String Theory and M-Theory

186 Strings with space-time supersymmetry
anticommute to P − , and the anticommutator of Q + and Q − gives the transverse
momenta.
PROBLEM 5.9
(i) Show that trF ∧ F is closed and gauge invariant.
(ii) This quantity is a characteristic class proportional to c2, the second
Chern class. Since it is closed, in a local coordinate patch one can
write trF ∧ F = dω3, where ω3 is a Chern–Simons three-form. Show
that
ω3 = tr A ∧ dA + 2
A ∧ A ∧ A .
3
(iii) Similarly, one can write trF 4 = dω7. Find ω7.
PROBLEM 5.10
Check the identity in Eq. (5.122) for SO(N).
PROBLEM 5.11
(i) Using the definition of Y in Eq. (5.125), obtain an expression for Y4.
(ii) Apply the descent formalism to obtain a formula for G2 in Eq. (5.132).
PROBLEM 5.12
Prove the relations given in Eqs (5.139)–(5.141).
PROBLEM 5.13
Verify that the identity (5.138) is satisfied for the gauge group E8 × E8.
PROBLEM 5.14
There is no string theory known with the gauge groups E8 × U(1) 248 or
U(1) 496 . Nevertheless, the anomalies cancel in these cases as well. Prove
that this is the case. Hint: infer the result from the fact that the anomalies
cancel for E8 × E8.
PROBLEM 5.15
Prove that TrF 4 = 1
100 (TrF 2 ) 2 for the adjoint representation of E8. Hint:
use the Spin(16) decomposition 248 = 120 + 128.