Holomorphic curves from matrices

TL;DR

This paper constructs holomorphic membranes in flat space using matrix theory, linking geometric quantization techniques with solutions representing embedded membranes, and develops a perturbative expansion for these matrices.

Contribution

It introduces a method to represent holomorphic membranes via infinite matrices derived from geometric quantization, including a perturbative expansion using Bergman projection.

Findings

Matrices correspond to holomorphic membranes embedded in space.

Perturbative expansion matches geometric quantization results.

First two terms relate to standard and metaplectic corrected geometric quantization.

Abstract

Membranes holomorphically embedded in flat noncompact space are constructed in terms of the degrees of freedom of an infinite collection of 0-branes. To each holomorphic curve we associate infinite-dimensional matrices which are static solutions to the matrix theory equations of motion, and which can be interpreted as the matrix theory representation of the holomorphically embedded membrane. The problem of finding such matrix representations can be phrased as a problem in geometric quantization, where epsilon -> l_P^3/R plays the role of the Planck constant and parametrizes families of solutions. The concept of Bergman projection is used as a basic tool, and a local expansion for the action of the projection in inverse powers of curvature is derived. This expansion is then used to compute the required matrices perturbatively in epsilon. The first two terms in the expansion correspond to the standard geometric quantization result and to the result obtained using the metaplectic correction to geometric quantization.