# Duality and Mock Modularity

### Submission summary

 As Contributors: Pavel Putrov Arxiv Link: https://arxiv.org/abs/2004.14387v3 (pdf) Date accepted: 2020-10-29 Date submitted: 2020-10-22 21:21 Submitted by: Putrov, Pavel Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory Mathematical Physics Approach: Theoretical

### 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.

Published as SciPost Phys. 9, 072 (2020)

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

### List of changes

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.

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

### Submission & Refereeing History

Resubmission 2004.14387v3 on 22 October 2020
Submission 2004.14387v2 on 21 July 2020

## Reports on this Submission

### Report

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

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
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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