T-Duality in $(2,1)$ Superspace

TL;DR

This paper derives T-duality rules for (2,1) supersymmetric sigma-models, clarifying their geometric structure and showing that obstructions to T-duality are less restrictive than previously thought.

Contribution

It provides the first explicit T-duality transformation rules in (2,1) superspace and develops a geometric gauging approach for these models.

Findings

Derived T-duality transformation rules in (2,1) superspace

Clarified the geometric structure of (2,1) sigma models

Found that obstructions to (2,1) T-duality are weaker than in other models

Abstract

We find the T-duality transformation rules for 2-dimensional (2,1) supersymmetric sigma-models in (2,1) superspace. Our results clarify certain aspects of the (2,1) sigma model geometry relevant to the discussion of T-duality. The complexified duality transformations we find are equivalent to the usual Buscher duality transformations (including an important refinement) together with diffeomorphisms. We use the gauging of sigma-models in (2,1) superspace, which we review and develop, finding a manifestly real and geometric expression for the gauged action. We discuss the obstructions to gauging (2,1) sigma-models, and find that the obstructions to (2,1) T-duality are considerably weaker.

Equations

$$S^0=S_{kin}^0+S_{WZ}^0.$$

$$S_{kin}^0=\frac{1}{2}{\int }_{\Sigma }\ast tr(h^{-1}{\varphi }^{\ast }g)$$

$$L_{kin}^0=\frac{1}{2}{g}_{ij}(x(\sigma ))d{x}^i\wedge \ast d{x}^j.$$

$$S_{WZ}^0={\int }_{\Gamma }{\varphi }^{\ast }H,$$

$$S_{WZ}^0=\frac{1}{3}{\int }_{\Gamma }{H}_{ijk}d{x}^i\wedge d{x}^j\wedge d{x}^k.$$

$$H=db,$$

$$S_{WZ}^0=\frac{1}{2}{\int }_{\Sigma }{b}_{ij}(x(\sigma ))d{x}^i\wedge d{x}^j.$$

$$[{\mathcal{L}}_K,{\mathcal{L}}_L]=f_{KL}{}^M{\mathcal{L}}_M.$$

$$\delta x^i={\lambda }^K{\xi }_K^i(x)$$

$${\iota }_{K}H=d{u}_K$$

$$c_{KL}\equiv {\iota }_K{u}_L$$

$${\mathcal{L}}_{K}b=0,$$

$$u_K={\iota }_{K}b,$$

$$\delta A^M=d{\lambda }^K-f_{KL}{}^M{A}^K{\lambda }^L.$$

$${\mathcal{L}}_K{u}_L=f_{KL}{}^M{u}_M$$

$${\iota }_K{u}_L=-{\iota }_L{u}_K$$

$$c_{KL}=-c_{LK}.$$

$$D_a{x}^i\equiv {\partial }_a{x}^i-A_a^K{\xi }_K^i$$

$$F^M=d{A}^M-\frac{1}{2}{f}_{KL}{}^M{A}^K\wedge A^L,$$

$$S=S_{kin}+S_{WZW}.$$

$$S_{kin}=\frac{1}{2}{\int }_{\Sigma }{g}_{ij}D{x}^i\wedge \ast D{x}^j,$$

$$S_{WZW}={\int }_{\Gamma }\left(\frac{1}{3}{H}_{ijk}D{x}^i\wedge D{x}^j\wedge D{x}^k+F^K\wedge u_{Ki}D{x}^i\right),$$

$$S_{WZW}={\int }_{\Sigma }\left(\frac{1}{2}{b}_{ij}d{x}^i\wedge d{x}^j+A^K\wedge u_K+\frac{1}{2}{c}_{KL}{A}^K\wedge A^L\right),$$

$${\mathcal{L}}_K{\xi }^L=0,{\mathcal{L}}_K{u}_L=0,{\mathcal{L}}_K{c}_{LM}=0.$$

$${\iota }_K{\iota }_{L}H={\mathcal{L}}_K{u}_L-d{\iota }_K{u}_L$$

$${\iota }_K{\iota }_{L}H=-d{c}_{KL},$$

$${\iota }_K{\iota }_L{\iota }_{M}H=0.$$

$$S_{LM}={\int }_{\Sigma }{A}^K\wedge d{\hat{x}}_K.$$

$${\xi }_K^i\frac{\partial }{\partial x^i}=\frac{\partial }{\partial x^K},$$

$$\delta S={\int }_{\Sigma }{c}_{(KL)}d{\lambda }^K\wedge A^L.$$

Full text

1. 1 Introduction

2. 2 The gauged sigma model and T-duality

3. 2.1 The gauged bosonic sigma model

4. 2.2 Dualisation

5. 2.3 Duality as a quotient of a higher dimensional space

6. 3 The $(2,1)$ sigma model in superspace

7. 4 Isometries in the $(2,1)$ sigma model

8. 5 The $(2,1)$ gauge multiplet and gauge symmetries

9. 6 The gauged $(2,1)$ sigma model

10. 7 T-duality of $(2,1)$ supersymmetric theories

11. 7.1 Generalities

12. 7.2 Computations

13. 7.2.1 T-duality from the gauged Lagrangian (6.5)

14. 7.2.2 T-duality from the geometric form (6.8) of the gauged Lagrangian.

15. 7.3 The dual geometry

16. 8 Comparison to the Buscher rules

17. 8.1 T-duality on the complex plane

18. 8.2 T-duality on a torus

19. 9 Geometry and obstructions for $(2,1)$ T-duality

20. 10 Conclusion

21. A Review of chiral and vector representations

22. B Gauge invariance and hermiticity of the action

23. C Calculation of $A_{-}(\varphi)$

24. D Reduction

25. D.1 Reduction of a Kähler $(2,2)$ sigma model to $(2,1)$ superspace

26. D.2 Reduction of a general $(2,1)$ sigma model to $(1,1)$ superspace

1 Introduction

Supersymmetric nonlinear sigma models with $D$-dimensional target spaces have a rich structure, which makes them good tools for studying various geometries. The target-space geometries are constrained by the number of supersymmetries; in particular, there is a direct correspondence between target-space complex structures and world-volume supersymmetries. For two dimensional $(p,q)$ supersymmetric models, the relationship between geometry and supersymmetry is particularly rich [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. The $(2,2)$ models of [2] have generalised Kähler geometry [12, 13]. A more general complex geometry with torsion arises for $(2,0)$ supersymmetry [4], and for $(2,1)$ supersymmetry [6], while the general geometry for $(p,q)$ supersymmetric models for all $p,q$ was found in [6]; see also [7, 8]. The $(2,1)$ supersymmetric models [5] will be the focus of this paper and are relevant for supersymmetric compactifications of ten-dimensional superstring theories as well as for critical superstrings with $(2,1)$ supersymmetry [14], which have interesting applications [15, 16, 17, 18]. Their target space geometries include the generalised Kähler geometries of the $(2,2)$ models as special cases. The reduction of $(2,2)$ models to $(2,1)$ superspace was discussed in ref. [19]. The $(2,1)$ superspace formulation was first given in [20].

T-duality relates two-dimensional sigma-models that have different target space geometries but which define the same quantum field theory; for a review and references, see [21]. When the target space of a model has isometry group $U(1)^{d}$, its T-dual is found by gauging the isometries and adding Lagrange multiplier terms (plus an important total derivative term) [22, 23, 24]. Integrating out the Lagrange multipliers constrains the (world-sheet) gauge fields to be trivial and so gives back the original model, while integrating out the gauge fields yields the T-dual theory, with the dual geometry given by the Buscher rules [22]. Various gaugings in and out of superspace have been described in [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37].

The starting point for T-duality is the gauging of the sigma model, and extended supersymmetry imposes restrictions on the gauging. In particular, the isometries must be compatible with the supersymmetries, i.e.  holomorphic with respect to all the associated complex structures [38, 32, 39]. For $(2,2)$ supersymmetry, the gauging was discussed in [38, 32, 26, 28, 37, 27], while the gauging of $(2,1)$ supersymmetric models was given in [34, 35] for the superspace formulation of [20] and in [32] for the formulation of [7, 8].

The supersymmetric T-duality transformations have an interesting geometric structure. For sigma models with Kähler target geometry, the T-duality changes the Kähler potential by a Legendre transformation [40, 24]. In general, duality can change the representation of the supersymmetry [40]. T-duality for $(2,2)$ supersymmetric sigma models has been studied in [24, 41, 42, 43, 44].

Here we will use the results of [34, 35] to analyse T-duality for $(2,1)$ supersymmetric models in $(2,1)$ superspace [20]. Adding Lagrange multiplier terms to the gauged theory and integrating out the gauge multiplets gives a dual geometry, with a $(2,1)$ supersymmetric version of the T-duality transformation rules. The supersymmetric gauging involves a complexification of the action of the isometry group, resulting in a T-duality transformation that is a complexification of the usual T-duality rules. The complexified T-duality we find is equivalent to a real Buscher T-duality combined with a diffeomorphism; this is the same mechanism that was previously found for the $(2,2)$ supersymmetric T-duality (see [22, 42]).

For bosonic and $(1,1)$ sigma-models with Wess-Zumino term, there are geometric and topological obstructions to gauging in general [29, 30]. For T-duality, however, the obstructions are considerably milder [45, 36]. Here we will extend this discussion to $(2,1)$ models, analysing the obstructions to gauging and T-duality. Moreover, we will interpret our results for T-duality in terms of generalised moment maps and a generalised Kähler quotient.

The paper is organised as follows. In section 2, we first review the general gauged sigma model and the obstructions to its gauging. We then summarise the formulation of T-duality of [36] in terms of a lift to a higher-dimensional sigma model and show that the obstructions to T-duality are much milder than the obstructions to gauging – one can T-dualise an ungaugable sigma model. In particular, we recall and emphasize that the Buscher rules are modified when the sigma-model Lagrangian is invariant only up to a total derivative term under the isometry used for the T-duality [45, 36]. In section 3 we give the superspace description of the $(2,1)$ models. Section 4 discusses the isometries of $(2,1)$ models in superspace. In section 5, we review the superspace description of the $(2,1)$ Yang-Mills supermultiplet. In section 6 we review the results of [34, 35] on the gauging of the $(2,1)$ models. We discuss T-duality for the $(2,1)$ sigma models in section 7, and derive the duality transformations of the potentials for the $(2,1)$ geometries with torsion. We find the duality transformations for the metric and $b$-field, which give a complex version of the Buscher rules. In section 8, we explain how our complexified T-duality transformations give the real Buscher rules combined with diffeomorphisms, and illustrate this with some examples. In section 9, we adapt the general results of ref. [36] to the geometry and T-dualisation of $(2,1)$ models, including the cases for which there are obstructions to the gauging and for which the standard T-dualisation procedure fails. Section 10 contains a summary of our results. Some technical details are collected in four appendices.

