Knot Invariants from Four-Dimensional Gauge Theory

By D. Gaiotto, E. Witten. 2011

177 citations recorded by the source corpus.

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 BPS monopoles and the relation between Chern-Simons gauge theory and Virasoro conformal blocks.

Subfields: Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/db11924db94a8845e4ecc79e5dbbb97fe2e266e8

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Citations in the corpus

Elliptic stable envelopes (2016, background): “The relation of this theory to knot theory and invariants of quantum groups has long been predicted by physicists [22, 53, 60].”
Exact Kähler potential from gauge theory and mirror symmetry (2012, methodology): “The problem of summing over all worldsheet instantons defines an interesting class of topological invariants, known as the Gromow-Witten invariants [4, 5].”
Fivebranes and Knots (2011, background): “For a direct verification of this for the special case corresponding to the Jones polynomial (that is, G = SU(2) with loop operators associated to the two-dimensional representation of G) see [57].”
Analytic continuation of Liouville theory (2011, methodology): “y conditions required at the ends of I are those of [18]. These boundary conditions have recently been reconsidered and the relation between Liouville and Chern-Simons theory developed in more detail [21]. Given these facts, instead of looking for complex solutions of Liouville theory on , we can lo”