These additional exercises follow the numeration in the book: so Ex. x.y.zz is relevant to Sec. x.y. Some are quite easy, some more challenging (just like for the problems in the book). I will keep adding more from time to time. [If you have any suggestions for exercises that you think might be interesting and relevant for the book, please write to me and I might decide to include them here.]

Chapter 3

Ex. 3.2.20 What happens to (3.2.35) when \(d\) is odd?

Ex. 3.2.21 Generalizing (3.2.23) and (3.2.57), show that \[ \frac1{l!}\gamma_{m_1\cdots m_l} \alpha_k \gamma^{m_l\cdots m_1} = (-1)^{kl} \sum_{p=0}^l (-1)^p {d-k\choose l-p}{k \choose p}\alpha_k \] Follow the basis method outlined for Exercise 3.2.4.

Ex. 3.3.5 Work out the analogue of (3.3.53) in \(d=3\) Lorentzian dimensions.

Chapter 4

Ex. 4.1.21 Use (4.1.115) to show the Bochner identity: \[ \alpha^m \Delta \alpha_m = -\frac12 \nabla^2 |\alpha|^2 + \nabla_m \alpha_n \nabla^m \alpha^n + R_{mn} \alpha^m \alpha^n\,.\] In particular integrate this to show that for a compact Riemannian manifold i) if \(R_{mn}\) is positive as a quadratic form, there are no harmonic one-forms; ii) if \(R_{mn}=0\), any harmonic one-form is covariantly constant.

Ex. 4.1.22 Show that the exterior differential and its adjoint commute with the Laplace–de Rham operator (4.1.112).

Ex. 4.1.23 On a compact \(M\) with \( R_{mn}\ge K g_{mn}\), prove the Lichnerowicz bound \[\lambda \ge K d\] for the eigenvalues \(\lambda\) of the Laplace operator. [Use the previous two exercises, taking \(\alpha = {\rm d} f\); use the inequality \(\mathrm{Tr}(m)^2 \ge \frac1d (\mathrm{Tr}(m))^2\), valid for any \(d\)-dimensional matrix \(m\); integrate over \(M\).]

Ex. 4.1.24 i) Check that the Moebius transformations (2.2.24) are conformal isometries of the \(S^2\) metric (4.1.47), as implicit in the sentence around (4.1.71). ii) Check that the local vector fields \(\ell_m \equiv -z^{m+1} \partial_z\) are conformal Killing vectors. Compute their Lie bracket with the Virasoro algebra (1.1.19). iii) Show that the \(\ell_m\) that do not diverge around \(z=0\) or \(z=\infty\) are \(\ell_1\), \(\ell_0\), \(\ell_{-1}\). (As usual, consider a coordinate \(z'=1/z\)). Compare this result with that for i).

Ex. 4.1.25 By using (4.1.43) (on the N chart, say), find that a basis of Killing vector fields for \(S^2\) is \[ -\sin\phi\partial_\theta - \cos\phi\cot\theta \partial_\phi \ ,\qquad \cos\phi\partial_\theta - \sin\phi\cot\theta \partial_\phi \ ,\qquad \partial_\phi\,, \] and that a basis of conformal Killing vectors is \[ -\sin\phi\cos\theta\partial_\theta - \frac{\cos\phi}{\sin\theta} \partial_\phi \ ,\qquad \cos\phi\cos\theta\partial_\theta - \frac{\sin\phi}{\sin\theta} \partial_\phi \ ,\qquad \sin \theta\partial_\theta\,. \]

Ex. 4.1.26 Working in frame indices, compute the Ricci tensor in terms of the Riemann tensor in \(d=2\). Show that \( R_{ab}=\frac12 g_{ab}R\). (You can also obtain this relation by varying (1.1.11).) Interpret this physically.

Ex. 4.1.27 Working in frame indices, compute the Ricci tensor in terms of the Riemann tensor in \(d=3\). You should get six equations for six variables, that can be inverted to obtain Riemann in terms of Ricci; for example you should find \(R_{1212}=\frac12 R-R_{33}\). More generally check that \[ \frac14 \epsilon^{acd} \epsilon^{bef} R_{cdef}= -R^{ab}+\frac12 g^{ab}R\,. \] Check that you can also find this relation by imposing that the Weyl tensor (4.1.28) vanishes. Interpret this result physically.

