Exact WKB and Abelianization for the $$T_3$$ T 3 Equation
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Communications in
Mathematical Physics
Exact WKB and Abelianization for the T3 Equation Lotte Hollands1 , Andrew Neitzke2 1 Department of Mathematics, Heriot-Watt University, Edinburgh, UK 2 Department of Mathematics, University of Texas at Austin, Austin, USA.
E-mail: [email protected] Received: 4 September 2019 / Accepted: 2 June 2020 Published online: 13 October 2020 – © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract: We describe the exact WKB method from the point of view of abelianization, both for Schrödinger operators and for their higher-order analogues (opers). The main new example which we consider is the “T3 equation,” an order 3 equation on the thricepunctured sphere, with regular singularities at the punctures. In this case the exact WKB analysis leads to consideration of a new sort of Darboux coordinate system on a moduli space of flat SL(3)-connections. We give the simplest example of such a coordinate system, and verify numerically that in these coordinates the monodromy of the T3 equation has the expected asymptotic properties. We also briefly revisit the Schrödinger equation with cubic potential and the Mathieu equation from the point of view of abelianization. Contents 1.
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Introduction . . . . . . . . . . . . . . . . . . 1.1 Exact WKB . . . . . . . . . . . . . . . . 1.2 Abelianization . . . . . . . . . . . . . . . 1.3 Voros symbols for Schrödinger equations 1.4 Exact WKB for order 3 equations . . . . . 1.5 The T3 equation . . . . . . . . . . . . . . 1.6 Integral equations and analytic structures . 1.7 Supersymmetric QFT . . . . . . . . . . . 1.8 Some questions . . . . . . . . . . . . . . Exact WKB for Schrödinger Equations . . . . 2.1 WKB solutions . . . . . . . . . . . . . . 2.2 Abelianization . . . . . . . . . . . . . . . 2.3 Gluing across the Stokes graph . . . . . . 2.4 W-framings . . . . . . . . . . . . . . . . 2.5 Spectral coordinates and their properties . 2.6 Integral equations . . . . . . . . . . . . .
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L. Hollands, A. Neitzke
Exact WKB for Schrödinger Operators with Cubic Potential . . . . . 3.2 The spectral coordinates . . . . . . . . . . . . . . . . . . . . . 3.3 Analytic continuation . . . . . . . . . . . . . . . . . . . . . . . 3.4 The regular pentagon . . . . . . . . . . . . . . . . . . . . . . . 3.5 Integral equations for spectral coordinates . . . . . . . . . . . . Exact WKB for the Mathieu Equation . . . . . . . . . . . . . . . . . 4.1 Exponential coordinate . . . . . .
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