Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 306 / 2 / d

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- Solution d

Solution

Transforming the point-split normal ordering changes the subtraction as well as the two derivatives. Expanding

w (z \pm δ /2)

to third order gives

T (w) = (\frac{d w}{d z})^{- 2} [T (z) - \frac{1}{12} {w, z}],

where the Schwarzian derivative is

R (w; z) = \frac{1}{12} {w, z} = \frac{1}{12} (\frac{w ^{'''} ( z )}{w ^{'} ( z )} - \frac{3}{2} \frac{w ^{''} ( z ) ^{2}}{w ^{'} ( z ) ^{2}}) .

The inhomogeneous Schwarzian term means that

T

is not a primary operator. It reflects the central charge

c = 1

of one free scalar.

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