Monodromy defects in Chern-Simons theory and Holography

Abstract

Wilson loop operators in Chern-Simons theory have revealed profound links between quantum field theory, the fractional quantum Hall effect, topology, conformal field theory, and string theory. In Chern-Simons theories with charge conjugation symmetry, we construct a new class of observables: codimension-two monodromy defects around which fields return to themselves up to charge conjugation. Whereas Wilson loops are labeled by integrable representations of an untwisted affine Lie algebra, monodromy defects are labeled by those of the corresponding twisted affine algebra. The modular and fusion data of these two algebras determine the exact correlation functions of Wilson lines and monodromy defects, which together furnish a $\mathbb{Z}_2$-crossed braided tensor category. The spectrum of line defects in Chern-Simons theory thus gives a physical realization of every algebra in Kac's classification of affine Lie algebras: untwisted for Wilson loops, twisted for monodromy defects. We also determine the exact 't Hooft expansion of monodromy defects in $SU(N)_k$ Chern-Simons theory and identify their holographic duals in topological string theory. The insertion of the lightest monodromy defect has a striking effect: it replaces the resolved conifold background of the Gopakumar-Vafa duality by a specific orientifold of the resolved conifold, transmuting the dual theory of oriented strings into one of unoriented strings. Each excited monodromy defect is then realized as a collection of branes in the orientifold background, with the brane content determined by the representation of the twisted affine algebra that labels the defect.