Symmetries of Prym theta functions for real curves

Studying of the realness conditions for Riemann theta function, and the corresponding real subvarieties of Jacobians, was pioneered by Dubrovin and Natanzon, with applications to the sine-Gordon equation. For the curves possessing involutions the realness conditions have been first studied by Novikov and Veselov, and led to real subvarieties of Prym varieties. They had applications to the Shroedinger equation. Studying of Hitchin systems leads to realness conditions for Prym theta functions corresponding to a principally polarized Abelian variety different from the Prymian but isogenic to it. In the case of real curves of separating type Prym theta functions were studied in the Lecture Notes by Fay. In the talk, we will present symmetries of Prym theta functions for real curves of both separating and non-separating types. Fixed points of the symmetries are exactly the points where the Abel–Prym map can be reversed by means of the Riemann vanishing theorem.