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12) The orbifold blocks for two complex fermions with symmetry ψ → −ψ are given by (see for example , Eq. 13) (τ ) η2 The treatment of the two fermions coming from The 00 block in the a, b sector is simply Z . a b ϑ = 1,9 ⊕ 1,9 ⎞3/4 ⎠ . 14) is slightly more delicate. (τ ) . 15) Let us now analyze the invariant states in the NS sector. 16) where n i , m i > 0 are half-integers. The above states have the sign ±1, respectively, under the Z2 symmetry generator g which exchanges the two lattices.
After a Poisson resummation over m one finds : ∞ ∞ n g (r ) F(λ) = g=0 r ∈H2 (M,Z) d=1 dλ 1 2 sin d 2 2g−2 e−dr ·t . 23). This expression, which takes into account the spectrum of “electric” states associated to D2-D0 branes, is valid in the large radius limit of the CY compactification, since in this region the lightest states are indeed the D2 and D0 branes and their bound states, while the D4 and D6 “magnetic” states are heavy. 29) leads to strong structural predictions for the topological string amplitudes Fg when written in terms of GV invariants.
30) are given by r · t = mt + + nt − + q · t, r t = (2m + 1)t + + (2n + 1)t − + q · t. 32) is due to the shift by β = δ, and we have 48 A. Klemm, M. Mariño relabelled n → n + 1. 11). 30) can be simplified as follows. First, one notices that cg2 (1/4 + p/2) equals (−1) p+1 cg3 (1/4 + p/2). This is easy to see by noting that they are the coef2 1 1 ficients of modular forms related by q 2 → −q 2 . Therefore, the J = 2 and J = 3 contributions are equal and add up. We will call their contribution the contribution of the twisted sector, while the contribution from J = 1 will be called the contribution of the untwisted sector.