Type Ii Dp Brane Boundary State

Fix $\mathrm{\hslash }=c=1$, ${\alpha }^{\mathrm{\prime }}>0$ with dimension length squared, and $T=(2\pi {\alpha }^{\mathrm{\prime }}{)}^{-1}$. Let $W={\mathbb{R}}^{1,p}\times \{X^a=y_0^a\}$ be the static timelike plane from the preceding Dp boundary-data definition; target coordinates $x^{\mu },y_0^a$ have length dimension and target momenta $p^{\mu }$ have inverse-length dimension. Write $x_{\perp }=(x^{p+1},\dots ,x^9)$ and $p_{\parallel }=(p^0,\dots ,p^p)$ for the normal position and tangent momentum coordinates.

In the free closed-string mode convention, a flat Dp boundary state is a distributional functional $\mathrm{\mid }B;y_0⟩$ on the algebraic finite-excitation closed-string state domain whose matter-coordinate component obeys

$$({\alpha }_n^{\mu }+S^{\mu }{}_{\nu }{\tilde{\alpha }}_{-n}^{\nu })\mathrm{\mid }{B}_X;y_0⟩=0,n\in \mathbb{Z}\setminus \{0\},$$

where $\eta =\mathrm{diag}(-1,+1,\dots ,+1)$, the dimensionless modes satisfy $[{\alpha }_m^{\mu },{\alpha }_n^{\nu }]=m{\eta }^{\mu \nu }{\delta }_{m+n,0}$, the left and right families commute and have the same algebra, and $S^{\mu }{}_{\nu }=+1$ in directions tangent to $W$ and $-1$ in directions normal to $W$. The zero-mode distribution is annihilated by every tangent momentum operator $p^{\alpha }$ and every normal position operator $X^a-y_0^a$. Equivalently, its target-position factor is proportional to $1_{x_{\parallel }}\otimes {\delta }^{(9-p)}(x_{\perp }-y_0)$, where $1_{x_{\parallel }}$ is the constant distribution along $W$. Thus it is translation invariant along $W$ with an undifferentiated delta factor normal to $W$; support on $W$ alone would not impose these zero-mode conditions. The matter factor may be represented formally by

$$\mathrm{\mid }{B}_X;y_0⟩={\mathcal{N}}_p{\delta }^{(9-p)}(x_{\perp }-y_0){\delta }^{(p+1)}(p_{\parallel })\exp \text{\hspace{0.17em}⁣}(-\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}{\alpha }_{-n}\cdot S{\tilde{\alpha }}_{-n})\mathrm{\mid }0⟩,$$

where ${\mathcal{N}}_p\mathrm{\ne }0$ is a fixed overall normalization datum, not evaluated here.

For the Ramond–Neveu–Schwarz (RNS) worldsheet fermions, the Neveu–Schwarz (NS) sector has antiperiodic fermions and modes $r\in \mathbb{Z}+\frac{1}{2}$, while the Ramond (R) sector has periodic fermions and modes $r\in \mathbb{Z}$, including Clifford zero modes. With ${\eta }_s\in \{-1,+1\}$ labeling the boundary spin structure, use the fermion gluing convention

$$({\psi }_r^{\mu }-i{\eta }_s{S}^{\mu }{}_{\nu }{\tilde{\psi }}_{-r}^{\nu })\mathrm{\mid }{B}_{\psi },{\eta }_s⟩=0.$$

The fermion modes satisfy $\{{\psi }_r^{\mu },{\psi }_s^{\nu }\}={\eta }^{\mu \nu }{\delta }_{r+s,0}$, with an independent right-moving copy. The formal fermion oscillator factors are

$$\mathrm{\mid }{B}_{\psi },{\eta }_s{⟩}_{\mathrm{N}\mathrm{S}}=\exp \text{\hspace{0.17em}⁣}(i{\eta }_s\sum _{r\in \mathbb{Z}+1\mathrm{/}2,r>0}{\psi }_{-r}\cdot S{\tilde{\psi }}_{-r})\mathrm{\mid }0{⟩}_{\psi ,\mathrm{N}\mathrm{S}}\mathrm{.}$$ $$\mathrm{\mid }{B}_{\psi },{\eta }_s{⟩}_{\mathrm{R}}=\exp \text{\hspace{0.17em}⁣}(i{\eta }_s\sum _{n=1}^{\mathrm{\infty }}{\psi }_{-n}\cdot S{\tilde{\psi }}_{-n})\mathrm{\mid }{B}_{\psi },{\eta }_s{⟩}_{\mathrm{R}}^{(0)}\mathrm{.}$$

