Abelian Yang-Baxter Deformations and TsT transformations

We prove that abelian Yang-Baxter deformations of superstring coset sigma models are equivalent to sequences of commuting TsT transformations, meaning T dualities and coordinate shifts. Our results extend also to fermionic deformations and fermionic T duality, and naturally lead to a TsT subgroup of the superduality group OSp(d_b,d_b|2d_f). In cases like AdS_5 x S^5, fermionic deformations necessarily lead to complex models. As an illustration of inequivalent deformations, we give all six abelian deformations of AdS_3. We comment on the possible dual field theory interpretation of these (super-)TsT models.


Introduction
Integrability is a key feature of the string σ model on AdS 5 × S 5 in the context of the AdS/CFT correspondence [1]. Progress in this field has led to substantial improvements in our understanding of both sides of this duality [2,3,4]. One way to further extend our understanding is to study deformations that extend beyond the maximally symmetric example of AdS 5 × S 5 and its lower dimensional cousins, while preserving integrability. The primary example of this is a string on the Lunin-Maldacena background [5,6,7], dual to real β deformed planar SYM. On the string side, this theory can be obtained by so-called TsT transformations -sequences of T dualities and shifts in commuting directions, also known as Melvin twists. More recently it was realised in the manifestly integrability preserving framework of Yang-Baxter deformations. The purpose of this paper is to elucidate the connection between these two approaches.
Yang-Baxter σ models were introduced as deformations of principal chiral models based on R operators solving the modified classical Yang-Baxter equation [8], preserving their integrability [9]. This notion was generalised to symmetric space coset σ models in [10] and then further to the supercoset σ model describing the AdS 5 × S 5 superstring [11]. 1 By a simple limit this deformation procedure can be extended to solutions of the classical Yang-Baxter equation [24]. These equations admit many solutions, and correspondingly there are many different integrable deformations of the AdS 5 × S 5 string. In terms of general structure, at the level of symmetries, deformations based on the modified classical Yang-Baxter equation correspond to quantum deformations [25], while deformations based on the classical Yang-Baxter equation result in Drinfeld twists [26], see also [17]. At the level of string theory, the condition that the backgrounds of these models solve the supergravity equations of motion requires the associated R operator to be unimodular [27]. All Yang-Baxter deformations of the string preserve κ symmetry however [11,27], meaning that their backgrounds necessarily solve a set of modified supergravity equations [28,29], guaranteeing scale but not Weyl invariance.
The structure described above matches with previously established results. Namely, the η deformation of the string -based on the modified classical Yang-Baxter equation -was originally constructed using a non-unimodular R operator, and indeed the associated background does not solve the supergravity equations [30], but rather the modified ones [28], see also [31]. Still, alternative R operators exist [25,32]. These appear to give inequivalent backgrounds, yet the same S matrix [32]. None of the studied R operators is unimodular, however, and it is not known whether a unimodular one exists. 2 For classical Yang-Baxter deformations the situation is more diverse. R operators of this type can be divided into abelian and non-abelian, depending on whether the associated generators all mutually commute or not. In the non-abelian class, bosonic jordanian R operators are not unimodular, and indeed the associated backgrounds solve the modified supergravity equations [37], but not the regular ones [38,37]. In fact, many jordanian deformations are closely related to the η model, as they can be obtained from it by singular boosts [37]. Further bosonic jordanian examples were recently investigated in [39]. The conformal symmetry of AdS 5 is large enough, however, to admit other, unimodular non-abelian R operators [27].
In contrast to non-abelian ones, abelian R operators are always unimodular, meaning any such operator maps a solution of supergravity to a solution of supergravity. Various abelian deformations were studied at the bosonic level, see e.g. [40,41,42,43], including the Lunin-Maldacena background mentioned above [44]. More recently some examples have been studied to quadratic order in fermions, both as singular boosts of the η model [30,37] and directly [38]. These individual examples all fit the proposal of one of the present authors [42], that abelian Yang-Baxter deformations are equivalent to sequences of commuting TsT transformations.
The objective of this paper is to get closer to a complete understanding of this abelian class of Yang-Baxter deformations, by giving a general proof of the equivalence between abelian Yang-Baxter deformations and (sequences of commuting) TsT transformations. This proof relies on always being able to find a group parameterisation such that the Maurer-Cartan forms manifest a set of chosen commuting isometries. For completeness, upon complexification we can extend our proof to include R operators based on anticommuting supercharges. These are equivalent to a generalised fermionic version of TsT transformations, which we introduce. Furthermore, in order to explore the various possible abelian deformations/TsT transformations and to get a better idea of their general structure, we consider AdS 3 -the simplest nontrivial non-compact example -which admits six inequivalent abelian deformations.
This paper is organised as follows. In section 2 we establish our conventions for the type IIB superstring in AdS 5 × S 5 and its integrable deformations based on the classical Yang-Baxter equation. Bosonic and fermionic T duality is introduced in section 3, where we also briefly discuss the duality groups O(d, d) and OSp(d b , d b |2d f ) respectively. We prove equivalence between abelian deformations and TsT transformations in section 4. In the last section we address the fact that there are different inequivalent commuting subalgebras in noncompact cosets, illustrating this with a discussion of all inequivalent abelian deformations of AdS 3 . In the conclusions we indicate some open questions and comment on the possible dual field theory interpretation of these deformed models.

