Moscow, January 21 – 27, 2018

Organizers

- Skoltech, Center for Advanced Studies
- Laboratory of Representation Theory and Mathematical Physics, Higher School of Economics
- Landau Institute for Theoretical Physics
- Interdisciplinary Scientific Center Poncelet

Organizing and Program Committee

Alexander Belavin,

Mikhail Bershtein,

Boris Feigin,

Aleksei Litvinov,

Andrei Marshakov,

Sessions wil be held

- on January 21, 25, 26, 27 at the Independent University of Moscow (11 Bolshoy Vlasyevskiy Pereulok, Moscow),
- on January 22, 23 at the Skoltech (Nobel Street, Building 3, Moscow),
- on January 24 at the Faculty of Mathematical of the Higher School of Economics (6 Usacheva Street, Moscow)

**Program: **

**Konstantin Aleshkin**(Landau Inst. & SISSA),**Alexander Belavin**(Landau Inst. & Kharkevich Inst.) – Superstring theory and Calabi-Yau manifolds**Rinat Kashaev**(Univ. of Geneva) – Quantum dilogarithm: applications to quantum topology**Alexei Kitaev**(Caltech) – SYK model**Pavel Putrov**(IAS Princeton) – Topological strings and Chern–Simons theory**Lev Spodyneiko**(Landau Inst. & Caltech) – K3 surface and string theory

**
Preliminary Timetable** // last updated January 12

January 21, Sunday // Independent University of Moscow | ||

14.00-16.00 | L.Spodyneiko | Introduction to compactifications on Calabi-Yau manifolds |

16.30-18.00 | M.Bershtein | Introduction to Kahler geometry |

January 22, Monday // Skoltech | ||

11.00-13.00 | A.Kitaev | SYK model, 1 |

14.00-15.30 | A.Kitaev | SYK model, 2 |

16.00-18.00 | R.Kashaev | Quantum dilogarithm: applications to quantum topology, 1 |

January 23, Tuesday // Skoltech | ||

10.30-12.00 | K.Aleshkin, A.Belavin | Introduction to Calabi-Yau manifolds |

12.30-14.00 | Seminar on A.Kitaev lectures | |

15.00-16.30 | A.Kitaev | SYK model, 3 |

17.00-18.30 | P.Putrov | Topological strings and Chern–Simons theory, 1 |

January 24, Wednesday // Higher School of Economics | ||

15.30-17.00 | R.Kashaev | Quantum dilogarithm: applications to quantum topology, 2 |

17.30-19.00 | R.Kashaev | Quantum dilogarithm: applications to quantum topology, 3 |

January 25, Thursday // Independent University of Moscow | ||

10.00-11.30 | L.Spodyneiko | K3 surface and string theory, 1 |

12.00-13.30 | P.Putrov | Topological strings and Chern–Simons theory, 2 |

14.30-16.00 | A.Kitaev | SYK model, 4 |

16.30-18.00 | Seminar on P.Putrov lectures | |

January 26, Friday // Independent University of Moscow | ||

10.00-11.30 | L.Spodyneiko | K3 surface and string theory, 2 |

12.00-13.30 | R.Kashaev | Quantum dilogarithm: applications to quantum topology, 4 |

14.30-16.00 | K.Aleshkin, A.Belavin | Superstring theory and Calabi-Yau manifolds, 1 |

16.30-18.00 | R.Kashaev | Quantum dilogarithm: applications to quantum topology, 5 |

January 27, Saturday // Independent University of Moscow | ||

10.00-11.30 | P.Putrov | Topological strings and Chern–Simons theory, 3 |

12.00-13.30 | L.Spodyneiko | K3 surface and string theory, 3 |

12.00-13.30 | Seminar on A.Kitaev lectures | |

14.30-16.00 | P.Putrov | K3 surface and string theory, 4 |

16.30-18.00 | K.Aleshkin, A.Belavin | Superstring theory and Calabi-Yau manifolds, 2 |

16.30-18.00 | Seminar on P.Putrov lectures |

Course annotations

∙Konstantin Aleshkin, Alexander Belavin Superstring theory and Calabi-

Yau manifolds

Superstring theory at = 10, as we know, is currently the main candidate to the Grand

Unified Theory, i.e the theory unifying the Gravity and the Standard model of elementary

particles.

In order to solve various phenomenological problems, including the hierarchy problem,

this theory, after 6 of the 10 dimensions being compactified, should possess Supersymmetry

at small scales. (The hierarchy problem is a question of why the mass of Higgs’ boson is

several orders of magnitude less than Plank’s constant.)

Compactification to the so-called Calabi-Yau manifolds is the only way to solve this,

as well as others, question of Fundamental physics.

