Representation stacks, D-branes and noncommutative geometry

TL;DR

This paper establishes a correspondence between points of a representation stack of an affine algebra and algebra morphisms to Azumaya algebras, linking representation theory with D-branes and noncommutative geometry.

Contribution

It provides a rigorous proof connecting the points of the representation Artin-stack to algebra morphisms into Azumaya algebras, advancing the mathematical understanding of D-branes in noncommutative geometry.

Findings

Points of the representation stack correspond to algebra morphisms R-->A with A Azumaya of degree n.

Connects algebraic geometry with D-branes and noncommutative geometry frameworks.

Enhances the mathematical foundation of noncommutative geometric models in string theory.

Abstract

In this note we prove that the spec(C)-points of the representation Artin-stack [rep_n R/PGL_n] of n-dimensional representations of an affine algebra R correspond to algebra morphisms R-->A where A is an Azumaya algebra of degree n over C. We connect this to the study of D-branes and Azumaya noncommutative geometry, developed by Chien-Hao Liu and Shing-Tung Yau in a series of papers from arXiv:0709.1515 to arXiv:1003.1178