List of Footnotes

1 Non-perturbative in the sense that their mass goes like 1∕gs, where gs is the string-coupling constant.
2 The discovery of the RNS model of interacting bosons and fermions in d = 10 critical dimensions is due to joining the results of the original papers [432Jump To The Next Citation Point, 404Jump To The Next Citation Point]. This was developed further in [405, 232].
3 The calculations of beta functions in general nonlinear sigma models were done in [17, 215]. For a general discussion of string theory in curved backgrounds see [124] or the discussions in books [260Jump To The Next Citation Point, 425Jump To The Next Citation Point].
4 Polyakov used the formulation of classical string theory in terms of an auxiliary world sheet metric [116, 176] to develop the modern approach to the path integral formulation of string theory in [427, 428].
5 Recently, it was pointed out in [390] that there may exist quantum-mechanically consistent superstrings in d = 3. It remains to be seen whether this is the case.
6 The existence of kappa symmetry as a fermionic gauge symmetry was first pointed out in superparticle actions in [169, 170, 451, 171]. Though the term kappa symmetry was not used in these references, since it was later coined by Townsend, the importance of WZ terms for its existence is already stated in these original works.
7 For a proper definition of these superfields, see Appendix A.1.
8 See Appendix A.1 for a better discussion of what this means.
9 See [490, 421] for reviews and textbooks on what an effective field theory is and what the principles behind them are.
10 I will introduce this notion more thoroughly in Section 3.4 and Appendix A.
11 Here, 𝒩 stands for the number of world volume supersymmetries.
12 The first examples of this phenomena were reported by Nambu [401] and Goldstone [242].
13 For earlier reviews on D-brane effective actions and on M-brane interactions, see [320Jump To The Next Citation Point] and [101], respectively.
14 Since I am not considering supersymmetric branes at this point, |Qp | = Tp is not a necessary condition.
15 This is the correct way to compute the energy momentum tensor due to the coupling of branes to gravity. The energy carried by such a brane must be localised on its p+ 1 dimensional world volume.
16 The importance of these assumptions will be stressed when discussing the regime of validity of brane effective actions in Section 3.7.
17 There actually exist further gravitational interaction terms necessary for the cancellation of anomalies [253], but we will always omit them in our discussions concerning D-brane effective actions.
18 For a discussion on the interpretation of an M5-brane as a ‘D-brane’ for an open membrane, see [55].
19 Relevant work on the subject includes [24, 77, 16, 75].
20 For a discussion of the supersymmetric and kappa invariant M5-brane covariant action propagating in super-Poincaré, see [144Jump To The Next Citation Point].
21 Following the same philosophy as for their bosonic truncations, this functional dependence can be derived from the double dimensional reduction of the supersymmetric M2-brane action to be discussed in Section 3.4.2 [477, 439]. This also provides a derivation of the WZ couplings to be constructed in this subsection. Of course, this consideration would only apply to the D2-brane, but T-duality would allow one to extend this conclusion for any Dp-brane [292, 331]
22 For p = 1, B2 is the NS-NS 2-form, whereas for p = 2, B3 = A3 is the d = 11 3-form gauge potential.
23 The derivation of the property (146View Equation) is made more manifest in formalisms in which the tension is generated dynamically by the addition of an auxiliary volume density [86Jump To The Next Citation Point, 356Jump To The Next Citation Point, 94Jump To The Next Citation Point].
24 I will prove this explicitly in Section 5.1.
25 All our charge conjugation matrices are antisymmetric and unitary, i.e., T C = − C and † C C = 𝟙. Furthermore, all Clifford matrices satisfy the symmetry relation T − 1 Γm = − CΓ mC.
26 For earlier work, see [4], which extended the original Volkov–Akulov approach in [489].
27 I have assumed both the background and the brane preserve some supersymmetry.
28 There exists some similar phenomena on the M5-brane dynamics with the self-dual 3-form field strength. See [74] for a discussion on the emergence of noncommutative gauge theories when the self-dual 3-form field strength is close to its critical value.
