Chan-Paton Factors

We can supplement open strings with Neumann boundary conditions by Chan-Paton degrees of freedom. Since the end-points of an open string act like particles, we can add additional internal degrees of freedom localised at the endpoints, which are separate from the string dynamics.

Suppose each end-point can in $N$ linearly independent states $|j⟩$ indexed by $j=1,\dots ,N$. Then, open string states are obtained by acting with raising operators on ground states

$|0;k;i,j⟩$

corresponding to strings with state $|i⟩$ at $\sigma =0$ and $|j⟩$ at $\sigma =\pi$. These strings will be called as $ij$-strings. The additional discrete labels are known as Chan-Paton factors.

We now introduce a complete set of ground states,

$|0;k;a⟩={\sum }_{i,j=1}^N{\lambda }_{ij}^a|0;k;i,j⟩$

where ${\lambda }_{ij}^a$ denotes a basis of Hermitian matrices indexed by $a=1,\dots ,N^2$ and normalised such that $\text{Tr}({\lambda }^a{\lambda }^b)={\delta }^{ab}$. Inner products are then invariant under transformations ${\lambda }^a\mapsto U{\lambda }^a{U}^{†}$, where $U$ is an $N\times N$ unitary matrix. Thus, open strings states now transform in the adjoint representation of $U(N)$.

When $a=1$, we have massless states ${\lambda }_{ij}^a{\alpha }_{-1}^{\mu }|0;k;i,j⟩$ which correspond to excitations of a non-abelian $U(N)$ gauge field $A_{\mu }=A_{\mu }^a{\lambda }^a$. In this way, open strings with Chan-Paton factors naturally produce $U(N)$ gauge fields with end-points of the string behaving as point particles transforming in the fundamental and anti-fundamental representations.

All strings states are neutral under the diagonal $U(1)\subset U(N)$ corresponding to the generator ${\lambda }_{ij}^a={\delta }_{ij}/\sqrt{N}$. Equivalently, the adjoint representation of $U(1)$ is trivial and these gauge fields decouple from the spectrum.