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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
Article

Abstract

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.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
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.

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Copyright information

© Springer-Verlag 1982

Authors and Affiliations

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

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