T-duality, quotients and generalized Kahler geometry

In this paper we reopen the discussion of gauging the two-dimensional off-shell (2,2) supersymmetric sigma models written in terms of semichiral superfields. The associated target space geometry of this particular sigma model is generalized Kahler (or bi-hermitean with two non-commuting complex structures). The gauging of the isometries of the sigma model is now done by coupling the semichiral superfields to the new (2,2) semichiral vector multiplet. We show that the two moment maps together with a third function form the complete set of three Killing potentials which are associated with this gauging. We show that the Killing potentials lead to the generalized moment maps defined in the context of twisted generalized Kahler geometry. Next we address the question of the T-duality map, while keeping the (2,2) supersymmetry manifest. Using the new vector superfield in constructing the duality functional, under T-duality we swap a pair of left and right semichiral superfields by a pair of chiral and twisted chiral multiplets. We end with a discussion on quotient construction.


Introduction and Summary
The geometry of the target space of two-dimensional sigma models is dictated by the amount of preserved world-sheet supersymmetry and by the representation of the sigma model fields. In the physics literature it has been known for quite a while [1] that (2,2) supersymmetric sigma models give rise to special geometry manifolds. These are called bi-hermitean manifolds, and are endowed with a Riemannian metric g, a closed three-form H = 3dB, and two complex structures J (±) . The metric is hermitean with respect to both complex structures, and J (±) are covariantly constant with respect to connections that have torsion determined by H. More recently, it has been shown that the superfield representations needed for a complete description of the (2,2) supersymmetric sigma model include, beside the better known chiral and twisted chiral superfields, the semichiral superfields [2]. With only chiral and twisted chiral among the sigma model fields, th! e bi-hermitean geometry acquires an almost product structure, with the two complex structures commuting. In the case when the sigma model fields include the left and right semichiral superfields, the commutator of the two complex structures no longer vanishes. It is this latter case that we address in this paper.
In the mathematics literature, the study of the generalized Calabi-Yau manifolds, which include a non-trivial B-field, lead to the development of generalized complex geometry [3]. Its main object is the generalized complex structure defined on the direct sum of the tangent and cotangent bundles T ⊕ T * . A special case of generalized complex geometry is the generalized Kähler geometry, which has two commuting generalized complex structures J 1 , J 2 , and a positive definite metric G = −J 1 J 2 . Gualtieri [4] has shown the equivalence of the data which define the bi-hermitean geometry with those of the generalized Kähler geometry. Therefore these two notions are interchangeable. Special cases of the generalized Kähler geometry include symplectic and Kähler geometry. For recent work on related topics see [5].
An interesting question arises in the presence of isometries. In the symplectic case it is possible to talk about a Hamiltonian reduction, by defining the moment map (a function which is preserved by the action of the isometry group and follows from the requirement that the symplectic form is preserved) and restricting to the subspace of constant moment map. Similarly it is possible to define a Kähler quotient. The basic object associated with the quotient construction is the moment map. There are several proposals for the moment map definition in the context of generalized complex geometry. On the other hand, from the sigma model perspective, there is a natural place to look for the moment map, and that is the gauged sigma model. The moment map (sometimes referred to as Killing potential) appears in the off-shell (2,2) supersymmetric gauged sigma model action, multiplying the gauge superfield strengths. We are interested in matching the sigma model construction of the mom! ent map with the appropriate mathematical definition. This paper is a follow up to [6], and here we give the answer to the open questions of that previous work. The new ingredient is the use of the appropriate (2,2) semichiral vector multiplet [7,8] for the gauging of the (2,2) supersymmetric semichiral sigma model. This is the subject of Section 2. We reduce the gauged action to (1,1) superspace. From the manifestly (1,1) supersymmetric gauged action we identify a set of three Killing potentials which multiply the various gauge superfield strengths.
In Section 3 we show how the Killing potentials relate to the moment maps. We have done this by starting from the reduced (1,1) action, and inquiring what are the conditions which insure its invariance under the second set of non-manifest (1,1) supersymmetries. Besides the usual bihermitean geometry requirements, we found a set of conditions which express the two moment maps in terms of the Killing potentials. In the process we discovered that the third Killing potential is instrumental in fixing a certain ambiguity in the definition of the moment maps (from general arguments, the moment maps are defined up to a function σ such that dσ is invariant under the action of the isometry group). After this ambiguity was fixed in the way we described, then we were able to prove the existence of two generalized moment maps, defined for the twisted generalized Kähler structure in [9], one for each generalized complex structure.
