String Theory and M-Theory

Homework Problems 185
PROBLEM 5.2
In order to obtain a nontrivial massless limit of Eq. (5.21), it is useful to
first restore the auxiliary field e(τ) described in Chapter 2.
(i) Re-express the massive D0-brane action with the auxiliary field e(τ).
(ii) Find the massless limit of the D0-brane action. 6
(iii) Verify the κ symmetry of the massless D0-brane action.
PROBLEM 5.3
Prove that, for a pair of Majorana spinors, Θ1 and Θ2, the flip symmetry is
given by
¯Θ1Γµ1···µnΘ2 = (−1) n(n+1)/2 ¯ Θ2Γµ1···µnΘ1,
as asserted at the end of Exercise 5.2.
PROBLEM 5.4
Derive the relevant Fierz transformation identities for Majorana–Weyl spinors
in ten dimensions and use them to prove that
Γ µ dΘ d ¯ ΘΓµdΘ = 0.
PROBLEM 5.5
Verify that the action (5.41) with Ω2 given by Eq. (5.55) is invariant under
supersymmetry transformations.
PROBLEM 5.6
Prove the identity
{Γµ1ν1, Γµ2ν2} = −2ηµ1µ2ην1ν2 + 2ηµ1ν2ην1µ2 + 2Γµ1ν1µ2ν2,
invoked in Exercise 5.5.
PROBLEM 5.7
Verify that the action (5.62) is supersymmetric.
PROBLEM 5.8
Construct the conserved supersymmetry charges for open strings in the
light-cone gauge formalism of Section 5.3 and verify that they satisfy the
supersymmetry algebra. Hint: the 16 supercharges are given by two eightcomponent
spinors, Q + and Q − . The Q + s anticommute to P + , the Q − s
6 This is sometimes called the Brink–Schwarz superparticle.