Related papers: Classification of super-modular categories by rank
We develop abstract nonsense for module categories over monoidal categories (this is a straightforward categorification of modules over rings). As applications we show that any semisimple monoidal category with finitely many simple objects…
We construct all integral non-group-theoretical modular categories of dimension $p^2q^2$, where $p$ and $q$ are distinct prime numbers, establishing that a necessary and sufficient condition for their existence is that $p \mid q+1$, and…
We prove that a non-pointed maximally non-self-dual (MNSD) modular category of Frobenius-Perron (FP) dimension less than $2025$ has at most two possible types, and all these types can be realized except those of FP dimension $675$, $729$…
Working over an arbitrary field, we define compact semisimple 2-categories, and show that every compact semisimple 2-category is equivalent to the 2-category of separable module 1-categories over a finite semisimple tensor 1-category. Then,…
We use the theory of Auslander Buchweitz approximations to classify certain resolving subcategories containing a semidualizing or a dualizing module. In particular, we show that if the ring has a dualizing module, then the resolving…
The complete classification of WZNW modular invariant partition functions is known for very few affine algebras and levels, the most significant being all levels of $A_1$ and $A_2$ and level 1 of all simple algebras. Here, we address the…
Let $\Lambda=\Bbb Z[t,t^{-1}]$ be the ring of Laurent polynomials over $\Bbb Z$. We classify all $\Lambda$-modules $M$ with $|M|=p^n$, where $p$ is a primes and $n\le 4$. Consequently, we have a classification of Alexander quandles of order…
A generalization of an argument due to Etingof-Nikshych-Ostrik yields a highly efficient necessary criterion for integral modular categorification. This criterion allows us to complete the classification of categorifiable integral modular…
We consider a subclass of the class of group-theoretical fusion categories: To every finite group $G$ and subgroup $H$ one can associate the category of $G$-graded vector spaces with a two-sided $H$-action compatible with the grading. We…
We develop a functorial theory of spinor and oscillator representations parallel to the theory of Schur functors for general linear groups. This continues our work on developing orthogonal and symplectic analogues of Schur functors. As…
In this paper, we study a family of fusion and modular systems realizing fusion categories Grothendieck equivalent to the representation category for $so(2p+1)_2$. These categories describe non-abelian anyons dubbed `metaplectic anyons'. We…
We define the class of rigid Frobenius algebras in a (non-semisimple) modular category and prove that their categories of local modules are, again, modular. This generalizes previous work of A. Kirillov, Jr. and V. Ostrik [Adv. Math. 171…
We classify indecomposable commutative separable (special Frobenius) algebras and their local modules in (untwisted) group-theoretical modular categories. This gives a description of modular invariants for group-theoretical modular data. As…
We study locally finitary realizations of simple transitive module categories of infinite rank over the monoidal category $\mathscr{C}$ of finite dimensional modules for the complex Lie algebra $\mathfrak{sl}_2$. Combinatorics of such…
We introduce a diagrammatic monoidal category, the spin Brauer category, that plays the same role for the spin and pin groups as the Brauer category does for the orthogonal groups. In particular, there is a full functor from the spin Brauer…
We investigate the unflavoured Schur indices of class $\mathcal S$ theories of modest rank, and in the case of $\mathcal{N}=4$ super Yang-Mills theory with special unitary gauge group of somewhat more general rank, with an eye towards their…
We classify semisimple rigid monoidal categories with two isomorphism classes of simple objects over the field of complex numbers. In the appendix written by P.Etingof it is proved that the number of semisimple Hopf algebras with a given…
We give a complete classification of pointed fusion categories over $\mathbb{C}$ of global dimension $p^3$ for $p$ any odd prime. We proceed to classify the equivalence classes of pointed fusion categories of dimension $p^3$ and we…
In this paper we classify all semisimple tensor categories with the same fusion rules as $\operatorname{Rep}(SO(4))$, or one of the associated truncations. We show that such categories are explicitly classified by two non-zero complex…
In this paper, we develop several general techniques to investigate modular invariants of conformal field theories whose algebras of the holomorphic and anti-holomorphic sectors are different. As an application, we find all such…