Non-abelian fermionic T-duality in supergravity
aa r X i v : . [ h e p - t h ] F e b Non-abelian fermionic T-duality in supergravity
Lev Astrakhantsev א , ב , ד , Ilya Bakhmatov א , ש , Edvard T. Musaev ב , ש א Institute for Theoretical and Mathematical Physics, Moscow State University, Russia ב Moscow Institute of Physics and Technology, Dolgoprudny, Russia ד Institute of Theoretical and Experimental Physics, Moscow, Russia ש Kazan Federal University, Institute of Physics, Kazan, Russia
Abstract
Field transformation rules of the standard fermionic T-duality require fermionicisometries to anticommute, which leads to complexification of the Killingspinors and results in complex valued dual backgrounds. We generalize thefield transformations to the setting with non-anticommuting fermionic isome-tries and show that the resulting backgrounds are solutions of double field the-ory. Explicit examples of non-abelian fermionic T-dualities that produce realbackgrounds are given. Some of our examples can be bosonic T-dualized intousual supergravity solutions, while the others are genuinely non-geometric.Comparison with alternative treatment based on sigma models on supercosetsshows consistency. [email protected] [email protected] [email protected] Introduction 22 Fermionic T-duality 5 p -brane backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 A very fruitful approach to the analysis of the structure of physical theories is thatbased on their symmetries. It allows to overcome difficulties related to a possible badchoice of the degrees of freedom and to search for a better one. A textbook example isthe reformulation of the Maxwell theory in terms of four-dimensional tensors rather thanthree-dimensional field strength vectors, which makes manifest the Lorentz symmetryof the Maxwell equations. String theory and supergravity possess a wealth of dualitysymmetries, which relate background field configurations that are equivalent from thepoint of view of the string. Restricting the narrative to perturbative dualities of the d =
10 superstring sigma model, one recalls that it enjoys bosonic T-duality (abelian,non-abelian and more generally Poisson-Lie) and fermionic T-duality symmetries. Thestandard abelian bosonic T-duality transformation starts with the string in a backgroundthat has d commuting Killing vectors, representing the isometry group U(1) d . Gaugingeach symmetry and introducing d Lagrange multipliers to preserve the amount of theworldsheet degrees of freedom, we can rewrite the model in the first order formalism.2ntegrating out the Lagrange multipliers we recover the initial theory, while integratingout the corresponding gauge fields leads to the same sigma model, however on a differentbackground [1, 2]. The two backgrounds are related by the so called Buscher rules, whichin particular mix the metric and the 2-form b -field degrees of freedom [3, 4]. The Buscherprocedure can be generalized to non-abelian isometry groups, in which case the symmetryis referred to as non-abelian T-duality [5] and Poisson-Lie T-duality [6,7]. For more detailson T-duality symmetry and its global structure see e.g. [8–12].An extension of this idea to the superspace setting, while conceptually straightfor-ward, was not developed until much later [13]. Instead of a Killing vector isometry oneassumes invariance of the background superfields under a shift of a fermionic coordinatein superspace. This implies the existence of an unbroken supersymmetry, parameterizedby a Killing spinor field. Starting from the superstring action in a manifestly spacetimesupersymmetric formalism such as Green-Schwarz or Berkovits pure spinor sigma model,fermionic version of the Buscher procedure yields the transformation rules for the super-gravity component fields [13]. These rules only affect the dilaton and the field strengthfields from the Ramond-Ramond sector, not affecting the initial values of the metric andthe NS-NS 2-form. Originally introduced as a building block in the AdS × S T-self-duality scheme, fermionic T-duality is a component in the string theory description of theamplitude/Wilson loop correspondence and the dual superconformal invariance of super-Yang-Mills scattering amplitudes [13–16]. For a review of fermionic T-duality and somerelated developments see [17].Results of [18] on self-duality of integrable Green-Schwarz sigma models under fermionicT-duality suggest, that such self-duality could imply integrability [19–23]. In particularthis is true for sigma models on supercosets based on AdS p × S p , for p = , ,
5, which areboth fermionic T-self-dual and integrable [24, 25]. Robust support for this point couldcome from the AdS × CP sigma model, which is known to be integrable [26, 27] (for areview see [28]). However, despite many attempts, T-self-duality of AdS × CP has notbeen shown to the moment [29, 30]. Part of the reason is that the duality transforma-tion is highly restricted by the commutativity constraints. Effectively these restrict theT-duality to complexified fermionic directions in superspace, which leads to complexifiedsupergravity backgrounds. As a matter of fact, even a single fermionic T-duality withrespect to a Majorana Killing spinor of d =
10 supergravity is non-abelian by default,in the sense that the supersymmetry generator does not have a vanishing bracket withitself { Q a , ¯ Q b } = ( Γ m ) ab P m . Complexifying the Killing spinor allows to satisfy the abelianconstraint and thus make fermionic T-duality consistent. In order to end up with a real3ackground one usually performs a chain of fermionic T-dualities, arranged in such a waythat the imaginary components of fields generated in the process cancel out [13, 31, 32].A natural goal is to modify or extend the abelian fermionic T-duality procedure in sucha way, that real backgrounds would be easier to access. A generalization of fermionic T-duality transformation was proposed by considering the most general fermionic symmetryof type II supergravity [33]. It was shown that the transformation can potentially givereal backgrounds, however, to our knowledge, no examples have been presented so far. Analternative approach, which we pursue in this work, is to relax the abelian constraint anddevelop the non-abelian fermionic T-duality. This makes using real (Majorana) Killingspinors possible.However, this comes at a price. As we will see, a natural arena for the non-abelianfermionic T-dual backgrounds appears to be double field theory (DFT). The superstringdynamics on such backgrounds is well-defined in the same sense in which it is well-definedon backgrounds of the generalized supergravity [34, 35]. The latter are known to form asubset among solutions to DFT equations of motion. As an illustration of this statementwe compare our results with the recent unified treatment