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Register a new userThe planar limit of ${cal N}=2$ chiral correlators
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We derive the planar limit of 2- and 3-point functions of single-trace chiral primary operators of ${cal N}=2$ SQCD on $S^4$, to all orders in the t Hooft coupling. In order to do so, we first obtain a combinatorial expression for the planar free energy of a hermitian matrix model with an infinite number of arbitrary single and double trace terms in the potential; this solution might have applications in many other contexts. We then use these results to evaluate the analogous planar correlation functions on ${mathbb R}^4$. Specifically, we compute all the terms with a single value of the $zeta$ function for a few planar 2- and 3-point functions, and conjecture general formulas for these terms for all 2- and 3-point functions on ${mathbb R}^4$.
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We consider a family of $mathcal{N}=2$ superconformal field theories in four dimensions, defined as $mathbb{Z}_q$ orbifolds of $mathcal{N}=4$ Super Yang-Mills theory. We compute the chiral/anti-chiral correlation functions at a perturbative level, using both the matrix model approach arising from supersymmetric localisation on the four-sphere and explicit field theory calculations on the flat space using the $mathcal{N}=1$ superspace formalism. We implement a highly efficient algorithm to produce a large number of results for finite values of $N$, exploiting the symmetries of the quiver to reduce the complexity of the mixing between the operators. Finally the interplay with the field theory calculations allows to isolate special observables which deviate from $mathcal{N}=4$ only at high orders in perturbation theory.
We obtain the perturbative expansion of the free energy on $S^4$ for four dimensional Lagrangian ${cal N}=2$ superconformal field theories, to all orders in the t Hooft coupling, in the planar limit. We do so by using supersymmetric localization, after rewriting the 1-loop factor as an effective action involving an infinite number of single and double trace terms. The answer we obtain is purely combinatorial, and involves a sum over tree graphs. We also apply these methods to the perturbative expansion of the free energy at finite $N$, and to the computation of the vacuum expectation value of the 1/2 BPS circular Wilson loop, which in the planar limit involves a sum over rooted tree graphs.
We compute the planar limit of both the free energy and the expectation value of the $1/2$ BPS Wilson loop for four dimensional ${cal N}=2$ superconformal quiver theories, with a product of SU($N$)s as gauge group and bi-fundamental matter. Supersymmetric localization reduces the problem to a multi-matrix model, that we rewrite in the zero-instanton sector as an effective action involving an infinite number of double-trace terms, determined by the relevant extended Cartan matrix. We find that the results, as in the case of $mathcal{N}=2$ SCFTs with a simple gauge group, can be written as sums over tree graphs. For the $widehat{A_1}$ case, we find that the contribution of each tree can be interpreted as the partition function of a generalized Ising model defined on the tree; we conjecture that the partition functions of these models defined on trees satisfy the Lee-Yang property, i.e. all their zeros lie on the unit circle.
We study the theory of a single fundamental fermion and boson coupled to Chern-Simons theory at leading order in the large $N$ limit. Utilizing recent progress in understanding the Higgsed phase in Chern-Simons-Matter theories, we compute the quantum effective potential that is exact to all orders in the t Hooft coupling for the lightest scalar operator of this theory at finite temperature. Specializing to the zero temperature limit we use this potential to determine the phase diagram of the large $N$ ${cal N}=2$ supersymmetric theory with this field content. This intricate two dimensional phase diagram has four topological phases that are separated by lines of first and second order phase transitions and includes special conformal points at which the infrared dynamics is governed by Chern-Simons theory coupled respectively to free bosons, Gross-Neveu fermions, and to a theory of Wilson-Fisher bosons plus free fermions. We also describe the vacuum structure of the most general ${cal N} = 1$ supersymmetric theory with one fundamental boson and one fundamental fermion coupled to an $SU(N)$ Chern-Simons gauge field, at arbitrary values of the t Hooft coupling.
We consider a new large-N limit, in which the t Hooft coupling grows with N. We argue that a class of large-N equivalences, which is known to hold in the t Hooft limit, can be extended to this very strongly coupled limit. Hence this limit may lead to a new way of studying corrections to the t Hooft limit, while keeping nice properties of the latter. As a concrete example, we describe large-N equivalences between the ABJM theory and its orientifold projection. The equivalence implies that operators neutral under the projection symmetry have the same correlation functions in two theories at large-N. Usual field theory arguments are valid when t Hooft coupling $lambdasim N/k$ is fixed and observables can be computed by using a planar diagrammatic expansion. With the help of the AdS/CFT correspondence, we argue that the equivalence extends to stronger coupling regions, $Ngg k$, including the M-theory region $Ngg k^5$. We further argue that the orbifold/orientifold equivalences between certain Yang-Mills theories can also be generalized. Such equivalences can be tested both analytically and numerically. Based on calculations of the free energy, we conjecture that the equivalences hold because planar dominance persists beyond the t Hooft limit.
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