String Theory and M-Theory

606 Black holes in string theory
S-duality symmetry of the dual heterotic string theory in four dimensions.
The modular parameter of this symmetry is τ, and it transforms nonlinearly
under SL(2, ) transformations in the usual way. Its real part, τ1, which is
an axion-like field, arises from a duality transformation of the two-form B
in four dimensions. Accordingly, the ten-dimensional interaction gives rise
to a four-dimensional term of the form
1
τ1 (trR ∧ R − trF ∧ F ) . (11.189)
8π
The normalization is fixed by the requirement that this should be well defined
up to a multiple of 2π when τ1 is shifted by an integer, since such
shifts are part of the SL(2, ) group. To get the rest of the group working,
specifically the transformation τ → −1/τ, it is necessary to add higher-order
terms by the replacement
τ = 1 24
1
log q → log η(τ) = log ∆(q). (11.190)
2πi 2πi 2πi
In the heterotic viewpoint, the corrections given by this substitution have the
interpretation as instanton contributions due to Euclideanized NS5-branes
wrapping the six-torus.
It follows that the S-duality invariant and supersymmetric completion of
the trR ∧ R term is25 1
Im
16π2
log ∆(q)tr [(R − iR ⋆ ) ∧ (R − iR ⋆ )] . (11.191)
The factor involving the curvatures is part of d4θW 2 , and its coefficient
determines F1 to be
F1 = i
log ∆(q). (11.192)
128π
This shows that F1 is independent of the K3 moduli. Moreover, Fh = 0
for h > 1. As a result, one finds that the prepotential for this case takes a
particularly simple form, namely
F (X, W 2 ) = − 1
2 CabX a X b
X 1
X 0
W 2
− log ∆(q). (11.193)
128πi
Using these formulas one can solve the attractor equations and the Legendre
transformation obtaining
φ0 = −2π
p1 . (11.194)
25 In terms of two-forms, R ∗ is defined by a duality transformation of the Lorentz indices
(R ∗ ) mn = 1
2 εmn pqR pq .
q0