2 The gauged sigma model and T-duality

2.1 The gauged bosonic sigma model

The two-dimensional sigma model with $D$-dimensional target space $M$ is a theory of maps $\phi:\Sigma\to M$, where $\Sigma$ is a 2-dimensional manifold. The action is the sum of a kinetic term $S^{0}_{kin}$ and a Wess-Zumino term $S^{0}_{WZ}$,

[TABLE]

Given a metric $g$ on $M$ and a metric $h$ on $\Sigma$, the kinetic term can be written as

[TABLE]

where the Hodge dual on $\Sigma$ for the metric $h$ is denoted by $*$ and $\phi^{*}g$ is the pull-back of $g$ to $\Sigma$. If $x^{i}$ ($i=1,\ldots D$) are coordinates on $M$ and $\sigma^{a}$ are coordinates on $\Sigma$, the map is given locally by functions $x^{i}(\sigma)$ and $tr(h^{-1}\phi^{*}g)=h^{ab}(\phi^{*}g)_{ab}=h^{ab}g_{ij}\partial_{a}x^{i}\partial_{b}x^{j}$, so that the Lagrangian 2-form can be written locally as

[TABLE]

Here and in what follows, the pull-back $\phi^{*}(dx^{i})=\partial_{a}x^{i}d\sigma^{a}$ will be written as $dx^{i}$, and it should be clear from the context whether a form on $M$ or its pull-back is intended.

The Wess-Zumino term is constructed using a closed 3-form $H$ on $M$. We write

[TABLE]

where $\Gamma$ is any 3-manifold with boundary $\Sigma$. This can be written in terms of local coordinates as

[TABLE]

Locally, $H$ is given in terms of a 2-form potential $b$ with

[TABLE]

and the Wess-Zumino term can be written locally in terms of a 2-form Lagrangian on a patch in $\Sigma$

[TABLE]

The functional integral involving the Wess-Zumino term (2.4) is well-defined and independent of the choice of $\Gamma$ provided $\frac{1}{2\pi}H$ represents an integral cohomology class111When the third cohomology group $\mathbb{H}^{3}$ of M is nontrivial, this leads to a quantisation condition for H; if $\mathbb{H}^{3}$ is trivial, then (2.6) is globally defined and there is no quantisation condition. on $M$.

The conditions for gauging isometries of this model were derived in [29, 30] and will now be briefly reviewed. Suppose there are $d$ Killing vectors $\xi_{K}$ ($K=1,\ldots d$) with ${\cal L}_{K}g=0$, ${\cal L}_{K}H=0$, where ${\cal L}_{K}$ is the Lie derivative with respect to $\xi_{K}$. The $\xi_{K}$ generate an isometry group with structure constants $f_{KL}{}^{M}$, with

[TABLE]

Then under the transformations

[TABLE]

with constant parameters $\lambda^{K}$, the action (2.1) changes by a surface term if $\iota_{K}H$ is exact, so that the equation

[TABLE]

is satisfied for some (globally defined) 1-forms $u_{K}$. The $u_{K}$ are defined by (2.10) up to the addition of exact forms. Thus the transformations (2.9) are global symmetries provided $\iota_{K}H$ is exact. When this is the case, the functions

[TABLE]

are globally defined. We note that in the special case in which the $b$-field is invariant,

[TABLE]

we have

[TABLE]

but in general $u_{K}\neq\iota_{K}b$.

The gauging of the sigma-model [29, 31, 30] consists in promoting the symmetries (2.9) to local ones, with parameters that are now functions $\lambda^{K}(\sigma)$, by seeking a suitable coupling to connection 1-forms $A^{K}$ on $\Sigma$ transforming as

[TABLE]

The conditions for gauging to be possible found in [29, 30] are that (i) $\iota_{K}H$ is exact, (ii) a 1-form $u_{K}=u_{Ki}dx^{i}$ satisfying (2.10) can be chosen that satisfies the equivariance condition

[TABLE]

(so that $\iota_{K}H$ represents a trivial equivariant cohomology class [46]), and (iii)

[TABLE]

so that the globally defined functions (2.11) are skew,

[TABLE]

Defining the covariant derivative of $x^{i}$ by

[TABLE]

and the field strength

[TABLE]

the gauged action is [29]

[TABLE]

The gauged metric term is minimally coupled:

[TABLE]

whereas the gauged Wess-Zumino-Witten term involves a non-minimal term:

[TABLE]

with $\partial\Gamma=\Sigma$. It was shown in [29, 30] that this is closed and locally can be written as

[TABLE]

with $u_{K}=u_{Ki}dx^{i}$. If the gauge group $G$ acts freely on $M$, then the gauged theory (2.20) gives a quotient sigma model with target space $M/G$ (the space of gauge orbits) on fixing a gauge and eliminating the gauge fields using their equations of motion.

2.2 Dualisation

A general method of dualisation of the ungauged sigma model (2.1) on $(M,g,H)$ is to gauge an isometry group $G$ as above and add the Lagrange multiplier term $\int_{\Sigma}F^{K}\hat{x}_{K}$ involving $d$ scalar fields $\hat{x}_{K}$. The Lagrange multiplier fields $\hat{x}_{K}$ impose the constraint that the gauge fields $A$ are flat and so pure gauge locally, so that (at least locally) one recovers the ungauged model. (If $\Sigma$ is simply connected, e.g. if $\Sigma=S^{2}$ or $\Sigma=\mathbb{R}^{2}$, then $A$ is pure gauge and one recovers precisely the ungauged model.) Alternatively, fixing the gauge by a suitable constraint on the coordinates $x^{i}$ and integrating out the gauge fields $A$ gives a dual sigma model whose coordinates now include the fields $\hat{x}_{K}$. This method applies quite generally, including the cases of non-Abelian or non-compact $G$.

In general, the two dual sigma models are distinct in the quantum theory. However for special cases, the two dual sigma models can define the same quantum theory, in which case the two dual theories are said to be related by a T-duality. T-dual theories arise for isometry groups $G$ that are compact and Abelian so that $G=U(1)^{d}$ with the action of $G$ defining a torus fibration on $M$ for which the torus fibres are the orbits of $G$. There are also further restrictions on the torus fibration; see e.g. [36]. The classic example is that in which $M$ is a torus $T^{d}$, with the natural action of $G=U(1)^{d}$ on the torus.

String theory backgrounds require sigma models that define conformally invariant quantum theories. For a sigma model on $(M,g,H)$ to define a conformal field theory in general requires the addition of a coupling to a dilaton field $\Phi$ on $M$ through a Fradkin-Tseytlin term, and the T-duality then takes a sigma model on $(M,g,H,\Phi)$ to a dual one $(M^{\prime},g^{\prime},H^{\prime},\Phi^{\prime})$ on a manifold $M^{\prime}$ (which in general is different from $M$), with the two sigma models defining the same conformal field theory. A proof of the quantum equivalence of T-dual CFT’s was given in [24].

For applications to T-duality, we focus on the case of Abelian isometries. We derive dual pairs of geometries for general Abelian isometry groups (including non-compact groups or ones that act with fixed points). It is convenient to refer to all of these as T-dualities, although not all lead to full quantum equivalence between dual theories, so not all are proper T-dualities in the strict sense. Our main interest will be in dual pairs that define equivalent quantum theories, but the same formulae apply to the more general class of dual theories.

For Abelian $G$, $f_{KL}{}^{M}=0$, so that, assuming $u$ satisfies the equivariance condition (2.15),

[TABLE]

Starting from (2.10) and (2.24), the identity

[TABLE]

implies $\iota_{K}\iota_{L}H$ is exact, with

[TABLE]

and

[TABLE]

To dualise the ungauged sigma model (2.1) on $(M,g,H)$ with respect to $d$ Abelian isometries, one gauges the isometries as above and adds the following Lagrange multiplier term involving $d$ scalar Lagrange multiplier fields $\hat{x}_{K}$ [49, 22, 23, 47, 24, 48]

[TABLE]

This differs from the expression $\int_{\Sigma}F^{K}\hat{x}_{K}$ by a surface term that is crucial for quantum equivalence [24, 48]. The $\hat{x}_{K}$ impose the constraint that the gauge fields $A$ are flat. For compact $G$, the holonomies $e^{i\oint A}$ around non-contractible loops on $\Sigma$ are eliminated by requiring the $\hat{x}_{K}$ to be periodic coordinates of a torus $T^{d}$ so that the winding modes of the $\hat{x}_{K}$ set the holonomies $e^{i\oint A}$ to the identity. Then the gauge field is trivial for any $\Sigma$ and $A$ can be absorbed by a gauge transformation, recovering the original ungauged model. Alternatively, fixing a gauge and integrating out the gauge fields $A$ gives the T-dual sigma model. In adapted coordinates $x^{i}=(x^{K},y^{\mu})$ in which

[TABLE]

one can fix the gauge setting the $x^{K}$ to constants and this gives a dual geometry with coordinates $(\hat{x}_{K},y^{\mu})$.

One of the conditions for gauging to be possible was that $c_{KL}=-c_{LK}$. If one relaxes this constraint, then (2.20) is no longer gauge invariant, with its gauge variation depending on the constants $c_{(KL)}$ and given by

[TABLE]

Remarkably, this variation can then be cancelled by the variation of (2.28) by requiring that

[TABLE]

so that $\hat{x}_{K}$ can be thought of as a compensator field, transforming as a shift under the gauge symmetry. This was first observed in [45] for the special case of a single isometry and extended to the general case in [36]. Furthermore, it was shown in [36] that introducing the fields $\hat{x}_{K}$ through (2.28) allows all three conditions for gauging listed above to be relaxed and replaced by one much milder condition. This allows the gauging and T-dualisation of ungaugable sigma models; we next review the construction of [36].

