and the φI satisfying the attractor equations qI = ∂FE /∂φI .

4 Duality Invariant OSV Integral

The quantity Σ given above can be used to define a duality invariant version of the OSV integral for supersymmetric black holes. The OSV conjecture expresses microscopic state degeneracies d(q, p) in terms of macroscopic data [16],

d(p, q) dφeπ [FE(p,φ)−qI φI ], (31)

where FE(p, φ) was defined in (29). Electric/magnetic duality is, however, not manifest in (31). A duality invariant version of the OSV integral can be constructed using (23), with Υ set to its attractor value Υ = −64. It reads [4]

where = |det ImFKL| (non-holomorphic corrections to the coupling functions

can also be incorporated into (32)). Integrating (32) over fluctuations I − ¯ I in

δ(Y Y )

saddle-point approximation results in [4]

d(p, q) dφ Δ(p, φ)eπ [FE(p,φ)−qI φI ], (33)

which is a modified version√ of the OSV integral (31), containing a non-trivial integration measure factor which is necessary for consistency with electric/magnetic duality. Evaluating (33) further in saddle-point approximation precisely yields the macroscopic entropy (27). The presence of the measure factor in (33) has been confirmed in the recent works [19, 7].

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