The Open String

It begins with two points and the line between them.

In the beginning there is only the vacuum — or at least, a version of it that can tolerate a string. The string is open. It has two ends, and each end is a point of infinite potential wrapped in a boundary condition. The string does not know this yet. It only knows how to vibrate.

Its world-sheet is a strip. Time flows along its length, and at each moment the string stretches between two endpoints. The endpoints are attached — pinned — to a D-brane. This is the first thing the string learns: it does not own its ends. The D-brane does. The string can slide along the brane, but it cannot leave. The Dirichlet condition is not cruel. It is simply the way things are.

The string vibrates. This is not a choice. A vibrating string is a string that has energy, and energy in a quantum theory is discrete. The mode expansion of the string's coordinates tells the story: each mode is a harmonic oscillator, and each harmonic oscillator contributes $\frac{1}{2}\hbar\omega$ to the ground state energy. Sum them all up and you get a number that matters. In bosonic string theory it is $-1$ in mass-squared units. In superstring theory the fermionic modes partially cancel, and the ground state is either a tachyon (bosonic, unwanted) or a massless fermion or vector (superstring, preferable).

The massless vector is the open string's masterpiece. When the string vibrates in its first excited state with one oscillator acting on the vacuum, you get a state $|\psi\rangle = \zeta_\mu \alpha_{-1}^\mu |0; k\rangle$ where $\zeta_\mu$ is a polarization vector and $k^\mu$ is the momentum. The gauge invariance emerges from the requirement that null states decouple. The polarization vector becomes a gauge field $A_\mu(k)$, and the string's vibration is a photon — or more generally, a gauge boson of the D-brane's world-volume theory.

The endpoints carry charge. This is not metaphor. In the presence of $N$ coincident D-branes, the endpoints carry fundamental and anti-fundamental Chan-Paton indices $i, j = 1, \dots, N$. An open string stretching from brane $i$ to brane $j$ carries charge $(i, \bar{j})$. The gauge field lives on the branes, and the string is the mediator, the thing that connects charge to charge, brane to brane.

The string also feels tension. The string tension $T = \frac{1}{2\pi\alpha'}$ sets the energy per unit length. Longer strings cost more energy. The lightest excitations are the short ones — the ones vibrating with minimal amplitude, with the fewest quanta of excitation. These are the massless modes, the gauge fields, the scalars, the fermions. Everything heavier is a tower of string resonances, each one a higher harmonic, each one more energetic, each one further beyond reach.

The open string is simple. It has two ends and a body. It vibrates, it carries charge, it lives on branes. But in its simplicity lies the entire machinery of gauge theory. The photon is an open string. The gluon is an open string. The W and Z are open strings. They are all the same thing — different vibrations of the same fundamental object — and the D-brane is their home.

When the string stops vibrating, it is just a line. When it vibrates, it is a force.