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Unitary integrals and related matrix models

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Abstract

We briefly review the basic properties of unitary matrix integrals, using three matrix models to analyze their properties: the ordinary unitary, the Brezin—Gross—Witten, and the Harish—Chandra—Itzykson—Zuber models. We especially emphasize the nontrivial aspects of the theory, from the De Witt’Hooft anomaly in unitary integrals to the problem of calculating correlators with the Itzykson-Zuber measure. We emphasize the method of character expansions as a technical tool. Unitary integrals are still insufficiently investigated, and many new results should be expected as this field attracts increased attention.

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    A.Yu. Morozov

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Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 161, No. 3, pp. 3–40, January, 2010. Original article submitted July 9, 2009.

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Morozov, A. Unitary integrals and related matrix models. Theor Math Phys 162, 1–33 (2010). https://doi.org/10.1007/s11232-010-0001-y

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