to cause $p$ to enter $C$). We will refer in the sequel to these operations as ``moving $p$ using $C$\". Suppose that increasing the height of $C$ moves $p$ toward a boundary $B$ of $D$. \n\nOur approach will use the results of Masur-Zorich~\\cite{MZ} that we summarized in Proposition \\ref{P:MZ}. In the sequel, we will say that a saddle connection ``leads out of $\\cC$\" if it borders a cylinder in $\\cC$ on exactly one side. \n\n\\begin{sublem}\\label{SL:MZ1}\nThe boundary $B$ cannot contain two saddle connections of multiplicity one. \n\\end{sublem}\n\n\\begin{proof}\nSuppose not to a contradiction. If both saddle connections in $B$ lead out of $\\cC$ then we increase the height of $C$ to move $p$ through $B$ and out of $\\cC$. Otherwise, by Proposition \\ref{P:MZ} the boundary $B$ of $D$ is as in one of the subfigures in Figure~\\ref{F:3hat2}\n\n\\begin{figure}[h!]\n\\includegraphics[width=.9\\linewidth]{3hat.pdf}\n\\caption{Possible configurations of $D$ and $B$}\n\\label{F:3hat2}\n\\end{figure}\n\nWe see that the left two configurations are impossible, as shown in Figure~\\ref{F:LeavingEC1}. In that figure, we shear $C$ and increase its height so that the marked point leaves $\\cC$. In the right configuration of Figure~\\ref{F:LeavingEC1} this is easy since there is a saddle connection in $B$ that leads out of $\\cC$. In the left configuration one of the two simple cylinders bordering $B$ is not $C$ (we have drawn this cylinder as short and sheared) and passing through it leads out of $\\cC$ by Proposition \\ref{P:MZ}.\n\n\\begin{figure}[h!]\n\\includegraphics[width=.9\\linewidth]{Fig2.pdf}\n\\caption{Shear $C$ and increase its height so $p$ exits $\\cC$}\n\\label{F:LeavingEC1}\n\\end{figure} \n\nTherefore, $D$ and its boundary $B$ are arranged as in Figure~\\ref{F:D-prime-configuration}. Let $D'$ be the simple cylinder that borders $D$ along $B$.\n\n\\begin{figure}[h!]\n\\includegraphics[width=.4\\linewidth]{Fig1.pdf}\n\\caption{The configuration of $D$ and its boundary $B$}\n\\label{F:D-prime-configuration}\n\\end{figure}\n\nLet $B'$ be the boundary of $D$ opposite $B$. Notice that $B'$ cannot consist of a single multiplicity two saddle connection since then $\\cQ = \\cQ(2, -1^2)$, which is rank one. Moreover, $B'$ cannot consist of two multiplicity one saddle connections that bound a simple cylinder (as in Figure~\\ref{F:D-prime-configuration}) since then $\\cQ = \\cQ(2, 2)$, which is also rank one. Therefore, Proposition \\ref{P:MZ} implies that either $B'$ contains a saddle connection that leads out of $\\cC$ or $B'$ borders two distinct simple cylinders in $\\cC$. The possibilities are shown in Figure~\\ref{F:LeavingEC2}. In that figure we see that when $D' \\ne C$ it is possible to shear $C$ and increase its height to move the marked point $p$ out of $\\cC$ as we did in Figure~\\ref{F:LeavingEC1}. It is important that $D' \\ne C$ since the marked point will pass through $D'$ as it moves to leave $\\cC$.\n\n\\begin{figure}[h!]\n\\includegraphics[width=.9\\linewidth]{Fig5.pdf}\n\\caption{Three configurations of $D$ and moving $p$ out of $\\cC$ when $D' \\ne C$. In the leftmost, the bottom saddle connection could also be two distinct saddle connections. }\n\\label{F:LeavingEC2}\n\\end{figure} \n\nTherefore, we may suppose that $D' = C$. Recall that there is a cross curve $v_1$ in $C$ and a positive real constant $c$ so that $v_2 = c v_1$ where $v_2$ is a saddle connection joining a singularity on $B'$ to $p$. We now decrease the height of $C$ so that the imaginary part of the period of $v_1$ changes from positive to negative. As we see in Figure~\\ref{F:LeavingEC3}this ``overcollapse of $C$\" can be performed in a way that moves the marked point out of $\\cC$ (in the rightmost figure we must first make the bottom cylinder that the marked point passes through sufficiently small; this may be achieved without affecting the position of the marked point since $v_2 = cv_1$). One subtle point is that when $C$ overcollapses, new cylinders in $\\cC$ could appear. In particular, we may worry that at the end of the ``overcollapse\" $p$ hasn't left $\\cC$, but has instead entered a newly formed cylinder in $\\cC$. In general this can happen, but in this case since $C$ borders an equivalent cylinder $D$ on both its boundaries the only newly formed cylinder in $\\cC$ is the one at the top of the figures in Figure~\\ref{F:LeavingEC3}. In particular, we see that the newly formed cylinder does not affect $p$ exiting $\\cC$. \n\n\\begin{figure}[h!]\n\\includegraphics[width=\\linewidth]{Fig6.pdf}\n\\caption{Moving $p$ out of $\\cC$ when $D' = C$. (This figure is drawn with $c=1$.)}\n\\label{F:LeavingEC3}\n\\end{figure}\n\nTherefore, we have a contradiction in all cases and so $B$ cannot consist of two saddle connections of multiplicity one. \n\\end{proof}\n\n\\begin{sublem}\\label{SL:MZ2}\nThe cylinder $D$ is simple and borders an equivalent cylinder along $B$.\n\\end{sublem}\n\\begin{proof}\nSuppose first to a contradiction that $B$ consists of one multiplicity two saddle connection. By Proposition \\ref{P:MZ} we have that $D$ must look like one of the cylinders in Figure~\\ref{F:LeavingEC4}. The leftmost configuration only occurs in $\\cQ(-1^4)$, which is rank one and hence precluded. In the middle configuration the boundary $B'$ opposite $B$ is a single multiplicity one saddle connection, which must lead out of $\\cC$ by Proposition \\ref{P:MZ}. Therefore, increasing the height of $C$ moves the marked point through $B$ and then out the opposite boundary of $D$ and hence out of $\\cC$. In the rightmost configuration we increase the height of $C$ to move the marked point through $B$ and then into a situation where as the height of $C$ increases the marked point moves towards a boundary with two multiplicity one saddle connections. As in Sublemma~\\ref{SL:MZ1}, specifically Figure \\ref{F:LeavingEC1}, one of these saddle connections either leads out of $\\cC$ or borders a cylinder other than $C$, which in turn leads out of $\\cC$, so we reach a contradiction. \n\n\\begin{figure}[h!]\n\\includegraphics[width=.9\\linewidth]{Fig8.pdf}\n\\caption{Moving $p$ out of $\\cC$ when $B$ contains one multiplicity two saddle connection}\n\\label{F:LeavingEC4}\n\\end{figure}\n\nBy Sublemma~\\ref{SL:MZ1}, $B$ cannot consist of two multiplicity one saddle connections. Therefore, by Proposition \\ref{P:MZ}, it must consist of a single multiplicity one saddle connection. If $B$ does not border another cylinder in $\\cC$ then by increasing the height of $C$ we may move the marked point through $B$ and out of $\\cC$, which is a contradiction (see the rightmost figure in Figure~\\ref{F:LeavingEC1} for a similar case). Therefore, $B$ borders another cylinder in $\\cC$ and hence by Proposition \\ref{P:MZ}, $D$ is simple. \n\\end{proof}\n\nLet $D'$ be the other cylinder in $\\cC$ which borders $B$. Because $C$ is a simple cylinder or a simple envelope, $D'\\neq C$. As $p$ moves into $D'$ it moves towards a boundary $B''$ of $D'$. We will reach a contradiction by moving $p$ out of $\\cC$. Since all argument are very similar to those already given in this step, we will only sketch them here.\n\nIf $B''$ consists of one multiplicity two saddle connection then we proceed as in the middle left subfigure of Figure~\\ref{F:FinalFig}. If $B''$ borders a simple cylinder on both boundaries then call this cylinder $D''$. If $D'' = C$ then we may reduce the height of $C$ to move $p$ out of $\\cC$ as in the leftmost subfigure of Figure~\\ref{F:FinalFig} (if there is another cylinder bordering $D'$ then we first make it short so that we do not need to overcollapse $C$ to move $p$ out of $\\cC$). Otherwise, we move $p$ out of $\\cC$ as in middle right subfigure of Figure~\\ref{F:FinalFig}. In all other cases we proceed as in the rightmost subfigure of Figure~\\ref{F:FinalFig}.\n\\begin{figure}[h!]\n\\includegraphics[width=\\linewidth]{FinalFig.pdf}\n\\caption{Moving $p$ out of $\\cC$ when $D$ is simple and borders an equivalent cylinder $D'$}\n\\label{F:FinalFig}\n\\end{figure}\n\\end{proof} \n\nIn the next two sections we will prove the following two results, which we will use in our proof of Theorem~\\ref{T:Q}. Note that the three strata that appear in the next theorem are exactly the higher rank strata of minimal dimension (they have dimension 4 and rank 2), and that all rank 1 strata are hyperelliptic. \n\n\\begin{thm}\\label{T:findcyl} \nIf $\\cQ$ is non-hyperelliptic, and $$\\cQ\\notin \\{\\mathcal{Q}(3, -1^3), \\mathcal{Q}(5, -1), \\mathcal{Q}(1, -1^5)\\},$$ then there exists $(Q, q)\\in \\cQ$ with a simple cylinder, or a pair of disjoint simple envelopes, such that degenerating any one of these cylinders gives $(Q', q', S)$, where $(Q', q')\\in \\cQ'$ and $\\cQ'$ is non-hyperelliptic.\n\\end{thm}\n\n\\begin{thm}\\label{T:basecase} \n$\\mathcal{Q}(3, -1^3),\\mathcal{Q}(5, -1)$ and $\\mathcal{Q}(1, -1^5)$ do not have periodic points. \n\\end{thm}\n\n\\begin{proof}[Proof of Theorem~\\ref{T:Q}] \nBy using induction on $\\dim \\cQ$, Proposition \\ref{P:induct} and Theorems \\ref{T:findcyl} and \\ref{T:basecase} immediately give the result when $\\cQ$ is not hyperelliptic. \n\nWhen $\\cQ$ is hyperelliptic, every surface in $\\cQ$ covers a surface in a genus 0 stratum $\\cQ_0$, which is not hyperelliptic and has the same rank. Let $n$ be the number of points that are not zeros or poles over which these covering maps are branched. (One can show $n\\in \\{0,1, 2\\}$.) Any $\\cQ$-periodic point gives rise to a periodic point over $\\cQ_0^{*n}$. This can be considered as a point marking over $\\cQ_0$, and in this point marking the point arising from the $\\cQ$-periodic point is not free. By Corollary \\ref{C:nothypbig} this means we get a $\\cQ_0$-periodic point, which is a contradiction. \n\\end{proof}\n\n\n\n\\section{Proof of Theorem \\ref{T:findcyl}}\\label{S:FindCyl}\n\nTo read this section it is necessary to be comfortable with the results of \\cite{MZ} that are recalled in Proposition \\ref{P:MZ}. Assumption \\ref{A} is still in effect; otherwise $\\cH(\\emptyset)$ would be a counterexample to Lemma \\ref{L:firstcyl} and $\\cQ=\\cH^{odd}(4)$ would be a counterexample to Theorem \\ref{T:findcyl}. \n\n\\begin{lem}\\label{L:firstcyl}\nAny $\\cQ$ other than $\\cQ(-1^4)$ contains a surface with a simple cylinder or two disjoint simple envelopes. \n\\end{lem}\n\n\\begin{proof}\nPick disjoint cylinders $C, D$ on a surface in $\\cQ$ where all parallel saddle connections are hat homologous. If both are simple envelopes, or if either is a simple cylinder, we are done, so we assume otherwise.\n\nBy Proposition \\ref{P:MZ}, since $C$ is not a simple envelope and is not simple, it has exactly two distinct saddle connections on one side, and cutting these saddle connections disconnects the surface into two components. Furthermore, the component $R$ not containing $C$ has trivial holonomy. \n\nIf $R$ contains a cylinder $C'$ hat homologous to $C$, then $C'$ must be simple. Otherwise, \\cite{SW2} gives a cylinder $C'$ on $R$ not parallel to $C$, which must be simple.\n\n(The use of \\cite{SW2} can be avoided by an argument using square-tiled surfaces. The use of translation surface with boundary can be avoided by gluing together the two boundary saddle connections of $R$. We are using the fact that on a generic Abelian differential, every cylinder is simple.)\n\\end{proof}\n\n\n\\begin{lem}\\label{L:secondcyl}\nSuppose $(Q', q')$ is in a hyperelliptic component other than $\\cQ(-1^4)$, $S$ is a set of non-singular points of the metric, and $c$ is a saddle connection on $(Q', q', S)$. Assume all points of $S$ are endpoints of $c$ (so $|S|\\in \\{0,1,2\\}$). Then, possibly after moving $(Q', q', S)$ in its stratum in such a way that $c$ remains a saddle connection, there is a simple cylinder $C'$ on $(Q', q')$ disjoint from $c$ and that does not contain any point of $S$ on its boundary. \n\\end{lem}\n\n\\begin{proof} \nWe assume all parallel saddle connections on $(Q', q')$ are hat homologous. Notice that since $\\cQ$ is a hyperelliptic component no cylinder is a simple envelope since every cylinder must be fixed by the hyperelliptic involution and a simple envelope does not admit an involution. \n\nFirst suppose $S$ is non-empty. The proof of Lemma \\ref{L:firstcyl} gives a cylinder $C'$ as desired, except that it may not be disjoint from $c$. However, moving the marked points we may make $c$ disjoint from $C'$. (Note that since $C'$ is simple it cannot cover the whole surface.)\n\nSo assume $S$ is empty. Let $C'$ be any cylinder disjoint from $c$. If $C'$ is simple we are done. Since the component is hyperelliptic, $C'$ can't be a simple envelope. If $C'$ has two distinct saddle connections on each side then by Proposition \\ref{P:MZ}, $C'$ disconnects the surface. Cut the two saddle connections on the opposite side from $c$. By Proposition \\ref{P:MZ}, the component $R$ not containing $C'$ or $c$ has trivial holonomy. As in Lemma \\ref{L:firstcyl}, we can find a simple cylinder in $R$. \n\nThe case that remains is that every cylinder $C'$ disjoint from $c$ is an envelope that has two multiplicity one saddle connections on one boundary. In this case Proposition \\ref{P:MZ} gives that the complement of $C'$ is connected and has trivial holonomy. In particular, there are only two poles, so we can't have two disjoint cylinders $C'$ of this type.\n\n\\begin{figure}[h!]\n\\includegraphics[width=.8\\linewidth]{Example.pdf}\n\\caption{}\n\\label{F:Example}\n\\end{figure}\n\nAssume $c$ is horizontal, and nudge the surface so that it becomes square-tiled. If there is more than one horizontal cylinder, by the previous comment at least one of them must not be as in the previous paragraph, so we are done (after nudging the surface again to restore the fact that all parallel saddle connections are hat homologous). So assume there is just one horizontal cylinder. If a saddle connection other than $c$ appears both on the top and the bottom of this cylinder, then we can find a transverse simple cylinder disjoint from $c$, as in Figure \\ref{F:Example} (left). So assume this is not the case. If two distinct saddle connections other than $c$ appear on the same side of the cylinder, we can nudge them to create a second horizontal cylinder while keeping the existing horizontal cylinder and $c$ horizontal, as in Figure \\ref{F:Example} (right). (One can keep the surface horizontally periodic after the nudge by making it still be square-tiled, i.e. have rational period coordinates.) \n\n\\begin{figure}[h!]\n\\includegraphics[width=\\linewidth]{BadStrebel.pdf}\n\\caption{}\n\\label{F:BadStrebel}\n\\end{figure}\nWe are now in one of the four cases illustrated in the top of Figure \\ref{F:BadStrebel}. Except that the left case is in $\\cQ(-1^4)$ and hence excluded by our assumption that $(Q', q')\\notin \\cQ(-1^4)$, and the right case is excluded since there is a marked point. For the middle two cases, the bottom of the figure shows how to find a simple cylinder disjoint from $c$. \n\\end{proof}\n\n\n\\begin{lemma}\\label{L:s-not-fixed}\nLet $(Q', q')\\in \\cQ'$, and assume $\\cQ'$ is hyperelliptic. Let $c$ be a saddle connection. Let $(Q, q)$ be the quadratic differential that arises from slitting $c$ and gluing in a simple cylinder. Then $(Q, q)$ belongs to a hyperelliptic component of a stratum of quadratic differentials iff $c$ is fixed by the hyperelliptic involution.\n\\end{lemma}\n\nWe omit the proof, which follows easily from Lemma \\ref{L:R-H}. \n\n\\begin{lemma}\\label{L:two-cylinders-no-hyp}\nSuppose $C_1$ and $C_2$ are disjoint simple cylinders on $(Q, q)$ such that collapsing either $C_i$ does not create marked points and gives a surface in a hyperelliptic component. Then $(Q, q)$ is in a hyperelliptic component or ${\\mathcal{Q}}(5, -1)$. \n\\end{lemma}\n\n\n\\begin{proof}\nDegenerating both $C_i$ gives a hyperelliptic surface $(Q', q')$ with no marked points and with two saddle connections, $c_1$ and $c_2$. (If marked points were created, then even after moving the marked points there would have to be a hyperelliptic involution fixing the set of marked points. This implies the degeneration is in $\\cH(0,0)$, contradicting Assumption \\ref{A}.) Since gluing in a cylinder to either $c_i$ gives a surface in a hyperelliptic component, Lemma \\ref{L:s-not-fixed} gives that each $c_i$ is fixed by the involution. Hence Lemma \\ref{L:s-not-fixed} gives that $(Q,q)$ is hyperelliptic. \n\n\\begin{figure}[h]\n\\includegraphics[width=.3\\linewidth]{Pillow.pdf}\n\\caption{If $c_1$ and $c_2$ are fixed by different hyperelliptic involutions, then up to $GL(2, \\bR)$ the situation is as illustrated here. Compare to Figure \\ref{F:Prym3A}.