qKZ and Bethe ansatz
We will look at how the quantum Knizhnik-Zamolodchikov (qKZ) equation arises in the study of quantum integrable systems and how quantum spin chains help to solve the qKZ equation.
In the course we will cover the KZ equation and its quantum analogue. On the side of integrable systems we will study the Heisenberg spin chains and Gaudin models.
In the first half of the course, we will consider the case of gl(2). In the second part, we will consider the generalisation to the case of gl(n).
In the course we will cover the KZ equation and its quantum analogue. On the side of integrable systems we will study the Heisenberg spin chains and Gaudin models.
In the first half of the course, we will consider the case of gl(2). In the second part, we will consider the generalisation to the case of gl(n).
Lecturer
Date
4th March ~ 30th May, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Friday | 13:30 - 15:05 | A3-2-303 | ZOOM 11 | 435 529 7909 | BIMSA |
Prerequisite
Linear algebra and other standard courses of the first two years of a mathematics or physics department. Basics of Lie algebra and its representation theory.
Syllabus
- Introduction to spin chain and solution of XXX Heisenberg spin chain using Bethe ansatz.
[Z] section 2.1, [S] sections 1, 4.
- Spin chain and relation to vertex models
[Z] section 3.1, 3.2, [S] section 1.2, section 6
[Z] section 2.1, [S] sections 1, 4.
- Spin chain and relation to vertex models
[Z] section 3.1, 3.2, [S] section 1.2, section 6
Reference
[Z] A. Zabrodin, "Lectures on Bethe ansatz and quantum integrable systems", http://crei.skoltech.ru/app/data/uploads/sites/42/2023/04/backint3.pdf
[S] N. Slavnov, "Algebraic Bethe ansatz", https://arxiv.org/pdf/1804.07350
M. Jimbo and T, Miwa, "Algebraic Analysis of Solvable Lattice Models"
H. M. Babujian and R. Flume, "Off-Shell Bethe Ansatz Equation for Gaudin Magnets and Solutions of Knizhnik-Zamolodchikov Equations"
P. I. Etingof, I. Frenkel, A. A. Kirillov, Lectures on Representation Theory and Knizhnik–Zamolodchikov Equations
V. Tarasov, A. Varchenko "Jackson Integral Representations for Solutions to the Quantized Knizhnik-Zamolodchikov Equation", https://arxiv.org/abs/hep-th/9311040
V. Tarasov, A. Varchenko "Solutions to the Quantized Knizhnik-Zamolodchikov Equation and the Bethe Ansatz", https://arxiv.org/abs/hep-th/9411181
[S] N. Slavnov, "Algebraic Bethe ansatz", https://arxiv.org/pdf/1804.07350
M. Jimbo and T, Miwa, "Algebraic Analysis of Solvable Lattice Models"
H. M. Babujian and R. Flume, "Off-Shell Bethe Ansatz Equation for Gaudin Magnets and Solutions of Knizhnik-Zamolodchikov Equations"
P. I. Etingof, I. Frenkel, A. A. Kirillov, Lectures on Representation Theory and Knizhnik–Zamolodchikov Equations
V. Tarasov, A. Varchenko "Jackson Integral Representations for Solutions to the Quantized Knizhnik-Zamolodchikov Equation", https://arxiv.org/abs/hep-th/9311040
V. Tarasov, A. Varchenko "Solutions to the Quantized Knizhnik-Zamolodchikov Equation and the Bethe Ansatz", https://arxiv.org/abs/hep-th/9411181
Audience
Advanced Undergraduate
, Graduate
, Postdoc
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Andrii Liashyk is a researcher in the field of integrated systems, mainly quantum ones. He received his degree from the Center for Advanced Study at Skoltech in 2020. In 2022 he joined BIMSA as a Assistant Professor.