Virasoro conformal block#Fusing matrix

Using operator product expansions (OPEs),

an $N$-point function on the sphere can be written as a combination of three-point structure constants, and universal quantities called $N$-point conformal blocks.P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, 1997, {{ISBN|0-387-94785-X}}

Given an $N$-point function,

there are several types of conformal blocks, depending on which OPEs are used. In the case $N=4$, there are three types of conformal blocks, corresponding to three possible decompositions of the same four-point function. Schematically, these decompositions read

:$\backslash left\backslash langle\; V\_1V\_2V\_3V\_4\backslash right\backslash rangle$

=\sum_s C_{12s}C_{s34}\mathcal{F}^{\text{(s-channel)}}_s = \sum_t C_{14t}C_{t23}\mathcal{F}^{\text{(t-channel)}}_t = \sum_u C_{13u}C_{24u}\mathcal{F}^{\text{(u-channel)}}_u\ ,

where $C$ are structure constants and $\backslash mathcal\{F\}$ 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 $z\_i$; right-moving Virasoro conformal blocks are the same functions of $\backslash bar\; z\_i$. The factorization of a conformal block into Virasoro conformal blocks is of the type

:$\backslash mathcal\{F\}^\{\backslash text\{(s-channel)\}\}\_\{s\_L\backslash otimes\; s\_R\}(\backslash \{z\_i\backslash \})\; =\; \backslash mathcal\{F\}^\{\backslash text\{(s-channel,\; Virasoro)\}\}\_\{s\_L\}(\backslash \{z\_i\backslash \})\backslash mathcal\{F\}^\{\backslash text\{(s-channel,\; Virasoro)\}\}\_\{s\_R\}(\backslash \{\backslash bar\; z\_i\backslash \})\backslash \; ,$

where $s\_L,s\_R$ are representations of the left- and right-moving Virasoro algebras respectively.

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.{{Cite arXiv |eprint = 1708.00680|last1 = Teschner|first1 = Joerg|title = A guide to two-dimensional conformal field theory|year = 2017|class = hep-th}}

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.

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 $\backslash Delta\_i,$ on their positions $z\_i,$ and on the s-channel conformal dimension $\backslash Delta\_s$. It can be written as $\backslash mathcal\{F\}^\{(s)\}\_\{\backslash Delta\_s\}(\backslash Delta\_i|\backslash \{z\_i\backslash \}),$ where the dependence on the Virasoro algebra's central charge is kept implicit.

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 $N$-point block on the sphere, global Ward identities reduce the dependence on the $N$ field positions to a dependence on $N-3$ cross-ratios. In the case $N=4,$

:$\backslash mathcal\{F\}^\{(s)\}\_\{\backslash Delta\_s\}(\backslash \{\backslash Delta\_i\backslash \}|\backslash \{z\_i\backslash \})=\; z\_\{23\}^\{\backslash Delta\_1-\backslash Delta\_2-\backslash Delta\_3+\backslash Delta\_4\}\; z\_\{13\}^\{-2\backslash Delta\_1\}\; z\_\{34\}^\{\backslash Delta\_1+\backslash Delta\_2-\backslash Delta\_3-\backslash Delta\_4\}\; z\_\{24\}^\{-\backslash Delta\_1-\backslash Delta\_2+\backslash Delta\_3-\backslash Delta\_4\}\backslash mathcal\{F\}^\{(s)\}\_\{\backslash Delta\_s\}\; (\backslash \{\backslash Delta\_i\; \backslash \}|z),$

where $z\_\{ij\}=z\_i-z\_j,$ and

:$z=\; \backslash frac\{z\_\{12\}z\_\{34\}\}\{z\_\{13\}z\_\{24\}\}$

is the cross-ratio, and the reduced block $\backslash mathcal\{F\}^\{(s)\}\_\{\backslash Delta\_s\}(\backslash \{\backslash Delta\_i\backslash \}|z)$ coincides with the original block where three positions are sent to $(0,\backslash infty,\; 1),$

:$\backslash mathcal\{F\}^\{(s)\}\_\{\backslash Delta\_s\}(\backslash \{\backslash Delta\_i\backslash \}|z)=\; \backslash mathcal\{F\}^\{(s)\}\_\{\backslash Delta\_s\}(\backslash \{\backslash Delta\_i\backslash \}|z,0,\backslash infty,1).$

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

:$\backslash mathcal\{F\}^\{(s)\}\_\{\backslash Delta\_s\}(\backslash \{\backslash Delta\_i\backslash \}|z)\; \backslash underset\{z\backslash to\; 0\}\{=\}\; z^\{\backslash Delta\_s-\backslash Delta\_1-\backslash Delta\_2\}\backslash left(1\; +\; \backslash sum\_\{n=1\}^\backslash infty\; c\_n\; z^n\backslash right),$

for some coefficients $c\_n.$ On the other hand, s-channel blocks have complicated singular behaviours at $z=1,\backslash infty$: it is t-channel blocks that are simple at $z=1$, and u-channel blocks that are simple at $z=\backslash infty.$

