Curved 11D Supergeometry

TL;DR

This paper develops a systematic method to compute the theta-expansion of eleven-dimensional supergeometry, providing explicit expressions up to fifth order and exact solutions at linear order, with applications to supermembrane coupling and M(atrix) theory.

Contribution

It introduces a recursive procedure for the theta-expansion of supervielbein and superforms in 11D supergravity, enabling explicit calculations to high order.

Findings

Explicit expressions for supervielbein and three-form up to order θ^5.

Exact all-order solutions in theta at linear order in supergravity fields.

Complete linear coupling of supermembrane to supergravity backgrounds.

Abstract

We examine the theta-expansion of the eleven-dimensional supervielbein. We outline a systematic procedure which can be iterated to any order. We give explicit expressions for the vielbein and three-form potential components up to order ${\cal O}(\th^5)$. Furthermore we show that at each order in the number of supergravity fields, in a perturbative expansion around flat space, it is possible to obtain exact expressions to all orders in theta. We give the explicit expression at linear order in the number of fields and we show how the procedure can be iterated to any desired order. As a byproduct we obtain the complete linear coupling of the supermembrane to the background supergravity fields, covariantly in component form. We discuss the implications of our results for M(atrix) theory.