5.5 Dyons

Dyons are on-shell supersymmetric D3-brane configurations describing a (p,q) string bound state ending on the brane. They are described by the array
D3 : 1 2 3 x x x x x x F : x x x 4 x x x x x (314 ) D1 : x x x 4 x x x x x .
Since the discussion is analogous to the one for BIons, I shall be brief. The ansatz is as in Eq. (289View Equation) but including some magnetic components for the gauge field. This is both because a (p,q) 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
∫ ๐’ž ∧ โ„ฑ . (315 ) 2
This shows that magnetic components in โ„ฑ couple to electric components in C2. Altogether, the dyonic ansatz is
μ μ i i X = σ , X = c , X4 (σa ) = y(σa) , V0 = V0(σa) , Va = Va(σb). (316 )

Supersymmetry analysis:
In this case, the matrix elements ๐’ข μν + โ„ฑ μν are

( − 1 F ) ๐’ขμν + โ„ฑμν = 0b , − F0a δab + ∂ay∂by + Fab − det(๐’ขμν + โ„ฑμν) = det(δab + ∂ay∂by − F0aF0b + Fab), (317 )
while the induced gamma matrices are exactly those of Eq. (291View Equation). Due to the electric and magnetic components of the gauge field, the bosonic kappa matrix has a quadratic term in F μν
1 Γ κ =--โˆ˜--------------๐œ€ μ0...μ3 (γμ0...μ3 iτ2 + 6Fμ0μ1γμ3μ4 τ1 + 3Fμ0μ1Fμ2μ3 iτ2) . (318 ) 4! − det(๐’ข + โ„ฑ)
To correctly capture the supersymmetries preserved by such a physical system, we impose the projection conditions
Γ 0123iτ2๐œ– = ๐œ–, (319 ) Γ 0y(cosα τ3 + sin α τ1)๐œ– = ๐œ–, (320 )
on the constant Killing spinor ๐œ–, describing a D3-brane and a (p,q)-string bound state, respectively. Defining a 1 abc B = 2๐œ– Fbc as the magnetic field and inserting Eqs. (319View Equation) and (320View Equation) into the resulting kappa symmetry preserving condition, one obtains
โˆ˜-------------- − det(๐’ข + โ„ฑ )๐œ– = [1 + Γ aΓ 0∂ay(cosα τ3 + sinα τ1) − Γ aΓ 0τ3F0a ab a +Γ F0a∂by(cosα τ3 + sinα τ1) − Γ Γ 0Ba τ1 +Ba ∂ay (cos ατ3 + sin ατ1) + BaF0ai τ2]. (321 )
This equation is trivially satisfied when the following BPS conditions hold
F = cos α∂ y, Ba = sin αδab∂ y. (322 ) 0a a b

Hamiltonian analysis:
Following [225Jump To The Next Citation Point], the Hamiltonian constraint can be solved and rewritten as a sum of positive definite terms34

2 โƒ— 2 โƒ— 2 โƒ— 2 โƒ— โƒ— 2 โƒ— โƒ— 2 โƒ— โƒ— 2 โ„ฐ = 1 + |∇y | + |E | + |B | + (E ⋅ ∇y ) + (B ⋅∇y ) + |E × B | = (1 + sin α โƒ—E ⋅ โƒ—∇y + cosα โƒ—B ⋅ โƒ—∇y )2 + |Eโƒ— − sin α โƒ—∇y |2 + |โƒ—B − cosα โƒ—∇y |2 2 2 + |cosα Eโƒ— ⋅ โƒ—∇y − sin α โƒ—B ⋅∇โƒ—y | + |Eโƒ— × โƒ—B | (323 )
where the last equality holds for any angle α. This allows one to derive the bound
โ„ฐ2 ≥ (1 + sin α โƒ—E ⋅∇โƒ—y + cosα โƒ—B ⋅ โƒ—∇y )2. (324 )
Thus, the total energy satisfies
E ≥ E0 + sin αZel + cosαZmag , (325 )
with
∫ ∫ Zel = โƒ—E ⋅ โƒ—∇y, Zmag = โƒ—B ⋅ โƒ—∇y. (326 ) D3 D3
The bound (324View Equation) is extremised when
tan α = Zelโˆ•Zmag, (327 )
for which the final energy bound reduces to
โˆ˜ ----------- E ≥ E + Z2 + Z2 . (328 ) 0 el mag
Here E0 corresponds to the energy of the vacuum configuration (infinite D3-brane). The bound (328View Equation) is saturated when
โƒ— โƒ— โƒ— โƒ— E = sin α ∇y, B = cos α∇y. (329 )
These are precisely the conditions (322View Equation) derived from supersymmetry considerations, confirming their BPS nature. Using the divergence free nature of both โƒ—E and โƒ—B, y must be harmonic, i.e.,
2 ∇ y = 0. (330 )
The interpretation of the isolated point singularities in this harmonic function as the endpoints of (p,q ) string carrying electric and magnetic charge is analogous to the BIon discussion. In fact, all previous results can be understood in terms of the SL (2,โ„ค) symmetry of type IIB string theory. In particular, a (1,0 ) string, or fundamental string, is mapped into a (p,q) string by an SO (2) transformation rotating the electric and magnetic fields. The latter is a non-local transformation in terms of the gauge field V, but leaves the energy density (323View Equation) invariant
( E ′a ) ( cosα − sinα ) ( Ea ) B ′a = sinα cosα Ba . (331 )
Applying this transformation to the BIon solution, one reproduces Eq. (329View Equation).
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