Subfactors and Fusion Categories (14w5083)

Arriving in Banff, Alberta Sunday, April 13 and departing Friday April 18, 2014


Vaughan Jones (Vanderbilt University)

( Australian National University)


(Northwestern University)

(Indiana University)


Before turning to the major problems in fusion categories and subfactors, let us first mention a few results about finite groups which we would like to emulate. The most famous is the classification of finite simple groups. Another classic problem in finite groups is the classification of multiply transitive group actions. The third result is more modern, the McKay correspondence and the classification of groups with a 2-dimensional representation (or more generally, the finite subgroups of another compact Lie group).

Note that in all of these cases the classification results come hand-in-hand with interesting examples: the sporadic finite simple groups, doubly transitive Frobenius groups, and the Platonic solid groups. We anticipate that in the study of fusion categories and subfactors the discovery of new examples should be closely related to classification programs. Indeed, most of the new examples of subfactors and fusion categories are “multiply transitive” in an appropriate sense, or come from analogues of classifying the finite subgroups of Lie groups.

A close analogue of a finite group with a representation of small dimension (as in the McKay correspondence) is a fusion category with an object of small dimension or a subfactor of small index. In fact, that classification of subfactors of index less than 4 has a McKay-like ADE classification, proved in the 80s. One major program in subfactor theory has been to push this classification to higher indices. This is analogous to finding finite subgroups of other small Lie groups, for example the higher Platonic solid groups in dimension 4. The current state of the art is a complete classification up to index 5. This classification has yielded three subfactors which were not known by other means (the Haagerup, Asaeda-Haagerup, and extended Haagerup subfactors). These examples are all constructed by direct calculation, using either the technology of planar algebras or connections for bimodules. One major goal is finding a natural theoretical home for these exotic examples. There has been recent progress in this direction in work of Izumi, Evans-Gannon, and Grossman-Snyder.

One analogue of transitivity for group actions is supertransitivity of subfactors. Recall that the only 6- or more transitive group actions are the symmetric and alternating families, the only 4- or 5- transitive group actions are the Mathieu groups, and there are several interesting families of 2- and 3- transitive group actions. A fusion category is supertransitive if there is an object whose small tensor powers are as simple as possible. We have few highly transitive examples with dimension greater than 2. The exotic Asaeda-Haagerup and extended Haagerup subfactors have supertransitivity 6 and 8 respectively, and examples of higher supertransitivity are not known.

Over the last year, there has been significant progress studying "near-group" subfactors and fusion categories. A near-group fusion category has one more object than a finite group representation category. (There is also a related class where there is a finite group sub-category constituting half the simple objects.) Izumi has given a prescription for constructing near-group categories. The construction requires solving certain polynomial equations whose variables are indexed by a finite group, and he gave explicit solutions for 3-supertransitive subfactors based on Z\_3 (which is the Haagerup subfactor), Z\_4, Z\_2xZ\_2, and Z\_5. Recently, Evans and Gannon found exact and numeric solutions for more groups, and suggested that the Haagerup subfactor thus lies in an infinite family. They also deal with a different class of 2-supertransitive near-group examples, and explicitly produce 40 new subfactors!

One of the major theoretical deficits in our current understanding of fusion categories is the lack of a good extension theory. Recent beautiful results of Etingof, Nikshych and Ostrik give a homotopy-theoretic approach to finding graded extensions; that is, categories graded by a finite group with a fixed fusion category in the trivial grading. The general case, however, remains very poorly understood and is a major goal in the field. Similarly, we do not yet have a good notion of a fusion category being "simple". The near-group examples described above, and others recently constructed by Izumi, appear to be excellent test cases --- they certainly look like they should count as extensions of simpler categories, and the challenge is to find a framework which explains this.

One of the key tools for studying a group is understanding its subgroups. One analogue in subfactor theory is the study of intermediate subfactors or lattices thereof. This has been an active area of research in subfactors, with a paper of Bisch-Jones in Inventiones and another of Grossman-Jones in JAMS. Nonetheless these ideas have not been explored in detail on the fusion category side. One unexplained and tantalizing observation of Grossman-Izumi is that the Haagerup and Asaeda-Haagerup subfactors appear naturally in certain classes of quadrilaterals of subfactors.

Another interpretation of “subgroup” is a module category over a fusion category. This leads to the maximal atlas, and Brauer-Picard groupoid. Etingof-Nikshych-Ostrik use this to give a homotopy-theoretic description of some extension problems in fusion categories. Grossman-Snyder explore the maximal atlas of the Haagerup and Asaeda-Haagerup subfactors, but there is much more work to be done in this direction.

As an example of the interactions between the ideas above, at a recent conference it was suggested that an Izumi near-group subfactor based on the group Z\_2xZ\_4 might provide the crucial missing case of Grossman and Snyder's recent analysis of the maximal atlas of the highly supertransitive Asaeda-Haagerup subfactor. If this could be achieved it would provide a uniform explanation for what is currently one of the most exotic objects in subfactor theory.

A significant remaining piece of work in the subject is to extend the current classification results, in several directions. We would really like to have the classification of small index subfactors extended out to index $3+\sqrt{5}$, and potentially even to 6. Similarly, there are very good prospects for classifying fusion categories with small global dimension (ambitiously out to 60); inconclusive previous attempts in this direction already produced a number of potential new examples. The case of modular categories is especially interesting from the point of view of topological quantum computing; the classification is understood up to rank 4, and it seems likely that we can go further. There are powerful number theoretic approaches for understanding integral modular tensor categories, for example by restricting the prime factors of the global dimension.

The proposed subjects, subfactors and fusion categories, allow many fruitful interactions with other subjects. Three of the most prominent are number theory, free probability and conformal field theory. The main connection with number theory is that dimensions of objects in fusion categories (and thus the indices of subfactors) must be algebraic integers in cyclotomic fields. This result has been surprisingly useful in the study of subfactors. Ng has made some exciting progress recently, using the rotation operator in planar algebras to give an alternative proof of this result. Free probability and subfactors have been brought into contact recently via the work of Guionnet-Jones-Shlyakhtenko. These ideas have not yet spread much to the tensor category community, although free products already play an important role there. Finally, many known examples of fusion categories can be enriched to give conformal field theories. Since the subject of conformal field theories is so rich it will be important to discover whether all unitary fusion categories are related to conformal field theories. Recent work of Evans-Gannon gives tantalizing clues that the Haagerup subfactor could come from a conformal field theory, while work of Kawahigashi-Longo has proved classification results for small conformal field theories analogous to the small index subfactor classification.