Third Winter School-Conference
“String theory, Integrable Models and Representation Theory”,
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 will 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 (r.303, Nobel Street, 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 23

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	K.Aleshkin, A.Belavin	Superstring theory and Calabi-Yau manifolds, 1
14.30-16.00	R.Kashaev	Quantum dilogarithm: applications to quantum topology, 4
16.30-18.00	R.Kashaev	Quantum dilogarithm: applications to quantum topology, 5
January 27, Saturday // Independent University of Moscow
10.00-11.30	L.Spodyneiko	K3 surface and string theory, 3
10.00-11.30		Seminar on A.Kitaev lectures
12.00-13.30	P.Putrov	Topological strings and Chern–Simons theory, 3
14.30-16.00	P.Putrov	Topological strings and Chern–Simons 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 abstracts

1. Konstantin Aleshkin, Alexander Belavin / Superstring theory and Calabi-Yau manifolds
   
   Superstring theory at d = 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 N = 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].

2. Rinat Kashaev / Quantum dilogarithm: applications to Quantum Topology
   
   1. Quantum Teichmuller 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 Teichmuller theory.
   2. R. Kashaev, Combinatorics of the Teichmuller TQFT.
   3. M. Marino, Spectral theory and mirror symmetry [arXiv:1506.07757].

3. 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].

4. 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].

5. 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