3 Brane Effective Actions

This review is concerned with the dynamics of low energy string theory, or M-theory, in the presence of brane degrees of freedom in a regime in which the full string (M-) theory effective action9 reduces to

S ≈ SSUGRA + Sbrane. (17 )
The first term in the effective action describes the gravitational sector. It corresponds to 𝒩 = 2 d = 10 type IIA/IIB supergravity or 𝒩 = 1 d = 11 supergravity, for the systems discussed in this review. The second term describes both the brane excitations and their interactions with gravity.

More specifically, I will be concerned with the kinematical properties characterising S brane when the latter describes a single brane, though in Section 7, the extension to many branes will also be briefly discussed. From the perspective of full string theory, it is important to establish the regime in which the full dynamics is governed by Sbrane. This requires one to freeze the gravitational sector to its classical on-shell description and to neglect its backreaction into spacetime. Thus, one requires

|T background| ≫ |T brane|, (18 ) mn mn
where Tmn stands for the energy-momentum tensor. This is a generalisation of the argument used in particle physics by which one decouples gravity, treating Newton’s constant as effectively zero.

Condition (18View Equation) is definitely necessary, but not sufficient, to guarantee the reliability of Sbrane. I will postpone a more thorough discussion of this important point till Section 3.7, once the explicit details on the effective actions and the assumptions made for their derivations have been spelled out in Sections 3.13.6.

Below, I focus on the identification of the degrees of freedom and symmetries to describe brane physics. The distinction between world volume and spacetime symmetries and the preservation of spacetime covariance and supersymmetry will lead us, once again, to the necessity and existence of kappa symmetry.

 3.1 Degrees of freedom and world volume supersymmetry
  3.1.1 Supergravity Goldstone modes
 3.2 Bosonic actions
 3.3 Consistency checks
  3.3.1 M2-branes and their classical reductions
  3.3.2 T-duality covariance
  3.3.3 M5-brane reduction
 3.4 Supersymmetric brane effective actions in Minkowski
  3.4.1 D-branes in flat superspace
  3.4.2 M2-brane in flat superspace
 3.5 Supersymmetric brane effective actions in curved backgrounds
  3.5.1 M2-branes
  3.5.2 D-branes
  3.5.3 M5-branes
 3.6 Symmetries: spacetime vs world volume
  3.6.1 Supersymmetry algebras
  3.6.2 World volume supersymmetry algebras
 3.7 Regime of validity

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