String Theory Seminars

Aspects of Hecke Symmetry

by Dr. Madhusudan Raman (TIFR, Mumbai)

Thursday, November 1, 2018 from to (Asia/Kolkata)
at A 304
Motivated by their appearance in supersymmetric gauge and string theories, we study the relations governing quasi-automorphic forms associated to certain discrete subgroups of SL(2,R) called Hecke groups. The Eisenstein series associated to a Hecke group H(m) satisfy a set of coupled linear differential equations, which are natural analogues of the well-known Ramanujan identities for modular forms of SL(2,Z). We derive these dentities by appealing to a correspondence with the generalized Halphen system. Each Hecke group is then associated to a (hyper-)elliptic curve, whose oefficients are found to be determined by an anomaly equation. The Ramanujan identities admit a natural geometrical interpretation as a vector field on the moduli space of this curve. They also allow us to associate a higher- order non-linear differential equation to each Hecke group. These equations are higher-order analogues of the well-known Chazy equation, and we show that they are solved by the quasi-automorphic weight-2 Eisenstein series associated to H(m) and its Hecke orbits, thereby generalizing a result of Takhtajan.