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This paper develops a detailed lattice-continuum correspondence for all common examples of Abelian gauge theories, with and without matter. These rules for extracting a continuum theory out of a lattice one represent an elementary way to rigorously define continuum gauge theories. The focus is on (2+1)D but the techniques developed here work in all dimensions. The first half of this paper is devoted to pure Maxwell theory. It is precisely shown how continuum Maxwell theory emerges at low energies in an appropriate parameter regime of the $\mathbb{Z}_K$ lattice gauge theory at large $K$. The familiar features of this theory -- its OPE structure, the distinction between compact and noncompact degrees of freedom, infrared "particle-vortex" dualities, and the Coulomb law behavior of its Wilson loop -- are all derived directly from the lattice. The rest of the paper studies gauge fields coupled to either bosonic or fermionic matter. Scalar QED is analyzed at the same level of precision as pure Maxwell theory, with new comments on its phase structure and its connections to the topological BF theory. Ordinary QED (Dirac fermions coupled to Maxwell theory) is constructed with an eye toward

Author

Radicevic, Djordje

Published

2021

Language

EN