String Theory and M-Theory

6.2 D-branes in type II superstring theories 209
is a 2 symmetry at the self-dual radius Rsd = √ α ′ . Let us now examine
the same T-duality transformation for type II superstring theories. It will
turn out that the type IIA theory is mapped to the type IIB theory and
vice versa. Of course, if several directions are compactified on circles it is
possible to carry out several T-dualities. In this case an even number of
transformations gives back the same type II theory that one started with
(on the dual torus). This is a symmetry if the torus is self-dual.
Returning to the case of a single circle, imagine that the X 9 coordinate of
a type II theory is compactified on a circle of radius R and that a T-duality
transformation is carried out for this coordinate. The transformation of the
bosonic coordinates is the same as for the bosonic string, namely
X 9 L → X 9 L and X 9 R → −X 9 R, (6.57)
which interchanges momentum and winding numbers. In the RNS formalism,
world-sheet supersymmetry requires the world-sheet fermion ψ 9 to
transform in the same way as its bosonic partner X 9 , that is,
ψ 9 L → ψ 9 L and ψ 9 R → −ψ 9 R . (6.58)
This implies that after T-duality the chirality of the right-moving Ramondsector
ground state is reversed (see Exercise 6.5). The relative chirality of the
left-moving and right-moving ground states is what distinguishes the type
IIA and type IIB theories. Since only one of these is reversed, it follows that
if the type IIA theory is compactified on a circle of radius R, a T-duality
transformation gives the type IIB theory on a circle of radius R.
In the light-cone gauge formulation, only X i and ψ i , i = 1, . . . , 8, are
independent dynamical degrees of freedom. In this case a T-duality transformation
along any of those directions works as described above, but one
along the x 9 direction is more awkward to formulate.
Now let us examine what happens to type II Dp-branes when the theory
is T-dualized. Since the half-BPS Dp-branes of the type IIA theory have
p even, while the half-BPS Dp-branes of the type IIB theory have p odd,
these D-branes are mapped into one another by T-duality transformations.
A similar statement can also be made for the non-BPS Dp-branes. The
relevant analysis is the same as for the bosonic string. Let us review the
analysis for a pair of flat parallel Dp-branes that fill the dimensions x µ , with
µ = 0, . . . , p, and have definite values of the other transverse coordinates.
An open string connecting these two Dp-branes satisfies Neumann boundary
conditions in p + 1 dimensions
∂σX µ |σ=0 = ∂σX µ |σ=π = 0, µ = 0, . . . , p, (6.59)