On non-Abelian T-duality for non-semisimple groups
aa r X i v : . [ h e p - t h ] M a r APCTP Pre2018-001
On non-Abelian T-duality for non-semisimple groups
Moonju Hong a , Yoonsoo Kim a , Eoin ´O Colg´ain ba Department of Physics, Postech, Pohang 37673, Korea b Asia Pacific Center for Theoretical Physics, Postech, Pohang 37673, Korea
Abstract
We revisit non-Abelian T-duality for non-semisimple groups, where it is well-known that a mixed gravitational-gauge anomaly leads to σ -models that arescale, but not Weyl-invariant. Taking into account the variation of a non-localanomalous term in the T-dual σ -model of Elitzer, Giveon, Rabinovici, Schwimmer& Veneziano, we show that the equations of motion of generalized supergravityfollow from the σ -model once the Killing vector I is identified with the trace ofthe structure constants. As a result, non-Abelian T-duals with respect to non-semisimple groups are solutions to generalized supergravity. We illustrate ourfindings with Bianchi spacetimes. Introduction
Following Buscher’s seminal work on T-duality [1], a generalisation to non-Abelian isometrieswas quickly proposed [2] . One striking feature of non-Abelian T-duality is that it breaksisometries, but it also turned out to be novel in other ways. In particular, it was demonstratedthat non-Abelian T-duality was not a symmetry of conformal field theory, but rather asymmetry between different theories [4]. Moreover, it was noted by Gasperini, Ricci &Veneziano that the procedure failed to provide a valid supergravity solution for BianchiV [5] and Bianchi III [6] cosmological models. It was subsequently realised that structureconstants with a non-vanishing trace were related to a mixed gravitational-gauge anomaly[7, 8], 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 [9] and became a powerfulsolution generating technique [10], especially for AdS/CFT geometries, where implicationsfor the dual CFTs were explored [11]. However, the fate of the non-Abelian T-dual geometryof Bianchi V and III remained a puzzle.In recent years, we have witnessed further interest in non-Abelian T-duality, driven byswift developments in integrable σ -models [12]. We recall that non-Abelian T-duals ariseas limits of λ -deformations of AdS p × S p geometries [13], while homogeneous Yang-Baxterdeformations [14] can be understood as non-Abelian T-duality transformations [15, 16] .One important by-product of Yang-Baxter deformations was the discovery that there areintegrable deformations that are not solutions of usual supergravity, but a modification,called generalized supergravity [21] (see also [22]). The modification is specified by a Killingvector I and bona fide supergravity solutions correspond to I = 0.With the advent of generalized supergravity [21] and the knowledge that homogeneousYang-Baxter deformations are non-Abelian T-duality transformations [16], it would be sur-prising if non-Abelian T-duals for non-semisimple groups did not also solve the equationsof motion (EOMs) of generalized supergravity (as originally anticipated in [15]). For theBianchi V geometry this was confirmed recently [23]. Here, we extend this observation toBianchi VI h , a one-parameter family of groups that include Bianchi III ( h = 0) and BianchiV ( h = 1) as special cases. Since Bianchi IV and VII h give rise to singular supergravitysolutions [24], this exhausts all Bianchi cosmologies based on non-semisimple groups.For non-Abelian T-duals of Bianchi cosmologies, we observe that the Killing vector I ofgeneralized supergravity is simply the trace of the structure constants. While this agreementmay be coincidental, to better understand this feature we return to the T-dual σ -model ofElitzer, Giveon, Rabinovici, Schwimmer & Veneziano (EGRSV), which includes the con-tribution from a non-local anomalous term [8]. A key observation of EGRSV was thatnon-vanishing β -functions for Bianchi V could be cancelled by the variation of an additionalnon-local term with respect to the conformal factor, which they demonstrated explicitly forBianchi V. Here, we confirm this result for Bianchi III, before providing proof that for any See [3] for earlier examples. Yang-Baxter deformations can be understood as open-closed string maps [17] where the r -matrix is thenoncommutativity parameter Θ, Θ = r [18, 19] (see [20] for an earlier observation in a restricted setting). β -functions of the EGRSV σ -model agree with the equations ofmotion of generalized supergravity once the Killing vector is identified with the trace of thestructure constants, I i = f jji .To get a better grasp of this claim, let us recall that the EGRSV σ -model is scale invariant,but not Weyl-invariant. Being scale invariant, it is known that the one-loop β -functions musttake the form [25] β G µν = R µν − H µρσ H ρσν + ∇ µ X ν + ∇ ν X µ , (1.1) β B µν = − ∇ ρ H ρµν + X ρ H ρµν + ∇ µ Y ν − ∇ ν Y µ , (1.2)for arbitrary vectors, X and Y . We recall that for X µ = ∂ µ Φ, Y µ = 0, we recover the usualone-loop β -functions of supergravity [26], where it should be noted that the contributiondue to the dilaton is a classical contribution at the same order as the one-loop quantum contributions of the G µν and B µν couplings. This happens because the term R (2) Φ is scalenon-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 [21] and