Superstring Theory and Sigma Models

Lecturers: Litvinov Alexey, Tseytlin Arkady

Annotation

(Super) String theory is a candidate theory which unifies all forces including gravity in a single framework. Despite there is no experimental evidence that string theory gives a correct description of our world and many features of string theory, such as extra dimensions, may look fabulous, its conclusions and mathematical structure are so impressive that they deserve to be studied. This includes a special superstring theory defined on a product of Anti de Sitter space and a 5-sphere, which is related to the t'Hooft limit of N=4 supersymmetric Yang-Mills theory in four dimensions, due to the duality known as AdS/CFT correspondence.

String theories are described by worldsheet sigma models. There are a number of examples of superstrings, one of them being $AdS^5×S_5$ superstring, whose worldsheet theories are integrable. It is precisely this property - the integrability which enables us to study the AdS/CFT correspondence in such a great generality. At the same time sigma models play an important role in other branches of theoretical physics including condensed matter physics and high-energy physics. When considered on positive curvature homogeneous spaces these theories serve as prototypical examples exhibiting QCD-like behaviour such as asymptotic freedom and existence of instanton solutions.

Along with the foundations of superstrings the course gives a systematic introduction to integrable two-dimensional superstring sigma models and their role in string theory and AdS/CFT correspondence.

Course plan

Review of bosonic string theory: "old" quantization, light-cone quantization, Polyakov's approach.
Superstring theory: worldsheet vs target space supersymmetry, NSR approach and GSO projection, the idea of GS superstring.
Sigma-models in string theory, field theory and condensed matter physics, examples of sigma models: O(N), CP(N) etc.
Principal chiral field model, Zakharov-Mikhailov integrability, local and ultra-local Poisson brackets, local integrals of motion.
Sigma models on homogeneous spaces. Integrability of sigma models on symmetric spaces.
Quantum sigma models I: dimensional regularization, minimal subtraction scheme, external field method, beta functions in PCM and O(N) models, asymptotic freedom.
Quantum sigma models II: Riemann normal coordinates, loop expansion, scheme dependence, scale vs conformal invariance.
Topological effects: theta-term, instantons, WZ term, classical integrability, Wess-Zumino-Witten model, generalized connection, parallelization, non-abelian bosonization.
Integrable deformations of sigma-models I: Yang-Baxter deformation of PCM and symmetric space sigma-models (eta-deformation), classical integrability, “sausage” model of Fateev-Onofri-Zamolodchikov, renormalization of deformed models, UV expansion.
Integrable deformations of sigma-models II: gauged WZW model and coset conformal field theories, GKO construction, free field representation of coset models, Witten "cigar", current deformations of gWZW model, classical integrability, renormalizability, relation with YB deformed sigma models.
Introduction to Green-Schwarz superstring in $AdS_5xS^5$ background: Lie superalgebras, superalgebra $PSU(2,2|4)$, sigma models on supermanifolds, $PSU(2,2|4)/Sp(1,1)xSP(2)$ sigma model, integrability.
Deformations of string sigma-models: YB deformation of $PSU(2,2|4)/Sp(1,1)xSP(2)$ sigma model, integrability, conformal invariance.

Literature

M. Green, J. Schwarz and E. Witten, Superstring Theory, Cambridge University Press
J. Polchinski, String Theory, Cambridge University Press
K. Becker, M. Becker and J. Schwarz, String Theory and M-Theory, Cambridge University Press
R. Blumenhagen, D. Lust, S. Theisen, Basic Concepts of String Theory, Springer
G. Arutyunov, S. Frolov, Foundations of the AdS5xS5 Superstring, arxiv.org/abs/0901.4937
B. Hoare, Integrable Deformations of Sigma Models, arxiv.org/abs/2109.14284
K. Zarembo, Integrability in Sigma-Models, arxiv.org/abs/1712.07725