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Paper: Driezen, S., Sevrin, A. & Thompson, D. (2016). Aspects of the doubled worldsheet. Journal of High Energy Physics, 2016(12) http://dx.doi.org/10.1007/JHEP12(2016)082

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Received: September 22, 2016 Accepted: December 12, 2016 Published: December 16, 2016 JHEP12(2016)082 Aspects of the doubled worldsheet

Sibylle Driezen, Alexander Sevrin1 and Daniel C. Thompson Theoretische Natuurkunde, Vrije Universiteit Brussel and The International Solvay Institutes, Pleinlaan 2, B-1050 Brussels, Belgium E-mail: [email protected], [email protected], [email protected]

Abstract: We clarify the relation between various approaches to the manifestly T-duality symmetric string. We explain in detail how the PST covariant doubled string arises from an unusual gauge ﬁxing. We pay careful attention to the role of “spectator” ﬁelds in this process and also show how the T-duality invariant doubled dilaton emerges naturally. We extend these ideas to non-Abelian T-duality and show they give rise to the duality invariant formalism based on the semi-Abelian Drinfeld Double. We then develop the N = (0, 1) supersymmetric duality invariant formalism.

Keywords: String Duality, Superspaces, Superstrings and Heterotic Strings, Supersym- metry and Duality

ArXiv ePrint: 1609.03315

1Also at: the Physics Department, Universiteit Antwerpen, Campus Groenenborger, 2020 Antwerpen, Belgium.

Open Access, c The Authors. doi:10.1007/JHEP12(2016)082 Article funded by SCOAP3. JHEP12(2016)082 ] and 3 , 2 ]. The study 3 , 2 the doubled worldsheet ] to use such a formalism 5 , 4 8 10 ] and Tseytlin [ 1 ) seeks to promote the U-duality 18 EFT 16 ( 25 5 14 – 1 – 5 1) case 22 13 , 11 ]. These ideas have also been explored in the context 8 18 20 = (0 N exceptional field theory 22 ), whose origins date to the seminal works of Tseytlin [ DFT ( ] where 1 24 10 , 9 ]. This approach was derived from the perspective of closed string field theory 7 , 6 ), of strings goes back to pioneering work of Duff [ The idea of a T-duality invariant worldsheet description, 4.2 The Tseytlin4.3 formulation Component form 3.1 Relation to Poisson Lie doubled formalism 4.1 The covariant formulation 2.1 Deriving the2.2 covariant doubled string Gauge symmetries2.3 and the origin Equations of of2.4 PST motion symmetry in the Gauge PST fixing2.5 doubled and formalism the dilaton A comment on chiral gauging DWS has been the developmentdouble of field theory a spacetimeSiegel T-duality [ invariant theory, oftenon a now torus dubbed byof Hull M-theory and [ Zwiebach [ string and M-theory maywhich be they promoted may be to used manifest to symmetries determine and the indeed structure( the of the extent underlying to theory. of this approach was reignitedto following define the strings proposal in of a Hull class [ of non-geometric backgrounds known as T-folds. Parallel to this 1 Introduction A central theme in recent years has been to understand the ways in which dualities of A Conventions B PST symmetry in the 5 Discussion and open problems 3 Application to non-abelian T-duality 4 Towards the supersymmetric doubled string Contents 1 Introduction 2 Bosonic abelian doubled string JHEP12(2016)082 ] 11 (1.3) (1.4) (1.5) (1.1) (1.2) DFT . Just i x -dimensional d exceptional field and program of West [ 11 . The T-duality group, , given in this basis by, } η E i -dual coordinates ˜ ˜ as, x d = , i , x O { and η ! ]. i T 1 = x 14 − 1 – O I , . − 12 X . . b g ! − double field theory HO ! S 0 1 , T 0 b 1 which one can think of as giving rise to a 1 1 O 0 0 -dimensional torus, we have a 2 1 in the internal toroidal directions are united 1 b g − − 1 d generalised metric = 1