2.3 Duality as a quotient of a higher dimensional space

It is natural to seek to interpret the Lagrange multiplier fields $\hat{x}_{K}$ as $d$ extra coordinates, so that we have a sigma model with $D+d$ dimensional target space $\hat{M}$ with coordinates $\hat{x}^{\alpha}=(x^{i},\hat{x}_{K})$, where $\alpha=1,\ldots D+d$. Then the gauged action plus the Lagrange multipler term can be viewed as a gauge-invariant sigma model on $\hat{M}$, and this can be compared with the standard form of the gauged sigma model (2.20) reviewed above. In particular, the terms linear in $A$ in the sum of the Wess-Zumino term (2.23) and the Lagrange multiplier term (2.28) are

[TABLE]

which suggests introducing a modified 1-form

[TABLE]

on $\hat{M}$. If the condition that $u_{K}$ is a globally defined one-form is dropped, the constraint that $du_{K}$ is a globally defined closed 2-form suggests interpreting $u_{K}$ as a connection one-form on a $U(1)^{d}$ bundle over $M$. If $\hat{x}_{K}$ are taken as fibre coordinates, then $\hat{u}_{K}$ can be globally defined one-form on $\hat{M}$; this is the starting point for the construction of [36].

The space $\hat{M}$ with coordinates $\hat{x}^{\alpha}=(x^{i},\hat{x}_{K})$ is then a bundle over $M$ with projection $\pi:\hat{M}\to M$ which acts as $\pi:(x^{i},\hat{x}_{K})\mapsto x^{i}$. A (degenerate) metric $\hat{g}$ and closed 3-form $\hat{H}$ can be chosen on $\hat{M}$ with no $\hat{x}_{K}$ components, i.e.

[TABLE]

where $\pi^{*}$ is the pull-back of the projection. The pull-back will often be omitted in what follows, so that the above conditions will be abbreviated to $\hat{g}=g,\hat{H}=H$. Then the only non-vanishing components of $\hat{g}_{\alpha\beta}$ are $g_{ij}$, $\partial/\partial\hat{x}_{K}$ is a null Killing vector, and the only non-vanishing components of $\hat{H}_{\alpha\beta\gamma}$ are $H_{ijk}$.

We consider the general set-up with $d$ commuting vector fields on $M$ preserving $H$. This implies that there are local potentials $u_{K}$ with $\iota_{K}H=du_{K}$, but they need not be global 1-forms, and need not satisfy (2.15) or (2.16).

We lift the Killing vectors $\xi_{K}$ on $M$ to vectors $\hat{\xi}_{K}$ on $\hat{M}$ with

[TABLE]

for some $\Omega_{KL}$ to be determined below. As the metric $g$ and the torsion 3-form $H$ are both independent of the coordinates $\hat{x}_{K}$, the $\hat{\xi}_{K}$ are Killing vectors on $\hat{M}$:

[TABLE]

As $du=d\hat{u}$, with $\hat{u}$ given in (2.32), it follows that

[TABLE]

where $\hat{\iota}_{K}$ denotes the interior product with $\hat{\xi}_{K}$. From (2.32), we find

[TABLE]

If we now choose

[TABLE]

then

[TABLE]

and the functions on $\hat{M}$ defined by

[TABLE]

are found to satisfy

[TABLE]

where the functions $c_{KL}$ are defined in (2.11).

Next, the Lie derivative of the potentials $\hat{u}_{K}$ with respect to $\hat{\xi}$ is now zero:

[TABLE]

so the $\hat{u}_{K}$ are equivariant. Finally, if

[TABLE]

then the isometry group generated by the $\hat{\xi}_{K}$ is Abelian,

[TABLE]

Note that this condition implies that $\hat{\iota}_{K}\hat{\iota}_{L}\hat{\iota}_{M}\hat{H}=0$.

The target space $\hat{M}$ has dimension $D+d$, where $D$ is the dimension of $M$ and $d$ is the dimension of the Abelian gauge group $G$. The gauged model on $\hat{M}$ gives, on eliminating the gauge fields, a quotient sigma model with target space given by the space of orbits, $\hat{M}/G$, which is also of dimension $D$. The T-dual geometry is given by this quotient space.

In summary, if we start from a geometry $(M,g,H)$ preserved by $d$ commuting Killing vectors $\xi_{K}$, then on a patch $U$ of $M$ we can find local potentials $u_{K}$ satisfying $du_{K}=\iota_{K}H$ and lift them to Killing vectors $\hat{\xi}_{K}$ and potentials $\hat{u}_{K}$ on a patch of $\hat{M}$. If the torsion 3-form $H$ on $M$ satisfies $\iota_{K}\iota_{L}\iota_{M}H=0$, then there are no further local obstructions to gauging the isometries on $\hat{M}$ generated by $\hat{\xi}_{K}$, even when there are local obstructions to gauging the isometries on $M$ generated by $\xi_{K}$. For the gauged action on $\hat{M}$ to be globally defined, one needs to specify the bundle over $M$ by giving the transition functions for the coordinates $\hat{x}_{K}$, require that the $\hat{\xi}_{K}$ are globally defined vector fields on $\hat{M}$ and also that the $\hat{u}_{K}$ are globally defined 1-forms on $\hat{M}$. In the overlaps $U\cap U^{\prime}$ of patches $U,U^{\prime}$ on $M$, the potentials $u_{K}$ satisfying $du_{K}=\iota_{K}H$ are related by $u^{\prime}_{K}=u_{K}+d\alpha_{K}$ for some transition functions $\alpha_{K}$, so that the $u_{K}$ are components of a connection on $M$ with field strength given by $\iota_{K}H$. The $\hat{x}_{K}$ are then fibre coordinates with $\hat{u}_{K}=u_{K}+d\hat{x}_{K}$ globally defined on $\hat{M}$. If the Killing vectors $\xi_{K}$ can be normalised so that $\frac{1}{2\pi}\iota_{K}H$ all represent integral cohomology classes, then the bundle can be taken to be a $U(1)^{d}$ bundle with fibres $(S^{1})^{d}$, while otherwise it is a line bundle with fibres $\mathbb{R}^{d}$. Details of the global structure are given in [36]. For T-duality, we require that the fibres be circles. Generalisations to cases in which the $\hat{\xi}_{K}$, the $\hat{u}_{K}$ or both are only locally defined, or in which $\iota_{K}\iota_{L}\iota_{M}H\neq 0$, were discussed in [36, 50, 51, 52]; such T-dualities, when they can be defined, typically lead to non-geometric backgrounds.

We remark on an important observation made in [45] for a single isometry and in [36] for the general case: when (2.12) is not satisfied, that is, when ${\cal L}_{K}b\neq 0$ and hence when $u_{K}\neq\iota_{K}b$, the Buscher rules [22] are modified. For a single isometry in adapted coordinates $x^{i}=(x^{0},y^{\mu})$, $\xi=\partial/\partial x^{0}$, the dual geometry has coordinates $(\hat{x}^{0},y^{\mu})$ and the modified Buscher rules are:

[TABLE]

[TABLE]

The usual Buscher rules are recovered when $u_{\mu}=b_{0\mu}$. Geometric formulae for the duality transformations for the tensors $g,H$ (without using adapted coordinates) for arbitrary numbers of isometries are given in [36].

Finally, the global issues which may arise when T-dualising are dealt with in the standard way. Suppose the coordinate $x^{0}$ is periodic with $x^{0}\sim x^{0}+2\pi$, and the metric contains the radii: $g_{00}=R^{2}$. Here, as throughout the paper, we have set the string tension $T=1$, but to keep track of dimensions, we can introduce it by rescaling the metric $g_{ij}\to Tg_{ij}$, so $g_{00}=TR^{2}$; then the radius in dimensionless units is $\sqrt{T}R$. After we gauge and introduce the dual coordinate $\hat{x}^{0}$, we can insure the holonomies of the gauge fields are trivial and hence the model is equivalent to the original ungauged model by insisting that $\hat{x}^{0}$ is periodic with $\hat{x}^{0}\sim\hat{x}^{0}+2\pi$. Consider the functional integral given by

[TABLE]

where $S$ is the gauged sigma model action (2.20), and $S_{LM}$ is the Lagrange multiplier term (2.28). Then (2.46) is invariant under large gauge transformations for compact world-sheets $\Sigma$ of arbitrary topology, and using the Buscher rules we have

[TABLE]

The analysis of the geometry, gauging and T-duality given in this section for bosonic sigma models readily extends to (1,1) supersymmetric sigma models formulated in (1,1) superspace: the geometry of the gauging is just as in the bosonic case. For such (1,1) models to have $(2,1)$ supersymmetry requires the existence of a complex structure with certain restrictions on the geometry. For the gauging to be possible with manifest $(2,1)$ supersymmetry requires the Killing vectors to be holomorphic. The geometry of the gauged $(2,1)$ sigma models and their application to T-duality will be analysed in the following sections.

3 The $(2,1)$ sigma model in superspace

The $(2,1)$ superspace is parametrised by two Bose coordinates $\sigma^{{{=\hskip-3.75pt{\vrule width=0.5pt,height=5.0pt,depth=1.0pt}}\hskip 3.75pt}},\sigma^{=}$, a complex Fermi chiral spinor coordinate $\theta^{+},\bar{\theta}^{+}$, and a single real Fermi coordinate $\theta^{-}$ of the opposite chirality. It is natural to define the complex conjugate left-handed spinor derivatives

[TABLE]

as well as a real right-handed spinor derivative

[TABLE]

These spinor derivatives satisfy the algebra

[TABLE]

We denote by $M$ the $D$ real dimensional target space manifold of the sigma model and pick local coordinates $x^{i}$, $i=1,\ldots D$ in which the metric and torsion potential are $g_{ij}$ and $b_{ij}$. It was shown in [4, 2, 25, 6] that invariance of the (1,1) supersymmetric sigma model action under a second (right-handed) chiral supersymmetry requires that

(i) $D$ is even

(ii) $M$ admits a complex structure222Supersymmetric models with almost complex structures were considered in [53, 54]; they obey a modified supersymmetry algebra and are not considered here. $J^{i}{}_{j}$

(iii) the metric is hermitian with respect to the complex structure and

(iv) the complex structure $J^{i}{}_{j}$ is covariantly constant with respect to the connection $\nabla^{+}=\nabla+\frac{1}{2}g^{-1}H$ with torsion $\frac{1}{2}g^{-1}H$.

