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+{"text":"\\section{Introduction}\\label{sec:intro}\n\t\n\tWhen handling real-world optimization problems by means of mathematical modeling, it is often necessary to treat inexact or uncertain input data due to various measurement errors or estimations. Throughout the years, several approaches for solving optimization problems with imprecise data have emerged, based on different sources of uncertainty and different requirements imposed on the solutions. This paper adopts the approach of interval linear programming, which can be considered a special case of multi-parametric optimization with no dependencies among the coefficients. Interval-valued coefficients have been used for modeling uncertainty in many practical applications, such as portfolio selection \\cite{Lai:2002, Kumar:2016}, environmental management \\cite{Tan:2010, Cheng:2015} or group decision making \\cite{Groselj:2017}.\n\t\n\tIn interval optimization, it is assumed that the coefficients may perturb independently within the given lower and upper bounds. However, rather than focusing on the worst case and trying to find a stable solution (as in robust optimization), we aim to cover the properties of all possible scenarios. This approach leads to questions such as describing the set of feasible solutions over all scenarios \\cite{Oettli:1964, Rohn:InexactLP:2006} and the set of all optimal solutions \\cite{Allahdadi:2013:opt}, computing the optimal value range \\cite{Mraz:1998, Rohn:InexactLP:2006b, Hladik:2009} or characterizing the duality gaps \\cite{Novotna:2017}. For an overview of other results in interval linear programming see the survey by \\cite{Hladik:2012}.\n\t\n\tThis paper discusses an ever-present issue in interval optimization and interval analysis in general: \\emph{the dependency problem}. Since we do not allow any dependencies among the coefficients of an interval program and assume that the values of the coefficients perturb independently, any transformation that changes the program and duplicates some of the interval coefficients will also create new possible scenarios. Due to the dependency problem, transforming an interval program into a desired form by applying the standard transformations used in classical linear programming may change its feasible or optimal solutions and other important properties. Therefore, it may be necessary to study different types of programs separately, since the results obtained for one type do not have to hold for other types.\n\t\n\tWe examine the effects of the transformations on the optimal solution set and the optimal value range in order to identify the transformations that are also applicable to interval linear programs while preserving the considered properties. This will allow us to directly generalize the results known for one type of programs to other forms, as well as highlight the cases in which the particular properties may differ. We will also study the problem on a special class of interval linear programs with interval coefficients only appearing in the objective function and the right-hand-side vector in order to derive stronger results. These types of programs arise in applications, where the coefficient matrix represents some known relations, such as transportation problems \\cite{Safi:2013, Cerulli:2017:inttransp} or network flow problems \\cite{Hashemi:2006}.\n\t\n\tThe paper is structured as follows: Section~\\ref{sec:ilp} introduces the basic notions and terminology of interval linear programming. In Section~\\ref{sec:transf}, we review the standard transformations used in classical linear programming and discuss their applicability on interval programs. Sections~\\ref{sec:optsol} and~\\ref{sec:optval} explore the effects of the transformations on the set of all optimal solutions and optimal values of an interval linear program. Section~\\ref{sec:concl} summarizes and concludes the paper. \n\t\n\t\\section{Interval Linear Programming}\\label{sec:ilp}\n\tGiven two real matrices $\\lb{A}, \\ub{A} \\in \\mathbb{R}^{m\\times n}$ satisfying $\\lb{A} \\le \\ub{A}$, we define an \\emph{interval matrix} as the set \\[\\bint{A} = [\\lb{A}, \\ub{A}] = \\{A \\in \\mathbb{R}^{m \\times n} : \\lb{A} \\le A \\le \\ub{A}\\}.\\]\n\tThe matrices $\\lb{A}, \\ub{A}$ are the \\emph{lower} and \\emph{upper bound} of $\\bint{A}$, respectively. Alternatively, we can also define an interval matrix by its \\emph{center} $A_c = \\frac{1}{2}(\\ub{A} + \\lb{A})$ and \\emph{radius} $A_\\Delta = \\frac{1}{2}(\\ub{A} - \\lb{A})$.\n\tThroughout the paper, we denote interval objects by bold lowercase (for one-dimensional intervals and interval vectors) and bold uppercase (for interval matrices) letters. The symbol $\\mathbb{IR}$ is used to denote the set of all (real) intervals.\n\t\n\tFurthermore, let us extend the classical definition of linear programs to the case of uncertain interval coefficients. We define an \\emph{interval linear program} (ILP) as a family of linear programs\n\t\\begin{equation}\\label{eq:ilp:def}\n\t\\{ \\text{minimize } c^T x \n\t\\text{ subject to} \\  x \\in \\mathcal{M}(A,b) : A \\in \\bint{A}, b \\in \\bint{b}, c \\in \\bint{c} \\}\n\t\\end{equation}\n\twhere $\\mathcal{M}(A,b)$ denotes the feasible set described by some linear constraints. For the sake of brevity, we usually write ILPs in the form of linear programs with interval coefficients (see \\eqref{eq:ilp:A}--\\eqref{eq:ilp:C} below for examples). A particular linear program in the family is called a~\\emph{scenario}. Given an interval linear program, we can define its \\emph{dual} ILP as the family of all dual linear programs.\n\t\n\tSimilarly, we can also generalize the notion of feasibility and optimality to the interval case. We say that a vector $x \\in \\mathbb{R}^n$ is a \\emph{(weakly) feasible\/optimal} solution to the interval linear program, if there exists a scenario determined by the triplet $A \\in \\bint{A}$, $b \\in \\bint{b}, c \\in \\bint{c}$ such that $x$ is a feasible\/optimal solution for the scenario (see \\cite{Rohn:InexactLP:2006} and \\cite{Hladik:2017} for an overview of the feasibility properties of interval linear systems). \n\tThe \\emph{(weakly) feasible\/optimal set} is the set of all weakly feasible\/optimal solutions of an ILP.\n\tWe often omit the word ``weakly'', since no confusion should arise in the paper.\n\t\n\tConsidering the optimal objective values, we may be interested in the best and the worst value that can be achieved as optimal for some scenario of the given ILP. For a minimization program, these bounds are defined as \n\t\\begin{align*}\n\t&\\lb{f}(\\bint{A}, \\bint{b}, \\bint{c}) = \\inf \\{f(A,b,c): A \\in \\bint{A}, b \\in \\bint{b}, c \\in \\bint{c}\\},\\\\\n\t&\\ub{f}(\\bint{A}, \\bint{b}, \\bint{c}) = \\sup \\{f(A,b,c): A \\in \\bint{A}, b \\in \\bint{b}, c \\in \\bint{c}\\},\n\t\\end{align*}\n\twhere $f(A,b,c)$ denotes the optimal value of the corresponding linear program, allowing the values $-\\infty$ and $\\infty$ for unbounded programs or infeasible programs.\n\tThe interval $[\\lb{f}(\\bint{A}, \\bint{b}, \\bint{c}), \\ub{f}(\\bint{A}, \\bint{b}, \\bint{c})]$ is called the \\emph{optimal value range}.\n\t\n\tDue to the dependency problem, we usually assume that an~ILP is given either in the general form\n\t\\begin{equation*}\n\t\\begin{array}{lrrrr@{\\ }l}\n\t\\text{minimize } & \\multicolumn{5}{l}{\\bint{c_1}^T x_1 + \\bint{c_2}^T x_2} \\\\\n\t\\text{subject to } & \\bint{A}x_1 &+ & \\bint{B}x_2 & = & \\bint{b_1},\\\\\n\t&  \\bint{C}x_1 & + & \\bint{D}x_2 & \\le & \\bint{b_2},\\\\\n\t& &  \\multicolumn{2}{r}{x_1, x_2} &\\ge & 0,\n\t\\end{array}\n\t\\end{equation*}\n\tor in one of the following forms often encountered in optimization problems:\n\t\\begin{align}\n\t&\\text{minimize } \\bint{c}^T x \\text{ subject to } \\bint{A}x = \\bint{b},\\;x \\ge 0, \\tag{I}\\label{eq:ilp:A}\\\\ \n\t&\\text{minimize } \\bint{c}^T x \\text{ subject to } \\bint{A}x \\le \\bint{b}, \\tag{II}\\label{eq:ilp:B}\\\\\n\t&\\text{minimize } \\bint{c}^T x \\text{ subject to } \\bint{A}x \\le \\bint{b},\\;x \\ge 0. \\tag{III}\\label{eq:ilp:C}\n\t\\end{align}\n\t\n\t\\section{Transformations of Interval Linear Programs}\\label{sec:transf}\n\tLet us now briefly review the basic transformations that are used in linear programming to convert a given program into the desired form. While some of them can also be applied to interval programs, other transformations may change the set of feasible or optimal solutions and affect the properties of the program. The effects of these transformations on the set of all optimal solutions and optimal values will be discussed in more detail in Sections~\\ref{sec:optsol} and~\\ref{sec:optval}.\n\t\n\t\\paragraph{Changing the objective.} A commonly used trick in linear programming is to switch the objective from maximization to minimization (or vice versa) by taking the opposite of the objective function. Since this transformation does not duplicate any coefficients, it can also be directly applied to interval programs: an interval objective $\\max \\bint{c}^T x$ can be equivalently rewritten as $- (\\min -\\bint{c}^T x)$.\n\t\n\t\\paragraph{Adding slack variables.} In order to convert inequality constraints to equations in a linear program, sign-restricted slack variables are added to the constraints. Analogously, we can transform an interval inequality system $\\bint{A}x \\le \\bint{b}$ into an interval system \\[ \\bint{A}x + Iy = \\bint{b},\\; y \\ge 0. \\]\n\tAgain, the transformation does not duplicate any of the existing coefficients and only introduces some new crisp coefficients. Applying this transformation to a set of constraints results in an equivalent interval linear program, since each new scenario is equivalent to a corresponding scenario in the original interval system.\n\t\n\t\\paragraph{Splitting equations into inequalities.} When changing a general linear program into an inequality-constrained form, we split each equation into two opposite inequalities. However, splitting an interval equation into interval inequalities while preserving all properties is not always possible, due to the dependency problem. Breaking dependencies among multiple occurrences of an interval coefficient leads to generating new scenarios that do not have a counterpart in the original problem. More precisely, while an interval equation $\\bint{a}^T x = b$ consists of the scenarios $a^T x = b$ with all possible choices of $a \\in \\bint{a}$, the system of two interval inequalities $\\bint{a}^T x \\le b$, $\\bint{a}^T x \\ge b$ corresponds to the (possibly larger) set of all scenarios in the form $a_1^T x \\le b, a_2^T x \\ge b$ with coefficients $a_1, a_2 \\in \\bint{a}$.\n\t\n\t\\paragraph{Imposing non-negativity.} Another transformation used in linear programming is substituting the difference of two non-negative variables for a free variable. Along with introducing two new variables for each original free variables, the operation also duplicates the corresponding coefficients. This, again, leads to breaking the dependency in case of interval constraints: By replacing a constraint $\\bint{a}^T x = \\bint{b}$ with $\\bint{a}^T x^+ - \\bint{a}^T x^- = \\bint{b}$ and $x^+, x^- \\ge 0$, we create two independent occurrences of the interval coefficient $\\bint{a}$, which can take on different values in a particular scenario.\n\t\n\t\\section{Optimal Solution Set under Transformations}\\label{sec:optsol}\n\t\\subsection{General Case}\\label{ssec:gen}\n\tIn this section, we discuss the two types of transformations that can possibly change the properties of an interval linear program: splitting equation constraints and imposing non-negativity on the variables. First, we present examples of ILPs showing that these transformations can change the optimal solution set. Note that in both cases, the transformed interval program contains all of the original scenarios, therefore, the original optimal set is always a subset of the transformed one.\n\t\n\tWe have seen in Section~\\ref{sec:transf} that splitting an interval equation into two inequalities can cause a dependency problem and possibly change the feasible or optimal solutions of an ILP. However, it was proved by \\cite{Li:2015} that this is not the case for the feasible set.\n\t\n\t\\begin{theorem}[\\cite{Li:2015}]\\label{thm:Li}\n\t\tA vector $x \\in \\mathbb{R}^n$ is a weakly feasible solution of an~interval linear system $\\bint{A}x = \\bint{b}$ if and only if it is a weakly feasible solution of the system $\\bint{A}x \\le \\bint{b}, \\bint{A}x \\ge \\bint{b}$.\n\t\\end{theorem}\n\t\n\tEven though the transformation preserves the weakly feasible solution set, the following example shows that creating new scenarios may, in fact, lead to expanding the set of optimal solutions.\n\t\n\t\\begin{example}\\label{ex:optsol}\n\t\tConsider the interval linear program\n\t\t\\begin{equation}\\label{eq:ex}\n\t\t\\begin{array}{lr@{\\ }l}\n\t\t\\text{minimize } & \\multicolumn{2}{l}{-x_1} \\\\\n\t\t\\text{subject to } & [0,1]x_1 - x_2 = 0,\\\\\n\t\t&  x_2 \\le 1,\\\\\n\t\t&  x_1, x_2 \\ge 0.\n\t\t\\end{array}\n\t\t\\end{equation}\n\t\tThe optimal solution set of~\\eqref{eq:ex} consists of all values with $x_1 \\in [1, \\infty)$ and $x_2 = 1$. By splitting the equation $[0,1]x_1 - x_2 = 0$ into two inequalities, we allow the value for each coefficient to be chosen independently. This results in introducing additional scenarios into the problem, such as the scenario \n\t\t\\begin{equation*}\n\t\t1x_1 - x_2 \\le 0,\\quad 0x_1 - x_2 \\ge 0\n\t\t\\end{equation*}\n\t\twith the optimal solution $(0,0)$. Therefore, the optimal solution sets of the two problems are not equivalent.\n\t\\end{example}\n\t\n\tThe second type of transformation affected by the dependency problem is restricting the sign of the variables. However, unlike all of the other transformations, replacing a free variable by a difference of two non-negative variables does not even preserve the feasible solution set of an interval linear program, as already noted by \\cite{Hladik:2012}. This implies that the set of optimal solutions may also change, in general.\n\t\n\t\\subsection{Special Case: Fixed Coefficient Matrix}\\label{ssec:spec}\n\tThis section is devoted to a special class of interval linear programs, in which uncertainty only affects the objective function and the right-hand-side vector. We show that, unlike general interval linear programs, programs with a fixed coefficient matrix allow for all types of transformations while preserving the optimal solution set. Theorem~\\ref{thm:spec:AC} presents the result for the transformation of splitting an equation into two opposite inequalities, i.e. converting a program of type~\\eqref{eq:ilp:A} with a fixed matrix into a program of type~\\eqref{eq:ilp:C}.\n\t\n\t\\begin{theorem}\\label{thm:spec:AC}\n\t\tLet $\\mathcal{S}(A,\\bint{b},\\bint{c})$ be the optimal solution set of an interval linear program of type~\\eqref{eq:ilp:A} with a fixed coefficient matrix given by the triplet $(A, \\bint{b}, \\bint{c})$. Then, $\\mathcal{S}(A,\\bint{b},\\bint{c})$ is equal to the optimal solution set of the problem\n\t\t\\begin{equation}\\label{eq:transfAC}\n\t\t\\begin{array}{l@{\\hskip 8pt}r@{\\ }l}\n\t\t\\textnormal{minimize} & \\multicolumn{2}{l}{\\bint{c}^T x}\\\\\n\t\t\\textnormal{subject to} & A x &\\le \\bint{b}_1,\\\\\n\t\t& -A x &\\le -\\bint{b}_2,\\\\\n\t\t& x &\\ge 0.\n\t\t\\end{array}\n\t\t\\end{equation}\n\t\twith $\\bint{b}_1 = \\bint{b}_2 = \\bint{b}$.\n\t\\end{theorem}\n\t\\begin{proof}\n\t\tLet $\\mathcal{S}(A',\\bint{b'},\\bint{c})$ denote the optimal solution set of program~\\eqref{eq:transfAC}. Clearly, the inclusion $\\mathcal{S}(A,\\bint{b},\\bint{c}) \\subseteq \\mathcal{S}(A',\\bint{b'},\\bint{c})$ holds, since \\eqref{eq:transfAC} contains all the scenarios of \\eqref{eq:ilp:A}.\n\t\t\n\t\tOn the other hand, let $x'$ be an optimal solution of ILP \\eqref{eq:transfAC} for a scenario determined by the coefficient vectors $c \\in \\bint{c}$ and $b_1, b_2 \\in \\bint{b}$ satisfying \n\t\t\\begin{equation}\\label{eq:transf:scenario}\n\t\tAx' \\le b_1,\\; -Ax' \\le -b_2.\n\t\t\\end{equation}\n\t\tSince we have, $b_2 \\le Ax' \\le b_1$, there exists $b_3 \\in [b_2, b_1] \\subseteq \\bint{b}$ with $Ax' = b_3$. We claim that $x'$ is also an optimal solution of program~\\eqref{eq:ilp:A} for the scenario\n\t\t\\begin{equation*}\n\t\t\\begin{array}{lrl@{\\ }l}\n\t\t\\text{minimize} & \\multicolumn{2}{l}{c^T x} \\\\\n\t\t\\text{subject to} & Ax &=b_3,\\\\\n\t\t& x & \\ge 0.\n\t\t\\end{array}\n\t\t\\end{equation*}\n\t\tSuppose, for the sake of contradiction, that there exists another feasible solution $x^*$ with $c^T x^* < c^T x'$. By the choice of $b_3$, the vector $x^*$ is also feasible for scenario~\\eqref{eq:transf:scenario}. However, since the objective function is the same for both problems, this contradicts the assumption that $x'$ is optimal in scenario~\\eqref{eq:transf:scenario}.\n\t\\end{proof}\n\t\n\tNote that while the transformation preserves the optimal solutions, it may still change other properties of the program, such as the existence of infeasible scenarios or the range of optimal values, which will be discussed in more detail in Section~\\ref{sec:optval}. Theorem~\\ref{thm:spec:nonneg} shows an analogous result for the transformation of free variables into non-negative variables.\n\t\n\t\\begin{theorem}\\label{thm:spec:nonneg}\n\t\tLet $\\mathcal{S}(A, \\bint{b}, \\bint{c})$ denote the optimal solution set of an interval linear program of type~\\eqref{eq:ilp:B} with a fixed coefficient matrix and let $\\mathcal{S}(A', \\bint{b}, \\bint{c}')$ be the optimal solution set of the program\n\t\t\\begin{equation}\\label{eq:transfBC}\n\t\t\\begin{array}{lr@{\\ }l}\n\t\t\\textnormal{minimize} & \\multicolumn{2}{l}{\\bint{c}_1^T x^+ - \\bint{c}_2^T x^-}\\\\\n\t\t\\textnormal{subject to} & A x^+ - A x^- &\\le \\bint{b},\\\\\n\t\t& x^+, x^- &\\ge 0\n\t\t\\end{array}\n\t\t\\end{equation}\n\t\twith $\\bint{c}_1 = \\bint{c}_2 = \\bint{c}$. Then, the following properties hold:\n\t\t\\begin{enumerate}\n\t\t\t\\item If $x \\in \\mathcal{S}(A,\\bint{b},\\bint{c})$, then there exists $(x^+, x^-) \\in \\mathcal{S}(A', \\bint{b}, \\bint{c}')$ with $x = x^+ - x^-$.\n\t\t\t\\item Conversely, if $(x^+, x^-) \\in \\mathcal{S}(A', \\bint{b}, \\bint{c}')$, then $x^+ - x^- \\in \\mathcal{S}(A,\\bint{b},\\bint{c})$.\n\t\t\\end{enumerate}\n\t\\end{theorem}\n\t\\begin{proof}\n\t\tLet $x \\in \\mathcal{S}(A,\\bint{b},\\bint{c})$ be an optimal solution for a scenario $c \\in \\bint{c}$, $b \\in \\bint{b}$. Decompose $x$ into the non-negative vectors $x^+ = \\max(0,x)$ and $x^- = -\\min(0,x)$, where the operations $\\max$ and $\\min$ are understood entry-wise. The vectors satisfy $x = x^+ - x^-$ and it is easy to see that the pair $(x^+, x^-)$ is optimal in ILP~\\eqref{eq:transfBC} for the scenario determined by the objective vector $(c, -c)$ and the right-hand side $b$.\n\t\t\n\t\tNow, let $(x^+, x^-) \\in \\mathcal{S}(A', \\bint{b}, \\bint{c}')$ be optimal for some $b \\in \\bint{b}$ and $c_1, c_2 \\in \\bint{c}$. By duality in linear programming, there exists a dual feasible vector $y$ satisfying\n\t\t\\begin{eqnarray*}\n\t\t\t&&c_1^T x^+ - c_2^T x^- = b^T y,\\\\\n\t\t\t&&A x^+ - Ax^- \\le b,\\; x^+ \\ge 0,\\; x^- \\ge 0,\\\\\n\t\t\t&&A^T y \\le c_1,\\; -A^T y \\le -c_2,\\; y \\le 0.\n\t\t\\end{eqnarray*}\n\t\tFrom the dual feasibility constraints, we have $A^T y = c_3$ for some $c_3 \\in [c_2, c_1] \\subseteq \\bint{c}$. We will prove that $x = x^+ - x^-$ is optimal for the scenario with objective vector $c_3$ and right-hand-side vector $b$ in ILP~\\eqref{eq:ilp:B}, by showing that it satisfies the system\n\t\t\\begin{equation*}\n\t\tc_3^T x = b^T y,\\; Ax \\le b,\\; A^T y = c_3,\\; y \\le 0.\n\t\t\\end{equation*}\n\t\tNamely, it remains to show that $c_3^T (x^+ - x^-) = b^T y$ holds. Complementary slackness implies that for each $i \\in \\{1, \\dots, n\\}$ we have\n\t\t\\begin{eqnarray*}\n\t\t\t&&(x^+)_i = 0\\;\\vee\\;(c_1-A^T y)_i = 0, \\text{ and}\\\\\n\t\t\t&&(x^-)_i = 0\\;\\vee\\;(A^T y-c_2)_i = 0.\n\t\t\\end{eqnarray*}\n\t\tIf $(x^+)_i > 0$ and $(x^-)_i > 0$ for some index $i$, then we have $(c_1)_i = (c_2)_i = (c_3)_i$ by the choice of $c_3$. For $(x^+)_i = 0$ and $(x^-)_i > 0$ we have $(c_2)_i = (c_3)_i$, and analogically, $(c_1)_i = (c_3)_i$ for the symmetric case. By substituting into the equation\n\t\t\\begin{equation*}\n\t\tc_1^T x^+ - c_2^T x^- = b^T y,\n\t\t\\end{equation*}\n\t\twe can see that the desired constraint is satisfied in all cases. Therefore, $x^+ - x^-$ is an optimal solution of program~\\eqref{eq:ilp:B}.\n\t\\end{proof}\n\t\n\tFigure~\\ref{fig:gen} provides an overview of the transformations preserving the feasible and optimal solution set of general interval linear programs, as presented in Section~\\ref{ssec:gen}. Note that ILPs of type~\\eqref{eq:ilp:C} are a special case of type~\\eqref{eq:ilp:B} and the transformation of type~\\eqref{eq:ilp:A} to type~\\eqref{eq:ilp:B} can be obtained by transitivity. As shown in Figure~\\ref{fig:spec}, all of the considered transformations preserve the optimal solution set for ILPs with a fixed coefficient matrix. Apart from the direct consequences of Theorem~\\ref{thm:spec:AC} and Theorem~\\ref{thm:spec:nonneg}, the remaining results follow again by transitivity.\n\t\n\t\\begin{figure}[b]\n\t\t\\begin{minipage}[t]{0.49\\textwidth}\n\t\t\t\\begin{tikzpicture}[scale=0.5]\n\t\t\t\\tikzstyle{sipka}=[-{>[length=7pt, width=5pt]}]\n\t\t\t\\tikzstyle{rect}=[draw,rectangle,minimum height=12ex,inner sep=3pt,text width=50pt, text height=2.25ex, align=center]\n\t\t\t\\node[rect] (a) at (0,4) {Type (I): \\small{$\\bint{A} x=\\bint{b}$, $x\\ge 0$}};\n\t\t\t\\node[rect] (b) at (-3.8,-1.2) {Type (II): \\small{$\\bint{A}x \\le \\bint{b}$}};\n\t\t\t\\node[rect] (c) at (3.8,-1.2) {Type (III): \\small{$\\bint{A}x \\le \\bint{b}$, $x \\ge 0$}};\n\t\t\t\n\t\t\t\\draw[sipka,dashed] (a.south west) -- (b.north);\n\t\t\n\t\t\t\\draw[sipka,dashed] (a.south east) -- (c.north);\n\t\t\t\\draw[sipka] ([xshift=-15pt]c.north) -- ([xshift=-15pt]a.south east);\n\t\t\n\t\t\t\\draw[sipka] ([yshift=-5pt]c.west) -- ([yshift=-5pt]b.east);\n\t\t\t\\node at (0,1) {$\\min\\,\\bint{c}^T x$};\n\t\t\t\\end{tikzpicture}\n\t\t\t\\caption{Transformations preserving the feasible (dashed arrows) and optimal (solid arrows) solution set of a general interval linear program.}\\label{fig:gen}\n\t\t\\end{minipage}\n\t\t\\hspace{0.02\\textwidth}\n\t\t\\begin{minipage}[t]{0.49\\textwidth}\n\t\t\t\\begin{tikzpicture}[scale=0.5]\n\t\t\t\\tikzstyle{sipka}=[-{>[length=7pt, width=5pt]}]\n\t\t\t\\tikzstyle{rect}=[draw,rectangle,minimum height=12ex,inner sep=3pt,text width=50pt, text height=2.25ex, align=center]\n\t\t\t\\node[rect] (a) at (0,4) {Type (I): \\small{$A x=\\bint{b}$, $x\\ge 0$}};\n\t\t\t\\node[rect] (b) at (-3.8,-1.2) {Type (II): \\small{$Ax \\le \\bint{b}$}};\n\t\t\t\\node[rect] (c) at (3.8,-1.2) {Type (III): \\small{$Ax \\le \\bint{b}$, $x \\ge 0$}};\n\t\t\t\n\t\t\t\\draw[sipka] (a.south west) -- (b.north);\n\t\t\t\\draw[sipka] ([xshift=15pt]b.north) -- ([xshift=15pt]a.south west);\n\t\t\t\\draw[sipka] (a.south east) -- node[above,sloped] {Thm.~\\ref{thm:spec:AC}} (c.north);\n\t\t\t\\draw[sipka] ([xshift=-15pt]c.north) --  ([xshift=-15pt]a.south east);\n\t\t\t\\draw[sipka] ([yshift=5pt]b.east) -- node[above] {Thm.~\\ref{thm:spec:nonneg}} ([yshift=5pt]c.west);\n\t\t\t\\draw[sipka] ([yshift=-5pt]c.west) -- ([yshift=-5pt]b.east);\n\t\t\t\\node at (0,1) {$\\min\\, \\bint{c}^T x$};\n\t\t\t\\end{tikzpicture}\n\t\t\t\\caption{Transformations preserving the optimal solution set of an interval linear program with a fixed coefficient matrix.}\\label{fig:spec}\n\t\t\\end{minipage}\n\t\\end{figure}\n\t\n\t\\section{Optimal Values under Transformations}\\label{sec:optval}\n\tLet us now discuss the effects of transformations on the set of all optimal values of an interval linear program and the optimal value range. Recall that the optimal value range refers to the interval $[\\lb{f}, \\ub{f}]$, which is the smallest interval that encloses all optimal values. This interval may differ from the set of optimal values, since not all values in the range have to be attained as optimal for a scenario. Again, we consider the two transformations that can possibly lead to a dependency problem: splitting equations into inequalities and substituting a difference of two non-negative variables for a free variable.\n\t\n\t\\subsection{General Case}\\label{ssec:gen:val}\n\tWe have already seen in Section~\\ref{ssec:gen} that splitting an equation into two opposite inequalities may change the optimal solution set of an interval linear program. This also holds for the optimal value range, namely, the transformation can change the worst-case bound ($\\ub{f}$ for a minimization program). Moreover, this is caused not only by creating an infeasible scenario resulting in $\\ub{f} = \\infty$, but can also happen due to an expansion of the set of finite optimal values.\n\t\\addtocounter{example}{-1}\n\t\\begin{example}[continued]\n\t\tThe optimal value range of the interval linear program\n\t\t\\begin{equation*}\n\t\t\\begin{array}{lr@{\\ }l}\n\t\t\\text{minimize } & \\multicolumn{2}{l}{-x_1} \\\\\n\t\t\\text{subject to } & [0,1]x_1 - x_2 = 0,\\\\\n\t\t&  x_2 \\le 1,\\\\\n\t\t&  x_1, x_2 \\ge 0.\n\t\t\\end{array}\n\t\t\\end{equation*}\n\t\tis the interval $(-\\infty, -1]$. However, by splitting the equation $[0,1]x_1 - x_2 = 0$ into two inequalities, the solution $(0,0)$ becomes optimal and, thus, the value~$0$ belongs to the set of optimal values (and the optimal value range) of the transformed program. This shows that $\\ub{f} = -1$ is no longer the worst-case optimal value.\n\t\\end{example}\n\t\n\tOn the other hand, we will now show that the transformation preserves the best-case optimal value~$\\lb{f}$. The proof is a consequence of a result on a unified approach for computing the optimal value range by~\\cite{Hladik:2009}. For the purposes of the following theorem, let us consider an interval linear program in the general form\n\t\\begin{equation*}\n\t\\begin{array}{lrrrr@{\\ }l}\n\t\\text{minimize } & \\multicolumn{5}{l}{\\bint{c_1}^T x_1 + \\bint{c_2}^T x_2} \\\\\n\t\\text{subject to } & \\bint{A}x_1 &+ & \\bint{B}x_2 & = & \\bint{b_1},\\\\\n\t&  \\bint{C}x_1 & + & \\bint{D}x_2 & \\le & \\bint{b_2},\\\\\n\t& &  \\multicolumn{2}{r}{x_1, x_2} &\\ge & 0.\n\t\\end{array}\n\t\\end{equation*}\n\tLet $\\bint{b} = (\\bint{b_1}, \\bint{b_2})$, $\\bint{c} = (\\bint{c_1}, \\bint{c_2})$ and let $\\mathcal{M}$ denote the set of all feasible solutions. Furthermore, let $\\mathcal{N}$ be the weakly feasible set of the dual ILP. Then, we can apply the formulas presented in Theorem~\\ref{thm:hladik:f} to compute the optimal value range.\n\t\n\t\\begin{theorem}[\\cite{Hladik:2009}]\\label{thm:hladik:f}\n\t\tWe have\n\t\t\\begin{equation}\\label{eq:lb}\n\t\t\\lb{f} = \\inf \\{c_c^T x - c_\\Delta^T \\lvert x \\rvert : x \\in \\mathcal{M}\\}.\n\t\t\\end{equation}\n\t\tIf $\\ub{f} < \\infty$, then\n\t\t\\begin{equation}\\label{eq:ub}\n\t\t\\ub{f} = \\sup \\{b_c^T y + b_\\Delta^T \\lvert y \\rvert : y \\in \\mathcal{N}\\}.\n\t\t\\end{equation}\n\t\\end{theorem}\n\t\n\tTheorem~\\ref{thm:optval:AC} proves that the transformation of splitting an equation into two opposite inequalities does not change the best-case optimal value of an interval linear program. The result implies that it is possible to convert an equation-constrained ILP of type~\\eqref{eq:ilp:A} into an inequality constrained ILP of type~\\eqref{eq:ilp:C}, while preserving the best optimal value.\n\t\n\t\\begin{theorem}\\label{thm:optval:AC} \n\t\tTransforming $\\bint{A}x = \\bint{b}$ into $\\bint{A}x \\le \\bint{b}, \\bint{A}x \\ge \\bint{b}$ does not change the best optimal value $\\lb{f}$ of a minimization ILP.\n\t\\end{theorem}\n\t\\begin{proof}\n\t\tBy Theorem~\\ref{thm:hladik:f}, the best optimal value $\\lb{f}$ of a general interval linear program can be found by minimizing the fixed objective function $c_c^T x - c_\\Delta^T \\lvert x \\rvert$ over the set of all weakly feasible solutions. Since the applied transformation of splitting an equation into two inequalities does not change the weakly feasible set of an interval system (see Theorem~\\ref{thm:Li}), the value of $\\lb{f}$ remains the same.\n\t\\end{proof}\n\t\n\tBy the property of strong duality in classical linear programming, we can also derive analogous results for the second considered transformation. For this case, let us first show that substituting the difference of two non-negative variables for a free variables can change the best-case bound $\\lb{f}$, as well as the set of optimal values.\n\t\n\t\\begin{example}\n\t\tConsider the dual interval linear program to~\\eqref{eq:ex}, which can be rewritten into a minimization form as\n\t\t\\begin{equation}\\label{eq:ex:2}\n\t\t\\begin{array}{lrrrr@{\\ }l}\n\t\t\\text{minimize } & \\multicolumn{5}{l}{-y_2} \\\\\n\t\t\\text{subject to } & [0,1]y_1 & & & \\le & -1,\\\\\n\t\t&  -y_1 & + & y_2 & \\le & 0,\\\\\n\t\t& &  \\multicolumn{2}{r}{y_2} &\\le & 0.