Algebraic geometry and path integrals for closed strings

The pth loop contribution to the partition function for closed strings is studied by applying recent mathematical results on the geometry of the moduli space Mp of smooth algebraic curves of genus p. By reasoning on determinants of operators and line bundles over Mp, we get a geometric explanation of the critical dimensions 26 and 10. The extension of path integrals for strings to the compactified moduli space Mp of stable curves is also discussed. While the Weil-Peterson measure has a continuous extension on Mp, the bosonic path integral has a bad behaviour on the boundary Mp - Mp. Instead, the functional approach to the spinning string of Ramond-Neveu-Schwarz seems to yield a finite pth loop contribution to the partition function. © 1986.