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Duality and Mock Modularity

by Atish Dabholkar, Pavel Putrov, Edward Witten

Submission summary

As Contributors: Pavel Putrov
Arxiv Link: (pdf)
Date submitted: 2020-07-21 02:00
Submitted by: Putrov, Pavel
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


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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Submission & Refereeing History

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Submission 2004.14387v2 on 21 July 2020

Reports on this Submission

Anonymous Report 1 on 2020-9-5 Invited Report


The paper is surely top 10% and to be published.

The topic is very interesting and opens a new and simple approach to the interpretation of the modular properties of Vafa-Witten partition functions in terms of a new holomorphic anomaly equation for the four dimensional gauge theory.
The solution of the problem in terms of anti-instantons contributions is elegant and very natural from the physics view point.

In the text the contribution of point-like instantons, which is known to the experts, is not explained as carefully as it could to make the text accessible at a broader level.

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