Ex. 4.3.16 Along the lines explained around (4.3.34), the Euler class in \(d=4\) can be written as \(\epsilon^{abcd} R_{ab} \wedge R_{cd}\), with \(R_{ab}\) the two-form associated to the Riemann tensor. To write this more explicitly, use \( e_a \wedge e_b \wedge e_c \wedge e_d = \epsilon^{abcd} \mathrm{vol}\) and \(\epsilon^{a_1 \ldots a_4} \epsilon_{b_1 \ldots b_4} = 4!\delta^{[a_1}_{[b_1}\ldots \delta^{a_4]}_{b_4]} \) to show that it is proportional to \[ ( R^2 - 4 R_{mn}R^{mn} + R_{mnpq} R^{mnpq})\mathrm{vol}\,. \]

Ex. 4.3.17 As we saw in the previous exercise, the integral \(E=\int_{M_4} {\rm d}^4 x\sqrt{g}( R^2 - 4 R_{mn}R^{mn} + R_{mnpq} R^{mnpq}) \) is a topological invariant in four dimensions. Use this to show Berger's identity \[ 4 R R_{mn} -8 R_{mp} R_n{}^p +8 R_{pmnq}R^{pq} +4 R_{mpqr} R_{n}{}^{pqr} = g_{mn}( R^2 - 4 R_{pq}R^{pq} + R_{pqrs} R^{pqrs})\,. \] (Hint: vary \(E\) with respect to the metric as if we were an action, with suitable integration by parts to write \(\delta E = \int \sqrt{g} \delta g_{mn} E^{mn}\); since \(E\) is a topological invariant, \(E_{mn}\) should be identically zero. Use (4.1.21) repeatedly to re-express all terms with two covariant derivatives.) Notice that taking a trace gives no new identity.

Chapter 5

Ex. 5.5.9 Define the analogue of (5.5.6) for \({\rm SU}(3)\)-structures in \(d=7\).

Chapter 7

Ex. 7.3.5 Recall from Section 3.4.6 that in \(d=3\) Euclidean dimensions the bilinears of a spinor \(\eta\) define a vielbein. Assuming that \(\eta\) is a Killing spinor (7.3.8), compute the exterior differentials \({\rm d}\) of the one-forms of this vielbein. Show that if \(\mu\) is purely imaginary we recover the Maurer–Cartan equations (4.4.6) for a three-sphere \(S^3\). What do we obtain instead when \(\mu\) is real?

Chapter 8

Ex. 8.1.5 Use (8.1.30) to show that the deformation of the Ricci scalar reads \(\delta R = h^{mn} R_{mn}- \nabla^2 h + \nabla^m \nabla^n h_{mn}\).

Chapter 9

Ex. 9.2.6 Work out the analogue of (9.2.5) for M2- and M5-branes.

Ex. 9.2.7 Along the lines of Ex. 9.2.2, find a brane system that realizes the Bogomolny equations \[F = *_3 D \phi\] with \(\phi\) a single real scalar in three dimensions in the adjoint of the gauge group, and \(D={\rm d} + [A,\,]\) the gauge covariant derivative.

Chapter 10

Ex. 10.3.10 Find the analogue of (10.3.76) for \({\rm AdS}_3\). [The result of Ex. 3.3.5 might be useful.]

Ex. 10.5.12 Work out the generalization of section 5.3.4 for deformations of a generalized complex structure \({\cal J}\) in terms of the generalized Dolbeault differential in (10.5.110).

Ex. 10.6.3\({}^*\) Check that the M-theory supersymmetry transformations (10.6.13) and the \(G\) equations in (10.6.12) together imply many components of the Einstein equations. When are all components implied? [This is much easier than the type II computation in Sec. 10.1.4: define a \({\cal D}_M\) such that \( \delta \psi_M = {\cal D}_M \epsilon\), and proceed as in (5.7.5) with \(D\to{\cal D}\). It might be convenient to rewrite \( \iota_M G = \frac12 [\Gamma_M,G]\). The computation was originally performed in [hep-th/0212008, App. B], but for a recent version more in line with the techniques in the book see [2112.10795, App. B.5]].

Ex. 10.6.4 Work out the analogue of (10.5.1) for the generalized \(G_2\)-structures defined in (10.6.6).

Ex. 10.6.5 Derive the pure spinor equations (10.3.20) for AdS\(_4\) from those in (10.6.7) for Mink\(_3\).

Chapter 11

Ex. 11.3.5 Derive (11.3.8) from the pure spinor equations, either in their original form (10.3.20) or using the Dolbeault variant (10.5.138).