2024/2025
Научно-исследовательский семинар "Интегрируемость в квантовой теории поля 1"
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
1, 2 модуль
Охват аудитории:
для всех кампусов НИУ ВШЭ
Преподаватели:
Алфимов Михаил Николаевич
Язык:
английский
Кредиты:
6
Контактные часы:
60
Course Syllabus
Abstract
This course is organized in the form of weekly seminars, where we are going to discuss the integrability structures appearing in quantum field theory. These structures nowadays are present in numerous examples, such as sigma models, supersymmetric gauge theories, string theories, gauge/string dualities, scattering amplitudes and correlation functions etc. As pedagogical examples of the integrable systems solved by the Bethe Ansatz method Bose gas and Principal Chiral Field models will be considered in the first part of the course together with the foundations of the AdS/CFT correspondence for the case of 4-dimensional superconformal gauge theory. In the second part of the course there will be given an introduction into the applications of the theory of integrable systems to the study of the spectrum of $\mathcal{N}=4$ supersymmetric Yang\Ч Mills theory and dual superstring theory on the $\text{AdS}_5\times\text{S}^5$ background and we will study integrable deformations of sigma models. The course is intended for PhD and Master students. Postdocs and Bachelor students are also welcome.
Learning Objectives
- Study examples of integrable quantum field theories: Bose gas model and Principal Chiral Field.
- Study AdS/CFT correspondence with the example of AdS_5 x S^5 string background.
- Get acquainted with classically integrable sigma models.
Expected Learning Outcomes
- Able to solve the supergravity equations, which yield the AdS_5 x S^5 superstring background as a key element of one of the examples of AdS/CFT duality.
- Able to transform the Y-system for N=4 SYM into the T-system and solve it.
- Became aware of the notions of flatness condition, monodromy operator and quasimomenta together with the conserved charges and their analytic structure.
- Can obtain AdS_5 x S^5 superstring theory worldsheet S-matrix from Zamolodchikov-Faddeev algebra.
- Can obtain the Asymptotic Bethe Ansatz equations, their thermodynamic limit and Thermodynamic Bethe Ansatz equations for the PCF model.
- Familiar with examples of calculation of the characteristics of the N=4 SYM spectrum using the Quantum Spectral Curve method.
- Knows how to derive Bethe Ansatz equations, their thermodynamic limit and Thermodynamic Bethe Ansatz equations.
- Knows how to derive the Quantum Spectral Curve equations for N=4 SYM and AdS_5 x S^5 superstring theory.
- Studied the notion of integrable deformation of sigma models and learned several examples of such models including O(N) models and others.
- Understands the 1-loop integrability of N=4 SYM, asymptotic spectrum and Thermodynamic Bethe Ansatz equations of this theory.
- Understands the transition from Y- to T-system and how to solve it.
Course Contents
- Principal Chiral Field (PCF) Model.
- Y- and T-system (Hirota equations) for PCF.
- AdS/CFT correspondence.
- Classical integrability of sigma models.
- Integrable structure of N=4 SYM.
- Y- and T-system for the spectrum of N=4 SYM.
- Quantum Spectral Curve for N=4 SYM and AdS_5 x S^5 superstring.
- Application of the QSC method for the SL(2) sector of N=4 SYM.
- S-matrix for the AdS_5 x S^5 superstring.
- Integrable deformations of sigma models.
- The model of Bose gas.
Bibliography
Recommended Core Bibliography
- Ahn, C., & Nepomechie, R. I. (2010). Review of AdS/CFT Integrability, Chapter III.2: Exact world-sheet S-matrix. https://doi.org/10.1007/s11005-011-0478-9
- Gromov, N. (2017). Introduction to the Spectrum of N=4 SYM and the Quantum Spectral Curve.
- Gromov, N., Kazakov, V., & Vieira, P. (2008). Finite Volume Spectrum of 2D Field Theories from Hirota Dynamics. https://doi.org/10.1088/1126-6708/2009/12/060
- Gromov, N., Kazakov, V., Leurent, S., & Volin, D. (2011). Solving the AdS/CFT Y-system. https://doi.org/10.1007/JHEP07(2012)023
- Gromov, N., Kazakov, V., Leurent, S., & Volin, D. (2013). Quantum spectral curve for AdS_5/CFT_4. https://doi.org/10.1103/PhysRevLett.112.011602
- Gromov, N., Kazakov, V., Leurent, S., & Volin, D. (2014). Quantum spectral curve for arbitrary state/operator in AdS$_5$/CFT$_4$. https://doi.org/10.1007/JHEP09(2015)187
- Gromov, N., Kazakov, V., Sakai, K., & Vieira, P. (2006). Strings as Multi-Particle States of Quantum Sigma-Models. https://doi.org/10.1016/j.nuclphysb.2006.11.018
- Korepin, V. E., Izergin, A. G., & Bogoliubov, N. M. (1993). Quantum Inverse Scattering Method and Correlation Functions.
- Minahan, J. A., & Zarembo, K. (2002). The Bethe-Ansatz for N=4 Super Yang-Mills. https://doi.org/10.1088/1126-6708/2003/03/013
- Tseytlin, A. A. (2010). Review of AdS/CFT Integrability, Chapter II.1: Classical AdS5xS5 string solutions.
- V. A. Fateev, & A. V. Litvinov. (2018). Integrability, duality and sigma models. Journal of High Energy Physics, 2018(11), 1–29. https://doi.org/10.1007/JHEP11(2018)204
Recommended Additional Bibliography
- A. V. Litvinov, & L. A. Spodyneiko. (2018). On dual description of the deformed O(N) sigma model. Journal of High Energy Physics, 2018(11), 1–29. https://doi.org/10.1007/JHEP11(2018)139
- Kazakov, V. (2018). Quantum Spectral Curve of $\gamma$-twisted ${\cal N}=4$ SYM theory and fishnet CFT. https://doi.org/10.1142/S0129055X1840010X
- Rej, A. (2009). Integrability and the AdS/CFT correspondence. https://doi.org/10.1088/1751-8113/42/25/254002