Modular Data for the Extended Haagerup Subfactor

Abstract

A. We compute the modular data (that is, the S and T matrices) for the centre of the extended Haagerup subfactor [BMPS12]. The full structure (i.e. the associativity data, also known as 6-j symbols or F matrices) still appears to be inaccessible. Nevertheless, starting with just the number of simple objects and their dimensions (obtained by a combinatorial argument in [MW14]) we nd that it is surprisingly easy to leverage knowledge of the representation theory of SL(2, Z) into a complete description of the modular data. We also investigate the possible character vectors associated with this modular data. IThe extended Haagerup subfactor provides perhaps the strangest currently known example of a quantum symmetry.Fusion categories provide a suitable axiomatization for the notion of quantum symmetry: they are the nitely semisimple rigid tensor categories. The fundamental examples are the representation categories of nite groups (over C), but there are many others. The semisimpli ed representation category of a quantum enveloping algebra U q g at a suitable root of unity gives another source of examples.The remarkable discovery of an interesting classi cation of nite depth subfactors above index 4, initiated by Haagerup . Some of these fusion categories are distinctly di erent from those arising from nite groups or quantum groups: in particular the fusion categories coming from the Haagerup and extended Haagerup subfactors cannot be de ned over any cyclotomic eld [MS12].Since the discovery of these examples, there has been some progress towards organising them. In particular, the theory of quadratic categories has been developed, particularly by Izumi [Izu01,Izu15] and Evans-Gannon [EG11, EG14]. These are categories with a group of invertible objects, and under the action of this group by left and right tensor product, just one other double coset. The category of N − N bimodules of the Haagerup subfactor is a quadratic category. While the fusion categories coming from the Asaeda-Haagerup subfactor are not quadratic, work of Grossman-Izumi-Snyder [GIS15] shows that they are Morita equivalent to quadratic categories. This leaves us with the following remarkable observation: the extended Haagerup fusion categories are the only known fusion categories not known to be related to nite groups, quantum groups, and quadratic categories. While this almost surely only re ects our feeble ability to discover and construct fusion categories, nevertheless these categories remain uniquely interesting objects.Every fusion category has a braided centre, which is a modular tensor category. This paper tackles the problem of describing the braided centre of the extended Haagerup categories. While we do not give a full description (in particular the associators), we produce the modular data, that is, the S and T matrices.Recently, Morrison-Walker discovered [MW14] that a purely combinatorial argument determines the number of simple objects, and their dimensions, in the centre of extended Haagerup. This paper uses that just that informat...

Auto-equivalences of the modular tensor categories of type $A$, $B$, $C$ and $G$

“…We also notice that the only other quantum group modular tensor category that could possibly have a reverse braided auto-equivalence is C(f 4 , 4). This auto-equivalence has 12 non-fixed simple objects, and we find that the 12×12 block of restricted modular data for C(f 4 , 4) exactly appears in the modular data for the centre of extended Haagerup [23]. Further, we can identify Z 4 ⋉Z 5 as the group component of the modular grafting, and construct a fairly natural looking formula for the modular data of centre of extended Haagerup, starting from the modular data of C(f 4 , 4) and the Drinfeld centre of Z 4 ⋊Z 5 .…”

Anyonic Chains, Topological Defects, and Conformal Field Theory

“…By its nature, our work leads to many questions, including • What is the space of 2D CFTs we can generate from the anyonic chain procedure? Is it possible to associate an RCFT with the (extended) Haagerup subfactor (see, e.g., [46][47][48] for some important work in this direction) thus realizing an old dream of Vaughan Jones? • We saw that when we take k → ∞ in the C in = Rep su(2) int k case, we could end up with an irrational output theory [45].…”

Eigenvalues of rotations and braids in spherical fusion categories

“…Given a spherical fusion category C one can compute the FS indicators from the matrix induced from the central induction functor as well as T for the center. Gannon and Morrison have outlined an effective procedure for finding a finite set of possibilities for the central induction and T matrix pairs for the center using the fusion rule of C alone, which often (but not always) yields a unique such pair [GM16]. Given an induction and T -matrix pair that does not actually correspond to a categorification of a given fusion rule, there seems to be no reason a priori why these numbers should live in the right number field suitable for applying the Galois automorphisms in the first place, or that the multiplicities of the eigenvalues should add up to the number as determined by the fusion rules.…”

“…However, it is desirable to be able to compute the rotation eigenvalues from as little information as possible since there are many situations where the full modular data for the center is not easily accessible. For example, in [GM16], Gannon and Morrison describe a procedure for computing possible modular data for the Drinfel'd centers Z(C) of categorifications C of a given fusion ring. Their algorithm effectively produces finitely many possibilities for the forgetful functor Z(C) → C and the T -matrix of Z(C), but finding the S-matrix requires solving a generally large system of quadratics which may or may not be possible.…”