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Duality and Mock Modularity
by Atish Dabholkar, Pavel Putrov, Edward Witten
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Submission summary
Authors (as registered SciPost users):  Pavel Putrov 
Submission information  

Preprint Link:  https://arxiv.org/abs/2004.14387v2 (pdf) 
Date submitted:  20200721 02:00 
Submitted by:  Putrov, Pavel 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
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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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Reports on this Submission
Anonymous Report 2 on 2020109 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2004.14387v2, delivered 20201009, doi: 10.21468/SciPost.Report.2066
Report
The partition function of twisted $N=4$ super YangMills (SYM) theory on a manifold $X$ was studied in a paper by Vafa and Witten in 1994. It was calculated for a number of examples of $X$ in that paper by reducing the path integral to an integral over the moduli space of instantons summed over all instanton numbers. For all $X$ the partition function $Z(\tau)$ calculated in this manner is a holomorphic function.
Sduality of $N=4$ SYM implies that the VafaWitten partition function should have good modular properties. In particular, for gauge group $SO(3)$ the partition function for $X=\mathbb{C} \mathbb{P}^2$ is expected to be invariant under the action of a certain subgroup of $SL(2,\mathbb{Z})$ on the gauge coupling $\tau$. On the other hand, the abovementioned calculation in this case gave rise to a function $Z(\tau)$ which is not a modular function, and therefore seems to be in conflict with Sduality of the theory. It was recognized in the VafaWitten paper that the function has (what is today called) a mock modular property  it can be completed to a modular function by adding a specific nonholomorphic correction term to $Z$, thus giving rise to a holomorphic anomaly. However, the physical origin of this correction term remained mysterious.
In the current paper the authors resolve this puzzle by explaining the physical origin of mock modularity. Using this idea they calculate the holomorphic anomaly  in two different ways. The first way is a careful reconsideration of the Coulomb branch integral (à la MooreWitten) for this theory; the boundary terms in this integral precisely reproduce the holomorphic anomaly. The second way is by constructing a dual twodimensional sigma model with noncompact target space which also has the same holomorphic anomaly. The 2d and the 4d theories are related in that they both arise from the same 6d $(2,0)$ theory on an M5 brane in different limits.
The paper contains new interesting results, and it is very clearly written. It will pay dividends to those who read it with some care  along the way there is also a nice review of the three possible twistings of $N=4$ SYM, of the brane construction of the theory, and an explicit construction of the offshell supersymmetry algebra that is needed for the localization calculations. A final section contains some generalizations of this story to other $X$ and other gauge groups. I have no doubt that it should be published.
I only have a few minor remarks:
 At various places in the introduction the phrase "modular function" (or "mock modular function") is used. I suppose the authors mean "a function which has modular properties".
This may give rise to a small confusion since the phrase also has a definition (invariant under modular transformation) which does not hold in the general context as used in the paper (e.g. for K3 the answer is a power of $\eta(\tau)$). I would suggest using (mock) modular formindeed, later in the paper this is used by the authorsor inserting a note saying that they are using the phrase loosely.
 Is it clear that there is no boundary term at $u=0$ in Section 3.3? (Footnote 9 seems to say that they need to cut out the origin, is this regulator consistent with the symmetries?) I would imagine so, but a brief note of explanation would be helpful.
 Typo: it should be $(2,0)$ in the penultimate paragraph on Page 11.
 It may be worth reminding the reader that $KX$ at the beginning of Section 5 stands for the canonical bundle of X (as defined earlier in the paper).
 One thing that confused me a bit is the penultimate paragraph on Page 5. The authors say that (a) there is full control of the 2d theory including the overall normalization (using a relation to the Hamiltonian interpretation), (b) the overall normalization is not under control in the 4d theory, and (c) the two theories come from the same 6d theory. If we track the 2d/4d duality through the M5 brane very carefully, can we keep track of the normalization, or at what step do we lose track of the normalization?
(This last point is probably just something I didn't understand personally, and this should not hold up publication of the paper but I would be happy if the authors could answer this.)
Anonymous Report 1 on 202095 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2004.14387v2, delivered 20200905, doi: 10.21468/SciPost.Report.1961
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 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.