# XVII DIAS-TH Winter School "Supersymmetry and Integrability"

31 January 2022 to 4 February 2022
Europe/Moscow timezone

## Supersymmetric field theory for beginners

I.L. Buchbinder (TSPU, Tomsk, Russia)

Abstract
This lecture course is devoted to an elementary introduction to the basic concepts of N = 1 supersymmetric classical field theory. It is assumed that the listeners have never studied this subject and this is their first acquaintance with supersymmetry. The course is intended for students familiar with the basic elements of Lie group theory and classical relativistic field theory.

Contents

Lecture 1. Basic idea of supersymmetry. Lorentz and Poincare groups. Two- component spinors. Lie algebra of the Poincare group.

Lecture 2. Super extension of the Poincare algebra. Supercharges. Anticommuting variables. Superspace and superfield. Chiral superfields.

Lecture 3. Superfield Lagrangians. Wess-Zumino model. Supersymmetric Yang-Mills theory.

Literature:

• I.L. Buchbinder, S.M. Kuzenko, Ideas and Methods of Supersymmetry and Supergravity or a Walk Through Superspace, IOP Publishing, Bristol and Philadelphia, 1998.

## Field theory in AdS Space (Higher Spins)

Ruben Manvelyan (Yerevan Physics Institute,Armenia)

1. Geometry of AdSD. Geodesics, trajectories, and one-parameter subgroups

2. Conformally coupled massless scalar in general gravitational background and restriction to AdS case. The Cauchy problem and the antipodal map.

3. Fronsdal Equation . General setup for higher spin equation of motion.

## Semiclassical quantization of invariant manifolds of Hamiltonian systems.

A.I.Shafarevich (MSU, Moscow, Russia)

1. Correspondence between invariant manifolds of the classical system and
spectral series of quantum operators. Quantization of rest points and
periodic trajectories.

2. Quantization of Lagrangian surfaces. Maslov canonical operator, Maslov
index and integrable systems.

3. Quantization of isotropic surfaces of incomplete dimension. Maslov
complex germ and spectral series corresponding to complex vector bundles
over isotropic manifolds.

## Introduction to twistors and supertwistors

S. Fedoruk (BLTP JINR, Dubna, Russia)

Lecture 1.
Conformal symmetry and twistors. Twistor space.
Penrose twistor transform and twistor formulation of massless particles.
Field twistor transform.
Twistorial description of higher spin particle.

Lecture 2.
Bitwistor formulation of massive particles and massless infinite spin
particles.
Conformal supersymmetry and supertwistors.
Twistor description of massless superparticles.

## Factorized Scattering Theory and the Bethe Ansatz

Gleb Arutyunov (Hamburg University)

Abstract
In these lectures we treat the elemental aspects of quantum integrable
models related to Factorized Scattering Theory and the Bethe Ansatz.

Factorized Scattering Theory
1.Classical scattering and integrability
2.Conservation laws and the Bethe wave function
3.S-matrix

Coordinate Bethe Ansatz
1.Periodicity condition for the Bethe wave function
2.Incarnations of the Lieb-Liniger model
3.Spin chain representation of permutation modules
4.Generalised Bethe hypothesis

Transfer Matrix Method
1.Transfer matrix
2.Diagonalisation of the transfer matrix by Lieb's method
3.Algebraic Bethe Ansatz
4.Nested Bethe Ansatz

Recommended literature:

1. C.N. Yang, Some exact results for the many body problems in one
dimension with
repulsive delta function interaction, Phys. Rev. Lett., 19:1312--1314,
1967.

2. C.N. Yang, S matrix for the one-dimensional N body problem with
repulsive or
attractive delta function interaction, Phys. Rev., 168:1920--1923, 1968.

3. A.B. Zamolodchikov and Al.B. Zamolodchikov,
Factorized S Matrices in Two-Dimensions as the Exact Solutions of
Certain Relativistic Quantum Field Models.
Annals Phys., 120:253--291, 1979.

4. L. D. Faddeev, How Algebraic Bethe Ansatz works for integrable model,
Les-Houches lectures, arXiv:hep-th/9605187.

5. M. Gaudin, The Bethe Wavefunction, Translated from French original La
fonction d'onde de Bethe' (1983) by J.-S. Caux, Cambridge University
Press, 2014.

6. G. Arutyunov, Elements of classical and quantum integrable systems,
Springer, 2019.

## Supersymmetric quantum mechanics

S. Sidorov (BLTP JINR, Dubna, Russia)

Abstract
Supersymmetric quantum mechanics is the simplest d=1 supersymmetric field
theory which displays salient features of higher-dimensional supersymmetric
theories via dimensional reduction. I will consider N=4, d=1 supersymmetry
that is related to N=1, d=4 supersymmetry (Lectures of I.L. Buchbinder).

Lecture 1. Introduction. N=4, d=1 supersymmetry. Scalar, сhiral and
harmonic superfields. Superfield Lagrangians.
Lecture 2. Deformed (Weak) supersymmetry. Model of harmonic oscillator.
Quantization.