294
Black holes in string theory
equations, so we shall only be concerned with the bosonic degrees of freedom.
All closed superstring theories have massless (bosonic) modes associated with the
graviton field G JLV (a symmetric, traceless, rank-2 tensor), the Kalb-Ramond field
BI-'v (an anti symmetric, rank-2 tensor) and the dilaton field",. (The expectation
value of the dilaton fixes the string coupling constant gs via gs = (e.).) In
type 11 superstring theories, these states arise in the NS-NS sector, i.e. they
are states which are constructed using the Neveu-Schwarz (half-integer-moded)
world-sheet fermion creation operators for both left- and right-movers. Massless
bosonic states also arise in the R-R sector: these are constructed using the
Ramond (integer-moded) world-sheet fermion creation operators for both leftand right-movers. In type IIA (in the light-cone gauge), the R-R states transform
as [I] = 8v and [3] = 56 representations of the (transverse) SO(8) group,
where [n] denotes the totally anti symmetric rank n tensor. They may, therefore,
conveniently be represented by form fields C(l) and C(3) where
1
C(n) == ,C"'IJL2 ••• /L .. dx"'l "dx1-'2 " ... "dx/LII.
(10.104)
n.
The representations [n] and [8 - n] of SO(8) are the same, since they related
by the eight-dimensional f-tensor, so we could as well represent these fields by
the forms C(7) and qS) respectively. The tree-level Weyl invariance of the string
world-sheet action is preserved in the quantum string theory provided that the
(renormalization-group)beta functions associated with these fields all vanish. The
resulting equations amount to spacetime field equations for the background fields
that would arise from the effective action:
SUA = ~{/dIOX(-G)I/2e-24tR(G)
2K IO
+ / [e- 24t (4dtj>" *dtIJ - !H(3)" * B(3) - !F(2) " • F(2)
I -
-
1
}
- ~F(4) " • F(4) - ~B(2) " F(4) " F(4)]
(10.105)
where G == det[Gl-'v],
2K2 10 - - (2;rr)7 a'4g s
2
(10.106)
is related to the ten-dimensional Newton constant by 2K?0 = 16;rrGIO; a' is
related to the string tension T by a' = ~ = !/~ (where Is is the string length
scale); the dilaton I-form is dtj> == BJLt/J dx/L, and the field-strength forms are
related to the potentials by
H(3) = dB(2) F(2) = dC(l) F(4) = dC(3)
F(4) = F(4) + C(l) " H(3)
(10.107)
Black holes in string theory
equations, so we shall only be concerned with the bosonic degrees of freedom.
All closed superstring theories have massless (bosonic) modes associated with the
graviton field G JLV (a symmetric, traceless, rank-2 tensor), the Kalb-Ramond field
BI-'v (an anti symmetric, rank-2 tensor) and the dilaton field",. (The expectation
value of the dilaton fixes the string coupling constant gs via gs = (e.).) In
type 11 superstring theories, these states arise in the NS-NS sector, i.e. they
are states which are constructed using the Neveu-Schwarz (half-integer-moded)
world-sheet fermion creation operators for both left- and right-movers. Massless
bosonic states also arise in the R-R sector: these are constructed using the
Ramond (integer-moded) world-sheet fermion creation operators for both leftand right-movers. In type IIA (in the light-cone gauge), the R-R states transform
as [I] = 8v and [3] = 56 representations of the (transverse) SO(8) group,
where [n] denotes the totally anti symmetric rank n tensor. They may, therefore,
conveniently be represented by form fields C(l) and C(3) where
1
C(n) == ,C"'IJL2 ••• /L .. dx"'l "dx1-'2 " ... "dx/LII.
(10.104)
n.
The representations [n] and [8 - n] of SO(8) are the same, since they related
by the eight-dimensional f-tensor, so we could as well represent these fields by
the forms C(7) and qS) respectively. The tree-level Weyl invariance of the string
world-sheet action is preserved in the quantum string theory provided that the
(renormalization-group)beta functions associated with these fields all vanish. The
resulting equations amount to spacetime field equations for the background fields
that would arise from the effective action:
SUA = ~{/dIOX(-G)I/2e-24tR(G)
2K IO
+ / [e- 24t (4dtj>" *dtIJ - !H(3)" * B(3) - !F(2) " • F(2)
I -
-
1
}
- ~F(4) " • F(4) - ~B(2) " F(4) " F(4)]
(10.105)
where G == det[Gl-'v],
2K2 10 - - (2;rr)7 a'4g s
2
(10.106)
is related to the ten-dimensional Newton constant by 2K?0 = 16;rrGIO; a' is
related to the string tension T by a' = ~ = !/~ (where Is is the string length
scale); the dilaton I-form is dtj> == BJLt/J dx/L, and the field-strength forms are
related to the potentials by
H(3) = dB(2) F(2) = dC(l) F(4) = dC(3)
F(4) = F(4) + C(l) " H(3)
(10.107)