Open string

An open string has a worldsheet with boundary. Its endpoint conditions may be Neumann boundary conditions, Dirichlet boundary conditions, or mixtures selected independently in target-space directions.

For an ordinary critical open string with all Neumann boundary conditions, there are 24 independent transverse creation oscillators at each positive mode number $n$. Summing their nonnegative occupation numbers gives one factor $(1-w^n{)}^{-24}$. The coefficient $d_p=[w^p]F(w)$ counts oscillator polarizations at level $p$, excluding continuous center-of-mass momentum and additional endpoint multiplicities. Cauchy coefficient formula gives $d_p=(2\pi i{)}^{-1}\oint F(w)w^{-p-1}dw$.

The Dedekind eta function modular transformation gives $F(e^{-t})\sim (t/2\pi {)}^{12}\exp (4{\pi }^2/t-t)$. The saddle-point approximation for coefficient extraction has $t_0=2\pi /\sqrt{p-1}$, phase $4\pi \sqrt{p-1}$ and curvature $(p-1{)}^{3/2}/\pi$. Its prefactor $t_0^{12}$ supplies $(p-1{)}^{-6}$ and the Gaussian width supplies $(p-1{)}^{-3/4}/\sqrt{2}$. This explains both the exponential growth and the power $-27/4$.

For the Nambu-Goto phase-space action, ${\dot{P}}_m={\partial }_{\sigma }{J}_m$ with $J_m=e{T}^2{X}_m^{\prime }+u{P}_m$. Thus ${\dot{\mathcal{P}}}_m=J_m(\pi )-J_m(0)$. Free endpoint variations force $J_m=0$ individually. A Dirichlet boundary condition allows an external support or D-brane to absorb momentum in the fixed direction.

The spatial integration-by-parts term in the Nambu-Goto phase-space action pairs endpoint displacement with momentum flux $J_m=e{T}^2{X}_m^{\prime }+u{P}_m$. Free endpoint displacements require zero flux. Fixed normal coordinates instead set those variations to zero, while tangential variations retain a Neumann boundary condition. Boundary conditions must also be preserved by the equations of motion. In a boundary-preserving gauge with $u=0$, mixed Neumann boundary conditions and Dirichlet boundary conditions implement a planar D-brane.

Free endpoint variations set the open string flux $e{T}^2{X}_m^{\prime }+u{P}_m$ to zero. In a boundary-adapted temporal gauge, freedom to vary time forces $u=0$ at the end. Nonzero lapse then gives $X^{\prime m}=0$, a Neumann boundary condition.

In a boundary-adapted temporal gauge, a free open string endpoint has $X^{\prime m}=0$. The Nambu–Goto phase-space constraints then imply $P^2=0$, while $\dot{X}=eP$ implies ${\dot{X}}^2=0$. For $X^0=t$, its spatial speed is one. This null boundary is compatible with a timelike interior worldsheet.

In conformal gauge, a free open-string coordinate with Neumann boundary conditions has a cosine expansion. Canonical quantization turns its nonzero Fourier coefficients into string oscillators.

On $0\le \sigma \le \pi$, a coordinate with Neumann boundary conditions at both ends has a constant position, linear time zero mode and cosine oscillators. A coordinate with Dirichlet boundary conditions at both ends has a linear spatial interpolation between endpoint positions and sine oscillators, with no momentum zero mode. Both have integer frequencies. This differs from an ND boundary condition, which applies different conditions at the two ends of the same coordinate and gives half-integer frequencies.

The Fourier kinetic term ${\sum }_{k>0}i{\alpha }_{-k}\cdot {\dot{\alpha }}_k/k$ supplies conjugate pairs, with ${\alpha }_{-k}={\alpha }_k^{\ast }$. Inverting its symplectic form yields the displayed Poisson brackets. Nonzero modes are independent of the center-of-mass pair, while ${\alpha }_0=\sqrt{2{\alpha }^{\prime }}p$ is the momentum zero mode. The imaginary factors are compatible with a real action up to a total derivative.

A transverse open string coordinate has an ND boundary condition when one endpoint obeys a Neumann boundary condition and the other a Dirichlet boundary condition. On a strip of width $\pi$, its nonconstant spatial eigenfunctions are $\cos ((n+1/2)\sigma )$ when the free endpoint is at zero. They solve the endpoint equations because their derivatives vanish at zero and their values vanish at $\pi$. Reversing the endpoints gives sine eigenfunctions with the same frequencies. The fixed endpoint removes the dynamical worldsheet zero mode.

For one free boson, the mixed ND open-string sector has strip vacuum energy $E_0=1/48$. The strip-to-plane transformation relates this energy to the plane Virasoro algebra zero mode by $E_0=h-c/24$, with central charge $c=1$. Hence its ground conformal weight is $h=1/48+1/24=1/16$. For $d$ independent ND coordinates the weights add to $d/16$. At $d=24$, the plane matter ground weight is $3/2$, while the strip oscillator vacuum energy is $1/2$. The plane physical condition $L_0^{\mathrm{plane}}-1=0$ is therefore equivalent to ${\alpha }^{\prime }{p}_{\parallel }^2+N+1/2=0$. The apparent different constants are coordinate/constraint conventions, not different mass spectra.