# Integrability and matrix models

@article{Morozov1993IntegrabilityAM, title={Integrability and matrix models}, author={Aleksey Morozov}, journal={Physics-Uspekhi}, year={1993}, volume={37}, pages={1 - 55} }

The theory of matrix models is reviewed from the point of view of its relation to integrable hierarchies. Discrete 1-matrix, 2-matrix, 'conformal' (multicomponent) and Kontsevich models are considered in some detail, together with the Ward identites ('W-constraints'), determinantal formulas and continuum limits, taking one kind of model into another. Subtle points and directions of the future research are also discussed.

## 237 Citations

Matrix Models as Integrable Systems

- Mathematics
- 1995

The theory of matrix models is reviewed from the point of view of its relation to integrable hierarchies. Determinantal formulas, relation to conformal field models, and the theory of generalized…

Generation of matrix models by Ŵ-operators

- Mathematics
- 2009

We show that partition functions of various matrix models can be obtained by acting on elementary functions with exponents of operators. A number of illustrations is given, including the Gaussian…

Matrix Model Tools and Geometry of Moduli Spaces

- Mathematics
- 1997

The relation between matrix models and geometrical structures on moduli spaces of algebraic curves is reviewed. The description of discretized moduli spaces and the generating function for…

Matrix model eigenvalue integrals and twist fields in the su(2)-WZW model

- Mathematics
- 2005

We propose a formula for the eigenvalue integral of the hermitian one matrix model with infinite well potential in terms of dressed twist fields of the su(2) level one WZW model. The expression holds…

M-theory of matrix models

- Computer Science
- 2006

Their M-theory unifies various branches of the Hermitian matrix model (including the Dijkgraaf-Vafa partition functions) with the Kontsevich τ-function, and the corresponding duality relations are reminiscent of instanton and meron decompositions, familiar from the Yang-Mills theory.

Observables and critical behaviour in fermionic matrix models

- Physics
- 1995

We review the properties of adjoint fermion one-, two-, and generic D-dimensional matrix models at large N. We derive and solve the complete sets of loop equations for the correlators of these models…

## References

SHOWING 1-10 OF 36 REFERENCES

MATRIX MODELS AS INTEGRABLE SYSTEMS: FROM UNIVERSALITY TO GEOMETRODYNAMICAL PRINCIPLE OF STRING THEORY

- Mathematics
- 1991

Matrix models are equivalent to certain integrable theories, partition functions being equal to certain τ-functions, i.e., the section of determinant bundles over infinite-dimensional Grassmannian.…

Topological quantum theories and integrable models.

- PhysicsPhysical review. D, Particles and fields
- 1991

It is found that in general the stationary-phase approximation presumes that the initial and final configurations are in different polarizations, as exemplified by the quantization of the SU(2) coadjoint orbit.

Conformal matrix models as an alternative to conventional multi-matrix models

- Computer Science
- 1993

Intersection theory, integrable hierarchies and topological field theory

- Physics
- 1991

The last two years have seen the emergence of a beautiful new subject in mathematical physics. It manages to combine a most exotic range of disciplines: two-dimensional quantum field theory,…

Landau-Ginzburg topological theories in the framework of GKM and equivalent hierarchies

- Mathematics
- 1993

We consider the deformations of “monomial solutions” to the Generalized Kontsevich Model [1, 2] and establish the relation between the flows generated by these deformations with those of N=2…

On connection between topological Landau-Ginzburg gravity and integrable systems

- Physics, Mathematics
- 1995

We study flows on the space of topological Landau-Ginzburg theories coupled to topological gravity. We argue that flows corresponding to gravitational descendants change the target space from a…