### 5.2 Intersecting M2-branes

As a first example of an excited configuration, consider the intersection of two M2-branes in a point
corresponding to the array
In the probe approximation, the M2-brane effective action describes the first M2-brane by fixing the static
gauge and the second M2-brane as an excitation above this vacuum by turning on two scalar fields
according to the ansatz
where runs over the spatial world volume directions and over the transverse directions not being
excited.

##### Supersymmetry analysis:

Given the ansatz (253), the induced metric components equal
(with ), whereas its determinant and the induced
gamma matrices reduce to
Altogether, the kappa symmetry preserving condition (214) is
If the excitation given in Eq. (253) must describe the array in Eq. (252), the subspace of Killing spinors
spanned by the solutions to Eq. (256) must be characterised by two projection conditions
one for each M2-brane in the array (252). Plugging these projections into Eq. (256)
one obtains an identity involving two different Clifford-valued contributions: the left-hand side is
proportional to the identity matrix acting on the Killing spinor, while the right-hand side involves some
subset of antisymmetric products of gamma matrices. Since these Clifford valued matrices are independent,
each term must vanish independently. This is equivalent to two partial differential equations
Notice this is equivalent to the holomorphicity of the complex function in terms of the
complex world space coordinates , since Eqs. (259) are equivalent to the Cauchy–Riemann
equations for .
When conditions (259) are used in the remaining left-hand side of Eq. (258), one recovers an identity.
Thus, the solution to Eq. (214) in this particular case involves the two supersymmetry projections (257)
and the BPS equations (259) satisfied by holomorphic functions .

##### Hamiltonian analysis:

Since this is the first non-trivial example of a supersymmetric soliton discussed in this
review, it is pedagogically constructive to rederive Eqs. (259) from a purely Hamiltonian point of
view [225]. This will also convince the reader that holomorphicity is the only requirement to be on-shell. To
ease notation below, rewrite Eq. (259) as
where standard vector calculus notation for is used, i.e., and .
Consider the phase space description for the M2-brane Lagrangian given in Eq. (226) in a Minkowski
background. The Lagrange multiplier fields impose the world space diffeomorphism constraints. In the
static gauge, these reduce to
where are the conjugate momenta to the eight world volume scalars describing transverse
fluctuations. For static configurations carrying no momentum, i.e., , the world space momenta will
also vanish, i.e., .
Solving the Hamiltonian constraint imposed by the Lagrange multiplier for the energy density
, one obtains [225]

This already involves the computation of the induced world space metric determinant and its rewriting in a
suggestive way to derive the bound
The latter is saturated if and only if Eq. (260) is satisfied. This proves the BPS character of the
constraint derived from solving Eq. (214) in this particular case and justifies that any solution
to Eq. (260) is on-shell, since it extremises the energy and there are no further gauge field
excitations.
Integrating over the world space of the M2 brane allows us to derive a bound on the charges carried by
this subset of configurations

stands for the energy of the infinite M2-brane vacuum, whereas is the topological charge
accounting for the second M2-brane in the system.
The bound (264) matches the spacetime supersymmetry algebra bound: the mass of the system is
larger than the sum of the masses of the two M2-branes. Field theoretically, the first M2-brane charge
corresponds to the vacuum energy , while the second corresponds to the topological charge
describing the excitation. When the system is supersymmetric, the energy saturates the
bound and preserves 1/4 of the original supersymmetry. From the world volume
superalgebra perspective, the energy is always measured with respect to the vacuum. Thus,
the bound corresponds to the excitation energy equalling . This preserves 1/2
of the world volume supersymmetry preserved by the vacuum, matching the spacetime 1/4
fraction.

For more examples of M2-brane solitons see [95] and for a related classification of D2-brane
supersymmetric soltions see [33].