Yang-Baxter Deformations
The Undeformed AdS 5 × S 5 Superstring Action Let us briefly introduce the conventions for the supercoset σ model with fields in which describes the Green-Schwarz type IIB superstring in AdS 5 × S 5 [45], see [2] for an extensive review. The argumentation in the section 4 will also hold for general bosonic symmetric space σ models and any supercoset σ models which can be described similarly to the AdS 5 × S 5 superstring. The string moving in a coset M = G/H is described by G valued fields g : Σ → G defined on the worldsheet Σ. The theory can be formulated in terms of the Maurer-Cartan forms taking values in the Lie algebra g of G Important for the integrability of the AdS 5 × S 5 superstring is the existence of the Z 4 -grading of g = su(2, 2|4): and for the supertrace of a matrix realisation of g STr(M (i) N (j) ) = 0 for m + n = 0 mod 4.
g (2) denotes the bosonic coset algebra, g (0) the little group algebra of the bosonic coset, and g (1) and g (3) are the odd parts of the algebra. 3 The action of the superstring in AdS 5 × S 5 in conformal gauge 4 takes the form [45] S ∝ d 2 σ L = d 2 σ STr(A + d − (A − )), (2.5) with the worldsheet light-cone components of A and the linear combinations of projection operators on the Z 4 -components Key features of the σ model (2.5) are κ symmetry and integrability. The latter is associated to a spectral parameter dependent Lax pair ± , (2.7) 3 We choose our superalgebra conventions as in [2], where elements of the algebra may be represented as an even supermatrix m η ϑ n with m, n : matrices built from c-numbers, η, ϑ Grassmann-valued matrices (2.4) Let us note, that we work with bosonic generators {h i } and fermionic generators {Q α } being even respectively odd supermatrices with only even entries, so that e. g.
are even supermatrices for a Grassmann-valued fields θ α . 4 This is purely a choice of convenience and does not affect our analysis.
where the flatness condition is equivalent to the equations of motion.
Let us now introduce integrable deformations of (super)coset σ models such as (2.5), based on solutions of the classical Yang-Baxter equation.

The Classical Yang-Baxter Equation and Linear R operators
The standard form of the classical Yang-Baxter equation (CYBE) defined on tensor products of an algebra or superalgebra g is [r 12 , r 13 ] + [r 12 , r 23 ] + [r 13 , r 23 ] = 0 for r ∈ g ⊗ g.
Deformations are formulated in terms of equivalent linear operators R : g → g. The transition from a graded skewsymmetric r matrix to an R operator is via the trace A simple solution of (2.9) over a given algebra g is the r matrix consisting of graded commuting generators. In the following we will call these r matrices abelian.

Deformations based on Solutions of the Classical Yang-Baxter Equation
Yang-Baxter deformations of coset σ models of the form of eqn. (2.5) are generated by skewsymmetric 5 linear R operators solving (2.9). A further ingredient is the "dressing" of the R operator R g = Ad −1 g • R • Ad g . The Yang-Baxter deformed action is given by [11,24] where we introduced the deformed currents J ± = 1 1±R g •d − (A ± ), and directly specified to the (unmodified) classical Yang-Baxter case. Note that we include deformation parameters already in the definition of R. These can take any real respectively Grassmannian value depending on the parity of the generating R operator, as the CYBE (2.9) is invariant under rescalings of R.
These deformations preserve the κ symmetry and integrability of the undeformed model (2.5). The associated deformed Lax pair is (2.11) These deformations break part of the global G symmetry g → g ′ g for g ′ ∈ G of the undeformed model. The unbroken symmetries are generated by the generators T for which [42] R

T Duality Groups and their TsT Subgroups
In this section we will briefly recall bosonic and fermionic T duality and the associated TsT transformations in the σ model context.

