Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 52 (String Theory)

The local gauge symmetries of the Polyakov action are worldsheet diffeomorphisms and Weyl transformations. Under a reparametrization ${\sigma }^{\alpha }\mapsto {\sigma }^{\prime \alpha }$, the embedding coordinates $X^{\mu }$ transform as scalars and the worldsheet metric as a covariant tensor; the contracted integrand is a scalar density. Infinitesimally, with the active convention, $\delta X=-{\xi }^{\alpha }{\partial }_{\alpha }X$ and $\delta \gamma =-{\mathcal{L}}_{\xi }\gamma$. A Weyl transformation sends ${\gamma }_{\alpha \beta }\mapsto e^{2\omega }{\gamma }_{\alpha \beta }$ and leaves $X$ fixed. In two dimensions the factors from $\sqrt{-\gamma }$ and ${\gamma }^{\alpha \beta }$ cancel. Target-space Poincaré transformations are additional global symmetries, rather than worldsheet gauge symmetries.

These gauge freedoms put the metric locally in conformal gauge, ${\gamma }_{\alpha \beta }={\eta }_{\alpha \beta }$. Its variation must still be imposed: it gives the worldsheet stress tensor constraints $T_{++}=T_{--}=0$. The field equation is the wave equation, and periodicity of a closed string gives independent left- and right-moving oscillator sets. In a normalization with ${\alpha }_0^{\mu }={\tilde{\alpha }}_0^{\mu }=\sqrt{{\alpha }^{\prime }/2}{k}^{\mu }$, their commutators and normal-ordered generators arewith the same formulas for the tilded sector and zero mixed commutators.

The closed-string physical-state Virasoro conditions in the critical flat-space theory areThey act on the oscillator Fock space, and physical spurious (null) states are quotiented out. Equivalently physical states are the suitable BRST cohomology classes. Positive modes constrain kets; their adjoints constrain bras, so one does not impose every positive and negative matter mode on the same ket. Subtracting the two zero-mode equations gives closed-string level matching, $N=\tilde{N}$, and adding their mass-shell content gives $M^2=4(N-1)/{\alpha }^{\prime }$. Quantum consistency fixes the intercept to one and, for just these flat coordinate fields, $d=26$: the matter central charge $d$ cancels the reparametrization ghost central charge $-26$. Other dimensions require additional worldsheet degrees of freedom or a noncritical completion.

For the residual symmetry, take ${\sigma }^{\pm }=\tau \pm \sigma$. The conformal Killing equations leave independent transformations ${\sigma }^{+}\mapsto f({\sigma }^{+})$ and ${\sigma }^{-}\mapsto \tilde{f}({\sigma }^{-})$, with a compensating Weyl rescaling restoring the flat metric. On the Euclidean plane the corresponding holomorphic vector fields are ${\ell }_n=-z^{n+1}{\partial }_z$ and their antiholomorphic partners. They satisfy the Witt algebra $[{\ell }_m,{\ell }_n]=(m-n){\ell }_{m+n}$. The conserved contour generators are the Virasoro modes $L_n=(2\pi i{)}^{-1}\oint z^{n+1}T(z)dz$. Their quantum Virasoro algebra iswith a second independent copy for the right movers. These are the generators of the residual conformal transformations, including their quantum central extension.

The two-dimensional quantum-gravity interpretation of strings follows from what is integrated, not from an analogy: at genus $h$, the Polyakov path integral integrates over both $X$ and the metric, divided by diffeomorphism and Weyl gauge volume. String perturbation theory sums these surface topologies, weighted by $g_s^{2h-2}$ before external-state coupling factors. Gauge fixing leaves worldsheet ghost determinants and integrals over worldsheet moduli. Thus it is two-dimensional quantum gravity coupled to the $d$ coordinate matter fields. A flat gauge is local; general higher-genus surfaces retain conformal-structure moduli and need not admit a globally flat representative.

Target-space gravity is a different conclusion. At $N=\tilde{N}=1$, the state ${ϵ}_{\mu \nu }{\alpha }_{-1}^{\mu }{\tilde{\alpha }}_{-1}^{\nu }|k⟩$ is massless, transverse in both indices, and has longitudinal gauge identifications. Its symmetric traceless transverse polarization is a graviton; the antisymmetric and scalar polarizations give the Kalb–Ramond field and dilaton. The graviton gauge identification is precisely the linearized metric diffeomorphism. Moreover the leading string-frame massless effective action contains $\int \sqrt{-G}{e}^{-2\Phi }R$, and worldsheet Weyl invariance with constant dilaton and zero two-form requires $R_{\mu \nu }=0$ at leading order in ${\alpha }^{\prime }$. Thus both the propagating massless spin-two state and its low-energy interactions reproduce gravity. Closed strings contain the graviton and yield Einstein gravity in their low-energy massless sector, together with other fields and higher-derivative corrections; the bosonic tachyon instability remains.

