arXiv subjects

Andrew Schopieray

Publications and source records attributed to Andrew Schopieray.

21 records · Page 2Linked to original sources

arXiv2018

Prime decomposition of modular tensor categories of local modules of Type D

Let $\mathcal{C}(\mathfrak{g},k)$ be the unitary modular tensor categories arising from the representation theory of quantum groups at roots of unity for arbitrary simple finite-dimensional complex Lie algebra $\mathfrak{g}$ and positive integer levels $k$. Here we classify nondegenerate fusion subcategories of the modular tensor categories of local modules $\mathcal{C}(\mathfrak{g},k)_R^0$ where $R$ is the regular algebra of Tannakian $\text{Rep}(H)\subset\mathcal{C}(\mathfrak{g},k)_\text{pt}$. For $\mathfrak{g}=\mathfrak{so}_5$ we describe the decomposition of $\mathcal{C}(\mathfrak{g},k)_R^0$ into prime factors explicitly and as an application we classify relations in the Witt group of nondegenerately braided fusion categories generated by the equivalency classes of $\mathcal{C}(\mathfrak{so}_5,k)$ and $\mathcal{C}(\mathfrak{g}_2,k)$ for $k\in\mathbb{Z}_{\geq1}$.

math.QA↗

arXiv2017

Level bounds for exceptional quantum subgroups in rank two

There is a long-standing belief that the modular tensor categories $\mathcal{C}(\mathfrak{g},k)$, for $k\in\mathbb{Z}_{\geq1}$ and finite-dimensional simple complex Lie algebras $\mathfrak{g}$, contain exceptional connected étale algebras at only finitely many levels $k$. This premise has known implications for the study of relations in the Witt group of nondegenerate braided fusion categories, modular invariants of conformal field theories, and the classification of subfactors in the theory of von Neumann algebras. Here we confirm this conjecture when $\mathfrak{g}$ has rank 2, contributing proofs and explicit bounds when $\mathfrak{g}$ is of type $B_2$ or $G_2$, adding to the previously known positive results for types $A_1$ and $A_2$.

math.QA↗

arXiv2016

Classification of $\mathfrak{sl}_3$ relations in the Witt group of nondegenerate braided fusion categories

The Witt group of nondegenerate braided fusion categories $\mathcal{W}$ contains a subgroup $\mathcal{W}_\text{un}$ consisting of Witt equivalence classes of pseudo-unitary nondegenerate braided fusion categories. For each finite-dimensional simple Lie algebra $\mathfrak{g}$ and positive integer $k$ there exists a pseudo-unitary category $\mathcal{C}(\mathfrak{g},k)$ consisting of highest weight integerable $\hat{g}$-modules of level $k$ where $\hat{\mathfrak{g}}$ is the corresponding affine Lie algebra. Relations between the classes $[\mathcal{C}(\mathfrak{sl}_2,k)]$, $k\geq1$ have been completely described in the work of Davydov, Nikshych, and Ostrik. Here we give a complete classification of relations between the classes $[\mathcal{C}(\mathfrak{sl}_3,k)]$, $k\geq1$ with a view toward extending these methods to arbitrary simple finite dimensional Lie algebras $\mathfrak{g}$ and positive integer levels $k$.

math.QA↗