The Notion of Bosonic and Fermionic T duality
Consider a generic (classical 6 ) string σ model of the form where we work in conformal gauge for the sake of convenience, and understand Z M as with some bosonic fields X µ and some fermionic Grassmann-valued fields θ ∆ . We refer to the parity of the coordinate Z M as s(M). E MN (Z) is the background field describing the coupling between the fields 7 with parity s(E MN ) = s(M) + s(N), so that s(L) = 0. Now we assume the model has a manifest isometry and choose the associated coordinate to be Z 1 , meaning the symmetry is realised as a shift of Z 1 . We write . Z 1 can be either bosonic or fermionic 8 . This allows us to rewrite the Lagrangian by introducing gauge fields A ± : where the Lagrange multiplierZ 1 ensures A ± = ∂ ± Z 1 by its equations of motion. Integrating out A ± instead ofZ 1 yields the action of the dual model with the dual backgroundĒ given bȳ For T duality along a bosonic isometry we reproduce Buscher's T duality rules [46]. For details on topological considerations and fermionic T duality and its implications in general we refer to e.g. [47,48]. 9 6 A dilaton φ enters the string action at a higher order in the coupling α ′ . At the classical level the dilaton has to be introduced in the corresponding supergravity (e.g. the RR-forms appear always as e φ F µ 1 ...µp ). As we will not do explicit field redefinitions, we neglect it and its behaviour under T duality from the start. Working at the classical level we also disregard any prefactors of the action and are only interested in its schematical form. 7 E MN could be decomposed into its graded symmetric (metric-like) and graded skewsymmetric part: E MN = G MN + B MN . But only the order θ 0 terms in G µν respectively B µν would have a direct physical interpretation as the components of metric and B field. We stick to the quite abstract 'background' E MN as it is practical and sufficient for our further considerations. 8 In the fermionic case the generator Q dual to the isometry coordinate has to anticommute with itself in order to correspond to a shift isometry. In other words, fermionic T duality requires a supercharge Q with Q 2 = 0. We will come back to this point below. 9 Note that our conventions for the σ model (3.1) differ from [47], leading to some different signs in (3.2). Furthermore note that, as defined, along a fermionic isometry coordinate only T 4 , not T 2 , is manifestly the identity operation. T 2 is a trivial and physically irrelevant coordinate redefinition of the background, Z 1 → (−1) s(1) Z 1 , however.

The O(d, d) Group of Bosonic T duality
Now we assume the model has d commuting bosonic isometries and choose the associated coordinates to be a T duality transformation along X i can be represented for every i ∈ {1, ..., d} as where E i is the D × D-matrix with every element being zero, except for (E i ) ii = 1. Other transformations, that even leave the Lagrangian invariant, are GL(d)-transformations of the isometry directions if we also transform E accordingly. Let A ∈GL(d) and

This can be represented by fractional linear action (3.3) on E of the group element
(3.5) Both G T i and G GL are elements of O(D, D), where we understand its elements as 2D × 2Dmatrices G fulfilling the pseudo-orthogonality relation The form of (3.4) and (3.5) suggests that we can write these as elements of O(d, d) 10 Note that det g T i = −1, so in fact bosonic T duality transformations itself are not in the component connected to the identity, in contrast to g GL . But we can generate further elements of the component connected to the identity of O(d, d) by a product of some general linear transformations and an even number of T duality transformations.

