### 5.5 Dyons

Dyons are on-shell supersymmetric D3-brane configurations describing a string bound state ending on the brane. They are described by the array
Since the discussion is analogous to the one for BIons, I shall be brief. The ansatz is as in Eq. (289) but including some magnetic components for the gauge field. This is both because a string is seen as a dyonic particle on the brane and a D-string is electrically charged under the RR two form. The latter can be induced from the Wess–Zumino coupling
This shows that magnetic components in couple to electric components in . Altogether, the dyonic ansatz is

##### Supersymmetry analysis:
In this case, the matrix elements are

while the induced gamma matrices are exactly those of Eq. (291). Due to the electric and magnetic components of the gauge field, the bosonic kappa matrix has a quadratic term in
To correctly capture the supersymmetries preserved by such a physical system, we impose the projection conditions
on the constant Killing spinor , describing a D3-brane and a -string bound state, respectively. Defining as the magnetic field and inserting Eqs. (319) and (320) into the resulting kappa symmetry preserving condition, one obtains
This equation is trivially satisfied when the following BPS conditions hold

##### Hamiltonian analysis:
Following [225], the Hamiltonian constraint can be solved and rewritten as a sum of positive definite terms

where the last equality holds for any angle . This allows one to derive the bound
Thus, the total energy satisfies
with
The bound (324) is extremised when
for which the final energy bound reduces to
Here corresponds to the energy of the vacuum configuration (infinite D3-brane). The bound (328) is saturated when
These are precisely the conditions (322) derived from supersymmetry considerations, confirming their BPS nature. Using the divergence free nature of both and , must be harmonic, i.e.,
The interpretation of the isolated point singularities in this harmonic function as the endpoints of string carrying electric and magnetic charge is analogous to the BIon discussion. In fact, all previous results can be understood in terms of the symmetry of type IIB string theory. In particular, a string, or fundamental string, is mapped into a string by an transformation rotating the electric and magnetic fields. The latter is a non-local transformation in terms of the gauge field , but leaves the energy density (323) invariant
Applying this transformation to the BIon solution, one reproduces Eq. (329).