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Abstract In a N $$ \mathcal{N} $$ = 2 superconformal gauge theory with matter hypermultiplets transforming in the symmetric and anti-symmetric representations of SU(N), we study the integrated correlators of two Coulomb-branch operators and two moment-map operators using localization. In the corresponding matrix model we identify the operator associated with the integrated insertions of moment-map operators and provide for it an exact expression valid for all values of the coupling constant in the planar limit. This allows us to study the integrated correlators at strong-coupling where we show that they behave as the 2-point functions of the Coulomb-branch operators, up to an overall constant dependent only on the conformal dimensions of the latter. The strong-coupling relation between integrated correlators and 2-point functions turns out to be the same as in N $$ \mathcal{N} $$ = 4 SYM at large N, despite the reduced amount of supersymmetry in our theory.

Abstract In the context of integrated correlators in N $$ \mathcal{N} $$ = 4 SYM, we study the 2-point functions of local operators with a superconformal line defect. Starting from the mass-deformed N $$ \mathcal{N} $$ = 2* theory in presence of a 1 2 $$ \frac{1}{2} $$ -BPS Wilson line, we exploit the residual superconformal symmetry after the defect insertion, and show that the massive deformation corresponds to integrated insertions of the superconformal primaries belonging to the stress tensor multiplet with a specific integration measure which is explicitly derived after enforcing the superconformal Ward identities. Finally, we show how the Wilson line integrated correlator can be computed by the N $$ \mathcal{N} $$ = 2* Wilson loop vacuum expectation value on a 4-sphere in terms of a matrix model using supersymmetric localization. In particular, we reformulate previous matrix model computations by making use of recursion relations and Bessel kernels, providing a direct link with more general localization computations in N $$ \mathcal{N} $$ = 2 theories.

Abstract We study the 3-point functions of gauge-invariant scalar operators in four dimensional N $$ \mathcal{N} $$ = 2 superconformal quiver theories using supersymmetric localization in the planar limit of a large number of colors. By exploiting a web of nontrivial relations, we show that the 3-point functions can be expressed in terms of the 2-point functions through exact Ward-like identities that are valid for all values of the coupling constant. In this way, using recent results about the 2-point functions, we are able to obtain the asymptotic strong-coupling expansion of the 3-point functions and of the corresponding structure constants in the planar limit. Our results extend to sub-leading orders what has been recently found at leading order, where a precise match with calculations within the AdS/CFT correspondence at the supergravity level is possible. Therefore, our findings can be interpreted also as a prediction for the sub-leading string corrections to these holographic calculations.

Abstract We study 4-dimensional N $$ \mathcal{N} $$ = 2 superconformal quiver gauge theories obtained with an orbifold projection from N $$ \mathcal{N} $$ = 4 SYM, and compute the 2- and 3-point correlation functions among chiral/anti-chiral single-trace scalar operators and the corresponding structure constants. Exploiting localization, we map the computation to an interacting matrix model and obtain expressions for the correlators and the structure constants that are valid for any value of the ’t Hooft coupling in the planar limit of the theory. At strong coupling, these expressions simplify and allow us to extract the leading behavior in an analytic way. Finally, using the AdS/CFT correspondence, we compute the structure constants from the dual supergravity theory and obtain results that perfectly match the strong-coupling predictions from localization.

Abstract We study the 3-point functions of single-trace scalar operators in a four-dimensional N $$ \mathcal{N} $$ = 2 SYM theory with gauge group SU(N) and matter in the symmetric plus anti-symmetric representation, which has a vanishing β-function. By mapping this computation to the matrix model arising from localization on a 4-sphere we are able to resum the perturbative expansion in the large-N ’t Hooft limit and derive the behavior of the correlators at strong coupling. Finally, by combining our results on the 3-point functions with those on the 2-point functions that have been recently found, we obtain the normalized 3-point coefficients of this conformal field theory at strong coupling and find that they depend in a simple way on the conformal dimensions of the single-trace operators.

