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

The X-Ray Position and Infrared Counterpart of the Eclipsing X-Ray Pulsar OAO 1657-415

We have measured the precise position of the 38-s eclipsing X-ray pulsar OAO 1657-415 with the Chandra X-Ray Observatory: RA = 17h00m48.90s, Dec = -41d39m21.6s, equninox J2000, error radius = 0.5 arcsec. Based on the previously measured pulsar mass function and X-ray eclipse duration, this 10.4-d high-mass X-ray binary is believed to contain a B supergiant companion. Deep optical imaging of the field did not detect any stars at the Chandra source position, setting a limit of V>23. However, near-IR imaging revealed a relatively bright star (J=14.1, H=11.9, K_s=10.7) coincident with the Chandra position, and we identify this star as the IR counterpart of OAO 1657-415. The IR colors and magnitudes and the optical non-detections for this star are all consistent with a highly reddened B supergiant (A_V= 20.4 +/- 1.3) at a distance of 6.4 +/- 1.5 kpc. This implies an X-ray luminosity of 3e36 erg/s (2-10 keV). IR spectroscopy can verify the spectral type of the companion and measure its radial velocity curve, yielding a neutron star mass measurement.Comment: 4 pages. ApJ in press (Vol. 573, July 10 issue

Quantum criticality in a generalized Dicke model

We employ a generalized Dicke model to study theoretically the quantum criticality of an extended two-level atomic ensemble interacting with a single-mode quantized light field. Effective Hamiltonians are derived and diagonalized to investigate numerically their eigenfrequencies for different quantum phases in the system. Based on the analysis of the eigenfrequencies, an intriguing quantum-phase transition from a normal phase to a superradiant phase is revealed clearly, which is quite different from that observed with a standard Dicke model.Comment: 6 pages, 3 figure

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