Duality and mock modularity

Atish Dabholkar, Pavel Putrov, Edward Witten

Research output: Contribution to journalArticlepeer-review

15 Scopus citations


We derive a holomorphic anomaly equation for the Vafa-Witten partition function for twisted four-dimensional N = 4 super Yang-Mills theory on CP2 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.

Original languageEnglish (US)
Article number072
JournalSciPost Physics
Issue number5
StatePublished - Nov 13 2020
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Physics and Astronomy


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