Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 306 (String Theory)

The Polyakov action contains the dilaton couplingFor a constant dilaton ${\Phi }_0$, the Gauss-Bonnet theorem gives $S_{\Phi }={\Phi }_0\chi (\Sigma )$, so the path-integral weight contributes $e^{-{\Phi }_0\chi }=g_s^{-\chi }$ with string coupling $g_s=e^{{\Phi }_0}$. A connected closed oriented genus-$g$ worldsheet has $\chi =2-2g$. Including one conventional factor of $g_s$ for each of $n$ external closed-string vertices gives the string genus expansionFor four external states on the sphere, this is $g_s^2$.

The quadratic worldsheet action makes each $X^{\mu }$ a free boson. Its Green function in the complex plane inverts the operator in the action and uses $\partial \bar{\partial }\log |z-w{|}^2=2\pi {\delta }^{(2)}(z-w)$ with the corresponding convention. The result isAn additive constant is physically irrelevant because it can be absorbed into the zero mode.

Split $X^{\mu }=x_0^{\mu }+X^{\prime \mu }$. Integrating the constant mode gives momentum conservation,For the nonzero modes, Wick theorem and the propagator giveThus, up to source-independent numerical normalization, the four-point amplitude isThe product is the Koba-Nielsen factor.

For the Möbius transformation $z^{\prime }=(az+b)/(cz+d)$ with $ad-bc=1$,The power of $|c{z}_i+d|$ contributed by every pair containing $i$ iswhere momentum conservation and the mass-shell condition $p_i^2=4/{\alpha }^{\prime }$ were used. The Koba-Nielsen factor therefore contributes $|c{z}_i+d{|}^4$ at each insertion, exactly cancelling the transformed measure. Hence the remaining integral is $SL(2,\mathbb{C})$ invariant, and division by its volume removes the residual conformal-gauge redundancy.

Because the gamma function has simple poles at the nonpositive integers and no zeros, the $s$-dependent numerator $\Gamma (-{\alpha }^{\prime }s/4)$ has poles atorFactorization of a scattering amplitude identifies each pole with an intermediate on-shell state. The Type II superstring mass spectrum therefore contains a massless level at $n=0$ and an infinite equally spaced tower in squared mass for $n\ge 1$. There is no negative-$M^2$ pole, consistently with the absence of a tachyon in Type II superstring theory.

The term ${\hat{p}}^{,2}/2$ is the target-space zero-mode kinetic energy. The constant $-1/12$ is the zero-point energy of the left- and right-moving oscillators: each chiral boson contributes $-1/24$. Finally,count the energies of the two independent sets of string oscillators; one excitation of mode $n$ has energy $n$.

For $n>0$, define occupation numbers $N_n,{\tilde{N}}_n\in {\mathbb{N}}_0$. A basis iswhere $\hat{p}|p;0,0⟩=p|p;0,0⟩$ and positive oscillator modes annihilate the vacuum. Since ${\alpha }_{-n}{\alpha }_n$ has eigenvalue $n{N}_n$, the simultaneous eigenvalues are

Put $\tau ={\tau }_1+i{\tau }_2$ and $q=e^{2\pi i\tau }$. The torus partition function is the traceThe continuum momentum states have density $V/(2\pi )$, so their contribution isEach left-moving oscillator contributes ${\sum }_{N\ge 0}{q}^{nN}=(1-q^n{)}^{-1}$, with the complex conjugate for the right mover. The zero-point factor combines these products into the Dedekind eta functionTherefore the torus partition function of a free boson is

For a compact boson $X\sim X+2\pi R$, a closed spatial circuit of the worldsheet may wind around the target circle:Single-valued target-space wavefunctions quantize the zero-mode momentum as $p=n/R$, $n\in \mathbb{Z}$. In the conventionthe Hamiltonian and worldsheet momentum becomeThe $nw$ term is the zero-mode contribution to closed-string level matching.

The momentum integral is replaced by the sum over momentum and winding modes. Combining the zero modes with the oscillator trace givesEquivalently,with the left/right convention of part (iv). The answer is invariant under the T-duality $R\leftrightarrow 1/R$ accompanied by $n\leftrightarrow w$ in these units.

An operator $\mathcal{O}(w,\bar{w})$ is a primary operator of conformal weights $(h,\tilde{h})$ when its OPEs with the stress tensors arewith no more singular terms. Its scaling dimension and two-dimensional spin are

Normal ordering defines a composite field by subtracting all singular self-contractions before bringing its constituent points together. For example,This makes $T$ and $\bar{T}$ well-defined local operators and is equivalent in oscillator language to placing creation operators to the left of annihilation operators.

