Ivancevic_Applied-Diff-Geom

1230 Applied Differential Geometry: A Modern IntroductionThis suggests an identification of the form f h (t A ) ∼ F top,h (t A ) and g top ∼(CX 0 ) −1 . For later purposes, we need precise relations between super–gravity and topological string quantities, including numerical coefficients.These can be determined by studying the limit of a large Calabi–Yau space.In the super–gravity notation, the genus 0 and 1 terms in the largevolume are given byF ( CX Λ , 256 ) = C 2 D ABCX A X B X CX 0− 1 6 c X A2AX 0 + · · ·= (CX 0 ) 2 D ABC t A t B t C − 1 6 c 2At A + · · · ,∫where c 2A = c 2 ∧ α A ,Mwith c 2 being the second Chern class of M, and C ABC = −6D ABC arethe 4–cycle intersection numbers. These terms are normalized so that themixed entropy S BH is given by (6.340). On the other hand, the topologicalstring amplitude in this limit is given byF top = − (2π)3 igtop2 D ABC t A t B t C − πi12 c 2At A + · · · (6.342)The normalization here is fixed by the holomorphic anomaly equations in[Bershadsky et. al. (1994)], which are nonlinear equations for F top,h .Comparing the one–loop terms in (6.341) and (6.342), which are independentof g top , we findF (CX Λ , 256) = − 2iπ F top(t A , g top ).Given this, we can compare the genus 0 terms to findThis impliesg top = ± 4πiCX 0 .ln Z BH = −π Im [ F (CX Λ , 256) ] = F top + ¯F topZ BH (φ Λ , p Λ ) = |Z top (t A , g top )| 2 , witht A = pA + iφ A /πp 0 + iφ 0 /π , g 4πitop = ±p 0 + iφ 0 /π .and