Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 46 / 1

Actions and equations. Use a mostly-plus Minkowski metric and define the induced worldsheet metric $h_{ab}={\partial }_{a}X\cdot {\partial }_{b}X$. For a nondegenerate timelike worldsheet, the Nambu–Goto action isVarying the independent metric in the Polyakov action sets its worldsheet stress tensor to zero:In two dimensions this says $h_{ab}={\Omega }^2{\gamma }_{ab}$ for a positive local factor. The factor drops out of $\sqrt{-\gamma }{\gamma }^{ab}$, and substitution gives the Nambu–Goto action. Conversely, any nondegenerate induced metric solves the auxiliary-metric equation up to a Weyl transformation. This classical equivalence of Polyakov and Nambu–Goto actions is a statement about classical embeddings; quantum equivalence additionally requires treatment of the metric measure and anomaly.

Variation of $X^m$ in the metric action, followed by elimination of $\gamma$, gives the Nambu–Goto equations of motionA closed string has periodic $X$ and no spatial endpoint variation.

For a background metric $G$, Kalb–Ramond field $b$, and dilaton $\Phi$, one consistent Lorentzian convention isThe orientation fixes the two-form sign; here the Lorentzian curvature convention is chosen so Wick rotation gives the positive Euclidean dilaton term $S_{E,\Phi }=\int \sqrt{\gamma }\Phi R^{(2)}/(4\pi )$. This avoids hiding the convention in the topology argument. For constant vacuum value ${\Phi }_0$, the Gauss-Bonnet theorem gives $S_{E,\Phi }={\Phi }_0\chi$. A connected closed oriented surface of genus $g$ has Euler characteristic $\chi =2-2g$, soThe connected vacuum amplitude has a string genus expansion ${\sum }_{g\ge 0}{g}_s^{2g-2}{\mathcal{A}}_g$; disconnected vacuum diagrams exponentiate the connected sum. Each additional handle supplies a factor $g_s^2$. This is dilaton Euler-characteristic weighting.

The rotating circle. For the specified embedding, the squared spatial speed and tangent length are both $1/2$, and their spatial inner product is zero. Including $X^0=t$ therefore givesThe metric is constant. Its equation reduces to $\ddot{X}-X^{\prime \prime }=0$, satisfied by the left- and right-moving trigonometric components and the linear time component. On a constant-time slice with $0\le \sigma <2\pi$, the proper length of a string isIt is time-independent.

Let $R(\theta )$ be an ordinary two-dimensional rotation matrix. Rotate the first coordinate pair by $R(t)$ and the second by $R(-t)$. In these time-dependent Cartesian coordinates both pairs become $(\cos \sigma ,\sin \sigma )/2$. A further fixed orthogonal change of basis givesThus every spatial slice is a planar circle of radius $1/\sqrt{2}$; its plane rotates in the ambient four-space. This is a rigid circular string in four spatial dimensions, not a pulsating circle. The rotating axes establish its spatial shape, not a transformation to an inertial spacetime frame.

The momentum density obtained from the Nambu–Goto action is $P_m=-T\sqrt{-h}{h}^{ta}{\partial }_a{X}_m$. Here $P_0=-T$, hence the conserved target-space energy isThe material velocity is transverse to the tangent and has magnitude $1/\sqrt{2}$. Its Lorentz factor is $\sqrt{2}$, so the excess over $TL$ is kinetic energy. The spatial circle being stationary in rotating axes does not eliminate this energy.

Constraints and endpoints. Varying the multipliers in the Nambu-Goto phase-space action imposesThese are first-class constraints generating normal and tangential worldsheet diffeomorphisms. They remove the two longitudinal embedding degrees of freedom; they are not extra physical force laws. Hamilton's equations areFor an open string, integration by parts leaves the spatial boundary variationAllowed variational open-string boundary conditions must make this vanish for every permitted endpoint displacement, and be preserved by the evolution. The bracketed expression is the open-string endpoint momentum flux. For free displacements, set that flux to zero. In a boundary-preserving gauge with $u=0$ and finite nonzero $e$, this is the Neumann boundary condition $X^{\prime m}=0$ at each end. This shows that free-end string boundary conditions are consistent.

At such an end, $C=0$ gives $P^2=0$, and $\dot{X}=eP$ gives ${\dot{X}}^2=0$. For a nontrivial endpoint trajectory with nonzero time velocity, its spatial speed is therefore one. This null motion of a free string endpoint follows from the boundary condition together with the constraints, not merely from the bulk wave equation. The induced metric may degenerate right at a free end; the result is understood as the endpoint limit or in the Polyakov description.

For a fixed spatial $p$-plane, split coordinates into $X^A$ along its $(p+1)$-dimensional worldvolume, including time, and $X^i$ transverse to it. Set $X^{\prime A}=0$ and $X^i=y^i$ at the ends. Tangential variations are free and their momentum flux vanishes; normal variations vanish, so their flux need not vanish. Evolution preserves the fixed normal values with $P^i=0$ at the boundary in the same gauge. These mixed Neumann boundary conditions and Dirichlet boundary conditions consistently restrict the endpoint worldlines to the plane.

The massless open-string excitations then separate into a gauge vector along the worldvolume and transverse scalar fields. A scalar displacement changes the corresponding $y^i$; its interpretation is a fluctuation of the plane's embedding. These worldvolume fields from open-string massless states support the interpretation as a dynamical planar D-brane. In the bosonic theory the still lower ground state is a tachyon, indicating an unstable brane; the interpretation does not require pretending that this tachyon is a stable massless field. Appropriate superstring sectors can remove that instability.