Correlation Functions of Large N Chern-Simons-Matter Theories and Bosonization in Three Dimensions

TL;DR

The paper analyzes the large-N limit of N complex scalars in 2+1 dimensions coupled to a U(N) Chern-Simons theory, computing exact planar correlation functions across scalar and gauge-invariant operators. By matching the results to the Maldacena–Zhiboedov high-spin framework, it derives a precise mapping between the bosonic CS-vector model and the Legendre-transformed fermionic CS theory, effectively realizing a 3D bosonization duality in the planar limit. The work provides explicit relations between the couplings, notably tenilde N = 2N sin(πλ)/(πλ) and tenilde λ = tan(πλ/2), and demonstrates how all higher-spin correlators are governed by these parameters. It also discusses the critical fixed point and finite-N generalizations, while noting tensions with thermal free energy results and outlining potential tests and extensions of the duality.

Abstract

We consider the conformal field theory of N complex massless scalars in 2+1 dimensions, coupled to a U(N) Chern-Simons theory at level k. This theory has a 't Hooft large N limit, keeping fixed λ= N/k. We compute some correlation functions in this theory exactly as a function of λ, in the large N (planar) limit. We show that the results match with the general predictions of Maldacena and Zhiboedov for the correlators of theories that have high-spin symmetries in the large N limit. It has been suggested in the past that this theory is dual (in the large N limit) to the Legendre transform of the theory of fermions coupled to a Chern-Simons gauge field, and our results allow us to find the precise mapping between the two theories. We find that in the large N limit the theory of N scalars coupled to a U(N)_k Chern-Simons theory is equivalent to the Legendre transform of the theory of k fermions coupled to a U(k)_N Chern-Simons theory, thus providing a bosonization of the latter theory. We conjecture that perhaps this duality is valid also for finite values of N and k, where on the fermionic side we should now have (for N_f flavors) a U(k)_{N-N_f/2} theory. Similar results hold for real scalars (fermions) coupled to the O(N)_k Chern-Simons theory.

Figures (16)

• Figure 1: Bootstrap equation for the scalar self-energy. A filled circle denotes the full scalar propagator.
• Figure 2: Connected diagrams in scalar 4-point function.
• Figure 3: 1-loop diagrams that include the $A^2\phi^2$ vertex, up to reflections.
• Figure 4: Bootstrap equation for the connected 4-point function, when $q^\pm=0$.
• Figure 5: The vertex $\langle J^{(0)} \phi \phi^\dagger\rangle$. A cross denotes a $J^{(0)}$ insertion in the free theory, and a circled cross denotes the exact vertex. The hatched ellipse denotes diagrams in which the 4 scalar lines are connected.