Macdonald Index and Chiral Algebra

For any 4d N=2 SCFT, there is a subsector described by a 2d chiral algebra. The vacuum character of the chiral algebra reproduces the Schur index of the corresponding 4d theory. The Macdonald index counts the same set of operators as the Schur index, but the former has one more fugacity than the latter. We conjecture a prescription to obtain the Macdonald index from the chiral algebra. The vacuum module admits a filtration, from which we construct an associated graded vector space. From this grading, we conjecture a notion of refined character for the vacuum module of a chiral algebra, which reproduces the Macdonald index. We test this prescription for the Argyres-Douglas theories of type $(A_1, A_{2n})$ and $(A_1, D_{2n+1})$ where the chiral algebras are given by Virasoro and su(2) affine Kac-Moody algebra. When the chiral algebra has more than one family of generators, our prescription requires a knowledge of the generators from the 4d.Comment: 25 pages, v2: major revision. Clarified the prescription to get the Macdonald grading; v3: corrected hyperlinks to the references. To appear in JHE

N=1 Deformations and RG Flows of N=2 SCFTs

We study certain N=1 preserving deformations of four-dimensional N=2 superconformal field theories (SCFTs) with non-abelian flavor symmetry. The deformation is described by adding an N=1 chiral multiplet transforming in the adjoint representation of the flavor symmetry with a superpotential coupling, and giving a nilpotent vacuum expectation value to the chiral multiplet which breaks the flavor symmetry. This triggers a renormalization group flow to an infrared SCFT. Remarkably, we find classes of theories flow to enhanced N=2 supersymmetric fixed points in the infrared under the deformation. They include generalized Argyres-Douglas theories and rank-one SCFTs with non-abelian flavor symmetries. Most notably, we find renormalization group flows from the deformed conformal SQCDs to the $(A_1, A_n)$ Argyres-Douglas theories. From these "Lagrangian descriptions," we compute the full superconformal indices of the $(A_1, A_n)$ theories and find agreements with the previous results. Furthermore, we study the cases, including the $T_N$ and $R_{0,N}$ theories of class $\mathcal{S}$ and some of rank-one SCFTs, where the deformation gives genuine N=1 fixed points.Comment: 45 pages, v2: added a paragraph on SUSY enhancement from the index. Minor corrections. To appear in JHE

New N=1 Dualities from M5-branes and Outer-automorphism Twists

We generalize recent construction of four-dimensional $\mathcal{N}=1$ SCFT from wrapping six-dimensional $\mathcal{N}=(2,0)$ theory on a Riemann surface to the case of $D$-type with outer-automorphism twists. This construction allows us to build various dual theories for a class of $\mathcal{N}=1$ quiver theories of $SO-USp$ type. In particular, we find there are five dual frames to $SO(2N)/USp(2N-2)/G_2$ gauge theories with $(4N-4)/4N/8$ fundamental flavors, where three of them are non-Lagrangian. We check the dualities by computing the anomaly coefficients and the superconformal indices. In the process we verify that the index of $D_4$ theory on a certain three punctured sphere with $Z_2$ and $Z_3$ twist lines exhibits the expected symmetry enhancement from $G_2 \times USp(6)$ to $E_7$.Comment: 56 pages, 29 colored figures; v2: minor corrections, references adde

Four-dimensional Lens Space Index from Two-dimensional Chiral Algebra

We study the supersymmetric partition function on $S^1 \times L(r, 1)$, or the lens space index of four-dimensional $\mathcal{N}=2$ superconformal field theories and their connection to two-dimensional chiral algebras. We primarily focus on free theories as well as Argyres-Douglas theories of type $(A_1, A_k)$ and $(A_1, D_k)$. We observe that in specific limits, the lens space index is reproduced in terms of the (refined) character of an appropriately twisted module of the associated two-dimensional chiral algebra or a generalized vertex operator algebra. The particular twisted module is determined by the choice of discrete holonomies for the flavor symmetry in four-dimensions.Comment: 47 pages; v2: minor corrections, published versio

The Seiberg-Witten Kahler Potential as a Two-Sphere Partition Function

Recently it has been shown that the two-sphere partition function of a gauged linear sigma model of a Calabi-Yau manifold yields the exact quantum Kahler potential of the Kahler moduli space of that manifold. Since four-dimensional N=2 gauge theories can be engineered by non-compact Calabi-Yau threefolds, this implies that it is possible to obtain exact gauge theory Kahler potentials from two-sphere partition functions. In this paper, we demonstrate that the Seiberg-Witten Kahler potential can indeed be obtained as a two-sphere partition function. To be precise, we extract the quantum Kahler metric of 4D N=2 SU(2) Super-Yang-Mills theory by taking the field theory limit of the Kahler parameters of the O(-2,-2) bundle over P1 x P1. We expect this method of computing the Kahler potential to generalize to other four-dimensional N=2 gauge theories that can be geometrically engineered by toric Calabi-Yau threefolds.Comment: 12 pages + appendix; v2: minor corrections, reference adde