By Kimball

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6 2 1+α (64) If we change the separation of the black hole and black ring while preserving the axial symmetry, that is, if we vary α , then the symmetry requires J1 to be conserved. Once again we see that this means that JT must remain fixed. δ αR Fig. 3 The configuration black ring with an off-set black hole on its axis. The parameter, α , is related to the angle of approach, δ , by α ≡ cot δ R Black Holes, Black Rings and their Microstates 25 The constancy of JT along with (62) imply that as the black hole is brought near the black ring, the embedding radius of the latter, R, must change according to: R2 = l 6p L4 ∑ nI −1 JT − nI NIBH 1 + α2 .

2 Homology and Cohomology The multi-center Gibbons-Hawking (GH) metrics also contain 12 N(N − 1) topologically non-trivial two-cycles, Δi j , that run between the GH centers. These two-cycles can be defined by taking any curve, γi j , between y(i) and y( j) and considering the U(1) fiber of (69) along the curve. This fiber collapses to zero at the GH centers, and so the curve and the fiber sweep out a 2-sphere (up to Z|q j | orbifolds). See Fig. 4. These spheres intersect one another at the common points y( j) .

Bena and N. P. Warner where ∇ × ξ = −∇H. (82) Hence, ξ is a vector potential for magnetic monopoles located at the singular points of H. To determine how these fluxes thread the two-cycles, we need the explicit forms for the vector potential, B, and to find these we first need the vector fields, vi , that satisfy: 1 ∇ × vi = ∇ . (83) ri One then has: N A= ∑ q j v j, ξ= j=1 N ∑ h j v j. (84) j=1 If we choose coordinates so that y(i) = (0, 0, a) and let φ denote the polar angle in the (x, y)-plane, then: (z − a) vi · dy = + ci d φ , (85) ri where ci is a constant.

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