75,060 research outputs found
Apollonian circle packings: Dynamics and Number theory
We give an overview of various counting problems for Apollonian circle
packings, which turn out to be related to problems in dynamics and number
theory for thin groups. This survey article is an expanded version of my
lecture notes prepared for the 13th Takagi lectures given at RIMS, Kyoto in the
fall of 2013.Comment: To appear in Japanese Journal of Mat
Dynamics on geometrically finite hyperbolic manifolds with applications to Apollonian circle packings and beyond
We present recent results on counting and distribution of circles in a given
circle packing invariant under a geometrically finite Kleinian group and
discuss how the dynamics of flows on geometrically finite hyperbolic
manifolds are related. Our results apply to Apollonian circle packings,
Sierpinski curves, Schottky dances, etc.Comment: To appear in the Proceedings of ICM, 201
Line defects and 5d instanton partition functions
We consider certain line defect operators in five-dimensional SUSY gauge
theories, whose interaction with the self-dual instantons is described by 1d
ADHM-like gauged quantum mechanics constructed by Tong and Wong. The partition
function in the presence of these operators is known to be a generating
function of BPS Wilson loops in skew symmetric tensor representations of the
gauge group. We calculate the partition function and explicitly prove that it
is a finite polynomial of the defect mass parameter , which is an essential
property of the defect operator and the Wilson loop generating function. The
relation between the line defect partition function and the qq-character
defined by N. Nekrasov is briefly discussed.Comment: 17 pages, 1 figure; typos fixed, references corrected; version to be
published in JHE
Positivity and periodicity of -systems in the WZW fusion ring
We study properties of solutions of -systems in the WZW fusion ring
obtained by the Kirillov-Reshetikhin modules. We make a conjecture about their
positivity and periodicity and give a proof of it in some cases. We also
construct a positive solution of the level restricted -system of
classical types in the fusion rings. As an application, we prove some
conjectures of Kirillov and Kuniba-Nakanishi-Suzuki on the level restricted
-systems.Comment: 29 pages;Table 1 reproduced from arXiv:math/9812022 [math.QA]; v2 :
no changes in main results, paper reorganized, introduction rewritten,
notations polished, typos corrected, references added; v3 : typos corrected;
v4 : minor change
A Proof of the KNS conjecture : case
We prove the Kuniba-Nakanishi-Suzuki (KNS) conjecture concerning the quantum
dimension solution of the -system of type obtained by a certain
specialization of classical characters of the Kirillov-Reshetikhin modules. To
this end, we use various symmetries of quantum dimensions. As a result, we
obtain an explicit formula for the positive solution of the level
restricted -system of type which plays an important role in
dilogarithm identities for conformal field theories.Comment: 13 pages, v3. published version, minor update (references added,
typos corrected
Light scalar mesons as tetraquarks within QCD Sum Rules
We examine the interpretation of the light scalar meson nonet as tetraquark
states using QCD sum rules. With the interpolating current for the tetraquark
states composed of scalar diquark and scalar antidiquark, first, we construct
the QCD sum rules by means of the operator product expansion up to the
operators of dimension 8 and show that there is no evidence of the coupling of
the tetraquark states to the light scalar meson nonet. In order to have a
stable sum rule, we propose a "good" interpolating current for the tetraquarks
based on chirality arguments which includes scalar and pseudoscalar
diquark--antidiquarks with equal weights. In particular, for the lowest
tetraquark --meson, we perform detail analysis of the QCD sum rule and
obtain mass for the around 780 MeV.Comment: 4 pages, 1 figure, Talk at the Yukawa International Seminar (YKIS)
2006, "New frontiers in QCD", Kyoto, Japa
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