Liouville Correlation Functions from Four-dimensional Gauge Theories

Abstract: 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 N=2 SCFTs recently defined by one of the authors. We conduct extensive tests of the conjecture at genus 0,1.

• The paper demonstrates that Liouville conformal blocks correspond to the instanton part of the Nekrasov partition functions from 4D N=2 SCFTs.
• It details a mapping between SCFT parameter spaces and geometric structures by sewing Riemann surfaces with punctures.
• Numerical verifications in the SU(2) case show that DOZZ three-point functions combined with conformal blocks match gauge theory predictions.

Overview of Liouville Correlation Functions from Four-dimensional Gauge Theories

This paper presents a conjecture linking Liouville theory conformal blocks and correlation functions on Riemann surfaces with Nekrasov partition functions derived from a class of four-dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs). These SCFTs arise naturally through the compactification of the six-dimensional $(2,0)$ theory of type $A_1$ on Riemann surfaces with genus $g$ and $n$ punctures.

Key Concepts and Methods

• Mapping SCFTs to Geometric Structures: The mapping of SCFTs to geometrical constructs is achieved by associating the complex structure moduli space of a punctured Riemann surface to the parameter space of the theory. The paper exploits the sewing of Riemann surfaces using three-punctured spheres, corresponding to Lagrangian descriptions of SCFTs.
• Nekrasov Partition Function: The Nekrasov partition function, essential in calculating non-perturbative effects in gauge theories, is expressed via a sum over Young tableaux. Each SCFT is characterized by a specific class of Lagrangians, each associated with a distinct sewing pattern of the Riemann surface.
• Liouville Theory: The paper conjectures that Liouville theory correlation functions and conformal blocks correspond to the instanton part of these partition functions, while the full partition function corresponds to specific Liouville correlators.

Numerical Results and Conjectures

1. Conformal Block Identification: It is shown that Nekrasov's instanton partition function can be identified with the conformal block of the Virasoro algebra for sphere and torus configurations, specifically at genus $g=0,1$.
2. Liouville Correlation Functions: The paper argues that the absolute value squared of the Nekrasov partition function, integrated over the Coulomb branch moduli, reconstructs the Liouville correlation functions. The integration incorporates both tree-level and one-loop contributions.
3. Verification of Equivalence: For the $SU(2)$ case, the DOZZ three-point functions, when combined with the conformal blocks, coincide with the structure obtained from the Nekrasov partition function. This is verified through calculations for multiple configurations, including $n=3,4$ punctured cases.

Practical and Theoretical Implications

The implications of this work extend to both mathematical physics and the study of dualities in quantum field theories:

• Higher Genus Extensions: While verification is meticulous for lower genera, the framework is conjectured for higher genus with multiple punctures, offering potential applications in more complex topological field theories.
• Path to Quantum Geometry: Proposed interpretations suggest a link to quantum Teichmüller theory, indicating a deeper connection between two-dimensional CFTs and four-dimensional gauge theories.
• Duality and Moduli Spaces: The findings contribute to understanding the dualities in $\mathcal{N}=2$ gauge theories, particularly the role of modular transformations on the torus and crossing symmetry in Liouville theory.

Speculation and Future Directions

Looking forward, several speculative insights emerge:

• Quantum Seiberg-Witten Curves: The introduction of higher-order Virasoro descendants and their insertion into the correlation functions suggest potential avenues to define quantum corrections to classical Seiberg-Witten geometries.
• Generalization to Other Lie Algebras: Extending the analyses to $A_{N-1}$ theories or exploring connections with Toda theories which use W-algebras suggests a rich field for future research.
• Cross-disciplinary Connections: The intriguing appearance of Liouville theory invites a deeper exploration into the web of string dualities and how such classical constructs may emerge from modern quantum theoretical frameworks.

In conclusion, this paper opens novel pathways by intertwining Liouville theory with $\mathcal{N}=2$ gauge theories, potentially revolutionizing our understanding of non-perturbative effects in both dimensions, offering a robust platform for further theoretical exploration.