D-brane

A D-brane is a hypersurface on which open-string endpoints obey Neumann boundary conditions tangentially and Dirichlet boundary conditions transversely. Its massless open-string vector mode is a worldvolume gauge field.

A type-IIB D-brane with five spatial dimensions. Its RR charge is magnetic with respect to the two-form potential $C_2$, or electric with respect to the dual six-form potential. It is distinct from the NS5-brane, whose charge is magnetic under the NS-NS two-form and whose string-frame tension has a different power of the coupling.

A Euclidean type-IIB D-brane localized in all spacetime directions, coupling electrically to the RR scalar $C_0$. It is an instanton, not a Lorentzian particle with negative spatial dimension. The magnetic partner of that scalar charge is a D7-brane.

A space-filling type-IIB D-brane. The associated ten-form potential is nonpropagating; its charge is not an additional local massless polarization. A consistent compact configuration must cancel any net space-filling RR charge, for example through appropriate other charged objects or projections.

A type-IIB D-brane with seven spatial dimensions. Its RR charge is magnetic with respect to $C_0$, or electric with respect to the dual eight-form potential. Its transverse geometry is codimension two, so the simple asymptotically flat $r^{p-7}$ harmonic formula for lower-dimensional D-branes does not apply unchanged.

For the asymptotically flat extremal D$p$ solution with $p<7$, $H=1+Q/r^{7-p}$ is harmonic away from the source and $e^{\Phi }=g_s{H}^{(3-p)/4}$. The electric potential can be chosen as $C_{0\cdots p}=g_s^{-1}(H^{-1}-1)$; magnetic descriptions use the dual flux, and a D3-brane has self-dual five-form flux. An equally oriented probe has DBI density $-{\mu }_p{g}_s^{-1}{H}^{-1}$ and Wess-Zumino density $+{\mu }_p{g}_s^{-1}(H^{-1}-1)$, whose sum is position independent. This proves the corresponding parallel D-brane no-force identity in the classical background. The supergravity approximation requires controlled curvature and coupling; it does not resolve every brane core.

Identical, parallel, equally oriented type-II D-branes preserve the same sixteen spacetime supercharges. Their annulus oscillator sum has the displayed numerator, which vanishes by the Jacobi abstruse identity. In the closed-string channel this is cancellation of attractive NS-NS exchange and repulsive RR exchange. The force vanishes in the flat supersymmetric configuration; anti-oriented branes, relative angles or supersymmetry-breaking backgrounds need not have this cancellation.

A D3-brane has three spatial dimensions. The low-energy fields of an isolated flat D3-brane form an abelian theory with sixteen linearly realized supersymmetry generators. Its transverse scalar fields describe its geometric position, and its gauge field can carry the charges of strings ending on it.

A D1-brane has one spatial dimension and couples to the two-form Ramond–Ramond potential. Its tension differs from the fundamental string tension by a factor $g_s^{-1}$ at zero axion.

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.

A massless open-string vector polarization tangential to a planar D-brane becomes a worldvolume gauge field. Transverse polarizations become scalar fields describing fluctuations of its position. This supports the interpretation of a Dirichlet plane as a dynamical extended object. The unprojected bosonic open-string tachyon additionally signals that its brane ground state is unstable; the massless field interpretation does not remove that instability.

An endpoint label attached to an open string. With several coincident branes, two endpoint labels organize the massless open-string fields as matrices and give nonabelian gauge symmetry. Strings between different stacks carry representations associated with the two endpoint groups. These labels are physical internal degrees of freedom, not extra spacetime coordinates.

An open string stretched a distance $L$ between parallel D-branes has squared mass $M^2=T^2{L}^2+(N-1)/{\alpha }^{\prime }$. The stretching energy makes strings between separated branes massive.

For two static parallel D-branes separated transversely by $d$, the open bosonic string mass spectrum becomes $M^2=d^2/(4{\pi }^2{\alpha }^{\prime 2})+(N-1)/{\alpha }^{\prime }$. The classical stretching contribution is the square of the string tension times the separation, while the oscillator vacuum contributes the intercept term. Thus the ground state is a tachyon for $d<2\pi \sqrt{{\alpha }^{\prime }}$, massless at equality, and massive above it. Its real rest energy, when stable, is $\sqrt{d^2/(4{\pi }^2{\alpha }^{\prime 2})-1/{\alpha }^{\prime }}$; the vacuum correction belongs to energy squared and is not an additive fixed subtraction from the stretching energy. Angled branes change oscillator moding and require a different formula.

A commuting diagonal expectation value has one transverse eigenvalue vector for each brane. In the covariant derivative normalization $D_a\varphi ={\partial }_a\varphi +i[A_a,\varphi ]$, the off-diagonal gauge field has mass $m^2={\sum }_I({\varphi }_1^I-{\varphi }_2^I{)}^2$. The corresponding stretched string has mass $|X_1-X_2|/(2\pi {\alpha }^{\prime })$ in its massless oscillator sector. Equality fixes the displayed scalar-position normalization, up to a common coordinate origin. The trace describes center-of-mass motion and the eigenvalue difference describes separation.

When $n$ parallel D-branes coincide, oriented strings joining distinct branes become massless alongside strings whose ends lie on one brane. Their Chan-Paton labels fill the adjoint representation of $U(n)$, enhancing $U(1{)}^n$ to $U(n)$.

For coincident D3-branes in the critical 26-dimensional bosonic string theory, the massless open-string oscillator has a four-dimensional gauge vector and 22 transverse scalar fields. Chan-Paton factors put all these fields in the adjoint of $U(k)$ for $k$ coincident branes. Ignoring the tachyon, their leading interactions are the dimensional reduction of 26-dimensional Yang-Mills theory. This contains a Yang-Mills sector but is not pure four-dimensional Yang-Mills theory and is not the supersymmetric six-scalar D3-brane theory of the ten-dimensional superstring.

For a D-brane in metric $G$, pulled-back B-field $B$ and worldvolume field strength $F$, tangential open-string coordinates obey the mixed condition $G_{ab}{\partial }_{\sigma }{X}^b+(B+2\pi {\alpha }^{\prime }F{)}_{ab}{\partial }_{\tau }{X}^b=0$. Only the gauge-invariant combination $B+2\pi {\alpha }^{\prime }F$ is physical.