Topological-Holomorphic $\mathcal{N}=4$ Gauge Theory: From Langlands Duality of Holomorphic Invariants to Mirror Symmetry of Quasi-topological Strings

Abstract

We perform a topological-holomorphic twist of $\mathcal{N}=4$ supersymmetric gauge theory on a four-manifold of the form $M_4={\Sigma }_1\times {\Sigma }_2$ with Riemann surfaces ${\Sigma }_{1,2}$, and unravel the mathematical implications of its physics. In particular, we consider different linear combinations of the resulting scalar supercharges under $S$-duality, where this will allow us to derive novel topological and holomorphic invariants of $M_4$ and their Langlands duals. As the twisted theory can be topological along ${\Sigma }_1$ whence we can dimensionally reduce it to 2d, via the effective sigma-model on ${\Sigma }_2$, we can also relate these 4d invariants and their Langlands duals to the mirror symmetry of Higgs bundles and that of quasi-topological strings described by the sheaf of chiral differential operators. As an offshoot, we would be able to obtain a fundamental understanding from 4d gauge theory, why chiral differential operators are purely perturbative objects.

30 pp. Further clarifications