We assume that these conditions are satisfied so that the sigma model has $(2,1)$ supersymmetry. We choose a complex coordinate system $z^{\alpha}$, $\bar{z}^{\bar{\beta}}=(z^{\beta})^{*}$, $(\alpha,\bar{\beta}=1\ldots\frac{1}{2}D)$ in which the line element is $ds^{2}=2g_{\alpha\bar{\beta}}dz^{\alpha}d\bar{z}^{\bar{\beta}}$ and the complex structure is constant and diagonal,

[TABLE]

The supersymmetric sigma model can then be formulated in $(2,1)$ superspace in terms of scalar superfields $\varphi^{\alpha}$, $\bar{\varphi}^{\bar{\alpha}}=(\varphi^{\alpha})^{*}$, which are constrained to satisfy the chirality conditions

[TABLE]

The lowest components of the superfields $\varphi^{\alpha}|_{\theta=0}=z^{\alpha}$ are the bosonic complex coordinates of $M$. The most general renormalizable and Lorentz invariant $(2,1)$ superspace action written in terms of chiral scalar superfields is [20]

[TABLE]

where

[TABLE]

and

[TABLE]

Here F is a holomorphic section, as it is defined only up to the addition of a complex constant. Since F depends only on chiral superfields, the integration in (3.8) is over $\theta^{+}$ only (and not over $\bar{\theta}^{+}$). The term $S_{2}$ is the analogue of the F-term in four dimensional supersymmetric field theories. In particular, this term can spontaneously break supersymmetry, it is not generated in sigma model perturbation theory if it is not present at tree level, and it is subject to a nonrenormalisation theorem, so that it is not corrected from its tree level value (up to possible wave-function renormalisations).

The $(2,1)$ sigma model geometry is sometimes referred to as strong Kähler with torsion (or SKT for short). It is determined locally by the complex vector field $k_{\alpha}(z,\bar{z})$ with complex conjugate

[TABLE]

The metric, torsion potential and torsion are given by

[TABLE]

where $b^{\prime}$ is the torsion potential in a gauge where it is purely $(1,1)$. If the torsion $H=0$, the manifold $M$ is Kähler with $k_{\alpha}={\textstyle{\frac{1}{2}}}\frac{\partial}{\partial z^{\alpha}}K(z,\bar{z})$ where $K(z,\bar{z})$ is the Kähler potential, and the $(2,1)$ supersymmetric model actually has $(2,2)$ supersymmetry, while for $H\neq 0$, $M$ is a hermitian manifold with torsion of the type introduced in [4, 2].

The torsion potential $b_{ij}$ is only defined up to an antisymmetric tensor gauge transformation of the form

[TABLE]

The (1,1)-form potential

[TABLE]

can be transformed to a (2,0)+(0,2) form by a gauge transformation

[TABLE]

where

[TABLE]

The geometry (3.10) is preserved by the transformation

[TABLE]

provided $\tau_{\alpha}$ satisfies

[TABLE]

for some arbitrary real $\chi$. This implies that $\tau$ is of the form

[TABLE]

for some holomorphic $\vartheta_{\alpha}$. The symmetry (3.15) is the analogue of the generalised Kähler transformation discussed in [2]. It leaves the metric and torsion invariant, but changes $b_{ij}$ by an antisymmetric tensor gauge transformation of the form (3.11).

4 Isometries in the $(2,1)$ sigma model

For the application to T-duality discussed in the following sections, we shall be interested in Abelian groups of isometries. For completeness, however, we discuss the general case of non-Abelian isometry groups.

Let $G$ be a group of isometries of $M$ generated by Killing vector fields $\xi^{i}_{K}$ that preserve the metric and 3-form $H$, ${\cal L}_{K}g=0$, ${\cal L}_{K}H=0$, and satisfy the algebra (2.8). This symmetry will be consistent with (2,1) supersymmetry if

[TABLE]

This allows us to write the symmetry of the $(2,1)$ supersymmetric model in (2,1) superspace as

[TABLE]

The constraint (4.1) is the condition that the $\xi^{i}_{K}$ are holomorphic Killing vectors333 A discussion of sigma models with non-holomorphic isometries can be found in ref. [39]. with respect to the complex structure $J_{ij}$, giving

[TABLE]

in complex coordinates. If the torsion vanishes, then $M$ is Kähler, and the Kähler 2-form $\omega$ (with components $\omega_{ij}\equiv g_{ik}J^{k}{}_{j}$) is closed. For every holomorphic Killing vector $\xi^{i}_{K}$, the 1-form with components $\omega_{ij}\xi^{j}_{K}$ is closed and locally there are functions $X_{K}$ such that $\omega_{ij}\xi^{j}_{K}=\partial_{i}X_{K}$; in complex coordinates, this equation becomes $\xi_{K\alpha}=i\partial_{\alpha}X_{K}$. The functions $X_{K}$ are sometimes called Killing potentials and play a central role in gauging the supersymmetric sigma-models without torsion [55, 25]. When $X_{K}$ are globally defined equivariant functions (i.e. ${\cal L}_{K}X_{M}=0$), they are referred to as moment maps and the gauging implements the Kähler quotient construction.

When the torsion does not vanish, this generalises straightforwardly [32]. The locally defined 1-form $u_{K}$ satisfies $\iota_{K}H=du_{K}$. If, in addition, (4.1) holds, then the 1-form with components $\nu_{i}\equiv\omega_{ij}(\xi^{j}_{K}+u^{j}_{K})$ satisfies $\partial_{[\alpha}\nu_{\beta]}=0$, so that there are generalised Killing potentials such that [32]

[TABLE]

The $X_{K}$ and $Y_{K}$ are locally defined functions on $M$; $Y_{K}$ simply reflects the ambiguity in the definition of $u_{K}$ in (2.10), and locally the $\partial_{\alpha}Y_{K}$ term can be absorbed into the definition of $u_{K}$.

Under the rigid symmetries (4.2), the variation of the action in (3.7) is

[TABLE]

where the Lie derivative of $k_{\alpha}$ is

[TABLE]

The variation of the superpotential term (3.8) in the action is

[TABLE]

so it will be left invariant by the isometries provided the holomorphic function $F(\varphi)$ is invariant up to constants, i.e.  if the equations

[TABLE]

are satisfied for some complex constants $e_{K}$.

In general, the isometry symmetries will not leave the potential $k_{\alpha}$ invariant, but will change it by a gauge transformation of the form (3.15)-(3.17), so that the action (3.7) is unchanged. The geometry and Killing potentials then determine the quantity ${\cal L}_{K}k_{\alpha}$ appearing in the variation (4.5) to take the form

[TABLE]

for some real functions $\chi_{K}$ and holomorphic 1-forms $\vartheta_{K\alpha}$,

[TABLE]

In ref. [34], the following explicit expressions for $\chi$ and $\vartheta$ were found:

[TABLE]

[TABLE]

Using (4.3), (4.4), and (4.6), it is straightforward to check that (4.11) and (4.12) satisfy (4.9) and (4.10) respectively. It follows that the action of the Lie bracket algebra on the vector potential $k_{\alpha}$ reduces to

[TABLE]

as it must (cf. (2.8)). The obstructions to gauging of the supersymmetric sigma model (without superpotential) were analysed in [34, 35] following [29, 30, 32]. It was found that, in order for the gauging to be possible, the following two conditions must hold:

[TABLE]

Condition (ii) is the statement that the generalised Killing potentials must be equivariant. If they are also globally defined, then they are sometimes referred to as generalised moment maps.

Observe that, together with the relation (4.4), the expression (4.12) for $\vartheta_{J\alpha}$ implies

[TABLE]

(as can be seen by contracting $\vartheta_{J\alpha}$ with $\xi^{\alpha}_{I}$ and symmetrising with respect to $I$ and $J$), so that condition (i) above is equivalent to

[TABLE]

where the functions $c_{IJ}$ were defined in (2.11); compare eq. (2.17).

For the gauging of the superpotential term term (3.8) to be possible, it is necessary that the constants $e_{K}$ defined in (4.8) vanish, so that the holomorphic function $F(\varphi)$ is invariant under the isometry symmetries,

[TABLE]

Consider the case of gauging one isometry that acts in adapted coordinates $(\varphi^{0},\varphi^{\mu})$ as a shift in $i(\varphi^{0}-\bar{\varphi}^{0})$, so that $\varphi^{0}\to\varphi^{0}+i\lambda$ , $\bar{\varphi}^{0}\to\bar{\varphi}^{0}-i\lambda$. The Killing vector $\xi$ then has components $(i,-i,0,\dots)$, with

[TABLE]

Then the condition (4.16) implies that

[TABLE]

which, combined with (4.12) implies that

[TABLE]

5 The $(2,1)$ gauge multiplet and gauge symmetries

We now promote the isometries (4.2) to local ones in which the constant parameters $\lambda^{K}$ are replaced by $(2,1)$ superfields $\Lambda^{K}$,

[TABLE]

These transformations preserve the chirality constraints (3.5) only if the $\Lambda^{K}$ are chiral,

[TABLE]

Under a finite transformation,

[TABLE]

where

[TABLE]

is the generator of the infinitesimal diffeomorphism with parameter $\Lambda\cdot\xi$.

