On Partition Functions of Hyperbolic Three-Geometry and Associated Hilbert Schemes
A. A. Bytsenko, E. Elizalde
备注：
17 pages, accepted contribution to K. Alladi, G.V. Milovanovic and M. Th. Rassias, Eds., "Analytic Number Theory, Approximation Theory and Special Functions", Springer, New York (to appear)
Highest-weight representations of infinite dimensional Lie algebras and Hilbert schemes of points are considered, together with the applications of these concepts to partition functions, which are most useful in physics. Partition functions (elliptic genera) are conveniently transformed into product expressions, which may inherit the homology properties of appropriate (poly)graded Lie algebras. Specifically, the role of (Selberg-type) Ruelle spectral functions of hyperbolic geometry in the calculation of partition functions and associated $q$-series are discussed. Examples of these connection in quantum field theory are considered (in particular, within the AdS/CFT correspondence), as the AdS$_{3}$ case where one has Ruelle/Selberg spectral functions, whereas on the CFT side, partition functions and modular forms arise. These objects are here shown to have a common background, expressible in terms of Euler-Poincaré and Macdonald identities, which, in turn, describe homological aspects of (finite or infinite) Lie algebra representations. Finally, some other applications of modular forms and spectral functions (mainly related with the congruence subgroup of $SL(2, {\mathbb Z})$) to partition functions, Hilbert schemes of points, and symmetric products are investigated by means of homological and K-theory methods.
On palindromic numerators of bigraded symmetric orbifold Hilbert series and Kostka-Foulkes polynomials
School of Physics and Mandelstam Institute for Theoretical Physics, University of the Witwatersrand（威特沃特斯兰德大学物理学院与曼德尔斯坦理论物理研究所）；National Institute of Theoretical and Computational Sciences（国家理论与计算科学研究所）
The star-triangle relation, lens partition function, and hypergeometric sum/integrals
本文的目的是考虑椭圆超几何和/积分恒等式的双曲极限，以及第二作者以前得到的统计力学的相关格模型。从这个极限得到的双曲和/积分恒等式，在所谓的规范/YBE对应中有两个重要的物理应用。对于统计力学，这个恒等式等价于杨-巴克斯特方程的星-三角关系形式的新解，它直接将Faddeev-Volkov模型推广到离散和连续自旋变量的情况。在规范理论方面，这个恒等式代表了某些三维$\mathcal N = 2$超对称规范理论的透镜配分函数的对偶性。
