Matrix Models 2025

Table of Contents

Course plan

Lecture 1. Introduction, motivation and course outline

Historical motivation (non-perturbative string theory description) along the lines of [FGZ95, pp.4–12]. Modern development according to [Mor25].

Lecture 2. Feynman graphs in QFT

Feynman graphs, Wick theorem, asymptotic series, connected and disconnected diagrams [Ski18, pp.13–26].

Lecture 3. Ribbon Feynman graphs. QCD. Large limit

Ribbon Feynman graphs — [FGZ95, sec.1.2]. To understand the connection with fancy, though accurate eq. (2.39) and (2.40) from [Ski18] (Lecture 2) see also [EKR15, sec.2.1], especially eqns. (2.1.20) and (2.1.21). QCD, large limit and planar diagrams — [Ton18, pp.305–313].

Lecture 4. Eigenvalue matrix models. Leading behaviour

You'll learn quick and dirty hack of obtaining leading behaviour of one-point correlators (as well as directly related one-point eigenvalue distributions). Main reference — [Zee10, pp.396–402]. For more convincing Vandermonde determinant derivations see footnote 8 in [FGZ95], as well as Appendix C.

Just in case of questions about the so-called Haar measure, see Example 5 here.

For more convincing Coloumb gas method explanations see also [FGZ95, sec.2.1].

Lecture 5. Loop equations. Subleading behaviour

Subleading behaviour can be obtained by straightforward expansion around saddle point from previous lecture. The recipe is as follows: expand the exponent around equillibrium . You'll have something like

(1)

so will be the propagator and all the rest — vertices. Next, you can calculate all the correlators using this Feyman diagrams technique. This procedure is quite tedious and even if you are patient enough to calculate first few orders, you may notice the beautiful pattern behind, which can be discovered independently and which is the subject of the current lecture.

Main goal is to understand [Mor25, sec.2.2]. It can be done with the help of [Mor94, sec.2.2], elementary matrix calculus [PP12] could also be of use. As a glimpse into the realm of AMM-CEO topological recursion describe the recursive procedure for obtaining all the correlators of the matrix model with the generating function given as sum over Young diagrams

(2)

by substituting it in the Virasoro constraints. For a result obtained from somewhat different persperctive see eq. (4.1.10) in [EKR15].

Lecture 6. Origin of loop equations. QCD

Makeenko-Migdal loop equations in QCD are well described in section 1 and appendix A of [Mig83]. Loop equations in 2 quantum gravity [Kaz89].

Lecture 7. -representation

The subject of the lecture is [Mor25, sec.2.3]. For the origin of this approach, as well as the notion of and matrix calculus derivation, see [MS09]. Grading and dilatation operator are also explained in my notes.

Lecture 8. Integrability

We need to understand [Mor25, sec.2.4]. For details consult [Mir94, secs.4.2, 4.5].

Lecture 9. Superintegrability

Lecture 10. Multi-matrix models. string theory

Lecture 11. Matrix model in external field. Kontsevich model

Lecture 12. Unitary models. Itzykson-Zuber integral. Large 2 lattice QCD

Problem set (not complete yet!)

Problem 1. (Matrix) models

For the Gaussian Hermitian matrix model find the generating function

(3)

where

(4)

via

a) Ward identities

(5)

b) -representation

(6)

where

(7)

and derivative by is understood as multiplication by .

c) Wick's theorem

(8)

d) Graphs/triangulations counting

(9)

where is actually a vector , except for the restriction sum is taken only over partitions of by natural numbers excluding 1 and 2, and is a number of (possibly disconnected) ribbon graphs with -valent vertices, that can be drawn on a surface of genus .

e) Determinantal representation

(10)

where

(11)

f) Character expansion

(12)

where is a Schur polynomial in symmetric sum variables .

Problem 2. Connected vs. disconnected

Find the expression for the generating function of cumulants up to 8-th order (the least order, where disconnected diagrams appear in ). Check for that defined as

(13)

actually count connected only ribbon graphs.

Grading scheme

Both talks and HW are graded on 5-point scale at first. Talks grade is just an average of individual talks grades. Homework grade is calculated as follows

(14)

Next, raw grade is calculated

(15)

and converted in final grade on 10-point scale

(16)

This function almost perfectly maps unsatisfactory, satisfactory, good, and excellent grades from one scale to another.

References

[FGZ95]
P. Francesco, P. Ginsparg, and J. Zinn-Justin, “2D gravity and random matrices,” Phys. Rep., vol. 254, no. 1-2, pp. 1–133, 1995, doi: 10.1016/0370-1573(94)00084-g. arXiv: hep-th/9306153.
[Mor25]
A. Morozov, “Integrability and matrix models,” in Encyclopedia of mathematical physics, in Encyclopedia of mathematical physics. , Elsevier, 2025, pp. 168–174. doi: 10.1016/b978-0-323-95703-8.00040-9. arXiv: 2212.02632.
[Ski18]
D. Skinner, “Quantum field theory ii,” 2018. Available: https://www.damtp.cam.ac.uk/user/dbs26/AQFT.html
[EKR15]
B. Eynard, T. Kimura, and S. Ribault, “Random matrices,” 2015. arXiv: 1510.04430.
[Ton18]
D. Tong, “Gauge theory,” 2018. Available: https://www.damtp.cam.ac.uk/user/tong/gaugetheory/gt.pdf
[Zee10]
A. Zee, Quantum field theory in a nutshell, 2nd ed. in In a nutshell Princeton. Princeton University Press, 2010.
[Mor94]
A. Y. Morozov, “Integrability and matrix models,” Phys.-Uspekhi, vol. 37, no. 1, pp. 1–55, 1994, doi: 10.1070/pu1994v037n01abeh000001. arXiv: hep-th/9303139.
[PP12]
K. B. Petersen and M. S. Pedersen, “Matrix cookbook,” 2012. Available: https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf
[Mig83]
A. Migdal, “Loop equations and 1n expansion,” Phys. Rep., vol. 102, no. 4, pp. 199–290, 1983, doi: 10.1016/0370-1573(83)90076-5.
[Kaz89]
V. Kazakov, “The appearance of matter fields from quantum fluctuations of 2d-gravity,” Mod. Phys. Lett. A, vol. 04, no. 22, pp. 2125–2139, 1989, doi: 10.1142/s0217732389002392.
[MS09]
A. Morozov and S. Shakirov, “Generation of matrix models by $ w$-operators,” J. High Energy Phys., vol. 2009, no. 04, p. 064, 2009, doi: 10.1088/1126-6708/2009/04/064. arXiv: 0902.2627.
[Mir94]
A. Mironov, “2D gravity and matrix models i: 2d gravity,” Int. J. Modern Phys. A, vol. 09, no. 25, pp. 4355–4405, 1994, doi: 10.1142/s0217751x94001746. arXiv: hep-th/9312212.