arXiv subjects
Publications and source records attributed to Paul Bruillard.
We present a correspondence between multiplicity-free, self-dual, fusion rings and a digraph, hypergraph pair $(D,H)$. This correspondence is used to provide a complete characterization of all fusion rings corresponding to graphical properties of $D$. Further, we exploit this correspondence to provide a complete list of all non-isomorphic, self-dual, multiplicity-free fusion rings of rank at most 8.
Graph eigenvalues play a fundamental role in controlling structural properties, such as bisection bandwidth, diameter, and fault tolerance, which are critical considerations in the design of supercomputing interconnection networks. This motivates considering graphs with optimal spectral expansion, called Ramanujan graphs, as potential candidates for interconnection networks. In this work, we explore this possibility by comparing Ramanujan graph properties against those of a wide swath of current and proposed supercomputing topologies. We derive analytic expressions for the spectral gap, bisection bandwidth, and diameter of these topologies, some of which were previously unknown. We find the spectral gap of existing topologies are well-separated from the optimal achievable by Ramanujan topologies, suggesting the potential utility of adopting Ramanujan graphs as interconnection networks.
We develop categorical and number theoretical tools for the classification of super-modular categories. We apply these tools to obtain a partial classification of super-modular categories of rank $8$. In particular we find three distinct families of prime categories in rank $8$ in contrast to the lower rank cases for which there is only one such family.
We discuss several useful interpretations of the categorical dimension of objects in a braided fusion category, as well as some conjectures demonstrating the value of quantum dimension as a quantum statistic for detecting certain behaviors of anyons in topological phases of matter. From this discussion we find that objects in braided fusion categories with integral squared dimension have distinctive properties. A large and interesting class of non-integral modular categories such that every simple object has integral squared-dimensions are the metaplectic categories that have the same fusion rules as $SO(N)_2$ for some $N$. We describe and complete their classification and enumeration, by recognizing them as $\mathbb{Z}_2$-gaugings of cyclic modular categories (i.e. metric groups). We prove that any modular category of dimension $2^km$ with $m$ square-free and $k\leq 4$, satisfying some additional assumptions, is a metaplectic category. This illustrates anew that dimension can, in some circumstances, determine a surprising amount of the category's structure.
We pursue a classification of low-rank super-modular categories parallel to that of modular categories. We classify all super-modular categories up to rank=$6$, and spin modular categories up to rank=$11$. In particular, we show that, up to fusion rules, there is exactly one non-split super-modular category of rank $2,4$ and $6$, namely $PSU(2)_{4k+2}$ for $k=0,1$ and $2$. This classification is facilitated by adapting and extending well-known constraints from modular categories to super-modular categories, such as Verlinde and Frobenius-Schur indicator formulae.
We give a complete classification of modular categories of dimension $p^3m$ where $p$ is prime and $m$ is a square-free integer. When $p$ is odd, all such categories are pointed. For $p=2$ one encounters modular categories with the same fusion ring as orthogonal quantum groups at certain roots of unity, namely $SO(2m)_2$. As an immediate step we classify a more general class of so-called even metaplectic modular categories with the same fusion rules as $SO(2N)_2$ with $N$ odd.
We study spin and super-modular categories systematically as inspired by fermionic topological phases of matter, which are always fermion parity enriched and modelled by spin TQFTs at low energy. We formulate a $16$-fold way conjecture for the minimal modular extensions of super-modular categories to spin modular categories, which is a categorical formulation of gauging the fermion parity. We investigate general properties of super-modular categories such as fermions in twisted Drinfeld doubles, Verlinde formulas for naive quotients, and explicit extensions of $PSU(2)_{4m+2}$ with an eye towards a classification of the low-rank cases.
We survey a number of classification tools developed in recent years and employ them to classify pseudo-unitary rank 5 premodular categories up to equivalence.
We consider the classification problem for rank 4 premodular categories. We uncover a formula for the 2nd Frobenius-Schur indicator of a premodular category, and complete the classification of rank 4 premodular categories (up to Grothendieck equivalence).
