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Gauged (2,2) Sigma Models and Generalized Kahler Geometry


hep-th/0610116

MCTP-06-21 UMDEPP 06-050 NSF-KITP-06-70

Gauged (2, 2) Sigma Models

arXiv:hep-th/0610116v2 24 Aug 2007

and Generalized K¨ ahler Geometry

Willie Merrell1, Leopoldo A. Pando Zayas2 and Diana Vaman2
1

Department of Physics University of Maryland

College Park, MD 20472
2

Michigan Center for Theoretical Physics Ann Arbor, MI 48109

Randall Laboratory of Physics, The University of Michigan

Abstract We gauge the (2, 2) supersymmetric non-linear sigma model whose target tures. The bihermitian geometry is realized by a sigma model which is written perspective of the gauged sigma model action and from the integrability condition for a Hamiltonian vector ?eld. We show that for a concrete example, the uct structure, the moment map can be used together with the corresponding generalized almost complex structure. Lastly, we discuss T-duality at the level example. Killing vector to form an element of T ⊕ T ? which lies in the eigenbundle of the SU (2) × U (1) WZNW model, as well as for the sigma models with almost prodspace has bihermitian structure (g, B, J± ) with noncommuting complex struc-

in terms of (2, 2) semi-chiral super?elds. We discuss the moment map, from the

of a (2, 2) sigma model involving semi-chiral super?elds and present an explicit

1

Introduction

Super?eld representations and geometry: The connection between geometry and supersymmetry has been an evolving and recurring theme since the early days of supersymmetry. The earliest example was the connection between K¨ ahler geometry and supersymmetric non linear sigma models introduced in a seminal paper by Zumino [1] who considered the conditions for existence of N = (2, 2) on two-dimensional nonlinear ? sigma models. This notion was further developed by Alvarez-Gaum? e and Freedman who showed that further extensions of supersymmetry to N = (4, 4) required the sigma model metric to be hyperk¨ ahler [2]. Other important geometric structures were understood in this context including the moment map as well as symplectic and K¨ ahler reductions. Many of these structures were devoloped independently in the mathematical and physics literature. The use of a Legendre transform and a symplectic quotient in the study of hyperk¨ ahler geometry arose from their use in supersymmetric sigma models [3–5]. In the context of hyperk¨ ahler geometry a comprehensive review was presented in [6]. An important step in understanding the general structure behind geometry and supersymmetry was taken in [7] which presented a classi?cation of the geometries consistent with extended supersymmetry paying particular attention to the type of super?eld representations involved. Perhaps the clearest example of the connection between geometry and the super?eld representations arose out of the study of two[8]. This work introduced twisted chiral super?elds on the supersymmetry side and bihermitian geometry on the mathematical side. Subsequently, Buscher, Lindstr¨ om, and Roˇ cek [9] expanded in this direction. These results showed that the underlying element in the relation between the amount of supersymmetry and di?erent versions of complex geometry is to a large extent determined by the super?eld representations involved. Generalized complex geometry and (2, 2) supersymmetry: What would complex geometry look like if instead of considering structures associated with the tangent bundle T one considers structures associated with the direct product of the tangent and cotanformulation of generalized complex geometry [10]. Generalized complex geometry naturally contains complex, symplectic, K¨ ahler and bihermitian geometries as particular gent bundle T ⊕ T ? ? This question has recently been posed by Hitchin leading to the dimensional N = (2, 2) supersymmetric models discovered by Gates, Hull, and Roˇ cek

1

cases. A fairly complete account can be found in Gualtieri’s thesis [11]. Several geometric concepts, like the moment map, reduction of generalized complex and generalized K¨ ahler geometries, and others are currently being developed [12–17]. A number of works have sought to clarify the connection of generalized complex geometry to supersymmetry. A very interesting analysis presented in [18] showed how the integrability conditions of the generalized complex structure could be understood, at the nonlinear sigma model level, as the conditions for a manifestly (1, 1) supersymmetric model to be (2, 2) supersymmetric. In the language of representations it has also become increasingly evident that semi-chiral super?elds play a central role [18–22]. The role of semi-chiral super?elds and our work: The mathematical literature seems to allow for some ambiguities in the de?nition of some of the geometric structures involved. In particular, several groups have proposed de?nitions of the reduction of generalized complex geometry and moment map [14–16,23]. On the physics side, these concepts are related to gauging some of the symmetries of the nonlinear sigma model. The above situation motivates us to approach these concepts from the physics point of view. We also believe that understanding nonlinear sigma models and the gauging of some of the symmetries in the most general context is an important problem in physics. We have heavily relied on previous works that addressed such questions in the context of complex and K¨ ahler geometry and partially in the case of bihermitian geometry [24–27]. A concept that does not seem to be intrinsic to the mathematical literature but that we will investigate, following in part the work of Roˇ cek and Verlinde [28], is T-duality in the presence of semi-chiral super?elds. More generally, the goal of this paper is to lay the groundwork for exploring the connection between generalized complex geometry and supersymmetry in terms of gauged nonlinear sigma models with super?elds in the semi-chiral representation. The paper is organized as follows. In section 2 we discuss the gauging of the sigma model with semi-chiral super?elds under the simplifying assumption that the K¨ ahler potential is invariant under the action of the U (1) symmetry. We reduce the action to (1, 1) superspace and ?nd, via comparison with [24], the moment map and the one form required for gauging a sigma model with B ?eld . We analyze the example of moment map and one form and point out an ambiguity which arises in the presence the SU (2) ? U (1) WZNW model and verify the identi?cations we have made for the

of semi-chiral representations. We conclude section 2 with a description of a gauging based on the prepotential. In section 3 we brie?y review the mathematical literature 2

on Hamiltonian action and moment map for generalized complex geometry. We also compare the mathematical de?nition with the physical de?nition of the moment map and explicitly discus the example of the SU (2) ? U (1) WZNW model. In section 4 we address T-duality in (2,2) superspace. We ?rst describe T-duality in case of chiral and twisted chiral super?elds following the formalism used in [28], but with a di?erent gauge ?xing procedure. Next, we use this gauge ?xing procedure to describe T duality in the case when semi-chiral super?elds are used. We ?nish by working out the 4d ?at space example. In section 5 we draw some conclusions and point out some interesting open questions.

2

Gauging a (2,2) sigma model and the moment map
One property of (2, 2) superspace is that the sigma model is given entirely in terms

of the K¨ ahler potential: S= ? 2 K ({Φ, Φ ? } ). d2 xD 2 D (2.1)

Geometric quantities associated with the sigma model, such as the target space metric and B ?eld, can be obtained in explicit form by performing a reduction to (1, 1) superspace. The speci?c properties of the sigma model target space geometry follow from the speci?c choice of N = 2 super?elds {Φ}. If {Φ} is a set of chiral ? ± Φ = 0, or twisted chiral super?elds D+ Φ = D ? ? Φ = 0, then the associsuper?elds D found among {Φ}, then the corresponding geometry goes by the name of “bihermitian geometry” with almost product structure. This particular geometry is characterized
2 by two commuting almost complex structures J± , J± = ?1, which are integrable and

ated target space geometry is K¨ ahler. If both chiral and twisted chiral super?elds are

covariantly constant with respect to a?ne connections with torsion, a metric that is

t bihermitian with respect to both complex structures J± gJ± = g , and a B -?eld. As a

consequence of the fact that [J+ , J? ] = 0, the metric acquires a block diagonal form, inducing a natural decomposition of the tangent space, along the chiral, respectively, twisted chiral components. It is the feature of having two commuting complex structures that distinguishes the almost product structure geometry among the class of bihermitian geometries. Lastly, if the set of super?elds which determine the K¨ ahler potential contains a more general (2, 2) semi-chiral super?eld, constrained only by a target space is bihermitian, with non-commuting complex structures. Recently, it has 3 single superspace covariant derivative D+ Φ = 0, or D? Φ = 0, then the sigma model

been shown [11] that the projection of the generalized K¨ ahler geometry onto the tanwhen investigating non-linear sigma models with (2, 2) supersymmetry. gent bundle yields the bihermitian data (g, J± , B ) found by Gates, Hull and Roˇ cek [8] If the sigma model target space has an isometry group, then a generic Killing vector can be decomposed in a basis of the Killing vectors kA which generate the Lie algebra of the isometry group
i ξ = ξ A kA = ξ A kA ?i ,

[kA , kB ] = fAB C kC ,

L ξ g = 0.