Bosonic TsT Transformations
Now we introduce TsT transformations in the above framework. These gained some attention in the context of the AdS/CFT correspondence, as a particular TsT transformation of the AdS 5 × S 5 background gives a supergravity background dual to β deformed SYM [5]. To do TsT transformations we need at least two isometries, which we parameterise by X 1 and X 2 in the following. A single TsT transformation is generated by a T duality transformation on the X 1 , a shift 11X and then a T duality transformation on theX 1 direction back. In the above group language, in the minimal d = 2 setting this looks like (3.12) Generic TsT transformations can be understood as the straightforward generalisation to fractional linear transformations of the type (3.3) with the generating group element where Γ is an antisymmetric d × d-matrix. This can be seen as

OSp(d b , d b |2d f ) as the Superduality Group
Consider a background E with d b bosonic and d f fermionic isometries and d = d b + d f . Let us write our coordinates as 11 Note that this a quite specific transformation. Generic coordinate transformations would also lead to contributions in the other blocks of an O(d, d) element in comparison to (3.12)). Shifts in the "other" direction likē between two T duality transformations would lead to The background is transformed with (3.7) and (3.3) only in the isometry components as where B ij are components corresponding to the isometry directions of the B-field. While these coordinate shifts (3.8) look quite similar to the ones of TsT transformations, Θ shifts act very differently on the background. Θ shifts clearly generate physically equivalent models up to boundary terms, as H = dB remains invariant.
The matrix representation in the sense of (3.3) and (3.7) of a single T duality transformation (3.2) along the isometry coordinate Z a is 12 We can further consider GL With supertransposition defined as the "group element" of such a GL(d b |d f )-transformation with the action (3.3) on the background components E in the conventions of (3.1) is given similarly to (3.5) by It is easy to show that both (3.16) and (3.17) are elements of a group with elements fulfilling a modified pseudoorthogonality relation (in comparison to (3.6)) This is a representation 13 of the orthosymplectic group OSp(d b , d b |2d f ) and nicely generalises the O(d b , d b ) group of bosonic T duality. This group was previously introduced in [51], see also [52]. We will constrain further discussion of OSp(d b , d b |2d f ) to the generalisation of generic TsT transformations (3.13) of the bosonic case. 12 Note that det g Ta = −(−1) s(a) . 13 More commonly one defines OSp(m, m|2n) as the group constisting of (2m|2n) × (2m|2n)-supermatrices M preserving the supermetric J J and J from (3.18) are connected via a similarity transformation

Fermionic Generalisation of TsT Transformations
Although along a fermionic coordinate g 2 T = 1, the structure of the superduality group (3.18) does not become more complicated, since as mentioned above T 2 α is only a coordinate transformation θ α → −θ α . As such we expect some fermionic analogue of the generic TsT transformation (3.13) to exist. For this we consider the (3.13)-like ansatz This lies in our representation (3.18) Similarly to the bosonic case above, group elements of this type form an abelian subgroup of OSp The group element (3.19) now corresponds to a sequence of Ts(T −1 ) transformations, with shifts defined as in (3.11). Purely fermionic Ts(T −1 ) transformations look like and indeed schematically T f s f f T −1 f give rise to symmetric, but off-diagonal entries in Λ f in (3.19). It turns out that the diagonal elements in Λ f cannot be understood as a type g T · g GL · g −1 T transformation. 14 From here on, we therefore understand generic Ts(T −1 ) transformations as group elements of OSp(d b , d b |2d f ) of the type (3.19) with generic symmetric, but off-diagonal Λ f . Let us note that there is no ambiguity for Ts(T −1 ) transformations "mixing" bosons and fermions: T f s f b T −1 f -and T b s b f T b -type transformations are equivalent and both correspond to the (skewsymmetric) odd part of Γ in (3.19). Of course Ts(T −1 ) transformations directly reduce to TsT transformations if the T duality is a bosonic one and so, for the sake of simplicity, we will refer to Ts(T −1 ) transformations as TsT transformations from now on. Both only differ by a trivial coordinate redefinition in any case.

Equivalence of Abelian Yang-Baxter Deformations and TsT Transformations
In this section we prove that any Yang-Baxter deformation generated by an abelian solution to the CYBE is equivalent to a TsT transformation at the level of the corresponding σ model. This equivalence was previously proposed in [42], and is supported by many examples, see e.g. [44,40,41], but a general proof is still missing. We will also extend this claim by considering r matrices built out of anticommuting supercharges. Using a parameterisation of the coset manifold with manifest shift invariance in d = d b + d f coordinates, we will prove that the (coordinate-dependent) TsT transformation behaviour (3.19) can be reproduced by an abelian R operator, and vice versa. As the Yang-Baxter deformed action (2.10) is independent of parameterisation this introduces a coordinate-independent notion of TsT transformations in the form of abelian Yang-Baxter deformations.

