ABJ Triality: from Higher Spin Fields to Strings

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.

• The paper establishes a triality linking ABJ three-dimensional gauge theories, Vasiliev's higher spin theory, and type IIA string theory via AdS/CFT correspondence.
• It employs both parity invariant and parity violating frameworks with U(M) Chan-Paton factors to incorporate supersymmetry and analyze phase transitions in holographic duals.
• The work introduces a robust framework for testing nontrivial field symmetries, paving the way for extended analytical and numerical studies in gauge/gravity duality.

Overview of "ABJ Triality: from Higher Spin Fields to Strings"

The paper "ABJ Triality: from Higher Spin Fields to Strings" presents a detailed exploration of the relationships, or "triality", between certain three-dimensional supersymmetric gauge theories and two different types of higher-dimensional theories: higher spin gauge theories and string theories. This work primarily focuses on the AdS/CFT correspondence, providing insights into how higher spin gauge theories in four-dimensional anti-de Sitter space (AdS$_4$) and type IIA string theory on AdS$_4 \times \mathbb{CP}^3$ are related to three-dimensional Chern-Simons-matter theories known as ABJ models.

Key Concepts and Results

1. Vasiliev's Higher Spin Theory: The paper discusses Vasiliev's higher spin gauge theory, which provides a candidate for a consistent quantum theory of a large spectrum of particles with increasing spins. Vasiliev's theory in AdS$_4$ is considered in both parity invariant and parity violating versions. The authors detail how supersymmetry and parity can be incorporated into the framework, extending the gauge symmetry to include $U(M)$ Chan-Paton factors.
2. ABJ Model: At the heart of the triality is the ABJ model, a three-dimensional theory with gauge group $U(N)_k \times U(M)_{-k}$ coupled to pairs of bifundamental matters. In the large $N$ limit with $M$ finite, this model is conjectured to have a holographic dual description in terms of higher spin fields.
3. Triality and Supersymmetry: The triality reflects a rich structure where one corner is the boundary CFT perspective (the ABJ model), another is the higher spin theory (a non-gravitational AdS dual), and the third is the string theory on AdS$_4 \times \mathbb{CP}^3$. The authors explore how the degrees of supersymmetry ($\mathcal{N}=6$, 4, 3, 2, and 1) affect the duals.
4. Holographic Duals and Phase Transitions: A compelling narrative is provided regarding the thermal phase transitions of these theories. In the free limit (Chern-Simons level $k \to \infty$), the partition function of the ABJ model on $S^2 \times S^1$ is analyzed, exhibiting phase transitions that mirror those expected based on the holographic duality.
5. Non-Abelian Generalization: The paper extends the discussion to non-Abelian cases by including higher spin fields with $U(M)$ Chan-Paton factors, revealing the structural role of 't Hooft coupling and its relation to the strength of non-abelian interactions.
6. Implications and Speculations: The authors speculate about how these relationships inform our understanding of string theory in the tensionless limit and propose a deeper connection between these theories and level-rank dualities in lower dimensions.

Implications and Future Directions

This study has several important theoretical implications:

• It enhances the understanding of the space of theorized dualities, particularly elucidating how interacting, three-dimensional gauge theories are related to higher spin theories and string theories.
• The work offers a new perspective on the large $N$ approximation and parity-violating theories within the context of holography.
• Importantly, it proposes a framework to test field theories with nontrivial symmetries against higher spin holography, thus setting the stage for future analytical and numerical investigations of three-dimensional gauge theories.

Future developments are likely to include:

• Further exploration into the detailed mechanism by which string theory reduces to Vasiliev's theory in appropriate limits.
• Applying these insights to find new solvable sectors or integrable structures within the larger framework of gauge/gravity duality.
• Exploring the extended landscape of holographic theories by considering modifications to boundary conditions and boundary field theories aligned with ABJ-type models.

In sum, this paper sets a crucial stage for an in-depth understanding of the connections between higher spin physics, gauge theories, and string theory through the lens of holography, providing a robust framework for future theoretical explorations and potential empirical tests.