Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 306 / 3

The unintegrated closed-string operator $c\tilde{c}\mathcal{O}$ must have total conformal weight $(0,0)$ and ghost number $(1,1)$. Since $c$ and $\tilde{c}$ have weights $(-1,0)$ and $(0,-1)$, the matter operator $\mathcal{O}$ must be a Virasoro primary operator of weightEquivalently, the full operator must be a BRST-closed operator and not a BRST-exact operator. For a momentum-dependent tensor operator, these requirements impose its target-space mass-shell, transversality, and gauge-equivalence conditions.

At closed-string level $N=\tilde{N}$, the bosonic string mass spectrum isThe state on the leading Regge trajectory uses only level-one oscillators and has maximal spin $J=N+\tilde{N}=2N$. Eliminating $N$ givesBecause $N$ is unbounded, the spectrum contains infinitely many particles of increasing mass and spin. The intercept $J=2$ at $M^2=0$ includes the graviton.

The Riemann sphere has three complex Conformal Killing vector fields, forming the Möbius group $PSL(2,\mathbb{C})$. Gauge fixing this residual conformal symmetry permits three insertion points to be fixed arbitrarily. Each unintegrated string vertex operator contains $c\tilde{c}$, and the three insertions exactly saturate the three holomorphic and three antiholomorphic ghost zero modes. Since every full vertex has weight $(0,0)$, the correlator is Möbius invariant and independent of the chosen three positions.

The free-boson worldsheet theory factorizes into independent holomorphic and antiholomorphic sectors. The factors $\partial X^{\mu }$ contract only in the holomorphic sector and produce one tensor $T^{\mu \kappa \rho }$, while the $\bar{\partial }{X}^{\nu }$ factors independently produce $T^{\nu \lambda \sigma }$. Their product is the closed-string version of left-right factorization, often summarized at tree level as closed-string kinematics being a square of open-string kinematics.

In one chiral sector there are three derivatives $\partial X$. A nonzero Wick contraction can pair two derivatives with each other and contract the remaining derivative with an exponential; this gives a metric times one momentum and hence the terms linear in $k$. Alternatively, all three derivatives can contract with exponential operators, giving three momenta and the term proportional to ${\alpha }^{\prime }{k}^3$. The same alternatives occur independently in the antiholomorphic sector.

An Einstein-Hilbert action contains two derivatives, so its cubic graviton vertex produces only the $k\times k$ part of the product of the two $T$ tensors. The string amplitude also contains cross terms of order ${\alpha }^{\prime }{k}^4$ and a term of order ${\alpha }^{\prime 2}{k}^6$. These cannot arise from pure Einstein gravity. They require higher-curvature corrections, schematicallywith index contractions and coefficients fixed by string amplitudes up to field redefinitions. The full bosonic-string effective action also contains the dilaton and Kalb-Ramond field. Thus Einstein gravity is only its leading low-energy approximation.