Here $\mathrm{\mid }0{⟩}_{\psi ,\mathrm{N}\mathrm{S}}$ is the NS matter-fermion ground state. On Ramond zero modes, write $\mathrm{\mid }A⟩\otimes \mathrm{\mid }\tilde{B}⟩$ for the left/right spinor ground-state basis and use the gluing bispinor

$$M({\eta }_s)=C{\mathrm{\Gamma }}^0{\mathrm{\Gamma }}^1\cdots {\mathrm{\Gamma }}^p\frac{1+i{\eta }_s{\mathrm{\Gamma }}_{11}}{1+i{\eta }_s},{\mathrm{\Gamma }}_{11}:={\mathrm{\Gamma }}^0{\mathrm{\Gamma }}^1\cdots {\mathrm{\Gamma }}^9,$$

where $C$ is the charge-conjugation matrix in the convention $C^T=-C$ and $({\mathrm{\Gamma }}^{\mu }{)}^T=-C{\mathrm{\Gamma }}^{\mu }{C}^{-1}$, and $\{{\mathrm{\Gamma }}^{\mu },{\mathrm{\Gamma }}^{\nu }\}=2{\eta }^{\mu \nu }$ defines the ten-dimensional Clifford generators; these conventions give ${\mathrm{\Gamma }}_{11}^2=1$, so ${\mathrm{\Gamma }}_{11}$ is the ten-dimensional chirality operator. The zero-mode state is $\mathrm{\mid }{B}_{\psi },{\eta }_s{⟩}_{\mathrm{R}}^{(0)}=M_{AB}({\eta }_s)\mathrm{\mid }A⟩\mathrm{\mid }\tilde{B}⟩$. The displayed factor fixes the chiral zero-mode solution in this convention; reversing the orientation of $W$ gives the antibrane and reverses the R–R component.

The reparametrization ghosts $b_n,c_n$ are Grassmann-odd integer modes with $\{b_m,c_n\}={\delta }_{m+n,0}$ and ghost numbers $-1,+1$, respectively; their formal boundary factor is

$$\mathrm{\mid }{B}_{bc}⟩=\exp \text{\hspace{0.17em}⁣}[\sum _{n=1}^{\mathrm{\infty }}(c_{-n}{\tilde{b}}_{-n}-b_{-n}{\tilde{c}}_{-n})]\frac{c_0+{\tilde{c}}_0}{2}{c}_1{\tilde{c}}_1\mathrm{\mid }1{⟩}_{bc},$$

where $\mathrm{\mid }1{⟩}_{bc}$ is the left/right global-conformal-invariant ghost vacuum, annihilated on each side by $b_n$ for $n\ge -1$ and $c_n$ for $n\ge 2$; the boundary factor obeys $(b_n-{\tilde{b}}_{-n})\mathrm{\mid }{B}_{bc}⟩=0$ and $(c_n+{\tilde{c}}_{-n})\mathrm{\mid }{B}_{bc}⟩=0$. The commuting superconformal ghosts ${\beta }_r,{\gamma }_r$ have conformal weights $3\mathrm{/}2,-1\mathrm{/}2$, respectively, and satisfy $[{\gamma }_r,{\beta }_s]={\delta }_{r+s,0}$ with the same NS/R moding as ${\psi }_r$. In the bosonization $\beta =e^{-\phi }\mathrm{\partial }\xi$, $\gamma ={\eta }_{\mathrm{g}\mathrm{h}}{e}^{\phi }$, where ${\eta }_{\mathrm{g}\mathrm{h}}$ and $\xi$ are anticommuting auxiliary fields, a picture label is the $\phi$-charge of the corresponding ghost vacuum. Use fixed total picture $-2$ and the representatives $(-1,-1)$ in NS–NS and $(-\frac{1}{2},-\frac{3}{2})$ in R–R. The formal superghost factors are

