In two-dimensional conformal field theory, Virasoro conformal blocks (named after Miguel Ángel Virasoro) are special functions that serve as building blocks of correlation functions. On a given punctured Riemann surface, Virasoro conformal blocks form a particular basis of the space of solutions of the conformal Ward identities. Zero-point blocks on the torus are characters of representations of the Virasoro algebra; four-point blocks on the sphere reduce to hypergeometric functions in special cases, but are in general much more complicated. In two dimensions as in other dimensions, conformal blocks play an essential role in the conformal bootstrap approach to conformal field theory.
Definition
Definition from OPEs Using operator product expansions (OPEs), an N {\displaystyle N} -point function on the sphere can be written as a combination of three-point structure constants, and universal quantities called N {\displaystyle N} -point conformal blocks. Given an N {\displaystyle N} -point function, there are several types of conformal blocks, depending on which OPEs are used. In the case N = 4 {\displaystyle N=4} , there are three types of conformal blocks, corresponding to three possible decompositions of the same four-point function. Schematically, these decompositions read
⟨ V 1 V 2 V 3 V 4 ⟩ = ∑ s C 12 s C s 34 F s (s-channel) = ∑ t C 14 t C t 23 F t (t-channel) = ∑ u C 13 u C 24 u F u (u-channel) , {\displaystyle \left\langle V_{1}V_{2}V_{3}V_{4}\right\rangle =\sum _{s}C_{12s}C_{s34}{\mathcal {F}}_{s}^{\text{(s-channel)}}=\sum _{t}C_{14t}C_{t23}{\mathcal {F}}_{t}^{\text{(t-channel)}}=\sum _{u}C_{13u}C_{24u}{\mathcal {F}}_{u}^{\text{(u-channel)}}\ ,}
where C {\displaystyle C} are structure constants and F {\displaystyle {\mathcal {F}}} are conformal blocks. The sums are over representations of the conformal algebra that appear in the CFT's spectrum. OPEs involve sums over the spectrum, i.e. over representations and over states in representations, but the sums over states are absorbed in the conformal blocks. In two dimensions, the symmetry algebra factorizes into two copies of the Virasoro algebra, called left-moving and right-moving. If the fields are factorized too, then the conformal blocks factorize as well, and the factors are called Virasoro conformal blocks. Left-moving Virasoro conformal blocks are locally holomorphic functions of the fields' positions z i {\displaystyle z_{i}} ; right-moving Virasoro conformal blocks are the same functions of z ¯ i {\displaystyle {\bar {z}}_{i}} . The factorization of a conformal block into Virasoro conformal blocks is of the type
F s L ⊗ s R (s-channel) ( { z i } ) = F s L (s-channel, Virasoro) ( { z i } ) F s R (s-channel, Virasoro) ( { z ¯ i } ) , {\displaystyle {\mathcal {F}}_{s_{L}\otimes s_{R}}^{\text{(s-channel)}}(\{z_{i}\})={\mathcal {F}}_{s_{L}}^{\text{(s-channel, Virasoro)}}(\{z_{i}\}){\mathcal {F}}_{s_{R}}^{\text{(s-channel, Virasoro)}}(\{{\bar {z}}_{i}\})\ ,}
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