Abstract We consider N $$ \mathcal{N} $$ = 2 superconformal quiver gauge theories in four dimensions and evaluate the chiral/anti-chiral correlators of single-trace operators. We show that it is convenient to form particular twisted and untwisted combinations of these operators suggested by the dual holographic description of the theory. The various twisted sectors are orthogonal and the correlators in each sector have always the same structure, as we show at the lowest orders in perturbation theory with Feynman diagrams. Using localization we then map the computation to a matrix model. In this way we are able to obtain formal expressions for the twisted correlators in the planar limit that are valid for all values of the ‘t Hooft coupling λ, and find that they are proportional to 1/λ at strong coupling. We successfully test the correctness of our extrapolation against a direct numerical evaluation of the matrix model and argue that the 1/λ behavior qualitatively agrees with the holographic description.

Abstract We consider the N $$ \mathcal{N} $$ = 2 SYM theory with gauge group SU(N) and a matter content consisting of one multiplet in the symmetric and one in the anti-symmetric representation. This conformal theory admits a large-N ’t Hooft expansion and is dual to a particular orientifold of AdS5 × S 5. We analyze this gauge theory relying on the matrix model provided by localization à la Pestun. Even though this matrix model has very non-trivial interactions, by exploiting the full Lie algebra approach to the matrix integration, we show that a large class of observables can be expressed in a closed form in terms of an infinite matrix depending on the ’t Hooft coupling λ. These exact expressions can be used to generate the perturbative expansions at high orders in a very efficient way, and also to study analytically the leading behavior at strong coupling. We successfully compare these predictions to a direct Monte Carlo numerical evaluation of the matrix integral and to the Padé resummations derived from very long perturbative series, that turn out to be extremely stable beyond the convergence disk |λ| < π 2 of the latter.

Abstract We study non-perturbative effects in supersymmetric U(N) gauge theories in eight dimensions realized by means of D(–1)/D7-brane systems with non-trivial world-volume fluxes turned on. Using an explicit string construction in terms of vertex operators, we derive the action for the open strings ending on the D(–1)-branes and exhibit its BRST structure. The space of vacua for these open strings is shown to be in correspondence with the moduli space of generalized ADHM gauge connections which trigger the non-perturbative corrections in the eight-dimensional theory. These corrections are computed via localization and turn out to depend on the curved background used to localize the integrals on the instanton moduli space, and vanish in flat space. Finally, we show that for specific choices of the background the instanton partition functions reduce to weighted sums of the solid partitions of the integers.

Abstract We consider a class of N $$ \mathcal{N} $$ = 2 conformal SU(N) SYM theories in four dimensions with matter in the fundamental, two-index symmetric and anti-symmetric representations, and study the corresponding matrix model provided by localization on a sphere S 4, which also encodes information on flat-space observables involving chiral operators and circular BPS Wilson loops. We review and improve known techniques for studying the matrix model in the large-N limit, deriving explicit expressions in perturbation theory for these observables. We exploit both recursive methods in the so-called full Lie algebra approach and the more standard Cartan sub-algebra approach based on the eigenvalue distribution. The sub-class of conformal theories for which the number of fundamental hypermultiplets does not scale with N differs in the planar limit from the N $$ \mathcal{N} $$ = 4 SYM theory only in observables involving chiral operators of odd dimension. In this case we are able to derive compact expressions which allow to push the small ’t Hooft coupling expansion to very high orders. We argue that the perturbative series have a finite radius of convergence and extrapolate them numerically to intermediate couplings. This is preliminary to an analytic investigation of the strong coupling behavior, which would be very interesting given that for such theories holographic duals have been proposed.

Abstract A generic half-BPS surface defect of N $$ \mathcal{N} $$ = 4 supersymmetric U(N ) Yang- Mills theory is described by a partition of N = n 1 + . . . + n M and a set of 4M continuous parameters. We show that such a defect can be realized by n I stacks of fractional D3-branes in Type II B string theory on a ℤ M orbifold background in which the brane world-volume is partially extended along the orbifold directions. In this set up we show that the 4M continuous parameters correspond to constant background values of certain twisted closed string scalars of the orbifold. These results extend and generalize what we have presented for the simple defects in a previous paper.

