Flat Dp Brane T Duality Map

In conformal gauge for the flat free coordinate $y$, the canonical momentum density is $P_y=(2\pi {\alpha }^{\mathrm{\prime }}{)}^{-1}{\mathrm{\partial }}_{\tau }{X}^y$. Define the dual coordinate locally by ${\mathrm{\partial }}_{\sigma }{Y}^y=2\pi {\alpha }^{\mathrm{\prime }}{P}_y={\mathrm{\partial }}_{\tau }{X}^y$ and ${\mathrm{\partial }}_{\tau }{Y}^y={\mathrm{\partial }}_{\sigma }{X}^y$. The free wave equation makes these two first-order equations compatible. They are the canonical transformation underlying the reflection of the right mover in [F2]; the global circle identification therefore has radius $R^{\mathrm{\prime }}$ from that supplier.

At a timelike endpoint, Neumann data are ${\mathrm{\partial }}_{\sigma }{X}^y=0$. By step 1.1 this is ${\mathrm{\partial }}_{\tau }{Y}^y=0$, so $Y^y$ is constant along that endpoint and obeys Dirichlet data. Conversely, fixed Dirichlet data for $X^y$ imply ${\mathrm{\partial }}_{\tau }{X}^y=0$, hence ${\mathrm{\partial }}_{\sigma }{Y}^y=0$, which is Neumann data for the dual coordinate.

For a diagonal constant Wilson line $A_y^{(a)}={\theta }_a\mathrm{/}(2\pi R)$ on a coincident-brane Chan–Paton space, [A1] shifts the total open-string momentum between endpoint labels $a,b$ by $\mathrm{\Delta }{p}_y=A_y^{(b)}-A_y^{(a)}=({\theta }_b-{\theta }_a)\mathrm{/}(2\pi R)$. Here total momentum means $p_y:={\int }_0^{\pi }{P}_{y}d\sigma$, where $P_y$ is the momentum density in step 1.1; for momentum label $n\in \mathbb{Z}$, $p_y=n\mathrm{/}R+\mathrm{\Delta }{p}_y$. Integrating ${\mathrm{\partial }}_{\sigma }{Y}^y=2\pi {\alpha }^{\mathrm{\prime }}{P}_y$ across $0\le \sigma \le \pi$ gives $Y^y(\pi )-Y^y(0)=2\pi {\alpha }^{\mathrm{\prime }}{p}_y=2\pi n{R}^{\mathrm{\prime }}+({\theta }_b-{\theta }_a)R^{\mathrm{\prime }}$. Modulo the dual circumference $2\pi R^{\mathrm{\prime }}$, the integer term is zero and the Wilson-line-dependent endpoint displacement is $({\theta }_b-{\theta }_a)R^{\mathrm{\prime }}$. Choosing the origin so the endpoint labelled $a$ is at $Y_a={\theta }_a{R}^{\mathrm{\prime }}$ therefore puts the endpoint labelled $b$ at $Y_b={\theta }_b{R}^{\mathrm{\prime }}$ modulo $2\pi R^{\mathrm{\prime }}$.

If the dualized coordinate is tangent to the original D$p$ worldvolume, [F1] assigns Neumann data there; step 2.1 changes that spatial tangent direction to a normal fixed coordinate, so the dual worldvolume loses one spatial dimension, D$p\mapsto$D$(p-1)$. If it is normal, the original Dirichlet coordinate becomes Neumann, adding one spatial worldvolume dimension, D$p\mapsto$D$(p+1)$.

In the right-moving Ramond ground module, [A2] implements the one-coordinate fermion reflection by $S_R=i{\mathrm{\Gamma }}_{11}{\mathrm{\Gamma }}^y$, with ${\mathrm{\Gamma }}^y$ spacelike. Since $\{{\mathrm{\Gamma }}_{11},{\mathrm{\Gamma }}^y\}=0$, we have ${\mathrm{\Gamma }}_{11}{S}_R=-S_R{\mathrm{\Gamma }}_{11}$. Thus if ${\mathrm{\Gamma }}_{11}\psi =\chi \psi$ for $\chi \in \{-1,+1\}$, then ${\mathrm{\Gamma }}_{11}(S_R\psi )=-\chi S_R\psi$. The left-moving ground module is fixed by [A2], so its chirality is unchanged; by [F6], the relative chirality assignments are exchanged between Type IIA and Type IIB.