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

Table of contents
- Solution iv

Solution

The holomorphic free boson contributes central charge

c_{X} = 1

, while one real chiral free fermion contributes

c_{ψ} = 1/2

. Equivalently, the double contractions in

T (z) T (w)

give

T (z) T (w) \sim \frac{3/4}{( z - w ) ^{4}} + \frac{2 T ( w )}{( z - w ) ^{2}} + \frac{\partial T ( w )}{z - w} .

Since the fourth-order coefficient is

c /2

,

c = 1 + \frac{1}{2} = \frac{3}{2} .

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