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Duality and Mock Modularity

by Atish Dabholkar, Pavel Putrov, Edward Witten

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Pavel Putrov
Submission information
Preprint Link:  (pdf)
Date accepted: 2020-10-29
Date submitted: 2020-10-22 21:21
Submitted by: Putrov, Pavel
Submitted to: SciPost Physics
Ontological classification
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.

Author comments upon resubmission

We thank the referees for the comments. We have addressed them as indicated in the ``list of changes''.

List of changes

In reply to Report 2:

1) A clarifying Footnote 2 added on page 2.

2) The text in the former Footnote 10 (which contained some relevant comments) was expanded and moved to the main text below eq. (3.29) on page 21.

3) The typo is corrected, we thank the referee for catching it.

4) A reminder about notation $KX$ is added on page 33.

5) Clarifying comments added on page 21.

In reply to Report 1:

6) A paragraph with a brief review of point-like instantons added on page 21.

Published as SciPost Phys. 9, 072 (2020)

Reports on this Submission

Anonymous Report 2 on 2020-10-26 (Invited Report)


The second version includes a change in the direction I suggested.

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Anonymous Report 1 on 2020-10-23 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2004.14387v3, delivered 2020-10-23, doi: 10.21468/SciPost.Report.2111


The authors have addressed all my previous comments satisfactorily.
In particular, in the normalization as defined in Footnote 2 the partition function for K3 has weight 0 and not -12 (as the partition function Z in the Vafa-Witten paper), so there is no confusion about the notation now.

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  • originality: -
  • clarity: -
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