Super Non-Abelian T-Duality

We analyse super non-Abelian T-duality for principal chiral models, symmetric space sigma models, and semi-symmetric space sigma models for general Lie supergroups. This includes T-duality along both bosonic and fermionic directions. As an example, we perform the explicit dualisation of the OSp(1|2) principal chiral model, and, whilst the target superspace of this model is a three-dimensional supergravity background, we find that its super non-Abelian T-dual falls outside the class of such backgrounds.


Introduction
The original 'radius Ñ 1/radius' duality of string theory discovered in [1,2] has evolved into a more general concept of T-duality. This is based on a very simple mechanism: two classically duality-equivalent string sigma models exist whenever we can formulate a firstorder Lagrangian which provides a set of equations of motion and Bianchi identities. Then, on the one hand, when imposing the Bianchi identities, one recovers the original model, while on the other hand, when imposing the equations of motion, one obtains the dual model.
Abelian T-duality [3,4], which is present when the string target space geometry has an Abelian isometry, is an exact symmetry of string theory that maps string backgrounds into dual ones corresponding to the same conformal field theory. In other words, it relates two world-sheet non-linear sigma models with equivalent physical properties, both at the classical level as well as the quantum level [5,6].
Instead, non-Abelian T-duality [7] 1 , which generalises the approach of [5] to string backgrounds that exhibit a bosonic non-Abelian group of isometries, does not represent an exact string symmetry [11]. Indeed, the non-Abelian T-dual model possesses a different, in general smaller, set of symmetries (local as well as non-local ones). Notable examples can be found in [7,[12][13][14]. If supersymmetry is part of the game, the T-dual model is, in general, less or non-supersymmetric, as can be inferred from studying the Killing spinor equations after dualisation [15]. Nevertheless, when starting with a string theory background with a semi-simple group of isometries, the T-dual model usually remains a string theory background essentially because non-Abelian T-duality preserves conformal invariance in this case. 2 Therefore, non-Abelian T-duality can fruitfully be used as a string/supergravity solution generating technique [15,14].
Abelian fermionic T-duality has been formulated in [23] as a natural generalisation of the Abelian bosonic T-duality to sigma models which possess a superisometry generated by an 'Abelian' Q supercharge (Q 2 " 0). The T-self-duality [24,25] of string theory on AdS 5ˆS 5 under a combination of four bosonic and eight fermionic Abelian T-duality transformations provides a dual string explanation of the duality between maximally-helicity-violating amplitudes and light-like Wilson loops in N " 4 supersymmetric Yang-Mills theory (see [26] for a nice review). Since under T-duality ordinary superconformal invariance acting on Wilson loops is mapped to dual superconformal invariance of scattering amplitudes, which ultimately can be thought as being generated by some non-local current, the investigation of T-self-duality is intimately linked to integrability. For the AdS 5ˆS 5 background this has been discussed in [24,25].