\n\t\t\\end{array}\n\t\t\\end{equation}\n\t\tThe optimal value range of program~\\eqref{eq:ex:2} is the interval $[1, \\infty)$. Let us now substitute the term $y_1^+-y_1^-$ with $y_1^+, y_1^- \\ge 0$ for the free variable $y_1$. Analogously to the previous example, the set of optimal values changes and the best-case bound $\\lb{f} = 1$ is no longer valid (again, the value $0$ becomes optimal).\n\t\\end{example}\n\t\n\tHowever, we can show that the transformation preserves the worst-case bound $\\ub{f}$. The proof uses the notion of strong feasibility, which also provides a characterization of finiteness of the bound $\\ub{f}$, as stated in Theorem~\\ref{thm:strfeas}. An interval linear program is said to be \\emph{strongly feasible}, if each scenario of the program is feasible.\n\t\n\t\\begin{theorem}[\\cite{Hladik:2009}]\\label{thm:strfeas}\n\t\tAn interval linear program (in the general form) is strongly feasible if and only if $\\ub{f} < \\infty$.\n\t\\end{theorem}\n\t\n\t\\begin{theorem}\n\t\tSubstituting $x = x^+ - x^-$ with $x^+, x^- \\ge 0$ for a free variable $x$ does not change the worst optimal value $\\ub{f}$ of a minimization ILP.\n\t\\end{theorem}\n\t\\begin{proof}\n\t\tLet us denote by $\\ub{f}$ and $\\ub{f}^\\pm$ the worst-case optimal value of the original program and the transformed program created by the introducing the substitution, respectively. \n\t\t\n\t\tFirst, assume that $\\ub{f} = \\infty$. Since all of the original scenarios are also included in the transformed ILP, the latter is a relaxation of the original program and $\\ub{f} \\le \\ub{f}^\\pm$ holds. Therefore, we also have $\\ub{f}^\\pm = \\infty$.\n\t\t\n\t\tFurther, let $\\ub{f} < \\infty$ hold for the original interval program. Then, the program is strongly feasible. Since the conditions for strong feasibility of the original and the transformed program are equivalent for both equation and inequality constraints (see \\cite{Hladik:2017}), the latter is also strongly feasible and,  by Theorem~\\ref{thm:strfeas}, the property $\\ub{f}^\\pm < \\infty$ holds. By formula~\\eqref{eq:ub} of Theorem~\\ref{thm:hladik:f} we can calculate the worst-case optimal values by optimizing the objective function $b_c^T y + b_\\Delta^T \\lvert y \\rvert$ over the dual feasible set of the respective ILPs. Note that applying the substitution to a free variable in the primal ILP corresponds to splitting an equation into two opposite inequalities in the dual ILP. As this preserves the set of feasible solutions (Theorem~\\ref{thm:Li}), the dual feasible sets of the original and the transformed program are equal, and thus $\\ub{f} = \\ub{f}^\\pm$. \n\t\\end{proof}\n\t\n\t\\subsection{Special Case: Fixed Coefficient Matrix}\\label{ssec:spec:val}\n\tIn Section~\\ref{ssec:gen:val} we have seen that the optimal values and bounds of the optimal value range may change under some transformations. Therefore, it is natural to ask whether these properties are preserved at least for ILPs with a fixed coefficient matrix. Unfortunately, even for this special class of programs, the transformations may change the optimal value range, as shown by the following trivial examples.\n\t\n\t\\begin{example}\\label{ex:optval:fixed}\n\t\tConsider the following interval linear programs:\n\t\t\\vskip\\abovedisplayskip\n\t\t\\noindent\n\t\t\\begin{subequations}\n\t\t\t\\begin{minipage}{.5\\linewidth}\n\t\t\t\t\\begin{equation}\\label{eq:ex:3a}\n\t\t\t\t\\begin{array}{lrrrr@{\\ }l}\n\t\t\t\t\\text{minimize } & \\multicolumn{5}{l}{[0,1]x} \\\\\n\t\t\t\t\\text{subject to } & x & \\ge & 1,\\\\\n\t\t\t\t\\end{array}\n\t\t\t\t\\end{equation}\n\t\t\t\\end{minipage\n\t\t\t\\begin{minipage}{.5\\linewidth}\n\t\t\t\t\\begin{equation}\\label{eq:ex:3b}\n\t\t\t\t\\begin{array}{lrrrr@{\\ }l}\n\t\t\t\t\\text{minimize } & \\multicolumn{5}{l}{-y} \\\\\n\t\t\t\t\\text{subject to } & y & = & [0,1].\\\\\n\t\t\t\t\\end{array}\\qquad\n\t\t\t\t\\end{equation}\n\t\t\t\\end{minipage}\n\t\t\\end{subequations}\n\t\t\\vskip\\belowdisplayskip\\noindent\n\t\tThe optimal value range of ILP~\\eqref{eq:ex:3a} is the interval $[0,1]$, for ILP~\\eqref{eq:ex:3b} it is the opposite interval $[-1,0]$. By substituting $x^+-x^-$ with non-negative variables $x^+, x^-$ for the free variable $x$ in \\eqref{eq:ex:3a}, we introduce an unbounded scenario (setting the objective to $0x^+ - 1x^-$) and the best-case bound of the optimal value range changes to $\\lb{f} = -\\infty$.\n\t\tSimilarly, by splitting the equation in \\eqref{eq:ex:3b} into two opposite inequalities $y \\le [0,1], y \\ge [0,1]$, we create an infeasible scenario leading to $\\ub{f} = \\infty$.\n\t\\end{example}\n\t\n\tHowever, note that there is an important difference between Example~\\ref{ex:optval:fixed} and the previous examples with an interval coefficient matrix. While in the examples of Section~\\ref{ssec:gen:val} we have seen that a transformation may cause a change in the set of optimal values, in programs \\eqref{eq:ex:3a} and \\eqref{eq:ex:3b} the transformations only change one of the bounds in the optimal value range due to infeasibility or unboundedness of a newly introduced scenario.\n\t\n\tLet us now consider the set of all finite optimal values of an interval linear program with a fixed coefficient matrix. The following theorems prove that even though the optimal value range may still change when transforming a~program with a fixed matrix, this can only be caused by the infinite optimal values for infeasible or unbounded scenarios and the finite optimal values remain the same.\n\t\n\t\\begin{theorem}\\label{thm:spec:optval}\n\t\tTransforming $Ax = \\bint{b}$ into $Ax \\le \\bint{b}, Ax \\ge \\bint{b}$ does not change the set of all finite optimal values of an ILP with a fixed coefficient matrix.\n\t\\end{theorem}\n\t\\begin{proof}\n\t\tClearly, all optimal values of the original program remain optimal in the transformed program. Let a solution $x^*$ be optimal for a scenario given by $c \\in \\bint{c}$ and $b_1, b_2 \\in \\bint{b}$ in the transformed program. It is easy to see that $x^*$ is also optimal for the scenario determined by $c$ and $b_3 = Ax^*$ of the original program, since it has the same objective function and a restricted feasible set. Therefore, the optimal value $c^T x^*$ of the transformed ILP is also optimal for the original ILP.\n\t\\end{proof}\n\t\n\t\\begin{theorem}\n\t\tSubstituting $x = x^+ - x^-$ with $x^+, x^- \\ge 0$ for a free variable $x$ does not change the set of all finite optimal values of an ILP with a fixed coefficient matrix.\n\t\\end{theorem}\n\t\\begin{proof}\n\t\tBy strong duality of linear programming, the sets of finite optimal values of an interval linear program and its dual are the same. As introducing the substitution in the primal ILP corresponds to the transformation of splitting an equation into two inequalities in the dual ILP, applying Theorem~\\ref{thm:spec:optval} yields the result.\n\t\\end{proof}\n\t\n\t\\section{Conclusion}\\label{sec:concl}\n\tWe addressed the dependency problem in transforming interval linear programs using the techniques known from classical linear programming. We showed that while it is possible to switch the objective of an interval linear program or add slack variables to convert inequalities into equations, other transformations are not always applicable to interval programs without affecting some of their properties.\n\t\n\tNamely, we considered three commonly used forms of interval linear programs. It was shown that the set of all optimal solutions may change, in general, under the transformations among these forms. Therefore, we also studied a special class of interval programs with a fixed coefficient matrix, for which we proved that all of the transformations preserve the optimal set.\n\t\n\tFurthermore, we also studied the effect of the transformations on the set of optimal values and the optimal value range of an interval linear program. We proved that the best-case optimal value $\\lb{f}$ of a minimization program remains the same when an equation is split into two opposite inequalities, while the worst-case optimal value $\\ub{f}$ is preserved when substituting the difference of two non-negative variables for a free variable. The complementary results do not hold, even in the case of a fixed coefficient matrix. However, the set of all finite optimal values does not change for transformations on the special class of programs.\n\t\n\tThe results allow us to generalize the theory that was derived for a particular form of ILPs to all other types that can be obtained by a transformation respecting the studied properties. We believe that they also provide a better insight into the nature of the dependency problem in interval optimization.\n\t\n\t\n\n\n\n\n\t\n\n\n\n\n\\bibliographystyle{abbrv}     \n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe versatile technique of positron annihilation makes use of the fact that positrons ($e^+$) are trapped at  free \nvolume-type defects which allows their detection by a specific variation of the positron-electron annihilation characteristics \\cite{Hautojaervi79, Brossmann12, Krause-Rehberg99, Puska94}. \nWhereas the kinetics of $e^+$ trapping at vacancy-type point defects can be well described by \nrate theory (so-called simple trapping model), it is well known that for trapping at extended defects like\ngrain boundaries, interfaces, voids, clusters, or precipitates, diffusion limitation of the trapping process may be an issue. \nDiffusion-limited positron trapping at interfaces and grain boundaries has been quantatively modeled by several groups,\nranging from entirely diffusion-controlled trapping \\cite{Brandt72}, diffusion-reaction controlled trapping including detrapping\n\\cite{Dupasquier93, Wuerschum96, Koegel96, dryzek1999}, up to diffusion-reaction controlled trapping at grain boundaries and competetive transition-limited trapping at point defects in crystals \n\\cite{Cizek02, Oberdorfer09, dryzek1998, Koegel96}.\n\nCompared to grain boundaries, diffusion-limited $e^+$ trapping at voids and clusters has not been studied in such detail \ndespite the undoubted relevance of positron annihilation for studying this important class of defects \\cite{Nieminen79a, Bentzon90, Eldrup03}.\nOne approach to deal with diffusion-limited trapping is based on effective diffusion-trapping rates \nwhich then allow an implementation in standard rate theory  (e.g., \\cite{Bentzon90}).\nDiffusion-limited trapping at point-like defects was studied by Dryzek \\cite{dryzek1998diffusion}\nfor the one-dimensional case. A full treatment of $e^+$ trapping and annihilation in voids\nin the framework of diffusion-reaction theory was given by Nieminen et al. \\cite{Nieminen79}.\nThis treatment of Nieminen et al. \\cite{Nieminen79} is conceptionally analogous to the subsequent work of Dupasquier et al. \n\\cite{Dupasquier93} for diffusion-limited $e^+$ trapping at grain boundaries, both of which lead to solutions exclusively in terms of infinite series.\n\nAnother treatment  of the diffusion-reaction problem of $e^+$ trapping at grain boundaries\nwas given by W\\\"urschum and Seeger \\cite{Wuerschum96} which yields closed-form expressions\nfor the mean $e^+$ lifetime and the intensity of the annihilation component associated with \nthe trapped state. This approach is applied in the present work to the diffusion-reaction problem \nof $e^+$ trapping and annihilation in spherical extended  defects (voids, clusters, precipitates).\\footnote{For the sake of\nsimplicity, representatively for all kinds of  spherical extended defects (voids, clusters, or precipitates) \nthe term voids is used in the following.} \nFollowing our earlier further work on grain boundaries \n\\cite{Oberdorfer09}, now in addition competitive reaction rate-limiting trapping at point defects is taken into account.\nThe present  treatment yields closed-form expressions of the major $e^+$ annihilation parameters\nfor this application-relevant case of competitive\n$e^{+}$ trapping in voids and point defects.\nThese closed-form expressions allow deeper insight in the physical details of\n$e^+$ annihilation characteristics as well as an assessment of the so far often used  approach based on\neffective diffusion-trapping rates. Above all, the results can be conveniently applied for\nthe analysis of experimental data.\n\nIn a further part, the model presented here and the previous model on positron trapping at grain boundaries\nare merged in order to study precipitates embedded in matrix.  \nHere, diffusion- and reaction limited trapping is considered for both the trapping\nfrom the matrix into the  precipitate$-$matrix interface and for  \nthe trapping from inside the precipitates into the interfaces.\n\n\\section{The Model}\nThe model describes positron ($e^+$) trapping and annihilation in  voids in the general case that both the $e^+$ diffusion and the transition reaction has to be taken into account (so called diffusion-reaction controlled trapping process).\nIn order to cover more complex cases, competitive transition-limited trapping at vacancy-type points defects is also considered\n(see Fig. \\ref{fig:1}).\nThis procedure follows our earlier study \nwhere concomitant positron trapping at grain boundaries and at point defects in crystallites has been considered \\cite{Oberdorfer09}.\n\n\n\nThe behavior of the positrons is described by their bulk (free) lifetime $\\tau_f$,  \nby their lifetime ($\\tau_t)$ in the  voids, \nby their lifetime ($\\tau_v)$ in the vacancy-type point defects in the lattice (matrix),\nand by their bulk diffusivity $D$.\nTrapping at the point defects of the matrix is characterized by the specific $e^+$ trapping rate $\\sigma_v$ (unit s$^{-1}$), as usual.\nThe  voids are considered as spherical-shaped extended defects (radius $r_0$) with a specific trapping rate $\\alpha$ (unit m\\,s$^{-1}$) which is related to the surface area of the void. In units of s$^{-1}$ the specific trapping rate of voids reads\n\\begin{equation}\n\\label{eq:sigma_t}\n\\sigma_t= \\frac{\\alpha 4\\pi r_0^2}{\\Omega},\n\\end{equation}\nwhere $\\Omega$ denotes the atomic volume.\n \nThe temporal and spatial evolution of the density  $\\rho_l$ of free positrons in the lattice is governed by:\n\\begin{equation}\n\\label{differential_eq}\n\\frac{\\partial \\rho_l}{\\partial t}=\nD\\nabla^{2}\\rho_l-\\rho_l\\left(\\frac{1}{\\tau_f}+\n\\sigma_v C_v\\right)\n\\end{equation}\nwhere $C_v$ denotes the concentration of vacancy-type point defects in the matrix.\nThe positrons trapped in the voids are described in terms of their density $\\rho_t$ obeying the rate equation\n\\begin{equation}\n\\label{eq:rho_t}\n\\frac{\\mathrm{d}\\rho_t}{\\mathrm{d}t}=\n\\alpha \\rho_l(r_{0},t)-\\frac{1}{\\tau_t}\\rho_t.\n\\end{equation}\nThe temporal evolution of the number $N_v$ of $e^+$ trapped in the point defects in the lattice is given by\n\\begin{equation}\n\\frac{\\mathrm{d}N_v}{\\mathrm{d}t}=\n-\\frac{1}{\\tau_v} N_v+\\sigma_v C_v N_f \\, ,\n\\end{equation}\nwhere the number $N_f$ of positrons in the free state follows from integration of $\\rho_l$:\n\\begin{equation}\nN_f=\\int \\rho_l \\mathrm{d}V.\n\\end{equation}\n\n\nThe continuity of the ${\\rm e}^{+}$ flux at the boundary between the lattice and the void surface is expressed by\\footnote{Note the negative sign in contrast to the model of $e^+$ trapping at grain boundaries (e.g., \\cite{Oberdorfer09}) where the corresponding continuity equation refers to the outer boundary.}\n\\begin{equation}\nD\\nabla \\rho_l\\Big|_{r=r_{0}}-\\alpha \\rho_l(r_{0},t)=0.\n\\end{equation}\n\nThe outer radius $R$ of the diffusion sphere is related to the void concentration \n\\begin{equation}\n\\label{eq:C_t}\nC_t = \\frac{3 \\Omega}{4 \\pi R^3} \\, .\n\\end{equation}\nThe outer boundary condition\n\\begin{equation}\n\\frac{\\partial \\rho_l}{\\partial r}\\Big|_{r=R}=0\n\\end{equation}\nreflects the vanishing $e^+$ flux through the outer border ($r = R$) of the diffusion sphere. \nThis boundary condition is the same as applied earlier in a quite\ndifferent diffusion-reaction model of ortho-para conversion of positronium at reaction cerntres \\cite{Wuerschum95}.\n\nAs initial condition we adopt the picture that at $t=0$ all thermalized positrons are in the free state and homogeneously distributed in the lattice, i.e., initial density $\\rho_l = \\rho_l (0)$,  $\\rho_t(0)=0$, $N_v (0) = 0$.\nUnder this initial condition the solution of Eq. (\\ref{differential_eq}) exhibits spherical symmetry.\n\nUp to this point the above formulated diffusion-reaction problem is identical to that of Nieminen et al. \\cite{Nieminen79}\napart from the additional rate-limited trapping at vacancy-type point defects which is considered here.\nHowever, compared to \\cite{Nieminen79}, in the following part of the present work the time dependence is handled by means of Laplace transformation which will lead to the more convenient\nclosed-form solutions. Applying the Laplace transformation \n\\begin{eqnarray}\n\\nonumber\n\\tilde{\\rho}_{l,t}(p)=\\int\\limits _{0}^{\\infty}\\exp(-pt)\\rho_{l,t}(t)\\mathrm{d}t,\n\\end{eqnarray}\n\\begin{equation}\n\\tilde{N}_{v,f}(p)=\\intop_{0}^{\\infty}\\exp(-pt)N_{v,f}(t)\\mathrm{d}t\n\\end{equation}\nleads to the basic equations\n\\begin{equation}\n\\label{differential_eq_r}\n\\frac{\\mathrm{d^{2}}\\tilde{\\rho}_l}{\\mathrm{d}r^{2}}+\\frac{2}{r}\n\\frac{\\mathrm{d}\\tilde{\\rho}_l}{\\mathrm{d}r}-\\gamma^{2}\\tilde{\\rho}_l\n=-\\frac{\\rho_l(0)}{D}\n\\end{equation}\nwith\n\\begin{eqnarray}\n\\label{eq:gamma}\n\\gamma^{2}=\\gamma^{2}(p)=\\frac{\\tau_f^{-1}+\\sigma_v C_v+p}{D} \\, ,\n\\end{eqnarray}\nand\n\\begin{equation}\n\\label{tilde_rho_t}\n\\tilde{\\rho}_t=\n\\frac{\\alpha \\tilde{\\rho}_l(r_{0},p)}{\n\\tau_{t}^{-1}+p} \\: ,\n\\end{equation}\n\\begin{equation}\n\\label{N_v}\n\\tilde{N}_v=\n\\frac{\\sigma_v C_v}{\\tau_v^{-1}+p} \\times \\int\\limits^{R}_{r_0} 4 \\pi r^2 \\tilde{\\rho_l}(r,p) {\\rm d} r\\: ,\n\\end{equation}\nwith the boundary conditions\n\\begin{equation}\n\\label{boundary_condition1}\nD\\,\\frac{\\mathrm{d}\\tilde{\\rho}_l}{\\mathrm{d}r}\\Bigg|_{r=r_{0}}-\n\\alpha \\tilde{\\rho}_l(r_{0},p)=0\n\\end{equation}\nand\n\\begin{equation}\n\\label{boundary_condition2}\n\\frac{\\mathrm{d}\\tilde{\\rho}_l}{\\mathrm{d}r}\\Bigg|_{r=R}=0 \\, .\n\\end{equation}\n\nThe solution of the differential equation (\\ref{differential_eq_r}) satisfying equations\n[Eq. (\\ref{boundary_condition1})] and [Eq. (\\ref{boundary_condition2})] can be written as\n\\begin{equation}\n\\label{solution_rho_tilde}\n\\tilde{\\rho}_l(r,p)=A\\, i_{0}^{(1)}(\\gamma r)+ B\\, i_{0}^{(2)}(\\gamma r) +\n\\frac{\\rho_l(0)}{\\tau_f^{-1}+\\sigma_v C_v + p}\n\\end{equation}\nwith\n\\begin{eqnarray}\n\\nonumber\nA:=& \\displaystyle{\\alpha~\\frac{\\rho_l(0)}{\\tau_f^{-1}+\\sigma_v C_v+p}~\\times~\\frac{i_{1}^{(2)}(\\gamma R)}{-D \\gamma F_1 + \\alpha F_2}} \\, , \\\\\n\\label{AB}\nB:= & \\displaystyle{\\alpha~\\frac{\\rho_l(0)}{\\tau_f^{-1}+\\sigma_v C_v+p}~\\times~\\frac{i_{1}^{(1)}(\\gamma R)}{D \\gamma F_1 - \\alpha F_2}}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\nonumber\nF1= i_1^{(2)}(\\gamma r_0) i_1^{(1)}(\\gamma R) - i_1^{(1)}(\\gamma r_0)  i_1^{(2)}(\\gamma R) \\, , \\\\\nF2= i_0^{(2)}(\\gamma r_0)  i_1^{(1)}(\\gamma R) - i_0^{(1)}(\\gamma r_0)  i_1^{(2)}(\\gamma R) \\, .\n\\end{eqnarray}\n\n \n$i_{n}^{(1)}$ and $i_{n}^{(2)}$ ($n = 0,1$) denote the modified spherical Bessel functions of order $n$\n\\cite{olver2010nist}\n\\begin{eqnarray}\n\\nonumber\ni_{n}^{(1)}(z):=(\\frac{\\pi}{2z})^{1\/2}I_{n+1\/2}(z), \\,     \n\\end{eqnarray}\n\\begin{equation}\ni_{0}^{(1)}=\\frac{\\sinh z}{z},\\quad                        \ni_{1}^{(1)}=\\frac{\\cosh z}{z}-\\frac{\\sinh z}{z^2}\n\\end{equation}\n\\begin{eqnarray}\n\\nonumber\ni_{n}^{(2)}(z):=(\\frac{\\pi}{2z})^{1\/2}I_{-n-1\/2}(z), \\,    \n\\end{eqnarray}\n\\begin{equation}\ni_{0}^{(2)}=\\frac{\\cosh z}{z},\\quad                        \ni_{1}^{(2)}=\\frac{\\sinh z}{z}-\\frac{\\cosh z}{z^2}\n\\end{equation}\nwhere $I_{\\pm n\\pm 1\/2}(z)$ represents the Bessel function.\n\nBasis for analyzing positron annihilation experiments is the total probability $n(t)$ that a\n$e^+$ implanted at $t=0$ has not yet been annihilated at time $t$.\nHere $n(t)$ is given by the number density of $e^+$ per lattice sphere at time\n$t$:\n\\begin{equation}\nn(t)= \\frac{1}{\\frac{4}{3}\\pi (R^3 - r_{0}^{3})\\rho_l(0)} \\times\n\\left\\{\n\\int\\limits _{r_{0}}^{R}4\\pi r^{2}\\rho_l(r,t)\\mathrm{d}r\n+4\\pi r_{0}^{2}\\rho_t (t) + N_v (t)\n\\right\\}       \\:.\n\\end{equation}\nThe Laplace transform of $n(t)$ can be calculated taking into account the solution of  $\\tilde{N}_v$ [Eq. (\\ref{N_v})]\nand the solution of the differential equation (\\ref{solution_rho_tilde}) which yields\n\\begin{equation}\n\\tilde{n}(p)=\\frac{1}{\\frac{4}{3}\\pi (R^3-r_{0}^{3})\\rho_l(0)}\n\\times\n\\left\\{\n \\left(1+\\frac{\\sigma_v C_v}{\\tau_v^{-1}+p}\\right)\n \\int\\limits^{R}_{r_0} 4\\pi r^{2} \\tilde{\\rho}_l(r,p) {\\rm d} r\n+ 4\\pi r^{2}_{0} \\tilde{\\rho}_t(p)\n\\right\\}       \\:.\n\\end{equation}\n\nSolving the integral after substituting $\\tilde{\\rho_t}(p)$ by Eq. (\\ref{tilde_rho_t}),  \ninsertion of $A$ and $B$ [Eq. (\\ref{AB})],\nyields after some algebra\n\n\\begin{equation}\n\\label{Laplace_n}\n\\tilde{n}(p)=\\frac{1}{t_{fc}^2 t_{v}  t_{t}} \\Biggl\\{t_{vc} t_{fc}  t_{t} + \\frac{K (t_{fc}  t_{v} - t_{vc}  t_{t})\n\\Bigl(\\gamma \\hat{R} - \\tanh(\\gamma \\hat{R}) [1- \\gamma^2 r_0 R]\\Bigr)}\n{\\gamma \\hat{R}- \\tanh(\\gamma \\hat{R}) [1-\\gamma^2 r_0 R] + \\frac{\\alpha r_0}{D}[\\gamma R - \\tanh(\\gamma \\hat{R})]} \\Biggr\\} \n\\end{equation}\nwith \n\\begin{equation}\n\\label{eq:K}\nK=\\frac{3\\alpha r_0^2}{R^3-r_0^3} \\, ,\n\\end{equation}\n\\begin{equation}\n\\label{eq:R}\n\\hat{R} = R-r_0 \\, ,\n\\end{equation}\nand the abbreviations\n\\begin{eqnarray}\n\\nonumber\nt_{t}=\\tau_t^{-1}+p \\, ; & t_{v}=\\tau_v^{-1}+p \\,; \\\\\nt_{vc}=\\tau_v^{-1}+\\sigma C+p \\, ; & t_{fc}=\\tau_f^{-1}+\\sigma C+p \\, .\n\\end{eqnarray}\nThe Laplace transform $\\tilde{n}(p)$ [Eq. (\\ref{Laplace_n})] represents the\nsolution of the present diffusion and trapping model from which\nboth the mean positron lifetime and the positron lifetime spectrum\ncan be deduced.\nThe mean positron lifetime $\\overline{\\tau}$ is obtained  by\ntaking the Laplace  transform  at $p = 0$:\n\\begin{equation}\n\\label{tauq_n_tilde}\n\\overline{\\tau} = \\tilde{n}(p=0) =  \\int\\limits^{\\infty}_{0}  n(t)\ndt \\, .\n\\end{equation}\nThe positron lifetime spectrum follows from  $\\tilde{n}(p)$ by means of Laplace inversion. The single poles $p = - \\lambda_i$\nof  $\\tilde{n}(p)$ in the complex $p$ plane define the decay rates $\\lambda_i (i=0,1,2,\\dots)$ of the positron lifetime spectrum:\n\\begin{equation}\n\\label{spectrum}\nn(t) = \\sum_{i=0}^{\\infty}I_i \\exp (-\\lambda_i t)  \\, ,\n\\end{equation}\nwhere $I_i$ denote the relative intensities.\n\n\n\n\\section{Analysis}\nAt first, we consider the most important case that $e^+$ trapping  exclusively occurs at  voids, i.e., \nwe omit $e^+$ trapping at point defects in the lattice ($C_v=0$).\nFor this case, we present the solution of the general diffusion-reaction theory (Sect.~\\ref{sec:general}) and compare \nit with the limiting cases of entirely reaction-controlled trapping (Sect.~\\ref{sec:rate_limit})\nand entirely diffusion-controlled trapping (Sect.~\\ref{sec:diffusion_limit}). Finally, the case of competitive \nreaction-controlled trapping at lattice defects is considered (Sect.~\\ref{sec:vacancies})\nand an extension to larger precipitates is presented for describing precipitate$-$matrix composite structures\n(Sect.~\\ref{sec:extended}).\n\n\\subsection{\\label{sec:general}\nGeneral case with trapping at voids, exclusively ($C_v=0$)}\n\nFor negligible trapping at vacancies within the lattice ($C_v=0$), \nthe diffusion-reaction model according to Eq.~(\\ref{Laplace_n})  yields\nfor positron trapping in voids as the single type of trap:\n\\begin{equation}\n\\label{eq:n}\n\\tilde{n}(p) = \\frac{1}{\\tau_f^{-1}+p} \\Biggl\\{ 1+\\frac{K(\\tau_f^{-1}-\\tau_t^{-1})}{(\\tau_t^{-1}+p)(\\tau_f^{-1}+p)} \\times \n\\frac{\\gamma \\hat{R}-\\tanh(\\gamma \\hat{R}) [1-\\gamma^2 r_0 R]}{\\gamma \\hat{R}-\\tanh(\\gamma \\hat{R}) [1-\\gamma^2 r_0 R]+ \\frac{\\alpha r_0}{D}[\\gamma R- \\tanh(\\gamma \\hat{R})]}\n \\Biggr \\} \n\\end{equation}\nand, hence, for the mean positron lifetime \n\\begin{equation}\n\\label{eq:tauq}\n\\overline{\\tau}=\\tilde{n}(0)= \\tau_f \\Biggl\\{1+ K (\\tau_t-\\tau_f) \\times \n\\frac{\\gamma_0 \\hat{R}-\\tanh(\\gamma_0 \\hat{R}) [1-\\gamma_0^2 r_0 R]}{\\gamma_0 \\hat{R}-\\tanh(\\gamma_0 \\hat{R}) [1-\\gamma_0^2 r_0 R]+ \\frac{\\alpha r_0}{D}[\\gamma_0 R- \\tanh(\\gamma_0 \\hat{R})]}\n \\Biggr \\} \\, .\n\\end{equation}\nThe pole of Eq.~(\\ref{eq:n}) for $p = - \\tau_t^{-1}$ corresponds to the positron lifetime  component $\\tau_t$ \nof the void-trapped state for which the following intensity is obtained:\n\\begin{equation}\n\\label{eq:I_t}\nI_t= \\frac{K}{\\tau_f^{-1}-\\tau_t^{-1}}\\times \n\\frac{\\gamma_t \\hat{R}-\\tanh(\\gamma_t \\hat{R}) [1-\\gamma_t^2 r_0 R]}{\\gamma_t \\hat{R}-\\tanh(\\gamma_t \\hat{R}) [1-\\gamma_t^2 r_0 R]+ \\frac{\\alpha r_0}{D}[\\gamma_t R- \\tanh(\\gamma_t \\hat{R})]} \\,.\n\\end{equation}\nIn equations (\\ref{eq:n}), (\\ref{eq:tauq}), (\\ref{eq:I_t}):\n\\begin{equation}\n\\label{eq:gamma_0_t}\n\\gamma^2=\\frac{\\tau _f^{-1}+p}{D}; \\, \n \\gamma_0^2=\\frac{\\tau _f^{-1}}{D}; \\, \n \\gamma_{t}^2=\\frac{\\tau _f^{-1}-\\tau_{t}^{-1}}{D} \\, .\n\\end{equation}\n\nIn addition to the annihilation component $\\tau_t^{-1}$ of the void-trapped state, \n$\\tilde{n}(p)$ [Eq.~(\\ref{eq:n})] comprises a sequence of first-order poles $p = - \\lambda_{0,j}$ for \n$\\lambda_{0,j} > \\tau_f^{-1}$. These  components $\\lambda_{0,j}$,  \nwhich define the fast decay rates ($\\lambda_{0,j} > \\tau_f^{-1}$) of the $e^+$ lifetime spectrum, \nare given by the solutions of the transcendental equation \n\\begin{equation}\n\\tan(\\gamma^{\\star} \\hat{R}) = \\frac{\\gamma^{\\star} (\\alpha r_0 R + D \\hat{R})}{D(1+\\gamma^{\\star 2} r_0 R) + \\alpha r_0}\n\\label{eq:transcendent}\n\\end{equation}\nwith\n\\begin{equation}\n\\gamma^{\\star 2} = \\frac{\\lambda_{0,j}-\\tau_f^{-1}}{D} \n\\label{eq:gamma'}\n\\end{equation}\nin agreement with the aforementioned earlier work of Nieminen et al. \\cite{Nieminen79}.\\footnote{Eq. (\\ref{eq:transcendent}) \nis identical to the corresponding eq. (15) in the work of Nieminen et al. when $\\nu$ in \\cite{Nieminen79}\nis identified with $4\\pi r_0^2 \\alpha$.}$^,$\\footnote{We note that   \nthe same problem was treated in the framework of a more general theoretical approach by K\\\"ogel \\cite{Koegel96}. The \nquoted specific function  in dependence of $\\gamma \\hat{R}$ [eq. (75) in \\cite{Koegel96}], which\ndetermines the mean $e^+$ lifetime and the intensity of the trap component,\n however, is not readily applicable.\n}\nAs usual for this kind of diffusion-reaction problem (see, e.g. \\cite{Oberdorfer09}),  \nthe intensities of these decay rates rapidly decrease.\nExperimentally only a single fast decay rate can be resolved in addition to the decay rate $\\tau_t^{-1}$ of the trapped state. An experimental two-component $e^+$ lifetime spectrum is practically entirely  defined by \n$\\overline{\\tau}$ [Eq.~(\\ref{eq:tauq})] and by $\\tau_t$ with the corresponding intensity $I_t$ [Eq.~(\\ref{eq:I_t})].\n\nThe appearance of a second-order pole in Eq.~(\\ref{eq:n}) at $p = - \\tau_f^{-1}$ (i.e., $\\gamma = 0$) is spurious. Closer inspection by applying Taylor expansion shows that the intensity associated with this pole cancels.\n\n\nFollowing the consideration of Dryzek \\cite{dryzek2002}, in analogy to the mean $e^+$ lifetime [Eq. (\\ref{eq:tauq})]\na respective relation for the mean line shape parameter $\\overline{S}$ of \nDoppler broadening of the positron-electron annihilation\ncan be given:\n\\begin{equation}\n\\label{eq:Doppler}\n\\overline{S}=  S_f \\Biggl\\{1+ K (S_t-S_f) \\times \n\\frac{\\gamma_0 \\hat{R}-\\tanh(\\gamma_0 \\hat{R}) [1-\\gamma_0^2 r_0 R]}{\\gamma_0 \\hat{R}-\\tanh(\\gamma_0 \\hat{R}) [1-\\gamma_0^2 r_0 R]+ \\frac{\\alpha r_0}{D}[\\gamma_0 R- \\tanh(\\gamma_0 \\hat{R})]}\n \\Biggr \\} \\, ,\n\\end{equation}\nwhere $S_f$ and $S_t$ denote the line shape parameters of the free and trapped state, respectively.\n\nFor the sake of completeness, we quote $\\tilde{n}(p)$ without derivation for the case that \nat time zero positrons are homogeneously distributed in the voids and the lattice, \ni.e., for the initial condition $\\rho_t (0) = r_0 \\rho_l (0)\/3$: \n\\begin{eqnarray}\n\\label{eq:n_homogen}\n\\nonumber\n\\tilde{n}(p) =  \\displaystyle{\\frac{1}{\\tau_f^{-1}+p} \\Biggl\\{ 1+ \\frac{r_0^3}{R^3} \\times\n\\frac{\\tau_f^{-1}-\\tau_t^{-1}}{\\tau_t^{-1} + p} +}\n \\frac{3 \\alpha r_0^2}{R^3} \\times \\frac{\\tau_f^{-1}-\\tau_t^{-1}}{(\\tau_t^{-1}+p)(\\tau_f^{-1}+p)} \n\\times \\\\\n \\displaystyle{\\frac{\\gamma \\hat{R}-\\tanh(\\gamma \\hat{R}) [1-\\gamma^2 r_0 R]}{\\gamma \\hat{R}-\\tanh(\\gamma \\hat{R}) [1-\\gamma^2 r_0 R]+ \\frac{\\alpha r_0}{D}[\\gamma R- \\tanh(\\gamma \\hat{R})]}\n \\Biggr \\}} \\, .\n\\end{eqnarray}\nEq.