Multiple D-branes

Consider the T-dual perspective. The $ij$-strings now satisfy the boundary conditions

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

Thus, there are now $N$ independent D-branes situated at positions $x^d=R^{\prime }{\theta }_i$ on the dual circle, up to a common additive constant. In the T-dual picture, $ij$-strings satisfy Dirichlet boundary conditions at both ends:

• Endpoint $\sigma =0$ ending on the D-brane at position $x^d=R^{\prime }{\theta }_i$.
• Endpoint $\sigma =\pi$ ending on the D-brane at position $x^d=R^{\prime }{\theta }_j$.

(diagram)

The mass-shell condition is now

$m^2={\left(\frac{{\theta }_i-{\theta }_j+2\pi n}{2\pi R}\right)}^2+\frac{2}{{\alpha }^{\prime }}(N-a)$.

In the original perspective, this simply reflects the shift in the contribution from the KK momentum. In T-dual perspective, the shift reflects additional contributions to the potential energy of $ij$-strings stretching between separated D-branes.

Let us look at the massless sector. Assuming the angles ${\theta }_1,\dots ,{\theta }_N$ are all distinct, the only massless states correspond to $i=j$ and $N=1$. In the original picture they are fluctuations of gauge field for the gauge symmetry $U(1{)}^N\subset U(N)$ left unbroken by the Wilson line. From the T-dual perspective, the D-branes are separated and the massless states have the following interpretation:

• Massless states ${\alpha }_{-1}^m|0;k;ii⟩$ correspond to fluctuations of a $U(1)$ gauge field $(A_m{)}_{ii}$ on the D-brane at $x^d={\theta }_{i}R$.
• Massless states ${\alpha }_{-1}^d|0;k;ii⟩$ correspond to fluctuations of a scalar field $(A_d{)}_{ii}$ on the D-brane at $x^d={\theta }_{i}R$.

A constant gauge connection $(A_d{)}_{ii}=-{\theta }_i/2\pi R$ corresponds in the T-dual picture to translating the $i$-th parallel D-brane by $R^{\prime }{\theta }_i$. It is therefore reasonable that fluctuations of $(A_d{)}_{ii}$ correspond to fluctuations in the position of the $i$-th D-brane. Therefore, D-branes are dynamical objects.

In the limit that all ${\theta }_j$ coincide, the $U(N)$ gauge symmetry in the original picture is restored and in the T-dual picture, there are $N$ coincident D-branes. Thus, $N$ coincident D-branes support a $U(N)$ gauge theory with connection $(A_m{)}_{ii}$. The remaining $N^2$ scalar fields $(A_d{)}_{ii}$ are more mysterious.