Monodromies of Second Order $q$-difference Equations from the WKB Approximation

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Detalles Bibliográficos
Autores principales:	Del Monte, Fabrizio, Longhi, Pietro
Formato:	Preprint
Publicado:	2024
Materias:	Mathematical Physics High Energy Physics - Theory
Acceso en línea:	Acceder al recurso
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_version_	1866917681725177856
author	Del Monte, Fabrizio Longhi, Pietro
author_facet	Del Monte, Fabrizio Longhi, Pietro
contents	This paper studies the space of monodromy data of second order $q$-difference equations through the framework of WKB analysis. We compute the connection matrices associated to the Stokes phenomenon of WKB wavefunctions and develop a general framework to parameterize monodromies of $q$-difference equations. Computations of monodromies are illustrated with explicit examples, including a $q$-Mathieu equation and its degenerations. In all examples we show that the monodromy around the origin of $\mathbb{C}^*$ admits an expansion in terms of Voros symbols, or exponentiated quantum periods, with integer coefficients. Physically these monodromies correspond to expectation values of Wilson line operators in five dimensional quantum field theories with minimal supersymmetry. In the case of the $q$-Mathieu equation, we show that the trace of the monodromy can be identified with the Hamiltonian of a corresponding $q$-Painlevé equation.
format	Preprint
id	arxiv_https___arxiv_org_abs_2406_00175
institution	arXiv
publishDate	2024
record_format	arxiv
spellingShingle	Monodromies of Second Order $q$-difference Equations from the WKB Approximation Del Monte, Fabrizio Longhi, Pietro Mathematical Physics High Energy Physics - Theory This paper studies the space of monodromy data of second order $q$-difference equations through the framework of WKB analysis. We compute the connection matrices associated to the Stokes phenomenon of WKB wavefunctions and develop a general framework to parameterize monodromies of $q$-difference equations. Computations of monodromies are illustrated with explicit examples, including a $q$-Mathieu equation and its degenerations. In all examples we show that the monodromy around the origin of $\mathbb{C}^*$ admits an expansion in terms of Voros symbols, or exponentiated quantum periods, with integer coefficients. Physically these monodromies correspond to expectation values of Wilson line operators in five dimensional quantum field theories with minimal supersymmetry. In the case of the $q$-Mathieu equation, we show that the trace of the monodromy can be identified with the Hamiltonian of a corresponding $q$-Painlevé equation.
title	Monodromies of Second Order $q$-difference Equations from the WKB Approximation
topic	Mathematical Physics High Energy Physics - Theory
url	https://arxiv.org/abs/2406.00175

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