0 b g – 2 – − 1 2 b = g = = − 2 = g S IJ IJ η

Ω P H → H -regular coordinates = and Ω are central to the recent proposal of Born geome- d η IJ , doubled worldsheet H S such that H , η = S and NS two-form fields 1 ) in this case, acts on this we refer the reader to the review articles [ ij Z g ; EFT ]. d, d gauge invariance of the theory requires a constraint, also known as the section 16 and , is that in order to make the duality act as a linearly realised symmetry, the dimen- generalised metric 15 EFT In all of these duality symmetric approaches, there is a price to pay; action principles A common theme of the Our focus in this note will be on the worldsheet rather than spacetime so for further introduction to DFT 1 based on the doubledand or extended spacetimes requirecondition, supplementary constraints. that In essentially declares the field content ofthe the theory to depend on only a The existence of the objectstries [ This doubled space isbasis as, also equipped with a natural symplectic product Ω given in this From the generalised metricproduct we structure see that“chiral the structure” specified doubled by space the is projection equipped operators, with an almost where the group element preserves the inner product, on the doubled spacewhich is parametrised O( by coordinates extended spacetime consisting of as position is conjugate toto momenta winding one of can the think string.ground of metric For these strings extra in coordinates ainto as curved a conjugate background, the components of the back- and collaborators. theory sionality of spacetime is augmentedstance, in by the the case introduction of T-duality of of strings additional on coordinates. a For in- group to a manifest symmetry of a spacetime action and in the JHEP12(2016)082 ] ). 32 , 1.6 (1.8) (1.6) (1.7) 31 to depend exactly half H I X ..., ) follows. Without + 1.6 J ], bosons X , the worldsheet bosons 3 d τ , . ]. ∂ 2 26 – IJ which are indeed compatible ..., DWS = 0 η 22 I J + H X X ] or by holomorphic factorisation J σ | = X possibly depending on some other ]). However multi-loop calculations ∂ 27 ∂ = J 35 + ∂ H I ] construction for chiral bosons. The ) ] globally, one recovers the generalised geome- J IJ − X 17 30 P σ H ( ∂ I ] and [ ) may be integrated and using a gauge in- In the case of X IJ | both of which we shall detail later. One can 34 = 1.8 2 , ]. Whilst this is certainly a viable route, one – 3 – in the present context (other attempts to make H ]. Ways in which the section condition can be consistently , I IJ 33 σ ∂ 21 29 X 2 and the worldsheet theory by allowing , , σ d = 0 20 ∂ DFT 28 J -model in the doubled spacetime, Z − X σ 2 1 DFT = ) give rise to the desired chirality constraints of eq. ( has been handled. σ ∂ τ 2 = ( J d I I ) f Z + parametrising a base manifold over which the doubled torus is Hull DWS and a generalised metric = 1 4 P S y ( I I = X δX Tseytlin S ] reformulation of supergravity of [ 19 , with coordinates To explain this let us momentarily restrict ourselves to a simple case; a doubled torus 18 Upon solving this section condition for type II DFT [ 2 d 2 on the internal coordinates areare found at in [ best very difficult without Lorentz covariance. try [ relaxed are of great interest and connect to gauged supergravities see [ variance of the form Despite its apparent non-covariancetechniques, one for instance can one-loop still betaand employ functions shown of some this to conventional action give field havewith rise been theory the calculated to equations [ background that fieldmore follow equations precise from the for linkage between which essentially employs aequations of Floreanini-Jackiw motion [ that follow from eq. ( brackets and then performingof canonical quantisation the [ resulting partitionshould very function much [ likeintroducing to extra have field an content actionLorentz this principle invariance is leading from possible to which the only eq. action at ( pioneered the by expense Tseytlin of [ sacrificing manifest in which, andand in also the a following, topologicalthen the term implement ellipsis involving the indicate Ω constraints terms supplementary to depending the on action the for spectators instance by using Dirac Chiral scalars areconstraints notoriously are tricky first objects orderclass to constraints differential describe, and equations can the and notis main in be to reason simply imposed the consider easily is terminology a with that of non-linear Lagrange these Dirac multipliers. second One approach T “spectator” coordinates trivially fibered with vanishingterms connection. of the In chiral this projections situation by, the constraints are given in are required to obeyare a left-movers chirality and constraint half meaning arephysical that right-movers central of and charge. the thereby 2 give Inthe the this chirality correct constraint note contribution of we to will the be focussed on the variety of ways in which physical spacetime’s worth of coordinates. JHEP12(2016)082 ) ). 1.9 1.8 (1.9) (1.11) (1.12) (1.10) ] and we 40 J ) X = . ∂ = + P ( . IJ ]. In the present context η , δf I 36 ) I [ = 0 ) X 3 J X = = X ∂ ∂ + + .... + P D ( P + J ( I f f ] wherein covariant gauge fixing choices J ) f | ) = = − = ∂ ∂ X 39 ∂ P , | ) invariance. This approach can be extended to ) one recovers the Tseytlin action eq. ( = ( , Z − ∂ τ + 38 ; ( J − I f ) = 0 P X d, d – 4 – ( X , = = | ∂ = IJ f ∂ η I IJ = 0 − ) , δf P H J ) X ( ] or via holomorphic factorisation of a partition func- I | X f = f ( X ∂ 27 − | I = | = − ∂ D ] wherein an axial gauge fixing gives rise to the non-covariant P J ( I 37 = Λ = σ ∂ ) 2 f f 1)) and higher genus worldsheets. I I + | , d = = X X P ∂ ∂ δ δ ( Z = (1 + 1 2 N = 1) 2) PST ) and more recently explored in [ ]. S 1.8 42 , 41 ]. However the implementation of these constraints at the level of the action has = 1 supersymmetry; one promotes partial derivatives to super covariant derivatives 43 This work arose out of an ongoing attempt to better understand the supersymmetric Whilst this overall picture is correct, the literature has been rather sketchy in places A further approach is to include extra fields so as to furnish the action with a gauge ] however at the cost of loosing manifest O( A different approach based upon gauging the the symmetries generated by the constraints was followed 5 N 3 ˇ tion [ rarely been considered; there is no knownin covariant [ formalismsuperspace in (at the least to style of eq. ( acting on superfields, Previous work in the literatureeither has via followed the Dirac route brackets of as imposing the in constraints [ by hand will recover a covariant version ofSevera the [ Poisson-Lie duality symmetric action of Klimˇc´ıkand doubled formalism. It isto quite clear in this case how to generalise the chirality constraints of this idea was suggested in [ action eq. ( were adopted (though those are notthis directly approach relevant to by the making presentbe discussion). direct able We the further to linkage clarify toThese some the ideas surprising PST will features form then concerning of be the the origin generalised action of to and the the will PST then case symmetries. of non-Abelian T-dualities [ about many of the detailsworldsheet concerning and the in derivations particular of has omitted thethe a covariant terms forms careful in treatment of ellipsis of the in spectatoroutstanding doubled the fields issues above (i.e. discussion). and exactly by In giving theformalism the following achieved we by complete will adopting derivation resolve an of many unusual the of gauge covariant these fixing bosonic in doubled a Buscher procedure. A version Upon using the second symmetry to fix The symmetries of this action are, redundancy which promotes the second classof constraint to the a first Pasti-Sorokin-Tonin class (PST) one.this approach This leads is to to the chiral a spirit doubled fields action, JHEP12(2016)082 . ] ): 5 1 db − (2.5) (2.1) (2.2) (2.3) (2.4) g = ( . The x ij H g : b y , ) by = g ( y . = ij K ∂ g , T ˜ B K ∂ y | = endowed with a metric T ∂ -model Lagrange density y and | σ = + do not depend on M ∂ B = b + B , . x β E B = and | T = X ∂ y . In an obvious matrix notation the g } = ) and introduce adapted coordinates N ∂ kβ d ∂ − B g T D | . − = y jk ˜ | ≤ = B B g AB , the non-linear ∂ b T ,D d } ij + , x b + + 1) supersymmetry on the worldsheet. It turns β , = ,D y ··· ∂x − . We denote the inverse of ∂ – 5 – , = ∂y AB ij β 1 = b g ··· − jβ -dimensional manifold = (0 , ∂y x g + = A 1 ∈ { M ∂ D ˜ ij ∇ iβ B N ij T X α g b ) as, g isometries ( | T x ∈ { , = | x = ∂ α d = = = | A 2.2 = ∂ y i i 1) system. Note that a Hamiltonian perspective was given , ∂ = ij ˜ ). We introduce “connections” B B , + A 1 E L + x X x = = . Locally we introduce the Kalb-Ramond 2-form = (0 g = H 1 E ∂ ∇ N − T g E x T | = = , such that the background fields x ∂ } 1 | = − , d = ∇ g g L ( = ··· , j i L 1 δ = ∈ { ] in which only bosonic degrees of freedom are doubled making supersymmetry less kj Here we take a step in this direction by carefully analysing the simplest supersymmetric i 44 g and a closed 3-form , i ik where, which are adapted coordinatethat representations are horizontal of and invariant (pull with respect backsWith to of) these the Killing we one-forms vectors may generating detailed rewrite the in isometry. eq. [ ( A special role is accordedg to We now assume the existencex of spectator coordinates are called Lagrange density becomes, Our starting point isg some compact Choosing local coordinates is given by, evident; here we will instead work in superspace. 2 Bosonic abelian doubled2.1 string Deriving the covariant doubled string out that even a firstextra order constraints formulation are comes similar accompaniedlevel in by of the external nature components constraints. to of the These nilpotencysimply superfields be constraints they are imposed on algebraic using superfields, and, Lagrangelike as so multipliers. a description consequence, at We of they do the can the givein both [ the PST and the Tseytlin constant generalised metric i.e.attempts assuming no to dependance generalise on thisessential spectator appear for coordinates. the to doubled Naive fail worldsheet badly to have and a addressing lifemodel, this in i.e. short superstring the one theory. coming which seems exhibits an and even a non-covariant Tseytlin style action has only been considered for the case of a JHEP12(2016)082 one (2.6) (2.7) (2.8) (2.9) A (2.11) (2.10) (2.12) | = ˜ B T y , should satisfy = M, = A x 1 − ) should be added + ˜ K ∂ = x A T NE T = y ) one indeed obtains x | ∂ = (˜ − ζ , | = ∂ ) we impose the gauge ]. This is very reminis- 2.8 ∂ = ], − K 3 + ∂ , ) is already “doubled” as − 2.8 ]. Just as the Floreanini- . = 2 = ) = 47 ˜ − | = B , ˜ | 30 2.8 B = appear. This was suggested ˜ T A = K ˜ A 46 T E x | A = y . x = | (˜ T = ∂ A ˜ = = B = ∂ , − 0 + 0 = − y 1 = x , A | − = | | = = A . and subsequently integrating out → B | ∂ → = transforming as, ˜ T B ∂ A NE fA ˜ = x = + = + − 4 ≡ = A T ∂ ] we expect the same for Tseytlin’s action. In ˜ x ∂ x = = = ∂ = – 6 – 49 − A A + ˜ ˜ , ζ , y N and = = y which are inert under the gauge transformations. = | + = 36 + = ˜ x = ζ , A B x | x fA | A x = = ∇ T | = ∂ = A ˜ x = K ∂ ∂ | ∇ = 0 − T ∂ M, and the dual coordinates ˜ | x = y 1 | E = + x − A → ∂ T ˜ x E = x | + = = | 0 = = gauge fields A = ∇ A ˜ B ˜ d M + E E → x T | | | T = = ˜ x = ] for a detailed development) where it was shown that by making the | = A ∇ , 48 −B Lagrange multipliers ˜ ∇ 1 − d = = E ] which is important to keep in mind as we treat boundary contributions in what follows. 45 = dual )[ ˜ ˜ E L gauged 2.8 ] (see also [ L Starting from the gauge invariant Lagrange density in eq. ( Let us now turn to the manifest T-dual invariant or doubled formulation of the model. Note that in order to avoid nontrivial holonomies around non-contractible loops, ˜ 37 4 = 0, integrating by parts on the Lagrange multiplier term and integrating out the gauge fixing condition, appropriate periodicity conditions.to In eq. ( addition, a surface term cent of the Floreanini-JackiwJackiw formulation formalism can of be a covariantized [ the chiral next boson we [ show howa by Lorentz making invariant doubled a worldsheet judicious formulation. gauge choice in eq. ( In fact our startingboth point, the the original gauged Lagrange coordinates in density [ eq. ( non-Lorentz covariant gauge choice recovers Tseytlin’s non-Lorentz covariant doubled formulation [ together with a shiftout in in the a dilaton path that integral. is seen when the dualisation procedure is carried with, The dual background fields are given by the Buscher rules [ Integrating over the Lagrange multipliersaway the sets gauge the fields field and strengths wex to recover the zero, original so model. wefields However can yields making gauge the the gauge dual choice model, together with The gauge invariant Lagrange density is then, For this we introduce 2 In order to obtain the T-dual model we gauge the isometries, JHEP12(2016)082 ) ) is 2.8 y , (2.17) (2.13) (2.14) (2.15) (2.20) (2.16) (2.18) (2.19) 2.8 = 5 is of the ) we are ξ ˆ K∂ T 2.12 y | = ∂ ) transformations X + R | | = = ; . B ∇ η − B d, d ˜ 1 x , T P | − = X H T = , ∂ | X = − ∇ | → O f , 1 2 = J . x B f , 1 | ∇ = = B − − = f f ∇ = g = + | , = = ˆ σ , T B f A ∂ ∂ ∂ X X ) | η = E 1 1 ∂ f → 1 2 ) where the gauge parameter = ∂ T ( − 2 ˜ = ξ X = = − = | 2.7 | = = − + X ∂ X J → O x = ∇ 1 2 = J – 7 – ∂ X = ˜ 1 + , σ Ω 0 , ˜ x , ) and ( | − = ˜ T x f A , = 0. From this we immediately identify the residual X g = | σ ! = X | 2.6 = = | f ˜ ∇ → = B B = ˆ ]. ∇ = A ∂ 1 HO ˜ | A ∂ x + = ∂

