N = 1 Geometric Supergravity and Chiral Triples on Riemann Surfaces

Abstract

We construct a global geometric model for the bosonic sector and Killing spinor equations of four-dimensional N = 1 supergravity coupled to a chiral non-linear sigma model and a Spinc 0 structure. The model involves a Lorentzian metric g on a four-manifold M, a complex chiral spinor and a map ϕ : M →Mfrom M to a complex manifold M endowed with a novel geometric structure which we call a chiral triple. Using this geometric model, we show that if M is spin then the Kähler-Hodge condition on M suffices to guarantee the existence of an associated N = 1 chiral geometric supergravity. This positively answers a conjecture proposed by D. Z. Freedman and A. V. Proeyen. We dimensionally reduce the Killing spinor equations to a Riemann surface X, obtaining a novel system of partial differential equations for a harmonic map with potential ϕ : X → M. We characterize all Riemann surfaces X admitting supersymmetric solutions with vanishing superpotential, showing that such solutions ϕ are holomorphicmaps satisfying a certain condition involving the canonical bundle of X and the chiral triple of the theory. Furthermore, we determine the biholomorphism type of all Riemann surfaces carrying supersymmetric solutions with complete Riemannian metric and finite-energy scalar map ϕ. © Springer-Verlag GmbH Germany, part of Springer Nature 2019