}\n\\label{F:Pillow}\n\\end{figure}\n\nThis proof works as long as the hyperelliptic involution on $(Q', q')$ is unique, which is true for all strata but $\\cQ(-1^4)$ and $\\cH(\\emptyset)$, the later of which cannot arise here. If $(Q', q')\\in \\cQ(-1^4)$, there is the possibility that $c_1$ is fixed by one hyperelliptic involution, and $c_2$ by another, as in Figure \\ref{F:Pillow}. In this case $(Q', q')\\in \\mathcal{Q}(5, -1)$. \n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem \\ref{T:findcyl}]\nLet $\\cQ$ be non-hyperelliptic and not $\\mathcal{Q}(3, -1^3),$ $\\mathcal{Q}(5, -1)$, or $\\mathcal{Q}(1, -1^5)$. \n\nLet $C_1$ be a cylinder that is either simple or a simple envelope on some $(Q, q)\\in \\cQ$. This exists by Lemma \\ref{L:firstcyl}. If possible, pick $C_1$ so that degenerating it produces marked points; otherwise we will assume that degenerating any simple cylinder or simple envelope does not produce marked points.\n\nLet $(Q_1, q_1, S_1)$ be the result of degenerating $C_1$. The cylinder $C_1$ becomes a saddle connection $c_1$ on $(Q_1, q_1, S_1)$. We may assume $(Q_1, q_1)$ is in a hyperelliptic component. \n\n\\bold{Case 1: $(Q_1, q_1)\\notin \\cQ(-1^4)$.} Let $C_2$ be the cylinder on $(Q_1, q_1, S_1)$ given by Lemma \\ref{L:secondcyl}, which is disjoint from $c_1$. There is a corresponding cylinder on $(Q,q)$, which we also call $C_2$. Let $(Q_2, q_2, S_2)$ be the result of degenerating $C_2$ on $(Q,q)$. We may assume $(Q_2, q_2)$ is in a hyperelliptic component. Since surfaces in hyperelliptic components don't contain simple envelopes, we get that both $C_i$ are simple. \n\nWe now claim that $S_1$ is empty. Otherwise, since $C_2$ does not contain any points of $S_1$ in its boundary, we see that degenerating $C_1$ on $(Q_2, q_2)$ (with marked points $S_2$ forgotten) produces a surface with marked points. Since $(Q_2, q_2)$ is contained in a hyperelliptic component, every element of the resulting stratum of surfaces with marked points must have an involution fixing the set of marked points. This implies the resulting stratum is $\\cH(0,0)$, which contradicts Assumption \\ref{A}. \n\nNow assume $S_1$ is empty. Our choice of $C_1$ implies that $S_2$ is empty. Hence Lemma \\ref{L:two-cylinders-no-hyp} contradicts the assumption that $\\cQ$ is non-hyperelliptic. \n\n\\begin{figure}[h!]\n\\includegraphics[width=.2\\linewidth]{NoChange.pdf}\n\\caption{Gluing a simple envelope onto a pillowcase with no marked points does not change the stratum.}\n\\label{F:NoChange}\n\\end{figure}\n\n\\bold{Case 2: $(Q_1, q_1)\\in \\cQ(-1^4)$.} \nIf $S_1=\\emptyset$, as in Figure \\ref{F:NoChange} we get that $C_1$ is a simple cylinder. As in Figure \\ref{F:SmallCases} (top left) we conclude that $\\cQ=\\cQ(2,-1,-1)$. This contradicts our assumption that $\\cQ$ is not hyperelliptic. \n\n\\begin{figure}[h!]\n\\includegraphics[width=.65\\linewidth]{SmallCases.pdf}\n\\caption{}\n\\label{F:SmallCases}\n\\end{figure}\n\n\nIf $|S_1|=2$, then $C_1$ was simple and $\\cQ=\\cQ(4, -1^4)$. See Figure \\ref{F:SmallCases} (top right). One can find a pair of disjoint simple envelopes $E, E'$ as in the figure such that degenerating either gives a surface in $\\cQ(3, -1^3)$. \n\nIf $|S_1|=1$, and $C_1$ was a simple envelope, then $\\cQ=\\cQ(1, -1^5)$. See Figure \\ref{F:SmallCases} (bottom left).\n\nIf $|S_1|$=1 and $C_1$ was simple, $\\cQ= \\cQ(3, -1^3)$. See Figure \\ref{F:SmallCases} (bottom right).\n\\end{proof}\n\n\n\n\n\\section{Proof of Theorem \\ref{T:basecase}}\\label{S:BaseCase}\n\nThe case of $\\mathcal{Q}(1, -1^5)$ follows from \\cite{Apisa}, which in particular classifies $\\cH(2)=\\tilde{\\mathcal{Q}}(1, -1^5)$-periodic points. So we need only treat $\\mathcal{Q}(3, -1^3)$ and $\\mathcal{Q}(5, -1)$.\n\nThe following is a slightly more general version of Apisa~\\cite[Lemma 6.1]{Apisa}. The proof is identical. \n\n\\begin{lemma}\\label{calc1}\nLet $(X, \\omega)$ be a translation surface in an affine invariant submanifold $\\mathcal{M}$. Let $C$ be a horizontal cylinder, and let $\\cD_1, \\cD_2$ be two vertical distinct $\\mathcal{M}$-equivalence classes of cylinders such that \n\\begin{enumerate}\n\\item The intersection of $\\cD_i$ with the interior of $C$ is connected for $i = 1, 2$.\n\\item All cylinders in each $\\cD_i$ have fixed rational ratio of heights and circumferences in $\\cM$. \n\\item All cylinders $\\mathcal{M}$-parallel to $C$ have circumference and height equal to that of $C$ in $\\cM$. \n\\end{enumerate}\nIf $p$ is an $\\mathcal{M}$-periodic point in the interior of $C$, then up to relabelling $\\cD_1$ and $\\cD_2$, the point $p$ is at the center of the rectangle given by the intersection of $\\cD_1$ and $C$. Furthermore, removing $\\cD_1$ and $\\cD_2$ divides $C$ into two rectangles of equal size. \n\n\\begin{figure}[H]\n \\centering\n \\resizebox{.6\\linewidth}{!}{\\begin{tikzpicture}\n \\draw[dashed] (0,3) -- (0,2);\n \\draw (0,2) -- (0,0) -- (7,0);\n \\draw[dashed] (1,3)--(1,0);\n \\draw (1,2) -- (4,2);\n \\draw[dashed] (4,3) -- (4,0);\n \\draw[dashed] (5,3) -- (5,0);\n \\draw (5,2) -- (7,2);\n \\draw[black, fill] (.5, 1) circle[radius = 1.6pt];\n \\node at (.5, .75) {$p$};\n \\draw (-1,0) -- (-1,1) -- (-1.25, 1) -- (-.75, 1) -- (-1, 1) -- (-1,0) -- (-1.25, 0) -- (-.75, 0); \\node at (-1.25, .5) {$h$};\n \\draw (-2,0) -- (-2,2) -- (-2.25, 2) -- (-1.75, 2) -- (-2, 2) -- (-2,0) -- (-2.25, 0) -- (-1.75, 0); \\node at (-2.25, 1) {$1$};\n \n\t\t\\node at (.5, 3) {$\\cD_1$}; \\node at (4.5, 3) {$\\cD_2$}; \\node at (7.5, 1) {$C$};\n\t\t\\draw (0, -1) -- (.5, -1) -- (.5, -.75) -- (.5, -1.25) -- (.5, -1) -- (0, -1) -- (0, -1.25) -- (0, -.75); \\node at (.25, -1.25) {$p \\ell_1$};\n\t\t\\draw (0, -2) -- (1, -2) -- (1, -1.75) -- (1, -2.25) -- (1, -2) -- (0, -2) -- (0, -1.75) -- (0, -2.25); \\node at (.5, -2.25) {$ \\ell_1$};\n\t\t\\draw (1, -1) -- (4, -1) -- (4, -.75) -- (4, -1.25) -- (4, -1) -- (1, -1) -- (1, -1.25) -- (1, -.75); \\node at (2.5, -1.25) {$a$};\n\t\t\\draw (4, -2) -- (5, -2) -- (5, -1.75) -- (5, -2.25) -- (5, -2) -- (4, -2) -- (4, -1.75) -- (4, -2.25); \\node at (4.5, -2.25) {$ \\ell_2$};\n\t\t\\draw (5, -1) -- (8, -1) -- (8, -.75) -- (8, -1.25) -- (8, -1) -- (5, -1) -- (5, -1.25) -- (5, -.75); \\node at (6.5, -1.25) {$b$};\n \\end{tikzpicture}}\n \\caption{Lemma \\ref{calc1} asserts that, after scaling so $C$ has height 1, we have $a = b$ and $p = h = \\frac{1}{2}$.}\n \\label{fig:shearing}\n\\end{figure}\n\\end{lemma}\n\nThe lemma allows for multiple intersections of cylinders in $\\cD_i$ and $C$, but requires that the union of these intersections be a single rectangle for each $i$. \n\n\nWe will apply Lemma~\\ref{calc1} to the vertically and horizontally periodic surfaces in Figure \\ref{F:Prym3A}. Note that our convention is that, except where indicated otherwise, opposite edges are identified when giving polygonal presentations for surfaces. \n\n\\begin{figure}[H]\n \\begin{subfigure}[b]{0.31\\textwidth}\n \\centering\n \\resizebox{.8\\linewidth}{!}{\\begin{tikzpicture}\n \t\t\\draw (0,0) -- (0,1) -- (1,1) -- (1,3) -- (3,3) -- (3,2) -- (2,2) -- (2,0) -- (0,0);\n\t\t\\draw[dotted] (1,0) -- (1,1) -- (2,1);\n\t\t\\draw[dotted] (1,2) -- (2,2) -- (2,3);\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t\\draw[black, fill] (.5,.5) circle[radius=1pt];\n\t\t\\draw[black, fill] (1.5,.5) circle[radius=1pt];\n\t\t\\draw[black, fill] (1.5,2.5) circle[radius=1pt];\n\t\t\\draw[black, fill] (2.5,2.5) circle[radius=1pt];\n \\draw[black] (1.5,1.5) circle[radius=2pt];\n\t\t\\draw[black] (1.5,3) circle[radius=2pt]; \n\t\\end{tikzpicture}\n }\n \n \\label{SF:Prym3}\n \\end{subfigure}\n \\qquad\n \\begin{subfigure}[b]{0.4\\textwidth}\n \\centering\n \\resizebox{.8\\linewidth}{!}{\\begin{tikzpicture}\n \t\t\\draw (0,0) -- (0,2) -- (-1,2) -- (-1,3) -- (1,3) -- (1,4) -- (2,4) -- (2,2) -- (3,2) -- (3,1) -- (1,1) -- (1,0) -- (0,0);\n\t\t\\draw[dotted] (0,1) -- (1,1) -- (1,3) -- (2,3) -- (2,2) -- (0,2) -- (0,3) -- (2,3) -- (2,1);\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t\\draw[black, fill] (-.5,2.5) circle[radius=1pt];\n\t\t\\draw[black, fill] (1,2.5) circle[radius=1pt];\n\t\t\\draw[black, fill] (1,1.5) circle[radius=1pt];\n\t\t\\draw[black, fill] (2.5,1.5) circle[radius=1pt];\n\t\n\t\t%\n\t\t\\draw[black] (.5,.5) circle[radius=2pt];\n\t\t\\draw[black] (.5,2) circle[radius=2pt];\n\t\t\\draw[black] (1.5,2) circle[radius=2pt];\n\t\t\\draw[black] (1.5,3.5) circle[radius=2pt]; \n\t\n \n \n\t\\end{tikzpicture}\n }\n \n \\label{SF:Prym4}\n \\end{subfigure}\n\\caption{$\\tilde{\\mathcal{Q}}(3, -1^3)$ (left) and $\\tilde{\\mathcal{Q}}(5, -1)$ (right).} \n\\label{F:Prym3A}\n\\end{figure}\n\n\\noindent \\textbf{Periodic Points in $\\mathcal{Q}(3, -1^3)$: } Consider the surface on the left in Figure~\\ref{F:Prym3A}. Letting $C$ be either the top or bottom horizontal cylinder, and $\\cD_1$ and $\\cD_2$ be the two equivalence classes of vertical cylinders, Lemma~\\ref{calc1} implies that any periodic point contained in one of these two horizontal cylinders must be at the points indicated with solid dots. Letting $C$ be the middle vertical cylinder and $\\cD_1$ and $\\cD_2$ be the two equivalence classes of horizontal cylinders, Lemma~\\ref{calc1} implies that any periodic point in $C$ must at the points indicated with circles. Since any of the solid dots can be moved into the middle vertical cylinder by a Dehn twist in horizontal cylinders, the solid dots are not periodic points (since they don't move onto circled points). \n\nNote the three cylinders labelled $C$ in the preceding paragraph cover the whole surface except for vertical saddle connection in the middle vertical cylinder and two horizontal saddle connections on the top and bottom horizontal cylinder. However, a point on these saddle connections can be moved off it by a Dehn twist in the simple horizontal or vertical cylinder whose core curve crosses the saddle connection. We conclude that any periodic point must lie in the orbit of the points marked with circles and hence be a fixed point of the involution. This proves Theorem~\\ref{T:basecase} for $\\tilde{\\mathcal{Q}}(3, -1^3)$. \n\n\\noindent \\textbf{Periodic Points in $\\mathcal{Q}(5, -1)$: } Consider the surface on the right in Figure \\ref{F:Prym3A}. Similarly to the previous case, setting $C$ to be either of the two middle horizontal cylinders and the $\\cD_i$ to be the two vertical equivalence classes, Lemma~\\ref{calc1} implies that any periodic point in the union of these cylinders must be one of the solid dots. Similarly, setting $C$ to be either of the two middle vertical cylinders and the $\\cD_i$ to be the two horizontal equivalence classes, Lemma~\\ref{calc1} implies that that any periodic point in the union of these cylinders must be one of the circled points.\n\nThe central point of the surface is fixed by the involution and is hence a periodic point; let us exclude this from our discussion. Any other point can be moved to the interior of one of the four cylinders labelled $C$ in the previous paragraph using Dehn twists. To conclude our analysis of $\\tilde{\\mathcal{Q}}(5, -1)$ it suffices to show that none of the eight solid or circled points drawn on the surface on the right in Figure \\ref{F:Prym3A} are periodic. \n\nBy using Dehn twists and symmetry, it suffices to show that the point $p$ in Figure \\ref{F:Prym4Special} is not periodic. \n\n\\begin{figure}[h]\n \\centering\n \\resizebox{0.45\\linewidth}{!}{\\begin{tikzpicture}\n \t\t\\draw (0,0) -- (0,3) -- (-1,3) -- (-1,5) -- (2,5) -- (2,6) -- (4,6) -- (4,3) -- (5,3) -- (5,1) -- (2,1) -- (2,0) -- (0,0);\n\t\t\\node at (-.25, .5) {$a$};\n\t\t\\node at (-.25, 2) {$1$};\n\t\t\\node at (-1.25, 4) {$1$};\n\t\t\\node at (-.5, 5.5) {$a$};\n\t\t\\node at (1, 5.5) {$1$};\n\t\t\\node at (2.5, 6.5) {$a$};\n\t\t\\node at (3.5, 6.5) {$1-a$};\n\t\t\\draw[dashed] (-1,4) -- (0, 5);\n\t\t\\draw[dashed] (-1,3) -- (1,5);\n\t\t\\draw[dashed] (0,3) -- (3,6);\n\t\t\\draw[dashed] (0,2) -- (4,6);\n\t\t\\draw[dashed] (0,1) -- (4,5);\n\t\t\\draw[dashed] (0,0) -- (4,4);\n\t\t\\draw[dashed] (1,0) -- (4,3);\n\t\t\\draw[dashed] (3,1) -- (5,3);\n\t\t\\draw[dashed] (4,1) -- (5,2);\n\t\t\\draw[black, fill] (2,3.9) circle[radius=1pt];\n\t\t\\node at (2.25, 3.9) {$p$};\n\t\\end{tikzpicture}}\n\\caption{} \n\\label{F:Prym4Special}\n\\end{figure}\n\nIn this figure, for any $1>a>0$ the slope 1 direction decomposes into four cylinders. With $a=\\frac12$, $p$ is on the boundary of one of the cylinders, but for nearby $a$ it is not. By continuity, after changing $a$ the point $p$ does not have rational height in the slope 1 cylinder in which it lies, showing that $p$ is not a periodic point by \\cite[Lemma 5.5]{Apisa}. This proves Theorem~\\ref{T:basecase} for $\\tilde{\\mathcal{Q}}(5, -1)$\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nGraph embeddings and linear layouts of graphs play an important role\nin graph drawing, parallel processing, matrix computation, VLSI\ndesign, and permutation sorting. A linear layout prescribes the\norder in which the vertices are processed and the embedding of the\nedges reveals structural properties of the given graph. A particular\nexample is a book embedding in which the edges are assigned to pages\nsuch that edges in the same page nest and do not cross.\nEquivalently, the vertices are visited in the linear order and the\nedges are processed in stacks. The concept of a book embedding of a\ngraph was introduced by Ollmann~\\cite{ollmann1973book} and by Kainen~\\cite{Kainen74}\nand can be formalized as follows.\nA $k$-page book embedding of a\ngraph $G=(V, E)$ is defined by a linear order of the vertices of $G$ and\na partition of the edges into $k$ sets $E_1, \\dots E_k$, so that the vertices of $G$\nare placed on a line in the given order and edges in $E_i$ are drawn on page $i$\n(typically with circular arcs), so that no two edges on the same page cross.\nThe book thickness of the graph $G$ is the smallest number of pages\nneeded, also known as stack number or page number.\n\n\n\nThe book thickness of planar graphs has been studied for over 40 years.\n Bernhart and Kainen~\\cite{bk-btg-79} characterized the graphs with book\n thickness one as the outerplanar graphs and the graphs with book thickness\n two as the sub-Hamiltonian planar graphs. Deciding whether a general planar\n graph has book thickness two is NP-hard~\\cite{chung1987embedding}.\n It is known that planar graphs require 3 pages and a series of improvements\n brought down the upper bound from 9~\\cite{buss1984pagenumber},\n to 7~\\cite{heath1984embedding}, and 4~\\cite{Yanna89}.\nAlthough in an earlier version of his 1989 paper Yannakakis in 1986~\\cite{YannaSTOC86}\n claimed 4 pages are necessary, and later Dujmovic and Wood in 2007~\\cite{dujmovic2007graph}\n also conjectured the same lower bound, there is still no conclusive evidence that this is indeed the case.\n\n\nMore recently there has been a greater interest in studying non-planar graphs which\nextend planar graphs by restrictions on crossings. A particular\nexample are {\\em 1-planar graphs} which can be drawn in the plane with at\nmost one crossing per edge. Such graphs were first defined by Ringel in the context of simultaneously drawing a planar graph\nand its dual~\\cite{ringel-65}. In many respects, 1-planar\ngraphs\ngeneralize planar graphs. There are 1-planar\nembeddings as witnesses of 1-planarity, in which the crossings are\ntreated as special vertices of degree four, and which then result in planarizations. \nLike $n$-vertex planar graphs which have at most $3n-6$ edges,\n$n$-vertex 1-planar graphs have at most $4n-8$ edges~\\cite{PT97}.\n Both planar and 1-planar 3-connected\ngraphs admit straight-line drawings in $O(n^2)$ area (with the exception of one edge in the outer face for the densest 1-planar graphs)~\\cite{ABK13}.