Form fields and D-branes
297
The string world-sheet XIJ.("C, 0') has an analogous coupling to the Kalb-Ramond
field BIJ.II:
s . . . . . f BIJ.IIEa~BaXIJ.B~XII d2~
(10.117)
where ~O.l are the two world-sheet coordinates ("C,O') respectively, Ea~ is the
antisymmetric tensor with EOI = -E 10 = -I. The gauge transformation
B(2) -+ B(2) + dA(1)
(10.118)
leaves the 3-form field strength H(3) = dB(2) invariant and, analogously to
(10.114),
dH(3) = 0
(10.119)
Evidently, the string, and (some of) its excitations, have a non-zero value
of the NS-NS 'electric' charge associated with the BIJ.II gauge field. The
generalization of (10.116) is that the electric charge Q I of the (one-dimensional)
string, associated with the 2-form gauge potential that is enclosed by the sevendimensional hypersurface S7, is given by
Q\ = ( ·H(3).
(10.120)
157
For a general anti symmetric (p + I )-dimensional tensor gauge field, the
generalization of (10.112) and (10.117) is a term in the action of the form
S"'" f CIJ.Io1J.2 .... Jl.p+IEaoal ... apBaoXIJ.1BaIXIJ.2 ... BapXIJ.P+1 dP+\~. (10.121)
This describes the coupling of the C(p+1) form gauge field to the (p + 1)dimensional world volume of an extended object having p spatial dimensions
(a p-brane) that has a non-zero value for the NS-NS or R-R electric charge
associated with the NS-NS or R-R sector gauge form fields described in the
previous section. Under a gauge transformation, a gauge form field transforms
as
C(p+l) -+ C(p+l) + dA(p)
(10.122)
and the field strength F(p+2) = dC(p+1) is invariant. The electric charge of the
p-brane associated with the gauge potential, that is enclosed by the hypersurface
S8-p is, therefore, generally given by
(10.123)
Qp = Is · F(p+2).
58-p
In addition to the field strength H(3) deriving from the Kalb-Ramond NSNS form field B(2), which we have already noted is coupled 'electrically' to
the string world sheet, the effective action for type HA superstrings (10.105)