(2.2)

The in?nitesimal transformation of the sigma-model ?elds is given by
i δφi = ?A kA ,

(2.3)

where ?A are rigid in?nitesimal parameters. For a sigma model with isometries, there are additional geometric data. These follow from the integrability conditions associated with the additional requirements that the action of the Killing vector leave invariant not just the metric, but the ?eld strength of the B-?eld H , and the symplectic forms ω± = gJ± : L ξ H = 0, L ξ ω ± = 0. (2.4)

From the condition that H is invariant, it follows that Lξ H = diξ H + iξ dH = diξ H = 0. Since the two-form iξ H is closed, locally it can be written as iξA H = duA , (2.6) (2.5)

where the one-form u is determined up to an exact, Lie-algebra valued one-form. The ambiguity in u can be ?xed, up to U (1) factors in the Lie algebra, by requiring that it is equivariant LA uB = fAB C uC . Besides this one form u, the other geometric datum associated with the existence of

an isometry group is the moment map (also known as Killing potential). From the conT ±H (X, Y, Z ), it follows that ω± ξ ? J± u is closed. Therefore, locally one ?nds T d?± = ω± ξ ? J± u,

dition that the symplectic form is invariant under ξ , and from dω± (J± X, J± Y, J± Z ) = (2.7)

where ?± are the moment maps. This expression is the generalization for a manifold with torsion of the integrability condition for a Hamiltonian vector ?eld ξ . Since ξ is also Killing, it follows that Lξ J± = 0. 4

The relevance of these two quantities, the one-form u and the moment map ?, becomes obvious when constructing the gauged sigma model, by promoting the rigid (global) isometries 2.3 to local ones. This is accompanied, in the usual manner, by
A C B C which transforms as δAA ? = ?? ? + fAB A? ? . For (1, 0) or (1, 1) supersymmetric i introducing a compensating connection (gauge potential) ?? φi ?→ ?? φ = ?? φi + AA ? kA ,

sigma models, the bosonic gauge connection becomes part of a corresponding (1, 0) or (1, 1) super Yang-Mills multiplet. Promoting the partial derivatives to gauge covariant derivatives is not enough in the presence of a B -?eld [24, 27, 29]. New terms, which depend on the u one-form and the moment map, must be added to the sigma model action. For a bosonic, (1, 0) or (1, 1) supersymmetric sigma-model, adding only udependent terms su?ces: S=
i A B d2 xd2 θ gij ?+ φi ?? φj + Bij D+ φj D? φj ? 2uiA AA (+ D?) φ + A+ A? c[AB ] , (2.8)

variant derivatives, while c[AB] = k[iA uiB] .

where D± are ?at superspace covariant derivatives and ?± are superspace gauge coWhen the sigma model has additional supersymmetries, then the gauged sigma

model action acquires new terms, which are moment map dependent. The additional supersymmetries, which at the level of N = 1 superspace are nonlinearly realized, are of the form δφi = ?J i j D+ φj . (2.9) The supersymmetry algebra requires that the (1, 1) tensor(s) J i j be identi?ed with the almost complex structure(s). The gauged (2,2) sigma model action typically contains a term δS = d2 xd2 θS?, (2.10)

where ? is the moment map, and S is a super-curvature that appears in the (2,2) ? ? }). superalgebra (more precisely in the super-commutators {?+ , ? The above formulae show that the moment map and the one-form u are necessary ingredients to gauge the sigma model. Intuitively, the one-form u is needed to gauge the sigma model with B ?eld and the moment map is needed to gauge a sigma model with extended supersymmetry. Alternatively, one could choose instead to remain at the level of (2, 2) superspace and perform the gauging there, without ever descending onto the (1, 1) superspace. In term, which is also moment-map dependent [24, 25]. 5 the process, the K¨ ahler potential of the ungauged sigma model K ({Φ}) acquires a new

We shall be interested in gauging (2, 2) sigma models whose target space has a bihermitian structure, with non-commuting complex structures. The natural starting point for us is the N = 2 superspace formulation of a sigma-model written in terms of (2, 2) semi-chiral super?elds: left chiral: ? + X = 0, D D? Y = 0 . (2.11)

right antichiral: with the contraints1 on X and Y .

We begin by making the observation that the following transformations are consistent

Y → (F + G)Y + W + Z,

X → (A + B )X + C + D,

(2.12)

where A, C are chiral super?elds, B, D are twisted anti chiral super?elds, F, W are anti chiral, and G, Z are twisted chiral. When these transformations correspond to gauge transformations they can be properly accounted for using both the chiral and twisted chiral vector multiplets. For simplicity we will consider only the gauge transformations where the semichiral super?elds are multiplied and shifted by chiral and anti-chiral super?elds2 . In this paper we shall follow two complementary approaches to constructing the gauged action in (2, 2) superspace. The ?rst method involves descending at the level of (1, 1) superspace by following the usual route of substituting the Grassmann integration ? → DD ? , and by the subsequent replacement of the ordinary by di?erentiation dθdθ ? → ?? ? . This is superspace covariant derivatives by gauge covariant derivatives D D equivalent to gauging by minimal coupling, if the K¨ ahler potential is invariant under the action of the isometry generators. The second method [25] uses the prepotential of the gauge multiplet V explicitly in the K¨ ahler potential to restore the invariance of the action under local transformations. For simplicity we restrict ourselves to U (1) isometries. As such we can go to a coordinate system where the isometry is realized by a shift of some coordinate. This ? Y, Y ? ) will be independent of a certain linear implies that the K¨ ahler potential K (X, X, combination of the left and right semi-chiral super?elds. For example, for ? Y +Y ? , X + Y ). K = K (X + X,
1

(2.13)

This is not the most general set of transformations consistent with the constraints on X and Y . However, these are the only transformations relevant to our considerations. 2 We thank S. Gates for various clari?cations on this point.

6

we can immediately read o? the Killing vector associated with the isometry. In this ? ? ? ? (2.14) ?i ? ?i +i ?. ?X ?Y ?X ?Y From (2.12) we see that this is an example of a K¨ ahler potential, with a U(1) isometry ξ=i which can be gauged using only the (un-twisted) (2,2) super Yang-Mills multiplet. 2.1 Gauging and the reduction to (1,1) superspace Let us consider the ?rst of the two approaches to gauging which we have outlined before. Since we are interested in extracting the geometric data (including those associated with isometries) from the sigma model, and these are most easily seen in the language of (1, 1) superspace, here we describe the bridge from (2, 2) to (1, 1) superspace, following [8] closely. We begin by recording the (2, 2) gauge covariant supersymmetry algebra for the ?A , AA , AA ): (un-twisted) super Yang-Mills multiplet (AA , A
α α

case it takes the form

[?α , ?β } = 0, ? β } = 2i(γ c )αβ ?c + 2g [Cαβ S ? i(γ 3 )αβ P ]ξ, [?α , ? ? β ξ, [?α , ?b } = λ(γb )α β W [?a , ?b } = ?iλ?ab W ξ,

(2.15)

where the bosonic two-dimensional indices are a, b, c = { , }, and the Grassmann odd two-dimensional spinor indices are α = ±. The skew-symmetric tensor Cαβ is used for raising and lowering indices in superspace. Having in mind the gauging of a certain isometry of a (2,2) sigma model, we used the Killing vector ξ to denote the couplings of the sigma model super?elds to the (2,2) super Yang-Mills multiplet. Also, following from the Bianchi identities, one has the following set of constraints ? α, ? β, ?α S = ?iW ?α P = ?(γ 3 )α β W ? β = 0, ? ?α W ?α d = (γ c )β α ?c Wβ ,

?α Wβ = iCαβ d ? (γ 3 )αβ W + (γ a )αβ ?a S ? i(γ 3 γ a )αβ ?a P.

