4.3 Calibrations
In the absence of WZ couplings and brane gauge field excitations, the energy of a brane configuration
equals its volume. The problem of identifying minimal energy configurations is equivalent to that of
minimising the volumes of
-dimensional submanifolds embedded in an n-dimensional ambient space. The
latter is a purely geometrical question that can, in principle, be mathematically formulated independently of
supersymmetry, kappa symmetry or brane theory. This is what the notion of calibration achieves. In this
subsection, I review the close relation between this mathematical topic and a subset of supersymmetric
brane configurations [235, 228, 2]. I start with static brane solitons in
, for which the
connection is more manifest, leaving their generalisations to the appropriate literature quoted
below.
Consider the space of oriented p dimensional subspaces of
, i.e., the Grassmannian
. For
any
, one can always find an orthonormal basis
in
such that
is a basis in
so that its co-volume is
A
-form
on an open subset
of
is a calibration of degree
if
- for every point
, the form
satisfies
for all
and such that the
contact set
is not empty.
One of the applications of calibrations is to provide a bound for the volume of
-dimensional
submanifolds of
. Indeed, the fundamental theorem of calibrations [289] states
Theorem:
Let
be a calibration of degree p on
. The
-dimensional submanifold
, for which
is volume minimising. One refers to such minimal submanifolds as calibrated submanifolds, or as
calibrations for short, of degree
.
The proof of this statement is fairly elementary. Choose an open subset
of
with boundary
and assume the existence of a second open subset
in another subspace
of
with the
same boundary, i.e.,
. By Stokes’ theorem,
where
is the volume form constructed out of the dual basis
to
.
Two remarks can motivate why these considerations should have a relation to brane solitons and
supersymmetry:
- For static brane configurations with no gauge field excitations and in the absence of WZ couplings, the
energy of the brane soliton equals the volume of the brane submanifold embedded in
. Thus,
bounds on the volume correspond to brane energy bounds, which are related to supersymmetry
saturation, as previously reviewed. Indeed, the dynamical field
does mathematically describe
the map from the world volume
into
. The above bound can then be re-expressed as
where
stands for the pullback of the
-form
.
- There exists an explicit spinor construction of calibrations emphasising the connection between
calibrated submanifolds, supersymmetry and kappa symmetry.
Let me review this spinor construction [159, 287]. For
, the
-form calibration takes
the form
where the set
(
) stands for the transverse scalars to the brane parameterising
,
is a constant real spinor normalised so that
and
are antisymmetrised products of
Clifford matrices in
. Notice that, given a tangent
-plane
, one can write
as
where the matrix
is evaluated at the point to which
is tangent. Given the restriction on the values of
,
It follows that
for all
. Since
is also closed, one concludes it is a calibration. Its contact
set is the set of
-planes for which this inequality is saturated. Using Eq. (239), the latter is equally
characterised by the set of
-planes
for which
Because of Eq. (241) and the fact that
, the solution space to this equation is always half the
dimension of the spinor space spanned by
for any given tangent
-plane
. However, this solution
space generally varies as
varies over the contact set, so that the solution space of the set is generally
smaller.
So far the discussion involved no explicit supersymmetry. Notice, however, that the matrix
in
Eq. (240) matches the kappa symmetry matrix
for branes in the static gauge with no gauge field
excitations propagating in Minkowski. This observation allows us to identify the saturation of the
calibration bound with the supersymmetry preserving condition (214) derived from the gauge fixing
analysis of kappa symmetry.
Let me close the logic followed in Section 4 by pointing out a very close relation between the
supersymmetry algebra and kappa symmetry that all my previous considerations suggest. Consider a single
infinite flat M5-brane propagating in
Minkowski and fix the extra gauge symmetry of the PST
formalism by
(temporal gauge). The kappa symmetry matrix (157) reduces to
where all
indices stand for world space M5 indices. Notice that the structure of this matrix is
equivalent to the one appearing in Eq. (216) for
by identifying
Even though, this was only argued for the M5-brane and in a very particular background, it does provide
some preliminary evidence for the existence of such connection. In fact, a stronger argument can be
provided by developing a phase space formulation of the kappa symmetry transformations that allows one
to write the supersymmetry anticommutator as [278
]
This result has not been established in full generality but it agrees with the flat space case [165] and those
non-flat cases that have been analysed [438, 437]. I refer the reader to [278
] where they connect the
functional form in the right-hand side of Eq. (245) with the kappa symmetry transformations for fermions
in its Hamiltonian form.
The connection between calibrations, supersymmetry and kappa symmetry goes beyond the arguments
given above. The original mathematical notion of calibration was extended in [277, 278] relaxing its first
condition
. Physically, this allowed one to include the presence of non-trivial potential
energies due to background fluxes coming from the WZ couplings. Some of the applications
derived from this notion include [231, 229, 230, 373, 139]. Later, the notion of generalised
calibration was introduced in [344
], where it was shown to agree with the notion of calibration
defined in generalised Calabi–Yau manifolds [267] following the seminal work in [298]. This
general notion allows one to include the effect of non-trivial magnetic field excitations on the
calibrated submanifold, but still assumes the background and the calibration to be static. Some
applications of these notions in the physics literature can be found in [344, 377, 413]. More
recently, this formalism was generalised to include electric field excitations [376], establishing a
precise correspondence between generic supersymmetric brane configurations and generalised
geometry.
Summary:
A necessary condition for a bosonic brane configuration to preserve supersymmetry is to solve
the kappa symmetry preserving condition (214). In general, this is not sufficient for being an on-shell
configuration, though it can be, if there are no gauge field excitations. Solutions to Eq. (214) typically
impose a set of constraints on the field configuration, which can be interpreted as BPS equations
by computing the Hamiltonian of the configuration, and a set of projection conditions on the
constant parts
of the background Killing spinors
. The energy bounds saturated when
the BPS equations hold are a field theory realisation of the algebraic bounds derived from the
supersymmetry algebra. An attempt to summarise the essence of these relations is illustrated in
Figure 6.