use an explicit solution tofix the vectors. However, provided one includes the anomaly term in the EGRSV σ -model,which appears at the same order as the dilaton, and simply varies it with respect to theconformal factor, we shall see that the equations of motion of generalized supergravity canbe derived.The structure of this short note is as follows. In section 2 we review non-Abelian T-dualitywith respect to both semisimple and non-semisimple groups. In section 3, we introduceBianchi cosmologies and describe non-Abelian T-dualities of both Bianchi I and BianchiII spacetimes, noting in the former case that the matrix inversion inherent to non-AbelianT-duality is simply three commuting Abelian T-dualities. In section 4, we demonstrate thatnon-Abelian T-duals of Bianchi VI h cosmologies lead to generalized supergravity solutionsonce the Killing vector is identified with the trace of the structure constants. In the specialcase where h = −
1, the trace of the structure constants vanishes and we find a solutionto usual supergravity. In section 5, we explain why I i = f jji . The EOMs of generalizedsupergravity can be found in the appendix. In this section we quickly review non-Abelian T-duality. We will do this in two stages: first,we introduce the transformation for semisimple groups without isotropy, before explaininghow the T-dual σ -model is modified for non-semisimple groups. Following [5], we will tailorthe discussion to Bianchi cosmologies from the outset. Consider the 2D string σ -model S = 12 π Z d z (cid:2) ∂X µ ( G µν + B µν ) ¯ ∂X ν + 2Φ ∂ ¯ ∂σ (cid:3) , (2.1)2here X µ , µ = 0 , . . . , d , denote the target spacetime coordinates, the couplings G µν , B µν aresymmetric and anti-symmetric, respectively, and correspond to the target space metric andNSNS two-form. σ is the worldsheet conformal factor, ∂ ¯ ∂σ = 14 √ hR (2) , (2.2)where h is the worldsheet metric and R (2) the worldsheet curvature with Φ denoting thescalar dilaton.Let us now assume that the target space has an isometry group, where the generatorsof the corresponding Lie algebra can be expressed in terms of the Killing vectors K i of thetarget space geometry, K i = K mi ∂ m , i = 1 , . . . , d, (2.3)where d , in addition to being the dimension of the space, is also the dimension of the Liealgebra. The Lie algebra is fully specified by the structure constants,[ K j , K k ] = f ijk K i . (2.4)Dual to the Killing vectors K i , one can define Maurer-Cartan one-forms σ i , which satisfy arelated differential condition: d σ i = 12 f ijk σ j ∧ σ k . (2.5)For Lie algebras of dimension three, d = 3, we have a complete classification due toBianchi [27] and it is this setting in which we will consider non-Abelian T-duality. Infact for each family of symmetries, one can define a corresponding “Bianchi cosmology”,which is a 4D spacetime, parametrised by coordinates ( t, ~x ), where the internal 3D spaceexhibits the symmetries of the corresponding Bianchi class and whatever warp factors appearonly depend on the time direction. Within this class of geometries, the usual Friedmann-Robertson-Walker (FRW) solutions correspond to Bianchi I, V and IX.Following [5], we can factorise the spacetime metric and NSNS two-form so that all spatialdependence x i is contained in a dreibein e im , G mn ( t, ~x ) = e im ( ~x ) γ ij ( t ) e jn ( ~x ) , B mn ( t, ~x ) = e im ( ~x ) β ij ( t ) e jn ( ~x ) , γ ij = γ ji , β ij = − β ji , (2.6)where we have isolated the spatial directions m = 1 , ,
3. We are assuming that B m = G m = 0. At this stage, one performs non-Abelian T-duality by gauging the isometries [28] ∂X m → ∂X m + A l K ml , ¯ ∂X m → ¯ ∂X m + ¯ A l K ml , (2.7)in the process introducing a set of pure gauge potentials A l , ¯ A l . The new action S ′ is theoriginal action plus an additional contribution, S ′ = S + 12 π Z d z (cid:18) A l K ml e im ( γ ij + β ij ) e jn ¯ ∂X n + ∂X m e im ( γ ij + β ij ) e jn ¯ A l K nl + A l K ml e im ( γ ij + β ij ) e jn ¯ A k K nk + ˜ X i F i (cid:19) , (2.8)3here F i is the field strength corresponding to A i , ¯ A i , F i = ∂ ¯ A i − ¯ ∂A i + f ijk A j ¯ A k and wehave added the Lagrange multiplier to enforce the condition F i = 0. Strictly speaking,integration on ˜ X only ensures that A, ¯ A are pure gauge in spherical worldsheets, in whichcase, one recovers the original σ -model. Instead of integrating out the Lagrange multiplier,one can integrate out the gauge potentials to get a dual σ -model, before subsequently fixingthe residual gauge symmetry. A convenient gauge choice is simply to take X m to be constant,which yields the dual action˜ S = 12 π Z d z h ∂X G ¯ ∂X + ∂ ˜ X i M ij ¯ ∂ ˜ X j + (2Φ + ln det M ) ∂ ¯ ∂σ i , (2.9)where we have defined the matrix M = ( γ + β + κ ) − (2.10)in terms of the anti-symmetric matrix κ : κ ij ≡ f kij ˜ X k . (2.11)As is evident from the dual action (2.9), the T-dual metric ˜ G and NSNS two-form ˜ B areread off from the symmetric and anti-symmetric components of M , ˜ G + ˜ B = M and it isworth noting that the Lagrange multipliers become the T-dual coordinates. Furthermore,the dilaton shift Φ → Φ + 12 ln det M, (2.12)is the result of a Jacobian factor from integrating out the gauge fields [1]. When κ = 0,so that all the structure constants vanish, this transformation reduces to the usual Buscherrules for Abelian T-duality. It is clear that one can generate complicated geometries throughthis map, so for simplicity we focus on examples where β = 0.The above treatment is adequate for semisimple groups. For non-semisimple groups, thereis a non-local anomalous contribution to the σ -model [8], S non-local = − π f jji Z d z (cid:18) ∂ A i + 1¯ ∂ ¯ A i (cid:19) √ hR (2) (2.13)which once the gauge fields are integrated out leads to a new dual action,˜ S = 12 π Z d z h ∂X G ¯ ∂X + ( ∂ ˜ X i − f kki ∂σ ) M ij ( ¯ ∂ ˜ X j + f llj ¯ ∂σ ) + (2Φ + ln det M ) ∂ ¯ ∂σ i . (2.14)It should be noted that irrespective of the group, the transformation of the metric, NSNStwo-form and dilaton follow the same prescription, but there is a difference in the dual σ -model, which will be important later. 