OPE Selection Rules for Schur Multiplets in 4D N=2\mathcal{N}=2 Superconformal Field Theories

We compute general expressions for two types of three-point functions of (semi-)short multiplets in four-dimensional $\mathcal{N}=2$ superconformal field theories. These (semi-)short multiplets are called "Schur multiplets" and play an important role in the study of associated chiral algebras. The first type of the three-point functions we compute involves two half-BPS Schur multiplets and an arbitrary Schur multiplet, while the second type involves one stress tensor multiplet and two arbitrary Schur multiplets. From these three-point functions, we read off the corresponding OPE selection rules for the Schur multiplets. Our results particularly imply that there are non-trivial selection rules on the quantum numbers of Schur operators in these multiplets. We also give a conjecture on the selection rules for general Schur multiplets.Comment: 39 page

On the chiral algebra of Argyres-Douglas theories and S-duality

We study the two-dimensional chiral algebra associated with the simplest Argyres-Douglas type theory with an exactly marginal coupling, i.e., the $(A_3,A_3)$ theory. Near a cusp in the space of the exactly marginal deformations (i.e., the conformal manifold), the theory is well-described by the $SU(2)$ gauge theory coupled to isolated Argyres-Douglas theories and a fundamental hypermultiplet. In this sense, the $(A_3,A_3)$ theory is an Argyres-Douglas version of the $\mathcal{N}=2$ $SU(2)$ conformal QCD. By studying its Higgs branch and Schur index, we identify the minimal possible set of chiral algebra generators for the $(A_3,A_3)$ theory, and show that there is a unique set of closed OPEs among these generators. The resulting OPEs are consistent with the Schur index, Higgs branch chiral ring relations, and the BRST cohomology conjecture. We then show that the automorphism group of the chiral algebra we constructed contains a discrete group $G$ with an $S_3$ subgroup and a homomorphism $G\to S_4 \times {\bf Z}_2$. This result is consistent with the S-duality of the $(A_3,A_3)$ theory.Comment: 25 pages, 2 figure

Two-dimensional crystal melting and D4-D2-D0 on toric Calabi-Yau singularities

We construct a two-dimensional crystal melting model which reproduces the BPS index of D2-D0 states bound to a non-compact D4-brane on an arbitrary toric Calabi-Yau singularity. The crystalline structure depends on the toric divisor wrapped by the D4-brane. The molten crystals are in one-to-one correspondence with the torus fixed points of the moduli space of the quiver gauge theory on D-branes. The F- and D-term constraints of the gauge theory are regarded as a generalization of the ADHM constraints on instantons. We also show in several examples that our model is consistent with the wall-crossing formula for the BPS index.Comment: 72 pages, 44 figure

Topological strings and 5d T_N partition functions

We evaluate the Nekrasov partition function of 5d gauge theories engineered by webs of 5-branes, using the refined topological vertex on the dual Calabi-Yau threefolds. The theories include certain non-Lagrangian theories such as the T_N theory. The refined topological vertex computation generically contains contributions from decoupled M2-branes which are not charged under the 5d gauge symmetry engineered. We argue that, after eliminating them, the refined topological string partition function agrees with the 5d Nekrasov partition function. We explicitly check this for the T_3 theory as well as Sp(1) gauge theories with N_f = 2, 3, 4 flavors. In particular, our method leads to a new expression of the Sp(1) Nekrasov partition functions without any contour integrals. We also develop prescriptions to calculate the partition functions of theories obtained by Higgsing the T_N theory. We compute the partition function of the E_7 theory via this prescription, and find the E_7 global symmetry enhancement. We finally discuss a potential application of the refined topological vertex to non-toric web diagrams.Comment: 79 pages, 27 figures; v2: minor improvements, references adde