Flow lines on the moduli space of rank $2$ twisted Higgs bundles

TL;DR

This paper explores the structure of gradient flow lines on the moduli space of rank 2 twisted Higgs bundles, revealing an algebro-geometric classification linked to secant varieties and providing a Morse-theoretic compactification with algebraic interpretation.

Contribution

It introduces a novel algebro-geometric classification of flow lines on Higgs bundle moduli spaces using secant varieties and connects Morse theory with algebraic geometry.

Findings

Flow line spaces classified by secant varieties of embedded curves.

Morse-theoretic compactification corresponds to Bertram's secant variety resolution.

Provides a geometric interpretation of flow lines in Higgs bundle moduli spaces.

Abstract

This paper studies the gradient flow lines for the $L^2$ norm square of the Higgs field defined on the moduli space of semistable rank $2$ Higgs bundles twisted by a line bundle of positive degree over a compact Riemann surface $X$. The main result is that these spaces of flow lines have an algebro-geometric classification in terms of secant varieties for different embeddings of $X$ into the projectivisation of the negative eigenspace of the Hessian at a critical point. The Morse-theoretic compactification of spaces of flow lines given by adding broken flow lines then has a natural algebraic interpretation via a projection to Bertram's resolution of secant varieties.