[Paper Review] On a class of selection rules without group actions in field theory and string theory

Justin Kaidi, Yuji Tachikawa+1 more|arXiv (Cornell University)|Jan 31, 2024

Distributed and Parallel Computing Systems4 citations

TL;DR

This paper introduces a class of exact tree-level selection rules in quantum field and string theories that arise not from group actions on fields, but from fusion algebras or conjugacy classes in non-Abelian groups. These rules are violated at higher loop orders and eventually reduce to standard group selection rules via abelianization, offering a framework for non-invertible symmetries in spacetime from worldsheet non-invertible symmetries.

ABSTRACT

We discuss a class of selection rules which i) do not come from group actions on fields, ii) are exact at tree level in perturbation theory, iii) are increasingly violated as the loop order is raised, and iv) eventually reduce to selection rules associated with an ordinary group symmetry. We start from basic field-theoretical examples in which fields are labeled by conjugacy classes rather than representations of a group, and discuss generalizations using fusion algebras or hypergroups. We also discuss how such selection rules arise naturally in string theory, such as for non-Abelian orbifolds or other cases with non-invertible worldsheet symmetries.

Motivation & Objective

• To identify and formalize a class of selection rules in quantum field theories that do not originate from group actions on fields.
• To explain how such selection rules emerge from fusion algebras or conjugacy classes in non-Abelian groups.
• To demonstrate that these rules are exact at tree level but increasingly violated at higher loop orders.
• To show that at infinite loop order, these rules reduce to standard selection rules associated with the abelianization of the underlying group.
• To connect these selection rules to physical realizations in string theory, such as non-Abelian orbifolds and worldsheet non-invertible symmetries.

Proposed method

• The paper constructs selection rules using fusion algebras with fusion rules $ ab = \sum_c N^c_{ab} c $, where fields are labeled by elements of the algebra.
• It imposes the condition $ e \prec \overline{a_1} \cdots \overline{a_n} a_{n+1} \cdots a_m $ for non-vanishing tree-level amplitudes.
• The method generalizes to cases where fields are labeled by conjugacy classes $[g_i]$ of a group $G$, requiring $ \tilde{g}_1 \cdots \tilde{g}_n = e $ for some $ \tilde{g}_i \in [g_i] $ in the Lagrangian.
• It uses the abelianization $ \mathrm{Ab}[G] = G/[G,G] $ to describe the limiting behavior at infinite loop order.
• The framework is applied to concrete examples: ADE orbifolds, Ising theory, $ S^1/\mathbb{Z}_2 $, and Tambara-Yamagami categories.
• It classifies low-rank fusion algebras with conjugate pair length >1, including non-fusion-category-realizable cases, and computes their quantum dimensions and commutant sets.

Experimental results

Research questions

• RQ1 How can selection rules in quantum field theories be defined without relying on group actions on fields?
• RQ2 What is the structure of selection rules that are exact at tree level but violated at higher loop orders?
• RQ3 How do such selection rules reduce to standard group selection rules at infinite loop order?
• RQ4 What is the role of fusion algebras and conjugacy classes in generating these selection rules?
• RQ5 How do these selection rules manifest in string theory, particularly in non-Abelian orbifolds and worldsheet non-invertible symmetries?

Key findings

• The paper identifies a class of selection rules based on fusion algebras that are exact at tree level but increasingly violated at higher loop orders.
• For theories labeled by conjugacy classes of a non-Abelian group $ G $, the selection rule $ \tilde{g}_1^{-1} \cdots \tilde{g}_n^{-1} \tilde{g}_{n+1} \cdots \tilde{g}_m = e $ holds at tree level for some $ \tilde{g}_i \in [g_i] $, but is violated at loop level.
• At infinite loop order, the selection rules reduce to those of the abelianization $ \mathrm{Ab}[G] = G/[G,G] $, recovering standard group selection rules.
• The paper constructs and classifies 8 rank-6 and 5 rank-7 fusion algebras with conjugate pair length >1, some of which cannot be realized by fusion categories.
• The total quantum dimensions for the rank-6 examples range from $ \mathcal{D}^2 \approx 18.9282 $ to $ 36.7792 $, and for rank-7 examples from $ \approx 21.1231 $ to $ 42 $, with $ \mathrm{Com}(A)^2 = \mathrm{Com}(A)^\infty $ in all cases.
• The commutant sets $ \mathrm{Com}(A)^2 $ and $ \mathrm{Com}(A)^\infty $ are found to be equal to the full set of labels $ \{1,2,3,4,5,6,7\} $ in all rank-7 cases.