Logarithmic Donaldson-Thomas Theory

University Of Texas At Austin, Austin TX

Investigators

Abstract

This award supports research on the mathematical side of the interaction between theoretical particle physics and mathematics. This interaction started with the advent of string theory in the 1980's and has become a huge source of inspiration in mathematics, often connecting hitherto disconnected areas of research. The Donaldson-Thomas invariants under investigation in this project are a mathematical version of the count of certain D-branes, which are higher-dimensional objects in string theory providing boundary conditions for open strings. The importance of such counts comes from the multitude of relations to other mathematical objects and notions, such as quiver representations, instantons, stability conditions, and Gromov-Witten invariants. Donaldson-Thomas-type invariants are also central to a long list of questions and conjectures posed both by physicists and mathematicians. This project will develop and test a theory of a logarithmic framework for Donaldson-Thomas invariants. The logarithmic version is designed for treating situations relative to a divisor and for studying degenerating families. This extension is of importance for the interpretation of the invariants, for computations, for new applications, and as a proof-theoretic tool, thus greatly expanding the scope of Donaldson-Thomas theory. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.