The $(2,1)$ super Yang-Mills multiplet is given in $(2,1)$ superspace by a set of Lie-algebra valued super-connections ${\cal A}_{(2,1)}=(A_{+},\bar{A}_{+},A_{-},A_{{{=\hskip-3.75pt{\vrule width=0.5pt,height=5.0pt,depth=1.0pt}}\hskip 3.75pt}},A_{=})$, with $A_{\bullet}=A_{\bullet}^{K}T_{K}$, where the Lie algebra generators $T_{K}$ are hermitian and satisfy the algebra $[T_{K},T_{L}]=if_{KL}^{~{}~{}~{}M}T_{M}$. These connections can be used to define gauge covariant derivatives $\nabla_{\bullet}\equiv D_{\bullet}-iA_{\bullet}$, which are constrained by the conditions:

[TABLE]

as well as $\nabla_{+}^{2}=\bar{\nabla}_{+}^{2}=0$. The remaining relations among the derivatives follow from these conditions and the Bianchi identities, e.g.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The conditions (5.5) were introduced in [34, 35]. Their consequences (5.6)-(5.9) correct statements in [34, 35].

The constraints (5.5) can be solved to give all connections in terms of a scalar prepotential $V$ and the spinorial connection $A_{-}$. In the chiral representation, the spinorial derivatives that appear in the algebra (5.5) are given by

[TABLE]

where $V=\bar{V}$ is hermitian, and the spinor connection $A_{-}$ is hermitian up to a similarity transformation because we are in chiral representation444Real representations are reviewed in Appendix A.: $\bar{A}_{-}=e^{V}(A_{-}+iD_{-})e^{-V}$. We then find

[TABLE]

so that

[TABLE]

The field strengths are obtained from (5.5) and (5.10),

[TABLE]

[TABLE]

Again, these are not complex conjugates because we are in a chiral representation. Note that if instead we used the anti-chiral representation, we would have $\bar{W}=-iD_{+}A_{-}$, and $W=\{e^{V}\bar{D}_{+}e^{-V},D_{-}-iA_{-}\}$.

We now turn to the gauge transformations of the $(2,1)$ Yang-Mills supermultiplet. Under a finite gauge transformation, the hermitian superfield prepotential $V$ transforms as

[TABLE]

For infinitesimal $\Lambda$, this yields

[TABLE]

In chiral representation, the superconnection $A_{-}$ transforms as

[TABLE]

The antichiral representation would be reached from this by a similarity transformation with $e^{V}$, giving the antichiral representation covariant derivative

[TABLE]

The spinor covariant derivative $\nabla_{-}$ is real in the sense that after taking the adjoint one is in the antichiral representation: $\bar{\nabla}_{-}=e^{V}\nabla_{-}e^{-V}$. In particular,

[TABLE]

For fields in a linear representation of the gauge group, the Lie algebra generators act in that representation. For the superfields $\varphi,\bar{\varphi}$, the symmetry is realised non-linearly, with the Lie algebra element $T_{K}$ generating the transformation $\varphi\to\varphi+\lambda^{K}\xi_{K}(\varphi)$.

Covariant derivatives can act on different representations of the group. This action is encoded in the matrix used to represent the generators of the Lie algebra; they can also act nonlinearly on the superfields $\varphi,\bar{\varphi}$. In this case, the covariant derivative uses this non-linear realisation:

[TABLE]

The scalar superfields $\varphi,\bar{\varphi}$ transform under the local isometry symmetries as in (5.1). Following [25], we define the chiral-representation version of $\bar{\varphi}$ as

[TABLE]

where

[TABLE]

Then the superfields $\varphi,\tilde{\varphi}$ satisfy the covariant chirality constraints

[TABLE]

and transform under the isometry symmetries as

[TABLE]

Here $\bar{\xi}^{\bar{\alpha}}_{K}(\tilde{\varphi})$ is obtained from $\bar{\xi}^{\bar{\alpha}}_{K}(\bar{\varphi})$ by replacing $\bar{\varphi}$ with $\tilde{\varphi}$. Note that the transformation of $\tilde{\varphi}$ involves the parameter $\Lambda$, while that for $\bar{\varphi}$ involves $\bar{\Lambda}$. The covariant derivatives of $\tilde{\varphi}$ are in chiral representation, and hence are given by:

[TABLE]

When gauging one translational isometry, we again choose adapted coordinates $\varphi^{0}$ in which the Killing vector has components $(i,-i,0,\dots)$ and acts as in (4.18). Then the above relations simplify: the only fields that transform are

[TABLE]

and minimal coupling is simply given by

[TABLE]

As explained above, since the transformation $\delta A_{-}=D_{-}\Lambda$ involves the chiral parameter $\Lambda$, it is necessarily complex, and $A_{-}$ is not real; however, the combination

[TABLE]

has the real transformation

[TABLE]

and is real – see Appendix A for details.

6 The gauged $(2,1)$ sigma model

The gauged $(2,1)$ sigma model in superspace was studied in [34, 35] and the full nonpolynomial gauged action was constructed in [35] using the methods of ref. [25]. We now briefly summarise the main results of the analysis, referring the reader to these papers for the derivations and further details of the construction.

Under the infinitesimal rigid transformations (4.2), the variation of the $(2,1)$ full superspace Lagrangian

[TABLE]

is given by (4.5)

[TABLE]

Invariance of the action requires (4.9):

[TABLE]

with $\chi_{K}$ a real function and $\vartheta_{K\alpha}$ a holomorphic 1-form which were shown in ref. [34] to take the explicit forms (4.11) and (4.12) respectively. The variation of the superpotential term (3.8) is given in (4.7), which vanishes provided the function $F$ satisfies (4.17), i.e.  if it is invariant under the rigid isometries (4.2).

Now consider promoting the rigid isometries to local symmetries (5.1). The variation of the $(2,1)$ superpotential term (3.8) is given by

[TABLE]

and this will vanish provided the function $F(\varphi)$ is itself invariant under the local isometries, i.e.  (4.17) holds for such isometries; in the following we will assume that this is the case and concentrate on the gauging of the full superspace term (3.7) in the action.

The main result of [35] is that the $(2,1)$ superspace action (3.7) can be gauged provided the geometric condition (4.14) holds, in which case the gauge invariant superspace Lagrangian for the gauged $(2,1)$ sigma model is (to all orders in the gauge coupling, which we have absorbed into the gauge fields)

[TABLE]

The operator $L\equiv iV^{K}\bar{\xi}^{\bar{\alpha}}_{K}\frac{\partial}{\partial\bar{\varphi}^{\bar{\alpha}}}$ is the one defined in (5.22), and the expression $\frac{1}{L}(e^{L}-1)$ in the Lagrangian can be defined by its Taylor series expansion in $L$ or equivalently by $\int_{0}^{1}dt\,e^{tL}$. The gauge invariance of the action obtained from integrating the Lagrangian (6.5) over superspace is proven in Appendix B for the case of a single isometry.

The full gauged sigma model $(2,1)$ superspace action is then

[TABLE]

for an invariant superpotential $F(\varphi)$: ${\cal L}_{K}F=0$.

This form of the gauged action was given in [35], but is not immediately comparable to the more geometric gauged action given for the bosonic model in (2.21),(2.23). As shown in Appendix B, using the relations

[TABLE]

we can rewrite the gauged Lagrangian as (for the case of a single isometry – the general case is similar):

[TABLE]

Because this Lagrangian is geometric, some properties that are hard to see in (6.5) are more transparent in this form. For example, the hermiticity of the action follows directly: the combination $A_{-}-{\textstyle{\frac{i}{2}}}D_{-}V$ (5.28) is real, with the real transformation (5.29)

[TABLE]

Since (2.24) and (4.14) imply that $u_{\alpha}$ and $X$ are invariant under (rigid) gauge transformations, we have, for any real function $f$, $f({\cal L}_{\xi+\bar{\xi}})X=0$. In particular, this implies

[TABLE]

where $\bar{L}=-iV\xi^{\alpha}\frac{\partial}{\partial\varphi^{\alpha}}$ and the holomorphy of $\xi$ implies $[L,\bar{L}]=0$. Similarly

[TABLE]

The hermiticity of the action then follows.

7 T-duality of $(2,1)$ supersymmetric theories

7.1 Generalities

The generalisation of T-duality [56, 57, 49, 22, 23, 47, 48, 21] to conformally invariant sigma models which admit isometries, and its explicit form in $(2,2)$ superspace, were elucidated in ref. [24]. In this section we generalise the construction to the superspace formulation of the $(2,1)$ supersymmetric sigma models reviewed above. The general procedure which defines the dual pairs locally is the same as in refs. [57, 49, 22, 23, 47, 24, 48, 21]. First, gauge the sigma model isometries and add a Lagrange multiplier term constraining the gauge multiplet to be flat. Second, eliminate the gauge fields by solving their field equations. Classically, this ensures the equivalence of the dual models, modulo global issues that arise in the case of compact isometries if the gauge fields have nontrivial holonomies along noncontractible loops. However, as already explained at the end of section 2, these issues are taken care of by giving the Lagrange multipliers suitable periodicities and adding a total derivative term, so that the holonomies are constrained to be trivial [24]; we will assume that this can also be done in the $(2,1)$ supersymmetric case at hand. Quantum mechanically, the duality in the $(2,2)$ case receives corrections from the Jacobian obtained upon integrating out the gauge fields, which at one loop leads to a simple shift of the dilaton [22].

For an Abelian gauging, the field strengths $W,\bar{W}$ in chiral representation given in (5.13,5.14) are:

[TABLE]

The condition that the gauge multiplet is pure gauge can be imposed by constraining $W,\bar{W}$ to vanish by adding to the Lagrangian (6.8) a term

[TABLE]

To this we add a total derivative term, which is important for constraining the holonomies of the flat connections correctly, to obtain

[TABLE]

where $\Theta=-i\bar{D}_{+}\Psi_{-},\bar{\Theta}=-iD_{+}\bar{\Psi}_{-}$ are chiral (respectively antichiral) Lagrange multiplier superfields. The full action to consider is then

[TABLE]

Integrating out the Lagrange multipliers $\Theta$ or $\Psi_{-}$ gives

[TABLE]

which implies that $V$ and $A_{-}$ are pure gauge (with the boundary terms constraining the holonomies):

[TABLE]

The term $S_{\Theta}$ then vanishes, and we recover the original sigma model with action (3.6)-(3.8). Alternatively, integrating out the gauge fields gives the T-dual theory.

