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.
-point function on the sphere can be written as a combination of three-point structure constants, and universal quantities called
The sums are over representations of the conformal algebra that appear in the CFT's spectrum.
Left-moving Virasoro conformal blocks are locally holomorphic functions of the fields' positions
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.
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
leave the correlation function invariant, and therefore relate different bases of conformal blocks with one another.
, such that The fusing matrix is a function of the central charge and conformal dimensions, but it does not depend on the positions
is a double gamma function, Although its expression is simpler in terms of momentums
Up to normalization, the fusing matrix coincides with Ruijsenaars' hypergeometric function, with the arguments
[6] The fusing matrix has several different integral representations, and obeys many nontrivial identities.
Such generators correspond to basis states in the Verma module with the conformal dimension
In Alexei Zamolodchikov's recursive representation of four-point blocks on the sphere, the cross-ratio
of the poles are the dimensions of degenerate representations, which correspond to the momentums The residues
are positive, due to this combination's interpretation in terms of sums of states in the pillow geometry.
And the block's prefactors can be interpreted in terms of the conformal transformation from the sphere to the pillow.
Using the S-matrix, constraints on a CFT's spectrum can be derived from the modular invariance of the torus partition function, leading in particular to the ADE classification of minimal models.
This relation maps the modulus of the torus to the cross-ratio of the four points' positions, and three of the four fields on the sphere have the fixed momentum
:[15][16] where The recursive representation of one-point blocks on the torus is[17] where the residues are Under modular transformations, one-point blocks on the torus behave as where the modular kernel is[18][19] For a four-point function on the sphere where one field has a null vector at level two, the second-order BPZ equation reduces to the hypergeometric equation.
Another basis is made of the two t-channel conformal blocks, The fusing matrix is the matrix of size two such that whose explicit expression is Hypergeometric conformal blocks play an important role in the analytic bootstrap approach to two-dimensional CFT.
then certain linear combinations of s-channel conformal blocks are solutions of the Painlevé VI nonlinear differential equation.
[22] The relevant linear combinations involve sums over sets of momentums of the type
This allows conformal blocks to be deduced from solutions of the Painlevé VI equation and vice versa.
limits, mysterious on the conformal field theory side, is explained naturally in the context of four dimensional gauge theories, using blowup equations,[25][26] and can be generalized to more general pairs
For example: In a theory whose symmetry algebra is larger than the Virasoro algebra, for example a WZW model or a theory with W-symmetry, correlation functions can in principle be decomposed into Virasoro conformal blocks, but that decomposition typically involves too many terms to be useful.
Instead, it is possible to use conformal blocks based on the larger algebra: for example, in a WZW model, conformal blocks based on the corresponding affine Lie algebra, which obey Knizhnik–Zamolodchikov equations.