[Paper Review] Bootstrapping Veneziano Amplitude of Vasiliev Theory and $3D$ Bosonization

Zhijin Li|arXiv (Cornell University)|Jun 13, 2019

Black Holes and Theoretical Physics9 citations

TL;DR

This paper develops a bootstrap approach to compute four-point correlation functions in three-dimensional CFTs with slightly broken higher spin symmetry, using an approximate higher spin Ward identity to derive a recursion relation in the 't Hooft coupling λ. The method yields exact λ-expansions terminating at λ², and the results are shown to match those from 3D bosonization duality via Legendre transformation, providing new evidence for the duality at the spinning four-point level.

ABSTRACT

Three-dimensional conformal field theories (CFTs) with slightly broken higher spin symmetry provide an interesting laboratory to study general properties of CFTs and their roles in the AdS/CFT correspondence. In this work we compute the four-point functions at arbitrary 't Hooft coupling $λ$ in the CFTs with slightly broken higher spin symmetry. We use a bootstrap approach based on the approximate higher spin Ward identity. We show that the bootstrap equation is separated into two parts with opposite parity charges, and it leads to a recursion relation for the $λ$ expansions of the correlation functions. The $λ$ expansions terminate at order $λ^2$ and the solutions are exact in $λ$. Our work generalizes the approach proposed by Maldacena and Zhiboedov to four-point correlators, and it amounts to an on-shell study for the $3D$ Chern-Simons vector models and the Vasiliev theory in $AdS_4$. Besides, we show that the same results can also be obtained rather simply from bosonization duality of $3D$ Chern-Simons vector models. The odd term at order $O(λ)$ in the spinning four-point function relates to the free boson correlator through a Legendre transformation. This provides new evidence on the $3D$ bosonization duality at the spinning four-point function level. We expect this work can be generalized to a complete classification of general four-point functions of single trace currents.

Motivation & Objective

• To compute four-point correlation functions in 3D CFTs with slightly broken higher spin symmetry at arbitrary 't Hooft coupling λ.
• To extend the Maldacena-Zhiboedov bootstrap approach from three- to four-point functions using an approximate higher spin Ward identity.
• To establish a connection between the resulting λ-expansions and the 3D bosonization duality via Legendre transformation.
• To demonstrate that the spinning four-point function at O(λ) is related to the free boson correlator through a Legendre transformation, offering new evidence for the duality.

Proposed method

• The authors use an approximate higher spin Ward identity derived from the slightly broken higher spin symmetry in 3D CFTs.
• They separate the bootstrap equation into components with opposite parity charges to simplify the solution of the integro-differential equation.
• The method leads to a recursion relation for the λ-expansion of correlation functions, which terminates at order λ².
• The solution is exact in λ, with coefficients determined order-by-order using conformal integration techniques and the D̄-function formalism.
• The authors compute spinning four-point functions via conformal integrals involving tensor structures, using dimensional regularization to handle divergences.
• They validate the results by showing equivalence to predictions from 3D bosonization duality, particularly through Legendre transformation of the scalar and spinning correlators.

Experimental results

Research questions

• RQ1 How can the four-point correlation functions in 3D CFTs with slightly broken higher spin symmetry be computed at arbitrary 't Hooft coupling λ using bootstrap methods?
• RQ2 What is the structure of the approximate higher spin Ward identity for spinning four-point functions, and how can it be solved order-by-order in λ?
• RQ3 How do the λ-expansions of the four-point functions relate to the 3D bosonization duality between Chern-Simons vector models?
• RQ4 What is the role of the Legendre transformation in connecting the spinning four-point function at O(λ) to the free boson correlator?
• RQ5 Can the conformal integrals with tensor structures be evaluated consistently to yield finite, physical results despite intermediate divergences in dimensional regularization?

Key findings

• The four-point functions in 3D CFTs with slightly broken higher spin symmetry are computed via a bootstrap approach based on the approximate higher spin Ward identity, yielding exact λ-expansions that terminate at order λ².
• The bootstrap equation splits into two parts with opposite parity, enabling a systematic solution via recursion relations in powers of λ.
• The spinning four-point function at O(λ) is shown to be related to the free boson correlator through a Legendre transformation, providing new evidence for 3D bosonization duality.
• The conformal integrals with tensor structures are evaluated using dimensional regularization, and the resulting divergences cancel in the final physical expressions, indicating consistency of the method.
• The same results are reproduced independently using the 3D bosonization duality, confirming the consistency of the bootstrap approach with known dualities in Chern-Simons vector models.
• The D̄-functions arising in the conformal integrals are expanded in ε near zero, and the final expressions involve hypergeometric functions and logarithmic terms that are finite and physically meaningful.