(2.16)

According to our previous discussion on gauging methods, we take our ?rst step towards constructing the gauged (2,2) sigma model begin by making the substitution ? 2 θ = 1 [?α ?α ? ? β? ?β +? ? β? ? β ?α ?α ], d2 θd 8 7 (2.17)

where we have used the conventions of [30]. In order to reduce the action written in (2, 2) superspace to (1, 1) superspace we need to express the (2, 2) gauge covariant derivatives in terms of two copies of the (1, 1) 1 i ?α = √ ? α ), ?α = √ ? α ). ? ( ?α + ? ? ( ?α ? ? 2 2 The (1, 1) derivatives satisfy the following algebra: ? α, ? ? β } = 2i(γ c )αβ ?c ? 2iλ(γ 3 )αβ P ξ, [? λ ? β ξ, ? α , ?b } = √ (γ b )α β W [? 2 ? α, ? ? β } = 2i(γ c )αβ ?c ? 2iλ(γ 3 )αβ P ξ, [? λ ? α , ?b } = √ ? β ξ, [? (γ b )α β W 2 ? α, ? ? β } = ?2iλCαβ Sξ, [? ?β = where W
i ?β √ (W 2

derivatives:

(2.18)

(2.19)

Next we consider the measure of the (2, 2) action (2.17) and we rewrite it in terms of the (1, 1) derivatives ? α? ? α? ? β? ? β = 2 ?α ?α ? ? β? ? β + 2? ? β? ? β ?α ?α + (...)ξ + total derivative ? on a potential which is invariant under the isometry, that is, satis?es ξK = 0. Reducing the (2, 2) Lagrangian amounts to evaluating L= ? 2 θK = 1 ? ? 2? ? 2 K (X, X, ? Y, Y ? ), d2 θd 4 (2.21) (2.20)

?β = ? Wβ ) and W

1 ?β √ (W 2

+ Wβ ).

? α? ? α? ? β? ? β and 2?α ?α ? ? β? ? β + 2? ? β? ? β ?α ?α are equivalent when acting Therefore ?

? Y, Y ? obey (2.11), but with the ordinary superwhere the semi-chiral super?elds X, X, space derivative D± replaces by the gauge-covariant derivatives (2.15). Then, using the relation ? α? ? α = ?2i? ? +? ? ? ? 2iλP ξ, ? (2.22)

right semi-chiral super?elds into (1,1) super?elds ? = X |, We end up with: ? ? X |, Ψ=?

? +? ? ? K . Additionally, we must decompose the (2,2) left and we only need to evaluate ? χ = Y |, ? + Y |. Υ=? (2.23)

? +? ? ?K = ? ? + ?I mII ′ ? ? ? χI ′ + ΥI ′ nI ′ I ΨI + ΨI (2ωIJ ? ? + ?J + ipII ′ ? ? + χI ′ ) ? + ? ? 8

i +2iλ(S + iP )Ki ξ i ? 2iλ(S ? iP )K? iξ ? i′ ? + ΦT · E · ? ? ? Φ + S+I uII ′ S?I ′ ? 2iλSKi′ ξ i′ + 2iλSK? = ? i′ ξ i +2iλ(S + iP )Ki ξ i ? 2iλ(S ? iP )K? iξ , ?

′ ? i′ i′ ? J′ ? I +ΥI i′ ξ + (2ωI ′ J ′ ?? χ ? iqI ′ I ?? ? ) ? 2iλSKi′ ξ + 2igSK?

?

(2.24)

derivatives of the K¨ ahler potential, are the same as in [19]. Also, analogous to [19] S+I uII
′ ′ ′ II ′ II ′ ? J ? J′ = ΥI + ? 2u ωIJ ?+ ? ? iu PIJ ′ ?+ χ II ′ II ′ ? J′ ? B = ΨA ? + 2u ωI ′ J ′ ?? χ ? iu qI ′ J ?? ?

where we have used the notation Ki = ??i K, Ki′ = ?χi′ K and that the U(1) Killing ′ ? ? i′ ?′ ? vector is ξ = ξ i??i + ξ i ?χi′ + ξ i ?? i + ξ ?χ ?? ?i . The index I is a collective index: I = {i, i}, ? χ, χ and Φ = {φ, φ, ?}. The matrices m, n, ω, p, q , expressed in terms of the second order

uII S?I ′

E = g+B =

2iωuq m ? 4ωuω ′ pt uq 2ipt uω ′

.

(2.25)

At a ?rst glance it appears that we have an asymmetric coupling of the ?eld strength P between the ?elds ?I and χI . However, this is just an artifact of our choice in evaluating the covariant derivatives. Note that
i i i ξK (?, χ) = 0 → Ki ξ i + K? i′ ξ = 0. i ξ + Ki′ ξ + K? ?
′ ′

?′

(2.26)

This means that the reduced Lagrangian is given by ? α? ?α ? ? + ΦT · E · ? ? ? Φ + S+I uII ′ S?I ′ L = ?

i i ? i +2iλ(S + P )Kiξ i ? 2iλ(S ? P )K? iξ 2 2 i i ′ ? i′ ?2ig (S + P )Ki′ ξ i + 2ig (S ? P )K? i′ ξ 2 2
? ?′

? α? ?α ? ? + ΦT · (g + B ) · ? ? ? Φ + S+I uII ′ S?I ′ = ?
i i i +2iλS (Ki ξ i ? K? i′ ξ ) i ξ ? Ki′ ξ + K? ?
′ ′

i i i ?λP (Ki ξ i + K? i′ ξ ) . i ξ ? Ki′ ξ ? K?

?′

(2.27)

This is the gauged sigma model we were after, and it is one of our main results. To understand the various terms that appear in (2.27), it is useful to compare this action with (2.8), given that both actions represent gauged sigma models with manifest ? α ΦT · g · ? ? αΦ + (1,1) supersymmetry. This explains the obvious common elements ? dependent terms. How about in our case? First, we notice that since we have assumed 9

? α ΦT · B · D ? α Φ. The gauging of the B -?eld terms is done in (2.8) by including the uD

that ξK = 0, in other words that the minimal coupling prescription will su?ce, this is indeed what the gauged sigma model Lagrangian (2.27) re?ects. The extra terms ? (? ΦA+) . As required for the gauging of the B -?eld terms can be combined into iξ B · D

a consequence of the assumption ξK = 0, we ?nd that Lξ B = 0. This is a stronger condition than Lξ H = 0, and it implies the latter. Since Lξ B = 0, we ?nd u = ?iξ B + dσ,

(2.28)

where dσ is an exact one-form invariant under the action of the isometry group. This is exactly what is required to match the minimal coupling of the B -?eld terms against the u-terms in (2.8). The cAB terms in (2.8) vanish in the case of a U (1) gauging. Otherwise, they, too, could be recognized in the minimal coupling gauging of (2.8). We shall see that the ambiguity in de?ning u, namely the exact one-form dσ , is re?ected in (2.27) in the term which multiplies the ?eld strength P . The expression ?λd(KI ξ I ? KI ′ ξ I ) is d(σ ). We verify that it is invariant under the U(1) action: Lξ dσ = d(iξ dσ ) = d (ξ I ?I + ξ I ?I ′ )(ξ J ?J ? ξ J ?J ′ )K = 2d (ξ I ?I + ξ I ?I ′ )ξ I ?J K
′ ′ ′ ′ ′

=


= 2d (?ξ I ?I ξ I ?I ′ + ξ I ?I ′ ξ I ?I )K

= 0, (2.29)

where in the last step we used that we can go to a coordinate system where the U(1) action is realized by a shift of some coordinate, which implies [ξ I ?I , ξ I ?I ′ ] = 0. The remaining terms in (2.27), such as those dependent on the auxiliary super?elds S± and which have no counterpart in (2.8), are present because our starting point was a (2,2) supersymmetric action with o?-shell (2,2) super?elds. Lastly, we recognize in the terms proportional to the super?eld strength S , a linear combination of the moment maps. Their presence is required to insure the invariance of the gauged sigma model action. While the expression proportional to S in (2.27) is not immediately relatable to the moment map given in (2.7), it does have a form similar to that given in [31, 24, 32] for the moment map. There the moment map is identi?ed as the imaginary part of the holomorphic transformation of the K¨ ahler potential under the action of the Killing vector. Thus we conclude with the identi?cations:
i i i Moment map ? Ki ξ i ? K? i′ ξ i ξ ? Ki′ ξ + K? i i i σ ? Ka ξ i + K? i′ ξ . i ξ ? Ki′ ξ ? K? ?
′ ′

?



