Duality and Mock Modularity

Abstract

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.

arXiv comment50 pages, 1 figure; v2: misprints corrected, clarifications added in Section 5.1; v3: minor clarifications added

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• New Modular Invariants in $\mathcal{N}=4$ Super-Yang-Mills Theory
• Vafa-Witten invariants from exceptional collections
• The joy of factorization at large $N$: five-dimensional indices and AdS black holes
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• On the topological and crosscap entropies in non-oriented RCFTs
• Chern characters for supersymmetric field theories