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SUMMARY:WKB analysis via topological recursion for hypergeometric differen
 tial equations - Yumiko Takei (National Institute of Technology(KOSEN)\, I
 baraki College)
DTSTART:20220912T133000Z
DTEND:20220912T143000Z
UID:TALK178097@talks.cam.ac.uk
DESCRIPTION:The exact WKB analysis is a method to analyze differential equ
 ations with a small parameter.&nbsp\;The main ingredient of the exact WKB 
 analysis is a formal solution\, called a WKB solution.&nbsp\;When we study
  differential equations by using the exact WKB analysis\,&nbsp\;the so-cal
 led Voros coefficients play an important role.&nbsp\;The Voros coefficient
  is defined as a contour integral of the logarithmic derivative of WKB sol
 utions.&nbsp\;\nOn the other hand\, the topological recursion introduced b
 y B. Eynard and N. Orantin&nbsp\;([EO])&nbsp\;is a recursive algorithm to 
 construct a formal solution to the loop equations that the correlation fun
 ctions of the matrix model satisfy.&nbsp\;\nThe quantization scheme connec
 ts WKB solutions with the topological recursion. It is found that WKB solu
 tions can be constructed via the topological recursion&nbsp\;([BE] etc.).&
 nbsp\;\nIn this talk\, we prove that the Voros coefficients for hypergeome
 tric differential equations are&nbsp\;described by the generating function
 s of free energies defined in terms of the topological recursion.&nbsp\;Fu
 rthermore\, as its applications&nbsp\;we show the following objects can be
  explicitly computed for hypergeometric equations: (i) three-term differen
 ce equations that the generating function of free energies satisfies\, (ii
 ) explicit form of the free&nbsp\;energies\, and (iii) explicit form of Vo
 ros coefficients ([IKT]\,[T]).&nbsp\;\nReferences\n[BE]&nbsp\;V.Bouchard a
 nd B.Eyanard\,&nbsp\;Reconstructing WKB from topological recursion\,&nbsp\
 ;Journal de l'Ecole polytechnique -- Mathematiques\,&nbsp\;4 (2017)\, 845-
 -908.&nbsp\;\n[EO] B.Eynard and&nbsp\; N.Orantin\,&nbsp\;Invariants of alg
 ebraic curves and topological expansion\,&nbsp\;Communications in Number T
 heory and Physics\,&nbsp\;1&nbsp\;(2007)\, 347--452.\n[IKT] K.Iwaki\, T.Ko
 ike and Y.-M.Takei\,&nbsp\;Voros coefficients for the hypergeometric diffe
 rential equations&nbsp\;and Eynard-Orantin's topological recursion\,&nbsp\
 ;\nPart I\, arXiv:1805.10945\,\n& Part II\, Journal of Integrable Systems\
 ,&nbsp\;3&nbsp\;(2019)\, 1--46.&nbsp\;\n[T] Y.-M.Takei\, Voros Coefficient
 s and the Topological Recursion for a Class of the&nbsp\;Hypergeometric Di
 fferential Equations associated with the Degeneration of the 2-dimensional
  Garnier System\, arXiv: 2005.08957.&nbsp\;\n&nbsp\;\n&nbsp\;
LOCATION:No Room Required
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