Constructive Tensor Field Theory: The $T^4_3$ Model

TL;DR

This paper constructively develops a superrenormalizable rank-three tensor field theory with quartic interactions on U(1)^3, demonstrating its analyticity and Borel summability through advanced multiscale techniques.

Contribution

It introduces a constructive approach to a tensor field theory requiring renormalization, extending the multiscale loop vertex expansion to matrix intermediate fields.

Findings

The tensor field theory is superrenormalizable with power counting similar to ^2.

Established analyticity and Borel summability of the model.

Developed new estimates for intermediate field integration of matrix type.

Abstract

We build constructively the simplest tensor field theory which requires some renormalization, namely the rank three tensor theory with quartic interactions and propagator inverse of the Laplacian on $U(1)^3$. This superrenormalizable tensor field theory has a power counting almost similar to ordinary $\phi^4_2$. Our construction uses the multiscale loop vertex expansion (MLVE) recently introduced in the context of an analogous vector model. However to prove analyticity and Borel summability of this model requires new estimates on the intermediate field integration, which is now of matrix rather than of scalar type.