\nHowever, there is a major difference in the complexity of the\nrecognition of planar and 1-planar graphs, which can be done in linear time\nfor planar graphs while it is $NP$-hard for 1-planar graphs~\\cite{GB07,KorzhikMohar13}. \nOn the other hand, there is a cubic time recognition algorithm for hole-free map graphs~\\cite {CGP06},\n which for 3-connected graphs coincide with \\textit{planar-maximal}\n 1-planar graphs (i.e., where no edge can be added without creating more crossing).\n\n\n\n\n\n\n\n\nIn this paper, we address the problem of book embedding of 1-planar graphs.\n Recently Bekos \\textit{et al.}~\\cite{BBKR15} gave a constant upper bound of 39 on the book\n thickness of 1-planar graphs.\nHere we prove that 1-planar graphs have book thickness at most 16 and 3-connected 1-planar\n graphs have book thickness at most 12.\n If the planar skeleton is Hamiltonian, then four pages suffice, and we have found 1-planar\n graphs which need four pages.\n\n\n\\section{Preliminaries}\n\n\n\n\n\n A \\emph{drawing} of a graph $G$ is a mapping of $G$ into the plane such that vertices\n are mapped to distinct points and edges are Jordan arcs between their endpoints.\n A drawing is \\emph{planar} if the edges do not cross and it is\n \\emph{1-planar} if each edge is crossed at most once.\nHence in a 1-planar drawing the crossing edges come in pairs.\nFor example, $K_5$ and $K_6$ are 1-planar graphs.\nAn \\emph{embedding} of a graph is planar (resp. 1-planar) if it admits a\n planar (resp. 1-planar) drawing. An embedding specifies\n the \\emph{faces}, which are topologically connected regions.\n The unbounded face is the \\emph{outer face}.\nAccordingly, a 1-\\emph{planar embedding} $\\mathcal{E}(G)$ specifies the\nfaces in a 1-planar drawing of $G$ including the outer face. A\n1-planar embedding is a witness for 1-planarity. In particular,\n$\\mathcal{E}(G)$ describes the pairs of crossing edges, the faces where\nthe edges cross, and the \\emph{planar} edges.\n\n\\begin{comment}\nEach pair of \\emph{crossing edges} $(a,c)$ and $(b,c)$ induces a\n\\emph{crossing point} $p$. Call the segment of an edge between the\nvertex and the crossing point a \\emph{half-edge}. Each half-edge is\n\\emph{impermeable}, analogous to the edges in planar drawings, in\nthe sense that no edge can cross such a half-edge without violating\nthe 1-planarity of the embedding. The non-crossed edges are called\n\\emph{planar}.\n\n\nA \\emph{planarization} $G^{\\times}$ is obtained from $\\mathcal{E}(G)$ by\nusing the crossing points as regular vertices and replacing each\ncrossing edge by its two half-edges. A 1-planar embedding $\\mathcal{E}(G)$\nand its planarization share equivalent embeddings, and each face is\ngiven by a list of edges and half-edges defining it, or\nequivalently, by a list of vertices and crossing points of the edges\nand half edges.\n\\end{comment}\n\n\nAugment a given 1-planar embedding $\\mathcal{E}(G)$ by adding as many edges to $\\mathcal{E}(G)$ as possible\n so that $G$ remains a simple graph and the newly added edges are planar in $\\mathcal{E}(G)$. We call such\n an embedding a \\textit{planar-maximal} embedding of $G$ and the operation \\textit{planar-maximal\n augmentation}.\nThen each pair of crossing edges is augmented to a $K_4$.\n The \\emph{planar skeleton} $\\mathcal{P}(\\mathcal{E}(G))$ consists of the planar edges of a\n planar-maximal augmentation. It is a planar embedded graph, since all pairs of crossing edges are\n omitted. Note that the planar augmentation and the planar skeleton are defined for an embedding,\n not for a graph.\n\n\n\n\\begin{comment}\n A graph may have different embeddings which give rise to different configurations\n and augmentations. The notion of planar-maximal embedding is different from the notions of maximal\n 1-planar embeddings and maximal 1-planar graphs, which are such that the addition of any edge\n violates 1-planarity (or simplicity)~\\cite{BEGGHR12}.\n\n\n The following claim, proven in many earlier\n papers ~\\cite{thomassen88,FM07,s-rm1pg-10,HELP12,BEGGHR12}, shows that a crossing pair of\n edges induces a $K_4$ in planar-maximal embedding, since missing edges of a $K_4$ can be added\n without inducing new crossings.\n\\begin{lemma}\n\\label{lem:induced-K4} Let $\\mathcal{E}(G)$ be a planar-maximal 1-planar embedding of a graph $G$ and let\n $(a,c)$ and $(b,d)$ be two crossing edges. Then the four vertices $\\{ a, b, c, d\\}$ induce a $K_4$.\n\\end{lemma}\n\\end{comment}\n\n\n\nThe \\emph{normal form} for an embedded 3-connected 1-planar graph $\\mathcal{E}(G)$ is obtained by first\n adding the four planar edges to form a $K_4$ for each pair of crossing edges while routing them close\n to the crossing edges and then removing old duplicate edges if necessary. Such an embedding of a\n 3-connected 1-planar graph is a {normal embedding} of it. A \\textit{normal planar-maximal augmentation}\n for an embedded 3-connected 1-planar graph is obtained by first finding a normal form of the embedding\n and then by a planar-maximal augmentation. \n\nGiven a 1-planar embedding $\\mathcal{E}(G)$, the normal planar-maximal augmentation of $\\mathcal{E}(G)$\n can be computed in linear time~\\cite{ABK13}. We say that an embedded 3-connected 1-planar graph\n is a \\textit{normal planar maximal} 1-planar graph if a normal planar maximal augmentation of the\n graph yields the same graph. In a 3-connected normal planar-maximal 1-planar graph, each pair of\n crossing edges $(a,c)$ and $(b,d)$ crosses each other either inside or outside the boundary of the\n quadrangle $abcd$ of the planar edges, and these define the so-called \\textit{augmented X-} and\n \\textit{augmented B-configurations}~\\cite{ABK13}.\n\n\n\n\n\n\nFor a 3-connected 1-planar graph $G$, Alam \\textit{et al.}~\\cite{ABK13} proved the following:\n\n\\begin{lemma}~\\cite{ABK13}\n\\label{lem:embedding} Let $G$ be a 3-connected 1-planar graph with a 1-planar embedding $\\mathcal{E}(G)$.\n Then the normal planar-maximal augmentation of $\\mathcal{E}(G)$ gives a planar-maximal 1-planar embedding\n $\\mathcal{E}(G^*)$ of a supergraph $G^*$ of $G$ so that $\\mathcal{E}(G^*)$ contains at most one augmented\n B-configuration in the outer face and each augmented X-configuration in $\\mathcal{E}(G^*)$ contains no vertex\n inside its skeleton.\n\\end{lemma}\n\n\n\n\n\n\n\n\\begin{comment}\n\n\n\n\nA book embedding of a graph $G=(V,E)$ is associated with a linear layout $L$ of the vertices of $G$,\n which is an ordering of $V$; i.e., a one-to-one function from $V$ to $\\{1, \\ldots, n\\}$, where $n=|V|$.\n We say that two edges $(a, b), (c, d)\\in E$ conflict in the layout $L$ if $L(a)< L(c) < L(b)< L(d)$ or\n $L(c) < L(a) < L(d) < L(b)$. Equivalently if we place the vertices on a horizontal line ordered by $L$\n and draw the edges above this line, two edges conflict if and only if they intersect. A \\textit{book\n embedding} of $G$ on $k$ pages consists of a linear layout $L$ of its vertices, and a coloring of\n its edges with $k$ colors (the ``pages'') so that conflicting edges in $L$ receive different colors (pages).\n\n\nRecall that a graph $G$ has a one page book embedding if and only if\n$G$ is a subgraph of an outerplanar graph, and $G$ has a two page\nbook embedding if and only if $G$ is planar subhamiltonian, i.e.,\n$G$ is a subgraph of a planar graph with a Hamiltonian cycle~\\cite{bk-btg-79}.\n\n\n\\end{comment}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Book Embeddings of 3-Connected 1-Planar Graphs}\n\\label{sec:3-connected}\n\n\nIf a graph can be embedded in a given number of pages, the same is true for its subgraphs. Given an\n embedded 3-connected 1-planar graph $G$, we therefore assume that $G$ is a normal planar\n maximal 1-planar graph. Lemma~\\ref{lem:embedding} implies that the planar skeleton of a normal\n planar maximal 3-connected 1-planar graph $G$ contains only triangular and quadrangular faces.\n Furthermore if we remove exactly one crossing edge (arbitrarily) from each pair of crossing edges in\n $G$, then the resulting graph is a maximal planar graph.\n\n\n\nOur algorithm uses the a ``peeling technique'' similar to Yannakakis~\\cite{Yanna89} and iteratively\n removes the vertices on the outer cycle of the planar skeleton $\\mathcal{P}(\\mathcal{E}(G))$ of $G$. This partitions the\n vertices of $G$ into \\textit{levels} according to their ``distance'' from the outer face of the planar\n skeleton $\\mathcal{P}(\\mathcal{E}(G))$. Vertices on the outer face of $\\mathcal{P}(\\mathcal{E}(G))$ are at level 0. Deleting these\n vertices from $\\mathcal{P}(\\mathcal{E}(G))$ yields the level 1 graph; the vertices that lie now on the outer face are at\n level 1. In general,\n the level $t$ graph is obtained by deleting all vertices at levels less than $t$; the vertices that lie on\n the outer face of this graph are at level $t$. The edges of $G$ (including the crossing edges) are\n partitioned into \\textit{level edges} at level $i$, edges that connect vertices at the same level $i$,\n and \\textit{binding edges}, edges that connect vertices at different levels. The fact that a level $i$\n vertex is not on the outer face after deleting the first $i-2$ levels implies that every level $i$ vertex\n lies in the interior of some cycle composed of level $i-1$ vertices. This means in particular that a\n level $i$ vertex cannot be adjacent to a level $j$ vertex with $j < i-1$ and binding edges connect\n only consecutive levels.\n \n\nSimilar to Yannakakis~\\cite{Yanna89} we first place level 0 vertices in the clockwise order (cw-order)\n as they appear on the outer cycle, assigning the edges on the outer cycle on the same page.\n Then we place the level 1 vertices and assign the following edges to some pages:\n (i) the level edges of each cycle on the outer boundary of the level 1 graph\n (ii) the binding edges between levels 0 and 1\n (iii) the crossing edges either at level $0$ or binding between level 0 and 1.\nLevel 1 vertices are placed in such a way that the vertices on each level 1 cycle are\n in the counterclockwise order (ccw-order) around the cycle.\n Now the rest of the graph is in the interior of level 1 cycles. The algorithm takes each\n level 1 cycle in turn and lays out its interior in a similar way.\n\n\nWe therefore next consider a \\textit{2-level subgraph} $H$ of $G$ defined as follows. The vertices of $H$\n are the vertices on a level $i$ cycle $C_i$ and all the level $i+1$ vertices $V_{i+1}$ interior to $C_i$.\n The edges of $H$ are all the planar and crossing edges inside the region between $C_i$ and the outer\n boundaries of all the level $i+1$ components inside $C_i$ (including the edges on $C_i$ and the level\n $i+1$ boundaries). Fig.~\\ref{fig:2-level} shows a 2-level subgraph inside a cycle $C_i=AB\\ldots Z$.\n We denote this 2-level subgraph of $H$ inside $C_i$ as $H(C_i)$. We assume that $C_i$ has already\n been embedded where the vertices of $C_i$ are placed in the cw (or ccw, resp.)\n order around $C_i$. We then extend this embedding to a book embedding of $H(C_i)$, by\n placing the remaining vertices of $H(C_i)$ and assign the remaining edges of $H(C_i)$ to seven pages.\nThe book embedding of $G$ is obtained by repeatedly computing the book embeddings of\n $H(C_i)$ and reusing the same seven pages for all odd (even) $i$.\n\n\n\n\n\\subsection{Drawing 2-Level Subgraphs}\n\nIn this section we prove the following lemma.\n\n\n\\begin{lemma}\n\\label{lem:2-level}\nLet $H(C_i)$ be a 2-level subgraph of $G$ inside\na level $i$ cycle $C_i$. Then there exists a book embedding\n$\\Gamma$ of $H(C_i)$ on seven pages where the vertices of\n $C_i$ are placed in the cw (or ccw) order around $C_i$.\n\\end{lemma}\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{new-illustration-graph.pdf}\n\\caption{A 2-level subgraph $H(C_i)$ of $G$ inside the level-$i$ cycle $C_i=AB\\ldots Z$, which is\n drawn with thick black edges. The outer boundary of the level $i+1$ component is drawn with thick\n blue edges. The red dashed edges are the crossing edges taken in the set $X$.}\n\\label{fig:2-level}\n\\end{figure}\n\n\nWe give a construction of a book embedding where the vertices of $C_i$ are placed in the cw-order\n (for ccw-order we flip the embedding of $H(C_i)$).\n\n\n\nLet $v_1, \\ldots, v_t$ be the vertices of $C_i$ in the cw-order around $C_i$. For the remaining part\n of this section, we call the vertices on $C_i$ as the \\textit{outer vertices} and the level $i+1$ vertices\n of $H(C_i)$ as the \\textit{inner vertices}.\nWe first obtain a planar graph $H'$ from $H(C_i)$ by removing exactly one edge from each\n pair $\\langle (a,b), (c,d)\\rangle$ of crossing edges. Let $X$ be the set of crossing edges that we remove.\n From each crossing edge pair $\\langle (a,b), (c,d)\\rangle$, we take one edge\n to be in $X$ as follows; see Fig.~\\ref{fig:2-level}.\n\n\\noindent\n\\textbf{Case S1.} If both $(a,b)$, $(c,d)$ are level edges at level $i$, then we take the edge adjacent to the\n vertex farthest from $v_1$ in cw-order on $C_i$ to be in $X$. In particular if the two level $i$ edges\n forming the crossing pair are $(v_p, v_r)$, $(v_q, v_s)$ with $p$ 99\\%) are multiply-adds as shown in the table included in Figure~\\ref{fig:motivation}.\nFor instance, each single image classification with AlexNet~\\cite{wrpn} requires a total of 2682 million operations, of which 99.86\\% (2678 million) are multiply-adds.\nTo this end, the compute units of \\bitfusion can dynamically fuse or decompose to match the bitwidth of each individual multiply-add operand without requiring the operands to be encoded in the same bitwidth.\n\nSecond, energy consumption for DNN acceleration is usually dominated by data accesses to on-chip storage and off-chip memory~\\cite{tetris:asplos:2017, eyeriss:isca:2016, eyeriss:jssc:2017}.\nTherefore, \\bitfusion comes with encoding and memory access logic that stores and retrieves the values in the lowest required bitwidth.\nThis logic reduces the overall number of bits read or written to on-chip and off-chip memory, proportionally reducing the energy dissipation of memory accesses.\nFurthermore, this strategy increases the effective on-chip storage capacity.\n\n\nThird, \\bitfusion builds upon the extensive prior work that shows DNNs can operate with reduced bitwidth without degradation in classification accuracy~\\cite{dorefa:arxiv:2016, qnn:arxiv:2016, li2016ternary, zhu2016trained, envision:isscc:2017, stripes:micro:2016}.\nThis opportunity exists across different classes of real-world DNNs, \nas shown in Figure~\\ref{fig:motivation}. \nOne category is Convolutional Neural Networks (CNNs) that usually use convolution and pooling layers followed by a stack of fully-connected layers.\nAlexNet, Cifar-10, LeNet-5, ResNet-18, SVHN, and VGG-7 in Figure~\\ref{fig:motivation} belong to this category.\nRecurrent Neural Networks (RNN) are another sub-class of DNNs that use recurrent layers including Long Short Term Memory (LSTM) and vanilla RNN layers to extract \\emph{temporal} features from time-varying data.\nThe RNN and LSTM benchmark DNNs in Figure~\\ref{fig:motivation} represent these categories.\nFurthermore, as the table in Figure~\\ref{fig:motivation} shows, most operations in DNNs ($>$ 99\\%), regardless of their categories, are multiply-adds.\nAs Figure~\\ref{fig:motivation}(a) illustrates, on average, 97.3\\% of multiply-adds require four or fewer bits and even in some DNNs a large fraction of the operations can be done with bitwidth equal to one.\nMore interestingly, the bitwidths vary within and across DNNs to guarantee no loss of accuracy.\nSuch a variation is not limited to the intermediate operands and exists in trained weights as illustrated in Figure~\\ref{fig:motivation}(b).\nTo exploit this property, a programmable accelerator needs to offer bit-level flexibility at runtime, which leads us to \\bitfusion.\n\nTo harvest the aforementioned opportunities, this paper makes the following contributions and realizes a new dimension in the design of DNN accelerators.\n\n\n\\begin{enumpacked}\n\t\\item \\textbf{Dynamic bit-level fusion and decomposition.