Gravity and Strings

598 Appendix A
There is another invariant that we can build in four dimensions,
Tr (Fµν ⋆ F µν ) ∼− 1
2 √ |g| K IJɛ µνρσ F I µν F J ρσ, (A.46)
but itisatotal derivative and does not contribute to the Aµ equations of motion. In the
Euclidean signature, the integral of the above term is (up to numerical factors) the instanton
number, atopological invariant that does contribute to the Euclidean path integral.
Sometimes it is useful to work with differential forms. Thus, we define the Lie-algebravalued
1- and 2-forms,
A ≡ Aµdx µ , F = 1
2 Fµνdx µ ∧ dx ν ≡ dA− A ∧ A, (A.47)
where we have defined the exterior covariant derivative D. The kinetic term for A can now
be written as the d-form (in d dimensions)
d d x |g| TrAdjF 2
∼ TrAdj(F ∧ ⋆ F), (A.48)
and the four-dimensional topological term can be rewritten as
TrAdj(F ∧ F). (A.49)
d 4 x |g| TrAdj(F ⋆ F) ∼
Now we define the Chern–Simons 3-form
ω3 = 1
3! ω3 µνρdx µ ∧ dx ν ∧ dx ρ
2
≡ TrAdj A ∧ dA− 3 A ∧ A ∧ A , (A.50)
or, in components, using the property Eq. (A.24) and the normalization K IJ = δIJ for a
compact group:
ω3 µνρ =−3! A I [µ∂ν A I ρ] − 1
3 f IJKA I [µ A J ν A K
ρ] . (A.51)
The Chern–Simons 3-form has the very important property 7
dω3 = TrAdj(F ∧ F), (A.52)
which makes it evident that the topological term F ∧ F is a total derivative.
A.2.3 SO(n+, n−) gauge theory
The group SO(n+, n−) is defined as the group of n × n (where n = n+ + n−)real matrices8 ˆ â ˆb that act on (contravariant) n-dimensional vectors by
ˆV ′â = ˆ â ˆb ˆV ˆb , (A.53)
7 To prove it one simply has to realize that the trace over the exterior product of four As vanishes because
complete antisymmetry in four indices is the opposite to cyclic symmetry. (For three indices, complete
antisymmetry and cyclic symmetry are the same thing and this is why ω3 can be defined at all.)
8 We use hats to avoid confusion with (Lorentzian) tangent-space indices. When n+ = 1 and n− = d − 1 they
are, of course, identical.