A weighted topological quantum field theory for Quot schemes on curves

TL;DR

This paper develops a weighted topological quantum field theory for Quot schemes on curves, unifying previous theories and enabling explicit, virtual-class-free computations of intersection numbers.

Contribution

It introduces a new weighted TQFT that incorporates the degree of vector bundles, generalizing existing theories and avoiding virtual classes.

Findings

Constructed a geometric weighted TQFT for Quot schemes

Unified Witten's TQFT with quantum cohomology TQFT of Grassmannians

Provided explicit formulas for enumerating points of Quot schemes

Abstract

We study Quot schemes of vector bundles on algebraic curves. Marian and Oprea gave a description of a topological quantum field theory (TQFT) studied by Witten in terms of intersection numbers on Quot schemes of trivial bundles. Since these Quot schemes can have the wrong dimension, virtual classes are required. But Quot schemes of general vector bundles always have the right dimension. Using the degree of the general vector bundle as an additional parameter, we construct a weighted TQFT containing both Witten's TQFT and the small quantum cohomology TQFT of the Grassmannian. This weighted TQFT is completely geometric (no virtual classes are needed), can be explicitly computed, and recovers known formulas enumerating the points of finite Quot schemes.