+ 53 1 2 T ˆ 2 σ x should be suitably chosen so as to have nowhere vanishing P = O = − | − = = H f → B = T X A ∇ | = = 0 X = σ E A = = ∇ H∇ = T f f )). A full and detailed discussion of the residual symmetries will be | H → H J X = = = | through its equations of motion, ∂ ∂ = , σ 1 2 ∇ A | 1 column matrix of scalar fields. Implementing this gauge fixing in eq. ( = 1 2 − σ × ( d = f ( ξ is some scalar field. In writing the gauge fixing choice as in eq. ( is a = f A ξ doubled Now the strategy is clear. We adopt this gauge choice and parameterize the gauge Note that a Lagrange density somewhat similar to this one has been obtained in the context of heterotic L 5 This action is nowacting (almost) as, manifestly invariant under global O( strings compactified on a Narain torus [ where, where, yields, after a little manipulation, the desired covariant doubled Lagrange density, where and eliminating as well. fields as, the above gauge choice simplifies to gauge symmetry. It isform given by eqs. ( given in section 2.2.invariant In under, addition to this, one verifies that the Lagrange density eq. ( emphasising that the function derivatives — we will discusscoordinate this transformation, requirement further in the discussion section. Making a where JHEP12(2016)082 : ) is Z ; P } → K (2.22) (2.21) = d, d , σ | = ] evaluates σ 5 { which implies ) and passing : ij P 2.7 b ) invariant Lagrange . → − Z T ; ij ) and ( N b 1 : d, d 2.6 . In addition, for the term − . Making use of the identity J P I f , ) N g η invariant, we require that = 1 4 0 µ (Ω f − − dx and thus both the Tseytlin and PST → µ ) non-invariant background field M = I A ) 1 Z J ; H → P ·H·P − I X : g | = P T d, d ∂ ) by demanding that the periodicities of the P , f − Z ) M ; P ), namely B-field shifts and Buscher dualities, f – 8 – 1 4 ) subgroup of the duality group preserves Ω, but with Z ; Z d, d P· J − ( , appears automatically. One might be tempted to d, X + Λ( d, d X J coordinates the coefficient of the topological term is M → I X 1 = n I ] this topological term ensures invariance under certain ∂ − ) = 5 ]. Ω 0 X contributing a sign in the path integral. B-field shifts have → P 8 T , X = N g I 4 Z X ∂ 2 1 | π = X + → ∂ : P − to this contribution and thus leave the path integral invariant. X invariant and explicitly given by, :( P Z K π is P = and again the path integral is invariant. for the gauged Lagrangian to have definite parity. In terms of the n ˆ K ]). We will see later that when generalised to non-Abelian T-duality it } 1) ˜ x 54 which − are preserved [ we see = Ω. Evidently the GL( − to have definite parity we should also insist that ˆ η K X j x, O − x Ω leaving the one-form gauge connection = ) is given by, T = ∂ } i | O = x ·P | } → { 2.14 = The action of parity is slightly non-standard. Since parity acts as The Lagrange density governing the spectator coordinates is altered as well — a fact Note that other than the initial integration by parts on the Lagrange multiplier term , σ ˜ x η ∂ = ij σ x, density. Upon gauge fixingto the the original second gauge order symmetryeq. formalism, ( eqs. the ( residual gauge symmetry extended by the symmetry P· actions have definite parity. 2.2 Gauge symmetries andIn the this origin section we of investigate PST the symmetry symmetries of the manifest O( { { doubled space we have E that the generalised metric must transform as replaced by the effect of addingFor 2 T-dualities that simplymultiplied swap by ( often ignored in the literature. Indeed the O( unless for the remaining componentsone of needs O( to exerciseto more the care. sum of Properlywinding products normalised sector of evaluates this winding to topological numbers around term canonically [ dual cycles and in a fixed ignore such a piece howeverof this the term is partition vital function forlarge and instance in gauge in [ getting transformations aemphasized that correct in factorisation are [ usedwill to no define longer the remainStrictly topological quantum speaking and theory this rather topological (as play term originally spoils the the role invariance of of a the potential action for under O( a WZ term. coordinates (and, see footnote 4, inboundary a contribution), careful we gauging have any notand boundary discarded the terms any topological from total term, this derivatives are in canceled this by manipulation a This invariance is further reduced to O( JHEP12(2016)082 ) ). = 2.7 = 2.17 (2.26) (2.24) (2.28) (2.25) (2.27) (2.23) A ), ( , 2.6 f A | = ∂ , ) = f ( | df = dε A , was given in eq. ( − f , ) . This explains the origin A = J = f , ∂ ). = = X 0 ξ EA 2.8 and = A , − + , ξ , | ∇ , = f = → ˜ S x S ˜ + x | S ) is invariant under these transfor- = δ δ J δ νµ δ δ δA P ( ∂ F | | = = 2.8 = . So one sees that this is not a new = f ν = ξ ξ ξ ξ ξ ξ ε ). In the remainder we explain how the = , we introduce the variations, f , A. − ∂ ≡ + + + } µ µ ) + ) 2.7 ∂ ∂ = f , A µ µ κ = + , gauged | ν µ µ , = S -model in eq. ( A A µ ξ ξ L δ A X | = A = µ µ σ δA | ν σ A 0 = – 9 – ξ ξ T | 2 ξ = = µ = ( ( f d ) is quite mysterious. It looks as if it is unrelated ∇ ). However the appearance of the PST symmetry ξ ξ | ) and ( E ∈ { = = µ f , R − − ∂ ∂ − → ∂ δf µ + δA 2.6 P -model in eq. ( = | , , 2.18 1.10 = = = = = , = = ∂ σ µ µ f J ξ κ = | | µ ˜ ε x = = A = S ξ | δx δ ∂ A = µ ε , ) survive provided we assign the following transformation ξ ξ ξ is the Lie derivative along δA + , f δA and = = ) − L − ξ f f ε | and a (trivial) equations of motion symmetry. 2.25 L = ( X = = = δf ∂ , ε δ µ | µ ˜ µ = x A + ξ δ δx A µ ν x δA ξ ≡ ∂ 1 column matrix of arbitrary functions of − = ε × − 0 = ν d x , ζ A f µ → ∂ x = and ) is a 2 f A µν F . The residual gauge symmetry is now, However the situation changes when making the gauge choice Given an infinitesimal vector, f A = Introducing the parameters and the symmetries inrules eq. to ( where we introduced thesymmetry: action it is a combinationwith of parameter a (field dependent) gauge transformation eqs. ( ∂ One easily verifies thatmations. the gauged This is not so surprising as these transformations can be rewritten as, where in the second order formulationto eq. the ( original gauge symmetryPST eqs. symmetry ( originates from the gauged of the first of thewhich symmetries acts of as, eq. ( where Λ( JHEP12(2016)082 ) f ). ) is f 2.18 2.23 (2.30) (2.31) (2.32) (2.33) (2.29) Γ( df . Λ = v X drops out and the = ˜ x , ∇ δ + . S P δ δA X f f ), follow as equations of | = | f = = , of the doubled action ( = ∂ ∂ ∇ since then 1.6 f − X . f∂ κ − f = P | , = = ˜ x f S X H ∂ f∂ δ δ | = = 2 | = f∂ = ∂ = 0 v | f ∇ ∂ = abelian gauge symmetries. Imposing − = ∂ − X − p d | P = dependent term in corresponds to a genuine gauge symme- p f∂ κ X ) | H = ∇ = ε = κ f b A − ∂ ( = ∇ 2 I = + + 2 ∂ + P ) the ,P , v f + H = Γ – 10 – = | = f = J f A I ∂ v = 2.16 = 0 = X = f | ∂ ∂ + = X f∂ is a trivial equations of motion symmetry. Eliminating | ) that incorporates the connection. There is also an = eliminates half of the gauge fields but introduces one which leads to the Tseytlin doubled formulation. | exactly reduce to the PST tranformations in eq. ( gauge fields and = = ∇ | f ∂ Λ = + | = κ = ∂ | τ f − d = 1.6 ∇ J f ∂ P A + p , = | | fA = = f f P − ∂ | = and ∂ f = = = Λ) ∂ ∂ | ∂ x v = ε A , ( , ˜ v ε ε 2 2 leaving one unfixed gauge symmetry which appears as the PST = − ε , d − x f = X = = = = = 0 = ˜ x fA ∇ δx δ δf | δA = + ∂ P H | = , to the chirality constraint, ∂ P 0 = ] and we adapt this to the doubled string taking into account the twisted nature of The equations of motion that follows from a variation in Concluding: we intially had 2 55 through its equations of motion, eq. ( jectors i.e. the covariant versioninhomogeneous of solution eq. ( of the form Λ allows the equations of motion to be recast as, The homogenous solution Λ = 0 corresponds exactly, after making use of the chiral pro- Introducing a one-form with components, motion in this approach. Forin a [ single chiral bosonthe a constraints. clear explanation of this was provided can be expressed as, gauge symmetry inallowing the it way to outlined be used above. to put The2.3 PST symmetry Equations acts of asWe motion can in a the now shift PST see on doubled how formalism the desired chirality constraints, eq. ( A transformations rules for the gauge choice new degree of freedom So one sees that thetry symmetry while parameterized the by one parameterized by one rewrites the transformation rules as, JHEP12(2016)082 ] X , with = (2.40) (2.37) (2.38) (2.39) (2.34) (2.35) (2.36) = ,A . | A ] = X A , [ . = L R ,A i ) | = − X A [ e | = ) L | X ∇ R = , | i = − 0 − f∂ P T ∇ e ( = − ] T ∂ fη . H | P X = 0 | , = ( = T , − f T f ) ∂ ] | f∂ = H η = X . That this does not give rise − X f∂ ∂ = ∂ , = T | | | P = = = ) X = ∂ f∂ ∂ X ,A ) = H ∇ ,A | − = | | | f = = f = − q . , ∂ ( A A = = P [ I ∇ | [ = ( ∂ = L ! L − 0 f∂ f R R f P | − | det 1 i T = i = ( f A = ∂ ) − ∂ − 2 T Λ = ) e , e − | η ] = ∂ yields an equation of motion, f = f∂ ] f − P ∂ | T f = | = = | ; A = ) f − = + ∂ Λ ∂ ∂ X b,c f ) det P f ( = dA [ | f∂ | p = = − H = = ][ – 11 – H gh = ∂ | ∂ v − = ∇ fA + L f ∂ ) | |

