Three-Point Functions in ABJM and Bethe Ansatz

TL;DR

This paper introduces an integrability-based method to compute structure constants in ABJM theory, utilizing a nested Bethe ansatz and matrix product states, providing new tools for analyzing three-point functions at weak coupling.

Contribution

It develops a nested Bethe ansatz for an alternating SU(4) spin chain and proposes a novel overlap formula for sub-determinant operators in ABJM theory.

Findings

Structure constants expressed as Bethe state overlaps.

Conjecture of determinant operator as an integrable matrix product state.

Evidence supporting integrability of sub-determinant operators.

Abstract

We develop an integrability-based framework to compute structure constants of two sub-determinant operators and a single-trace non-BPS operator in ABJM theory in the planar limit. In this first paper, we study them at weak coupling using a relation to an integrable spin chain. We first develop a nested Bethe ansatz for an alternating SU(4) spin chain that describes single-trace operators made out of scalar fields. We then apply it to the computation of the structure constants and show that they are given by overlaps between a Bethe eigenstate and a matrix product state. We conjecture that the determinant operator corresponds to an integrable matrix product state and present a closed-form expression for the overlap, which resembles the so-called Gaudin determinant. We also provide evidence for the integrability of general sub-determinant operators. The techniques developed in this paper can be applied to other quantities in ABJM theory including three-point functions of single-trace operators.