Abstract
The Abelian Chern-Simons gauge theory is constructed on the three-dimensional spacetime lattice. This proposal introduces both lattice and dual lattice, and the gauge field on the dual lattice is expressed in terms of the gauge field on the original lattice. This treatment circumvents the issue of forward/backward difference, which is the common problem that many previous proposals have, and also avoids the duplication problem, which prevents people from introducing the dual lattice. The form of the lattice action is very simple, and is symmetric with respect to the three spacetime dimensions. These features make it straightforward to calculate the expectation values of Wilson loops, and the results agree with the topological field theory in continuous spacetime. Generalizations to multiple types of lattices are also discussed.
Highlights
As the simplest topological field theory, Chern-Simons theory [1] with Abelian gauge group has a clear geometrical meaning: it counts the linking number of loops [2,3]
The lattice approach makes it possible to study Chern-Simons theory with discrete gauge groups like Zn, which has no clear analog in continuous spacetime
A new definition of lattice Abelian Chern-Simons theory is proposed. This proposal introduces both lattice and dual lattice, and the gauge field on the dual lattice is expressed in terms of the gauge field on the original lattice
Summary
As the simplest topological field theory, Chern-Simons theory [1] with Abelian gauge group has a clear geometrical meaning: it counts the linking number of loops [2,3]. A new definition of lattice Abelian Chern-Simons theory is proposed This proposal introduces both lattice and dual lattice, and the gauge field on the dual lattice is expressed in terms of the gauge field on the original lattice. The matter field can be regarded as a product of the parton fields on the vertexes of a lattice cell, and the Wilson loop consists of multiple lines on the lattice This regularizes the divergence in the continuous theory. The form of the action is symmetric with respect to the three spacetime dimensions, and the calculation of Wilson loops is very straightforward. This construction works for both compact and noncompact gauge groups.
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