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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Toda systems and hypergeometric equations

Authors: Chang-Shou Lin, Zhaohu Nie and Juncheng Wei
Journal: Trans. Amer. Math. Soc. 370 (2018), 7605-7626
MSC (2010): Primary 35J47, 33C20
Published electronically: August 9, 2018
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Abstract: This paper establishes certain existence and classification results for solutions to $ \mathrm {SU}(n)$ Toda systems with three singular sources at 0, 1, and $ \infty $. First, we determine the necessary conditions for such an $ \mathrm {SU}(n)$ Toda system to be related to an $ n$th order hypergeometric equation. Then, we construct solutions for $ \mathrm {SU}(n)$ Toda systems that satisfy the necessary conditions and also the interlacing conditions from Beukers and Heckman. Finally, for $ \mathrm {SU}(3)$ Toda systems satisfying the necessary conditions, we classify, under a natural reality assumption, that all the solutions are related to hypergeometric equations. This proof uses the Pohozaev identity.

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Additional Information

Chang-Shou Lin
Affiliation: Department of Mathematics, Taida Institute of Mathematical Sciences, National Taiwan University, Taipei 106, Taiwan

Zhaohu Nie
Affiliation: Department of Mathematics and Statistics, Utah State University, Logan, Utah 84322-3900

Juncheng Wei
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada

Keywords: Toda systems, singular sources, $W$-invariants, hypergeometric equations, interlacing conditions, monodromy, Pohozaev identity
Received by editor(s): October 10, 2016
Published electronically: August 9, 2018
Additional Notes: The second author thanks the University of British Columbia, Wuhan University, and the National Taiwan University for hospitality during his visits in 2015 and 2016, where part of this work was done. He also acknowledges the Simons Foundation through Grant #430297.
The research of the third author is partially supported by NSERC of Canada.
Article copyright: © Copyright 2018 American Mathematical Society

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