Donaldson-Witten theory, surface operators and mock modular forms

TL;DR

This paper studies the $u$-plane integral in Donaldson-Witten theory with surface operators, revealing connections to mock modular forms and localization phenomena on certain Kähler surfaces.

Contribution

It introduces a new formulation of the $u$-plane integral using indefinite theta functions and demonstrates localization at the cusp for specific Kähler surfaces.

Findings

Reexpression of the integrand as a total derivative involving mock modular forms.

Localization of the integral at the cusp at infinity for certain Kähler surfaces.

Connection between surface operators, ramified invariants, and mock modular forms.

Abstract

We revisit the $u$-plane integral of the topologically twisted $\mathcal{N}=2$ super Yang-Mills theory, the Donaldson-Witten theory, on a closed four-manifold $X$ with embedded surfaces that support supersymmetric surface operators. This integral mathematically corresponds to the generating function of the ramified Donaldson invariants of $X$. By including a $\overline{\mathcal{Q}}$-exact deformation to the $u$-plane integral we are able to re-express its integrand in terms of a total derivative with respect to an indefinite theta function, a special kind of mock modular form. We show that for specific K\"ahler surfaces of Kodaira dimension $-\infty$ the integral localizes at the cusp at infinity of the Coulomb branch of the theory.