More generally, T-self-duality under combinations of bosonic and fermionic T-dualities
has been proved for string sigma models on AdS dˆS dˆM 10´2d in [27]. Furthermore, whilst T-self-duality for the AdS 4ˆC P 3 string sigma model is also expected to hold, its proof remains an open problem. In particular, the existence of T-self-duality for this background is supported by the amplitudes/Wilson loops duality in three-dimensional Aharony-Bergman-Jafferis-Maldacena theory [28][29][30][31] as well as by the integrability emerging on the string theory sides [32][33][34][35] and on the gauge theory side [36,37]. However, the proof of 1 See also [8][9][10] for previous results. 2 In the case of non-semi-simple groups, this is no longer true [16,17] due to the appearance of a mixed gauge-gravitational anomaly [11,18]. As shown in [19,20], such backgrounds are solutions to generalised supergravity field equations [21,22].
T-self-duality in this case is hindered by the appearance of singularities in the T-duality transformations of the fields [38][39][40][41][42], and a stringy explanation of the amplitudes/Wilson loops duality in three dimensions remains open.
Evidently, Abelian fermionic T-duality should be generalised to a non-Abelian setting when non-Abelian isometries form supergroups. This can be achieved by following a prescription similar to the one introduced in [7] for purely bosonic non-Abelian isometries.
However, the procedure of integrating out degrees of freedom to obtain the dual action appears more complicated because of the nilpotent nature of the fermionic coordinates involved in the dualisation [43]. This is one of the reasons why the studies in this direction have been rather limited so far. Dualisation of fermionic directions has been discussed somewhat implicitly in connection with λ-deformed sigma models [44], introduced in [45][46][47], and with the construction of Yang-Baxter-deformed sigma models as non-Abelian T-duals of deformed coset superspaces [48][49][50]. In particular, it was proved in [49] that for a type II Green-Schwarz superstring, non-Abelian T-duality always produces a dual background which again satisfies the torsion constraints. This was shown under the assumption that one can pick the background B 2 -field that is invariant under the dualised isometries. This is possible e.g. for the Ramond-Ramond backgrounds described by semi-symmetric coset superspaces, but, in general, it is not the case for Neveu-Schwarz-Neveu-Schwarz backgrounds. In ten-dimensional supergravity, a non-Abelian version of fermionic T-duality has been formulated in [51] but is limited, however, to fermionic isometries whose algebra closes on Abelian bosonic isometries. These lead to non-geometric solutions to double field theory subject to an extra constraint whose origin remains unclear. Generalisations of Tduality to supermanifolds and super Lie groups has also been explored in the context of Poisson-Lie T-duality [52][53][54][55][56], originally introduced as an extension of standard bosonic dualisation [57,58].
Despite these efforts, a clear picture of non-Abelian T-duality for supergroups is still missing. We shall refer to this as super non-Abelian T-duality in the following. One of the main open problems is to understand better the interplay between bosonic and fermionic non-Abelian T-duality. In particular, it would be interesting to understand whether the combination of fermionic and bosonic non-Abelian T-dualities can provide a more extensive tool for generating new string/supergravity solutions.
Inspired by these questions, in this paper we shall carry out a systematic study of super non-Abelian T-duality, when the dualisation is generically done along the directions corresponding to a non-Abelian subsupergroup of the supergroup of isometries. Using the current algebra approach, we perform in full generality the super non-Abelian T-duality of principal chiral models on supergroup manifolds, as well as of symmetric and semi-symmetric supercoset models. For all these models, we construct the T-dual action and identify the corresponding equations of motion.
A well-known feature of bosonic T-duality is that integrability is preserved. We investigate this issue in one particular example, the OSpp1|2q principal chiral model which describes a three-dimensional anti-de Sitter supergravity geometry. For ordinary three-dimensional anti-de Sitter geometry realised as an SLp2, Rq principal chiral model, it has been shown [12] that the T-dualisation of the whole SLp2, Rq isometry group leads to a dual geometry that describes a black hole that is asymptotically anti-de Sitter, though the three-dimensional matter-coupled gravity in which this black hole may arise as a solution is unknown yet. Aimed at investigating whether this black hole has a conventional supersymmetric generalisation, we study super non-Abelian T-duality of the OSpp1|2q model whose geometry is that of the N " 1 AdS 3 superspace. Dualising either the whole OSpp1|2q supergroup or its maximal bosonic SLp2, Rq subgroup, we argue that the T-dual model does not satisfy the three-dimensional supergravity constraints [65][66][67][68], thus falling outside the class of supergravity backgrounds. The reason for this is the fact that the super non-Abelian-T-duality procedure gives rise to dual supervielbeins and a B 2 -field with a complicated space-time dependence that prevents them to satisfy the three-dimensional supergravity constraints (even after any possible redefinition of supervielbeins and connection). This seems to be a first example in which super non-Abelian T-duality does not generate supergravity solutions. Whether this failure is a peculiarity of this model or arises more generally remains an open question that we plan to address in a near future. We should note that this result does not contradict the result of [49] since, as will be explained in detail below, the OSpp1|2q sigma model under consideration does not describe a (three-dimensional) Green-Schwarz superstring.
This paper is organised as follows. In Section 2, after a brief review of principal chiral models and supercosets, we summarise the general procedure for dualising (part of) their superisometries that we are going to apply in the rest of the paper. This is the direct generalisation of the procedure for purely bosonic non-Abelian T-duality [5,7] based on gauging the subsupergroup involved in the dualisation. In Section 3, we focus on the super non-Abelian T-duality of principal chiral models. After deriving the T-dual action in general and studying its symmetries, we investigate in detail the OSpp1|2q principal chiral model describing the super AdS 3 geometry. In particular, we argue that the target superspace of the T-dual model obtained by dualising either the whole supergroup or its maximal bosonic subgroup does not satisfy the three-dimensional supergravity constraints.
In Section 4, we give preliminary results regarding the application of super non-Abelian T-duality to supercosets. Since for these models the dualisation procedure can be affected by the non-invertibility of the gauge equations of motion required to remove the original coordinates in favor of a consistent set of dual ones, we cannot determine the T-dual action in general. At this point, a case-by-case study would be necessary which is, however, beyond the scope of the present paper. Finally, in Section 5, we close by collecting some comments and a list of possible future directions. Several appendices follow, which contain some technical details and a brief summary of three-dimensional supergravity (see Appendix D).

Preliminaries
We begin with a short review of the classes of two-dimensional sigma models which will be considered in the present paper. This primarily helps us fixing notation and conventions.
We consider a generic Lie supergroup G with Lie superalgebra g. We denote the number of bosonic and fermionic generators of g by n b and n f , respectively. In addition, we assume that we are given an inner product x´,´y on g, that is, a non-degenerate Ad-invariant graded-symmetric bilinear form.
Generally, to define two-dimensional sigma models involving G, we consider elements g P C 8 pΣ, Gq where C 8 pΣ, Gq is the set of smooth maps from a two-dimensional Lorentzian manifold Σ, the world-sheet, to G. We then introduce the (pull-back to Σ via g of the) Maurer-Cartan form on G by j :" g´1dg P Ω 1 pΣ, gq (2.1) and which satisfies the Maurer-Cartan equation Here, 'd' is the exterior derivative on Σ and Ω p pΣ, gq are the g-valued differential p-forms on Σ. In addition, the Lie bracket r´,´s of g is extended to Ω p pΣ, gq in the usual way.
The current in (2.1) is invariant under the global (left) G-action g Þ Ñ g´1 0 g for all g 0 P G , whereas under the global (right) G-action g Þ Ñ gg 0 for all g 0 P G , (2.3b) it transforms adjointly j Þ Ñ g´1 0 jg 0 .
Furthermore, if '‹' denotes the Hodge star operator on Σ with respect to the world-sheet metric, we have ‹ 2 " id on Ω 1 pΣ, gq. Therefore, we have the decomposition Ω 1 pΣ, gq -Ω 1 pΣ, gq ' Ω 1 pΣ, gq into self-dual and anti-self-dual g-valued one-forms on Σ. We shall make use of the projectors P˘: Ω 1 pΣ, gq Ñ Ω 1 pΣ, gq with P˘:" 1 2 pid˘‹q . where j is the current in (2.1), the wedge product is understood and '‹' is the Hodge star operator on Σ with respect to the world-sheet metric. The model describes the dynamics of n b bosonic and n f fermionic degrees of freedom. The equation of motion following from (2.6) is and together with the Maurer-Cartan equation (2.2), this forms what is known as the first-order system of the model. Lax connection. Importantly, the first-order system, consisting of (2.7) and (2.2), is equivalent to is called the Lax connection with z a complex spectral parameter [69]. To derive the firstorder system from (2.9), we note that ‹ 2 " id and ρ^‹σ "´‹ρ^σ for ρ, σ P Ω 1 pΣ, gq.
As is well-known, a flat Lax connection always yields an infinite number of conservation laws [60], thus ensuring the integrability of the model (for a nice recent review, see [70]).

