The semi-chiral quotient, hyperkähler manifolds and T-duality

Abstract

We study the construction of generalized Kähler manifolds, described purely in terms of N = (2, 2) semichiral superfields, by a quotient using the semichiral vector multiplet. Despite the presence of a b-field in these models, we show that the quotient of a hyperkähler manifold is hyperkähler, as in the usual hyperkähler quotient. Thus, quotient manifolds with torsion cannot be constructed by this method. Nonetheless, this method does give a new description of hyperkähler manifolds in terms of two-dimensional N = (2, 2) gauged non-linear sigma models involving semichiral superfields and the semichiral vector multiplet. We give two examples: Eguchi-Hanson and Taub-NUT. By T-duality, this gives new gauged linear sigma models describing the T-dual of Eguchi-Hanson and NS5-branes. We also clarify some aspects of T-duality relating these models to N = (4, 4) models for chiral/twisted-chiral fields and comment briefly on more general quotients that can give rise to torsion and give an example. 1 crichigno@max2.physics.sunysb.edu Recent developments in both physics and mathematics are renewing the interest in general d = 2, N = (2, 2) sigma models. From the physics perspective, these models describe string compactifications with NS-NS fluxes and, from the mathematics perspective, they provide a useful tool in exploring aspects of Generalized Complex Geometry. This is an example of the interesting interplay between geometry and supersymmetry, initiated by Zumino in the classic work [1]. It is well known by now that the conditions under which d = 2, N = (1, 1) sigma models (with no Wess-Zumino term) admit an extended supersymmetry, can be solved by requiring the target space to be Kähler, for the case of N = (2, 2), and hyperkähler for N = (4, 4) [2]. The action for the sigma model is then simply given by the Kähler potential K(Φ a ,Φ a ) of the target space, with the complex coordinates Φ a identified with N = (2, 2) chiral superfields satisfyingD + Φ a =D − Φ a = 0. These ideas lead to a variety of applications of supersymmetric methods to Kähler geometry. An example of this is the hyperkähler quotient [3,4]. This method is based on the gauging of isometries along directions parametrized by chiral superfields and provides a powerful method for constructing potentials describing hyperkähler manifolds.

Extended supersymmetry of semichiral sigma models in 4D

“…9 This is equivalent to dualising the scaling symmetry of (4.1). These isometries both commute with the extra supersymmetry [11,12]. 10 In pure gauge they become…”

Dynamics of two-dimensional N = 2 , 2 $$ \mathcal{N}=\left(2,\;2\right) $$ theories with semichiral superfields I

“…Eguchi-Hanson space is the simplest example of the ALE spaces, and can be constructed via hyperkähler quotient in terms of semichiral superfields [22]:…”

“…Taub-NUT space is an example of the ALF space, and can be constructed by semichiral GLSM as follows [22]:…”

T-duality in (2, 1) superspace

“…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].…”