New results on superconformal quivers

TL;DR

The work shows that every superconformal quiver satisfies $c=a$ and systematically classifies completely chiral quivers into three- and four-block families. It introduces shrinking (and orbifolding) as a constructive method to generate new rank-2 quivers from known del Pezzo quivers, revealing that all 3-block and many 4-block models originate from del Pezzo quivers or their shrunk variants. The classification reduces to solving diophantine equations that fix the allowed quiver data and $R$-charges, with a finite set of models (16 three-block cases, plus shrunk-by-four-block cases) and a consistent prediction that rank-2 chiral quivers are bounded by the del Pezzo quiver with 11 nodes. The results reinforce the geometric engineering viewpoint, suggesting a broad correspondence between quiver data and CY singularities, and hint at a string-theoretic interpretation of shrinking via fluxes or discrete torsion. Overall, the paper provides a coherent framework linking central charges, RG structure, dualities, and geometric realizations of a wide class of four-dimensional superconformal quivers.

Abstract

All superconformal quivers are shown to satisfy the relation c = a and are thus good candidates for being the field theory living on D3 branes probing CY singularities. We systematically study 3 block and 4 block chiral quivers which admit a superconformal fixed point of the RG equation. Most of these theories are known to arise as living on D3 branes at a singular CY manifold, namely complex cones over del Pezzo surfaces. In the process we find a procedure of getting a new superconformal quiver from a known one. This procedure is termed "shrinking" and, in the 3 block case, leads to the discovery of two new models. Thus, the number of superconformal 3 block quivers is 16 rather than the previously known 14. We prove that this list exausts all the possibilities. We suggest that all rank 2 chiral quivers are either del Pezzo quivers or can be obtained by shrinking a del Pezzo quiver and verify this statement for all 4 block quivers, where a lot of "shrunk'' del Pezzo models exist.

Figures (5)

• Figure 1: One of the quivers associated to $dP_0 \equiv \mathbb{CP}^2$. This is the only root of its associated Duality Tree, also called the Markov Tree.
• Figure 2: General 3-block chiral quiver diagram. $\alpha, \beta, \gamma$ are the number of nodes in each block. $a, b, c$ are the number of bifundamental connecting the nodes.
• Figure 3: 4-block chiral quiver diagram.
• Figure 4: Intersection between the solution to equation $X + Y + Z = \sqrt{X Y Z}$ and the region $X \leq Y \leq Z \leq X + Y$ for a fixed value of $X$. In this figure $X = 5$.
• Figure 5: Effect of the increase of the value of $X$. There is a maximum value of $X$ over which the solution to $X + Y + Z = \sqrt{X Y Z}$ doesn't pass over the region containing the minimal solutions.