Brane effective actions capture the relevant dynamics of M-theory or string theory in some appropriate regimes of validity. Thus, they contain reliable information about its spectrum and its dynamics in those regimes. In this section, I will develop the tools to study the world volume realisation of supersymmetric states carrying the extra bosonic (topological) charges appearing in the maximal supersymmetry algebras introduced in Sections 3.6.1 and 3.6.2.

One such realisation is in terms of classical bosonic on-shell configurations. As it often occurs with supersymmetric configurations, instead of focusing on the integration of the equations of motion, I will focus on the conditions ensuring preservation of supersymmetry and on their physical interpretation. In particular,

- I will argue the existence of a necessary condition that any bosonic supersymmetric configuration must satisfy involving the kappa symmetry matrix and the background Killing spinors .
- I will review the Hamiltonian formulation for brane effective actions to compute the energy of
these configurations. The latter will minimise the energy for a given set of charges carried by the
state. The existence of energy bounds can be inferred from merely algebraic considerations and I
will discuss their field theory realisations as BPS bounds
^{30}. Furthermore, the relation between their saturation and the solution to the necessary kappa symmetry condition will also be explained. - I will discuss the relation between these physical considerations and the mathematical notion of calibration, which is a purely geometric formulation of the problem of finding volume minimising surfaces. Since the latter corresponds to a subset of bosonic brane supersymmetric configurations, this connection will allow us to review the notion of generalised calibration, which, in physical terms, includes world volume gauge field excitations.

The framework and set of relations covered in this section are summarised in Figure 6.

4.1 Supersymmetric bosonic configurations and kappa symmetry

4.2 Hamiltonian formalism

4.2.1 D-brane Hamiltonian

4.2.2 M2-brane Hamiltonian

4.2.3 M5-brane Hamiltonian

4.3 Calibrations

4.2 Hamiltonian formalism

4.2.1 D-brane Hamiltonian

4.2.2 M2-brane Hamiltonian

4.2.3 M5-brane Hamiltonian

4.3 Calibrations

Living Rev. Relativity 15, (2012), 3
http://www.livingreviews.org/lrr-2012-3 |
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