In Section 4 we turn to the subject of T-duality. Our starting point is the gauged (2,2) semichiral sigma model action. We construct the duality functional in (2,2) superspace. Under T-duality, a pair of left and right semichiral superfields along the isometry direction and their antifields are replaced by a pair of chiral and twisted chiral superfields, and their antifields. We work out the T-dual of the torus T 4 and show, at the level of (2,2) superspace, the usual map of the radius of the compact T-duality direction R ↔ 1/R. Section 4.1 is dedicated to spelling out the role of the moment map in the T-duality procedure. Lastly, in section 5, we discuss the quotient construction, and work out one explicit example.
2 The gauged (2,2) sigma model reduced to (1,1) superspace We begin by recalling the new gauged (2,2) supersymmetry algebra, which defines the new semichiral vector multiplet [7,8] (our notation follows [7]): where the gauged supercovariant derivatives are defined as The notation is such that D α ,D a are the usual (2,2) supercovariant derivatives, Γ α is the superconnection, and ξ is the generator of the U(1) gauge transformation 1 . The associated Bianchi identities are: The constraints preserving the semichiral representation are solved by and the standard constraints (γ a ) αβ [∇ α ,∇ β } = −4i∇ a allow for solving the vector superfield gauge potential, Γ a , in terms of the fermionic superfield gauge potential Γ α .
As discussed in [10,6], the gauging of the sigma model can be done most straightforwardly at the level of (2,2) superspace. Here the sigma-model is defined entirely by the Kähler potential, which is a functional of the (2,2) superfields. The (2,2) superfields needed for a complete description of the two-dimensional off-shell (2,2) supersymmetric sigma models are [2]: In the case we are interested in, the Kähler potential depends on left and right semichiral superfields and their antifields 2 Next, one uses that the Grassmann integration is equivalent to differentiation. In order to couple the matter fields to the vector superfield, the supercovariant derivatives D α ,D α are replaced by the gauged supercovariant derivatives ∇ α ,∇ α . Lastly, we descend to the level of (1,1) superspace by replacing the (2,2) gauged supercovariant derivatives by two copies of (1,1) derivatives. The final step is to keep only one of the two (1,1) supersymmetries manifest, by reducing along the direction of the other (1,1). This will give the manifestly (1,1) supersymmetric gauged sigma model.
More concretely, the two (1,1) gauge supercovariant derivatives are defined bŷ It is important to keep in mind that from the point of view of the (1,1) gauged sigma model, thẽ ∇ α derivatives act as the generators of the additional, non-manifest (1,1) supersymmetry.
The (1, 1) gauge supercovariant derivatives obey the algebra The (2, 2) fermionic measure is evaluated using the (2, 2) gauge supercovariant derivatives Using the relation we can show that This allows us to rewrite the fermionic measure in terms of the (1, 1) derivatives as The implicit assumption here is that the Kähler potential that we are gauging is invariant under the symmetry transformation, i.e. it satisfies ξK = 0 (there is of course the possibility that the Kähler potential is invariant up to general Kähler transformations; the extension to this case, though relatively straightforward, is not addressed in this paper).
We now reduce the manifestly (2,2) supersymmetric action to (1, 1) superspace by evaluating the (1, 1) derivatives∇ α∇ α onto the Kähler potential. After some algebra, we obtaiñ where we have kept the notation of [12]: I = (i,ī), ϕ = X|, χ = Y |, etc... By inspecting the resulting (1,1) sigma model action, we see that, as expected, we have the same metric and NS-NS two-form obtained in [12]. However, there are some differences with respect to the case when the gauging of the U(1) isometry is done by using the usual (2,2) super Yang-Mills multiplet [6]. These differences are visible in the terms which depend on the superfield strengths. We shall focus on this aspect in the next section.

