Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 53 / 1 / Solution

Use target-space signature $(-,+,\dots ,+)$ and let a dot between target vectors denote contraction with the Minkowski metric. Varying the string embedding map in the Polyakov action, with periodic variations around the closed string, givesThus the embedding equation of motion isThe independent worldsheet metric variation uses $\delta \sqrt{-\gamma }=-\frac{1}{2}\sqrt{-\gamma }{\gamma }_{ab}\delta {\gamma }^{ab}$. It sets the worldsheet stress tensor to zero:In conformal gauge, ${\gamma }_{ab}=e^{2\omega }\mathrm{diag}(-1,1)$, the Weyl factor cancels. Define worldsheet derivatives ${\partial }_{\pm }=({\partial }_{\tau }\pm {\partial }_{\sigma })/2$. The equations and Virasoro constraints reduce toEquivalently, $\ddot{X}-X^{\prime \prime }=0$, $\dot{X}\cdot X^{\prime }=0$ and ${\dot{X}}^2+X^{\prime 2}=0$. The free wave equation permits independent periodic left- and right-moving fields.

Take $0\le \sigma <2\pi$ and the uncompactified closed-string mode expansionThe Virasoro generators are the Fourier coefficients of the two quadratic worldsheet stress tensors. Their quantum expressions areThe canonical commutation relations are $[{\alpha }_m^{\mu },{\alpha }_n^{\nu }]=m{\eta }^{\mu \nu }{\delta }_{m+n,0}$, with an identical tilded algebra and commuting chiral sectors. Normal ordering places annihilation modes to the right. It gives $L_0={\alpha }^{\prime }{p}^2/4+N$, where $N={\sum }_{n>0}{\alpha }_{-n}\cdot {\alpha }_n$, and similarly on the other side. In old covariant string quantization, physical string states obeymodulo physical null string states. Negative generators are adjoint constraints on bras, not further annihilation conditions on the same ket. In critical bosonic string theory, $d=26$ and the string intercept is $a=1$. These closed-string physical-state Virasoro conditions implyThus the massless first closed-string level has $N=\tilde{N}=1$, and its general state isTo derive the polarization tensor constraints, the oscillator algebra gives $[L_n,{\alpha }_m^{\mu }]=-m{\alpha }_{n+m}^{\mu }$. Applying $L_1$ to this state replaces ${\alpha }_{-1}^{\mu }$ by ${\alpha }_0^{\mu }$; the independent tilded condition replaces the other oscillator by ${\tilde{\alpha }}_0^{\nu }$. All higher positive generators annihilate the state automatically. ThereforeThere is no condition that the tensor trace vanish: each chiral side has only one oscillator, so neither $L_2$ nor ${\tilde{L}}_2$ imposes a trace condition. The null-state quotient of a string also identifiesFor example, $L_{-1}({\xi }_{\nu }{\tilde{\alpha }}_{-1}^{\nu }|0,p⟩)$ is a momentum-longitudinal state with zero inner product against every transverse physical polarization. This explains the string-state gauge redundancy, rather than merely counting covariant components. For an ordinary nonzero null momentum, choose a frame adapted to $p$. The constraints remove one null direction in each index, and these null-state identifications remove the remaining longitudinal directions. The representatives are then ${\zeta }_{ij}$, $i,j=1,\dots ,24$.

Decompose that transverse matrix into its symmetric trace-free part, antisymmetric part and trace. They describe the graviton, Kalb–Ramond field and dilaton, respectively. Their particle polarization counts areThese are representations of the rotational part $SO(24)$ of the massless little group. The dilaton corresponds to the transverse trace; a covariant scalar polarization can be written proportional to ${\eta }_{\mu \nu }-p_{\mu }{\ell }_{\nu }-{\ell }_{\mu }{p}_{\nu }$, with ${\ell }^2=0$ and $p\cdot \ell =1$. It is transverse in both indices and has trace 24. The nonzero-momentum assumption is appropriate for propagating particle states; zero momentum requires separate treatment of gauge and background zero modes.

In a general background, these fields replace the flat-space free action by the string nonlinear sigma model. A convenient Euclidean convention isHere ${ϵ}^{ab}$ is the orientation density; reversing its convention changes the displayed sign of the two-form coupling. The metric tensor is $G_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }+\cdots$, $B$ is the antisymmetric Kalb–Ramond field, and $\Phi$ is the dilaton. Their worldsheet couplings reproduce the corresponding massless closed-string vertex operators when expanded about flat space. The Polyakov path integral integrates $e^{-S_E}$ over embedding fields and metrics modulo worldsheet diffeomorphisms and Weyl transformations; gauge fixing supplies ghost determinants and moduli integrations. Quantum consistency of a general background additionally requires vanishing sigma-model beta functions, not just insertion of arbitrary backgrounds into a free theory.

Finally, separate the constant dilaton mode, $\Phi ={\Phi }_0+\hat{\Phi }$. By the Gauss-Bonnet theorem, its action isFor a connected oriented surface of genus $g$, its Euler characteristic is $\chi =2-2g$. Defining the string coupling by $g_s=e^{{\Phi }_0}$ givesThus the string genus expansion is a topology sum with two extra powers of $g_s$ per handle. An amplitude with $n$ canonically normalized external closed-string vertices has the additional vertex normalization $g_s^n$, so its overall factor is $g_s^{2g-2+n}$. The curvature term, rather than a guessed coupling inserted at each internal interaction, supplies this dependence directly from the functional integral.