+ = P = f A − ]+ X − dA d ∂ ∂ P f = H ; ( = ][ = ∂ ( = X , A X H f ∇ f − A d 0 = 0 = T [ f∂ = + [ = ) | L = = P δ X Z ( ∂ R f∂ f∂ 1 i f | should not be considered dynamical but rather it is a fixed = | | = = H 1 = f A − = ∂ ∂ ∇ ∂ | T e f = ] ] ) det Z + ∂ p q = X ( dc P f come in handy to show that this equation can be recast as δ ( | = ][ ] 1 = = 2 dA ∂ = ∇ ) db ][ ] Λ = f P f + | | ][ | δ = = = dA = P ∂ ( ][ ∂ ( dA dA dA | f = ][ ][ ][ 1 = X X X ∂ dA = d d d ][ [ [ [ ∂ | X = ) we have, Z Z Z − d ∂ [ f ( = = . Hence, Z T ] = f f 0 = | f Performing the variation with respect to = = [ = = ∂ ∂ | Z = Z X At this stage thebackground function object that definestegral a and gauge since fixing. thisA is The just delta an function algebraic restricts equation the one path can in- solve it by replacing and insert the gauge fixing condition and Jacobian, In our above derivations we introduced a gauge fixing condition, Let us consider how this should be done in a path integral. We begin with the ill-defined, and hence follows as ato consequence extra of dynamical the equations field equation is for a manifestation2.4 of the PST gauge Gauge symmetry. fixing and the dilaton Here the projectors which is of the correct form to be gauged away with Γ δ trivially closed. However this is a pure gauge piece; under the residual gauge symmetry JHEP12(2016)082 1 ). − g 2.45 gauge (2.44) (2.42) (2.43) (2.45) (2.41) → ξ g R , and the ghost Lagrangian is X , giving rise to a determinant f ] does not depend on the gauge = | A = f c . [ ∇ | = , Z should be invariant. For + ] A∂ f P φ [ g . = 2 g . , , f f b ∂ Z mode, − | ] ε = = = e | ∂ ∂ A df b c f f g [ | − = = = = ln det + ln det , in the covariant fixing we have a Gaussian Z ∂ ∂ ∂ ∂ c p 1 1 2 1 4 single X = − | − − = ) − – 12 – − g PST 1 components of a gauge field in the Gaussian term b c ∇ f f φ φ | | | | f b ∂ = = = = − | vol ∂ ∂ = ∂ ∂ = P ∂ two 0 f = Φ = φ | ε ε ε = = Z ∂ 1 2 1 2 ε , gh . The determinant of the metric enters with half the power = = = = L 2 d X − δb δc δf ) δ f = f∂ | = and we integrate over a ∂ ( f × = 1 2 f∂ − | ) = in much the same way as one averages over gauge choices to obtain g , it can be fixed without the need for further ghost terms. f giving essentially a factor of det( f = AgA∂ We can see that in the above derivation it is this doubled dilaton that emerges auto- To progress to the doubled formalism we now need to integrate out the gauge fields gA in this path integral. As is well known, under T-duality, the dilaton receives a shift | = term factor det( and thus will give riseNote to that even a if Fradkin we Tseytlin beginthe coupling with doubled of a dilaton non-flat to geometry will the in not doubled which be. the dilaton normal eq. dilaton ( is constant matically in the covariant doubled formalismBuscher for procedure elementary on reasons; integrates whereas out inA a tradition this means that T-dual dilaton is given by On the other hand a T-duality invariant “doubled dilaton” is given by A which in the Buscher procedureGaussian can integral be over attributed the tothat gauge the determinant the fields. that string comes A frame from useful the supergravity mnemonic measure to obtain the correct shift is in which we divide by theshift volume on of the PST group. Since the PST symmetry acts a simple can now be re-interpreted asfixing saying nothing choice. more than Wefunction can then simplyin choose QED. That to is integrate we over can choices consider, of the gauge fixing given by The PST symmetry, which extends to the ghost sector as, in which we made the final change of variables JHEP12(2016)082 X = H == y ∂ (2.49) (2.50) (2.48) (2.46) (2.47) ∂ A priori and X ], two of the | = . | = 49 I ∂ ) X | = ∂ ) results in a variation ) are physical. − P ( 1.6 , , A.13 | = ...... , == 6= 0. That these derivatives | = == h , + + h h | =0 − = J J = X X ε ε -model and performed an unusual sigma-model from the outset and I X X | = h = = = σ (not to be confused with the usual X h = ∂ = ∇ ∂ ∂ | = ∂ ∇ + I ∂ X I − − P == X h = | X = h = | = | | = h = = ∇ == ε doubled I ∂ δ h h ) ∇ X | + = = y and h = ( ∂ ∂ IJ X | | = = = ∇ | IJ H = | ε ε – 13 – = ∂ H σ h + + 2 , − σ d | I 2 = P = ) | ε d ε = | Z X = = ε = ∂ ∂ Z 1 2 ∂ = -model on the doubled space, 2 1 correspond to independent symmetries however as we shall = = + = σ | X | = P = = δ ( ε | = | == = | = δh δh Hull gauged h and S S − = ε I X | = ∂ = I X | h = ∇ We start with a Hull style Performing a gauge variation,and integrating making by use parts of all the terms identities obeyed containing by the projectors eq. ( In fact, though theirthese derivates structure are is not informedare at not by all actually the covariant covariant makes usual assurprising. the e.g. conformal We fact continue covariant that regardless derivative, the of following this construction and works consider even the more “gauged” action, and “covariant” derivatives, soon see gauge invariance willaction force that this them putative to symmetry be does0); related. not this correspond It is to is a one curious rigidWe of invariance that proceed (unless the in by features the introducing that ungauged worldsheet gauge makes metric the fields components) following with gauging the procedure usual rather conformal atypical. transformation rules, such that only the field configurationsthe obeying gauge the parameters constraint eq. ( in which the ellipseswant indicate to spectator furnish the terms action that with will a play gauge no invariance, role in what follows. We present authors emphasizedcan that be PST obtained by style gaugingBeltrami actions a parametrisation for chiral for the (super)-conformal corresponding (supersymmetric) symmetry gaugecurrent field. and chiral case by This bosons although then approach also in specifying works a in a spectators the rather for surprising simplicity). way which we will now illustrate (suppressing In the derivation above we startedgauge with the fixing usual in string asigma Buscher model procedure whose to equationsadopt obtain a of the different motion manifestly tactic implyinvoke namely Lorentz to chirality the covariant begin doubled conditions. constraints with via a One a might gauging wish procedure. to In a previous paper, [ 2.5 A comment on chiral gauging JHEP12(2016)082 ). ) is . 1.9 (3.1) (3.2) (3.3) (3.4) ab (2.51) (2.52) (2.53) δ are the 2.50 = 0, one = a of isome- ]. − b L 6 52 J J specified by P T – a X X H ) | T G 50 = | T = ∂ ) ∂ I Tr + ) − P X P , = = c ∂ ∂ . T + + c | P = P ab | ε ( = I if ∂ . = X − ] = P = ] = . ( ∂ == b h IJ IJ f f ,A ,T | η η = | = = = a | = ∂ ∂ T | A [ = == [ , Ag . h h = , ε | b = − = = -dimensional group space − , = | ε ε L = d c ε 2 == ab A L ∂g = − = + 2 E ∧ | ∂ = | a = J = J b , h ) | = L – 14 – X − L X h f f = which minimally couples through the introduction Dg | bc | ======∂ a ∂ ∂ ∂ I A G f L → ) | + = 1 2 ) the Lorentz covariant action PST action of eq. ( -model on a X P = ∂ , ε | ( σ = ∂g | I = = ∂ = | = X 2.50 a invariance that we can gauge by introducing a connection − = 1 | h = = P | L ∂ = ( F == G IJ IJ h | H H = | | = = = dg , dL ∂ ∂ h 1 | = = in the algebra of − ε ε g a b T − − T a = Tr iA is a constant (or possibly spectator dependant) matrix and the ab = ab iδ E A gauged − L -model has a global with conventions, δ = σ 1 2 a G In this work we restrict our attention to the cases in which the structure constants of the group dualised L 6 ∈ are traceless; this iscurved background to which avoid upon thedoes dualisation occurrence not can obey of the give a (super)gravity rise mixed equations. to gravitational-gauge For a anomaly discussion Weyl of when anomaly this coupled and i.e. to related a a issues dual see background [ that one-form of covariant derivatives, The connection has a field strength, g This inclusion after. Let usthe consider Lagrange a density, in which pull back to the worldsheet of the left invariant Maurer-Cartan forms for a group element 3 Application to non-abelianLet T-duality us now considertries, the and generalisation for of clarity these we ideas ignore to spectator a fields non-Abelian first group and then give the result with their and noting that the quadratic termimmediately in recovers gauge fields from vanishes eq. by virtue ( of ( must enforce, It is easy to seeidentifications that and these the are definitions consistent of withgauge the the invariant. gauge projectors Solving transformations we the rules. find first With that of these indeed these action relations eq. with ( a Beltrami parametrisation To cancel this we see that the gauge variation parameters are not independent and one of the Lagrange density, JHEP12(2016)082 ] = (3.9) (3.5) (3.6) (3.7) (3.8) ,A | (3.14) (3.12) (3.13) (3.10) (3.11) = A . = = L ) A . Then the action T + 1 P = H ( T , DED T = | ) . = L 1 to enforce a flat connection a A f f − D.D | ˜ , L g = = , = b + b which entered the action as a ∂ ∂ | c ∧ ∂h . = = a 1 2 1 f , ˜ gT c L ∂v v . ˜ ) a L − A = vF L ∧ = − g , T h ∂ T ( ∧ ∂ b a ∧ | = = 1 − a ba b L A L − c L ED L ) ) D | a = ab FA T = − = Tr( Ah F which obeys | f = bc = 1 L P a = = 2 f vh . ab + − a A ab 1 H − 1 2 h ˜ ( B L − ) = E D + = | b − ( L h T = → v T L – 15 – ,D , | dv ∧ = v f , A → f f | | ∂ ) = ba ab = = A | ! = = a v DEL T = ∂ ∂ L D a a D ∂ ∂ ˜ | a L ( T = a A ˜ L L = 1 2 d A g , A ∧ A AB A − 1