Moment maps
In the case of Kähler geometry, which is the target space geometry associated with a sigma model derived from a (2,2) chiral superfield-dependent Kähler potential [13], the gauging of an isometry requires that the generator of the isometry preserves not only the metric (i.e. it is Killing) but the complex structure as well (i.e. it is holomorphic). As a consequence, the isometry generator preserves the symplectic form ω = gJ. Therefore, implies that i ξ ω is locally exact. This defines the moment map also referred to as the Hamiltonian function for symplectic manifolds, and the Killing potential for Kähler manifolds [14]. In the latter case, by going to the holomorphic coordinate base which diagonalizes the complex structure, and using that ω = 2i∂ ij Kdφ i ∧ dφj, one finds This can be integrated in the case of an U(1) isometry to yield up to a constant.
Studying (2,2) supersymmetric two-dimensional sigma models, Gates, Hull and Rocek [1] showed that their target space admits a bi-hermitean metric (hermitean with respect to two complex structures). The complex structures are covariantly constant with respect to a torsion-full connection. The torsion is related to the field strength of a two-form potential, the B field. In the mathematics literature, the bi-hermitean geometry is known as generalized Kähler geometry [4].
If the (2,2) supersymmetric sigma model employs only chiral and twisted chiral superfields, the two complex structures commute. This type of geometry is referred to as an almost product structure space [1]. As in the previous case, the moment map follows from requiring that the isometry generator preserve the anti-symmetric two-forms ω (±) = gJ (±) . This means that In the case of generalized Kähler geometry, ω (±) is no longer a closed form, rather in the presence of a non-trivial B-field it satisfies Then from (22) it follows that where When µ ± can be defined globally they are called moment maps. Since the isometry generator ξ preserves the complex structures, it respects the natural decomposition of the tangent space induced by the chiral φ i and twisted chiral ψ i ′ coordinates. For ξ = ξ i ∂ i + ξī∂ī, the gauging of the sigma model is done by coupling with an ordinary (2,2) vector multiplet. Forξ =ξ i ′ ∂ i ′ +ξī ′ ∂ī′, the gauging is done by coupling with a twisted (2,2) vector multiplet [15] 3 . Following an off-shell (2,2) supersymmetric sigma model analysis, Hull, Papdopoulos and Spence [15] showed that the moment maps are identified with the Killing potentials iξī∂īK and respectively iξī ′ ∂ī′K. In terms of the significance of the moment maps for the generalized Kähler geometry, it can be shown that is either the sum or the difference of the two moment maps µ + , µ − which defines an eigenvector of the generalized complex structure J 1/2 [6], i.e. (ξ ± i 2 (dµ + ± dµ − )) ∈ T ⊕ T * lies in the eigenbundle of J 1/2 . Lastly, we turn to the generic case of bi-hermitean geometry with non-commuting complex structure, which is realized by a semichiral superfield sigma-model [12].
In [6] it was found by studying a certain example of generalized Kähler geometry, the SU (2) × U (1) WZNW sigma model, that the two a priori distinct moment maps are indeed distinct. This point deserves a further clarification since the on-shell (2,2) supersymmetric sigma model analysis in [16] points out to the existence of a unique moment map, with µ + and µ − being identified. In this paper we extend the investigation opened in [6] of an off-shell supersymmetric gauged (2,2) sigma model, by appropriately coupling the semichiral superfields with the newly found (2,2) semichiral vector multiplet [7,8]. In the process we shall find that besides the two moment maps there is a third function, called σ in [6], which together with the two distinct moment maps forms the complete set of three Killing potentials.
The connection between the moment maps and the gauged sigma model action was previously discussed in [15]. The idea is to start from the reduced (1,1) supersymmetric sigma-model action, and require that it is invariant under the additional, non-manifest (1,1) supersymmetries generated by∇ ± . These act on the (1,1) sigma-model superfields as where Φ stands for the sigma-model superfields ϕ I , χ I ′ [12].
The action of the non-manifest supersymmetries on the gauge superconnections is inferred from: Further using that S −iP is a twisted chiral superfield and that T is chiral, we find the non-manifest supersymmetry variation of the field strength superfields: Let us now concentrate on the invariance of the manifestly (1,1) supersymmetric gauged sigma model action under the additional (26,27,28) supersymmetries. In the case we are investigating we have assumed that the Kähler potential is strictly invariant under the action of the U(1) isometry generator ξ.
