We prove conjecture due to Erickson-Semenoff-Zarembo and Drukker-Gross which relates supersymmetric circular Wilson loop operators in the ${\mathcal N=4}$ supersymmetric Yang-Mills theory with a Gaussian matrix model. We also compute the partition function and give a new matrix model formula for the expectation value of a supersymmetric circular Wilson loop operator for the pure ${\mathcal N=2}$ and the ${\mathcal N=2^*}$ supersymmetric Yang-Mills theory on a four-sphere. A four-dimensional ${\mathcal N=2}$ superconformal gauge theory is treated similarly.

Localization of Gauge Theory on a Four-Sphere and Supersymmetric Wilson Loops

The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N=4 super Yang-Mills theory in four dimensions. The key ingredients are electric-magnetic duality of gauge theory, mirror symmetry of sigma-models, branes, Wilson and 't Hooft operators, and topological field theory. Seemingly esoteric notions of the geometric Langlands program, such as Hecke eigensheaves and D-modules, arise naturally from the physics.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cntp/2007/0001/0001/CNTP-2007-0001-0001-a001.pdf

Electric-Magnetic Duality And The Geometric Langlands Program

On the Structure of the Topological Phase of Two-dimensional Gravity

We construct a cubic field theory which provides all genus amplitudes of the topological A-model for all non-compact toric Calabi-Yau threefolds. The topology of a given Feynman diagram encodes the topology of a fixed Calabi-Yau, with Schwinger parameters playing the role of Kahler classes of the threefold. We interpret this result as an operatorial computation of the amplitudes in the B-model mirror which is the quantum Kodaira-Spencer theory. The only degree of freedom of this theory is an unconventional chiral scalar on a Riemann surface. In this setup we identify the B-branes on the mirror Riemann surface as fermions related to the chiral boson by bosonization.

/pdf/the-topological-vertex-3lrad1kwkt.pdf

The Topological Vertex

By Vyjayanthi Chari and Andrew Pressley: 651 pp., £22.95 (US$34.95), isbn 0 521 55884 0 (Cambridge University Press, 1994).

A guide to quantum groups

We consider Wilson loop observables for Chern-Simons theory at large N and its topological string dual and extend the previous checks for this duality to the case of links. We find an interesting structure involving representation/spin degeneracy of branes ending on branes which features in the large-N dual description of Chern-Simons theory. This leads to a refinement of the integer invariants for links and knots. We illustrate our results with explicit computations on the Chern-Simons side.

/pdf/knots-links-and-branes-at-large-n-4sgk0os2pp.pdf

Knots, links and branes at large N

Massless particles in arbitrary representations of the Lorentz group

We make a precision test of a recently proposed conjecture relating Chern–Simons gauge theory to topological string theory on the resolution of the conifold First, we develop a systematic procedure to extract string amplitudes from vacuum expectation values (vevs) of Wilson loops in Chern–Simons gauge theory, and then we evaluate these vevs in arbitrary irreducible representations of SU(N) for torus knots We find complete agreement with the predictions derived from the target space interpretation of the string amplitudes We also show that the structure of the free energy of topological open string theory gives further constraints on the Chern–Simons vevs Our work provides strong evidence towards an interpretation of knot polynomial invariants as generating functions associated to enumerative problems

http://cds.cern.ch/record/436014/files/0004196.pdf

Polynomial invariants for torus knots and topological strings

We consider Wilson loop observables for Chern-Simons theory at large N and its topological string dual and extend the previous checks for this duality to the case of links. We find an interesting structure involving representation/spin degeneracy of branes ending on branes which features in the large N dual description of Chern-Simons theory. This leads to a refinement of the integer invariants for links and knots. We illustrate our results with explicit computations on the Chern-Simons side.

/pdf/knots-links-and-branes-at-large-n-1klts8hjh0.pdf

José M. F. Labastida

Papers

Knots, links and branes at large N

Massless particles in arbitrary representations of the Lorentz group

Polynomial invariants for torus knots and topological strings

Knots, links and branes at large N

Topological gravity in two dimensions