Gauge/Liouville Triality

TL;DR

The paper demonstrates a remarkable triality linking q-deformed Liouville conformal blocks, a 3d ${\cal N}=2$ U($N$) gauge theory ${\cal G}_{3d}$ in the ${\cal M}_q$ Omega-background, and a 5d ${\cal N}=1$ gauge theory ${\cal G}_{5d}$. DF integrals for the Liouville blocks reproduce the ${\cal G}_{3d}$ partition function, and residues of these integrals yield the Nekrasov instanton sums of ${\cal G}_{5d}$ on the ${\cal M}_{q,t}$ background, with pole data labeled by partitions. By tuning Coulomb moduli to integer masses, the paper proves the equality ${\cal Z}_{3d}={\cal Z}_{5d}$ and with the Liouville blocks, establishing a manifest 3d–5d–Liouville triality that generalizes aspects of AGT to five dimensions. The results are grounded in M-theory constructions and geometric transitions, highlighting vortex-instanton correspondences and spectral dualities that unify CFT data with gauge-theory partition functions. The work provides a concrete, computable bridge between conformal blocks, 3d vortex dynamics, and 5d instanton counting, with potential implications for higher-dimensional AGT and topological-string realizations.

Abstract

Conformal blocks of Liouville theory have a Coulomb-gas representation as Dotsenko-Fateev (DF) integrals over the positions of screening charges. For q-deformed Liouville, the conformal blocks on a sphere with an arbitrary number of punctures are manifestly the same, when written in DF representation, as the partition functions of a class of 3d U(N) gauge theories with N=2 supersymmetry, in the Omega-background. Coupling the 3d gauge theory to a flavor in fundamental representation corresponds to inserting a Liouville vertex operator; the two real mass parameters determine the momentum and position of the puncture. The DF integrals can be computed by residues. The result is the instanton sum of a five dimensional N=1 gauge theory. The positions of the poles are labeled by tuples of partitions, the residues of the integrand are the Nekrasov summands.

Figures (3)

• Figure 1: The "comb" diagram and parameters of the associated block.
• Figure 2: The four possible vertices, $V_{++}, V_{--}, V_{+-}$ and $V_{-+}$. Poles in the graph are the vertical lines going up, where $e^p$ goes to infinity. Zeros are the vertical spikes running down, where $e^{p}$ goes to zero. In $V_{++}$ both $\Sigma_+$ and $\Sigma_-$ have a pole. In $V_{+-}$, $\Sigma_+$ gets a pole and a zero.
• Figure 3: Graph of a Riemann surface $\Sigma_{+}\cup {\Sigma_-}$ corresponding to $M=3$, and the 5-point conformal block.