} \n\t%\n\tThe paper introduces and explores the dimension of bit-level flexible DNN accelerator architectures, \\bitfusion, that dynamically matches bit-level composable processing engines to the varying bitwidths required by DNN layers. \n\t%\n\tBy offering this flexibility, \\bitfusion aims to minimize the computation and communication required by a DNN at the bit granularity on a per layer basis.\n\t\n\t\\item \\textbf{Microarchitecture design for bit-level composability.} To explore \\bitfusion, we design and implement a DNN accelerator using a novel bit-flexible computation unit, called \\bricks.\n\t%\n\tThe accelerator supports both feed-forward (CNN) and recurrent (LSTM and RNN) layers.\n\t%\n\tA 2D array of \\bricks constructs a fusible processing engine that can perform the DNN computation at various bitwidths.\n\t%\n\tThe microarchitecture also comes with a storage logic that allows feeding the \\bricks with different bitwidth operands.\n\t\t\t\n\t\\item \\textbf{Hardware-software abstractions for bit-flexible acceleration.} To enable DNN applications to take advantage of these unique bit-level fusion capabilities, we propose a block-structured instruction set architecture, called \\fusionisa. To amortize the cost of programmability, \\fusionisa expresses operations of DNN layers as bit-flexible instruction blocks with iterative semantics.\n\\end{enumpacked}\n\nThese three contributions define the novel architecture of \\bitfusion, a possible microarchitecture implementation, and the hardware-software abstractions to offer bit-level flexibility.\nOther complementary and inspiring works have explored bit serial computation~\\cite{stripes:micro:2016, tartan:arxiv:2017} without exploring the fusion dimension.\nIn contrast, \\bitfusion \\emph{spatially} fuses a group of \\bricks together, to collectively execute operations at different bitwidths.\nUsing eight real-world feed-forward and recurrent real-world DNNs, we evaluate the benefits of \\bitfusion.\nWe implemented the proposed microarchitecture in Verilog and synthesized in 45~nm technology.\nUsing the synthesis results and cycle accurate simulation, we compare the benefits of \\bitfusion to two state-of-the-art DNN accelerators, Eyeriss~\\cite{eyeriss:isca:2016} and Stripes~\\cite{stripes:micro:2016}.\nThe latter is an optimized bit-serial architecture.\nIn the same area, frequency, and technology node, \\bitfusion offers \\eyerissPerfAvg speedup and \\eyerissEnergyAvg energy savings over Eyeriss.\nCompared to Stripes~\\cite{stripes:micro:2016}, \\bitfusion provides \\stripesPerfAvg speedup and \\stripesEnergyAvg energy reduction at 45~nm node when \\bitfusion area and frequency are set to those of Stripes.\nScaling to GPU technology node of 16~nm, \\bitfusion provides a \\bitfusionPerfAvgOverTX speedup over the Jetson TX2 mobile GPU.\nFurther, \\bitfusion almost matches the performance of a 250-Watt \\titanxp, which uses $8$-bit vector instructions, while \\bitfusion merely consumes 895 milliwatts of power.\n\\section{\\bitfusionsection Architecture}\n\\label{sec:arch}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.95\\linewidth]{fig\/bitfusion.pdf}\n\t\\vspace{-0.25ex}\n\t\\caption{Dynamic composition of \\bricks (BBs) in a \\fusionunit to construct Fused Processing Engines (Fused-PE), shown as F-PE.}\n\t\\label{fig:bitfusion}\n\t\\vspace{-3ex}\n\\end{figure}\n\n\nTo minimize the computation and communication at the finest granularity, \\bitfusion dynamically matches the architecture of the accelerator to the bitwidth required for the DNN, which may vary layer by layer, without any loss in accuracy.\nAs such, \\bitfusion is a collection of bit-level computational elements, called \\bricks, that dynamically compose to \\emph{logically} construct Fused Processing Engines (\\fusedpe) that execute DNN operations with the required bitwidth.\nSpecifically, \\fusedpes provide bit-level flexibility for multiply-adds, which are the dominant operations across all types of DNNs.\nBelow, we discuss how \\bricks can be dynamically fused together to support a range of bitwidths, yet provide a significant increase in parallelism when operating at lower bitwidths.\n\n\\vspace{-1ex}\n\\subsection{Bit-Level Flexibility via Dynamic Fusion}\nAs depicted in Figure~\\ref{fig:bitfusion}, \\bitfusion arranges the \\bitbricks in a 2-dimensional \\emph{physical} grouping, called \\fusionunit. \nEach \\bitbrick in a \\fusionunit can perform individual binary (0, +1) and ternary (-1, 0, +1) multiply-add operations.\nAs Figure~\\ref{fig:bitfusion} shows, the \\bricks \\emph{logically} fuse together at run-time to form Fused Processing Engines (\\fusedpes) that match the bitwidths required by the multiply-add operations of a DNN layer.\nThe \\bricks in a \\fusionunit multiply an incoming variable-bitwidth input (input forward) to a variable-bitwidth weight (from WBUF) to generate the product.\nThe \\fusionunit then adds the product to an incoming partial sum to generate an outgoing partial sum (\\texttt{Psum forward} in Figure~\\ref{fig:bitfusion}(a)).\n\nFigures~\\ref{fig:bitfusion}(b),~\\ref{fig:bitfusion}(c), and~\\ref{fig:bitfusion}(d) show three different ways of logically fusing \\bricks to form (b) 16 \\fusedpes that support ternary (binary); (c) four \\fusedpes that support mixed-bitwidths (2-bits for weights and 8-bits for inputs), (d) one \\fusedpe that supports 8-bit operands, respectively.\nFor binary or ternary operations (Figures~\\ref{fig:bitfusion}(b)), each \\fusedpe contains a single \\brick, offering the highest parallelism.\nThe \\fusionunit then adds the results from all \\fusedpes and the incoming partial sum to generate a single outgoing partial sum.\nFigure~\\ref{fig:bitfusion}(c) shows four \\bricks fused together in a column to form a \\fusedpe that can multiply 2-bit weights with 8-bit inputs.\nThe bitwidths of operands supported by a \\fusedpe depend on the spatial arrangement of \\bricks fused together.\nAlternatively, by varying the spatial arrangement of the four fused \\bricks, the \\fusedpe can support 8-bit\/2-bit, 4-bit\/4-bit, and 2-bit\/8-bit configurations for inputs\/weights.\nFinally, up to 16 \\bricks can fuse together to construct a single \\fusedpe that can operate on 8-bit operands for the multiply-add operations (Figure \\ref{fig:bitfusion}(d)).\nThe \\bricks fuse together in powers of 2. That is, a single \\fusionunit with 16 \\bricks can offer 1, 2, 4, 8, and 16 \\fusedpes with varying operand bitwidths.\nDynamic composability of the \\fusionunits at the bit level enables the architecture to expose the maximum possible level of parallelism with the finest granularity that matches the bitwidth of the DNN operands.\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.8\\linewidth]{fig\/architecture.pdf}\n\t\\caption{\\bitfusion systolic architecture comprising a collection of \\bricks (BBs) that can fuse to form \\fusedpes.}\n\t\\label{fig:arch}\n\t\\vspace{-3.5ex}\n\\end{figure}\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.7\\linewidth]{fig\/systolic-array-execution.pdf}\n\t\\vspace{-0.5ex}\n\t\\caption{Bit-Flexible matrix-vector multiplication.}\n\t\\label{fig:systolic-execution}\n\t\t\\vspace{-1ex}\n\\end{figure}\n\n\\vspace{-1ex}\n\\subsection{Accelerator Organization}\nTwo insights guide the architecture design of \\bitfusion.\nFirst, DNNs offer high degrees of parallelism and benefit significantly from increasing the number of \\fusionunits available within the accelerator's area budget.\nTherefore, it is essential to minimize the overhead of control in the accelerator by not only maximizing the number of \\fusionunits but also minimizing the overhead of dynamically constructing \\fusedpes, thereby integrating the maximum number of \\bitbricks in the area budget.\nSecond, on-chip SRAM and register-file accesses dominate the energy consumption when accelerating DNNs~\\cite{tetris:asplos:2017, eyeriss:isca:2016, eyeriss:jssc:2017}.\nTherefore, it is essential to reduce the number of bits exchanged with on-chip and off-chip memory while maximizing data reuse.\n\n\\niparagraph{\\bitfusion Systolic array.}\nWith these insights, we employ a 2-dimensional systolic array of \\fusionunits as the architecture for \\bitfusion, as shown in Figure~\\ref{fig:arch}.\nThe systolic organization reduces the overhead of control by sharing the control logic across the entire systolic array.\nMore importantly, systolic execution alleviates the need for provisioning control for each \\fusedpe as a dataflow architecture would have required.\nAs such, the systolic architectures fit the most number of \\bitbricks in a given area budget.\nThus, the entire systolic array composed of \\fusedpes acts as a single compute unit that can execute, for example, a single matrix-vector multiplication operation with various bitwidths, which also sets the level of parallelism.\nIn addition, the systolic organization of \\fusionunits enforces sharing of input data across columns of the array and accumulates partial results across rows of the array to minimize access to on-chip memory.\nAs depicted in Figure~\\ref{fig:arch}, the input buffers (\\code{IBUFs}) only located at the borders and feed the rows simultaneously.\nSimilarly, the output buffers (\\code{OBUFs}) reside on the bottom and collect the flowing results, which is accumulated by each column's accumulator.\nAs shown in Figure~\\ref{fig:arch}, each column harbors a pooling and an activation unit before its output buffer.\nFinally, the systolic organization also eliminates the need for local buffers for input, output, or partial results within \\fusionunits.\nAs such, each \\fusionunit is accompanied by only a weight buffer (\\code{WBUF}).\nUsing \\fusedpes as the building blocks, the performance of the systolic array maximally matches the bitwidths, with the highest performance at binary and ternary settings.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.8\\linewidth]{fig\/bitbrick.pdf}\n\t\\vspace{-0.5ex}\n\t\\caption{A single \\brick. (HA: Half Adder, FA: Full Adder.)}\n\t\\label{fig:bitbrick}\n\t\\vspace{-3.5ex}\n\\end{figure}\n\n\\begin{figure*}[h]\n\\begin{minipage}{0.65\\linewidth}\n\t\\centering\n\t\\hspace{-1ex}\n\t\\includegraphics[height=2.2in]{fig\/bitbrick-4bx4b.pdf}\n\t\\caption{Using \\bitbricks to execute $4$-bit multiplications.}\n\t\\label{fig:bitbrick-4bx4b}\n\\end{minipage}\n\\begin{minipage}{0.3\\linewidth}\n\t\\centering\n\t\\hspace{-1ex}\n\t\\includegraphics[height=1.4in]{fig\/bitbrick-4bx2b-algorithm.pdf}\n\t\\caption{Two $4$-bit $\\times$ $2$-bit multiplications decomposed to four $2$-bit multiplications followed by the accumulation (summation) logic.}\n\t\\label{fig:bitbrick-4bx2b-algorithm}\n\\end{minipage}\n\\vspace{-3.5ex}\n\\end{figure*}\n\n\\niparagraph{Memory organization.}\nDepending on the number of \\fusedpes and their organization, the buffers must supply different number of operands with various bitwidths.\nAs such, we augment the input and the weight buffers with a register that holds a row of data that is gradually fed to the \\fusedpes according to their bitwidth.\nAs illustrated in Figure~\\ref{fig:arch}, a series of multiplexers after the register make this data infusion possible.\nThe benefit of this design is avoiding multiple accesses to the data array of the buffer which conserves energy.\nWith this design, at each cycle, the systolic array consumes a vector of inputs and matrix of weights to produce a vector of outputs with the fewest accesses to the buffers and the minimal bitwidth possible.\n\n\\vspace{-1ex}\n\\subsection{\\bitfusionsubsection Execution Model}\n\nFigure~\\ref{fig:systolic-execution} illustrates the \\bitfusion systolic execution in the mixed-bitwidth mode using when an input vector is multiplied to a weight matrix.\nThe input vector has $4\\times N$ 8-bit elements that are being multiplied to a matrix with $4\\times N \\times M$ 2-bit elements.\nAs such, the 16-\\bitbricks in a \\fusionunit logically compose to form four 8$\\times$2 \\fusedpes.\nBoth input and weight buffers provide 32 bits per access.\nThe read values are split into 8-bit input values and 2-bit weight values in the output register of each buffer using its accompanying multiplexers as mentioned before.\nThe input values are shared across the \\fusionunits of each row and weight values are specific to each \\fusedpe.\nAs such, all of the $4\\times N \\times M$ \\fusedpes work in parallel while only a single 32-bit value is read from the input and weight buffers.\nExploiting the lower bitwidth of weights, \\bitfusion increases the level of parallelism by 4$\\times$ while reducing the number of accesses to the weight buffer data arrays by the same factor of four.\nAs discussed above, each \\fusionunit adds the results of its \\fusedpes with its incoming partials results and forwards the partial output to the \\fusionunit underneath it.\nAs shown in Figure~\\ref{fig:systolic-execution}, we support 32-bit bitwidth for the partial and final results to avoid any inaccuracies.\n\\section{Related Work}\n\\label{sec:related}\n\nA growing body of related works develop DNN accelerators.\n\\bitfusion fundamentally differs from prior work as it introduces and explores a new dimension of bit-level composable architectures that can dynamically match the bitwidth required by DNN operations.\n\\bitfusion aims to minimize both computations and communications in the finest granularity possible without compromising on the DNN accuracy.\nBelow, we discuss the most related work.\n\n\\niparagraph{Precision flexibility in DNNs.}\n\\stripes~\\cite{stripes:micro:2016} and Tartan~\\cite{tartan:arxiv:2017} use bit-serial compute units to provide precision flexibility for inputs at the cost of additional area overhead.\nBoth works provide performance and efficiency benefits that are proportional to the precision reduction for inputs.\nWe directly compare the benefits of \\bitfusion to \\stripes in Section~\\ref{sec:eval}.\nUNPU~\\cite{unpu:isscc:2018} fabricates a bit-serial DNN accelerator at 65~nm, similar to \\stripes~\\cite{stripes:micro:2016}.\n\\loom~\\cite{loom:arxiv:2017} uses bit-serial computation for precision flexibility.\n\\deeprecon~\\cite{deeprecon:IJCNN:2017} skips stages of a fully-pipelined floating-point-multiplier to perform either one 16-bit, two 12-bit, or four 8-bit multiplications.\nIn contrast, the \\fusionunits are spatial designs that use combinational logic to dynamically compose and decompose 2-bit multipliers (\\bricks) to construct variable bitwidth multiply-add units.\nMoons et al. propose aggressive voltage scaling techniques at low precision for increased energy efficiency at constant throughput by turning off parts of the multiplier~\\cite{envision:isscc:2017, moons:vlsi:2016}.\nAs such, they do not offer fusion capabilities.\nTPU~\\cite{tpu:isca:2017} proposes a systolic architecture for DNNs and supports 8-bit and 16-bit precision.\nThis work, on the other hand, proposes an architecture that dynamically composes low-bitwidth compute units (\\bricks) to match the bitwidth requirements of DNN layers.\n\n\\niparagraph{Binary DNN accelerators.}\nSeveral inspiring works have explored ASIC and FPGA accelerators optimized for Binary DNNs.\nFINN~\\cite{finn:fpga:2017} uses FPGAs for accelerating Binary DNNs, while YodaNN~\\cite{yodann:arxiv:2017} and BRein~\\cite{brein:isscc:2017} propose an ASIC accelerator for binary DNNs.\nKim, et al.~\\cite{binarydecompose:dac:2017} decompose the convolution weights for binary CNNs to improve performance and energy efficiency.\nThe above works focus solely on binary DNNs to achieve high performance at the cost of classification accuracy.\n\\bitfusion, on the other hand, flexibly matches the bitwidths of DNN operations for performance\/energy benefits without losing accuracy.\n\n\\niparagraph{Sparse Accelerators for DNNs.}\nEIE~\\cite{eie:isca:2016}, Cambricon-X~\\cite{cambricon-x:micro:2016}, Cnvlutin~\\cite{cnvlutin:isca:2016}, and SCNN~\\cite{scnn:isca:2017} explore the sparsity in the DNN layers and use zero-skipping to provide performance and energy-efficiency benefits.\nOrthogonal to the works above, \\bitfusion explores the dimension of bit-flexible accelerators for DNNs.\n\n\n\\niparagraph{Other ASIC accelerators for DNNs.