Knot invariants from four-dimensional gauge theory

Abstract

It has been argued based on electric-magnetic duality and other ingredients that the Jones polynomial of a knot in three dimensions can be computed by counting the solutions of certain gauge theory equations in four dimensions. Here, we attempt to verify this directly by analyzing the equations and counting their solutions, without reference to any quantum dualities. After suitably perturbing the equations to make their behavior more generic, we are able to get a fairly clear understanding of how the Jones polynomial emerges. The main ingredient in the argument is a link between the four-dimensional gauge theory equations in question and conformal blocks for degenerate representations of the Virasoro algebra in two dimensions. Along the way we get a better understanding of how our subject is related to a variety of new and old topics in mathematical physics, ranging from the Bethe ansatz for the Gaudin spin chain to the M -theory description of Bogomol'nyi-Prasad-Sommerfield (BPS) monopoles and the relation between Chern-Simons gauge theory and Virasoro conformal blocks. e-print archive: http://lanl.arXiv.org/abs/1106.4789v1

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

“…When the underlying 3d N = 4 theory is an A-type quiver gauge theory, the resulting superpotential coincides with the Yang-Yang functional for a rational Gaudin model [91,92]. In this case, the very A-twisted Landau-Ginzburg model appeared in recent work on knot homology [59,93]. More generally, the superpotential appears to govern the physics of an M2-M5 brane system that has appeared in many physical constructions of knot homology, related to the classic M5-M5' construction of [94,95].…”

“…(This is a particular instance of NekrasovShatashvili duality [91].) Notably, the same superpotential appeared in [59], in the study of M2/M5 brane systems. The physical reason for this is fairly clear: the 3d A n -type quiver gauge theory can be engineered from a system of intersecting D3-NS5 branes, and both M2-M5 and D3-NS5 systems are dual to a common D2-NS5 system.…”

“…In particular, each critical point σ ν is associated with infinitely many critical values, differing by integer multiples of m ρ . This ambiguity reflects the internal U(1) ρ grading in the category of boundary conditions for T 2d ; an extended discussion of such a phenomenon can be found in [59,Section 4.1.4]. Similarly, the difference of critical values W (σ ν ) − W (σ ν ) is modified by q m ρ for some q ∈ Z, reflecting the structure of central charges in a theory with flavor symmetry that we described abstractly in section 7.7.2.…”

“…The paper [59] studied braid actions in the A-twisted Landau-Ginzburg theory coming from varying mass parameters. One claim of that paper was that these actions, at the categorical level, should provide a physical construction of Khovanov knot homology.…”

“…(Such affine braid group actions have played a central role in constructions of knot homology, both in mathematics and physics, cf. [54][55][56][57][58][59].) In the 2d reductions of 3d gauge theories that we study in section 7, two commuting braid-group actions will appear.…”