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

Joseph Ben Geloun, Arnauld Solente

TL;DR

The work extends Tensor Field Theory by allowing tensor fields of varying orders $d'\le d$ and develops a rigorous group-theoretic framework to enumerate unitary invariant observables. It generalizes fixed-order counting to multiple orders via colored bipartite graphs and a Burnside-orbit-stabilizer approach, yielding a compact counting formula that encompasses prior results as special cases. A central contribution is the CountMultiOrder theorem, which expresses the number of inequivalent multi-order contractions as a Burnside sum over color-type partitions, and reduces to the known $Z^d_n$ formula $\displaystyle Z^d_n=\sum_{\mu\vdash n} {\rm Sym}(\mu)^{d-2}$ in fixed-order limits. The paper also provides concrete applications, including a third-order tensor with matrices and vectors that produce 20 distinct orbits, along with combinatorial proofs and computational implementations. These results open avenues for exploring renormalization-group behavior, matter couplings in theory space, and potential connections to Topological Field Theory.

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.

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

TL;DR

The work extends Tensor Field Theory by allowing tensor fields of varying orders and develops a rigorous group-theoretic framework to enumerate unitary invariant observables. It generalizes fixed-order counting to multiple orders via colored bipartite graphs and a Burnside-orbit-stabilizer approach, yielding a compact counting formula that encompasses prior results as special cases. A central contribution is the CountMultiOrder theorem, which expresses the number of inequivalent multi-order contractions as a Burnside sum over color-type partitions, and reduces to the known formula in fixed-order limits. The paper also provides concrete applications, including a third-order tensor with matrices and vectors that produce 20 distinct orbits, along with combinatorial proofs and computational implementations. These results open avenues for exploring renormalization-group behavior, matter couplings in theory space, and potential connections to Topological Field Theory.

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 . Here, we propose an extended theoretical framework for TFT that incorporates tensor fields of varying orders , satisfying . 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.
Paper Structure (23 sections, 16 theorems, 101 equations, 8 figures, 3 tables)

This paper contains 23 sections, 16 theorems, 101 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Counting unitary tensor invariants: a review
  3. Tensor contractions and unitary group invariance
  4. Unitary invariants as colored bipartite graphs
  5. Counting invariants
  6. Counting formula
  7. Comments on low orders
  8. Counting unitary contractions of multiple-orders
  9. Encoding a bipartite colored graph
  10. Tensor contractions of multiple-orders
  11. Unitary and group action invariances
  12. Invariance under the unitary group
  13. Invariance under group action
  14. Counting unitary invariants
  15. An application
  16. ...and 8 more sections

Key Result

Proposition 1

Let $d,n \in {\mathbb N}^*$, $n$ tensors $T$ and $n$ tensors $\overline{T}$, each of order $d$ and each defined over a complex vector space $E$ of dimension $N$. Denote by $U(N)$ the unitary group, and consider the natural action of $U(N)^{\otimes d}$ on tensors of order $d$ via the fundamental repr

Figures (8)

  • Figure 1: Graph associated with the vector contraction $I( \sigma_1; \phi)$, $\sigma_1=(1)(2)(3)$.
  • Figure 2: Graph associated with the matrix contraction $I( \sigma; M)$, with $\sigma=( \sigma_1=(1)(2), \sigma_2=(12))$.
  • Figure 3: Illustration of $T_{\color{blue}a\color{black}b\color{red}c}T_{\color{blue}d\color{black}e\color{red}f}\overline{T}_{\color{blue}a\color{black}b\color{red}f}\overline{T}_{\color{blue}d\color{black}e\color{red}c}$ (left), $M_{\color{red}a\color{blue}b}M_{\color{red}c\color{blue}d}M_{\color{red}e\color{blue}f}\overline{M}_{\color{red}a\color{blue}d}\overline{M}_{\color{red}e\color{blue}b}\overline{M}_{\color{red}c\color{blue}f}$ (middle) and $T_{\color{red}a\color{blue}b\color{OliveGreen}c\color{black}d}\overline{T}_{\color{red}a\color{blue}b\color{OliveGreen}c\color{black}d}$ (right).
  • Figure 4: Illustration of an edge-colored bipartite graph.
  • Figure 5: Four graphs representing multiple-order contractions of tensors issued from figure \ref{['fig:IllusPregraph']}: the order-3 tensors are not connected to the vector and matrix fields.
  • ...and 3 more figures

Theorems & Definitions (39)

  • Definition 1
  • Definition 2: Tensor contraction
  • Proposition 1
  • Definition 3: Group action
  • Theorem 1: Number of orbits of unitary invariants of order $d$ BenGeloun:2013lim
  • proof
  • Definition 4: Colored vertex and type
  • Definition 5: Colored section, color multiplicity and chromatic index
  • Definition 6: Compatible colored sets of vertices
  • Definition 7: Colored bijection and bipartite colored graph
  • ...and 29 more