Generalized Kähler Manifolds and Off-shell Supersymmetry

Journal of High Energy Physics, 2009

We introduce and study the notion of a biholomorphic gerbe with connection. The biholomorphic gerbe provides a natural geometrical framework for generalized Kähler geometry in a manner analogous to the way a holomorphic line bundle is related to Kähler geometry. The relation between the gerbe and the generalized Kähler potential is discussed.

Journal of High Energy Physics, 2007

The geometry of the target space of an N = (2, 2) supersymmetry sigma-model carries a generalized Kähler structure. There always exists a real function, the generalized Kähler potential K, that encodes all the relevant local differential geometry data: the metric, the B-field, etc. Generically this data is given by nonlinear functions of the second derivatives of K. We show that, at least locally, the nonlinearity on any generalized Kähler manifold can be explained as arising from a quotient of a space without this nonlinearity.

Journal of High Energy Physics, 2008

We use the new N = (2, 2) vector multiplets to clarify T-dualities for generalized Kähler geometries. Following the usual procedure, we gauge isometries of nonlinear σ-models and introduce Lagrange multipliers that constrain the field-strengths of the gauge fields to vanish. Integrating out the Lagrange multipliers leads to the original action, whereas integrating out the vector multiplets gives the dual action. The description is given both in N = (2, 2) and N = (1, 1) superspace.

Journal of High Energy Physics, 2007

We introduce two new N = (2, 2) vector multiplets that couple naturally to generalized Kähler geometries. We describe their kinetic actions as well as their matter couplings both in N = (2, 2) and N = (1, 1) superspace.

Communications in Mathematical Physics, 2011

In this paper we consider pseudo-bihermitian structures -pairs of complex structures compatible with a pseudo-Riemannian metric. We establish relations of these structures with generalized (pseudo-) Kähler geometry and holomorphic Poisson structures similar to that in the positive definite case. We provide a list of compact complex surfaces which could admit pseudo-bihermitian structures and give examples of such structures on some of them. We also consider a naturally defined null plane distribution on a generalized pseudo-Kähler 4-manifold and show that under a mild restriction it determines an Engel structure.

Journal of High Energy Physics, 2013

Semichiral sigma models with a four-dimensional target space do not support extended N = (4, 4) supersymmetries off-shell [1], . We contribute towards the understanding of the non-manifest on-shell transformations in (2, 2) superspace by analyzing the extended on-shell supersymmetry of such models and find that a rather general ansatz for the additional supersymmetry (not involving central charge transformations) leads to hyperkähler geometry. We give non-trivial examples of these models.

Journal of High Energy Physics, 2008

Journal of High Energy Physics, 2007

We revisit our earlier work on the AKSZ-like formulation of topological sigma model on generalized complex manifolds, or Hitchin model, [20]. We show that the target space geometry geometry implied by the BV master equations is Poissonquasi-Nijenhuis geometry recently introduced and studied by Stiénon and Xu (in the untwisted case) in [44]. Poisson-quasi-Nijenhuis geometry is more general than generalized complex geometry and comprises it as a particular case. Next, we show how gauging and reduction can be implemented in the Hitchin model. We find that the geometry resulting form the BV master equation is closely related to but more general than that recently described by Lin and Tolman in [40, 41], suggesting a natural framework for the study of reduction of Poissonquasi-Nijenhuis manifolds.

Journal of High Energy Physics, 2008

We show how to carry out the gauging of the Poisson sigma model in an AKSZ inspired formulation by coupling it to a generalization of the Weil model worked out in ref. [31]. We call the resulting gauged field theory, Poisson-Weil sigma model. We study the BV cohomology of the model and show its relation to Hamiltonian basic and equivariant Poisson cohomology. As an application, we carry out the gauge fixing of the pure Weil model and of the Poisson-Weil model. In the first case, we obtain the 2-dimensional version of Donaldson-Witten topological gauge theory, describing the moduli space of flat connections on a closed surface. In the second case, we recover the gauged A topological sigma model worked out by Baptista describing the moduli space of solutions of the so-called vortex equations.

Journal of High Energy Physics, 2006

BiHermitian geometry, discovered long ago by Gates, Hull and Roček, is the most general sigma model target space geometry allowing for (2, 2) world sheet supersymmetry. By using the twisting procedure proposed by Kapustin and Li, we work out the type A and B topological sigma models for a general biHermtian target space, we write down the explicit expression of the sigma model's action and BRST transformations and present a computation of the topological gauge fermion and the topological action.

