Donaldson-Witten theory, surface operators and mock modular forms

Georgios Korpas

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arXiv · 1810.07057

Donaldson-Witten theory, surface operators and mock modular forms

Georgios Korpas

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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.

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Georgios Korpas. 2018-10-16. Donaldson-Witten theory, surface operators and mock modular forms. https://arxiv.org/abs/1810.07057

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