Abstract
We study a four-dimensional U(1) gauge theory with the θ angle, which was originally proposed by Cardy and Rabinovici. It is known that the model has the rich phase diagram thanks to the presence of both electrically and magnetically charged particles. We discuss the topological nature of the oblique confinement phase of the model at θ = π, and show how its appearance can be consistent with the anomaly constraint. We also construct the SL(2, ℤ) self-dual theory out of the Cardy-Rabinovici model by gauging a part of its one-form symmetry. This self-duality has a mixed ’t Hooft anomaly with gravity, and its implications on the phase diagram is uncovered. As the model shares the same global symmetry and ’t Hooft anomaly with those of SU(N) Yang-Mills theory, studying its topological aspects would provide us more hints to explore possible dynamics of non-Abelian gauge theories with nonzero θ angles.
Highlights
The violation of Bianchi identity at the lattice scale
We study a four-dimensional U(1) gauge theory with the θ angle, which was originally proposed by Cardy and Rabinovici
In the continuum formulation, we introduce the θ angle as a coupling to the instanton density, but such topologies are suffered from lattice discretization and lose some of important features valid in the continuum (See refs. [9, 10] for recent developments on topologies of lattice U(1) gauge theories)
Summary
We give a brief review of the work by Cardy and Ravinobici on the lattice U(1) gauge theory with the θ angle [7, 8]. This model is expected to show the rich phase structure due to the various types of charge, monopole, and dyon condensations [7]. The local dynamics of this model enjoys the SL(2, Z) self-duality, which constrains possible structures of the phase diagram [8]
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