Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 306 (String Theory)

Varying the Polyakov action with respect to the worldsheet metric givesIn conformal gauge, variation of $X$ gives the wave equationwhile the Virasoro constraints become $\dot{X}\cdot X^{\prime }=0$ and ${\dot{X}}^2+X^{\prime 2}=0$. The boundary variation is $-T\int d\tau ,X^{\prime }\cdot \delta X$ at each endpoint. It vanishes through a Neumann boundary condition $X^{\prime \mu }=0$, a Dirichlet boundary condition $\delta X^{\mu }=0$, or a direction-by-direction mixture of the two.

Translations $\delta X^{\mu }=a^{\mu }$ and Lorentz transformations $\delta X^{\mu }={\omega }^{\mu }{}_{\nu }{X}^{\nu }$ leave the action invariant because it depends only on derivatives and Lorentz contractions. Noether theorem gives the stated currents, whose equations are ${\partial }_{\alpha }{P}^{\alpha \mu }=0$ and ${\partial }_{\alpha }{J}^{\alpha \mu \nu }=0$. For $0\le \sigma \le \pi$, the conserved open-string charges areThe endpoint fluxes vanish for the allowed boundary conditions.

Orthogonality of the cosine modes gives ${\int }_0^{\pi }\cos (n\sigma )d\sigma =0$ and ${\int }_0^{\pi }\cos (n\sigma )\cos (m\sigma )d\sigma =\pi {\delta }_{nm}/2$. Since the supplied expansion uses ${\alpha }^{\prime }{p}^{\mu }\tau$ rather than the more usual $2{\alpha }^{\prime }{P}^{\mu }\tau$,With ${\alpha }_{-n}={\alpha }_n^{\ast }$ for a real embedding,The first term is orbital angular momentum and the second is the contribution of the string oscillators.

A rigidly rotating stretched solution iswith all other coordinates constant. Each coordinate obeys the wave equation and $X^{\prime }=0$ at $\sigma =0,\pi$. Directly, $\dot{X}\cdot X^{\prime }=0$ andso both Virasoro constraints hold. Its energy and planar angular momentum areConsequently $J=M^2/(2\pi T)={\alpha }^{\prime }{M}^2$, the classical leading open-string Regge trajectory.

The Green function for the normalization in the question isDifferentiating at distinct points gives the holomorphic two-point function

In $T(z)\partial X(w)$, a single Wick contraction can be made with either factor in $T$. Using the two-point function and expanding the remaining field around $w$ givesThis operator product expansion says that $\partial X$ is a holomorphic primary operator of conformal weight $(1,0)$.

A primary operator of conformal weight $(h,\tilde{h})$ transforms under $z\mapsto w(z)$ asThe derivative $\partial X$ is a primary of weight $(1,0)$, and $\bar{\partial }X$ is a primary of weight $(0,1)$.

Transforming the point-split normal ordering changes the subtraction as well as the two derivatives. Expanding $w(z\pm \delta /2)$ to third order giveswhere the Schwarzian derivative isThe inhomogeneous Schwarzian term means that $T$ is not a primary operator. It reflects the central charge $c=1$ of one free scalar.

In the Polyakov path integral, fix worldsheet diffeomorphism and Weyl symmetry to conformal gauge. The Faddeev-Popov determinant supplies the worldsheet ghosts, and anomaly cancellation selects $D=26$. Insert $m$ integrated closed-string tachyon vertex operators $e^{i{p}_i\cdot X}$ on the sphere. The zero mode of $X$ gives ${\delta }^{(26)}({\sum }_i{p}_i)$, while Gaussian contractions give the Koba-Nielsen factor ${\prod }_{j<l}|z_j-z_l{|}^{{\alpha }^{\prime }{p}_j\cdot p_l}$. Dividing by the conformal Killing group $SL(2,\mathbb{C})/{\mathbb{Z}}_2$ and using the sphere power of the string coupling gives the displayed amplitude with $g_s^{m-2}$.

Write the position-dependent factor as $e^f$, withIn the fixed-angle hard-scattering limit the integral is governed by stationary points. Differentiation gives the scattering equationsOnly $m-3$ complex equations are independent after quotienting by $SL(2,\mathbb{C})$.

Fix $z_1=0$, $z_2=1$, and $z_4=\infty$, and write $z_3=z$. Its scattering equation isso in the hard limitEvaluating the Koba-Nielsen factor at this saddle and using $s+t+u=O(m_{\mathrm{tachyon}}^2)$ yields, with the logarithms defined by analytic continuation,The omitted terms grow more slowly than the displayed fixed-angle $s\log s$ terms.

The string embedding coordinates form a nonlinear sigma model whose target-space metric is the spacetime metric $G_{\mu \nu }$. A consistent quantum string background must preserve worldsheet Weyl invariance, so all sigma-model beta functions must vanish, and the total matter-plus-ghost central charge must cancel. For a bosonic string with only a metric, this requires $D=26$ and, to leading order in ${\alpha }^{\prime }$, ${\beta }_{\mu \nu }^G={\alpha }^{\prime }{R}_{\mu \nu }+O({\alpha }^{\prime 2})=0$; thus the target metric must be Ricci-flat at leading order.

For the stereographic coordinate $Z=\tan (\theta /2)e^{i\phi }$, the round metric isSubstitution into the sigma-model action givesso

Set $Z=\lambda \delta Z$. Expansion about the north pole givesThus $⟨\delta Z(k)\delta \bar{Z}(-k)⟩=1/k^2$ up to the Fourier convention, and the leading four-point vertex is $-2{\lambda }^2$ times the scalar product of the two differentiated momenta, summed over assignments of differentiated legs.

Contracting the two undifferentiated fields in this vertex gives a one-loop tadpole proportional toIt produces the divergent metric correction $-2{\lambda }^{2}I\partial \delta Z\partial \delta \bar{Z}$, up to the allowed overall-normalization ambiguity. Geometrically this is the one-loop term ${\beta }_{\mu \nu }^G\propto {\alpha }^{\prime }{R}_{\mu \nu }$. Since the round sphere has positive Ricci curvature, the isolated $S^2$ sigma model is not conformal. Therefore $S^2\times N$ is not a bosonic-string background unless contributions from $N$, a dilaton, an antisymmetric tensor, or further corrections cancel the sphere beta function.