5D AGT conjecture for circular quivers

TL;DR

This work extends the AGT correspondence to $q$-Virasoro blocks on elliptic surfaces and tests it against 5D circular quiver gauge theories and Shiraishi defect functions. It develops DF-like integral representations for both generic and degenerate ($q$-deformed) torus blocks, establishes precise equivalences with 5D instanton partition functions (including a minimal $m=2$ torus case) up to universal prefactors, and derives a parallel between Shiraishi defects and degenerate torus blocks via explicit coefficient matching. The results reveal correction factors $\sigma(p,x)$ and $\rho(p)$ that capture BPS/topological-string contributions and suggest multiple future directions, such as BPZ-type equations for Shiraishi functions, new bases/series expansions, and elliptic $SL(2,\mathbb{Z})$ identities at higher Chern–Simons levels. Collectively, the paper provides a concrete bridge between $q$-deformed conformal blocks on the torus and 5D circular-quiver physics, extending the reach of the AGT framework beyond genus-zero and linear-quiver cases.

Abstract

The best way to represent generic conformal blocks is provided by the free-field formalism, where they acquire a form of multiple Dotsenko-Fateev-like integrals of the screening operators. Degenerate conformal blocks can be described by the same integrals with special choice of parameters. Integrals satisfy various recurrent relations, which for the special choice of parameters reduce to closed equations. This setting is widely used in explaining the AGT relation, because similar integral representations exist also for Nekrasov functions. We extend this approach to the case of q-Virasoro conformal blocks on elliptic surface -- generic and degenerate. For the generic case, we check equivalence with instanton partition function of a 5d circular quiver gauge theory. For the degenerate case, we check equivalence with partition function of a defect in the same theory, also known as the Shiraishi function. We find agreement in both cases. This opens a way to re-derive the sophisticated equation for the Shiraishi function as the equation for the corresponding integral, what seems straightforward, but remains technically involved and is left for the future.

Figures (3)

• Figure 1: Circular quiver of $m$ gauge groups of type $SU(N)$.
• Figure 2: Linear quiver of $m-1$ gauge groups of type $SU(2)$. The theory has 4 fundamental matter multiplets shown in rectangular boxes.
• Figure 3: Topological string geometry that engineers the gauge theory with circular quiver $SU(2)^2$.