Topological Charge of Lattice Abelian Gauge Theory

In the gauge-invariant construction of abelian chiral gauge theories on the lattice based on the Ginsparg-Wilson relation, the gauge anomaly is topological and its cohomologically trivial part plays the role of the local counter term. We give a prescription to solve the local cohomology problem within a finite lattice by reformulating the Poincar\'e lemma so that it holds true on the finite lattice up to exponentially small corrections. We then argue that the path-integral me...

A detailed comparison is made between the field-theoretic and geometric definitions of topological charge density on the lattice. Their renormalizations with respect to continuum are analysed. The definition of the topological susceptibility, as used in chiral Ward identities, is reviewed. After performing the subtractions required by it, the different lattice methods yield results in agreement with each other. The methods based on cooling and on counting fermionic zero modes...

We extend the definition of L\"uscher's lattice topological charge to the case of $4$d $SU(N)$ gauge fields coupled with $\mathbb{Z}_N$ $2$-form gauge fields. This result is achieved while maintaining the locality, the $SU(N)$ gauge invariance, and $\mathbb{Z}_N$ $1$-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 pur...

This paper has been withdrawn by the author.

We revisit the issue of the geometrical separability of the Hilbert space of physical states on lattice Abelian theories in the context of entanglement entropy. We discuss the conditions under which vectors in the Hilbert space, as well as the gauge invariant algebra, admit a tensor product decomposition with a geometrical interpretation. With the exception of pure gauge lattices with periodic boundary conditions which contain topological degrees of freedom, we show that the ...

In this contribution we give an introduction to the foundations and methods of lattice gauge theory. Starting with a brief discussion of the quantum mechanical path integral, we develop the main ingredients of lattice field theory: functional integrals, Euclidean field theory and the space-time discretization of scalar, fermion and gauge fields. Some of the methods used in calculations are reviewed and illustrated by a collection of typical results.

Motivated by the connection between gauge field topology and the axial anomaly in fermion currents, I use the fourth power of the naive Dirac operator to define a local lattice measure of topological charge. For smooth gauge fields this reduces to the usual topological density. For typical gauge field configurations in a numerical simulation, however, quantum fluctuations dominate, and the sum of this density over the system does not generally give an integer winding. On cool...

In a somewhat overlooked work by Seiberg, a definition of the topological charge for SU(N) lattice fields was given. Here, it is shown that Seibergs and L\"{u}schers charge definition are related up to the section of the bundle. With the continued interest in baryon number violating processes, Seibergs paper is useful since it allows for a Chern-Simons number also.

We propose an approach of lattice gauge theory based on a homotopic interpretation of its degrees of freedom. The basic idea is to dress the plaquettes of the lattice to view them as elementary homotopies between nearby paths. Instead of using a unique $G$-valued field to discretize the connection 1-form, $A$, we use an $\AG$-valued field $U$ on the edges, which plays the role of the 1-form $\ad_A$, and a $G$-valued field $V$ on the plaquettes, which corresponds to the Farada...

In recent years, attempts to generalize lattice gauge theories to model topological order have been carried out through the so called $2$-gauge theories. These have opened the door to interesting new models and new topological phases which are not described by previous schemes of classification. In this paper we show that we can go beyond the $2$-gauge construction when considering chain complexes of abelian groups. Based on elements of homological algebra we are able to grea...