Lectures on Two-Loop Superstrings
  In these lectures, recent progress on multiloop superstring perturbation theory is reviewed. A construction from first principles is given for an unambiguous and slice-independent two-loop superstring measure on moduli space for even spin structure. A consistent choice of moduli, invariant under local worldsheet supersymmetry is made in terms of the super-period matrix. A variety of subtle new contributions arising from a careful gauge fixing procedure are taken into account.   The superstring measure is computed explicitly in terms of genus two theta-functions and reveals the importance of a new modular object of weight 6. For given even spin structure, the measure exhibits a behavior under degenerations of the worldsheet that is consistent with physical principles. The measure allows for a unique modular covariant GSO projection. Under this GSO projection, the cosmological constant, the 1-, 2- and 3- point functions of massless supergravitons vanish pointwise on moduli space. A certain disconnected part of the 4-point function is shown to be given by a convergent integral on moduli space. A general consistent formula is given for the two-loop cosmological constant in compactifications with central charge c=15 and with N=1 worldsheet supersymmetry. Finally, some comments are made on possible extensions of this work to higher loop order. 
Eric D'Hoker and D.H. Phong