a − = = − − v − L | dual h a = = a = = ( a = ˜ L L ∂ d L A L A . Since the non-Abelian term in the field strength [ | d → T = Ω L = A EL g | A → T = | local transformations, T = . Obtaining the non-Abelian T-dual is then achieved by gauge a L H + L c L 1 2 L v G c = − ab = if L − gauged H | L = T = L ab 1 2 F to the identity and integrating out the gauge fields to yield, = g L Notice that the pull back of Ω If we define, Now we invoke the covariant gauge fixing choice, this implies a three-form flux purely topological term inKalb-Ramond the potential. Abelian case Since, is no-longer topological, instead it serves as a then one finds a doubled action, and integrate out thevanishes field in this gaugedescribed the earlier. manipulations are actually quite similar to the Abelian case in which fixing After integrationLagrange by density, parts of the Lagrange multiplier term one finds a gauged In addition we introduce awhich Lagrange is multiplier gauge term invariant provided Tr the Lagrange multipliers transforms in the adjoint, in which we have definedis the invariant under adjoint action the We see then that the gauging replaces the Maurer-Cartan forms with, JHEP12(2016)082 . ]. ˜ G 42 , (3.18) (3.15) (3.16) (3.17) (3.21) (3.19) (3.20) and 41 G ∇ = L , ) , , b + , , and repeating c b ! c P y a = T ˜ T H ˇ ca L b ( T N b y , a 1 T ac | = = N − = 1 ∇ = ˇ if L − L ab ab − g Eg K ∂ ] . f f Π 1 c | − T = = b , − T y a ∂ ∂ a G

| δ , = decompose as, 1 2 ˜ Π) bc ∂ b ˜ ]. = = f C ˜ T − i + + i T a B 62 b deﬁned as in the Abelian case in b | E C ∇ = = − ˜ ) , then the generators of the double T ] = 1 | [( L ˆ ˜ b G K ) a − σ AB ˜ T T 2 a − NL h , ,P d iF P a T , the group of + ( y T H ! ν Z [ , | ( b = y ] = G T ∂ T = = B M those of , | 1 ∂ ∈ ab ∇ = T ˜ = 0 M + S c b µ − , 1 a – 16 – L g i y y g ab ˜ b A − T | f f = ˜ = = T ˜ g f , T | T i = = ∂ | g E b = ∂ ∂ a . The statement of Poisson-Lie T-duality then is the a i.e., µν ˜ ˜ L ˜ 1 2 and T

