String Theory and M-Theory

Homework Problems 245
theory for radius R agrees with the spectrum of the type IIB theory for
radius R = α ′ /R.
PROBLEM 6.2
Equations (6.38) and (6.39) describe a generalization of the result of Exercise
6.2 from a U(1) gauge field to a U(N) gauge field
A = − 1
2πR diag(θ1, θ2, . . . , θN),
where the θ s are again constants. Derive these equations.
PROBLEM 6.3
The T-duality rules for R–R sector tensor fields can be derived by taking into
account that the field strengths are constructed as bilinears in Majorana–
Weyl spinors in the covariant RNS approach. Explicitly,
Fµ1...µn = ¯ ψLΓµ1...µnψR.
(i) Explain why n is even for the type IIA theory and odd for the type
IIB theory.
(ii) Explain why (in differential form notation) Fn = ⋆F10−n.
(iii) Show that, for both the type IIA and type IIB theories, the number of
independent components of the tensor fields agrees with the number
of degrees of freedom of a tensor product of two Weyl–Majorana
spinors in ten dimensions.
PROBLEM 6.4
Show that the Dirac equations for ψL and ψR in the previous problem imply
that the field equations and Bianchi identities for the field strengths are
satisfied, that is,
∂ [µF µ1...µn] = 0, ∂ µ Fµµ2...µn = 0.
Also, show that, when these equations for Fn are re-expressed as equations
for F10−n, the field equation and Bianchi identity are interchanged.
PROBLEM 6.5
Derive the T-duality transformation formulas for NS–NS background fields
in (6.97). You may ignore the dilaton term and set hαβ = ηαβ. Verify that if
the transformation is repeated a second time, one recovers the original field
configuration.