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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.

The string nonlinear sigma model regards the target-space metric, the Kalb–Ramond field and the dilaton as couplings of the two-dimensional worldsheet theory. Schematically its Euclidean terms areQuantization produces running background couplings and a possible worldsheet Weyl anomaly. Gauge-fixed matter bosons contribute central charge $D$, while the reparameterization bc ghost system contributes $-26$ in each chirality. In the critical flat background cancellation therefore requires $D=26$. In a curved background, vanishing improved sigma-model beta functions further imposes field equations for $G,B,\Phi$. For example, the leading metric condition isThese are spacetime conditions because the couplings are functions of the embedding coordinates, not additional worldsheet coordinates.

The leading equations arise by varying the string-frame massless effective actionThe factor $e^{-2\Phi }$ is the sphere contribution in the string genus expansion. Higher sigma-model orders generate higher-derivative terms in the ${\alpha }^{\prime }$ expansion; higher genera give string-loop corrections. Thus conformal consistency of the quantum worldsheet determines the low-energy target equations and their action. This is a massless-sector truncation: the bosonic tachyon is not removed by writing an action only for $G,B,\Phi$.

For a brane embedding $X^{\mu }(\xi )$, the D-brane worldvolume pullbacks areThe dilaton is evaluated at the embedding. When $T_p$ denotes the physical tension in a background with constant mode ${\Phi }_0$, the convention is $\tilde{\Phi }=\Phi (X)-{\Phi }_0$, so $e^{-\tilde{\Phi }}=1$ in that vacuum. Equivalently one can use $\tilde{\Phi }=\Phi (X)$ and a coefficient with the constant $e^{-{\Phi }_0}$ not already included. The physics is the same; the two conventions must not count that constant factor twice.

Several features motivate the Dirac-Born-Infeld action. Without gauge or two-form fields, it is the relativistic induced-volume action for a tensionful extended object. An open-string boundary couples to its gauge potential $A$, so a bulk transformation $B\mapsto B+d\Lambda$ is canceled by $A\mapsto A-P[\Lambda ]/(2\pi {\alpha }^{\prime })$. Thus only $P[B]+2\pi {\alpha }^{\prime }F$ can enter. For slowly varying constant field strength, the open-string disk partition function resums its dependence into the determinant; expansion gives the Maxwell kinetic term. The disk has Euler characteristic one, giving $e^{-\Phi }$ rather than the closed-string sphere's $e^{-2\Phi }$. Worldvolume covariance and the induced metric then combine these ingredients into the determinant expression.

Explicitly, in flat background and static gauge $X^a={\xi }^a$ with transverse fluctuations $X^I$, the weak-field expansion starts asThe linear term in the antisymmetric matrix $F$ vanishes; the quadratic term follows from expanding $\frac{1}{2}\mathrm{Tr}\log (1+2\pi {\alpha }^{\prime }{\eta }^{-1}F)$. Thus the DBI action contains both the scalar and gauge kinetic energies, while higher powers and derivatives are string-scale corrections.

For two coincident D-branes, an oriented open string has independent labels $r,s\in \{1,2\}$ at its ends. Its Chan-Paton factors are the matrix units $E_{rs}$. The four massless vector modes span the Lie algebra of $U(2)$: diagonal strings supply the two Abelian fields, and oppositely oriented interbrane strings supply the off-diagonal modes. Joining strings multiplies these matrices, producing the non-Abelian commutators. Transverse vector polarizations become adjoint scalar fields. This is Coincident-D-brane gauge enhancement.

The massless gauge/scalar dynamics reduce to dimensionally reduced Yang-Mills theory in a flat background with constant dilaton and vanishing pulled-back $B$, at energies and field variations well below the string scale. More precisely, derivative corrections satisfy ${\alpha }^{\prime }{E}^2\ll 1$, and one expands in small dimensionless field strengths and scalar gradients rather than retaining the full nonlinear determinant. Weak string coupling suppresses loop corrections. After subtracting the constant brane vacuum energy and using the scalar normalization fixed below, the coefficient isA bosonic brane additionally has an open-string tachyon; the displayed action governs its massless sector and is not a stable complete truncation. For supersymmetric branes the corresponding leading action also has the massless fermions.

There is a sign qualification in the printed scalar potential. With the stated $D_a\varphi ={\partial }_a\varphi +i[A_a,\varphi ]$, take $A_a,{\varphi }^I$ Hermitian and $[{\varphi }^I,{\varphi }^J{]}^2$ to mean the literal matrix product. Then dimensional reduction hasConsequently $F_{IJ}{F}^{IJ}=-[{\varphi }^I,{\varphi }^J{]}^2$. The consistent low-energy action isThis is the Hermitian commutator potential in D-brane Yang-Mills theory. Its potential energy is $-\sum \mathrm{Tr}[{\varphi }^I,{\varphi }^J{]}^2/(4g_{\mathrm{YM}}^2)\ge 0$, because a Hermitian-matrix commutator is anti-Hermitian. For a concrete check, take ${\varphi }^1=v{\sigma }_1$, ${\varphi }^2=v{\sigma }_2$ with Pauli matrices. Then $[{\varphi }^1,{\varphi }^2{]}^2=-4v^{4}I$, so the literal printed plus sign would make the static energy negative and unbounded below. The printed plus sign is consistent only if its square denotes the positive norm $[{\varphi }^I,{\varphi }^J{]}^{†}[{\varphi }^I,{\varphi }^J]$, rather than a literal square. The gauge-boson mass calculation below uses the unambiguous kinetic terms.

In an ordinary separated-brane vacuum the scalar expectation values commute and can be simultaneously diagonalized:Their eigenvalue vectors specify the transverse positions, up to the normalization to be determined. The trace encodes the center of mass; their difference encodes separation. When the two vectors coincide, every $U(2)$ transformation commutes with the vacuum. When they are distinct, only independent diagonal phases do, so the unbroken group is $U(1)\times U(1)$. The off-diagonal vectors acquire mass by the Higgs mechanism.

Write the off-diagonal entries of $A_a$ as $W_a$ and $W_a^{\ast }$ and put $\Delta v^I=v_1^I-v_2^I$. At a constant diagonal vacuum, $D_a{\varphi }^I=i[A_a,⟨{\varphi }^I⟩]$, soThe quadratic off-diagonal gauge kinetic term is $-f_{ab}^{\ast }{f}^{ab}/(2g_{\mathrm{YM}}^2)$ with $f_{ab}={\partial }_a{W}_b-{\partial }_b{W}_a$, and the scalar kinetic term supplies $-{\sum }_I(\Delta v^I{)}^2{W}_a^{\ast }{W}^a/g_{\mathrm{YM}}^2$. Comparing these with the complex massive-vector action givesHere the term W-boson means this broken-generator vector, not specifically an electroweak particle. The normalization places the gauge coupling outside the action, so no extra $g_{\mathrm{YM}}$ multiplies these $v_r^I$; rescaling to canonically normalized scalar fields moves that coupling into the usual Higgs mass formula.

The corresponding open-string vector oscillator stretched over distance $L=|X_1-X_2|$ has classical stretching mass $TL=L/(2\pi {\alpha }^{\prime })$. Its unstretched mass contribution vanishes because it is the vector level, rather than the bosonic tachyon level. Equating it with the gauge-theory mass givesThe component relation follows by rotational covariance and a common choice of transverse origin; a uniform translation adds the same constant to both brane coordinates. This is the D-brane scalar-position normalization. Together with the string tension determined in Question 1, it fixes the exact normalization of scalar eigenvalues as brane positions.