Models of String Theory and 2-D Quantum Gravity are Solved Exactly

Abstract

In how many ways may a quadrilateral be dissected into a given number of smaller quadrilaterals? Or a hexagon into triangles? Or, for that matter, any polygon into given numbers of other polygons? Such questions about enumeration of dissections of a planar surface attracted the attention of mathematicians—most notably, of Leonhard Euler—in the middle of the 18th century. Mathematicians solved many cases of such problems using methods of graph theory in the 1960s. In 1978, however, physicists, in their attempts to enumerate Feynman diagrams in certain theories, independently developed a rather elegant and general method that can be used to enumerate dissections of any surface. Physical insights into the divergences in the solutions obtained by those general methods have now led to exact solutions—the first nontrivial exact solutions to date—of simple models of quantum gravity in two dimensions and of simple string theories. String theorists hope that the solutions will provide much-needed insights into the structure of those theories, just as solving the Schrödinger equation for simple, one-dimensional potentials yields insights into quantum dynamics: For example, the solutions for one-dimensional potentials show features, such as the existence of the zero-point energy and the possibility of tunneling, that distinguish quantum mechanical behavior from the classical one. In the absence of any exact solutions, attempts to understand the content of string theories had been limited to perturbation expansions in the string coupling constant. The new, exact solutions show that string theories have features, not fully understood yet, that no perturbation expansion in the coupling constant could unravel.