Room 318

Aspects of Hecke symmetry

Madhusudhan Raman

TIFR, Mumbai

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 m
coupled linear differential equations, which are natural analogues of the
well-known Ramanujan identities for modular forms of SL(2,Z). We prove
these identities by appealing to a correspondence with the generalized
Halphen system. Each Hecke group is then associated to a (hyper-)elliptic
curve, whose coefficients 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 non-linear differential equation of order m to
each Hecke group. These equations are higher-order analogues of the 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. We
conclude by demonstrating that these non-linear equations possess the
Painlevé property.