Source identity and kernel functions for Inozemtsev-type systems

Source identity and kernel functions for elliptic Calogero-Sutherland type systems

Kernel functions related to quantum many-body systems of Calogero-Sutherland type are discussed, in particular for the elliptic case. The main result is an elliptic generalization of an identity due to Sen that is a source for many such kernel functions. Applications are given, including simple exact eigenfunctions and corresponding eigenvalues of Chalykh-Feigin-Veselov-Sergeev-type deformations of the elliptic Calogero-Sutherland model for special parameter values.

On the Effective Action of Dressed Mean Fields for N = 4 Super-Yang-Mills Theory

On the basis of the general considerations such as $R$-operation and Slavnov-Taylor identity we show that the effective action, being understood as Legendre transform of the logarithm of the path integral, possesses particular structure in ${\cal N} = 4$ supersymmetric Yang-Mills theory for kernels of the effective action expressed in terms of the dressed effective fields. These dressed effective fields have been introduced in our previous papers as actual variables of the effective action. The concept of dressed effective fields naturally appears in the framework of solution to Slavnov-Taylor…

Heat-kernels on the discrete circle and interval

As is known, the free heat-kernel on the integers (a modified Bessel function) is turned into the periodic free heat-kernel on the discrete circle by factoring, giving a pre-image sum. I generalise existing treatments by making the functions periodic up to a phase, thus introducing an extra parameter into the analysis. Identifying the classical paths form with the conventional eigenfunction expression, I find a combinatorial trace identity which allows various Bessel identities to be extracted, such as a generalisation of the Jacobi-Anger expansion.The free Dirichlet, Neumann and hybrid Dirich…

Gap Probabilities for Double Intervals in Hermitian Random Matrix Ensembles as $τ$-Functions -- Spectrum Singularity case

The probability for the exclusion of eigenvalues from an interval $(-x,x)$ symmetrical about the origin for a scaled ensemble of Hermitian random matrices, where the Fredholm kernel is a type of Bessel kernel with parameter $ a $ (a generalisation of the sine kernel in the bulk scaling case), is considered. It is shown that this probability is the square of a $τ$-function, in the sense of Okamoto, for the Painlevé system \PIII. This then leads to a factorisation of the probability as the product of two $τ$-functions for the Painlevé system \PIIIdash. A previous study has given a formula of thi…

来自Virasoro TQFT的非有理Verlinde公式

A non-rational Verlinde formula from Virasoro TQFT

本研究利用Virasoro TQFT推导了适用于中心荷c≥25的非有理Virasoro-Verlinde公式，证明其对角化Virasoro 6j符号，并给出在边界Liouville CFT、纯3d引力及AdS3/CFT2中的三项应用，为非有理2d CFT融合规则研究提供新结果。

Self-similar behavior of the exchange-driven growth model with product kernel

We study the self-similar behavior of the exchange-driven growth model, which describes a process in which pairs of clusters, consisting of an integer number of monomers, interact through the exchange of a single monomer. The rate of exchange is given by an interaction kernel $K(k,l)$ which depends on the sizes $k$ and $l$ of the two interacting clusters and is assumed to be of product form $(k\,l)^λ$ for $λ\in [0,2)$. We rigorously establish the coarsening rates and convergence to the self-similar profile found by Ben-Naim and Krapivsky [7]. For the explicit kernel, the evolution is linked to…