Because of this assumption, the first term in the gauged action is actually obtained by minimal coupling. In other words, since L ξ g = L ξ B = 0, then the kinetic terms and the B-field dependent terms in the sigma-model are gauged in the same way, by minimal coupling. We have introduced the notation µ 1 , µ 2 for the terms which multiply the superfield strengths S, T 2 in (29), even though we have their concrete expression in terms of derivatives of the Kähler potential from (17). The reason for our feigned ignorance is that we want to be able to show the rapport between µ 1 , µ 2 and the moment maps. This will become transparent once we require that (29) has the additional (1,1) supersymmetries.
The invariance of (29) is conditioned, among other things, by the cancellation of the terms in δS which are proportional to the superfield strengths S, P, T 1 , T 2 . Those terms which are proportional to S are Therefore we find that Similarly, the terms which are which are proportional to T 2 are which implies that the action is invariant provided that In order for these two sets of equations to be satisfied, σ must be such that To complete our investigation of the relationship between the Killing potentials µ 1 , µ 2 , σ and the moment maps µ + , µ − , we recall that we have worked under the assumption that the Kähler potential is invariant under the action of the isometry generator ξK = 0. With the metric and B-field determined by the invariant Kähler potential, then L ξ g = L ξ B = 0. As a consequence, the one form u defined in (25) can be explicitly solved where dσ is an exact one-form, invariant under ξ. What (31) and (33) show is that and that 2µ 1 and 2µ 2 are equal to the sum and respectively the difference of the moment maps µ ± .
The supersymmetry variations which are proportional to the superfield strengths P and T 1 give rise to an equivalent set of constraints. There are three terms which are proportional to each of these superfield strengths. Two of these terms are obvious, coming from supersymmetry variations of (δS)µ 1 and P (δσ), and similarly for the terms proportional to T 1 . The third term will arise from the supersymmetry variations of∇ − Φ · (g + B) ·∇ + Φ, where we keep the contributions coming from the second and third term in (27). After partial integration, these terms combine by using the anticommutator {∇ + ,∇ − }.
Of course, in addition to these constraints, in order to ensure the invariance of the action under the non-manifest (1,1) supersymmetries, the metric and B field must satisfy the usual requirements which define the bi-hermitean geometry. A perhaps unexpected requirement emerging from our supersymmetry analysis is that E = g + B ought to be bi-hermitean. This is, however, in complete agreement with the manifestly (2,2) supersymmetric origin of the (1,1) action (29). From a (2,2) superspace perspective, the complex structures, the metric and the B field arise from second order derivatives of the Kähler potential [12]. These explicit expressions enable the check that indeed E = g+B is bi-hermitean. These expressions should also allow a demonstration that the constraints, equations (31, 33), are also satisfied. While we were unable to show this in general, we have observed that they hold in the flat space and SU (2) ⊗ U (1) examples.

Generalized Moment Maps
In [6,16] an effort was made to check whether the moment maps obtained from the sigma model correspond to the moment maps used in [17,9] as part of the definition of generalized moment maps. The equations derived at the end of Section 3 allow us to extend these previous attempts to (2,2) supersymmetric sigma models with semichiral superfields, i.e. sigma models with three Killing potentials: the two moment maps and the function σ. More explicitly, the equations (31) and (33) can be rewritten as From these equations it follows where we have used that u = −ξB + 1 2 dσ. As in [16], (38) can be written in terms of J 2 , one of the generalized complex structures given in [4], to show that that ξ + u − idµ 1 is an eigenvector of J 2 . This corresponds to the definition of a generalized moment map for twisted generalized Kähler geometry [9]. The proof is given by noting that since then (38) can be written as J 2 (ξ + u) = dµ 1 . This is equivalent to the equation J 2 (ξ + u − idµ 1 ) = i(ξ + u − idµ 1 ) which verifies the claim. Similarly, dµ 2 is used in the construction of a second twisted generalized moment map, eigenvector of J 1 .