Symmetric space sigma model
Lie algebra decomposition. We now consider sigma models defined on coset superspaces G{H where H is a Lie subsupergroup of G which arises as the fixed point set of an automorphism of G of order 2. Such coset spaces are known as symmetric spaces. At the level of the Lie superalgebras, this means that the Lie superalgebra g of G decomposes as gh ' m with h :" 1 2 p1`σq looomooon ": P h pgq and m :" 1 2 p1´σq looomooon where h is the Lie superalgebra of H and σ : g Ñ g is an automorphism of g with σ 2 " 1.
Elements of h are said to be of homogeneity 0 and elements of m of homogeneity 2, respectively and we denote the homogeneity of a homogeneous element by |´|. Furthermore, we assume that the inner product x´,´y is compatible with the decomposition (2.10), that is, we take xU, V y " 0 for |U |`|V | ‰ 0 mod 4 for any two homogeneous elements U, V P g.
Sigma model action. In order to formulate the sigma model action, we first observe that under the decomposition (2.10) the current j " g´1dg decomposes as j " A`m , A :" P h pjq P Ω 1 pΣ, hq and m :" P m pjq P Ω 1 pΣ, mq .
where ∇ A has been defined in (2.4). This model describes the dynamics of n b " n G and together with the equation of motion (2.14) they constitute the first-order system of a symmetric space sigma model. Importantly, this first-order system is equivalent to dJpzq`1 2 rJpzq, Jpzqs " 0 , (2.17a) where Jpzq :" A`1 2 pz 2`z´2 q m´1 2 pz 2´z´2 q ‹m (2.17b) is the Lax connection with z a complex spectral parameter.
Evidently, this Lax connection depends on the choice of the coset representative. However, in order to generate an infinite tower of conserved charges which are manifestly gauge invariant, it is preferable to consider a Lax connection that is invariant under H-gauge transformations. This requirement is satisfied by Furthermore, upon defining z 1 :"´logpzq and expanding this expression around z 1 " 0, we obtain J 1 pzpz 1 qq " ‹L N z 1`O pz 12 q , (2.19) where the coefficent L N is the Noether current (2.15). Thus, at order z 1 the flatness of J 1 implies the conservation of the Noether current.

Semi-symmetric space sigma model
Lie algebra decomposition. Let us now come to sigma models on semi-symmetric spaces, that is coset superspaces G{H where H is a Lie subsupergroup of G which arises as the fixed point set of an automorphism of G of order 4. This means that the Lie superalgebra g of G decomposes as where h is the Lie superalgebra of H and σ : g Ñ g is an automorphism of g with σ 4 " 1 and rh, hs Ď h , rp, hs Ď p , rm, hs Ď m , rq, hs Ď q , rq, qs Ď m . (2.20c) Elements of h are said to be of homogeneity 0, elements of p of homogeneity 1, elements of m of homogeneity 2, and elements of q of homogeneity 3, respectively. Furthermore, we assume that the inner product x´,´y is compatible with the decomposition (2.20), that is, we take xU, V y " 0 for |U |`|V | ‰ 0 mod 4 for any two homogeneous elements U, V P g.

Lax connection. The Maurer-Cartan equation (2.2) decomposes under (2.20) as
and together with the equations of motion (2.23) they constitute the first-order system of a semi-symmetric space sigma model. Importantly, this first-order system is equivalent to is the Lax connection with z being a complex spectral parameter [74]. 1 As before, this Lax connection depends on the choice of coset representative but again, there is a Lax connection that is invariant under H-gauge transformations, Upon setting z 1 :"´logpzq and expanding this expression around z 1 " 0, we obtain where now L N is the Noether current (2.24).
1 Generally, such Lax connections and flatness conditions arise for coset spaces that admit a Zm-

Topological deformations
We may also consider deformations of the models (2.6), (2.13), and (2.22) that do neither alter the equations of motion nor the Noether symmetries, thus preserving integrability.
Due to these properties, such type of deformations are referred to as topological, and they have first been studied in the literature [76,77,48,49] in connection with Yang-Baxter deformations of two-dimensional sigma models [78][79][80][81].
Two-cocycles and derivations. Let Ω P H 2 pgq be a Lie superalgebra two-cocycle (we may also restrict to a Lie subsuperalgebra of g) 1 , that is, Ω P Ź 2 g˚and it is subject to the cocycle condition for all U, V, W P g where |´| denotes the Graßmann degree. By the Riesz representation theorem, there is a unique endomorphism D : g Ñ g such that Whilst the graded anti-symmetry of Ω is equivalent to the cocycle condition (2.29) is equivalent to DprU, V sq " rDpU q, V s`rU, DpV qs for all U, V P g , (2.32) that is, D is a derivation for the Lie bracket r´,´s. Furthermore, upon writing g " e V for g P G and V P g, we can extend D to a left-invariant vector field on G by It then follows that for all g P G and for all U P g. See Appendix A for details. The Ad-invariance of x´,´y then immediately yields for all g P G and for all U, V P g. Moreover, we shall also make use of the notation for all g P G and for all A, U P g. Note that to show this one uses that g´1Dpgq "´Dpg´1qg for all g P G as directly follows from (2.33).
Topological deformations. Next, if we extend Ω and D in the usual fashion to Ω p pΣ, gq, we can define a deformed model by As mentioned before, we may consider the restriction Ω| k of Ω to some Lie subsuperalgebra k of g. If K denotes the Lie supergroup associated with k, we may consider the change g Þ Ñ k´1g for k P C 8 pΣ, Kq and then deform any of the sigma model actions (2.6), (2.13), 1 Indeed, Ω| h " 0 is equivalent to D| h : h Ñ m, and the other conditions then follow from the derivation property of D and the requirement of preserving the commutations relation (2.10b) and (2.20c). 2 Another deformation one may consider is by using the Kac-Moody two-cocycle on the based loop algebra L0g given by ş Σ ş 1 0 dt x dJ dt , Jy with J : ΣˆS 1 Ñ g with dJ`1 2 rJ, Js " 0 and Jpt " 0q " Jpt " 1q " j. Such a deformation also renders the theory integrable. or (2.22) by adding ζ 2 ş Σ Ω| k pk´1dk, k´1dkq with Ω| k P H 2 pkq. In fact, under the assumption that Ω| k is non-degenerate, homogeneous Yang-Baxter deformations have been shown in [77] to be equivalent to the non-Abelian T-dual of the deformed model when T-duality is performed, with Ω|´1 k solving the (classical) Yang-Baxter equation and the deformation parameter given by ζ´1.

