WKB solutions of difference equations and reconstruction by the topological recursion. (18th December 2017)

• Record Type: Journal Article
• Authors: Marchal, Olivier
• Is Part Of: Nonlinearity. Volume 31:Number 1(2018:Jan.)
• Page Start: 226
• Page End: 262
• Publication Date: 2017-12-18
• Subjects: WKB expansions -- topological recursion -- Gromov–Witten invariants -- difference systems -- determinantal formulas -- topological type property; 39A13 -- 14H70 -- 14N35 -- 53D45; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aa92ed
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: The purpose of this article is to analyze the connection between Eynard–Orantin topological recursion and formal WKB solutions of a ħ -difference equation: Ψ ( x + ħ ) = ( e ħ d d x ) Ψ ( x ) = L ( x ; ħ ) Ψ ( x ) with L ( x ; ħ ) ∈ G L 2 ( ( C ( x ) ) [ ħ ] ) . In particular, we extend the notion of determinantal formulas and topological type property proposed for formal WKB solutions of ħ -differential systems to this setting. We apply our results to a specific ħ -difference system associated to the quantum curve of the Gromov–Witten invariants of P 1 for which we are able to prove that the correlation functions are reconstructed from the Eynard–Orantin differentials computed from the topological recursion applied to the spectral curve y = cosh − 1 ⁡ x 2 . Finally, identifying the large x expansion of the correlation functions, proves a recent conjecture made by Dubrovin and Yang regarding a new generating series for Gromov–Witten invariants of P 1 .