ABJM Wilson Loops in Arbitrary Representations

TL;DR

This paper analyzes vacuum expectation values of circular half BPS Wilson loops in arbitrary representations within ABJM theory, utilizing the Fermi gas formalism to derive exact and numerical results, and explores instanton effects and their consistency with topological string predictions.

Contribution

It introduces a method to compute Wilson loop VEVs in arbitrary representations using the Fermi gas formalism, including a determinant expression for non-hook representations, and connects instanton effects with topological string theory.

Findings

Hook representation VEVs reduce to elementary integrals.

Non-hook VEVs expressed as determinants of hook VEVs.

Results align with topological string instanton effects.

Abstract

We study vacuum expectation values (VEVs) of circular half BPS Wilson loops in arbitrary representations in ABJM theory. We find that those in hook representations are reduced to elementary integrations thanks to the Fermi gas formalism, which are accessible from the numerical studies similar to the partition function in the previous studies. For non-hook representations, we show that the VEVs in the grand canonical formalism can be exactly expressed as determinants of those in the hook representations. Using these facts, we can study the instanton effects of the VEVs in various representations. Our results are consistent with the worldsheet instanton effects studied from the topological string and a prescription to include the membrane instanton effects by shifting the chemical potential, which has been successful for the partition function.