Start by using the metric equation of the Polyakov action to retain the Virasoro constraints after choosing conformal gauge. In the light-cone gauge in string theory, make $X^{+}$ proportional to worldsheet time on a patch with nonzero $k^{+}$. The constraints determine $X^{-}$ from the transverse coordinates, so its oscillators are not independent physical excitations. Neumann boundary conditions make each transverse coordinate a cosine standing wave on the open interval. Quantize these waves as independent harmonic oscillators and build their Fock space on the momentum-labelled vacuum. There are $d-2=24$ transverse directions in the critical bosonic string, and the normal-ordering intercept is one. Thus physical oscillator states have nonnegative occupation numbers $r_{n,i}$ and level $p={\sum }_{n,i}n{r}_{n,i}$, with $M^2=(p-1)/{\alpha }^{\prime }$. Continuous center-of-mass momentum is not part of the discrete oscillator degeneracy.

For each mode $n$ and transverse direction, summing its occupation numbers gives ${\sum }_{r\ge 0}{w}^{nr}=(1-w^n{)}^{-1}$. Therefore the open-string oscillator degeneracy generating function isThis is the Cauchy coefficient formula. The original PDF's power $w^{-p}$ is an index error: it extracts $d_{p-1}$, and would give zero for the vacuum $p=0$, which actually has one oscillator state. Any positively oriented circle strictly inside the unit disk works.

Put $w=e^{-t}$ with $t>0$. Expand the logarithm and sum the geometric series to obtainSince $t/(e^{mt}-1)\le 1/m$, dominated convergence gives $t\log F(e^{-t})\to 24{\sum }_{m\ge 1}{m}^{-2}=4{\pi }^2$. Thus the leading singular term is $4{\pi }^2/t$.

The subleading term printed in the PDF also needs correction: the product implies $+12\log ϵ$, rather than $-6\log ϵ$. To obtain the coefficient needed for the final power law, use $\tau =it/(2\pi )$ and the Dedekind eta function. Its definition gives $F(e^{-t})=e^{-t}\eta (\tau {)}^{-24}$. The modular identity $\eta (-1/\tau )=\sqrt{-i\tau }\eta (\tau )$, recorded in NIST's eta transformation formulas, and the product at large imaginary argument giveSince $t=-\log (1-ϵ)=ϵ+{ϵ}^2/2+O({ϵ}^3)$, this yieldsThe printed logarithmic correction cannot be used to derive the stated $p^{-27/4}$ prefactor; the corrected correction does.

For the saddle-point approximation, change the coefficient contour to $t=-\log w$. Its local contribution isSet $q=p-1$. The positive saddle is $t_0=2\pi /\sqrt{q}$, with $S(t_0)=4\pi \sqrt{q}$ and $S^{\prime \prime }(t_0)=q^{3/2}/\pi$. Along the local vertical steepest-descent direction $t=t_0+iy$, the exponent is $S(t_0)-S^{\prime \prime }(t_0)y^2/2$ to quadratic order. The factor $(t/2\pi {)}^{12}$ varies negligibly over its Gaussian width. HenceThis proves the large-level degeneracy of an open bosonic string, including the otherwise unspecified leading constant. The factor $p^{-6}$ comes from the eta prefactor and $p^{-3/4}$ from the saddle width. As allowed, no bound on the remote part of the contour is needed.

At high mass, $p\simeq {\alpha }^{\prime }{M}^2$, so the density grows exponentially as $e^{{\beta }_{H}M}$ within units $k_B=\hslash =c=1$. A canonical sum with $e^{-\beta M}$ cannot converge for $\beta <{\beta }_H$, regardless of polynomial prefactors. This suggests a Hagedorn threshold and a breakdown or transition of the ordinary dilute-string thermal description. The power prefactor can make the partition sum itself finite at the threshold while sufficiently high energy moments diverge; the asymptotic count alone does not prove an absolute upper temperature. These are properties of the formal high-level spectrum, separate from the tachyon instability of the bosonic vacuum.

