SciPost Phys. 13, 092 (2022) ·
published 11 October 2022

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We present exact expressions for certain integrated correlators of four superconformal primary operators in the stress tensor multiplet of $\cN=4$ supersymmetric YangMills (SYM) theory with classical gauge group, $G_N$ $= SO(2N)$, $SO(2N+1)$, $USp(2N)$. These integrated correlators are expressed as twodimensional 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 GoddardNuytsOlive duality. The integrated correlators can also be formally written as infinite sums of nonholomorphic 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 $G_{N1}$. These ``Laplacedifference 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 YangMills quantum field theory, together with various multiinstanton 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 nonholomorphic Eisenstein series with halfinteger 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.
SciPost Phys. 4, 012 (2018) ·
published 27 February 2018

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First we compute the $\mbox{S}^2$ partition function of the supersymmetric $\mathbb{CP}^{N1}$ model via localization and as a check we show that the chiral ring structure can be correctly reproduced. For the $\mathbb{CP}^1$ case we provide a concrete realisation of this ring in terms of Bessel functions. We consider a weak coupling expansion in each topological sector and write it as a finite number of perturbative corrections plus an infinite series of instantonantiinstanton contributions. To be able to apply resurgent analysis we then consider a nonsupersymmetric deformation of the localized model by introducing a small unbalance between the number of bosons and fermions. The perturbative expansion of the deformed model becomes asymptotic and we analyse it within the framework of resurgence theory. Although the perturbative series truncates when we send the deformation parameter to zero we can still reconstruct nonperturbative physics out of the perturbative data in a nice example of Cheshire cat resurgence in quantum field theory. We also show that the same type of resurgence takes place when we consider an analytic continuation in the number of chiral fields from $N$ to $r\in\mathbb{R}$. Although for generic real $r$ supersymmetry is still formally preserved, we find that the perturbative expansion of the supersymmetric partition function becomes asymptotic so that we can use resurgent analysis and only at the end take the limit of integer $r$ to recover the undeformed model.
Dr Dorigoni: "We thank the referee for the t..."
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