Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 306 / 4

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.