[Paper Review] Duality Constraints on String Theory: Instantons and spectral networks

Chuan-Tsung Chan, Hirotaka Irie+1 more|arXiv (Cornell University)|Aug 29, 2013

Black Holes and Theoretical Physics116 references3 citations

TL;DR

This paper investigates non-perturbative duality constraints in $(p,q)$ minimal string theory using spectral networks and matrix model techniques. It shows that $p-q$ duality imposes strict constraints on non-perturbative contour ambiguities and D-instanton fugacities, ruling out ghost D-instantons and meta-stable vacua inconsistent with duality, thus establishing duality as a fundamental non-perturbative selection principle in string theory.

ABSTRACT

We study an implication of $p-q$ duality (spectral duality or T-duality) on non-perturbative completion of $(p,q)$ minimal string theory. According to the Eynard-Orantin topological recursion, spectral $p-q$ duality was already checked for all-order perturbative analysis including instanton/soliton amplitudes. Non-perturbative realization of this duality, on the other hand, causes a new fundamental issue. In fact, we find that not all the non-perturbative completions are consistent with non-perturbative $p-q$ duality; Non-perturbative duality rather provides a constraint on non-perturbative contour ambiguity (equivalently, of D-instanton fugacity) in matrix models. In particular, it prohibits some of meta-stability caused by ghost D-instantons, since there is no non-perturbative realization on the dual side in the matrix-model description. Our result is the first quantitative observation that a missing piece of our understanding in non-perturbative string theory is provided by the principle of non-perturbative string duality. To this end, we study Stokes phenomena of $(p,q)$ minimal strings with spectral networks and improve the Deift-Zhou's method to describe meta-stable vacua. By analyzing the instanton profile on spectral networks, we argue the duality constraints on string theory.

Motivation & Objective

• To investigate the implications of $p-q$ duality on the non-perturbative completion of $(p,q)$ minimal string theory.
• To identify how non-perturbative duality acts as a constraint rather than just a dictionary between dual theories.
• To resolve the issue of contour ambiguity and meta-stable vacua in matrix models through duality consistency.
• To establish a quantitative link between spectral networks, Stokes phenomena, and non-perturbative duality in minimal string theory.
• To demonstrate that ghost D-instantons and inconsistent non-perturbative completions are ruled out by duality.

Proposed method

• Utilizes Eynard-Orantin topological recursion to verify spectral $p-q$ duality in all-order perturbative amplitudes.
• Applies the Deift-Zhou steepest descent method to analyze Stokes phenomena and non-perturbative ambiguities in isomonodromy systems.
• Constructs spectral networks to visualize anti-Stokes lines and track instanton profiles across different vacua.
• Employs Baker-Akhiezer functions and classical monodromy matrices to describe multi-cut boundary conditions and Stokes multipliers.
• Introduces weaving techniques for proper spectral networks to resolve solvability of Riemann-Hilbert problems.
• Uses resolvent dynamics and eigenvalue profiles to compare $(p,2)$ and $(2,p)$ systems under duality.

Experimental results

Research questions

• RQ1 How does non-perturbative $p-q$ duality constrain the choice of non-perturbative contour ambiguities in matrix models?
• RQ2 What role do spectral networks play in resolving Stokes phenomena and identifying instanton profiles in minimal string theory?
• RQ3 Why are certain meta-stable vacua—induced by ghost D-instantons—ruled out by non-perturbative duality?
• RQ4 How does the duality between $(p,2)$ and $(2,p)$ minimal string models constrain the structure of instanton contributions?
• RQ5 What is the relationship between the large $N$ limit and the non-commutativity of two integrals in the context of duality constraints?

Key findings

• Non-perturbative $p-q$ duality acts as a constraint on the D-instanton fugacity and contour ambiguity, excluding inconsistent non-perturbative completions.
• Ghost D-instantons, which lead to meta-stable vacua, are prohibited because they lack a non-perturbative realization on the dual side.
• The $(5,2)$ and $(2,5)$ systems show explicit duality constraints: the $(5,2)$ system with a large instanton is inconsistent with the $(2,5)$ system's small instanton profile.
• The monodromy matrix transformation under duality is derived as $c_{n,l,j} = ho^{(l-j)/2}$, with $ ho = e^{2 au i/p}$, confirming duality consistency in Stokes data.
• The classical Baker-Akhiezer function and monodromy matrices are enhanced via weaving of proper spectral networks to resolve solvability of the Riemann-Hilbert problem.
• The non-commutativity of two integrals in the large $N$ limit is shown to be incompatible with duality unless contour ambiguities are constrained.