Wilson Loops, Geometric Transitions and Bubbling Calabi-Yau's

TL;DR

This paper explores new bulk descriptions of Wilson loops in Chern-Simons theory using brane configurations and bubbling Calabi-Yau geometries, linking knot invariants to topological transitions.

Contribution

It introduces a novel geometric transition approach to represent Wilson loops via bubbling Calabi-Yau manifolds, expanding the understanding of their topological and holographic duals.

Findings

Wilson loops described by brane configurations in resolved conifold

Bubbling Calabi-Yau geometries encode Wilson loop representations

Confirmed the unknot correspondence to all orders in genus expansion

Abstract

Motivated by recent developments in the AdS/CFT correspondence, we provide several alternative bulk descriptions of an arbitrary Wilson loop operator in Chern-Simons theory. Wilson loop operators in Chern-Simons theory can be given a description in terms of a configuration of branes or alternatively anti-branes in the resolved conifold geometry. The representation of the Wilson loop is encoded in the holonomy of the gauge field living on the dual brane configuration. By letting the branes undergo a new type of geometric transition, we argue that each Wilson loop operator can also be described by a bubbling Calabi-Yau geometry, whose topology encodes the representation of the Wilson loop. These Calabi-Yau manifolds provide a novel representation of knot invariants. For the unknot we confirm these identifications to all orders in the genus expansion.