First regulate the Gaussian functional integral, separating any constant zero mode. Integration by parts gives $S[\varphi ]=\frac{1}{2}(\varphi ,A\varphi )$ with positive kinetic operator $A=-\Delta$ on the nonzero modes. Complete the square:A translation of the regulated integration variables then yieldsThe sign is essential: the kernel in the negative source exponent is the negative of the positive Gaussian covariance. The source-independent prefactor is formally $\mathcal{N}=\int D\chi e^{-(\chi ,A\chi )/2}\propto (\det {}^{\prime }A{)}^{-1/2}$, with the prime denoting the excluded zero mode and normalization depending on the regulated measure. This is the Euclidean scalar source functional and Laplacian sign convention.

Consequently ${\Delta }_{z}G(z,w)={\delta }^{(2)}(z-w)$. In the plane,up to harmonic additions fixed by boundary and infrared conditions. Away from $w$ this logarithm is harmonic. Integrating its outward normal derivative around a small circle centered at $w$ gives $2\pi$, establishing the distributional delta coefficient. The arbitrary infrared length $L$ cancels for a source of zero total integral. On a compact regulated worldsheet the zero-mode-projected equation has the corresponding constant subtraction from the delta function.

For the string action $I_E=(4\pi {\alpha }^{\prime }{)}^{-1}\int (\partial X{)}^2$, the nonzero-mode covariance isCompute initially with Euclidean target coordinates and restore target Lorentzian contractions by analytic continuation. Treat the physical tachyon vertices as normal-ordered exponentials; this removes coincident self-contractions. Completing the Gaussian square with the imaginary delta-function sources, or applying Wick contractions, givesafter fixing the vertex normalization. Each cross-contraction has the sign $i^2=-1$, canceling the negative logarithmic covariance, which explains the positive exponent.

The constant target-space mode must also be integrated:This momentum-conservation factor is momentum dependent and cannot literally be absorbed into the PDF's momentum-independent $\mathcal{N}$. Its intended position integral is the amplitude with this overall delta function stripped off, restricted to conserving momenta. In that convention,The logarithmic infrared constant only gives a fixed normalization on the common external mass shell, since ${\sum }_{r<s}{k}_r\cdot k_s=-\frac{1}{2}{\sum }_r{k}_r^2$. Thus the Gaussian determinant and normalization are independent of the scattering invariants. To make the tree position integral finite one also divides by the residual Möbius gauge volume, equivalently fixes three insertions with their ghost factor. The four-position expression is its formal unfixed representation.

Under $z\mapsto (az+b)/(cz+d)$ with determinant one,At insertion $r$, the Koba-Nielsen factor therefore contributes $|c{z}_r+d{|}^{-{\alpha }^{\prime }{\sum }_{s\ne r}{k}_r\cdot k_s}$. Momentum conservation makes the exponent ${\alpha }^{\prime }{k}_r^2$. On the tachyon mass shell $k_r^2=4/{\alpha }^{\prime }$, it is four and cancels the measure's exponent $-4$. Hence the full integration density is Möbius invariant. The correlator factor alone is covariant; the density including $d^2{z}_r$ is invariant.

The supplied Virasoro–Shapiro amplitude has generic simple poles when $-1-{\alpha }^{\prime }s/4=-n$, givingand likewise in the other channels. These are exchanges of closed-string states with equal left and right levels $n$: the tachyon, the massless sector and the massive tower. At special kinematics zeros or crossed-channel coincidences need to be handled by the full constrained amplitude; numerator poles are not three independent sequences of unrelated singularities.

To isolate the three-tachyon coupling, first use momentum conservation and $k_r^2=-{\mu }^2$:At the tachyon pole $s={\mu }^2=-4/{\alpha }^{\prime }$, put $x={\alpha }^{\prime }t/4$ and $y={\alpha }^{\prime }u/4$. The relation gives $x+y=-3$. Thus $-1-y=2+x$ and $2+y=-1-x$, and the entire $t,u$ gamma-function ratio cancels to one. The remaining numerator is $\Gamma (-{\alpha }^{\prime }(s-{\mu }^2)/4)\sim -4/[{\alpha }^{\prime }(s-{\mu }^2)]$, while its denominator tends to $\Gamma (1)=1$. HenceWith the exchange convention ${\hat{A}}_4\sim g_{TTT}^2/({\mu }^2-s)$ and canonically normalized external legs, factorization therefore gives $\boxed{g_{TTT}^2=4C/{\alpha }^{\prime }}$, or $g_{TTT}=2\sqrt{C/{\alpha }^{\prime }}$ up to its unobservable overall sign. A convention using the opposite propagator denominator or factors of $i$ changes the corresponding amplitude phase. Because $C$ itself is unspecified, the residue fixes this relation, not an absolute numerical coupling independent of normalization.

