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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 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 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 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.

A D-brane is a localized RR-charged source whose long-distance fields solve supergravity. The extremal charged $p$-brane solutions carry the same RR charges, brane tension and preserved supersymmetry as the open-string construction. For the standard asymptotically flat cases $p<7$, the D-brane supergravity solution in the string frame isThe function $H$ is a positive harmonic function away from the source, in $9-p$ transverse dimensions. In an electric gauge, $C_{0\cdots p}=g_s^{-1}(H^{-1}-1)$, with its orientation chosen to match the source. Magnetic branes use the dual flux; for the D3-brane, the five-form includes both electric and magnetic pieces and is self-dual. Killing spinors obey one brane projector, leaving sixteen supersymmetries. Aligned multicenter harmonic functions describe several branes with no static binding force.

To check this identification directly, insert an equally oriented static probe into the solution. Put ${\mu }_p=(2\pi {)}^{-p}{\alpha }^{\prime -(p+1)/2}$, so the asymptotic tension is ${\mu }_p/g_s$. The induced-volume factor is $H^{-(p+1)/4}$; combined with $e^{-\Phi }$, its DBI action density is $-{\mu }_p{g}_s^{-1}{H}^{-1}$. Its Wess-Zumino brane coupling contributes $+{\mu }_p{g}_s^{-1}(H^{-1}-1)$. Hence the total static Lagrangian density is the constant $-{\mu }_p/g_s$, independently of position. This gives the parallel D-brane no-force identity in the spacetime description.

The supergravity background is the back-reacted, coarse long-distance description, while the boundary-condition description exposes the light open-string degrees of freedom. Controlled use of supergravity requires curvature small in string units and suitable local coupling. Singular cores and the special codimension-two or domain-wall cases are not automatically resolved by the elementary harmonic ansatz.

Take $0\le \sigma \le \pi$ and split the coordinates into $a=0,\dots ,p$ along the branes and $I=p+1,\dots ,25$ transverse to them. In conformal gauge the wave equation permits a separated mode $e^{-i\omega \tau }f(\sigma )$ with $f^{\prime \prime }+{\omega }^{2}f=0$. For Neumann boundary conditions at both ends, $f^{\prime }(0)=f^{\prime }(\pi )=0$ selects $\cos m\sigma$ and integer $m$. The zero-frequency solution is independent of $\sigma$, with a constant position and a term linear in time. Therefore the open-string mode expansion isThe canonical momentum density integrated over this interval is $k^a$.

For Dirichlet boundary conditions, subtract the static interpolation between fixed endpoint positions $y_0^I$ and $y_{\pi }^I$. The remaining field vanishes at both ends, selecting $\sin m\sigma$ with integer $m$ and excluding a momentum worldsheet zero mode. The expansion isHere ${\alpha }_m^{I†}={\alpha }_{-m}^I$. There is no factor $i$ in the sine series with this reality convention: changing $m$ to $-m$ reverses both the sine and the denominator. These are the integer-mode expansion with parallel D-brane endpoints; they are not the half-integer modes of a coordinate with a Neumann condition at one endpoint and a Dirichlet condition at the other.

The endpoints can move on $(p+1)$-dimensional worldvolumes but are fixed in the transverse directions. Thus these are open strings attached to D-branes, or stretched between parallel D-branes at $y_0$ and $y_{\pi }$. The tangential momenta describe propagation on the brane; the transverse positions describe its embedding. Let $L=|y_{\pi }-y_0|$. The linear spatial worldsheet zero mode contributes $L^2/(4{\pi }^2{\alpha }^{\prime })$ to $L_0$. Quantization with string intercept one givesThe first term is also the square of the classical stretched-string energy $TL$, since the string tension is $T=1/(2\pi {\alpha }^{\prime })$.

For two coincident D3-branes, label the ends of an oriented open string by $i,j\in \{1,2\}$. The four endpoint combinations give Chan-Paton factors ${\lambda }_{ij}$, which form the full algebra of $2\times 2$ matrices. Under a change of brane basis these transform as $\lambda \mapsto U\lambda U^{-1}$, the adjoint action of $U(2)$. At $L=0$, the first oscillator level is massless. Its tangential polarization supplies a four-dimensional vector $A_a$, while the 22 transverse polarizations supply real adjoint scalar fields ${\Phi }^I$. This is the bosonic D3-brane low-energy field content.

The endpoint matrices multiply in their boundary order in disk amplitudes. The difference between the two orders in the three-vector interaction gives the matrix commutator; the kinematic factor is the one-derivative Yang-Mills theory cubic vertex. Factorization and the vector gauge redundancy then supply the quartic interaction with the same commutator. Equivalently, the leading low-energy action is the dimensional reduction of 26-dimensional Yang-Mills theory, withFor Hermitian scalar fields the last displayed term corresponds to a nonnegative potential, because their commutators are anti-Hermitian. Higher-derivative terms are suppressed at energies well below $1/\sqrt{{\alpha }^{\prime }}$. Ignoring the tachyon, the gauge sector is four-dimensional $U(2)$ Yang-Mills theory, accompanied by 22 adjoint scalar fields. It is not pure Yang-Mills theory, nor the supersymmetric six-scalar D3-brane theory of a ten-dimensional superstring.

Separate the two branes and use the D-brane scalar-position normalizationA Yang-Mills gauge transformation preserves this expectation value only when it is diagonal if $y_1\ne y_2$. Thus the unbroken group is $U(1)\times U(1)$. The scalar field kinetic term supplies a mass for the off-diagonal vectors, sinceSumming over $I$ gives $m_{12}^2=L^2/(4{\pi }^2{\alpha }^{\prime 2})$, exactly the mass of the stretched $N=1$ string. The two diagonal gauge vectors remain massless. This identifies brane separation with an adjoint Higgs mechanism, rather than an explicit deletion of the off-diagonal strings.

For the ground state of an open string joining the two different branes, $N=0$, so the bosonic stretched-string tachyon threshold isAt equality it is massless, and above it the stretched ground state has positive mass squared. The same-brane ground strings still have $M_0^2=-1/{\alpha }^{\prime }$; this calculation removes only the stretched-string instability, not the original instability of the bosonic branes.