?′

(2.30) (2.31)

?′

10

These identi?cations, and especially the rapport between (2.30) and (2.7), will be veri?ed in the next section. 2.2 An example: the SU (2) × U (1) WZNW model In this section we apply our previous construction of a (2,2) gauged sigma model to a concrete example: the SU (2) × U (1) WZNW model. The (2, 2) supersymmetric SU (2) × U (1) WZNW sigma model was ?rst formulated in terms of semi-chiral

super?elds in [33]. These authors discovered non-commuting complex structures on try. However, this duality functional allows to map between two seemingly di?erent per?elds and the another description in terms of semi-chiral super?elds. The explicit SU (2) × U (1) and constructed a duality functional that does not change the geome-

descriptions, one for SU (2) × U (1) described in terms of chiral and twisted chiral su-

form of the K¨ ahler potential was given in [34, 35]. A discussion on the various dual descriptions which can be obtained by means of a Legendre transform can be found in [35]. The SU (2) × U (1) K¨ ahler potential is 1 ? + η )(φ + η K = ?(φ ?) + (? η + η )2 ? 2 2
η ?+η

dxln(1 + ex/2 ),

(2.32)

? + η, φ + η ? + φ = D? η = 0. Because K = K (φ where D ?, η + η ?) we cannot directly

gauge the theory, using only the coupling with the (un-twisted) (2,2) super Yang-Mills multiplet. However, there is an easy remedy to this problem, namely we shall use a dual description, found via a Legendre transform [35]: ? η, η K (r, r ?, η, η ?) = K (φ, φ, ?) ? rφ ? r ?φ, (2.33)

? + r = 0, and φ is unconstrained. By integrating over r , we where r is semi-chiral, D recover the previous K¨ ahler potential. On the other hand, by integrating over φ, that is eliminating it from its equation of motion, we ?nd a K¨ ahler potential K = K (r + η, r ?+ η ?, η + η ?) (up to terms that represent a generalized K¨ ahler transform 1 η2 + 1 η ?2 ). 2 2 This is an example of a “duality without isometry” [35], where the K¨ ahler potential of a semi-chiral super?eld sigma model can be mapped via Legendre transforms into four di?erent, but equivalent expressions, all involving only semi-chiral super?elds. The new form taken by the SU (2) × U (1) K¨ ahler potential ? = (? K r+η ?)(r + η ) ? 2 11
η ?+η

dxln(1 + ex/2 )

(2.34)

indicates that the U (1) isometry is realized by the transformations r → r + i?, η → η + (i?), (2.35)

where ? is a constant real parameter. However, when promoting this symmetry to a local one, according to our previous discussion, ? is to be interpreted as a chiral super?eld, and ? ? as an anti-chiral super?eld. The K¨ ahler potential is left invariant under the action of the (2,2) Killing vector ξ=i ? ? ? ? ?i ?i +i . ?r ?r ? ?η ?η ? (2.36)

From (2.25) we can now calculate the B ?eld, its ?eld strength and their contractions with the Killing vector: B = (1 ? 2f )(dr ∧ dη ? + dr ? ∧ dη ) iξ B = i(1 ? 2f )dη ? ? i(1 ? 2f )dη + i(1 ? 2f )dr ? ? i(1 ? 2f )dr H = dB = 2( ?f ?f dr ? dr ?) ∧ dη ∧ dη ? ?η ?η ? (2.37)

iξ H = d(2if [?dr + dr ? ? dη + dη ?]) = du, where f = f (? η + η) =
1 exp[ 2 (? η + η )] . 1 1 + exp[ 2 (? η + η )]

(2.38)

minimal coupling [27,29]. As discussed before, it implies that u = ?iξ B + dσ where dσ

We also ?nd that Lξ B = 0, in accord to the expectation that the gauging is done via is an exact one-form, invariant under the action of the Killing vector. As to the term proportional to P in (2.27) we ?nd that is equal to 2iλσ , where dσ = d(? r?r+η ? ? η ). Indeed, this one-form satis?es the condition iξ dσ = 0. Next, we show how the term proportional to S corresponds to the moment map. 2.2.1 The Moment Map Here we verify that the term proportional to the super-curvature S in (2.27) i(Kr ? Kr ?+ η + η ? ? 2ln(1 + exp( ? ? Kη + Kη ? ) = 2i r + r 12 η+η ? )) ≡ M 2 (2.39)

is a certain linear combination of the two moment maps of the bihermitian geometry. We recall their de?nition
j gij ξ j ± ui = I± i ?j ?± .

(2.40)

Before we consider (2.40) we must ?rst address the ambiguity in the expression for the one form u. The one-form u is de?ned only up to an exact one form that satis?es constrained only by Cr ? Cr ? ? Cη + Cη ? = 0. However, our previous considerations have Lξ dσ = 0: u = 2if [?dr + dr ?? dη + dη ?]+ di(Cr r + Cr ?+ Cη η + Cη ?) with Cr,r ?r ?η ?,η,η ? constants,

eliminated most of the freedom in dσ , given that, from the gauged action we have the moment maps we ?nd the following relationship with M : M = ?(?+ + ?? ). 2.3 Alternative gauging procedure: the prepotential In section 2.1 we gauged the sigma model by replacing the Grassmann integration measure with gauge supercovariant derivatives and thus reducing the (2,2) action to a gauged action with (1,1) manifest supersymmetry. Here we take the alternative approach of using the gauge prepotential super?eld V to arrive at a gauge-invariant K¨ ahler potential. This procedure is done in (2,2) superspace, and all supersymmetries remain manifest. Therefore this gauging method has the advantage of facilitating the discussion of duality functionals, which we will address in the next section. In simple cases, the gauging is done by adding the prepotential V to the appropriate ? Y+ combination of super?elds in the K¨ ahler potential: for the example K = K (X + X, ? , X + Y ), the global symmetry is promoted to a local one via the substitution Y ? Y +Y ? , X + Y ) → K (X + X ? + V, Y + Y ? + V, X + Y + V ), K (X + X, (2.42) identi?ed Cr = ?1, Cr ? = 1, Cη = ?1, Cη ? = 1. Armed with the concrete expressions of (2.41)

if the gauging is done using the (un-twisted) (2,2) super Yang-Mills prepotential, in other words, if the gauge parameter is a chiral super?eld. On the other hand, if the gauge parameter is a twisted chiral super?eld, then we must use the gauge prepotential associated with a twisted (2,2) super Yang-Mills multiplet Vt . For example, we could ? Y +Y ?,X + Y ? ) by gauge K = K (X + X, ? Y +Y ?,X + Y ? ) → K (X + X ? + Vt , Y + Y ? + Vt , X + Y ? + V t ). K (X + X, (2.43)

13

For concreteness we continue to address only the gauging done using the coupling to the (un-twisted) (2,2) super Yang-Mills multiplet. In general, the isometry transformations of a given super?eld are given by:
?ξ ? ? → e?i? X → ei?ξ X ? X X,

(2.44)

where ξ denotes the isometry generator and ? is a real valued constant parameter. When promoting this global symmetry to a local one, the gauge parameter ? becomes a chiral super?eld, and ? ? an anti-chiral super?eld. The invariance is restored by introducing the gauge prepotential super?eld V , transforming as V → V + i(? ? ? ?). We include V through the replacement: ? → eiV ξ X. ? X been restored. Although we have used the whole Killing vector ξ in constructing the ?eld that transforms properly (2.46), to be more speci?c, it is only the part of the Killing vector that induces a transformation with the anti-chiral gauge parameter which contributes ? Y +Y ? , X + Y ), X, Y ? to this de?nition. In the example that we gave, K = K (X + X, ? Y , with an anti-chiral parameter. transform with a chiral gauge parameter, and X, ? The Killing vector will generally factorize ξ = ξc + ξc ? such that ξc and ξc induce a chiral parameter, respectively an anti-chiral parameter gauge transformation. In the
? ? SU (2) ? U (1) example we have ξc ? i ?η . ? = ?i ? r ?