M ∂ G T i ˆ ] = a h 1 K | ) can be understood in this context. To do so we remind b | T = B = G ∈ = − − ˜ T = L + T L g | , = B 1 i -models, a 3.12 = A b + L + − ab ˜ σ T T L , g is a Lie algebra that can be decomposed as the sum of two T [ | Ω = h η b a | | T = T = that are maximally isotropic with respect to an inner product T D T = ˜ b Π, for ˜ ,P , h L + Π) B EL a ˜ G c 1 2 1 2 1 a b, B ˜ | T T = − AB may have arbitrary dependence on the coordinates − a, c − + η L = ab E ( G ⊕ g ∇ = = L if = a σ L B 2 T = = L η d 1 H obey | ] = T = − ∇ b D T } L Z g L | a ∇ = ,T 1 2 are the generators of ˜ a T = L , a E,M,N,K T + 1 2 [ ). a S T T = { 2.21 The Drinfeld double It is quite straightforward to extend these considerations to include a fibration and L . If = A and tilde analogues, ˜ equivalence between the two and the Jacobi identityWe places also need further to constraints define on some the matrices admissible for choices of T The structure constants of the double [ The result we obtained inthe eq. ( reader of a little technology — the Drinfeldsub double algebras [ h·|·i eq. ( 3.1 Relation toThere Poisson Lie is doubled an formalism existingfact formulation also for accommodates a a non-Abelian further T-duality generalisation double known formalism, as which Poisson in Lie T-duality [ in which we defined, and the modified Lagrangian on the base involves spectator coordinates. Starting with the Lagrangian, in which the above procedure yields the doubled action, JHEP12(2016)082 ) ) of S d, d 3.21 (3.23) (3.24) (3.25) (3.22) (3.26) 8 , -models . ], i σ i l , l σ 7 | 57 a = ). The case ∂ T ∂ 1 a 1 − 3.9 c − l l iL ˜ | L is just the O( − + |H| ∧ l -models correspond P b i b σ ] by constructing an σ ˜ . B T ∂ L ˜ |H a S 1 ) given by [ T l ) is no more than the 42 ∧ respectively (a háˇcekis | will give the action − g , i = deformations. dv l ] ( a ) and eq. ( ∂ h a b g 1 |H| ˜ 41 1 G ) λ 3.22 L dl Σ − A c g 1 − Z 3.1 a ( l T − ab h 1 h 1 2 A f − and f and f T ] and references within. | = = dl, l ia η = A i− 1 63 G ] ∂ = 3 ∂ L − = i dl l AB Σ C [ 1 | Z L H = − dg dl 1 2 1 a ∧ 1 -dimensional non-Abelian Lie group and − T − d + dl, l B g a l 1 h i L gives the dual action l 3 − + iL l ). All that remains is to understand the WZ ∧ = [ ˜ M and integrating out ˜ g g | + ∂ a g Z ). However since A 1 – 17 – dl a ˜ ). One can now see that all the terms involving a G T L − 1 3.23 ˜ 1 = ˜ 1 l T 12 T | − ∈ l a − a l + CD h ˜ g 3.11 g L + iv η 3 a i P i C l M = |H Z = idv l ) are dual was established in [ and is a suitable three-manifold whose boundary is the world- ∂ AB = dl 1 1 = ˜ F 3 12 1 G ∂ − = exp( introduced above). 1 l − 3.21 M + = l ∈ dg − i ˜ l g L 1 i l g h ] |H| τ − l is the algebra of some f f g ∂ dl | = | 1 1 = = are Abelian corresponds to a dualisation of non-isometric G ∂ + − ∂ ∂ we have an Abelian double and the dual pairs of − from 1 l with ˜ | d G ), we see that, − Σ l ˇ l ˜ L gg g σ Z h dl, l ( gg d ∂ (1) 1 Σ 1 2 1 T then, 1 nor refer to the left-invariant one-forms of − Z = ˜ U − l − D ˜ [ g ˜ l l − 1 2 G | ˇ gg L 1 h = − Σ = dl ˜ = ˜ 1 g Z G is group element of the Drinfeld double, ) directly match those in eq. ( , the double is said to be semi-Abelian and the two dual models in eq. ( l ] and is essentially a doubled action in a Tseytlin style non-covariant gauge. − 1 2 l d and l = = ) and doing the converse with h PST 56 − L = (1) 3.12 dl G 1 u 3.21 Let us now restrict our attention to the semi-Abelian double appropriate for non- There also exists a PST version of the doubled action eq. ( That the actions in eq. ( If − For a brief summaryTo of the this best direction of the our reader knowledge may this consult has [ not appeared in the literature and we are grateful to K. Sfetsos PLT l in ( 7 8 PLT = S S ˜ term for which we observe, for sharing his notes in which it was derived. adjoint action, coinciding with the definitionH in eq. ( Abelian T-duality. The firstdouble thing as to note is that if we express the group element on the in which we parametrised ˜ schein [ Parametrising eq. ( in which coset generalised metric and sheet Σ. This action can be thought of as deforming the chiral WZW model of Sonnen- where neither and has recently foundclasses of new integrable applications models in in two the dimensions context known as ofaction on the the relation Drinfeld between double given certain by, used to distinguish to Abelian T-duals. If G reduce exactly to non-Abelian T-dual related actions of eq. ( where JHEP12(2016)082 , , ] ] | = −− 64 26 67 A F – – (4.1) (4.2) (4.3) (4.4) ) de- y 23 64 ( and ] use the E − = 67 , = F . In order to , 1) worldsheet S 66 , , E | − L = ) F y ( , S = = (0 and using Lagrange | = L F − N + A 9 ˜ x . − − A | D = and | J + = i A x , − ) f A , + 2 1) gauge multiplet which consists of y , − ( − S D ED J this reduces to the current case. | L = = (0 = = x + N − − i A -model is given by, x J σ − + 2 whose dynamics is governed by − – 18 – y x E D ˜ x , | EA = | = | ), thus confirming what started off as a topological = 4 electrically gauged supergravities. The works [ = f A , A ∂ i ∂ | = N i A − ∂ 3.14 = 2 x we introduce gauge fields = | = + 2 ε L | ∂ = x T + A − E x = → | = x E D x | J = gives the original model back. Motivated by the non-supersymmetric i ∂ x = 2 we impose flatness. The gauged L x . Introducing Lagrange multipliers which constrain all fieldstrengths is a set of adapted coordinates such that the background field − 1) superspace (conventions can be found at the beginning of appendix A). , A x To close this section let us finally note that actions of this style have been used in [ Note that we could as well have introduced the full and = (0 9 ] have the chirality constraint as supplementary to the action and those of [ = case we impose the gauge choice, A to zero, one finds that upon making a field redefinition on ˜ where, Integrating over ˜ where pends only on the spectatorgauge coordinates the isometries multipliers ˜ For simplicity we restrictstraightforwardly ourselves be to generalized a tothe trivial a Lagrange bundle non-trivial density, structure. bundle structure. All The results starting can point rather is supersymmetry. While extremely simple itin models already with exhibits more all supersymmetry. subtletiesN We which will also keep occur supersymmetry manifest by working in 4.1 The covariant formulation 4 Towards the supersymmetricA doubled supersymmetric string first order manifesting. T-dual invariant worldsheet Even formulation a is still non-covarianthere lack- Tseytlin a type first description step has by not constructing been the given simplest yet. model We which provide has an 65 Tseytlin style formulation. Itthe will spacetime be violation of of interest to sectionand make the condition generalised more leading notions precise to of the gaugeddependence Poisson-Lie linkage supergravities on duality between whose as both worldsheet in coordinates generalised [ and metric their has duals. term in the Abelianin doubled the theory non-Abelian is doubled precisely theory. what is needed asto a describe potential strings for whose doubled theto target WZ give space a is world a sheet twisted description torus of and have been conjectured which is in agreement with eq. ( JHEP12(2016)082 ]; and 59 (4.5) (4.6) (4.7) (4.8) (4.9) , ), one (4.10) (4.11) (4.12) (4.13) X − | 58 = ψ ∂ . − ]). Using 4.10 ) one gets, − . However ) iθ y 60 A ( 4.5 η P + S X L x ) and ( + | = = | ∂ 4.9 = | , = x J f f J , 1 | − = f . f x − ∂ D ··· − g 1) scalar superfields. The | one obtains the constraint, → = D , − , f J | f and ˜ = − + − − J f f x 1 D − D | − D = (0 = − J , ∂ + D directly be solved for g i , f f A N ) i . | − 2 f f = − , f | J − d ∂ − = ( ) − − ε ∂ J D , y X . f ( − ) − D − ∂ 1 2 ) that the constraint can be solved for half cannot S 1 i A J f D ( , D − L 1 − = 0 − ε f g f + − f + 2 ). Repeatedly using eqs. ( + 4.10 A − − − g − P + ˜ f = | = i J J − ∂ D = D ˜ – 19 – x 4.9 1 is a set of − , → J J ∂ f i − ) i are given by, − D − ) one solves for g → in the first order Lagrange density eq. ( y A − + ( − D fA g A D | 1 2 ˜ A = 2 x = S A 4.8 1 − x J − L − η P − − 1 g J ,A remain undetermined but they will not play any role in − X = + fD . As such this constraint can simply be imposed using La- ) − | g = ˜ f f X ψ J ) is an algebraic constraint on the components of the su- | ( = f − − i ∂ | ε f A x E D ∂ 1 f = f | 4.9 D | D − = ∂ = = f + and f ∂ ∂ + 2 + D | − 1 2 = i ∂ x ψ x + ∂ D − η P − → = 2 − = + x Ψ − L A on this one gets, D f x E D f | − = = − , one readily verifies using eq. ( ∂ f f D − D i ∂ | = = i ˜ ψ ∂ ∂ − is an arbitrary function and = 2 +2 iθ f L = 2 + x L = ˜ ˜ rewrites this as, together with the constraint given in eq. ( this in the equations of motion eq. ( where the terms following what follows. Using this to eliminate perfields. Indeed writingx the superspaceof components the of component fields grange multipliers. This is verya reminiscent of systematic the treatment nilpotent superfield and constraints review ([ of nilpotent superfields can be found in [ Acting with Despite appearances, eq. ( which, because of theone fermionic notes nature that of by multiplying the equations of motion by In addition the action is invariant under, as