N = 1 $$ \mathcal{N}=1 $$ theories of class S k $$ {\mathcal{S}}_k $$

Abstract

We construct classes of N = 1 superconformal theories elements of which are labeled by punctured Riemann surfaces. Degenerations of the surfaces correspond, in some cases, to weak coupling limits. Different classes are labeled by two integers (N, k). The k = 1 case coincides with A N −1 N = 2 theories of class S and simple examples of theories with k > 1 are Z k orbifolds of some of the A N −1 class S theories. For the space of N = 1 theories to be complete in an appropriate sense we find it necessary to conjecture existence of new N = 1 strongly coupled SCFTs. These SCFTs when coupled to additional matter can be related by dualities to gauge theories. We discuss in detail the A 1 case with k = 2 using the supersymmetric index as our analysis tool. The index of theories in classes with k > 1 can be constructed using eigenfunctions of elliptic quantum mechanical models generalizing the Ruijsenaars-Schneider integrable model. When the elliptic curve of the model degenerates these eigenfunctions become polynomials with coefficients being algebraic expressions in fugacities, generalizing the Macdonald polynomials with rational coefficients appearing when k = 1.

E‐String Theory on Riemann Surfaces

“…One can note that the index forms characters of a larger global symmetry SO (12), which cannot be realized from the 6d construction. However, the 6d picture suggests that besides SO (11) one can also have SU(2) × SO (9) and USp(4) × SO (7) as global symmetries at special points on the conformal manifold.…”

“…One can engineer these models as follows. [8,12] We start with a compactification of two M5 branes probing Z 2 singularity and put these on a sphere with two maximal, having SU(2) 2 symmetry, and two minimal, having U(1) symmetry, punctures. These four punctured sphere has a Lagrangian description as an SU(2) 2 gauge theory which happens to be identical to Figure 3 but without the flip fields.…”

“…[55] The derivation of the Lagrangian is similar to the derivation of Lagrangian for the E 6 MN model. [56] There is one theory in this list of theories which stands out, and it was denoted T A in [8] (see also [12]). The construction of the T A model is described in great detail in [8] and we refer the reader there for all the properties of this model.…”

“…[58][59][60] It will be interesting to understand the details of such a TFT in our case and its relation to surface defects [61] in the four dimensional models and to integrable models (for example the eight parameter elliptic relativistic generalization of the Heun equation [62][63][64] ). 12 …”

D-brane instantons as gauge instantons in orientifolds of chiral quiver theories

“…In addition, these theories have also appeared recently in the context of class S k theories in [40][41][42]. 6 Even restricting to D-branes at toric singularities, it is possible to have multiple USp nodes, and also…”