Modified instanton sum in QCD and higher-groups

TL;DR

This work shows that restricting the instanton sum in SU($N$) gauge theories to sectors with instanton number multiples of $p$ can be achieved by coupling to a non-propagating topological sector described by a $2\pi$-periodic scalar $\chi$ and a $U(1)$ 3-form gauge field $c^{(3)}$, leaving local YM dynamics unchanged. The global structure acquires higher-form symmetries ${\mathbb{Z}}_{N}^{[1]}$ and ${\mathbb{Z}}_{p}^{[3]}$ that form a nontrivial 4-group extension, and the periodicity of the YM theta angle is effectively shortened to $2\pi/p$, yielding an $Np$-branch vacuum structure constrained by a mixed anomaly. Large-$N$ analysis and semiclassical ${\mathbb{R}}^3\times S^1$ studies confirm the branch structure and identify the topological susceptibility as an order parameter for the ${\mathbb{Z}}_{p}^{[3]}$ symmetry; introducing fermions enlarges the discrete chiral symmetry to ${\mathbb{Z}}_{2Np}$ and produces $Np$ vacua with universes distinguished by a mixed anomaly with the 3-form symmetry, precluding dynamical domain walls between universes. The paper also connects the 4d construction to 2d charge-$p$ models via torus compactification, and discusses generalized QCD with fundamental fermions, highlighting the role of universes and external probes. Together, these results reveal a deep link between topological sector sums, higher-form symmetries, and vacuum structure in gauge theories, with potential implications for confinement, dualities, and UV completions.

Abstract

We consider the $SU(N)$ Yang-Mills theory, whose topological sectors are restricted to the instanton number with integer multiples of $p$. We can formulate such a quantum field theory maintaining locality and unitarity, and the model contains both $2π$-periodic scalar and $3$-form gauge fields. This can be interpreted as coupling a topological theory to Yang-Mills theory, so the local dynamics becomes identical with that of pure Yang-Mills theory. The theory has not only $\mathbb{Z}_N$ $1$-form symmetry but also $\mathbb{Z}_p$ $3$-form symmetry, and we study the global nature of this theory from the recent 't Hooft anomaly matching. The computation of 't Hooft anomaly incorporates an intriguing higher-group structure. We also carefully examine that how such kinematical constraint is realized in the dynamics by using the large-$N$ and also the reliable semiclassics on $\mathbb{R}^3\times S^1$, and we find that the topological susceptibility plays a role of the order parameter for the $\mathbb{Z}_p$ $3$-form symmetry. Introducing a fermion in the fundamental or adjoint representation, we find that the chiral symmetry becomes larger than the usual case by $\mathbb{Z}_p$, and it leads to the extra $p$ vacua by discrete chiral symmetry breaking. No dynamical domain wall can interpolate those extra vacua since such objects must be charged under the $3$-form symmetry in order to match the 't Hooft anomaly.

Figures (1)

• Figure 1: The vacuum structure of $SU(N)$ SYM theory where instanton sum is restricted to multiples of charge-$p$ ($N=4, p=2$ in the figure). The theory has $Np$ vacua. These vacua split naturally to two sets, denoted by the blue circles and the red squares. The vacua for which $n_1 -n_2 =0$ mod $p$ are relative superselection sectors. There are dynamical domain walls in between. The vacua for which $n_1 -n_2 \neq 0$ mod $p$ are different universes. There exists no dynamical domain walls that can connect them, and no tunneling in between exists even when the theory is compactified. Only external probes charged under $\mathbb{Z}_{p}^{[3]}$ can connect them. When mass deformation is added, this structure extrapolates to $Np$ branches of generalized YM theory.