Тождество Фея: от Математической Физики через Геометрию к Теории Чисел

We give a survey of various avatars of the trisecant Fay identity which appear in the context of Integrable systems (as forms of the Associative Yang-Baxter Equation) and as conditions on generating functions for period polynomials of (quasi-)modular forms and group cocycle conditions for some multiparametric modular groups.