Free real scalar CFT on fuzzy sphere: spectrum, algebra and wavefunction ansatz

TL;DR

This paper introduces a model for free scalar CFT on the fuzzy sphere, demonstrating accurate reproduction of the spectrum, algebra, and ground state wavefunctions, with implications for non-commutative quantum field theories.

Contribution

The paper presents a simple, tunable model for free scalar CFT on the fuzzy sphere, including a wavefunction ansatz with high fidelity to the true ground state.

Findings

Model reproduces operator spectrum and correlation functions

Wavefunction overlap exceeds 0.99 for certain parameters

Discusses algebraic structures and implications for non-commutative QFTs

Abstract

We introduce a simple model to realize the free real scalar CFT on the fuzzy sphere. The model is structurally similar to the original model that realizes the 3D Ising CFT on the fuzzy sphere. Owing to the shift symmetry of the free scalar, the free scalar CFT fixed point in our model can be accessed with only a single tuning parameter-the conformal coupling. We numerically demonstrate that our model correctly reproduces the operator spectrum, correlation functions, and, crucially, the harmonic oscillator algebra of the real scalar CFT. We also examine the fuzzy sphere algebra, a generalization of the Girvin-MacDonald-Platzman algebra, and discuss its potential implications for defining quantum field theories on non-commutative geometries. Finally, we propose wavefunction ansatz for the ground states of both the free scalar and Ising CFTs, which exhibit remarkable agreement with the CFT ground state wavefunctions of the fuzzy sphere model. For instance, the wavefunction overlap between our Ising ansatz and the Ising CFT ground state exceeds $0.99$ for $N=28$ fermions, suggesting a promising direction for tackling the 3D Ising CFT.