Note on lattice description of generalized symmetries in SU(N)/ZNSU(N)/\mathbb{Z}_N gauge theories

Topology and generalized symmetries in the $SU(N)/\mathbb{Z}_N$ gauge theory are considered in the continuum and the lattice. Starting from the $SU(N)$ gauge theory with the 't~Hooft twisted boundary condition, we give a simpler explanation of the van~Baal's proof on the fractionality of the topological charge. This description is applicable to both continuum and lattice by using the generalized L\"uscher's construction of topology on the lattice. Thus we can recover the $SU(N)/\mathbb{Z}_N$ principal bundle from lattice $SU(N)$ gauge fields being subject to the $\mathbb{Z}_N$-relaxed cocycle condition. We explicitly demonstrate the fractional topological charge, and verify an equivalence with other constructions reported recently based on different ideas. Gauging the $\mathbb{Z}_N$ $1$-form center symmetry enables lattice gauge theories to couple with the $\mathbb{Z}_N$ $2$-form gauge field as a simple lattice integer field, and to reproduce the Kapustin--Seiberg prescription in the continuum limit. Our construction is also applied to analyzing the higher-group structure in the $SU(N)$ gauge theory with the instanton-sum modification.Comment: 22 pages, 2 figures, to appear in PR

Lattice realization of the axial U(1)U(1) noninvertible symmetry

In $U(1)$ lattice gauge theory with compact $U(1)$ variables, we construct the symmetry operator, i.e.\ the topological defect, for the axial $U(1)$ noninvertible symmetry. This requires a lattice formulation of chiral gauge theory with an anomalous matter content and we employ the lattice formulation on the basis of the Ginsparg--Wilson relation. The invariance of the symmetry operator under the gauge transformation of the gauge field on the defect is realized, imitating the prescription by Karasik in continuum theory, by integrating the lattice Chern--Simons term on the defect over \emph{smooth\/} lattice gauge transformations. The projection operator for allowed magnetic fluxes on the defect then emerges with lattice regularization. The resulting symmetry operator is manifestly invariant under lattice gauge transformations. In an appendix, we give another way of constructing the symmetry operator on the basis of a 3D $\mathbb{Z}_N$ topological quantum field theory, the level-$N$ BF theory on the lattice.Comment: 47 pages, 5 figures, the final version to appear in PTE

Lattice construction of mixed 't Hooft anomaly with higher-form symmetry

In this talk, we give the lattice regularized formulation of the mixed 't Hooft anomaly between the $\mathbb{Z}_N$ $1$-form symmetry and the $\theta$ periodicity for $4$d pure Yang-Mills theory, which was originally discussed by Gaiotto $\textit{et al.}$ in the continuum description. For this purpose, we define the topological charge of the lattice $SU(N)$ gauge theory coupled with the background $\mathbb{Z}_N$ $2$-form gauge fields $B_p$ by generalizing L\"uscher's construction of the $SU(N)$ topological charge. We show that this lattice topological charge enjoys the fractional $1/N$ shift completely characterized by the background gauge field $B_p$, and this rigorously proves the mixed 't Hooft anomaly with the finite lattice spacings. As a consequence, the Yang-Mills vacua at $\theta$ and $\theta+2\pi$ are distinct as the symmetry-protected topological states when the confinement is assumed.Comment: 8 pages, 2 figures, talk presented at the 40th International Symposium on Lattice Field Theory (Lattice2023), July 31st - August 4th, 2023, Fermi National Accelerator Laborator

Topology of SU(N) lattice gauge theories coupled with ℤ N 2-form gauge fields

Abstract We extend the definition of Lüscher’s lattice topological charge to the case of 4d SU(N) gauge fields coupled with ℤ N 2-form gauge fields. This result is achieved while maintaining the locality, the SU(N) gauge invariance, and ℤ N 1-form gauge invariance, and we find that the manifest 1-form gauge invariance plays the central role in our construction. This result gives the lattice regularized derivation of the mixed ’t Hooft anomaly in pure SU(N) Yang-Mills theory between its ℤ N 1-form symmetry and the θ periodicity