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.

We provide a unified construction of the symplectic forms which arise in the solution of both N=2 supersymmetric Yang-Mills theories and soliton equations. Their phase spaces are Jacobian-type bundles over the leaves of a foliation in a universal configuration space. On one hand, imbedded into finite-gap solutions of soliton equations, these symplectic forms assume explicit expressions in terms of the auxiliary Lax pair, expressions which generalize the well-known Gardner-Faddeev-Zakharov bracket for KdV to a vast class of 2D integrable models; on the other hand, they determine completely the effective Lagrangian and BPS spectrum when the leaves are identified with the moduli space of vacua of an N=2 supersymmetric gauge theory. For SU(N c ) with N f ≤ N c + 1 flavors, the spectral curves we obtain this way agree with the ones derived by Hanany and Oz and others from physical considerations.

The two-loop contribution to the Type IIB low energy effective action term $D^4 R^4$, predicted by SL(2,Z) duality, is compared with that of the two-loop 4-point function derived recently in superstring perturbation theory through the method of projection onto super period matrices. For this, the precise overall normalization of the 4-point function is determined through factorization. The resulting contributions to $D^4 R^4$ match exactly, thus providing an indirect check of SL(2,Z) duality. The two-loop Heterotic low energy term $D^2F^4$ is evaluated in string perturbation theory; its form is closely related to the $D^4 R^4$ term in Type II, although its significance to duality is an open issue.Comment: 44 pages, LaTeX, epsfig, 1 figur

The N-point amplitudes for the Type II and Heterotic superstrings at two-loop order and for N ≤ 4 massless NS bosons are evaluated explicitly from first principles, using the method of projection onto super period matrices introduced and developed in the first five papers of this series. The gauge-dependent corrections to the vertex operators, identified in paper V, are carefully taken into account, and the crucial counterterms which are Dolbeault exact in one insertion point and de Rham closed in the remaining points are constructed explicitly. This procedure maintains gauge slice independence at every stage of the evaluation. Analysis of the resulting amplitudes demonstrates, from first principles, that for N ≤ 3, no two-loop corrections occur, while for N = 4, no two-loop corrections to the low energy effective action occur for R 4 terms in the Type II superstrings, and for F 4 , F 2 F 2 , F 2 R 2 , and R 4 terms in the Heterotic strings.

We develop a Hamiltonian theory for 2D soliton equations. In particular, we identify the spaces of doubly periodic operators on which a full hierarchy of commuting flows can be introduced, and show that these flows are Hamiltonian with respect to a universal symplectic form ω = 1 2 Res ∞ < Ψ * 0 δL ∧ δΨ 0 > dk. We also construct other higher order symplectic forms and compare our formalism with the case of 1D solitons. Restricted to spaces of finite-gap solitons, the universal symplectic form agrees with the symplectic forms which have recently appeared in non-linear WKB theory, topological field theory, and Seiberg-Witten theories. We take the opportunity to survey some developments in these areas where symplectic forms have played a major role.* Research supported in part by the National Science Foundation under grant DMS-95-05399. g i=1 dλ(z i ) when finite-gap solitons are imbedded in the space of doubly periodic operators. Here we show that it is a symplectic form in its own right on L(b), and that with respect to this form, the hierarchy of 2D flows is Hamiltonian. Their Hamiltonians are shown to be nH n+m , where H s are the coefficients of the expansion of the quasi-momentum in terms of the quasi-energy.• Our formalism is powerful enough to encompass many diverse symplectic structures for 1D solitons. For example, ω reduces to the Gardner-Faddeev-Zakharov symplectic