Knot invariants from four-dimensional gauge theory

Davide Gaiotto, Edward Witten

Research output: Contribution to journalArticlepeer-review

66 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)935-1086
Number of pages152
JournalAdvances in Theoretical and Mathematical Physics
Issue number3
StatePublished - Jun 2012
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • General Physics and Astronomy


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