#### 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.

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