Conformal blocks and Casimir equations

Use four identical real scalar primaries in Euclidean dimension d at least three, with unit two-point coefficient and the reduced four-point function defined by the preceding distance prefactor. In the channel that fuses points one and two, each exchanged symmetric traceless primary contributes its normalized block. For identical bosonic scalars, only even integer spin appears in this OPE. Different global-symmetry sectors or nonidentical external operators require additional coefficients and channel bookkeeping.

Choose the identity block to be one and scalar exchanged blocks to have leading behavior equal to u to half their dimension in the simultaneous OPE limit. Spinning blocks need a fixed leading angular polynomial and tensor normalization as well; a positive rescaling can be absorbed consistently into the coefficient convention. Mixing a block formula from one convention with coefficients from another changes the correlator. The full family, rather than an isolated leading power, is what transforms consistently under channel comparisons.

Apply the quadratic conformal Casimir built from generators acting on the first pair of insertions. Every state in an irreducible exchanged family has the same Casimir eigenvalue, so its block is an eigenfunction of a differential operator in cross-ratios. This turns a sum over descendants into a differential-equation problem. The eigenvalue depends on the exchanged dimension and rotation representation, while the differential operator also knows the external dimensions and spacetime dimension.

The eigenvalue equation alone does not select a unique physical block. One must impose the OPE asymptotic behavior and the proper branch, distinguishing the desired dimension from its shadow partner. Singular solutions or wrong boundary conditions can share the same eigenvalue. The Casimir is not a positive Hamiltonian or a state norm, so a negative eigenvalue is not itself a unitarity violation. Positivity constraints must instead be applied to the radial representation and its descendants.

In two dimensions the global conformal algebra factorizes into holomorphic and antiholomorphic parts. A holomorphic global block with identical external weights can be written with a hypergeometric function. This is a useful analytic benchmark, but a global block includes only global descendants and generally differs from a full Virasoro block. The central charge and the extra stress-tensor descendants enter the latter and cannot be inferred from a global expression alone.

At a particular exchanged holomorphic weight the hypergeometric series reduces to a logarithm, allowing a direct truncation check. Its branch is fixed initially in the real interval between zero and one; continuing past singularities requires a prescription. The function lab can show a simple polynomial and its derivative, but it does not solve Casimir equations, choose a shadow branch or generate CFT blocks. Any polynomial used as a local approximation needs an error estimate appropriate to the actual block.