W_3 irregular states and isolated N=2 superconformal field theories

TL;DR

The paper extends the AGT correspondence to irregular punctures for the ${\mathcal{W}}_{3}$ algebra by constructing irregular states via colliding a full SU(3) puncture with $n$ simple punctures on a punctured Riemann surface. These states become simultaneous eigenvectors of the higher modes $L_n$ and $W_k$ (specifically $L_n,\dots,L_{2n}$ and $W_{2n},\dots,W_{3n}$) and yield Seiberg–Witten data for isolated ${\mathcal{N}}=2$ SCFTs with $SU(3)$ flavor symmetry, realized as IR fixed points on SU(3) linear quivers. The analysis systematically builds irregular states for Virasoro and ${\mathcal{W}}_{3}$ algebras, derives their Ward-identity constraints, and links them to corresponding SW curves, BPS quivers, and Hitchin-system perspectives, while identifying normalization ambiguities and limitations in determining lower non-negative modes. The results reveal a landscape of ${\mathcal{W}}(A_{1},C_{0,1,\{n+1\}})$ and ${\mathcal{W}}(A_{2},C_{0,1,\{n\}})$ AD-type theories, enriching the dictionary between 2d irregular conformal blocks and 4d isolated SCFTs with concrete SW data and quiver realizations.

Abstract

We explore the proposal that the six-dimensional (2,0) theory on the Riemann surface with irregular punctures leads to a four-dimensional gauge theory coupled to the isolated N=2 superconformal theories of Argyres-Douglas type, and to two-dimensional conformal field theory with irregular states. Following the approach of Gaiotto-Teschner for the Virasoro case, we construct W_3 irregular states by colliding a single SU(3) puncture with several regular punctures of simple type. If n simple punctures are colliding with the SU(3) puncture, the resulting irregular state is a simultaneous eigenvector of the positive modes L_n, ..., L_{2n} and W_{2n}, ..., W_{3n} of the W_3 algebra. We find the corresponding isolated SCFT with an SU(3) flavor symmetry as a nontrivial IR fixed point on the Coulomb branch of the SU(3) linear quiver gauge theories, by confirming that its Seiberg-Witten curve correctly predicts the conditions for the W_3 irregular states. We also compare these SCFT's with the ones obtained from the BPS quiver method.

Figures (1)

• Figure 1: The collision of vertex operators $V_1(z_1),\cdots{\:\!\!},V_{n+1}(z_{n+1})$.