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

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Solution

In any two-dimensional conformal field theory,

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

(1)

Double contractions in the bosonic beta-gamma system give

c_{β γ} = 2 (6 h^{2} - 6 h + 1) .

(2)

The polynomial is unchanged by

h \mapsto 1 - h

, as required when the roles of the two fields are interchanged. If the fields anticommute, a closed fermionic contraction contributes an additional minus sign, giving the fermionic bc-system result

c_{b c} = - 2 (6 h^{2} - 6 h + 1) .

(3)

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

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