$$\mathrm{\mid }{B}_{\beta \gamma },{\eta }_s{⟩}_{\mathrm{N}\mathrm{S}}=\exp \text{\hspace{0.17em}⁣}[i{\eta }_s\sum _{r\in \mathbb{Z}+1\mathrm{/}2,r>0}({\gamma }_{-r}{\tilde{\beta }}_{-r}-{\beta }_{-r}{\tilde{\gamma }}_{-r})]\mathrm{\mid }P=-1⟩\mathrm{\mid }\tilde{P}=-1⟩,$$ $$\mathrm{\mid }{B}_{\beta \gamma },{\eta }_s{⟩}_{\mathrm{R}}=\exp \text{\hspace{0.17em}⁣}[i{\eta }_s\sum _{n=1}^{\mathrm{\infty }}({\gamma }_{-n}{\tilde{\beta }}_{-n}-{\beta }_{-n}{\tilde{\gamma }}_{-n})]\exp (i{\eta }_s{\gamma }_0{\tilde{\beta }}_0)\mathrm{\mid }P=-\frac{1}{2}⟩\mathrm{\mid }\tilde{P}=-\frac{3}{2}⟩\mathrm{.}$$

Use the reduced superghost algebra with the auxiliary ${\xi }_0$ omitted. On a chiral picture vacuum $\mathrm{\mid }P⟩$, impose ${\beta }_r\mathrm{\mid }P⟩=0$ for $r\ge -P-\frac{1}{2}$ and ${\gamma }_r\mathrm{\mid }P⟩=0$ for $r\ge P+\frac{3}{2}$, with the appropriate integer or half-integer moding. Thus ${\beta }_0\mathrm{\mid }-\frac{1}{2}⟩=0$ and ${\tilde{\gamma }}_0\mathrm{\mid }\tilde{P}=-\frac{3}{2}⟩=0$ in the displayed R pictures; ${\gamma }_0$ and ${\tilde{\beta }}_0$ are their respective polynomial creators. Picture is fixed on each such module: $\beta$ and $\gamma$ preserve it. In bosonized terms the picture operator is the sum of superghost charge and auxiliary ${\eta }_{\mathrm{g}\mathrm{h}},\xi$ charge, so the $\phi$-charge description of a vacuum is not a parity rule for its excited states.

Their oscillator boundary conditions are $({\beta }_r+i{\eta }_s{\tilde{\beta }}_{-r})\mathrm{\mid }{B}_{\beta \gamma },{\eta }_s⟩=0$ and $({\gamma }_r+i{\eta }_s{\tilde{\gamma }}_{-r})\mathrm{\mid }{B}_{\beta \gamma },{\eta }_s⟩=0$.

The Gliozzi–Scherk–Olive (GSO) projection acts on the full matter-plus-superghost module. Let $N_{\psi ,j}$ count nonzero matter-fermion creators modulo two, and $N_{\mathrm{s}\mathrm{g}\mathrm{h},j}$ count all superghost creators modulo two, for $j=L,R$; the latter includes powers of ${\gamma }_0$ on the left and ${\tilde{\beta }}_0$ on the right in R. Counts are taken on finite ordered Fock monomials. Left and right odd modes use the graded tensor product. Define commuting involutions ${\mathsf{q}}_j$ by

$$\begin{array}{ccc}& {\mathsf{q}}_L& {\mathsf{q}}_R\\ \mathrm{N}\mathrm{S}(-1,-1)& (-1{)}^{N_{\mathrm{s}\mathrm{g}\mathrm{h},L}}& (-1{)}^{N_{\mathrm{s}\mathrm{g}\mathrm{h},R}}\\ \mathrm{R}(-\frac{1}{2},-\frac{3}{2})& (-1{)}^{N_{\mathrm{s}\mathrm{g}\mathrm{h},L}}& -(-1{)}^{N_{\mathrm{s}\mathrm{g}\mathrm{h},R}}\mathrm{.}\end{array}$$

