### 5.3 Intersecting M2 and M5-branes

As a second example of BPS excitation, consider the 1/4 BPS configuration
corresponding to the brane array
The idea is to describe an infinite M5-brane by the static gauge and to turn on a transverse scalar field
to account for the M2-brane excitation. However, is not enough to support an M2-brane
interpretation, since the latter is electrically charged under the 11-dimensional supergravity three form
. Thus, the sought M5-brane soliton must source the components. From the Wess–Zumino
coupling
one learns that the magnetic components, where hatted indices stand for world space directions
different from , i.e., , must also be excited.
The full ansatz will assume delocalisation along the direction, so that the string-like excitation in
the direction can be viewed as a membrane:

##### Supersymmetry analysis:

The M5-brane kappa symmetry matrix (157) in the temporal gauge
reduces to
For the subset of configurations described by the ansatz (268), it follows
This reduces Eq. (269) to
To solve the kappa symmetry preserving condition (214), I impose two projection conditions
on the constant Killing spinors . The eight supercharges satisfying them match the ones preserved by
. Using Eq. (272) in Eq. (271), keeps a non-trivial dependence on . Requiring
its coefficient to vanish gives rise to the BPS condition
Overall, the kappa symmetry preserving condition (214) reduces to the purely algebraic condition
To check this holds, notice the only non-vanishing components of are
This allows us to compute the determinant
which becomes a perfect square once the BPS equation (273) is used
This shows that Eq. (274) holds automatically. Thus, the solution to the kappa symmetry preserving
condition (214) for the ansatz (268) on an M5-brane action is solved by the supersymmetry projection
conditions (272) and the BPS equation (273). Since the soliton involves a non-trivial world
volume gauge field, the Bianchi identity must still be imposed. This determines the
harmonic character for the excited transverse scalar in the four dimensional world space

##### Hamiltonian analysis:

The Hamiltonian analysis for this system was studied in [225] following the M5-brane
phase space formulation given in Eq. (229). For static configurations, the Hamiltonian constraint can be
solved by the energy density as
where
and world space indices were denoted by latin indices . It was noted in [225] that by
introducing a unit length world space 5-vector , i.e., , the energy density could be written
in the suggestive form
The unit vector provides a covariant way of introducing a preferred direction in the 5-dimensional world
space. Choosing and , to match the delocalisation direction in our bosonic ansatz, one
derives the inequality
The latter is saturated if and only if
and
where is only defined on the 4-dimensional subspace , orthogonal to , and is its Hodge
dual. This confirms the BPS nature of Eq. (273). Since is closed, is harmonic in
.
To regulate the divergent energy, one imposes periodic boundary conditions in the 5-direction
making the orbits of the vector field have finite length . Then, the total energy satisfies
where is the topological charge
The tension of the soliton, i.e., energy per unit of length, equals . It is bounded by . It
only equals the latter for configurations satisfying Eq. (284). Singularities in the harmonic function match
the strings found in [301]. To check this interpretation, consider a solution with a single isolated point
singularity at the origin. Its energy can be rewritten as the small radius limit of a surface integral over a
3-sphere surrounding the origin. Since is constant on this integration surface, one derives the string
tension [225]
is the string charge. Even though this tension diverges, it does so consistently, being the boundary of a
semi-infinite membrane.