# On non-Abelian T-duality for non-semisimple groups

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## Abstract

We revisit non-Abelian T-duality for non-semisimple groups, where it is well-known that a mixed gravitational-gauge anomaly leads to \(\sigma \)-models that are scale, but not Weyl-invariant. Taking into account the variation of a non-local anomalous term in the T-dual \(\sigma \)-model of Elitzer, Giveon, Rabinovici, Schwimmer and Veneziano, we show that the equations of motion of generalized supergravity follow from the \(\sigma \)-model once the Killing vector *I* is identified with the trace of the structure constants. As a result, non-Abelian T-duals with respect to non-semisimple groups are solutions to generalized supergravity. We illustrate our findings with Bianchi spacetimes.

## 1 Introduction

Following Buscher’s seminal work on T-duality [1, 2], a generalisation to non-Abelian isometries was quickly proposed [3].^{1} One striking feature of non-Abelian T-duality is that it breaks isometries, but it also turned out to be novel in other ways. In particular, it was demonstrated that non-Abelian T-duality was not a symmetry of conformal field theory, but rather a symmetry between different theories [6]. Moreover, it was noted by Gasperini, Ricci and Veneziano that the procedure failed to provide a valid supergravity solution for Bianchi V [7] and Bianchi III [8] cosmological models. It was subsequently realised that structure constants with a non-vanishing trace were related to a mixed gravitational-gauge anomaly [9, 10], which explained why non-Abelian T-duality was no longer a symmetry of supergravity. Eventually, non-Abelian T-duality was extended to the RR sector [11, 12] and became a powerful solution generating technique [13, 14, 15, 16, 17, 18, 19, 20, 21, 22], especially for AdS/CFT geometries, where implications for the dual CFTs were explored [23, 24, 25, 26, 27, 28]. However, the fate of the non-Abelian T-dual geometry of Bianchi V and III remained a puzzle of sorts.

In recent years, we have witnessed further interest in non-Abelian T-duality, driven by swift developments in integrable \(\sigma \)-models [29, 30, 31, 32, 33, 34]. We recall that non-Abelian T-duals arise as limits of \(\lambda \)-deformations of \(AdS_p \times S^p\) geometries [35, 36], while homogeneous Yang–Baxter deformations [37] can be understood as non-Abelian T-duality transformations [38, 39, 40, 41, 42].^{2} One important by-product of Yang–Baxter deformations was the discovery that there are integrable deformations that are not solutions of usual supergravity, but a modification, called *generalized supergravity* [48] (see also [49]). The modification is specified by a Killing vector *I* and bona fide supergravity solutions correspond to \(I = 0\).

With the advent of generalized supergravity [48] and the knowledge that homogeneous Yang–Baxter deformations are non-Abelian T-duality transformations [39, 40, 41, 42], it would be surprising if non-Abelian T-duals for non-semisimple groups did not also solve the equations of motion (EOMs) of generalized supergravity (as originally anticipated in [38]). For the Bianchi V geometry this was explicitly confirmed in [50]. Here, we extend the analysis to Bianchi \(\hbox {VI}_h\), a one-parameter family of groups that include Bianchi III \((h=0)\) and Bianchi V \((h=1)\) as special cases. Since Bianchi IV and VII\(_h\) give rise to singular supergravity solutions [51], this exhausts all Bianchi cosmologies based on non-semisimple groups.

As explained, on its own the observation that non-Abelian T-duals of Bianchi spacetimes are solutions to generalized supergravity may not be new. However, working case by case in this paper we observe that the Killing vector *I* of generalized supergravity is simply the trace of the structure constants. We are unaware of any explicit statement of this fact in the literature. Since Bianchi spacetimes are representative examples where the duality group acts without isotropy, this is expected to constitute a general result for all non-Abelian T-duals with respect to non-semisimple groups without isotropy. In effect this result refines the connection between non-Abelian T-duality and generalized supergravity. To explain this fact, we return to the T-dual \(\sigma \)-model of Elitzer, Giveon, Rabinovici, Schwimmer and Veneziano (EGRSV), which includes the contribution from a non-local anomalous term [10]. A key observation of EGRSV was that non-vanishing \(\beta \)-functions for Bianchi V could be cancelled by the variation of an additional non-local term with respect to the conformal factor, which they demonstrated explicitly for Bianchi V. Here, we confirm this result for Bianchi III, before providing proof that *for any background the one-loop* \(\beta \)*-functions of the EGRSV* \(\sigma \)*-model agree with the equations of motion of generalized supergravity once the Killing vector is identified with the trace of the structure constants*, \(I^{i} = f^{j}_{j i}\). The obvious implication is that non-Abelian T-duality with respect to non-semisimple groups, where the group acts without isotropy, results in solutions to generalized supergravity. This provides a new perspective on the connection between generalized supergravity and non-Abelian T-duality. In some sense, the first solution to generalized supergravity appeared in [10], but the theory was not recognised at the time.

