Introduction to String Theory and D–Branes - School of Natural ...

V a (x). But we can now relate this component to another object which has an index which is contravariant
under general coordinate transformations, V µ . These objects are related by our handy vielbiens: V a (x) =
e a µ (x)V µ .
2.14.3 The Lorentz Group as a Gauge Group
The standard covariant derivative of e.g., a contravariant vector V µ , has a counterpart for V a = e a µ V µ :
DνV µ = ∂νV µ + Γ µ νκV κ ⇒ DνV a = ∂νV a + ω a bνV b ,
where ω a bν is the spin connection, which we can write as a 1–form in either basis:
ω a b = ω a bµdx µ = ω a bµe µ c e c νdx ν = ω a bce c .
We can think of the two Minkowski indices (a, b) from the space tangent structure as labeling components
of ω as an SO(1, D−1) matrix in the fundamental representation. So in an analogy with Yang–Mills theory,
ωµ is rather like a gauge potential and the gauge group is the Lorentz group. Actually, the most natural
appearance of the spin connection is in the structure equations of Cartan. One defines the torsion T a , and
the curvature R a b, both 2–forms, as follows:
T a ≡ 1
2 T a bce a ∧ e b = de a + ω a b ∧ e b
R a b ≡ 1
2 Ra bcde c ∧ e d = dω a b + ω a c ∧ ω c b . (109)
Now consider a Lorentz transformation e a → e ′a = Λ a be b . It is amusing to work out how the torsion changes.
Writing the result as T ′a = Λ a bT b , the reader might like to check that this implies that the spin connection
must transform as (treating everything as SO(1, D − 1) matrices):
ω → ΛωΛ −1 − dΛ · Λ −1 , i.e., ωµ → ΛωµΛ −1 − ∂µΛ · Λ −1 , (110)
or infinitesimally we can write Λ = e −Θ , and it is:
A further check shows that the curvature two–form does
δω = dΘ + [ω, Θ] . (111)
R → R ′ = ΛRΛ −1 , or δR = [R, Θ] , (112)
which is awfully nice. This shows that the curvature 2–form is the analogue of the Yang–Mills field strength
2–form. The following rewriting makes it even more suggestive:
R a b = 1
2 Ra bµνdx µ ∧ dx ν , R a bµν = ∂µω a bν − ∂νω a bµ + [ωµ, ων] a
b .
2.14.4 Fermions in Curved Spacetime
Another great thing about this formalism is that it allows us to nicely discuss fermions in curved spacetime.
Recall first of all that we can represent the Lorentz group with the Γ–matrices as follows. The group’s
algebra is:
[Jab, Jcd] = −i(ηadJbc + ηbcJad − ηacJbd − ηdbJac) , (113)
with Jab = −Jba, and we can define via the Clifford algebra:
{Γ a , Γ b } = 2η ab , J ab = − i a b
Γ , Γ
4
, (114)
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