Gravity and Strings

542 The extended objects of string theory
be different in each case. The d = 4 solution associated with the heterotic string was found
in [453] and also has a wormhole interpretation.
19.2.8 The D7-brane and holomorphic (d − 3)-branes
The D7-brane solution Eqs. (19.64) is just the simplest of a very rich family of solutions of
the action Eq. (19.68), which share many interesting properties and some pathologies that
can be eliminated after a careful analysis. They are the subject of this section.
The SL(2, R)/SO(2) σ-model of Eq. (19.68) is invariant under transformations of the
whole group SL(2, R), Eqs. (11.205) and (11.206), but only the discrete subgroup SL(2, Z)
is supposed to relate equivalent (dual) type-IIB theories. On the other hand, as discussed
in Section 11.4.1, only the modular group G ≡ PSL(2, Z) = SL(2, Z)/{±I2×2} acts on ˆτ.
In conclusion, type-IIB S duality tells us, then, that values of the ˆτ field that are related by
modular transformations must be considered equivalent and should be identified. The same
will be true in the cases in which τ can be viewed as the modular parameter of a torus. 12
Thus τ, which in principle takes values in the whole complex upper half plane H, can be
restricted to take values in the fundamental domain of the modular group in H, which we
are going to discuss now.
The modular group G is generated by the elements T and S
S =
0 −1
, T =
1 0
1 1
, (19.73)
0 1
whose actions on τ are T (τ) = τ + 1andS(τ) =−1/τ. Observe that S2 =−I2×2 ∼ I2×2
in G and also (ST) 3 = (T −1S) 3 ∼ I in G. Thus S and ST generate two cyclic subgroups of
orders 2 and 3, respectively.
The fundamental domain of G in H can be defined as the quotient H/G and corresponds
to the region |τ|≥1 and − 1
1
≤ Re(τ) ≤ with the lines Re(τ) =−1 and Re(τ) =+1
2 2 2 2 identified
by a T transformation and with the arc of unit radius eiθ , θ ∈ [π/3, 2π/3] joining
the fundamental domain corners e 2πi
3 and e πi
3 identified with itself (“orbifolded”) according
to e iθ ∼ e i(π−θ) (the S transformation) (see Figure 19.2).
H/G has, therefore, two special points associated with the two cyclic subgroups gener-
ated by S and ST: τ = i, which is invariant under S; and τ = ρ ≡ e 2πi
3 ,which is invariant
under ST.Ifweconsider its compactification H/G inwhich the point at infinity is added,
then a third special point appears: ∞ itself, which is invariant under the infinite subgroup
of integer powers of T and can be understood as an infinite-order orbifold point.
Since the fundamental domain in which τ takes values is topologically non-trivial, we
expect τ(x), which maps the transverse space on the fundamental domain, to be a multivalued
function of x whose monodromies are in G. On the other hand, the real part of τ has,
therefore, the typical behavior of an axion field and takes values in a circle.
12 We remove all the hats henceforth, since the results of this section will be valid for many cases and dimensions
apart from the N = 2B, d = 10 case. The results for D7-branes in d = 10 dimensions will be valid for
(d − 3)-branes in d dimensions.