*X*and

*Y*. We recall that for \(X_{\mu } = \partial _{\mu } \Phi \), \(Y_{\mu } =0\), we recover the usual one-loop \(\beta \)-functions of supergravity [56], where it should be noted that the contribution due to the dilaton is a

*classical*contribution at the same order as the one-loop

*quantum*contributions of the \(G_{\mu \nu }\) and \(B_{\mu \nu }\) couplings. This happens because the term \(R^{(2)} \Phi \) is scale non-invariant at the classical level, while the other couplings lose scale invariance at one-loop. At this point, we could adopt the strategy employed in [48] and use an explicit solution to fix the vectors. However, provided one includes the anomaly term in the EGRSV \(\sigma \)-model, which appears at the same order as the dilaton, and simply varies it with respect to the conformal factor, we shall see that the equations of motion of generalized supergravity (A.1)–(A.3) can be derived. To be more precise, the EOMs for \(G_{\mu \nu }\) and \(B_{\mu \nu }\) follow from the action, while as we demonstrate in the “Appendix”, the dilaton \(\beta \)-function follows from the divergence of the Einstein equation, just mirroring the story for usual supergravity [57]. This ensures that even for \(\sigma \)-models formulated in flat two-dimensional spacetime that the \(\beta \)-function for the dilaton can be recovered. Note, while the EOMs of generalized supergravity (including the RR sector) can be derived from kappa-symmetry [49], without assuming fermions or the existence of a fermionic symmetry, simply revisiting the analysis of EGSRV and extending it to arbitrary spacetimes, we show that one can derive the EOMs of generalized supergravity. This provides a purely bosonic derivation.

^{3}

The structure of this short note is as follows. In Sect. 2 we review non-Abelian T-duality with respect to both semisimple and non-semisimple groups. In Sect. 3, we introduce Bianchi cosmologies and describe non-Abelian T-dualities of both Bianchi I and Bianchi II spacetimes, noting in the former case that the matrix inversion inherent to non-Abelian T-duality is simply three commuting Abelian T-dualities. In Sect. 4, we demonstrate that non-Abelian T-duals of Bianchi VI\(_h\) cosmologies lead to generalized supergravity solutions once the Killing vector is identified with the trace of the structure constants. In the special case where \(h=-1\), the trace of the structure constants vanishes and we find a solution to usual supergravity. In Sect. 5, we explain why \(I^{i} = f^{j}_{j i}\). The EOMs of generalized supergravity can be found in the “Appendix”.

## 2 Review of non-Abelian T-duality

*h*is the worldsheet metric and \(R^{(2)}\) the worldsheet curvature with \(\Phi \) denoting the scalar dilaton.

*d*, in addition to being the dimension of the space, is also the dimension of the Lie algebra. The Lie algebra is fully specified by the structure constants,

*M*, \(\tilde{G} + \tilde{B} = M\) and it is worth noting that the Lagrange multipliers become the T-dual coordinates. Furthermore, the dilaton shift,

## 3 Bianchi cosmology

Having explained the fundamental map (2.10) at the heart of Buscher procedure, we put it to work on Bianchi cosmologies. Our main focus will be discussing non-Abelian T-duals of Bianchi cosmologies where the trace of the structure constants is non-vanishing. Before discussing these more exotic examples, we will begin by introducing the Bianchi spacetimes and discussing both Abelian and non-Abelian T-duality in the simpler setting of Bianchi I and Bianchi II.

We will follow the description of the spacetimes presented in [51], where a general family of supergravity solutions, namely spacetimes supported both by the scalar dilaton \(\Phi \) and NSNS two-form *B*, were presented. As stated earlier, since the NSNS two-form only complicates the non-Abelian T-duality, we will further restrict the solutions presented in [51] to \(B=0\). With this restriction, the solutions are expected to agree with [8]. Before beginning, we warn the reader that our dilaton is not the dilaton \(\phi \) presented in [51], but instead \(\Phi = - \frac{1}{2} \phi \).

