Геометрия и комбинаторика алгебр Годена тема диссертации и автореферата по ВАК РФ 01.01.06, доктор наук Рыбников Леонид Григорьевич

• Рыбников Леонид Григорьевич
• доктор наук
• 2021

Оглавление диссертации доктор наук Рыбников Леонид Григорьевич

Table of Contents

Introduction

1 Gaudin model and its generalizations

1.1 Gaudin Hamiltonians from the center at the critical level

1.2 Inhomogeneous Gaudin model

1.3 Shift of argument subalgebras

1.4 Uniqueness of Gaudin algebras

2 Bethe ansatz conjecture

2.1 Diagonalization problem

2.2 Bethe ansatz equations for sl2

2.3 Opers on the projective line

2.4 Bethe Ansatz Conjecture

2.5 A possible topological meaning of BAC for shift of argument subalgebras

3 Compacification problem

3.1 Degenerate Gaudin models

3.2 Limits of shift of argument subalgebras

3.3 Monodromy problem

4 Monodromy problem and combinatorics of crystals

4.1 Cactus group

4.2 Crystals and their tensor products

4.3 Etingof's monodromy conjecture

4.4 Gaudin algebras and the monodromy of their eigenlines

4.5 Operadic coverings and coboundary categories

4.6 Shift of argument algebras

4.7 Completion of the main proof

4.8 Another monodromy result

5 Further development

5.1 Schubert intersections and Opers

5.2 Weyl groups and quantum cohomology

References

Acknowledgements

Appendix A Article 1. Uniqueness of higher Gaudin hamiltonians

Appendix B Article 2. Limits of Gaudin Systems: Classical and

Quantum Cases

Appendix C Article 3. Opers with irregular singularity and spectra

of the shift of argument subalgebra

Appendix D Article 4. Limits of Gaudin algebras, quantization of

bending flows, Jucys-Murphy elements and Gelfand-Tsetlin bases

Appendix E Article 5. Cactus group and monodromy of Bethe vectors

Appendix F Article 6. A proof of the Gaudin Bethe Ansatz conjecture

Appendix G Article 7. Crystals and monodromy of Bethe vectors

Введение диссертации (часть автореферата) на тему «Геометрия и комбинаторика алгебр Годена»

Introduction

The main subject of the present thesis is the Bethe ansatz problem for the Gaudin quantum magnet chain. The main results of the thesis fall into the following three directions:

1. (joint with Boris Feigin and Edward Frenkel) Proof of the Bethe Ansatz Conjecture which states the completeness of the algebraic Bethe ansatz for the Gaudin model in the Feigin-Frenkel form.

2. (joint with Alexander Chervov and Gregorio Falqui) Classification of degenerations of the Gaudin model. Presenting Deligne-Mumford compactification of the moduli space of rational curves with marked points as the parameter space for Gaudin models.

3. (joint with Iva Halacheva, Joel Kamnitzer and Alex Weekes) Indexing of solutions of algebraic Bethe ansatz for the Gaudin model by Kashiwara crystals and description of the monodromy of the solutions in combinatorial terms.

The results of the present thesis were published in the following articles:

1. L. Rybnikov, Uniqueness of higher Gaudin hamiltonians, Reports on Mathematical Physics 61 (2008) No 2, pp. 247-252. [R08]

2. A. Chervov, G. Falqui, L. Rybnikov, Limits of Gaudin Systems: Classical and Quantum Cases, SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 5 (2009), 029 [CFR09]

3. B. Feigin, E. Frenkel, L. Rybnikov, Opers with irregular singularity and spectra of the shift of argument subalgebra, Duke Mathematical Journal 155 (2010), No 2, pp. 337-363. [FFR10]

4. A. Chervov, G. Falqui, L. Rybnikov, Limits of Gaudin algebras, quantization of bending flows, Jucys-Murphy elements and Gelfand-Tsetlin bases, Letters in mathematical physics 91 (2010), No 2, pp. 129-150. [CFR10]

5. L. Rybnikov, Cactus group and monodromy of Bethe vectors, International Mathematics Research Notices 2018 No 1, pp. 202-235. [R18]

6. L. Rybnikov, A proof of the Gaudin Bethe Ansatz conjecture, International Mathematics Research Notices 2020 No 22, pp. 8766-8785. [R20]

7. I Halacheva, J Kamnitzer, L Rybnikov, A Weekes, Crystals and monodromy of Bethe vectors, Duke Mathematical Journal 169 (2020) No 12, pp. 2337-2419. [HKRW]

The thesis consists of the Introduction, the list of references and the papers (1-7) from the above list. The rest of the Introduction is organized as follows. In Section 1 we introduce the Gaudin algebras, show how they help to construct quantum integrals of Hamiltonian systems and outline the uniqueness result of [R08]. In Section 2 we state the Bethe ansatz conjecture for Gaudin model in the Feigin-Frenkel form and explain its proof following [FFR10, R20]. In Section 3 we explain the compactification procedure for the parameter space of Gaudin algebras and extend Bethe ansatz conjecture to this compactification, following [CFR09, CFR10, R18]. In Section 4 we describe the monodromy of solutions of Bethe ansatz along the compactified parameter space, following [HKRW]. In Section 5 we outline some directions of further development of the results of [HKRW].

Список литературы диссертационного исследования доктор наук Рыбников Леонид Григорьевич, 2021 год

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