To provide a tad more explanation to what suresh1 says in his reply, since the mechanism here is as simple as it is fundamental to all things supersymmetric and index-theoretic:
It is a general fact that "the partition function of a supersymmetric system is a topological invariant". This was first discovered in the context of supersymmetric quantum mechanics, where the phenomenon is most clearly visible, but precisely the same general mechanism controls all supersymmetric quantum field theories and string theories, too.
Namely,consider a system with Hamiltonian \(H\) that "has a supercharge" hence for which there is an odd-graded self-adjoint operator \(D\) such that \(H = \tfrac{1}{2}\{D,D\} = D^2\)
Then consider the Euclideanized partition function of the system
\(Z_t = \mathrm{sTr}(\exp(-t H))\)
where we take the super-trace, hence the trace over even-graded states minus that over odd-graded states.
Now the point to notice is that all eigenstates of H of non-vanishing eigenvalue appear in “supermultiplet” pairs of the same eigenvalue: if \(|\psi\rangle\) has eigenvalue E>0 under H, then
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\(D |\psi\rangle \neq 0\);
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also \(D |\psi\rangle\) has eignevalue E (since [H,D]=0).
Therefore all eigenstates for non-vanishing eigenvalues appear in pairs whose members have opposite sign under the supertrace. So only states with \(H |\psi\rangle = 0\) contribute to the supertrace. But if H and D are hermitean operators for a non-degenerate inner product, then it follows that \((H |\psi\rangle = 0) \Leftrightarrow (D|\psi\rangle = 0)\) and hence these are precisely the states which are also annihilated by the supercharge (are in the kernel of D), hence are precisely only the supersymmetric states.
On these now the weight \(\exp(-t D^2) = 1\) and hence the supertrace over this “Euclidean propagator” simply counts the number of supersymmetric states, signed by their fermion number.
Notice that this result is "topological" in that it does not actually depend on the geometric background encoded by the Hamiltonian (it's energy spectrum and hence the couplings which this encodes) In particular it does not depend on the worldline length t itself.
It seems that this simple argument goes back at least all the way to
- H. MacKean, Isadore Singer, Curvature and eigenvalues of the Laplacian, J. Diff. Geom. 1 (1967)
It became popular with the advent of supersymmetric quantum mechanics due to
- Luis Alvarez-Gaumé, Supersymmetry and the Atiyah-Singer index theorem, Comm. Math. Phys. Volume 90, Number 2 (1983), 161-173. (Euclid)
In the mathematics literature this was then picked up in
- Ezra Getzler, Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem, Comm. Math. Phys. 92 (1983), 163-178. (pdf)
and so today a standard textbook account for this is
(see around prop. 3.48, 3-50). Mathematicians speak of the "heat kernel technique for index theory" or the like. For more see on the nLab at index.
Now to come to the string: superstring propagation may be thought of as essentially being supersymmetric quantum mechanics on loop space (this is how Witten was brought to supersymmetric QM in the first place). The Dirac operator now is the Dirac-Ramond operator. The partition function of the superstring is hence formally directly analogous, just richer, to the above. In particular if a sector has a supercharge, then only the supersymmetric ground states contribute to the partition function.
In particular for the type II superstring in the NS-R sector or else for the heterotic string, of the two possible worldsheet Dirac-Ramond supercharges only one exists (the right-moving one, say). Hence the partition function of the superstring in this case -- called the Witten genus -- may be thought of as being a generating functional for counting of one-half supersymmetric string states, by precisely the above kind of argument.
But one-half supersymmetric string states have an important interpretation in terms of the effective target space supergravity theory: theory are the BPS-states. Hence the partition function of the one-half-supersymmetric superstring (type II in NS-R sector else heterotic) counts BPS-states in target space.
Lecture notes expanding on this include for instance
This is finally where the black hole entropy comes in: suitably extremal black holes in supergravity (= strongly coupled strings) correspond indeed to BPS states in the weakly coupled theory (certain half-supersymmetric D-brane configurations). The former are counted by the Witten genus. But since -- by the above argument -- that does not actually depend on "geometric" data such as the coupling constant, one knows that it remains the same even in the strong coupling regime in which there is a black hole.
This is what one means when one says that BPS states are "protected" from changes of the coupling constant.
More pointers are on the nLab at black holes in string theory
Q&A (4872)