of bosonic and fermionic T-dualities that appeared in [36]. The authors show explicitly that the Green-Schwarzsuperstring on a coset superspace can be T-dualized along both bosonic and fermionicisometries. The known non-abelian bosonic T-duality transformation rules are re-derived,and the conditions are mentioned under which the fermionic transformations reproducethe known formulae of abelian fermionic T-duality. In section 4 we compare our resultsto those of [36] and observe that for the particular case of supercoset backgrounds theseare consistent, and hence the non-abelian fermionic T-duality transformation rules thatwe analyse indeed keep the sigma model on a supercoset invariant.We start with a brief recap of abelian fermionic T-duality in section 2, then describeour proposed non-abelian extension. This is followed by a concise review of double fieldtheory in section 3. We check that the dilaton field equation is satisfied by a non-abelianfermionic T-dual background. Then in section 4 we make connection with the non-abelianduality transformation that was derived for generic supercosets. Several explicit examplesof non-abelian fermionic T-duals follow in section 5, and we formulate some conclusionsin section 6. 4 Fermionic T-duality
Fermionic T-duality requires invariance of the background superfields under the shiftisometry of some fermionic direction. Such shift symmetry is equivalent to an unbrokensupersymmetry, which in type II supergravity is defined by a pair of Killing spinors ǫ, ˆ ǫ .Depending on whether we are in Type IIA or Type IIB theory these would be of theopposite or the same chirality respectively. To avoid confusions it is important to mentionthat ǫ, ˆ ǫ is a pair of Killing spinors that fix a single fermionic direction in the N = ( , ) or N = ( , ) d =
10 superspace. Anticommutation constraint for the pair is given byvanishing of the Killing vector field˜ K m = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ ǫ ¯ γ m ǫ − ˆ ǫγ m ˆ ǫ ( IIA ) ǫ ¯ γ m ǫ + ˆ ǫ ¯ γ m ˆ ǫ ( IIB ) ⎫⎪⎪⎪⎬⎪⎪⎪⎭ ! = . (2.1)Our spinor and gamma matrix conventions are summarized in the Appendix A. As demon-strated there, ˜ K m is simply proportional to a commutator of the supersymmetry trans-formation with itself, [ δ ǫ, ˆ ǫ , δ ǫ, ˆ ǫ ] . It is important to observe that in a Majorana basis where γ = − ¯ γ =
1, ˜ K is a simple sum of squares of all components of a spinor. Thus theabelian constraint cannot be satisfied by the standard Killing spinors of type II super-gravity, which are real in this representation. As a result, abelian constraint for fermionicT-duality necessitates complexification of the Killing spinors.The resulting abelian fermionic T-dual background can be deduced via the fermionicversion of the Buscher procedure : e φ ′ F ′ = e φ F + i ǫ ⊗ ˆ ǫC ,φ ′ = φ +
12 log C. (2.2)The background fields that undergo the transformation are the RR bispinor F and thedilaton φ , while the scalar parameter C is defined by the system of PDEs in terms of theKilling spinors [13]: ∂ m C = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ i ( ǫ ¯ γ m ǫ + ˆ ǫγ m ˆ ǫ ) ( IIA ) ,i ( ǫ ¯ γ m ǫ − ˆ ǫ ¯ γ m ˆ ǫ ) ( IIB ) . (2.3) Despite its name, fermionic T-duality does not affect the background fermionic fields. Note also thatthe only NSNS field affected is the dilaton; in particular, the metric is invariant. E.g. in type IIB, F α ˆ β = ( γ m ) α ˆ β F m + ( γ m m m ) α ˆ β F m m m +
12 15! ( γ m ...m ) α ˆ β F m ...m .
5e denote ∂ m C = iK m because the above expressions are very similar to ˜ K m .In what follows we will see how this transformation can be modified to include thenon-abelian case. So what happens when the supersymmetry transformation violates the abelian con-straint? Focusing on Type IIB case for definiteness, we have that both K m = ǫ ¯ γ m ǫ − ˆ ǫ ¯ γ m ˆ ǫ and ˜ K m = ǫ ¯ γ m ǫ + ˆ ǫ ¯ γ m ˆ ǫ are nonzero. If the field transformation (2.2) is formally applied inthis case, it would not map a supergravity solution to a solution. Let us however take acloser look at K and ˜ K . It is easy to see that they are orthogonal, ˜ K m K m =
0, by invokingthe Fierz identities for the chiral d =
10 spinors ǫ and ˆ ǫ . Since ˜ K m is a Killing vector,this implies that K m can indeed be represented by a derivative of a scalar as in (2.3),up to terms that vanish identically upon contraction with ˜ K m . Moreover, Killing spinorequations can be employed to check that ˜ K m is divergence free: ∇ m ˜ K m = ǫ ¯ γ m ∇ m ǫ + ǫ ¯ γ m ∇ m ˆ ǫ = ǫ ¯ γ m [ / H m ǫ + e φ ( / F ( ) + / F ( ) + / F ( ) ) ¯ γ m ˆ ǫ ] − ˆ ǫ ¯ γ m [ / H m ˆ ǫ + e φ ( / F ( ) − / F ( ) + / F ( ) ) ¯ γ m ǫ ] = . (2.4)One has to take into account that ǫ / Hǫ =
0, ¯ γ m / F ( ) ¯ γ m = − / F ( ) , ¯ γ m / F ( ) ¯ γ m = − / F ( ) ,¯ γ m / F ( ) ¯ γ m = C : ∂ m C = iK m − ib mn ˜ K n , ˜ ∂ m C = i ˜ K m , (2.5)where ˜ ∂ m denotes derivative with respect to the dual coordinates ˜ x m of double fieldtheory, and the b mn term is added in order to make the two equations consistent. Indeed,as we briefly review in the next section, consistency of the double field theory formulationrequires the doubled coordinates X M = ( x m , ˜ x m ) dependence of all fields to comply withthe section constraint. For the field C , which is the only function of dual coordinates6ere, the weak and the strong section constraints read ∂ m C ˜ ∂ m C = , ∂ m ˜ ∂ m C = . (2.6)One can immediately notice, that the former is satisfied due to the identity K m ˜ K m ≡ ∂ m ˜ ∂ m C = i∂ m ˜ K m = i ∇ m ˜ K m − i ˜ K m ∂ m g g , (2.7)where the first term on the right hand side vanishes because ˜ K m is a Killing vector. Thelast term can be turned to zero by choosing adapted coordinates, where g is independentof the direction of the isometry that is given by the vector field ˜ K m . It is temptingto include the factor g / into the definition of ˜ ∂ m C to make the above hold for anycoordinate choice. However, as we show further, explicit examples and comparison withthe sigma model approach for coset spaces requires the definition as in (2.5). Note, thatthis does not cause problems with the section constraint of DFT as the transformation(2.5) is defined only for backgrounds with specific fermionic and coordinate isometries.To summarize, our proposed non-abelian fermionic T-duality prescription results indouble field theory backgrounds, where explicit dependence on dual coordinate enters viathe second equation in (2.5). In the next section we will review the double field theoryequations of motion in order to see that these are actually solutions of DFT. We will provide a short overview of the necessary concepts of double field theory, whichenables us to quickly get to the equations of motion. For more detailed description of theconstruction see the original works [37–44] and reviews [45–47].
