Multi-Instantons, Multi-Axions, and Non-Invertible Symmetries in 4d QFT

TL;DR

The paper develops a systematic, Lagrangian-based framework for non-invertible symmetries in 4d QFTs with multiple instantons and multiple axions. By combining partial gauging of 3d CS/BF theories with a generalized half-space gauging construction, it realizes anomaly inflow that cancels arbitrary ABJ anomaly matrices $K_{ij}$, producing non-invertible 0-form and 1-form symmetries and clarifying their action on local and extended operators, including 't Hooft lines and axion strings. It extends the analysis to multi-gauge sectors, fractional and CFU instantons, and demonstrates how the Witten effect is manifested in these settings, thereby unifying several prior constructions. The framework yields concrete, Lagrangian descriptions and computational handles that bridge formal QFT with phenomenological axion models, with potential implications for BSM scenarios and axion cosmology.

Abstract

We study non-invertible global symmetries in 4d quantum field theories, aiming to generalize existing discussions to theories with multiple instantons and axions, and to make the subject more accessible to particle phenomenology. Building on both the Adler-Bell-Jackiw (ABJ) anomaly construction and the half-space gauging approach, we identify the 3d topological quantum field theories required to describe non-invertible symmetries in the presence of multi-instanton effects. To this end, we introduce a method we call partial gauging and show that partial gauging of 3d Chern-Simons theories naturally leads to anomaly inflow actions with general ABJ anomaly matrices. We further generalize the half-space gauging construction to the case of multiple gauge sectors and compute correlation functions of boundary line operators. This enables us to analyze the action of non-invertible operators on various species of 't Hooft lines and to interpret the results in terms of the Witten effect. For theories with multiple axions, we determine both the non-invertible 0-form and 1-form symmetries, as well as their actions on axion strings and 't Hooft lines, thereby generalizing the previously known notion of the non-invertible Gauss law. Our framework unifies several previously disparate constructions, provides a concrete Lagrangian-based formulation of non-invertible symmetries, and naturally extends to theories with multiple instantons and axions, which are of both phenomenological and theoretical interest. These results may also find broader applications in beyond-the-Standard-Model scenarios.

Figures (3)

• Figure 1: The action of the non-invertible SDO on (a) the 't Hooft line and (b) the axion string worldsheet. One spatial dimension is suppressed so that the 2-manifold $\Sigma_{2}$ supporting the non-invertible SDO is represented as a 1-dimensional object in the figure.
• Figure 2: Braiding of abelian anyons in 3d. In the discussion below, we consider the braiding of two $\left[V^p\right]^s$, i.e., the $\mathbb{Z}_N^{(1)}$ symmetry-generating lines of $\mathcal{A}^{pN,1}$.
• Figure 3: Illustration of the pushed-off loop and ribbon. For each boundary loop $\Sigma_1\subset\Sigma_3$, take a pushed-off copy $\tilde{\Sigma}_1$. Then choose a surface $\Sigma_2$ with $\partial \Sigma_2=\tilde{\Sigma}_1$ (note that $\Sigma_2\not\subset\Sigma_3$). Finally, choose a ribbon $R_2$ with $\partial R_2=\tilde{\Sigma}_1-\Sigma_1$.