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Extended Heun Hierarchy in Quantum Seiberg-Witten Geometry

Peng Yang, Yi-Rong Wang, Kilar Zhang

Abstract

We investigate the quantum geometry of the Seiberg-Witten curve for $\mathcal{N}=2$, $\mathrm{SU(2)}^n$ linear quiver gauge theories. By applying the Weyl quantization prescription to the algebraic curve, we derive the corresponding second-order differential equation and demonstrate that it is isomorphic to the Extended Heun Equation with $n+3$ regular singular points. The physical parameters of the gauge theory are linked to the canonical coefficients of the Heun equation via a polynomial representation of the Seiberg-Witten curve. This framework provides the necessary mathematical foundation to apply non-perturbative gauge-theoretic techniques, such as instanton counting, to spectral problems in gravitational physics, most notably for higher-dimensional black holes.

Extended Heun Hierarchy in Quantum Seiberg-Witten Geometry

Abstract

We investigate the quantum geometry of the Seiberg-Witten curve for , linear quiver gauge theories. By applying the Weyl quantization prescription to the algebraic curve, we derive the corresponding second-order differential equation and demonstrate that it is isomorphic to the Extended Heun Equation with regular singular points. The physical parameters of the gauge theory are linked to the canonical coefficients of the Heun equation via a polynomial representation of the Seiberg-Witten curve. This framework provides the necessary mathematical foundation to apply non-perturbative gauge-theoretic techniques, such as instanton counting, to spectral problems in gravitational physics, most notably for higher-dimensional black holes.
Paper Structure (12 sections, 59 equations, 1 figure)

This paper contains 12 sections, 59 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Classical Seiberg-Witten Geometry for Linear Quivers
  3. Quantum Geometry and the Extended Heun Equation
  4. Weyl Quantization and the Schrödinger Form
  5. Analytic Structure and Rational Representation
  6. Implications for Black Hole Spectroscopy
  7. The Gauge/Gravity Dictionary in Higher Dimensions
  8. Quantization Condition via the Nekrasov-Shatashvili Limit
  9. Conclusion
  10. Detailed Derivation of the Quantum Curve
  11. Singularity Structure and Potential Coefficients
  12. Derivation of the Polynomial Numerator

Figures (1)

  • Figure 1: The Riemann surface for the $\mathrm{SU}(2)^n$ linear quiver viewed as a gluing of $n+1$ three-punctured spheres. The tubes represent the gauge nodes $\mathrm{SU}(2)_k$, and the marked points correspond to the mass singularities $m_i$ discussed in the text.