Super non-Abelian T-duality
We are now interested in studying non-Abelian T-duality of the sigma models from the preceeding sections. The general procedure that we shall apply is a straightforward generalisation of the well-known prescription of purely bosonic non-Abelian T-duality [5,7] to supergroups, and we shall refer to it as super non-Abelian T-duality in the following.
Gauging. Following [5,7], the first step to dualise one of the previously discussed sigma models is to gauge a Lie subsupergroup K of G, corresponding to the directions that we wish to dualise. If k denotes the Lie superalgebra of K, the gauging is obtained by introducing a k-valued connection one-form ω P Ω 1 pΣ, kq. Consequently, the current (2.1) generalises to j ω :" g´1ωg`g´1dg for g P C 8 pΣ, Gq . (2.39) Evidently, F jω " g´1F ω g and so, the flatness of j ω is equivalent to the flatness of ω.
Furthermore, j ω is invariant under the local (left) K-action g Þ Ñ k´1g and ω Þ Ñ k´1ωk`k´1dk for all k P C 8 pΣ, Kq . (2.40) Adding Lagrange multipliers. To implement F ω " 0, we introduce Lagrange multipliers represented by Λ P C 8 pΣ, kq, and we define the master action which is manifestly K-gauge invariant. This form of the master action will be the starting point of our T-dualisation procedure.
T-dualisation. Upon integrating out Λ in (2.41), we recover the original sigma model action. Indeed, the equation of motion for Λ is the flatness condition F ω " 0, and hence, ω is locally pure gauge. Generally, however, ω might have non-trivial holonomies around non-contractible loops which prevents us from having ω be pure gauge globally. However, we shall ignore this technicality by assuming that the topology of Σ is suitably chosen.
Hence, we have ω pure gauge globally and so, we may fix a K-gauge in which it vanishes identically.
On the other hand, if we integrate out ω, we obtain the T-dual model. Since the latter will still be invariant under K-gauge transformation, we may fix a gauge to remove dimpKq coordinates. For instance, for principal chiral models, this can be used to gauge away dimpKq original coordinates, thus arriving at a gauge-fixed T-dual action with the same number of degrees of freedom as the original one, with the key difference that now the dimpKq Lagrange multipliers play the role of dual coordinates. In particular, when taking K " G, gauge invariance can be used to set the original group element g " 1, and, consequently, the T-dual model depends only on the dual coordinates Λ.
In the case of sigma models defined on coset superspaces G{H, the reasoning is similar with the only difference being that the original model describes dimpGq´dimpHq degrees of freedom. The gauge invariance inherited by the T-dual model has to be used to fix also some of the Lagrange multipliers [85,14], while the remaining ones provide a set of dual coordinates.
It should be stressed that this procedure can be complicated by certain uniqueness is- before gauge fixing them [85,14,49]. Instead, for sigma models defined on supergroups the problem is made more severe by the appearance of fermionic coordinates, which are not invertible by their own nature. It turns out that in general switching on the fermionic Lagrange multipliers does not help in making the equations invertible, and more sophisticated approaches need to be used. In the case of purely fermionic cosets, this has been largely discussed in [43]. We shall come back to this in Section 4 and in a forthcoming paper.
In what follows, we are going to discuss the T-dualisation of the three classes of models introduced above. Our primary focus is to identify explicitly the T-dual actions, study the associated Lax pair and the fate of integrability. In a simple example, corresponding to the principal chiral model describing an AdS 3 supergravity background, the question whether super non-Abelian T-duality leads to a dual background which still solves the supergravity torsion constraints is carefully investigated.

General derivation of the T-dual model
We shall gauge a subsupergroup K of G so that the master action (2.43) becomes with j ω as given in (2.39).
T-dual sigma model action.
The T-dual model is obtained by integrating out the ω field. In particular, upon varying (3.1) with respect to ω, we obtain Upon making use of (2.5) and (2.36), this equation can be solved for j ω as Therefore, with the help of the identity which is a consequence of (2.31), the T-dual action that follows from (3.1) upon substitut-

5)
T-dual equation of motion and Lax connection. Using (3.4), the expansion and δDΛ " ad δΛ , some algebra shows that the equation of motion following from the T-dual wherej :" j ω with j ω given in (3.3). Consequently,j satisfies (3.2). Upon combining this with (3.7) and making use of the Jacobi identity and the properties of D, we obtain d‹j " 0 . where Dpg 0 q was defined in (2.33). Indeed, this follows immediately from (2.37). The Noether current associated with this symmetry is then given bỹ J N " ‹DΛpjq . (3.11)

