On the non-commutative geometry of topological D-branes

TL;DR

This paper explores the noncommutative geometric framework of topological D-branes, linking A-infinity categories with noncommutative symplectic supergeometry to describe their dynamics and moduli spaces.

Contribution

It introduces a novel description of topological D-brane systems using noncommutative symplectic supergeometry and constructs extended moduli spaces as non-commutative algebraic superschemes.

Findings

A formulation of D-brane systems via Z2-graded quivers.

Construction of a generating function for disk correlators as quiver necklaces.

Development of non-commutative algebraic superschemes for moduli spaces.

Abstract

This is a noncommutative-geometric study of the semiclassical dynamics of finite topological D-brane systems. Starting from the formulation in terms of A -infinity categories, I show that such systems can be described by the noncommutative symplectic supergeometry of Z2-graded quivers, and give a synthetic formulation of the boundary part of the generalized WDVV equations. In particular, a faithful generating function for integrated correlators on the disk can be constructed as a linear combination of quiver necklaces, i.e. a function on the noncommutative symplectic superspace defined by the quiver's path algebra. This point of view allows one to construct extended moduli spaces of topological D-brane systems as non-commutative algebraic `superschemes'. They arise by imposing further relations on a Z2-graded version of the quiver's preprojective algebra, and passing to the subalgebra preserved by a natural group of symmetries.