In the special case of one isometry, we can choose local complex coordinates $\{z^{\alpha},\bar{z}^{\bar{\alpha}}\}=\{z^{0};\bar{z}^{\bar{0}};z^{\mu};\bar{z}^{\bar{\mu}}\}$, with $\mu,\bar{\mu}=1,\ldots\frac{D}{2}-1$, such that the isometry acts by a translation leaving $(z^{0}+\bar{z}^{0})$ invariant. Moreover we can use a diffeomorphism combined with a $b$ field gauge transformation to arrange for the metric and $b$ field to depend only on $(z^{0}+\bar{z}^{0})$ and on the set of coordinates $\{z^{\mu};\bar{z}^{\bar{\mu}}\}$, but to be independent of $i(z^{0}-\bar{z}^{0})$; however, if we use a geometric formulation, there is no need to do so. The indices $\mu,\bar{\mu}$ now run over the ‘spectator’ coordinates transverse to $z^{0},\bar{z}^{\bar{0}}$. As reviewed in the previous section, the requirement of $(2,1)$ supersymmetry restricts the admissible isometries to those that act holomorphically on chiral superfields.

7.2 Computations

Recall that the geometry constrains ${\cal L}_{K}k_{\alpha}$ to take the form ${\cal L}_{K}k_{\alpha}=i\partial_{\alpha}\chi_{K}+\vartheta_{K\alpha}$ with

[TABLE]

(as follows from (4.9), (4.10), (4.11) and (4.12)) .

We can perform the T-duality starting from either of the two forms of the gauged Lagrangian, (6.5) or (6.8); for completeness, we consider both.

7.2.1 T-duality from the gauged Lagrangian (6.5)

In addition to the $X_{K}$, we define new potentials

[TABLE]

Since the Abelian isometries are independent, we focus on only one of them for the sake of clarity and henceforth drop the index $K$. Splitting the indices as $\alpha=(0,\mu),$ and using coordinates adapted to the isometry $\xi^{0}=i$ and $\xi^{\mu}=0$, from (7.7), we find

[TABLE]

it can be checked that $\vartheta_{\alpha}$ is holomorphic, $\partial_{\bar{\beta}}\vartheta_{\alpha}=0$. This is the set-up in the case where the obstructions to gauging vanish (cf. eq. (4.14)). However, as we shall see in section 9, this is not the most general situation in which T-duality is possible.

Now consider the general gauged Lagrangian in (6.5) with a single translational isometry. In adapted coordinates we have

[TABLE]

and the gauge invariance of the Lagrangian is shown in Appendix B.

For later purposes, we may rewrite (6.5) as

[TABLE]

Here $k_{\alpha},\bar{k}_{\bar{\alpha}},X,Z$ are all functions of $\varphi,\tilde{\varphi}\equiv e^{L}\bar{\varphi}$, whereas $\bar{\vartheta}$ is a function of $\bar{\varphi}$.

We add to the general Lagrangian (7.11) the invariant $L_{\Theta}$ (7.3) and consider $L_{T}=L_{g}+L_{\Theta}$ where $\Theta$ and $\bar{\Theta}$ are chiral and antichiral superfield Lagrange multipliers. As discussed above, integrating out $\Theta$ and $\bar{\Theta}$ sets the field strengths (7.1) to zero (modulo boundary terms):

[TABLE]

so that the gauge multiplet is pure gauge: $A_{-}=D_{-}\Lambda$, $V=i(\bar{\Lambda}-\Lambda)$ (7.6). Shifting $\varphi\to\varphi+i\Lambda$, we recover the original ungauged action (3.7).

To find the dual action, we integrate out the gauge fields instead. Specialising once again to one isometry, the variation of $V$ gives an expression for $A_{-}$; however, since $A_{-}$ enters as a Lagrange multiplier, it drops out of the final Lagrangian.555For completeness, we give the calculation of $A_{-}$ in Appendix C. The variation of $A_{-}$ implies

[TABLE]

which should be solved for $V=V(\Theta+\bar{\Theta},\varphi,\bar{\varphi})$. We can eliminate all dependence on $\varphi^{0},\bar{\varphi}^{\bar{0}}$ by choosing the gauge $\varphi^{0}=0$; in this gauge, $\tilde{\varphi}^{\bar{0}}=V$, and

[TABLE]

Then (7.13) implies

[TABLE]

We also need the following expression for $D_{-}V$, which we find by differentiating (7.13) and using the last equation in (7.7) (which gives $X,_{0}=-g_{0\bar{0}}$):

[TABLE]

Using these results, we now evaluate $L_{g}+L_{\Theta}$ from (7.11) and (7.3) to find the dual Lagrangian:

[TABLE]

where $V(\varphi,\bar{\varphi},\Theta+\bar{\Theta})$ is found by solving (7.13).

7.2.2 T-duality from the geometric form (6.8) of the gauged Lagrangian.

Using the equivalent geometric form of the gauged Lagrangian (6.8) together with the Lagrange multiplier term (7.3) instead, the dual Lagrangian reads

[TABLE]

7.3 The dual geometry

From (7.17), we can identify the components of the dual vector potential $k^{D}$ as follows

[TABLE]

Note that $\bar{k}^{D}_{\bar{\mu}}$ differs from the complex conjugate of $k^{D}_{\mu}$ by a a complex transformation of the form (3.15)-(3.17), so $k^{D}$ differs from a real vector by such a transformation. Likewise, from (7.18) we read off

[TABLE]

Here $\bar{\hat{k}}^{D}_{\bar{\mu}}$ is the complex conjugate of $\hat{k}^{D}_{\mu}$ so $\hat{k}^{D}$ is a real vector. Formulae (LABEL:dualks2) and (LABEL:dualks2r) only differ by terms that do not affect the metric and $b$ field. We can calculate the components of the dual metric $g^{D}$ and of the dual $b$-field $b^{D}$. Using (3.10), we find

[TABLE]

and

[TABLE]

In the case of $N$ Abelian isometries the expressions for the dual geometry involve $N\times N$ matrices replacing some entries, for example $g_{0\bar{0}}\to(g+b)_{mn}$ as in the bosonic case [48].

8 Comparison to the Buscher rules

The results (7.21),(7.22) for the (2,1) duality transformations are similar but not identical to the Buscher transformations in the modified form (2.45). In the Buscher duality (2.45), a coordinate $x^{0}$ is replaced by a dual coordinate $\hat{x}^{0}$ (e.g. if $x^{0}$ is a coordinate on a circle of radius $R$, $\hat{x}^{0}$ is a coordinate on the dual circle of radius $2\pi/RT$, again reinstating the string tension to keep track of dimensions), whereas in the (2,1) duality transformations, a complex coordinate $z^{0}=\varphi^{0}|_{\theta=0}$ is replaced by a dual complex coordinate $\hat{z}^{0}=\Theta|_{\theta=0}$. This arises because the $(2,1)$ gauging involves the action of the complexification of the isometry group. As was explained in [22, 42] for the $(2,2)$ case, the complex duality transformation consists of a T-duality and a diffeomorphism: it gives a T-duality transformation of the imaginary part of the coordinate $z^{0}$ and a coordinate transformation of the real part. Writing $z^{0}=y^{0}+ix^{0}$, $\hat{z}^{0}=\hat{y}^{0}+i\hat{x}^{0}$, the (2,1) duality transformation consists of a T-duality transformation in which the coordinate $x^{0}$ is replaced by a dual coordinate $\hat{x}^{0}$ (so that if $x^{0}$ is a coordinate on a circle of radius $R$, $\hat{x}^{0}$ is a coordinate on the dual circle of radius $2\pi/RT$), while $\hat{y}^{0}$ is related to $y^{0}$ by a coordinate transformation (so that if $y^{0}$ is a coordinate on a circle of radius $R$, $\hat{y}^{0}$ is a different coordinate on the same circle of radius $R$).

To see this, we start from the constraint (7.13):

[TABLE]

Then setting $\theta=0$ and choosing the Wess-Zumino gauge in which $V|_{\theta=0}=0$, this implies

[TABLE]

which gives

[TABLE]

The solution of this gives $\hat{y}^{0}$ as a function of $y^{0},z^{\mu},\bar{z}^{\bar{\mu}}$, so that the complex duality transformation gives the coordinate transformation

[TABLE]

together with the T-duality transformation replacing $x^{0}$ with the dual coordinate $\hat{x}^{0}$.

This can also be understood by comparing our (2,1) superspace analysis with the corresponding computation in (1,1) superspace which gives the Buscher duality (2.45): the equivalence of the two calculations is guaranteed, and so will relate the (2,1) duality transformations to the Buscher ones. The explicit calculations are carried out in Appendix D. We now illustrate this discussion with two simple and instructive examples.

8.1 T-duality on the complex plane

Our first simple example is the complex plane dualised with respect to the isometry given by a rotation about the origin666This example is interesting in that it shows that the flat plane, which, when regarded as a string background, has no winding modes, is formally dual to a singular geometry with no normalizable (radial) momentum modes and only winding modes.

[TABLE]

for real $\lambda$. The adapted coordinates are $\varphi=\ln z$, transforming under the isometry by an imaginary shift $\varphi\to\varphi+i\lambda$. The metric is given by

[TABLE]

for which the potential can be taken to be

[TABLE]

In this case, the Lagrangian is invariant, and $\vartheta=\chi=0$, so (7.2.1) gives

[TABLE]

On gauging, this becomes $X=-e^{\varphi+\bar{\varphi}+V}$ and, on choosing the gauge $\varphi=0$, this reduces to

[TABLE]

so that (8.1) implies

[TABLE]

Then using (LABEL:dualks2r), we have

[TABLE]

and

[TABLE]

In real coordinates $\varphi=y+ix$, the line element (8.6) is

[TABLE]

and the isometry is generated by $\partial_{x}$. Dualizing gives

[TABLE]

To compare this line element to (8.12), we write $\Theta=\hat{y}+i\hat{x}$, and use the coordinate transformation (8.1):

[TABLE]

Then (8.12) becomes:

[TABLE]

which does indeed match (8.14).

8.2 T-duality on a torus

Consider a flat torus $S^{1}\times S^{1}$ parametrised by a single complex coordinate $z$ and let $\varphi$ be the $(2,1)$ superfield such that $\varphi|\equiv z$. For simplicity, we consider the case of a single holomorphic isometry and we suppress all spectator fields. We take the flat metric on the torus to be

[TABLE]

with

[TABLE]

The coordinate $x$ that we are dualizing is scaled so that its periodicity is

[TABLE]

so it parameterises a circle of circumference $2\pi R$ and $R$; the coordinate $y$ can have any periodicity:

[TABLE]

so the circumference of the corresponding circle is $\tau R$.