4 Bianchi Cosmology
Having explained the fundamental map (2.10) at the heart of Buscher procedure, we putit to work on Bianchi cosmologies. Our main focus will be discussing non-Abelian T-dualsof Bianchi cosmologies where the trace of the structure constants is non-vanishing. Beforediscussing these more exotic examples, we will begin by introducing the Bianchi spacetimesand discussing both Abelian and non-Abelian T-duality in the simpler setting of Bianchi Iand Bianchi II.We will follow the description of the spacetimes presented in [24], where a general family ofsupergravity solutions, namely spacetimes supported both by the scalar dilaton Φ and NSNStwo-form B , were presented. As stated earlier, since the NSNS two-form only complicatesthe non-Abelian T-duality, we will further restrict the solutions presented in [24] to B = 0.With this restriction, the solutions are expected to agree with [6]. Before beginning, we warnthe reader that our dilaton is not the dilaton φ presented in [24], but instead Φ = − φ .To begin, let us consider the Bianchi spacetime [24],d s = − a a a e − d t + a σ + a σ + a σ , (3.1)where a i and Φ are functions of t . Note, it is more usual to fix the gauge so that g tt = −
1. Incontrast, the above form is unorthodox, but the advantage of the rescaled temporal directionis that the dilaton equation is simplified. Before discussing it, let us note that when I = 0and B = 0, the EOM for the NSNS two-form (A.1) is trivially satisfied, so we only need todiscuss the Einstein equation (A.2) and the dilaton EOM (A.3).Setting I = B = 0 in the Einstein equation (A.2) and contracting, we get, R + 2 ∇ Φ = 0 . (3.2)Combining with the dilaton equation (A.3), we can eliminate the Ricci scalar R to get ∇ µ ∇ µ Φ − ∂ µ Φ ∂ µ Φ = 1 √− g ∂ µ ( e − √− gg µν ∂ ν Φ) = 0 . (3.3)Since Φ is assumed to be only a function of t , given the above spacetime (3.1), we arriveat an easily solved equation: ∂ t Φ = 0 , ⇒ Φ = βt, (3.4)where we have exploited the freedom to shift Φ to remove a constant. We now turn to theEinstein equation. To solve the Einstein equation, we need to consider a given Bianchi classwith specific Maurer-Cartan one-forms. For Bianchi I spacetimes, the Maurer-Cartan formsare simply σ = d x, σ = d y, σ = d z , which are all closed and therefore from (2.5) all thestructure constants vanish. In this case, the functions a i are [24] a i = e p i t , (3.5)5here p i are constants and we have absorbed additional constants by redefining the coordi-nates x, y and z . The final equation is the Einstein equation in the time direction, whichholds once the constants we have introduced satisfy the following equation E tt = X i 00 0 e p . (3.8)We observe that it is symmetric, so the inverse is also symmetric. As a result, we will notgenerate a B -field and the components of the inverse matrix correspond to the T-dual metric:d s = − e p + p + p − β ) t d t + e − p t d x + e − p t d y + e − p t d z , (3.9)where we have introduced ( ˜ X , ˜ X , ˜ X ) = ( x, y, z ) as the dual coordinates, in line with usualpractice. However, it should be noted that we have the freedom to label the dual coordinatesas we please and we are still guaranteed to produce a solution. Recall that we have gaugefixed so the original coordinates disappeared in the transformation leaving the Lagrangemultipliers to take their place. There appears to be a mistake in [24]. The three-form H = d B = Aσ ∧ σ ∧ σ , so A = 0 should recoverour result, but instead the quoted result is P i 6o read off the transformation for the dilaton, we can either use the shifted expression(2.12), or simply note that the density e − √− g is invariant. Using the latter, while ne-glecting the g tt term as it does not change, we get e ( p + p + p − β ) t = e − e − ( p + p + p ) t , (3.10)which implies the T-dual dilaton isΦ = ( β − p − p − p ) t. (3.11)Substituting the new metric (3.9) and dilaton (3.11) into the supergravity EOMs, we findthat the EOMs are satisfied provided (3.6) holds. This confirms that we have generated anew solution from old.While we have not performed a genuine non-Abelian T-duality, through this example wehave 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 sameoperation will be at the heart of the subsequent examples, but anti-symmetric components,essentially introduced via κ (2.11) will generate a B -field. For Bianchi II spacetimes, we consider the same metric (3.1), but now the one-forms are σ = d x − z d y, σ = d y, σ = d z. (3.12)The structure constants are obtained from the differential