~(\\ref{eq:n_homogen}) includes in the limiting case  of negligible trapping ($\\alpha = 0$)\nas  mean $e^+$ lifetime $\\overline{\\tau} = \\tilde{n}(0) = \n[(R^3-r_0^3) \\tau_f + r_0^3 \\tau_t]\/R^3$ the expected volume-averaged mean value of $\\tau_f$ and $\\tau_t$.  \n\n\n\\subsection{\\label{sec:rate_limit}\nLimiting case of entirely reaction limited trapping ($C_v=0$)}\n\nIf the e$^+$ diffusivity is high ($\\gamma \\hat{R} \\ll 1$), the hyperbolic tangent \nin Eq.~(\\ref{eq:n})  can be expanded.\nExpansion up to the third order \n\\begin{equation}\n\\tanh(z)\\approx z-\\frac{z^3}{3}\n\\end{equation}   \nyields the mean $e^+$ lifetime \n\\begin{equation}\n\\label{eq:tauq_rate}\n\\overline{\\tau}=\\displaystyle{\n\\tau_f \\frac{1+ K \\tau_t}{1+K \\tau_f}}\n\\end{equation}\nand for the $e^+$ lifetime component $\\tau_t$ the intensity \n\\begin{equation}\n\\label{eq:I_t_rate}\nI_t=\\displaystyle{\n\\frac{K }{\\tau_f^{-1}+ K -\\tau_t^{-1}}\n}\n\\end{equation}\nwith $K$ according to Eq.~(\\ref{eq:K}).\nEquations (\\ref{eq:tauq_rate}) and (\\ref{eq:I_t_rate}) are the well-known solutions of the simple trapping model\nwhen we identify $K$ for vanishing defect volume with the trapping rate $\\sigma_t C_t$ \n[equations (\\ref{eq:sigma_t}) and (\\ref{eq:C_t})].\nNote that the standard trapping model does not take into account the finite defect volume (here $4 \\pi r_0^3\/3$)\nand, therefore, does not contain the subtrahend $r_0^3$ as in Eq. (\\ref{eq:K}). With this subtrahend,\nequations (\\ref{eq:tauq_rate}) and (\\ref{eq:I_t_rate}) correctly contain \nthe exact values  $\\overline{\\tau} = \\tau_t$ and $I_t = 1$ \nas limiting case for $R=r_0$.\n\n\n\\subsection{\\label{sec:diffusion_limit}\nLimiting case of entirely diffusion limited trapping ($C_v=0$)}\n\nThe present solution includes in the limiting special case $\\alpha \\rightarrow \\infty$ the relationships\nfor an entirely diffusion-limited trapping, i.e., for Smoluchowski-type boundary condition \n\\begin{equation}\n\\label{eq:Smoluchowski}\n\\rho_l (r_0, t) = 0 \\, .\n\\end{equation}\nIn this limit one obtains from the Laplace transform [Eq. (\\ref{eq:n})]\nthe mean $e^+$ lifetime \n\\begin{equation}\n\\label{eq:tauq_diffusion}\n\\overline{\\tau}=\\tau_f \\Biggl \\{ 1+ \\frac{3r_{\\rm 0}D}{R^3-r_{\\rm 0}^3}(\\tau_t-\\tau_f) \n\\frac{\\gamma_0 \\hat{R}-\\tanh(\\gamma_0 \\hat{R})[1-\\gamma_0^2 r_0 R]}{\\gamma_0 R-\\tanh(\\gamma_0 \\hat{R})} \\Biggr\\}\n\\end{equation}\nand for the trap component $\\tau_t$ the intensity\n\\begin{equation}\n\\label{eq:I_t_diffusion}\nI_t=\\frac{3 r_0 D}{R^3-r_0^3} \\times \\frac{1}{\\tau_f^{-1}-\\tau_t^{-1}} \\times \\frac{\\gamma_t \\hat{R}-\\tanh(\\gamma_t \\hat{R})[1-\\gamma_t^{2}r_0 R]}{\\gamma_t R-\\tanh(\\gamma_t \\hat{R})}\n\\end{equation}\nwith $\\gamma_0$, $\\gamma_t$ according to eq. (\\ref{eq:gamma_0_t}).\n\n\n \\subsection{\\label{sec:vacancies}\nGeneral case with voids {\\em and} lattice vacancies}\n\nThe positron annihilation characteristics of diffusion-reaction\ncontrolled trapping at voids and concomitant transition-limited\ntrapping at point defects in the lattice is given by Eq.\n(\\ref{Laplace_n}) in combination with Eq. (\\ref{tauq_n_tilde}) and\nEq. (\\ref{spectrum}).\nThe mean positron lifetime [Eq. \\ref{tauq_n_tilde}], obtained from\nEq. (\\ref{Laplace_n}) for $p=0$,  reads in the general case:\n\\begin{eqnarray}\n\\label{eq:tau_q_general}\n\\nonumber\n\\overline{\\tau}=  \\displaystyle{\n\\frac{1}{(\\tau_f^{-1}+\\sigma_v C_v)^{2}} }\n\\Biggl\\{(\\tau_f^{-1}+\\sigma_v C_v)  (\\tau_v^{-1}+\\sigma_v C_v) \\tau_v+  \\\\\n\\displaystyle{\n\\frac{K \\Bigl((\\tau_f^{-1}+\\sigma_v C_v)  \\tau_{t} - (\\tau_v^{-1}+\\sigma_v C_v)  \\tau_{v} \\Bigr)\\Bigl(\\gamma_0 \\hat{R} - \\tanh(\\gamma_0 \\hat{R})[1- \\gamma^2 r_0 R]\\Bigr)}\n{\\gamma_0 \\hat{R}- \\tanh(\\gamma_0 \\hat{R})[1-\\gamma_0^2 r_0 R] + \\frac{\\alpha r_0}{D}[\\gamma_0 R - \\tanh(\\gamma_0 \\hat{R})]}\\Biggr\\}\n}\n\\end{eqnarray}\nwith\n\\begin{equation}\n\\label{eq:gamma_v}\n\\gamma_0^2=\\frac{\\tau_f^{-1}+\\sigma_v C_v}{D} \\, .\n\\end{equation}\n\nIn addition to the pole $p = - \\tau_t^{-1}$ which characterizes the void trapped state,\n$\\tilde{n}(p)$ [Eq. (\\ref{Laplace_n})] contains the further defect-related pole  \n$p = - \\tau_v^{-1}$ for the vacancy-type defect in the lattice.\nFrom the residues of  $\\tilde{n}(p)$ [Eq. (\\ref{Laplace_n})], \nthe corresponding relative intensities\n\\begin{equation}\n\\label{eq:I_t_general}\nI_t=\\frac{K}{\\tau_f^{-1}+\\sigma_v C_v-\\tau_t^{-1}}\\times\\frac{\\gamma_t \\hat{R} - \\tanh(\\gamma_t \\hat{R})[1- \\gamma_t^2 r_0 R]}{\\gamma_t \\hat{R}- \\tanh(\\gamma_t \\hat{R})[1-\\gamma_t^2 r_0 R] + \n\\frac{\\alpha r_0}{D}[\\gamma_t R - \\tanh(\\gamma_t \\hat{R})]}\n\\end{equation}\nand \n\\begin{eqnarray}\n\\nonumber\nI_v=\n\\frac{\\sigma_v C_v}{\\tau_f^{-1}+\\sigma_v C_v-\\tau_v^{-1}}  \\Biggl\\{1- \\frac{K}{\\tau_f^{-1}+\\sigma_v C_v-\\tau_v^{-1}} \\times \\\\\n\\label{eq:I_v} \n\\frac{\\gamma_v \\hat{R} - \\tanh(\\gamma_v \\hat{R})[1- \\gamma_v^2 r_0 R]}{\\gamma_v \\hat{R}- \\tanh(\\gamma_v \\hat{R})[1-\\gamma_v^2 r_0 R] + \\frac{\\alpha r_0}{D}(\\gamma_v R - \\tanh(\\gamma_v \\hat{R})]}\\Biggr\\}\n\\end{eqnarray}\nare deduced with \n\\begin{equation}  \n\\label{eq:gamma_t_v}\n\\gamma_{t,v}^2=\\frac{\\tau _f^{-1}+\\sigma_v C_v-\\tau_{t,v}^{-1}}{D} \\, .\n\\end{equation}\n\n\n\\subsection{\\label{sec:extended}\nExtended model for larger preciptates  with  $e^+$-trapping from both sides of precipitate$-$matrix interface}\nThe model presented above describes $e^+$ annihilation from a trapped state ($\\tau_t$)\nin spherical defects. Particularly, for larger precipitate sizes a situation may prevail where \n$e^+$ annihilation inside the precipitates occurs from a free state with a characteristic $e^+$ lifetime $\\tau_p$\nand where also from this free precipitate state positrons may  get trapped into the spherical interfacial shell between the precipitate and the surrounding matrix. This means that the precipitates are characterized by two compoments, one corresponding to the precipitate volume ($\\tau_p$) and one corresponding to the trapped state in the matrix$-$precipitate interface ($\\tau_t$).\n\n\nThe present model can be extended in a straight forward manner to this case under the reasonable assumption that the $e^+$ trapping from inside the precipitates is entirely reaction controlled. This is pretty well fulfilled as long as\nthe precipitate diameter is remarkably lower than the $e^+$ diffusion length in the precipitate.\\footnote{A further model extension avoiding this\nconstraint will be outlined below.} \nIn this case the extension can be described by an additional rate equation for the temporal evolution of the number $N_p$ of $e^+$ \ninside the precipitates \n\\begin{equation}\n\\label{eq:N_p}\n\\frac{\\mathrm{d}N_p}{\\mathrm{d}t}=\n-\\Bigl( \\frac{1}{\\tau_p} + \\frac{3\\beta}{r_0} \\Bigr) N_p \\, ,\n\\end{equation}\nwhere $\\beta$ denotes the specific trapping rate (in units of m\/s) at the spherical interfacial shell.\nThis trapping from inside the precipitates, which occurs in addition to the diffusion- and reaction-limited trapping\ninto the interfacial shell from the surrounding matrix, has to be taken into account in the rate equation for \n$\\rho_t$ (Equation \\ref{eq:rho_t})\nby the additional summand $\\beta \\rho_p(t)$ with the number density\n$\\rho_p = 3 N_p \/ (4 \\pi r_0^3)$ of $e^+$ in the precipitate.\n\nAssuming a homogeneous distribution of $e^+$ at time zero  in the matrix and the \nprecipitate ($\\rho_l (0) = \\rho_p (0)$) without  $e^+$ in the trapped state ($\\rho_t (0) = 0$) for $t=0$,\none obtains with the Laplace transform of eq.~(\\ref{eq:N_p})\n\\begin{equation}\n\\label{eq:Laplace_N_p}\n\\tilde{N_p} = \\displaystyle{\\frac{N_p(0)}{\\tau_p^{-1} + \\frac{3 \\beta}{r_0} + p}}\n\\end{equation}\nthe additional summand\n\\begin{equation}\n\\label{eq:n_extension}\n\\Bigl( \\frac{r_0}{R} \\Bigr)^3 \n\\displaystyle{\\Bigl(\\frac{\\frac{3 \\beta}{r_0}}{\\tau_t^{-1} +p} +1 \\Bigr)\n\\frac{1}{\\tau_p^{-1} + \\frac{3 \\beta}{r_0} + p}}\n\\end{equation}\nin  eq. (\\ref{Laplace_n}) of $\\tilde{n}(p)$. \nMoreover, in the bracket of eq. (\\ref{Laplace_n}) the first summand  is extended by the weighting factor \n$[ 1 - (r_0\/R)^3]$ and  the  trapping rate $K$ (Equation \\ref{eq:K}) in the second summand is replaced by \n$3 \\alpha r_0^2\/R^3$.\n\nFor \\underline{$C_v = 0$} this leads to the mean $e^+$ lifetime \n\\begin{eqnarray}\n\\label{eq:tauq_extended}\n&\\overline{\\tau}= \\tau_f \\Biggl\\{\\displaystyle{ \\Bigl[1 - \\Bigl(\\frac{r_0}{R} \\Bigr)^3 \\Bigr]} + \n\\nonumber \\\\\n& \\displaystyle{\\frac{3 \\alpha r_0^2}{R^3} \n\\times (\\tau_t-\\tau_f) \\times \n\\frac{\\gamma_0 \\hat{R}-\\tanh(\\gamma_0 \\hat{R}) [1-\\gamma_0^2 r_0 R]}{\\gamma_0 \\hat{R}-\\tanh(\\gamma_0 \\hat{R}) [1-\\gamma_0^2 r_0 R]+ \\frac{\\alpha r_0}{D}[\\gamma_0 R- \\tanh(\\gamma_0 \\hat{R})]} \\Biggr \\} } + \\nonumber \\\\\n& \\displaystyle{\\Bigl(\\frac{r_0}{R} \\Bigr)^3 \\times \\tau_t \\times \\frac{\\tau_t^{-1} + \\frac{3 \\beta}{r_0}}{\\tau_p^{-1} + \\frac{3 \\beta}{r_0}}}\n\\, , \n\\end{eqnarray}\nas compared to eq. (\\ref{eq:tauq}).\nEq.~(\\ref{eq:tauq_extended}) includes in the limiting case  of negligible trapping ($\\alpha = \\beta = 0$)\nas  mean $e^+$ lifetime $\\overline{\\tau} = \n[(R^3-r_0^3) \\tau_f + r_0^3 \\tau_p]\/R^3$ the expected volume-averaged mean value of $\\tau_f$ and $\\tau_p$.  \n\nThe additional pole for $p = - (\\tau_p^{-1} + 3 \\beta \/r_0)$ of  $\\tilde{n}(p)$ yields the intensity \nof the $e^+$ lifetime component $\\tau_p$ in the precipitate:\n\\begin{equation}\nI_p = \\Bigl(\\frac{r_0}{R} \\Bigr)^3 \\times \n\\Biggl(1 - \\displaystyle{\\frac{\\frac{3 \\beta}{r_0}}{\\tau_p^{-1} + \\frac{3 \\beta}{r_0} - \\tau_t^{-1}} \\Biggr)}\n\\, . \n\\label{eq:I_p}\n\\end{equation}\nApart from the weighting prefactor ($(r_0\/R)^3$), $I_p$ corresponds to the solution of the simple trapping model.\\footnote{\nNote that $I_p$ characterizes the free state in the precipitate.} Without trapping ($\\beta = 0$), $I_p$ simply takes  the form of the weighting prefactor $(r_0\/R)^3$.\n\nSince $e^+$ trapping into the precipitate$-$matrix interface occurs both from inside the precipitate and from the surrounding matrix, the intensity  \nof the trap component $\\tau_t$ is given by the sum\n\\begin{equation}\nI_t = I_t^{precip} + I_t^{matrix} \\text{ with } \nI_t^{precip} = \\Bigl(\\frac{r_0}{R} \\Bigr)^3 \\times \n\\displaystyle{\\frac{\\frac{3 \\beta}{r_0}}{\\tau_p^{-1} + \\frac{3 \\beta}{r_0} - \\tau_t^{-1}} }\n\\, ,\n\\label{eq:I_t_tot}\n\\end{equation}\nwhere $I_t^{matrix}$ corresponds to the intensity $I_t$ according to eq.~(\\ref{eq:I_t}) with  \n$K$   replaced by $3 \\alpha r_0^2\/R^3$.\\footnote{The identical equation for $I_t$ (equation~\\ref{eq:I_t_tot})  follows \nfrom the root $p = - \\tau_t$ of the Laplace transform $\\tilde{n}(p)$ in which the above mentioned extensions \nof eq. (\\ref{Laplace_n}) are taken into consideration.}\n \nWe note that the two $e^+$ trapping processes into the precipitate$-$matrix interface, namely that from inside the precipitate and that from the surrounding matrix, are completely decoupled. The trapping process from inside the precipitate can, therefore, be treated independently.\nThis also means that the process has not to be restricted to the case of entirely reaction-controlled trapping as given above, but \nthat $e^+$ trapping at the precipitate$-$matrix interface from inside the spherical precipitates can also be treated \nin the framework of diffusion-reaction theory. Hence, the available solutions for \ndiffusion- and reaction-limited trapping at grain boundaries (GBs) of spherical crystallites \\cite{Wuerschum96, Oberdorfer09} can \nbe directly applied. For this purpose the solutions for the GB-model have simply to be weighted by the factor $(r_0\/R)^3$ which denotes the volume fraction of the precipitates.\\footnote{Given the above initial condition $\\rho_l (0) = \\rho_p (0)$ and $\\rho_t (0) = 0$.}\n\nFor instance, for the mean $e^+$ lifetime, the last summand in Eq.~(\\ref{eq:tauq_extended}), i.e., the rate-equation solution has \nto be replaced by that calculated for diffusion- and reaction-limited trapping at GBs \\cite{Wuerschum96, Oberdorfer09},\nyielding:\n\\begin{eqnarray}\n\\label{eq:tauq_double_extended}\n&\\overline{\\tau}= \\tau_f \\Biggl\\{\\displaystyle{ \\Bigl[1 - \\Bigl(\\frac{r_0}{R} \\Bigr)^3 \\Bigr]} + \n\\nonumber \\\\\n& \\displaystyle{\\frac{3 \\alpha r_0^2}{R^3} \n\\times (\\tau_t-\\tau_f) \\times \n\\frac{\\gamma_0 \\hat{R}-\\tanh(\\gamma_0 \\hat{R}) [1-\\gamma_0^2 r_0 R]}{\\gamma_0 \\hat{R}-\\tanh(\\gamma_0 \\hat{R}) [1-\\gamma_0^2 r_0 R]+ \\frac{\\alpha r_0}{D}[\\gamma_0 R- \\tanh(\\gamma_0 \\hat{R})]} \\Biggr \\} } + \\nonumber \\\\\n& \\displaystyle{\\Bigl(\\frac{r_0}{R} \\Bigr)^3 \\Biggl\\{ \n\\tau_p +  (\\tau_t-\\tau_p) \\times \n\\frac{3 \\beta L(\\gamma'_0 r_0)}{r_0 \\gamma'_0\n\\Bigl( \\beta + \\gamma'_0 D L (\\gamma'_0 r_0) \\Bigr) }\\Biggr\\}}\n\\, , \n\\end{eqnarray}\nwith $\\gamma'_0 = (\\tau_p D)^{-1\/2}$,  $\\gamma_0 = (\\tau_f D)^{-1\/2}$\n and the Langevin function\n\\begin{equation}\nL (z) = \\coth z - \\frac{1}{z} \\, .\n\\label{eq:langevin}\n\\end{equation}\n\nLikewise the intensity component $I_t^{precip}$\n of the rate-equation solution in Eq.~(\\ref{eq:I_t_tot}) has to be replaced by \\cite{Wuerschum96, Oberdorfer09}:\n\\begin{equation}\n\\label{eq:I_p_diff}\nI_t^{precip} = \\Bigl(\\frac{r_0}{R} \\Bigr)^3 \\times \\frac{3 \\beta}{r_0(\\tau_p^{-1}-\\tau_t^{-1})} \\times\n\\Biggl\\{\n\\frac{\\gamma'_t D L(\\gamma'_t r_0)}{\n \\beta + \\gamma'_t D L(\\gamma'_t r_0)  }\n\\Biggr\\}\n\\, \n\\end{equation}\nwith \n\\begin{equation}\n\\gamma'^2_t = \\frac{\\tau_p^{-1} - \\tau_t^{-1}}{D} \\, .\n\\label{eq:gamma'_t}\n\\end{equation}\nFor the sake of completeness we quote the mean $e^+$ lifetime for reaction-controlled trapping from both in- and outside:\n\\begin{equation}\n\\label{eq:tauq_extended_rate}\n\\overline{\\tau}= \\displaystyle{\\tau_t \\Bigl(1 - \\Bigl[\\frac{r_0}{R} \\Bigr]^3 \\Bigr) \n\\frac{\\tau_t^{-1} + \\frac{3 \\alpha r_0^2}{R^3-r_0^3}}{\\tau_f^{-1} + \\frac{3 \\alpha r_0^2}{R^3-r_0^3}}\n+ \\tau_t \\Bigl(\\frac{r_0}{R} \\Bigr)^3  \\, \\, \\frac{\\tau_t^{-1} + \\frac{3 \\beta}{r_0}}{\\tau_p^{-1} + \\frac{3 \\beta}{r_0}}}\n\\, . \n\\end{equation}\nA further extension for taken into account additional $e^+$ trapping at point defects\ninside the matrix (Sect.~\\ref{sec:vacancies})\nand inside the precipitates (in analogy to the GB model \\cite{Oberdorfer09}) is straightforward, so that \nthe corresponding equations have not to be stated explicitly.\n\n\n\\section{\\label{discussion} Discussion}\n\\subsection{\\label{sec:voids}\nVoids, clusters, small precipitates}\n\nThe presented model with the exact solution of diffusion-reaction\ncontrolled trapping at voids (or other extended spherical defects like clusters and small precipitates)\nand competitive transition-limited\ntrapping at vacancy-type defects yields closed-form expressions\nfor the mean positron lifetime $\\overline{\\tau}$ [Eq. (\\ref{eq:tau_q_general})]\nand for the relative intensities $I_t$ [Eq. (\\ref{eq:I_t_general})]\nand $I_v$ [Eq. (\\ref{eq:I_v})] of the $e^+$ lifetime components\n$\\tau_t$ and $\\tau_v$ of the void and the vacancy\ntrapped states, respectively.\n\nWe start the discussion considering exclusively diffusion-reaction\ncontrolled trapping at voids  (Sect. \\ref{sec:general}).\nThe model contains as limiting cases both the solution of the simple trapping model \n(Sect. \\ref{sec:rate_limit}) and the one of the entirely diffusion-limited trapping (Sect. \\ref{sec:diffusion_limit}).\nThe mean $e^+$ lifetime $\\overline{\\tau}$ [Eq. (\\ref{eq:tauq})] and the intensity \n$I_t$ [Eq. (\\ref{eq:I_t})] in dependence of the radius $R$ of the diffusion sphere\nare compared in Fig. \\ref{fig:2} with the two limiting cases. Note, that $R$ is related to the the void concentration \n[Eq. (\\ref{eq:C_t})].\nFor illustration the following characteristic e$^+$ annihilation parameters are used:\na free $e^+$ lifetime  $\\tau_f=160$~ps as typical for aluminium, \na $e^+$ lifetime $\\tau_t=400$~ps as typical for voids \\cite{Eldrup03},\na $e^+$ diffusion coefficient $D=2 \\times 10^{-5}$~m$^2$s$^{-1}$, a void radius $r = 3$~nm, and  \na specific e$^+$ trapping rate  $\\alpha = 3 \\times 10^3$~ms$^{-1}$\nreported by Dupasquier et al. \\cite{Dupasquier93} for interfaces in Al. For surfaces of Al \na value $\\alpha = 7.6 \\times 10^3$~ms$^{-1}$ was calculated by Nieminen and Lakkonnen \\cite{Nieminen79a}.\nUsing an atomic volume $\\Omega$ for Al of $\\Omega^{-1} = 6 \\times 10^{28}$~m$^{-3}$,   \n$\\alpha = 3 \\times 10^3$~ms$^{-1}$ corresponds to  a trapping rate $\\sigma_t = 2\\times 10^{16}$ s$^{-1}$ \n[Eq. \\ref{eq:sigma_t}] which is similar to that deduced by Bentzon and Evans \\cite{Bentzon90} for voids in Mo.\\footnote{\nA value $\\sigma_t = 4\\times 10^{16}$ s$^{-1}$ is deduced from \nthe trapping rate of $3.2 \\times 10^9$~s$^{-1}$ at 300~K and \na void number density of $5.3 \\times 10^{21}$~m$^{-3}$\nquoted in \\cite{Bentzon90}.}\n\nBoth $\\overline{\\tau}$ (Fig. \\ref{fig:2}a) and $I_t$ (Fig. \\ref{fig:2}b) exhibit the characteristic sigmoidal increase\nfrom the free state to the saturation-trapped state with decreasing $R$, i.e., increasing void concentration $C_t$.\nCompared to the exact solution of the present model, the standard trapping model \nand the limiting case of entirely diffusion-limited trapping \nshow qualitatively the same trend for $\\overline{\\tau}$ and $I_t$.\nHowever, both special cases systematically overestimate $\\overline{\\tau}$ and $I_t$, i.e., predict \nstronger trapping since either the rate-limiting effect or the diffusion-limiting effect are neglected in these\napproximations. For instance, if one would determine  the void concentration from a typical, experimentally measured intensity $I_t$ of\n45\\,\\% \\cite{Nambissan1989}, a concentration 36\\,\\% too low would be deduced from the standard trapping model \ncompared to the exact theory for the parameter set according to Fig. \\ref{fig:2}b.\n\nThe deviations of the two limiting cases from the exact solution become even more clear when \nthe ratios of the trap component intensities of the limiting and exact solution is considered\nas shown in the upper part of Fig. \\ref{fig:3}.\nThe deviation from the exact solution substantially increases with decreasing intensity, i.e., with\ndecreasing void concentration. In this low concentration regime, the deviations \nattain a factor of ca. 1.5 (reaction limit) or larger than 3 (diffusion limit)\nfor the present set of parameters, i.e., the entirely diffusion-limiting case\ndeviates in this example more strongly than the reaction-limited case. \nDiffusion limitation gets even more pronounced when $e^+$ diffusivity is reduced, e.g., due to\nscattering at lattice imperfections.\nRegarding the opposite side of high defect concentrations,  \nFig. \\ref{fig:3} (upper part) nicely demonstrates that deviations from the exact theory  \nvanishes upon approaching $e^+$ saturation trapping since in this regime  \nkinetic effects tends to become irrelevant.\n\n\n\n\\subsubsection{\\label{sec:effective_rate} Comparison with effective rate approach}\nNext we compare the present model with approximations according to which\ndiffusion limitation is taking account in the standard trapping model by means of a diffusion-limited trapping rate \n\\cite{Seeger74,Bentzon90}:\n\\begin{equation}\nK_{diff} = \\frac{4\\pi r_0 D}{\\Omega} \\times C_t \\,.\n\\label{eq:K_diffusion}\n\\end{equation}\nThe case of both transition- and diffusion-limited trapping, is treated in this approximation by means of\nthe effective trapping rate \\cite{Seeger74,Bentzon90}\n\\begin{equation}\nK_{eff} = \\frac{K_{diff} \\sigma_t C_t}{K_{diff} + \\sigma_t C_t} \n\\label{eq:K_eff}\n\\end{equation}\nwith $\\sigma_t$ and $C_t$ according to equations (\\ref{eq:sigma_t}) and  (\\ref{eq:C_t}), respectively.\nWe note that the diffusion-limited trapping rate according to eq. (\\ref{eq:K_diffusion}) is also included in the present model; in fact $K_{diff}$ is identical to the pre-factor of $I_t$ for entirely diffusion limited trapping [Eq. (\\ref{eq:I_t_diffusion})] when the \nsubtrahend $r_0^3$ in the nominator, which is associated with the defect volume,  is omitted.\n\nIn figure \\ref{fig:4} the concentration dependence of the relative intensity \n$I_t$ of the $e^+$ lifetime component $\\tau_t$ \nin voids is shown for the exact models of diffusion-reaction  [Eq. (\\ref{eq:I_t})] or \nentire diffusion limitation [Eq. (\\ref{eq:I_t_diffusion})] in comparison with the corresponding approximations\nusing the above mentioned effective or  diffusion-trapping rates [Equations (\\ref{eq:K_diffusion}), (\\ref{eq:K_eff})] \nwith the simple trapping  model [Eq. (\\ref{eq:I_t_rate})].\nAlthough the effective-rate approximations of the diffusion limitation describe the sigmoidal curve fairly well,\ndeviations from the exact diffusion models are also apparent, e.g., for the example, $I_t = 45$\\,\\%, mentioned above \nthe deviation in concentration is ca. 7\\,\\% compared to the exact diffusion-reaction theory.\n\nThe deviations become clearer once more when we consider the intensity ratio of the effective-rate model and the exact\ntheory, as plotted in the lower part of Fig. \\ref{fig:3}.\nRemarkably, since the effectice trapping rate $K_{eff}$ is lower than both the reaction-trapping rate \n$\\sigma_t C_t$ and the diffusion-trapping rate $K_{diff}$,\nthe intensity $I_t$ deduced from the effective trapping model is smaller than the exact value.\nDeviations from the full model occur throughout the entire intensity regime, although these \ndeviations  are less pronounced compared to the two limiting cases \n(fully reaction- or diffusion limited, upper part of Fig. \\ref{fig:3}). \nFor applications in the analysis of experimental data, \nthe accuracy of the effective rate approach [equation~(\\ref{eq:K_eff})] can be assessed \nby plotting the intensity ratio (lower part of Fig. \\ref{fig:3}) for the respective parameter set.\nIrrespectively whether deviations of the effective-rate approach are strong or  minor only, \nthe present model founded on diffusion-reaction theory is that which covers the underlying physics\nmost accurately.\n  \n\n\\subsubsection{\\label{sec:competitive} Competitive trapping at point defects}\nNow, we discuss the general case that in addition to diffusion-reaction\ncontrolled trapping at voids also competitive transition-limited trapping\nat vacancy-type defects in the lattice occurs (Sect. \\ref{sec:vacancies}).\nThe relative intensities of the void component $I_t$  [Eq. (\\ref{eq:I_t_general})] and \nof the vacancy component $I_v$ [Eq. (\\ref{eq:I_v})] is plotted in figure \\ref{fig:5}\nin dependence of void concentration $C_t$ (a) and vacancy concentration $C_v$ (b), \nfor a given fixed $C_v$ or $C_t$, respectively. For the \nvacancy-type defect a $e^+$ lifetime component $\\tau_v=250$~ps and \na specific trapping rate $\\sigma_v=4\\times10^{14}$~s$^{-1}$ \\cite{Schaefer87} is assumed; the other parameters \nare the same as used above.\nThe competitive $e^+$ trapping at voids and  vacancy-type defects becomes evident.\nFor a given vacancy concentration the\nintensity $I_t$ of the void increases and the\nintensity $I_v$ of the vacancy  component decreases with increasing \nvoid concentration due to the increasing fraction of e$^+$ that reaches the\nvoids (Fig.~\\ref{fig:5}.a). Likewise, for a given void concentration,\n$I_v$  increases and  $I_t$ decreases with increasing vacancy concentration\n(Fig. \\ref{fig:5}.b).\n\n\n\n\\subsubsection{\\label{sec:gb} Comparison with $e^+$ trapping at grain boundaries}\nIn the end of this subsection (\\ref{sec:voids}), the results of the present model on diffusion-reaction limited $e^+$ trapping\nat extended spherical defects will briefly be compared with the corresponding model \nof $e^+$ trapping at grain boundaries of spherical crystallites with radius $R$ \\cite{Wuerschum96, Oberdorfer09}.\nWhereas in the latter case the surface of the diffusion sphere  \nwith area $4 \\pi R^2$ acts as $e^+$ trap, in the present case with voids of radius $r_0$, the trapping active area \n$4 \\pi r_0^2$ is much smaller. Moreover,  the trapping rate $3 \\alpha \/R$ for grain boundary trapping \\cite{Oberdorfer09}\ndecreases much more slowly with increasing $R$ compared to the trapping rate $3 \\alpha r_0^2 \/ (R^3-r_0^3)$ of \nspherical extended defects with radius $r_0$ [Eq. \\ref{eq:K}].\nThis is the reason why diffusion limitation \naffects the kinetics of $e^+$ trapping at grain boundaries more strongly than in the case of voids\nwhich is nicely  demonstrated in Fig. \\ref{fig:6} where the exact solutions are compared with those of infinite diffusivities.\nIn Fig. \\ref{fig:6} the mean $e^+$ lifetime according to the exact solutions  and those of the standard rate theory  \nfor the two types of extended traps are plotted.\nThe exact solution for $e^+$ trapping at grain boundaries of spherical crystallites with radius $R$ reads \n\\cite{Wuerschum96, Oberdorfer09}\n\\begin{equation}\n\\label{eq:tauq_GB}\n\\overline{\\tau} = \\tau_f +  (\\tau_t-\\tau_f) \\times \n\\frac{3 \\alpha L(\\gamma_0 R)}{R \\gamma_0\n\\Bigl\\{ \\alpha + \\gamma_0 D L(\\gamma_0 R) \\Bigl\\}} \\text{ with } L (z) = \\coth z - \\frac{1}{z}\\, .\n\\end{equation}\nThe more stronger deviation between the exact solution and the rate theory in the case of grain boundary trapping is obvious (Fig. \\ref{fig:6}).\n\n\\subsection{\\label{sec:composite}\nLarger precipitates: $e^+$-trapping from both sides of precipitate$-$matrix interface }\n\nIn Sect.~(\\ref{sec:extended}) we extended the model for applying it to larger precipitates taking into account free $e^+$ annihilation\nwithin the precipitate. The $e^+$ trapping from the precipitate into the precipitate$-$matrix interface is handled \neither by rate theory, for special cases where the precipitate radius is well below the $e^+$ diffusion length, or else \nby diffusion-reaction theory, for the more general case that the precipitate radius is in the range of or larger than \nthe $e^+$ diffusion length.\nWith this extension the present model is applicable to a wide variety of structurally complex scenarios, namely to all type\nof composite structures where spherical precipitates are embedded in a matrix irrespective of the size and the number density of \nthe precipitates.\n\nWhereas for extended defects with smaller size, which were discussed in Sect.~(\\ref{sec:voids}),\nthe deviations between the exact model and the rate theory may be of less relevance since the \n trapping active area $4 \\pi r_0^2$ is small,  \nfor larger precipitates the diffusion-limitation in any case gets relevant owing\nto the much larger trapping active area, similar as for $e^+$ trapping at GBs \n(see Fig.~\\ref{fig:6}).\nThis is demonstrated in Fig.~\\ref{fig:7}, where the variation of the mean $e^+$ lifetime with radius $R$\nis compared for four different solutions, namely diffusion-limitation of trapping into the \nprecipitate$-$matrix interface from both the matrix and the precipitate, from the matrix only, and for\nentirely reaction-limited trapping from both sides with standard-trapping rate or with effective diffusion-limited trapping rate.\nThe latter is obtained by replacing in equation~(\\ref{eq:tauq_extended_rate}) the standard-trapping rates by \nthe  effective diffusion-limited trapping rate according to equation~(\\ref{eq:K_eff}), i.e., \n$3 \\alpha r_0^2 (R^3-r_0^3)^{-1}$ by $3 \\alpha D r_0^2 R^{-3} (\\alpha r_0 +D)^{-1}$ and \n$3 \\beta r_0^{-1}$ by $3 \\beta D r_0^{-1} (\\beta r_0 +D)^{-1}$.\n\nIn contrast to the case of small extended defects (Fig.~\\ref{fig:6}), for larger precipitates \n(example $r_0=100$~nm) substantial deviations between the solutions occur for the entire concentration regime  \nif the diffusion-limitation is neglected (Fig.~\\ref{fig:7}). \nEven the rate approach with effective diffusion-limited trapping rate,\nwhich at least for small extended defects is a reasonable approximation (Sect.~\\ref{sec:effective_rate}, Fig.~\\ref{fig:4}),\nturns out to be completely inadequate for the larger precipitate size.\nThe deviations are much less if the diffusion-limitation is only neglected\nfor the trapping from the precipitate into the interface, since the precipitate size (in contrast to the precipitate distance) remains in the range of the $e^+$ diffusion length independent of the \nprecipitate concentration. Anyhow, for a precise description even for such small precipitate sizes, \nthe exact theory of diffusion- and reaction controlled trapping has to be applied for\nthe trapping from the interior of the precipitates.\n\nFinally, we compare this model with that presented by Dryzek \\cite{dryzek1999, dryzek2016}\nfor studying recrystallization in highly deformed metals. In that case recrystallized grains are embedded in a highly deformed matrix. Diffusion-limited $e^+$ trapping occurs from the grains into the matrix,\nwhereas within the matrix saturation trapping of $e^+$ prevails due to the high defect density.