Lectures on String Theory

– 94 –
d(d−1)
2
local Lorentz transformations which should allow to remove the additional
(unphysical) components of the vielbein:
}{{} d 2 d(d − 1)
−
2
vielbein } {{ }
local Lorentz
=
d(d + 1)
2 } {{ }
metric
Thus, the vielbein brings new degrees of freedom, but they can be removed by a
new symmetry which is the local Lorentz transformations. On the other hand, the
vielbein allows one to introduce coupling of the gravitational degrees of freedom with
spinors.
Spinor representations
(Pseudo-)orthogonal groups (SO(d − 1, 1)) SO(d) in addition to the usual tensor
representations have also spinor representations. These are not single-valued but
rather double-valued representations of SO(d).
Consider, for instance, the group SO(3). This group has the so-called universal
covering group which is SU(2). The relation between them is as follows
SO(3) → SU(2)/Z 2 .
The group SO(3) is not simply connected, while SU(2) is. Here
Z 2 =
{( ) ( )}
1 0 −1 0
,
0 1 0 −1
is the center of SU(2), i.e. a set of matrices from SU(2) which commute with all other
SU(2)-matrices. Due to existence of the discrete center Z 2 representations of SU(2)
split into two different classes of integer and half-integer spin. Only representations
with integer spin are those of (single-valid) SO(3). Representations of SU(2) with
half-integer spin are spinor (double-valid) representations of SO(3). Indeed, rotation
by the angle φ around the axes given by a fixed 3dim unit vector n = (n 1 , n 2 , n 3 ),
n 2 i = 1, corresponds the transformation
(
g(φ, n) = exp − i ) ( cos
φ
2 φ n iσ i =
− in 2 3 sin φ − (in
2 1 + n 2 ) sin φ )
2
(−in 1 + n 2 ) sin φ cos φ + in 2 2 3 sin φ ,
2
where σ i are three Pauli matrices. One can easily verify that g(φ, n) ∈ SU(2).
Rotation by the angle φ = 2π is an identity transformation in SO(3) but it is not
the identity in SU(2). Indeed, one can see that
g(φ + 2π, n) = −g(φ, n) ,
g(φ + 4π, n) = g(φ, n) .