Tau functions of the UC hierarchy as partition functions of matrix models

Chuanzhong Li^a, A. D. Mironov^bcd, A. Yu. Orlov<sup>cdef

a</sup> College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, China
^b Lebedev Physical Institute, Russian Academy of Sciences, Moscow, Russia
^c National Research Centre "Kurchatov Institute", Moscow, Russia
^d Institute for Information Transmission Problems (Kharkevich Institute), Russian Academy of Sciences, Moscow, Russia
^e Shirshov Institute of Oceanology, Russian Academy of Sciences, Moscow, Russia
^f National Research University Higher School of Economics, Moscow, Russia

Abstract: We present a family of matrix models such that their partition functions are tau functions of the universal character (UC) hierarchy. We find new matrix models associated with the product of two spheres with embedded graphs via a gluing matrix. We also generalize these studies to the multi-matrix model case, which corresponds to the multi-component UC hierarchy.

Keywords: matrix models, GinUE, partition functions, Schur functions, universal character, UC hierarchy, tau function, embedded graphs, corner matrices.

Received: 06.03.2025
Revised: 10.09.2025

DOI: 10.4213/tmf10973

 English version: Theoretical and Mathematical Physics, 2025, 225:3, 2105–2127

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