conditions on the one-forms,d σ = σ ∧ σ ⇒ f = 1 . (3.13)Since the structure constant is traceless, the T-dual geometry will be a solution to usualsupergravity. We recall that the dilaton is the same as Bianchi I, Φ = βt , and from [24] wefind that the functions a i in general can be written as a = e Φ (cid:16) p cosh( p t ) (cid:17) / ,a = e Φ cosh( p t ) / e p t ,a = e Φ cosh( p t ) / e p t , (3.14)where some unnecessary constants have been absorbed into coordinates. One can check thatEOMs require the condition p p − p = 4 β . (3.15)7t turns out that performing T-duality corresponds to inverting the matrix, γ + κ = a a x − x a , (3.16)with a i as presented above. In contrast to the previous example, our metric now has ananti-symmetric component. We have once again chosen to label the Lagrange multipliers( ˜ X , ˜ X , ˜ X ) = ( x, y, z ), so from κ (2.11) the 23-component of the matrix is x .From the inverse matrix, we read off the dual metric and B -field,d s = − a a a e − βt d t + 1∆ (cid:0) ( a a + x )d x + a a d y + a a d z (cid:1) ,B = − a ∆ x d y ∧ d z, (3.17)where we have defined ∆ = a ( a a + x ) . (3.18)Having started with Φ = βt , the transformed dilaton isΦ = βt − 12 log ∆ . (3.19)Substituting the explicit values for a i , one finds that EOMs are satisfied through (3.15). As outlined in the introduction, our main motivation is to show that there are Bianchicosmologies outside of the (Ricci-flat) Bianchi V class [23] that give rise to generalizedsupergravity solutions under non-Abelian T-duality transformations. We focus on BianchiVI h cosmologies as these are the only non-singular solutions where the structure constantshave a non-vanishing trace. Interestingly, VI h is a one-parameter family of groups, whichcovers both Bianchi III and V, but also includes one group ( h = − 1) where the trace ofthe structure constants vanishes. Therefore, for Bianchi VI h h = − − we anticipatethat the T-dual will be a genuine supergravity solution. We begin with the generic case. h We can import the solution directly from [24]. The spacetime takes the same form (3.1), butwe redefine the Maurer-Cartan one-forms as σ = d x, σ = e hx d y, σ = e x d z, (4.1)8nd the dilaton is unchanged (3.4). The functions appearing in the metric may be expressedas [24]: a = e Φ (cid:18) p h + 1 (cid:19) ( h h +1)2 sinh( p t ) − ( h h +1)2 e ( h − h +1) p t ,a = e Φ (cid:18) p h + 1 (cid:19) h ( h +1) sinh( p t ) − h ( h +1) e p t ,a = e Φ (cid:18) p h + 1 (cid:19) h +1) sinh( p t ) − h +1) e − p t , (4.2)where p i denote constants and we have absorbed redundant constants. From these expres-sions, it is clear that the h = − h + h + 1)( h + 1) p = p + 4 β . (4.3)The Maurer-Cartan one-forms satisfy the differential conditions:d σ = 0 , d σ = hσ ∧ σ , d σ = σ ∧ σ , (4.4)and the corresponding Killing vectors are respectively, K = ∂ x − z∂ z − hy∂ y , K = ∂ y , K = ∂ z . (4.5)The Killing vectors satisfy the Lie algebra[ K , K ] = hK , [ K , K ] = K . (4.6)From either the differential forms or the Killing vectors, one can easily identify the structureconstants f = h, f = 1 . (4.7)Immediately, one notes that the trace is zero when h = − 1. With the structure constants athand, we are set to perform the non-Abelian T-duality. As prescribed earlier, all we have todo is invert the matrix, γ + κ = a hy z − hy a − z a . (4.8)and extract the symmetric and anti-symmetric components:d s = − a a a e − βt d t + 1∆ (cid:18) a a d x + ( z + a a )d y − hyz d y d z + ( h y + a a )d z (cid:19) B = − 1∆ d x ∧ ( hya d y + za d z ) , (4.9)9here we have defined ∆ = h y a + a ( z + a a ) . (4.10)The change in the dilaton is once again easily read off, giving usΦ = βt − 12 log ∆ . (4.11)As with the original Bianchi V T-dual [5], or the later Bianchi III T-dual [6], the metric,NSNS two-form and dilaton do not satisfy the usual supergravity EOMs on their own. Thiswas the original puzzle posed a quarter century ago. However, once complemented with theappropriate Killing vector, in this case I = − ( h + 1) ∂ x , (4.12)it is a straightforward exercise to check that the generalized supergravity EOMs are satisfied.Recalling that VI h includes Bianchi III and V as special cases, our analysis reduces to theearlier result of [23] when h = 1. It is worth noting that I is simply the trace of the structureconstants, I = f ii , where we have used the fact that ˜ X = x . − Once again, we can reproduce the solution from [24]. Up to constants, which can be absorbed,the functions may be expressed as a = √ p exp (cid:20) e p t + (cid:16) p β (cid:17) t (cid:21) , a = a = √ p e ( p + β ) t , Φ = βt. (4.13)The remaining equations are satisfied provided,2 p p + p = 4 β . (4.14)The non-Abelian T-dual geometry follows from inverting the matrix γ + κ = a − y zy a − z a . (4.15)The T-dual metric and B -field are read off from the symmetric and anti-symmetric compo-nents of the inverse matrix, respectively,d s = − a a e − βt d t + 1∆ (cid:18) a d x + (cid:18) a + z a (cid:19) d y + 2 yza d x d y + (cid:18) a + y a (cid:19) d z (cid:19) ,B = 1∆ d x ∧ ( y d y − z d z ) , (4.16)10here we have defined ∆ = a a + y + z . (4.17)We remark that the resulting geometry has no obvious isometries and all symmetries appearto be broken. The dilaton is again easily determined,Φ = βt − 12 ln( a ∆) . (4.18)It is worth noting in