Communications in Mathematical Physics

, Volume 148, Issue 3, pp 469–485 | Cite as

The solution space of the unitary matrix model string equation and the Sato Grassmannian

  • Konstantinos N. Anagnostopoulos
  • Mark J. Bowick
  • Albert Schwarz


The space of all solutions to the string equation of the symmetric unitary one-matrix model is determined. It is shown that the string equation is equivalent to simple conditions on pointsV1 andV2 in the big cell Gr(0) of the Sato Grassmannian Gr. This is a consequence of a well-defined continuum limit in which the string equation has the simple form
matrices of differential operators. These conditions onV1 andV2 yield a simple system of first order differential equations whose analysis determines the space of all solutions to the string equation. This geometric formulation leads directly to the Virasoro constraintsL n (n≧0), whereL n annihilate the two modified-KdV τ-functions whose product gives the partition function of the Unitary Matrix Model.


Differential Equation Neural Network Complex System Partition Function Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Mehta, M.L.: Random Matrices. 2nd Edition, Boston: Academic Press 1991Google Scholar
  2. 2.
    Dyson, F.J.: Statistical theory of the energy levels of complex systems I. J. Math. Phys.3, 140–156 (1962)CrossRefGoogle Scholar
  3. 3.
    Dyson, F.J.: Statistical theory of the energy levels of complex systems II. J. Math. Phys.3, 157–165 (1962)CrossRefGoogle Scholar
  4. 4.
    Dyson, F.J.: Statistical theory of the energy levels of complex systems III. J. Math. Phys.3, 166–175 (1962)CrossRefGoogle Scholar
  5. 5.
    't Hooft, G.: A planar diagram theory for strong interactions. Nucl. Phys.B72, 461–473 (1974)CrossRefGoogle Scholar
  6. 6.
    Gross, D., Witten, E.: Possible third-order phase transition in the large-N lattice gauge theory. Phys. Rev.D21, 446–453 (1980)Google Scholar
  7. 7.
    Wadia, S.R.: Dyson-Schwinger equations approach to the large-N limit: Model systems and string representation of Yang-Mills theory. Phys. Rev.D24, 970–978 (1981)Google Scholar
  8. 8.
    Brézin, E., Kazakov, V.: Exactly solvable field theories of closed strings. Phys. Lett.B236, 144–149 (1990)CrossRefGoogle Scholar
  9. 9.
    Douglas, M., Shenker, S.: Strings in less than one dimension. Nucl. Phys.B335, 635–654 (1990)CrossRefGoogle Scholar
  10. 10.
    Gross, D., Migdal, A.: Nonperturbative two-dimensional quantum gravity. Phys. Rev. Lett.64, 127–130 (1990)CrossRefGoogle Scholar
  11. 11.
    Gross, D., Migdal, A.: A nonperturbative treatment of two-dimensional quantum gravity. Nucl. Phys.B340, 333–365 (1990)CrossRefGoogle Scholar
  12. 12.
    Bowick, M.J., Brézin, E.: Universal scaling of the tail of the density of eigenvalues in random matrix models. Phys. Lett.B268, 21–28 (1991)CrossRefGoogle Scholar
  13. 13.
    Periwal, V., Shevitz, D.: Unitary matrix models as exactly solvable string theories. Phys. Rev. Lett.64, 1326–1329 (1990)CrossRefGoogle Scholar
  14. 14.
    Periwal, V., Shevitz, D.: Exactly solvable unitary matrix models: multicritical potentials and correlations. Nucl. Phys.B344, 731–746 (1990)CrossRefGoogle Scholar
  15. 15.
    Neuberger, H.: Scaling regime at the large-N phase transition of two-dimensional pure gauge theories. Nucl. Phys.B340, 703–720 (1990)CrossRefGoogle Scholar
  16. 16.
    Neuberger, H.: Non-perturbative contributions in models with a non-analytic behavior at infiniteN. Nucl. Phys.B179, 253–282 (1981)CrossRefGoogle Scholar
  17. 17.
    Demeterfi, K., Tan, C.I.: String equations from unitary matrix models. Mod. Phys. Lett.A5, 1563–1574 (1990)CrossRefGoogle Scholar
  18. 18.
    Martinec, E.: On the origin of integrability in matrix models. Commun. Math. Phys.138, 437–449 (1991)CrossRefGoogle Scholar
  19. 19.
    Bowick, M.J., Morozov, A., Shevitz, D.: Reduced unitary matrix models and the hierarchy of τ-functions. Nucl. Phys.B354, 496–530 (1991)CrossRefGoogle Scholar
