Holomorphic supercurves and supersymmetric sigma models

Abstract

We introduce a natural generalisation of holomorphic curves to morphisms of supermanifolds, referred to as holomorphic supercurves. More precisely, supercurves are morphisms from a Riemann surface, endowed with the structure of a supermanifold which is induced by a holomorphic line bundle, to an ordinary almost complex manifold. They are called holomorphic if a generalised Cauchy-Riemann condition is satisfied. We show, by means of an action identity, that holomorphic supercurves are special extrema of a supersymmetric action functional.

Divergence theorems and the supersphere

“…Modern monographs on the general theory of supermanifolds include [Var04] and [CCF11]. Following the conventions of [Gro11] and [Gro13], we denote the (super) tangent sheaf of M , i.e. the sheaf of superderivations of O M , by SM := Der(O M ).…”

“…Maps with flesh allow for having "odd component fields" and are deeply related to inner Hom objects in the category of supermanifolds [SW11]. For details on the derived differential calculus, see [Gro11], while an application is given in Sec. 3.1 below.…”

Killing vector fields and harmonic superfield theories

“…Tensors on SM are similarly endowed to that sheaf by R L -multilinear extension. For details, consult [Gro11].…”

“…Following the conventions of [Gro11], we denote the (super) tangent sheaf, i.e. the sheaf of superderivations of O M , by SM := Der(O M ).…”

“…The pullback of tensors on N is defined as follows. Following the conventions used in [Gro11], let E ∈ End O N (SN ) and…”