Natural Parameterisation with Manifest Shift Isometries
The starting point of our proof is to choose a natural parameterisation of the coset manifold where we have shift isometries in the coordinates associated to (anti)commuting generators t a , namely g = exp(Z a t a )ḡ(Z a ). There the Z a are the d = d b + d f isometry coordinates and Z a are the remaining coordinates, Z M = (Z a , Z a ) = (X i , θ α , Z a ).ḡ is assumed to be chosen in a way that the metric is nondegenerate, so we can consider (4.1) to be a valid parameterisation of the coset manifold. This is motivated for instance by the group parameterisations of AdS N in Poincaré coordinates as where p µ respectively D are the momentum respectively dilatation generators of the conformal algebra so(2, N − 1). There we have N − 1 isometries parameterised by X µ , as [p µ , p ν ] = 0 by means of the conformal algebra. This type of group parameterisation should always be possible for general group and coset manifolds and any choice of (anti)commuting generators t a in the symmetry algebra. Let us sketch a proof for the bosonic case. We assume that we have a geometry with d commuting Killing vector fields. Then there are coordinates Z M = (X i , Y i ) in which these vector fields are ∂ ∂X i , thus the commuting isometries are parameterised by X i . In particular, the background and a choice of a local frame e µ a with a corresponding spin connection ω µ ab are independent of the X i . The Maurer-Cartan form on a coset manifold (see e.g. [45]) decomposes into with coset generators P a and isotropy generators J ab , so in our case

The flatness of A implies that
For every Y these span a d-dimensional commuting algebra. It follows there is similarity transformation with a group valued function g 2 (Y) where the h i are the constant commuting generators of the algebra corresponding to the Lie algebra of the commuting Killing vector fields. 15 Note that we use the notation h i for a general set of commuting generators, which in the non-compact case will generically not be the Cartan generators. Now consider a group parameterisationg = exp(X i h i )g 2 (Y) withÃ = −g −1 dg. It follows thatÃ Again from the flatness of A follows that so that g 1 is generated by the h i . It follows that a group parameterisation of the form exists for any choice of commuting generators h i .

Bosonic Abelian Yang-Baxter Deformations
Now consider a generic abelian r matrix that consists some bosonic commuting generators h i of the global symmetry algebra of the coset model with a (real) antisymmetric d × d parameter matrixΓ ij . Consider a parameterisation of the form (4.1), Due to the fact that the h i commute, the Maurer-Cartan form becomes and the Lagrangian is manifestly shift-invariant in the X i -coordinates. With this we see that the abelian r matrix (4.5) is actually built from some components of the conserved currents with respect to the global symmetry of the coset σ model, The corresponding dressed r matrix then is and the associated linear R operator can be expressed nicely in terms of the Maurer-Cartan form components it follows that The Yang-Baxter deformed Lagrangian (2.10) then becomes with the general coordinates X M = (X i , Y i ) and the deformed background This directly corresponds to the O(d, d) group element (3.13) describing a generic bosonic TsT transformation.

Inclusion of Fermions
A generic abelian graded skewsymmetric r matrix over a Lie superalgebra in our conventions is built out of (anti)commuting even (odd) generators ..., d b and α, β = 1, ..., d f , We should emphasize that su(2, 2|4) and psu(2, 2|4) do not contain real supercharges that anticommute with themselves, so these fermionic extensions of abelian r matrices do not exist for the real AdS 5 × S 5 superstring, or its AdS 3 and AdS 2 cousins. To consider them we need to work with the complexified model. The r matrices are then complex and break reality of the action, but are otherwise admissible. With some care 16 regarding the Grassmann-valued fields θ the proof works in the same way as in the bosonic case. First we choose a group parameterisation with manifest isometries corresponding to the (anti)commuting generators and express the R g operator corresponding to (4.12) by some components of the Maurer-Cartan form.
The undeformed background E MN is given terms of the components of the Maurer-Cartan form in the conventions of (3.1) and (2.5) by In the same way as in the bosonic case the abelian Yang-Baxter deformation results in a deformed backgroundẼ In other words, we directly reproduce the generic TsT transformation behaviour (3.19) of the superduality group OSp(d b , d b |2d f ), and vice versa.
The direct approach via a natural parameterisation with manifest isometries like (4.1) is useful to see the TsT behaviour of abelian Yang-Baxter deformations as in (3.13), in particular to determine its effect on the concrete background. The abelian Yang-Baxter deformation in the form (2.10) on the other hand, gives a coordinate-independent representation of 16 This is rather tedious with our conventions, as for the fermionic Maurer-Cartan components It is important to pay attention to some subtleties of the graded tensor product in the definition of r g = (Ad −1 g ⊗ Ad −1 g ) · r which match the above ambiguity and lead to the desired R g operator in (4.14).
TsT transformations (in contrast to the OSp(d b , d b |2d f )-approach). Moreover this manifestly shows that every TsT transformation of such a (super)coset gives an integrable model with (2.11) as the associated Lax pair. Abelian Yang-Baxter deformed models correspond to supergravity solutions by construction, as T duality and thus TsT transformations map two supergravity solutions to each other [53], also in the fermionic case [47]. 17 This matches the analysis of [27], as any abelian r matrix is unimodular.

