Knot invariants from four-dimensional gauge theory
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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.
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