Sigma models and complex geometry (original) (raw)
Communications in Mathematical Physics, 2014
Generalized Kähler geometry is the natural analogue of Kähler geometry, in the context of generalized complex geometry. Just as we may require a complex structure to be compatible with a Riemannian metric in a way which gives rise to a symplectic form, we may require a generalized complex structure to be compatible with a metric so that it defines a second generalized complex structure. We explore the fundamental aspects of this geometry, including its equivalence with the bi-Hermitian geometry on the target of a 2-dimensional sigma model with (2, 2) supersymmetry, as well as the relation to holomorphic Dirac geometry and the resulting derived deformation theory. We also explore the analogy between pre-quantum line bundles and gerbes in the context of generalized Kähler geometry.
A potential for generalized Kähler geometry
We show that, locally, all geometric objects of Generalized Kähler Geometry can be derived from a function K, the "generalized Kähler potential". The metric g and two-form B are determined as nonlinear functions of second derivatives of K. These nonlinearities are shown to arise via a quotient construction from an auxiliary local product (ALP) space.
Generalized Kahler geometry and manifest N=(2,2) supersymmetric nonlinear sigma-models
Journal of High Energy Physics
Generalized complex geometry is a new mathematical framework that is useful for describing the target space of N = (2, 2) nonlinear sigma-models. The most direct relation is obtained at the N = (1, 1) level when the sigma model is formulated with an additional auxiliary spinorial field. We revive a formulation in terms of N = (2, 2) semi-(anti)chiral multiplets where such auxiliary fields are naturally present. The underlying generalized complex structures are shown to commute (unlike the corresponding ordinary complex structures) and describe a Generalized Kähler geometry. The metric, B-field and generalized complex structures are all determined in terms of a potential K.
Gauged (2,2) sigma models and generalized Kähler geometry
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.
Generalized Kähler geometry and manifest Script N = (2,2) supersymmetric nonlinear sigma-models
Journal of High Energy Physics, 2005
Generalized complex geometry is a new mathematical framework that is useful for describing the target space of N = (2, 2) nonlinear sigma-models. The most direct relation is obtained at the N = (1, 1) level when the sigma model is formulated with an additional auxiliary spinorial field. We revive a formulation in terms of N = (2, 2) semi-(anti)chiral multiplets where such auxiliary fields are naturally present. The underlying generalized complex structures are shown to commute (unlike the corresponding ordinary complex structures) and describe a Generalized Kähler geometry. The metric, B-field and generalized complex structures are all determined in terms of a potential K.
A topological sigma model of biKähler geometry
Journal of High Energy Physics, 2006
BiKaehler geometry is characterized by a Riemannian metric g ab and two covariantly constant generally non commuting complex structures K ± a b , with respect to which g ab is Hermitian. It is a particular case of the biHermitian geometry of Gates, Hull and Roceck, the most general sigma model target space geometry allowing for (2, 2) world sheet supersymmetry. We present a sigma model for biKaehler geometry that is topological in the following sense: i) the action is invariant under a fermionic symmetry δ; ii) δ is nilpotent on shell; iii) the action is δ-exact on shell up to a topological term; iv) the resulting field theory depends only on a subset of the target space geometrical data. The biKaehler sigma model is obtainable by gauge fixing the Hitchin model with generalized Kaehler target space. It further contains the customary A topological sigma model as a particular case. However, it is not seemingly related to the (2, 2) supersymmetric biKaehler sigma model by twisting in general.
Pseudo-Hyperkähler Geometry and Generalized Kähler Geometry
Letters in Mathematical Physics, 2011
We discuss the conditions for extra supersymmetry in N = (2, 2) supersymmetric nonlinear sigma models described by one left and one right semi-chiral superfield and carrying a pair of non-commuting complex structures. Focus is on linear non-manifest transformations of these fields that have an algebra that closes off-shell. We solve the conditions for invariance of the action and show that a class of these solutions correspond to a bihermitian metric of signature (2, 2) and a pseudo-hyperkähler geometry of the target space. This is in contrast to the usual sector of bi-hermitian geometry with commuting complex structures where extra supersymmetries lead to bi-hypercomplex target space geometry.
Generalized Kähler Geometry from Supersymmetric Sigma Models
Letters in Mathematical Physics, 2006
We give a physical derivation of generalized Kähler geometry. Starting from a supersymmetric nonlinear sigma model, we rederive and explain the results of Gualtieri [10] regarding the equivalence between generalized Kähler geometry and the bi-hermitean geometry of Gates-Hull-Roček . When cast in the language of supersymmetric sigma models, this relation maps precisely to that between the Lagrangian and the Hamiltonian formalisms. We also discuss topological twist in this context.
The Generalised Complex Geometry of (p, q) Hermitian Geometries
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
Linearizing generalized Kähler geometry
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.