\nIn this sense, the model of Dryzek represents an extension of the diffusion-reaction theory for trapping at grain boundaries, where instead of GBs a surrounding deformed matrix is considered. The model presented here, represents a further extension where  diffusion- and reaction-controlled trapping also from the matrix into the interfaces is considered.  \n\n \n\n\\section{\\label{conclusion} Conclusion}\n\nThe present model with the exact solution\nof  the diffusion-reaction theory for the $e^+$ trapping at \n extended spherical defects  \nand competitive transition-limited trapping at  atomic defects \nyields a basis for the quantitative description of the\n $e^+$  behaviour in materials with complex defect structure.\nIt could be shown that the model includes  as special cases\nthe simple trapping model and the entirely diffusion-limited trapping,\nbut both of these limiting cases represent approximations, only.\nFor the full model, closed-form expressions were obtained for\nthe mean positron lifetime  $\\overline{\\tau}$ and for the intensities\nof the e$^+$ lifetime components associated with\ntrapping. This exact model allowed a quantitative assessment of  the usual approach, \nwhich takes  diffusion limitation for the trapping at voids into account \nby effective diffusion-trapping rates. The present closed-form solutions also  \nrenders this effective rate approach unnecessary.\n\nThe presented theory  goes even much  far beyond existing models,  \nsince it is not only applicable to small extended defects (such as voids or clusters), but also \nto larger precipitates where positron trapping from the precipitates into the\nprecipitate$-$matrix interface is taken into consideration.\nTherefore, the model presents the basis for studying all type\nof composite structures where spherical precipitates are embedded in a matrix irrespective of their size and  their number density.\n\n\n\\begin{acknowledgments}\nThe senior author (R.W.) dedicates this work to Alfred Seeger \nwhose numerous pioneering works also included modeling of positron annihilation.\nThis work was performed in the framework of the inter-university cooperation of TU Graz and Uni Graz on natural science (NAWI\nGraz).\n\\end{acknowledgments}\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nConformal field theories (CFTs) in two dimensions are of interest for various areas of physics, from condensed matter physics to string theory. In string theory they naturally arise on the worldsheet of the string. In the context of holographic duality, certain two-dimensional CFTs are also known to be \\emph{dual to} string theories on three-dimensional anti de Sitter spacetimes~\\cite{Maldacena:1997re,Giveon:1998ns,Kutasov:1999xu}. An important instance of the $AdS_3\/CFT_2$ duality is obtained by studying string theory on $AdS_3\\times S^3\\times T^4$. In the case of pure NSNS flux the string is described by a Wess-Zumino-Witten (WZW) model on the group $F=SL(2,\\mathbb R)\\times SU(2)$, see~\\cite{Maldacena:2000hw,Maldacena:2000kv,Maldacena:2001km} and references there. We will be interested mainly in this setup. The CFT description of the worldsheet theory allows to make precise statements about the AdS\/CFT duality in this case. A recent example is the duality between the symmetric product orbifold CFT and the string WZW model at level $k=1$~\\cite{Eberhardt:2018ouy}.\n\nGenerally speaking it is interesting to understand the conformal manifold of a CFT, i.e. the space of marginal deformations generated by adding a local perturbation to the Lagrangian. When applied to the WZW model under study, the marginal deformations correspond to deformations of the supergravity background. They give us (at least in principle) a way to go beyond the usual $AdS_3\/CFT_2$ duality and extend it to cases in which e.g. the supersymmetry or the conformal symmetry of the dual $CFT_2$ are broken. Local marginal deformations of WZW models were studied by Chaudhuri and Schwartz (CS)~\\cite{Chaudhuri:1988qb}. They found that {a necessary condition for} local operators constructed out of the chiral and antichiral currents as \n\\begin{equation}\nO(\\sigma,\\bar \\sigma)= c^{a b}J_a(\\sigma)\\bar J_{ b}(\\bar \\sigma)\\,,\n\\end{equation}\nwith $c^{ab}$ some constant coefficients, {to give a marginal deformation is that $c^{ab}$ satisfy}\n\\begin{equation}\\label{eq:CS-weak-intro}\nC\\cdot C+ \\bar C\\cdot \\bar C=0\\,,\n\\end{equation}\nwhere we have defined $C^{abc}\\equiv c^{da}c^{eb}f_{de}^{\\ c}$, and $\\bar C^{abc}\\equiv c^{ad}c^{be} f_{de}^{\\ c}$, and the product is obtained using the Killing metric $K_{ab}$, e.g. $C\\cdot C\\equiv C^{abc}C^{def}  K_{ad} K_{be}K_{cf}$.\nEquation~\\eqref{eq:CS-weak-intro} is quartic in $c^{ab}$, and it involves also the structure constants of the algebra of $F$, the Lie group of the WZW model. We will call it the \\emph{weak} CS condition. CS were interested in the case of CFTs where the group $F$ is compact. In that case~\\eqref{eq:CS-weak-intro} reduces to \n\\begin{equation}\\label{eq:CS-strong-intro}\nC^{abc}=0\\,,\\qquad\\text{and}\\qquad\n\\bar C^{abc}=0\\,,\n\\end{equation}\nwhich is an equation quadratic in $c^{ab}$ that we will call the \\emph{strong} CS condition. It imposes, in fact, a stronger constraint, since it is only in the case of compact groups that it is equivalent to~\\eqref{eq:CS-weak-intro}. CS also showed that solutions of the strong condition correspond to abelian subalgebras of $Lie(F)$, since when~\\eqref{eq:CS-strong-intro} holds it is always possible to identify linear combinations of $J_a,\\bar J_a$ such that their OPEs do not have the term involving the structure constants --- the so-called ``no simple-pole condition'', cf.~\\eqref{eq:JJ-OPE}. {In that case the correlation functions of $O$ are the same as for an $O$ constructed out of free bosons, which in turn implies that the deformation can be completed to all orders in conformal perturbation theory in the deformation parameter. For deformations satisfying only the weak CS condition on the other hand there is no guarantee that they remain marginal beyond lowest order in the deformation parameter.}\nIn the literature on marginal deformations of WZW models, see e.g.~\\cite{Chaudhuri:1992yca,Hassan:1992gi,Kiritsis:1993ju,Tseytlin:1993hm,Forste:2003km,Israel:2004vv,Israel:2004cd,Orlando:2006cc,Detournay:2005fz,Fredenhagen:2007rx}, we did not find examples that satisfy the weak CS condition but not the strong one. Here we will construct such examples by involving sufficient components of the chiral and antichiral currents of $SL(2,\\mathbb R)$. In this sense our results identify new directions to explore the conformal manifold.\n\n\nIn~\\cite{Hassan:1992gi,Henningson:1992rn,Kiritsis:1993ju} it was argued that $O(d,d)$ transformations provides the correct language to obtain the exact (in the deformation parameter) version of the CFTs deformed by the abelian current-current operators. Indeed, such  transformations do not break the isometries involved in the deformation and one can show that the derivative of the action with respect to the deformation parameter is given by\n\\begin{equation}\n\\frac{dS}{d\\eta}=-\\frac{T}{2}\\int d^2\\sigma\\,J^\\eta\\bar J^\\eta\\,,\n\\end{equation}\nwhere $J^\\eta$ and $\\bar J^\\eta$ are (anti)chiral currents of the \\emph{deformed} theory corresponding to the isometries involved in the deformation. This makes it clear that the infinitesimal deformation can be integrated to a finite one.\nThe relevant so-called $\\beta$-shifts of $O(d,d)$, corresponding to a simple shift of the $B$-field of the dual model obtained by performing T-duality on two $U(1)$ isometries, are also known as TsT (T-duality, shift, T-duality) transformations~\\cite{Lunin:2005jy,Frolov:2005ty,Frolov:2005dj,Alday:2005ww}. A TsT transformation exploits an abelian $U(1)^2$ global symmetry of the sigma model to construct a deformation parameterised by a continuous deformation parameter. Since the deformation is constructed by exploiting T-duality, on-shell the deformed model is equivalent to the undeformed one, and the deformation can be equivalently understood as a twist in the boundary conditions in the compact direction of the worldsheet.  \n\nTsT transformations are known to belong to a larger class of deformations of sigma models, usually called Yang-Baxter (YB) deformations. They first appeared in the context of integrable models, since the deformations do not break the classical integrability of the original model~\\cite{Klimcik:2008eq,Delduc:2013qra,Kawaguchi:2014qwa}. YB deformations are particularly interesting in the context of the $AdS\/CFT$ correspondence, since they can be used to generate string backgrounds deforming the standard ones appearing in the $AdS_{d+1}\/CFT_d$ dualities. Particularly important cases are those for which integrability techniques may be applied. In the case of $AdS_5\/CFT_4$ it was proposed that the deformations of the $AdS_5\\times S^5$ background should correspond to non-commutative deformations of $\\mathcal N=4$ super Yang-Mills~\\cite{Matsumoto:2014gwa,vanTongeren:2015uha,vanTongeren:2016eeb,Araujo:2017jkb}.\nThe name YB comes from the fact that the deformation is controlled by an object $R$ which is an element of $\\mathfrak g\\wedge \\mathfrak g$ (where $\\mathfrak g$ is the algebra of isometries of the starting background) and solves the classical Yang-Baxter equation\\footnote{There is also a version of these models where $R$ solves instead the \\emph{modified} CYBE~\\cite{Klimcik:2008eq,Delduc:2013qra}. The story in that case is quite different and we will not consider it here.} (CYBE) on $\\mathfrak g$. The simplest solutions to the CYBE are the so-called abelian $R$-matrices, e.g. $R=T_1\\wedge T_2$ with $T_i\\in \\mathfrak g$ and $[T_1,T_2]=0$. In this case the CYBE is trivially satisfied because the relevant structure constants vanish. Abelian YB deformations were shown to be equivalent to TsT transformations in~\\cite{Osten:2016dvf}. On compact algebras, the CYBE only admits abelian solutions. On non-compact algebras, instead, more interesting non-abelian solutions (i.e. $R$-matrices constructed out of generators of a non-abelian subalgebra) are possible. \n\nOriginally, in the construction of the YB-deformed sigma models, the CYBE was necessary in order to preserve the classical integrability. Later it was understood that YB deformations may be obtained from non-abelian T-duality (NATD)~\\cite{Hoare:2016wsk,Borsato:2016pas}. That interpretation revealed a consistent generalisation of what is known about TsT transformations, since it became clear that YB deformations correspond to a shift of the $B$-field of the dual (undeformed) sigma model; the deformed model is then obtained by applying NATD on the subalgebra of $\\mathfrak g$ where $R$ is non-degenerate. After restricting the domain, $R$ may be inverted and its inverse $R^{-1}$ is a Lie algebra 2-cocycle. The shift of the dual $B$-field is given by this 2-cocycle. In other words, it is possible to go beyond the construction related to integrable models, and understand the CYBE as being a constraint necessary to shift the dual $B$-field without modifying its field strength $H=dB$. Consistently with this interpretation, in~\\cite{Lust:2018jsx,Sakamoto:2018krs} it was proposed  to identify YB deformations with the $\\beta$-shifts of a larger group extending the known $O(d,d)$ group of abelian T-duality, that in~\\cite{Lust:2018jsx} was dubbed ``non-abelian T-duality group''.\nThe  logic of NATD\/$\\beta$-shifts may be used to construct YB deformations of generic sigma models~\\cite{Lust:2018jsx,Sakamoto:2018krs,Borsato:2018idb}, beyond those for which YB deformations were first introduced, the Principal Chiral Model and (super)cosets.\\footnote{Ref.~\\cite{Bakhmatov:2017joy} put forward a first proposal for a set of transformation rules to go beyond these cases.}   Here we will use the transformation rules of~\\cite{Borsato:2018idb}, which  were derived from the NATD construction and have the advantage of being applicable to a generic background with isometries  (even when the initial $G-B$ is not invertible, as is the case for the background metric and Kalb-Ramond field that we have to consider in this paper).\n\nBecause of their realization via NATD, YB models will be Weyl invariant at least to one loop in $\\sigma$--model perturbation theory (and exactly in the deformation parameter), provided that the Lie algebra on which the $R$-matrix is non-trivial is unimodular (i.e. the trace of its structure constants vanish). In this case the deformed background solves the standard supergravity equations of motion. When the algebra is not unimodular there is a potential Weyl anomaly \\cite{Alvarez:1994np,Elitzur:1994ri}. In that case the resulting background solves instead a generalization of the standard supergravity equations \\cite{Arutyunov:2015mqj,Wulff:2016tju} controlled by a Killing vector field $K$. Even in these non-unimodular cases, it can happen that there is actually no Weyl anomaly. This is reflected in the fact that the generalized supergravity equations can have ``trivial'' solutions, i.e. solutions with $K\\neq0$ but where nevertheless the other fields solve the standard supergravity equations \\cite{Wulff:2018aku}. In \\cite{Wulff:2018aku} it was shown that this can happen if $K$ is null. In appendix \\ref{app:trivial} we show that this condition can be weakened and $K$ does not have to be null if the one-form {$\\rm X$ appearing in the generalized supergravity equations takes the form $\\rm X=d\\phi+\\tilde K$ with $\\phi$ the dilaton and $\\tilde K$ another Killing vector}.\nWe will see that YB deformations of the $AdS_3\\times S^3$ WZW model give rise also to ``trivial'' solutions, both ones with $K^2=0$ and with $K^2\\neq0$. We will also find some examples with a genuine anomaly, corresponding to $K$ not being null (and $\\tilde K$ not defining a Killing vector).\n\nAs we will discuss in more detail in section~\\ref{sec:YB}, at leading order in the deformation parameter YB deformations correspond to current-current deformations. We are therefore led to study YB deformations of strings on backgrounds containing an $AdS_3$ subspace, expecting to find marginal deformations of the corresponding WZW model. Particularly  interesting for us are the deformations that do not solve the strong version of the CS condition, but only the weak one. We will construct explicitly such examples. Such possibilities are allowed because we exploit also the non-compact part of the current algebra to generate the deformations. \n\nThe paper is organised as follows. In section~\\ref{sec:WZW} we review some aspects of the $SL(2,\\mathbb R)\\times SU(2)$ WZW model and of marginal current-current deformations that are important for our discussion. In section~\\ref{sec:YB} we review the transformation rules of YB deformations and explain in which cases we can understand them as compositions of simpler YB transformations. We will also explain the connection to the marginal current-current deformations. Using the classification of $R$-matrices in appendix~\\ref{app:R-matrices}, we later study deformations of $AdS_3$ and $AdS_3\\times S^3$. We give our conclusions in section~\\ref{sec:disc}.\nAppendix~\\ref{app:field-red} collects some details on the field redefinition used in section~\\ref{sec:YB}, and  appendix~\\ref{app:onshell} discusses the on-shell equivalence of the YB models to the undeformed ones. In appendix~\\ref{app:sl3} we consider the case of the $\\mathfrak{sl}_3$ algebra, which is separate from the rest of the paper. In appendix~\\ref{app:trivial} we extend the triviality condition of~\\cite{Wulff:2018aku}.\n\n\n\n\\section{Wess-Zumino-Witten model and marginal current-current deformations}\\label{sec:WZW}\nIn this section we review certain aspects of WZW models and their marginal current-current deformations. Although the discussion can be made general, for concreteness we will take the example of the $SL(2,\\mathbb R)\\times SU(2)$ WZW model, since it is important for string theory applications and it already contains all the salient features.\n\n\\subsection{The $AdS_3\\times S^3$ sigma model}\\label{sec:sigmamodel}\nWe start with a sigma model describing the propagation of a string in $AdS_3\\times S^3$, that can be viewed (after adding four free bosons) as the bosonic sector of the superstring . The sigma model action is\\footnote{\nWe work with a Lorentzian worldsheet and we introduce worldsheet coordinates $\\sigma^\\pm=\\sigma^0\\pm\\sigma^1$, so that $\\eta^{+-}=\\eta^{-+}=-2$, $\\epsilon^{+-}=-\\epsilon^{-+}=-2$ and $d^2 \\sigma=\\frac12d\\sigma^+d\\sigma^-$. We also use the standard notation $\\sigma,\\bar \\sigma$ in place of $\\sigma^+,\\sigma^-$, as well as $\\partial =\\partial_\\sigma,\\bar \\partial=\\partial_{\\bar \\sigma}$.}\n\\begin{equation}\nS=\\frac{k}{2\\pi}\\int d^2\\sigma\\ \\left(\\frac{-\\partial x^-\\bar\\partial x^++\\partial z\\bar\\partial z}{z^2}+\\frac{1}{4} \\partial\\phi_i\\bar\\partial\\phi_i+\\frac{1}{2} \\partial\\phi_2\\bar \\partial\\phi_1\\sin \\phi_3\\right)\\,.\n\\end{equation}\nHere we are considering the pure NSNS background, and $k$ will be the level of the WZW model. The string tension is $T=k\/\\pi$, and the metric and $B$-field appearing in the sigma model action follow from $S=T\\int  d^2 \\sigma\\ L=\\frac{T}{2}\\int d^2 \\sigma\\ \\partial x^m (G_{mn}-B_{mn})\\bar \\partial x^n$ and are\\footnote{In our conventions $B=\\frac{1}{2} B_{nm}dx^m\\wedge dx^n$.}\n\\begin{equation}\n\\begin{aligned}\nds^2&=ds_{\\text{AdS}_3}^2+ds_{\\text{S}^3}^2=\n\\frac{-dx^+dx^-+dz^2}{z^2}\n+\\frac{1}{4}\\left[d\\phi_3^2+\\cos^2\\phi_3d\\phi_1^2+(d\\phi_2+\\sin \\phi_3 d\\phi_1)^2\\right]\\,,\\\\\nB&=\\frac{dx^+\\wedge dx^-}{2z^2}-\\frac{1}{4}\\sin \\phi_3 d\\phi_1\\wedge d\\phi_2\\,.\n\\end{aligned}\n\\end{equation}\n$AdS_3$ is parameterised by the boundary coordinates $x^\\pm$ and the radial coordinate $z$, while the angles $\\phi_i$ parameterise the sphere. The $AdS_3$ metric admits the following Killing vectors \n\\begin{equation}\n\\begin{aligned}\n&k_{0}^m\\partial_m=x^+\\partial_{x^+}+\\tfrac{1}{2}z\\partial_{z}\\,,\\qquad\n&&k_{+}^m\\partial_m=\\partial_{x^+}\\,,\\qquad\n&&&k_{-}^m\\partial_m=-(x^+)^2\\partial_{x^+}-z^2\\partial_{x^-}-x^+z\\partial_{z}\\,,\\\\\n&\\bar k_{0}^m\\partial_m=-x^-\\partial_{x^-}-\\tfrac{1}{2}z\\partial_{z}\\,,\\qquad\n&&\\bar k_{-}^m\\partial_m=\\partial_{x^-}\\,,\\qquad\n&&&\\bar k_{+}^m\\partial_m=-(x^-)^2\\partial_{x^-}-z^2\\partial_{x^+}-x^-z\\partial_{z}\\,.\n\\end{aligned}\n\\end{equation}\nThey satisfy $[k_a^m\\partial_m,k_b^n\\partial_n]=-f_{ab}^{\\ c}k_c^p\\partial_p$ (and similarly for $\\bar k_a$), where $f_{ab}^{\\ c}$ are the structure constants of the algebra of $SL(2,\\mathbb R)$\n\\begin{equation}\n[S_0,S_\\pm]=\\pm S_\\pm\\,,\\qquad[S_+,S_-]=2S_0\\,.\n\\label{eq:sl2}\n\\end{equation}\nIn these formulas and in the following we use a bar to distinguish the right copy of the algebra from the left copy.\\footnote{We will interchangeably place the bar on an object or on its index, in other words $\\bar k_a$ or $k_{\\bar a}$ have the same meaning. For readability sometimes we will prefer the former.}\nFor the sphere we have two copies of $SU(2)$, whose algebra is generated by $T_a$ ($a=1,2,3$) with commutation relations $[T_a,T_b]=-\\epsilon_{abc}T_c$.\nWe will not write explicitly all Killing vectors of $S^3$ since we will not need them. For our purposes it will be enough to use the two commuting Killing vectors\n\\begin{equation}\nk_1=-\\partial_{\\phi_1}\\qquad\\mbox{and}\\qquad\\bar k_2=\\partial_{\\phi_2}\\,.\n\\end{equation}\n\nThe sigma-model action is invariant under the transformations generated by the above Killing vectors, although in certain cases the $B$-field is not invariant but changes by a total derivative. Therefore in general the corresponding Noether currents are given by\n\\begin{equation}\n\\mathcal J_{A,\\pm}=k_A^{m}(G_{mn}\\pm B_{mn})\\partial_{\\pm}x^n+j_{A,\\pm}\\,,\n\\label{eq:Noether}\n\\end{equation}\nwhere $j_{A,\\pm}$ is defined by looking at the variation of the Lagrangian $\\delta_A L=\\varepsilon\\partial_ij_A^i$ under the infinitesimal global transformation.\nBecause of our choice of gauge, in the $AdS_3$ part only $j_-^i$ and $\\bar j_+^i$ are non-zero, and we also have $j_1^i=\\bar j_2^i=0$. In the following we will ignore the transformations generated by $S_-,\\bar S_+$, since for our discussion it will be enough to focus on the (maximal solvable) subalgebra generated by\n\\begin{equation}\\label{eq:subalgebra}\nS_0,\\quad S_+,\\quad \\bar S_0,\\quad \\bar S_-,\\quad T_1,\\quad \\bar T_2\\,.\n\\end{equation} \nAll Noether currents that we will need to consider will therefore have $j_{A,\\pm}=0$.\nLet us anticipate that these Noether currents are not always equal to the chiral (resp. antichiral) currents of the WZW description, which we shall denote by $J$ (resp. $\\bar J$) and write explicitly in the next subsection. They agree up to ``improvement terms'' that do not spoil the current conservation, of the type $\\epsilon^{ij}\\partial_jc$ for some $c$. Restricting to the generators in~\\eqref{eq:subalgebra}, for $AdS_3$ we have\n\\begin{equation}\\label{eq:noether-chiralAdS3}\n\\begin{aligned}\n&\\mathcal J_{0,+}=J_0-\\tfrac{1}{2}\\partial \\log z\\,,\\qquad\n&\\mathcal J_{0,-}=+\\tfrac{1}{2}\\bar\\partial \\log z\\,,\\qquad\n&\\mathcal J_{+,+}=J_+\\,,\\qquad\n&\\mathcal J_{+,-}=0\\,,\\\\\n&\\bar{\\mathcal J}_{0,-}=\\bar J_0+\\tfrac{1}{2}\\bar\\partial \\log z\\,,\\qquad\n&\\bar{\\mathcal J}_{0,+}=-\\tfrac{1}{2}\\partial \\log z\\,,\\qquad\n&\\bar{\\mathcal J}_{-,-}=\\bar J_-\\,,\\qquad\n&\\bar{\\mathcal J}_{-,+}=0\\,,\n\\end{aligned}\n\\end{equation}\nwhile for $S^3$\n\\begin{equation}\\label{eq:noether-chiralS3}\n\\begin{aligned}\n&\\mathcal J_{1,+}=J_1+\\tfrac{1}{4}\\partial\\phi_1\\,,\\qquad\n&\\mathcal J_{1,-}=-\\tfrac{1}{4}\\bar\\partial\\phi_1\\,,\\\\\n&\\bar{\\mathcal J}_{2,-}=\\bar J_2-\\tfrac{1}{4}\\bar\\partial\\phi_2\\,,\\qquad\n&\\bar{\\mathcal J}_{2,+}=\\tfrac{1}{4}\\partial\\phi_2\\,.\n\\end{aligned}\n\\end{equation}\nThis fact will later play an important role in our discussion.\n\n\n\\subsection{The $SL(2,\\mathbb R)\\times SU(2)$ WZW model}\nThe action of the WZW model is $S_{WZW}=S_1+k\\Gamma$ where $k$ is the level and\n\\begin{equation}\nS_1[g] = \\frac{k}{4 \\pi} \\int_{\\partial\\mathcal B} d^2\\sigma \\mathrm{Tr}(\\partial^ig^{-1}\\partial_i g )\\,,\\qquad\n\\Gamma[g] = -\\frac{1}{6\\pi}\\int_{\\mathcal B}d^3\\sigma \\epsilon^{ijk}\\mathrm{Tr}(g^{-1} \\partial_ig g^{-1} \\partial_jg g^{-1} \\partial_kg)\\,.\n\\end{equation}\nHere $g$ is an element of a group $G$, depending on coordinates on $\\mathcal B$, whose boundary $\\partial\\mathcal B$ is the worldsheet of the string.\nIn the following we will take the action\\footnote{The relative minus sign is needed to get the correct sign in front of the $S^3$ metric. In the supersymmetric case it is naturally accounted for by the supertrace.} $S_{WZW}[g_{\\rm a}]-S_{WZW}[g_{\\rm s}]$ with\n\\begin{equation}\ng_{\\rm a} = e^{x^+S_+}z^{2S_0}e^{-x^-S_-}\\in SL(2,\\mathbb R)\\,,\\qquad\ng_{\\rm s} = e^{\\phi_1T_1}e^{\\phi_3T_3}e^{\\phi_2T_2}\\in SU(2)\\,.\n\\end{equation}\nWe realise the generators of the algebra of $SL(2,\\mathbb R)$ in terms of the Pauli matrices as $S_0=\\sigma_3\/2$, $S_+=(\\sigma_1+i\\sigma_2)\/2$, $S_-=(\\sigma_1-i\\sigma_2)\/2$, and similarly for $SU(2)$ we take $T_a=\\tfrac{i}{2}\\sigma_a$.\nThe Killing form is related to the trace in this representation as $K_{ab}=f_{ac}^{\\ d}f_{bd}^{\\ c}=4\\mathrm{Tr}(S_aS_b)$, and similarly for $T_a$. We will use the bilinear form induced by the trace, rather than the Killing form, to raise and lower algebra indices.\n\nThe equations of motion for the action $S_{WZW}[g]$ imply chirality for the current $J=\\partial g g^{-1}$, and equivalently antichirality for the current $\\bar J=- g^{-1} \\bar \\partial g$, i.e. $\\bar \\partial J=0$, $\\partial \\bar J=0$.\nWe decompose the currents as $J=J_aS^a$ for $AdS_3$ and $J=J_aT^a$ for $S^3$, and similarly for $\\bar J$, where $S^0=2S_0, S^\\pm=S_\\mp$ and $T^a=-2T_a$. Thanks to these definitions the component $J_a$ of the chiral current corresponds to the action of the generator $S_a$ (or $T_a$ in the case of the sphere) from the left, while the component $\\bar J_a$ of the antichiral current corresponds to the action of the same generator from the right. The same holds for the corresponding Killing vectors $k_a$ and $\\bar k_a$. In particular we have $k_a^m\\partial_mg_{\\rm a}=+S_ag_{\\rm a}$, $\\bar k_{a}^m\\partial_mg_{\\rm a}=-g_{\\rm a}S_{a}$ for $AdS$ and $k_1^m\\partial_mg_{\\rm s}=-T_1g_{\\rm s}$, $\\bar k_{2}^m\\partial_mg_{\\rm s}=+g_{\\rm s}T_{2}$ for the sphere.\\footnote{The relative minus sign between $AdS_3$ and $S^3$ is again related to the fact that we are not using the supertrace in the action and in order to define the components of the currents.}  In our  parameterisation the components of the $SL(2)$ currents read \n\\begin{equation}\n\\begin{aligned}\n&J_0={\\phantom{-}}\\frac{z \\partial z-x^+\\partial x^-}{z^2}\\,,\n&& J_-=\\frac{x^+ (x^+\\partial x^--2 z \\partial z)}{z^2}+\\partial x^+\\,,\n&&& J_+=-\\frac{\\partial x^-}{z^2}\\,,\n\\\\\n&\\bar J_0=-\\frac{z \\bar\\partial z-x^-\\bar\\partial x^+}{z^2}\\,,\n&&\\bar J_+=\\frac{x^-(x^-\\bar\\partial x^+-2 z\\bar \\partial z)}{z^2}+\\bar\\partial x^-\\,,\n&&&\\bar J_-=-\\frac{\\bar\\partial x^+}{z^2}\\,,\n\\end{aligned}\n\\end{equation}\nwhile for the $SU(2)$ currents we have\n\\begin{equation}\n\\begin{aligned}\n&J_1=\\tfrac{1}{2} (-\\partial \\phi_1 - s_3\\partial\\phi_2 )\\,,\n&& J_2=\\tfrac{1}{2} (-c_1 c_3 \\partial \\phi_2 - s_1\\partial \\phi_3 )\\,,\n&&& J_3=\\tfrac{1}{2} (-c_1 \\partial\\phi_3 + c_3 s_1\\partial\\phi_2 )\\,,\n\\\\\n& \\bar J_2= \\tfrac{1}{2} (+\\bar\\partial \\phi_2 + s_3\\bar\\partial \\phi_1 )\\,, \n&&\\bar J_1=\\tfrac{1}{2} (+c_2 c_3 \\bar\\partial \\phi_1 + s_2\\bar\\partial \\phi_3 )\\,,\n&&&\\bar J_3=\\tfrac{1}{2} (+c_2 \\bar\\partial \\phi_3 - c_3 s_2\\bar\\partial \\phi_1 )\\,,\n\\end{aligned}\n\\end{equation}\nwhere we use the shorthand notation $s_i=\\sin\\phi_i$, $c_i=\\cos\\phi_i$.\nThe (anti)chiral currents appear also when computing the Noether currents from the action $S_{WZW}=S_1+k\\Gamma$. In fact, invariance of the WZW action under left transformations $g\\to (1+\\varepsilon_L+\\ldots)g$ implies the conservation of the Noether current\n$\\mathscr J^i=\\tfrac{1}{4} (\\eta^{ij} -\\epsilon^{ij})\\partial_j gg^{-1}$\nwhich is related to the chiral current as\n\\begin{equation}\\label{eq:noether-WZW}\n\\mathscr J=(\\mathscr J_+,\\mathscr J_{-})=(J,0)\\,. \n\\end{equation}\nSimilarly, from the right transformations $g\\to g(1+\\varepsilon_R+\\ldots)$ one finds the Noether current $\\bar{\\mathscr J}^i=-\\tfrac{1}{4} (\\eta^{ij} +\\epsilon^{ij})g^{-1}\\partial_j g$\nrelated to the antichiral current as\n\\begin{equation}\\label{eq:noetherbar-WZW}\n\\bar{\\mathscr J}=(\\bar{\\mathscr J}_{+},\\bar{\\mathscr J}_{-})=(0,\\bar J)\\,.\n\\end{equation}\nConservation of the Noether current $\\partial_i\\mathscr J^i=0$ (respectively $\\partial_i\\bar{\\mathscr J}^i=0$) implies chirality of $J$ (respectively antichirality of $\\bar J$).\nAs we have already pointed out, in general these Noether currents are not the same as those of the sigma model description, which we denoted by $\\mathcal J$.\n\n\\subsection{Marginal deformations}\\label{sec:marginal}\nIn~\\cite{Chaudhuri:1988qb} Chaudhuri and Schwartz considered two-dimensional CFTs with $J_a,\\bar J_{a}$ satisfying current algebra relations\\footnote{Since we are not normalising the currents with an explicit $k$ and we raise\/lower indices with the bilinear form induced by the trace, as opposed to the Killing form, certain factors differ from \\cite{Chaudhuri:1988qb}.}\n\\begin{equation}\\label{eq:JJ-OPE}\n\\begin{aligned}\nJ_a(\\sigma)J^b(\\sigma')\\sim \\frac{i\\ \\delta_a^b}{2k(\\sigma-\\sigma')^2}+ \\frac{if_{ac}^{\\ b}J^c}{2k(\\sigma-\\sigma')}\\,,\\\\\n\\bar J_{ a}(\\bar \\sigma)\\bar J^{ b}(\\bar \\sigma')\\sim \\frac{i\\ \\delta_{ a}^{ b}}{2k(\\bar \\sigma-\\bar \\sigma')^2}+ \\frac{if_{ a c}^{\\  b}\\bar J^{ c}}{2k(\\bar \\sigma-\\bar \\sigma')}\\,,\n\\end{aligned}\n\\end{equation}\nwhere we use $\\sim$ since we are omitting regular terms and $f_{ab}^{\\ c}$ are structure constants of a Lie algebra $\\mathfrak f$. \nThe authors of~\\cite{Chaudhuri:1988qb} were interested in exploring the space of marginal deformations induced by dimension $(1,1)$ operators of the type\n\\begin{equation}\ng\\mathcal O(\\sigma,\\bar \\sigma)= g c^{a b}J_a(\\sigma)\\bar J_{ b}(\\bar \\sigma)\\,,\n\\end{equation}\nwhere $c^{ab}$ are constant coefficients. The above operator is ``integrably'' or exactly marginal (i.e. can be completed to all orders in conformal perturbation theory in $g$) if it has no anomalous dimension, and they found that {a necessary condition for this to hold is that}\n\\begin{equation}\\label{eq:CS-weak}\nC^{abc}C^{def}  K_{ad} K_{be}K_{cf}\n+ \\bar C^{abc}\\bar C^{def}  K_{ad} K_{be}K_{cf}=0\\,,\n\\end{equation}\nwhere $K_{ab}$ is the Killing form and we have defined \n\\begin{equation}\\label{eq:def-C-Cbar}\nC^{abc}\\equiv c^{da}c^{eb}f_{de}^{\\ c}\\,,\\qquad\n\\bar C^{abc}\\equiv c^{ad}c^{be} f_{de}^{\\ c}\\,.