(2.45)

(2.46)

? transforms in the same way as in the global case and thus the invariance has Now X

Therefore we de?ne

? = eL X, ? L = iV ξc X ?.

(2.47)

? , transforms under the gauge transformation in the exact same way The new ?eld, X ? did under the global isometry. Therefore by replacing X ? in the K¨ as X ahler poten? we insure that the transformation of the K¨ tial by X ahler potential under the local transformation is the same as for the global isometry, namely it is a generalized K¨ ahler transformation. Of course, the other semi-chiral super?eld Y undergoes a similar treatment: ? = eL Y. Y 14 (2.48)

If the K¨ ahler potential remains invariant under the action of the Killing vector i.e. ? ? ) = 0, the minimal coupling perscription is given by replacing X ? with X ? ξK (X, X, Y, Y ? . Speci?cally, the gauged (2,2) Lagrangian is given by the replacement and Y with Y ? Y, Y ? ) → K (X, X, ? Y ?,Y ? ). K (X, X, Lagrangian as
L ? Y ?,Y ? ) = eL K (X, X, ? Y, Y ? ) = K (X, X, ? Y, Y ? ) + e ? 1 LK K (X, X, L L e ? 1 ? Y, Y ?) + = K (X, X, V M, L

(2.49)

? Y ?,Y ? ) = eL K (X, X, ? Y, Y ? ) to rewrite the At this point we can use the relation K (X, X,

(2.50)

where in M = iξc ?K we recognized the same object which we have identi?ed from the gauged (1,1) action as the moment map (3.4). Next, we address the case of a K¨ ahler potential which under the action of the isomtery generator transforms with terms that take the form of generalized K¨ ahler transformations ?(X ? ) + g (Y ) + g ? ). ξK = f (X ) + f ?(Y (2.51)

The trick is to introduce new coordinates and add them to the K¨ ahler potential in such a way that the new K¨ ahler potential is invariant under the transformation generated ? + α = D? β = 0. We by the new Killing vector. Speci?cally we introduce α, β with D ?) = K (X, X, ? ? Y, Y ? , α, α ? Y, Y ?) ? α ? α K ′ (X, X, ? , β, β ??β?β ξ ′ = ξ + f (X ) ? ?(X ? ) ? + g (Y ) ? + g ?) ? . +f ?(Y ? ?α ?α ? ?β ?β (2.52)

construct the new K¨ ahler potential and Killing vector

and we can proceed as before. We replace all ?elds which transform with the parameter


Now the new K¨ ahler potential K ′ is invariant under the new Killing vector Lξ′ K ′ = 0

′ ? ? with the combination which transforms with the ?eld ? by using eL where L′ = iV ξc ?. ? Y, α Next we de?ne the tilde versions of X, ? , β as follows

? = eL′ X, ? X

? = eL′ Y, Y

α ? = eL α ?



? = eL′ β. β

(2.53)

The gauged Lagrangian is obtained by the same substitution as before. Finally we get ? β ?) = K (X, X, ??β ? ? Y ?,Y ? , α, α ? Y ?,Y ?) ? α ? α K ′ (X, X, ? , β, ??β 15

L ?(X ? ) + g (Y )) ? Y, Y ? ) ? i e ? 1 V (f = eL K (X, X, L L ?(X ? Y, Y ? ) + e ? 1 (LK ? iV f ? ) ? iV g (Y )) = K (X, X, L L ? Y, Y ? ) + e ? 1 V M. = K (X, X, L

(2.54)

3
3.1

Eigenspaces of Generalized Complex Structures
Hamiltonian action and moment map in the mathematical literature In the context of generalized complex geoemtry, the origin of subsequent de?nitions

of the Hamiltonian action can be found in Gualtieri’s thesis [11] where it was shown that certain in?nitesimal symmetries preserving the generalized complex structure J can be extended to second order. Intuitively, given a Hamiltonian action on a generalized complex manifold, the moment map is a quantity that is constant along the action of the group elements. More formal de?nitions of moment map were given, for example, in [14–16, 36]; in [16] Hu considered the Hamiltonian group globally. For concreteness here we will explore one of the de?nitions put forward by Lin and Tolman [14] in the simplest setting without H -twisting, namely, de?nition 3.4: Let a compact Lie group G with Lie algebra g act on a manifold M , preserving a √ generalized complex structure J . Let L ∈ T ⊕ T ? denote the ?1-eigenbundle of J .

A generalized moment map is a smooth function ? : M → g ? so that √ (i) ξM ? ?1 d?ξ lies in L for all ξ ∈ g , where ξM denotes the induced vector ?eld on M. (ii) ? is equivariant. In subsequent works, the de?nition of Hamiltonian action was generalized to include

the H -twisted case [15, 16]. In [17], the authors arrived at a de?nition of moment map in terms of the action of a Lie algebra on a Courant algebroid. In what follows we will explore the particular de?nition cited above, and compare it with the expressions that we gave for the moment map in the previous sections. We leave for a future publication the issue of the equivalence of the various de?nitions given in the math literature, and their relationship with the physical point of view advocated in this paper, via the gauging of the (2,2) sigma model.

16

3.2

Generalized Kahler geometry and the eigenvalue problem In a series of papers [19, 20] the authors established that chiral, twisted chiral,

and semi-chiral super?elds are the most generic o?-shell multiplets for N = (2, 2) supersymmetric non-linear sigma models. The use of these (2, 2) multiplets yields generalized K¨ ahler geometries. To practically use the above de?nition of moment map in the case of K¨ ahler geometry we recall that, according to Gualtieri (see Chapter 6 in [11]), the generalized complex structures of the generalized K¨ ahler geometry take the following expressions: J1/2 = 1 2 1 0 B 1
?1 ?1 J+ ± J? ?(ω+ ? ω? ) t ?(J+

1

0

ω+ ? ω?

±

t J? )

?B 1

(3.1)

where g is a K¨ ahler metric, which is bihermitian with respect to both complex structures B -?eld of the sigma model for section 3.4. J± , while B is a 2-form ?eld. We leave a discussion about its relationship with the First, we shall derive the conditions for a generic element of (ξ, ±id?) ∈ T ⊕ T ?

to be an eigenvector of the generalized complex structures. By identifying ξ ∈ T

with a Killing vector, we solve for the one form d? ∈ T ? . Next, after verifying that d? is an exact one-form, we shall compare it with the the moment map and enquire

whether these expressions are compatible. We discuss two concrete settings: the almost product structure spaces, with their commuting complex structures, and as an example of bihermitian geometry we turn to the SU (2) × U (1) WZNW sigma model. lies in the eigenbundle of J1 is We begin with some formal statements. The condition that an element of T ⊕ T ? ξ icd? ξ icd?

J1

= ai

,

(3.2)

where c = ±1, a = ±1. After a bit of massaging, we ?nd that this eigenvalue problem is equivalent to the following linear homogeneous equation system3 (J+ ? ai)(Γ ? ξ ) = 0
For the eigenvalue problem associated with the other generalized almost complex structure J2 , we ?nd a similar linear homogeneous system: (J+ ? ai)(Γ ? ξ ) = 0, (J? + ai)(Γ + ξ ) = 0.
3

17

(J? ? ai)(Γ + ξ ) = 0, where Γ = G?1 (Bξ ? icd?)