On Inequivalent TsT Transformations
In this section we want to illustrate the fact that there are different inequivalent sets of commuting shift isometries and thus TsT transformations on non-compact backgrounds. For completeness we start with TsT transformation of S 3 .

Sphere S 3
We have seen in the previous section that a natural parameterisation of the background with d commuting isometries is g = exp(X i h i )ḡ with a choice of d commuting generators {h i }. As S N and its isometry group O(N + 1) is compact, any other choice of the commuting generators {k i } is connected via a similarity transformation with a group element S related to the {h i } as k i = Sh i S −1 . Exactly as in (4.3) the corresponding group element yields the same background as g because S is constant. We work with generators n ij of so(N + 1), satisfying [n ij , n kl ] = δ il n jk − δ jl n ik − δ ik n jl + δ jk n il i, j, k, l = 1, ..., N + 1.
The TsT deformed three-sphere looks like 3) 17 In terms of the action on the background fields, the standard treatment of T duality for a supergravity background coupling to a Green-Schwarz superstring [54,55] does not admit an immediate O(d, d)-like formulation of TsT transformations. However, an appropriate extension to the Ramond-Ramond forms exists [56,57,58]. The action of the superduality group OSp(d b , d b |2d f ) on the supergravity fields has not been investigated yet to our knowledge. For fermionic T duality transformations themselves some progress was made in [59] in the canonical formulation. TsT transformations including fermions were studied previously in [50] for deformations of S 5 in the σ model approach.

Anti-de Sitter Space AdS 3
In the non-compact case there are inequivalent choices of commuting generators. We will only explicitly discuss the inequivalent deformations of AdS 3 , where this undertaking is greatly simplified due to the structure of so (2,2). This gives some insight in the various possible abelian Yang-Baxter deformations of AdS 5 .
From the point of view of the Yang-Baxter deformations the overall scaling of the r matrix only contributes to the deformation parameter, so for each factor in (5.5) we only need to consider det s < 0, det s > 0 or det s = 0. These three classes of generators are clearly inequivalent to each other under similarity transformationss = SsS −1 with S ∈ SL(2, R). SL(2, R) moreover acts transitively on each class (up to rescaling). Convenient representants are We can now combine these sl(2, R) generators of both copies in so(2, 2) to a generic r matrix. Exchanging the two copies of sl(2, R) is an outer automorphism of so(2, 2) The physical interpretation is either With use of (5.6) we are left with six types of abelian r matrices, namely: • h 1 ∧ h 2 corresponds to the (non-compact) Cartan r matrix r = −γm 01 ∧ D. A convenient parameterisation is given by g = exp (θm 01 + ln(z)D) exp((uz)p 0 ), corresponding to the metric (ds) 2 = −(zdu) 2 + (uz) 2 (dθ) 2 + (d ln(z)) 2 of hyperpolar Poincaré coordinates. A coordinate change u → x/z yields ln(z) and the boost-angle θ as isometry coordinates. The associated Yang-Baxter deformed background reads in terms of the original hyperpolar Poincaré coordinates.
translates to the (compact) Cartan r matrix r = −γ m 03 ∧ m 12 leading to a TsT transformation corresponding to time shifts and spatial rotations. These are natural in global coordinates, where both isometries are manifest. With a group parameterisation g = exp(φm 03 + θm 12 ) exp(ρm 23 ) the undeformed and deformed backgrounds are • a 1 ∧ a 2 corresponds tor = −γp + ∧ p − ∝ r = −γp 0 ∧ p 1 . With group parameterisation g = exp(−x 0 p 0 + x 1 p 1 ) z D the undeformed and deformed backgrounds are The manifest isometry coordinates for the remaining three r matrices are not very intuitive as the r matrices mix the generators corresponding to costumary choices of coordinates (like global or Poincaré coordinates). We therefore give the deformed backgrounds in light-cone Poincaré coordinates (group parameterisation g = exp( 2 ,

