Black Hole Entropy Associated with Supersymmetric Sigma Model

By means of an identity that equates elliptic genus partition function of a supersymmetric sigma model on the $N$-fold symmetric product $S^N X$ of $X$ ($S^N X=X^N/S_N$, $S_N$ is the symmetric group of $N$ elements) to the partition function of a second quantized string theory, we derive the asymptotic expansion of the partition function as well as the asymptotic for the degeneracy of spectrum in string theory. The asymptotic expansion for the state counting reproduces the logarithmic correction to the black hole entropy.Comment: 11 pages, no figures, version to appear in the Phys. Rev. D (2003

Splitting fields of elements in arithmetic groups

We prove that the number of unimodular integral matrices in a norm ball whose characteristic polynomial has Galois group different than the full symmetric group is of strictly lower order of magnitude than the number of all such matrices in the ball, as the radius increases. More generally, we prove a similar result for the Galois groups associated with elements in any connected semisimple linear algebraic group defined and simple over a number field $F$. Our method is based on the abstract large sieve method developed by Kowalski, and the study of Galois groups via reductions modulo primes developed by Jouve, Kowalski and Zywina. The two key ingredients are a uniform quantitative lattice point counting result, and a non-concentration phenomenon for lattice points in algebraic subvarieties of the group variety, both established previously by the authors. The results answer a question posed by Rivin and by Jouve, Kowalski and Zywina, who have considered Galois groups of random products of elements in algebraic groups.Comment: submitte

Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks

AbstractFuchsian groups (acting as isometries of the hyperbolic plane) occur naturally in geometry, combinatorial group theory, and other contexts. We use character-theoretic and probabilistic methods to study the spaces of homomorphisms from Fuchsian groups to symmetric groups. We obtain a wide variety of applications, ranging from counting branched coverings of Riemann surfaces, to subgroup growth and random finite quotients of Fuchsian groups, as well as random walks on symmetric groups. In particular, we show that, in some sense, almost all homomorphisms from a Fuchsian group to alternating groups An are surjective, and this implies Higman's conjecture that every Fuchsian group surjects onto all large enough alternating groups. As a very special case, we obtain a random Hurwitz generation of An, namely random generation by two elements of orders 2 and 3 whose product has order 7. We also establish the analogue of Higman's conjecture for symmetric groups. We apply these results to branched coverings of Riemann surfaces, showing that under some assumptions on the ramification types, their monodromy group is almost always Sn or An. Another application concerns subgroup growth. We show that a Fuchsian group Γ has (n!)μ+o(1) index n subgroups, where μ is the measure of Γ, and derive similar estimates for so-called Eisenstein numbers of coverings of Riemann surfaces. A final application concerns random walks on alternating and symmetric groups. We give necessary and sufficient conditions for a collection of ‘almost homogeneous’ conjugacy classes in An to have product equal to An almost uniformly pointwise. Our methods involve some new asymptotic results for degrees and values of irreducible characters of symmetric groups

A Generalization of Haldane state-counting procedure and π\pi-deformations of statistics

We consider the generalization of Haldane's state-counting procedure to describe all possible types of exclusion statistics which are linear in the deformation parameter $g$. The statistics are parametrized by elements of the symmetric group of the particles in question. For several specific cases we determine the form of the distribution functions which generalizes results obtained by Wu. Using them we analyze the low-temperature behavior and thermodynamic properties of these systems and compare our results with previous studies of the thermodynamics of a gas of $g$-ons. Various possible physical applications of these constructions are discussed.Comment: 17 pages, latex, 6 figures small corrections were made, reference and acknowledgments are adde

Permutation Patterns, Reduced Decompositions with Few Repetitions and the Bruhat Order

This thesis is concerned with problems involving permutations. The main focus is on connections between permutation patterns and reduced decompositions with few repetitions. Connections between permutation patterns and reduced decompositions were first studied various mathematicians including Stanley, Billey and Tenner. In particular, they studied pattern avoidance conditions on reduced decompositions with no repeated elements. This thesis classifies the pattern avoidance and containment conditions on reduced decompositions with one and two elements repeated. This classification is then used to obtain new enumeration results for pattern classes related to the reduced decompositions and introduces the technique of counting pattern classes via reduced decompositions. In particular, counts on pattern classes involving 1 or 2 copies of the patterns 321 and 3412 are obtained. Pattern conditions are then used to classify and enumerate downsets in the Bruhat order for the symmetric group and the rook monoid which is a generalization of the symmetric group. Finally, motivated by coding theory, the concepts of displacement, additive stretch and multiplicative stretch of permutations are introduced. These concepts are then analyzed with respect to maximality and distribution as a new prospect for improving interleaver design

Large Component QCD and Theoretical Framework of Heavy Quark Effective Field Theory

Based on a large component QCD derived directly from full QCD by integrating over the small components of quark fields with $|{\bf p}| < E + m_Q$, an alternative quantization procedure is adopted to establish a basic theoretical framework of heavy quark effective field theory (HQEFT) in the sense of effective quantum field theory. The procedure concerns quantum generators of Poincare group, Hilbert and Fock space, anticommutations and velocity super-selection rule, propagator and Feynman rules, finite mass corrections, trivialization of gluon couplings and renormalization of Wilson loop. The Lorentz invariance and discrete symmetries in HQEFT are explicitly illustrated. Some new symmetries in the infinite mass limit are discussed. Weak transition matrix elements and masses of hadrons in HQEFT are well defined to display a manifest spin-flavor symmetry and $1/m_Q$ corrections. A simple trace formulation approach is explicitly demonstrated by using LSZ reduction formula in HQEFT, and shown to be very useful for parameterizing the transition form factors via $1/m_Q$ expansion. As the heavy quark and antiquark fields in HQEFT are treated on the same footing in a fully symmetric way, the quark-antiquark coupling terms naturally appear and play important roles for simplifying the structure of transition matrix elements, and for understanding the concept of `dressed heavy quark' - hadron duality. In the case that the `longitudinal' and `transverse' residual momenta of heavy quark are at the same order of power counting, HQEFT provides a consistent approach for systematically analyzing heavy quark expansion in terms of $1/m_Q$. Some interesting features in applications of HQEFT to heavy hadron systems are briefly outlined.Comment: 59 pages, RevTex, no figures, published versio