Lectures on Two-Loop Superstrings

TL;DR

This work provides a first-principles, slice-independent construction of the two-loop superstring measure for even spin structures, built from the super period matrix and a careful treatment of odd supermoduli. It delivers an explicit genus-2 chiral measure in terms of theta constants with a new modular object Ξ6[δ], establishes a unique modular covariant GSO projection, and shows the cosmological constant and low-point graviton amplitudes vanish pointwise on moduli space. The analysis includes rigorous degeneration checks, finiteness of the disconnected 4-point piece, and a complete discussion of the two-loop cosmological constant for c=15 and N=1, with extensions to compactifications. Overall, the lectures advance our understanding of higher-loop amplitudes in Type II and Heterotic strings and outline the challenges and prospects for extending these results to higher genus. The results have implications for non-renormalization theorems, effective actions, and the role of supersymmetry in stabilizing quantum corrections.

Abstract

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.