Multiple-Order Tensor Field Theory: Enumeration of unitary invariant observables

TL;DR

This paper extends Tensor Field Theory by developing a group-theoretic framework to systematically enumerate unitary invariant observables for tensor fields of varying orders, unifying and expanding existing counting methods.

Contribution

It introduces a new theoretical framework and computational tools for counting tensor invariants of different orders, revealing previously unknown integer sequences.

Findings

Unified enumeration method for tensor invariants of varying orders

Recovery of known counting results as special cases

Discovery of new integer sequences related to tensor invariants

Abstract

In Tensor Field Theory (TFT), observables are defined through tensor field contractions that produce unitary invariants for complex-valued tensor fields. Traditionally, these observables are constructed using tensor fields of a fixed order $d$. Here, we propose an extended theoretical framework for TFT that incorporates tensor fields of varying orders $d'$, satisfying $d' \leq d$. We then establish a comprehensive group-theoretic formalism that enables the systematic enumeration of these complex TFT observables. This approach encompasses existing counting methods and therefore recovers known results in specific limiting cases. Additionally, we provide computational tools to facilitate the enumeration of these invariants, unveiling novel integer sequences that have not been documented elsewhere.