Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 306 (String Theory)

The Euler-Lagrange equation of the gauge-fixed Polyakov action is the two-dimensional wave equationIts general solution is a sum of left- and right-moving functions. Periodicity in $\sigma$ and a Fourier expansion give the closed-string mode expansionThe zero modes satisfy ${\alpha }_0^{\mu }={\tilde{\alpha }}_0^{\mu }=\sqrt{{\alpha }^{\prime }/2},p^{\mu }$. Here $x^{\mu }$ and $p^{\mu }$ are the center-of-mass position and momentum, while the nonzero string oscillators describe shape excitations.

A classical string must also satisfy the equations obtained by varying the worldsheet metric before imposing conformal gauge. Vanishing of the worldsheet stress-energy tensor givesIn modes these are the Virasoro constraintsfor every integer $m$. Reality of $X^{\mu }$ also requiresTogether with periodicity and the wave equation, these conditions remove the unphysical longitudinal worldsheet excitations.

For a target circle of radius $R$, maps may wind, so the boundary condition isSingle-valued target-space wavefunctions quantize the center-of-mass momentum as $p_Y=n/R$, with $n\in \mathbb{Z}$. The compact boson expansion becomesEquivalently, its left- and right-moving zero-mode momenta areThe integers $n$ and $w$ are the momentum and winding modes.

Let $M$ be the mass measured in the uncompactified 25-dimensional spacetime. The quantum zero-mode constraints areThe string level operatorshave nonnegative integer eigenvalues. Adding the constraints and using $(p_L^2+p_R^2)/2=n^2/R^2+w^2{R}^2/{\alpha }^{\prime 2}$ givesTheir difference gives closed-string level matching, $N-\tilde{N}+nw=0$ with the present left-right convention. The normal-ordering constant of a string $a$ is the regularized zero-point energy generated when oscillator products are normal ordered.

The quadratic kinetic operator of the beta-gamma system couples $\beta$ only to $\gamma$. Its Green-function equation isin the stated normalization. Since $\bar{\partial }(1/(z-w))=2\pi {\delta }^{(2)}(z-w)$, the singular part isThe inverse kinetic matrix has no $\beta \beta$ or $\gamma \gamma$ entry, so those two operator product expansions are nonsingular. For commuting fields, reversing the order gives $\gamma (z)\beta (w)\sim 1/(z-w)$ after expanding about $w$.

Applying Wick theorem to the stated holomorphic stress-energy tensor givesandThese are exactly the stress-tensor OPEs of primary operators. Hence $\gamma$ has holomorphic conformal weight $h$ and $\beta$ has weight $1-h$.

In any two-dimensional conformal field theory,Double contractions in the bosonic beta-gamma system giveThe polynomial is unchanged by $h\mapsto 1-h$, as required when the roles of the two fields are interchanged. If the fields anticommute, a closed fermionic contraction contributes an additional minus sign, giving the fermionic bc-system result

The $D$ free embedding scalars contribute $c_X=D$. The ordinary diffeomorphism ghosts contribute $c_{bc}=-26$. The fermionic $(m,n)$ pair has $h_n=0$ and contributes $-2$. Each of the $D/2$ fermionic $({\chi }_a,{\psi }^a)$ pairs has $h_{\psi }=1/2$ and contributes $+1$. Each bosonic $({\beta }_r,{\gamma }^r)$ pair has $h_{\gamma }=-1/2$ and contributes $+11$, and there are two such pairs. The total holomorphic central charge is thereforeCancellation of the worldsheet Weyl anomaly requires $c_{\mathrm{tot}}=0$, soThe antiholomorphic sector gives the same condition.

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.

In old covariant quantization, an open-string physical string state obeyswith analogous left- and right-moving conditions for a closed string. A spurious string state is a Virasoro descendant orthogonal to every physical state. A null string state is both physical and spurious; it has zero norm and zero inner product with every physical state, so quotienting by null states removes gauge redundancy without removing observable states.

Let $|s⟩=L_{-1}|\chi ⟩$ with $L_n|\chi ⟩=0$ for every $n\ge 0$. Since $[L_0,L_{-1}]=L_{-1}$, $|s⟩$ has $L_0$ eigenvalue one. Moreover,and for $n>1$,It is therefore physical. It is a Virasoro descendant and hence spurious, while its norm isThus it is null.

SetIts $L_0$ eigenvalue is $-1+2=1$. The Virasoro algebra givesso $L_1|s⟩=0$. At critical central charge $c=26$,so $L_2|s⟩=0$. All $L_n$ with $n\ge 3$ annihilate it directly. Hence $|s⟩$ is physical and, being a positive-level Virasoro descendant, spurious; it is therefore a null state.

The zero-mode condition is automatic:Using $[L_m,{\alpha }_n^{\mu }]=-n{\alpha }_{m+n}^{\mu }$ and oscillator annihilation of the vacuum giveswhich vanishes when $t_{\mu \nu }{\alpha }_0^{\nu }=-v_{\mu }$. Similarly,which vanishes when $v\cdot {\alpha }_0=-t^{\mu }{}_{\mu }/2$. Higher positive Virasoro modes annihilate a level-two state. These are precisely the stated physical-state conditions.

Using ${\alpha }_0^2=-2$, tracelessness and transversality of ${ϵ}_{\mu \nu }$ givesand $v\cdot {\alpha }_0=-t^{\lambda }{}_{\lambda }/2$, so the constraints hold.

On the momentum vacuum,andThereforewhereThe vacuum has $L_0|0,p⟩={\alpha }_0^2|0,p⟩/2=-|0,p⟩$ and is annihilated by positive modes, so part c proves that $|n⟩$ is null. The physical content is consequently the transverse traceless tensor ${ϵ}_{\mu \nu }$.