M2-branes wrapped on holomorphic curves

TL;DR

This paper advances the understanding of supergravity solutions for membranes wrapped on holomorphic curves by classifying generalised calibrations and exploring supersymmetry constraints in flux backgrounds.

Contribution

It provides a first step towards classifying generalised calibrations for wrapped membranes in non-vanishing flux spacetimes, extending beyond Kähler calibrations.

Findings

Supersymmetry imposes constraints on the Hermitean metric.

Allowed calibrations satisfy specific constraint equations.

Kähler calibrations are a subset of possible solutions.

Abstract

The generalised calibration for a wrapped membrane is gauge equivalent to the supergravity three-form under which the membrane is electrically charged. Given the relevant calibration, one can go a long way towards constructing the supergravity solution for the wrapped brane. Applications of this method have been restricted since generalised calibrations have not yet been completely classified in spacetimes with non-vanishing flux. In this paper, we take a first step towards such a classification by studying membranes wrapping holomorphic curves. Supersymmetry preservation imposes a constraint on the Hermitean metric in the embedding space and it is found that this can be expressed as a restriction on possible generalised calibrations. Allowed calibrations in a particular spacetime are simply those which satisfy the constraint equation relevant to that background; in particular, we see that the previously considered Kahler calibrations are just a subclass of possible solutions.