Two-Loop Superstrings on Orbifold Compactifications

TL;DR

This paper constructs the two-loop superstring measure on orbifold compactifications, demonstrating the vanishing of the cosmological constant in certain models and the non-vanishing in others, advancing understanding of superstring compactifications.

Contribution

It provides a first-principles construction of the two-loop measure for superstrings on orbifolds, including generalizations to NS backgrounds and applications to specific models.

Findings

Orbifold by a single Z2-twist yields a vanishing cosmological constant.

A Kachru-Kumar-Silverstein type model with a Z2-twist by fermion parity has a non-zero cosmological constant.

The measure is unambiguous and gauge slice independent.

Abstract

The two-loop chiral measure for superstring theories compactified on $\bZ_2$ reflection orbifolds is constructed from first principles for even spin structures. This is achieved by a careful implementation of the chiral splitting procedure in the twisted sectors and the identification of a subtle worldsheet supersymmetric and supermoduli dependent shift in the Prym period. The construction is generalized to compactifications which involve more general NS backgrounds preserving worldsheet supersymmetry. The measures are unambiguous and independent of the gauge slice. Two applications are presented, both to superstring compactifications where 4 dimensions are $\bZ_2$-twisted and where the GSO projection involves a chiral summation over spin structures. The first is an orbifold by a single $\bZ_2$-twist; here, orbifolding reproduces a supersymmetric theory and it is shown that its cosmological constant indeed vanishes. The second model is of the type proposed by Kachru-Kumar-Silverstein and additionally imposes a $\bZ_2$-twist by the parity of worldsheet fermion number; it is shown here that the corresponding cosmological constant does not vanish pointwise on moduli space.