}\nDaDianNao~\\cite{dadiannao:micro:2014} uses eDRAM to eliminate off-chip accesses and provide high performance and efficiency for DNNs.\nPuDianNao~\\cite{pudiannao:asplos:2015} is an accelerator designed for machine learning, but does not support CNNs.\nMinerva~\\cite{minerva:isca:2016} proposes operation pruning and data quantization techniques to reduce power consumption for ASIC acceleration.\n\\eyeriss~\\cite{eyeriss:isca:2016, eyeriss:jssc:2017} presents an optimized row-stationary dataflow for DNNs to improve efficiency.\nTetris~\\cite{tetris:asplos:2017} and Neurocube~\\cite{neurocube:isca:2016} propose 3-D stacked DNN accelerators to provide high bandwidth for DNN operations.\nISAAC~\\cite{isaac:isca:2016}, PipeLayer~\\cite{pipelayer:hpca:2017}, and Prime~\\cite{prime:isca:2016} use resistive RAM (ReRAM) for accelerating DNNs.\nGanax~\\cite{ganax:isca:2018} uses a SIMD-MIMD architecture to support DNNs and generative models.\nSnapea~\\cite{snapea:isca:2018} employs early termination to skip computations.\n\n\\niparagraph{Instruction Sets for DNNs.}\nCambricon~\\cite{cambricon:isca:2016} provides an ISA to express the different computations in a DNN using vector and matrix operations without significant loss in efficiency over DaDianNao.\nDnnWeaver~\\cite{dnnweaver:micro:2016} proposes a coarse grained ISA to express layers of DNNs, which are first translated to micro-codes for FPGA acceleration.\nUnlike prior work, the \\fusionisa proposed in the work is designed to enable bit-level flexibility for accelerating DNNs.\nFurther, the \\fusionisa uses \\code{loop} instructions with iterative semantics to significantly reduce instruction footprint.\n\n\\niparagraph{Code optimization techniques.}\nAlwani, et. al~\\cite{fused-cnn} propose layer-fusion, that combines multiple convolutional layers to save off-chip accesses for FPGA acceleration of CNNs.\nEscher~\\cite{escher:fccm:2017} proposes a CNN FPGA accelerator using flexible buffering that balances the off-chip accesses for inputs and weights in CNNs.\nThe above works have inspired the code-optimizations explored in this paper, however, the key contribution of this work is a bit-level flexible DNN accelerator.\n\n\n\\niparagraph{Software techniques for Binary\/XNOR DNNs.}\nQNN~\\cite{qnn:arxiv:2016} shows that efficient GPU kernels for XNOR-based binary DNNs can provide up to 3.4$\\times$ improvement in performance.\nXNOR-Net~\\cite{xnornet:arxiv:2017} shows that specialized libraries for Binary\/XNOR-nets can achieve 58$\\times$ performance on CPUs.\nIn contrast, \\bitfusion is an ASIC accelerator architecture that supports a wide range of bitwidths (binary to 16-bits) for DNNs with no accuracy loss.\n\n\\niparagraph{Core Fusion and CLPs.} Core Fusion~\\cite{corefusion} and CLPs~\\cite{tflex} are dynamically configurable chip multiprocessors that a group of independent processors can fuse and form a more capable CPU.\nIn contrast to these inspiring works, \\bitfusion performs the composition in the bit level rather than at the level of full-fledged cores.\n\n\\subsection{Methodology}\n\\label{sec:methodology}\n\n\\niparagraph{Benchmarks.}\nTable~\\ref{tab:benchmarks} shows the list of 8 CNN and RNN benchmarks from diverse domains including image classification, object and optical character recognition, and language modeling.\nThe selected DNN benchmarks use a diverse size of input data, which allows us to evaluate the effect of input data size on the \\bitfusion architecture.\n\\bench{AlexNet}~\\cite{alexnet, wrpn}, \\bench{SVHN}~\\cite{svhn:nips:2011, qnn:arxiv:2016}, \\bench{CIFAR10}~\\cite{cifar10, qnn:arxiv:2016}, \\bench{LeNet-5}~\\cite{lenet-5, li2016ternary}, \\bench{VGG-7}~\\cite{vgg, li2016ternary}, \\bench{ResNet-18}~\\cite{resnet, wrpn} are popular and widely-used CNN models.\nAmong them, \\bench{AlexNet} and \\bench{ResNet-18} benchmarks are image classification applications that have different network topologies that use the ImageNet dataset.\nThe \\bench{SVHN} and \\bench{LeNet-5} benchmarks are optical character recognition applications that recognize the house numbers from the house view photos and handwritten\/machine-printed characters, respectively.\n\\bench{CIFAR10} and \\bench{VGG-7} are object recognition applications based on the CIFAR-10 and ImageNet dataset, respectively.\nThe \\bench{RNN}~\\cite{qnn:arxiv:2016} and \\bench{LSTM}~\\cite{lstm, qnn:arxiv:2016} are recurrent networks that perform language modeling on the Penn TreeBank dataset~\\cite{penn-treebank}. \nIn Table~\\ref{tab:benchmarks}, the ``Multiply-Add Operations'' column shows the required number of Multiply-Add operations for each model and the ``Model Weights'' column shows the size of model parameter.\nNote that the multiply-add operations and model weights have variable bitwidths as presented in Figure~\\ref{fig:motivation}.\n\n\\niparagraph{Reduced bitwidth DNN models.}\n\\bitfusion aims to accelerate the inference of a wide range of DNN models with varying bitwidth requirements, with \\emph{no loss in classification accuracy}.\nThe benchmarks, listed in Table~\\ref{tab:benchmarks}, employ the model topologies proposed in prior work~\\cite{dorefa:arxiv:2016, qnn:arxiv:2016, li2016ternary, wrpn} that train low bitwidth DNNs and achieve the same accuracy as the 32-bit floating-point models.\nWe did not engineer these quantized DNNs and merely took them from the existing deep learning literature~\\cite{dorefa:arxiv:2016, qnn:arxiv:2016, li2016ternary, wrpn}.\nBenchmarks \\bench{Cifar-10}, \\bench{SVHN}, \\bench{LSTM}, and \\bench{RNN} use the quantized models presented in~\\cite{qnn:arxiv:2016}.\nBenchmarks \\bench{LeNet-5} and \\bench{VGG-7} use ternary (+1,0,-1) networks~\\cite{li2016ternary}.\n\\bench{AlexNet} and \\bench{ResNet-18} use the $4$-bit $2\\times$ wide models presented in~\\cite{wrpn} that double the number of channels for convolution and fully-connected layers.\nWe use the regular \\bench{AlexNet} and \\bench{ResNet-18} models for \\eyeriss and the GPU baselines, and use their $2\\times$ wide quantized models for \\bitfusion and \\stripes.\n\n\\begin{table}\n\t\\centering\n\t\\caption{Evaluated CNN\/RNN benchmarks.}\n\t\\vspace{-1ex}\n\t\\includegraphics[width=0.95\\linewidth]{fig\/benchmarks.pdf}\n\t\\vspace{-1ex}\n\t\\label{tab:benchmarks}\n\\end{table}\n\n\\begin{table}\n\t\\centering\n\t\\caption{Evaluated ASIC and GPU platforms. *\\stripes entries per-tile.}\n\t\\vspace{-1ex}\n\t\\includegraphics[width=0.95\\linewidth]{fig\/platforms.pdf}\n\t\\vspace{-3ex}\n\t\\label{tab:platforms}\n\\end{table}\n\n\\niparagraph{Accelerator development and synthesis.}\nWe use RTL-Verilog to implement the configuration of the \\bitfusion architecture and verify the design through extensive RTL-simulations.\nWe synthesize \\bitfusion at 45nm technology node using \\bench{Synopsys Design Compiler (L-2016.03-SP5)} and a commercial standard-cell library.\n\\bench{Design Compiler} provides the chip area, achievable frequency, and dynamic\/static power, which we use to estimate the performance and energy-efficiency of the \\bitfusion accelerator.\n\n\\niparagraph{Simulation infrastructure for \\bitfusion.}\nWe compile each DNN benchmark to the instructions of the \\fusionisa (Section~\\ref{sec:isa}).\nWe develop a cycle-accurate simulator that takes the \\fusionisa instructions for the given DNN and simulates the execution to calculate the cycle counts as well as the number of accesses to on-chip buffers (IBUF, OBUF, and WBUF in Figure~\\ref{fig:arch}) and off-chip memory.\nWe verify the cycle counts of the simulator against our Verilog implementation of the \\bitfusion architecture.\nUsing the frequency defined in Table~\\ref{tab:platforms} and the cycle counts, the simulator measures the execution time of the \\bitfusion architecture.\nTo evaluate the energy efficiency, we model the energy consumption for on-chip buffers for the \\bitfusion accelerator using the results from CACTI-P~\\cite{cactip}.\n\n\\niparagraph{Comparison with Eyeriss.}\nTo measure the performance and energy dissipation of our comparison point, \\eyeriss, we use their open-source simulation infrastructure~\\cite{tetris:asplos:2017}.\nThe resulting area and energy metrics are shown in Table~\\ref{tab:platforms}.\nAs mentioned, we use the same area budgets as \\eyeriss, which is 1.1 mm$^2$ for compute units and 5.87 mm$^2$ for chip to synthesize \\bitfusion, shown in Table~\\ref{tab:platforms}.\nWe use a total 112~KB SRAM for on-chip buffers (IBUF, OBUF, and WBUF in Figure~\\ref{fig:arch}).\nEyeriss operates on the 16-bit operands and \\bitfusion supports flexible bitwidths from 2, 4, 8, to 16 bits.\n\n\\niparagraph{Comparison with Stripes.}\nThe authors of Stripes graciously shared their simulator~\\cite{stripes:micro:2016}.\nTheir power estimation tools were in 65~nm node, which we scaled to 45~nm.\n\\stripes operates on 16-bit inputs and variable-bitwidth weights (1 through 16), using Serial Inner-Product units (SIPs).\n\\stripes is organized into 16 tiles each of which has \\xx{4096} SIPs.\nFor a fair comparison, we replace the \\xx{4096} SIPs in each tile of \\stripes with our proposed \\bitfusion systolic array with 512 \\fusionunits, each with 16 \\bricks to match the same budget of \\xx{1.1mm$^2$} for compute, which is the area after scaling to 45~nm and use the same total on-chip memory.\n\n\\niparagraph{Comparison with GPUs.}\nWe use two GPUs (\\titanxp and Tegra X2) based on Nvidia's Pascal architecture to compare with \\bitfusion.\nTable~\\ref{tab:platforms} shows the details of the two GPUs.\nWe use Nvidia's custom \\bench{TensorRT 4.0}~\\cite{tensorrt} library compiled with the latest CUDA~\\xx{9.0} and cuDNN~\\xx{7.1} which support 8-bit quantized calculations, the smallest possible in the architecture.\nAcross GPU platforms, we use 1,000 warm-up batches, followed by 10,000 batches to measure performance and use the average.\nFor a head-to-head comparison, we conservatively scale \\bitfusion to 16~nm technology node assuming a \\xx{$0.86\\times$} voltage scaling and \\xx{$0.42\\times$} capacitance scaling according to the methodology presented in~\\cite{dark_silicon:isca}.\nHowever, we assume the same frequency of \\xx{500 MHz} as \\eyeriss and do not increase the \\bitfusion frequency.\nThe scaled \\bitfusion architecture has 4096 \\fusionunits with \\xx{896 KB} SRAM and has a total chip area of \\xx{5.93 mm$^2$} and consumes 895~milliwatts of power.\nAs a point of reference, \\titanxp in the same 16~nm node, has a chip area of 471~mm$^2$ and has a TDP of 250~Watts, as summarized in Table~\\ref{tab:platforms}.\n\\section{Acknowledgments}\nWe thank Amir Yazdanbaksh, Divya Mahajan, Jacob Sacks, and Payal Preet Bagga for insightful discussions and comments.\nThis work was in part supported by NSF awards CNS\\#1703812, ECCS\\#1609823, Air Force Office of Scientific Research (AFOSR) Young Investigator Program (YIP) award \\#FA9550-17-1-0274, and gifts from Google, Microsoft, Xilinx, and Qualcomm.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{intro}\n\nThe Schr\\\" odinger solutions belong to a class of solutions\nto the gravitational equations of motion which asymptotically do not\npreserve the Lorentz symmetry. They, however, do respect some\nnon-relativistic symmetries. The deviation from the relativistic\nsymmetry is parametrized by the Schr\\\" odinger scaling\nexponent $z$, or the dynamical exponent.\n\nThe Schr\\\" odinger solution was first obtained by Son\\cite{Son:2008ye}\nas well as Balasubramanian and Mcgreevy\\cite{Balasubramanian:2008dm}.\nThey assumed the stress tensor consisting of the cosmological constant\nterm and the pressure-less dust.\nThe Schr\\\" odinger solution possesses the Galilean\nboost invariance by assigning a specific transformation property to\none of the light-like directions\\cite{Son:2008ye,\n Balasubramanian:2008dm}(non-relativistic metrics in higher\nderivative gravity were discussed in \\cite{Adams:2008zk}).\n\nIn this note we analyse this question in more detail by spanning\nthe entire coupling parameter space of Lovelock theories in various\ndimensions. Up to four space-time dimensions, the Lovelock\naction is identical to the Einstein Hilbert action with the\ncosmological constant, but from five dimensions onwards the Lovelock\naction has additional Gauss-Bonnet term in the action. This term can \nbe added in four dimensions as well but being\ntotal derivative term it does not contribute to the dynamics. In five\nand higher dimensions the Gauss-Bonnet term does contribute to the\ndynamics. Similarly the cubic order Lovelock term can be added from\nsix dimensions onwards but it contributes to dynamics only from seven\ndimensions onwards. \n\nWe will show that the Schr\\\" odinger metric is generically not a\nsolution to the Lovelock equations of motion, however, it exists as a\nsolution on a co-dimension $1$ locus in the Lovelock coupling space. We\nshow that the Schr\\\" odinger solution exists precisely on the same\nlocus on which the Lifshitz solution is known to exist\\footnote{For closely\n related solutions of Kasner type in the Lovelock theory, see\n \\cite{Camanho:2015yqa}.}. In our computation we restrict ourselves\nto the Lovelock terms up to cubic order in the curvature tensor but we\ngeneralize our analysis to arbitrary dimensions. The co-dimension 1\nlocus on which we get the Schr\\\" odinger solution is interesting from\nanother point of view. It is known that the Lovelock theories can be\nwritten in terms of the parity preserving Chern-Simons theory.\nHowever, this representation exists only for specific values of the\nLovelock couplings. The Chern-Simons formulation exists at a point on\nthis co-dimension $1$ locus on which we find the Schr\\\" odinger\nsolutions. We present these solutions in the Chern-Simons gauge field\nforms as well.\n\nThe Schr\\\"odinger solutions are relevant from the point of view of\napplication to holographically dual condensed matter physics systems.\nIt then naturally raises a question of relevance of these higher\ndimensional solutions to $2+1$ and $3+1$ dimensional condensed matter\nsystems. In this regard it is worth pointing out that unlike the AdS\nand Lifshitz holography which relates $D$ dimensional theory of\ngravity to $D-1$ dimensional field theory, the Schr\\\"odinger\nholography relates $D$ dimensional theory of gravity to $D-2$\ndimensional field theory. Therefore, $4+1$ and $5+1$ dimensional\nLovelock theories are relevant to $2+1$ and $3+1$ dimensional boundary\nphysics. Higher dimensional theories can be dimensionally reduced to\nlower dimensional theories. Such higher dimensional theories\ntypically give rise to scalar-tensor theories of gravity which are\neither referred to as Galileon or Horndeski\ntheories\\cite{Nicolis:2008in,Deffayet:2009wt,Deffayet:2009mn,Deffayet:2011gz}.\nFor example, let us consider $D=d+n+1$ dimensional theory of gravity\nwith the cosmological constant, the Einstein-Hilbert, the Gauss-Bonnet\nterm\n\\begin{equation}\n \\label{eq:9}\n S= \\int d^Dx\\sqrt{-g} \\left[ R-2 \\Lambda + a_2 \\mathcal{L}_2\\right]\\ ,\n\\end{equation}\nwhere, $\\mathcal{L}_2$ is the Gauss-Bonnet term. We will dimensionally\nreduce it down to $d+1$ dimensions by using an $n$-dimensional compact\nmanifold $\\tilde{K}_n$ such that\n\\begin{equation}\n \\label{eq:10}\n ds^2_D = d\\bar{s}^2_{d+1}+ e^{\\phi} d\\tilde{K}_n^2\\ .\n\\end{equation}\nThis is a simple but consistent diagonal toroidal compactification\nwhich give rise to one extra scalar degree of freedom, that is the\nsize of the internal space. All terms with a tilde refers to internal\n$n$ dimensional space, while terms with a bar refers to the\n$d+1$ dimensional space-time. As we integrate out the internal space the\neffective action looks like \\cite{Charmousis:2012dw}\n\\begin{equation}\n \\label{eq:11}\n \\begin{split}\n \\bar{S}_{(d+1)} = \\int d^{d+1} x \\sqrt{-\\bar{g}}\\, e^{{n \\over 2}\\phi}\n &\\bigg\\{ \\bar{R}- 2\\Lambda + a_2 \\bar{\\mathcal{G}} + {n\\over 4}\n (n-1)(\\partial \\phi \\partial \\phi) - a_2 n(n-1)\n \\bar{G}^{\\mu\\nu} \\partial_{\\mu}\\phi \\partial_{\\nu} \\phi \\\\\n & -{a_2 \\over 4} n(n-1)(n-2)\\left[(\\partial \\phi \\partial \\phi) \\nabla^2\\phi-\n {(n-1)\n \\over 4} (\\partial \\phi \\partial \\phi)^2\\right] \\\\\n & +e^{-\\phi} \\tilde{R} \\big[ 1+ a_2 \\bar{R} +4 a_2\n (n-2)(n-3)(\\partial \\phi \\partial \\phi)\\big]+ a_2\n \\tilde{\\mathcal{G}}e^{-2\\phi } \\bigg\\}\\ ,\n\\end{split}\n\\end{equation}\nwhere, $\\bar{\\mathcal{G}}$ is $d+1$ dimensional Gauss-Bonnet term,\n$\\tilde{\\mathcal{G}}$ is $n$ dimensional Gauss-Bonnet term and\n$\\bar{G}^{\\mu\\nu}$ is the Einstein tensor of $d+1$ dinemsional space.