Miura opers and critical points of master functions

Evgeny Mukhin; Alexander Varchenko

Open Mathematics (2005)

  • Volume: 3, Issue: 2, page 155-182
  • ISSN: 2391-5455

Abstract

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Critical points of a master function associated to a simple Lie algebra 𝔤 come in families called the populations [11]. We prove that a population is isomorphic to the flag variety of the Langlands dual Lie algebra t 𝔤 . The proof is based on the correspondence between critical points and differential operators called the Miura opers. For a Miura oper D, associated with a critical point of a population, we show that all solutions of the differential equation DY=0 can be written explicitly in terms of critical points composing the population.

How to cite

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Evgeny Mukhin, and Alexander Varchenko. "Miura opers and critical points of master functions." Open Mathematics 3.2 (2005): 155-182. <http://eudml.org/doc/268827>.

@article{EvgenyMukhin2005,
abstract = {Critical points of a master function associated to a simple Lie algebra \[\mathfrak \{g\}\] come in families called the populations [11]. We prove that a population is isomorphic to the flag variety of the Langlands dual Lie algebra \[^t \mathfrak \{g\}\] . The proof is based on the correspondence between critical points and differential operators called the Miura opers. For a Miura oper D, associated with a critical point of a population, we show that all solutions of the differential equation DY=0 can be written explicitly in terms of critical points composing the population.},
author = {Evgeny Mukhin, Alexander Varchenko},
journal = {Open Mathematics},
keywords = {82B23; 17B67; 14M15},
language = {eng},
number = {2},
pages = {155-182},
title = {Miura opers and critical points of master functions},
url = {http://eudml.org/doc/268827},
volume = {3},
year = {2005},
}

TY - JOUR
AU - Evgeny Mukhin
AU - Alexander Varchenko
TI - Miura opers and critical points of master functions
JO - Open Mathematics
PY - 2005
VL - 3
IS - 2
SP - 155
EP - 182
AB - Critical points of a master function associated to a simple Lie algebra \[\mathfrak {g}\] come in families called the populations [11]. We prove that a population is isomorphic to the flag variety of the Langlands dual Lie algebra \[^t \mathfrak {g}\] . The proof is based on the correspondence between critical points and differential operators called the Miura opers. For a Miura oper D, associated with a critical point of a population, we show that all solutions of the differential equation DY=0 can be written explicitly in terms of critical points composing the population.
LA - eng
KW - 82B23; 17B67; 14M15
UR - http://eudml.org/doc/268827
ER -

References

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  14. [14] E. Mukhin and A. Varchenko: “Discrete Miura Opers and Solutions of the Bethe Ansatz Equations”, math.QA/0401137, (2004), pp. 1–26. 
  15. [15] E. Mukhin and A. Varchenko: “Miura Opers and Critical Points of Master Functions”, math.QA/0312406, (2003), pp. 1–27. Zbl1108.82011
  16. [16] E. Mukhin and A. Varchenko: “Multiple orthogonal polynomials and a counterexample to Gaudin Bethe Ansatz Conjecture”, math.QA/0501144, (2005), pp. 1–40. 
  17. [17] N. Reshetikhin and A. Varchenko: “Quasiclassical asymptotics of solutions to the KZ equations”, In: Geometry, topology & physics. Conf. Proc. Lecture Notes Geom. Topology, VI, Internat. Press, Cambridge, MA, 1995, pp. 293–322. Zbl0867.58065
  18. [18] I. Scherbak and A. Varchenko: “Critical point of functions, sl 2 representations and Fuchsian differential equations with only univalued solutions”, math.QA/0112269, (2001) pp. 1–25. 
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