AdS 5
The conformal symmetry of AdS 5 does not decompose nicely as in the AdS 3 case, and we will not give an extensive list of inequivalent TsT transformations here. To illustrate the extent of the full list, note that we could for instance consider abelian Yang-Baxter deformations based on the subalgebras so(2, 4) ⊃ so(2, 2) ⊕ so(2) space ≃ sl(2, R) ⊕ sl(2, R) ⊕ so(2) space , so(2, 4) ⊃ so(2) time ⊕ so(4) ≃ so(2) time ⊕ su(2) ⊕ su(2), or so(2, 4) ≃ conf(1, 3) ⊃ span(p µ ) or span(k µ ), (5.13) leading to many tens of inequivalent deformations already. A method to obtain and classify all inequivalent commuting subalgebras of so(2, 4) and thus also abelian Yang-Baxter deformations was proposed in principle in [61]. In addition to pure AdS 5 deformations we could of course mix AdS 5 and S 5 directions.

Conclusion and Outlook
In this paper we proved that abelian Yang-Baxter deformations are equivalent to sequences of commuting TsT transformations. This proof is completely generic and holds for any group or (semi-)symmetric coset σ model, including fermions to all orders. We included the fermionic generalisation of these transformations, which however typically requires complexification. Including fermionic transformations naturally leads to a TsT subgroup of the superduality group OSp(d b , d b |2d f ) generalising the bosonic T duality group O(d b , d b ).
For illustrative purposes we moreover presented all six possible inequivalent abelian deformations of AdS 3 . In terms of the so(2, 2)-generators the associated r matrices are given by m 01 ∧ D, m 03 ∧ m 12 , One natural question to ask is what the dual field theory interpretation of Yang-Baxter deformations is. For r matrices solving the regular classical Yang-Baxter equation -which includes the present abelian ones -these duals are generically conjecture to be noncommutative versions of supersymmetric Yang-Mills theory [26], provided they exist. This conjecture relies on the twisted symmetry structure of the gravitational models, whose realisation on the hypothetical field theory side requires a nontrivial star product. Several abelian deformed theories are known to fit this description, notably the gravity duals of β deformed SYM [5] and canonical spacelike noncommutative SYM [62,63]. As discussed in [26], the situation is less clear for the naive time-like noncommutative version of SYM and the related abelian deformation of AdS 5 × S 5 for example. The generalisation from the β to the γ i deformation [7] shows subtleties as well, though at least in the spectrum a notion of duality appears to remain, see e.g. [64,65,66]. It is important to understand in which (isolated) cases, and how, the general dual field theory picture breaks down.
In principle we can formally extend the conjecture of [26] to our fermionic TsT transformations, replacing field products in the SYM Lagrangian by star products built on the twist e iγr , where r is associated r matrix. As such r matrices are not real, however, this would be a complex deformation of SYM. Moreover, manifest conformal invariance would be broken, cf. eqn. (2.12). 19 In particular such star products introduce new, possibly dimensionful, couplings in the theory. On the gravity side it would be useful to gain a better understanding of the action of fermionic TsT transformations on the supergravity fields (and their reality). Duals of mixed bosonic-fermionic deformations could be defined similarly, though the nature of their deformation parameter is slightly odd.
There are a number of further open questions. First, it would be interesting to consider classical solutions and associated integrable classical mechanical models for these abelian deformed models, as well as non-abelian ones, as done for the β deformation [6], and the η model in e.g. [67,68,69,70,71]. Second, given the classical equivalence between the η and λ models via Poisson-Lie duality (cf. footnote 1), we might wonder whether similar dual theories exist for CYBE-based deformations. Third, non-Cartan abelian deformations (and non-abelian ones) invariably break the isometries required to fix the standard BMN light cone gauge of the exact S matrix approach to the quantum string σ model [2]. In other words, the effect of these deformations at the quantum level is mysterious, in contrast to the β deformation for example [65].
Recently, hints of generalised TsT structures have been found also in non-abelian cases [39,27]. It would be interesting to try and extend our approach here, especially to the unimodular (supergravity) cases described in [27].