\nThe effective action written above can be related to the so called\nGalileon action, with the Galileon field is realized as the scalar\nparametrising the volume of the internal space. As we have reduced\nfrom the Einstein Gauss Bonnet action, in the reduced action all the\nterms are up to quartic order in derivatives of the metric or the\nscalar or of both, but the equations of motion following from it will\nstill be of second order. The term of the form\n$(\\partial \\phi \\partial \\phi) \\nabla^2\\phi$ is often called the DGP\nterm\\cite{Dvali:2000hr} appearing in the decoupling limit of the DGP\nmodel, and the term of the form $(\\partial \\phi \\partial \\phi)^2$ is\nthe standard Galileon term\\cite{Nicolis:2008in}. As we will argue\nthat the Schr\\\"odinger solution exists in a co-dimension 1 subspace of\nLovelock moduli space. If we restrict to only the Gauss-Bonnet term\nthen it exists on a subspace which relates $a_2$ to $\\Lambda$. We\nwill get back to the issue of dimension reduction in this context in\nthe discussion section. As long as $n\\leq 2$, neither the DGP term\nnor the Galileon term appear in the dimensionally reduced theory. In\naddition for Ricci-flat compact spaces the lower dimensional action,\nup to the addition of higher derivative curvature terms, has a\nfamiliar form.\n\nThis note is organized in the following manner. We will first give\nbasics of the Lovelock theory in arbitrary dimensions. Most of the\ninformation in this section is not new but is useful to fix the\nnotation. In the next section we will look at various solutions to\nthe Lovelock equations of motion. It is well known that the AdS\nsolution generically exists for arbitrary values of Lovelock couplings\nin any dimension. This feature is not shared by the Schr\\\"odinger\nsolution. We present the solution for general value of $D$ and in\nexplicit form for dimensions $D=5,6,7$. We also comment on the\nsolutions with anisotropic scaling in spatial direction and their\nrelation to AdS$\\times R$ type solutions. In section 4, we analyse\nbranches of the AdS solution\\cite{Kofinas:2007ns, Jatkar:2015ffa}.\nOur interest in presenting this result is to emphasize that the\nnon-relativistic solutions exists only when the discriminant vanishes.\nDegeneracy in the configuration\nspace\\cite{Dotti:2007az,Dehghani:2010kd} has been well studied for\nLifshitz solutions\\cite{Kachru:2008yh,Taylor:2008tg}. We show that\nthis degeneracy is responsible for unconstrained $z$ for Schr\\\"\nodinger case as well. Unlike Lifshitz, in the case of Schr\\\" odinger\nsolutions this degeneracy extends beyond the Lovelock moduli space.\nIn this sense our results provide a template for a dynamical exponent $z$\nfor which all values are equally likely. Any suitable value of $z$\nthen can be obtained by either appropriately modifying the couplings\nin this theory or by adding new interactions.\n\nIt is known that the Lovelock theory in odd space-time dimensions can\nbe written in the Chern-Simons form and in even space-time dimensions\nin the Born-Infeld form exactly when the discriminant vanishes\n\\cite{Kofinas:2007ns}. We discuss the relation between vanishing\ndiscriminant and locus of non-relativistic solutions in section 5 and\nwrite down the Schr\\\" odinger solution in the Chern-Simons gauge field\nform in odd space-time dimensions and in the Born-Infeld gauge field\nform in even space-time dimensions. Finally, we point out the\nrelation with the causality and stability constraints obtained in the\nLovelock theories in higher dimensions \\cite{deBoer:2009gx,\n Camanho:2009hu, Camanho:2010ru}. Finally we summarize our results\nand speculate about their applications. Various technical details are\nrelegated to Appendix. Appendix \\ref{apdx1} contains details of\nLovelock equations of motion. Appendix \\ref{apdx2} recounts details of\nAdS and Lifshitz solutions, which are given for the purpose of\ncomparison with the Schr\\\" odinger solution. Appendix \\ref{apdx3}\ncontains spin connections and curvature tensors for Schr\\\" odinger\nsolution.\n\n\\section{The Lovelock Gravity Theory}\n\nLet us consider following action\n\\begin{equation}\n \\label{eq:1}\n I = \\frac{1}{16\\pi G}\\int_{M}d^Dx\n \\sum_{p=0}^{[(D-1)\/2]} a_p\\mathcal{L}_p\\ ,\n\\end{equation}\nwhere $G$ is the $D$ dimensional Newton's constant, $a_p$ are coupling\nconstants with $a_0=-2\\Lambda=(D-1)(D-2)\/\\ell^2$, $a_1=1$, $a_2$ is\nthe Gauss-Bonnet coupling etc.\\footnote{It is important to note that the mass \ndimensions of various parameters appearing in the action,\neq.(\\ref{eq:1}) are as follows \n$[G]=D-2, \\ [\\Lambda] = [a_0] = 2, \\ [a_1]=0, \\ [a_2] = - 2, \\ [a_3]=\n-4, \\ \\cdots .$\nand the parameter $\\ell$ has dimensions of length.}, and\n$\\mathcal{L}_p$ are terms in the Lagrangian density of the Lovelock\naction, \n\\begin{equation}\n \\label{eq:2}\n \\mathcal{L}_p = \\frac{1}{2^p} \\sqrt{-g}\n \\delta_{\\mu_1\\mu_2\\cdots\\mu_{2p}}^{\\nu_1\\nu_2\\cdots\\nu_{2p}} \nR_{\\nu_1\\nu_2}^{\\mu_1\\mu_2}\\cdots\nR_{\\nu_{2p-1}\\nu_{2p}}^{\\mu_{2p-1}\\mu_{2p}}\\ ,\n\\end{equation}\nwhere $\\delta_{\\mu_1\\mu_2\\cdots\\mu_{2p}}^{\\nu_1\\nu_2\\cdots\\nu_{2p}}$\nis totally antisymmetric product of $2p$ Kronecker deltas normalized\nto take values $0$ and $\\pm 1$, and hence is completely antisymmetric\nin all its upper and lower indices separately. It can also be\nconsidered as the determinant of an $(2p \\times 2p)$ matrix whose\n$(ij)$-th element is given by $\\delta^{\\nu_i}_{\\mu_j}$.\n\nWe are using notation of \\cite{Kofinas:2007ns} and the equation of\nmotion can be written in the compact form as\n\\begin{equation}\n \\label{eq:3}\n E_\\mu^\\nu = \\sum_{p=0}^{[(D-1)\/2]}\\frac{a_p}{2^p}\n \\delta_{\\mu\\mu_1\\mu_2\\cdots\\mu_{2p}}^{\\nu\\nu_1\\nu_2\\cdots\\nu_{2p}}\n R_{\\nu_1\\nu_2}^{\\mu_1\\mu_2}\\cdots \nR_{\\nu_{2p-1}\\nu_{2p}}^{\\mu_{2p-1}\\mu_{2p}}=0 \\ .\n\\end{equation}\nIn this way of writing the Lovelock terms make is obvious that up to\n$D=4$ only relevant couplings are $a_0$ and $a_1$. The term with the\ncoupling $a_2$ is topological in $D=4$. In $D=5,6$ the Gauss-Bonnet\nterm is dynamical. In these dimensions we will explore the parameter\nspace spanned by $a_0$ and $a_2$ to find the range of values for which\nthe Lifshitz solution is possible. In $D=6$, the term with coupling\n$a_3$ can be written down but like the Gauss-Bonnet in $D=4$, this\nterms is topological in $D=6$ and hence does not affect the equation of\nmotion. However, this term becomes relevant in $D>6$. We will\nexplore the three dimensional parameter space spanned by $a_0$, $a_2$\nand $a_3$ and find conditions for Lifshitz solutions.\n\nWe will start with the study of solution to the $D=5$ equations of motion.\nWe therefore write the action of pure gravity in $D$-dimensions as\n\\begin{equation} \\label{action1}\n I=\\int d^{D} x~ \\sqrt{-g}~ \\left[R-2 \\Lambda + \\mathcal{L}_{hd}\\right]\n\\end{equation}\nwhere $\\Lambda$ is the cosmological constant and $ \\mathcal{L}_{hd}$\nis the Lagrangian for the higher derivative terms of the Lovelock form.\n\nAs mentioned earlier in $D=5$ space-time dimensions the higher\nderivative Lagrangian density contains the quadratic Lovelock term,\nalso known as the Gauss-Bonnet term, which appears in the Lagrangian\nwith the coupling constant, $a_2$,\n\\begin{equation}\\label{deflgb}\n\\begin{aligned}\n \\mathcal{L}_{hd} = &a_2\\mathcal{L}_2, ~~ \\text{where}, \\\\\n \\mathcal{L}_2= &(R^2-4 R_{\\mu\\nu}R^{\\mu\\nu} +\n R_{\\mu\\nu\\rho\\sigma}R^{\\mu\\nu\\rho\\sigma}) \n\\end{aligned}\n\\end{equation}\n\nFor $D=7$ space-time dimensions we will have, in addition to the\nGauss-Bonnet term, the cubic Lovelock term $\\mathcal{L}_3$,\n\\begin{equation} \\label{deflll}\n \\mathcal{L}_{hd}= a_2\\mathcal{L}_2+a_3\\mathcal{L}_3\n\\end{equation}\nwhere $a_3$ is the coupling constant of the cubic Lovelock term. The\ncubic term is explicitly written as\n\\begin{equation}\n\\begin{aligned}\n \\mathcal{L}_3= &2\n R^{\\mu\\nu\\sigma\\kappa}R_{\\sigma\\kappa\\rho\\tau}{R^{\\rho\\tau}}_{\\mu\\nu}\n + 8 {R^{\\mu\\nu}}_{\\sigma\\rho}\n {R^{\\sigma\\kappa}}_{\\nu\\tau}{R^{\\rho\\tau}}_{\\mu\\kappa}\n +24 R^{\\mu\\nu\\sigma\\kappa}R_{\\sigma\\kappa\\nu\\rho}R^{\\rho}_{\\mu} +3\n RR_{\\mu\\nu\\rho\\sigma}R^{\\mu\\nu\\rho\\sigma} \\\\\n& +24 R^{\\mu\\nu\\sigma\\kappa} R_{\\sigma\\mu}R_{\\kappa\\nu} + 16\nR^{\\mu\\nu}R_{\\nu\\sigma}R^{\\sigma}_{\\mu} -12 RR^{\\mu\\nu}R_{\\mu\\nu}+R^3\n \\end{aligned}\n\\end{equation}\n\nThe equation of motion that follows from here are as written below,\n\\begin{equation}\\label{eom1}\n G^{(1)}_{\\mu\\nu}+a_2\n G^{(2)}_{\\mu\\nu}+a_3 G^{(3)}_{\\mu\\nu} - \\Lambda g_{\\mu\\nu}=0\n\\end{equation}\nwhere the explicit forms of $G^{(1)}_{\\mu\\nu} $, $G^{(2)}_{\\mu\\nu} $\nand $G^{(3)}_{\\mu\\nu} $ are given in appendix \\ref{apdx1}.\nThe equations of motion in $D=5$ are obtained by setting $a_3=0$ in\n(\\ref{eom1}). We will now study specific solutions to the equations of\nmotion. \n\n\\section{Solutions to the Lovelock Gravity}\n\nIn this section we will analyse solutions to the Lovelock gravity\nequations of motion. \nIn Appendix \\ref{apdx2} we will summarise known results about the AdS\nand Lifshitz solutions. In particular the AdS solution is possible\nfor generic values of the Lovelock couplings $a_2$, $a_3$, etc. On the \nother hand the Lifshitz solutions exist only on the co-dimension one subspace.\n As we will see below the Schr\\\" odinger solutions can be obtained only on\nthe same co-dimension one locus in the parameter space.\nFurthermore, as we will see, on this locus the solution\nadmits arbitrary dynamical scaling exponent $z$, which is also true for \nthe Lifshitz solutions. \n\n\n\\subsection{The Schr\\\" odinger Solution} \\label{schsold5}\n\nWe consider the Schr\\\" odinger space-times as solutions of the higher\nderivative Lovelock gravity theories. This is an example of\nnon-relativistic space-time, known apart from the Lifshitz solution.\nWe will look for the Schr\\\" odinger solution to the Lovelock gravity\nequations of motion in arbitrary dimensions but we will restrict to\nterms cubic in curvature tensor. Generalization to higher order\nLovelock terms is tedious but straightforward.\n\nThe metric ansatz for the Schr\\\" odinger solution looks like\n\\begin{equation} \\label{Schmet1}\n ds^2= L_{\\text{sch}}^2\\bigg[-{dt^2 \\over r^{2z}}+{dr^2\\over\n r^2}+{2\\over r^2}dtd\\xi+{1\\over r^2} \\sum_{i=1}^{D-3} dx_i^2\\bigg] \n\\end{equation}\nNote that this metric for the Schr\\\" odinger solution also has two\nparameters, $z$ which is the Schr\\\" odinger exponent and\n$L_{\\text{sch}}$ which is the ``Schr\\\" odinger radius''. \nWe will first state the results for arbitrary $D$ but restricting upto cubic\nLovelock terms and then write down explicit expressions for $D=5,6,7$.\n\nThe Schr\\\" odinger space-time solution in general $D$ dimensions\nexists subject to following two constraints,\n\\begin{equation}\\label{eq:8}\n\\begin{split}\n\\Lambda =& - {(D-1)(D-2) \\over 4L_{\\text{Sch}}^2} \\bigg( 1 -\n(D-3)(D-4)(D-5)(D-6) {a_3\\over L_{\\text{sch}}^4 }\\bigg)\\ , \\\\\na_2 =&{L_{\\text{Sch}}^2 \\over 2 (D-3)(D-4)}+ {3(D-5)(D-6) \\over\n 2L_{\\text{Sch}}^2} a_3\\ .\n\\end{split}\n\\end{equation}\nThe dynamical exponents $z$ is unconstrained. If we eliminate\n$L_{\\text{sch}}$, then it gives one relation between the parameters in\nthe Lovelock action. Thus the Schr\\\" odinger solutions exist on\nco-dimension one subspace of the Lovelock moduli space.\n\n\nIn $D=5$ space-time as we have the Gauss-Bonnet term in the Lovelock\naction besides the Einstein-Hilbert and the cosmological constant term.\nThe constraint (\\ref{eq:8}) corresponds to \n\\begin{equation}\n \\begin{split} \\label{eq:sch5sol1}\n\\Lambda = -{3 \\over L_{\\text{Sch}}^2} ~~ \\text{and} ~~ \n a_2 = \\frac{L_{\\text{Sch}}^2}{4}\\ \\Longrightarrow\\ a_2\\Lambda = -3\/4\\ .\n \\end{split} \n\\end{equation}\nThe non-zero components of the Ricci tensor and the Ricci scalar $R$\nfor the metric of Schr\\\" odinger space-time are given by\n$R_{tt} = 2 \\left(z^2+1\\right)\/ r^{2 z},\\ R_{t\\xi} = R_{rr} =R_{x_ix_i}\n= -4\/r^2$; and $R= -20\/L_{\\text{Sch}}^2$.\nIn $D=6$ space-time the Gauss-Bonnet term is important but the\ncurvature cubed Lovelock term being a total derivative is not. \nThe constraint eq.(\\ref{eq:8}) becomes\n$\\Lambda = -{5\/L_{\\text{Sch}}^2}$ and $a_2 = L_{\\text{Sch}}^2\/12$.\nWe again write down the components of the Ricci tensor\n$R_{tt} = \\left(2z^2+z+2\\right) r^{-2 z}$, $R_{t\\xi} = R_{rr} =R_{x_ix_i}\n= -5\/r^2$\n and the Ricci scalar $R= -(30 \/ L_{\\text{Sch}}^2)$.\n\nIn $D=7$ space-time apart from the the Gauss-Bonnet\nterm, the cubic order Lovelock term will also be important and hence\nthe action will contain three parameters, $\\Lambda$, $a_2$ and $a_3$.\nThe constraint (\\ref{eq:8}) in this case takes the form\n\\begin{equation} \\label{sch7sol1}\n \\begin{split} \n \\Lambda = -{15 \\over 2L_{\\text{sch}}^2}\\left(1-{24 \\over\n L_{\\text{sch}}^4}a_3\\right) ~~ \\text{and} ~~ \n a_2 = \\frac{L_{\\text{sch}}^2}{24} + 3 {a_3\\over L_{\\text{sch}}^2}\\ .\n \\end{split} \n\\end{equation}\nThe Ricci tensor components are\n$ R_{tt} = 2\\left(z^2+z+1\\right) r^{-2 z},\\ R_{t\\xi} = R_{rr} =R_{x_ix_i}\n= -6\/r^2$,\n and the Ricci Scalar becomes $R= -(42 \/ L_{\\text{Sch}}^2)$. \n\n\n\n\\subsection{Other Solutions} \\label{secaltsol}\n\nLifshitz solutions\\footnote{See eq.(\\ref{lifmet2}) for Lifshitz metric in appendix \\ref{apdx2}, where we discuss Lifshitz solutions briefly.}, where the time coordinate ($t$) scaled differently\ncompared to the other coordinates, are known to be solutions of Lovelock gravity.\n As an alternative one can consider\na different version of the Lifshitz solutions where instead of the\ntime coordinate one of the spatial coordinates may scale differently\ncompared to others. We call it ``spatial Lifshitz'' space-time. More\nspecifically, we take in $D=5$ dimensions the following metric\n\\begin{equation}\\label{lifaltmet}\n ds^2= L_{\\text{Lif}}^2 \\left({-dt^2+dr^2 + dx_1^2 \n + dx_2^2 \\over r^2}+{dx_3^2 \\over r^{2z}}\\right) .\n\\end{equation}\nIt is obvious from the metric in eq.(\\ref{lifaltmet}) that the $x_3$\ncoordinate scales differently compared to the other coordinates with a\nLifshitz exponent parametrized by $z$.\n\nFollowing the similar procedure as in the previous two subsections, we\ncan obtain this ``spatial Lifshitz'' as a solution to the Lovelock action\neq.(\\ref{action1}). In $D=5$, for the Gauss-Bonnet case\neq.(\\ref{deflgb}), we find the solution to be identical to both the\nSchr\\\" odinger case eq.(\\ref{eq:sch5sol1}), and the Lifshitz case\neq.(\\ref{Lif5sol2}). Similarly, for $D=7$ dimensions in the cubic\nLovelock theory we find this ``spatial Lifshitz'' as a solution, again,\nidentical to both the Schr\\\" odinger case eq.(\\ref{sch7sol1}), and the\nLifshitz case eq.(\\ref{lif7sol1}). In all these cases, this solution\nexists at the same point in the coupling space at which the Schr\\\"\nodinger and Lifshitz solution exist. Thus we see that the special\npoint continues to be relevant as long as one direction, whether\nspatial or temporal, has anisotropic scaling property. In case of\nmultiple anisotropic directions also one can show that such solutions\nexist at special points in the coupling space but generically this\npoint is different from the one under consideration.\n\nIt is also worth mentioning that the situation here is analogous to\nwhat happens in the case of Schr\\\" odinger solutions\ndiscussed earlier, namely the dynamical exponent $z$ remains\nunconstrained for this solution as well. This, in particular, allows\nus to consider a special case when the dynamical exponent for the\n``spatial Lifshitz'' solution vanishes, i.e., $z=0$. A vanishing $z$\nin eq.