We consider the $(2,1)$ sigma model whose target space has the above geometry (with zero $b$-field). This is defined by the potential

[TABLE]

The isometry is generated by

[TABLE]

and the Killing potential is

[TABLE]

The Lie derivative of $k$ is zero, so we are in the simple case with $\chi=\vartheta_{\alpha}=0$. The T-dual metric is then

[TABLE]

where

[TABLE]

and the dual $b$-field is zero. Eq. (8.24) looks like the metric that would result from T-dualising on both circles, but to see whether this is the case, we need to be careful with the periodicities. From the T-duality, we know that

[TABLE]

so the circumference of the $\hat{x}$ circle is $\frac{2\pi}{R}$ and we find the dual radius $\hat{R}=\frac{1}{R}$ as expected. The constraint (8.1), together with (8.23) and (8.25), gives

[TABLE]

so the periodicity of $\hat{y}$ is $\hat{y}\sim\hat{y}+R^{2}\tau$. Using the dual metric (8.24), the circumference of the circle parameterised by $\hat{y}$ is $R^{-1}R^{2}\tau=R\tau$ which is the same as that of the original circle parameterised by $y$. The T-duality has implemented the change of variables (8.27) from $y$ to $\hat{y}$ and this diffeomorphism preserves the circumference of the circle. Rewriting (8.24) in terms of $\hat{x}$ and the original coordinate $y$, we find

[TABLE]

which is the result of the standard Buscher rules for T-duality in the $x$-circle. Thus we see that the $(2,1)$ T-duality, which appears to give a T-duality in two directions, in fact gives a T-duality in just one direction, combined with a diffeomophism whose role is to maintain the extra supersymmetry and the complex geometry.

9 Geometry and obstructions for $(2,1)$ T-duality

We start by recalling the results of [36] reviewed in section 2. For a sigma model with Abelian isometries generated by Killing vectors $\xi_{K}$, the conditions for gauging are that the $u_{K}$ are globally defined 1-forms that are invariant, ${\cal L}_{K}u_{L}=0$, and satisfy $\iota_{K}u_{L}=-\iota_{L}u_{K}$. For T-duality, we require none of these conditions but only that $\iota_{K}\iota_{L}\iota_{M}H=0$, and we introduce a bundle $\hat{M}$ over $M$ with fibre coordinates $\hat{x}_{K}$. The metric and $H$-flux are defined by (2.33) and we take

[TABLE]

We lift the Killing vectors $\xi_{K}$ on $M$ to Killing vectors $\hat{\xi}_{K}$ on $\hat{M}$ satisfying (2.34) and (2.38). The space $\hat{M}$ can be chosen so that $\hat{u}$ is invariant and globally defined on $\hat{M}$ with (2.39) satisfied, so that the only condition necessary for gauging and hence for T-duality is (2.43). The T-dual space is then $\hat{M}/G$ where $G$ is the Abelian gauge group generated by the $\hat{\xi}_{K}$.

For the $(2,1)$ supersymmetric sigma model to be defined on $M$, $M$ has to be complex with the geometry reviewed in section 3 and the Killing vectors must be holomorphic. Then there are generalised Killing potentials $Y_{K}+iX_{K}$ satisfying (4.4). This can be written as

[TABLE]

with real 1-forms $u_{K}=u_{iK}dx^{i}$, $\xi_{K}=g_{ij}\xi^{j}_{K}dx^{i}$. Locally, we can absorb $Y_{K}$ into a redefinition of $u_{K}$ as discussed below (4.4). For gauging of the sigma model on $M$ to be possible, the final form of $u_{K}$ that arises after absorbing all the $dY_{K}$ terms should be a globally defined one-form; for T-duality, this is not necessary as the $u_{K}$ do not need to be globally defined. If the $(2,1)$ sigma-model on $M$ allows a (1,1) gauging777A (1,1) gauging is the same as the bosonic gauging discussed in section 2., then the gauging will be $(2,1)$ supersymmetric provided the Killing vectors are holomorphic and the potentials $X_{K}$ are globally defined scalars which are invariant: ${\cal L}_{K}X_{L}=0$. The $(2,1)$ gauging is defined by restricting to the subspace $X=0$ and taking a quotient by $G$ to give $X^{-1}(0)/G$.

For $(2,1)$ T-duality, introducing $n$ extra coordinates $\hat{x}_{K}$ ($K=1,\dots,n$) would in general be inconsistent with supersymmetry; for example, if $n$ is odd, $\hat{M}$ would have odd dimension and so cannot be complex. Instead, we introduce $n$ complex coordinates $\Theta_{K}$ corresponding to the chiral Lagrange multiplier fields introduced in section 5. This leads to a complex manifold $\check{M}$ with holomorphic coordinates $z^{a}=(\varphi^{\alpha},\Theta_{K})$ that is a bundle over $M$ with projection $\check{\pi}:\check{M}\to M$ with $\check{\pi}:(\varphi^{\alpha},\Theta_{K})\mapsto\varphi^{\alpha}$. A metric $\check{g}$ and closed 3-form $\check{H}$ can be chosen on $\check{M}$ with no $\Theta_{K}$ components, i.e.

[TABLE]

where $\check{\pi}^{*}$ is the pull-back of the projection.

Writing

[TABLE]

we identify the coordinates $\hat{x}_{K}$ with the extra coordinates needed for the (1,1) T-duality. Then

[TABLE]

and we take the Killing vectors on $\check{M}$ to be the $\hat{\xi}_{K}$ given by (2.34) and (2.38). Now if (9.2) holds on $M$ (with $Y_{K}$ absorbed into $u_{K}$), then on $\check{M}$ we have

[TABLE]

and, since $\partial\bar{\Theta}=\bar{\partial}\Theta=0$, this can be rewritten as

[TABLE]

where

[TABLE]

It follows that $\hat{u}$ can be chosen so that it is globally defined on $\check{M}$ and invariant. Then $d\check{X}_{K}$ will be invariant under the action of the Killing vectors $\hat{\xi}$, so that

[TABLE]

for some constants $C_{LK}$. Introducing the Killing vectors on $\check{M}$ given by

[TABLE]

we have that the $\check{X}_{K}$ are invariant:

[TABLE]

Then the bundle $\check{M}$ can be defined so that the $\hat{u}_{K}$ and $\check{X}_{K}$ are globally defined (so that the transition functions for $u_{K}$ on $M$ determine those of $\hat{x}_{K}$ and the transition functions for $X_{K}$ on $M$ determine those of $\hat{y}_{K}$). As $\hat{u}_{K}$ and $\check{X}_{K}$ are globally defined and invariant under the action of $\check{\xi}$, the isometries generated by $\check{\xi}$ can be gauged provided (2.43) holds, giving a $(2,1)$ supersymmetric gauged sigma model on $\check{M}$.

The gauging imposes the generalised moment map constraints

[TABLE]

which is precisely the condition (7.13) obtained previously. This defines a $D+n$ dimensional subspace $\check{X}^{-1}(0)$ of the $D+2n$ dimensional space $\check{M}$. The gauging then gives the quotient $\check{X}^{-1}(0)/G$, which is of dimension $D$; this is the T-dual target space.

10 Conclusion

The $(p,q)$ supersymmetric sigma models (with $p\geq 1,q\geq 1$) are a special class of $(1,1)$ models which have extra geometric structure arising from the $(p-1)+(q-1)$ complex structures of the target space. The T-duality of such models can be analysed using a $(1,1)$ supersymmetric gauging by coupling to $(1,1)$ vector multiplets and adding $(1,1)$ supersymmetric Lagrange multiplier terms, and this gives the standard Buscher rules when the $b$-field is invariant, and modified Buscher rules when only $H=db$ is invariant. However, considerable insight arises from performing T-duality in $(p,q)$ superspace by coupling to $(p,q)$ vector multiplets and adding $(p,q)$ supersymmetric Lagrange multiplier terms. For example, for sigma models with Kähler target space, the T-duality is realised as a Legendre transformation of the Kähler potential [40, 24].

Here we used the $(2,1)$ superspace formulation of [20] and the gauging found in [34, 35] to find the T-dual $(2,1)$ geometry, realised as a transformation from the potential $k$ to a dual potential $k^{D}$ that can be viewed as a generalisation of the Legendre transformation of the Kähler case. As for the $(2,2)$ case, the supersymmetric gauging requires a gauging of a complexification of an isometry group. If the group $G$ of real isometries to be gauged is generated by Killing vectors $\xi^{i}_{K}$, the group which is actually gauged in the $(2,1)$ supersymmetric gauging is the complexification $G^{C}$ of $G$ generated by the $\xi^{i}_{K}$ together with the vector fields $J^{i}{}_{j}\xi^{j}_{K}$. The vector fields $J^{i}{}_{j}\xi^{j}_{K}$ are in general not Killing vectors, so that they generate diffeomorphisms that are not isometries. This gives a good local picture of the gauging. Global issues are discussed in [58]. For T-duality with $G$ Abelian, the dual metric $g^{D}$ and the dual torsion potential $b^{D}$ given by eqs. (7.21)-(7.22) involve a duality in complex coordinates, which does not look like the standard Buscher rules. However, as discussed in section 8, just as for the (2,2) supersymmetric case, the T-duality transformations in fact correspond to the usual Buscher rules for the real isometry group $G$, together with diffeomorphisms that introduce the Killing potentials as new coordinates.

The geometry of the T-duality was discussed in sections 2 and 9. The $(1,1)$ T-duality is understood through the construction of a ‘doubled’ manifold $\hat{M}$ with $n$ extra coordinates $\hat{x}_{K}$ arising from the Lagrange multipliers, so that the coordinates $x^{K}$ of the torus fibres generated by the $n$ Killing vectors $\xi_{K}$ are doubled to give a ‘doubled torus’ with $2n$ coordinates $x^{K},\hat{x}_{K}$. The action of the isometry group $G$ is lifted to $\hat{M}$, and the T-duality space is the quotient $\hat{M}/G$.

