From Painlevé equations to ${\\cal N}=2$ susy gauge theories: prolegomena

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Summary

The research paper explores the connection between Painlevé equations and N=2 supersymmetric gauge theories in the Nekrasov-Shatashvili (NS) regime. The authors study the linear problems associated with Painlevé III and III1 equations, finding that they reduce to the modified Mathieu equation and the Doubly Confluent Heun Equation, respectively. These equations are related to the quantization of Seiberg-Witten differentials for SU(2) gauge theories with N=0 and N=2 flavors. The authors also discuss the isomonodromy of the Painlevé flow and its implications for the connection problem.

Highlights

• The Painlevé III equation is connected to the modified Mathieu equation, which is the quantization of the Seiberg-Witten differential for pure SU(2) gauge theory.
• The Painlevé III1 equation is connected to the Doubly Confluent Heun Equation, which is the quantization of the Seiberg-Witten differential for SU(2) gauge theory with N=2 flavors.
• The authors use the isomonodromy of the Painlevé flow to study the connection problem.
• The connection problem is related to the computation of the dual instanton period.
• The authors discuss the implications of their results for the study of N=2 supersymmetric gauge theories.
• The research provides a new perspective on the connection between Painlevé equations and gauge theories.
• The authors use a combination of mathematical and physical techniques to study the connection problem.

Key Insights

• The connection between Painlevé equations and N=2 supersymmetric gauge theories is a deep and fascinating area of research. The authors' work provides a new perspective on this connection, highlighting the importance of the isomonodromy of the Painlevé flow.
• The modified Mathieu equation and the Doubly Confluent Heun Equation are two important examples of linear problems associated with Painlevé equations. The authors' work shows that these equations are connected to the quantization of Seiberg-Witten differentials for SU(2) gauge theories.
• The connection problem is a fundamental problem in the study of Painlevé equations. The authors' work provides a new approach to this problem, using the isomonodromy of the Painlevé flow to compute the dual instanton period.
• The research has important implications for the study of N=2 supersymmetric gauge theories. The authors' work provides a new tool for studying the connection between Painlevé equations and gauge theories, and for computing the dual instanton period.
• The authors' use of a combination of mathematical and physical techniques is a key strength of the research. The work demonstrates the power of interdisciplinary approaches to solving complex problems.
• The research is part of a broader effort to understand the connection between Painlevé equations and gauge theories. The authors' work provides a new perspective on this connection, and highlights the importance of the isomonodromy of the Painlevé flow.
• The authors' work has the potential to lead to new insights and discoveries in the study of N=2 supersymmetric gauge theories. The research provides a new tool for studying the connection between Painlevé equations and gauge theories, and for computing the dual instanton period.

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Citation

Fioravanti, D., & Rossi, M. (2024). From Painlevé equations to ${\cal N}=2$ susy gauge theories: prolegomena (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2412.21148