Conformal four-point correlators of the three-dimensional Ising transition via the quantum fuzzy sphere
Chao Han, Liangdong Hu, W. Zhu, Yin-Chen He
Abstract
In conformal field theory (CFT), the four-point correlator is a fundamental object that encodes CFT properties, constrains CFT structures, and connects to the gravitational scattering amplitude in holography theory. However, the four-point correlator of CFTs in dimensions higher than two-dimensional remains largely unexplored due to the lack of nonperturbative tools. In this paper, we introduce a new approach for directly computing four-point correlators of three-dimensional (3D) CFTs. Our method employs the recently proposed fuzzy (noncommutative) sphere regularization, and we apply it to the paradigmatic 3D Ising CFT. Specifically, we have computed three different four-point correlators: $\ensuremath{\langle}\ensuremath{\sigma}\ensuremath{\sigma}\ensuremath{\sigma}\ensuremath{\sigma}\ensuremath{\rangle}, \ensuremath{\langle}\ensuremath{\sigma}\ensuremath{\sigma}\ensuremath{\epsilon}\ensuremath{\epsilon}\ensuremath{\rangle}$, and $\ensuremath{\langle}\ensuremath{\sigma}\ensuremath{\sigma}{T}_{\ensuremath{\mu}\ensuremath{\nu}}{T}_{\ensuremath{\rho}\ensuremath{\eta}}\ensuremath{\rangle}$. Additionally, we verify the crossing symmetry of $\ensuremath{\langle}\ensuremath{\sigma}\ensuremath{\sigma}\ensuremath{\sigma}\ensuremath{\sigma}\ensuremath{\rangle}$, which is a notable property arising from conformal symmetry. Remarkably, the computed four-point correlators exhibit continuous crossing ratios, showcasing the continuum nature of the fuzzy sphere regularization scheme. This characteristic renders them highly suitable for future theoretical applications, enabling further advancements and insights in 3D CFT.