The existence of energy bounds in supersymmetric theories can already be derived from purely superalgebra considerations. For example, consider the M-algebra (171). Due to the positivity of its left-hand side, one derives the energy bound

where the charge conjugation matrix was chosen to be and the spacetime indices were split as . For simplicity, let us set the time components and to zero. The superalgebra reduces to The bound (215) is now equivalent to the statement that no eigenvalue of can exceed unity. Any bosonic charge (or distribution of them) for which the corresponding satisfies defines a projector . The eigenspace of with eigenvalue 1 coincides with the one spanned by the Killing spinors determining the supersymmetries preserved by supergravity configurations corresponding to individual brane states. In other words, there is a one-to-one map between half BPS branes, the charges they carry and the precise supersymmetries they preserve. This allows one to interpret all the charges appearing in in terms of brane excitations: the 10-momentum describes massless superparticles [93], the 2-form charges M2-branes [90, 91], whereas the 5-form charges , M5-branes [464]. This correspondence extends to the time components . These describe branes appearing in Kaluza–Klein vacua [311, 481]. Specifically, is carried by type IIA D6-branes (the M-theory KK monopole), while can be related to type IIA D8-branes.That these algebraic energy bounds should allow a field theoretical realisation is a direct consequence of the brane effective action global symmetries and Noether’s theorem [406, 407]. If the system is invariant under time translations, energy will be preserved, and it can be computed using the Hamiltonian formalism, for example. Depending on the amount and nature of the charges turned on by the configuration, the general functional dependence of the bound (215) changes. This is because each charge appears in multiplied by different antisymmetric products of Clifford matrices. Depending on whether these commute or anticommute, the bound satisfied by the energy changes, see for example a discussion on this point in [394]. Thus, one expects to be able to decompose the Hamiltonian density for these configurations as sums of the other charges and positive definite extra terms such that when they vanish, the bound is saturated. More precisely,

- For non-threshold bound states, or equivalently, when the associated Clifford matrices anticommute, one expects the energy density to satisfy
- For bound states at threshold, or equivalently, when the associated Clifford matrices commute, one expects

In both cases, the set involves non-trivial dependence on the dynamical fields and their derivatives. Due to the positivity of the terms in the right-hand side, one can derive lower bounds on the energy, or BPS bounds,

being saturated precisely when are satisfied, justifying their interpretation as BPS equations [107, 429]. Thus, saturation of the bound matches the energy with some charges that may usually have some topological origin [165].In the current presentation, I assumed the existence of two non-trivial charges, and . The argument can be extended to any number of them. This will change the explicit saturating function in Eq. (215) (see [394]), but not the conceptual difference between the two cases outlined above. It is important to stress that, just as in supergravity, solving the gravitino/dilatino equations, i.e., , does not guarantee the resulting configuration to be on-shell, the same is true in brane effective actions. In other words, not all configurations solving Eq. (214) and saturating a BPS bound are guaranteed to be on-shell. For example, in the presence of non-trivial gauge fields, one must still impose Gauss’ law independently.

After these general arguments, I review the relevant phase space reformulation of the effective brane Lagrangian dynamics discussed in Section 3.

As in any Hamiltonian formulation^{31},
the first step consists in breaking covariance to allow a proper treatment of time evolution. Let me split the
world volume coordinates as for and rewrite the bosonic D-brane
Lagrangian by singling out all time derivatives using standard conjugate momenta variables

The modified conjugate momenta and determining all these constraints are defined in terms of the original conjugate momenta as

stands for the pullback to the world volume of the contraction of along the vector field . Equivalently, . is defined analogously. Notice stands for the Hodge dual in the -dimensional D-brane world space.In practice, given the equivalence between the Lagrangian formulation and the one above, one solves the equations of motion on the subspace of configurations solving Eq. (214) in phase space variables and finally computes the energy density of the configuration by solving the Hamiltonian constraint, i.e., , which is a quadratic expression in the conjugate momenta, as expected for a relativistic dynamical system.

The Hamiltonian formulation for the M2-brane can be viewed as a particular case of the analysis provided above, but in the absence of gauge fields. It was originally studied in [88]. One can check that the full bosonic M2-brane Lagrangian is equivalent to

where the modified conjugate momentum is related to the standard conjugate momentum by where describes the Hodge dual computed in the 2-dimensional world space spanned by . Notice no dynamically-generated tension was considered in the formulation above.As before, one usually solves the equations of motion in the subspace of phase space configurations solving Eq. (214), and computes its energy by solving the Hamiltonian constraint, i.e., .

It turns out the Hamiltonian formulation for the M5-brane dynamics is more natural than its Lagrangian
one since it is easier to deal with the self-duality condition in phase space [92]. One follows the same
strategy and notation as above, splitting the world volume coordinates as with .
Since the Hamiltonian formulation is expected to break into , one works in the
gauge . It is convenient to work with the world space metric and its inverse
^{32}.
Then, the following identities hold

It was shown in [92] that the full bosonic M5-brane Lagrangian in phase space equals

where and are the conjugate momenta to and the 2-form Notice the last equation is equivalent to , from which we conclude , using the Bianchi identify for . The last three functionals appearing in Eq. (229) correspond to constraints generating time translations, world space diffeomorphisms and the self-duality condition. The following definitions were used in the expressions above As for D-branes and M2-branes, in practice one solves the equations of motion in the subspace of phase space configurations solving Eq. (214) and eventually computes the energy of the system by solving the quadratic constraint coming from the Hamiltonian constraint .
Living Rev. Relativity 15, (2012), 3
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