this case that the trace of the structure constants is zero. For thisreason, we expect a supergravity solution and it can be checked that the supergravity EOMsare satisfied, once one imposes (4.14), in line with our expectations. In the previous section we have shown that non-Abelian T-duals of Bianchi VI h cosmologicalmodels lead to solutions to generalized supergravity where the Killing vector I is the traceof the structure constants. At this stage the observation that I is the trace of the structureconstants may simply be a coincidence. In this section we dispel this notion by returning tothe T-dual σ -model of EGRSV [8] and show that a non-local anomalous term in the σ -modelcaptures the modification in the equations of generalized supergravity. Before doing this ingeneral, we will present the analysis for Bianchi III. For completeness, we revisit the BianchiV analysis of [8] in the appendix. The EOMs of generalized supergravity can be found alsoin the appendix. In this section, we revisit the analysis of [8] but for Bianchi III. We will also work in theaccustomed gauge, i. e. g tt = − s = − d t + t ( σ + σ ) + σ . (5.1)where we have defined Maurer-Cartan one-forms: σ = d x, σ = d y, σ = e − x d z, (5.2)The one-forms satisfy the following differential conditions,d σ = 0 , d σ = 0 , d σ = − σ ∧ σ . (5.3)so the only structure constant is f = 1 . (5.4)11s explained previously, non-Abelian T-duality reduces to inverting the matrix, γ + κ = t − z z t . (5.5)The resulting solution to generalized supergravity isd s = − d t + t t + z (d x + d z ) + d y ,B = zt + z d x ∧ d z, Φ = − 12 ln( t + z ) . (5.6)Here the Killing vector I that completes the solution is self-selecting; although ∂ y is Killing,we have not deformed this direction and this leaves I = c ∂ x , where c is a constant. Thecorrect constant of proportionality follows from the trace of the structure constant, I = ∂ x , (5.7)and it can be checked that this constitutes a solution to generalized supergravity. We wouldnow like to confirm that one arrives at the same constant from considering the variationof the T-dual action (2.14) with respect to the background conformal factor σ , followinganalysis presented in [8].We recall the T-dual σ -model [8] S = 12 π Z d z h ( − ∂t ¯ ∂t + ( ∂ ˜ X j − ˜ cδ j ∂σ ) M jk ( ¯ ∂ ˜ X k + ˜ cδ k ¯ ∂σ ) + ln det M ∂ ¯ ∂σ i , (5.8)where we have defined ˜ c = f ii . We can further decompose the action as follows: S = 12 π Z d z (cid:16) − ∂t ¯ ∂t + ∂ ˜ X j M jk ¯ ∂ ˜ X k + ln det M ∂ ¯ ∂σ (cid:17) ,S = − ˜ c π Z d zσ (cid:16) ¯ ∂ ( M j ∂ ˜ X j ) − ∂ ( M k ¯ ∂ ˜ X k ) (cid:17) ,S = − ˜ c π Z d zM ∂σ ¯ ∂σ. (5.9)We will now consider the variation of the total action S = S + S + S with respect tothe conformal factor, following [8]. To leading order in σ the variation δ σ S = 0, so we willignore this term. It is worth noting that we will not be doing a quantum calculation here, butsimply importing the known one-loop result for S and combining it with the variation of S ,which is a classical contribution. It should be borne in mind that the dilaton contribution tothe one-loop β -functions is also purely classical [26] for reasons explained in the introduction.12he variation of S with respect to the conformal factor gives π δS δσ = 12 (cid:16) β I =0 G µν + β I =0 B µν (cid:17) ∂X µ ¯ ∂X ν + 12 β I =0Φ ∂ ¯ ∂σ, (5.10)where X µ ≡ { t, ˜ X j } and β I =0 B µν , β I =0 G µν and β I =0Φ are the usual supergravity one-loop β -functions[26], essentially the EOMs of generalized supergravity evaluated at I = 0. From the variationof the second term S with respect to σ , one finds δ σ S = − ˜ c π Z d zδσ (cid:20) ¯ ∂ (cid:18) ( t ∂x − z∂z ) t + z (cid:19) − ∂ (cid:18) ( t ¯ ∂x + z ¯ ∂z ) t + z (cid:19)(cid:21) = − ˜ c π Z d zδσ (cid:20) − t ( t − z )( t + z ) ( ¯ ∂t∂x − ¯ ∂x∂t ) + 2 t z ( t + z ) ( ¯ ∂x∂z − ¯ ∂z∂x )+ 4 t z ( t + z ) ( ¯ ∂t∂z + ¯ ∂z∂t ) + 2( z − t )( t + z ) ¯ ∂z∂z − z ( t + z ) ¯ ∂∂z (cid:21) . (5.11)At this point we can use the EOM to replace the second derivative term. To leading orderin σ , the EOM takes the form, ∂ ( M ij ¯ ∂ ˜ X j ) + ¯ ∂ ( M ji ∂ ˜ X j ) − δM jk δ ˜ X i ∂ ˜ X j ¯ ∂ ˜ X k = 0 , (5.12)where we have focused on the spatial terms. From the EOMs, we find¯ ∂∂z = 2 tz ( t + z ) ( ¯ ∂t∂x − ¯ ∂x∂t )+ z ( t + z ) ( ¯ ∂z∂z − ¯ ∂x∂x ) − ( z − t ) t ( t + z ) ( ¯ ∂t∂z + ¯ ∂z∂t ) , (5.13)and substitute it back into the above expression to get, δ σ S = − ˜ c π Z d zδσ (cid:20) − t ( t + z ) ¯ ∂z∂z + 2 z ( t + z ) ¯ ∂x∂x + 2 zt ( t + z ) ( ¯ ∂t∂z + ¯ ∂z∂t ) − t ( t + z ) ( ¯ ∂t∂x − ¯ ∂x∂t ) + 2 t z ( t + z ) ( ¯ ∂x∂z − ¯ ∂z∂x ) (cid:21) . (5.14)Now, let us compare to the contribution due to I = c ∂ x coming from the EOMs ofgeneralized supergravity. We are particularly interested in comparing the terms that dependon the constant c . These contribute to the one-form X (A.4) in the following way: X = c ( t + z ) (cid:0) t d x + z d z (cid:1) . (5.15)13e now reproduce the EOMs of generalized supergravity focusing on the terms due to theKilling vector: β B tx = − c t ( t + z ) , β B xz = c t z ( t + z ) ,β G tz = − c zt ( t + z ) , β G xx = − c z ( t + z ) , β G zz = c t ( t + z ) ,β Φ = − t c ( c − . (5.16)As advertised earlier, we find a solution to generalized supergravity when c = 1, but it isinteresting that the dilaton EOM is also satisfied