Chambered invariants of real Cauchy-Riemann operators

TL;DR

This paper introduces three new chambered invariants for real Cauchy-Riemann operators, linking symplectic geometry and algebraic invariants through counts of pseudo-holomorphic sections and solutions to vortex equations.

Contribution

It constructs novel chambered invariants associated with real Cauchy-Riemann operators, connecting symplectic counts to algebraic geometric invariants.

Findings

Defined three chambered invariants: $n_{Bl}$, $n_{1,2}$, $n_{2,1}$.

Connected invariants to counts of pseudo-holomorphic sections and vortex solutions.

Conjectured relations to algebraic invariants like Pandharipande-Thomas and Donaldson-Thomas.

Abstract

Motivated by counting pseudo-holomorphic curves in symplectic Calabi-Yau $3$-folds, this article studies a chamber structure in the space of real Cauchy-Riemann operators on a Riemann surface, and constructs three chambered invariants associated with such operators: $n_{\mathrm{Bl}}$, $n_{1,2}$, $n_{2,1}$. The first of these invariants is defined by counting pseudo-holomorphic sections of bundles whose fibres are modeled on the blow-up of $\mathbf{C}^2/\{\pm 1\}$. The other two are defined by counting solutions to the ADHM vortex equations. We conjecture that $n_{1,2}$ and $n_{2,1}$ are related to putative symplectic invariants generalizing the Pandharipande-Thomas and rank $2$ Donaldson-Thomas invariants in algebraic geometry.