well. The equations of motion for Lagrange density becomes so, The residual gauge invariance is given by, where JHEP12(2016)082 - ) σ , 4.14 (4.18) (4.14) (4.15) (4.16) (4.17) X | = effectively ∂ ˆ + − D. P , which trans- − + f θ | 1 = ∂ = . X defined in eq. ( − f − µ − D D − SD + ) invariant. For pedagog- − D fP + 4.13 − , , θ ) in the absence of the Lagrange D Λ η P σ | f = was randomly chosen, we expect ∂ , iε X = 4.13 i 2 2 ˆ f ∂ ] we get that the Lagrange density D S ∂ − = 0 = − 49 + ) 2 D ) , X X f ). Note that because of the presence of )[ ˆ i ) X 2 D − σ τ y − ∂ ( ( D 4.9 − f S D H , enforces the constraint, + + Λ( + + L X , = X P f X Ψ – 20 – ( + σ f P ˆ D f ∂ − − i 2 X → → → − D − D θ f + + X + D i components of the Lagrange multiplier Ψ 2 Even after we gauge fixed the gauged non-linear ≡ Ψ P X + d τ = + f iε µ = − η P 2 η ∂ ∂ ∂ + X only + = 1 column matrix of arbitrary functions of Ψ ˆ D + | − X = i × 2 | θ P = As the gauge fixing function ∂ d ∂ − + − − ) in the same way as θ = P Z i 2 ; f L | ε + = ) is a 2 ∂ ε d, d − f D = = ≡ X δf δ ˆ ) becomes, D constraint itself. Asgrange a multiplier that result, renders oneical the is purpose whole then Lagrangian we ( guaranteed illustratethe this a appendix. in transformation the of simplest the case La- of constant background fields in one produces only terms that areor proportional to derivatives the thereof. constraint Moreover, this property is shared by the variation of the that it can beTo shifted see this in first an consider the arbitrary actionmultiplier way defined term. which by eq. After is ( some the significant effort origin one of determines the that under PST the symmetry. variation, where Λ( model there is a residual gauge invariance left: 4.13 The covariant action has two classes of symmetries, where, We now pass to aa Lorentz Tseytlin non-covariant gauge like for formulation.eq. the ( PST Choosing symmetry in order to recover 4.2 The Tseytlin formulation The PST symmetry appear in the lagrangian. The residual gauge symmetry forms under O( which is equivalent tothe the projection constraint operator in eq. ( where the topological term has been dropped. The Lagrange multiplier Ψ JHEP12(2016)082 (4.22) (4.23) (4.24) (4.25) (4.26) (4.27) (4.19) (4.20) (4.21) , ), which ) . τ ( 4.15 F = 0 = + . Ψ + + Ψ P + = 0 − P . However multiplying θ − + + θ + Ψ Ψ + X + + ˆ P P X X D − − − ˆ + , D θ θ ) + . X τ + . SP SD − − − X Λ( X SP D ˆ τ ˆ − D D D = 0 + iD + + + . P . iD S ∂ − P ) one obtains, X iP − − − SP − X ˆ = 0 + = 0 ) D D D , − − X D 4.23 τ + i 4 + X X X − + P | iD P ˆ − = P – 21 – D SD = 0 − − − ∂ − D θ − θ + Λ( read, SD + − X − + ˆ − D DS = ˆ X Ψ P D P = X D i + D + + . Using the residual gauge invariance, eq. ( X ) one also gets, X + P → → + τ iP P ) implies, ˆ − D − and − iP X + + + X θ D θ σ 4.19 P + ) gives, Ψ + + 4.19 X = ) allows one to solve for for P ) as expected. ˆ ) and using eq. ( X S ∂ D X ˆ 4.25 D − − 4.22 4.27 ˆ DS 4.22 X D i ˆ τ gives, DS + ∂ i + P − − θ + ˆ X D θ ) and ( σ on eq. ( X on eq. ( σ + S ∂ − 4.24 P P S ∂ − − X ) an arbitrary function of τ X τ ∂ ( τ ∂ F The first equation in eq. ( which combined with eq. ( So the equations ofstraints motion eqs. of ( the model in the Tseytlin gauge indeed reproduce the con- From the first equation in eq. ( Acting with this equation with Acting with this function can be put to zero leaving us with, assumes now the form, The second of these equations immediately implies, with The equations of motion for Ψ JHEP12(2016)082 (4.31) (4.32) (4.33) (4.28) (4.29) (4.30) . derivative projection − ) to pass to σ Ξ − ∂ + P 4.30 ηP to find that the + σ Ψ ∂ . i i 2 − + Ξ = = 0 . − S + Ξ − =0 ) are exactly the component S Ψ − = 0 D θ X − + | + = X P ˆ D 4.32 | = SD − SP . + S∂ ∂ − − D − − − D Ξ D ηD = 0 i i 2 2 , S − ), and ( . X ,P − Ξ | + + = 1 2 ∂ X X X 4.30 = 0 SD | − σ σ = + − ) = 0 P − ), ( S∂ S∂ S∂ Ξ − and the equation of motion eq. ( – 22 – iD = X − − − + Ξ σ + + P X D D D S 4.29 . We use that − i i i 2 2 2 − X X − S ∂ SP σ σ D Ξ D − − + + + ∂ + − D S∂ − gives an equation of motion that is a total P − − − H . Note the presence of a four-fermi interaction term that i D − i θ Ξ Ξ Ξ X = X X | | | =0 + σ = = = = θ + iD S ∂ | ∂ ∂ ∂ ∂ − − 1 4 ( X X + Ξ + D | − − − − − = ≡ = P − P P P ∂ Ξ X P | X = τ X = = η ∂ 2 0 = η∂ − and X Ξ σ i ∂ =0 − 1 4 θ | S = − D projection of this equation fixes the Lagrange multiplier however the =0 θ + ≡ L| P − S − D 5 Discussion and openIn problems this paperT-duality we symmetric have worldsheet theory clarified and many shownthrough how a missing novel such gauge a details fixing formulation choice. in can This be the procedure obtained allowed construct us to of make the generalisation the to manifestly in which we used that the final line. Together thecontent of equations the ( superspace equations, The provides a fermion equation of motion, The variation with respect to the fermion is more intricate and yields, follows. As above, the variation with respect to the Lagrange multiplier enforces, The variation with respectwhich, to using the residual gauge redundancy, can be integrated to yield, Here, and inD the following component expressions,would have we been hard adopt to the guess from implicit the bosonic notation case; that this term will prove essential in what For convenience we nowsuperfield give expansion the resultslagrangian can expanded be into expressed as, components as defined by the 4.3 Component form JHEP12(2016)082 ], 1) 2) , , 70 . = (0 = (2 N N DFT 1) and , ], it is very hard to 33 – = (1 -functions in a perhaps β 31 N ] to curved backgrounds. = 0. Put another way [ 68 ]. It will also be of interest to du 61 . Whilst the non-covariant Tseytlin DFT 2) remains less obvious but should be an , has nowhere vanishing first derivatives so as to – 23 – = (2 f N was via a gauge fixing, the interpretation here is that the 1) case is already under study and will directly follow from , f looks to require the introduction of a globally defined exact | = (1 = N fA = ∂ to a compact Riemann surfaces of non-vanishing Euler character. The = 2 ] it is sufficient to work with a closed form R . In fact this is too strong, as is known from previous studies of the PST = 69 df fA | = = ∂ u This formulation may have great utility; by calculating the Our discussion has been local in nature and there are sensitive issues, even in the The natural next direction here is to extend this work to both the project G020714N andsel a through postdoctoral the fellowship, StrategicJ.P. and Research Ang, by Program the David “High-Energy Vrije Berman,discussions, Universiteit Physics”. Chris and Brus- to Blair We K. are and Sfetsosmulation grateful for Martin of discussions to Roˇcekand Poisson-Lie and for duality. sharing We with numerous would us illuminating like his to work on thank the the PST Simons for- Center for Geometry a valuable starting point for the calculation of duality covariantAcknowledgments corrections to This work is supportedInteruniversity in Attraction part Pole by P7/37, the Belgian and Federal in Science part Policy by Office the through the “FWO-Vlaanderen” through which relate to thestyle target action allows space for formulation suchextend of progress this to to higher be loops madeof — at Feynman the 1-loop diagrams non-Lorentz invariant order taxing structure [ at makesof the best. this regularisation trouble. Using Optimistically the we covariant hope formulation that may the alleviate techniques some in this paper could prove to be gauge fixing choice adopted can notpossible be resolution globally is extended and to isthis find only will a locally be suitable well defined. an global interesting A fixing topic or for to further work investigation naive patchwise. manner Understanding one could hope to find background field equations for the generalised metric still requires in theallow manipulations such that terms to appearnecessitates in a the the denominator existence ofextend of fractions from in a a nowhere PSTappearance vanishing approach. vector of Since field, the this it function is not obvious how to choice form formalism [ the residual gauge invariancecome is from sufficient integrating to the eliminate equation cohomological of contributions motion that to produce the constraint. However one exciting arena to makethis a direction direct have link recentlyconsider been to spacetime reported Hitchin’s supersymmetry by generalised generalising one geometry. the of result Initial us of results [ [ in bosonic theory, that will havein to full be addressed Polyakov if sum the over derivation genus used is at to the be quantum implemented level. At first sight our gauge fixing supersymmetry. The essential reasonchiral for bosons the whose complexity chirality comes is from mis-aligned having with in that the ofsupersymmetry. theory the The supersymmetry. the techniques outlined within. The the supersymmetric case in the most minimal, but still non-trivial, extension to JHEP12(2016)082 } , d O ∈ (A.9) (A.4) (A.5) (A.6) (A.7) (A.8) (A.1) (A.2) (A.3) ··· , 1 ∈ { i , i , x ! 1 − 1 − b g − b , 1 ) b g − . We write the adapted coordi- C } 1 σ . d b g ) multiplet, − + and their T-duals ˜ g Z − i − − ; x . τ g EA D , ) as, X . ( . d, d

