[Paper Review] Holomorphic reduction of N=2 gauge theories, Wilson-'t Hooft operators, and S-duality

Anton Kapustin|ArXiv.org|Dec 12, 2006

Black Holes and Theoretical PhysicsPhysics and Astronomy35 references89 citations

TL;DR

This paper introduces a holomorphic reduction of N=2 gauge theories on a product of two Riemann surfaces, C × Σ, resulting in a topological B-model on C with target space the moduli of nonabelian vortex equations on Σ. It establishes that Wilson-'t Hooft operators in the twisted theory form a commutative algebra independent of the gauge coupling, and proposes an N=2 analog of geometric Langlands duality via S-duality group actions on the derived category of the moduli space.

ABSTRACT

We study twisted N=2 superconformal gauge theory on a product of two Riemann surfaces Sigma and C. The twisted theory is topological along C and holomorphic along Sigma and does not depend on the gauge coupling or theta-angle. Upon Kaluza-Klein reduction along Sigma, it becomes equivalent to a topological B-model on C whose target is the moduli space MV of nonabelian vortex equations on Sigma. The N=2 S-duality conjecture implies that the duality group acts by autoequivalences on the derived category of MV. This statement can be regarded as an N=2 counterpart of the geometric Langlands duality. We show that the twisted theory admits Wilson-'t Hooft loop operators labelled by both electric and magnetic weights. Correlators of these loop operators depend holomorphically on coordinates and are independent of the gauge coupling. Thus the twisted theory provides a convenient framework for studying the Operator Product Expansion of general Wilson-'t Hooft loop operators.

Motivation & Objective

• To generalize the physical derivation of geometric Langlands duality from N=4 to N=2 supersymmetric gauge theories.
• To construct a twisted N=2 theory on C × Σ that is topological on C and holomorphic on Σ, independent of gauge coupling and theta-angle.
• To identify Wilson-'t Hooft loop operators in the twisted theory, labeled by electric and magnetic weights, and study their operator product expansion (OPE).
• To propose an N=2 counterpart of geometric Langlands duality, where the S-duality group acts by autoequivalences on the derived category of the moduli space of nonabelian vortex equations.
• To establish a framework for computing exact OPE algebras of general Wilson-'t Hooft operators using a twist that allows arbitrary electric and magnetic charges.

Proposed method

• Twist the N=2 superconformal gauge theory on C × Σ such that it becomes topological on C and holomorphic on Σ, with no dependence on the gauge coupling or theta-angle.
• Perform Kaluza-Klein reduction along C to obtain a 2d topological B-model on C with target space M_V, the moduli space of nonabelian vortex equations on Σ.
• Identify Wilson-'t Hooft operators as BRST-invariant loop operators of the form γ × p, where γ is a loop in C and p is a point in Σ, labeled by pairs (μ, ν) in the coweight and weight lattices modulo the Weyl group.
• Use the holomorphic dependence of correlators on Σ to compute the OPE algebra of Wilson-'t Hooft operators exactly, as they are independent of the gauge coupling.
• Show that the S-duality group acts on the derived category of M_V via autoequivalences, generalizing the geometric Langlands correspondence to N=2 theories.
• Construct an analog of the Hitchin fibration for M_V using gauge-invariant polynomials in the Higgs field, showing that the base is a cone of dimension 18(g−1) with 7(g−1) constraints.

Experimental results

Research questions

• RQ1 How can the N=4 twisted theory's derivation of geometric Langlands duality be generalized to N=2 gauge theories?
• RQ2 What is the structure of the operator product expansion (OPE) algebra for general Wilson-'t Hooft operators in N=2 gauge theories, including both electric and magnetic charges?
• RQ3 Can a holomorphic-topological twist of N=2 theories on C × Σ lead to a consistent framework for computing exact OPEs independent of the gauge coupling?
• RQ4 How does S-duality act on the derived category of the moduli space of nonabelian vortex equations in N=2 theories?
• RQ5 What is the geometric structure of the moduli space M_V of nonabelian vortex equations, and how does it compare to the Hitchin moduli space in N=4 theories?

Key findings

• The twisted N=2 theory on C × Σ is topological on C and holomorphic on Σ, and its observables are independent of the gauge coupling and theta-angle.
• Wilson-'t Hooft loop operators are labeled by pairs (μ, ν) in the coweight and weight lattices modulo the Weyl group, and their correlators depend holomorphically on Σ.
• The OPE algebra of Wilson-'t Hooft operators is commutative and independent of the gauge coupling, allowing exact semiclassical computation.
• For purely electric (Wilson) operators, the OPE algebra matches the fusion algebra of irreducible representations of G.
• For purely magnetic ('t Hooft) operators in the adjoint matter case, the OPE algebra matches the fusion algebra of irreducible representations of the Langlands dual group L G.
• The S-duality group acts by autoequivalences on the derived category of the moduli space M_V of nonabelian vortex equations, providing an N=2 analog of geometric Langlands duality.