The Fusion Algebra of Bimodule Categories

Abstract

We establish an algebra-isomorphism between the complexified Grothendieck ring F of certain bimodule categories over a modular tensor category and the endomorphism algebra of appropriate morphism spaces of those bimodule categories. This provides a purely categorical proof of a conjecture by Ostrik concerning the structure of F. As a by-product we obtain a concrete expression for the structure constants of the Grothendieck ring of the bimodule category in terms of endomorphisms of the tensor unit of the underlying modular tensor category.

On representing some lattices as lattices of intermediate subfactors of finite index

“…In view of recent development using category theory (cf. [12]), both conjectures can in fact be stated in categorical terms, and we do not know any counter examples in the categorical setting. In Proposition 5.17 we prove that a weaker version of Conjecture 5.12 implies Conjecture 5.1, and from this we are able to prove Conjecture 5.1 for modular tensor category from SU(n) at level k (cf.…”

Topological defect lines and renormalization group flows in two dimensions

“…There is a straightforward generalization when the non-diagonal theory is defined through an automorphism of the fusion rules for the bulk local operators [7]. For blockdiagonal modular invariant theories, the fusion relations of TDLs can be noncommutative (see, for example, [15] for related discussions).…”

Anomalies and bounds on charged operators

“…An invariant way to characterize a global symmetry and its 't Hooft anomaly is by the associated (invertible) topological defect lines L [11][12][13][14][15][16][17][18][19][20][21][22][23]. See [24,25] for modern applications of topological defect lines to renormalization group flows and gauging.…”

“…To the left of R = √ 2, the bounds become flat, because a gap of (∆ gap ) L = 1 4 can also be interpreted as a gap of any smaller value. 21 To the right of R = 2, the bounds become flat and unsaturated.…”