In a four-point block that obeys a BPZ differential equation, $z=0,1,\backslash infty$ are regular singular points of the differential equation, and $\backslash Delta\_s-\backslash Delta\_1-\backslash Delta\_2$ is a characteristic exponent of the differential equation. For a differential equation of order $n$, the $n$ characteristic exponents correspond to the $n$ values of $\backslash Delta\_s$ that are allowed by the fusion rules.

Permutations of the fields $V\_i(z\_i)$ leave the correlation function

:$\backslash left\backslash langle\backslash prod\_\{i=1\}^NV\_i(z\_i)\backslash right\backslash rangle$

invariant, and therefore relate different bases of conformal blocks with one another. In the case of four-point blocks, t-channel blocks are related to s-channel blocks by

:$\backslash mathcal\{F\}^\{(t)\}\_\{\backslash Delta\}(\backslash Delta\_1,\backslash Delta\_2,\backslash Delta\_3,\backslash Delta\_4|z\_1,z\_2,z\_3,z\_4)\; =\; \backslash mathcal\{F\}^\{(s)\}\_\{\backslash Delta\}(\backslash Delta\_1,\; \backslash Delta\_4,\; \backslash Delta\_3,\backslash Delta\_2|z\_1,z\_4,z\_3,z\_2),$

or equivalently

:$\backslash mathcal\{F\}^\{(t)\}\_\{\backslash Delta\}(\backslash Delta\_1,\backslash Delta\_2,\backslash Delta\_3,\backslash Delta\_4|z)\; =\; \backslash mathcal\{F\}^\{(s)\}\_\{\backslash Delta\}\; (\backslash Delta\_1,\; \backslash Delta\_4,\; \backslash Delta\_3,\; \backslash Delta\_2|\; 1-z).$

The change of bases from s-channel to t-channel four-point blocks is characterized by the fusing matrix (or fusion kernel) $F$, such that

:$\backslash mathcal\{F\}^\{(s)\}\_\{\backslash Delta\_s\}(\backslash \{\backslash Delta\_i\backslash \}|\backslash \{z\_i\backslash \})$

= \int_{i\mathbb{R}} dP_t\ F_{\Delta_s,\Delta_t}\begin{bmatrix} \Delta_2 & \Delta_3 \\ \Delta_1 & \Delta_4 \end{bmatrix} \mathcal{F}^{(t)}_{\Delta_t}(\{\Delta_i\}|\{z_i\}).

The fusing matrix is a function of the central charge and conformal dimensions, but it does not depend on the positions $z\_i.$ The momentum $P\_t$ is defined in terms of the dimension $\backslash Delta\_t$ by

:$\backslash Delta\; =\; \backslash frac\{c-1\}\{24\}-P^2.$

The values $P\backslash in\; i\backslash mathbb\{R\}$ correspond to the spectrum of Liouville theory.

We also need to introduce two parameters $Q,b$ related to the central charge $c$,

:$c=\; 1+6Q^2,\; \backslash qquad\; Q=b+b^\{-1\}.$

Assuming $c\backslash notin\; (-\backslash infty,\; 1)$ and $P\_i\backslash in\; i\backslash R$, the explicit expression of the fusing matrix is{{Cite arXiv |eprint = 1202.4698|last1 = Teschner|first1 = J.|last2 = Vartanov|first2 = G. S.|title = 6j symbols for the modular double, quantum hyperbolic geometry, and supersymmetric gauge theories|year = 2012|class = hep-th}}

:$\backslash begin\{align\}$

F_{\Delta_s,\Delta_t} &\begin{bmatrix} \Delta_2 & \Delta_3 \\ \Delta_1 & \Delta_4 \end{bmatrix} = \\

&= \left(\prod_{\pm}\frac{\Gamma_b(Q\pm 2P_s)}{\Gamma_b(\pm 2P_t)}\right) \frac{\Xi_+(P_1,P_4,P_t)\Xi_+(P_2,P_3,P_t)}{\Xi_-(P_1,P_2,P_s)\Xi_-(P_3,P_4,P_s)} \times

\\

&\quad \times \int_{\frac{Q}{4}+i\R}du \ S_b \left (u-P_{12s} \right ) S_b \left (u-P_{s34} \right ) S_b \left (u-P_{23t} \right )S_b \left (u-P_{t14} \right )