when I = 0. The key point is that theterms due to the Killing vector I (5.16) are precisely of the same form as the terms comingfrom the variation of the σ -model action (5.14) once one sets c = ˜ c . In other words, for thisexample of a non-Abelian T-dual of a Bianchi III spacetime, the variation of the S termin the action recovers the equations of motion of generalized supergravity evaluated on thesame solution provided the Killing vector I is simply the trace of the structure constants. It is easy to extend the analysis above to the general case. To do so, we recall that M jk = G jk + B jk . (5.17)In terms of the metric and NS two-form, the equation of motion (5.12) is¯ ∂∂ ˜ X i + ∂ ˜ X j ¯ ∂ ˜ X k (Γ ijk − H ijk ) = 0 , (5.18)where we have introduced the Christoffel symbol and field strength H = d B . Replacing f jji → I i , we can write the anomaly term as δ σ S = − π Z d zδσI i (cid:16) ¯ ∂ ( M ji ∂ ˜ X j ) − ∂ ( M ik ¯ ∂ ˜ X k ) (cid:17) , = − π Z d zδσI i (cid:18) B ji ∂ ¯ ∂ ˜ X j + [ ∂ k G ij − ∂ j G ik − ( ∂ k B ij + ∂ j B ik )] ∂ ˜ X j ¯ ∂ ˜ X k (cid:19) , (5.19)= − π Z d zδσI i (cid:18) B ji (cid:18) H jkl − Γ jkl (cid:19) + ∂ l G ik − ∂ k G il − ( ∂ l B ik + ∂ k B il ) (cid:19) ∂ ˜ X k ¯ ∂ ˜ X l . where in the last line we have used the equation of motion. These terms precisely match theequations of motion of generalized supergravity (A.1) - (A.2) once one uses the fact that I is Killing and that equation (2.13) of [21], namely the relation I k H kij + ∂ i Z j − ∂ j Z i = 0 , (5.20)where we have defined X k = I k + Z k . 14 Discussion The purpose of this note is to confirm that non-Abelian T-duality with respect to non-semisimple groups leads to solutions to generalized supergravity, in line with earlier expec-tations [15]. By illustrating this for Bianchi VI h spacetimes, we have extended an earlierresult [23] to all non-singular Bianchi cosmology solutions to supergravity (with zero NSNStwo-forms).Based on explicit solutions, we observed that the Killing vector is simply the trace of thestructure constants. To better understand this fact, we returned to the T-dual σ -model ofEGRSV [8] and noted that the variation of a non-local anomalous action with respect to theconformal factor σ precisely matches the EOMs of generalized supergravity once the Killingvector I is identified with the trace of the structure constants. While the σ -model is scaleinvariant, it has recently been suggested that Weyl invariance can be restored through a shiftin the dilaton in a doubled formalism [30] and it would also be interesting to understandthis directly at the level of the EGRSV σ -model to see if Weyl invariance can be restored.Throughout this work, we have driven home the message that non-Abelian T-duality issimply a matrix inversion. We recall that all Yang-Baxter deformations can be understood asopen-closed string maps [18], which are also simple matrix inversions. It would be interestingto revisit statements in the literature connecting Yang-Baxter deformations to non-AbelianT-duality [15, 16] in order to better understand this relation, especially in light of the ob-servation that all transformations can be reduced to matrix inversions. This suggests thereshould exist a simple overarching description.Finally, it would be interesting to better understand non-Abelian T-duality. One distinc-tive feature of the transformation is that it decompactifies geometries, thereby obscuring theAdS/CFT interpretation. Viewing the transformation in two steps, this is easy to see why.Firstly, the metric γ in the transformation (2.10) is defined with respect to the Maurer-Cartanone-forms and not the coordinates. This means in the case of Bianchi IX that one replaces acompact space, for example a (constant radius) three-sphere, with R . The anti-symmetricterms κ ij = ǫ ijk x k , where we have used the structure constants of SU (2) symmetry, simplybreak the translation symmetry leaving a residual SU (2) symmetry from the rotation gener-ators. In short, non-Abelian T-duality, especially for a three-sphere, looks like a deformationof a flat space geometry and not a compact geometry. It would be extremely interesting ifone could decompose non-Abelian T-duality into two steps: an initial step where the originalgeometry is “flattened” and a second step that determines what deformations of this flat-tened geometry give rise to supergravity solutions. This may shed light on various puzzlingaspects of non-Abelian T-duality. Acknowledgements We acknowledge T. Araujo, N. S. Deger, D. Giataganas, M. M. Sheikh-Jabbari & H. Yavar-tanoo for collaboration on related topics. We thank B. Hoare, D. C. Thompson & J. Nian15or discussion, as well as Y. Lozano & K. Yoshida for comments on preliminary drafts. A Generalized Supergravity EOMs For completeness we review the EOMs of generalized supergravity [21]. For the purposes ofthis paper, it is enough to restrict our attention to the NS sector. The EOMs take the form, β B µν = − ∇ ρ H ρµν + X ρ H ρµν + ∇ µ X ν − ∇ ν X µ , (A.1) β G µν = R µν − H µρσ H ρσν + ∇ µ X ν + ∇ ν X µ , (A.2) β Φ = R − H + 4 ∇ µ X µ − X µ X µ , (A.3)where we have defined X µ ≡ ∂ µ Φ + ( g νµ + B νµ ) I ν . (A.4)It should be noted that usual supergravity is recovered when I = 0. A.1 Bianchi V In this section, we dissect the calculation presented