Tachyon Condensation on the Elliptic Curve

TL;DR

This paper employs matrix factorizations to analyze topological B-branes on the cubic curve, revealing their RR charges, moduli dependence, and tachyon condensation processes, including explicit constructions of higher-rank branes.

Contribution

It introduces a systematic approach to study tachyon condensation and higher-dimensional branes using matrix factorizations on elliptic curves, with explicit examples.

Findings

Explicit construction of all rank-two matrix factorizations.

Demonstration of moduli dependence and cohomology jumps.

Systematic method for building higher-dimensional branes from minimal ones.

Abstract

We use the framework of matrix factorizations to study topological B-type D-branes on the cubic curve. Specifically, we elucidate how the brane RR charges are encoded in the matrix factors, by analyzing their structure in terms of sections of vector bundles in conjunction with equivariant R-symmetry. One particular advantage of matrix factorizations is that explicit moduli dependence is built in, thus giving us full control over the open-string moduli space. It allows one to study phenomena like discontinuous jumps of the cohomology over the moduli space, as well as formation of bound states at threshold. One interesting aspect is that certain gauge symmetries inherent to the matrix formulation lead to a non-trivial global structure of the moduli space. We also investigate topological tachyon condensation, which enables us to construct, in a systematic fashion, higher-dimensional matrix factorizations out of smaller ones; this amounts to obtaining branes with higher RR charges as composites of ones with minimal charges. As an application, we explicitly construct all rank-two matrix factorizations.