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Fermigas correlators of ADHM theory and triality symmetry
by Yasuyuki Hatsuda, Tadashi Okazaki
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Submission summary
As Contributors:  Tadashi Okazaki 
Preprint link:  scipost_202108_00070v1 
Date submitted:  20210830 14:43 
Submitted by:  Okazaki, Tadashi 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We analytically study the Fermigas formulation of sphere correlation functions of the Coulomb branch operators for 3d $\mathcal{N}=4$ ADHM theory with a gauge group $U(N)$, an adjoint hypermultiplet and $l$ hypermultiplets which can describe a stack of $N$ M2branes at $A_{l1}$ singularities. We find that the leading coefficients of the perturbative grand canonical correlation functions are invariant under a hidden triality symmetry conjectured from the twisted Mtheory. The triality symmetry also helps us to fix the nexttoleading corrections analytically.
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Reports on this Submission
Anonymous Report 3 on 2021929 (Invited Report)
Report
The authors consider N=4 3d supersymmetric gauge theory known as ADHM theory on a threesphere spacetime. The theory has U(N) gauge group, one adjoint and a certain number of fundamental hypermultiplets. Using the known socalled Fermigas approach they rewrite the partition function of the theory and certain correlation functions in terms of a system of N noninteracting fermions and consider its large N limit (or, equivalently, the limit of large chemical in the grand canonical ensemble). The authors find agreement with some predictions from the holographically dual description in Mtheory (in particular agreement with the triality symmetry that exchanges three complex lines in the spacetime of Mtheory) and numerical results from a previous work by other authors.
I believe that the results and the techniques in this manuscript will be interesting to other researchers working on localization in supersymmetric gauge theories, matrix models, AdS/CFT correspondence and related topics. The paper is generally well written. I would like to recommend it for publication.
Requested changes
I have the following minor suggestions which I think can improve the readability of the paper, particularly for nonspecialists:
1) From the expressions (2.8) it seems that the Wigner transform of the Hamiltonian $H_W$ is generically complex valued (in particular its classical part) . Later $(2\pi\mu/\epsilon_1H_W)$ appears as the argument of the functions like Heaviside step function and Dirac delta function, which are ordinarily defined for a real argument only. I think it would be better if the authors add a clarification about interpretation of such expressions.
2) In the beginning of Section 3 the authors use subscript $n_*$. I suggest that the authors add a comment on what is its meaning, what is the range of the sum in (3.1), and why only $n_*=0$ appears in (3.2).
3) In the formulas like (3.52) (similarly in (3.61)) the authors may consider indicating dependence of $\langle \mathcal{O}\rangle$ on $N$ inside the sum more explicitly , otherwise the formula looks a little confusing.
4) I find that the manuscript in some places (for example around pages 12, 1821) is quite heavy on technical details which are rather elementary (like calculation of integrals). I think moving them to Appendix might make the reading of the paper more enjoyable. But I leave it up to the authors.
Report 2 by Jihwan Oh on 2021928 (Invited Report)
Report
In this nice and rigorous paper, the authors compute 3sphere correlation functions of Coulomb branch operators in 3d N=4 ADHMlike gauge theory. The key advance made was an analytical proof of numerical results, which were derived in arXiv:2004.13810. The result further sheds light on the structure of underlying twisted Mtheory background and reconfirms triality property of the algebra of operators in a novel way using correlation function. The authors used wellestablished technique Fermi gas technique masterfully and the result that they obtain is novel and powerful. The paper is expected to generate a new direction in a growing literature of twisted Mtheory.
Overall, this paper has top quality. Therefore, without further editing, I recommend to publish it.
Anonymous Report 1 on 2021926 (Invited Report)
Report
In the paper the authors analyzed the $S^3$ partition function of the 3d N=4 supersymmetric U(N) gauge theory with $l$ fundamental matter multiplets and one adjoint matter multiplet which can be interpreted as N M2branes placed on an omegadeformed background, where the omega deformation parameters are identified with the mass parameter of the adjoint matter multiplet. They demonstrated that the leading and subleading part of the grand potential in the limit of large chemical potential, which correspond to the leading and subleading part of the large N free energy, is invariant under the triality symmetry in the omega deformation parameters as suggested from the Mtheory background. The authors also evaluated some correlation functions of the same theory and confirmed that the leading part of these quantities are also invariant under the triality symmetry. They further determined the explicit expressions of the subleading part of the correlation functions by requiring the triality symmetry.
The analyses of the correlation function are completely original results of the paper. Also, although the large N expansion of the $S^3$ partition function was already obtained in a previous research in a dual description by an ABJMlike theory, the reinterpretation in the omegadeformed Mtheory is new. For these reasons I recommend this paper to be published.