*t*. Note, it is more usual to fix the gauge so that \(g_{tt} = - 1\). In contrast, the above form is unorthodox, but the advantage of the rescaled temporal direction is that the dilaton equation is simplified. Before discussing it, let us note that when \(I = 0\) and \(B=0\), the EOM for the NSNS two-form (A.1) is trivially satisfied, so we only need to discuss the Einstein equation (A.2) and the dilaton EOM (A.3).

*R*to get

*t*, given the above spacetime (3.1), we arrive at an easily solved equation:

*x*,

*y*and

*z*. The final equation is the Einstein equation in the time direction, which holds once the constants we have introduced satisfy the following equation:

^{4}

### 3.1 Bianchi I: Abelian as a non-Abelian T-duality

The solution (3.7) is the simplest Bianchi I solution supported only by a scalar field. The solution has three Killing vectors, \(\partial _{x}, \partial _y\) and \(\partial _z\), which generate translations in the spatial directions. The Killing vectors commute with one another so all structure constants associated with the Lie algebra vanish.

*B*-field and the components of the inverse matrix correspond to the T-dual metric:

While we have not performed a genuine non-Abelian T-duality, through this example we have recast Abelian T-duality, or more accurately three commuting T-dualities, as a non-Abelian duality transformation where all the structure constants vanish. As we have seen, the Buscher procedure for this simple example reduces to inverting a matrix. This same operation will be at the heart of the subsequent examples, but anti-symmetric components, essentially introduced via \(\kappa \) (2.11) will generate a *B*-field.

### 3.2 Bianchi II: semisimple warm-up

*x*.

*B*-field,

## 4 Solutions to generalized supergravity

As outlined in the introduction, our main motivation is to show that there are Bianchi cosmologies outside of the (Ricci-flat) Bianchi V class [50] that give rise to generalized supergravity solutions under non-Abelian T-duality transformations. We focus on Bianchi VI\(_h\) cosmologies as these are the only non-singular solutions where the structure constants have a non-vanishing trace. Interestingly, VI\(_h\) is a one-parameter family of groups, which covers both Bianchi III and V, but also includes one group (\(h = -1\)) where the trace of the structure constants vanishes. Therefore, for Bianchi VI\(_h\) \(h\ne -1\) we expect to find non-Abelian T-duals that are solutions to generalized supergravity, but for VI\(_{-1}\) we anticipate that the T-dual will be a genuine supergravity solution. We begin with the generic case.

### 4.1 Bianchi VI\(_{h}\)

*I*is simply the trace of the structure constants, \(I^{1} = f^{i}_{i1}\), where we have used the fact that \(\tilde{X}_1 = x\).

### 4.2 Bianchi \(\hbox {VI}_{-1}\)

*B*-field are read off from the symmetric and anti-symmetric components of the inverse matrix, respectively,

## 5 Relation to generalized supergravity

In the previous section we have shown that non-Abelian T-duals of Bianchi VI\(_h\) cosmological models lead to solutions to generalized supergravity where the Killing vector *I* is the trace of the structure constants. At this stage the observation that *I* is the trace of the structure constants may simply be a coincidence. In this section we dispel this notion by returning to the T-dual \(\sigma \)-model of EGRSV [10] and show that a non-local anomalous term in the \(\sigma \)-model captures the modification in the equations of generalized supergravity. Before doing this in general, we will present the analysis for Bianchi III. For completeness, we revisit the Bianchi V analysis of [10] in the “Appendix”. The EOMs of generalized supergravity can be found also in the “Appendix”.

### 5.1 Bianchi III

*I*that completes the solution is self-selecting; although \(\partial _{y}\) is Killing, we have not deformed this direction and this leaves \(I = c \, \partial _{x}\), where

*c*is a constant. The correct constant of proportionality follows from the trace of the structure constant,

*c*. These contribute to the one-form

*X*(A.4) in the following way:

*I*(5.16) are precisely of the same form as the terms coming from the variation of the \(\sigma \)-model action (5.14) once one sets \( c = \tilde{c}\). In other words, for this example of a non-Abelian T-dual of a Bianchi III spacetime, the variation of the \(S_1\) term in the action recovers the equations of motion of generalized supergravity evaluated on the same solution provided the Killing vector

*I*is simply the trace of the structure constants.

