Dp Boundary State Cylinder Scope

Here the cylinder worldsheet is the annulus $\mathrm{\Sigma }=S^1\times [0,1]$ with its two boundary circles mapped to two parallel static flat Dp-branes at transverse positions $0$ and $Y\in {\mathbb{R}}^{9-p}$. The conformal structure on $\mathrm{\Sigma }$ is parameterized by a positive annulus modulus, with numerical normalization fixed separately in each source. In the closed channel, the interval coordinate is propagation time and closed-string states propagate between the boundary sources. In the open channel, the circle coordinate is Euclidean time and the trace runs over open-string states whose endpoints lie on the two branes.

Miao Li records the following closed-channel expression in his source convention:

In this expression, $t_{\mathrm{L}\mathrm{i}}>0$ is Li's integration parameter in equation (2.1); $F_{\mathrm{L}\mathrm{i}}$ and ${\tilde{F}}_{\mathrm{L}\mathrm{i}}$ denote only the left- and right-moving fermion-number operators whose parity insertions appear there. Li calls ${\mathrm{\Delta }}_{\mathrm{L}\mathrm{i}}$ the closed-string propagator, but equation (2.1) does not specify it as a particular Virasoro-zero-mode combination. The source variables have been subscripted here only to distinguish the two papers' notation. Li's parity insertions are not identified with the sector-dependent $G_{\mathrm{N}\mathrm{S}}$ and $G_{\mathrm{R}}$ defined in the boundary-state item. Li states in section 2 that he works in Euclidean spacetime and obtains Minkowski results by Wick rotation; equation (2.1) is recorded here in that Euclidean source convention, without a separate local continuation argument.

Separately, Polchinski–Chaudhuri–Johnson's equation (2.23) sets $q=e^{-\pi t_{\mathrm{P}\mathrm{C}\mathrm{J}}}$ in their cylinder calculation, and equation (2.31) writes the closed-channel overlap as

Here $t_{\mathrm{P}\mathrm{C}\mathrm{J}}>0$ is their dimensionless open-channel parameter in equation (2.23), where $q=e^{-\pi t_{\mathrm{P}\mathrm{C}\mathrm{J}}}$; its reciprocal appears in the closed-channel exponent. The symbols $L_0$ and ${\tilde{L}}_0$ are the left- and right-moving Virasoro zero modes in their bosonic-string convention. Equation (2.31) occurs in their bosonic-string cylinder calculation, whose surrounding massless exchange is expressed using the Green function in $25-p$ transverse dimensions. It is cited here only to record that source's own propagator and modulus notation, not as a Type II calculation. No operator identification between ${\mathrm{\Delta }}_{\mathrm{L}\mathrm{i}}$ and $L_0+{\tilde{L}}_0$, and no modulus map between $t_{\mathrm{L}\mathrm{i}}$ and $t_{\mathrm{P}\mathrm{C}\mathrm{J}}$, is asserted.

Li's equations (2.2)–(2.4) separately give the open-string one-loop representation and the massless closed-channel residue. His equation (2.5) explicitly reports

where ${\alpha }^{\mathrm{\prime }}>0$ has length-squared dimension and $T_{p,\mathrm{L}\mathrm{i}}$ is the source's Dp-brane tension parameter in its massless-exchange convention, of dimension ${\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^{3-p}$. This normalization comes from the individual exchange contributions, even though the NS–NS and R–R contributions cancel in their sum. This is a source-reported coefficient result, not a finiteness assertion for the full modulus integral or its massless limit. No identification with a tension density in another bulk-field convention, or with the unevaluated overall factor ${\mathcal{N}}_p$ in the formal boundary-state definition, is made here. That factor still has no local absolute-normalization calculation, and no full string duality is asserted.

The tube sketch labels the closed-channel reading of the same cylinder; it records the two boundary sources and the propagation direction only, and is not a proof of the channel transformation.

Type Ii Dp Brane Boundary State