Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 306 (String Theory)

The induced worldsheet metric isThe Nambu–Goto action iswhich is minus the string tension times the invariant area of the string worldsheet. Varying the independent metric in the Polyakov action givesIn two dimensions this says $g_{\alpha \beta }=e^{2\omega }{\gamma }_{\alpha \beta }$ for an undetermined Weyl transformation factor. Substitution gives $\sqrt{-g}{g}^{\alpha \beta }{\gamma }_{\alpha \beta }=2\sqrt{-\det \gamma }$, and hence $S_{\mathrm{P}}=S_{\mathrm{NG}}$.

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Sincethe determinant variation givesIntegrating by parts and taking $\delta X^{\mu }$ to vanish on the boundary yieldsThe principle of stationary action therefore gives

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In static gauge, take $\sigma \in [0,2\pi )$ and use the circular ansatzIts Nambu–Goto Lagrangian isThe conserved energy givesfor $R(0)=R_0$ and $\dot{R}(0)=0$. Equivalently,whose solution isThe signed continuation is periodic; the geometric radius is $|R(\tau )|$, and the loop collapses whenever the cosine vanishes.

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For the circular embedding,On $R=R_0\cos (\tau /R_0)$ this becomesThus the worldsheet metric is conformally flat between collapse events and becomes degenerate at the instant when the loop shrinks to zero size.

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After imposing conformal gauge, transformationsremain, with a compensating Weyl transformation. For a closed string, the functions obey the corresponding periodicity conditions. Introduce target-space light-cone coordinates $X^{\pm }=(X^0\pm X^{D-1})/\sqrt{2}$. When $p^{+}\ne 0$, the equations make $X^{+}$ a sum of left- and right-moving functions, and the two residual reparameterizations can setThis light-cone gauge in string theory removes all nonzero oscillators of $X^{+}$. Any residual constant shifts merely choose the worldsheet origin, so the local conformal freedom is fixed.

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The Virasoro constraints determine $X^{-}$ in terms of the $d=D-2$ transverse coordinates $X^i$. Their closed-string expansion introduces independent left- and right-moving string oscillators satisfyingFor $n>0$, ${\alpha }_n^i$ and ${\tilde{\alpha }}_n^i$ annihilate the momentum vacuum, while ${\alpha }_{-n}^i$ and ${\tilde{\alpha }}_{-n}^i$ create excitations. A general state is a product of these creation operators acting on $|p⟩$, subject to closed-string level matching $N=\tilde{N}$.

Normal ordering the quantum constraints introduces the normal-ordering constant of a string $a$. Requiring the Lorentz generators $J^{-i}$ to obey the Lorentz algebra without an anomalous term fixesThus Lorentz invariance resolves both the ordering ambiguity and the critical dimension of string theory.

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The bosonic string mass spectrum isWith Lorentz invariance, $a=1$ and $D=26$, so $N=0$ is tachyonic, $N=1$ is massless, and higher levels are massive. To display the degeneracies in general notation, put $d=D-2$. The numbers of states in one chiral sector at levels $N=0,1,2,3$ areLevel matching pairs any left state with any right state at the same level, so the total closed-string degeneracies areAt $D=26$ these are $1$, $576$, $104976$, and $10240000$. If “first four levels” is taken to exclude the ground state, the next chiral degeneracy iswhich is $25650$ at $d=24$, giving total degeneracy $25650^2=657922500$.

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Under a finite conformal coordinate change $z^{\prime }=f(z)$ and ${\bar{z}}^{\prime }=\bar{f}(\bar{z})$, a primary operator of conformal weight $(h,\tilde{h})$ obeysEquivalently, its operator product expansion with the holomorphic stress-energy tensor has singular partThe antiholomorphic stress tensor gives the analogous formula with $\tilde{h}$.

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After integration by parts, the quadratic operator in the action is $K=-{\pi }^{-1}\partial \bar{\partial }$. Its Green function $G(z,w)=⟨X(z,\bar{z})X(w,\bar{w})⟩$ therefore satisfiesUsing the given distributional identity giveswhere the additive constant depends on the infrared convention and cancels from neutral correlators.

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For this normalization,Contracting once and twice withgivesIt is therefore a primary operator with

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Because normal ordering removes contractions within either exponential, Wick theorem sums all cross-contractions intoUnder $z\mapsto \lambda z$, this scales as ${\lambda }^{-k^2}$. Each operator has scaling dimension $\Delta =h+\tilde{h}=k^2/2$, so a two-point function of two such primaries must scale as ${\lambda }^{-2\Delta }={\lambda }^{-k^2}$. The two calculations agree.

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Choose a reference metric in each conformal class. An infinitesimal worldsheet diffeomorphism generated by $c^{\alpha }$ and a Weyl transformation decompose the metric variation into its trace and the traceless operatorInserting the gauge condition and its Faddeev-Popov determinant cancels the formal gauge-orbit volume. Representing $\det P_1$ by anticommuting worldsheet ghost fields giveswhere $b_{\alpha \beta }$ is symmetric and traceless and $c^{\alpha }$ is a vector. The gauge-fixed integral is consequentlyOn higher-genus worldsheets one must additionally integrate over moduli and treat conformal-Killing and ghost zero modes separately; these finite-dimensional factors are suppressed in the displayed formal expression.

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In Euclidean conformal gauge, complex coordinates reduce the worldsheet ghost action toup to a convention-dependent common normalization absorbed into the fields. Inverting $\bar{\partial }$ and $\partial$ giveswith mixed holomorphic-antiholomorphic correlators equal to zero. These are the standard local-plane propagators; compact worldsheets require the zero-mode qualifications described in part (a).

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