Topological quantum field theory for dormant opers

TL;DR

This paper develops a new enumerative geometric framework for counting dormant G-opers using a semisimple 2D TQFT, extending intersection theory on moduli stacks of opers.

Contribution

It introduces the notion of dormant faithful twisted G-opers and constructs a moduli stack with a perfect obstruction theory, leading to a novel 2D TQFT for counting G-do'pers.

Findings

Constructed a compact moduli stack with perfect obstruction theory.

Developed a semisimple 2D TQFT for counting G-do'pers.

Established an analogue of the Witten-Kontsevich theorem for G-do'pers.

Abstract

The purpose of the present paper is to develop the enumerative geometry of dormant $G$-opers for a semisimple algebraic group $G$. In the present paper, we construct a compact moduli stack admitting a perfect obstruction theory by introducing the notion of a dormant faithful twisted $G$-oper (or a "$G$-do'per", for short). The resulting virtual fundamental class induces a semisimple $2$d TQFT (= $2$-dimensional topological quantum field theory) counting the number of $G$-do'pers. This $2$d TQFT gives an analogue of the Witten-Kontsevich theorem describing the intersection numbers of psi classes on the moduli stack of $G$-do'pers.