Open string stretched between D-branes

An open string stretched a distance $L$ between parallel D-branes has squared mass $M^2=T^2{L}^2+(N-1)/{\alpha }^{\prime }$. The stretching energy makes strings between separated branes massive.

For two static parallel D-branes separated transversely by $d$, the open bosonic string mass spectrum becomes $M^2=d^2/(4{\pi }^2{\alpha }^{\prime 2})+(N-1)/{\alpha }^{\prime }$. The classical stretching contribution is the square of the string tension times the separation, while the oscillator vacuum contributes the intercept term. Thus the ground state is a tachyon for $d<2\pi \sqrt{{\alpha }^{\prime }}$, massless at equality, and massive above it. Its real rest energy, when stable, is $\sqrt{d^2/(4{\pi }^2{\alpha }^{\prime 2})-1/{\alpha }^{\prime }}$; the vacuum correction belongs to energy squared and is not an additive fixed subtraction from the stretching energy. Angled branes change oscillator moding and require a different formula.

A commuting diagonal expectation value has one transverse eigenvalue vector for each brane. In the covariant derivative normalization $D_a\varphi ={\partial }_a\varphi +i[A_a,\varphi ]$, the off-diagonal gauge field has mass $m^2={\sum }_I({\varphi }_1^I-{\varphi }_2^I{)}^2$. The corresponding stretched string has mass $|X_1-X_2|/(2\pi {\alpha }^{\prime })$ in its massless oscillator sector. Equality fixes the displayed scalar-position normalization, up to a common coordinate origin. The trace describes center-of-mass motion and the eigenvalue difference describes separation.

When $n$ parallel D-branes coincide, oriented strings joining distinct branes become massless alongside strings whose ends lie on one brane. Their Chan-Paton labels fill the adjoint representation of $U(n)$, enhancing $U(1{)}^n$ to $U(n)$.

For coincident D3-branes in the critical 26-dimensional bosonic string theory, the massless open-string oscillator has a four-dimensional gauge vector and 22 transverse scalar fields. Chan-Paton factors put all these fields in the adjoint of $U(k)$ for $k$ coincident branes. Ignoring the tachyon, their leading interactions are the dimensional reduction of 26-dimensional Yang-Mills theory. This contains a Yang-Mills sector but is not pure four-dimensional Yang-Mills theory and is not the supersymmetric six-scalar D3-brane theory of the ten-dimensional superstring.