Third School-Conference “String theory, Integrable Models and Representation Theory”,
Moscow, January 21 – 27, 2018


  • 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)


  • 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
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.
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.
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].
∙ Alexei Kitaev The SYK model
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
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.
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