  20. 20.
    Crnković, Č., Douglas, M., Moore, G.: Physical solutions for unitary matrix models. Nucl. Phys.B360, 507–523 (1991)CrossRefGoogle Scholar
  21. 21.
    Crnković, Č., Moore, G.: Multicritical multi-cut matrix models. Phys. Lett.B257, 322–328 (1991)CrossRefGoogle Scholar
  22. 22.
    Crnković, Č., Douglas, M., Moore, G.: Loop equations and the topological structure of multi-cut models. Preprint YCTP-P25-91 and RU-91-36Google Scholar
  23. 23.
    Hollowood, T., Miramontes, L., Pasquinucci, A., Nappi, C.: Hermitian vs. Anti-Hermitian 1-matrix models and their hierarchies. Preprint IASSNS-HEP-91/59 and PUPT-1280Google Scholar
  24. 24.
    Douglas, M.R.: Strings in less than one dimensions and the generalized KdV hierarchies. Phys. Lett.B238, 176–180 (1990)CrossRefGoogle Scholar
  25. 25.
    Schwarz, A.: On solutions to the string equation. Mod. Phys. Lett.A6, 2713–2725 (1991)CrossRefGoogle Scholar
  26. 26.
    Dijkgraaf, R., Verlinde, H., Verlinde, E.: Loop equations and Virasoro constraints in non-perturbative two-dimensional quantum gravity. Nucl. Phys.B348, 435–456 (1991)CrossRefGoogle Scholar
  27. 27.
    Fukuma, M., Kawai, H., Nakayama, R.: Continuum Schwinger-Dyson equations and universal structures in two-dimensional quantum gravity. Int. J. Mod. Phys.A6, 1385–1406 (1991)CrossRefGoogle Scholar
  28. 28.
    Kac, V., Schwarz, A.: Geometric interpretation of the partition function of 2D gravity. Phys. Lett.B257, 329–334 (1991)CrossRefGoogle Scholar
  29. 29.
    Fukuma, M., Kawai, H., Nakayama, R.: Infinite-dimensional Grassmannian structure of two-dimensional gravity. Preprint UT-572-TOKYOGoogle Scholar
  30. 30.
    Schwarz, A.: On some mathematical problems of 2D-gravity andW h-gravity. Mod. Phys. Lett.A6, 611–616 (1991)CrossRefGoogle Scholar
  31. 31.
    Fukuma, M., Kawai, H., Nakayama, R.: Explicit solution forp−q duality in two dimensional gravity. Preprint UT-582-TOKYOGoogle Scholar
  32. 32.
    Anagnostopoulos, K.N., Bowick, M.J., Ishibashi, N.: An operator formalism for unitary matrix models. Mod. Phys. Lett.A6, 2727–2739 (1991)CrossRefGoogle Scholar
  33. 33.
    Peterson, D.H., Kac, V.G.: Infinite flag varieties and conjugacy theorems. Proc. National Acad. Sci. USA80, 1778–1782 (1983)Google Scholar
  34. 34.
    Kac, V.G., Peterson, D.H.: Lectures on the infinite Wedge representation and the MKP hierarchy. Sém. Math. Sup.,102, pp. 141–184. Montréal: Presses University Montréal 1986Google Scholar
  35. 35.
    Kac, V.G., Wakimoto, M.: Exceptional hierarchies of soliton equations. Proc. Symposia Pure Math.49, 191–237 (1989)Google Scholar
  36. 36.
    Moore, G.: Geometry of the string equations. Commun. Math. Phys.133, 261–304 (1990)CrossRefGoogle Scholar
  37. 37.
    Moore, G.: Matrix models of 2D gravity and isomonodromic deformation. Prog. Theor. Phys. Suppl.102, 255–285 (1990)Google Scholar
  38. 38.
    Brézin, E., Itzykson, C., Parisi, G., Zuber, J.B.: Planar diagrams. Commun. Math. Phys.59, 35–51 (1978)CrossRefGoogle Scholar
  39. 39.
    Myers, R.C., Periwal, V.: Exact solutions of critical self dual unitary matrix models. Phys. Rev. Lett.65, 1088–1091 (1990)CrossRefGoogle Scholar
  40. 40.
    The most appropriate exposition for our purposes is given in Mulase, M.: Category of Vector Bundles on Algebraic Curves and Infinite Dimensional Grassmannians. Int. J. Math.1, 293–342 (1990)CrossRefGoogle Scholar
  41. 41.
    Segal, G., Wilson, G.: Loop groups and equations of KdV type. Publ. I.H.E.S.61, 5–65 (1985)Google Scholar
  42. 42.
    Watterstam, A.: A solution to the string equation of unitary matrix models. Phys. Lett.B263, 51–58 (1991)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Konstantinos N. Anagnostopoulos
    • 1
  • Mark J. Bowick
    • 1
  • Albert Schwarz
    • 2
  1. 1.Physics DepartmentSyracuse UniversitySyracuseUSA
  2. 2.Department of MathematicsUniversity of CaliforniaDavisUSA

Personalised recommendations