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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.

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 brane is an extended object with $p$ spatial dimensions. Its history is a $(p+1)$-dimensional worldvolume embedded in target spacetime. A point particle has $p=0$, a string has $p=1$, and a membrane has $p=2$. Geometric brane actions depend on the induced worldvolume metric; D-branes also carry open-string endpoint and gauge-field data.

Under T-duality, the positions $y_i$ of D-branes transverse to a circle of radius $R$ become diagonal Wilson lines on branes wrapping the dual circle $R^{\prime }={\alpha }^{\prime }/R$. Their holonomies are $\exp (i2\pi R^{\prime }{A}_i)=\exp (i{y}_i/R)$. An Open string stretched between D-branes with displacement $y_j-y_i+2\pi Rw$ becomes a string with shifted compact momentum $w/R^{\prime }+A_j-A_i$. The stretching contribution and the shifted-momentum contribution to its squared mass are identical.

A D0-brane is a point-like D-brane in space, with a one-dimensional worldvolume consisting only of time. It belongs to type IIA superstring theory and couples electrically to its one-form Ramond–Ramond potential. “Zero” counts spatial dimensions, rather than asserting zero mass or absence of dynamics.

Its mass scales as $1/(g_s\sqrt{{\alpha }^{\prime }})$, so it is heavy at weak string coupling. Unlike an ordinary perturbative string excitation, its open-string endpoints are fixed in every spatial direction for a static brane, while the brane's own position is a dynamical collective coordinate. The low-energy description of several nearby D0-branes is quantum mechanics of their worldvolume fields; it is not a field theory with an additional extended spatial direction.^[ref]

A D2-brane is a membrane-like D-brane with two spatial directions and a three-dimensional worldvolume including time. It is a charged extended object of type IIA superstring theory, coupling electrically to a three-form Ramond–Ramond potential.

The three-form can be integrated over its worldvolume to give the charge coupling, just as a one-form electromagnetic potential is integrated along a charged particle's trajectory. Open strings ending on the membrane supply a worldvolume gauge field and fields describing transverse motion. The relation between this three-dimensional gauge field and a scalar is useful in connecting the D2 description with an M2-brane, while the bulk and worldvolume descriptions address different degrees of freedom.

A D4-brane is a D-brane with four spatial directions and a five-dimensional worldvolume, belonging to type IIA superstring theory. It supports the low-energy modes of open strings ending on it, while also coupling to bulk closed-string fields.

Its electric coupling uses a five-form Ramond–Ramond potential, dual to the three-form potential coupled electrically to a D2-brane. Thus brane dimension determines which differential-form field measures its charge, not merely the size of a geometric object. At long wavelengths, an isolated brane's slowly varying fields are described by a Dirac-Born-Infeld action, with transverse scalar fields encoding its position and fluctuations.

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 D6-brane is a D-brane with six spatial dimensions. Its seven-dimensional worldvolume includes time, leaving three transverse directions in ten-dimensional type IIA superstring theory. Open-string endpoints can move along the brane while satisfying fixed-position Dirichlet boundary conditions in the transverse directions.

It couples electrically to the seven-form description of a Ramond–Ramond potential, equivalently carrying magnetic charge for the IIA one-form potential. This electric–magnetic distinction relates the D6-brane to the D0-brane: their relevant field strengths are dual in ten dimensions. The open-string description supplies worldvolume fields, whereas the closed-string description treats the same brane as a source for bulk fields.

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.

A D8-brane is a D-brane with eight spatial directions and a nine-dimensional worldvolume. In ten-dimensional type IIA superstring theory, it has one transverse spatial direction, so it acts as a domain wall. Its charge is associated with a nine-form Ramond–Ramond potential.

The dual zero-form field strength is the Romans mass; its value can jump across a charged D8-brane. Consequently a D8-brane cannot simply be inserted into the elementary massless M-theory circle duality dictionary used for lower-dimensional IIA branes. Its gravitational matching and scalar-field profiles must be treated consistently on both sides of the wall.^[ref]

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 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.

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.

An M-brane is a brane in M-theory. Its elementary membrane and five-brane carry electric and magnetic charges for the three-form potential of eleven-dimensional supergravity. Compactification and wrapping relate them to strings and D-branes.