Only the fermionic part of $T$ contracts with $\psi$. Applying Wick theorem to $T_{\psi }(z)=-\frac{1}{2}:\psi \partial \psi :(z)$ and using $\psi (z)\psi (w)\sim 1/(z-w)$ givesThere is no singular OPE with $\bar{T}(\bar{z})$. Thus $\psi$ is a primary operator with

The holomorphic free boson contributes central charge $c_X=1$, while one real chiral free fermion contributes $c_{\psi }=1/2$. Equivalently, the double contractions in $T(z)T(w)$ giveSince the fourth-order coefficient is $c/2$,

The free equations of motion are $\bar{\partial }\psi =0$ and $\partial \bar{\partial }X=0$, henceThus the superconformal current $G=i\psi \partial X$ is holomorphic. Differentiating the boson OPE gives $\partial X(z)\partial X(w)\sim -1/(z-w{)}^2$. The double contraction in $G(z)G(w)$ is consequently $1/(z-w{)}^3$, while the single contractions combine into twice the full stress tensor. ThereforeThe leading coefficient equals $2c/3=1$ for $c=3/2$.

Extract a mode usingIn the radially ordered double contour for the anticommutator, the simple pole $2T(w)/(z-w)$ gives $2L_{m+n}$. Expanding $z^{m+1/2}$ about $w$, the third-order pole contributes one half of its second derivative,The remaining contour is nonzero only for $m+n=0$. Restoring the general leading OPE coefficient $2c/3$ givesThis is the fermionic relation in the N=1 super-Virasoro algebra.

In Euclidean worldsheet signature, the string nonlinear sigma model action can be writtenThe factor of $i$ in the B-field term is absent in Lorentzian signature. Here $G$ is the target metric, $B$ the Kalb–Ramond field, and $\Phi$ the dilaton.

The worldsheet metric is a gauge variable, so quantum consistency requires the Weyl transformation to remain non-anomalous. The background fields are couplings of a two-dimensional quantum field theory, and their sigma-model beta functions multiply the trace of the worldsheet stress tensor. Requiring every beta function to vanish gives, at the first nontrivial order in the derivative or ${\alpha }^{\prime }$ expansion, precisely the stated metric, B-field and dilaton equations. They are also the Euler-Lagrange equations of the leading spacetime string-frame effective action

Define the leading right-hand side of the dilaton equation byTake a divergence of the metric equation. The contracted Bianchi identity, commutation of covariant derivatives on ${\nabla }_{\nu }\Phi$, the B-field equation ${\nabla }^{\lambda }{H}_{\lambda \mu \nu }=2{\nabla }^{\lambda }\Phi H_{\lambda \mu \nu }$, and the supplied H-field identity giveThe metric equation contracted with ${\nabla }^{\rho }\Phi$ saysSubstitution cancels the curvature, Hessian and H-field terms in ${\nabla }_{\nu }C$, leavingThus $C$ is constant on each connected component. The remaining constant is fixed by the central-charge deficit; it vanishes in the critical 26-dimensional bosonic theory at this order.

First, the displayed equations are only the one-loop sigma-model beta functions. Higher worldsheet loops generate additional covariant terms with more curvatures, H-fields and derivatives, each accompanied by higher powers of ${\alpha }^{\prime }$. Second, string scattering amplitudes contain momentum corrections from the finite string length and from integrating out the infinite tower of massive string states. Their low-energy expansion produces the same higher-derivative spacetime operators, such as curvature-squared and higher-curvature terms. Both arguments require ${\alpha }^{\prime }$ corrections even though local field redefinitions can move individual correction terms between equations.

Let $a=0,\dots ,p$ label directions tangent to the D-brane and $i=p+1,\dots ,25$ label transverse directions. The endpoints obey Dirichlet conditions transversely,and the boundary variation of the metric and constant B-field terms gives mixed tangential conditionsup to the orientation sign at the two ends. A worldvolume gauge potential contributes $2\pi {\alpha }^{\prime }{F}_{ab}$ in the same place, so the gauge-invariant condition contains $\mathcal{F}=b+2\pi {\alpha }^{\prime }F$.

From the brane perspective, the pullback of the background B-field is therefore indistinguishable locally from a constant worldvolume electromagnetic field strength, modulo its two-form gauge symmetry and a compensating transformation of the brane gauge field. This is the open-string boundary condition in a B-field; sufficiently general constant $b$ also induces the familiar noncommutative deformation of D-brane endpoint coordinates.