Example: OSpp1|2q principal chiral model
As an explicit example of super non-Abelian T-duality, we shall now consider the principal chiral model for the orthosymplectic group OSpp1|2q. The interest in this supergroup manifold is due to its interpretation as an N " 1 supersymmetric AdS 3 , and as such, it represents an appropriate background for three-dimensional supergravity [86,66,67,[87][88][89], as we are going to discuss.
In the absence of supersymmetry, non-Abelian T-duality of the SLp2, Rq principal chiral model, which describes AdS 3 , has been performed in [12]. There, it has been shown that the T-dualisation of an SLp2, Rq subgroup of the isometry group SLp2, RqˆSLp2, Rq yields a three-dimensional metric corresponding to a black hole space-time and a non-trivial B 2field. To the best of our knowledge, it is still unknown whether this black hole and B 2 -field configuration is a solution of a matter-coupled gravity and whether it can be embedded into a three-dimensional supergravity.
It is then interesting to investigate what happens in the case of the AdS 3 supergeometry.
To this end, we shall discuss the T-dualisation of an OSpp1|2q subgroup of the group of superisometries OSpp1|2qˆOSpp1|2q. We will also comment about the T-dualisation of the maximal bosonic subgroup SLp2, Rq Ď OSpp1|2q. We will find that in both cases, the target superspaces and B 2 -fields of the T-dual models fall outside the class of possible solutions of a three-dimensional N " 1 supergravity. This indicates that the black hole configuration obtained by the non-Abelian T-dualisation of the SLp2, Rq principal chiral model in [12] does not seem to have a conventional supergravity extension.
Orthosymplectic algebra. To begin with, we fix our conventions for the orthosymplectic algebra ospp1|2q. In spinorial notation, it is described by a set of two fermionic generators Q α and three bosonic generators L αβ " L βα for α, β, . . . " 1, 2 subject to the following non-trivial commutation relations rL αβ , L γδ s "´i`ε γpα L βqδ`εδpα L βqγ˘, rQ α , Q β s " L αβ , rL αβ , Q γ s "´iε γpα Q βq . (3.12) The parentheses on the right-hand sides denote normalised symmetrisation of the enclosed indices and i is the imaginary unit. In addition, ε αβ "´ε βα with ε αγ ε γβ " δ α β and δ α β is the Kronecker symbol. Furthermore, we introduce an inner product x´,´y on ospp1|2q by xQ α , Q β y :" iε αβ , xL αβ , L γδ y :" ε αpγ ε δqβ , and xQ α , L βγ y :" 0 . Sigma model action. Since H 2 pospp1|2qq vanishes [82], we work with the undeformed action (2.6). In particular, we parametrise g P G as g :" g b g f with g b :" e x αβ L αβ , g f :" e´F , and F :" θ α Q α , (3.14) with three bosonic coordinates x αβ " x βα and two fermionic coordinates θ α . The corresponding current j " g´1dg ": j αβ L αβ`j α Q α can be split as where (see also Appendix B) j b :" g´1 b dg b and j f :"´∇ jb F´1 2 ad F p∇ jb Fq´1 6 ad 2 F pdFq (3.15b) and the covariant derivative is as defined in (2.4). Explicitly, where θ 2 :" θ α θ α . Consequently, In order to compute e αβ explicitly, we use the Baker-Campbell-Hausdorff formula together with the identities (C.5) specialised to U " x αβ L αβ and V " dU . Indeed, j b " e´U de U " pk`1q! ad k U pdU q, and so Note that the series in (3.18a) are in powers of x 2 only.
2˙. T-dual sigma model action. We now perform super non-Abelian T-duality along OSpp1|2q corresponding to the left action (2.3a). The general expression for the T-dual action of the undeformed principal chiral model is given in (3.5) with DΛ replaced with adΛ. Moreover, since we are gauging the whole group, we can fix a gauge in which g " 1.

