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 -point function on the sphere can be written as a combination of three-point structure constants, and universal quantities called -point conformal blocks.
Given an -point function, there are several types of conformal blocks, depending on which OPEs are used. In the case , there are three types of conformal blocks, corresponding to three possible decompositions of the same four-point function. Schematically, these decompositions read
where are structure constants and 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 ; right-moving Virasoro conformal blocks are the same functions of . The factorization of a conformal block into Virasoro conformal blocks is of the type
where are representations of the left- and right-moving Virasoro algebras respectively.
Definition from Virasoro Ward identities
Conformal Ward identities are the linear equations that correlation functions obey, as a result of conformal symmetry.
In two dimensions, conformal Ward identities decompose into left-moving and right-moving Virasoro Ward identities. Virasoro conformal blocks are solutions of the Virasoro Ward identities.
OPEs define specific bases of Virasoro conformal blocks, such as the s-channel basis in the case of four-point blocks. The blocks that are defined from OPEs are special cases of the blocks that are defined from Ward identities.
Properties
Any linear holomorphic equation that is obeyed by a correlation function, must also hold for the corresponding conformal blocks. In addition, specific bases of conformal blocks come with extra properties that are not inherited from the correlation function.
Conformal blocks that involve only primary fields have relatively simple properties. Conformal blocks that involve descendant fields can then be deduced using local Ward identities. An s-channel four-point block of primary fields depends on the four fields' conformal dimensions on their positions and on the s-channel conformal dimension . It can be written as where the dependence on the Virasoro algebra's central charge is kept implicit.
Linear equations
From the corresponding correlation function, conformal blocks inherit linear equations: global and local Ward identities, and BPZ equations if at least one field is degenerate.
In particular, in an -point block on the sphere, global Ward identities reduce the dependence on the field positions to a dependence on cross-ratios. In the case
where and
is the cross-ratio, and the reduced block coincides with the original block where three positions are sent to
Singularities
Like correlation functions, conformal blocks are singular when two fields coincide. Unlike correlation functions, conformal blocks have very simple behaviours at some of these singularities. As a consequence of their definition from OPEs, s-channel four-point blocks obey
for some coefficients On the other hand, s-channel blocks have complicated singular behaviours at : it is t-channel blocks that are simple at , and u-channel blocks that are simple at