Double field theory is an approach to supergravity that makes T-duality manifest atthe level of the action by doubling the spacetime coordinates. It introduces the usual‘momentum’ (spacetime) coordinates x m together with new ‘winding’ coordinates ˜ x m X M = ( x m , ˜ x m ) and also the covariant constraint η MN ∂ M ● ∂ N ● = , η MN = ⎡⎢⎢⎢⎢⎣ δ mn δ nm ⎤⎥⎥⎥⎥⎦ . (3.1)The condition, called the section constraint, effectively eliminates half of the coordinatesand ensures closure of the algebra of local coordinate transformations [48].The action of ten-dimensional supergravity on such doubled space can be made mani-festly covariant under the global O ( d, d ; R ) T-duality rotations as well as the local gener-alized diffeomorphisms, which include standard diffeomorphisms, gauge transformationsof the Kalb-Ramond b -field, and transformations exchanging momentum and windingcoordinates. The action is background independent and takes the form S = S NSNS + S RR = ∫ d x d ˜ x ( e − d R ( H , d ) + ( / ∂χ ) † S / ∂χ ) , (3.2)where the NSNS degrees of freedom are encoded by the invariant dilaton d and thegeneralized metric H MN with its spin representative S ∈ Spin ( d, d ) , while the RR fieldstrengths are contained in the spinorial variable χ .Let us start with the NSNS fields. The invariant dilaton d is simply d = φ −
14 log g, (3.3)where g = det g mn . The generalized metric of DFT is an element of the coset spaceO ( d, d )/ O ( d ) × O ( d ) and in terms of the background fields is defined as follows H MN = ⎡⎢⎢⎢⎢⎣ g mn − b mp g pq b qn b mp g pl − g kp b pn g kl ⎤⎥⎥⎥⎥⎦ . (3.4)Varying the action (3.2) with respect to the dilaton field d and choosing the representationof R so that it resembles the dilaton equation of the usual supergravity, we obtain thefirst equation of motion: R ( H , d ) ≡ H MN ∂ M ∂ N d − ∂ M ∂ N H MN − H MN ∂ M d∂ N d + ∂ M H MN ∂ N d + H MN ∂ M H KL ∂ N H KL − H MN ∂ M H KL ∂ K H NL = . (3.5)This is the equation that we will later check for specific non-abelian fermionic T-dualbackgrounds to ensure that they are DFT solutions. The above equation does not contain8R fields as they do not couple to the dilaton in the DFT action.Next consider the variation with respect to the generalized metric H MN . For the NSNSpart of resulting field equation one obtains the following expression: δS NSNS = ∫ d x d ˜ x e − d δ H MN R MN . (3.6)Here R MN is the DFT analogue of Ricci curvature [49] and reads R MN ≡ ( δ PM − H PM ) K P Q ( δ QN + H QN ) + ( δ PM + H PM ) K P Q ( δ QN − H QN ) , (3.7)with K MN ≡ ∂ M H KL ∂ N H KL − ( ∂ L − ( ∂ L d )) ( H LK ∂ K H MN ) + ∂ M ∂ N d − ∂ ( M H KL ∂ L H N ) K + ( ∂ L − ( ∂ L d )) ( H KL ∂ ( M H N ) K + H K ( M ) ∂ K H LN ) ) . (3.8)In contrast to the dilaton equation, equations of motion for the generalized metric containcontributions from the RR fields coming from the variation of the field S that appearsin the RR part of the action. The RR potentials of Type II theory in the democraticformulation are encoded in the Spin ( , ) spinor χ , such that the corresponding fieldstrengths read ∣ F ⟩ ≡ ∣ / ∂χ ⟩ = ∑ p = p ! F i ...i p ψ i ⋯ ψ i p ∣ ⟩ , ⟨ F ∣ ≡ ⟨ / ∂χ ∣ = ∑ p = p ! ⟨ ∣ ψ i p ⋯ ψ i F i ...i p . (3.9)Here the gamma matrices ( ψ a , ψ a ) of Spin ( , ) are defined in the usual way (up torescaling) { ψ a , ψ b } = δ ab . (3.10)To define Dirac conjugation one introduces the matrix A = ( ψ − ψ ) ( ψ − ψ ) ⋯ ( ψ − ψ ) : ⟨ / ∂χ ∣ = ∑ p = p ! ⟨ ∣ Aψ i p ⋯ ψ i F i ...i p . (3.11)Finally, the kinetic operator K = A − S is written in terms of the Spin(10,10) image of thegeneralized metric, and it also contributes to the variation with respect to H MN .The complete equations of motion of generalized metric of DFT for the action (3.2)9ecome (see [43] for technical details) e − d R MN + E MN = , (3.12)where the symmetric ‘stress-tensor’ E MN with upper indices is defined as E MN = H P ( M / ∂χ Γ N ) P K / ∂χ = − H P ( M / ∂χ Γ N ) P / ∂χ. (3.13)Explicitly in terms of the background RR field strengths this takes the following form E MN = − √ − g ⎡⎢⎢⎢⎢⎢⎣ F m F n + F mpq F npq + F mpqrs F npqrs − g mn ∑ i = , ∣ F ( i ) ∣ F nmp F p + F nmpqr F pqr F mnp F p + F mnpqr F pqr F m F n + F mpq F npq + F mpqrs F npqrs − g mn ∑ i = , ∣ F ( i ) ∣ ⎤⎥⎥⎥⎥⎥⎦ , (3.14)where we see that the diagonal blocks are exactly the stress energy tensor for associateddifferential forms. Consider now the dilaton equation of double field theory and perform non-abelianfermionic T-duality transformation, which maps d → d + log C . The dilaton equationacquires additional terms∆ R = H MN ∂ M ∂ N log C − H MN ∂ M log C∂ N log C − H MN ∂ M d ∂ N log C + ∂ M H MN ∂ N log C. (3.15)The above must be zero for the dual background to solve the dilaton equation. Assumingthat the initial background does do not depend on dual coordinates, we write the aboveexplicitly as∆ R = g mn C − ∂ mn C − g mn C − ∂ m C∂ n C − g mn C − ∂ m d∂ n C + C − ∂ m g mn ∂ n C + b mn