The Schwinger Dyson equations and the algebra of constraints of random tensor models at all orders

TL;DR

This paper extends the Schwinger Dyson equations and the algebra of constraints for random tensor models to all orders in 1/N, revealing a complex algebraic structure indexed by D-colored graphs.

Contribution

It completes the formulation of the algebra of constraints for random tensor models at all orders, generalizing previous large N results to include full 1/N corrections.

Findings

The full algebra of constraints is indexed by D-colored graphs.

The leading order D-ary tree algebra forms a Lie subalgebra of the full constraints algebra.

The work generalizes the Virasoro algebra to arbitrary dimensions in tensor models.

Abstract

Random tensor models for a generic complex tensor generalize matrix models in arbitrary dimensions and yield a theory of random geometries. They support a 1/N expansion dominated by graphs of spherical topology. Their Schwinger Dyson equations, generalizing the loop equations of matrix models, translate into constraints satisfied by the partition function. The constraints have been shown, in the large N limit, to close a Lie algebra indexed by colored rooted D-ary trees yielding a first generalization of the Virasoro algebra in arbitrary dimensions. In this paper we complete the Schwinger Dyson equations and the associated algebra at all orders in 1/N. The full algebra of constraints is indexed by D-colored graphs, and the leading order D-ary tree algebra is a Lie subalgebra of the full constraints algebra.