Take ${\eta }^{\alpha \beta }=\mathrm{diag}(-1,1)$ with coordinate zero equal to $\tau$ and coordinate one to $\sigma$. In the supplied representation setThey satisfy the Clifford algebra $\{{\rho }^{\alpha },{\rho }^{\beta }\}=2{\eta }^{\alpha \beta }1$. For anticommuting spinor components and an ordinary matrix $B$, relabelling the two summed indices givesNow $C^T=-C$, whereas $C{\rho }^0=-1$ and $C{\rho }^1=\mathrm{diag}(-1,1)$ are symmetric. Therefore the Majorana Grassmann bilinear interchange yieldsThese signs require fermionic, Grassmann-odd fields; commuting spinors would have the opposite interchange signs.

In flat conformal gauge, suppress the common factor $(4\pi {\alpha }^{\prime }{)}^{-1}$ and write the combined density as $\mathcal{L}=-{\partial }_{\alpha }X\cdot {\partial }^{\alpha }X+\bar{\psi }\cdot {\rho }^{\alpha }{\partial }_{\alpha }\psi$. The supersymmetry variation including its odd parameter is an even derivation. From $({\rho }^{\alpha }{)}^{T}C=-C{\rho }^{\alpha }$ we obtainAlso $(C{\rho }^{\alpha }{\rho }^{\beta }{)}^T=-C{\rho }^{\beta }{\rho }^{\alpha }$, so interchanging odd spinors gives $\bar{\chi }{\rho }^{\alpha }{\rho }^{\beta }ϵ=\overline{ϵ}{\rho }^{\beta }{\rho }^{\alpha }\chi$. The two density variations, with target contractions understood, are thereforeIn the second term the symmetric second derivative annihilates the antisymmetric part of ${\rho }^{\beta }{\rho }^{\alpha }$. Using $2{\eta }^{\alpha \beta }-{\rho }^{\beta }{\rho }^{\alpha }={\rho }^{\alpha }{\rho }^{\beta }$ now gives the exact off-shell identityThus the action is invariant when this boundary flux vanishes and the transformation preserves the field boundary conditions, for example with suitable falloff or compatible supersymmetric endpoint conditions. On a spatial circle, periodic fields admit the constant parameter; antiperiodic fermions with periodic bosons require the local superconformal treatment, since a nonzero constant parameter would make the boson variation antiperiodic. The constant parameter and ordinary derivatives describe rigid supersymmetry in flat gauge; a locally supersymmetric curved-worldsheet formulation includes the worldsheet gravitino. The result is the rigid worldsheet supersymmetry boundary term without using field equations.

For the commutator on $X$, keep the order of the odd parameters:For the fermion, the order is equally important. Moving ${ϵ}_2$ through both odd factors in $({\overline{ϵ}}_1{\partial }_{\alpha }\psi ){ϵ}_2$ incurs two minus signs. ConsequentlyTo check the spinor matrix explicitly, set ${ϵ}_1=(a,b{)}^T$, ${ϵ}_2=(c,d{)}^T$, with all four components anticommuting and monomials ordered $a,b,c,d$. ThenDirect multiplication gives ${\rho }^{\alpha }M+M{\rho }^{\alpha }=2V^{\alpha }1$. HenceVariation of the fermion action gives the Dirac equation ${\rho }^{\alpha }{\partial }_{\alpha }\psi =0$. On this equation the second term vanishes, provingThis is on-shell closure of rigid worldsheet supersymmetry; the action's invariance was off shell, but its fermionic transformation algebra closes in this field content only on shell.

The associated superconformal current is proportional to $\psi \cdot \partial X$, of conformal weight $3/2$. Its fermionic modes $G_r$ extend the stress-tensor generators to the N=1 super-Virasoro algebra:The modes are half-integral in the Neveu–Schwarz sector and integral in the Ramond sector. Their anticommutator giving a Virasoro generator is the conformal-mode counterpart of the translation commutator just derived. Ten coordinate bosons and ten Majorana fermions give matter central charge $c=10+10/2=15$, canceled by the usual reparametrization and superconformal ghosts in the critical RNS string.