For the Nambu-Goto phase-space action, ${\dot{P}}_m={\partial }_{\sigma }{J}_m$ with $J_m=e{T}^2{X}_m^{\prime }+u{P}_m$. Thus ${\dot{\mathcal{P}}}_m=J_m(\pi )-J_m(0)$. Free endpoint variations force $J_m=0$ individually. A Dirichlet boundary condition allows an external support or D-brane to absorb momentum in the fixed direction.

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.

The common Neumann boundary conditions define a $(p+1)$-dimensional timelike worldvolume: one time direction and $p$ spatial directions. The remaining Dirichlet boundary conditions fix its transverse position. An open string therefore ends on a D-brane of spatial dimension $p$, a D$p$-brane. For the critical bosonic string theory there are $25-p$ transverse directions.

If the Dirichlet constants agree at the two endpoints, both ends lie on the same planar D-brane. If they differ, the string stretches between two parallel D-branes with the same orientation but different positions. Neumann directions are tangent to the brane; Dirichlet directions are transverse to it. The brane is dynamical: tangential massless open-string polarizations supply a gauge field, while transverse ones describe its displacement, as in worldvolume fields from open-string massless states. The unprojected bosonic brane also has a tachyon; the boundary-condition interpretation does not make that ground state supersymmetrically stable.

A supersymmetric type-II D-brane carries a Ramond–Ramond potential charge and saturates the corresponding generalized BPS bound in supersymmetry: its brane tension equals the charge-dependent central/tensorial-charge bound in the relevant normalization. It is therefore a BPS state, preserving sixteen of the thirty-two spacetime supercharges. Identical parallel branes with the same orientation impose the same supersymmetry projector. This statement does not apply to arbitrary unstable non-BPS branes or to the bosonic branes of the preceding parts.

One direct demonstration of the parallel D-brane no-force identity is the annulus vacuum amplitude. For separation $r$, its oscillator-dependent part is proportional toHereThe first two terms describe the projected Neveu–Schwarz sector, and the last is the oppositely signed Ramond sector contribution. The Jacobi abstruse identity gives $f_3^8-f_4^8-f_2^8=0$ for every $t$. ThereforeIn the closed-string channel, attractive graviton and dilaton exchange cancel repulsive Ramond–Ramond potential exchange. The classical probe calculation in the D-brane supergravity solution gives the same cancellation. The common supersymmetry allows separation to remain a flat modulus; relative angles, reversed RR charge or supersymmetry-breaking backgrounds can remove this protection. For a brane–antibrane pair the RR interaction reverses and does not cancel the NS-NS attraction. The charge and half-supersymmetry interpretation is established in Dirichlet-Branes and Ramond-Ramond Charges.

For $N$ parallel type-II D-branes, an oriented open string has endpoint labels $i,j$, its Chan-Paton factors. At coincidence, the massless states form $N\times N$ matrices, giving adjoint fields of $U(N)$. Tangential polarizations produce the Yang-Mills theory gauge field; transverse polarizations produce $9-p$ adjoint scalar fields. The leading low-energy action is the dimensional reduction of ten-dimensional super Yang-Mills theory to $p+1$ dimensions. Expanding the Dirac-Born-Infeld action gives the gauge and scalar kinetic terms, while dimensional reduction supplies the Hermitian commutator potential in D-brane Yang-Mills theory:The sign is important: for Hermitian $\varphi$, the commutator is anti-Hermitian. Zero potential requires commuting scalars, whose simultaneous eigenvalues are brane positions through the D-brane scalar-position normalizationThis makes the Higgs mechanism geometrical. Separating branes gives masses to strings joining different positions; bringing them together gives Coincident-D-brane gauge enhancement.

Two parallel D3-branes in type IIB superstring theory give four-dimensional N=4 super Yang-Mills theory with gauge group $U(2)$, containing a gauge field, six real adjoint scalar fields and four adjoint Weyl fermions. On its Coulomb branch, chooseFor distinct position vectors, $U(2)$ breaks to $U(1)\times U(1)$. The off-diagonal vector multiplets are the two orientations of a string joining the branes, andThe same mass follows from the scalar covariant-derivative term, $m_W^2={\sum }_I({\varphi }_1^I-{\varphi }_2^I{)}^2$. The overall $U(1)$ describes center-of-mass motion; the relative $SU(2)$ sector is Higgsed to its Cartan $U(1)$. At coincidence the off-diagonal states become massless and the full $U(2)$ gauge symmetry is restored.