Holomorphic blocks and the 5d AGT correspondence

TL;DR

The paper surveys holomorphic block factorisation of supersymmetric partition functions across 3d, 4d, and 5d, and develops a coherent picture in which 3d/5d partition functions are interpreted as correlators of a q,t-deformed Virasoro symmetry. It introduces 5d holomorphic blocks 𝔅^{5d} built from 1-loop, classical, and instanton data and shows how they glue to reproduce partition functions on S^5, S^3_b, and other toric Sasaki–Einstein manifolds, with Higgsing revealing codimension-two defects as vortex sectors. The chiral side, via 𝑉𝑖𝑟_{q,t} and q-DF integrals, links instanton-vortex counting to deformed conformal blocks, including Gaiotto–Whittaker states and Nekrasov-type decompositions; dual 5d theories are connected by refined topological strings and dual quiver presentations. Altogether, the work extends the AGT program to 5d with deformed Virasoro symmetry, clarifies the role of defects, and provides a calculational bridge between gauge theories, q-deformed CFTs, and topological strings.

Abstract

We review the holomorphic block factorisation of partition functions of supersymmetric theories on compact manifolds in various dimensions. We then show how to interpret 3d and 5d partition functions as correlation functions with underlying symmetry given by a deformation of the Virasoro algebra.

Figures (6)

• Figure 1: The Hanany-Witten brane setup for a linear quiver. On the RHS a surface operator corresponding to a position-dependent vev is engineered by an extra D2 brane.
• Figure 2: Higgsing a linear quiver and dual CFT interpretation. The first column represents an $2+SU(2)\times SU(2)+2$ linear quiver and its dual CFT 5-point block, the instanton partition functions involves summing over two 2-vectors of Young tableaux associated to the two $SU(2)$ nodes. The second column represents the analytic continuation of the mass/momenta to the Higgsing/degeneration condition. Correspondently the sum over the second vector of Young tableaux reduces to a sum over a column-diagram. In the third column the bulk theory is decoupled and the instanton partition function reduces to a vortex partition function which is mapped to the conformal block with 3 non-degenerate and one degenerate-primaries.
• Figure 3: The comb conformal diagram for the $M+2$ block.
• Figure 4: The toric diagram of the CY geometry engineering either $3+SU(3)+3$ or $2+SU(2)^2+2$.
• Figure 5: Crossing symmetry requires the equality of correlations involving chiral blocks in the s-channel (on the LHS) and in the u-channel (on the RHS).