Cotangent bundle over all the compact Hermitian symmetric spaces and projective superspace

Abstract

Abstract. We construct the N = 2 supersymmetric nonlinear sigma model on the cotangent bundle over the compact Hermitian symmetric space E7/E6 × U (1) by using the projective superspace formalism which is an off-shell superfield formulation in four-dimensional spacetime. We also give a simple formula giving the hyper-Kähler potential of the cotangent bundle over all the compact Hermitian symmetric spaces. IntroductionSupersymmetry (SUSY) has intimate relations to the complex geometry [1]. In particular, the target space of SUSY nonlinear sigma models (NLSMs) possessing 8 supercharges must be hyper-Käbler manifold [2]. It means that SUSY is a powerful tool to construct a new hyperKäbler manifold. In other words, a new hyper-Kḧaler manifold can be obtained through a new SUSY NLSM with 8 supercharges. One of the most convenient formulations to construct SUSY NLSMs with 8 supercharges is the projective superspace formalism [3,4,5,6] being an off-shell superfield formulation in theories with 8 supercharges 1 .The projective superspace formalism is a powerful devise to construct N = 2 SUSY NLSMs. With the use of this formalism, first the SUSY NLSMs on the tangent bundles over the compact and non-compact classical Hermitian symmetric as well as, using the generalized Legendre transform [4], the cotangent bundles corresponding to the hyperkähler metrics have been constructed [9,10,11,12]. Developed the results in [9,10,11,12], the cotangent bundle over the compact type of E 6 /SO(10) × U (1) have been also constructed [13]. After these works, the closed form expression of the cotangent bundle over any compact Hermitian symmetric space was obtained [14], which gives a form of the cotangent bundle presented in a matrix form. We elaborated on this result to provide a new closed formula for cotangent bundle action, and applied it to construction of the N = 2 SUSY NLSM on the cotangent bundle over the compact E 7 /E 6 × U (1) [15]. In this proceeding, we review our work based on [15].