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

TL;DR

This paper explores a nonperturbative approach to 2D quantum gravity with a complex action, demonstrating a series representation over moduli spaces of punctured spheres and connecting it to erl summability and the Friedan-Shenker framework.

Contribution

It introduces a novel nonperturbative formulation of 2D gravity involving erl summable series over moduli spaces of punctured spheres, linking it to the Friedan-Shenker approach.

Findings

The specific heat exhibits physically consistent behavior.

The series can be expressed as an integral over infinitely punctured spheres.

Conjecture that the expansions can be derived via the Duistermaat-Heckman theorem.

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 \`a 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.