Abstract
We derive a holomorphic anomaly equation for the Vafa-Witten partition function for twisted four-dimensional \mathcal{N} =4𝒩=4 super Yang-Mills theory on \mathbb{CP}^{2}ℂℙ2 for the gauge group SO(3)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 SS-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)(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.
Highlights
If the T 2 is very small compared to X, as already stated, we reduce to gauge theory on X
In the opposite limit that X is very small compared to T 2, we reduce to a supersymmetric field theory on T 2
In forthcoming work, Manschot and Moore have analyzed the Coulomb branch integral and the associated mock modularity in the N = 2∗ theory, which is closely related to N = 4 super Yang-Mills, which we study in the present paper
Summary
In a sigma-model with target W , supersymmetric localization reduces the computation of the elliptic genus to an integral over the space of constant maps from T 2 to W This space of constant maps plays the role of the Coulomb branch in the gauge theory. A computation using only the lowest order terms in the effective action on the Coulomb branch of the gauge theory or the target space of the sigma-model will not show the holomorphic anomaly. In forthcoming work, Manschot and Moore have analyzed the Coulomb branch integral and the associated mock modularity in the N = 2∗ theory, which is closely related to N = 4 super Yang-Mills, which we study in the present paper Their calculation might lead to a way to resolve the normalization issue mentioned in the last paragraph.
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