Lectures on String Theory

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Finally, we discuss the completeness condition for the ρ-matrices. The matrices ρ a ,
¯ρ and the identity matrix I form a basis in the space of all 2 × 2-matrices. Any
2 × 2-matrix M can be expanded as
M = q a ρ a + ¯q¯ρ + qI .
The coefficients of this expansion are found as follows
q a = 1 2 Tr(Mρa ) , ¯q = 1 2 Tr(M ¯ρ) , q = 1 2 Tr(M) .
Plugging this back we obtain
2M = Tr(Mρ a )ρ a + Tr(M ¯ρ)¯ρ + Tr(M)I
or, more explicitly,
2M αβ = M γδ
[
(ρ a ) δγ ρ a αβ + ¯ρ δγ ¯ρ αβ + δ δγ δ αβ
]
.
Since M is an arbitrary matrix from here we derive the following completeness relation
(ρ a ) δγ ρ a αβ + ¯ρ δγ ¯ρ αβ + δ δγ δ αβ = 2δ αγ δ βδ .
Spin connection
We would like to introduce a new local symmetry which is the Lorentz symmetry.
However, we have to guarantee that the the theory we are after should be invariant
w.r.t. these local transformations. As for the case of any local gauge invariance,
the local Lorentz invariance can be achieved by introducing q gauge field ωα a b (x) for
SO(d − 1, d). Here a, b are SO(d − 1, d)-indices and α is the “curved” vector index.
Under the local Lorentz transformations with the matrix Λ this field transforms as
follows
ω α → Λω α Λ −1 − ∂ α ΛΛ −1 .
The gauge field of the local Lorentz symmetry is usually called the spin connection.
The spin connection plays the same role for the “flat” indices as the Christoffel
connection plays for the “curved” ones. We have the following substitution of the
basic objects in the theory
(
) (
)
h aβ (x), Γ δ αβ(x) → e a α(x), ωα a b(x) .
Introduction of the spin connection should not change the gravitational content of
the theory. This means that the spin connection should not be a new independent
field, rather it should be determined in terms of vielbein. The simplest and elegant
way to do it is to notice that we have the covariant derivatives
D α V β = ∂ α V β + Γ β αδ V δ
D α V a = ∂ α V a + ω a α bV b