Non-Perturbative String Theory

Abstract

The conventional approach to two dimensional gravity and to string theory is perturbative with respect to fluctuations of the topology. One sums over two dimensional geometries by first performing the functional integral for fixed topology (genus= number of handles) and then summing over genus. However, this sum is very badly behaved. The higher terms grow as factorials of the genus, and the positivity of these terms renders the series non Borel summable.[1]This situation is made worse by the lack of an adequate nonperturbative framework for the theory. Such a framework, (for example, a useful formulation of second quantized string theory) should be capable of reproducing the topological series as an asymptotic expansion, valid in the perturbative domain; but it should also provide a physical picture and a mathematical framework valid for strong coupling. It is essential that we develop nonperturbative methods if we are to relate unified string theories to the real world. At the perturbative level of string theory there are many too many possible worlds, i.e. classical vacua about which consistent perturbative expansions can be made. All of them have undesired features, such as unbroken supersymmetry and massless dilatons. One must hope that nonperturbative physics will lift the degeneracy and break the unwanted symmetries. This is strongly suggested by the divergence and non Borel summability of perturbation theory which can be taken as an indication of the nonperturbative instability of the classical vacua [1].