C − ∂ m ˜ ∂ n C − b mn C − ∂ m C ˜ ∂ n C − b mn C − ∂ m d ˜ ∂ n C + C − ∂ m b mn ˜ ∂ n C + ( g mn − b mk b kn ) C − ˜ ∂ mn C − ( g mn − b mk b kn ) C − ˜ ∂ m C ˜ ∂ n C. (3.16)10ecall now the relation between the derivatives of the function C = C ( x m , ˜ x m ) and thequadratic expressions built from the Killing spinors, which we denoted K m and ˜ K m : ∂ m C = iK m − ib mn ˜ K n , ˜ ∂ m C = i ˜ K m . (3.17)Substituting this into the expression for ∆ R one obtains∆ R = ig mn C − ∂ m K n + g mn C − K m K n − ig mn C − ∂ m dK n + C − i∂ m g mn K n + C − ib mn ∂ m ˜ K n + ( g mn − b mk b kn ) C − i ˜ ∂ m ˜ K n + C − g mn ˜ K m ˜ K n . (3.18)One can check that the first line vanishes using the Fierz identities and the Killing spinorequations for ǫ, ˆ ǫ . This is to be expected since the first line contains the new terms inthe dilaton equation that appear when performing abelian fermionic T-duality. Termsin the second line are more subtle. Firstly, note that ˜ ∂ m ˜ K n ≡ ∂ m ˜ K n ≡ ∂ m ˜ K n = ∂ m ˜ ∂ n C = ˜ ∂ n ∂ m C = ˜ ∂ n ( K m − b mk ˜ K k ) ≡ . (3.19)Finally, the last term g mn ˜ K m ˜ K n vanishes since it is proportional to terms of the form ǫγ m ǫ ǫγ m ǫ .We conclude that the transformation (2.2) with (2.5) always generates backgroundsthat solve the generalized dilaton equation of double field theory. This is true even ifnon-trivial dependence on the dual coordinates is generated. Although being a stronghint, this is not enough to claim that such transformation always gives solutions to allDFT equations, including the RR sector. Instead of going through tedious computationsto show that strictly and in full, we provide a set of examples supporting the statementand establish a relation to the approach based on supercoset sigma models. The dualbackgrounds in the examples also solve the Einstein DFT equation (3.12), although thishas not been checked in general. A proof that non-abelian fermionic T-dual backgroundsalways solve the complete set of the double field theory equations we postpone to a futurework. 11 Sigma model perspective
The field theory description of non-abelian fermionic T-duality transformations pre-sented above together with examples of the next section provides strong evidence thatit always yields a solution to double field theory equations of motion. It is natural toask how string theory sigma model behaves under such transformation of its backgroundfields. To see that we turn to a generic scheme of non-abelian duality transformationsof the supercoset sigma model considered in [36]. Avoiding the full discussion and refer-ring the reader for details to the original paper, we focus only on the transformation ofbackground fields, which has been shown to leave the sigma model invariant.While the results of [36] are valid for both supergroup and supercoset models, let usfor simplicity restrict our consideration to the former. Then, denoting an element of thesupergroup g ∈ G and generators of the corresponding superalgebra as T A ,P one definesthe supervielbein E A = E MA dz M in the usual way g − dg = E A T A , (4.1)where z M = ( x m , θ µ , ˆ θ ˆ µ ) are coordinates of the d = N =
A = ( a, α, ˆ α ) . Lowest order θ = ˆ θ = e am and the doubletof gravitini ψ αm , ψ ˆ αm , however the most relevant for the discussion here is the spinor-spinorblock ⎛⎝ E αµ E ˆ αµ E α ˆ µ E ˆ α ˆ µ ⎞⎠ . (4.2)It is always possible to represent the fermionic isometry that we are T-dualizing bythe shift of a certain superspace fermion, θ → θ + ρ , where ρ is a constant Grass-mann number. Then the above components of the supervielbein appear in the expres-sions E α M δZ M ∣ θ = ˆ θ = = ǫ α ρ and E ˆ α M δZ M ∣ θ = ˆ θ = = ǫ ˆ α ρ , where we defined ǫ α = E α ∣ θ = ˆ θ = and ǫ ˆ α = E ˆ α ∣ θ = ˆ θ = . Since E α M δZ M ∣ θ = ˆ θ = and E ˆ α M δZ M ∣ θ = ˆ θ = in principle correspond to the localsupersymmetry parameters, we conclude that ǫ α and ǫ ˆ α are commuting Killing spinors ofthe supergravity background that is invariant under the shifts of θ .We are now in a position to interpret the non-abelian T-duality rule for the RR bispinorderived in [36] F ′ α ˆ α = ( Λ F ) α ˆ α + i Λ E α M N MN E ˆ α N ∣ θ = ˆ θ = . (4.3)Here Λ is a Lorentz transformation corresponding to a bosonic (non-abelian) T-duality12nd in what follows we set Λ =
1. In the second term one finds the matrix N MN whoseinverse is defined by N MN = E MA E N B ( G AB − B AB + ˜ z C f ABC ) . (4.4)Dual (super-)coordinates ˜ z A appear as Lagrange multipliers as in the standard T-dualityprocedure . The superalgebra structure constants f ABC are defined as usual in terms of thegenerators T A . Finally, for the case of supergroups we consider here, G AB , B AB are con-stant fields that are tangent superspace representation of the Green-Schwarz superfields G MN , B MN , i.e. G MN = E MA E N B G AB ,B MN = E MA E N B B AB . (4.5)It is easy to check that the above reproduces the standard abelian fermionic T-dualityprescription. Assuming that the initial background has vanishing gravitino, only thespinor-spinor part of the supervielbein in (4.3) contributes: δ F α ˆ α = i E α N E ˆ α ∣ θ = ˆ θ = = i ǫ α ( − B ∣ θ = ˆ θ = ) − ˆ ǫ ˆ α . (4.6)Note that the 1’s above are fermionic indices. Dual coordinates ˜ z M do not appear dueto the anticommutativity constraint f A =
