Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 306 / 2 / b

Table of contents
- Solution b

Solution

Applying Wick theorem to the stated holomorphic stress-energy tensor gives

T (z) γ (w) \sim \frac{hγ ( w )}{( z - w ) ^{2}} + \frac{\partial γ ( w )}{z - w},

and

T (z) β (w) \sim \frac{( 1 - h ) β ( w )}{( z - w ) ^{2}} + \frac{\partial β ( w )}{z - w} .

These are exactly the stress-tensor OPEs of primary operators. Hence

γ

has holomorphic conformal weight

h

and

β

1 - h

.

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