T-duality
Next we discuss T-duality. We follow the basic procedure outlined in [18]. First we gauge the U(1) isometry of the sigma model using the prepotentials of the gauge multiplet. Then we add the Lagrange multipliers which will force the field strength of the gauge multiplet to vanish. In the last step leading to the duality functional, we use the gauge freedom to gauge away the appropriate superfields. By solving the Lagrange multiplier constraints and substituting back into the duality functional we return to the original action. The dual action is obtained by imposing the prepotential equations of motion. We will work out one concrete example, T 4 , and observe the characteristic interchange of the S 1 radius R ↔ 1/R in the T-dual actions.
In our discussion of T-duality we will maintain manifest the (2,2) supersymmetry. The gauging of the (2,2) supersymmetric sigma model action [7] is done at the level of the Kähler potential by replacing the left and right semichiral superfields X and Y by: where V 1 and V 2 are the prepotentials of the semichiral vector multiplet. The prepotentials are worth a brief review as their transformations are important for the discussion of T-duality and quotients. The form of the gauge covariant derivative algebra requires that the fermionic gauge potentials satisfy Γ + = D +V1 and Γ − = D −V2 . Since the gauge covariant derivatives are invariant under δΓ α = D α L, the prepotentials share a common transformation by a real scalar superfield denoted L, One can also note, that since left semichiral superfields are in the kernel ofD + and right semichiral superfields are in the kernel ofD − that V 1 and V 2 can transform by left and right semichiral superfields respectively, i.e.
where Λ is left semichiral and U is right semichiral. The substitutions given in (40) as part of the gauging prescription replace the field X, which satisfies the regular semichiral constraintD + X = 0, withX, which satisfies the gauge covariant semichiral constraint ∇ +X = 0, and likewise for Y . The covariantly semichiral superfieldsX andỸ as well as the covariantly semichiral constraints are then invariant with respect to (42) and transform covariantly with respect to (41). Here we assume that the gauging is done such that the Kähler potential is left unchanged under a gauge transformation (in a more general case, the Kähler potential may change by a general Kähler transformation, which leaves the sigma model metric invariant). The replacement in (40) ensures the invariance of the gauged Kähler potential K g = K(X,X,X,Ỹ ).
Because the isometry being gauged is a U(1), we can realize it as a shift in the superfields. The Kähler potential dependence on the fields is This would result in the gauge potentials For concreteness, let's assume that the U(1) generator is and accordingly, On the tangent bundle T we can define three other vectors, which together with with ξ form a vector basis: The gauged Kähler potential can then be rewritten as which emphasizes the role of µ 1 , µ 2 and σ as Killing potentials. In addition, this expression of the gauged Kähler potential makes manifest the dependence on only three of the four prepotentials, besides the three Killing potentials.
In order to select the appropriate supersymmetry representation of the Lagrange multipliers, we first solve the gauge superfield strengths in terms of the prepotentials As mentioned in the previous section, T is a chiral superfield and S − iP is twisted chiral.
Next we add the Lagrange multipliers which enforce the condition that the gauge field is pure gauge (i.e. its field strength vanishes) to obtain where in the second step we substituted the superfield strengths in terms of the prepotentials, and integrated by parts twice. Therefore, φ is chiral and ψ is twisted chiral. Lastly, because the prepotential gauge transformation is a shift by a semichiral superfield we can choose the gauge in which X = 0 and Y = 0. This yields the duality functional To see how we recover the original Kähler potential we study the constraints imposed by the Lagrange multipliers. The φ and ψ equations of motion require Plugging this back into (52) we obtain the original potential.
If on the other hand, we impose the equations of motion of the prepotentials, solve for V 1 and V 2 and substitute back into (52), we obtain the T-dual Kähler potential. This duality replaces a pair of left and right semichiral superfields with a pair of chiral and twisted chiral superfields.
We would like to mention that the duality functional obtained before appears to be related to the Legendre transforms described in [19]. The authors of [19] began by writing the Kähler potential as K = K(V,V , W,W ) − (XV + Y W + c.c), where X, Y are left, right semichiral superfields and V, W are unrestricted. If the Kähler potential has an isometry, resulting in a dependence of only three real independent linear combinations of the unconstrained complex V and W , then by integrating out the semichiral superfields, one is left with a Kähler potential expressed in terms of chiral and twisted chiral superfields.