\n\\end{equation}\nWe will call~\\eqref{eq:CS-weak} the \\emph{weak} Chaudhuri-Schwartz (CS) condition. Ref.~\\cite{Chaudhuri:1988qb} considered only the case of compact algebras, meaning that the Killing form $K_{ab}$ is negative definite and can be taken to be diagonal. In this case~\\eqref{eq:CS-weak} becomes a sum of squares of $C^{abc}$ and $\\bar C^{abc}$, and it holds if and only if\n\\begin{equation}\\label{eq:CS-strong}\nC^{abc}=0\\,,\\qquad\\text{and}\\qquad\n\\bar C^{abc}=0\\,.\n\\end{equation}\nWe will call~\\eqref{eq:CS-strong} the \\emph{strong} CS condition, because it is a stronger constraint in the case of non-compact algebras.\nIn~\\cite{Chaudhuri:1988qb} it was also shown that the strong condition is equivalent to being able to rewrite \n\\begin{equation}\n\\mathcal O(\\sigma,\\bar \\sigma)= \\tilde c^{a b}\\tilde J_a(\\sigma)\\tilde{\\bar{J}}_{ b}(\\bar \\sigma)\\,,\n\\end{equation}\nwhere $\\tilde J_a(\\sigma),\\tilde{\\bar{J}}_{ b}(\\bar \\sigma)$ are linear combinations of the original $J_a(\\sigma)$, ${\\bar{J}}_{ b}(\\bar \\sigma)$ such that \n\\begin{equation}\n\\begin{aligned}\n\\tilde J_a(\\sigma)\\tilde J^b(\\sigma')\\sim \\frac{i\\ \\delta_a^b}{2k(\\sigma-\\sigma')^2}\\,,\\qquad\n\\tilde{\\bar J}_{ a}(\\bar \\sigma)\\tilde{\\bar J}^{ b}(\\bar \\sigma')\\sim \\frac{i\\ \\delta_{ a}^{ b}}{2k(\\bar \\sigma-\\bar \\sigma')^2}\\,,\n\\end{aligned}\n\\end{equation}\ni.e. the structure constants for this particular set of currents vanish. {The absence of a simple pole in these OPEs means that they are the same as similar ones for free bosons, which in turn means that the $\\beta$-function for the deformation parameter $g$ vanishes and the deformation is exactly marginal. In other words, deformations corresponding to abelian subalgebras, which is the only possibility in the compact case, are exactly marginal.} When the Lie algebra $\\mathfrak f$ is non-compact {it is possible to find deformations that satisfy only the weak CS condition}, as we will see. {A priori they are not guaranteed to be marginal beyond lowest order, and indeed we will find both examples which are and those which are not.}\n\nIn fact a \\emph{sufficient} condition on $c^{ab}$ such that the weak CS~\\eqref{eq:CS-weak} holds is that the coefficients $c^{ab}$ identify two solvable subalgebras of $\\mathfrak f$ (one corresponding to $J_a$ and one to $\\bar J_b$). This follows directly from Cartan's criterion for a solvable Lie algebra $\\mathfrak h$\n\\begin{equation}\n\\mathfrak h \\text{ solvable } \\iff \\mathrm{Tr}(ab)=0, \\qquad \\forall a\\in\\mathfrak h, b\\in [\\mathfrak h,\\mathfrak h]\\,.\n\\end{equation}\nIf we are in such a situation then the two terms in~\\eqref{eq:CS-weak} separately vanish because\n\\begin{equation}\nC^{abc}C^{def}  K_{cf}=0\\,,\\qquad\\qquad\n \\bar C^{abc}\\bar C^{def}  K_{cf}=0\\,.\n\\end{equation}\nIn  the case of the $SL(2,\\mathbb R)\\times SU(2)$ WZW model, we may for example identify the two solvable subalgebras generated by $\\{S_0,S_+,T_1\\}$ and $\\{\\bar S_0,\\bar S_-,\\bar T_2\\}$. Then if we call $Y_a\\equiv\\{J_0,J_+,J_1\\}$ and $\\bar Y_a=\\{\\bar J_0,\\bar J_-,\\bar J_2\\}$ the list of the corresponding (anti)chiral currents, an operator\n\\begin{equation}\n\\mathcal O(\\sigma,\\bar \\sigma)=  c^{a b}Y_a(\\sigma)\\bar Y_{ b}(\\bar \\sigma)\\,,\n\\end{equation}\nwill be {marginal to lowest order} for \\emph{generic} coefficients $c^{ab}$. Notice that generically $c^{ab}$ will not solve the strong CS condition \\eqref{eq:CS-strong}.\n\nAll the solutions to the weak CS condition that we will generate from the CYBE on $\\mathfrak g=\\mathfrak f_{\\mbox{\\tiny L}}\\oplus \\mathfrak f_{\\mbox{\\tiny R}}=\\mathfrak{sl}(2,\\mathbb R)_{\\mbox{\\tiny L}}\\oplus \\mathfrak{su}(2)_{\\mbox{\\tiny L}}\\oplus \\mathfrak{sl}(2,\\mathbb R)_{\\mbox{\\tiny R}}\\oplus \\mathfrak{su}(2)_{\\mbox{\\tiny R}}$ will be of this type. Indeed to solve the CYBE it is enough to look at the subalgebra  generated by  $\\{S_0,S_+,T_1,\\bar S_0,\\bar S_-,\\bar T_2\\}$. When they come from the YB construction, the coefficients $c^{ab}$ will obviously not be generic, and we will relate them to certain components of the $R$-matrix, see the discussion at the end of section~\\ref{sec:YB-CS}. The YB construction has the advantage of giving a way to go beyond the infinitesimal deformation driven by $\\mathcal O(\\sigma,\\bar \\sigma)$, and gives a sigma-model action that is exact in the deformation parameter.\n\nAs we have argued, we expect the CYBE to give solutions to the weak CS condition in more generic situations. In appendix~\\ref{app:sl3} we discuss a solution of the CYBE that provides coefficients $c^{ab}$ that solve the weak CS condition without identifying solvable subalgebras.\n\n\\section{Yang-Baxter and current-current deformations}\\label{sec:YB}\n\n\\subsection{Yang-Baxter deformations}\nWe now review the transformation rules for the target space fields for YB deformations derived in~\\cite{Borsato:2018idb}.\nGiven an initial sigma model with metric and Kalb-Ramond fields $G_{mn}$, $B_{mn}$, the background of the YB deformed model is given by\n\\begin{equation}\n\\tilde G-\\tilde B = (G-B)[1+\\eta\\Theta(G-B)]^{-1}\\,,\n\\label{eq:GBtilde}\n\\end{equation}\nwhere for simplicity we are suppressing all spacetime indices. Here $\\Theta^{mn}=k^m_AR^{AB}k^n_B$ is a tensor constructed out of the Killing vectors $k^m_A$ and of $R^{AB}$, which is a solution to the CYBE on the Lie algebra $\\mathfrak g$\n\\begin{equation}\\label{eq:CYBE}\nR^{D[A}R^{|E|B}f_{DE}^{C]}=0\\,.\n\\end{equation}\nIn our case $\\mathfrak g= \\mathfrak f_{\\mbox{\\tiny L}}\\oplus \\mathfrak f_{\\mbox{\\tiny R}}$ is the sum of a left and a right copy of $\\mathfrak f=\\mathfrak{sl}(2)\\oplus \\mathfrak{su}(2)$, and $G,B$ were given in section~\\ref{sec:sigmamodel}. \nThe derivation of~\\cite{Borsato:2018idb} assumes that the $B$-field is invariant under the isometries used in the deformation, i.e. the ones appearing in $\\Theta$. This is ensured by picking the form of $B$ in section~\\ref{sec:sigmamodel} and using only the isometries generated by (\\ref{eq:subalgebra}), which is enough to generate any Yang-Baxter deformation (see appendix \\ref{app:R-matrices}).\n\nThe deformation produces also a shift of the dilaton calculated from the determinant\\footnote{In the supersymmetric case the determinant is replaced by the superdeterminant.}\n\\begin{equation}\ne^{-2\\tilde\\Phi}=e^{-2\\Phi}\\det[1+\\eta\\Theta(G-B)]\\,,\n\\label{eq:Phitilde}\n\\end{equation}\nwhere $\\Phi$ is the dilaton of the original background (in our case $\\Phi=0$).\nIn general YB backgrounds are solutions to the  equations of generalised supergravity~\\cite{Arutyunov:2015mqj,Wulff:2016tju}, so that in addition to the usual fields one may have also a vector $K$ computed as\\footnote{Eq.~\\eqref{eq:K} may be obtained from the formula derived in~\\cite{Borsato:2018idb} (i.e. $K^m=\\eta \\Theta^{mn}n_n=\\eta k^m_AR^{AB}n_B$ with $n_A=f_{AB}^{\\ B}$) after using the identity $R^{AB}f_{AB}^{\\ C}=-2f_{AB}^{\\ A}R^{BC}$, which is a consequence of the CYBE. It is also easy to check that~\\eqref{eq:K} agrees with $K^m=\\eta \\nabla_n^{(0)}\\Theta^{mn}$ proposed in~\\cite{Araujo:2017jkb}, where $\\nabla_n^{(0)}$ is the covariant derivative of the original undeformed background. Indeed, using first the Killing equation for $k_A^m$ and then the anti-symmetry of $R$, we have $\\nabla_m^{(0)}\\Theta^{mn}=k_A^mR^{AB}\\nabla_m^{(0)}k_B^n=R^{AB}k_A^m\\partial_mk_B^n$. Knowing that Killing vectors satisfy $[k_A^m\\partial_m,k_B^n\\partial_n]=-f_{AB}^{\\ C}k_C^p\\partial_p$ we obtain again~\\eqref{eq:K}.}\n\\begin{equation}\\label{eq:K}\nK^m = -\\frac{\\eta}{2}R^{AB}f_{AB}^{\\ C}k_C^m\\,,\n\\end{equation}\nwhich is a Killing vector of the YB background $\\nabla_{(m}K_{n)}=0$.\nFor such generalised supergravity solutions the role of (the derivative of) the dilaton is replaced by the vector\\footnote{This expression applies in a gauge where $B$ is invariant under the isometry generated by $K$, $\\mathcal L_KB=0$. Here we stick with the original notation of~\\cite{Wulff:2016tju}. In~\\cite{Borsato:2018idb} and~\\cite{Wulff:2018aku} $X_m$ was used instead, but there is a risk of confusing it with $X_m={\\rm X}_m+K_m$ of~\\cite{Wulff:2016tju}.}\n\\begin{equation}\\label{eq:X}\n{\\rm X}_m=\\partial_m\\tilde\\Phi -B_{mn}K^n\\,.\n\\end{equation}\nWhen $K^m$ vanishes one goes back to a standard supergravity solution. From~\\eqref{eq:K} the relation to the unimodularity condition of~\\cite{Borsato:2016ose} is manifest. There exist also so-called ``trivial solutions'' of generalised supergravity~\\cite{Wulff:2018aku}, i.e. when $K$ does not vanish but it decouples from the equations. A trivial solution is therefore both a solution of the generalised and the standard supergravity equations. Later we will encounter examples of this type.\n\n\n\\vspace{12pt}\n\n{Let us comment on the fact that the YB transformations constructed in~\\cite{Borsato:2018idb} were derived by assuming a group of left isometries for the sigma model. This is necessary in order to apply the NATD construction and twist the model with the corresponding Killing vectors $k_A$. The isometries that we will exploit here to deform $AdS_3\\times S^3$, corresponding to the generators in~\\eqref{eq:subalgebra}, belong both to the left and to the right copy of the symmetry group of the WZW model. The reason why we can apply the above rules of YB transformations is that the corresponding sigma model may be constructed as a coset on $SO(2,2)\/SO(1,2)\\times SO(4)\/SO(3)$ with a WZ term. For example, focusing on $AdS_3$, we may relate the generators of $\\mathfrak{sl}(2,\\mathbb R)$ to those of the conformal algebra as\n\\begin{equation}\n\\begin{aligned}\n&S_0=+\\tfrac{1}{2}(D-J_{01}),\\qquad\n&&S_+=p_+,\\qquad\n&&&S_-=k_-,\\\\\n&\\bar S_0=-\\tfrac{1}{2}(D+J_{01}),\\qquad\n&&\\bar S_+=k_+,\\qquad\n&&&\\bar S_-=p_-,\n\\end{aligned}\n\\end{equation}\nwhere e.g. $p_\\pm=\\tfrac12 (p_0\\pm p_1)$.\nThen we obtain the wanted sigma model action from $S=\\tfrac{k}{2\\pi}\\int d^2 \\sigma \\mathrm{Tr}[g^{-1}\\partial g(P+b)g^{-1}\\bar \\partial g]$ where $g=\\exp(x^+p_++x^-p_-)\\exp(D\\log z)$, $P$ projects on the generators of the coset $p_i-k_i$ and $D$, and finally  $b(p_\\pm-k_\\pm)=\\pm(p_\\pm-k_\\pm)$ produces the $B$-field. In this formulation the isometries that we want to exploit, generated by $S_0,S_+,\\bar S_0,\\bar S_-$, act from the left as $g\\to hg$ and leave also the $B$-field invariant.\n}\n \n\\vspace{12pt}\n\nFor later convenience, let us say at this point that it is easy to check that when an $R$-matrix is given by the sum of two $R$-matrices, the corresponding background can be understood as the composition of two successive YB transformations.\nThis is easily seen using the following identity valid when $\\Theta=\\Theta_1+\\Theta_2$\n\\begin{equation}\n[1+\\eta\\Theta_1(G-B)]^{-1}\\left[1+\\eta\\Theta_2(G-B)[1+\\eta\\Theta_1(G-B)]^{-1}\\right]^{-1}=[1+\\eta\\Theta(G-B)]^{-1}\\,,\n\\end{equation}\nwhich holds without assuming any property\\footnote{Obviously we need to assume invertibility of the above operators.} for $\\Theta_i$, neither antisymmetry nor CYBE.\nThanks to this formula it is straightforward to argue that the background metric and $B$-field of a YB deformation generated by $\\Theta=\\Theta_1+\\Theta_2$ are equivalent to those coming from the composition of two successive deformations, e.g. first one generated by $\\Theta_1$ and then one generated by $\\Theta_2$ (or vice versa). The same holds for the transformation rule of the dilaton, and of the vector $K$, which is linear in $\\Theta$ (or equivalently $R$). Obviously, the interpretation as YB deformations in the intermediate steps will be possible only if $\\Theta_1,\\,\\Theta_2$ separately solve the CYBE, and if the isometries needed to implement the second deformation are not broken by the first one. In this case we will say that $\\Theta$ is ``decomposable''. Apart from these subtleties, it will often prove useful to interpret a deformation generated by $\\Theta=\\Theta_1+\\Theta_2$ as a composition of two transformations.\n\nLater we will encounter examples in which $\\Theta_1$ generates the undeformed background \\emph{up to} a ($\\eta$-dependent) field redefinition. In this case one can say that $\\Theta=\\Theta_1+\\Theta_2$ is equivalent to the YB deformation generated by $\\Theta_2$ alone \\emph{only if} the field redefinition  $x^m\\to x^m(x')=x'{}^m+\\eta f^m(x')$ needed to trivialise $\\Theta_1$ is compatible with $\\Theta_2$. It is easy to convince oneself that the necessary compatibility condition is\n\\begin{equation}\\label{eq:compTheta2}\nA^{-1}\\Theta_2(x'{}^m+\\eta f^m(x'))A^{-T}=\\Theta_2(x'{}^m)\\,,\n\\end{equation}\nwhere $A^m_{\\ n}=\\frac{\\partial x^m}{\\partial x^{'n}}$, and we are writing the explicit dependence of $\\Theta_2$ on the coordinates. \n\n\n\\subsection{Relation to marginal current-current deformations}\\label{sec:YB-CS}\nBefore discussing YB deformations of $AdS_3\\times S^3$, let us make a simple observation: at leading order in the deformation parameter, the YB deformation is of the form $\\mathcal J\\mathcal J$, where $\\mathcal J$ are the Noether currents of the sigma model. This is straightforwardly checked by expanding the sigma model action $S=\\tfrac{T}{2}\\int d^2 \\sigma\\ \\partial x^m (\\tilde G_{mn}-\\tilde B_{mn})\\bar \\partial x^n$ to lowest order in the deformation parameter\\footnote{Recall that we are restricting to the case when the isometries used to construct the deformation leave not just the action but also the Lagrangian invariant, so that $j_{Ai}=0$. This was the assumption also in the derivation done in~\\cite{Borsato:2018idb}.}\n\\begin{equation}\\label{eq:expand-YB}\nS=S_0-\\eta\\frac{T}{2}\\int d^2\\sigma\\,R^{AB}\\mathcal J_{A+}\\mathcal J_{B-} +\\mathcal{O}(\\eta^2)\\,.\n\\end{equation}\nWhile this is true for a generic sigma model, this observation is particularly interesting when the original sigma model is related to a WZW model. If the Noether currents $\\mathcal J_{Ai}$ coincided with the chiral Noether currents of the WZW model $\\mathscr J_{Ai}=\\{\\mathscr J_{ai},\\bar{\\mathscr J}_{ai}\\}$, then we would automatically obtain a current-current deformation of the type $J\\bar J$. In fact, from~\\eqref{eq:noether-WZW} and ~\\eqref{eq:noetherbar-WZW} one immediately finds that\\footnote{Notice that the $R$-matrix may have also non-vanishing components with both indices in the left (or both in the right) copy of the algebra, but these will not contribute in the final expression, and the contributions coupling currents with the same chirality cancel out.} $\\int d^2\\sigma \\ \\epsilon^{ij}\\, R^{BA}\\mathscr J_{Ai} \\mathscr J_{Bj}=4\\int d^2\\sigma\\ R^{a\\bar b} J_a \\bar J_{\\bar b}$. \nAs we have seen, though, in general $\\mathcal J_{A}^i= \\mathscr J_{A}^i+\\epsilon^{ij}\\partial_jc_A$, and the  discussion is more subtle because~\\eqref{eq:expand-YB} will contain additional terms together with the wanted $J\\bar J$ ones. We are about to show that for YB deformations of the $AdS_3\\times S^3$ sigma model these additional terms can be removed by proper field redefinitions. We can therefore relate YB deformations to the deformations of the type $c^{a b}J_a\\bar J_{ b}$ considered by CS. From this discussion it is also clear that we should identify the coefficients $c^{a b}$ of CS with an ``off-diagonal'' block of the $R$-matrix. More explicitly, since $R\\in \\mathfrak g\\wedge \\mathfrak g$ we are solving the CYBE on an algebra which is the sum of a left and a right copy $\\mathfrak g=\\mathfrak f_{\\mbox{\\tiny L}}\\oplus \\mathfrak f_{\\mbox{\\tiny R}}$, the $R$-matrix can be decomposed as\n\\begin{equation}\nR=\\left(\n\\begin{array}{cc}\nR_{\\mbox{\\tiny L}\\sL} & R_{\\mbox{\\tiny L}\\mbox{\\tiny R}}\\\\\nR_{\\mbox{\\tiny R}\\mbox{\\tiny L}} & R_{\\mbox{\\tiny R}\\sR}\n\\end{array}\n\\right)\n\\end{equation}\nwith $R_{\\mbox{\\tiny L}\\sL}^{T}=-R_{\\mbox{\\tiny L}\\sL}, R_{\\mbox{\\tiny R}\\sR}^{T}=-R_{\\mbox{\\tiny R}\\sR}$ and $R_{\\mbox{\\tiny L}\\mbox{\\tiny R}}^T=-R_{\\mbox{\\tiny R}\\mbox{\\tiny L}}$. The relation to the coefficients of CS is therefore $c^{a b}=R_{\\mbox{\\tiny L}\\mbox{\\tiny R}}^{ab}$.\nWe can therefore generate solutions to the (weak) CS condition from solutions of the CYBE, and we will find several non-trivial examples in the following.\nObviously abelian $R$-matrices ($R=a\\wedge b, [a,b]=0$) will give solutions of the strong CS condition. When dealing with non-compact algebras the CYBE allows also for solutions that are not of the abelian type. Some of them will give coefficients $c^{ab}$ that do not solve the strong CS condition. They all solve the weak CS condition as already explained at the end of section~\\ref{sec:marginal}. \n\nIt would be very interesting to understand more deeply the relation between the space of solutions of the CYBE~\\eqref{eq:CYBE} and the weak CS condition~\\eqref{eq:CS-weak} in the case of a generic algebra $\\mathfrak g=\\mathfrak f_{\\mbox{\\tiny L}}\\oplus \\mathfrak f_{\\mbox{\\tiny R}}$. {It is interesting to notice that in order to solve the CYBE one may need also components of the diagonal blocks $R_{\\mbox{\\tiny L}\\sL}$ and $R_{\\mbox{\\tiny R}\\sR}$, while in the weak CS condition these will not enter. In fact, given $R_{\\mbox{\\tiny L}\\mbox{\\tiny R}}^{a\\bar b}=c^{a \\bar b}$ and taking the CYBE on mixed left\/right indices (where we use an explicit bar for indices of the right copy of the algebra), one gets for example $c^{d\\bar a}R^{eb}f_{de}^{\\ c}+c^{b\\bar d}c^{c\\bar e}f_{\\bar d\\bar e}^{\\ \\bar a}+R^{dc}c^{e\\bar a}f_{de}^b=0$. Depending on the coefficients $c^{a\\bar b}$ one may also need non-vanishing left-left $R^{ab}$ components in order to solve this equation,\\footnote{Such an example is given by the fourth $R$-matrix in table~\\ref{tab:non-ab-sl2}.} but the CS condition is not sensitive to them.}\n\nLet us also comment that, differently from what was claimed in~\\cite{Araujo:2018rho}, the strong CS condition~\\eqref{eq:CS-strong} is \\emph{not} the CYBE, not even when one further imposes the unimodularity condition (and in fact our $\\mathbf R_9$ in table~\\ref{tab:non-ab-sl2-su2-r4} is a counter example to that claim).\n\n\n\\subsection{Field redefinition}\nIn order to display the current-current structure of the deformed model it is convenient to write the Lagrangian in the form\n\\begin{equation}\nL=L_0-\\tfrac12\\eta\\mathcal J_{A+}[(1+\\eta RM)^{-1}R]^{AB}\\mathcal J_{B-}\\,,\n\\label{eq:Lprime}\n\\end{equation}\nwhere $L_0$ is the undeformed Lagrangian and $M_{AB}=k_A^mk_B^n(G-B)_{mn}$. This follows directly from the form of $\\tilde G-\\tilde B$ in (\\ref{eq:GBtilde}) and the definition of the Noether currents in (\\ref{eq:Noether}) upon recalling that we will pick $R$ so that the last term in $\\mathcal J$ does not contribute. To compare this to the discussion of current-current deformations of the WZW model we need to perform a field redefinition that replaces the Noether currents in the above expression with the chiral currents.\\footnote{It is actually enough to look at the lowest order in $\\eta$ if we are considering infinitesimal deformations. {The action written in terms of the Noether currents as in~\\eqref{eq:Lprime} was given also in~\\cite{Araujo:2018rho} but the rewriting in terms of the chiral currents is missing there.}} In appendix~\\ref{app:field-red} we find such a field redefinition for a general deformation specifying to $AdS_3\\times S^3$ for concreteness.\nHere we will just say that for all deformations that we consider we find that the Lagrangian can be written in the form\n\\begin{equation}\nL=\nL_0\n-\\tfrac12\\eta\\hat J_a[(1+\\eta RM')^{-1}R]^{a\\bar a}\\hat{\\bar J}_{\\bar a}\\,,\n\\label{eq:LdefAdS3}\n\\end{equation}\nwhere $M'$ is a shorthand for $M$ after the field redefinition. $\\hat J_a,\\hat{\\bar J}_{\\bar a}$ are modifications of the chiral currents of the undeformed WZW model. Their explicit form is given in~\\eqref{eq:JhatAdS3} for deformations of $AdS_3$, and in section~\\ref{sec:def-AdS3S3} for deformations of $AdS_3\\times S^3$.\n\nAt leading order in $\\eta$ the above Lagrangian becomes\n\\begin{equation}\nL=\nL_0\n-\\tfrac12\\eta J_aR^{a\\bar a}\\bar J_{\\bar a}+\\mathcal{O}(\\eta^2)\\,,\n\\end{equation}\nso that the comparison to the current-current deformations considered by CS is now manifest.\n\n\\subsection{YB deformations of $AdS_3$}\\label{sec:def-AdS3}\nLet us start by looking at YB deformations that deform only $AdS_3$. We will start with the simplest ones which are TsT-transformations. They come from the abelian $R$-matrices of $\\mathfrak{sl}(2,\\mathbbm R)_{\\mbox{\\tiny L}}\\oplus \\mathfrak{sl}(2,\\mathbbm R)_{\\mbox{\\tiny R}}$ which (up to automorphisms) are\\footnote{From now on we will use the components $R^{AB}$ to construct $R=R^{AB}\\mathbf T_A\\wedge \\mathbf T_B\\in \\mathfrak g\\wedge \\mathfrak g$.\n} \\footnote{We could consider also $R=S_0\\wedge \\bar S_-$, but it is related to $R=S_+\\wedge \\bar S_0$ if we also exchange the left and right copy of the algebra.}\n\\begin{equation}\nS_+\\wedge\\bar S_-\\,,\\qquad S_0\\wedge\\bar S_0\\,,\\qquad S_+\\wedge\\bar S_0\\,.\n\\end{equation}\nFor the first one we obtain, from (\\ref{eq:GBtilde}) and (\\ref{eq:Phitilde}), the supergravity background of a deformation of $AdS_3\\times S^3\\times T^4$\n\\begin{equation}\n\\begin{aligned}\nds^2&=\n-\\frac{dx^+dx^-}{z^2-\\eta}\n+\\frac{dz^2}{z^2}+ds_{S^3}^2+ds_{T^4}^2\\,,\\\\\nB&=\\frac{dx^+\\wedge dx^-}{2(z^2-\\eta)}\n-\\frac{1}{4}\\sin \\phi_3 d\\phi_1\\wedge d\\phi_2\\,,\\qquad\\quad\ne^{-2\\tilde\\Phi}=1-\\frac{\\eta}{z^2}\\,.\n\\end{aligned}\n\\end{equation}\nIn this case the isometries involved in the deformation procedure correspond to Noether currents that agree with the (anti)chiral currents, see~\\eqref{eq:noether-chiralAdS3}. We therefore automatically get that to leading order the deformation of the Lagrangian is given by the marginal operator $\\eta J_+\\bar J_-$. In~\\cite{Giveon:2017nie} the above  background was argued to be the dual of the ``single-trace'' $T\\bar T$ deformation of the symmetric product orbifold CFT$_2$. This is also is accordance with the fact this particular YB deformation is just a TsT transformation involving the two boundary coordinates.\\footnote{We can have a TsT interpretation because we can implement the sequence T-duality, shift, T-duality in terms of the coordinates $x^0,x^1$, instead of the null coordinates $x^\\pm$.}\nAt finite order in the deformation parameter the Lagrangian is given by (\\ref{eq:LdefAdS3}), (\\ref{eq:JhatAdS3}) and (\\ref{eq:M})\n\\begin{equation}\nL=L_0-\\tfrac12\\frac{\\eta}{1-\\frac{\\eta}{z^2}}J_+\\bar J_-\\,.\n\\label{eq:TsT1}\n\\end{equation}\nThe derivative of the action with respect to the deformation parameter is given by\n\\begin{equation}\n\\frac{dS}{d\\eta}=-\\frac{T}{2}\\int d^2\\sigma \\ J_+^\\eta\\bar J_-^\\eta\\,,\n\\end{equation}\nwhere $J_+^\\eta=(1-\\eta z^{-2})^{-1}J_+$ and $\\bar J_-^\\eta=(1-\\eta z^{-2})^{-1}\\bar J_-$ are (anti)chiral currents of the \\emph{deformed} model.{ This background was analysed in~\\cite{Forste:1994wp}.}\nLet us also note that in this case the deformation parameter can be absorbed by a rescaling of the coordinates. There are therefore only three cases: $\\eta>0$, $\\eta=0$ and $\\eta<0$. The first of these is not globally well behaved since the dilaton becomes imaginary when crossing $z=\\sqrt\\eta$. For $\\eta<0$ the solution interpolates between two CFTs: the $SL(2,\\mathbbm R)$ WZW model and a linear dilaton background (plus two decoupled bosons). It would be interesting to study further the implications of the existence of this interpolating solution but we will not do so here. See also~\\cite{Giveon:2017nie} for a discussion on the different interpretations depending on the sign of the deformation parameter.\n\nFor the remaining two abelian $R$-matrices we obtain by a similar calculation the Lagrangians\\footnote{{The $J_0\\bar J_0$-deformation was considered also in~\\cite{Israel:2003ry}.}}\n\\begin{equation}\n\\begin{aligned}\n&L=L_0-\\tfrac12\\eta\\frac{(4+\\eta)z^2}{4z^2+\\eta(4+\\eta)x^+x^-}J_0 \\bar J_0\\,,\\\\\n&L=L_0-\\tfrac12\\eta\\frac{z^2}{z^2+\\eta  x^-}J_+ \\bar J_0\\,.\n\\end{aligned}\n\\end{equation}\nFor all the abelian examples one finds, as already mentioned, that\n\\begin{equation}\n\\frac{dS}{d\\eta}=-\\frac{T}{2}\\int d^2\\sigma \\ R^{a\\bar a}J_a^\\eta\\bar J_{\\bar a}^\\eta\\,,\n\\end{equation}\nwhere $J_a^\\eta,\\bar J_{\\bar a}^\\eta$ are (anti)chiral currents of the deformed model. For the last two abelian examples they are\n\\begin{equation}\n\\begin{aligned}\n&R=S_0\\wedge \\bar S_0: && \nJ_0^\\eta=\\alpha J_0\\,, \\qquad \n\\bar J_0^\\eta=\\alpha \\bar J_0\\,,\\qquad\n\\alpha\\equiv\\frac{4 z^2\\sqrt{1+\\frac{\\eta }{2}}}{4z^2+\\eta(4+\\eta)x^+x^-}\\,,\\\\\n&R=S_+\\wedge \\bar S_0: && J_+^\\eta=\\alpha J_+\\,, \\qquad \n\\bar J_0^\\eta=\\alpha \\bar J_0\\,,\\qquad\n\\alpha\\equiv\\frac{z^2}{z^2+\\eta  x^-}\\,.\n\\end{aligned}\n\\end{equation}\nFollowing~\\cite{Forste:1994wp}, the above result is another way to see that the YB model provides a deformation that is marginal exactly in the deformation parameter.\n\n\\vspace{12pt}\n\nIn order to find deformations that at least potentially are not TsT, one should look at the class of non-abelian $R$-matrices. The full list of non-abelian $R$-matrices of $\\mathfrak{sl}(2,\\mathbb{R})_{\\mbox{\\tiny L}}\\oplus\\mathfrak{sl}(2,\\mathbb{R})_{\\mbox{\\tiny R}}$ (up to $SL(2,\\mathbb{R})_{\\mbox{\\tiny L}}\\times SL(2,\\mathbb{R})_{\\mbox{\\tiny R}}$ automorphisms) is given in table~\\ref{tab:non-ab-sl2}. They are special cases of the $R$-matrices for $\\mathfrak{sl}(2,\\mathbbm R)_{\\mbox{\\tiny L}}\\oplus \\mathfrak{sl}(2,\\mathbbm R)_{\\mbox{\\tiny R}}\\oplus\\mathfrak{su}(2)_{\\mbox{\\tiny L}}\\oplus \\mathfrak{su}(2)_{\\mbox{\\tiny R}}$ classified in appendix \\ref{app:R-matrices}.\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|}\n\\hline\n& & & \\\\\n$R=R^{AB}\\mathbf T_A\\wedge \\mathbf T_B$ & Deformation?& $c^{ab}J_a\\bar J_b$ & Strong CS?\\\\\n\\hline\n$R_1=S_0\\wedge S_+$ & Trivial deformation & 0& Yes\\\\\n$R_2=(S_0\\mp\\bar S_-)\\wedge S_+$ & TsT & $\\pm J_+\\bar J_-$& Yes\\\\\n$R_3=(S_0-a\\bar S_0)\\wedge S_+$ & TsT & $aJ_+\\bar J_0$& Yes\\\\\n$R_4=(S_0-\\bar S_0)\\wedge (S_+\\pm\\bar S_-)$ & Not SUGRA & $J_+\\bar J_0\\pm J_0\\bar J_-$& No\\\\\n$R_5=S_0\\wedge S_+\\pm\\bar S_0\\wedge \\bar S_-+\\lambda {S_+\\wedge \\bar S_-}$ & {TsT} & $\\lambda {J_+\\bar J_-}$& Yes\\\\\n\\hline\n\\end{tabular}\n\\caption{Non-abelian $R$-matrices of $\\mathfrak{sl}(2)_{\\mbox{\\tiny L}}\\oplus\\mathfrak{sl}(2)_{\\mbox{\\tiny R}}$ up to $SL(2,\\mathbb{R})_{\\mbox{\\tiny L}}\\times SL(2,\\mathbb{R})_{\\mbox{\\tiny R}}$ inner automorphisms and swaps of $\\mbox{\\tiny L}\\leftrightarrow \\mbox{\\tiny R}$. For convenience we also write the marginal operators that they give rise to, and whether they satisfy the strong CS condition.}\n\\label{tab:non-ab-sl2}\n\\end{center}\n\\end{table}\nAnalysing the first example $R_1=S_0\\wedge S_+$ one finds that, after the field redefinition $x^+\\to x^+-\\frac{\\eta}{2}\\log z$, not only the leading order in $\\eta$ vanishes in the action, but also all the higher orders. This is obvious from eq. (\\ref{eq:LdefAdS3}) and the fact that $R$ with an anti-chiral index vanishes. In other words the deformation is trivial, since its effect is to give the undeformed $AdS_3$ background in new ($\\eta$-dependent) coordinates. To be more precise, from $R_1$ we get back undeformed $AdS_3$ \\emph{up to} a non-vanishing $K=-\\eta\\partial_{x^+}$, from (\\ref{eq:K}). This is of course a ``trivial solution''  of generalised supergravity (i.e. one that solves also standard supergravity upon dropping $K$, notice  that $K$ is null).\\footnote{Obviously a similar discussion holds also for $R=\\bar S_0\\wedge \\bar S_-$.}\n\nThe above result for $R_1=S_0\\wedge S_+$ turns out to be useful to analyse some of the following examples in the table. \nIt is easy to see that $R_2$ and $R_3$ in table~\\ref{tab:non-ab-sl2} are of the form $R= S_0\\wedge S_++R'$ and that in both cases $R'$ is compatible with the field redefinition  that trivialises the effect of the piece $S_0\\wedge S_+$, see~\\eqref{eq:compTheta2}. Therefore these two $R$-matrices are equivalent to two of the TsT transformations already discussed,\\footnote{Also in this case the equivalence holds up to a non-vanishing $K=-\\eta\\partial_{x^+}$ that is null and decouples from the equations of generalised supergravity.} generated by $S_+\\wedge \\bar S_-$ and $S_+\\wedge \\bar S_0$. Alternatively it is easy to see this directly from the Lagrangian in (\\ref{eq:LdefAdS3}) and (\\ref{eq:JhatAdS3}).