Donaldson-Witten theory, surface operators and mock modular forms
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...
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Zusammenfassung: | 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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DOI: | 10.48550/arxiv.1810.07057 |