Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 306 / 3 / i

Table of contents
- Solution i

Solution

An operator

O (w, \overset{w}{ˉ})

is a primary operator of conformal weights

(h, h)

when its OPEs with the stress tensors are

T (z) O (w, \overset{w}{ˉ}) \sim \frac{h O ( w , w ˉ )}{( z - w ) ^{2}} + \frac{\partial _{w} O ( w , w ˉ )}{z - w},

\overset{ˉ}{T} (\overset{z}{ˉ}) O (w, \overset{w}{ˉ}) \sim \frac{h O ( w , w ˉ )}{( z ˉ - w ˉ ) ^{2}} + \frac{\partial _{\overset{w}{ˉ}} O ( w , w ˉ )}{z ˉ - w ˉ},

with no more singular terms. Its scaling dimension and two-dimensional spin are

Δ = h + h, s = h - h .

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