Towards a 4d/2d correspondence for Sicilian quivers

TL;DR

This work extends the 4d/2d AGT correspondence to Sicilian ${SU(2)}$ quivers by formulating a CFT description for the elementary trifundamental half-hypermultiplet as a three-descendant three-point block of Liouville/Gaiotto states. It develops and applies a direct ${ m Sp}(1)/{ m SO}(4)$ instanton-counting framework for half-hypermultiplets, resolves UV/IR coupling mappings, and verifies the proposed correspondence in several examples including genus-two quivers, with consistency checks against decoupling limits and known bifundamental reductions. The results establish a concrete dictionary between Sicilian 4d theories and 2d conformal blocks, while highlighting subtleties in coordinate choices and spurious factors that arise in conformal-block realizations. The work opens avenues for geometric engineering and refined-vertex approaches to fully realize the SICILIAN AGT program and its operator content.

Abstract

We study the 4d/2d AGT correspondence between four-dimensional instanton counting and two-dimensional conformal blocks for generalized SU(2) quiver gauge theories coming from punctured Gaiotto curves of arbitrary genus. We propose a conformal block description that corresponds to the elementary SU(2) trifundamental half-hypermultiplet, and check it against Sp(1)-SO(4) instanton counting.

Figures (11)

• Figure 1: The double covering of the the $U(2)$ Gaiotto curve over the $Sp(1)$ Gaiotto curve relates the microscopic coupling $q_{U(2)}$ (which is the cross-ratio of the $U(2)$ Gaiotto curve) to the microscopic coupling $q_{Sp(1)} = 4q$. Note that, whereas a $U(2)$ gauge group is represented by a tube, an $Sp(1)$ gauge group is represented geometrically by a tube with a twist-line. In the double covering this twist-line gets the interpretation of a branch-cut.
• Figure 2: The Gaiotto curve for the $SU(2)$ trifundamental as a double cover over the $Sp(1)-SO(4)$ Gaiotto curve (on the left) and the non-existing $Sp(1)^3$ Gaiotto curve (on the right). On the right is illustrated why we cannot find a Gaiotto curve corresponding to the $Sp(1)$ trifundamental: it is not possible to close the three twist-lines (or branch-cuts) on the $Sp(1)$ curve.
• Figure 3: On the left is illustrated the Gaiotto curve for the conformal quiver gauge theory with three $SU(2)$ gauge groups that are all coupled by a trifundamental interaction and each individually to two massive fundamental hypermultiplets. In the decoupling limit where we take all masses $m_i$ of the fundamental hypermultiplets to infinity, we are left with the Gaiotto curve corresponding to the asymptotically free quiver gauge theory that couples the three $SU(2)$ gauge groups by a trifundamental half-hypermultiplet.
• Figure 4: The solutions to the Dirac equation in a given representation of the gauge group form a vector bundle ${\cal V}$ over the ADHM moduli space ${\cal M}$. A pseudo-real representation induces a real structure $\tau$ on the vector bundle ${\cal V}$ that splits it into two copies ${\cal V} = {\cal V}_{\mathbb{R}} \oplus i {\cal V}_{\mathbb{R}}$. The relevant solutions for a half-hypermultiplet are either parametrized by ${\cal V}_{\mathbb{R}}$ or $i {\cal V}_{\mathbb{R}}$.
• Figure 5: Decomposition of the sphere with six punctures into three-punctured spheres and tubes, and the corresponding conformal blocks.