(3.3)

(3.4)

Then, by solving (3.3) we ?nd ξ and Γ. The number of independent solutions is equal to the number of zero eigenvalues of J± ? ai. After identifying ξ with a certain Killing vector, we generically ?nd a corresponding Γ. This allows us to solve for ?: d? = ic(GΓ ? Bξ ). next sections we explore two concrete examples of bihermitian geometry. 3.3 Specialization to spaces with almost product structure In the case of a space with almost product structure, which is realized by a (2,2) sigma model written in terms of chiral and twisted chiral super?elds [8], we may choose to work in a coordinate system where the two commuting complex structures are diagonal: J+ = J1 0 0 J2 J? = J1 0 0 ?J2 . (3.6) (3.5)

To test the compatibility between this expression and the moment map (2.7), in the

In the same coordinate system, the metric and B -?eld are also block-diagonal: g = g1 0 0 g2 B= 0 ?b
t

b 0

.

(3.7)

decomposition for ξ ,Γ and d?. Speci?cally, ξ= ξ1 ξ2 Γ=

The expressions taken by g, B, J+ , and J? suggest that we should consider a similar Γ1 Γ2 d?1 d?2

d? =

.

(3.8)

Under this decomposition Γ1,2 , ξ1,2 are solutions to (3.3): (J1 ? ai)Γ1 = (J1 ? ai)ξ1 = 0

aiΓ2 = ?J2 ξ2 . 18

(3.9)

and (3.5) becomes d?1 = icg1 Γ1 ? icbξ2

d?2 = icg2 Γ2 + icbt ξ1 ,

(3.10)

When we specialize to the case where the Lie derivative of B with respect to ξ vanishes, that match up most closely with the generalized complex structures we de?ne ? = 1 (d?+ + d?? ), dM 2 where ?1 dM ?2 dM = ω1 ξ 1
t t ?J2 b ξ1

T How does this compare with the moment maps which are given by d?± = ω± ξ ? J± u?

Lξ B = 0, we can use that u = ?Bξ + dσ . Taking the appropriate linear combinations ? = 1 (d?+ ? d?? ) dM 2 ?1 dM ?2 dM
t ?J1 bξ2

(3.11)

,

=

ω2 ξ 2

.

(3.12)

? or dM ? . For The matching between (3.10) and (3.12) can be done using either dM ? . This is done provided that concreteness, we choose to match (3.10) against dM Γ1 = ξ1 = 0. The condition ξ1 = 0 is automatically satis?ed for almost product structure geometries, where J1,2 are both diagonal. This is so because the requirement that ξ is holomorphic (i.e. it leaves invariant the complex structures) implies that that either ξ1 or ξ2 vanish [24]. Next to complete the matching of (3.12) and (3.10) we need Γ2 = ±iJ2 ξ2 , but is exactly the expression of Γ2 which we get from (3.9). Now that we have veri?ed the compatibility of two moment map de?nitions, (3.5)

and (2.7), for the almost product structure geometry, we want to investigate their compatibility in a more generic case of bihermitian geometry. Since the complex structures do not commute in this case, it is di?cult to analyze what happens in general. However we can consider the concrete SU (2) ? U (1) example and see how things work out there. 3.4 The SU (2) × U (1) example In this case, the non-commuting complex structures, read o? from the supersymmetry transformations ? i ? ? 0 J+ = ? ? ?2i ? 0 of the non-linear sigma model [19, 33], are: ? ? i 0 2i(1 ? f ) 0 0 0 0 ? ? ? 0 ?2i(1 ? f ) ?i 0 0 ? ? J? = ? 0 ?i ? ? ?i 0 0 ?i 0 ? ? 0 0 0 0 0 i 2i 0 i 19 ?

? ? ?, ? (3.13) ?

in (2.37), and the metric takes the form ? ? 0 2 0 2(1 ? f ) ? ? ? ? 2 0 2(1 ? f ) 0 ?. ? g=? ? 0 2(1 ? f ) 0 2(1 ? f ) ? ? 2(1 ? f ) 0 2(1 ? f ) 0
T The moment map d?+ = ω+ ξ ? J+ u reads

where f = f (η + η ?). The U(1) Killing vector is ξ = (i, ?i, ?i, i). The B -?eld was given

(3.14)

d?+ = (?2f, ?2f, 0, 0) ? (?2f, ?2f, ?2f, ?2f ) ? (iCr ? 2iCη , ?iCr ? + 2iCη ?, ?iCη , iCη ?) = (iCr ? 2iCη , ?iCr ? + 2iCη ?, 2f ? iCη , 2f + iCη ? ). stants Cr,r ?,η,η ? satisfy the constraint Cr ? Cr ? ? Cη + Cη ? = 0. given by (ξ, Γ1,± ), where ξ = (i, ?i, ?i, i) and Γ1,+ = For a ?i eigenvector (a = 1), we ?nd Γ1,? = i, i, ?i 1+f , ?i , 1?f (3.17) ?i, ?i, i, i 1+f 1?f . (3.16) (3.15)
T wheare the last term on the ?rst line represents the ambiguity in u, J+ dσ . The con-

We ?nd that the solution to (3.3), corresponding to a +i eigenvector, (a = 1), is

to the +i eigenvalue and Γ2,? = (i, i, i, ?i) to the ?i eigenvalue. we get

the second generalized almost complex structure J2 : Γ2,+ = (?i, ?i, i, ?i) corresponds

for the same Killing vector ξ . For completeness we record the eigenvectors (ξ, Γ2,± ) of

From (3.12), substituting Γ1,± as well as the the metric, B -?eld, and Killing vector icBξ = c(?1 + 2f, 1 ? 2f, ?1 + 2f, 1 ? 2f ), (3.18)

icGΓ1,+ = c(?2f, 2f, ?4f, 0),

where we recall that c = ±1. We have also identi?ed the 2-form B in the generalized

almost complex structure with the B -?eld. Notice that in order to be able to recover an expression compatible with (3.15), we must take the sum ic(GΓ + Bξ ), and not the di?erence of the two terms in (3.18)! The reason for an apparent discrepancy between the two expressions that we have for the moment map, (2.7) and (3.5) lies 20

in the identi?cation of the sigma model B ?eld and the 2-form B that appears in the generalized almost complex structure (3.1). The agreement is restored upon making the identi?cation between minus the sigma model B -?eld and the object by the same name present in (3.1). It is essential that in replacing B → ?B in (3.1), with B the geometry objects. To complete our argument, we have to make the following assignments for the constants which enter in the one-form dσ : Cr = Cr ? = Cη = Cη ? = i. In conclusion, we still ?nd it possible to obtain the moment map from the condition that together with the Killing vector forms a pair (ξ, icd?) which lies in the eigenbundle of the generalized almost complex structure. However, we must exercise caution and interpret the 2-form B in (3.1) as minus the sigma-model B -?eld. We have also seen that the matching between (3.5) and (2.7) requires making use of the ambiguity in de?ning the one-form u. The exact, U(1) invariant one-form dσ required by the matching between the two moment map de?nitions led us to a di?erent one-form dσ than the one we identi?ed in Section 2.2 by matching u with the gauged sigma model action. sigma model B -?eld, we haven’t spoiled any of the properties of the generalized K¨ ahler