Journal of High Energy Physics, 2007

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

Communications in Mathematical Physics

We define (p, q) Hermitian geometry as the target space geometry of the two dimensional (p, q) supersymmetric sigma model. This includes generalised Kähler geometry for (2, 2), generalised hyperkähler geometry for (4, 2), strong Kähler with torsion geometry for (2, 1) and strong hyperkähler with torsion geometry for (4, 1). We provide a generalised complex geometry formulation of hermitian geometry, generalising Gualtieri's formulation of the (2, 2) case. Our formulation involves a chiral version of generalised complex structure and we provide explicit formulae for the map to generalised geometry. Contents

Journal of High Energy Physics, 2014

We show that in the context of two-dimensional sigma models minimal coupling of an ordinary rigid symmetry Lie algebra g leads naturally to the appearance of the "generalized tangent bundle" TM ≡ T M ⊕ T * M by means of composite fields. Gauge transformations of the composite fields follow the Courant bracket, closing upon the choice of a Dirac structure D ⊂ TM (or, more generally, the choide of a "small Dirac-Rinehart sheaf" D), in which the fields as well as the symmetry parameters are to take values. In these new variables, the gauge theory takes the form of a (non-topological) Dirac sigma model, which is applicable in a more general context and proves to be universal in two space-time dimensions: a gauging of g of a standard sigma model with Wess-Zumino term exists, iff there is a prolongation of the rigid symmetry to a Lie algebroid morphism from the action Lie algebroid M × g → M into D → M (or the algebraic analogue of the morphism in the case of D). The gauged sigma model results from a pullback by this morphism from the Dirac sigma model, which proves to be universal in two-spacetime dimensions in this sense.

Journal of High Energy Physics, 2017

Off-shell (4, q) supermultiplets in 2-dimensions are constructed for q = 1, 2, 4. These are used to construct sigma models whose target spaces are hyperkähler with torsion. The off-shell supersymmetry implies the three complex structures are simultaneously integrable and allows us to construct actions using extended superspace and projective superspace, giving an explicit construction of the target space geometries.

Journal of High Energy Physics, 2009

We investigate the target space geometry of supersymmetric sigma models in two dimensions with Euclidean signature, and the conditions for N = 2 supersymmetry. For a real action, the geometry for the N = 2 model is not the generalized Kähler geometry that arises for Lorentzian signature, but is an interesting modification of this which is not a complex geometry.

Journal of High Energy Physics, 2008

We discuss two dimensional N-extended supersymmetry in Euclidean signature and its R-symmetry. For N = 2, the R-symmetry is SO(2) × SO(1, 1), so that only an A-twist is possible. To formulate a B-twist, or to construct Euclidean N = 2 models with Hflux so that the target geometry is generalised Kahler, it is necessary to work with a complexification of the sigma models. These issues are related to the obstructions to the existence of non-trivial twisted chiral superfields in Euclidean superspace.

Symmetry, 2012

This is a review of how sigma models formulated in Superspace have become important tools for understanding geometry. Topics included are: The (hyper)kähler reduction; projective superspace; the generalized Legendre construction; generalized Kähler geometry and constructions of hyperkähler metrics on Hermitian symmetric spaces.

Journal of High Energy Physics, 2015

Briefly: using a novel (1, 1) superspace formulation of semichiral sigma models with 4D target space, we investigate if an extended supersymmetry in terms of semichirals is compatible with having a 4D target space with torsion. In more detail : semichiral sigma models have (2, 2) supersymmetry and Generalized Kähler target space geometry by construction. They can also support (4, 4) supersymmetry and Generalized Hyperkähler geometry, but when the target space is four dimensional indications are that the geometry is restricted to Hyperkähler. To investigate this further, we reduce the model to (1, 1) superspace and construct the extra (on-shell) supersymmetries there. We then find the conditions for a lift to (2, 2) super space and semichiral fields to exist. Those conditions are shown to hold for Hyperkähler geometries. The SU(2) ⊗ U(1) WZW model, which has (4, 4) supersymmetry and a semichiral description, is also investigated. The additional supersymmetries are found in (1, 1) superspace but shown not to be liftable to a (2, 2) semichiral formulation.

Journal of High Energy Physics

It is shown how extended supersymmetry realised directly on the (2, 2) semichiral superfields of a symplectic sigma model gives rise to a geometry on the doubled tangent bundle consisting of two Yano F structures on an almost para-hermitian manifold. Closure of the algebra and invariance of the action is discussed in this framework and integrability of the F structures is defined and shown to hold. The reduction to the usual (1, 1) sigma model description and identification with the bi-quaternionic set of complex structures and their properties is elucidated. The F structure formulation should be applicable to many other models and will have an equivalent formulation in Generalised Geometry.

Journal of High Energy Physics, 2016

We study a class of two-dimensional N = (2, 2) supersymmetric gauge theories, given by semichiral multiplets coupled to the usual vector multiplet. In the UV, these theories are traditional gauge theories deformed by a gauged Wess-Zumino term. In the IR, they give rise to nonlinear sigma models on noncompact generalized Kähler manifolds, which contain a three-form field H and whose metric is not Kähler. We place these theories on S 2 and compute their partition function exactly with localization techniques. We find that the contribution of instantons to the partition function that we define is insensitive to the deformation, and discuss our results from the point of view of the generalized Kähler target space.