A test of bosonization at the level of four-point functions in Chern-Simons vector models

TL;DR

This work computes the four-point function of the scalar primary in large-$N$ $U(N)_k$ Chern-Simons vector models with fundamental fermions, to all orders in the 't Hooft coupling $\lambda$, for a restricted external-momentum configuration. By solving the exact ladder diagram in light-cone gauge and using the Schwinger-Dyson framework, the authors obtain closed-form fermionic results for free, interacting, and critical theories, and then perform detailed comparisons with non-critical and critical bosonic theories under the bosonization duality. The key finding is a precise agreement between the fermionic and bosonic four-point functions in the appropriate dual limits, providing a nontrivial four-point check of the non-supersymmetric bosonization duality in three dimensions. The results demonstrate the power of exact planar techniques in CS vector models and suggest avenues for extending to general momenta and $1/N$ corrections, with potential bootstrap applications leveraging the slightly-broken higher-spin symmetry.

Abstract

We study four-point functions in Chern-Simons vector models in the large $N$ limit. We compute the four-point function of the scalar primary to all orders in the `t Hooft coupling $λ=N/k$ in $U(N)_k$ Chern-Simons theory coupled to a fundamental fermion, in both the critical and non-critical theory, for a particular case of the external momenta. These theories cover the entire 3-parameter "quasi-boson" and 2-parameter "quasi-fermion" families of 3-dimensional quantum field theories with a slightly-broken higher spin symmetry. Our results are consistent with the celebrated bosonization duality, as we explicitly verify by calculating four-point functions in the free critical and non-critical bosonic theories.

Figures (7)

• Figure 1: The diagrammatic definition of the exact ladder diagram, which is the shaded box. All propagators in this diagram are exact propagators. In light cone gauge, this diagram is sufficient to calculate all planar correlation functions of single trace operators.
• Figure 2: A contraction of the tree-level ladder diagram corresponding to Equation \ref{['contr']}. Note that there is no integration over momenta.
• Figure 3: The diagrammatic relation for $f^{(n)}$ in terms of $f^{(n-1)}$.
• Figure 4: Diagrams in the interacting theory.
• Figure 5: The plot of modulus of $H(p,\lambda)$ function and $\tan^3[\frac{\pi \lambda}{2}]$ vs $\lambda$ for momenta $p=1$. It is clear that the magnitude of $H(\lambda)$ rises slower than $\tan^3[\frac{\pi \lambda}{2}]$.