4

T Duality
T-duality can be implemented, while preserving the manifest (2,2) supersymmetries

of the sigma model, by performing a Legendre transformation of the K¨ ahler potential. This procedure amounts to starting from the gauged sigma model, introducing a Lagrange multiplier that enforces the condition that the gauge ?eld is pure gauge, and eliminating the gauge ?eld from its equation of motion. In terms of the geometric data, by descending to the level of (1,1) superspace, we ?nd that under T-duality, the metric and B-?eld transform according to the Buscher rules. Let us begin with some review material detailing the execution of T-duality in (2,2) superspace. The simplest example of T-duality involves a non-linear sigma model written in terms of either chiral or twisted chiral super?elds with an U(1) isometry. Under T-duality the chiral multiplets are mapped into twisted anti-chiral and vice-versa. Speci?cally, we choose a coordinate system such that the isometry is realized by a shift in a particular coordinate. Then the K¨ ahler potential has the form ? + Φ, Z a ), K = K (Φ 21 (4.1)

where Z a are spectator ?elds that can be either chiral or twisted chiral. According to ? + Φ with the discusion in Section 2.3, the gauged action is obtained by replacing Φ ? + Φ + V where V is the usual super?eld prepotential for the gauge multiplet. The Φ gauged K¨ ahler potential is ? + Φ + V, Z a ). Kg = K ( Φ (4.2)

To construct the duality functional we introduce a Lagrange multiplier that forces the gauge multiplet ?eld strength to vanish: ? (S ? iP ). KD = Kg + U (S + iP ) + U (4.3)

i ? ? equations of motion force V to be D+ D? V we see that the U and U Since (S + iP ) = 2 ? with Λ a chiral super?eld. For the next step, by choosing pure gauge, i.e., V = Λ + Λ,

? have been completely gauged away a gauge such that Φ + Φ Kg = K (V, Z a ) we arrive at the duality functional ? (S ? iP ). KD = K (V, Z a ) ? U (S + iP ) ? U theory is obtained by integrating out the gauge ?eld. Its equation of motion is ?K ? = 0, ? (Ψ + Ψ) ?V The dual potential (4.6) (4.5) (4.4)

? . The T-dual The original K¨ ahler potential is recovered by integrating out U and U

i ? ? , Z a ). where Ψ = 2 D+ D? U is a twisted anti-chiral super?eld. This de?nes V = V (Ψ+ Ψ

? = K (V, Z a ) ? (Ψ + Ψ) ? V K is the Legendre transform of the original potential (4.1).

(4.7)

When one introduces semi-chiral super?elds the story becomes somewhat more complicated. In [35], Grisaru et al. gave a detailed discussion of the various descriptions of a (2,2) sigma model, which can be obtained by means of a Legendre transform. Starting ? Y, Y ? ), with a (2,2) K¨ ahler potential written in terms of semi-chiral super?elds K (X, X, one constructs the duality functional ?r ? K (r, r ?, s, s ?) ? Xr ? X ? ? sY ? s ?Y 22 (4.8)

where r, r ?, s, s ? are unconstrained super?elds. Depending which ?elds are integrated out (X, Y ), (r, s), (r, Y ), (s, X ) one ?nds four equivalent formulations. In the absence of isometries, this amounts to performing a sigma-model coordinate transformation. The authors of [35] investigated the consequences that the existence of an isometry have on the duality functional. For instance if the K¨ ahler potential has a U(1) isometry ? X +Y ?,X ? + Y ), the duality functional reads K (r + r K = K (X + X, ?, r ? + s, r + s ?) ? ? ?Y ?Y ? )(r + r ? +Y ?Y ? )(r ? r ? . By (X + X ?)/2 + (X ? X ?)/2 ? (r + s ?)Y ? (? r + s )Y integrating over r ? r ?, ultimately leads to expressing X and Y as the sum and di?erence of a chiral and twisted chiral super?eld. In this case, the dual description of the sigma model involves chiral and twisted chiral super?elds. The SU (2) × U (1) WZNW model

has two such dual descriptions [33]. The geometry does not change as we pass from one description to the other, but the pair of complex structures does change, from non-commuting complex structures, to commuting ones. On the other hand, not all the dualities following from (4.8) can be derived from gauging an isometry. The reason is that Lagrange multipliers in (4.8) are semi-chiral super?elds. Following the discussion given at the beginning of this section, one would need a gauge multiplet with a semi-chiral ?eld strength, in order to cast the gauged action duality functional (4.5) into (4.8). However, no known (2, 2) gauge multiplet contains such a ?eld strength. Therefore we choose to pursue the construction of the T-dual action of a sigma model with semi-chiral multiplets following the steps which led to (4.5). We add Lagrange multiplier terms to the gauged action as described previously, and construct the duality functional as in [28]. However, a technical di?culty, related to gauge ?xing, prevents a straightforward application of this procedure. Let us explain. The U(1) invariant K¨ ahler potential, which generically takes the form given in (2.13), can be gauged by adding the prepotential V to the appropriate ?eld combina? + V, Y + Y ? + V, X + Y + V ). tions. The gauged K¨ ahler potential is Kg = K (X + X Because the semi-chiral super?eld is not generically reducible in terms of chiral and twisted chiral super?elds4 one cannot completely gauge away X or Y , as it was possible for the chiral and/or twisted chiral super?elds. Trying to gauge away X we could ?x set to zero. Since X has higher order components which are independent of the lower
4

X | = Dα X | = D 2 X | = 0, where | means evaluation with all the Grassmann variables

components we realized that we have not gauged away all the X components. The inWe thank Martin Roˇ cek for explaining this point to us.

23

dependent left over components form a (1, 1) Weyl spinor super?eld. We shall address the resolution to this question in the following section. 4.1 Dualizing With Chiral and Twisted Chiral Super?elds. For simplicity we will consider a K¨ ahler potential, parameterized by chiral and twisted chiral super?elds, which is strictly invariant under the isometry. The potential is given by (4.1). We begin in the slightly more general setting:
L ? + Φ, Z a ) + e ? 1 V M. Kg = K ( Φ L

(4.9)

? The moment map, M , is given by M = iξc ?K , and in this case ξc ? = ?i ? Φ ? . To con-

struct the duality functional we add Lagrange multiplier terms that force the super?eld strength to vanish. This gives the Lagrangian

L ? + Φ, Z a ) + e ? 1 V M + (Ψ ? + Ψ)V. KD = K ( Φ (4.10) L ? = 0, we choose the WessThe ?nal step is chosing a gauge. Instead of setting Φ + Φ

Zumino gauge for the prepotential V V | = Dα V | = D 2 V | = 0. (4.11)

This gauge choice will allow a better comparison with the semi-chiral case. To see that ? This implies that we do get back the original Lagrangian, we integrate out Ψ and Ψ. ? + Λ, V =Λ (4.12)

where Λ is a chiral super?eld. However, consistency with the gauge choice requires that V = 0 and this give us back the original K¨ ahler potential. To ?nd the dual potential we integrate out V . Since (V )3 = 0 in the Wess-Zumino gauge, this allows us to solve for V explicity. We obtain V =i ? +Ψ+M Ψ ξc ?M

2 ? ? = K (Φ ? + Φ, Z a ) + i (Ψ + Ψ + M ) . K 2 ξc ?M

(4.13)

The important thing to note here is that consistency of the solution for V with the gauge ?xing conditions require that V|=0=i ? +Ψ+M Ψ ? + Ψ)| = ?M | | ? (Ψ ξc ?M 24 (4.14)

It should be understood that this is a component equation, and not a super?eld equation. With this in hand we can show the following; ? ?2K ? ? Φ | = 0, ?Φ ? ?2K | = 0, ?Z a ? Φ ? ?2K ? ? Φ | = ?1. ?Ψ (4.15)

B ?eld. Let us demonstrate how this works with a simple example, speci?cally R → for one of the cycles on T 2 . The K¨ ahler potential and moment map are: K= R ? (Φ + Φ)2 2

the geometry has been replaced by Ψ| up to a surface term that comes from the new
1 R

The implication which follows from these equations is that the contribution of Φ| to

? + Φ). M = R (Φ The dual potential is ? + Ψ)2 ? (Φ ? + Φ)(Ψ ? + Ψ). ? = ? 1 (Ψ K 2R

(4.16)

(4.17)

