| Summary: | We derive a holomorphic anomaly equation for the Vafa-Witten partition
function for twisted four-dimensional $\mathcal{N} =4$ super Yang-Mills theory
on $\mathbb{CP}^{2}$ for the gauge group $SO(3)$ from the path integral of the
effective theory on the Coulomb branch. The holomorphic kernel of this
equation, which receives contributions only from the instantons, is not modular
but `mock modular'. The partition function has correct modular properties
expected from $S$-duality only after including the anomalous nonholomorphic
boundary contributions from anti-instantons. Using M-theory duality, we relate
this phenomenon to the holomorphic anomaly of the elliptic genus of a
two-dimensional noncompact sigma model and compute it independently in two
dimensions. The anomaly both in four and in two dimensions can be traced to a
topological term in the effective action of six-dimensional (2,0) theory on the
tensor branch. We consider generalizations to other manifolds and other gauge
groups to show that mock modularity is generic and essential for exhibiting
duality when the relevant field space is noncompact.
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