3.24b)
and the coefficient functions ip1`2x 4 q p2´x 2 qp1´2x 2 q 2 ,g 4 :" 4i p1´2x 2 q 2 , g 5 :" 2i 1´2x 2 ,g 6 :" 2ip1`x 2 q p2´x 2 qp1´2x 2 q ,g 7 :" 2i 2´x 2 , To apply these formuae to the OSpp1|2q case, we consider the expansionsẽ ":ẽ αβ L αβ`ẽ α Q α and Λ ":x αβ L αβ`θ α Q α , and compute rẽ,ẽs " p2iẽ γpα^ẽ γ βq´ẽα^ẽβ qL αβ`2 iẽ αβ^ẽ β Q α ": υ αβ L αβ`υ α Q α , (3.29) the equations in (3.22) imply that (3.30b) Therefore, (3.26) becomes SLp2, Rq T-dual model. Note that upon settingθ α " 0, the model (3.24) reduces to the T-dual model found in [12] when dualising the SLp2, Rq principal chiral model. In particular, the target space metricg αβ ,γδ forθ α " 0 in (3.24) was interpreted in [12] as a black hole that is asymptotically anti-de Sitter with the scalar curvaturẽ R "´2 4x 4´6x2`9 p1´2x 2 q 2 (3.32) for the Levi-Civita connection. Furthermore, we observe that theB 2 -fieldB αβ ,γδ forθ α " 0 in (3.24) has a non-constant field strength as follows from (3.28) together with (3.30a) after some algebra. Being non-constant, the curvature (3.32) and the field strength (3.33) cannot be a solution of Einstein gravity coupled only to a B 2 -field. Indeed, the equations of motion for this theory are where R µν is the Ricci tensor for the Levi-Civita connection of the metric g µν , ∇ µ the covariant derivative with respect to the Levi-Civita connection, and H µνκ the components of H 3 . The second equation implies that H µνκ " c a |g| ε µνκ with ε µνκ the Levi-Civita symbol and c P R some constant. Thus, H 3 must be constant. Moreover, the trace over the first equation yields that R " 1 4 H µνκ H µνκ " 3 2 c 2 and so, also the scalar curvature must be constant. Since neither the scalar curvature (3.32) is constant nor the three-form field strength (3.33) is of the desired form, we conclude that the bosonic T-dual model is not a solution of Einstein gravity coupled only to a B 2 -field. Therefore, the metricg αβ ,γδ and thẽ B 2 -fieldB αβ ,γδ may only be solutions in a theory in which gravity and the B 2 -field couple to some other matter fields. 1 It would be of interest to find this theory.
Supergravity constraints for the OSpp1|2q T-dual model. We will not elaborate on the above issue of the bosonic SLp2, Rq model here but instead ask the question whether the three-dimensional black hole of [12] can be part of a conventional three-dimensional supergravity. In other words, we shall study whether the target superspace of the super non- Abelian T-dual model obtained from the OSpp1|2q principal chiral model may be interpreted as a supersymmetric black hole which might be a solution of a three-dimensional N " 1 supergravity. For this to be possible, the target superspace of the T-dual model should respect an additional key feature: it should be understood as an appropriate supergravity background. Put differently, it should satisfy the superspace supergravity constraints, which in three-dimensions are off-shell constraints that do not imply the on-shell supergravity equations of motion. 1 Note that the T-dualisation also yields a non-trivial dilatonφ "´logp1´2x 2 q but the standard coupling of the dilaton to Einstein gravity and a B2-field does not solve this issue for the scalar curvature (3.32) and three-form field strength (3.33).
A brief review of three-dimensional supergravity is given in Appendix D. In particular, we are interested in the torsion constraints (D.12c) and the constraint (D.13) on the threeform field strength H 3 of the B 2 -field.
The target superspace of the original OSpp1|2q principal chiral model is an AdS 3 solution of three-dimensional N " 1 supergravity. As such, it satisfies the constraints (D.12c).
Indeed, the supervielbeins can be read off the flat current j " g´1dg given in (3.17).
In particular, expanding j ": j αβ L αβ`j α Q α and using the algebra In particular, this implies that the terms in (3.31a) proportional toẽ α^ẽβ have to sum up to a constant. From the expressions above, we see that the desiredẽ α^ẽβ term is contained in υ αβ defined in (3.29), which in turn appears in Z αβ . Therefore, focusing only on the υ αβ terms, we obtain (3.38) where the ellipses denotes all the other terms that are irrelevant to the argument. Now, it is easy to realise that in (3.38) there are non-constant terms proportional toẽ α^ẽβ such as e γ^ẽδx pγ pαx δq βq . However, such terms can never sum up to a constant. Consequently, we may conclude that the T-dual supervielbeinsẽ do not satisfy the torsion constraints of threedimensional supergravity. In order to improve the situation, we could consider performing a local OSpp1|2q-transformation of the formẽ Þ Ñẽ g :" g´1ẽg for g P C 8 pΣ, OSpp1|2qq.
Then, with Λ g :" g´1Λg and A :" g´1dg, the structure equation (3.26) becomes However, one quickly realises that whilst such transformations can in principle remove some of the unwanted terms in (3.38), they will never remove terms of the formẽ γ^ẽδx pγ pαx δq βq simply because A appears linearly in (3.39).
An additional confirmation of the incompatibility of the dual model with supergravity comes from considering the three-form curvatureH 3 , which should have the particular structure (D.13) in terms of an arbitrary scalar superfield. Using the formula (3.28) forH 3 together with (3.30a), it is not too difficult to find the explicit expression ofH 3 generalising (3.33) to the supersetting. In particular, it turns out that it contains a non-constant term which is cubic in the fermionic components of the supervielbein,  One may wonder whether this result is a peculiarity of dualising all of OSpp1|2q. In Appendix E, we comment on the maximal bosonic subgroup SLp2, Rq Ď OSpp1|2q and argue that also in this case the dualisation is not compatible with the supergravity constraints.
The above issue may be related to the fact that our starting point was not a threedimensional Green-Schwarz superstring sigma model on AdS 3 , while the fermionic non-Abelian T-duality of stringy sigma models was argued in [48,49] to produce dual superbackgrounds that satisfy supergravity constraints. The AdS 3 superstring sigma model is obtained by subtracting from the action (3.19) the term j α^‹ j α and adding, instead of it, the Wess-Zumino term associated with the worldvolume pull-back of H 3 in (3.37) with a relative coefficient k that ensures kappa-symmetry, (3.41) In the presence of such Wess-Zumino terms the gauging procedure is however more delicate [91] and the approach we took in this work would not be applicable. Note that the three-dimensional superstring sigma model under consideration is similar to ten-dimensional superstrings in an Neveu-Schwarz-Neveu-Schwarz background which are not described by sigma models on semi-symmetric Ramond-Ramond superbackgrounds. The B 2 -field of the AdS 3 superstring is invariant under the super non-Abelian isometry only modulo a gauge transformation, and, as such, it falls out of the class of examples considered in [49]. For this reason, the dualisation of the three-dimensional Green-Schwarz superstring deserves further study. Another possibility of tackling this issue might be to generalise the above three-dimensional superspace construction to a type IIB Green-Schwarz superstring in an This would require the consideration of a semi-symmetric space sigma model with an isometry supergroup PSUp1, 1|2qˆPSUp1, 1|2q rather than a higher-dimensional supergroup of the OSp series. The non-Abelian T-dualisation of the SLp2, Rq isometry subgroup in the bosonic string sigma model on this background was recently considered in [90], where it was shown that a dual background AdS 2ˆRˆS 3ˆC Y 2 is a solution of massive type IIA supergravity. It would certainly be of interest to generalise this example to its supernon-Abelian T-dualised counterpart, and we postpone the study of this problem to future work.

Super non-Abelian T-duality of supercoset models
We shall now move on and discuss the super non-Abelian T-duality procedure for symmetric and semi-symmetric spaces from Sections 2.2 and 2.3. As indicated, the dualisation procedure for sigma models on coset superspaces G{H is affected by additional technical complications related to certain uniqueness issues when trying to solve the equation of motion for ω for the projection of j ω onto h. This is simply due to the fact that the action for coset models does not involve the projection of j onto h. In addition, the non-invertibility can also be brought into the game by the fermionic coordinates which are intrinsically noninvertible. This peculiarity prevents us from presenting the super non-Abelian T-duality procedure in complete generality, as a case-by-case analysis would be necessary. For this reason, we shall only present the general approach, in particular pointing out where the non-invertibility arises, and postpone to future work the study of specific models of physical relevance. Nevertheless, we can still discuss various properties of the T-dual model such as integrability and the exchange of the equations of motion with the Maurer-Cartan equations which is an identifier pattern of dualisation.