As a concrete example of the T-duality map we consider the torus T 4 . Its (2,2) supersymmetric sigma model action is derived from the Kähler potential The duality functional is The dual potential, up to generalized Kähler gauge transformations, reads This is indeed is the potential for the T-dual T 4 , this time written in terms of chiral and twisted chiral superfields. As expected, the radius R of the dualized S 1 is mapped into 1/R.

T-duality and the Killing potentials
Having identified the Killing potentials of the theory in terms of the Kähler potential, we would like to highlight their role in the T-duality map. Let us recall the more familiar situation encountered when T-dualizing along a chiral superfield direction, and consider the U(1) Killing vector: From the invariance of the Kähler potential ξK = 0 we derive that This implies that µ(φ,φ) = −iξ i ∂ i K is the Killing potential. µ is also the moment map [14]. Next we turn to the T-duality functional [18] which is constructed starting from the gauged Kähler potential [10]: whereφ = e −iVξφ . This means that K g has the exact same dependence on (φ,φ) as K has on (φ,φ). By varying K g with respect to V we get whereξī = e −iVξξī . As explained previously, the duality functional is given by the gauged Kähler potential plus the Lagrange multipliers enforcing the pure gauge condition with ψ a twisted chiral superfield. The T-dual potential is obtained by imposing the prepotential equation of motion So, T-duality embeds the moment mapμ as the real part of the dual coordinate.
A similar story goes through when considering sigma models with semichiral superfields. Starting with a potential K = K(X, X,Ȳ , Y ) which is invariant under the action of the isometry generator ξ, ξK = 0, we can derive the same equations as those given above. In this case, however, there are three Killing potentials instead of one The potential is gauged by making the substitutions in (40): K g = K(X,X,X,Ỹ ). Just as above, we note that With the duality functional given by (51), the complex prepotential equations of motion are After rewriting them as it is clear that only three of the four equations of motion are independent, sincẽ This is, of course, the statement that the Kähler potential possesses an isometry. The content of the remaning three equation is as follows: -the Killing potentialμ 1 is mapped to the imaginary part of ψ: -μ 2 is mapped to the imaginary part of φ: -the third Killing potentialσ is mapped to the sum of the real parts of φ and ψ:

Quotients
Lastly we address the quotient construction for the semichiral sigma models. The quotient manifold is obtained by extremizing the gauged Kähler potential with respect to the three real linear independent combinations of the prepotentials for each of the isometry group generators. The dimension of the quotient manifold, which remains bi-hermitean, is dim M − 4dim G.
After adding the FI terms 4 to the gauged Kähler potential given in (49), and extremizing with respect toṼ ,Ṽ 1 andṼ 2 , one finds We end with three constraints, corresponding to the three Killing potentials that can be defined when gauging the semichiral sigma model. These are the equations which define the quotient. In practical terms, one solves them for the prepotentials, and substitutes back into the gauged Kähler potential to arrive at the quotient manifold potential. We recall that a similar constraint (involving the only Killing potential) defines the Kähler quotient on a Kähler manifold [14].
Next, solving for |A| we find Further substitutingṼ 1 in terms ofṼ 2 yields which may be solved directly forṼ 2 , giving e 2Ṽ 2 = (r 1 + r 2 )zz − r 2 xy ± ((r 1 + r 2 )zz − r 2 xy) 2 + 2(r 2 2 + r 2 0 )(zz − 1 2 xy)xy 2y(zz − 1 2 xy) The reality ofṼ 2 will require thatzz ≥ 1 2 xy indicating the presence of a boundary in the quotient target space. The solutions forṼ 1 andṼ follow with To complete the discussion of the quotient, we have to choose a gauge. Considering (74), we will pick the gauge where X ′ 1 = 1, X ′ 2 = X 2 X 1 , Y ′ 1 = 1 and Y ′ 2 = Y 2 Y 1 . Despite the complexity of the final answer, the gauge fixing step demonstrates that the dimension of the quotient manifold is smaller by 4 (which was the expected result since we quotient a U(1) isometry). Bi-hermiticiy of the quotient geometry is guaranteed since (2, 2) supersymmetry has been preserved. An interesting point is that since the quotient potential is more than quadratic in the fields, the quotient target space has non-trivial H flux.