While this looks as though both directions of T 2 were dualized, one must remember that the real part of Ψ| is proportional to R times the real part of Φ|. Only the direction parameterized by the imaginary part of Φ| was dualized. 4.2 Dualizing with semi-chiral super?elds Now we can give a straightforward extension of the previous discussion to the case when we dualize an isometry of a sigma model parametrized by semi-chiral super?elds. We start with equation (2.50), add the Lagrange multipliers enforcing that V is pure gauge, and choose the same gauge Wess-Zumino gauge as in the previous section. The dual potential Kahler potential is:
2 ? ? = K (X, X, ? Y, Y ? , Z a ) + i (Ψ + Ψ + M ) . K 2 ξc ?M

(4.18)

The analogue of (4.15) reads: ? (ξc )(ξc ?)K | = 0, ? ? (iξc ?K ) |=0 ?Z a ? ? (iξc ?K ) | = ?1. ?Ψ (4.19)

From (4.19) we see that the coordinates in the combination of semi-chiral super?elds corresponding to ξc have been replaced by coordinates in a twisted chiral super?eld in 25

the dual geometry. This is analogous to what happened in the case of chiral and twisted chiral super?elds. It was also expected from gauge ?xing considerations, although it was not a propri clear exactly how it would happen. We now have an explicit description of the T dual of a theory with semi-chiral super?elds at the manifest (2, 2) sigma model level. 4.3 An example: T-duality with semi-chiral super?elds in ?at space In this section we try to develop some intuition about the dualization prescription described in the previous section. Given that we perform a duality transformation by gauging away part of a certain combination of semi-chiral super?elds, and in doing so we trade it for a twisted chiral super?eld, it is not a priori obvious that this is equivalent to the Buscher rules. In particular, we would like to check this in a simple example, namely ?at space with a U(1) isometry. We start with four-dimensional ?at space as our simplest example because one needs both left and right pairs of chiral and anti-chiral super?elds in order to be able to eliminate the auxiliary components of the semi-chiral super?elds and obtain a sigmamodel action. Therefore we begin with the following (2,2) K¨ ahler potential ? )2 ? +Y ? )(X + Y ) ? R (Y + Y K = R (X 4 ?nd the sigma model metric (4.20)

? + X = D? Y = 0. By descending to the level of (1,1) superspace using [20], we where D ? 0 2R R 0 0 0 R ?

the bosonic components S=

? |, Y ? |, Y |. This gives us the action for where the rows and columns are labelled by X |, X ? + ? a (X ? +Y ? )?a (X + Y )), d2 σR(? a X?a X (4.22)

? ? 2R G=? ? R ? 0

? 0 R ? ?, 0 R ? ? R 0

(4.21)

where for simplicity we denoted by X the bosonic component of the (1,1) super?eld X |. Denoting Z = X + Y we notice that it is inert under the global shift symmetry.

26

? Z, Z ? ), we obtain the metric By performing a di?eomorphism transformation to (X, X, in canonical form ? 0 R 0 0 ?

The T-dual sigma model is obtained from the dual (2,2) Kahler potential given in (4.18). In this particular case, (4.18) reads: ? + R (X ? = R (X ? +Y ? )(X + Y ) ? R (Y + Y ? + X + 1 (Y ? )2 ? 1 ψ + ψ ? + Y )) K 4 3R 2 and the corresponding T-dual sigma-model metric is ? ?4R 5R 4R ?5R ? ? 5R ?4R ?5R 4R ? ? 4R ?5R ?4R 5R 9 ? G=? ? ?5R 4R 2 5R ?4R ? ? 5 ?4 ?5 4 ? ?4 5 4 ?5 equal to: 5 ?4 ? 5 ? ? ?5 4 ? ? ?, 4 ?5 ? ? 4 14 ? ?R R ? 14 4 ?R R ?4 ?
2

? ? ? R 0 0 0 ? ? G=? ? 0 0 0 R ?. ? ? 0 0 R 0

(4.23)

(4.24)

(4.25)

?. At ?rst sight this result ? Y ? , Y, ψ, ψ where the rows and columns are labelled by X, X, is puzzling, because we claim that we found the T-dual of a sigma model whose target space is ?at four-dimensional space. At the same time, the dual sigma-model involves six ?elds, and so, apparently the target space is six-dimensional. These two seemingly contradictory statements are reconciled when one takes a closer look at the T-dual metric, and ?nds that it actually describes a four dimensional subspace. This is obvious when expressing the previous T-dual metric in terms of the following coordinates: ?) ? W = Y ? X, ? W ? , ψ, ψ (X, X, ? ? 0 0 0 0 0 0 ? ? ? 0 0 0 0 0 0 ? ? ? ? 0 0 ?4R 5R ? 4 ? 5 9 ? ? G=? (4.26) ?, ? ? 2 0 0 5 R ? 4 R ? 5 4 ? ? ? 0 0 4 14 ? 4 ? 5 ? ? R R ? 14 4 0 0 ?5 4 ?R R

action. The ?nal step in getting the metric in its canonical form is to make a coordinate 27

? is also inert under the global U(1) where we make the observation that W = Y ? X

1 ψ: transformation to T = W ? R ?

0 0 0 0 0 0 0 0 0 0

0 0 5R 0 0

0 0 5R 0

0 0 0 0

0

This form of the T-dual metric makes it clear that the T-dual geometry is fourdimensional and that the Buscher rules, which in this case amount to R → 1/R, are obeyed. The reason why extracting the T-dual geometry required some work on our part is that the semi-chiral super?elds give rise to two (1,1) super?elds, one of them being auxiliary. As we gauge an isometry with (2,2) vector super?elds, the Lagrange multipliers enforcing the condition that the vector ?eld is pure gauge are twisted chiral super?elds (or chiral super?elds). To some extent, as we dualize, we exchange the coordinates in a combination of semi-chiral super?elds by coordinates in a twisted chiral super?elds. However, due to the presence of the auxiliary super?elds, even the coordinates which ? coordinates) end up mixing with are not directly a?ected by the duality (like the Z, Z the dualized coordinate.

? ? ? ? 9 ? G=? ? 2 ? ? ?

0 0 ?4R

?4R 0 0

9 R

? 0 ? ? 0 ? ? ?. 0 ? ? 9 ? R ? 0

?

(4.27)

5

Conclusions
In this paper we have continued the ongoing investigation of the connection be-

models. Speci?cally, we addressed aspects in the area concerning gauged sigma models with semi-chiral super?elds and its relation to various geometric structures on the mathematical side. We have given the form of the gauged action in (1, 1) and (2, 2) superspace and identi?ed the moment map as well as the one-form needed for the gauging the sigma model, as demonstrated by Hull and Spence in [27]. In two particular model written in terms of semi-chiral super?elds, we have found that the combination √ of the moment map and Killing vector associated to the isometry ξ ? ?1d?ξ does cases, namely spaces with almost product structure and the SU (2) × U (1) WZNW

tween generalized K¨ ahler geometry and two-dimensional N = (2, 2) non-linear sigma

lie in the eigenbundle of the generalized almost complex structures, as stipulated by the de?nitions in the mathematical literature. We leave for a future publication the 28

relationship between the physical moment map and the mathematical de?nition (in the general case). Finally, we have presented a description of T-duality for generic (2, 2) sigma models with manifest (2, 2) supersymmetry. It is interesting to note that T-duality, as we have described it, introduces a twisted chiral super?eld into the sigma model. We could also have described it in such a way that a chiral super?eld would be introduced. It would also be interesting to explore the changes which the sigma model geometry undergoes since it appears that the super?eld representations for the coordinates get mixed due to T-duality.

Acknowledgments
We are grateful to L. Anguelova, D. Belov, S.J. Gates, V. Mathai, M. Roˇ cek and B. Uribe for comments and suggestions. W.M. thanks the MCTP for hospitality during di?erent stages of this work. WM and LAPZ also thank the KITP for hospitality during the early stages of this work while both were participants of the program “Mathematical Structures in String Theory”. This work is partially supported by Department of Energy under grant DE-FG02-95ER40899 to the University of Michigan and by the National Science Foundation under rant No. PHY99-07949.

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