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Yang–Baxter Deformation of 2D Non-Linear Sigma Models: Towards Applications to AdS/CFT

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In mathematical physics, one of the fascinating issues is the study of integrable systems. In particular, non-perturbative techniques that have been developed have triggered significant insight for real physics. There are basically two notions of integrability:  classical integrability and  quantum integrability. In this book, the focus is on the former, classical integrability. When the system has a finite number of degrees of freedom, it has been well captured by the Arnold–Liouville theorem. However, when the number of degrees of freedom is infinite, as in classical field theories, the integrable structure is enriched profoundly. In fact, the study of classically integrable field theories has a long history and various kinds of techniques, including the classical inverse scattering method, which have been developed so far. In previously published books, these techniques have been collected and well described and are easy to find in traditional, standard textbooks. 
One of the intriguing subjects in classically integrable systems is the investigation of deformations preserving integrability. Usually, it is not considered systematic to perform such a deformation, and one must study systems case by case and show the integrability of the deformed systems by constructing the associated Lax pair or action-angle variables. 
Recently, a new, systematic method to perform integrable deformations of 2D non-linear sigma models was developed. It was invented by C. Klimcik in 2002, and the integrability of the deformed sigma models was shown in 2008. The original work was done for 2D principal chiral models, but it has been generalized in various directions nowadays. In this book, the recent progress on this Yang–Baxter deformation is described in a pedagogical manner, including some simple examples. Applications of Yang–Baxter deformation to string theory are also described briefly. 

Author(s): Kentaroh Yoshida
Series: SpringerBriefs in Mathematical Physics, 40
Publisher: Springer
Year: 2021

Language: English
Pages: 78
City: Singapore

Preface
Acknowledgements
Contents
1 Integrable Non-linear Sigma Models in (1+1)-Dimensions
1.1 What is Classical Integrability?
1.2 Principal Chiral Model
1.2.1 Classical Action and Symmetry
1.2.2 Target-Space Geometry
1.2.3 BIZZ Construction of Non-local Conserved Charges
1.2.4 Yangian Algebra
1.2.5 Lax Pair and Monodromy Matrix
1.3 Symmetric Coset Sigma Model
1.3.1 Symmetric Coset
1.3.2 Classical Action and Symmetry
1.3.3 The Proof of the Flatness Condition
1.3.4 Example: Coset Construction of S2
1.3.5 Another Form of Lax Pair
References
2 Yang–Baxter Sigma Models
2.1 Classical Yang–Baxter Equation and Linear R-Operator
2.2 Yang–Baxter Deformation of 2D Principal Chiral Model
2.2.1 Deformed Classical Action
2.3 Yang–Baxter Deformation of Symmetric Coset Model
2.3.1 Deformed Classical Action
2.3.2 Lax Pair
2.4 History of the Development of the Yang–Baxter Deformation
References
3 Recent Progress on Yang–Baxter Deformation and Generalized Supergravity
3.1 Introduction
3.2 Yang–Baxter Deformation of the AdS5timesS5 Superstring
3.2.1 Classical Integrability of the AdS5timesS5 Superstring
3.2.2 Yang–Baxter Deformed Action and Supercoset Construction
3.2.3 Unimodular Examples
3.3 Generalized Supergravity
3.3.1 Non-unimodular Example
3.3.2 Hoare-Tseytlin Conjecture
3.3.3 Non Yang–Baxter Solution
3.4 Other Topics
References