Integrable Spin Chain of Superconformal U(M)xU(N) Chern-Simons Theory

TL;DR

This paper demonstrates the integrability of the planar limit of N=6 superconformal U(M)xU(N) Chern-Simons theory, explores parity symmetry breaking in associated spin chains, and suggests boundary effects can reveal discrete holonomy.

Contribution

It constructs a parity non-invariant spin chain Hamiltonian for the theory and shows integrability persists despite parity breaking, extending understanding of integrable structures in superconformal Chern-Simons theories.

Findings

Integrability exists in the spectrum of single trace operators.

Parity symmetry can be broken without affecting integrability.

Open spin chains may detect discrete holonomy and parity breaking.

Abstract

N=6 superconformal Chern-Simons theory with gauge group U(M)xU(N)} is dual to N M2-branes and (M-N) fractional M2-branes, equivalently, discrete 3-form holonomy at C4/Zk orbifold singularity. We show that, much like its regular counterpart of M=N, the theory at planar limit have integrability structure in the conformal dimension spectrum of single trace operators. We first revisit the Yang-Baxter equation for a spin chain system associated with the single trace operators. We show that the integrability by itself does not preclude parity symmetry breaking. We construct two-parameter family of parity non-invariant, alternating spin chain Hamiltonian involving three-site interactions between 4 and 4* of SU(4). At weak `t Hooft coupling, we study the Chern-Simons theory perturbatively and calculate anomalous dimension of single trace operators up to two loops. The computation is essentially parallel to the regular case M=N. We find that resulting spin chain Hamiltonian matches with the Hamiltonian derived from Yang-Baxter equation, but to the one preserving parity symmetry. We give several intuitive explanations why the parity symmetry breaking is not detected in the Chern-Simons spin chain Hamiltonian at perturbative level. We suggest that open spin chain, associated with open string excitations on giant gravitons or dibaryons, can detect discrete flat holonomy and hence parity symmetry breaking through boundary field.