Dp Brane Worldvolume Fields

For one smooth flat Dp worldvolume $W$ from the preceding definition, the bosonic part of the leading massless open-string fields consists of a $U(1)$ connection $A$ on a Hermitian line bundle over $W$ and $9-p$ smooth transverse scalar fields specifying the embedding. In a fixed normalization, write the transverse displacement as $Y^a=2\pi {\alpha }^{\mathrm{\prime }}{\mathrm{\Phi }}^a$, where $Y^a:W\to \mathbb{R}$ has length dimension, ${\alpha }^{\mathrm{\prime }}$ has length squared, and ${\mathrm{\Phi }}^a$ has inverse-length dimension; the perturbed embedding is $X^a=y_0^a+Y^a$. Locally the connection is a real one-form in the convention $D=d-iA$, so its curvature is $F_A=dA$.

The open-string Ramond sector also contributes a massless fermion field $\mathrm{\Psi }$. Choose a real 32-dimensional module $S_{10}$ for the ten-dimensional Clifford algebra, with generators ${\mathrm{\Gamma }}^{\mu }$ satisfying $\{{\mathrm{\Gamma }}^{\mu },{\mathrm{\Gamma }}^{\nu }\}=2{\eta }^{\mu \nu }I$, and put ${\mathrm{\Gamma }}_{11}={\mathrm{\Gamma }}^0{\mathrm{\Gamma }}^1\cdots {\mathrm{\Gamma }}^9$. The field $\mathrm{\Psi }$ is a smooth section of the trivial bundle $W\times S_{10}^{\chi }$, where $S_{10}^{\chi }=\ker ({\mathrm{\Gamma }}_{11}-\chi I)$ is the 16-real-dimensional Majorana–Weyl spinor module with chirality $\chi \in \{-1,+1\}$ selected by the open-string GSO projection. For a general smooth free field the massless constraint is ${\mathrm{\Gamma }}^{\alpha }{\mathrm{\partial }}_{\alpha }\mathrm{\Psi }=0$, with $\alpha =0,\dots ,p$ tangent to $W$. Its complexified plane-wave solutions $\mathrm{\Psi }(x)=u{e}^{i{p}_{\alpha }{x}^{\alpha }}$, with real tangent momentum $p$ and nonzero polarization $u\in S_{10}^{\chi }{\otimes }_{\mathbb{R}}\mathbb{C}$, are exactly those with ${\mathrm{\Gamma }}^{\alpha }{p}_{\alpha }u=0$; this implies $p^2=0$. Real solutions are obtained by taking real parts. This records the field content and free massless constraint, not an interacting worldvolume action.

For $N\in \mathbb{N}$, $N\ge 1$, coincident branes with the same $W$ and $y_0$, the Chan–Paton gauge group is $U(N)$; the connection is a unitary connection on a rank-$N$ Hermitian bundle $E\to W$, and each transverse scalar is a Hermitian section of the adjoint bundle $\mathrm{End}(E)$. The fermion has the same adjoint Chan–Paton transformation law and, in the Hermitian-generator convention, is a smooth section of $W\times S_{10}^{\chi }{\otimes }_{\mathbb{R}}\mathrm{Herm}(E)$, where $\mathrm{Herm}(E)$ is the real bundle of Hermitian endomorphisms of $E$. This records the massless field content in the low-energy truncation only and asserts no exact non-Abelian DBI or fermion interaction action.

Dp Brane Open String Boundary Condition

String Tension Alpha Prime

Use length-valued coordinates $x^{\alpha }$ on $W$ and the dimensionless $U(1)$ gauge parameter in the convention $D_{\alpha }={\mathrm{\partial }}_{\alpha }-i{A}_{\alpha }$. Since $D_{\alpha }$ and ${\mathrm{\partial }}_{\alpha }$ both have dimension length${}^{-1}$, $A_{\alpha }$ has dimension length${}^{-1}$. Thus $A=A_{\alpha }d{x}^{\alpha }$ has dimensionless holonomy, and $F_A=dA=\frac{1}{2}{F}_{\alpha \beta }d{x}^{\alpha }\wedge d{x}^{\beta }$ has components $F_{\alpha \beta }$ of dimension length${}^{-2}$. A $U(N)$ connection in the same normalization has the same component dimensions; this statement concerns units only and does not assert a non-Abelian DBI action.

The plane-wave characterization follows directly from ${\mathrm{\Gamma }}^{\alpha }{\mathrm{\partial }}_{\alpha }(u{e}^{ip\cdot x})=i({\mathrm{\Gamma }}^{\alpha }{p}_{\alpha })u{e}^{ip\cdot x}$ and $({\mathrm{\Gamma }}^{\alpha }{p}_{\alpha }{)}^2=p^{2}I$ on the full Clifford module. It includes $p=0$, giving constant solutions; the zero field is allowed separately. No growth or Fourier-representability restriction is imposed on the general smooth free field. If such a field additionally defines a tempered distribution on the affine worldvolume, its Fourier transform is understood distributionally, with $\widehat{{\mathrm{\partial }}_{\alpha }\mathrm{\Psi }}=i{p}_{\alpha }\widehat{\mathrm{\Psi }}$. In this class the free equation is equivalent to ${\mathrm{\Gamma }}^{\alpha }{p}_{\alpha }\widehat{\mathrm{\Psi }}=0$ as a distribution; multiplying once more gives $p^2\widehat{\mathrm{\Psi }}=0$, hence support on the null cone. Null-cone support alone is not the spinor constraint, and the Fourier transform need not be an ordinary polarization function or an integral against a prescribed mass-shell measure.