Matrix Models, 2D Gravity and non-critical Strings

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  • Evgeny Sobko



Matrix Models, 2D Gravity and non-critical Strings

Evgeny Sobko (EIMI)

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Random matrices ubiquitously appear in various areas of mathematics and theoretical physics ranged from graph combinatorics and number theory to quantum chaos and black holes. Probably the most remarkable role they play - is in the context of quantum gravity and string theory where they set a frame-work for the numerous exact calculations giving us a powerful insight into the microscopic structure of space-time, and establishing at the same time deep interrelations between matrix integrals, graphs, algebraic curves, integrability and topology.

In the first part of this course, we will cover the basic techniques and ideas - a minimal toolkit necessary for any Matrix Model (MM) practitioner. First, we will show how the calculation of matrix integrals can be organized into the sum over topologies of certain graphs. This beautiful observation originally made by ’t Hooft in the context of QCD will allow us later to interpret matrix integrals at large N as partition functions of quantum gravity and strings. Then we will consider such standard MM techniques as Coulomb gas method, Loop Equation, Orthogonal Polynomials, and Character Expansion. Particularly, we will explain the central role of spectral curve - an algebraic curve encoding information about MM; clarify the connections of MMs with Integrable systems,Topological recursion of Eynard and Orantin, and resummation of asymptotic (trans) series. Numerous examples and exercises will point to further connections with combinatorics, integrability, limit shapes etc. In the end of the first part we will discuss, mainly in the overview format, the duality between c=1 Matrix Model and 2D String (Field) Theory.

Nowadays we see a significant resurgence of interest in MMs stimulated by new ideas coming from SYK, JT gravity, Black holes and powerful computers. The second part (next year) of this course will be aimed to give an overview of the most recent developments, ideas, and open questions. The plan of Part II is very preliminary and just illustrates some of the topics to be covered.

The course is aimed at students and researchers interested in Matrix Models. Prerequisite for Part I is minimal and includes just real and complex analysis, linear algebra, very basic knowledge of (matrix) group theory and geometry/topology. The knowledge of Quantum Mechanics is not necessary but would be very helpful.

Part I Introduction to Matrix Models

  • Topological (Large N) expansion
    Gaussian Matrix ensembles. Gaussian integrals and graph combinatorics. Ribbon graphs and maps. Wick’s Theorem. Topological expansion. Multimatrix Models, colored graphs and Ising model.

  • Coulomb gas method
    Saddle-point approximation. Coulomb gas of eigenvalues at large N. Spectral curve.

  • Loop equation
    Loop equation at finite N. Loop equation and Topological expansion. Loop equation and the glimpse of Topological recursion (very sketchy).

  • Orthogonal polynomials and Integrability
    Orthogonal polynomials as determinants. Recursion and derivation relations. MM Partition function as a tau-function, KP hierarchy and Hirota equation.

  • Itzykson-Zuber integrals, character expansion and Dually Weighted Graphs

  • Matrix Quantum Mechanics, 2d Quantum Gravity and c=1 String Theory
    MQM as a discretization of the Polyakov’s 1D string path integral. Free
    fermions and Large N (WKB) analysis. Double-scaling limit and c=1 String Field Theory. Compactification on the circle, vortexes, black hole.

Relevant reviews for Part I : [1]-[10]

Part II Recent developments

  • SU(N) Principal Chiral Model as a MM in 1+1 dimensions. Large N analysis and Double scaling. Comparison with c=1 MM.
    Based on [11] + work in progress

  • Matrix bootstrap
    Based on [12, 13, 14]

  • SYK and Tensor models. Chaos. Low energy SYK limit (=1D Schwarzian theory) as JT gravity on a disc.
    Based on : too many refs

  • JT gravity and Matrix Models.
    Based on [15] + ...

  • Machine Learning and Matrix Models
    Based on [16]

  • ...


[1] B. Eynard, T. Kimura and S. Ribault, “Random matrices,” [arXiv:1510.04430 [math-ph]].
[2] D. Anninos and B. Muhlmann, “Notes on matrix models (matrix musings),” J. Stat. Mech. 2008 (2020), 083109 doi:10.1088/1742-5468/aba499 [arXiv:2004.01171 [hep-th]].
[3] Alexander K. Zvonkin and Sergei K. Lando “Graphs on Surfaces and Their Applications” doi:10.1007/978-3-540-38361-1
[4] P. Di Francesco, “2D quantum gravity, matrix models and graph combinatorics,” [arXiv:math-ph/0406013 [math-ph]].
[5] P. H. Ginsparg and G. W. Moore, “Lectures on 2-D gravity and 2-D string theory,” [arXiv:hep-th/9304011 [hep-th]].
[6] P. Di Francesco, P. H. Ginsparg and J. Zinn-Justin, “2-D Gravity and random matrices,” Phys. Rept. 254 (1995), 1-133 doi:10.1016/0370-1573(94)00084-G [arXiv:hep-th/9306153 [hep-th]].
[7] M. Marino, “Les Houches lectures on matrix models and topological strings,” [arXiv:hep-th/0410165 [hep-th]].
[8] M. Marino, “Lectures on non-perturbative effects in large N gauge theories, matrix models and strings,” Fortsch. Phys. 62 (2014), 455-540 doi:10.1002/prop.201400005 [arXiv:1206.6272 [hep-th]].
[9] I. R. Klebanov, “String theory in two-dimensions,” [arXiv:hep-th/9108019[hep-th]].
[10] J. Polchinski, “What is string theory?,” [arXiv:hep-th/9411028 [hep-th]].
[11] V. Kazakov, E. Sobko and K. Zarembo, “Double-Scaling Limit in the Principal Chiral Model: A New Noncritical String?,” Phys. Rev.Lett. 124 (2020) no.19, 191602 doi:10.1103/PhysRevLett.124.191602 [arXiv:1911.12860 [hep-th]].
[12] H. W. Lin, “Bootstraps to strings: solving random matrix models with positivite,” JHEP 06 (2020), 090 doi:10.1007/JHEP06(2020)090 [arXiv:2002.08387 [hep-th]].
[13] X. Han, S. A. Hartnoll and J. Kruthoff, “Bootstrapping Matrix
Quantum Mechanics,” Phys. Rev. Lett. 125 (2020) no.4, 041601
doi:10.1103/PhysRevLett.125.041601 [arXiv:2004.10212 [hep-th]].
[14] V. Kazakov and Z. Zheng, “Analytic and Numerical Bootstrap for One-Matrix Model and ”Unsolvable” Two-Matrix Model,” [arXiv:2108.04830[hep-th]].
[15] P. Saad, S. H. Shenker and D. Stanford, “JT gravity as a matrix integral,” [arXiv:1903.11115 [hep-th]].
[16] X. Han and S. A. Hartnoll, “Deep Quantum Geometry of Matrices,” Phys. Rev. X 10 (2020) no.1, 011069 doi:10.1103/PhysRevX.10.011069
[arXiv:1906.08781 [hep-th]].

December 2021

November 2021

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