Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 354 / 2 / c / Solution

A pair of boundary circles has one dimensionless invariant under the conformal field theory group. If their radii are

R_{1}, R_{2}

and their center separation is

d = R_{1} + R_{2} + L,

a convenient invariant is the inversive distance

χ = \frac{d ^{2} - R _{1}^{2} - R _{2}^{2}}{2 R _{1} R _{2}} .

The choice between connected and disconnected bulk surfaces can depend only on

χ

. Let the transition occur at the single theory-independent numerical value

χ = X

, with

X > 1

. Then

(R_{1} + R_{2} + C)^{2} = R_{1}^{2} + R_{2}^{2} + 2 X R_{1} R_{2},

so

C (R_{1}, R_{2}) = R_{1}^{2} + R_{2}^{2} + 2 X R_{1} R_{2} - R_{1} - R_{2} .

All dependence on the two radii is fixed by conformal symmetry; only the pure number

X

requires the explicit minimal-surface calculation.

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