Abstract In conformal N $$ \mathcal{N} $$ = 2 Super Yang-Mills theory, the energy emitted by an accelerated heavy particle is computed by the one-point function of the stress tensor operator in the presence of a Wilson line. In this paper, we consider the theory on the ellipsoid and we prove a conjectured relation between the stress tensor one-point function and the first order expansion of the Wilson loop expectation value in the squashing parameter. To do this, we analyze the behavior of the Wilson loop for a small deformation of the background geometry and, at first order in the deformation, we fix the kinematics using defect CFT constraints. In the final part of the paper, we analyze the consequences of our results for the weak coupling perturbative expansion. In particular, comparing the weakly coupled matrix model with the ordinary Feynman diagram expansion, we find a natural transcendentality driven organization for the latter.

Abstract We consider the 1/2 BPS circular Wilson loop in a generic N $$ \mathcal{N} $$ = 2 SU(N) SYM theory with conformal matter content. We study its vacuum expectation value, both at finite N and in the large-N limit, using the interacting matrix model provided by localization results. We single out some families of theories for which the Wilson loop vacuum expectation values approaches the N $$ \mathcal{N} $$ = 4 result in the large-N limit, in agreement with the fact that they possess a simple holographic dual. At finite N and in the generic case, we explicitly compare the matrix model result with the field-theory perturbative expansion up to order g 8 for the terms proportional to the Riemann value ζ (5), finding perfect agreement. Organizing the Feynman diagrams as suggested by the structure of the matrix model turns out to be very convenient for this computation.

Abstract We consider conformal N = 2 $$ \mathcal{N}=2 $$ super Yang-Mills theories with gauge group SU(N) and N f = 2N fundamental hypermultiplets in presence of a circular 1/2-BPS Wilson loop. It is natural to conjecture that the matrix model which describes the expectation value of this system also encodes the one-point functions of chiral scalar operators in presence of the Wilson loop. We obtain evidence of this conjecture by successfully comparing, at finite N and at the two-loop order, the one-point functions computed in field theory with the vacuum expectation values of the corresponding normal-ordered operators in the matrix model. For the part of these expressions with transcendentality ζ(3), we also obtain results in the large-N limit that are exact in the ’t Hooft coupling λ.

Abstract We study half-BPS surface operators in supersymmetric gauge theories in four and five dimensions following two different approaches. In the first approach we analyze the chiral ring equations for certain quiver theories in two and three dimensions, coupled respectively to four- and five-dimensional gauge theories. The chiral ring equations, which arise from extremizing a twisted chiral superpotential, are solved as power series in the infrared scales of the quiver theories. In the second approach we use equivariant localization and obtain the twisted chiral superpotential as a function of the Coulomb moduli of the four- and five-dimensional gauge theories, and find a perfect match with the results obtained from the chiral ring equations. In the five-dimensional case this match is achieved after solving a number of subtleties in the localization formulas which amounts to choosing a particular residue prescription in the integrals that yield the Nekrasov-like partition functions for ramified instantons. We also comment on the necessity of including Chern-Simons terms in order to match the superpotentials obtained from dual quiver descriptions of a given surface operator.

Abstract We calculate the instanton partition function of the four-dimensional N = 2 ⋆ $$ \mathcal{N}={2}^{\star } $$ SU(N) gauge theory in the presence of a generic surface operator, using equivariant localization. By analyzing the constraints that arise from S-duality, we show that the effective twisted superpotential, which governs the infrared dynamics of the two-dimensional theory on the surface operator, satisfies a modular anomaly equation. Exploiting the localization results, we solve this equation in terms of elliptic and quasi-modular forms which resum all non-perturbative corrections. We also show that our results, derived for monodromy defects in the four-dimensional theory, match the effective twisted superpotential describing the infrared properties of certain two-dimensional sigma models coupled either to pure N = 2 $$ \mathcal{N}=2 $$ or to N = 2 ⋆ $$ \mathcal{N}={2}^{\star } $$ gauge theories.