Ivancevic_Applied-Diff-Geom

1228 Applied Differential Geometry: A Modern IntroductionIt is useful to rephrase the above results in the context of type IIB superstringsin terms of geometry of Calabi–Yau. In this case the attractor equationsfix the complex geometry of the Calabi–Yau. The electric/magenticcharges correlate with three cycles of Calabi–Yau. Choosing a symplecticbasis for the three cycles gives a choice of the splitting to electric and magneticcharges. Let A Λ denote a basis for the electric three cycles, B Σ thedual basis for the magnetic charges and Ω the holomorphic 3–form at theattractor point. Ω is fixed up to an overall multiplication by a complexnumber Ω → λΩ. There is a unique choice of λ such that the resulting Ωhas the property that∫∫p Λ = Re Ω = Re[CX Λ ], q Λ = Re Ω = Re[CF 0Λ ],A Λ B ΛwhereRe Ω = 1 (Ω + Ω).2In terms of this choice, the black hole entropy can be written asS BH = π ∫Ω ∧ Ω.4Higher–Order CorrectionsF −term corrections to the action are encoded in a string loop correctedholomorphic prepotential∞∑F (X Λ , W 2 ) = F h (X Λ )W 2h , (6.337)CYh=0where F h can be computed by topological string amplitudes (as we reviewin the next section) and W 2 involves the square of the anti–self dualgraviphoton field strength. This obeys the homogeneity equationX Λ ∂ Λ F (X Λ , W 2 ) + W ∂ W F (X Λ , W 2 ) = 2F (X Λ , W 2 ). (6.338)Near the black hole horizon, the attractor value of W 2 obeys C 2 W 2 = 256,and therefore the exact attractor equations read(p Λ = Re[CX Λ ], q Λ = Re[CF Λ X Λ , 256 )]C 2 . (6.339)This is essentially the only possibility consistent with symplectic invariance.It has been then argued that the entropy as a function of the charges isS BH = πi2 (q Λ ¯C ¯X Λ − p Λ ¯C ¯FΛ ) + π 2 Im[C3 ∂ C F ], (6.340)