On character expansion and Gaussian regularization of Itzykson-Zuber measure

Abstract

Character expansions are among the most important approaches to modern quantum field theory, which substitute integrals by combinations of peculiar special functions from the Schur-Macdonald family. These formulas allow various deformations, which are not transparent in integral formulation. We analyze from this point of view the Itzykson-Zuber integral over unitary matrices which is exactly solvable, but difficult to deform in $\beta$ and $(q,t)$ directions. Character expansion straightforwardly resolves this problem. However, taking averages with the so defined measure can look problematic, because integrals of individual expansion terms often diverge and well defined is only the sum of them. We explain a way to overcome this problem by Gaussian regularization, which can have a broad range of further applications.

13 pages

References in corpus (22)

• Instantons and Merons in Matrix Models
• On the complete perturbative solution of one-matrix models
• Superintegrability for ($\beta$-deformed) partition function hierarchies with $W$-representations
• Superintegrability summary
• Interpolating Matrix Models for WLZZ series
• On KP-integrable skew Hurwitz $\tau$-functions and their $\beta$-deformations
• Correlation Functions of Harish-Chandra Integrals over the Orthogonal and the Symplectic Groups
• Many-body integrable systems implied by WLZZ models
• Non-Abelian W-representation for GKM
• Superintegrability in $\beta$-deformed Gaussian Hermitian matrix model from $W$-operators
• Commutative subalgebras from Serre relations
• $(q,t)$-deformed (skew) Hurwitz $\tau$-functions
• Hunt for 3-Schur polynomials
• Position Space Equations for Banana Feynman Diagrams
• On W-operators and superintegrability for dessins d'enfant
• Cauchy formula and the character ring
• Bilinear character correlators in superintegrable theory
• Non-Linear Algebra and Bogolubov's Recursion
• Two $\beta$-ensemble realization of $\beta$-deformed WLZZ models
• $W_{1+\infty }$ and $\tilde{W}$ algebras, and Ward identities
• Deformation of superintegrability in the Miwa-deformed Gaussian matrix model
• Towards the theory of Yangians

Cited by in corpus (4)

• Commutative families in DIM algebra, integrable many-body systems and $q,t$ matrix models
• Generalized $\tilde{W}$ algebras
• Vogel's universality and the classification problem for Jacobi identities
• Itzykson-Zuber correlators from character expansion