\n\n\nThe last two $R$-matrices in table~\\ref{tab:non-ab-sl2}, instead, give backgrounds that are not of this type. From (\\ref{eq:K}) we find that they have $K=-\\eta(\\partial_{x^+}\\pm\\partial_{x^-})$ and $K=-\\eta(\\partial_{x^+}\\mp\\partial_{x^-})$ respectively, neither of which is null. The only way they can give solutions of standard supergravity is then, {as shown in appendix \\ref{app:trivial}, if ${\\rm X}=d\\phi+\\tilde K$ with $\\phi$ the dilaton and $\\tilde K$ an independent Killing vector field. We can extract $\\tilde K$ from the equation $dK+i_{\\tilde K}H=0$ in (\\ref{eq:trivial}) and for $R_4$ one finds\n$\\tilde K=\\eta(\\partial_{x^+}\\mp\\partial_{x^-})\\pm\\eta^2z^{-1}\\partial_z+\\mathcal O(\\eta^3)$ which, at lowest order, differs from $K$ only in the sign of the $\\partial_{x^+}$-term. However from the lowest order correction to the action $J_+\\bar J_0\\pm J_0\\bar J_-$ we read off the deformed metric \n\\begin{equation}\nz^{-2}(-dx^-dx^++dzdz)-\\eta z^{-4}(zdz(dx^-\\mp dx^+)+(\\pm x^+-x^-)dx^+dx^-)+\\mathcal O(\\eta^2)\\,,\n\\end{equation}\nwhich is only invariant under $\\delta x^+=\\eta\\epsilon$, $\\delta x^-=\\pm\\eta\\epsilon$ which is generated by $K$, and not under $\\delta x^+=\\eta\\epsilon$, $\\delta x^-=\\mp\\eta\\epsilon$, $\\delta z=\\pm\\eta^2\\epsilon z^{-1}$ which is generated by $\\tilde K$. Therefore $\\tilde K$ is not Killing and the $R_4$-background is not a solution to standard supergravity. Note that it only fails to be a solution at order $\\eta^2$ which is consistent with the fact that the leading-order deformation  satisfies the weak CS condition.}\n We will not consider this background further here since our interest is mainly in string theory applications. For $R_5$, instead, one can show that $\\tilde K$ is a Killing vector of the deformed metric and it satisfies~\\eqref{eq:trivial}, meaning that we get a ``trivial solution'' of generalised supergravity for generic $\\lambda$. Actually, using (\\ref{eq:LdefAdS3}) and (\\ref{eq:JhatAdS3}), for $R_5$ we find the Lagrangian\\footnote{For concreteness we take $R_5=S_0\\wedge S_++\\bar S_0\\wedge \\bar S_-+\\lambda {S_+\\wedge \\bar S_-}$, but similar results apply also for $R_5=S_0\\wedge S_+-\\bar S_0\\wedge \\bar S_-+\\lambda {S_+\\wedge \\bar S_-}$.}\n\\begin{equation}\nL=L_0+{\\frac{\\eta  z^2 (\\eta -4 \\lambda )}{2 \\left(\\eta  (\\eta -4 \\lambda )+4 z^2\\right)}}J_+\\bar J_-\\,,\n\\label{eq:lambda0}\n\\end{equation}\nwhich is exactly that of the TsT example in~\\eqref{eq:TsT1} with {$\\eta\\rightarrow \\eta\\lambda-\\eta^2\/4$}. This background therefore provides an example of the more general kind of trivial solution of the generalized supergravity equations discussed in appendix \\ref{app:trivial} for which $K^2\\neq0$.\n\n{This example also shows how the identification of the deformation parameters can be non-trivial. In fact, although at leading order the marginal deformation is given only by $\\eta\\lambda J_+\\bar J_-$, the deformation exact in $\\eta$ shows that the deformation parameter of the TsT is instead $\\eta\\lambda-\\eta^2\/4$, and in particular it does not vanish even when $\\lambda=0$.}\n\nIt is interesting to look at the marginal operators of the type $c^{ab}J_a\\bar J_b$  that are generated by each $R$-matrix. We write them for convenience in table~\\ref{tab:non-ab-sl2}. While they all solve the weak CS condition, they all solve also the strong CS condition {(which guarantees that they are exactly marginal)} except for the fourth one {(which we have seen fails to be marginal beyond lowest order)}. \n\n\\subsection{YB deformations of $AdS_3\\times S^3$}\\label{sec:def-AdS3S3}\nLooking at deformations of $AdS_3\\times S^3$ gives a richer set of possibilities. In this case we want to solve the CYBE on the algebra $\\mathfrak{sl}(2)_{\\mbox{\\tiny L}}\\oplus\\mathfrak{sl}(2)_{\\mbox{\\tiny R}}\\oplus\\mathfrak{su}(2)_{\\mbox{\\tiny L}}\\oplus\\mathfrak{su}(2)_{\\mbox{\\tiny R}}$.\n\n\nThe simplest example to start with is $R=S_+\\wedge \\bar T_2$. This is an abelian $R$-matrix and therefore corresponds to a TsT mixing $AdS_3$ and $S^3$. In~\\cite{Chakraborty:2018vja,Apolo:2018qpq} it was argued that this background is dual to the single-trace $T\\bar J$ deformation of the CFT$_2$. From the YB procedure one explicitly finds\n\\begin{equation}\n\\begin{aligned}\nds^2&=ds_{\\text{AdS}_3}^2+ds_{\\text{S}^3}^2+\\frac{\\eta}{4z^2}dx^-(d\\phi_2+2\\sin \\phi_3d\\phi_1),\\\\\nB&=B_0\n+\\frac{\\eta  dx^-\\wedge(d\\phi_2 +2 \\sin \\phi_3 d\\phi_1)}{8 z^2},\n\\qquad e^{-2\\Phi}=1.\n\\end{aligned}\n\\end{equation}\nFrom~\\eqref{eq:noether-chiralS3} we see that the Noether current $\\bar{\\mathcal J}_2$ differs from the antichiral current $\\bar J_2$, and therefore field redefinitions are needed in order to put the action in the form that makes the chirality structure manifest. After the field redefinition $x^+\\to x^+-\\frac{\\eta}{4}\\phi_2$, or equivalently by looking directly at (\\ref{eq:Ldef}), (\\ref{eq:Jhat}) and (\\ref{eq:Jbarhat}) the Lagrangian becomes\n\\begin{equation}\nL=L_0-\\tfrac12\\eta J_+\\bar J_2\\,.\n\\end{equation}\nThis deformation is special since the leading linear order is exact. Obviously $dS\/d\\eta=- \\tfrac{T}{2} \\int J_+\\bar J_{2}$.\n\nWe will now focus on non-abelian $R$-matrices. These are classified for $\\mathfrak{sl}(2)_{\\mbox{\\tiny L}}\\oplus\\mathfrak{sl}(2)_{\\mbox{\\tiny R}}\\oplus\\mathfrak{su}(2)_{\\mbox{\\tiny L}}\\oplus\\mathfrak{su}(2)_{\\mbox{\\tiny R}}$ in appendix~\\ref{app:R-matrices}. Since the CYBE on the two copies of $\\mathfrak{su}(2)$ implies that the subset of generators from this part of the algebra should be abelian, our classification is useful also to study deformations of the full $AdS_3\\times S^3\\times T^4$ or the more generic $AdS_3\\times S^3\\times S^3\\times S^1$. Every time an $\\mathfrak{su}(2)$ generator appears, it may as well be replaced by another compact generator, as long as all compact generators involved form an abelian subalgebra. For concreteness we will only look at deformations of $AdS_3\\times S^3$. The rank-2 non-abelian $R$-matrices are collected in table~\\ref{tab:non-ab-sl2-su2-r2}. Since they are special cases of the rank-4 $R$-matrices, listed in table~\\ref{tab:non-ab-sl2-su2-r4}, we will not consider them separately.\n\n\n\\begin{table}[ht]\n\\begin{center}\n\\begin{tabular}{|l|c|}\n\\hline\n&   \\\\\n$R=R^{AB}\\mathbf T_A\\wedge \\mathbf T_B$ & Deformation?\\\\\n\\hline\n&   \\\\\n$\\mathbf R_1=(S_0-\\bar S_0+T)\\wedge (S_+\\pm\\bar S_-)$ & $R_4$+TsT$\\implies$ not SUGRA  \\\\\n$\\mathbf R_2=(S_0-a\\bar S_0+T)\\wedge S_+$ & $R_3$+TsT$\\implies$ TsT \\\\\n$\\mathbf R_3=(S_0\\mp\\bar S_-+T)\\wedge S_+$ & $R_2$+TsT$\\implies$ TsT  \\\\\n\\hline\n\\end{tabular}\n\\begin{tabular}{|c|c|c|}\n\\hline\n& &  \\\\\n$R$& $c^{ab}J_a\\bar J_b$ & Strong CS?\\\\\n\\hline\n&  & \\\\\n$\\mathbf R_1$ &  $\\pm(J_0+aJ_1)\\bar J_-+J_+(\\bar J_0+b\\bar J_2)$ & No \\\\\n$\\mathbf R_2$ & $J_+(a\\bar J_0+b\\bar J_2)$& Yes \\\\\n$\\mathbf R_3$ & $J_+(\\pm\\bar J_-+b\\bar J_2)$& Yes \\\\\n\\hline\n\\end{tabular}\n\\caption{Rank-2 non-abelian $R$-matrices of $\\mathfrak{sl}(2)_{\\mbox{\\tiny L}}\\oplus\\mathfrak{sl}(2)_{\\mbox{\\tiny R}}\\oplus \\mathfrak{su}(2)_{\\mbox{\\tiny L}}\\oplus\\mathfrak{su}(2)_{\\mbox{\\tiny R}}$ up to $SL(2,\\mathbb{R})_{\\mbox{\\tiny L}}\\times SL(2,\\mathbb{R})_{\\mbox{\\tiny R}}\\times SU(2)_{\\mbox{\\tiny L}}\\times SU(2)_{\\mbox{\\tiny R}}$ inner automorphisms and swaps of $\\mbox{\\tiny L}\\leftrightarrow \\mbox{\\tiny R}$. With $T$ we denote a generic linear combination of $T_1, \\bar T_2$. }\n\\label{tab:non-ab-sl2-su2-r2}\n\\end{center}\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|l|c|}\n\\hline\n& \\\\\n$R=R^{AB}\\mathbf T_A\\wedge \\mathbf T_B$ & Deformation?\\\\\n\\hline\n&  \\\\\n$\\mathbf R_4=\\mathbf R_1+a T_1\\wedge \\bar T_2$ & $\\mathbf R_1+$TsT$\\implies$not SUGRA \\\\\n$\\mathbf R_5=\\mathbf R_2+b T_1\\wedge \\bar T_2$ & $\\mathbf R_2+$TsT$\\implies$TsT  \\\\\n$\\mathbf R_6=(S_0+T)\\wedge S_++T' \\wedge (\\bar S_0+T'')$ & SUGRA  \\\\\n$\\mathbf R_7=(S_0+T)\\wedge S_++T' \\wedge (\\bar S_-+T'')$ & TsT  \\\\\n$\\mathbf R_8=\\mathbf R_3+a T_1\\wedge \\bar T_2$ & $\\mathbf R_3+$TsT$\\implies$TsT \\\\\n$\\mathbf R_9=(S_0+\\bar S_0+T)\\wedge T'+S_+\\wedge \\bar S_-$ & SUGRA, not TsT \\\\\n$\\mathbf R_{10}={(S_0+T)\\wedge S_+\\pm(\\bar S_0+T')\\wedge \\bar S_-+\\lambda S_+\\wedge \\bar S_-}$ & {TsT}  \\\\\n\\hline\n\\end{tabular}\n\\begin{tabular}{|c|c|c|}\n\\hline\n&  & \\\\\n$R$& $c^{ab}J_a\\bar J_b$ & Strong CS?\\\\\n\\hline\n& &  \\\\\n$\\mathbf R_4$  & $(J_0+aJ_1)\\bar J_-+J_+(\\bar J_0+b\\bar J_2)+a J_1\\bar J_2$ & No \\\\\n$\\mathbf R_5$  & $J_+(a\\bar J_0+b\\bar J_2)+bJ_1\\bar J_2$ & Yes \\\\\n$\\mathbf R_6$ & $aJ_+\\bar J_2+bJ_1\\bar J_0+cJ_1\\bar J_2$ & Yes \\\\\n$\\mathbf R_7$  &  $aJ_+\\bar J_2+bJ_1\\bar J_-+cJ_1\\bar J_2$ & Yes \\\\\n$\\mathbf R_8$ & $J_+(\\bar J_-+b\\bar J_2)+aJ_1\\bar J_2$ & Yes \\\\\n$\\mathbf R_9$ & $aJ_0\\bar J_2+bJ_1\\bar J_0+cJ_1\\bar J_2+J_+\\bar J_-$ & No ($a\\neq 0$ or $b\\neq 0$)\\\\\n$\\mathbf R_{10}$ & ${cJ_+\\bar J_2+d J_1\\bar J_-+\\lambda J_+\\bar J_-}$ & {Yes} \\\\\n\\hline\n\\end{tabular}\n\\caption{Rank-4 non-abelian $R$-matrices of $\\mathfrak{sl}(2)_{\\mbox{\\tiny L}}\\oplus\\mathfrak{sl}(2)_{\\mbox{\\tiny R}}\\oplus \\mathfrak{su}(2)_{\\mbox{\\tiny L}}\\oplus\\mathfrak{su}(2)_{\\mbox{\\tiny R}}$ up to $SL(2,\\mathbb{R})_{\\mbox{\\tiny L}}\\times SL(2,\\mathbb{R})_{\\mbox{\\tiny R}}\\times SU(2)_{\\mbox{\\tiny L}}\\times SU(2)_{\\mbox{\\tiny R}}$ inner automorphisms and swaps of $\\mbox{\\tiny L}\\leftrightarrow \\mbox{\\tiny R}$. With $T,T',T''$ we denote  generic linear combinations of $T_1, \\bar T_2$.}\n\\label{tab:non-ab-sl2-su2-r4}\n\\end{center}\n\\end{table}\n\nTo compute the Killing vector $K$ that appears in the generalized supergravity equations we note that eq. (\\ref{eq:K}) implies that only the non-abelian generators matter and we can set $T=T'=T''=0$ (where these are   generic linear combinations of $T_1, \\bar T_2$). We find therefore the same answer as for the $SL(2,\\mathbbm R)$ case in the previous section and comparing to that analysis we see that only $\\mathbf R_4$ (and therefore $\\mathbf R_1$) does not give a supergravity solution. Since we are interested in string theory applications we will focus on the analysis of $\\mathbf R_5$ through $\\mathbf R_{10}$. Recall that in this way we automatically consider also the rank-2 $R$-matrices $\\mathbf R_2$ and $\\mathbf R_3$.  From (\\ref{eq:Ldef}), (\\ref{eq:Jhat}) and (\\ref{eq:Jbarhat}) it follows that (after the field redefinitions) the Lagrangian takes the form\n\\begin{equation}\nL=\nL_0\n-\\tfrac12\\eta\\hat J_a[(1+\\eta RM')^{-1}R]^{a\\bar a}\\hat{\\bar J}_{\\bar a}\\,,\n\\end{equation}\nwhere for $\\mathbf R_5$--$\\mathbf R_8$\n\\begin{equation}\n\\hat J_a=J_a+\\eta\\delta_a^0y_iY^i_AR^{A+}J_+\\,,\\qquad\n\\hat{\\bar J}_{\\bar a}=\\bar J_{\\bar a}-\\eta\\delta_{\\bar a}^{\\bar0}Y^i_AR^{A\\bar-}y_i\\bar J_-\\,,\n\\end{equation}\nwhile for $\\mathbf R_9$\n\\begin{equation}\n\\hat J_a=e^{\\delta_a}J_a\\,,\\qquad\n\\hat{\\bar J}_{\\bar a}=e^{\\bar\\delta_{\\bar a}}\\bar J_{\\bar a}\\,,\\qquad\n\\delta_+=-\\bar\\delta_{\\bar-}=-\\eta Y^i_AR^{A0}y_i\\,.\n\\end{equation}\nIt is not hard to show that $\\mathbf R_5$, $\\mathbf R_7$ and $\\mathbf R_8$ are in fact equivalent to TsT transformations. Consider the first one. The deformed Lagrangian is\n\\begin{equation}\nL=\nL_0\n-\\tfrac12\\eta[J_a+\\eta\\delta_a^0y_iY^i_AR^{A+}J_+][(1+\\eta RM')^{-1}R]^{a\\bar a}\\bar J_{\\bar a}\\,.\n\\end{equation}\nHowever, due to the form of the $R$-matrix and the fact that $M'_{+A}=0$ this is equal to\n\\begin{equation}\nL=\nL_0\n-\\tfrac12\\eta J_a[(1+\\eta RM')^{-1}R]^{a\\bar a}\\bar J_{\\bar a}\\,,\n\\end{equation}\nwhere furthermore we can replace $R$ by the abelian $R$-matrix obtained by dropping the $S_0\\wedge S_+$-term in $\\mathbf R_5$. This deformation therefore reduces to a sequence of commuting TsT transformations. The same conclusion applies to $\\mathbf R_8$ as is easily seen from the form of the $R$ matrix. For $\\mathbf R_7$ we have\n\\begin{equation}\nL=\nL_0\n-\\tfrac12\\eta[J_a+\\eta\\delta_a^0y_iY^i_AR^{A+}J_+][(1+\\eta RM')^{-1}R]^{a\\bar a}[\\bar J_{\\bar a}-\\eta\\delta_{\\bar a}^{\\bar0}Y^i_AR^{A\\bar-}y_i\\bar J_-]\\,,\n\\end{equation}\nbut again the terms with $a=0$ and $\\bar a=\\bar0$ drop out due to the form of the $R$-matrix and the fact that $M_{+A}=0$ and this reduces to\n\\begin{equation}\nL=\nL_0\n-\\tfrac12\\eta J_a[(1+\\eta RM')^{-1}R]^{a\\bar a}\\bar J_{\\bar a}\\,.\n\\end{equation}\nIt is also not hard to see, using again the form of the matrices $M$ and $R$, that one can again replace $R$ by the abelian $R$-matrix obtained by dropping the $S_0\\wedge S_+$-term in $\\mathbf R_7$. The resulting background is therefore also a TsT. \nThe fact that $\\mathbf R_5$, $\\mathbf R_7$ and $\\mathbf R_8$ are equivalent to TsT backgrounds may be argued also from the fact that the $R$-matrices are abelian up to the $S_0\\wedge S_+$-term, and that~\\eqref{eq:compTheta2} holds.\n\n\nFor $\\mathbf R_6$ the deformed Lagrangian is\n\\begin{equation}\nL=\nL_0\n-\\tfrac12\\eta[J_a+\\eta\\delta_a^0y_iY^i_AR^{A+}J_+][(1+\\eta RM')^{-1}R]^{a\\bar a}\\bar J_{\\bar a}\\,.\n\\end{equation}\nAgain, using the form of $R$ and the fact that $M_{+A}=0$, this simplifies to\n\\begin{equation}\nL=\nL_0\n-\\tfrac12\\eta J_a[(1+\\eta RM')^{-1}R]^{a\\bar a}\\bar J_{\\bar a}\\,.\n\\end{equation}\nHowever, in this case we cannot get an equivalent background simply by dropping the $S_0\\wedge S_+$-term in $\\mathbf R_6$.\\footnote{One way to see this is that $\\mathbf R_6-R_1$ is not compatible with the coordinate redefinition needed to undo the transformation with $R_1=S_0\\wedge S_+$ (see eq.~\\eqref{eq:compTheta2}), therefore one does not expect to be able to undo the effect of the $R_1$ piece of the $R$-matrix.} Let us work out the background for the simplest case, $T=T''=0$ and $T'=aT_1+b\\bar T_2$. One finds\n\\begin{equation}\nL\n=\nL_0\n-\\eta f^{-1}\n\\Big(\n2b\\eta J_+\\bar J_2\n+2ab\\eta(1-\\eta\\frac{x^-}{z^2})J_1\\bar J_2\n-\\tfrac12\\eta^2(a^2+b^2-2ab\\sin\\phi_3)J_+\\bar J_0\n+8aJ_1\\bar J_0\n\\Big)\n\\end{equation}\nwhere $f=16+\\eta^2(a^2+b^2-2ab\\sin\\phi_3)[1-\\eta x^-\/z^2]$. One can show that in fact the (anti)chiral currents entering this action $J_+$, $J_1$, $\\bar J_0$ and $\\bar J_2$ extend to chiral currents to all orders in $\\eta$, i.e. the corresponding isometries are not broken by the deformation. Since this is a characteristic feature of TsT backgrounds it is natural to guess that this background can be generated in that way. It is not hard to show this explicitly in the special cases $a=1$, $b=0$ and $a=0$, $b=1$ in which cases it is equivalent to the backgrounds generated by\n\\begin{equation}\nR=T_1\\wedge\\bar S_0-\\frac{\\eta^2}{16}S_+\\wedge\\bar S_0\\qquad\\mbox{and}\\qquad\nR={\\frac{4\\eta}{16+\\eta^2}S_+\\wedge\\bar T_2}-\\frac{\\eta^2}{16+\\eta^2}S_+\\wedge\\bar S_0\\,,\n\\end{equation}\nrespectively.\n\n$\\mathbf R_9$ is the only unimodular example, i.e. the only one with $K=0$. The deformed Lagrangian is\n\\begin{equation}\nL=\nL_0\n-\\tfrac12\\eta\\hat J_a[(1+\\eta RM')^{-1}R]^{a\\bar a}\\hat{\\bar J}_{\\bar a}\\,,\n\\end{equation}\nwith\n\\begin{equation}\n\\hat J_a=e^{\\delta_a}J_a\\,,\\qquad\n\\hat{\\bar J}_{\\bar a}=e^{\\bar\\delta_{\\bar a}}\\bar J_{\\bar a}\\,,\\qquad\n\\delta_+=-\\bar\\delta_{\\bar-}=-\\eta Y^i_AR^{A0}y_i\n\\end{equation}\nFor simplicity we will set $T=0$ and $T'=T_1$. We can argue that this example is not equivalent to a TsT as follows. To order $\\eta^2$ the Lagrangian is\n\\begin{equation}\nL=\nL_0\n-\\tfrac12\\eta(1+\\frac{\\eta}{z^2})J_+\\bar J_-\n+\\tfrac12\\eta J_1\\bar J_0\n-\\tfrac18\\eta^2J_0\\bar J_0\n-\\tfrac12\\eta^2\\frac{x^-}{z^2}J_1\\bar J_-\n+\\mathcal O(\\eta^3)\\,.\n\\end{equation}\nThe action is clearly not invariant under the isometry corresponding to constant shifts of $x^-$ and therefore the corresponding chiral current $J_+$ does not extend to the deformed theory. Instead the equations of motion lead to chiral currents\n\\begin{equation}\nJ_+^\\eta=(1+\\frac{\\eta}{z^2})J_+-\\eta\\frac{x^-}{z^2}J_1+\\mathcal O(\\eta^2)\n\\,,\\qquad\nJ_1^\\eta=J_1+\\mathcal O(\\eta^2)\\,,\n\\end{equation}\nwhile the remaining equations of motion read\n\\begin{equation}\n\\partial[(1+\\frac{\\eta}{z^2})\\bar J_-]-\\eta J_1\\bar J_-=\\mathcal O(\\eta^2)\\,,\\qquad\n\\partial[\\bar J_0+\\eta\\frac{x^-}{z^2}\\bar J_-]-\\eta J_+\\bar J_-=\\mathcal O(\\eta^2)\\,.\n\\end{equation}\nAt the same time we have\n\\begin{equation}\n\\frac{dS}{d\\eta}=\n-\\frac{T}{2}\\int\\,\\left(J_+^\\eta(1+\\frac{\\eta}{z^2})\\bar J_--J_1\\bar J_0+3\\eta\\frac{x^-}{z^2}J_1\\bar J_-+\\tfrac12\\eta J_0\\bar J_0+\\mathcal O(\\eta^2)\\right)\\,,\n\\end{equation}\nwhich clearly cannot be written as a bilinear in deformed chiral currents.\nExplicitly, we obtain the background\n\\footnote{Here we are writing the background that is obtained \\emph{before}  doing the field redefinition.}\n\\begin{equation}\n\\begin{aligned}\nds^2=&\n-\\frac{\\eta  d\\phi_1 (x^+ dx^-+x^-  dx^+)+4 dx^- dx^++\\left(\\eta -z^2\\right) d\\phi_1^2}{\\eta  (\\eta  x^- x^+-4)+4  z^2}\n+\\frac{dz^2}{z^2}\\\\\n&+\\frac{d\\phi_2^2+d\\phi_3^2}{4}\n-\\frac{2 d\\phi_2 \\sin \\phi_3 \\left(\\eta  x^- dx^++\\left(\\eta -z^2\\right) d\\phi_1\\right)}{\\eta  (\\eta  x^- x^+-4)+4 z^2},\\\\\nB=&\\frac{-4 dx^-\\wedge dx^+-2 \\sin \\phi_3 \\left( d\\phi_2\\wedge(\\eta  x^- dx^++\\left(\\eta -z^2\\right) d\\phi_1)\\right)+\\eta d\\phi_1\\wedge(  x^+  dx^--  x^-  dx^+)}{2 \\eta  (\\eta  x^- x^+-4)+8 z^2},\\\\\ne^{-2\\Phi}=&1+\\frac{\\eta(-4+\\eta x^+x^-)}{4z^2}.\n\\end{aligned}\n\\end{equation}\n{Finally, $\\mathbf R_{10}=R_5+T\\wedge S_++T'\\wedge \\bar S_-$. Since the $\\mathfrak{sl}(2,\\mathbb R)$ $R$-matrix $R_5$ does not break the isometries generated by $S_+,\\bar S_-$, we conclude that the additional terms in $\\mathbf R_{10}$ have the only effect of adding further TsT transformations on top of the background generated by $R_5$, which is itself a TsT background.}\n\n\nInterestingly, the matrices $\\mathbf R_4$ (which may be decomposed in terms of $\\mathbf R_1$) and $\\mathbf R_9$  give rise (at leading order in the deformation parameter) to marginal deformations of the WZW model that obviously satisfy the weak CS condition, but not the strong one.\n\n\n\\section{Discussion}\\label{sec:disc}\nIn this paper we have constructed YB deformations of strings on the pure NSNS $AdS_3\\times S^3\\times T^4$ background. Together with abelian YB deformations, which are known to reproduce TsT transformations, we also  constructed non-abelian YB deformations. While some non-abelian $R$-matrices give rise to backgrounds that cannot be obtained simply from TsT transformations, we found that  others generate again TsT backgrounds, or even no deformation at all.\\footnote{Except for the introduction of a (decoupled) non-vanishing vector $K$ in the equations of generalised supergravity.} We expect this to be related to the fact that the initial $G-B$ is degenerate.\n\nFor example, the Jordanian $R$-matrix $R_1=S_0\\wedge S_+$ gives back the undeformed $AdS_3$ background up to an $\\eta$-dependent field redefinition (and up to a non-vanishing $K=-\\eta \\partial_{x^+}$). Recalling that the YB deformation is equivalent to a shift of the $B$-field plus NATD, this observation suggests that $AdS_3$ with NSNS flux has a certain property of self T-duality, when we dualise the non-abelian algebra of isometries generated by $S_0,S_+$ and we also regularise the action by performing a $B$-field gauge transformation.\n\nAlthough we used our classification of $R$-matrices to deform, for concreteness, only the $AdS_3\\times S^3$ part of the background, our results may also be used to obtain deformations involving the $T^4$ of $AdS_3\\times S^3\\times T^4$, or even deformations of the more general $AdS_3\\times S^3\\times S^3\\times S^1$ background. Indeed, in all our expressions of the $R$-matrices the generators $T_1,\\bar T_2$ may be substituted with any other two commuting generators of the compact part of the isometry algebra.\\footnote{That is because the CYBE implies that the restriction of an $R$-matrix to a compact algebra must be abelian. We are therefore free to choose which abelian subalgebra we wish to consider.}\nLet us also mention that the string on these $AdS_3$ backgrounds is integrable~\\cite{Cagnazzo:2012se,Sundin:2012gc} and that our deformations preserve the classical integrability.\n\n\nTo leading order in the deformation parameter, all our YB deformations reduce to the marginal current-current deformations of the type considered by Chaudhuri and Schwartz in~\\cite{Chaudhuri:1988qb}. While they are all {marginal to lowest order}, since  they satisfy what we called the ``weak CS condition'', some of them do not satisfy the ``strong CS condition'' and the celebrated ``no simple-pole condition'' {which guarantee exact marginality. Indeed we found examples which failed to be marginal beyond lowest order (and at one-loop in $1\/k$), all involving the $R$-matrix $R_4$ in Table \\ref{tab:non-ab-sl2}, and one example, $\\mathbf R_9$ in Table \\ref{tab:non-ab-sl2-su2-r4}, which remains marginal to all orders in $\\eta$ at least up to one loop in $1\/k$.} The relation between the space of solutions of the CYBE and that of the weak CS condition is an interesting question. The former is a quadratic equation for $R$, while the latter is a quartic equation for the coefficients $c^{ab}$ of the current-current deformation, related to the left-right block of the $R$-matrix simply as $c^{ab}=R_{\\mbox{\\tiny L}\\mbox{\\tiny R}}^{ab}$. While we expect all solutions of the CYBE to generate solutions of the weak CS condition (including the trivial ones) it seems hard to prove this statement for a generic Lie algebra. In appendix~\\ref{app:sl3} we took a digression from the setup of the paper and we considered the CYBE on the $\\mathfrak{sl}_3$ algebra, finding again that it generates non-trivial solutions to the weak CS condition (that do not solve the strong one). We do not rule out the possibility of having solutions to the weak CS condition that cannot be ``completed'' to  a solution of the CYBE equation. \nIn section~\\ref{sec:YB} we actually discussed a more generic criterion (related to the solvability of the subalgebras involved) not requiring CYBE, to construct solutions of the weak CS condition.\n\nIn~\\cite{Azeyanagi:2012zd} a marginal deformation constructed out of abelian currents (a TsT transformation) was interpreted in terms of spectral flow. It would be interesting to understand if this can be generalised to the non-abelian set-up.\n\nWe would like to stress that we worked out the YB deformations in the sigma model description. It would be very interesting to understand how to formulate the YB deformation directly at the level of the WZW action. Such a construction was performed in~\\cite{Delduc:2014uaa,Delduc:2017fib,Delduc:2018xug} for $R$ a solution of the \\emph{modified} CYBE.\\footnote{There the special propriety $R^3=-R$ was used, so that we do not expect their results to be immediately applicable to the case of the homogeneous CYBE. Moreover, here we want $R$ to be a solution of the CYBE on $\\mathfrak f_{\\mbox{\\tiny L}}\\oplus \\mathfrak f_{\\mbox{\\tiny R}}$ that also couples the left and right copy of the algebra.} The formulation of the deformation of the WZW action may be obtained from the construction in terms of NATD, in the spirit of~\\cite{Hoare:2016wsk} and~\\cite{Borsato:2016pas}. An alternative may be to use the language of $\\mathcal E$ models~\\cite{Klimcik:1995dy,Stern:1998my,Klimcik:2015gba}, see~\\cite{Demulder:2018lmj} for a recent application in similar contexts.\n\nOne motivation to carry out this work came from the recent developments on the $T\\bar T$ deformation~\\cite{Smirnov:2016lqw,Cavaglia:2016oda} and its generalisations. The components $T,\\bar T$ of the stress-energy tensor of a (quite generic) two-dimensional relativistic field theory may be used to construct a ``double-trace'' operator generating an irrelevant perturbation of the theory. The deformation is solvable in the sense that the spectrum of the deformed theory may be computed \\emph{exactly} in the deformation parameter  as a function of the spectrum of the original undeformed theory. In~\\cite{Giveon:2017nie} a ``single-trace'' version of the $T\\bar T$ deformation of the symmetric product orbifold CFT was considered. It was argued that the irrelevant deformation of the ``spacetime'' CFT governed by\\footnote{We refer to~\\cite{Giveon:2017nie} for the connection to Little String Theories. The above operator should be compared to the original double-trace version studied in~\\cite{Smirnov:2016lqw,Cavaglia:2016oda} and given by $T(x)\\bar T(\\bar x)$, where $T(x)=\\sum_{i=1}^NT_i(x)$ and $\\bar T(\\bar x)=\\sum_{i=1}^N\\bar T_i(\\bar x)$.} $\\mathcal O(x)\\propto \\sum_{i=1}^NT_i(x)\\bar T_i(\\bar x)$, where $i$ labels each copy in the symmetric product,  corresponds to a marginal deformation of the dual WZW model that infinitesimally is just the current-current deformation  $J_+(\\sigma)\\bar J_-(\\bar \\sigma)$, where $J_+,\\bar J_-$ are the left and right $SL(2,\\mathbb R)$ currents generating shifts of the boundary coordinates $x^+,x^-$. \nAnother deformation, similar in spirit to the above one, was studied in~\\cite{Chakraborty:2018vja} and~\\cite{Apolo:2018qpq} after replacing the $\\bar T$ with an antichiral $U(1)$ current of the compact factor.\\footnote{This   deformation is in fact the single-trace version of the one first constructed in~\\cite{Guica:2017lia}.}\nIt was argued in~\\cite{Chakraborty:2018vja,Apolo:2018qpq} that the deformation of the dual WZW model is governed again by a marginal deformation bilinear in the currents (where now the antichiral current belongs to the compact part of the algebra). Such marginal deformations of the WZW model may be completed to finite values of the deformation parameter in terms of TsT (or equivalently certain $O(d,d)$) transformations. \nSince TsT deformations are a subclass of YB ones, it would be interesting to understand if it is possible to provide a holographic interpretation also for the YB deformations of $AdS_3\\times S^3\\times T^4$ considered here. (The connection to YB models was also pointed out the recent paper~\\cite{Araujo:2018rho}.) We expect our marginal deformations of the WZW model to correspond to deformations of the dual CFT$_2$ which generalise the (single trace version of the) $T\\bar T$ construction. \nIt would be very interesting to understand for example the case of the non-abelian $R$-matrix $\\mathbf R_9$, which gives rise to the marginal deformation $aJ_0\\bar J_2+bJ_1\\bar J_0+cJ_1\\bar J_2+J_+\\bar J_-$. The non-abelianity of the generators involved forbids the usual iteration of the infinitesimal deformation in order to obtain the exact one. The YB deformation, despite the non-abelianity, provides the realisation of the finite deformation on the worldsheet of the string.\n\n\n\\section*{Acknowledgements}\nWe thank  R. Conti, B. Hoare and D. Thompson for useful discussions. LW thanks the participants of the workshop ``A fresh look at $AdS_3\/CFT_2$'' in Villa Garbald, Castasegna, for stimulating discussions. The work of RB is supported by the Maria de Maeztu Unit of Excellence MDM-2016-0692, by FPA2017-84436-P, by Xunta de Galicia (ED431C 2017\/07), and by FEDER. His work was supported by the ERC advanced grant No 341222 while at Nordita.\n\n\\vspace{2cm}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA widely studied class of disordered systems in statistical physics\nconsist in adding random impurities to a field coupled with the order\nparameter. A textbook example of such a system is the random field Ising model (RFIM), introduced by \\cite{Larkin},  that has been a very useful playground for theoretical ideas. \nThe Hamiltonian of the standard RFIM reads\n\\begin{equation}\n{\\cal H} = - \\sum_{i<j} J_{ij} S_i S_j + \\sum_i h_i S_i \\; ,\n\\label{Hamiltonian}\n\\end{equation} \nwhere all the non-zero interactions are ferromagnetic, i.e. $J_{ij} \\ge 0$. The $N$ Ising spins $S_i=\\pm1$, $i=1,\\dots,N$, are placed at the\nvertices of a graph (lattice), and $h_i$ are quenched random\nmagnetic fields. The fact that all the interactions $J_{ij}$ are non-negative is fundamental, it means that in the absence of the fields there is no explicit frustration in the problem.\n\nThe case where the graph of interactions is a finite-dimensional lattice and where the fields are taken from a Gaussian distribution with zero mean and a\nvariance $H_R$ has received a lot of attention. Of particular interest\nis the phase diagram in the $T$-$H_R$ plane, where $T$ is the\ntemperature. Several authors have suggested, based on non-rigorous field\ntheoretic arguments, that there exists an equilibrium spin glass phase\nin the three-dimensional RFIM, that is a phase with a random frozen ordering\n\\cite{intermediate,intermediate2,CyranoBrezin,CyranoBrezin2,CyranoBrezin3}.