Cite as: doi:10.26081/K6Q31S

**
Mock theta functions were introduced by Ramanujan in his famous last letter to
Hardy in 1920 but were properly understood only recently with the work of
Zwegers in 2002. I will describe three manifestations of this apparently exotic
mathematics in three important physical contexts of holography, topology and
duality where mock modularity has come to play an important role.
**

**In particular, I will derive a holomorphic anomaly equation for the indexed
partition function of a two-dimensional CFT2 dual to AdS3 that counts the black
hole degeneracies, and for Vafa-Witten partition function for twisted four
dimensional N=4 super Yang-Mills theory on CP2 for the gauge group SO(3) that
counts instantons. The holomorphic kernel of this equation is not modular but
`mock modularâ€™ and one obtains correct modular properties only after including
certain `anomalousâ€™ nonholomorphic boundary contributions. This phenomenon can
be related to the holomorphic anomaly of the elliptic genus of a
two-dimensional noncompact supersymmetric sigma model, and in a simpler context
of quantum mechanics to the Atiyah-Patodi-Singer eta invariant. Mock
modularity is thus essential to exhibit modular symmetries expected from the
AdS3/CFT2 holographic equivalence in quantum gravity and the S-duality
symmetry of four-dimensional quantum gauge theories.
**

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