in [8] to recast the terms that cancelthe usual supergravity β -function as the modification inherent in generalized supergravity.Here, the T-dual solution is [8]d s = − d t + t x ( t + x ) d x + xt d y + t t + x d z ,B = 12( t + x ) d x ∧ d z, Φ = − 12 ln (cid:0) t ( t + x ) (cid:1) , (A.5)where the coordinates are related to the original Lagrange multipliers as follows:˜ X = z, ˜ X = √ x cos y, ˜ X = √ x sin y. (A.6)Here the structure constants are f = f = 1, so that equating I with the trace, I i = f jji ,we get I = − 2. Recalling that ˜ X = z , we expect that I = − ∂ z and it is easy to confirmthat this is a solution to generalized supergravity.Let us now consider the T-dual σ -model (2.14) and we again take f jj = ˜ c . To repeat theanalysis, one just needs the matrix M = 1 t ( t + x ) t − t √ x cos y − t √ x sin yt √ x cos y t + x sin y − x cos y sin yt √ x sin y − x cos y sin y t + x cos y . (A.7)16valuating the variation δ σ S , the result is [8] δ σ S = − ˜ c π Z d zδσ (cid:20) − t ( t + x ) ( ¯ ∂t∂z − ¯ ∂z∂t ) − t ( t + x ) ( ¯ ∂x∂z − ¯ ∂z∂x ) − t + x ) (cid:18) t ( ¯ ∂t∂x + ¯ ∂x∂t ) − t x ¯ ∂x∂x + 2 x ¯ ∂z∂z (cid:19) + 2 xt ¯ ∂y∂y (cid:21) , (A.8)where once again we have used the EOM to eliminate a second derivative. Now, assuming I = c ∂ z , let us compare to the EOMs of generalized supergravity. To do so, we note thecontribution of I to X , X = − c t + x ) (cid:0) d x − t d z (cid:1) , (A.9)which allows us to identify the contribution to the EOMs of generalized supergravity: β B tz = − c t ( t + x ) , β B xz = − c t ( t + x ) ,β G tx = ct ( t + x ) , β G xx = − c t x ( t + x ) , β G zz = c x ( t + x ) , β G yy = − c xt ,β Φ = − t c ( c + 2) . (A.10)One can quickly see that all the terms agree with c = ˜ c = − δ σ S cancelled the contributionfrom the one-loop β -functions [8], it was not appreciated at the time that the additionalterms agree with the EOMs of generalized supergravity [21] once the Killing vector I is thetrace of the structure constants. References [1] T. H. Buscher, “A Symmetry of the String Background Field Equations,” Phys. Lett.B , 59 (1987); T. H. Buscher, “Path Integral Derivation of Quantum Duality inNonlinear Sigma Models,” Phys. Lett. B , 466 (1988).[2] X. C. de la Ossa and F. Quevedo, “Duality symmetries from nonAbelian isometries instring theory,” Nucl. Phys. B , 377 (1993) [hep-th/9210021].[3] B. E. Fridling and A. Jevicki, “Dual Representations and Ultraviolet Divergences inNonlinear σ Models,” Phys. Lett. , 70 (1984); E. S. Fradkin and A. A. Tseytlin,“Quantum Equivalence of Dual Field Theories,” Annals Phys. , 31 (1985).[4] A. Giveon and M. Rocek, “On nonAbelian duality,” Nucl. Phys. B , 173 (1994)[hep-th/9308154]. 175] M. Gasperini, R. Ricci and G. Veneziano, “A Problem with nonAbelian duality?,” Phys.Lett. B , 438 (1993) [hep-th/9308112].[6] M. Gasperini and R. Ricci, “Homogeneous conformal string backgrounds,” Class. Quant.Grav. , 677 (1995) [hep-th/9501055].[7] E. Alvarez, L. Alvarez-Gaume and Y. Lozano, “On nonAbelian duality,” Nucl. Phys. B , 155 (1994) [hep-th/9403155].[8] S. Elitzur, A. Giveon, E. Rabinovici, A. Schwimmer and G. Veneziano, “Remarks onnonAbelian duality,” Nucl. Phys. B , 147 (1995) [hep-th/9409011].[9] K. Sfetsos and D. C. Thompson, “On non-abelian T-dual geometries with Ramondfluxes,” Nucl. Phys. B , 21 (2011) [arXiv:1012.1320 [hep-th]]; Y. Lozano, E. ´OColg´ain, K. Sfetsos and D. C. Thompson, “Non-abelian T-duality, Ramond Fields andCoset Geometries,” JHEP , 106 (2011) [arXiv:1104.5196 [hep-th]].[10] Y. Lozano, E. ´O Colg´ain, D. Rodr´ıguez-G´omez and K. Sfetsos, “Supersymmetric AdS via T Duality,” Phys. Rev. Lett. , no. 23, 231601 (2013) [arXiv:1212.1043 [hep-th]];G. Itsios, C. Nunez, K. Sfetsos and D. C. Thompson, “Non-Abelian T-duality and theAdS/CFT correspondence:new N=1 backgrounds,” Nucl. Phys. B , 1 (2013); A. Bar-ranco, J. Gaillard, N. T. Macpherson, C. N´u˜nez and D. C. Thompson, “G-structuresand Flavouring non-Abelian T-duality,” JHEP , 018 (2013) [arXiv:1305.7229 [hep-th]]; S. Zacar´ıas, “Semiclassical strings and Non-Abelian T-duality,” Phys. Lett. B ,90 (2014) [arXiv:1401.7618 [hep-th]]; E. Caceres, N. T. Macpherson and C. N´u˜nez,“New Type IIB Backgrounds and Aspects of Their Field Theory Duals,” JHEP ,107 (2014) [arXiv:1402.3294 [hep-th]]; P. M. Pradhan, “Oscillating Strings and Non-Abelian T-dual Klebanov-Witten Background,” Phys. Rev. D , no. 4, 046003 (2014)[arXiv:1406.2152 [hep-th]]; T. R. Araujo and H. Nastase, “Non-Abelian T-duality fornonrelativistic holographic duals,” JHEP , 203 (2015) [arXiv:1508.06568 [hep-th]]; H. Dimov, S. Mladenov, R. C. Rashkov and T. Vetsov, “Non-abelian T-dualityof Pilch-Warner background,” Fortsch. Phys. , 657 (2016) [arXiv:1511.00269 [hep-th]]; L. A. Pando Zayas, V. G. J. Rodgers and C. A. Whiting, “Supergravity solutionswith AdS from non-Abelian T-dualities,” JHEP , 061 (2016) [arXiv:1511.05991[hep-th]]; L. A. Pando Zayas, D. Tsimpis and C. A. Whiting, “Supersymmetric IIBbackground with AdS vacua from massive IIA supergravity,” Phys. Rev. D , no. 4,046013 (2017) [arXiv:1701.01643 [hep-th]][11] Y. Lozano and C. N´u˜nez, “Field theory aspects of non-Abelian T-duality and N =2 linear quivers,” JHEP , 107 (2016) [arXiv:1603.04440 [hep-th]]; Y. Lozano,N. T. Macpherson, J. Montero and C. N´u˜nez, “Three-dimensional N = 4 linear quiv-ers and non-Abelian T-duals,” JHEP , 133 (2016) [arXiv:1609.09061 [hep-th]];Y. Lozano, C. Nunez and S. Zacarias, “BMN Vacua, Superstars and Non-Abelian T-duality,” JHEP , 000 (2017) [arXiv:1703.00417 [hep-th]]; G. Itsios, Y. Lozano,18. Montero and C. Nunez, “The AdS non-Abelian T-dual of Klebanov-Witten as a N = 1 linear quiver from M5-branes,” JHEP , 038 (2017) [arXiv:1705.09661[hep-th]]; G. Itsios, H. Nastase, C. N´u˜nez, K. Sfetsos and S. Zacar´ıas, “Penrose lim-its of Abelian and non-Abelian T-duals of AdS × S and their field theory duals,”arXiv:1711.09911 [hep-th]; J. van Gorsel and S. Zacar´ıas, “A Type IIB Matrix Modelvia non-Abelian T-dualities,” JHEP , 101 (2017) [arXiv:1711.03419 [hep-th]].