Symmetric space sigma models
Gauged sigma model action. Starting point is the action (2.13). Upon gauging a subgroup K Ď G, we decompose the covariant current (2.39) as A ω :" P h pj ω q " A`P h pg´1ωgq and m ω :" P m pj ω q " m`P m pg´1ωgq , where A and m were introduced in (2.11) and the projectors P h and P m in (2.10), respectively. Consequently, the master action (2.43) becomes T-dual sigma model action. The variation of (4.2) with respect to ω yields ‹m ω`∇jωΛ´D pj ω q " 0 . and n K f from the Lagrange multipliers, respectively. In the following, we shall ignore this non-invertibility issue and proceed with (4.5). Note that the first equation of (4.3b) arises from varying (4.5) with respect toÃ. Note also that an advantage of this hybrid action is that it enjoys an H-gauge invariance by means of A Þ Ñ h´1Ãh`h´1dh andΛ Þ Ñ h´1Λh`h´1Dphq Furthermore, the variation with respect toÃ leads to Taking into account thatm satisfies the second equation in (4.3b) identically and using the equations of motion above and the Jacobi identity, after a short calculation we obtain ∇Ã‹m " 0 . as the T-dual Lax connection which ensures integrability.

Semi-symmetric space sigma models
Gauged sigma model action. Finally, we discuss super non-Abelian T-duality for semisymmetric space sigma models on G{H. Since the procedure is exactly the same as the one for symmetric space sigma models we can be rather brief.
Specifically, if we define R :"¨´1

R˜´∇ÃΛ
p`a dΛ q pm ω q ∇ÃΛ q`a dΛ p pm ω q¸. (4.13b) Upon subsituting these expressions into the third equation of (4.12b) and using the projectors (2.5), we can solve for m ω to obtain  We should now subsitute these expressions in the first equation in (4.12b) to obtain a linear equation forÃ. However, as in the case of symmetric space sigma models, this equation will not, in general, admit a unique as the linear operator that needs to be inverted may have a non-trivial kernel. This would require a case-by-case analysis and so, we proceed with a hybrid formulation by only inserting the solutions for m ω , p ω , and q ω into (4.11).
We find 14) The first equation in (4.12b) arises from varying this action with respect toÃ. Note that this action enjoys an H-gauge invariance by means of A Þ Ñ h´1Ãh`h´1dh andΛ Þ Ñ h´1Λh`h´1Dphq (4.15) for all h P C 8 pΣ, Hq. As before, the transformation rule forΛ follows directly from (2.43) using (2.12). Note again that h´1Dphq P m for all h P C 8 pΣ, Hq.

Conclusions
We have considered a general procedure of non-Abelian T-duality of sigma models on group manifolds, symmetric, and semi-symmetric spaces with the emphasis on the T-dualisation of sigma models whose target spaces are supermanifolds possessing non-Abelian superisometries. All these models are classically integrable, and since super non-Abelian T-duality exchanges the equations of motion with Bianchi identities (i.e. Maurer-Cartan equations), it is straightforward to write down the T-dual Lax connection. In general, super non-Abelian T-duality encounters a problem of integrating out all the variables of the initial model and getting the dual model described entirely with a dual set of coordinates.
We have considered in full detail the relatively simple example of the OSpp1|2q principal chiral model in which this problem does not occur. We have, however, argued that even though the target superspace of the initial model is an AdS 3 solution of three-dimensional N " 1 supergravity, the dual target superspace background does not satisfy the threedimensional supergraity constraints. Such model cannot even be a solution of a generalised supergravity [21,22] because also in that case the torsion supergravity constraints are the same. A reason for the breaking of such constraints may be related to the fact that the OSpp1|2q principal chiral model does not describe the dynamics of a string in three-dimensional N " 1 supergravity. The N " 1, d " 3 superstring action is of the Green-Schwarz type (see e.g. [92]) with the Wess-Zumino term which in the case of the AdS 3 superspace is a three-dimensional counterpart of an Neveu-Schwarz-Neveu-Schwarz background of ten-dimensional superstrings. To the best of our knowledge, non-Abelian Tduality of string sigma models in Neveu-Schwarz-Neveu-Schwarz super-backgrounds has not been studied in the literature, and this may be an interesting problem to address in future work.
Another point 1 that deserves further investigation concerns the extension of the non- Abelian T-duality group defined in [57,93,94] to our super-non-Abelian-T-duality setting.
Such a generalisation should exist as it would represent the non-Abelian analogue of the [95,96]. The latter extends the bosonic Opd b , d b q symmetry [6] to backgrounds with bosonic and fermionic Abelian isometries.

A. Cocycles and derivations
Consider the derivation D defined in (2.30) and let V P g. Then, using e s ad V pU q " it is easy to see that Dpe ad V pU qq " e ad V pDpU qq`8 ÿ k,l"0 1 pk`l`1q! pad k V˝a d DpV q˝a d l V qpU q (A.1) because of the derivation property of D. Next, since ş 1 0 dt p1´tq k t l " k!l! pk`l`1q! , we can rewrite the sum as 8 ÿ k,l"0 1 pk`l`1q! ad k V˝a d DpV q˝a d l V " Consequently, e ad VˆD pU q`" where we have inserted the definition (2.33). This verifies (2.34).

B. Fermionic current
Let us derive (3.15b) which is a special case of a more general formula [97,98]. We consider the one-parameter family (t P R) Consequently, where ∇ jb is as defined in (2.4). Therefore, since jpt " 1q " j. Upon using e s ad V pU q " ř 8 This establishes (3.15b).

