Abstract
It is well-known that if we gauge a ℤn symmetry in two dimensions, a dual ℤn symmetry appears, such that re-gauging this dual ℤn symmetry leads back to the original theory. We describe how this can be generalized to non-Abelian groups, by enlarging the concept of symmetries from those defined by groups to those defined by unitary fusion categories. We will see that this generalization is also useful when studying what happens when a non-anomalous subgroup of an anomalous finite group is gauged: for example, the gauged theory can have non-Abelian group symmetry even when the original symmetry is an Abelian group. We then discuss the axiomatization of two-dimensional topological quantum field theories whose symmetry is given by a category. We see explicitly that the gauged version is a topological quantum field theory with a new symmetry given by a dual category.
Highlights
Let us start by considering a two-dimensional theory T with Zn symmetry
Lakshya Bhardwaja and Yuji Tachikawab aPerimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5 Canada bKavli Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba 277-8583, Japan E-mail: lbhardwaj@pitp.ca, yuji.tachikawa@ipmu.jp. It is well-known that if we gauge a Zn symmetry in two dimensions, a dual Zn symmetry appears, such that re-gauging this dual Zn symmetry leads back to the original theory. We describe how this can be generalized to non-Abelian groups, by enlarging the concept of symmetries from those defined by groups to those defined by unitary fusion categories
In this paper we reviewed the notion of unitary fusion categories, or symmetry categories as we prefer to call them, and how they formalize the generalized notion of finite symmetries of a two-dimensional system
Summary
Let us start by considering a two-dimensional theory T with Zn symmetry. We can gauge it to get the gauged theory T /Zn. We will argue that there is a natural notion of gauging this symmetry formed by Wilson lines such that gauging T /G results back in the original theory T This raises the following question: how do we specify a generalized symmetry that a theory can admit? For the gauged theory T /G for possibly non-Abelian group G, the Wilson line operators form Rep(G), which is a symmetry category formed by the representations of G. There are vast number of symmetry categories not related to finite groups, formed by topological line operators of two-dimensional rational conformal field theories (RCFTs). Examples include the form of new symmetry categories C when we gauge a non-anomalous subgroup H of an anomalous finite group G, and the symmetry categories of RCFTs. Fourth, in section 6, we move on to the discussion of the axioms of two-dimensional TFTs whose symmetry is given by a symmetry category C.
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