Research Seminar "Integrable Quantum Field Theory 2"

2023/2024

Type: Optional course (faculty)

Delivered by: Faculty of Mathematics

When: 3, 4 module

Open to: students of all HSE University campuses

Instructors: Mikhail Alfimov

Language: English

ECTS credits: 6

Contact hours: 72

Full 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. 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 N=4 supersymmetric Yang-Mills theory and dual superstring theory on the AdS_5×S^5 background. The course is intended for PhD and Master students. Postdocs and Bachelor students are also welcome.

Learning Objectives

Understand the derivation of the AdS_5 x S^5 superstring theory S-matrix with the usage of the Zamolodchikov-Faddeev algebra.
Study the integrable structure of N=4 SYM and thus AdS_5 x S^5 superstring theory, including the 1-loop integrabiity, Thermodynamic Bethe Ansatz equations with the Y- and T-systems and the method of Quantum Spectral Curve.
Learn about the integrable deformations of sigma models, such as O(N) sigma models and others.

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.
Familiar with examples of calculation of the characteristics of the N=4 SYM spectrum using the Quantum Spectral Curve method.
Knows how to derive the Quantum Spectral Curve equations for N=4 SYM and AdS_5 x S^5 superstring theory.
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

Y- and T-system (Hirota equations) for PCF.
AdS/CFT correspondence.
Classical integrability of sigma models.
S-matrix for the AdS_5 x S^5 superstring.
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.

Assessment Elements

Talk at the research seminar
Taking part in the seminar discussions

Interim Assessment

2023/2024 4th module
0.3 * Taking part in the seminar discussions + 0.7 * Talk at the research seminar

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
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.

Recommended Additional Bibliography

Rej, A. (2009). Integrability and the AdS/CFT correspondence. https://doi.org/10.1088/1751-8113/42/25/254002

Course Syllabus