(\\ref{lifaltmet}) corresponds to the metric in the $x_3$\ndirection being invariant under scaling of the radial coordinate.\nSince the metric in the directions transverse to $x_3$ is simple AdS\nmetric in the Poincar\\'e coordinates, we obtain a $AdS_4 \\times R$\nsolution of the form\n\\begin{equation}\\label{ads4rmet}\n ds^2= L^2 \\left({-dt^2+dr^2 + dx_1^2 + dx_2^2 \\over r^2}+dx_3^2\\right) .\n\\end{equation}\nThis kind of anisotropic solution was studied earlier in\n\\cite{Jain:2014vka}, where additional matter in the form of a linearly\nvarying dilaton was coupled to two derivative Einstein gravity with\nnegative cosmological constant to construct such anisotropic\nsolutions.\n\nFinally we would like to mention that Lovelock equations of motion do\nnot support black brane solutions with either Lifshitz or Schr\\\" odinger\nasymptotic. \n\nWe illustrate this in five dimensions by considering a finite\ntemperature metric ansatz for the Lifshitz solutions of the following\nform\n\\begin{equation} \\label{lifmetfinT}\n ds^2= L_{\\text{lif}}^2\\bigg[-{f(r)dt^2 \\over r^{2z}}+{dr^2\\over\n r^2 f(r)}+{1\\over r^2} \\sum_{i=1}^{3} dx_i^2\\bigg] \n\\end{equation}\nwith $f(r) = 1+ c_1 r^{c_2}$, where $c_1$ and $c_2$ are constants.\nThe equations of motion are solved by this ansatz if $z=1$ and\n$c_2 =-2$ with arbitrary $c_1$. That is\nfinite temperature black brane solutions with only $AdS$ asymptotic\nare allowed. A similar analysis can be done for Schr\\\"odinger\nsolutions and again we find that there are no finite temperature\nSchr\\\"odinger black brane solutions to eq.(\\ref{eq:sch5sol1}). \n\n\n\n\\section{The Phase-Space of Solutions}\n\nIn this section we analyse the parameter space of the higher\nderivative theory and understand in some more detail the phase space\nof the solutions we obtained in the previous section. We are working\nwith the action eq.{\\eqref{action1}} and the number of \nparameters of the theory depend on $D$. For $D\\leq 4$,\nthe cosmological constant $\\Lambda$ is the only parameter. For $D=5,6$\nwe have $\\Lambda$ and the Gauss-Bonnet coupling constant $a_2$ and\nfrom $D=7$ onwards we also have $a_3$ the cubic term coupling.\nWhile AdS solutions is parametrized only by its radius\n$L_{\\text{AdS}}$, the Schr\\\" odinger and Lifshitz solution has two\nparameters, radius $L_{\\text{Sch}}$, respectively $L_{\\text{Lif}}$ and\nthe dynamical exponent $z$. \nNote that the Schr\\\" odinger solutions obtained\nin the last section and the Lifshitz solutions exist on the locus\nwhich does not pass through the origin of the Lovelock moduli space.\nThus these solutions would cease to exist if we turn off higher\nderivative couplings and they cannot be obtained perturbatively in Lovelock\ncouplings.\n\n \n\\subsection{The Phase-Space of Solutions in $D=5$}\n\nThe AdS solution in $D=5$ is written in eq.\\eqref{ads5sol2}.\nOne can re-express it as $L_{\\text{AdS}}$\nbeing determined in terms of $\\Lambda$. \n \\begin{equation}\n \\begin{aligned} \n {1 \\over L_{\\text{AdS}}^2}={6 \\pm \\sqrt{36+48 a_2\n \\Lambda} \\over 24 a_2} \n \\end{aligned} \n \\end{equation}\n which indicates that there are two branches for the AdS solution with\n different AdS-radius. This AdS solution exists when the term within\n the square root is non-negative \n \\begin{equation}\n \\Lambda \\ge -{3 \\over 4 a_2}.\n \\end{equation}\n There is one more constraint on the parameters for the existence of\n AdS solution coming from the demand that $L_{\\text{AdS}}^2>0$, which\n is $a_2 >0$.\n\nThe two branches of the AdS solutions meet at a point in the\n2-dimensional phase-space spanned by the parameters $\\{\\Lambda,~a_2\\}$, given by $\\Lambda = -3\/(4 a_2)$.\nThe Schr\\\" odinger solution and the Lifshitz solution in\n$D=5$ eq.\\eqref{eq:sch5sol1} and eq.\\eqref{Lif5sol2}\nexist at this point in the phase-space with arbitrary value of\nthe dynamical exponent $z$. We will discuss the\nrelation between unconstrained $z$ and the degeneracy of the\nconfiguration space later in this section.\n\n\\subsection{The Phase-Space of Solutions in $D=7$}\n\nThe AdS solution in $D=7$ dimensions is given in\neq.\\eqref{ads7sol1}. The AdS radius squared $L_{\\text{AdS}}^2$ can be\nexpressed in terms of the parameters $\\Lambda,~a_2$ and $a_3$. From\neq.\\eqref{ads7sol1} it is easy to see that one has to solve a cubic\nequation for $L_{\\text{AdS}}^2$ and the solutions are\n\\begin{equation} \\label{Ladsrel}\n\\begin{split}\n L_{\\text{AdS}}^2 &= \\frac{540 a_2 \\Lambda+s\\Lambda\n (s\\Lambda-15)+225}{3s \\Lambda^2} \\\\ \n L_{\\text{AdS}}^2 &= -\\frac{540 a_2 \\Lambda+(s\\Lambda+15)^2}{6\n \\Lambda^2 s}\\pm\\frac{i \\left(540 a_2 \\Lambda- s^2 \\Lambda^2\n +225\\right)}{2 \\sqrt{3} \\Lambda^2 s}\n\\end{split}\n\\end{equation}\nwhere \n\\begin{equation} \\label{defs}\ns^3 = \\frac{-135 \\left(6 \\Lambda \\left(15 a_2+6 a_3\n \\Lambda-\\sqrt{180 a_2 a_3 \\Lambda+36 a_3^2 \\Lambda^2+50\n a_3-240 a_2^3 \\Lambda-75 a_2^2}\\right)+25\\right)}{\\Lambda^3} \n\\end{equation}\nOne can see that the last two roots are complex conjugate of each\nother. Now demanding that the term within the square root in\neq.(\\ref{defs}) vanishes, {\\em i.e.}, \n\\begin{equation}\n180 a_2 a_3 \\Lambda+36 a_3^2 \\Lambda^2+50 a_3-240 a_2^3\n\\Lambda-75 a_2^2 = 0, \n\\end{equation}\nand also using $\\Lambda= {1 \\over L_{\\text{AdS}}^2}\\left(-15 + 180\n {a_2 \\over L_{\\text{AdS}}^2} -360 {a_3 \\over\n L_{\\text{AdS}}^4}\\right)$, one obtains a relation between $a_2$\nand $a_3$ \n\\begin{equation}\n a_2 = \\frac{L_{\\text{AdS}}^2}{24} + 3 {a_3\\over L_{\\text{AdS}}^2}\\ \n\\Longrightarrow\n \\Lambda = -{15\\over 2 L_{\\text{AdS}}^2} \\left(1-{24 a_3 \\over\n L_{\\text{AdS}}^4}\\right)\\ .\n\\end{equation}\nThese relations are same as those encountered in our study of Schr\\\"\nodinger and Lifshitz solutions in $D=7$.\nFor these values, the imaginary parts of the second and\nthird roots in eq.(\\ref{Ladsrel}) vanish and they become\nequal, whereas the first root remains different,\n\\begin{equation}\n L_{\\text{AdS}}^2 = \\frac{-2 \\sqrt{60 a_2 \\Lambda+25}-5}{\\Lambda}, ~\n L_{\\text{AdS}}^2 = \\frac{\\sqrt{60 a_2 \\Lambda+25}-5}{\\Lambda}, ~\n L_{\\text{AdS}}^2 =\\frac{\\sqrt{60 a_2 \\Lambda+25}-5}{\\Lambda}.\n\\end{equation}\nInterestingly, there is yet another choice which simplifies solutions\nto the cubic equation. Consider the choice $a_2= -5\/(12 \\Lambda)$,\nand set the discriminant equal to zero then the quantity $s$\nin (\\ref{defs}), vanishes and all three roots\nof the cubic become equal to $L_{\\text{AdS}}^2=-5\/\\Lambda$\nThis point is on the co-dimension one locus and admits\nsolutions of Schr\\\" odinger and Lifshitz kind. \n\n\\subsection{Degeneracy of the Configuration Space}\n\\label{sec:degen-conf-space}\n\nWe will now take up the issue of unconstrained\ndynamical exponent in the Lifshitz and the Schr\\\" odinger metrics in\neq.(\\ref{lifmet2}) and eq.(\\ref{Schmet1})respectively. It is known in\nthe literature, see \\cite{Dotti:2007az,Dehghani:2010kd}, that in pure\nLovelock theories, unconstrained dynamical exponent $z$ of the\nLifshitz solutions follows from the existence of degeneracy of the\nconfiguration space. The degeneracy of the configuration space\ncorresponds to complete arbitrariness in specifying the metric\ncomponent $g_{tt}(r)$ on this locus. Since this metric component is\ncompletely unconstrained, it naturally follows that the\ndynamical exponent is not constrained. While this result is known for\nthe Lifshitz metrics, we believe our results for the Schr\\\" odinger\nmetrics are new. More specifically, in $D=5$ space-time dimensions,\nthe statement of degeneracy in configuration space amounts to the fact\nthat a metric ansatz of the following form for the Schr\\\"odinger\ngeometry\n\\begin{equation} \\label{schmetdeg1}\nds^2= L_{\\text{sch}}^2\\bigg[-f(r){dt^2 \\over r^{2z}}+{dr^2\\over\n r^2}+{2\\over r^2}dtd\\xi+{1\\over r^2} \\sum_{i=1}^{2} dx_i^2\\bigg], \n\\end{equation}\nhappens to be a solution for the action in eq.(\\ref{action1}) with any\narbitrary choice of the function $f(r)$, at the same locus in the\nparameter space, eq.(\\ref{eq:sch5sol1}). \n\nThe degeneracy of the configuration space was studied mostly in the\ncontext of the Chern-Simons representation of the Lovelock theory.\nSince the Chern-Simons action does not have any free parameters,\nthis representation exists only at a point in the Lovelock moduli\nspace. However, in dimensions $D>6$ the Lifshitz and Schr\\\" odinger\nsolutions exist on a subspace which\nextends way beyond the Chern-Simons point. In order to\nunderstand the relation between the degeneracy of the configuration\nspace and the special locus better, we deform the five dimensional\nLovelock theory by adding $R^2$ and $R_{\\mu\\nu}R^{\\mu\\nu}$ terms to\nit. Since neither Lifshitz nor Schr\\\" odinger solution exist in the\nLovelock moduli space away from this locus, only way to study\ndependence of the degeneracy on the couplings is to expand the\ncoupling constant space by adding new terms.\n\nLet us deform the Lovelock action in $D=5$ (\\ref{action1}) by adding\n$R^2$ and $R_{\\mu\\nu}R^{\\mu\\nu}$ terms. This deformation of the Lovelock\ntheory is given in terms of two parameters $b_1$ and $b_2$,\n\\begin{equation}\n\\label{actgen1}\nI=\\int d^{5} x~ \\sqrt{-g}~ \\left[R-2 \\Lambda + a_2 (R^2-4\n R_{\\mu\\nu}R^{\\mu\\nu} + R_{\\mu\\nu\\rho\\sigma}R^{\\mu\\nu\\rho\\sigma})+b_1\n R^2 + b_2 R_{\\mu\\nu}R^{\\mu\\nu}\\right].\n\\end{equation}\nWe are now considering the most general action of gravity up to\nquadratic order in curvatures in $D=5$. This action now contains $4$\nparameters, $\\Lambda,~ a_2, ~ b_1$ and $b_2$. \n\nThe Lifshitz solution, eq.(\\ref{lifmet2}), as is known in the\nliterature \\cite{Dehghani:2010kd}, occurs at \n\\begin{equation}\n\\begin{split}\n\\lambda =& -\\frac{3}{L_{\\text{Lif}}^2}-b_1\\frac{2 z (z+3) (z\n (z+3)+6)}{L_{\\text{Lif}}^4} -b_2\\frac{ z (z+3)\n \\left(z^2+3\\right)}{L_{\\text{Lif}}^4}, \\\\\na_2 = &\\frac{L_{\\text{Lif}}^2}{4}-b_1 (z (z+3)+6)-\\frac{b_2}{2}\n\\left(z^2+3\\right).\n\\end{split}\n\\end{equation}\nWhen $b_1= b_2=0$ we get back eq.(\\ref{Lif5sol2}), the Lifshitz\nsolution exists for non-zero $b_1,b_2$ but with fixed\ndynamical exponent $z$, determined by the parameters of the theory.\nThis agrees with \\cite{Dehghani:2010kd} that the degeneracy in the\nconfiguration space for the Lifshitz solution occurs only at \n\\begin{equation}\n\\lambda = -\\frac{3}{L_{\\text{Lif}}^2}, ~~a_2 =\n\\frac{L_{\\text{Lif}}^2}{4}, ~~ b_1=b_2=0. \n\\end{equation}\nInterestingly, the Schr\\\"odinger solution with the metric ansatz,\neq.(\\ref{Schmet1}) in $D=5$, for a general theory of higher derivative\ngravity beyond Gauss-Bonnet theory with action eq.(\\ref{actgen1}), is\nobtained as \n\\begin{equation} \\label{sch5solgen}\n\\begin{split}\n\\Lambda =&-\\frac{3}{L_{\\text{sch}}^2}\n-b_1\\frac{80}{L_{\\text{sch}}^4}+b_2\\frac{4 \\left(3\n z^2-7\\right)}{L_{\\text{sch}}^4},\\\\ \na_2 =& \\frac{L_{\\text{sch}}^2}{4}-10 b_1 +b_2 \\left(z^2-3\\right).\n\\end{split}\n\\end{equation}\nWe recover eq.(\\ref{eq:sch5sol1}), as expected, when we put\n$b_1=b_2=0$. But, it is interesting to notice that in\neq.(\\ref{sch5solgen}), we get a solution with unconstrained $z$ when\n$b_2=0$ but $b_1 \\neq 0$. Which in turn means, if we go beyond the\nGauss-Bonnet theory and deform it with only $R^2$ term, but with no\n$R_{\\mu \\nu}R^{\\mu \\nu}$ term, we still obtain a Schr\\\"odinger\nsolution with arbitrary dynamical exponent. It is then natural to ask,\nif the degeneracy of the configuration space is still present at this\nlocus in parameter space beyond Gauss-Bonnet point, and it indeed\nturns out to be true. More specifically, a Schr\\\"odinger metric with\nthe ansatz \n\\begin{equation}\nds^2= L_{\\text{sch}}^2\\bigg[-f(r){dt^2 \\over r^{2z}}+{dr^2\\over\n r^2}+{2\\over r^2}dtd\\xi+{1\\over r^2} \\sum_{i=1}^{2} dx_i^2\\bigg],\n\\end{equation} \nis a solution to the equations of motion obtained from the action in\neq.(\\ref{actgen1}) with $b_2=0$, for \n\\begin{equation} \\label{sch5solgen1}\n\\Lambda =-\\frac{3}{L_{\\text{sch}}^2} -b_1\\frac{80}{L_{\\text{sch}}^4},~~ \na_2 = \\frac{L_{\\text{sch}}^2}{4}-10 b_1\n\\end{equation}\nwith arbitrary $f(r)$. We thus conclude that the\ndegeneracy in configuration space and the solutions with arbitrary\ndynamical exponent belong to the same locus on the parameter\nspace.\nThe special locus in Gauss-Bonnet theory on which both Lifshitz and\nSchr\\\"odinger solutions co-exist also a Chern-Simons description but the\ndegeneracy of the configuration space of Schr\\\" odinger solution is\nneither confined to Chern-Simons description nor to the Lovelock subspace.\nAlthough, we have carried out the study of degeneracy of the\nconfiguration space in $D=5$ for Gauss-Bonnet theory and\nits deformation to more general quadratic curvature \ntheories, similar analysis can be done in $D>5$ dimensions. \n\n\n\\section{Lovelock Gravity as $AdS$ Chern-Simons Gravity \nand Born-Infeld Gravity}\n\nThe Lovelock theory has the property that the action has general\ncovariance and the field equations contain at most two derivatives of\nthe metric. We parametrize the Lovelock theory using a set of real\ncoefficients $a_p, ~p=0,1, \\cdots, [D\/2]$ which are coupling constants\nof the higher derivative terms. It is convenient to adopt the\nfirst order approach, with the dynamical variables being the vielbein,\n$e^a = e^a_{\\mu} dx^{\\mu}$, and the spin connection,\n$\\omega^{ab} = {\\omega^{ab}}_{\\mu}~dx^{\\mu}$, obeying first order\nequations of motion. It is straightforward to solve the vanishing of\nthe torsion for the connection and eliminate them by writing them in\nterms of the vielbeins to obtain the standard second order form in\nterms of metric.\n\nThe action is constructed as a polynomial of degree $[D\/2]$ in\n$R^{ab}=(1\/ 2) {R^{ab}}_{\\mu\\nu} ~dx^{\\mu}\\wedge dx^{\\nu}$ \nand\n\\begin{equation} \\label{lifaction}\n I = \\frac{1}{16\\pi G}\\int_{M}d^Dx \\sum_{p=0}^{[D\/2]} a_p\n \\mathcal{L}_p,\\quad \\hbox{\\rm where},\\ \n\\mathcal{L}_p = \\epsilon_{a_1 \\cdots a_D} R^{a_1a_2} \\cdots\nR^{a_{2p-1}a_{2p}}~e^{a_{2p+1}} \\cdots e^{a_D} \n\\end{equation}\nImposing the condition that the theory possesses maximum possible\ndegrees of freedom determines all Lovelock couplings in terms\n$\\Lambda$ and $G_N$. The action in odd\ndimensions can then be written as a Chern-Simons action with \n$AdS$, $dS$ or Poincar\\'e\nsymmetry\\cite{Witten:1988hc,Zanelli:2005sa,Crisostomo:2000bb}, and in\neven dimensions as a Born-Infeld like\naction\\cite{Zanelli:2012px}\\footnote{Though having explicit torsion in\n the Lagrangian for $D=4k-1$ is possible with the same requirements,\n we will not consider them here.}.\n\n\\subsection{Odd Dimensions: Lovelock Gravity as\nChern-Simons Gravity} \\label{sec5a}\n\n\\subsubsection{The Chern-Simons Theory}\n\nIt is well known that gravity in $(2+1)$ dimensions can\nequivalently be written as a Chern-Simons theory for the gauge groups\n$ISO(2,1)$ or $SO(2,2)$, but with no propagating bulk degrees of\nfreedom. In higher dimensions, $D=2n-1,~n\\ge2$, the essential idea for\nconstructing a Chern-Simons theory is to utilize the fact that there\nexists a $2n$-form in $D=2n$,\n\\begin{equation}\nQ_{2n}({\\bf A}) =Tr[{\\bf F}^n] = Tr[\\underbrace{{\\bf F} \\wedge {\\bf F}\n \\wedge \\cdots \\wedge {\\bf F}}_{n -times}]\\ .\n\\end{equation}\nThis form is closed, i.e. $dQ_{2n} =0$, where ${\\bf A}$ is the Lie Algebra\nvalued connection 1-form \n${\\bf A} = A^a_{\\mu} {\\bf T}_a dx^{\\mu}$\nand ${\\bf F}=d{\\bf A}+{\\bf A}\\wedge{\\bf A}$ is the corresponding field strength or curvature 2-form,\nwith ${\\bf T}_a$ being the generators of the Lie algebra $g$ of the\ngauge group ${\\bf G}$\\cite{Troncoso:1999pk}. The fact that $Q_{2n}$ is\nclosed leads to the existence of a $(2n-1)$-form $L_{CS}^{2n-1}$ such\nthat \\begin{equation} d L_{CS}^{2n-1} = Q_{2n} = Tr[{\\bf F}^n] \\end{equation} which can always\nbe solved as \\begin{equation} L_{CS}^{2n-1} ({\\bf A}) = {1 \\over (n+1)!