Gauge Theory And Wild Ramification

TL;DR

This paper extends the gauge-theory realization of geometric Langlands to wild ramification by incorporating irregular singularities, Stokes phenomena, and isomonodromic deformation. It develops a local abelian model for wild singularities, embeds this into Hitchin moduli spaces, and shows how mirror symmetry between G and its Langlands dual ^LG persists in the wild setting. A central mechanism is viewing wild ramification data as an affine deformation of a cotangent bundle, enabling a D-module interpretation of A-branes and establishing a framework in which the duality commutes with isomonodromic deformations. The work also analyzes the action of the braid group arising from isomonodromy and discusses how relaxing regularity assumptions broadens the scope, extending the Langlands correspondence to more general irregular singularities.

Abstract

The gauge theory approach to the geometric Langlands program is extended to the case of wild ramification. The new ingredients that are required, relative to the tamely ramified case, are differential operators with irregular singularities, Stokes phenomena, isomonodromic deformation, and, from a physical point of view, new surface operators associated with higher order singularities.

Figures (7)

• Figure 1: For each pair $i,j$, there are $n-1$ Stokes rays of type $(ij)$ and an equal number of type $(ji)$. They alternate and are equally spaced, as shown here for $n=4$. To avoid clutter, only the Stokes rays associated with one pair $i,j$ are shown.
• Figure 2: A Riemann surface $C$, here taken to be of genus $g_C=1$, with an irregular singularity at a point $p$. A basepoint is taken at $q$. Show are the Stokes rays near $p$ and the important paths in defining the generalized monodromy data.
• Figure 3: A surface operator whose support is a two-manifold $D=\mathbb{R}\times L$ in a four-manifold $M=\mathbb{R}\times W$. Here $\mathbb{R}$ parametrizes the time direction, which runs vertically. By endowing the surface operator with time-dependent couplings, we define a flat connection on the bundle $\widehat{\mathcal{H}}\to {\fam\eusmfam X}$ of physical Hilbert spaces.
• Figure 4: We consider supersymmetric fields with a singularity along ${\mathbb{R}}\times p\subset M'={\mathbb{R}}\times C$.
• Figure 5: The dotted lines are the Stokes lines that divide the plane into $n-1$ positive sectors and $n-1$ negative sectors, as sketched here for $n=4$. The contour $\mathcal{C}$ runs from the origin to the point $z$; near the origin, it is a straight line in one of the positive sectors.