Action of T-duality

Recall the open string mode expansion

$X^{\mu }(\tau ,\sigma )=x^{\mu }+{\ell }^2{p}^{\mu }\tau +i\ell {\sum }_{n\ne 0}\frac{1}{n}{\alpha }_n^{\mu }{e}^{-in\tau }\cos (n\sigma )$.

We can decompose the mode expansion into the left- and right- moving components depending on the combinations ${\sigma }^{+}=\tau +\sigma$ and ${\sigma }^{-}=\tau -\sigma$.

$X_L^{\mu }({\sigma }^{+})=\frac{x^{\mu }+c^{\mu }}{2}+\frac{{\ell }^2}{2}{p}^{\mu }{\sigma }^{+}+\frac{i\ell }{2}{\sum }_{n\ne 0}\frac{1}{n}{\alpha }_n^{\mu }{e}^{-in{\sigma }^{+}}$

$X_R^{\mu }({\sigma }^{-})=\frac{x^{\mu }-c^{\mu }}{2}+\frac{{\ell }^2}{2}{p}^{\mu }{\sigma }^{-}+\frac{i\ell }{2}{\sum }_{n\ne 0}\frac{1}{n}{\alpha }_n^{\mu }{e}^{-in{\sigma }^{-}}$

where we have allowed an arbitrary constant shift $c^{\mu }$ that cancels out between the two.

Now, we will compactify on a circle by demanding $x^d\sim x^d+2\pi R$. Then, momentum must be quantised $p^d=\frac{n}{R}$ for $n\in \mathbb{Z}$. However, for the open string with Neumann boundary conditions, there is no conserved winding number (quantum number). Therefore, we cannot implement T-duality as a transformation that interchanges KK momentum and the winding number.

However, we also saw that T-duality could be implemented by a parity transformation on the right-moving mode expansion $X_R^d\to -X_R^d$. Thus, there are T-dual expansions

$X^d({\sigma }^{+},{\sigma }^{-})=X_L^d({\sigma }^{+})+X_R^d({\sigma }^{-})$

$X^{\prime d}({\sigma }^{+},{\sigma }^{-})=X_L^d({\sigma }^{+})-X_R^d({\sigma }^{-})$,

which provide equivalent descriptions of the same string theory at radii $R$ and $R’=\alpha /R$. This equivalence is local in nature from a world-sheet perspective and can be applied to the open string as well.

The T-dual mode expansion of the open string is

$X^{\prime d}(\tau ,\sigma )=c^d+{\ell }^2\frac{n}{R}\sigma +\ell {\sum }_{n\ne 0}\frac{1}{n}{\alpha }_n^d{e}^{-in\tau }\sin (n\sigma )$.

Note that there is no $\tau$ dependence in the zero-mode sector; there is no total momentum $p^{\prime d}=0$. Moreover, the endpoints of the string are fixed

$X^{\prime d}(\tau ,0)=c^d$

$X^{\prime d}(\tau ,\pi )=c^d+\frac{2\pi {\alpha }^{\prime }}{R}n$

with $X^{\prime d}(\tau ,\pi )-X^{\prime d}(\tau ,0)=\frac{2\pi {\alpha }^{\prime }}{R}n=2\pi R^{\prime }n$.

Note that $R^{\prime }={\alpha }^{\prime }/R$ is the radius of the T-dual circle.

Both endpoints have Dirichlet boundary conditions from the T-dual perspective and KK momentum $n$ becomes the winding number. The hyperplane $x^d=c^d$ where the open strings end, spanning the directions $x^m=\{x^0,x^1,\dots ,x^{d-1}\}$, is called a D-brane.

The mass-shell condition is $m^2={\left(\frac{n}{R}\right)}^2+\frac{2}{{\alpha }^{\prime }}(N-a)={\left(\frac{R^{\prime }}{{\alpha }^{\prime }}n\right)}^2+\frac{2}{{\alpha }^{\prime }}(N-a)$.

From the original perspective, the first term is the contribution to mass from the KK momentum. In the T-dual picture, it arises from the potential energy of wrapped strings beginning and ending on the D-brane.

We now consider the massless spectrum when $a=1$. For a generic radius, this consists of states with $n=0$ and $N=1$. In the original description, ${\alpha }_{-1}^{\mu }|0;k⟩$ correspond to excitations of a $U(1)$ gauge field $A_{\mu }$. In the T-dual perspective, the D-brane breaks translation invariance and we must decompose it into components:

• States ${\alpha }_{-1}^m|0;k⟩$ correspond to excitations of a $U(1)$ gauge field $A_m$ on the D-brane spanning the directions $x^m=\{x^0,x^1,\dots ,x^{d-1}\}$.
• State ${\alpha }_{-1}^d|0;k⟩$ corresponds to excitations of a scalar field $\varphi$ on the D-brane spanning teh directions $x^m=\{x^0,x^1,\dots ,x^{d-1}\}$.

We can argue that the scalar field corresponds to fluctuations of the position of the D-brane in the $x^d$-direction and therefore, D-branes are dynamical objects in string theory.