ABJ Triality: from Higher Spin Fields to Strings

TL;DR

The paper constructs and analyzes parity-violating, supersymmetric Vasiliev theories in AdS_4 with Chan-Paton factors, showing that suitably chosen boundary conditions preserve varying amounts of supersymmetry and yield holographic duals to 3D N-extended Chern-Simons vector models, including ABJ theory. It establishes a precise link between the bulk parity phase θ_0 and boundary correlator structures, and demonstrates a triality connecting ABJ theory, bulk Vasiliev theory, and type IIA string theory on AdS_4×CP^3. The authors develop a comprehensive framework for deconstructing SUSY-boundary conditions to map bulk deformations (double/triple traces, CS couplings) onto boundary Lagrangian parameters, and they analyze finite-temperature phase transitions in the ABJ setup via a matrix model. Collectively, the work provides a cohesive picture in which higher-spin bulk dynamics, open-closed string interactions, and strongly-coupled 3D CS-vector theories intertwine, suggesting deep geometric and holographic unity across these theories. The results pave the way for further nonperturbative field-theoretic checks and a deeper understanding of how Vasiliev theory embeds into string/M-theory regimes.

Abstract

We demonstrate that a supersymmetric and parity violating version of Vasiliev's higher spin gauge theory in AdS$_4$ admits boundary conditions that preserve ${\cal N}=0,1,2,3,4$ or 6 supersymmetries. In particular, we argue that the Vasiliev theory with U(M) Chan-Paton and ${\cal N}=6$ boundary condition is holographically dual to the 2+1 dimensional $U(N)_k\times U(M)_{-k}$ ABJ theory in the limit of large $N,k$ and finite $M$. In this system all bulk higher spin fields transform in the adjoint of the U(M) gauge group, whose bulk t'Hooft coupling is $\frac{M}{N}$. Analysis of boundary conditions in Vasiliev theory allows us to determine exact relations between the parity breaking phase of Vasiliev theory and the coefficients of two and three point functions in Chern-Simons vector models at large $N$. Our picture suggests that the supersymmetric Vasiliev theory can be obtained as a limit of type IIA string theory in AdS$_4\times \mathbb{CP}^3$, and that the non-Abelian Vasiliev theory at strong bulk 't Hooft coupling smoothly turn into a string field theory. The fundamental string is a singlet bound state of Vasiliev's higher spin particles held together by U(M) gauge interactions. This is illustrated by the thermal partition function of free ABJ theory on a two sphere at large $M$ and $N$ even in the analytically tractable free limit. In this system the traces or strings of the low temperature phase break up into their Vasiliev particulate constituents at a U(M) deconfinement phase transition of order unity. At a higher temperature of order $T=\sqrt{\frac{N}{M}}$ Vasiliev's higher spin fields themselves break up into more elementary constituents at a U(N) deconfinement temperature, in a process described in the bulk as black hole nucleation.

Figures (1)

• Figure 1: Eigenvalue distribution for wavy$(\lambda=5)$ and clumped$(\lambda=1)$ phases of Gross-Witten-Wadia model.