Non-planar corrections in orbifold/orientifold $$ \mathcal{N} $$ = 2 superconformal theories from localization

Abstract

We study non-planar corrections in two special $$ \mathcal{N} $$ N = 2 superconformal SU(N) gauge theories that are planar-equivalent to $$ \mathcal{N} $$ N = 4 SYM theory: two-nodes quiver model with equal couplings and $$ \mathcal{N} $$ N = 2 vector multiplet coupled to two hypermultiplets in rank-2 symmetric and antisymmetric representations. We focus on two observables in these theories that admit representation in terms of localization matrix model: free energy on 4-sphere and the expectation value of half-BPS circular Wilson loop. We extend the methods developed in arXiv:2207.11475 to derive a systematical expansion of non-planar corrections to these observables at strong ’t Hooft coupling constant λ. We show that the leading non- planar corrections are given by a power series in λ3/2/N2 with rational coefficients. Sending N and the coupling constant λ to infinity with λ3/2/N2 kept fixed corresponds to the familiar double scaling limit in matrix models. We find that in this limit the observables in the two models are related in a remarkably simple way: the free energies differ by the factor of 2, whereas the Wilson loop expectation values coincide. Surprisingly, these relations hold only at strong coupling, they are not valid in the weak coupling regime. We also discuss a dual string theory interpretation of the leading corrections to the free energy in the double scaling limit suggesting their relation to curvature corrections in type IIB string effective action.

Wilson loop correlators at strong coupling in $$ \mathcal{N} $$ = 2 quiver gauge theories

“…Specifically in this paper we derive an exact expression, valid for every value of the 't Hooft coupling, for the n-point Wilson loop correlator in the planar limit. Then, generalizing to the case of 4d N = 2 Z M quiver theory the techniques introduced in [41][42][43], we find the leading order of its strong coupling expansion, which agrees with the corresponding result in N = 4 SYM up to a numerical proportionality factor. To the best of our knowledge, this is the first example of an exact expression for a generic correlator among n coincident Wilson loops in the planar limit of a four dimensional N = 2 superconformal gauge theory for which also the strong coupling behaviour is analytically derived.…”

“…where ϕ (ℓ) (x) = J ℓ ( √ x). We observe that (B.9) can be obtained starting from the expression (C.1) of [43] n,m (1) ≡ w nm . Let us now move to analyse the properties of the coefficients (B.9) valid for any value of the coupling g. Following the same steps performed in appendix C of [43], one can firstly show that the coefficients w (ℓ) n,m (s α ) satisfy the differential equation…”

“…with n, m ≥ 0 and where ϕ (ℓ) (x) ≡ J ℓ ( √ x) with ℓ = 1, 2. These coefficients are the generalization for every s α of the w n,m introduced in [43] and their main properties have been collected in appendix B. Crucially for us, we can express the quantities S (P ) (s α ) appearing in (4.14) using the coefficients (4.20) and, for instance, for the first values of P we get…”

“…corrections to other observables in the Z 2 quiver theory have been, for instance, recently investigated in [43].…”

“…In all our analysis a crucial role has been played by the relation (5.20) that provides an analytic expression for the generating function G (0) (s α , −2π). From a purely mathematical point of view, this expression has been obtained by extending the analysis performed in [43] to a M -nodes quiver gauge theory. Moreover, since the generating function G (0) (s α , x) is not associated to any specific observable, the relation (5.20) could turn out to be useful also in the evaluation of the strong coupling regime of other type of correlators with operators belonging to conjugated twisted sectors, e.g.…”