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Exact results for duality-covariant integrated correlators in $\mathcal{N}=4$ SYM with general classical gauge groups

by Daniele Dorigoni, Michael B. Green, Congkao Wen

This is not the latest submitted version.

Submission summary

As Contributors: Congkao Wen
Preprint link: scipost_202203_00025v1
Date submitted: 2022-03-18 10:45
Submitted by: Wen, Congkao
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


We present exact expressions for certain integrated correlators of four superconformal primary operators in the stress tensor multiplet of $\cN=4$ supersymmetric Yang--Mills (SYM) theory with classical gauge group, $G_N$ $= SO(2N)$, $SO(2N+1)$, $USp(2N)$. These integrated correlators are expressed as two-dimensional lattice sums by considering derivatives of the localised partition functions, generalising the expression obtained for $SU(N)$ \mbg{gauge group} in our previous works. These expressions are manifestly covariant under Goddard-Nuyts-Olive duality. The integrated correlators can also be formally written as infinite sums of non-holomorphic Eisenstein series with integer indices and rational coefficients. Furthermore, the action of the hyperbolic Laplace operator with respect to the complex coupling $\tau=\theta/(2\pi) + 4\pi i /g^2_{_{YM}}$ on any integrated correlator for gauge group $G_N$ relates it to a linear combination of correlators with gauge groups $G_{N+1}$, $G_N$ and $G_{N-1}$. These ``Laplace-difference equations'' determine the expressions of integrated correlators for all classical gauge groups for any value of $N$ in terms of the correlator for the gauge group $SU(2)$. The perturbation expansions of these integrated correlators for any finite value of $N$ agree with properties obtained from perturbative Yang--Mills quantum field theory, together with various multi-instanton calculations which are also shown to agree with those determined by supersymmetric localisation. The coefficients of terms in the large-$N$ expansion are sums of non-holomorphic Eisenstein series with half-integer indices, which extend recent results and make contact with low order terms in the low energy expansion of type IIB superstring theory in an $AdS_5\times S^5/\mathbb{Z}_2$ background.

Current status:
Has been resubmitted

Submission & Refereeing History

Resubmission scipost_202203_00025v2 on 7 August 2022

Reports on this Submission

Anonymous Report 2 on 2022-6-15 (Invited Report)


1. The integrated correlators are rich observables, well-deserving of study. The authors give clear and compelling evidence for their intriguingly simple structure as they did for the SU(N) case in earlier work.

2. The computations are thorough.

3. The paper is clearly written.


1. Some features of the results are obscured.

2. The paper is a bit long.


Integrated four-point functions in N=4 super-Yang-Mills are, clearly, very intriguing observables, as originally shown by Binder/Chester/Pufu/Wang (ref. [3]) in 2019 for SU(N) SYM and then in subsequent works. There is good motivation to extend previous analyses to the non-simply-laced cases, which these authors do for classical gauge groups.

The results are as expected, for better or worse, but still rewarding. The punchline is that these correlators can be written in essentially similar form, have similar complexity, and obey similar Laplace difference equations which, conjecturally, determine all correlators starting solely from the SU(2) one. The results can be written in a rather uniform way across gauge groups---a very appealing result.

As is often the case, it is less exciting to see the details of the SO(N)/Sp(N) case once the SU(N) case is understood. On the other hand these results will likely be useful for future studies of the SO(N)/Sp(N) SYM theory. For example they should be usable as input in numerical studies of the conformal bootstrap of the SO(N)/Sp(N) theories, following Chester/Dempsey/Pufu's results for the SU(N) theory and Chester's work for SQCD.

The authors stick rather closely to their own blueprint from their earlier works while neglecting some subsequent developments. This could be improved.

A major outstanding question is that it would be nice to say more about the origin of the Laplace difference equation. the Lemma on p.12 is supported by evidence from the perturbative expansions, but for it to carry real weight one would like to know why these relations hold.

The paper deserves publication subject to these remarks.

Requested changes

1. The work [9] revealed several new things about the SU(N) correlators that ought to have been included in the optimal treatment of the SO(N)/Sp(N) cases. The spectral representation is functionally simpler and makes the meaning of various expressions (for example properties of the kernel, the existence of the lattice representation) physically transparent. This should be more than a sidenote. It is hard to understand why, for example, the (formal) representation as an infinite sum of Eisenstein series with positive integer index receives such emphasis---even in the abstract.

2. In Sec 4.2 where the integral kernels B are given, the authors explain how to compute them, but not what they are physically. It would have been nice to investigate this. They are presumably best thought of as modified Borel transforms of the perturbative expansion which manifestly preserve the GNO duality?

3. In the conclusion, the authors mention that it would be nice to find the same lattice representation of the integrated correlator with four mass derivatives. Why would this be expected to exist? Some justification could be provided either way. It seems that actually it may not exist based on previous computations by Chester et al.

  • validity: top
  • significance: good
  • originality: good
  • clarity: high
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  • grammar: perfect

Anonymous Report 1 on 2022-4-27 (Invited Report)


This paper is a continuation of the authors' previous works where an conjectural $SL(2,\mathbb{Z})$-invariant expression for the quantity $\Delta_\tau \partial^2_m \log Z_{S^4}$ of $\mathcal{N}=2^*$ $SU(N)$ theory was found and carefully checked; the novelty in this paper is the extension to other classical gauge groups.

The paper is mostly well-written but the referee wants to request a few improvements.

1) The main recursion relations, (1.12), (3.2) and (3.3) should be given a derivation which should be at least slightly more detailed. The referee understands that the method to be applied was explained in detail for the $SU(N)$ case in the authors' previous papers, but the manuscript in the current form is too terse. Having an extra appendix for the derivation would be nice.

2) The comments around (2.10) and (2.11) are somewhat misguided. It is well-known in the referee's opinion that the large-N expansion of SO and USp gauge theories are given in terms of possibly-unorientable worldsheets. This underlies Maldacena's duality for SO and USp gauge groups, where the holographic dual contains orientifolds. According to the standard review of AdS/CFT by MAGOO, , it goes back to Cicuta, .

The large N expansion is in terms of $N$ to the power of the Euler number of the worldsheet, and the Euler number is $2-2g$ for an oriented surface of genus $g$, but it can be an odd integer for unoriented worldsheets. Therefore, a better-motivated version of (2.11) would have a summation of the form $\sum_{k} N_G^{2-k}$, where $N_G$ is $N_{SU(N)}=N$, $N_{SO(n)}=n-2$ and $N_{USp(n)}=n+2$, and $k$ is restricted to even integers when $G=SU(N)$. The shift by $\pm2$ does not seem to follow from a group theoretical analysis of the large $N$ Feynman diagrams, but has a natural motivation in AdS/CFT.

What perplexes the referee is that this is essentially reviewed in the Appendix C of this manuscript, and still is not reflected in the comments around (2.10) and (2.11)!

3) The authors commented that the localization computations for the exceptional gauge groups are ill-understood. This is still true to some extent, but it should be noted that there is a way to compute instanton partition functions recursively using something called the blow-up equations, see e.g. . This should allow the authors to check their conjectural lattice-sum formula for $C_G$ by determining $B(t)$ from the perturbative part and comparing its prediction for the instanton contributions against the computations from the blow-up equations. The referee recommends the authors to contact the authors of the paper just mentioned, since setting up the computations using the blow-up equations requires a tedious amount of hard work.

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