0. Up to a conventional sign, the above isprecisely the transformation derived in [13] with C = B ∣ θ = ˆ θ = . It is important to realizethat C = B ∣ θ = ˆ θ = is the component of the superfield B MN and may not be constant.Indeed, one can compute its derivative using the definition of the spinor-spinor-vectorcomponent of the superfield strength H = dB : ∂ m B ∣ θ = ˆ θ = = ( H m − ∂ B m )∣ θ = ˆ θ = = ( E A m E B E C H ABC − ∂ ( E A B m A )) ∣ θ = ˆ θ = . (4.7)Using the type IIB supergravity constraints H αβc = i ( ¯ γ c ) αβ and H ˆ α ˆ βc = − i ( ¯ γ c ) ˆ α ˆ β , the firstterm above gives E A m E B E C H ABC ∣ θ = ˆ θ = = i ( ǫ ¯ γ m ǫ − ˆ ǫ ¯ γ m ˆ ǫ ) = iK m . The second term may be For doubled superspace constructions see [50–52]. They may depend on spectator fields in case when bosonic dimension of the supergroup is less thanten. For type IIA, the constraint is H αβc = − i (C Γ c Γ ) αβ where the full α = , . . . ,
32 Dirac spinors andthe corresponding gamma matrices are used [53–55]. Thus there is no relative sign in the definition of K m θ dependence of E aµ = i ( γ a ) ρµ θ ρ :2 ∂ B m = ∂ ( E a B ma ) = ∂ ( i ( γ a ) ρ θ ρ B ma ) = i ( γ n ) B mn . (4.8)Finally, (4.7) becomes ∂ m B ∣ θ = ˆ θ = = iK m − ib mn ˜ K n , (4.9)where the Kalb-Ramond field b mn is the lowest order component field of B mn , and we haveconverted the world fermionic indices on the gamma matrix to tangent space with thehelp of the Killing spinors, ( γ n ) ∣ θ = ˆ θ = = ǫ α ( γ n ) αβ ǫ β + ˆ ǫ ˆ α ( γ n ) ˆ α ˆ β ˆ ǫ ˆ β = ˜ K n . For the abelianfermionic isometry the second term does not contribute since ˜ K m = ǫ, ˆ ǫ that violate the anticommutativity relation, ˜ K ≠
0, so that both terms in (4.9) contribute.We should also take into account the non-vanishing structure constant f m = i ˜ K m ≠ δ F α ˆ α = i ǫ α C − ˆ ǫ ˆ α ,if we redefine C to take care of the new terms: C = ( B + ˜ z M f M )∣ θ = ˆ θ = = B ∣ θ = ˆ θ = + i ˜ x m ˜ K m , (4.10)where B ∣ θ = ˆ θ = satisfies (4.9). One immediately infers the dual coordinate derivative˜ ∂ m C = i ˜ K m . This can be viewed as one of the defining equations for C in the non-abeliancase in the constructive approach, i.e. if one starts from the Killing spinors. This equationholds beyond supercoset backgrounds as we have shown above. In the second definingequation for ∂ m C we have the standard contribution iK m plus extra terms: ∂ m C = iK m − ib mn ˜ K n + i ˜ x n ∂ m ˜ K n . (4.11)We have seen in the previous section that the derivative ∂ m C defined as above, howeverwithout the last term, ensures that the generalized dilaton equation of DFT holds. Withthe extra term i ˜ x n ∂ m ˜ K n that equation gets new contributions of different orders in ˜ x m ,contracted with expressions that depend purely on geometric coordinates. Hence, theonly condition for this to satisfy the generalized dilaton equation of motion is˜ x n ∂ m ˜ K n = C as in (2.5).At this point the origin of the above condition is not completely clear and more detailedanalysis of the sigma model is needed. From the field theory point of view this is simplyimposed by the equations of motion. In section 5 we will see that this condition indeedis satisfied for the non-trivial example of non-abelian fermionic T-duality of the D-branebackground. We conclude that the field transformations (2.2) with C defined by (2.5) forgeneral supersymmetric backgrounds agree with the results of [36] in the case of supercosetsigma models. As a proof-of-concept and an elementary example let us consider the flat empty space-time in d =
10, which is a maximally supersymmetric solution, i.e. one has 32 fermionicdirections given by the degenerate doublets of constant Killing spinors ( ǫ i , ) , ( , ˆ ǫ i ) , i ∈ { , . . . , } : ( ǫ i ) α = δ iα , ( ˆ ǫ j ) ˆ α = δ j ˆ α . (5.1)Since the space-time metric g mn is constant and the b -field is zero, the generalized Riccitensor can be written completely in terms of derivatives of the generalized dilaton R MN = ⎛⎝ ∂ m ∂ n − g ml g nk ˜ ∂ l ˜ ∂ k g ml g nk ∂ l ˜ ∂ k − ˜ ∂ m ∂ n g ml g nk ˜ ∂ l ∂ k − ∂ m ˜ ∂ n ˜ ∂ m ˜ ∂ n − g ml g nk ∂ l ∂ k ⎞⎠ d. (5.2)This means that the generalized Einstein equations (3.12) simplify a lot. Below we provideseveral examples of non-abelian fermionic T-duality of Minkowski spacetime based onvarious choices of the Killing spinors. These are selected so as to highlight certain genericproperties of non-abelian fermionic T-dual backgrounds. All the examples solve the fieldequations of the NSNS sector of double field theory. Example 1.
