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Suppose we quantize some classical CFT algebra given by generators which satisfy $$[l_n,l_m]=(n-m)l_{n+m},$$

$$[\overline{l}_n,\overline{l}_m]=(n-m)\overline{l}_{n+m},$$

$$[l_n,\overline{l}_m]=0.$$

Using OPE we can obtain the following relations for generators of Virasoro algebra :

$$[L_n,L_m]=(n-m)L_{n+m}+\frac{c}{12}n(n^2-1)\delta_{n+m}$$

and

$$[\overline{L}_n,\overline{L}_m]=(n-m)\overline{L}_{n+m}+\frac{\overline{c}}{12}n(n^2-1)\delta_{n+m}.$$

In many books/articles one can find that we always assume that $c=\overline{c}$, i.e. central charges for $L$'s and $\overline{L}$'s are equal. My question is: Why? Is there any physical or mathematical condition that must be satisfied and hence we need this assumption?

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We can have CFTs with $c \neq {\tilde c}$ as long as $$ c - {\tilde c} \in 24 {\mathbb Z} $$ This condition arises from modular invariance of the CFT when it is put on the torus.

PS - In radially quantized CFTs, the adjoint condition is $L_m^\dagger = L_{-m}$ and ${\bar L}_m^\dagger = {\bar L}_{-m}$.

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