[Paper Review] 2d QCD and Integrability, Part I: 't Hooft model

Federico Ambrosino, Shota Komatsu|arXiv (Cornell University)|Jan 1, 2023

Black Holes and Theoretical Physics1 citations

TL;DR

This paper establishes a deep connection between two-dimensional QCD in the large-Nc limit and integrable systems by showing that the 't Hooft equation for meson masses is equivalent to a TQ-Baxter equation, extending previous results beyond the special case of zero quark masses. The authors derive a spectral representation and inhomogeneous Fredholm formulation, enabling analytic study of the meson spectrum in the complex quark mass plane and revealing critical points and spontaneous PT-symmetry breaking for imaginary masses. This reformulation opens new avenues for linking gauge theories to integrable structures and topological strings.

ABSTRACT

We study analytical properties and integrable structures of the meson spectrum in large $N_c$ QCD$_2$. We show that the integral equation that determines the masses of the mesons, often called the 't Hooft equation, is equivalent to finding solutions to a TQ-Baxter equation. Using the Baxter equation, we extract systematic expansions of the energy levels as well as analytic asymptotic expressions for wavefunctions. Our analysis extends previous results for a special quark mass by Fateev et al. to arbitrary quark masses. This reformulation, together with its relation to an inhomogeneous Fredholm equation, is particularly suited for analytical treatments and makes accessible the analytic structure of the spectrum in the complex plane of the quark masses. We also comment on applications of our techniques to non-perturbative topological string partition functions.

Motivation & Objective

• To extend the integrable structure of the 't Hooft model beyond the special case of zero quark masses, where previous work by Fateev et al. found a TQ-system only for α₁ = α₂ = 0.
• To develop a systematic analytic framework for the meson spectrum in two-dimensional QCD with general quark masses by reformulating the 't Hooft equation as a TQ-Baxter equation.
• To explore the analytic structure of the meson spectrum in the complex plane of quark masses, identifying poles and critical points in the spectral function Ψ(ν).
• To connect the inhomogeneous 't Hooft equation to Fredholm integral equations and hypergeometric functions, enabling asymptotic and spectral sum analysis.
• To apply the developed techniques to non-perturbative topological string theory, particularly in the context of the TS/ST correspondence and mirror curve quantization.

Proposed method

• Reformulate the 't Hooft equation in momentum (ν) space using a spectral representation involving sinh functions and the auxiliary function f(ν).
• Derive the TQ-relation by showing that the eigenvalue problem maps to a Baxter TQ-system, with T(ν) and Q(ν) satisfying a functional equation involving sinh and integral kernels.
• Introduce an inhomogeneous Fredholm integral equation formulation to describe the spectral problem, allowing for analytic continuation into the complex mass plane.
• Use the Sokhotski–Plemelj theorem to handle singularities in the kernel and derive boundary terms that lead to the TQ-Baxter equation.
• Construct solutions to the TQ-system in terms of hypergeometric functions, enabling explicit computation of spectral sums and asymptotic expansions.
• Apply the formalism to topological string theory by relating the spectral determinant of the inhomogeneous problem to partition functions on toric Calabi–Yau threefolds.

Experimental results

Research questions

• RQ1 Can the integrable TQ-system structure found in the 't Hooft model at zero quark mass be extended to general quark masses?
• RQ2 What is the analytic structure of the meson spectrum when quark masses are analytically continued into the complex plane?
• RQ3 How do critical points in the spectrum emerge, particularly in the chiral limit and on the second Riemann sheet?
• RQ4 What is the role of PT symmetry in the context of imaginary quark masses, and how does spontaneous PT breaking manifest?
• RQ5 Can the inhomogeneous Fredholm formulation and TQ-system be used to compute spectral determinants relevant to topological string theory?

Key findings

• The 't Hooft equation is shown to be equivalent to a TQ-Baxter equation for general quark masses, extending the integrable structure beyond the α₁ = α₂ = 0 case.
• The spectral function Ψ(ν) exhibits poles that correspond to meson states, and their positions are determined by the zeros of the Q-function in the TQ-system.
• In the chiral limit (m₁, m₂ → 0), the model exhibits a critical point where the pion becomes massless, consistent with chiral symmetry restoration.
• For imaginary quark masses, the system displays spontaneous PT symmetry breaking, with the spectrum becoming complex, signaling a phase transition.
• The inhomogeneous Fredholm equation formulation allows for the computation of spectral sums and asymptotic expansions of eigenfunctions using hypergeometric functions.
• The framework provides a bridge to topological string theory, where the spectral determinant of the inhomogeneous problem matches the partition function of local P¹×P¹ geometry.