Integrability and matrix models

@article{Morozov1993IntegrabilityAM,
  title={Integrability and matrix models},
  author={Aleksey Morozov},
  journal={Physics-Uspekhi},
  year={1993},
  volume={37},
  pages={1 - 55}
}
  • A. Morozov
  • Published 24 March 1993
  • Engineering
  • Physics-Uspekhi
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. 
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References

SHOWING 1-10 OF 36 REFERENCES
MATRIX MODELS AS INTEGRABLE SYSTEMS: FROM UNIVERSALITY TO GEOMETRODYNAMICAL PRINCIPLE OF STRING THEORY
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.
TLDR
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.
Intersection theory, integrable hierarchies and topological field theory
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
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
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
...
...