### 5.2 General case

*I*is Killing. In particular, in addition to \(\nabla _{(i} I_{j)} = 0\), one should also use the fact that the Lie derivative of the NSNS two-form with respect to

*I*vanishes, \(\mathcal {L}_{I} B = 0\), or in the notation of generalized supergravity that,

## 6 Discussion

The main result of this note is that the EOMs of generalized supergravity, restricted to the NS sector, follow from the non-Abelian T-dual \(\sigma \)-model of Elitzur et al. [10]. Not only does this provide a purely bosonic derivation of generalized supergravity, but with the benefit of hindsight, we can now better understand the relation between generalized supergravity and non-Abelian T-duality. Concretely, we have explicitly confirmed that non-Abelian T-duality with respect to non-semisimple groups leads to solutions to generalized supergravity, in line with earlier expectations [38]. By illustrating this for Bianchi VI\(_h\) spacetimes, we have extended an earlier result [50] to all non-singular Bianchi cosmology solutions to supergravity (with zero NSNS two-forms).

Based on explicit solutions, we observed that the Killing vector is simply the trace of the structure constants. To better understand this fact, we returned to the T-dual \(\sigma \)-model of EGRSV [10] and noted that the variation of a non-local anomalous action with respect to the conformal factor \(\sigma \) precisely matches the EOMs of generalized supergravity once the Killing vector *I* is identified with the trace of the structure constants. While the \(\sigma \)-model is scale invariant, it has recently been suggested that Weyl invariance can be restored through a shift in the dilaton in a doubled formalism [61] and it would also be interesting to understand this directly at the level of the EGRSV \(\sigma \)-model to see if Weyl invariance can be restored.

In this work we have restricted our attention to the EGRSV \(\sigma \)-model and non-Abelian T-duality where the group acts *without isotropy*. It has been argued more generically, in particular for Yang–Baxter deformations based on *r*-matrix solutions to the homogeneous Classical Yang–Baxter equation, that the corresponding non-Abelian T-duals are solutions to generalized supergravity [38]. It is expected that one can generalise the analysis based on the EGRSV \(\sigma \)-model to non-Abelian T-duals with isotropy, where the Killing vector *I* should no longer be a constant, but instead should satisfy the Killing equation. We hope to return to this in future work.

Throughout this work, we have driven home the message that non-Abelian T-duality is simply a matrix inversion. We recall that all Yang–Baxter deformations can be understood as open-closed string maps [44, 45], which are also simple matrix inversions. It would be interesting to revisit statements in the literature connecting Yang–Baxter deformations to non-Abelian T-duality [38, 39, 40, 41, 42] in order to better understand this relation, especially in light of the observation that all transformations can be reduced to matrix inversions. This suggests there should exist a simple overarching description.

Finally, it would be interesting to better understand non-Abelian T-duality. One distinctive feature of the transformation is that it decompactifies geometries, thereby obscuring the AdS/CFT interpretation. Viewing the transformation in two steps, this is easy to see why. Firstly, the metric \(\gamma \) in the transformation (2.10) is defined with respect to the Maurer–Cartan one-forms and not the coordinates. This means in the case of Bianchi IX that one replaces a compact space, for example a (constant radius) three-sphere, with \(\mathbb {R}^3\). The anti-symmetric terms \(\kappa _{ij} = \epsilon _{ijk} x_k\), where we have used the structure constants of *SU*(2) symmetry, simply break the translation symmetry leaving a residual *SU*(2) symmetry from the rotation generators. In short, non-Abelian T-duality, especially for a three-sphere, looks like a deformation of a flat space geometry and not a compact geometry. It would be extremely interesting if one could decompose non-Abelian T-duality into two steps: an initial step where the original geometry is “flattened” and a second step that determines what deformations of this flattened geometry give rise to supergravity solutions. This may shed light on various puzzling aspects of non-Abelian T-duality.

## Footnotes

- 1.
- 2.
- 3.
While it is known that the supergravity equations of motion can be recovered from studying the low energy modes of string scattering, the analogous picture here is currently unclear.

- 4.
There appears to be a mistake in [51]. The three-form \(H = \text {d}B = A \sigma _1 \wedge \sigma _2 \wedge \sigma _3\), so \(A= 0\) should recover our result, but instead the quoted result is \(\sum _{i < j} p_i p_j = 0\). Since this implies the dilaton does not back-react, one concludes there is a typo.

## Notes

### Acknowledgements

We acknowledge T. Araujo, N. S. Deger, D. Giataganas, M. M. Sheikh-Jabbari and H. Yavartanoo for collaboration on related topics. We thank B. Hoare, D. C. Thompson and J. Nian for discussion, as well as Y. Lozano and K. Yoshida for comments on preliminary drafts. This research was supported by the Ministry of Science, ICT and Future Planning, Gyeongsangbuk-do and Pohang City.

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