Notes on gauging noninvertible symmetries. Part I. Multiplicity-free cases
Alonso Perez-Lona, Daniel Robbins, Eric Sharpe, Thomas Vandermeulen, Xiaolan Yu
Abstract
A bstract In this paper we discuss gauging noninvertible zero-form symmetries in two dimensions. We specialize to certain gaugeable cases, specifically, fusion categories of the form $$ \textrm{Rep}\left(\mathcal{H}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Rep</mml:mi> <mml:mfenced> <mml:mi>H</mml:mi> </mml:mfenced> </mml:math> for $$ \mathcal{H} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> </mml:math> a suitable Hopf algebra (which includes the special case Rep( G ) for G a finite group). We also specialize to the case that the fusion category is multiplicity-free. We discuss how to construct a modular-invariant partition function from a choice of Frobenius algebra structure on $$ {\mathcal{H}}^{\ast } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> . We discuss how ordinary G orbifolds for finite groups G are a special case of the construction, corresponding to the fusion category Vec( G ) = Rep(ℂ[ G ] * ). For the cases Rep( S 3 ), Rep( D 4 ), and Rep( Q 8 ), we construct the crossing kernels for general intertwiner maps. We explicitly compute partition functions in the examples of Rep( S 3 ), Rep( D 4 ), Rep( Q 8 ), and $$ \textrm{Rep}\left({\mathcal{H}}_8\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Rep</mml:mi> <mml:mfenced> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>8</mml:mn> </mml:msub> </mml:mfenced> </mml:math> , and discuss applications in c = 1 CFTs. We also discuss decomposition in the special case that the entire noninvertible symmetry group acts trivially.