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Duality and Mock Modularity
by Atish Dabholkar, Pavel Putrov, Edward Witten
Submission summary
As Contributors:  Pavel Putrov 
Arxiv Link:  https://arxiv.org/abs/2004.14387v2 (pdf) 
Date submitted:  20200721 02:00 
Submitted by:  Putrov, Pavel 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We derive a holomorphic anomaly equation for the VafaWitten partition function for twisted fourdimensional $\mathcal{N} =4$ super YangMills 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 antiinstantons. Using Mtheory duality, we relate this phenomenon to the holomorphic anomaly of the elliptic genus of a twodimensional 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 sixdimensional (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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Anonymous Report 1 on 202095 Invited Report
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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 VafaWitten partition functions in terms of a new holomorphic anomaly equation for the four dimensional gauge theory.
The solution of the problem in terms of antiinstantons contributions is elegant and very natural from the physics view point.
In the text the contribution of pointlike instantons, which is known to the experts, is not explained as carefully as it could to make the text accessible at a broader level.