} \\int_0^1\ndt~ Tr\\big[{\\bf A}(t d{\\bf A}+t^2 {\\bf A}^2)^{n-1}\\big] + \\alpha \\end{equation}\nwith $\\alpha$ being some arbitrary closed $(2n-1)$-form. This way one\nconstructs a Chern-Simons Lagrangian $L_{CS}^{2n-1} ({\\bf A})$ in\n$D=2n-1$ dimensions with an action \\begin{equation} \\label{csact} I_{CS} ({\\bf A})\n=\\int_{M_{2n-1}} L_{CS}^{2n-1} ({\\bf A}). \\end{equation}\n\n\\subsubsection{Connection with the Lovelock Gravity}\n\nIn odd dimensions, {\\em i.e.}, $D=2n-1$, it was argued that the requirement\nof having maximum possible degrees of freedom fixes the Lovelock\ncoefficients as\\cite{Troncoso:1999pk}\n\\begin{equation} \\label{coeflov}\na_p = {\\kappa L^{2p-D} \\over {D-2p}} {n-1 \\choose p}, \\quad 0\\le p \\le\nn-1 \n\\end{equation}\nleaving the action depending on only two parameters, gravitational\nconstant $\\kappa$ and the cosmological constant\n$\\Lambda$\\footnote{Note that $L$ is the length parameter related to\n cosmological constant as $\\Lambda = \\pm {(D-1)(D-2)\\over 2 L^2}$,\n where as the Newton's constant $G_N$ is related to $\\kappa$ through\n $\\kappa^{-1} = 2(D-2)! \\Omega_{D-2} G_N$.}.\nThe precise connection of Lovelock theories with Chern-Simons gravity\ntheories in odd dimensions ($D=2n-1$) is that the Lagrangian for the\nLovelock theory can be cast as a Chern-Simons theory for the group\n$AdS$. This can be demonstrated through the packaging of the Lovelock\nvielbeins $e^a$ and connections $\\omega^{ab}$ in the following\nconnection 1-form as \n\\begin{equation}\nW^{AB} = \\left[\n\\begin{array}{ll}\n\\omega^{ab} & {e^a \\over L}\\\\\n-{e^a \\over L} & 0\n\\end{array}\\right]\n\\end{equation}\nwhere the indices $a,b=1,\\cdots,D$ and $A,B=1,\\cdots,D+1$. Note that\nthe $A,~B$-indices are raised or lowered with respect to the $AdS$\nmetric \n\\begin{equation}\n\\Pi_{AB}=\\left[\n\\begin{array}{ll}\n\\eta_{ab} & 0\\\\\n0 & -1\n\\end{array}\\right]\n\\end{equation}\nThis connection defines a curvature 2-form, also called the $AdS$\ncurvature, as \n\\begin{equation} \\label{defFAB}\nF^{AB} = dW^{AB}+{W^A}_{C}\\wedge {W_{C}}^{B} = \n\\left[\n\\begin{array}{ll}\nR^{ab}+{e^a\\wedge e^b \\over L^2} & {T^a\\over L}\\\\\n-{T^a\\over L} & 0\n\\end{array}\\right]\n\\end{equation}\nwhere $R^{ab}=d\\omega^{ab}+{\\omega^a}_{c}\\wedge {\\omega_{c}}^{b}$ is\nthe curvature 2-form for the 1-form $\\omega^{ab}$, \nwhich is $(2n-1)$-dimensional and not to be confused with the\n$2n$-dimensional $AdS$ curvature 2-form $F^{AB}$. $T^a$ is the\ntorsion form and setting it to zero corresponds to imposing\ntorsion-free constraint. \n\nNext, using the invariant tensor for the $AdS$ group $\\epsilon_{A_1\n \\cdots A_{2n}}$ along with the $2n$-dimensional AdS curvature\n$F^{AB}$ one constructs the Euler form in $2n$-dimension\n\\begin{equation}\n\\mathcal{E}_{2n} = \\epsilon_{A_1 \\cdots A_{2n}} F^{A_1A_2} \\cdots\nF^{A_{2n-1}A_{2n}} \n\\end{equation}\nUsing the Bianchi identity for the $AdS$ curvature\n$F^{AB}$ we can show that this Euler density is closed,\n$d\\mathcal{E}_{2n}=0$,\nand eq.(\\ref{defFAB}) one\ncan write the $(2n-1)$-form $L_{CS}^{2n-1}$ in terms of the\n$(2n-1)$-dimensional curvature 2-form $R^{ab}$ and the vielbeins $e^a$\nsuch that \n\\begin{equation}\nL_{CS}^{2n-1} = \\sum_{p=0}^{[D\/2]} a_p ~\\epsilon_{a_1 \\cdots a_D}\nR^{a_1a_2} \\cdots R^{a_{2p-1}a_{2p}}~e^{a_{2p+1}} \\cdots e^{a_D}\\ , \n\\end{equation}\nand $d L_{CS}^{2n-1} = \\mathcal{E}_{2n}$. The coefficients $a_p$ are\ncompletely fixed here due to the relation between the\nChern-Simons density and the Euler density and they\nturn out to be exactly same as those in eq.(\\ref{coeflov}). The\nfield equations obtained from the action in eq.(\\ref{csact}) are \n\\begin{equation} \n\\begin{split}\n\\epsilon_{a_1a_2a_3\\cdots \\alpha_{2n-1}}F^{a_2a_3}\\cdots\nF^{a_{2n-2}a_{2n-1}}&=0,\\\\ \n\\label{eqCSgrav2}\n\\epsilon_{a_1a_2a_3\\cdots \\alpha_{2n-1}}F^{a_3a_4}\\cdots\nF^{a_{2n-3}a_{2n-2}} T^{a_{2n-1}}&=0. \n\\end{split}\n\\end{equation}\n\n\\subsection{Even Dimensions: Lovelock Gravity as Born-Infeld Gravity} \\label{sec5b}\n\nAs we have seen in odd dimensions there are gravity actions which are\ninvariant not just under Lorentz group but also under some its\nextensions, e.g. $AdS$ group $SO(D-1,2)$. On the contrary, this is\nnot possible in even dimensions, $D=2n$. However the\nrequirement of having maximum possible number of degrees of freedom\nfixes the Lovelock coefficients as\\cite{Troncoso:1999pk}\n\\begin{equation}\na_p= \\kappa {n \\choose p}, \\quad 0\\le p \\le n.\n\\end{equation}\nThe Lovelock action depends on two constants only, the\ngravitational constant and the cosmological constants, and the\nLagrangian, given in eq.(\\ref{lifaction}), becomes \n\\begin{equation}\n\\mathcal{L}= {\\kappa \\over 2n} \\epsilon_{a_1 \\cdots a_D} F^{a_1a_2}\n\\cdots F^{a_{D-1}a_{D}} \n\\end{equation}\nwhich is pfaffian of the two form $F^{ab}=R^{ab}+{e^a e^b \\over L^2}$\nand can be cast in Born-Infeld form\\cite{Troncoso:1999pk}\n\\begin{equation}\n\\mathcal{L}= 2^{n-1} (n-1)! \\kappa \\sqrt{det\\left(R^{ab}+{e^a e^b\n \\over L^2}\\right)}. \n\\end{equation}\nIt is important to note that the two forms $F^{ab}$ are no longer a part\nof any $AdS$ curvature. The field equations in even dimensions take\nthe form \n\\begin{equation}\n \\label{eqBIgrav1}\n \\begin{split}\n \\epsilon_{a b_1 \\cdots b_{D-1}}F^{b_1b_2} \\cdots F^{b_{D-3}b_{D-2}}\n~e^{b_{D-1}} &= 0 \\\\\n\\epsilon_{a b a_3 \\cdots a_{D}}F^{a_3a_4} \\cdots F^{a_{D-3}a_{D-2}}\nT^{a_{D-1}}e^{a_D} &= 0.\n \\end{split}\n\\end{equation}\n\n\n\\subsection{Schr\\\" odinger Space-Time as a Solution to Chern-Simons\n Gravity in $D=5$ Dimensions}\n\nWe will now explicitly show that the Schr\\\" odinger solution obtained\nearlier from the Lovelock action can also be seen as a solution to\nthe Chern-Simons gravity in odd dimensions. We will work in $D=5$\ndimensions. The metric in $5$-dimension looks like \n\\begin{equation}\\label{schmet5}\n ds^2= L_{\\text{sch}}^2\\bigg[-{dt^2 \\over r^{2z}}+{dr^2\\over\n r^2}+{2\\over r^2}dtd\\xi+{1\\over r^2} (dx^2+dy^2)\\bigg]\\ .\n\\end{equation}\nWe make the following choice for vielbeins corresponding to the \nmetric in eq.(\\ref{schmet5}),\n\\begin{equation} \\label{schviel}\n\\begin{split}\ne^1_t =- {L_{\\text{sch}} \\over r^z}, ~ e^1_\\xi= L_{\\text{sch}} r^{z-2},\n~ e^3_\\xi = L_{\\text{sch}} r^{z-2} ,~ e^2_r =e^4_x=e^5_y=\n{L_{\\text{sch}} \\over r}\n\\end{split}\n\\end{equation} The spin connections $\\omega^{ab}$ and the AdS curvature $F^{AB}$\nfor the Schr\\\" odinger metric are listed in appendix \\ref{apdx3}. It\nis easy to see that the $AdS$ curvature $F^{AB}$ does indeed satisfy\nthe field equations \neq.(\\ref{eqCSgrav2}).\n\n\n\\subsection{Relation with Causality and Stability Constraints}\n\\label{sec:relat-with-unit}\n\nStability analysis of Lovelock theories in higher dimensions has been\ncarried out in the past \\cite{deBoer:2009gx, Camanho:2009hu,\n Camanho:2010ru}. These studies derive constraints on the values of\nGauss-Bonnet coupling ($a_2$) and the cubic Lovelock coupling ($a_3$)\nby demanding causality and stability condition on the\nsolutions of the Lovelock theory in $D=7$. These two conditions are\nsatisfied in a region in the neighbourhood of the origin of the\n($a_2$, $a_3$) plane and at an isolated point, which in our choice of\nnormalization corresponds to ($a_2=L^2\/36,~ a_3= L^4\/648$). The\nLovelock parameters ($a_2$, $a_3$) used in this paper are related to the\nparameters ($\\beta_2$, $\\beta_3$) or ($\\lambda_1$, $\\lambda_2$) used\nin \\cite{deBoer:2009gx} in the following way \n$a_2 = \\beta_2 L^2 = (\\lambda_1\/12) L^2$ and $a_3=\n\\beta_3 L^4 = (\\lambda_2\/ 72) L^4$.\nIn terms of these parameters the isolated point, mentioned above,\ncorresponds to ($\\lambda_1=1\/3$, $\\lambda_2=1\/9$)\\footnote{Note that\n in \\cite{deBoer:2009gx} the cosmological constant is taken to be\n $\\Lambda=-15\/L^2$ and relation between $\\beta_i$ and $\\lambda_i$ is\n given for $L=1$. This reduces the coupling parameter space from\n three to two dimensions, therefore the Schr\\\" odinger or Lifshitz\n solutions exist only a point in the reduced parameter space\n $(\\beta_1, \\beta_2)$ or $(\\lambda_1,\\lambda_2)$.}. It is\ninteresting to note that this isolated apex point in the phase\ndiagram, see figure $1$ in \\cite{Camanho:2010ru}, is also the same\npoint where we have the Chern-Simons representation for the Lovelock\ntheory. The Schr\\\" odinger and Lifshitz solutions exist only at this\npoint in the Lovelock coupling space, which presumably also implies\nthat they also satisfy the causality and stability constraints. It\nwould be interesting to check this explicitly for these solutions.\n\n\\section{Discussion}\n\nWe studied the coupling constant parameter space of Lovelock gravity\ntheories in arbitrary dimensions, while restricting our analysis to the\nLovelock terms up to cubic in curvatures. We demonstrated that\nSchr\\\" odinger solutions exist on co-dimension 1 subspace\nin the parameter space. Similar results for Lifshitz solutions\nalready exist in the literature. Interestingly, both the solutions exist on\nthe same locus. We found that on this locus, both Schr\\\" odinger\nand Lifshitz exponents were completely unconstrained. Even if we\ncouple the Maxwell or Yang-Mills fields to these Lovelock theories,\nthe Schr\\\" odinger and the Lifshitz moduli space would continue to be\nthe same co-dimension 1 subspace. \n\nAs already mentioned earlier in the introduction, Schr\\\" odinger\nholography relates a theory of gravity to field theories living on a\nco-dimension 2 subspace. Therefore, $D=5$ and $D=6$ dimensional\nSchr\\\"odinger geometries are directly relevant for studying field\ntheories with this symmetry in 2+1 and 3+1 dimensions. However, higher\ndimensional Schr\\\"odinger geometries in $D \\ge 7$ dimensions are also\nrelevant for studying field theory systems in lower dimensions, since\nthe higher dimensional Schr\\\"odinger space-times can be first\ndimensionally reduced to lower dimensions and then we can analyse those\ndimensionally reduced theories. As was pointed out in eq.\\eqref{eq:11},\nthe dimensional reductions do generally lead to effective theories in\nthe lower dimensions similar to Galileon type theories. In fact,\nin the case when $n=1$ in eq.\\eqref{eq:11}, that is starting from $D=d+2$\ndimensions we come down to $D=d+1$ dimensions, the effective action\ntakes even simpler form\n \\begin{equation}\n \\label{eq:12}\n \\begin{split}\n \\bar{S}_{(d+1)} = \\int d^{d+1} x \\sqrt{-\\bar{g}}\\, e^{\\frac{\\phi}{2}}\n \\bigg\\{ \\bar{R}- 2\\Lambda + a_2 \\bar{\\mathcal{G}}\\ ,\n \\bigg\\} \n \\end{split} \n \\end{equation}\n and for the case when $n=2$, with simple toroidal reduction it becomes\n \\begin{equation}\n \\label{eq:13}\n \\begin{split}\n \\bar{S}_{(d+1)} = \\int d^{d+1} x \\sqrt{-\\bar{g}}\\, e^{\\frac{\\phi}{2}}\n \\bigg\\{ \\bar{R}- 2\\Lambda + a_2 \\bar{\\mathcal{G}} + {1 \\over 2}\n \\bar{g}^{\\mu\\nu}\\partial_\\mu \\phi \\partial_\\nu \\phi - 2 a_2\n \\bar{G}^{\\mu\\nu} \\partial_{\\mu}\\phi \\partial_{\\nu} \\phi\n \\bigg\\}\\ .\n \\end{split} \n \\end{equation}\n\n Since these effective actions are obtained are obtained by\n dimensionally reducing the higher dimensional theories, any solution\n of the higher dimensional theories continues to solve the equations\n of motion obtained from the reduced action.\n \n We have seen in section \\ref{schsold5} that for a specific value for\n the Gauss-Bonnet coupling constant $a_2$, given in\n eq.\\eqref{eq:sch5sol1}, we obtain a Schr\\\"odinger solution in $D=5$\n dimensions with an unconstrained exponent $z$. Similarly, starting in\n $D=7$ dimensions and for the sake of simplicity allowing only the\n Gauss-Bonnet term, that is assuming $a_3=0$, we can perform a\n toroidal compactification over a $2$ dimensional internal manifold\n and the resulting effective theory in $D=5$ dimension comes with\n action given in eq.\\eqref{eq:13}. The Schr\\\"odinger solution in $D=7$\n dimension, given in eq.\\eqref{sch7sol1} with $a_3=0$, also becomes a\n solution to the dimensionally reduced effective theory eq.\\eqref{eq:13},\n with unconstrained dynamical exponent $z$ for these particular values\n of the parameters $\\Lambda$ and $a_2$. Therefore, from the view point\n of effective lower dimensional theories with the action motivated by\n the dimensional reductions from higher dimensional theories, we can\n take a phenomenological bottom-up approach to study various aspects\n of field theoretical systems with Schr\\\"odinger symmetries. In this\n regard, our study in this paper provides a template for studying\n non-relativistic field theories with arbitrary dynamical exponent via\n holography, in theories of gravity coupled to matter systems through\n non-trivial but specific values of couplings of the Galileon terms,\n as dictated by eq.\\eqref{eq:sch5sol1} and eq.\\eqref{sch7sol1}. For\n $n\\leq 2$ there is further simplification because in that case\n neither the DGP term nor the Galileon type terms appear in the $d+1$\n dimensional theory as is evident from eq.\\eqref{eq:12} and\n eq.\\eqref{eq:13}. \n Starting from these dimensionally reduced theories, one can then\n deform them appropriately by adding new terms to the action and\n engineer non-relativistic solutions with particular fixed values of\n the dynamical exponent $z$. Analysis of these deformations leading\n to specific values of $z$ relevant for application to, say, the\n condensed matter systems is beyond the scope of this investigation.\n\nWe also pointed that the co-dimension 1 subspace of the Lovelock\ntheories on which Schr\\\" odinger and Lifshitz solutions exist also\nsupports Chern-Simons formulation in odd space-time dimensions and\nBorn-Infeld formulation in even space-time dimensions. We cast our\nnon-relativistic metric in the gauge connection form suitable for\nthese formulations. At this point it is interesting to note\nthat\\cite{Zanelli:2005sa,Zanelli:2012px} these gauge connection\nformulations have natural super-symmetric extension. It would be\ninteresting to explore super symmetric non-relativistic solutions in\nthe Lovelock theories.\n\nIt is known that the Chern-Simons point in the coupling space of the\nLovelock theories is maximally symmetric. However, most of the\nstudies at this point are concentrated either on the AdS type\nsolutions or on the black brane solutions. Neither of these solutions\ncan shed direct light on the possible values of the dynamical exponent\n$z$ that appears in the Schr\\\" odinger or Lifshitz solutions. \nUnconstrained dynamical exponent is related to the degeneracy of the\nconfiguration space \\cite{Dotti:2007az,Dehghani:2010kd}, which was\nstudied in the context of Chern-Simons formulation. We studied\nmodification of the Gauss-Bonnet theory by doing general deformation\nusing terms quadratic in curvature and found that the degeneracy of\nthe configuration space in case of the Schr\\\" odinger solution is not\nconfined to the Chern-Simons point but \nextends in the direction orthogonal to the Lovelock moduli space\ncorresponding to deformation by the Ricci scalar squared term. Thus\nthe unconstrained dynamical exponent is a result of the degeneracy and\nit has weak dependence on the special locus in the Lovelock moduli\nspace in the case of the Schr\\\" odinger solutions.\n\nAnother point worth mentioning is that we do not find hyper-scaling\nviolating solutions anywhere in the Lovelock moduli space, nor do we\nfind black brane solutions with either Schr\\\" odinger or Lifshitz type\nscaling. This in turn means it is harder to turn on temperature in\nthese geometries, however, these shortcomings can be remedied by\ndeforming away from the Lovelock moduli space.\n\n\n\n\\paragraph{\\large Acknowledgements}: We would like to thank Sujay\nAshok, Anirban Basu, Jyotirmoy Bhattacharya, Sayantani Bhattacharyya,\nAnshuman Maharana, K. Narayan, Gautam Mandal, S. Kalyana Rama, Ashoke\nSen and Sandip P. Trivedi for discussion and useful suggestions. One of\nus (DPJ) would like to thanks IMSc for hospitality during the course\nof this work. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}