Lectures on String Theory

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The “flat” and “curved” indices of the vector are related as V a = e a αV α . This way of
transforming “flat” to “curved” indices should be valid for any tensor, in particular,
one has to have
D α V a = e a βD α V β .
This is possible only if
D α e a β = ∂ α e a β − Γ λ αβe a λ + ω a α be b β = 0 .
This equation can be solved for ω α expressing it through the vielbein:
ω ab
α = 1 2 eβa (∂ α e b β − ∂ β e b α) − 1 2 eβb (∂ α e a β − ∂ β e a α) − 1 2 eλa e γb (∂ λ e γc − ∂ γ e λc )e c α .
Since the connection is completely expressed via the dynamical vielbein and, by this
means, is not an independent field, it is called sometimes composite.
Spin manifolds
On which manifolds one can introduce spinors? This is rather non-trivial question.
Vielbein can always be introduces locally in a coordinate patch U α . It is quite
rare that the vielbein can be also globally defined. In the latter case such manifolds
are called parallelizable. Examples of parallelizable manifolds are Lie groups. On the
other hand, two-sphere is not parallelizable because there is no globally defined vector
field (not talking about the vielbein) which vanishes nowhere. Thus, the vielbein
is defined locally and in the intersection of the coordinate patches U α ∩ U β one has
e (α) (x) = Λ (αβ) (x)e (β) (x) .
Here Λ (αβ) (x) is the local Lorentz transformation (i.e. a matrix from SO(d − 1, 1))
which is called the transition function. In the region of triple intersection U α ∩ U β ∩
U γ = U αβγ the transition functions should satisfy the following condition
Λ (αβ) Λ (βγ) Λ (γα) = 1 .
Now if we introduce locally a spinor field ψ (α) then passing from one coordinate patch
to another one the field must transform according to
ψ (α) (x) = ¯Λ (αβ) ψ (β) (x) .
Here ¯Λ (αβ) is the SO(d − 1, 1) matrix in the spinor representation. For spinorial
transition functions ¯Λ it also makes sense to require that in the triple intersection
region the following relation is satisfied
¯Λ (αβ) ¯Λ(βγ) ¯Λ(γα) = ±1 . (6.2)
However, since the spinor representation is double-valued, instead of ¯Λ one can
equally use −¯Λ. Thus, to define spinors on a non-parallelizable manifold one has