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2 and one double logarithmic solution Fˆ := Xˆ ρ−1 . We define the mirror map for the K3 as tˆa (z) = Xˆ a (z), Xˆ 0 a = 1, . . , ρ − 2. 7) The couplings C za ,z b transform like sections of the bundle Sym2 T ∗ M ⊗ L−2 , where M is the moduli space of complex structures on the K3 and L is the Kähler line bundle. t. 8) where ηˆ ab are the classical intersection numbers of the generators of the Picard lattice of the mirror K3, related to the ηi j above by ηi j = h 0,ρ−1 ⊕ ηˆ ab , h 0,ρ−1 = 01 .

For each of these blocks there is a different Narain lattice J with different α J , β J , and we will denote g J g = J (τ, α J , β J ). 8) J where g IJ = Pg (q) Y g−1 g J (τ ) f J (q). 9) In this equation, Pg (q) is a one-loop correlation function of the bosonic fields and is given by [37,3] e−π λ 2τ 2 2π η3 λ ϑ1 (λ|τ ) ∞ 2 (2π λ)2g Pg (q). 10) 32 A. Klemm, M. Mariño f J (q) is a modular form which depends on the details of the internal CFT. 9) is a moduli-dependent function related to the Kähler potential as K = − log Y .

It turns out that there are two natural lattice reductions to perform the computation: the geometric reduction, and the Borcherds-Harvey-Moore (BHM) reduction. We will present the results for the couplings in both reductions and we will also propose a type IIA interpretation of these results. 8) for the heterotic FHSV model. This is rather straightforward by using the results of the previous section. We have four orbifold blocks, but the first block (corresponding to h = g = 0) vanishes. 1) (1 − q n )−12 (1 + q n−1/2 )−12 .

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Communications In Mathematical Physics - Volume 280 by M. Aizenman (Chief Editor)

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