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

TY - UNPB

T1 - Flow lines on the moduli space of rank $2$ twisted Higgs bundles

AU - Wilkin, Graeme Peter Desmond

PY - 2024/8/23

Y1 - 2024/8/23

N2 - 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 compactification of spaces of flow lines given by adding broken flow lines then has a natural interpretation via a projection to Bertram's resolution of secant varieties.

AB - 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 compactification of spaces of flow lines given by adding broken flow lines then has a natural interpretation via a projection to Bertram's resolution of secant varieties.

KW - Higgs bundles

KW - Morse-Bott-Smale

KW - Gradient flow

U2 - 10.48550/arXiv.2408.13098

DO - 10.48550/arXiv.2408.13098

M3 - Preprint

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

PB - arXiv

ER -