= Z , 1 1 ; ! = ··· η , − − ! = ∂ = , ) ! 0 1 1 O i = 2 ˜ d, d D x x ! satisﬁes, ) transform non-linearly, 0 1 AB CD − = 1 y O

+ − ∈ { O( ( 0 – 24 –

η b 1 0

= ij D X µ = T b σ, σ = 2 − , = EB X O O ∈ µ → η D + ! y O ) + 1 ) plays a central role. In the present context X = ( τ y 0 , − 0 ( Z g ; = ij E H | g = 0 g σ d, d → ) = E y ! ( b ij − matrix with integer entries satisfying, E ) as, 1 0 1 d Z 2 ;

× = d, d d . The fermionic derivative H generalised metric O( − 1) superspace this is extended by adding one one-component real fermionic θ , ) is a 2 O ∈ Z = (0 ; In the current paper we use adapted coordinates The T-duality group O( N d, d however, the the background fields which transforms under the action of Writing together with spectator coordinates nates together with the dual ones into a single O( where, O( In coordinate Geometry and T-dualities” while part of this work wasA finalized. Conventions Throughout the paper we use worldsheet lightcone coordinates, and Physics for providing a stimulating environment during the conference “Generalized JHEP12(2016)082 + P (B.7) (B.2) (B.3) (B.4) (B.5) (B.6) (B.1) (A.10) (A.11) (A.12) (A.13) , f . = − ηµ 0) case. Under these σD + , | = − . , Ψ , i ∓ X . − = (0 − ν σ X SP ) + = | | D = N = y ρ ∂ + ∂ ∂ η∂ , | P + − = + ) = | P = σ ρ = . σ ( f Λ S , . , = | 1 η y . + = = ) ∂ S − ∂ ∂ µ X σ − f f | f X ( − S = − HO | | ε = = = = = , εν T D ∂ ∂ ∂ ∂ ( ην f∂ = | f H O = i − 1 − ,P − − = 2 η X = D , ρ ∂ 1) case + iD σ H 1 2 0 – 25 – | , = + Λ = η X f f = = | and in which the constraint, and its derivative are = S Ψ = = = = ) + = ∂ − ∂ ∂ − ε , δ + ρ = + ηµ = (0 δµ D P + = + P | H → H H = iε , ν f σ T N 1 ( = Ψ = 2Ψ δf iε − X P ∂ iδ f − = = Xη = = | = | D = = = ∂ − | + = Λ σ + L | P Λ = δ fD fP | H = − i∂ . Using this we introduce the orthogonal projection operators D 1 − = = = = + µ L 2 S we construct an almost product structure , H − P in which we defined transformation one finds, and the variation of the Lagrangian reads, The PST transformations in the case of constant backgrounds reduce to, which exactly replicate those already seen in the bosonic where we have defined Ψ given by, We assume constant background fields and for notation convenience define, In terms of these quantities we can recast the Lagrangian as, Some often used identities include, B PST symmetry in the such that and From transforms linearly, JHEP12(2016)082 ]. B 566 111 (B.8) ] SPIRE ]. Nucl. ] Class. IN , , ][ D 47 SPIRE Phys. Lett. (2011) 074 IN , Phys. Rept. [ (1993) 2826 , 06 (2005) 065 ]. Phys. Rev. Lett. ]. Phys. Rev. , 10 , D 48 arXiv:0904.4664 hep-th/0104081 . [ JHEP [ , (1990) 610 SPIRE hep-th/0605149 SPIRE [ IN | = JHEP IN | ][ = ]. , ][ Λ supergravity B 335 = Phys. Rev. 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