What is mock modular formulae?

Hi Sreehari, I came to hear of certain physical aspects of the Mock modular forms through an international workshop held at IMSc (Chennai, INDIA) in April, 2014. [1] Mathieu moonshine relates the geometry of K3 surfaces to the representations of the Mathieu group M24, and to a certain mock modular form. This can be generalized by Umbral moonshine, to a corresponding sequence of finite groups, and to the 23 Niemeier lattices. [2] There is a nice connection between the Wall-crossing phenomena in string theory and supersymmetric gauge theory, and the realization of modular or automorphic symmetries in the corresponding systems. For instance, in N=4 string compactifications, the degeneracies of supersymmetric black holes have been shown to be Fourier coefficients of mock modular forms. [3] These mock modular forms are interpreted as the supersymmetric partition functions of certain non-compact superconformal field theories. Such CFTs are significant in the physics of black holes and black strings, and appear to be very substantial for the above mentioned aspects. There is a general belief among scientists that these underlying structures, maybe originated in sting theory, might unify these completely orthogonal approaches and they should have implications for representation theory, number theory, geometry, gauge theory and sting theory. All the best.

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In mathematics, a mock modular form is the holomorphic part of a harmonic weak maass form, and a mock theta function is essentially a mock modular form of weight 1/2. The first examples of mock theta functions were described by Srinivasa Ramanujan in his last 1920 letter to G. H. Hardy and in his lost notebook. Sander Zwegers (2001 , 2002) discovered that adding certain non-holomorphic functions to them turns them into harmonic weak Maass forms.

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