The spinors are taken to be ( ǫ − iǫ , − ˆ ǫ − i ˆ ǫ ) . (5.3)Using the explicit realization of the gamma matrices, one can check that this directionis non-abelian, and the doubled spacetime derivatives of C (2.5) are ∂ C =
4, ˜ ∂ C = i ,15eading to C = ( x + i ˜ x ) . (5.4)One may recover the fermionic T-dual background using (2.2). The metric is still flat andno b field is generated. We do get a nontrivial dilaton φ =
12 log 4 ( x + i ˜ x ) , (5.5)and the RR fluxes (assuming that we are in type IIB): F = − i C − / ,F = F = − F = − F = F = − F = F = F = − C − / ,F = F = − F = F = − F = − F = F = F = − F = − F = F = − F = F = F = i C − / . (5.6)Overall this looks very reminiscent of abelian fermionic T-duality transformation. Themain difference is of course that this background is not a supergravity solution due to theexplicit dual coordinate dependence of C . However, it can be shown to satisfy equations ofmotion of double field theory. The easiest way to do this is to perform a bosonic T-dualityalong x , which is equivalent to replacing ˜ x ↔ x in all expressions. This will bring us tothe Type IIA theory, still with flat metric and no b field, the dilaton φ = log 4 ( x + ix ) andthe RR fluxes reshuffled according to F ′ α ˆ β = F α ˆ δ ( ¯ γ ) ˆ δ ˆ β . Explicit computation shows thatsupergravity field equations are then satisfied. This implies that (5.5), (5.6) is a complex-valued solution to the DFT equations of motion, which one can also verify directly, usingthe equations of section 3. It belongs to the geometric orbit of T-duality, i.e. one bosonicT-duality takes it to a (complex-valued) solution of supergravity. Example 2.
One may consider a fermionic T-duality transformation generated simplyby a single Majorana spinor of flat spacetime, which is also non-abelian: ( ǫ , ) . (5.7)Then the fermionic T-dual is Minkowski with non-trivial spacetime dependence of thedilaton, g mn = η mn , b mn = ,e φ = i ( x − ˜ x + x + ˜ x ) ,F RR = , (5.8)16hich cannot be T-dualized into any geometric background and hence is genuinely non-geometric following the nomenclature of [56]. The dilaton field has exotic dependence ondual coordinates, i.e. on combinations of the type x ± ˜ x , however the section constraint issatisfied. No RR flux is generated in this transformation, because we have chosen ˆ ǫ = ( e i π ǫ , ) , (5.9)which results in the dilaton e φ = − x + ˜ x − x − ˜ x . This real background is too asolution of double field theory, however it is still not completely satisfactory as it includesdependence on the dual time. In fact, it is impossible to avoid dependence on the dualtime without using linear combinations of Killing spinors with relative complex phaseshifts between different terms. The reason is that the supersymmetric Killing vector ˜ K m is always timelike or null in real valued supergravity [57, 58], which implies ˜ x dependenceby virtue of (2.5). Example 3.
