Matrix Representations of Holomorphic Curves on $T_{4}$

TL;DR

This paper develops a matrix representation for holomorphic membranes in complex tori, linking brane configurations to gauge fields and providing a natural deformation quantization scheme for symplectic surfaces.

Contribution

It introduces a novel matrix representation for membranes in complex tori and establishes an explicit, coordinate-invariant star-product for quantization.

Findings

Constructed matrix representations of membranes in complex tori.

Linked brane configurations to U(N) gauge fields with specific properties.

Provided an explicit associative star-product for quantizing symplectic surfaces.

Abstract

We construct a matrix representation of compact membranes analytically embedded in complex tori. Brane configurations give rise, via Bergman quantization, to U(N) gauge fields on the dual torus, with almost-anti-self-dual field strength. The corresponding U(N) principal bundles are shown to be non-trivial, with vanishing instanton number and first Chern class corresponding to the homology class of the membrane embedded in the original torus. In the course of the investigation, we show that the proposed quantization scheme naturally provides an associative star-product over the space of functions on the surface, for which we give an explicit and coordinate-invariant expression. This product can, in turn, be used the quantize, in the sense of deformation quantization, any symplectic manifold of dimension two.