String Theory and M-Theory

7.3 Toroidal compactification 281
Now one can compute the OPEs of these currents using the rules discussed
in Chapter 3. Defining J ± (z) = (J 1 (z) ± iJ 2 (z))/ √ 2, one obtains
J i (z)J j (w) ∼
δij (z − w) 2 + iεijk J k (w)
+ . . .
z − w
Defining the modes by
J i (z) =
n∈¢ J i nz −n−1
or J i
n =
dz
2πi znJ i (z),
as appropriate for h = 1 operators, it is possible to verify using the techniques
described in Chapter 3 that
i
Jm, J j
ijk k
n = iε Jm+n + mδ ij δm+n,0,
which is a level-one SU(2) Kac–Moody algebra. ✷
EXERCISE 7.4
T-duality, which inverts G, can be translated into transformations on the
background fields G and B. Show that G ↔ G−1 (a statement about 2n×2n
matrices) is equivalent to G + B ↔ 1
4 (G + B)−1 (a statement about n × n
matrices).
SOLUTION
In order to check this, a new metric G and tensor field B, which are related
to the old fields by
G + B = 1
4 (G + B)−1 ,
are introduced. Taking the symmetric and antisymmetric parts leads to
G = 1 −1 −1
(G + B) + (G − B)
8
and
B = 1 −1 −1
(G + B) − (G − B)
8
.
By simple manipulations, these can be rewritten in the form
G = 1 −1 −1
G − BG B
4
and
B = −G −1 B G.
Using these expressions for G and B and comparing Eqs (7.69) and (7.70)
one concludes that
G = G −1