Nonperturbative 2D Gravity, Punctured Spheres and $\Theta$-Vacua in String Theories

Abstract

We consider a model of 2D gravity with the coefficient of the Einstein-Hilbert action having an imaginary part $\pi /2$. This is equivalent to introduce a $\Theta$-vacuum structure in the genus expansion whose effect is to convert the expansion into a series of alternating signs, presumably Borel summable. We show that the specific heat of the model has a physical behaviour. It can be represented nonperturbatively as a series in terms of integrals over moduli spaces of punctured spheres and the sum of the series can be rewritten as a unique integral over a suitable moduli space of infinitely punctured spheres. This is an explicit realization à la Friedan-Shenker of 2D quantum gravity. We conjecture that the expansion in terms of punctures and the genus expansion can be derived using the Duistermaat-Heckman theorem. We briefly analyze expansions in terms of punctured spheres also for multicritical models.

12 pages + 2 figures available upon request. LaTeX File

References in corpus (2)

• Combinatorics of the Modular Group II: the Kontsevich integrals
• Polynomial averages in the Kontsevich model

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• Algebraic-geometrical formulation of two-dimensional quantum gravity