Dirac-Born-Infeld action ($-T_p\int e^{-\tilde{\Phi }}\sqrt{-\det (P[G]+P[B]+2\pi {\alpha }^{\prime }F)}$)

The Dirac-Born-Infeld action describes the leading derivative dynamics of one D-brane, including nonlinear dependence on slowly varying worldvolume field strengths. Its induced tensors are D-brane worldvolume pullbacks. The combination $P[B]+2\pi {\alpha }^{\prime }F$ is invariant under the combined two-form and boundary gauge transformation. The dilaton factor reflects a disk's Euler characteristic. Expanding about flat space with small field strengths and embedding gradients gives the worldvolume gauge and scalar kinetic terms. A coincident stack has a non-Abelian low-energy Yang-Mills action; this does not imply that a naive traced determinant is exact at every order.

With ${\mathbf{E}}_i=2\pi {\alpha }^{\prime }{F}_{0i}$, this normalization separates the nonlinear constitutive relation from the D3-brane tension. It is the normalized canonical electric momentum. The Gauss law constraint in gauge theory requires its divergence to vanish away from string endpoints. On the electric BIon locus it equals the scaled electric field, but this equality is not true for arbitrary configurations.

A supersymmetric spike of a D-brane combines a worldvolume gauge field with a transverse scalar profile. Its geometry and charge represent strings ending on the brane. Electric spikes represent fundamental strings; magnetic spikes on a D3-brane represent D1-branes. The original construction is Brane Dynamics From the Born-Infeld Action.

The DBI action resums field strengths but truncates their derivative expansion. A slowly varying large-charge throat may therefore be controlled even when its field strengths are not small. Large derivatives, large embedding curvature, string loops and gravitational backreaction restrict this description, and its ideal singular core is not uniformly controlled. The electric throat window $|N|g_s\gg 1$, $g_s^2|N|\ll 1$ and magnetic window $|N|\gg 1$, $g_s|N|\ll 1$ separate derivative and backreaction requirements.

For $X-X_{\infty }=c/r$, subtracting the planar D3-brane energy leaves $T_3\int |\nabla X{|}^2{d}^{3}x=4\pi T_3{c}^2/r_0$ outside a cutoff. Since $X_{\mathrm{cut}}-X_{\infty }=c/r_0$, its coefficient is the carried string tension. Electric charge quantization gives $c=\pi g_s|N|{\alpha }^{\prime }$ and coefficient $|N|T_{F1}$; magnetic charge gives $c=\pi |N|{\alpha }^{\prime }$ and coefficient $|N|T_{D1}$.

For one constant charge angle, the DBI Hamiltonian bound for an electric spike and its dyonic extension give the displayed equations away from sources. Each aligned branch preserves eight of the sixteen linearly realized worldvolume supersymmetries. The corresponding spacetime string–D3 configuration preserves eight of thirty-two. The bound and charge interpretation are derived in BPS Bounds for Worldvolume Branes.

For a static D3-brane with one scalar and no magnetic field, taking the Legendre transform of the DBI action gives the displayed identity. It bounds energy by the electric scalar charge and is saturated at $\mathbf{d}=\nabla X$. The opposite orientation uses the analogous identity with reversed signs. On saturation, the constitutive relation gives $\mathbf{E}=\mathbf{d}$ and the Gauss law constraint in gauge theory reduces to the Laplace equation for $X$.

For Hermitian scalar matrices, their commutator is anti-Hermitian, so $-\mathrm{Tr}[{\varphi }^I,{\varphi }^J{]}^2=\mathrm{Tr}[{\varphi }^I,{\varphi }^J{]}^{†}[{\varphi }^I,{\varphi }^J]\ge 0$. Dimensional reduction gives $F_{IJ}=i[{\varphi }^I,{\varphi }^J]$, fixing the minus sign before the literal commutator square inside the conventional overall-negative Yang-Mills action. Commuting scalar expectation values minimize this potential and describe ordinary separated branes. A plus sign instead requires interpreting the squared symbol as the positive Hermitian norm, not literal matrix multiplication.

The brane embedding pulls the spacetime metric and two-form back to the worldvolume. The dilaton is evaluated at that embedding. If the quoted tension includes the constant-background factor $e^{-{\Phi }_0}$, use $\tilde{\Phi }=\Phi (X)-{\Phi }_0$; otherwise use $\tilde{\Phi }=\Phi (X)$ and a tension coefficient independent of that factor. These are normalization conventions for the same Dirac-Born-Infeld action.