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Register a new userClassification of N=2 Superconformal Field Theories with Two-Dimensional Coulomb Branches
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We study the classification of 2-dimensional scale-invariant rigid special Kahler (RSK) geometries, which potentially describe the Coulomb branches of N=2 supersymmetric field theories in four dimensions. We show that this classification is equivalent to the solution of a set of polynomial equations by using an integrability condition for the central charge, scale invariance, constraints coming from demanding single-valuedness of physical quantities on the Coulomb branch, and properties of massless BPS states at singularities. We find solutions corresponding to lagrangian scale invariant theories--including the scale invariant G_2 theory not found before in the literature--as well as many new isolated solutions (having no marginal deformations). All our scale-invariant RSK geometries are consistent with an interpretation as effective theories of N=2 superconformal field theories, and, where we can check, turn out to exist as quantum field theories.
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We continue the classification of 2-dimensional scale-invariant rigid special Kahler (RSK) geometries. This classification was begun in [hep-th/0504070] where singularities corresponding to curves of the form y^2=x^6 with a fixed canonical basis of holomorphic one forms were analyzed. Here we perform the analysis for the y^2=x^5 type singularities. (The final maximal singularity type, y^2=x^3(x-1)^3, will be analyzed in a later paper.) These singularities potentially describe the Coulomb branches of N=2 supersymmetric field theories in four dimensions. We show that there are only 13 solutions satisfying the integrability condition (enforcing the RSK geometry of the Coulomb branch) and the Z-consistency condition (requiring massless charged states at singularities). Of these solutions, one has a marginal deformation, and corresponds to the known solution for certain Sp(2) gauge theories, while the rest correspond to isolated strongly interacting conformal field theories.
Under reasonable assumptions about the complex structure of the set of singularities on the Coulomb branch of $mathcal N=2$ superconformal field theories, we present a relatively simple and elementary argument showing that the scaling dimension, $Delta$, of a Coulomb branch operator of a rank $r$ theory is allowed to take values in a finite set of rational numbers$Deltain big[frac{n}{m}big|n,minmathbb N, 0<mle n, gcd(n,m)=1, varphi(n)le2rbig]$ where $varphi(n)$ is the Euler totient function. The maximal dimension grows superlinearly with rank as $Delta_text{max} sim r lnln r$. This agrees with the recent result of Caorsi and Cecotti.
Turning on N=2 supersymmetry-preserving relevant operators in a 4-dimensional N=2 superconformal field theory (SCFT) corresponds to a complex deformation compatible with the rigid special Kahler geometry encoded in the low energy effective action. Field theoretic consistency arguments indicate that there should be many distinct such relevant deformations of each SCFT fixed point. Some new supersymmetry-preserving complex deformations are constructed of isolated rank 1 SCFTs. We also make predictions for the dimensions of certain Higgs branches for some rank 1 SCFTs.
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 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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