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Flavor deformations and supersymmetry enhancement in $4d$ $\mathcal{N}=2$ theories
by Usman Naseer, Charles Thull
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Authors (as registered SciPost users):  Usman Naseer · Charles Thull 
Submission information  

Preprint Link:  https://arxiv.org/abs/2110.09329v2 (pdf) 
Date submitted:  20211215 23:22 
Submitted by:  Naseer, Usman 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We study $\mathcal{N}=2$ theories on fourdimensional manifolds that admit a Killing vector $v$ with isolated fixed points. It is possible to deform these theories by coupling positiondependent background fields to the flavor current multiplet. The partition function of the deformed theory only depends on the value of the background scalar fields at the fixed points of $v$. For a single adjoint hypermultiplet, the partition function becomes independent of the supergravity as well as the flavor background if the scalars attain special values at the fixed points. For these special values, supersymmetry at the fixed points enhances from the DonaldsonWitten twist to the Marcus twist or the VafaWitten twist of $\mathcal{N}=4$ SYM. Our results explain the recently observed squashing independence of $\mathcal{N}=2^*$ theory on the squashed sphere and provide a new squashing independent point. Interpreted through the AGTcorrespondence, this implies the $b$independence of torus onepoint functions of certain local operators in Liouville/Toda CFT. The positiondependent deformations imply relations between correlators of partially integrated operators in any $\mathcal{N}=2$ SCFT with flavor symmetries.
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Reports on this Submission
Anonymous Report 2 on 202221 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2110.09329v2, delivered 20220201, doi: 10.21468/SciPost.Report.4301
Report
This paper studies partition functions in 4dimensional N=2 supersymmetric theories with flavour deformations. The partition functions are computed using supersymmetric localization. In particular, it is found that when the deformation parameters are tuned to special values at fixed points, the partition functions simplify considerably. The underlying mechanism for the simplicity is the enhancement of supersymmetry.
The results help to clarify and generalize the recently observed phenomenon that N=2 partition functions simplify when some parameters take special values. Using the AGT correspondence, the authors comment on the applications of the results to certain onepoint functions in Liouville and Toda CFTs. Finally, the construction was applied to study partially integrated correlators in N=4 SYM, generalizing recent developments computing integrated correlators in N=4 SYM using supersymmetric localization. I think the new result from this current paper on the correlators in N=4 SYM still requires further development to make it useful in practice. In particular, one would like to express the constraints in terms of a single function T(U, V), as the authors also commented.
The paper is very wellwritten and clear, and as I summarized above the results of the paper are interesting and novel, so I suggest the paper be published in SciPost. I found a few punctuation typos:
The period of the sentence right before (3.4) should be removed or be comma?
Similarly
2. The period of Equation (3.18).
3. In equation (4.3), the period after \theta should be moved outside the big braket .
4. The period of Equation (4.11).
Anonymous Report 1 on 2022118 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2110.09329v2, delivered 20220118, doi: 10.21468/SciPost.Report.4178
Report
The paper under review is about fourdimensional N=2 superconformal theories placed on general backgrounds in presence of a flavor deformation. The main techniques are introduced at the first stages of the manuscript, in particular the prescription to place an extended supersymmetric theory on a curved manifold as well as the localized partition function on a general background.
The flavor deformations are implemented with a similar method, namely by coupling the flavor current multiplet to a background vector multiplet and by imposing the supersymmetry variation of the fermions to vanish. The authors concentrate on a specific matter content, the N=2* theory with a single adjoint hypermultiplet, where such mechanism generates a dynamical mass term for the hypermultiplet.
The main goal is then to identify specific values of the hypermultiplet mass where the localized partition function extremely simplifies. This method generalizes some previous results (mainly obtained on a four dimensional squashed sphere) to more general backgrounds. Following similar ideas from the threedimensional set up, the authors show that the simplification of the free energy at special values of the masses corresponds to a supersymmetry enhancement mechanism, proving that such values correspond to some well known twists of N=4 SYM.
The paper is clearly written and shows a good attention to the details. Using the present techniques some previous results in literature can be clarified and extended. Also, in the last section some further applications are discussed, opening new directions both in AGT context and for integrated correlators in N=4 SYM.
For these reasons, I recommend the paper under review for publication in SciPost.
Requested changes
1. At the beginning of section 3.3.1 I suggest to highlight (and clarify) the values for the mass $m_x$ and the definition of $m^*_{\epsilon_x,\epsilon'_x}$ (as well as $m^*_{\epsilon_y,\epsilon'_y}$), since there are frequent references to these values in the following.
2. Typos in the second line of formulas (2.18) and (3.10): I guess $\epsilon_x$ ($\epsilon'_x$) should be replaced by $\epsilon_y$ ($\epsilon'_y$)
3. Small typo in text: "respectivley" slightly before eq. (3.21)
4. Small typo in text: "paramters" 6 lines before the end of pag. 19