29 There are several claims in the literature advocating that extra massless degrees of freedom emerge in brane effective actions when the latter probe black holes very close to their horizons. See [322, 323, 313, 300] for interesting work in this direction.
30 BPS stands for Bogomolny, Prasad and Sommerfield and their work on stable solitonic configurations [107Jump To The Next Citation Point, 429Jump To The Next Citation Point].
31 For a complete and detailed discussion of the supersymmetric and kappa invariant D-brane Hamiltonian formalism, see [94Jump To The Next Citation Point], which extends the bosonic results in [356Jump To The Next Citation Point, 327] and the type IIB superMinkowski ones in [329]. Here I follow [94Jump To The Next Citation Point] even though the analysis is restricted to the bosonic sector.
32 This notation is introduced to emphasise that ij 𝒢5 does not correspond to the world space components of 𝒢μν, but to the inverse matrix of the restriction of 𝒢μν to the world space subspace.
33 There are many papers studying the dynamics of BIons, including [353, 31, 335] and [475], where the solution to the Born–Infeld action reviewed here is proven to solve the equations of motion derived from higher-order corrections to the effective action.
34 For simplicity I am setting the D3-brane tension to one.
35 This equation has a huge history in mathematical physics. For a self-contained presentation on all the mathematical developments regarding this equation, see [178]. For generalisations to higher dimensions, see [180, 179].
36 Strictly speaking, if the supertube cross-section is open, they can carry D2-brane charge. The arguments given above only apply to closed cross-sections. The reader is encouraged to read the precise original discussion in [380Jump To The Next Citation Point] concerning this point and the bounds on angular momentum derived from it.
37 For a list of reviews on this subject, see [382, 383, 65, 459, 36, 454].
38 By arbitrary, it is meant a general curve that is not self-intersecting and whose tangent vector never vanishes.
39 I do not write this term explicitly here because it will not couple to our D3-brane probes.
40 Dual giant gravitons are spherical rotating D3-branes in which the 3-sphere wrapped by the brane is in AdS5. See [264] for a proper construction of these configurations and some of its properties.
41 These configurations appeared in [64, 71], extending earlier seminal work [173, 53].
42 What is meant here by maximally entropic is that, given a large black hole, there may be more than one possible deconstruction of the total charge in terms of constituents with different charge composition. By maximally entropic I mean the choice of charge deconstruction whose moduli space of configurations carries the largest contribution to the entropy of the system.
43 This overall coupling constant is derived from the string tension 1∕2πα′ and the overall L2 ∼ gsN α′ scale from the AdS5 × S5 background geometry.
44 For an analysis of supersymmetric D5-branes in a supergravity background dual to 𝒩 = 1 SYM, see [408].
45 ΦI is the natural adjoint scalar field after dimensional reduction. The rescaling by 2πα ′ is to match the natural scalar fields appearing in the abelian description provided by the DBI action. A similar rescaling occurs for the fermions omitted below, Ψ = 2πα ′ψ.
46 There is a lot of work in this direction. For a review on the emergence of geometry and gravity in matrix models, in particular in the context of the IKKT matrix conjecture [316], see [465]. For more recent discussions, see [106].
47 Using T-duality arguments this would also include acceleration and higher-derivative corrections in the scalar sector m X describing the excitations of the D-brane along the transverse dimensions.
48 Eight is the number of transverse dimensions to the world volume of the M2-branes.
49 For a complete list of references, see [12Jump To The Next Citation Point].
50 The equivalence of the equations of motion obtained in the PST-formalism and the ones developed in the superembedding formalism was proven in [46].
51 The reader should keep in mind that the RR field strengths Rn with n ≥ 5 are non-physical, in the sense that they are Hodge duality related to the physical propagating degrees of freedom contained in the RR field strengths R10−n [185, 196].
52 The chirality can be ±1 or ±i, depending on the reality of the volume form eigenspace.
53 Same comments as above.