An Ran Chen's 2017 honours thesis in mathematics, on "On Generalized Frobenius-Schur Indicators for Spherical Fusion Categories", supervised by
Corey Jones, is available here.
Outline
Classically, for a finite group $G$ and a representation $V$ over $\mathbb C$ with character
$\chi$, we define the Frobenius-Schur indicator, as
$$
\nu(V) = \frac{1}{|G|}\sum_{g \in G} \chi(g^2)
$$
If $V$ is an irreducible representation, then this indicator helps us to determine the flavour of duality of $V$. Specifically, the Frobenius-Schur theorem states that $\nu(V)$ could only be 1, −1, or 0, and if
- $\nu(V) = 1$: $V$ is symmetrically self-dual
- $\nu(V) = -1$: $V$ is antisymmetrically self-dual
- $\nu(V) = 0$: $V$ is not self-dual
A great deal of work has been done to generalize the Frobenius-Schurc(FS) indicators for Hopf algebras (see [LM00], [KSZ06], [MN05], [Sch04], [NS08]) and for categories (see [FS03], [FGSV99]). This culminated in the definition of generalized FS indicators for pivotal categories (see [NS07]) and a formula of generalized FS indicators for spherical fusion categories, given by Ng and Schauenberg in [NS10].
In broad terms, the generalized FS indicators for fusion categories are the traces of generalized rotation operators on homspaces in the category. Generalized rotation operators have been studied extensively and play an important role in the study of subfactor planar algebra. V. Jones used these rotations to show that certain quadratic tangles are linearly independent [Jon12] and to construct annular structures of subfactors [Jon01], which played a crucial part in the classification of subfactors of index at most 5 (see [JMS14] for an overview).
Generalized FS indicators have proven to be a useful tool for analyzing fusion categories. One important application is in the proof of the congruence subgroup conjecture for spherical fusion categories, which states that the kernels of the modular representations of modular categories are congruence subgroups of $SL_2(\mathbb Z)$ (see [NS10]). The confirmation of this conjecture provides important insight on the relationship between rational conformal field theories and modular categories.
Generalized FS indicators are also useful for classification purposes, as they can be used to create bounds and have nice number theoretic properties. It was used by Bruillard, Ng, Rowell and Wang to show rank finiteness of modular tensor categories, which is that, up to equivalence, there are only finitely many modular categories of any fixed rank [BNRW16]. Furthermore, the indicators have been used to classify fusion categories of small rank (see [Ost14], [Lar15]).
The focus of this thesis is to give a self-contained derivation of the generalized Frobenius-Schur indicator formulas for spherical fusion categories given in [NS10]. This thesis will be presented as follows. The first two chapters give an introduction to the language of monoidal categories, focusing mostly on the theory needed in the remainder of the thesis. Specifically, chapter 1 defines pivotal and semisimple monoidal categories and chapter 2 defines braided monoidal categories, modular data and the Drinfeld center. In chapter 3, we define the induction functor to the Drinfeld center, and give a formula for the generalized Frobenius-Schur indicators in terms of the inductor functor and modular data of the center. Finally, in chapter 4, we follow the work of Barter, C. Jones and Tucker in [BJT16] and use the indicator formula to construct special torus link invariants for modular categories.