An example of a combination that provides more sensible, however stillcomplex background is: ǫ = √ ( ǫ + iǫ ) , ˆ ǫ = . (5.10)This leads to the following genuinely non-geometric solution g mn = η mn , b mn = ,e φ = − x − ˜ x + i ( x + ˜ x ) ,F ( p ) = . (5.11)Hence, non-abelian fermionic T-duality gives a genuinely non-geometric background ofDFT, which is complex and does not depend on dual time.17 .2 D p -brane backgrounds Moving on to less trivial examples, we can consider D p -brane solutions in d = ds ( p ) = H − ( p ) [ − dt + dy ( p ) ] + H ( p ) dx ( − p ) ,e − ( φ − φ ) = H p − ( p ) , C = e − φ ( H − ( p ) − ) dt ∧ dy ∧ . . . ∧ dy p , (5.12)where H ( p ) is a function of the transverse directions x ( − p ) , and is a harmonic function ina given dimension. The solutions preserve half of the maximum supersymmetry, due tothe BPS constraints being the projection conditions ( − Γ ..p O p ) ( ǫ ˆ ǫ ) = , (5.13)where O p depends on the dimension as well as on type IIA versus type IIB theory and issuch as to make the above operator a projector. Overall, there are 16 independent Killingspinors in the D p -brane background, parametrized by an arbitrary constant Majorana-Weyl spinor ˆ ǫ of appropriate chirality: ǫ = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ H − ( p ) iγ ...p ˆ ǫ ( p even , IIA ) ,H − ( p ) γ ... ¯ p ˆ ǫ ( p odd , IIB ) , ˆ ǫ = H − ( p ) ˆ ǫ . (5.14)Expressions for the Killing spinor bilinears K m , ˜ K m simplify substantially due to thegamma matrix identities ( γ ...p ) T ¯ γ a ( γ ...p ) = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ − γ a , a ≤ p,γ a , a > p, ( p even , IIA ) , (5.15) ( γ ... ¯ p ) T ¯ γ a ( γ ... ¯ p ) = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ ¯ γ a , a ≤ p, − ¯ γ a , a > p, ( p odd , IIB ) . (5.16)Recalling that we have alternating signs in the type IIA versus type IIB expressions for18 m (2.3) and ˜ K m (2.1), we end up with the same answer for any D p -brane, ∂ m C = ∂ m C = i δ ma ˆ ǫ γ a ˆ ǫ ⎫⎪⎪⎪⎬⎪⎪⎪⎭ m ≤ p, (5.17) ∂ m C = i δ am ˆ ǫ γ a ˆ ǫ ˜ ∂ m C = ⎫⎪⎪⎪⎬⎪⎪⎪⎭ m > p, (5.18)(replace γ a → ¯ γ a according to the chirality of ˆ ǫ ). It is noteworthy that the vielbein andthe metric have worked out precisely so that the right hand sides above are constant.We observe that the function C cannot depend on coordinates dual to the transversedirections. It may depend on coordinates dual to the isometric directions along theworldvolume. Hence, the combination ˜ x m ˜ K m is non-zero and in fact does not depend ongeometric (transverse, in this case) coordinates, thus satisfying the condition (4.12). Alsorecall that we had the condition ∂ m g ˜ ∂ m C = ǫ α = √ e iπ ( − δ α + iδ α + δ α + iδ α ) , (5.19)where the constants have been adjusted so as to yield a simple value for the dualityparameter: C = x + i ˜ x . (5.20)This appears in the dual dilaton e φ = e φ C , as well as in the RR fields F ( ) = − e − φ C / dx , (5.21) F ( ) = ie − φ C / [ dx ( H − dx + dx − dx ) − dx , + idx ( dx + dx ) + idx ( dx + dx )] , (5.22) F ( ) = − e − φ C / [ ∑ k = H ( δ k + CH ∂ k H ) dx k + dx ( dx − dx ) − idx ( dx ( dx + dx ) + dx ( dx + dx )) ] . (5.23)We have employed the obvious notation dx mn = dx m ∧ dx n , etc., in order to keep the19xpressions compact. This background is a DFT solution and can be mapped to a solutionof supergravity by a bosonic T-duality in x . To summarize, we have proposed and worked out a supergravity description of a non-abelian generalization of fermionic T-duality. The construction hinges on the observationthat relaxing the abelian constraint ˜ K m = x m ∂ n ˜ K m =
0, where ˜ K m is the supersymmetric Killing vector built out of theKilling spinors and ˜ x m are the dual (winding) coordinates. From the field theory point ofview, adding such term to the scalar function C as prescribed by the sigma model proce-dure violates equations of motion of DFT. Detailed analysis of the sigma model origin ofthis constraint is beyond the scope of the present paper.We have seen that non-abelian fermionic T-duality generates solutions of double fieldtheory equations of motion. Checking this for the explicit examples: empty Minkowskispacetime and D p -brane backgrounds, we find several essentially different cases summa-rized in Table 1. We observe that at least for the Minkowski spacetime it is impossible toFields Dependence on the dual time OrbitComplex no geometricReal yes non-geometricComplex no non-geometricTable 1: Typical backgrounds generated by a non-abelian fermionic T-duality.generate a real solution which does not depend on the dual time, when dualizing along asingle fermionic isometry. Dependence on the dual time makes the background effectivelycomplex due to the wrong sign of the kinetic terms of RR fields.Apart from the reality property of dual backgrounds, of interest is their dependenceon dual coordinates reflected by the last column of Table 1. One typically finds C to belinear function of the form αx m + β ˜ x n , where x m and ˜ x n are some particular geometricand dual coordinates. In the case m ≠ n the background belongs to the geometric orbit20f bosonic T-duality in the sense of [56]. Indeed, then a T-duality along ˜ x n would turnit into the geometric x n and all dual coordinate dependence is gone. Otherwise, when m = n it is impossible to remove the dependence on the dual coordinate by means of abosonic T-duality, and hence the corresponding background belongs to a non-geometricorbit and is referred to as genuinely non-geometric. Examples of such genuinely non-geometric backgrounds have been found in [56] in generalized Scherk-Schwarz reductionsof DFT for twist matrices violating the section constraint. In contrast, genuinely non-geometric backgrounds of the form we obtain respect the section constraint and are validstring theory backgrounds.For some examples that we consider one finds vanishing RR fields in the dual back-ground and linear dependence on the dual coordinate in the scalar function C . Although,the dilaton is related to C via the exponent, this reminds the double and exceptional fieldtheory description [59, 60] of generalized supergravity [34]. There one considers a solutionof the section constraint, where the metric and the background gauge fields depend onat most nine spacetime coordinates, while the dilaton is allowed to depend linearly onthe remaining tenth coordinate x . Then bosonic T-duality along x turns it into ˜ x andintroduces linear dependence on the dual coordinate. For the Green-Schwarz superstringone can check that such backgrounds are allowed by the kappa-symmetry [35, 61]. Thus itmay be interesting to inspect string behaviour on backgrounds produced by non-abelianfermionic T-duality.Another interesting further direction is to investigate possible relations between non-abelian fermionic T-dualities and deformations of supercoset sigma models correspond-ing to Dynkin diagrams with all fermionic simple roots. Bivector deformations of in-tegrable sigma models on supercosets are known to correspond to bosonic non-abelianT-dualities, at least for a certain class where the r -matrix is invertible [36, 62]. Integrable η -deformations of 10-dimensional backgrounds with an AdS factor have been found toproduce solutions of the ordinary supergravity equations when the deformation is donealong fermionic simple roots of the isometry superalgebra [63] (see [64] for a review).Finally, of particular interest is the admittedly long-standing problem of self-duality ofthe AdS × CP background. So far searches for a chain of fermionic and bosonic T-dualitythat map the background to itself have all failed [29, 30, 65], and one may hope that thenovel non-abelian fermionic T-duality might be helpful here. Indeed, conventionally onewould try to organise the chain such that each T-duality maps a solution to a solutionof supergravity equations of motion. As we have seen, this greatly restricts the possiblechoices of the Killing spinors, but there is no such restriction in the non-abelian case.21ince solutions of DFT equations of motion can be understood as proper background ofthe double or even the standard superstring, one could hope to organise such a chainwhich goes beyond the set of supergravity backgrounds and maps AdS × CP to itself. Acknowledgements
This work has been supported by Russian Science Foundation under the grant RSCF-20-72-10144.
