String Theory and M-Theory

600 Black holes in string theory
28 U(1) gauge fields. 22 These transform as a vector of the O(22, 6; ) duality
group. 22 of the gauge fields belong to 22 vector multiplets, while the other
6 belong to the supergravity multiplet. The allowed charges of these gauge
fields are given by sites of the Narain lattice, as described in Chapter 7.
Since this is an even lattice, a charge vector in this lattice squares to an
even integer. In other words, since the charges are encoded in the internal
momenta (pR, pL), where pL has 22 components and pR has six components,
The mass formula for these states is
p 2 R − p 2 L = 2N. (11.156)
1
4 α′ M 2 = 1
2 p2 R + NR = 1
2 p2 L + NL − 1, (11.157)
where NL and NR are the usual oscillator excitation numbers.
Dabholkar–Harvey states
Most states with masses given by Eq. (11.157) are unstable, but the BPS
states are stable. The BPS states, that is, the state belonging to short
supermultiplets for which the mass saturates the BPS bound, have NR = 0.
In this case
α ′ M 2 = 2p 2 R, (11.158)
while NL is arbitrary. This results in a whole tower of stable states, which
are sometimes called Dabholkar–Harvey states. For these states the levelmatching
condition reduces to
NL − 1 = N. (11.159)
For example, if there is winding number w and Kaluza–Klein excitation number
n on one cycle of the torus, then N = |nw|. In general, the degeneracy
of states for large N is given by
dN ≈ exp 4π √
N , (11.160)
resulting in a leading contribution to the black-hole entropy given by
S = log dN ≈ 4π √ N. (11.161)
22 We assume generic positions in the moduli space so that there is no enhanced gauge symmetry.