\nThese suggestions were disproved rigorously in \\cite{KrzakalaRicci10}\nfor the RFIM defined by Hamiltonian (\\ref{Hamiltonian}). In particular\n\\cite{KrzakalaRicci10} showed that for the RFIM (\\ref{Hamiltonian}) a special case of the Fortuin-Kasteleyn-Ginibre (FKG) inequality \\cite{FKG} implies that the spin glass susceptibility is upper-bounded by the ferromagnetic\nsusceptibility. Since the spin glass susceptibility diverges in the\nwhole spin glass phase, a spin glass phase can not exist away from the\nferromagnetic critical point\/line in the RFIM.\n\nThe field theoretic approach of\n\\cite{intermediate,intermediate2,CyranoBrezin,CyranoBrezin2,CyranoBrezin3},\nhowever, was not formulated with the Ising spin Hamiltonian\n(\\ref{Hamiltonian}) but instead with the soft-spin\ndescription of the random field model. This is the well-known\nGinzburg-Landau model (or the so-called\n$\\phi^4$-theory) which is defined by the following Hamiltonian\n\\begin{equation}\n{\\cal H}_N = - \\sum_{ij} J_{ij} \\phi_i \\phi_j - \\sum_i h_i \\phi_i\n+ \\sum_i r_i \\phi_i^2 + \\sum_i u_i \\phi_i^4 \\; ,\n\\label{Ham_LG}\n\\end{equation}\nwhere $\\phi_i$ are now real numbers, $\\phi_i \\in \\mathbb{R}$, and the interactions are ferromagnetic, $J_{ij} \\ge 0$ (this will be the case in the whole article).\n\nThe generalized model (\\ref{Ham_LG}) includes several special cases. \nThe Ising model is recovered in the limit where $r_i=-2u_i$ and $u_i\\to \\infty$.\nThe most common {\\it random field model} is obtained when $h_i$ are random variables while $r_i=r$ and $u_i=u$ are fixed, and $J_{ij}=0$ or $J_{ij}=1$ depending if the spins $ij$ interact or not. Another version that was considered in the literature, the {\\it random temperature model}, is when $r_i$ are random while $h_i=0$, $u_i=u$ and $J_{ij}\\in \\{0,1\\}$. The\nexistence of a spin-glass phase was also\npredicted in the random temperature model \\cite{MaRudnick,Tarjus02}, based again on some non-rigorous arguments using\nperturbation theory; this result was, however, questioned in \\cite{Sherrington90}.\n\nOur results work even for the slightly more general Hamiltonian\n\\begin{equation}\n{\\cal H}_N = - \\sum_{ij} J_{ij} \\phi_i \\phi_j + \\sum_i f_i(\\phi_i) \\; ,\\label{Ham_gen}\n\\end{equation}\nwhere $J_{ij} \\ge 0$ and the local constraining potentials $f_i()$ are arbitrary analytic functions, but for the requirement that the partition function\n\\begin{equation}\nZ_{N} = \\int_{-\\infty}^\\infty \\prod_{i=1}^N {\\rm d}\\phi_i\\;\ne^{-\\beta {\\cal H}_{N}(\\{\\phi_i\\}) } \\label{Z_N} \\; .\n\\end{equation}\nmust exists for any $N \\in \\mathbb{N}$.\nThis is the most minimalist requirement, since the non-convergence of the integral in (\\ref{Z_N}) would make the Gibbs-Boltzmann measure ill defined and the model would not be a physical one.\n\nThe Gibbs-Boltzmann average at temperature $T=\\beta^{-1}$ is defined by\n\\begin{equation}\n\\langle A\\rangle^{(N)} = \\frac{1}{Z_N} \\int_{-\\infty}^\\infty \\prod_{i=1}^N {\\rm d}\\phi_i A\\, e^{-\\beta {\\cal H}_{N}(\\{\\phi_i\\}) } \\; ,\n\\end{equation}\nThe superscript $(N)$ on the angular brackets will be written explicitly only when the size dependence is crucial, while the temperature dependence is always made implicit.\nConnected correlation functions are defined as\n\\begin{equation}\n\\langle A\\,B\\rangle_c = \\langle A\\,B\\rangle - \\langle A\\rangle\n\\langle B\\rangle \\; .\n\\end{equation}\n\nIt is worth noticing that the convergence of the integral in (\\ref{Z_N}) ensures that the partition function $Z_N$ is an analytic function of the coupling constants $J_{ij}$ for any finite value of $N$. Then the derivative\n\\begin{equation}\n\\beta^{-1} \\frac{\\partial \\ln Z_N}{\\partial J_{ij}} =\n\\langle \\phi_i \\phi_j \\rangle\n\\end{equation}\nexists as well for any pair of indices $i,j$, and this implies that single-variable marginal probability distributions have the first and the second moment, $\\langle \\phi_i \\rangle$ and $\\langle \\phi_i^2 \\rangle$.\nActually in soft-spin models used in the literature, such as the spherical model and the $\\phi^4$ model, single-variable marginal probabilities decay exponentially fast for large values of $\\phi_i$, and so all the moments $\\langle \\phi_i^k \\rangle$ exist. However, our proof only requires the first two moments to exist.\n\nNote also that any type of lattice can be encoded in model (\\ref{Ham_gen}) by setting $J_{ij}=0$ if spins $i$ and $j$ do not interact. \n\nThe main contribution of this paper is a rigorous proof that the soft-spin random-field random-temperature model defined by (\\ref{Ham_gen}) does not have a spin glass phase as long as the interactions are ferromagnetic (non-negative). This generalizes the result of \\cite{KrzakalaRicci10}. \n\n\n\\section{Definitions of susceptibilities}\n\nWe define the ferromagnetic and the spin glass phases using the\nproperties of the ferromagnetic and spin glass susceptibilities. \n\nThe order parameter that characterizes a ferromagnetic transition is\nthe magnetization $m= \\sum_i \\langle \\phi_i\\rangle\/N $. However, a non-zero\nmagnetization does not imply a ferromagnetic phase. Indeed, $m>0$ even at\nlarge temperatures when a uniform positive external magnetic field is applied. A convenient way to characterize the ferromagnetic phase is to\ndefine the ferromagnetic susceptibility as\n\\begin{equation}\n\\chi^0_F(N) = \\frac{1}{N} \\sum_{ij} \\langle \\delta\\phi_i\\, \\delta\\phi_j\\rangle \\; ,\n\\end{equation}\nwhere\n\\begin{equation}\n\\delta\\phi_i =\n\t\\frac{\\phi_i - \\langle \\phi_i \\rangle}\n\t{\\sqrt{\\langle \\phi_i^2 \\rangle - \\langle \\phi_i \\rangle^2}}\\;,\n\\label{fluct}\n\\end{equation}\nare the fluctuations with respect to the average values, normalized by the variances.\n\nIn the thermodynamic limit ($N \\to \\infty$), $\\chi^0_F(\\infty)$ is finite in the high temperature ($T > T_c$) paramagnetic phase and it diverges approaching the ferromagnetic critical point from above ($T \\searrow T_c$).\nRight at the critical point ($T=T_c$), $\\chi^0_F(N)$ diverges with $N\\to \\infty$ signaling that the system is critical, i.e.\\ has long range correlations between fluctuations of its variables. Unfortunately, the ferromagnetic susceptibility $\\chi^0_F(N)$ diverges with $N$ also in the whole low temperature ($T<T_c$) ferromagnetic phase: however this divergence is not due to criticality (i.e.\\ long range correlation of fluctuations), but only because below $T_c$ two ferromagnetic states coexist\\footnote{In the presence of two or more equivalent states, an appropriately chosen perturbation, although of infinitesimal strength, may induce a macroscopic change of state, thus leading to an infinite susceptibility.}.\n\nGiven that we are interested in finding critical points and critical lines where a phase transition takes place, we would like to measure an observable that diverges only at criticality, and so we consider the following {\\em ferromagnetic susceptibility}\n\\begin{equation}\n\\chi_F = \\lim_{h \\searrow 0} \\lim_{N \\to \\infty} \\chi_F(h,N)\n\t  = \\lim_{h \\searrow 0} \\lim_{N \\to \\infty} \\frac{1}{N}\n\t  \\sum_{ij} \\langle \\delta\\phi_i\\, \\delta\\phi_j \\rangle \\, ,\n\\label{chi_ferro}\n\\end{equation}\nwhere $h$ is an auxiliary uniform magnetic field (in practice one needs to add a term $-h \\sum_i \\phi_i$ in the Hamiltonian).\nDue to the order of the limits in (\\ref{chi_ferro}), below $T_c$, the infinitesimal external field $h$ makes the two ferromagnetic states no longer equivalent, and consequently $\\chi_F$ is finite everywhere, but at the critical point $T_c$ (which is indeed defined as the point where $\\chi_F$ diverges).\n\nIn general to define a susceptibility that diverges only when a critical state is present one should explicitly break (by adding infinitesimal perturbations) all the symmetries of the Hamiltonian. In our case, the Hamiltonian (\\ref{Ham_gen}) is very general, but the first term is invariant under the transformation $\\phi_i \\to -\\phi_i\\; \\forall i$. In case the potentials too are invariant under such a transformation, $f_i(\\phi)=f_i(-\\phi)$, then the infinitesimal auxiliary uniform field in (\\ref{chi_ferro}) is strictly required.\n\nThe spin glass phase is characterized by a freezing of spins in\nrandom directions \\cite{EA}, hence the {\\em spin glass susceptibility} is defined as\n\\begin{equation}\n\\chi_{SG} = \\lim_{h \\searrow 0} \\lim_{N \\to \\infty} \\chi_{SG}(h,N)\n\t   = \\lim_{h \\searrow 0} \\lim_{N \\to \\infty} \\frac{1}{N}\n\t   \\sum_{ij} \\langle \\delta\\phi_i\\, \\delta\\phi_j \\rangle^2 \\,.\n\\label{chi_SG}\n\\end{equation}\nAgain we use the infinitesimal auxiliary external field to break the $\\phi \\to -\\phi$ symmetry, if present.\nThe susceptibility $\\chi_{SG}$ is related closely to what is measured in simulations and experiments \\cite{HERTZ}, and it is predicted to diverge at the critical point in spin glass theories such as replica symmetry breaking \\cite{MF}, or the droplet description \\cite{DROPLET}.\nIn a spin glass phase $\\chi_{SG}$ is infinite, because of the presence of at least two states\\footnote{The number of states depends on the model and for some models, like the 3D Edwards-Anderson model, it is still a matter of debate.} related by symmetries, which are not broken by the auxiliary field.\nFor this reason we can define that a system is in a spin glass phase {\\it if and only if} the ferromagnetic susceptibility (\\ref{chi_ferro}) is finite, while the spin glass susceptibility (\\ref{chi_SG}) is infinite.\n\nMore precisely the computation of these two susceptibilities must proceed by first taking the thermodynamic limit in the presence of the external field, $\\chi_F(h,\\infty)$ and $\\chi_{SG}(h,\\infty)$, and then studying the limit $h \\searrow 0$ of these two functions. If such a limit exists, then we say that the susceptibility is finite and we are away from the critical point, while if a divergence is found while decreasing $h$, then we say that the susceptibility is infinite.\n\nIn the next Section we prove that $\\chi_{SG}(h,N) \\le \\chi_F(h,N)$, for any value of $h$ and $N$, thus excluding the possibility of a spin glass phase (defined by $\\chi_F<\\infty$ and $\\chi_{SG} \\to \\infty$) in the model (\\ref{Ham_gen}) in the absence of explicit frustration in the couplings.\n\n\n\\section{Results}\n\nWe start by proving a generalization of the 2nd Griffith's inequality \\cite{Griffiths67b}. The following Lemma is also a consequence of much more general FKG inequalities \\cite{FKG,FKG_gen}, we, however, find useful to present an independent and more elementary proof. \n\n\\begin{lemma}\nIn the model defined by the Hamiltonian (\\ref{Ham_gen}) with non-negative couplings, $J_{ij} \\ge 0 \\; \\forall i,j$, under the Gibbs-Boltzmann measure $e^{-\\beta {\\cal H}_N}\/Z_N$ the correlation between fluctuations of any two variables is non-negative and bounded by 1,\n\\begin{equation}\n0 \\le \\langle \\delta\\phi_i\\, \\delta\\phi_j \\rangle \\le 1 \\qquad \\forall i,j\\;. \n\\label{eq_lemma_FKG}\n\\end{equation}\n\\label{lemma_FKG}\n\\end{lemma}\n\n\\begin{proof}[Lemma \\ref{lemma_FKG}]\nLet us prove first the second inequality in (\\ref{eq_lemma_FKG}).\nFrom the definition (\\ref{fluct}) of the relative fluctuations we have that $\\langle \\delta\\phi_i^2 \\rangle = 1$ for any $i$.\nMoreover for any pair of indices $i,j$ we have that\n\\begin{equation}\n0 \\le \\langle (\\delta\\phi_i - \\delta\\phi_j)^2 \\rangle =\n\\langle \\delta\\phi_i^2 \\rangle + \\langle \\delta\\phi_j^2 \\rangle\n- 2 \\langle \\delta\\phi_i\\, \\delta\\phi_j \\rangle =\n2 (1 - \\langle \\delta\\phi_i\\, \\delta\\phi_j \\rangle)\n\\end{equation}\nfrom which $\\langle \\delta\\phi_i\\, \\delta\\phi_j \\rangle \\le 1$ follows.\n\nIn order to prove the first inequality in (\\ref{eq_lemma_FKG}) we notice that it is equivalent to the inequality\n\\begin{equation}\n\\langle \\phi_i \\phi_j \\rangle_c \\ge 0\\;,\n\\label{simpler}\n\\end{equation}\nthanks to the fact that all denominators in the definition (\\ref{fluct}) of $\\delta\\phi_i$ are positive and can be canceled without changing the sign of the correlation.\n\nThen we prove (\\ref{simpler}), by induction in the system size. In a system of $N=1$ spin \n\\begin{equation}\n\\langle \\phi_1^2 \\rangle^{(1)}_c \\ge 0 \\, ,\n\\end{equation} \nsince the variance is always non-negative. \nThen we assume the property to hold in a system of $N$ spins and we consider a system with $N+1$ spins. The Hamiltonian of that system is related to the $N$-spin system as\n\\begin{equation}\n      {\\cal H}_{N+1}   = {\\cal H}_{N} - \\sum_{i=1}^N J_{N+1,i} \\phi_{N+1} \\phi_i +f_{N+1}(\\phi_{N+1})  \\, .\n\\end{equation}\nWe denote \n\\begin{eqnarray}\n     P(x) = \\frac{1}{Z_{N+1}}   \\int_{-\\infty}^\\infty \\prod_{i=1}^N\n     {\\rm d}\\phi_i  \\hspace{7cm} \\\\ \\exp{\\left[-\\beta\n         {\\cal H}_{N}(\\{\\phi_i\\}) + \\beta \\sum_{i=1}^N J_{N+1,i} x \\phi_i\n         - \\beta f_{N+1}(x) \\right]}\\, . \\nonumber\n\\end{eqnarray}\nLet us denote the thermodynamic average in a modified external magnetic field as\n\\begin{equation}\n\\langle A \\rangle_x^{(N)}    =  \\frac{\\int \\prod_{i=1}^N {\\rm d}\\phi_i   A \\, e^{-\\beta {\\cal H}_{N}(\\{\\phi_i\\}) + \\beta \\sum_{i=1}^N J_{N+1,i} x \\phi_i   }}{\\int \\prod_{i=1}^N {\\rm d}\\phi_i \\,   e^{-\\beta {\\cal H}_{N}(\\{\\phi_i\\}) + \\beta \\sum_{i=1}^N J_{N+1,i} x \\phi_i  }}\\, .\n\\end{equation}\nThe connected correlation between spins $\\phi_{N+1}$ and $\\phi_i$ in the $N+1$-spin system can then be rewritten as\n\\begin{eqnarray}\n     \\langle \\phi_{N+1} \\phi_i \\rangle_c^{(N+1)} &=& \\int_{-\\infty}^\\infty  {\\rm d}x  \\,  x \\, P(x)   \\langle \\phi_i \\rangle_x^{(N)}   -   \\int_{-\\infty}^\\infty  {\\rm d}y \\,   y \\, P(y)  \\int_{-\\infty}^\\infty  {\\rm d}x\\,  P(x)   \\langle \\phi_i \\rangle_x^{(N)} \\nonumber \\\\  & =&   \\int_{-\\infty}^\\infty  {\\rm d}x  \\,  \\left[ x -  \\int_{-\\infty}^\\infty  {\\rm d}y \\,   y \\, P(y)   \\right]\\, P(x)   \\langle \\phi_i \\rangle_x^{(N)} \\, .\n\\end{eqnarray}\nWe can then use the following inequality: For any real non-decreasing function $g(x)$ such that \n\\begin{equation}\n     \\int_{-\\infty}^\\infty  {\\rm d}x  \\, g(x) = 0\\, ,\n\\end{equation} \nand any non-decreasing function $f(x)$ one has\n\\begin{equation}\n     \\int_{-\\infty}^\\infty  {\\rm d}x \\,  g(x) \\, f(x) \\ge 0 \\, . \\label{lemma2}\n\\end{equation}\nProof of this statement is elementary, function $g(x)$ has to be non-positive for some $x \\le x_0$ and non-negative for $x \\ge x_0$. Since $f(x)$ is non-decreasing one has \n\\begin{equation}\n    \\int_{-\\infty}^{x_0}  {\\rm d}x \\,  |g(x)| \\, f(x) \\le   \\int_{x_0}^\\infty  {\\rm d}x \\,  |g(x)| \\, f(x) \\, ,\n\\end{equation}\nfrom which (\\ref{lemma2}) follows.\nWe observe that $\\langle \\phi_i \\rangle_x^{(N)}$ is a non-decreasing function of $x$. Indeed\n\\begin{equation}\n        \\frac{{\\rm d} \\langle \\phi_i \\rangle_x^{(N)}}{{\\rm d}x} = \\beta \\sum_{j=1}^N J_{N+1,j}  \\langle \\phi_j \\phi_i \\rangle_c^{(N)} \\ge 0\\, ,\n\\end{equation}\nwhere the last inequality follows from the inductive assumption and since $J_{N+1,j}\\ge 0$.\nAnd that\n\\begin{equation}\n     \\int_{-\\infty}^\\infty  {\\rm d}x  \\,  \\left[ x -  \\int_{-\\infty}^\\infty  {\\rm d}y \\,   y \\, P(y)   \\right]\\, P(x)  =0\\, .\n\\end{equation}\nHence (\\ref{lemma2}) implies\n\\begin{equation}\n   \\langle \\phi_{N+1} \\phi_i \\rangle_c^{(N+1)} \\ge 0\\, .\n\\end{equation} \n\nWe proceed similarly for the correlation function between two spins that were already present in the $N$-spin system\n\\begin{eqnarray}\n    \\langle \\phi_i \\phi_j \\rangle_c^{(N+1)}   &=&\n    \\int_{-\\infty}^\\infty  {\\rm d}x  \\, P(x)   \\langle \\phi_i\n    \\phi_j \\rangle_x^{(N)} -  \\int_{-\\infty}^\\infty  {\\rm d}x  \\,\n    P(x)   \\langle \\phi_i \\rangle_x^{(N)} \\int_{-\\infty}^\\infty\n    {\\rm d}y  \\, P(y)   \\langle \\phi_j \\rangle_y^{(N)} \\nonumber \\\\ &=& \n\\int_{-\\infty}^\\infty  {\\rm d}x  \\, P(x)   \\langle \\phi_i \\phi_j \\rangle_{x,c}^{(N)}  + \\nonumber \\\\ && \\int_{-\\infty}^\\infty  {\\rm d}x  \\, P(x)   \\langle \\phi_i \\rangle_x^{(N)}\\left[  \\langle \\phi_j \\rangle_x^{(N)}   -  \\int_{-\\infty}^\\infty  {\\rm d}y  \\, P(y)   \\langle \\phi_j \\rangle_y^{(N)} \\right]\n\\end{eqnarray}\nwhere the first term is non-negative by the inductive assumption, and the second term is non-negative according to (\\ref{lemma2}), because $\\langle \\phi_i \\rangle_x^{(N)}$ and $\\langle \\phi_j \\rangle_x^{(N)}$ are non-decreasing functions of $x$ and \n\\begin{equation}\n    \\int_{-\\infty}^\\infty  {\\rm d}x  \\, P(x)  \\left[  \\langle \\phi_j \\rangle_x^{(N)}   -  \\int_{-\\infty}^\\infty  {\\rm d}y  \\, P(y)   \\langle \\phi_j \\rangle_y^{(N)} \\right] = 0\\, .\n\\end{equation}\nHence\n\\begin{equation}\n   \\langle \\phi_i \\phi_j \\rangle_c^{(N+1)}   \\ge 0.\n\\end{equation} \nThis concludes the proof of Lemma \\ref{lemma_FKG}. \\qed\n\\end{proof}\n\nBased on the previous Lemma we can now easily state the main result of the present paper.\n\n\\begin{theorem}\nIn the model defined by the Hamiltonian (\\ref{Ham_gen}) with non-negative couplings, $J_{ij} \\ge 0 \\; \\forall i,j$, under the Gibbs-Boltzmann measure $e^{-\\beta {\\cal H}_N}\/Z_N$, the spin glass susceptibility $\\chi_{SG}(h,N)$ is always upper-bounded by the ferromagnetic susceptibility $\\chi_F(h,N)$. Consequently the model does not posses a thermodynamic spin glass phase.\n\\label{main_lemma}\n\\end{theorem}\n\\begin{proof}[Theorem \\ref{main_lemma}]\nThe hypothesis of the present Theorem are the same as those of Lemma \\ref{lemma_FKG}, with the only difference that to properly define the susceptibilities we need to add the external auxiliary field term to the original model Hamiltonian.\nThen Lemma \\ref{lemma_FKG} can be used only if\n\\begin{equation}\nZ_N(h) = \\int_{-\\infty}^\\infty \\prod_{i=1}^N {\\rm d}\\phi_i\\;\ne^{-\\beta {\\cal H}_{N}(\\{\\phi_i\\}) + \\beta h \\sum_i \\phi_i}\\;\n\\end{equation}\nexists also for $h > 0$, and this is easy to prove. Indeed $Z_N(0)$ exists (otherwise the Gibbs-Boltzmann measure would be ill-defined) and also the first two derivatives of $Z_N(h)$ with respect to $h$ exist (because $\\langle \\phi_i \\rangle$ and $\\langle \\phi_i^2 \\rangle$ exist): so $Z_N(h)$ can be continued in a neighborhood on $h=0$, that we call $S_0$, and this is enough to take the limit $h \\searrow 0$ that is required to define properly the susceptibility. Please note that the region $S_0$ coincide with $\\mathbb{R}$ for all the models used in the literature, such as the spherical model and the $\\phi^4$ model.\n\nGiven that the hypothesis of Lemma \\ref{lemma_FKG} are satisfied in $S_0$, we can make use of inequalities in (\\ref{eq_lemma_FKG}) and find that\n\\begin{equation}\n\\langle \\delta\\phi_i \\delta\\phi_j \\rangle^2 \\le\n\\langle \\delta\\phi_i \\delta\\phi_j \\rangle \\quad \\Longrightarrow \\quad\n\\chi_{SG}(h,N) \\le \\chi_F(h,N)\n\\label{main}\n\\end{equation}\nfor any value of $N$ and $h \\in S_0$.\nEven in the thermodynamic limit the inequality holds\n\\begin{equation}\n\\chi_{SG}(h,\\infty) \\le \\chi_F(h,\\infty)\n\\end{equation}\nand so the spin glass susceptibility can not diverge if the ferromagnetic one stays finite.\n\nIn other words, from the definitions given in the previous Section it is clear that if a thermodynamic spin glass phase exists, then for a sufficiently large value of $N$ and a sufficiently small value of $h$ the spin glass susceptibility must be larger than the ferromagnetic one and this would violate the inequality in (\\ref{main}). Then we conclude that a thermodynamic spin glass phase does not exists in the model defined in the hypothesis. \\qed\n\\end{proof}\n \n \n\\section{Discussion}\n\nWe have shown rigorously that there is no spin glass phase in the scalar soft-spin random-field random-temperature Ginzburg-Landau model with ferromagnetic interactions defined by (\\ref{Ham_gen}).\nThis shows that with two-body interactions and a scalar order\nparameter one cannot obtain a genuine spin glass phase at\nequilibrium without {\\it explicit frustration} in the couplings\n(another possibility to frustrate the system is to impose a non-equilibrium value of magnetization, see \\cite{KrzakalaRicci10}).\n\nOur proof contradicts the conclusions of some works that used field theoretic arguments\n\\cite{intermediate,intermediate2,CyranoBrezin,CyranoBrezin2,CyranoBrezin3,MaRudnick,Tarjus02}.\nIt is yet to be discovered where the problem lies in those approaches.  \nOne possibility to consider is that the spin glass instability could be an artifact of some truncation in the perturbative expansion. For some of\nthese works the discrepancy may stem from the use of vectorial soft-spin models instead of scalar ones.\nAnother possibility, that is related to what was suggested recently in \\cite{TarjusRecent}, is that the observed \"replica symmetry\nbreaking\" instabilities arise only in disorder averaged quantities and never in the thermodynamic limit of a single instance quantities. These instabilities would then not be equivalent to the divergence of the spin glass susceptibility (which we prove impossible out of the ferromagnetic critical point), but they could instead be connected to some subtle non-self-averaging effects between different realizations of the system. Indeed all the above-mentioned works considered a \"replicated\" field theory, that is, a field theory averaged over\nmany realizations of the disorder. The divergences that they observed could hence be coming from strong sample to sample fluctuation. \nThe fact that some non-self-averaging is present in the RFIM has been suggested by Parisi and Sourlas \\cite{P-S}. They argue that the correlation function, or equivalently the ferromagnetic susceptibility, of the RFIM is non self-averaging in the critical region, and they argue that this was the source of the problems with perturbative expansions. Note, however, that such simple non-self-averaging effects can {\\it only}\ntake place {\\it at the ferromagnetic critical point} in any finite\ndimensional system. This is a straightforward\nconsequence of a theorem by Wehr and Aizenman \\cite{AW} who proved\nthat any extensive quantities (such as the ferromagnetic\nsusceptibility away from the critical point) is self-averaging in finite dimensional systems. In other words, if this effect was the one observed in the field theoretic approaches, it has to be limited to the ferromagnetic critical point\nitself.\n\nFinally, it would be very interesting to see if our proof can be generalized further. There are two interesting\ncounter-examples that seem to put strong limits to such\ngeneralizations. Matsuda and Nishimori (private communication) showed that a random field Ising model\nwith 3-spins interactions on the\nBethe lattice can have a spin glass phase. And so moving beyond pairwise interacting models seems impossible in full generality.\nMoreover Parisi (private communication) provided an interesting example of a\npairwise interacting $n=2$ component vector spin system where the two point connected correlation can be negative even if all couplings are positive.\nIt is a chain of spins with an external field that smoothly rotates by 180 degrees along the chain, such that the field on the last spin is opposite to field on the first spin.\nIf the field strength is strong enough, each spin will be mostly aligned along the local field and will thermally fluctuate around this position. \nHowever, given that the extremal spins are in opposite directions, their thermal fluctuations will be negatively correlated.\nThis is a very specific configuration which may not happen in typical samples, but its existence implies that the proof strategy presented in this paper cannot be straightforwardly generalized to vector spin models.\n\n\n\\vspace{5mm}\n\n\\noindent {\\bf Acknowledgment:} We thank G. Parisi, H. Nishimori, F. Toninelli and  for very useful comments and enlightening discussions. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\begin{figure*}[ht!]\n\\centering\n\\begin{tikzpicture}\n    \\node[rectangle, inner sep=10pt, draw, text width=1cm] at (-1, 1) (GD Teacher){GD Teacher};\n    \\node[rectangle, inner sep=5pt, draw] at (-1, -2) (GD Baseline){GD Baseline}; \n    \\node[rectangle, inner sep=5pt, draw] at (-1, -3) (GD Student){GD Student}; \n    \n    \\node[rectangle, inner sep=10pt, draw, text width=1cm] at (3.5, 1) (Baseline Teacher) {Baseline Teacher};\n    \\node[rectangle, inner sep=10pt, draw, text width=1cm] at (5.5, 1) (Adapted Teacher) {Adapted Teacher};\n    \n    \\node[rectangle, inner sep=5pt, draw] at (1.5, -1) (1){1}; \n    \\node[rectangle, inner sep=5pt, draw] at (3.5, -1) (2){2}; \n    \\node[rectangle, inner sep=5pt, draw] at (5.5, -1) (3){3}; \n    \\node[rectangle, inner sep=5pt, draw] at (1.5, -2) (4){4}; \n    \\node[rectangle, inner sep=5pt, draw] at (3.5, -2) (5){5}; \n    \\node[rectangle, inner sep=5pt, draw] at (5.5, -2) (6){6}; \n    \\node[rectangle, inner sep=5pt, draw] at (1.5, -3) (7){7}; \n    \\node[rectangle, inner sep=5pt, draw] at (3.5, -3) (8){8}; \n    \\node[rectangle, inner sep=5pt, draw] at (5.5, -3) (9){9}; \n    \n    \\draw[dotted] (0.5, 1.7) -- (0.5, -3.4);\n    \\node at (1.5, 1) {In-Domain};\n\n    \\draw[->] (GD Teacher) .. controls (1.5, 2) and (5.5, 2) .. (Adapted Teacher.north);\n    \\draw[->, dashed] (GD Teacher.west) .. controls (-3, -1.5) and (-3,-3) .. (GD Student.west);\n    \n    \\draw[->, dashed] (Baseline Teacher) -- (2.north);\n    \\draw[->, dashed] (Baseline Teacher.south) .. controls (4.5, -1) and (4.5, -2) .. (5.east);\n    \\draw[->, dashed] (Baseline Teacher.south) .. controls (4.5, -1) and (4.5, -2) .. (8.east);\n    \n    \\draw[->, dashed] (Adapted Teacher) -- (3.north);\n    \\draw[->, dashed] (Adapted Teacher.south) .. controls (6.5, -1) and (6.5, -2) .. (6.east);\n    \\draw[->, dashed] (Adapted Teacher.south) .. controls (6.5, -1) and (6.5, -2) .. (9.east);\n\n    \\draw[->] (GD Baseline) -- (4.west);\n    \\draw[->] (3.5, -2.5) -- (5.south);\n    \\draw[->] (GD Baseline.east) .. controls (1.5, -2.5) .. (3.5, -2.5) -- (5.5, -2.5) -- (6.south);\n    \n    \\draw[->] (GD Student) -- (7.west);\n    \\draw[->] (3.5, -3.5) -- (8.south);\n    \\draw[->] (GD Student.east) .. controls (1.5, -3.5) .. (3.5, -3.5) -- (5.5, -3.5) -- (9.south);\n    \n\\end{tikzpicture}\n\\caption{There are 9 possible configurations for training small, in-domain models with knowledge distillation and domain adaptation. Models trained on general-domain data are shown on the left, and in-domain models are shown on the right. Solid arrows represent domain adaptation via continued training. Dashed arrows represent improved optimization via sequence-level knowledge distillation. Configuration 1 is the model which is trained  on in-domain data with random initializations and without the assistance of a teacher. }\n\\label{fig:configs}\n\\end{figure*}\n\n\nMachine translation systems rely on large amounts of data to deduce the rules underlying translation from one language to another. This presents challenges in some important niche domains, such as patent and medical literature translation, due to the high cost of hiring experts to generate suitable training data. A cost-effective alternative is \\emph{domain adaptation}, which leverages large amounts of parallel documents from less difficult and readily-available domains, such as movie subtitles and news articles.\n\nDomain adaptation works well in practice. However, these large datasets, which we call \\emph{general domain} datasets, introduce some scalability problems. Large datasets require large models; neural machine translation systems can take days or weeks to train. Some models require gigabytes of disk space, making deployment to edge computing devices challenging. They can also require excessive compute during inference, making them slow and costly to scale up in production environments.\\cite{gordon_2019}\n\nTo alleviate these issues, \\emph{knowledge distillation} (aka Teacher-Student) \\cite{Hinton2015-on} is used to compress models into a manageable form. But although knowledge distillation is the most commonly used form of model compression in practice, it is also one of the least understood.\n\nIn this work, we perform a \\textbf{large-scale empirical analysis} to attempt to discover best practices when using knowledge distillation in combination with domain adaptation. Out of several common-sense configurations, we find that two stages of knowledge distillation give the best performance: one using general-domain data and another using in-domain data with an adapted teacher. We perform experiments on multiple language pairs (Russian-English, German-English, Chinese-English), domains (patents, subtitles, news, TED talks), and student sizes.\n\n\\section{Background}\n\n\\paragraph{Domain Adaptation} helps overcome a lack of quality training data in niche domains by leveraging large amounts of data in a more accessible general-domain. Domain adaptation is usually accomplished by \\emph{continued training} \\cite{Luong2015-ym,Zoph2016-pt}, which involves two steps:\n\n\\begin{enumerate}\n    \\item A model is randomly initialized and trained until convergence on the general-domain data.