Large-N Matrix models and Spectral Curves
Matrix models are ubiquitous both in Mathematics and in Theoretical Physics. The aim of this course is to introduce the main approaches to U(N)-invariant random matrix models: Coulomb gas, orthogonal polynomials, loop equations, topological recursion, tau-functions and integrable hierarchies. Most of these techniques are based on the fact that the eigenvalues of the random matrix behave as free fermions in a confining potential. The collective theory for these fermions in the large N limit is formulated in terms of a special Riemann surface - the spectral curve of the matrix model. Once the spectral curve is known, the topological recursion generates the whole 1/N expansion.
Keywords: U(N) matrix integrals, planar graphs, tau-functions, Hirota equations, Virasoro constraints, Coulomb gas, free fermions, bosonisation, 1/N, topological recursion.
Keywords: U(N) matrix integrals, planar graphs, tau-functions, Hirota equations, Virasoro constraints, Coulomb gas, free fermions, bosonisation, 1/N, topological recursion.
Lecturer
Ivan Kostov
Date
17th April ~ 14th June, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Wednesday,Friday | 09:50 - 11:25 | A3-1-301 | ZOOM 07 | 559 700 6085 | BIMSA |
Prerequisite
Graduate level knowledge linear algebra and complex analysis on the plane
Syllabus
I. U(N)-invariant matrix integrals
Hermitian matrix model: Spectral correlation functions, Dyson gas of eigenvalues, method of orthogonal polynomials, saddle-point equation for the spectral density at large N. Explicit solution for the spectral density via hyperelliptic curve. Period matrix and filling fractions.
2. The matrix model as a chiral CFT
Loop equations as Virasoro constraints. Fermionic Fock space representation in the form of a Jimbo-Miwa tau-function. SU(2) current algebra. Screening operator. Virasoro constraints as conformal Ward identities. Joint eigenvalue probabilities as fermionic correlation functions. Trace correlators as bosonic correlation functions.
3. Semi-classical, or 1/N, or topological, expansion.
Gaussian field on a hyper-elliptic Riemann surface. Branch points as primary conformal fields (twist operators). The case of a single branch point: Kontsevich-Witten tau-function. The case of several branch points: Givental-like decomposition of the partition function into a product of several Kontsevich-Witten tau-functions. Feynman rules. Eynard-Orantin topological recursion.
4. Matrix chains associated to semi-simple Lie algebras and their affine extensions. Loop equations and Virasoro constraints. Operator construction in terms of multi-component fermion fields. KP and Toda tau-functions. Spectral curves and topological expansion. Other methods: character expansion. Application for matrix models that cannot be reduced to eigenvalue integrals.
5. Matrix Quantum Mechanics. Integrable perturbations of the Gaussian potential. Toda integrable structure. Dispersionless limit of Toda flows.
Hermitian matrix model: Spectral correlation functions, Dyson gas of eigenvalues, method of orthogonal polynomials, saddle-point equation for the spectral density at large N. Explicit solution for the spectral density via hyperelliptic curve. Period matrix and filling fractions.
2. The matrix model as a chiral CFT
Loop equations as Virasoro constraints. Fermionic Fock space representation in the form of a Jimbo-Miwa tau-function. SU(2) current algebra. Screening operator. Virasoro constraints as conformal Ward identities. Joint eigenvalue probabilities as fermionic correlation functions. Trace correlators as bosonic correlation functions.
3. Semi-classical, or 1/N, or topological, expansion.
Gaussian field on a hyper-elliptic Riemann surface. Branch points as primary conformal fields (twist operators). The case of a single branch point: Kontsevich-Witten tau-function. The case of several branch points: Givental-like decomposition of the partition function into a product of several Kontsevich-Witten tau-functions. Feynman rules. Eynard-Orantin topological recursion.
4. Matrix chains associated to semi-simple Lie algebras and their affine extensions. Loop equations and Virasoro constraints. Operator construction in terms of multi-component fermion fields. KP and Toda tau-functions. Spectral curves and topological expansion. Other methods: character expansion. Application for matrix models that cannot be reduced to eigenvalue integrals.
5. Matrix Quantum Mechanics. Integrable perturbations of the Gaussian potential. Toda integrable structure. Dispersionless limit of Toda flows.
Reference
[1] B. Eynard, T. Kimura, and S. Ribault. Random matrices, https://arxiv.org/abs/1510.04430
[2] I. Kostov. Matrix models a conformal field theories. In E. B. et al., editor, proc. Les Houches 2004 “Applications of random matrices in physics”, pages 459–487. 2006 Springer, 2004.
[3] I. Kostov and N. Orantin. CFT and topological recursion. JHEP, 056, 06 2011, https://arxiv.org/pdf/1006.2028.pdf
[4] V. Kazakov, M. Staudacher, and T. Wynter. Character expansion methods for matrix models of dually weighted graphs. Commun. Math. Phys., 177:451–468, 1996. https://arxiv.org/pdf/hep-th/9502132.pdf
[5] I. Kostov. Integrable flows in c = 1 string theory. J. Phys., A36:3153–3172, 2003. https://arxiv.org/pdf/hep-th/0208034.pdf
[2] I. Kostov. Matrix models a conformal field theories. In E. B. et al., editor, proc. Les Houches 2004 “Applications of random matrices in physics”, pages 459–487. 2006 Springer, 2004.
[3] I. Kostov and N. Orantin. CFT and topological recursion. JHEP, 056, 06 2011, https://arxiv.org/pdf/1006.2028.pdf
[4] V. Kazakov, M. Staudacher, and T. Wynter. Character expansion methods for matrix models of dually weighted graphs. Commun. Math. Phys., 177:451–468, 1996. https://arxiv.org/pdf/hep-th/9502132.pdf
[5] I. Kostov. Integrable flows in c = 1 string theory. J. Phys., A36:3153–3172, 2003. https://arxiv.org/pdf/hep-th/0208034.pdf
Audience
Graduate
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Ivan Kostov obtained his PhD in 1982 from the Moscow State University, with scientific advisers Vladimir Feinberg and Alexander Migdal. Then he worked in the group of Ivan Todorov at the INRNE Sofia, and since 1990 as a CNRS researcher at the IPhT, CEA-Saclay, France. Currently he is emeritus DR CNRS at IPhT and a visiting professor at UFES, Vitoria, Brazil.