The R vacuum sign on the right is essential: the zero-mode superghost-number operator $-{\gamma }_0{\beta }_0$ has eigenvalue zero on a vacuum annihilated by ${\beta }_0$, and eigenvalue $-1$ on one annihilated by ${\gamma }_0$, since $[{\gamma }_0,{\beta }_0]=1$. Each creator changes its parity. These ${\mathsf{q}}_j$ are superghost-number parities, not picture parities.

Put ${\mathsf{f}}_j=(-1{)}^{N_{\psi ,j}+1}$ in NS. In R put ${\mathsf{f}}_j={\mathsf{J}}_j(-1{)}^{N_{\psi ,j}}$, where the two commuting Clifford gradings on a zero-mode coefficient matrix $X$ are

$${\mathsf{J}}_{L}X={\mathrm{\Gamma }}_{11}^{T}X,{\mathsf{J}}_{R}X=X{\mathrm{\Gamma }}_{11}\mathrm{.}$$

All these operators act trivially on the target-coordinate and $bc$ factors. With ${ϵ}_{\mathrm{I}\mathrm{I}}=+1$ in Type IIA and $-1$ in Type IIB, the full fixed-picture projectors are

$${\mathrm{\Pi }}_{\mathrm{N}\mathrm{S}\mathrm{N}\mathrm{S}}=\frac{1+{\mathsf{f}}_L{\mathsf{q}}_L}{2}\frac{1+{\mathsf{f}}_R{\mathsf{q}}_R}{2},{\mathrm{\Pi }}_{\mathrm{R}\mathrm{R}}=\frac{1+{ϵ}_{\mathrm{I}\mathrm{I}}{\mathsf{f}}_L{\mathsf{q}}_L}{2}\frac{1-{\mathsf{f}}_R{\mathsf{q}}_R}{2}\mathrm{.}$$

Their factors are commuting idempotents on the finite-excitation module. Equivalently the sector operators retained with eigenvalue $+1$ are $G_{\mathrm{N}\mathrm{S}}^{(j)}={\mathsf{f}}_j{\mathsf{q}}_j$, $G_{\mathrm{R}}^{(L)}={ϵ}_{\mathrm{I}\mathrm{I}}{\mathsf{f}}_L{\mathsf{q}}_L$ and $G_{\mathrm{R}}^{(R)}=-{\mathsf{f}}_R{\mathsf{q}}_R$. These are full operators on the stated picture modules; no additional matter-only projector is imposed.

The target-space chirality labels may conventionally be written ${\chi }_L=+1$, with ${\chi }_R=-1$ for Type IIA and ${\chi }_R=+1$ for Type IIB. They label the spacetime supersymmetry spaces; they do not identify every coefficient of an asymmetric-picture boundary functional with a fixed matter-chirality eigenspace. No map from that coefficient module to target supersymmetry parameters, or between picture representatives in BRST cohomology, is asserted here. Type IIA has opposite target chiralities and even $p$; Type IIB has equal target chiralities and odd $p$.

Write $\mathrm{\mid }B,{\eta }_s⟩$ for the tensor product of the four displayed factors before projection. In the NS–NS sector both chiral modes are NS; in R–R both are R. Applying the corresponding full projector to $\mathrm{\mid }B,+⟩$ gives

$$\mathrm{\mid }B{⟩}_{\mathrm{N}\mathrm{S}\mathrm{N}\mathrm{S}}=\frac{1}{2}(\mathrm{\mid }B,+{⟩}_{\mathrm{N}\mathrm{S}\mathrm{N}\mathrm{S}}-\mathrm{\mid }B,-{⟩}_{\mathrm{N}\mathrm{S}\mathrm{N}\mathrm{S}}),\mathrm{\mid }B{⟩}_{\mathrm{R}\mathrm{R}}=\frac{1}{2}(\mathrm{\mid }B,+{⟩}_{\mathrm{R}\mathrm{R}}+\mathrm{\mid }B,-{⟩}_{\mathrm{R}\mathrm{R}})\mathrm{.}$$

Reversing the brane orientation reverses the relative R–R sign.

