PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

Chapter 6
Concluding Remarks
Complete understanding of the microstates of an arbitrary black hole is one the motivations
behind a quantum theory of grvity. String theory is the leading candidate for a quantum gravity
theory and there is much progress in understanding the microscopics of supersymmetric black
holes.
In this thesis our aim was to calculate the microscopic degeneracy of Black holes, in N = 4
superstring theory, which preserve 1 -th of the supersymmetries. The formula was known [7] for
4
black holes which carry charges (Q, P ) subject to the condition, g.c.d(Q ∧ P ) = 1 [20, 48, 63].
We extended the result to black holes carrying arbitrary charge vectors. Although we could not
provide a first principle derivation of the result, it was done after some time in the reference [85].
There are some aspects of these formulae which require better understanding. We have
already seen the appearance of the modular group of the genus two surface, namely Sp(2, Z).
The partition function of the theory is given by a modular form of this group or one of its
subgroups. But the appearance of Sp(2, Z) is not expected because it is not a symmetry
group of the theory. The S-duality group is only a subgroup of this. The first step towards
explaining this has been taken in [10, 18]. The contour prescription which we discussed in the
introduction can also be derived [84] from the M-theory lift described in the last two papers.
It will be very interesting to see how much more can be learned from this picture.
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