Rank-finiteness for modular categories

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 $\mathbb{Z}[e^{\frac{2\pi i}{N}}]$ is identical to that of $N$.Comment: 25 pages (last version). Version 2: removed weakly integral rank 6 and integral rank 7 section, improved rank 5 classification up to monoidal equivalence. Version 3: removed rank 5 classification (note title change)--this will be published separately. Significantly improved expositio

On the Classification of Low-Rank Braided Fusion Categories

A physical system is said to be in topological phase if at low energies and long wavelengths the observable quantities are invariant under diffeomorphisms. Such physical systems are of great interest in condensed matter physics and computer science where they can be applied to form topological insulators and fault–tolerant quantum computers. Physical systems in topological phase may be rigorously studied through their algebraic manifestations, (pre)modular categories. A complete classification of these categories would lead to a taxonomy of the topological phases of matter. Beyond their ties to physical systems, premodular categories are of general mathematical interest as they govern the representation theories of quasi–Hopf algebras, lead to manifold and link invariants, and provide insights into the braid group. In the course of this work, we study the classification problem for (pre)modular categories with particular attention paid to their arithmetic properties. Central to our analysis is the question of rank finiteness for modular categories, also known as Wang’s Conjecture. In this work, we lay this problem to rest by exploiting certain arithmetic properties of modular categories. While the rank finiteness problem for premodular categories is still open, we provide new methods for approaching this problem. The arithmetic techniques suggested by the rank finiteness analysis are particularly pronounced in the (weakly) integral setting. There, we use Diophantine techniques to classify all weakly integral modular categories through rank 6 up to Grothendieck equivalence. In the case that the category is not only weakly integral, but actually integral, the analysis is further extended to produce a classification of integral modular categories up to Grothendieck equivalence through rank 7. It is observed that such classification can be extended provided some mild assumptions are made. For instance, if we further assume that the category is also odd–dimensional, then the classification up to Grothendieck equivalence is completed through rank 11. Moving beyond modular categories has historically been difficult. We suggest new methods for doing this inspired by our work on (weakly) integral modular categories and related problems in algebraic number theory. The allows us to produce a Grothendieck classification of rank 4 premodular categories thereby extending the previously known rank 3 classification

On classification of modular categories by rank

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.Comment: arXiv admin note: substantial text overlap with arXiv:1310.705

On the Classification of Weakly Integral Modular Categories

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.Comment: Version 2: fixed missing metadata, version 3: corrected incomplete introduction and added theorem numbers, version 4: 32 pages, significant cosmetic revisions, version 5: final revision pre-submissio

High contrast ultrasonic imaging of resin-rich regions in graphite/epoxy composites using entropy

This study compares different approaches for imaging a near-surface resin-rich defect in a thin graphite/epoxy plate using backscattered ultrasound. The specimen was created by cutting a circular hole in the second ply; this region filled with excess resin from the graphite/epoxy sheets during the curing process. Backscat-tered waveforms were acquired using a 4 in. focal length, 5MHz center frequency broadband transducer, scanned on a 100 × 100 grid of points that were 0.03 × 0.03 in. apart. The specimen was scanned with the defect side closest to the transducer. Consequently, the reflection from the resin-rich region cannot be gated from the large front-wall echo. At each point in the grid 256 waveforms were averaged together and subsequently used to produce peak-to-peak, Signal Energy (sum of squared digitized waveform values), as well as entropy images of two different types (a Renyi entropy, and a joint entropy). As the figure shows, all of the entropy images exhibit better border delineation and defect contrast than the either the peak-to-peak or Signal Energy. The best results are obtained using the joint entropy of the backscattered waveforms with a reference function. Two different references are examined. The first is a reflection of the insonifying pulse from a stainless steel reflector. The second is an approximate optimum obtained from an iterative parametric search. The joint entropy images produced using this reference exhibit three times the contrast obtained in previous studies