2020/2021

Research Seminar "Integrability in Quantum Field Theory 2"

Type: Optional course (faculty)

Delivered by: Faculty of Mathematics

Where: Faculty of Mathematics

When: 3, 4 module

Instructors: Mikhail Alfimov

Language: English

ECTS credits: 3

Contact hours: 36

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. 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 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

• 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

• Understands the 1-loop integrability of N=4 SYM, asymptotic spectrum and Thermodynamic Bethe Ansatz equations of this theory.
• Able to transform the Y-system for N=4 SYM into the T-system and solve it.
• Knows how to derive the Quantum Spectral Curve equations for N=4 SYM and AdS_5 x S^5 superstring theory.
• Familiar with examples of calculation of the characteristics of the N=4 SYM spectrum using the Quantum Spectral Curve method.
• Can obtain AdS_5 x S^5 superstring theory worldsheet S-matrix from Zamolodchikov-Faddeev algebra.
• Studied the notion of integrable deformation of sigma models and learned several examples of such models including O(N) models and others.

Course Contents

• S-matrix for the AdS_5 x S^5 superstring.
  Derivation of the S-matrix for the superstring sigma model on AdS_5 x S^5 from Zamolodchikov-Faddeev algebra.
• Integrable structure of N=4 SYM.
  Bethe equations for the XXX Heisenberg spin chain (1-loop spectrum of anomalous dimensions of the local operators in the SU(2) sector of N=4 SYM). Asymptotic Bethe equations for the spectrum of N-4 SYM. TBA equations for the spectrum of N=4 SYM.
• Y- and T-system for the spectrum of N=4 SYM.
  The corresponding Hirota equations and wronskian solution of these equations.
• Quantum Spectral Curve for N=4 SYM and AdS_5 x S^5 superstring.
  Derivation of AdS/CFT Quantum Spectral Curve for AdS_5 x S^5 superstring theory and N=4 SYM.
• Application of the QSC method for the SL(2) sector of N=4 SYM.
  Non-perturbative characteristics of the operator trajectories in the N=4 SYM.
• Integrable deformations of sigma models.
  Integrable deformations of the O(N) sigma models. q-deformed S-matrix. Eta-deformed AdS_5 x S^5 superstring theory and its S-matrix.

Assessment Elements

• Устный доклад на семинаре
• Участие в дискуссии на семинаре

Interim Assessment

• Interim assessment (4 module)
  0.7 * Устный доклад на семинаре + 0.3 * Участие в дискуссии на семинаре

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