N=1 theories of class S_k

TL;DR

The paper proposes ${\\cal S}_k$, a broad family of ${\\cal N}=1$ SCFTs obtained from twisted compactifications of 6d ${\\cal T}^N_k$ on punctured Riemann surfaces, with the surface degenerations mapping to weakly coupled limits and an index-based 2d TFT structure encoding their data.A ${\\mathbb Z}_k$ orbifold description of ${\\cal N}=2$ class S quivers is developed, yielding a 5d necklace quiver ${\\cal N}_{N,k}$ and linking the 6d origin to 4d ${\\cal N}=1$ cores through circle compactification and brane engineering.Building blocks include a gluing prescription for tubes, a free trinion as a basic three-punctured sphere, and an index avatar that expresses the theory as a 2d TFT of eigenfunctions of novel elliptic-difference operators, enabling computation of indices for general ${\\cal S}_k$ theories.The work analyzes RG flows closing minimal and maximal punctures, showing how discrete curvature data for the 6d global symmetries enters and how new strongly coupled SCFTs arise; these flows are dual to quiver tails and Argyres-Seiberg-like frames.A rich 5d/4d dictionary is developed, including surface defects defined by position-dependent vevs, six-dimensional boundary conditions via orbifold Nahm poles, and a web of dualities and flux correspondences that illuminate the structure of class ${\\cal S}_k$ theories and their holographic prospects.

Abstract

We construct classes of ${\cal N}=1$ 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} ${\cal N}=2$ 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 ${\cal N}=1$ theories to be complete in an appropriate sense we find it necessary to conjecture existence of new ${\cal N}=1$ 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.

Figures (24)

• Figure 1: The brane configuration which engineers the core ${\cal N}=1$ gauge theories. We draw a setup with $N=4$ D4 branes (or $kN=16$ fractional D4 branes) sitting at the locus of a $\mathbb Z_4$ singularity, intersected by 5 NS5 branes. The D4 branes in each segment between consecutive NS5 branes engineer the four necklaces of $SU(N)$ gauge groups. The 4-4 strings across the NS5 branes engineer the five sets of zig-zag chiral multiplets between the necklaces. The semi-infinite D4 branes can be either associated to the two necklaces of flavor groups or to five-dimensional necklace gauge theories coupled to the four-dimensional quiver theory.
• Figure 2: The necklace quiver ${\cal N}_{N,k}$ in five dimensions. It is a $\mathbb Z_k$ orbifold of the maximally supersymmetric YM, with $k$$SU(N)$ nodes. Notice that each link here represents a full bi-fundamental hypermultiplet and each node a full $SU(N)$ vector multiplet. There is $U(1)$ global symmetry associated with each link, rotating the bifundamental hypermultiplets. There is also a $U(1)$ symmetry, the instanton symmetry, associated with each gauge group node.
• Figure 3: The "honeycomb" bi-partite graph drawn on a cylinder and the corresponding quiver gauge theory. The top and bottom lines are identified. Each cell of the bi-partite graph maps to an $SU(N)$ gauge group with $3N$ flavors corresponding to the six edges of the cell. Each node of the bi-partite graph indicates a cubic superpotential term, with sign associated to the color of the node
• Figure 4: The six-dimensional lift of the brane configuration which engineers the core ${\cal N}=1$ gauge theories. We have $N$ M5 branes sitting at the locus of a $\mathbb Z_k$ singularity and wrapping a cylinder, intersected by 5 transverse M5 branes. The world volume theory of the $N$ M5 branes is the ${\cal T}^N_k$$(1,0)$ SCFT, with $5$ "minimal" defects (left). We expect that the de-coupling the four-dimensional degrees of freedom can be implemented by replacing the semi-infinite ends of the cylinder by maximal punctures on a sphere (right).
• Figure 5: A simple set of generators for the Abelian global symmetries of the core theories. Each arrow represents a generator, acting on the chiral multiplets crossed by the arrow with charge $\pm 1$, depending on the sign of the crossing. The sum of all generators is $0$. The vertical (red) arrows are chosen to generate the $U(1)_{\alpha}$ global symmetries associated to minimal punctures. The SE pointing (green) arrows, quotiented by their diagonal, generate the $\left[\frac{U(1)^k}{U(1)}\right]_\beta$ "intrinsic" symmetries. The SW pointing (blue) arrows, quotiented by their diagonal, generate the $\left[\frac{U(1)^k}{U(1)}\right]_\gamma$ "intrinsic" symmetries. Finally, the anti-diagonal combination of blue and green arrow give the generator of the $U(1)_t$ intrinsic symmetry.