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Flavor deformations and supersymmetry enhancement in $4d$ $\mathcal{N}=2$ theories

by Usman Naseer, Charles Thull

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Submission summary

Authors (as registered SciPost users): Usman Naseer · Charles Thull
Submission information
Preprint Link:  (pdf)
Date submitted: 2021-12-15 23:22
Submitted by: Naseer, Usman
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


We study $\mathcal{N}=2$ theories on four-dimensional manifolds that admit a Killing vector $v$ with isolated fixed points. It is possible to deform these theories by coupling position-dependent 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 Donaldson-Witten twist to the Marcus twist or the Vafa-Witten 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 AGT-correspondence, this implies the $b$-independence of torus one-point functions of certain local operators in Liouville/Toda CFT. The position-dependent 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 2022-2-1 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2110.09329v2, delivered 2022-02-01, doi: 10.21468/SciPost.Report.4301


This paper studies partition functions in 4-dimensional 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 one-point 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 well-written 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?

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).

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Anonymous Report 1 on 2022-1-18 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2110.09329v2, delivered 2022-01-18, doi: 10.21468/SciPost.Report.4178


The paper under review is about four-dimensional 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 three-dimensional 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

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