\\

& \qquad \times

S_b \left ( \tfrac{Q}{2}-u+P_{1234} \right ) S_b \left (\tfrac{Q}{2}-u+P_{st13} \right ) S_b \left(\tfrac{Q}{2}-u+P_{st24} \right)S_b \left(\tfrac{Q}{2}-u \right )

\end{align}

where $\backslash Gamma\_b$ is a double gamma function,

:$\backslash begin\{align\}$

S_b(x) &= \frac{\Gamma_b(x)}{\Gamma_b(Q-x)} \\[6pt]

\Xi_\epsilon(P_1,P_2,P_3) &=\prod_{\underset{\epsilon_1\epsilon_2\epsilon_3=\epsilon}{\epsilon_1,\epsilon_2,\epsilon_3=\pm }} \Gamma_b\left(\tfrac{Q}{2}+\sum_i\epsilon_iP_i\right) \\[6pt]

P_{ijk} &= P_i+P_j+P_k

\end{align}

Although its expression is simpler in terms of momentums $P\_i$ than in terms of conformal dimensions $\backslash Delta\_i$, the fusing matrix is really a function of $\backslash Delta\_i$, i.e. a function of $P\_i$ that is invariant under $P\_i\backslash to\; -P\_i$. In the expression for the fusing matrix, the integral is a hyperbolic Barnes integral. Up to normalization, the fusing matrix coincides with Ruijsenaars' hypergeometric function, with the arguments $P\_s,P\_t$ and parameters $b,b^\{-1\},P\_1,P\_2,P\_3,P\_4$. The fusing matrix has several different integral representations, and obeys many nontrivial identities.

In $N$-point blocks on the sphere, the change of bases between two sets of blocks that are defined from different sequences of OPEs can always be written in terms of the fusing matrix, and a simple matrix that describes the permutation of the first two fields in an s-channel block,{{Cite journal |doi = 10.1007/BF01238857|title = Classical and quantum conformal field theory|year = 1989|last1 = Moore|first1 = Gregory|last2 = Seiberg|first2 = Nathan|journal = Communications in Mathematical Physics|volume = 123|issue = 2|pages = 177–254|bibcode = 1989CMaPh.123..177M|s2cid = 122836843|url = http://projecteuclid.org/euclid.cmp/1104178762}}

:$\backslash mathcal\{F\}^\{(s)\}\_\{\backslash Delta\_s\}(\backslash Delta\_1,\backslash Delta\_2,\backslash Delta\_3,\backslash Delta\_4|z\_1,z\_2,z\_3,z\_4)\; =\; e^\{i\backslash pi(\backslash Delta\_s-\backslash Delta\_1-\backslash Delta\_2)\}\; \backslash mathcal\{F\}^\{(s)\}\_\{\backslash Delta\_s\}(\backslash Delta\_2,\backslash Delta\_1,\backslash Delta\_3,\backslash Delta\_4|z\_2,z\_1,z\_3,z\_4).$

The definition from OPEs leads to an expression for an s-channel four-point conformal block as a sum over states in the s-channel representation, of the type

{{cite journal |arxiv=0907.3946|last1=Marshakov|first1=A.|last2=Mironov|first2=A.|last3=Morozov|first3=A.|title=On Combinatorial Expansions of Conformal Blocks|journal=Theoretical and Mathematical Physics|year=2009|volume=164|pages=831–852|doi=10.1007/s11232-010-0067-6|s2cid=16017224}}

:$\backslash mathcal\{F\}^\{\backslash text\{(s)\}\}\_\{\backslash Delta\_s\}(\backslash \{\backslash Delta\_i\backslash \}|z)=\; z^\{\backslash Delta\_s-\backslash Delta\_1-\backslash Delta\_2\}\backslash sum\_\{L,L\text{'}\}\; z^$

The sums are over creation modes $L,L\text{'}$ of the Virasoro algebra, i.e. combinations of the type $L=\backslash prod\_i\; L\_\{-n\_i\}$ of Virasoro generators with $1\backslash leq\; n\_1\backslash leq\; n\_2\backslash leq\; \backslash cdots$, whose level is $|L|=\backslash sum\; n\_i$. Such generators correspond to basis states in the Verma module with the conformal dimension $\backslash Delta\_s$.

The coefficient $f\_\{12s\}^L$ is a function of $\backslash Delta\_1,\backslash Delta\_2,\backslash Delta\_s,L$, which is known explicitly. The matrix element $Q\_\{L,L\text{'}\}^s$ is a function of $c,\backslash Delta\_s,L,L\text{'}$ which vanishes if $|L|\backslash neq\; |L\text{'}|$, and diverges for $|L|=N$ if there is a null vector at level $N$.

