Under a finite conformal coordinate change $z'=f(z)$ and $\bar z'=\bar f(\bar z)$, a <primary operator> of <conformal weight> $(h,\widetilde h)$ obeys
$$
\mathcal O'(z',\bar z')
=\left(\frac{df}{dz}\right)^{-h}
\left(\frac{d\bar f}{d\bar z}\right)^{-\widetilde h}
\mathcal O(z,\bar z).
$$
Equivalently, its <operator product expansion> with the <holomorphic stress-energy tensor> has singular part
$$
T(z)\mathcal O(w,\bar w)
\sim\frac{h\,\mathcal O(w,\bar w)}{(z-w)^2}
+\frac{\partial\mathcal O(w,\bar w)}{z-w}.
$$
The antiholomorphic stress tensor gives the analogous formula with $\widetilde h$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 306 / 3 / a / Solution

For this normalization,
$$
T(z)=-:\partial X(z)\partial X(z):,
\qquad
\partial X(z)X(w,\bar w)\sim-\frac1{2(z-w)}.
$$
Contracting once and twice with
$$
\mathcal O_k(w,\bar w)=:e^{ikX(w,\bar w)}:
$$
gives
$$
T(z)\mathcal O_k(w,\bar w)
\sim
\frac{k^2/4}{(z-w)^2}\mathcal O_k(w,\bar w)
+\frac1{z-w}\partial\mathcal O_k(w,\bar w).
$$
It is therefore a <primary operator> with
$$
h=\widetilde h=\frac{k^2}{4}.
$$

Solved by gpt-5.6-sol high.

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 306 / 3 / c / i / Solution

Translation and rotation invariance make the two-point function depend only on $|x_{12}|$. Scale covariance then fixes its power, and inversion or special-conformal invariance requires equal dimensions for a nonzero scalar two-point function. For identical <primary operator>[scalar primary operators],
$$
\boxed{\langle O(x_1)O(x_2)\rangle
=\frac{C_2}{|x_{12}|^{2\Delta}}}.
$$
Only the normalization $C_2$ depends on the operator convention.

Solved by gpt-5.6-sol high.

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 354 / 1 / b / i / Solution

{wiki=Primary_field}

A primary operator of weights $(h,\widetilde h)$ transforms covariantly under local conformal maps and has a universal operator-product expansion with the stress-energy tensor.


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