The full formal state is the projected spin-structure sum of the displayed matter, fermion, $bc$-ghost, and $\beta \gamma$-ghost tensor products. With $L_0,{\tilde{L}}_0$ the dimensionless zero modes of the left and right stress tensors, its cylinder overlap uses the fixed propagator convention $\exp [-\pi (L_0+{\tilde{L}}_0)\mathrm{/}t]$ for modulus $t>0$. Because the zero modes are delta-supported and the oscillator product is formal, $\mathrm{\mid }B;y_0⟩$ is a distributional source for closed-string boundary data, not a normalizable particle state. This definition does not establish Becchi–Rouet–Stora–Tyutin (BRST) closure, the absolute value or physical normalization of ${\mathcal{N}}_p$, or equality of the full open- and closed-channel amplitudes.

Dp Brane Open String Boundary Condition

String Tension Alpha Prime

Miao Li's 1995 paper constructs the massless boundary-state sector and says that constructing an exact state including all oscillators is formidable. The cited later construction supplies formal oscillator and ghost factors; this item does not claim a local BRST calculation, a cylinder modular transform, or an absolute physical normalization.

For a precise meaning of the preceding distributional functional, use the mixed zero-mode representation obtained by Fourier-transforming the tangent target coordinates and retaining normal positions. Its coordinates $p_{\parallel }\in {\mathbb{R}}^{p+1}$ and $x_{\perp }\in {\mathbb{R}}^{9-p}$ commute. Set ${\mathcal{S}}_0=\mathcal{S}({\mathbb{R}}_{p_{\parallel }}^{p+1}\times {\mathbb{R}}_{x_{\perp }}^{9-p})$. Sectorwise, let ${\mathcal{F}}_{\mathrm{a}\mathrm{l}\mathrm{g}}$ be the algebraic span of finite ordered Fock monomials in the displayed matter, fermion, $bc$ and $\beta \gamma$ modules, including the algebraic $bc$ zero-mode module, the finite-dimensional R Clifford module and the specified polynomial R superghost zero-mode module. Fix its monomial basis $\{e_b\}$ and its finite-support coefficient-dual space ${\mathcal{F}}_{\mathrm{f}\mathrm{i}\mathrm{n}}^{\circ }={⨁}_b\mathbb{C}{e}_b^{\circ }$, where $e_b^{\circ }(e_{b^{\mathrm{\prime }}})={\delta }_{b{b}^{\mathrm{\prime }}}$. This identifies finite-excitation test labels with coefficient-extraction functionals. Each displayed GSO involution preserves finite support; let ${\mathrm{\Pi }}^{\circ }\mathrm{\ell }=\mathrm{\ell }\circ \mathrm{\Pi }$ be the transpose projector. The projected test space is

$${\mathcal{T}}_{\mathrm{\Pi }}={\mathcal{S}}_0{\otimes }_{\mathrm{a}\mathrm{l}\mathrm{g}}{\mathrm{\Pi }}^{\circ }{\mathcal{F}}_{\mathrm{f}\mathrm{i}\mathrm{n}}^{\circ },$$

with the NS–NS and R–R spaces taken in direct sum. Thus the projection acts on the complete correlated matter-superghost test module. One may equivalently evaluate the unprojected boundary series on ${\mathrm{\Pi }}^{\circ }\mathrm{\ell }$; this is coefficientwise the projected boundary series evaluated on $\mathrm{\ell }$.

