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

A Euclidean conformal transformation is a coordinate map whose Jacobian preserves angles:For ${\tilde{x}}^{\mu }=x^{\mu }+\delta {ϵ}^{\mu }+O({\delta }^2)$, its left side isTaking the trace identifies the infinitesimal scale change as $2\partial \cdot ϵ/d$. The traceless part must vanish, giving the conformal Killing equation

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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 scalar primary operators,Only the normalization $C_2$ depends on the operator convention.

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The same symmetries fix the scalar three-point function toEach insertion acquires total scaling weight $2{\Delta }_i$ from the two distances containing it, while special conformal transformations leave no independent cross-ratio for three points.

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Under the AdS-CFT correspondence, each single-trace operator is represented by one bulk field. The leading two-point Witten diagram is a bulk propagator joining the two boundary insertions, equivalently two bulk-to-boundary legs contracted through the quadratic bulk action. The leading connected three-point diagram has one bulk cubic vertex integrated over AdS and three bulk-to-boundary propagators ending at $x_1,x_2,x_3$. The bulk kinetic normalization fixes $C_2$, while the cubic coupling fixes $C_{123}$.

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For three identical operators, conformal invariance givesWhen $|x|\ll |y|$, $|y-x|=|y|[1+O(|x|/|y|)]$, soInserting the operator product expansion, the identity term has zero expectation with $O(y)$, while the $O(0)$ term giveswhich has precisely the required position dependence.

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Comparing the coefficients in the short-distance limit givesIf the operator is normalized so that $C_2=1$, its OPE coefficient equals the three-point normalization.

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