After imposing <conformal gauge>, transformations
$$
\sigma^+\mapsto f(\sigma^+),
\qquad
\sigma^-\mapsto\widetilde f(\sigma^-)
$$
remain, with a compensating <Weyl transformation>. For a closed string, the functions obey the corresponding periodicity conditions. Introduce target-space light-cone coordinates $X^\pm=(X^0\pm X^{D-1})/\sqrt2$. When $p^+\ne0$, the equations make $X^+$ a sum of left- and right-moving functions, and the two residual reparameterizations can set
$$
X^+(\tau,\sigma)=x^++2\alpha'p^+\tau.
$$
This <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.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 306 / 2 / a / Solution

In Euclidean <conformal gauge>, complex coordinates reduce the <worldsheet ghost action> to
$$
S_{\rm Gh}=\frac1{2\pi}\int d^2z\,
\left(b_{zz}\bar\partial c^z+
b_{\bar z\bar z}\partial c^{\bar z}\right)
$$
up to a convention-dependent common normalization absorbed into the fields. Inverting $\bar\partial$ and $\partial$ gives
$$
\boxed{\displaystyle
\langle b_{zz}(z)c^z(w)\rangle=\frac1{z-w}},
\qquad
\boxed{\displaystyle
\langle b_{\bar z\bar z}(\bar z)c^{\bar z}(\bar w)\rangle
=\frac1{\bar z-\bar w}},
$$
with 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).

Solved by gpt-5.6-sol high.

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 306 / 4 / b / Solution

{wiki}

Conformal gauge uses worldsheet diffeomorphisms and Weyl transformations to set $g_{\alpha\beta}=e^{2\phi}\eta_{\alpha\beta}$ locally.


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 Incoming links: Conformal gauge