The geometry of string perturbation theory

Abstract

This paper is devoted to recent progress made towards the understanding of closed bosonic and fermionic string perturbation theory, formulated in a Lorentz-covariant way on Euclidean space-time. Special emphasis is put on the fundamental role of Riemann surfaces and supersurfaces. The differential and complex geometry of their moduli space is developed as needed. New results for the superstring presented here include the supergeometric construction of amplitudes, their chiral and superholomorphic splitting and a global formulation of supermoduli space and amplitudes.

Comments on noncommutative gravity

“…From the calculation of the disk amplitude we get 18) with .19) Following the same procedure as above, it can be seen that this part of the three-graviton amplitude induces the following term in the effective action…”

“…We have to calculate the disk correlation function of three graviton vertex operators [17,18] 1) where the symmetric-traceless polarization tensor has to satisfy the transversality conditions p µ ε µν (p) and the momentum has to be on-shell, p 2 = 0, for the vertex operator to be a primary field with conformal weight (1,1). Naively, the coupling between three on-shell gravitons (or gauge fields) vanishes for kinematical reasons.…”

M-theory and the string genus expansion

“…One now calculates the measure on moduli space, being especially careful with the normalisation, [10,11]. So far the above has been exactly as for the string.…”

“…To calculate the measure on moduli space one typically writes the moduli in terms of quadratic differentials (see [10,11]). Then the norm of each quadratic differential is given by:…”

“…Following, [10] the space of Riemann surface metrics is parameterised byγ = exp(δv)e 2σγ . Whereγ is transverse to the orbits of Weyl(Σ) × Diff 0 (Σ) and exp(δv) denotes a diffeomorphism generated by the vector field δv.…”