In previous chapters, we have seen that string theory at the classical level shows promise of describing the Standard Model, and can realize at least one scenario for physics beyond: low-energy supersymmetry. But there are many puzzles, most importantly the existence of moduli and the related question of the cosmological constant. At tree level, in the Calabi–Yau solutions, the cosmological constant vanishes. But whether this holds in perturbation theory and beyond requires understanding of the quantum theory.

In studying string theory, we have certain tools:

1. (1) weak coupling expansions,

2. (2) long-wavelength (low-momentum, α′) expansions.

We have exploited both of these techniques up to this point. In analyzing string spectra, we have worked in a weak coupling limit. There are corrections to the masses and couplings, for example, in string perturbation theory, and most of the states that we have studied have finite lifetimes. At weak coupling, these effects are small, but at strong coupling, the theories presumably look dramatically different.

In asserting that Calabi–Yau vacua are solutions of the string equations, we used both types of expansions. We wrote the string equations both in lowest order in the string coupling, and also with the fewest number of derivatives (two). Even at weak coupling and in the derivative expansion, we can ask whether Calabi–Yau spaces are actually solutions of the string equations, both classically and quantum mechanically. For example, we have seen that, at lowest order in both expansions, there are typically many massless particles. We might expect tadpoles to appear for these fields, both in the α′ and in loops.