Communications in Mathematical Physics

, Volume 86, Issue 3, pp 321–326 | Cite as

On the positivity of the effective action in a theory of random surfaces

  • E. Onofri


It is shown that the functional\(S[\eta ] = \frac{1}{{24\pi }}\int {\left( {\frac{1}{2}\left| {\nabla \eta } \right|^2 + 2\eta } \right)d\mu _0 }\), defined onC functions on the two-dimensional sphere, satisfies the inequalityS[η]≧0 if η is subject to the constraint\(\int {(e^\eta - 1)d\mu _0 = 0}\). The minimumS[η]=0 is attained at the solutions of the Euler-Lagrange equations. The proof is based on a sharper version of Moser-Trudinger's inequality (due to Aubin) which holds under the additional constraint\(\int {e^\eta xd\mu _0 = 0}\); this condition can always be satisfied by exploiting the invariance ofS[η] under the conformal transformations ofS2. The result is relevant for a recently proposed formulation of a theory of random surfaces.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
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  1. 1.
    Onofri, E., Virasoro, M.: Nucl. Phys.B201, 159–175 (1982)Google Scholar
  2. 2.
    Berger, M. S.: J. Diff. Geom.5, 325–332 (1971)Google Scholar
  3. 3.
    Moser, J.: Indiana Univ. Math. J. 20 (11), 1077–1092 (1971)Google Scholar
  4. 4.
    Moser, J.: On a non-linear problem in differential geometry. In: Dynamical Systems, Peixoto, M. M. (ed.) New York: Academic Press Inc., 1973 pp. 273–280Google Scholar
  5. 5.
    Aubin, T.: J. Funct. Anal.32, 148–174 (1979)Google Scholar
  6. 6.
    Berger, M. S.: Non-linearity and functional analysis. New York: Academic Press Inc., 1977, Chap. 6 and references thereinGoogle Scholar
  7. 7.
    Kazdan, J. L., Warner, F. W.: Ann. Math.99, 14–47 (1974)Google Scholar
  8. 8.
    Gluck, H.: Bull. Am. Math. Soc.81, 313–329 (1975)Google Scholar
  9. 9.
    Polyakov, A. M.: Phys. Lett.103B, 207–210 (1981)Google Scholar
  10. 10.
    Hardy, H. G., Littlewood, J. E., Pólya, G.: Inequalities. Cambridge: Cambridge University Press, 2nd edn. 1952Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • E. Onofri
    • 1
  1. 1.Theory DivisionCERNGeneva 23Switzerland

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