[12] C. Klimcik, “Yang-Baxter sigma models and dS/AdS T duality,” JHEP , 051(2002) [hep-th/0210095]; C. Klimcik, “On integrability of the Yang-Baxter sigma-model,” J. Math. Phys. , 043508 (2009) [arXiv:0802.3518 [hep-th]]; K. Sfetsos, “In-tegrable interpolations: From exact CFTs to non-Abelian T-duals,” Nucl. Phys. B , 225 (2014) [arXiv:1312.4560 [hep-th]]; F. Delduc, M. Magro and B. Vicedo,“On classical q -deformations of integrable sigma-models,” JHEP , 192 (2013)[arXiv:1308.3581 [hep-th]]; F. Delduc, M. Magro and B. Vicedo, “An integrable de-formation of the AdS × S superstring action,” Phys. Rev. Lett. , no. 5, 051601(2014) [arXiv:1309.5850 [hep-th]]; T. Matsumoto and K. Yoshida, “Yang?Baxter sigmamodels based on the CYBE,” Nucl. Phys. B , 287 (2015) [arXiv:1501.03665 [hep-th]].[13] K. Sfetsos and D. C. Thompson, “Spacetimes for λ -deformations,” JHEP , 164(2014) [arXiv:1410.1886 [hep-th]]; S. Demulder, K. Sfetsos and D. C. Thompson, “Inte-grable λ -deformations: Squashing Coset CFTs and AdS × S ,” JHEP , 019 (2015)[arXiv:1504.02781 [hep-th]].[14] I. Kawaguchi, T. Matsumoto and K. Yoshida, “Jordanian deformations of the AdS xS superstring,” JHEP , 153 (2014) [arXiv:1401.4855 [hep-th]].[15] B. Hoare and A. A. Tseytlin, “Homogeneous Yang-Baxter deformations as non-abelianduals of the AdS sigma-model,” J. Phys. A , no. 49, 494001 (2016) [arXiv:1609.02550[hep-th]].[16] R. Borsato and L. Wulff, “Integrable Deformations of T -Dual σ Models,” Phys.Rev. Lett. , no. 25, 251602 (2016) [arXiv:1609.09834 [hep-th]]; B. Hoare andD. C. Thompson, “Marginal and non-commutative deformations via non-abelianT-duality,” JHEP , 059 (2017) [arXiv:1611.08020 [hep-th]]; J. Sakamoto andK. Yoshida, “Yang-Baxter deformations of W , × T , and the associated T-dual mod-els,” Nucl. Phys. B , 805 (2017) [arXiv:1612.08615 [hep-th]]; R. Borsato and L. Wulff,“On non-abelian T-duality and deformations of supercoset string sigma-models,” JHEP , 024 (2017) [arXiv:1706.10169 [hep-th]].[17] N. Seiberg and E. Witten, “String theory and noncommutative geometry,” JHEP ,032 (1999) [hep-th/9908142].[18] T. Araujo, I. Bakhmatov, E. ´O Colg´ain, J. Sakamoto, M. M. Sheikh-Jabbari andK. Yoshida, “Yang-Baxter σ -models, conformal twists, and noncommutative Yang-Mills theory,” Phys. Rev. D , no. 10, 105006 (2017) [arXiv:1702.02861 [hep-th]];19. Araujo, I. Bakhmatov, E. ´O Colg´ain, J. i. Sakamoto, M. M. Sheikh-Jabbari andK. Yoshida, “Conformal Twists, Yang-Baxter σ -models & Holographic Noncommuta-tivity,” arXiv:1705.02063 [hep-th].[19] T. Araujo, E. ´O Colg´ain, J. Sakamoto, M. M. Sheikh-Jabbari and K. Yoshida, “ I ingeneralized supergravity,” Eur. Phys. J. C , no. 11, 739 (2017) [arXiv:1708.03163[hep-th]].[20] S. J. van Tongeren, “Almost abelian twists and AdS/CFT,” Phys. Lett. B , 344(2017) [arXiv:1610.05677 [hep-th]].[21] G. Arutyunov, S. Frolov, B. Hoare, R. Roiban and A. A. Tseytlin, “Scale invarianceof the η -deformed AdS × S superstring, T-duality and modified type II equations,”Nucl. Phys. B , 262 (2016) [arXiv:1511.05795 [hep-th]].[22] L. Wulff and A. A. Tseytlin, “Kappa-symmetry of superstring sigma model and general-ized 10d supergravity equations,” JHEP , 174 (2016) [arXiv:1605.04884 [hep-th]].[23] J. J. Fernandez-Melgarejo, J. i. Sakamoto, Y. Sakatani and K. Yoshida, “ T -folds fromYang-Baxter deformations,” JHEP , 108 (2017) [arXiv:1710.06849 [hep-th]].[24] N. A. Batakis and A. A. Kehagias, “Anisotropic space-times in homogeneous stringcosmology,” Nucl. Phys. B , 248 (1995) [hep-th/9502007].[25] D. Friedan, “Nonlinear Models in Two Epsilon Dimensions,” Phys. Rev. Lett. , 1057(1980); T. L. Curtright and C. K. Zachos, “Geometry, Topology and Supersymmetryin Nonlinear Models,” Phys. Rev. Lett. , 1799 (1984); C. M. Hull, “ σ Model BetaFunctions and String Compactifications,” Nucl. Phys. B , 266 (1986); B. E. Fridlingand A. E. M. van de Ven, “Renormalization of Generalized Two-dimensional Nonlinear σ Models,” Nucl. Phys. B , 719 (1986);[26] C. G. Callan, Jr., E. J. Martinec, M. J. Perry and D. Friedan, “Strings in BackgroundFields,” Nucl. Phys. B , 593 (1985).[27] L. Bianchi, “Sugli spazii a tre dimensioni che ammettono un gruppo continuo di movi-menti”, Soc. Ital. Sci. Mem. di Mat. 11, 267 (1898)[28] M. Rocek and E. P. Verlinde, “Duality, quotients, and currents,” Nucl. Phys. B ,630 (1992) [hep-th/9110053].[29] E. Kasner, “Geometrical theorems on Einstein’s cosmological equations,” Am. J. Math. , 217 (1921).[30] J. i. Sakamoto, Y. Sakatani and K. Yoshida, “Weyl invariance for generalized super-gravity backgrounds from the doubled formalism,” PTEP2017