Geometry of gauged LG-model with compatible boundary conditions

TL;DR

This paper explores the geometric structure of open string Floer theory in gauged Landau-Ginzburg models, focusing on gauged Witten equations, Lagrangian submanifolds, and their intersections within a Hamiltonian framework.

Contribution

It introduces the geometric framework for gauged Landau-Ginzburg models with boundary conditions using gauged Witten equations and constructs Lefschetz thimbles from Lagrangian submanifolds.

Findings

Construction of Lefschetz thimbles from proper Lagrangian submanifolds.

Analysis of an energy functional related to gauged Witten equations.

Identification of critical points as Lagrangian intersections in reduced critical space.

Abstract

This paper introduces the geometry of the open string Floer theory of gauged Landau-Ginzburg model via gauged Witten equations. Given a $G$-invariant Morse-Bott holomorphic function $W$ on a Hamiltonian space $(M,\omega,G),$ Lefschetz thimbles are constructed from proper Lagrangian submanifolds of critical set of $W.$ We study an energy functional on path space whose gradient flow equation corresponds to the gauged Witten equations with temporal gauge on a strip end, and whose critical points are Lagrangian intersections in the reduced critical space of $W.$