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

TL;DR

The paper demonstrates that the ${ m N}=6$ ABJ theory with gauge group $U(M)\times\overline{U(N)}$ possesses an integrable spin-chain structure in the planar limit, mirroring the ABJM case but with parity-odd deformations encoded by $b=(M-N)/k$. Using a Yang-Baxter framework for an alternating SU(4) spin chain, it derives a parity-non-invariant two-parameter family of integrable Hamiltonians, and then shows that two-loop operator mixing in Chern-Simons theory yields a parity-invariant dilatation operator with an effective coupling $\lambda\bar{\lambda}$. This implies that parity breaking is not detectable in the closed spin chain at perturbative order; open spin chains attached to giant gravitons or dibaryons are proposed as potential avenues to observe the discrete holonomy. The work links integrability, discrete holonomy, and AdS4/CFT3 duality in ABJ, suggesting that boundary effects hold the key to accessing parity-violating dynamics across coupling regimes.

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.

Figures (8)

• Figure 1: Two loop contribution of scalar sextet interaction to anomalous dimension matrix of ${\cal O}$.
• Figure 2: Two loop contribution of gauge and fermion exchange interaction to anomalous dimension of ${\cal O}$.
• Figure 3: Two loop contribution of diamagnetic gauge interactions to wave function renormalization of $Y, Y^\dagger$. They contribute to $\mathbb{I}$ operator in the dilatation operator.
• Figure 4: Two loop contribution of paramagnetic gauge interactions to wave function renormalization of $Y, Y^\dagger$. They contribute to $\mathbb{I}$ operator in the dilatation operator.
• Figure 5: Two loop contribution of Chern-Simons interaction to wave function renormalization of $Y, Y^\dagger$. They contribute to $\mathbb{I}$ operators in the dilatation operator.