A Index summary and spinor conventions
Some of the indices that we use are: m, n ; a, b ∈ { , . . . , } Lorentz vector index ( world / tangent ) ,µ, ν ; α, β ∈ { , . . . , } Weyl spinor index ( world / tangent ) ,M, N ∈ { , . . . , } double spacetime coordinate index . Hatted spinor indices ˆ α, ˆ β correspond to the second spinor of N = ǫ = ⎡⎢⎢⎢⎢⎣ ǫ α ⎤⎥⎥⎥⎥⎦ , ˆ ǫ = ⎡⎢⎢⎢⎢⎣ ǫ ˆ α ⎤⎥⎥⎥⎥⎦ , (A.1)IIB: ǫ = ⎡⎢⎢⎢⎢⎣ ǫ α ⎤⎥⎥⎥⎥⎦ , ˆ ǫ = ⎡⎢⎢⎢⎢⎣ ˆ ǫ ˆ α ⎤⎥⎥⎥⎥⎦ . (A.2)We are using a Majorana-Weyl representation of the SO ( , ) gamma matrices, such thatΓ m = ⎡⎢⎢⎢⎢⎣ ( γ m ) αβ ( ¯ γ m ) αβ ⎤⎥⎥⎥⎥⎦ , C = ⎡⎢⎢⎢⎢⎣ c αβ ¯ c αβ ⎤⎥⎥⎥⎥⎦ , ψ = ⎡⎢⎢⎢⎢⎣ φ α χ α ⎤⎥⎥⎥⎥⎦ . (A.3)The charge conjugation matrix can be used to define Majorana conjugation, relatingcovariant and contravariant spinors, ¯ ψ = ψ T C . One can look up an explicit example of thegamma matrices in this representation e.g. in [31]. The important algebraic properties ofthe γ matrices are ( γ m ) αβ ( ¯ γ n ) βγ + ( γ n ) αβ ( ¯ γ m ) βγ = η mn δ αγ and ( γ m ) ( αβ γ mγ ) δ =
0. Both γ and ¯ γ are symmetric, as are γ m ...m ; γ m m m on the contrary are antisymmetric. In the22ain text we sometimes use the obvious notation γ a ¯ b...c = γ [ a ¯ γ b . . . γ c ] .Depending on chirality, one gets for the commutator of the supersymmetry transfor-mation with itself [ ¯ ǫQ, ¯ ǫQ ] = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ − ( ǫc ) α { Q α , Q β } ( ǫc ) β , for ǫ = ⎡⎢⎢⎢⎢⎣ ǫ α ⎤⎥⎥⎥⎥⎦ , − ( ǫ ¯ c ) α { Q α , Q β } ( ǫ ¯ c ) β , for ǫ = ⎡⎢⎢⎢⎢⎣ ǫ α ⎤⎥⎥⎥⎥⎦ . (A.4)Thus, for N = d =
10 supersymmetry with two independent charges Q, ˆ Q , employingthe fundamental relations of the supersymmetry algebra, { Q, Q } = ( C Γ m ) P m = { ˆ Q, ˆ Q } ,we have [ δ ǫ, ˆ ǫ , δ ǫ, ˆ ǫ ] = [ ¯ ǫQ, ¯ ǫQ ] + [ ¯ˆ ǫ ˆ Q, ¯ˆ ǫ ˆ Q ] = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ − ( ǫ ¯ γ m ǫ − ˆ ǫγ m ˆ ǫ ) P m , IIA , − ( ǫ ¯ γ m ǫ + ˆ ǫ ¯ γ m ˆ ǫ ) P m , IIB , (A.5)where we restrict to a gamma matrix representation with c αβ = δ βα , ¯ c αβ = − δ αβ . Thismotivates the definition of the vector field ˜ K m (2.1). Vanishing of ˜ K m is simply a criterionthat the supersymmetry which corresponds to the fermionic shift δ ǫ, ˆ ǫ is abelian.Supersymmetry transformations for type IIB are given by δψ m = ∇ m ǫ − / H m ǫ − e ϕ ( / F ( ) + / F ( ) + / F ( ) ) ¯ γ m ˆ ǫδ ˆ ψ m = ∇ m ˆ ǫ + / H m ˆ ǫ + e ϕ ( / F ( ) − / F ( ) + / F ( ) ) ¯ γ m ǫ,δλ = / ∂ϕ ǫ − / Hǫ + e ϕ ( / F ( ) + / F ( ) ) ˆ ǫ,δ ˆ λ = / ∂ϕ ˆ ǫ + / H ˆ ǫ − e ϕ ( / F ( ) − / F ( ) ) ǫ, (A.6)where / F ( n ) = n ! F m ...m n γ m ...m n , / H m = H mnk γ nk . (A.7) References [1] T. Buscher, “Path integral derivation of quantum duality in nonlinear sigmamodels,”
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