With Lebesgue measure in this mixed representation, the delta part of the matter zero-mode factor acts on $f\in {\mathcal{S}}_0$ by $⟨{\delta }^{(9-p)}(x_{\perp }-y_0){\delta }^{(p+1)}(p_{\parallel }),f⟩=f(0,y_0)$, so including the displayed overall factor gives ${\mathcal{N}}_{p}f(0,y_0)$. Fourier-normalization constants are included in the fixed factor ${\mathcal{N}}_p$. In the full position representation the same zero-mode distribution acts, up to that Fourier normalization, by $g\mapsto {\mathcal{N}}_p{\int }_{{\mathbb{R}}^{p+1}}g(x_{\parallel },y_0)d^{p+1}{x}_{\parallel }$. The annihilation conditions exclude both tangent dependence and normal delta derivatives: in mixed coordinates $z=(p_{\parallel },x_{\perp }-y_0)$ they read $z_{j}U=0$ for every component. A test function can be written as $f(0)\rho +{\sum }_j{z}_j{f}_j$, with $\rho (0)=1$ and Schwartz $\rho ,f_j$, by a local Taylor expansion and division away from the origin; hence $U(f)=U(\rho )f(0)$. Thus $U$ is a multiple of the displayed point delta in mixed coordinates, whose inverse tangent Fourier transform is the stated uniform distribution on $W$. Interpret each displayed oscillator exponential, including the Ramond $\beta \gamma$ zero-mode exponential, by its formal power series and pair coefficientwise with the algebraic Fock-basis monomials fixed by the displayed mode algebras and vacua. A fixed finite-excitation monomial receives contributions from only finitely many mode indices and powers, so this defines a linear functional on the unprojected coefficient test space and, after the displayed transpose projection, on ${\mathcal{T}}_{\mathrm{\Pi }}$. It asserts no convergence in a completed oscillator Hilbert space or continuity in an oscillator topology; the zero-mode part is a tempered distribution, and the full boundary state remains a formal, non-normalizable functional.

The projection identities can be checked within these algebraic modules. Put $G={\mathrm{\Gamma }}_{11}$ and $A=C{\mathrm{\Gamma }}^0\cdots {\mathrm{\Gamma }}^p$. The fixed charge-conjugation convention gives $G^T=-CG{C}^{-1}$ and hence $G^{T}M({\eta }_s)=(-1{)}^{p}M(-{\eta }_s)$, while $M({\eta }_s)G=M(-{\eta }_s)$. Nonzero fermion and superghost parities reverse the sign of their respective ${\eta }_s$-dependent exponentials. Consequently in R,

$${\mathsf{f}}_L{\mathsf{q}}_L\mathrm{\mid }B,{\eta }_s⟩=(-1{)}^p\mathrm{\mid }B,-{\eta }_s⟩,{\mathsf{f}}_R{\mathsf{q}}_R\mathrm{\mid }B,{\eta }_s⟩=-\mathrm{\mid }B,-{\eta }_s⟩,$$

so ${\mathrm{\Pi }}_{\mathrm{R}\mathrm{R}}\mathrm{\mid }B,+⟩=\frac{1}{4}(1+{ϵ}_{\mathrm{I}\mathrm{I}}(-1{)}^p)(\mathrm{\mid }B,+⟩+\mathrm{\mid }B,-⟩)$, which is the stated half-sum for the allowed parity of $p$. In NS each full chiral parity sends $\mathrm{\mid }B,{\eta }_s⟩$ to $-\mathrm{\mid }B,-{\eta }_s⟩$, giving the half-difference.

In particular, with $z={\gamma }_0{\tilde{\beta }}_0$, $v=\mathrm{\mid }-\frac{1}{2}⟩\mathrm{\mid }\tilde{P}=-\frac{3}{2}⟩$ and $P_{\pm }=(1\pm G)\mathrm{/}2$, the projected R zero-mode series is $A(P_{+}\cos z+P_{-}\sin z)v$. Its constant and $z^1$ coefficients are $A{P}_{+}$ and $A{P}_{-}$. Their opposite right-matter gradings are accompanied by right superghost parities $-1$ and $+1$; both have ${\mathsf{f}}_R{\mathsf{q}}_R=-1$ and are retained. This includes the admitted Type IIB $p=1$ case without removing either coefficient. These finite-coefficient identities impose no oscillator convergence, BRST, or picture-change theorem.