Similarly, the $(2,1)$ duality introduces $n$ extra complex coordinates $\Theta_{K}$ corresponding to each Killing vector, giving a space $\check{M}$ with an extra $2n$ real dimensions. The T-dual space is now the symplectic quotient of $\check{M}$ given by taking the quotient of the subspace $X=0$, giving a generalised moment map and a generalisation of the Kähler quotient construction. For generalised Kähler spaces, this reduces to the generalised moment map of [59], which was constructed from a $(2,2)$ gauging in [26].

Acknowledgments

We are indebted to Rikard von Unge for collaboration in the early stages of this project. M.A. wishes to thank Nathan Berkovits, Reimundo Heluani, Kentaro Hori, Nikita Nekrasov and Warren Siegel for discussions and helpful comments. We thank NORDITA, the GGI, the ICTP-SAIFR, the IMPA, Imperial College London, the Mainz ITP, and the SCGP for hospitality while this work was in progress. We have benefited from participation in several Simons Summer Workshops at the SCGP. This work was supported by the EPSRC programme grant “New Geometric Structures from String Theory” EP/K034456/1 and by STFC grant ST/L00044X/1. MR was supported in part by NSF grant # PHY1620628.

Appendix A Review of chiral and vector representations

The superspace constraints (5.5) can be solved in terms of unconstrained superfields. The chiral representation solution is given in (5.10) in terms of $V$ and $A_{-}$; in this representation, gauge covariant derivatives transform with the chiral parameter $\Lambda$. A real representation [60] can be found by writing

[TABLE]

with gauge transformations that depend on both $\Lambda$ and on new, hermitian parameters $K$:

[TABLE]

Note that this is compatible with (5.15). Performing a similarity transformation $\nabla_{\bullet}\to e^{\Omega}\nabla_{\bullet}e^{-\Omega}$ on the chiral representation derivatives (5.10) gives the hermitian vector representation derivatives

[TABLE]

We can use the $K$ transformation to make $\Omega$ real; in that case, $\Omega=\bar{\Omega}={\textstyle{\frac{1}{2}}}V$ and we find that

[TABLE]

is Hermitian. The higher order terms are absent in the Abelian case (compare (5.28)).

Appendix B Gauge invariance and hermiticity of the action

We now derive the following useful identities for the operator $\frac{e^{L}-1}{L}\equiv\int_{0}^{1}dt\,e^{tL}~{}$:

[TABLE]

which immediately implies the corollary

[TABLE]

We prove (B.1) as an operator relation by applying it to a test function $f(\varphi^{i})$ where $\varphi^{i}$ represents $\varphi^{\alpha},\bar{\varphi}^{\bar{\alpha}}$:

[TABLE]

where the dependence on the spectator fields is suppressed after the first line.

These identities help us prove several important relations. We start with the proof of gauge invariance of (6.5). Its gauge variation may be written as

[TABLE]

The first term in (B) is

[TABLE]

where a chiral term has been dropped since the measure $\int d^{2}\theta^{+}d\theta^{-}$ annihilates it. The second term is

[TABLE]

where we used that $\bar{\vartheta}_{\bar{0}}=0$, cf. (4.19). After partially integrating the $\chi$ terms in (B.5) and (LABEL:term2) we add them to the third term in (B) and find

[TABLE]

which cancels the fourth term in (B). The fifth term is zero since $X$ is equivariant. The sixth term is

[TABLE]

where we used (B.1). The $\Lambda$-term in (B.8) cancels the $\bar{\vartheta}$-term in (LABEL:term2), leaving

[TABLE]

From the definition (5.22) of $L$ we see that

[TABLE]

and we have

[TABLE]

which is antichiral and again annihilated by the measure.

As discussed at the end of Section 6, hermiticity of the gauged action follows immediately from the expression (6.8). We now prove that (6.5) is (6.8) (modulo total derivatives). We use the relations (6.7), which we rewrite here for convenience:

[TABLE]

We write out (6.5), using the definitions $\tilde{\varphi}=e^{L}\bar{\varphi}$, $e^{L}=1+{\frac{e^{L}-1}{L}}L$, and (B.13)

[TABLE]

Subtracting (6.8), we have:

[TABLE]

Applying the identity (B.2) on the two $D_{-}V$ terms in (B.15), we find, up to total derivatives

[TABLE]

This vanishes because of (B.12).

If we keep all the total derivative terms, we find that the difference between the two Lagrangians (6.5) and (6.8) is the total derivative

[TABLE]

where $Z\equiv X+2\bar{k}_{\bar{0}}=-k_{0}+\bar{k}_{\bar{0}}+\chi$ is defined in (7.8-7.2.1).

Appendix C Calculation of $A_{-}(\varphi)$

As observed in section 7.2.1, the spinor connection $A_{-}$ enters the action that we use for T-duality–the sum of (6.8) and (7.3)–as a Lagrange multiplier, and hence is not needed; for completeness we present its calculation here.

The expression for the potential $A_{-}$ is found from the $V$ field equation. The variation of $V$ in the sum of (6.8) and (7.3) is

[TABLE]

where $u,\bar{u}$ and $X$ now all depend on $\varphi$ and $\tilde{\varphi}$. The $V$ field equation results from setting the expression multiplying $\delta V$ to zero, and determines $A_{-}$ to be

[TABLE]

We note that, using the $A_{-}$ field equation (7.13), we have

[TABLE]

Since X is real and equivariant, $X_{0}=X_{\bar{0}}$, and (C.3) is indeed the complex conjugate of (C.2). Hence the combination $A_{-}-{\textstyle{\frac{i}{2}}}D_{-}V$ in (5.28), which is the average of (C.2) and (C.3), is manifestly real.

Appendix D Reduction

In this appendix we reduce $(2,2)$ models to $(2,1)$ superspace and $(2,1)$ models to $(1,1)$ superspace.

D.1 Reduction of a Kähler $(2,2)$ sigma model to $(2,1)$ superspace

The gauged $(2,2)$ action for chiral superfields $\phi$ reads [25]

[TABLE]

where $-{\textstyle{\frac{1}{2}}}X$ is the Killing potential for the isometry. To reduce to $(2,1)$ we write

[TABLE]

and act with $Q_{-}$ on $K$, splitting the fermionic measure as follows:

[TABLE]

We also write the gauge covariant derivative as

[TABLE]

which implies

[TABLE]

where a vertical bar denotes the reduction to $(2,1)$ superspace. Note that this is proportional to the real combination (5.28). We use the chiral constaint

[TABLE]

and keep the $(2,2)$ notation ($\phi,V$) for the $(2,1)$ superfields (that is, we don’t bother writing $\varphi:=\phi_{|}$, etc.).

This leads to the $(2,1)$ action

[TABLE]

As expected, this is the $(2,1)$ action (6.8) with $k_{\alpha}=K_{,\alpha}$ and $u_{\alpha}=0$ because the geometry is Kähler.

D.2 Reduction of a general $(2,1)$ sigma model to $(1,1)$ superspace

The general gauged $(2,1)$ model (6.8) is reduced to $(1,1)$ superspace using similar techniques to those in the previous subsection. However, the reduction of the gauge multiplet is somewhat different.

As above, we define

[TABLE]

and write the measure as

[TABLE]

We can define gauge covariant $(1,1)$ derivatives from the $(2,1)$ derivatives in section 5; however, since we need to distinguish them, we will write $\textstyle\nabla$ for the $(2,1)$ gauge covariant derivatives and $\nabla$ for the $(1,1)$ derivatives. In $(1,1)$ superspace, the natural group has a real superfield gauge parameter, and does not involve a complexification. Instead, the real part of the $(2,1)$ complex gauge parameter $\Lambda$ is used to gauge away $V_{|}$. This is the $(1,1)$ version of Wess-Zumino gauge; note that $V_{|}=D_{\pm}V_{|}=0$, but $(Q_{+}V)_{|}\neq 0$. In Wess-Zumino gauge, the $(1,1)$ objects that we find are independent of whether we are in chiral, anti-chiral, or real representation.

Similarly to (D.4), we define the $(1,1)$ gauge covariant derivative:

[TABLE]

where a vertical bar denotes the reduction to $(1,1)$ superspace. Here we have used (5.10). Because in Wess-Zumino gauge $(D_{+}V)_{|}=0$, we find

[TABLE]

We also reduce the field strengths $W$, $\bar{W}$ in (5.14),(5.13):

[TABLE]

Since in Wess-Zumino gauge $(D_{-}V)_{|}=0$, we find

[TABLE]

Then the real part of $W$ is just the $(1,1)$ gauge field strength:

[TABLE]

and the imaginary part is a real scalar field:

[TABLE]

We will use the following identity below:

[TABLE]

We now push in the generator of the nonmanifest left supersymmetry $Q_{+}$:

[TABLE]

where we have integrated by parts to eliminate the $D_{+}D_{-}\varphi$ and $D_{+}D_{-}\bar{\varphi}$ terms, and use the shorthand notation

[TABLE]

This is not yet manifestly left-right symmetric. We use (6.7):

[TABLE]

to rewrite this as

[TABLE]

Using the definition of gauge covariant derivatives:

[TABLE]

as well as (3.10) in the form

[TABLE]

we rewrite (D.2) as

[TABLE]

Except for the $sX$ term discussed below, this is precisely the general gauged sigma model (2.21),(2.23) in $(1,1)$ superspace.

Next, we reduce the Lagrange multiplier term $L_{\Theta}$ (7.3) to $(1,1)$ superspace:

[TABLE]

Combining (D.2) and (D.24), we find the usual real T-duality with in addition the $(1,1)$ superfield $s$ acting as a Lagrange multiplier to impose the condition

[TABLE]

This is a diffeomorphism that expresses $\Theta+\bar{\Theta}$ in terms of $\varphi,\bar{\varphi}$; because of the isometry in the original model, $\Theta+\bar{\Theta}$ does not depend on $x^{0}:={\textstyle{\frac{i}{2}}}(\bar{\varphi}^{\bar{0}}-\varphi^{0})$, which is the coordinate dual to $\hat{x}^{0}:={\textstyle{\frac{i}{2}}}(\bar{\Theta}-\Theta)$.