Such characteristics of the Theory as the number of quark-lepton generations are

defined by the topology of the corresponding Calabi-Yau manifold. And the dynamics

of supermultiplets of fundamental particles is given by the Geometry of the space of

parameters on which the manifold depends.

In these lectures we shall consider why 6 of the 10 dimensions should be compactified

to Calabi-Yau manifolds, which properties these manifolds possess, as well as a new way

to obtain that very Special K¨ahler Geometry, which is connected to these manifolds and

defines the structure of the theory of Fundamental particles.

Literature:

1. A. Belavin, L. Spodyneiko = 2 superconformal algebra in NSR string and Gepner

approach to space-time supersymmetry in ten dimensions [arXiv:1507.01911].

2. K. Aleshkin, A. Belavin Compactification and Killing spinors [on the School’s

site (in Russian)].

3. K. Aleshkin, A. Belavin Killing spinors and Calabi-Yau manifolds [on the School’s

site (in Russian)].

4. K. Aleshkin, A. Belavin Special geometry on the Calabi-Yau moduli space [on the

School’s site (in Russian)].

5. K. Aleshkin, A. Belavin A new approach for computing the geometry of the moduli

spaces for a Calabi-Yau manifold [arXiv:1706.05342].

∙ Rinat Kashaev Quantum dilogarithm: applications to Quantum Topology

1. Quantum Teichm¨uller theory, unitary projective representations of mapping class

groups of punctured surfaces in infinite-dimensional Hilbert spaces, quantum hyperbolic

invariants of mapping torii.

2. Teichm¨uller TQFT and quantum hyperbolic invariants of cusped 3-manifolds and

3-manifolds with weighted string links.

3. Trace class operators, Fredholm determinants and non-perturbative topological

string partition functions for toric Calabi-Yau three-folds.

Literature:

1. R. Kashaev, Lectures on quantum Teichm¨uller theory [link].

2. R. Kashaev, Combinatorics of the Teichm¨uller TQFT [link].

3. M. Marino, Spectral theory and mirror symmetry [arXiv:1506.07757].

1

∙ Alexei Kitaev The SYK model

Literature

1. A. Kitaev, S. J. Suh, The soft mode in the Sachdev-Ye-Kitaev model and its gravity

dual, [arXiv:1711.08467].

2. A Kitaev, Notes on ̃︁(2,R) representations, [arXiv:1711.08169].

∙ Pavel Putrov Topological strings and the Chern-Simons theory

In my mini-course I’m about to tell the following: two-dimensional topological quantum

field theories; two-dimensional supersymmetric sigma-models; topological twist of types

A and B; topological strings; mirror symmetry; the case of toric Calabi-Yau manofolds;

the connection of topological strings to Chern-Simons theory and matrix models.

Desirable preliminary knowledge: it is desirable to have an idea of differential and

complex geometry (cohomologies, K¨ahler manifolds etc.), gauge theories, sypersymmetric

theories.

Literature

1. K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil and

E. Zaslow, Mirror symmetry, Clay Mathematics Monographs 1. American Mathematical

Society, Providence, RI, 1.

2. R. Gopakumar and C. Vafa, On the gauge theory / geometry correspondence, Adv.

Theor. Math. Phys. 3, 1415 (1999) [arXiv:hep-th/9811131]
3. M. Aganagic, A. Klemm, M. Marino and C. Vafa, Matrix model as a mirror of

Chern-Simons theory, JHEP 0402, 010 (2004) [arXiv:hep-th/0211098]
4. E. Witten, Chern-Simons gauge theory as a string theory, Prog. Math. 133 (1995)

637 [arXiv:hep-th/9207094]
∙ Lev Spodyneiko K3 surface in String theory

The goal of the mini-course is to make listeners familiar with such important concepts

of string theory as moduli space, compactification, orbifolds, Gepner models and Witten

index on a simple example of evaluating the elliptic genus of the K3 surface.

1. K3-surface and a string on it.

2. Orbifolds and Gepner models for K3.

3. Elliptic genus and Moonshine.

Desired preliminary knowledge: The first lecture has a purely review spirit and

so does not demand any special knowledge (as well as does not give any new). Knowing

the models of a free boson and a free fermion on a circle and their partition functions is

enough for understanding the second and the third lectures.

Literature

1. P. Aspinwall, K3 Surfaces and String Duality, [arXiv:hep-th/9611137]
2. T. Eguchi, H. Ooguri, A. Taormina, Superconformal Algebras and String Compactification

on Manifolds with () Holonomy, Nucl. Phys. B315, 193-221.

3. T. Kawai, Y. Yamada, S. K. Yang, Elliptic Genera and = 2 Superconformal Field

Theory, Nucl. Phys. B414, 191-212.

Please find more details in Russian in this link