The feasibility of a classification-by-rank program for modular categories follows from the Rank-Finiteness Theorem. We develop arithmetic, representation theoretic and algebraic methods for classifying modular categories by rank. As an application, we determine all possible fusion rules for all rank=$5$ modular categories and describe the corresponding monoidal equivalence classes.
Motivated by physical and topological applications, we study representations of the group $\mathcal{LB}_3$ of motions of $3$ unlinked oriented circles in $\mathbb{R}^3$. Our point of view is to regard the three strand braid group $\mathcal{B}_3$ as a subgroup of $\mathcal{LB}_3$ and study the problem of extending $\mathcal{B}_3$ representations. We introduce the notion of a \emph{standard extension} and characterize $\mathcal{B}_3$ representations admiting such an extension. In particular we show, using a classification result of Tuba and Wenzl, that every irreducible $\mathcal{B}_3$ representation of dimension at most $5$ has a (standard) extension. We show that this result is sharp by exhibiting an irreducible $6$-dimensional $\mathcal{B}_3$ representation that has no extensions (standard or otherwise). We obtain complete classifications of (1) irreducible $2$-dimensional $\mathcal{LB}_3$ representations (2) extensions of irreducible $3$-dimensional $\mathcal{B}_3$ representations and (3) irreducible $\mathcal{LB}_3$ representations whose restriction to $\mathcal{B}_3$ has abelian image.
We prove a rank-finiteness conjecture for modular categories: up to equivalence, there are only finitely many modular categories of any fixed rank. Our technical advance is a generalization of the Cauchy theorem in group theory to the context of spherical fusion categories. For a modular category $\mathcal{C}$ with $N=ord(T)$, the order of the modular $T$-matrix, the Cauchy theorem says that the set of primes dividing the global quantum dimension $D^2$ in the Dedekind domain $\mathbb{Z}[e^{\frac{2πi}{N}}]$ is identical to that of $N$.
We classify all modular categories of dimension $4m$, where $m$ is an odd square-free integer, and all ranks $6$ and $7$ weakly integral modular categories. This completes the classification of weakly integral modular categories through rank $7$. Our results imply that all integral modular categories of rank at most $7$ are pointed (that is, every simple object has dimension $1$). All strictly weakly integral (weakly integral but non-integral) modular categories of ranks $6$ and $7$ have dimension $4m$, with $m$ an odd square free integer, so their classification is an application of our main result. The classification of rank $7$ integral modular categories is facilitated by an analysis of two actions on modular categories: the Galois group of the field generated by the entries of the $S$-matrix and the group of isomorphism classes of invertible simple objects. The interplay of these two actions is of independent interest, and we derive some valuable arithmetic consequences from their actions.
We classify integral modular categories of dimension pq^4 and p^2q^2 where p and q are distinct primes. We show that such categories are always group-theoretical except for categories of dimension 4q^2. In these cases there are well-known examples of non-group-theoretical categories, coming from centers of Tambara-Yamagami categories and quantum groups. We show that a non-group-theoretical integral modular category of dimension 4q^2 is equivalent to either one of these well-known examples or is of dimension 36 and is twist-equivalent to fusion categories arising from a certain quantum group.
It is a well-known result of Etingof, Nikshych and Ostrik that there are finitely many inequivalent integral modular categories of any fixed rank $n$. This follows from a double-exponential bound on the maximal denominator in an Egyptian fraction representation of 1. A naïve computer search approach to the classification of rank $n$ integral modular categories using this bound quickly overwhelms the computer's memory (for $n\geq 7$). We use a modified strategy: find general conditions on modular categories that imply integrality and study the classification problem in these limited settings. The first such condition is that the order of the twist matrix is 2,3,4 or 6 and we obtain a fairly complete description of these classes of modular categories. The second condition is that the unit object is the only simple non-self-dual object, which is equivalent to odd-dimensionality. In this case we obtain a (linear) improvement on the bounds and employ number-theoretic techniques to obtain a classification for rank at most 11 for odd-dimensional modular categories.
We study the linearized approximation of N=1 topologically massive supergravity around AdS3. Linearized gravitino fields are explicitly constructed. For appropriate boundary conditions, the conserved charges demonstrate chiral behavior, so that chiral gravity can be consistently extended to chiral supergravity.