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Register a new userUniversality of Toda equation in ${cal N}=2$ superconformal field theories
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We show that extremal correlators in all Lagrangian ${cal N}=2$ superconformal field theories with a simple gauge group are governed by the same universal Toda system of equations, which dictates the structure of extremal correlators to all orders in the perturbation series. A key point is the construction of a convenient orthogonal basis for the chiral ring, by arranging towers of operators in order of increasing dimension, which has the property that the associated two-point functions satisfy decoupled Toda chain equations. We explicitly verify this in all known SCFTs based on $mathrm{SU}(N)$ gauge groups as well as in superconformal QCD based on orthogonal and symplectic groups. As a by-product, we find a surprising non-renormalization property for the ${cal N}=2$ $mathrm{SU}(N)$ SCFT with one hypermultiplet in the rank-2 symmetric representation and one hypermultiplet in the rank-2 antisymmetric representation, where the two-loop terms of a large class of supersymmetric observables identically vanish.
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We study, using ADHM construction, instanton effects in an ${CN}=2$ superconformal $Sp(N)$ gauge theory, arising as effective field theory on a system of $N$ D-3-branes near an orientifold 7-plane and 8 D-7-branes in type I string theory. We work out the measure for the collective coordinates of multi-instantons in the gauge theory and compare with the measure for the collective coordinates of $(-1)$-branes in the presence of 3- and 7-branes in type I theory. We analyse the large-N limit of the measure and find that it admits two classes of saddle points: In the first class the space of collective coordinates has the geometry of $AdS_5times S^3$ which on the string theory side has the interpretation of the D-instantons being stuck on the 7-branes and therefore the resulting moduli space being $AdS_5times S^3$, In the second class the geometry is $AdS_5times S^5/Z_2$ and on the string theory side it means that the D-instantons are free to move in the 10-dimensional bulk. We discuss in detail a correlator of four O(8) flavour currents on the Yang-Mills side, which receives contributions from the first type of saddle points only, and show that it matches with the correlator obtained from $F^4$ coupling on the string theory side, which receives contribution from D-instantons, in perfect accord with the AdS/CFT correspondence. In particular we observe that the sectors with odd number of instantons give contribution to an O(8)-odd invariant coupling, thereby breaking O(8) down to SO(8) in type I string theory. We finally discuss correlators related to $R^4$, which receive contributions from both saddle points.
Aharony, Bergman, Jafferis and Maldacena have recently proposed a dual gravitational description for a family of superconformal Chern Simons theories in three spacetime dimensions. In this note we perform the one loop computation that determines the field theory superconformal index of this theory and compare with the index computed over the Fock space of dual supersymmetric gravitons. In the appropriate limit (large $N$ and large $k$) we find a perfect match.
We study the four-dimensional N=2 superconformal field theories that describe D3-branes probing the recently constructed N=2 S-folds in F-theory. We introduce a novel, infinite class of superconformal field theories related to S-fold theories via partial Higgsing. We determine several properties of both the S-fold models and this new class of theories, including their central charges, Coulomb branch spectrum, and moduli spaces of vacua, by bringing to bear an array of field-theoretical techniques, to wit, torus-compactifications of six-dimensional N=(1,0) theories, class S technology, and the SCFT/VOA correspondence.
We present a general method for computing the central charges a and c of N=2 superconformal field theories corresponding to singular points in the moduli space of N=2 gauge theories. Our method relates a and c to the U(1)_R anomalies of the topologically twisted gauge theory. We evaluate these anomalies by studying the holomorphic dependence of the path integral measure on the moduli. We calculate a and c for superconformal points in a variety of gauge theories, including N=4 SU(N), N=2 pure SU(N) Yang-Mills, and USp(2N) with 1 massless antisymmetric and 4 massive fundamental hypermultiplets. In the latter case, we reproduce the conformal and flavor central charges previously calculated using the gravity duals of these gauge theories. For any SCFT in the class under consideration, we derive a previously conjectured expression for 2a-c in terms of the sum of the dimensions of operators parameterizing the Coulomb branch. Finally, we prove that the ratio a/c is bounded above by 5/4 and below by 1/2.
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.
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