Literature
- [Mor25] Core resource. Extremely succinct, but covering most of the main ideas.
- [Mor94] Detailed review of integrability in matrix models and some other special topics.
- [Kaz21] Great video-lectures on Matrix Models from one of the greatest contributors to the field.
- [EKR15] Self-consistent text on various aspects of matrix models, including the ones we are interested in. Ribbon graphs and eigenvalue representation are explained very well.
- -representation and superintegrability are discussed in detail in [MO22].
- For matrix models in the context of 2 quantum gravity see [BIZ80; LZ04; Wit90].The latter one contains a lot of useful material on matrix models in general.
- Matrix models in the context of QCD are considered in the Chapter VII.4 of [Zee10]. They also appear in the large lattice gauge formulation of QCD [BG80; GW80].
Problem set
Matrix calculus
Evaluate
at the order , where is the measure on Hermitian matrices
Virasoro constraints
Using previous result obtain the expression for the Virasoro operators
annihilating the formal generating function
via Virasoro constraints
(Matrix) models
For the Gaussian Hermitian matrix model find the generating function
where
and is the number of parts equaling , up to 4-th order ( ) using the methods below. Compare the results.
Ward identities
-representation
where
and derivative by is understood as multiplication by .
Wick's theorem
Graphs/triangulations counting
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 .
Determinantal representation
where
Character expansion
(org-element-property :name datum)where is a Schur polynomial in symmetric sum variables .
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
actually count connected only ribbon graphs.
Grading
Both talk and HW are graded on 5-point scale at first. You have 9 exercises for the homework, so
Next, raw grade is calculated
and converted in final grade on 10-point scale
This function almost perfectly maps unsatisfactory, satisfactory, good, and excellent grades from one scale to another.
Individual results
| Talk | 1 | 2 | 3.1 | 3.2 | 3.3 | 3.4 | 3.5 | 3.6 | 4 | Grade | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Azheev | 5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 9 | |
| Egor | 5 | 1 | 1 | 2 | 5 | ||||||
| Hasib | 5 | 1 | 1 | 1 | 1 | 5 | |||||
| Kostya | 5 | 1 | 1 | 1 | 1 | 5 | |||||
| Lena | 5 | 1 | 1 | 1 | 1 | 5 | |||||
| Matthew | 5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 10 |