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
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,
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,
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 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
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 [88]. One can check that the full bosonic M2-brane Lagrangian is equivalent to
where the modified conjugate momentum 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
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Living Rev. Relativity 15, (2012), 3
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