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, the 2-form charges M2-branes [90, 91], whereas the 5-form charges , M5-branes . 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 . 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,
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,BPS equations [107, 429]. Thus, saturation of the bound matches the energy with some charges that may usually have some topological origin .
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 ), 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 formulation31, 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 to generate the tension dynamically [86, 356]. It is convenient to study the tensionless limit in these actions as a generalisation of the massless particle action limit. It was shown in  that can be written as a sum of constraints sources in Gauss’ law. Finally, and generate world-space diffeomorphisms and time translations, respectively.
The modified conjugate momenta and determining all these constraints are defined in terms of the original conjugate momenta as
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 . One can check that the full bosonic M2-brane Lagrangian is equivalent to
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 . 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  that the full bosonic M5-brane Lagrangian in phase space equals
Living Rev. Relativity 15, (2012), 3
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