Bootstrapping Tensor Integrals

TL;DR

This paper introduces a positivity-based bootstrap approach to analyze large N limits of random invariant tensors, successfully approximating moments and conjecturing new explicit formulas.

Contribution

It develops a novel bootstrap methodology combining Dyson-Schwinger equations and positivity constraints for tensor models, extending techniques from matrix models.

Findings

Models converge quickly and match known solutions

Conjecture of new explicit formulas for tensor moments

Support for conjectures via computational methods

Abstract

This work proposes a bootstrapping with positivity methodology to study random $U(N)^{D}$ invariant tensors in the large $N$ limit. As has been done for $U(N)$ invariant random matrices, we combine the Dyson-Schwinger equations and positivity constraints of moments to approximate the moments of such tensor models. As examples, we bootstrap the quartic and two hexic rank three tensor models. All models studied converge quickly, and for those which have known analytic formulae, they converge to such solutions. We conjecture new explicit formulae for all moments of the rank three quartic model and support this conjecture using bootstrapped results and explicit double-series computations with 'feyntensor'.