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In the simplest situation C is a smooth curve of genus g in the class r , then δ = 0 and n g (r ) = (−1)dim(MC ) χ (MC ). 4) This can be understood directly as follows. If C is smooth the Jac(C) is non-degenerate and carries Ig as su(2) L Lefshetz representation of the fibre. 26) gives –up to sign— the Euler number of the base. If the contribution to n g−δ (r ) comes only from an irreducible curve with δ nodes we can calculate in certain situations χ (C ( p) ) to obtain the BPS number. We now apply these ideas to D2 branes wrapping curves C in the fibre of the Calabi-Yau manifold M of the FHSV model.
9) is a moduli-dependent function related to the Kähler potential as K = − log Y . We will also define the holomorphic counterpart Pg (q) of Pg (q) by ∞ 2 2π η3 λ ϑ1 (λ|τ ) = (2π λ)2g Pg (q). 11) g=0 The quantities Pg (q) can be explicitly written in terms of generalized Eisenstein series. To do this, one uses the expansion ∞ 2π η3 z = − exp ϑ1 (z|τ ) ζ (2k) E 2k (τ )z 2k . 12) If we now introduce the polynomials Sk through: ∞ ∞ exp xn z n Sn (x1 , . . 13) n=0 we can easily check that Pg (q) is a quasimodular form of weight (2g, 0) given by Pg (q) = Sg xk = |B2k | E 2k (q) , k(2k)!
16) corresponds to the RR Wilson line background along the H 0 (E, Z) direction [18,4]. 19) matches with the geometric computation of BPS invariants proposed in . Moreover, in Sect. 6 we will find perfect agreement of the heterotic predictions with a B-model computation of Fg for g ≤ 4. 19) is similar to other results obtained for heterotic models, it has some additional properties that make it particularly simple. 22) where Ci j is the intersection matrix of 1,1 ⊕ E 8 (−1). 23) which can be proved by comparing the even part of the τ derivative of f 1 (q)P1 (q) with the even part of f 1 (q)P2 (q).