Unitary Integrals and Related Matrix Models

Abstract

Concise review of the basic properties of unitary matrix integrals. They are studied with the help of the three matrix models: the ordinary unitary model, Brezin-Gross-Witten model and the Harish-Charndra-Itzykson-Zuber model. Especial attention is paid to the tricky sides of the story, from De Wit-t'Hooft anomaly in unitary integrals to the problem of correlators with Itzykson-Zuber measure. Of technical tools emphasized is the method of character expansions. The subject of unitary integrals remains highly under-investigated and a lot of new results are expected in this field when it attracts sufficient attention.

27 pages

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Cited by in corpus (45)

• On AGT relation in the case of U(3)
• Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions
• Proving AGT conjecture as HS duality: extension to five dimensions
• On KP-integrable Hurwitz functions
• A direct proof of AGT conjecture at beta = 1
• Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid
• On Combinatorial Expansions of Conformal Blocks
• Correlators in tensor models from character calculus
• Integrability of Hurwitz Partition Functions. I. Summary
• Matrix Models for Random Partitions
• Towards a proof of AGT conjecture by methods of matrix models
• Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
• Sum rules for characters from character-preservation property of matrix models
• Superintegrability of Kontsevich matrix model
• Cut and join operator ring in Aristotelian tensor model
• On KP-integrable skew Hurwitz $\tau$-functions and their $\beta$-deformations
• Torus HOMFLY as the Hall-Littlewood Polynomials
• New Insights into Superintegrability from Unitary Matrix Models
• On determinant representation and integrability of Nekrasov functions
• Superintegrability in $\beta$-deformed Gaussian Hermitian matrix model from $W$-operators
• BPS coherent states and localization
• Commutative families in $W_{\infty }$, integrable many-body systems and hypergeometric $\tau$-functions
• Jucys-Murphy Elements and Unitary Matrix Integrals
• Commutative families in DIM algebra, integrable many-body systems and $q,t$ matrix models
• Unitary Chern-Simons matrix model and the Villain lattice action
• q-Painleve equation from Virasoro constraints
• Two $\beta$-ensemble realization of $\beta$-deformed WLZZ models
• 3-Schurs from explicit representation of Yangian $Y({\widehat{\mathfrak{gl}}}_1)$. Levels 1-5
• Beta Deformation and Superpolynomials of (n,m) Torus Knots
• Why Is Landau-Ginzburg Link Cohomology Equivalent To Khovanov Homology?
• Primitive factorizations, Jucys-Murphy elements, and matrix models
• Faces of matrix models
• Perturbation of multi-critical unitary matrix models, double scaling limits, and Argyres-Douglas theories
• Genus expansion of matrix models and $\hslash$ expansion of KP hierarchy
• A new approach to the characteristic polynomial of a random unitary matrix
• On character expansion and Gaussian regularization of Itzykson-Zuber measure
• Unitary matrix integral for QCD with real quarks and the GOE-GUE crossover
• Grassmannian Integrals as Matrix Models for Non-Compact Yangian Invariants
• On the Virasoro constraints for torus knots
• On an extension of the generalized BGW tau-function
• Vogel's universality and the classification problem for Jacobi identities
• Phases in WLZZ Matrix Models
• On the Integrable Structure of Super Yang-Mills Scattering Amplitudes
• On a Matrix Ensemble for Arbitrary Complex Quantum Systems
• Itzykson-Zuber correlators from character expansion