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Register a new userA new large-N limit and the planar equivalence outside the planar limit
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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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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 ${mathbb R}^4$.
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.
Noncompact SO(1,N) sigma-models are studied in terms of their large N expansion in a lattice formulation in dimensions d geq 2. Explicit results for the spin and current two-point functions as well as for the Binder cumulant are presented to next to leading order on a finite lattice. The dynamically generated gap is negative and serves as a coupling-dependent infrared regulator which vanishes in the limit of infinite lattice size. The cancellation of infrared divergences in invariant correlation functions in this limit is nontrivial and is in d=2 demonstrated by explicit computation for the above quantities. For the Binder cumulant the thermodynamic limit is finite and is given by 2/(N+1) in the order considered. Monte Carlo simulations suggest that the remainder is small or zero. The potential implications for ``criticality and ``triviality of the theories in the SO(1,N) invariant sector are discussed.
Minimal string theory has a number of FZZT brane boundary states; one for each Cardy state of the minimal model. It was conjectured by Seiberg and Shih that all branes in a minimal string theory could be expressed as a linear combination of the brane associated to the identity operator of the minimal model with complex shifts in the boundary cosmological constant. Subsequently it was found that this identification of FZZT branes does not hold exactly for some cylinder amplitudes but was spoiled by terms that are associated with vanishing worldsheet area and are therefore non-universal. In this paper we investigate this claim systematically, using both Liouville and matrix model methods, beyond the planar limit. We find that the aforementioned identification of FZZT branes is spoiled by terms that do not admit an interpretation as non-universal terms. Furthermore, the spoiling terms as computed using the matrix model are found to be in agreement with those coming from Liouville theory, which also suggests that these terms have universal meaning. Finally, we also investigate the identification of FZZT branes by replacing the boundary state with a sum of local operators. We find in this case that the brane associated with the identity operator appears to be special as it is the only one to correctly reproduce the correlation numbers for bulk operators on the torus.
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