\n    \\item A new model is initialized with the parameters resulting from Step 1 and trained until convergence on the in-domain dataset.\n\\end{enumerate}\n\nWe can consider domain adaptation as extracting a useful \\emph{inductive-bias} from the general-domain dataset, which is encoded and passed along to the in-domain model as a favorable weight initialization. While there are other methods of extracting inductive bias from general-domain datasets (including mixed fine-tuning \\cite{Chu2017-db} and cost weighting \\cite{Chen2017-vw}), continued training is most common and the focus of this paper. \n\n\\paragraph{Knowledge Distillation} is a method for improving the performance of under-parameterized ``Student\" models by exploiting the probability distribution of a more computationally complex ``Teacher\" network. Kim and Rush \\shortcite{Kim2016-st} presented an extension of knowledge distillation to machine translation in two flavors: word-level and sequence-level knowledge distillation.\n\n \\emph{Sequence-level knowledge distillation}, which is more general, involves three steps:\n \n \\begin{enumerate}\n     \\item A large Teacher network is randomly initialized and trained until convergence on the data.\n     \\item The source-side of the training data is decoded using the Teacher to produce ``distilled\" target data.\n     \\item A smaller Student model is randomly initialized and trained until convergence on the distilled source-target pairs (discarding the original target sequences in the data).\n \\end{enumerate}\n\nThe goal of knowledge distillation is to train the student model to mimic the teacher's probability distribution over translations. Since the teacher and the student are trained on the same dataset, they should be capable of learning the same distribution in theory. In practice, however, using the teacher as an additional training signal improves student test performance.\\footnote{Interestingly, this can be true even when the student has the same computational resources as the teacher \\cite{Furlanello2018-lp}} Explanations for this phenomenon include dark knowledge \\cite{Furlanello2018-lp}, mode reduction \\cite{Zhou2019-wp}, and regularization \\cite{Gordon2019-qs,Dong2019-ve}, but no definitive evidence has been given.\n\n Sequence-level knowledge distillation is widely used in both industry \\cite{Xia2019-iq} and research and is the second focus of this paper. \\footnote{Sequence-level knowledge distillation is also commonly used to train non-autoregressive machine translation models \\cite{Zhou2019-wp}.}\n\n\\section{Distilling and Adapting}\n\nHow domain adaptation and knowledge distillation would interact when applied in combination was not previously clear. Specifically, our research questions are:\n\n\\begin{itemize}\n    \\item Is a distilled model easier or harder to adapt to new domains?\n    \\item Should knowledge distillation be used on in-domain data? If so, how should the teacher be trained?\n\\end{itemize}\n\nTo answer these questions, we performed experiments on 9 possible configurations which are assigned configuration numbers in Figure \\ref{fig:configs}. For ease of reference, we will primarily refer to small, in-domain models by their configuration number and encourage readers to consult Figure \\ref{fig:configs}. Each configuration has two attributes of interest. \n\n\\paragraph{Distilling In-Domain Data} How is in-domain data pre-processed using knowledge distillation? Some models are trained with no pre-processing (configurations 1, 4, and 7), while others use a teacher to pre-process the in-domain training data. This teacher might be a baseline trained on in-domain data only (configurations 2, 5, and 8) or it can be trained on general-domain data and then adapted to in-domain via continued training (configurations 3, 6, and 9).\n\n\\paragraph{Initialization} How are models initialized? A model might be randomly initialized (configurations 1, 2, and 3), or it might be adapted from a model trained on general-domain data. This general-domain model might be a baseline trained directly on the general-domain data (configurations 4, 5, and 6) or it might be a student model trained on the output of a general-domain teacher (configurations 7, 8, 9).\n\n\n\\section{Experiments}\n\\subsection{Data}\n\\paragraph{General-Domain Data}\nWe train models in multiple settings: 3 language pairs (German-English, Russian-English, and Chinese-English) each with 1 general-domain dataset and 2 different in-domain datasets. \nThe general-domain datasets for each language are a concatenation of data from OpenSubtitles2018 \n\\cite{Tiedemann2016-ob,Lison2016-kr} (which contains translated movie subtitles) and the WMT 2017 datasets \\cite{Ondrej2017-dw} (which includes a variety of sources, including news commentary, parliamentary proceedings, and web-crawled data).\n\n\\paragraph{In-Domain Data}\nWe use the World International Property Organization (WIPO) COPPA-V2 dataset \\cite{Junczys-Dowmunt2018-oh} and the TED Talks dataset \\cite{Qi2018-le} as our two in-domain datasets. The WIPO data contains parallel sentences from international patent abstracts, while the TED Talks dataset consists of translated transcripts of public speeches.\n\n\\paragraph{Data Statistics} The size of each training dataset is presented in Table \\ref{tab:data-sizes}. General-domain datasets contain tens of millions of sentences, while in-domain datasets contain much less. German-English WIPO has an exceptional amount of training data (4.5 times more than the next biggest in-domain dataset) and helps qualify how our results might change when more in-domain data is available.\n\n\\begin{table}[]\n    \\centering\n    \\begin{tabular}{lrrr}\n        Language & General-Domain & WIPO & TED\\\\\n        \\hline\n        De-En & 28.3 M & 821 k & 15 k\\\\\n        Ru-En & 51.1 M & 29 k & 180 k\\\\\n        Zh-En & 35.9 M & 154 k & 169 k\\\\\n        \n    \\end{tabular}\n    \\caption{The number of training sentences in each dataset.}\n    \\label{tab:data-sizes}\n\\end{table}\n\n\\paragraph{Pre-processing} All datasets are tokenized using the Moses\\footnote{\\href{http:\/\/statmt.org\/moses}{statmt.org\/moses}} tokenizer. A BPE vocabulary \\cite{Sennrich2016-an} of 30,000 tokens is constructed for each language using the training set of the general-domain data. This BPE vocabulary is then applied to both in-domain and general-domain datasets. This mimics the typical scenario of a single, general-domain model being trained and then adapted to new domains as they are encountered. Note that re-training BPE on in-domain data to produce a different vocabulary would force us to re-build the model, making adaptation impossible.\n\n\\paragraph{Evaluation} The general-domain development set for each language contains newstest2016 concatenated with the last 2500 lines of OpenSubtitles2018. We reserve 3000 lines of WIPO to use as the in-domain development set. TED talks development sets are provided by the authors and contain around 2000 lines each. Evaluations of each model are performed by decoding the appropriate development set with a beam-search size of 10 and comparing to the reference using multi-bleu.perl from the Moses toolkit.\n\n\\subsection{Architectures and Training}\n A list of architecture sizes is provided in Table \\ref{tab:sizes}. Teachers are trained using the Large hyper-parameter settings, while we experiment with Medium, Small, and Tiny students for each configuration and language\/domain setting.\n \n All models are Transformers \\cite{Vaswani2017-vu}. We use the same hyper-parameters (which are based on a template from \\cite{Duh_undated-uo}\\footnote{\\href{https:\/\/git.io\/JvL85}{https:\/\/git.io\/JvL85}}) for every model, except those that affect the size of the model (Table \\ref{tab:sizes}). Models are trained either for 300,000 updates, 100 epochs, or until the model does not improve for 10 checkpoints (early-stopping), whichever comes first.\n \n \\begin{table}[]\n     \\centering\n     \\begin{tabular}{lccc}\n          Size & Layers & FF Size & Hidden Size \\\\\n          \\hline\n          Large & 12 & 2048 & 512 \\\\\n          Medium & 6 & 2048 & 512 \\\\\n          Small & 6 & 1024 & 256 \\\\\n          Tiny & 2 & 1024 & 256 \\\\\n     \\end{tabular}\n     \\caption{Hyper-parameters of various model sizes used in this work. For example, the Large Transformer model architecture uses 6 encoder and 6 decoder layers, a feed-forward hidden dimension of 2048 at each layer, and a word-embedding \/ hidden dimension of 512.}\n     \\label{tab:sizes}\n \\end{table}\n \n \n\n\\paragraph{Continued Training} Work by \\cite{Gordon2019-qs} suggests that students may benefit from training on some combination of the distilled and un-distilled reference dataset. We experimented with this by continuing to train each in-domain student model on the original, un-distilled dataset, using similar stopping criterion to the first round of training. This improved some models by up to 1 BLEU. Because of this, we recommend that any distilled model continue training on the original dataset as long as development accuracy improves. When continued training improves performance of a student, we show that score instead of the score without continued training.\n\n\\section{Recommendations}\n\\subsection{Adapt Teachers}\n    \nIn this section, we compare training in-domain models with no teacher (config 1), a teacher trained on in-domain data only (config 2), and a teacher adapted from the general domain (config 3). The performance of the two teachers in each language-pair and domain is listed in Table \\ref{tab:id-teachers}. It shows that adaptation greatly improves the performance of every in-domain teacher except German-English WIPO.\\footnote{German WIPO is also the largest in-domain dataset we test, which might make adaptation unnecessary. Another explanation might be that the German-English general-domain is not similar enough to the patent domain in this case to improve performance.}\n\nTable \\ref{tab:id-students} shows the results of using these teachers to distill the in-domain data before training student models in various settings. \\textbf{We see that in almost every case, using an adapted teacher gives the best or close to the best results.} This is somewhat expected since models with better development scores tend to make better teachers \\cite{Zhang-2018ak}. Although knowledge distillation is typically seen as ``simplifying\" data for students, in this case we suspect that the adapted teacher's knowledge about the general-domain is making its way to students via the distilled in-domain data.\n    \\begin{table}\n    \\centering\n    \\begin{tabular}{lllccc}\n    Domain & Size & Adapted From & de-en & ru-en & zh-en\\\\\n    \n\\hline\nted & Large & None                       & 29.25 & 19.38 & 14.79\\\\\n &  & Large                      & \\color{ForestGreen}37.64 & \\color{ForestGreen}26.57 & \\color{ForestGreen}20.45\\\\\n\\hline\nwipo & Large & None                      & 48.31 & 21.36 & 31.02\\\\\n &  & Large                     & \\color{ForestGreen} 48.56 & \\color{ForestGreen}37.08 & \\color{ForestGreen}36.80\\\\\n\n    \\end{tabular}\n    \\caption{BLEU development score of in-domain teachers. Adaptation drastically improves performance on every language pair and domain, except de-en WIPO.}\n    \\label{tab:id-teachers}\n    \\end{table}\n\n    \\begin{table}\n    \\centering\n    \\begin{tabular}{llcccc}\n    Domain & Size & Config \\# & de-en & ru-en & zh-en\\\\\n    \n &  & 1                                  & 27.73 & 19.34 & 15.17\\\\\nted & medium & 2                         & 29.11 & 20.31 & 15.71\\\\\n &  & 3                                  & \\textbf{29.54} & \\textbf{20.56} & \\textbf{15.90}\\\\\n\\hline\n &  & 1                                  & 27.89 & 18.42 & 14.87\\\\\n & small & 2                             & 28.93 & 19.65 & 14.95\\\\\n &  & 3                                  & \\textbf{29.52} & \\textbf{19.88} & \\textbf{15.79}\\\\\n\\hline\n &  & 1                                  & 25.78 & 17.48 & 13.03\\\\\n & tiny & 2                              & 27.20 & 17.87 & 13.39\\\\\n &  & 3                                  & \\textbf{27.58} & \\textbf{19.27} & \\textbf{13.74}\\\\\n\\hline\n &  & 1                                  & 48.89 & 24.45 & 30.13\\\\\nwipo & medium & 2                        & \\textbf{50.66} & \\textbf{24.62} & 32.13\\\\\n &  & 3                                  & 50.23 & 24.60 & \\textbf{33.16}\\\\\n\\hline\n &  & 1                                  & 47.94 & 21.91 & 30.66\\\\\n & small & 2                             & 49.46 & \\textbf{23.70} & 32.19\\\\\n &  & 3                                  & \\textbf{49.72} & 23.50 & \\textbf{32.61}\\\\\n\\hline\n &  & 1                                  & 44.15 & 21.39 & 27.67\\\\\n & tiny & 2                              & 48.03 & \\textbf{22.24} & 28.18\\\\\n &  & 3                                  & \\textbf{48.51} & 22.03 & \\textbf{29.88}\\\\\n\n\n    \\end{tabular}\n    \\caption{BLEU development scores for in-domain students with no teacher (config 1), an in-domain only teacher (config 2), or an adapted teacher continued from the general-domain (config 3). In almost every case, using an adapted teacher gives the best or close to the best results. }\n    \\label{tab:id-students}\n    \\end{table}\n\n\n\n\\subsection{Adapt the Best Student}\n\nWe also train small models directly on the general-domain data and adapt them to in-domain data. The possible configurations are random initialization (config 1), initializing from a baseline model trained on general-domain data (config 4), or initializing from a student model distilled from a general-domain teacher (config 7). Table \\ref{tab:gd} shows the performance of these models on the general-domain datasets, and Table \\ref{tab:id-adapted} shows their performance on in-domain datasets.\n\nWhile adapting teachers gives modest gains on in-domain datasets (0-2 BLEU), training small models directly on the general-domain data gives much more substantial gains (5-10 BLEU). We believe this is because a large amount of data is required to fully reveal the teacher's probability distribution over translations \\cite{Fang2019-lw}. While an adapted teacher might contain much information from the general-domain, it is unable to express that knowledge to students just by translating the smaller in-domain dataset. \\textbf{To get the full benefit of general-domain data, the small models must be directly trained on general-domain data.}\\footnote{A reasonable alternative to this might include data-free KD \\cite{Yin2019-bm}, which explores the teacher's probability distribution without any dependence on data.}\n\nWe also observe that Medium-sized models are not small enough to benefit from knowledge distillation in the general-domain, and so their general-domain scores do not improve with distillation. These distilled Medium-sized models (config 7) also tend to do slightly worse than their baseline counter-parts (config 4) on in-domain data. Indeed, Figure \\ref{fig:gd-id} shows that in-domain performance is roughly linearly related to general-domain performance regardless of whether distillation is applied before adaptation.\n\nThis implies that \\textbf{distillation does not interfere with the adaptability of a model}, so the model with the best general-domain performance should be adapted, regardless of whether distillation was applied. Adapting a distilled model can improve performance by up to 1 BLEU over adapting the baseline model without distillation.\n\n    \\begin{table}\n    \\centering\n    \\begin{tabular}{lccc}\n    Model & de-en & ru-en & zh-en\\\\\n    \n\\hline\nTeacher                                  & 41.08 & 32.25 & 47.17\\\\\n\\hline\nMedium Baseline                          & 39.86 & 30.81 & 45.40\\\\\nMedium Student                           &\\color{red} 39.40 & \\color{red} 30.65 & \\color{red} 45.11\\\\\n\\hline\nSmall Baseline                           & 36.78 & 27.54 & 42.09\\\\\nSmall Student                            & \\color{ForestGreen} 38.51 & \\color{ForestGreen}28.88 & \\color{ForestGreen}42.73\\\\\n\\hline\nTiny Baseline                            & 31.27 & 23.63 & 34.71\\\\\nTiny Student                             & \\color{ForestGreen}34.58 & \\color{ForestGreen}25.86 & \\color{ForestGreen}36.09\\\\\n\n    \\end{tabular}\n    \\caption{General-domain models, teachers and students. While knowledge distillation improves small and tiny models, it appears medium-sized models are not under-parameterized enough for knowledge distillation to improve performance.}\n    \\label{tab:gd}\n    \\end{table}\n\n    \\begin{table}\n    \\centering\n    \\begin{tabular}{llcccc}\n    Domain & Size & Config \\# & de-en & ru-en & zh-en\\\\\n    \n\n\\hline\n &  & 1                                  & 27.73 & 19.34 & 15.17\\\\\nted & medium & 4                         & \\textbf{36.94} & \\textbf{25.82} & 20.13\\\\\n &  & 7                                  & 35.93 & 25.43 & \\textbf{20.18}\\\\\n\\hline\n &  & 1                                  & 27.89 & 18.42 & 14.87\\\\\n & small & 4                          & 34.78 & 24.10 & 18.84\\\\\n &  & 7                                  & \\textbf{35.33} & \\textbf{24.30} & \\textbf{19.32}\\\\\n\\hline\n &  & 1                                  & 25.78 & 17.48 & 13.03\\\\\n & tiny & 4                           & 31.52 & 21.30 & 16.51\\\\\n &  & 7                                  & \\textbf{32.30} & \\textbf{21.65} & \\textbf{17.06}\\\\\n\\hline\n &  & 1                                  & \\textbf{48.89} & 24.45 & 30.13\\\\\nwipo & medium & 4                        & 48.58 & \\textbf{35.98} & \\textbf{35.33}\\\\\n &  & 7                                  & 48.53 & 35.55 & 35.27\\\\\n\\hline\n &  & 1                                  & 47.94 & 21.91 & 30.66\\\\\n & small & 4                         & 48.13 & \\textbf{35.30} & \\textbf{34.90}\\\\\n &  & 7                                  & \\textbf{48.31} & 35.18 & 34.52\\\\\n\\hline\n &  & 1                                  & 44.15 & 21.39 & 27.67\\\\\n & tiny & 4                          & 46.06 & 31.13 & 28.45\\\\\n &  & 7                                  & \\textbf{46.54} & \\textbf{31.74} & \\textbf{29.07}\\\\\n\n    \\end{tabular}\n    \\caption{In-domain models that are initialized randomly (config 1), initialized from a baseline trained on general-domain data directly (config 4), or initialized from a  general-domain student trained using a general-domain teacher (config 7). }\n    \\label{tab:id-adapted}\n    \\end{table}\n    \n    \\begin{figure}\n        \\centering\n        \\includegraphics[width=9cm]{images\/gd-vs-id.png}\n        \\caption{The BLEU of general-domain models vs. their corresponding in-domain scores when adapted to a different domain. We see that in-domain performance is roughly linearly related to general-domain performance regardless of whether distillation is applied before adaptation.}\n        \\label{fig:gd-id}\n    \\end{figure}\n\n\\subsection{Distill, Adapt, Distill}\nFinally, we test whether these two ways of improving small, in-domain models are orthogonal. We might hypothesize that training small models directly on general-domain data eliminates the need to adapt teachers or use an in-domain teacher at all. To test this, we also train adapted student models using a baseline teacher (config 8) and an adapted teacher (config 9).\n\n   \\begin{table}\n    \\centering\n    \\begin{tabular}{llcccc}\n    Domain & Size & Config \\# & de-en & ru-en & zh-en\\\\\n    \n\\hline\n &  & 7                                  & 35.93 & 25.43 & \\textbf{20.18}\\\\\nted & medium & 8                         & 35.23 & 25.18 & 19.96\\\\\n &  & 9                                  & \\textbf{36.65} & \\textbf{25.91} & 20.13\\\\\n\\hline\n &  & 7                                  & 35.33 & 24.30 & 19.32\\\\\n & small & 8                             & 35.11 & 23.97 & 19.17\\\\\n &  & 9                                  & \\textbf{35.57} & \\textbf{24.95} & \\textbf{19.48}\\\\\n\\hline\n &  & 7                                  & 32.30 & 21.65 & 17.06\\\\\n & tiny & 8                              & 32.21 & 21.45 & 16.72\\\\\n &  & 9                                  & \\textbf{33.12} &\\textbf{22.49} & \\textbf{17.54}\\\\\n\\hline\n &  & 7                                  & 48.53 & 35.55 & 35.27\\\\\nwipo & medium & 8                        & 49.07 & 34.71 & 35.09\\\\\n &  & 9                                  & \\textbf{49.82} & \\textbf{35.83} & \\textbf{36.48}\\\\\n\\hline\n &  & 7                                  & 48.31 & \\textbf{35.18} & 34.52\\\\\n & small & 8                             & \\textbf{48.79} & 34.27 & 34.89\\\\\n &  & 9                                  & 48.35 & 35.10 & \\textbf{35.55}\\\\\n\\hline\n &  & 7                                  & 46.54 & 31.74 & 29.07\\\\\n & tiny & 8                              & \\textbf{49.90} & 31.12 & 30.05\\\\\n &  & 9                                  & 49.70 & \\textbf{31.75} & \\textbf{31.82}\\\\\n\n    \\end{tabular}\n    \\caption{In-domain models which are initialized from a general-domain student and trained on in-domain data which is pre-processed either with no teacher (config 7), an in-domain only teacher (config 8), or an adapted teacher continued from general-domain data (config 9).}\n    \\label{tab:id-adapt-both}\n    \\end{table}\n\nTable \\ref{tab:id-adapt-both} shows that \\textbf{distilling a second time using in-domain data with an adapted teacher can further boost performance of an already distilled model}, while using an un-adapted in-domain teacher can sometimes hurt performance. \n\nThese results lead us to a general recipe for training small, in-domain models using knowledge distillation and domain adaptation in combination:\n\n\\begin{enumerate}\n    \\item Distill general-domain data to improve general-domain student performance.\n    \\item Adapt the best model from Step 1 to in-domain data.\\\\\n    (2-10 BLEU better than no adaptation)\n    \\item Adapt the teacher and distill again in-domain. \\\\\n    (0-2 BLEU better than no or non-adapted teacher)\n\\end{enumerate}\n\nFollowing this procedure will result in either configuration 6 or 9 as described in Figure \\ref{fig:configs}. And indeed, configuration 9 performs the best or near best (within 0.1 BLEU) in almost every case, as shown in Table \\ref{tab:best-configs}. For those Medium sized models which were not improved by distillation in the general-domain, configuration 6 performs the best. \n\n\n    \\begin{table}\n    \\centering\n    \\begin{tabular}{llcccc}\n    Domain & Size & Config \\# & de-en & ru-en & zh-en\\\\\n    \n\\hline\n & medium & 6                            & 36.80 & 26.26 & 20.13\\\\\nted & small & 6                          & 35.50 & 24.68 & 19.31\\\\\n & tiny & 6                              & 32.09 & 22.20 & 17.25\\\\\n\\hline\n & medium & 6                            & 48.31 & 35.82 & 36.58\\\\\nwipo & small & 6                         & 49.04 & 35.30 & 35.40\\\\\n & tiny & 6                              & 48.02 & 31.57 & 30.53\\\\\n\n    \\end{tabular}\n    \\caption{Development scores for models initialized from a model trained on general-domain data. The in-domain data is pre-processed with a teacher adapted from the general-domain (config 6).}\n    \\label{tab:config-6}\n    \\end{table}\n    \n    \nModels trained on German-English WIPO are an exception, with adaptation from the general-domain not improving performance. This is in line with the results from Table \\ref{tab:id-teachers} which shows adaptation does not improve teachers, either. We suspect this is because the German-English WIPO dataset is the biggest out of any in-domain dataset, making adaptation unnecessary. Future work might also benefit from a quantification of domain similarity between datasets \\cite{Britz2017-qg}, which would guide the use of domain adaptation in cases like these.\n    \n    \\begin{table}\n    \\centering\n    \\begin{tabular}{llccc}\n    Domain & Size & de-en & ru-en & zh-en\\\\\n    \n\\hline\n & medium & 4\/6 & 6 & 4\/6\/7\/9\\\\\nted & small & 6\/9 & 9 & 9\\\\\n & tiny & 9 & 9 & 9\\\\\n\\hline\n & medium & 2 & 4\/6\/9 & 6\\\\\nwipo & small & 3 & 4\/6 & 9\\\\\n & tiny & 8 & 7\/9 & 9\\\\\n\n    \\end{tabular}\n    \\caption{Best configurations for each setting. Scores within 0.1 BLEU of the best are also listed. Configuration 9 generally performs best, while configuration 6 is best for those medium-sized models which were not improved by distillation in the general-domain.}\n    \\label{tab:best-configs}\n    \\end{table}\n       \n\n\\subsection{Training Times}\nThe models trained in this work collectively required 10 months of single-GPU compute time. Table \\ref{tab:times} breaks this down by model size and dataset.\n\nWhile distilling twice might give the best performance, it also increases the amount of computation time required. Rather than training a single in-domain model, configuration 9 requires training a general-domain teacher, a general-domain student, and then adapting both. This can increase compute required to train models by 2-4x.\n\nA huge portion of computation was also spent on decoding the general-domain data using a teacher model for sequence-level knowledge distillation, which could take up to 24 days of GPU time (using a beam size of 10 and a batch size of 10). This can be arbitrarily sped up using multiple GPUs in parallel, but future work might explore how to distill teachers in a less expensive way.\n\n\\begin{table}[]\n    \\centering\n    \\begin{tabular}{lccc}\n        Model & General-Domain & In-Domain & Adapting \\\\\n        \\hline\n        Large & 2-4 days & 2-4 days & 7-48 hours \\\\ \n        Medium & 2-4 days & 2-4 days & 1-48 hours \\\\ \n        Small & 1-2 days & 1-2 days & 2-14 hours \\\\ \n        Tiny & 1 days & 1-24 hours & 2-24 hours \\\\\n        \\hline\n        Distilling & 10-24 days & 1-2 days &\n    \\end{tabular}\n    \\caption{Estimates of the computation time required for training randomly initialized models on just general-domain data or just in-domain data. We also show the time required for adapting general-domain models and distilling data using teachers.}\n    \\label{tab:times}\n\\end{table}\n\n\\section{Related Work}\nOur work is one the few that focuses specifically on training small, under-parameterized in-domain models. There is, however, similar work which is \\emph{not directly comparable} but uses knowledge distillation to adapt to new domains.\n\n\\paragraph{Knowledge Adaptation} uses knowledge distillation to transfer knowledge from multiple, labeled source domains to un-labeled target domains. This is in contrast to our setting, which has labels for both general-domain and in-domain data. Ruder et al. \\shortcite{Ruder2017-zt} introduced this idea as ``Knowledge Adaptation,\" using multi-layer perceptrons to provide sentiment analysis labels for unlabeled in-domain data. Similar work includes Iterative Dual Domain Adaptation \\cite{Zeng2019-eg} and Domain Transformation Networks \\cite{Wang2019-hs}. These ideas are not limited to machine translation; recent work by Meng et al. \\shortcite{Meng2020-yu} trains in-domain speech recognition systems with knowledge distillation, while Orbes-Arteaga et al. \\shortcite{Orbes-Arteaga2019-de} does similar work on segmentation of magnetic resonance imaging scans. \n\n\\paragraph{Compressing Pre-trained Language Models} Domain adaptation via continued training in NMT is closely related to the idea of pre-training a language model and fine-tuning to different tasks, which might come from different data distributions than the pre-training data. Because language models tend to be extremely cumbersome to train and evaluate, more focus is given to the compression aspect of knowledge distillation. Sanh et al. \\shortcite{Sanh2019-gl}, Sun et al. \\shortcite{Sun2019-io}, and Liu et al. \\shortcite{Liu2019-ho} independently showed that knowledge distillation could be used to compress pre-trained models without affecting downstream tasks. Tang et al. \\shortcite{Tang2019-qc} showed that task-specific information could be distilled from a large Transformer into a much smaller Bi-directional RNN. These methods might reasonably be extended to domain adaptation for NMT.\n\n\\section{Conclusion}\nIn this work, we conducted a large-scale empirical investigation to determine best practices when using sequence-level knowledge distillation and domain adaptation in combination. We found that adapting models from the general-domain makes them better teachers and that distilling using general-domain data does not impact a model's adaptability. This leads us to recommend distilling twice for best results: once in the general-domain to possibly improve student performance, and again using an adapted in-domain teacher. The results are robust among multiple language pairs, student sizes, in-domain settings.\n\\bibliographystyle{named}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}