Communications in Analysis and Geometry

Volume 28 (2020)

Number 4

The First of Two Special Issues in Honor of Karen Uhlenbeck’s 75th Birthday

Special-Issue Editors: Georgios Daskalopoulos (Brown University), Kefeng Liu, Chuu-Lian Terng (U. of Cal. Irvine), and Shing-Tung Yau

The KW equations and the Nahm pole boundary condition with knots

Pages: 871 – 942

DOI: https://dx.doi.org/10.4310/CAG.2020.v28.n4.a4

Authors

Rafe Mazzeo (Department of Mathematics, Stanford University, Stanford, California, U.S.A.)

Edward Witten (School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey, U.S.A.)

Abstract

It is conjectured that the coefficients of the Jones polynomial can be computed by counting solutions of the KW equations on a fourdimensional half-space, with certain boundary conditions that depend on a knot. The boundary conditions are defined by a “Nahm pole” away from the knot with a further singularity along the knot. In a previous paper, we gave a precise formulation of the Nahm pole boundary condition in the absence of knots; in the present paper, we do this in the more general case with knots included. We show that the KW equations with generalized Nahm pole boundary conditions are elliptic, and that the solutions are polyhomogeneous near the boundary and near the knot, with exponents determined by solutions of appropriate indicial equations. This involves the analysis of a “depth two incomplete iterated edge operator.” As in our previous paper, a key ingredient in the analysis is a convenient new Weitzenböck formula that is well-adapted to the specific problem.

Received 4 December 2017

Accepted 15 September 2018

Published 1 October 2020