Liouville Correlation Functions from Four-Dimensional Gauge Theories

  title={Liouville Correlation Functions from Four-Dimensional Gauge Theories},
  author={Luis F. Alday and Davide Gaiotto and Yuji Tachikawa},
  journal={Letters in Mathematical Physics},
We conjecture an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov partition function of a certain class of $${\mathcal{N}=2}$$ SCFTs recently defined by one of the authors. We conduct extensive tests of the conjecture at genus 0, 1. 

Figures and Tables from this paper

Localization of four-dimensional super Yang-Mills theories compactified on Riemann surface
We consider the partition function of super Yang-Mills theories defined on $T^2 \times \Sigma_g$. This path integral can be computed by the localization. The one-loop determinant is evaluated by theExpand
The Liouville side of the vortex
We analyze conformal blocks with multiple (semi-)degenerate field insertions in Liouville/Toda conformal field theories an show that their vector space is fully reproduced by the four-dimensionalExpand
On AGT conjecture
In these notes we consider relation between conformal blocks and the Nekrasov partition function of certain $ \mathcal{N} = 2 $ SYM theories proposed recently by Alday, Gaiotto and Tachikawa. WeExpand
Super Liouville conformal blocks from $ \mathcal{N} = 2 $ SU(2) quiver gauge theories
The conjecture about the correspondence between instanton partition functions in the N = 2 SUSY Yang-Mills theory and conformal blocks of two-dimensional conformal field theories is extended to theExpand
Gluing theory of Riemann surfaces and Liouville conformal field theory
We study the gluing theory of Riemann surfaces using formal algebraic geometry, and give computable relations between the associated parameters for different gluing processes. As its application toExpand
Instantons on ALE spaces and super Liouville conformal field theories
We provide evidence that the conformal blocks of $ \mathcal{N} = 1 $ super Liouville conformal field theory are described in terms of the SU(2) Nekrasov partition function on the ALE space $Expand
Liouville Quantum Gravity on the Riemann Sphere
In this paper, we rigorously construct Liouville Quantum Field Theory on the Riemann sphere introduced in the 1981 seminal work by Polyakov. We establish some of its fundamental properties likeExpand
Generalized matrix models and AGT correspondence at all genera
We study generalized matrix models corresponding to n-point Virasoro conformal blocks on Riemann surfaces with arbitrary genus g. Upon AGT correspondence, these describe four dimensional $Expand
Gauge theories on Ω-backgrounds from non commutative Seiberg-Witten curves
We study the dynamics of a $ \mathcal{N} = 2 $ supersymmetric SU(N) gauge theory with fundamental or adjoint matter in presence of a non trivial Ω-background along a two dimensional plane. TheExpand
Affine SL(2) Conformal Blocks from 4d Gauge Theories
We study Nekrasov’s instanton partition function of four-dimensional $${\mathcal{N}=2}$$ gauge theories in the presence of surface operators. This can be computed order by order in the instantonExpand


Seiberg-Witten theory and random partitions
We study \( \mathcal{N} = 2 \) supersymmetric four-dimensional gauge theories, in a certain 525-02 = 2 supergravity background, called theΩ-background. The partition function of the theory in theExpand
Modular bootstrap in Liouville field theory
Abstract The modular matrix for the generic 1-point conformal blocks on the torus is expressed in terms of the fusion matrix for the 4-point blocks on the sphere. The modular invariance of the toricExpand
Two and three point functions in Liouville theory
Abstract Based on our generalization of the Goulian-Li continuation in the power of the 2D cosmological term we construct the two- and three-point correlation functions for Liouville exponentialsExpand
Liouville theory revisited
We try to develop a coherent picture of Liouville theory as a two-dimensional conformal field theory that takes into account the perspectives of the path-integral approach, bootstrap, canonicalExpand
From Liouville theory to the quantum geometry of Riemann surfaces
The aim of this note is to propose an interpretation for the full (non-chiral) correlation functions of the Liouville conformal field theory within the context of the quantization of spaces ofExpand
Localization of Gauge Theory on a Four-Sphere and Supersymmetric Wilson Loops
We prove conjecture due to Erickson-Semenoff-Zarembo and Drukker-Gross which relates supersymmetric circular Wilson loop operators in the $${\mathcal N=4}$$ supersymmetric Yang-Mills theory with aExpand
Nonrational Conformal Field Theory
We introduce a formalism for the construction of correlation functions in certain classes of nonrational conformal field theories. An important role is played by non-degenerate hermitian forms on theExpand
ABCD of Instantons
We solve = 2 supersymmetric Yang-Mills theories for an arbitrary classical gauge group, i.e. SU(N), SO(N), Sp(N). In particular, we derive the prepotential of the low-energy effective theory, and theExpand
The Topological Vertex
We construct a cubic field theory which provides all genus amplitudes of the topological A-model for all non-compact toric Calabi-Yau threefolds. The topology of a given Feynman diagram encodes theExpand
N = 2 dualities
A bstractWe study the generalization of S-duality and Argyres-Seiberg duality for a large class of N = 2 superconformal gauge theories. We identify a family of strongly interacting SCFTs and use themExpand