C. Useful identities for ospp1|2q
Here we derive the set of identities that have been used in Section 3. (C.1) We wish to compute the action of the operator 1 1´ad U on V . We start by expanding the operator 1 1´ad U in powers of ad Uf . Since ad k Uf " 0 for k ą 2, we obtain Ub .

(C.2)
We then expand in powers of ad Ub and apply it to V b`Vf , For the bosonic part we note that, defining u 2 as in (3.18b) and using (3.12), we obtain Consequently, acting recursively with ad Ub , we find For the fermionic part, a similar calculation leads to ad 2k Ub pV f q "`1 2 u 2˘k η α Q α and ad 2k`1 Ub pV f q "´i` We are now ready to compute 1 1´ad U pV q using the expansion (C.2) and the previous identities. For the first term in the expansion we obtain From this structure, it easily follows that Simply repeating the same procedure a few more times, we obtain the second term in (C.2) according tô from which we easily find and the third term (C.11) In summary, (C.12b)

D. Three-dimensional supergravity
Cartan structure equations. Consider a pp, q|2nq-dimensional supermanifold M with metric g of Graßmann degree zero. The structure group of M is generically OSppp, q|2nq which is the group preserving the canonical graded symmetric bilinear form on R p,q|2n . In particular, we let A, B, . . . the R p,q|2n -indices, and we denote the canonical gradedsymmetric bilinear form on R p,q|2n by η AB . We have η AB " p´1q |A||B| η BA where |´| denotes the Graßmann degree. Note that OSppp, q|2nq is generated by 1 2 pp`qqpp`q´1q`np2n`1q bosonic generators and 2pp`qqn fermionic generators.
Let now E A be the supervielbeins on M . The structure functions f AB C are given by where r´,´s denotes the (graded) Lie bracket. Furthermore, we define the dual supervielbeins e A by E A e B " δ A B where δ A B is the Kronecker symbol and ' ' denotes the interior product. We set δ A B " p´1q |A||B| δ A B . Then, in the supervielbein basis we write where 'd' denotes the graded symmetric tensor product. The inverse η AB of η AB is given by Let now 'L X ' be the Lie derivative along a vector field X on M and 'd' the exterior derivative. Then, we have the standard Cartan formula L X ω " dpX ωq`X dω for any differential form ω on M , and since L X is a graded derivation with respect to the tensor product, these then imply that Here, '^' denotes the graded antisymmetric tensor product.
Next, we define the torsion and curvature two-forms, where ω A B " e C ω CA B is the connection one-form.
The Ricci tensor and the scalar curvature are then given by R AB :" p´1q |B||C| R ACB C and R :" η BA R AB , (D.7) where η AB is the inverse metric as defined in (D.3).
Connection one-form from metric compatibility. The metric compatibility is expressed by ω AB`p´1 q |A||B| ω BA " 0 with ω AB :" ω A C η CB . where p´1q |C|p|A|`|B|q F C AB " η CD F DA E η EB .
Three-dimensional supergravity constraints. Let M be a three-dimensional Lorentzian spin manifold. In that case, we have the factorisation T C M -S d S of the tangent bundle where S is the spin bundle. This is equivalent to picking a conformal structure on M . In an orthonormal frame, the components of the metric on M are pη ab q " diagp´1, 1, 1q with a, b, . . . " 0, 1, 2, and because of the identification T C M -S dS, we may write ε αpγ ε δqβ for η ab with the spinor indices α, β, . . . " 1, 2. Here, ε αβ "´ε βα is the standard symplectic structure on S with ε αγ ε γβ " δ α β and δ α β the Kronecker symbol.
If three-dimensional N " 1 supergravity contains a two-form superfield B 2 , its threeform field strength H 3 is constrained in such a way that its components are functions of a scalar superfield L and its derivatives (see e.g. [87]), where D α is the superspace covariant derivative and S is a scalar 'prepotential' superfield which appears in the solution of the torsion constraints (D.12c).

E. T-dualisation of SLp2, Rq Ď OSpp1|2q
Let us briefly discuss the dualisation of the maximal bosonic subgroup SLp2, Rq Ď OSpp1|2q of the OSpp1|2q principal chiral model. In particular, since H 2 pslp2, Rqq " 0, we consider the T-dual action (3.5) withΛ " g´1Λg and Λ ":x αβ L αβ . By construction, in this case, (3.5) possesses an SLp2, Rq gauge symmetry. Given the parametrisation (3.14) for g, we can exploit this gauge invariance to set g b " 1. Then, a short calculation reveals that Λ " g´1 f Λg f "`1`i 4 θ 2˘xαβ L αβ`i θ γx γ α Q α ": y αβ L αβ`χ α Q α . (E.1) This differs from theΛ used in the dualisation of OSpp1|2q in Section 3.2 by the changes x αβ Ñ y αβ andθ α Ñ χ α . Therefore, we can rephrase the procedure given there by simply making these replacements. In particular, upon inspecting (3.38) once these replacements are made, it is evident that we cannot satisfy the first torsion constraint in (D.12c), exactly as in the case when dualising all of OSpp1|2q. The same argument also applies to H 3 in (3.40).

F. Semi-symmetric space sigma model
Let us provide some more details on the T-dualisation of semi-symmetric space sigma models.
T-dual equations of motion. In order to compute the equations of motion (4.16), we recall the definition of the matrix R " pR ij q from (4.13a) and the operator S from (4.13d), respectively. A quick calculation shows that xR 12 pU q, V y "´xU, R 12 pV qy , xR 21 pU q, V y "´xU, R 21 pV qy , xR 11 pU q, V y "´xU, R 22 pV qy , (F.2a) and therefore, for all U, V P Ω p pΣ, gq. Therefore, applying the general relation δ`1 1˘S˘"¯1 1˘S˝δ S˝1 1˘S , the variation of the first term in (4.14) is δ B T, 1 1´S`P`p T q˘F "´xδT,my´BˆδS˝1 1´S˙p T q, Using all these formulae, it is now not too hard to derive the equations of motion (4.16).