Arising from a topological twist of $\mathcal{N} = 4$ super Yang-Mills theory are the Kapustin-Witten equations, family gauge-theoretic equations on four-manifold parametrized by $t\in\mathbb{P}^1$. The parameter corresponds to linear combination two charges in twist. When $t=0$ and is compact K\"ahler surface, become Simpson which was originally studied Hitchin Riemann as demonstrated independ...

We extend most of the results of generic vanishing theory to bundles of holomorphic forms and rank-one local systems, and more generally to certain coherent sheaves of Hodge-theoretic origin associated with irregular varieties. Our main tools are Saito’s mixed Hodge modules, the Fourier–Mukai transform for D-modules on abelian varieties introduced by Laumon and Rothstein, and Simpson’s harmonic...

The Hodge theorem also asserts that these two spaces are complex conjugates and hence the equality of the two dimensions. In this note we give a p-adic proof of (1.1) when X/C is a smooth projective threefold. Our approach is based on the following observation: one first notes a purely p-adic assertion that (1.1) holds when Hodge numbers are replaced by more delicate p-adic invariants introduce...

The Hodge theory of the de Rham complex in the setting of the Sobolev topology is studied. As a result, a new elliptic boundaryvalue problem is obtained. Next, the Hodge theory of the @-Neumann problem in the Sobolev topology is studied. A new @-Neumann boundary condition is obtained, and the corresponding subelliptic estimate derived. The classical Hodge theory on a domain in R N+1 (or, more g...

We propose a new, conjectural recursion solution for Hurwitz numbers at all genera. This conjecture is based on recent progress in solving type B topological string theory on the mirrors of toric Calabi–Yau manifolds, which we briefly review to provide some background for our conjecture. We show in particular how this B-model solution, combined with mirror symmetry for the one-leg, framed topol...

We propose a new, conjectural recursion solution for Hurwitz numbers at all genera. This conjecture is based on recent progress in solving type B topological string theory on the mirrors of toric Calabi–Yau manifolds, which we briefly review to provide some background for our conjecture. We show in particular how this B-model solution, combined with mirror symmetry for the one-leg, framed topol...