I know about the crossing symmetry in conformal field theory, which is a symmetry satisfied by the primary fields' conformal block. Is there a generalized crossing symmetry relation which holds also for conformal blocks of descendant fields?
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2 Answers
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This can be derived by the idea of OPE. Consider the following correlator: $$< {\phi _1}|{\phi _2}({z_1}){\phi _3}({z_2})|{\phi _4} > $$ Insert a complete set of basis, it can be written as : $$\sum_i<\phi_1|\phi_2(z_1)|\phi_i><\phi_i|\phi_3(z_2)|\phi_4>$$ Then we consider the limit $z_1\rightarrow z_2$. In this limit we can first do a OPE of $\phi_2$ and $\phi_3$, and it's easy to see that:$$\phi_2(z_1)\phi_3(z_2)\sim\sum_k<\phi_k|\phi_2(z_1-z_2)|\phi_3>\phi_k(z_2)$$ Therefore, we get another expression for the original correlator: $$\sum_k<\phi_1|\phi_k(z_2)|\phi_4><\phi_k|\phi_2(z_1-z_2)|\phi_3>$$
Finally we analytically continue the expression to outside the region of $z_1\rightarrow z_2$. This completes the proof of crossing symmetry relation for any fields from $\phi_1$ to $\phi_4$. $$\sum_i<\phi_1|\phi_2(z_1)|\phi_i><\phi_i|\phi_3(z_2)|\phi_4>=\sum_k<\phi_1|\phi_k(z_2)|\phi_4><\phi_k|\phi_2(z_1-z_2)|\phi_3>$$
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Yes, conformal blocks of descendent fields obey the same crossing symmetry relations as conformal blocks of the corresponding primary fields.
For example, in a case where a descendent's blocks are obtained by acting on a primary's blocks with a differential operator, you can apply that operator to the crossing symmetry relation involving the primary's blocks, and get the crossing symmetry relation for the descendent's blocks.
