Conformal-helicity duality & the Hilbert space of free CFTs

TL;DR

This paper constructs primary operators in free conformal field theories across dimensions 2, 3, and 4 using momentum space spinors, revealing a duality with particle phase space geometry and providing a new harmonic analysis framework.

Contribution

It introduces an explicit construction of primary operators via harmonic polynomials on phase space manifolds, generalizing the little group concept to N-particle systems in free CFTs.

Findings

Primary operators correspond to harmonics on Stiefel and Grassmannian manifolds.

The spectrum is described by harmonic polynomials on these manifolds.

Provides a method to construct these polynomials using $U(N)$ and $O(N)$ representation theory.

Abstract

We identify a means to explicitly construct primary operators of free conformal field theories (CFTs) in spacetime dimensions $d=2,~3$, and $4$. Working in momentum space with spinors, we find that the $N$-distinguishable-particle Hilbert space $\mathcal{H}_N$ exhibits a $U(N)$ action in $d=4$ ($O(N)$ in $d=2,3$) which dually describes the decomposition of $\mathcal{H}_N$ into irreducible representations of the conformal group. This $U(N)$ is a natural $N$-particle generalization of the single-particle $U(1)$ little group. The spectrum of primary operators is identified with the harmonics of $N$-particle phase space which, specifically, is shown to be the Stiefel manifold $V_2(\mathbb{C}^N) = U(N)/U(N-2)$ (respectively, $V_2(\mathbb{R}^N)$, $V_1(\mathbb{R}^N)$ in $d=3,2$). Lorentz scalar primaries are harmonics on the Grassmannian $G_2(\mathbb{C}^N) \subset V_2(\mathbb{C}^N)$. We provide a recipe to construct these harmonic polynomials using standard $U(N)$ ($O(N)$) representation theory. We touch upon applications to effective field theory and numerical methods in quantum field theory.