Up to $|L|=1$, this reads

:$$

\mathcal{F}^{\text{(s)}}_{\Delta_s}(\{\Delta_i\}|z)= z^{\Delta_s-\Delta_1-\Delta_2}

\Bigg\{ 1

+ \frac{(\Delta_s+\Delta_1-\Delta_2)(\Delta_s+\Delta_4-\Delta_3)}{2\Delta_s} z + O(z^2)\Bigg\}\ .

(In particular, $Q\_\{L\_\{-1\},L\_\{-1\}\}^s=\backslash frac\{1\}\{2\backslash Delta\_s\}$ does not depend on the central charge $c$.)

In Alexei Zamolodchikov's recursive representation of four-point blocks on the sphere, the cross-ratio $z$ appears via the nome

:$$

q = \exp -\pi \frac{F(\frac12,\frac12,1,1-z)}{F(\frac12,\frac12,1,z)} \underset{z\to 0}{=} \frac{z}{16}+\frac{z^2}{32}+O(z^3) \quad \iff \quad z = \frac{\theta_2(q)^4}{\theta_3(q)^4} \underset{q\to 0}{=} 16 q - 128 q^2 + O(q^3)

where $F$ is the hypergeometric function, and we used the Jacobi theta functions

:$$

\theta_2(q) = 2q^\frac14\sum_{n=0}^\infty q^{n(n+1)} \quad , \quad \theta_3(q) = \sum_{n\in{\mathbb{Z}}} q^{n^2}

The representation is of the type

:$$

\mathcal{F}^{(s)}_{\Delta}(\{\Delta_i\}|z) = (16q)^{\Delta -\frac14 Q^2} z^{\frac14 Q^2-\Delta_1-\Delta_2} (1-z)^{\frac14 Q^2-\Delta_1-\Delta_4} \theta_3(q)^{3Q^2-4(\Delta_1+\Delta_2+\Delta_3+\Delta_4)} H_{\Delta}(\{\Delta_i\}|q)\ .

The function $H\_\{\backslash Delta\}(\backslash \{\backslash Delta\_i\backslash \}|q)$ is a power series in $q$, which is recursively defined by

:$$

H_{\Delta}(\{\Delta_i\}|q) = 1 + \sum_{m,n=1}^\infty \frac{(16q)^{mn}}{\Delta-\Delta_{(m,n)}} R_{m,n} H_{\Delta_{(m,-n)}}(\{\Delta_i\}|q)\ .

In this formula, the positions $\backslash Delta\_\{(m,n)\}$ of the poles are the dimensions of degenerate representations, which correspond to the momentums

:$$

P_{(m,n)} = \frac12 \left(mb+nb^{-1}\right)\ .

The residues $R\_\{m,n\}$ are given by

:$$

R_{m,n} = \frac{2P_{( 0,0)} P_{( m,n)}}{\prod_{r=1-m}^m \prod_{s=1-n}^n 2P_{(r,s)}}

\prod_{r\overset{2}{=}1-m}^{m-1} \prod_{s\overset{2}{=}1-n}^{n-1} \prod_\pm (P_2\pm P_1 + P_{( r,s)}) (P_3\pm P_4 +P_{( r,s)})\ ,

where the superscript in $\backslash overset\{2\}\{=\}$ indicates a product that runs by increments of $2$. The recursion relation for $H\_\{\backslash Delta\}(\backslash \{\backslash Delta\_i\backslash \}|q)$ can be solved, giving rise to an explicit (but impractical) formula.{{cite arXiv |eprint=1406.4290|last1=Ribault|first1=Sylvain|title=Conformal field theory on the plane|class=hep-th|year=2014}}{{cite journal |arxiv=1502.07742|last1=Perlmutter|first1=Eric|title=Virasoro conformal blocks in closed form|journal=Journal of High Energy Physics|year=2015|volume=2015|issue=8|pages=88|doi=10.1007/JHEP08(2015)088|bibcode=2015JHEP...08..088P|s2cid=54075672}}

While the coefficients of the power series $H\_\{\backslash Delta\}(\backslash \{\backslash Delta\_i\backslash \}|q)$ need not be positive in unitary theories, the coefficients of $\backslash prod\_\{k=1\}^\backslash infty\; (1-q^\{2k\})^\{-\backslash frac12\}\; H\_\{\backslash Delta\}(\backslash \{\backslash Delta\_i\backslash \}|q)$ 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.

The recursive representation can be seen as an expansion around $\backslash Delta=\backslash infty$. It is sometimes called the $\backslash Delta$-recursion, in order to distinguish it from the $c$-recursion: another recursive representation, also due to Alexei Zamolodchikov, which expands around $c=\backslash infty$, and generates a series in powers of $z$.

The $c$-recursion can be generalized to $N$-point Virasoro conformal blocks on arbitrary Riemann surfaces. The $\backslash Delta$-recursion can be generalized to one-point blocks on the torus. In other cases, there are no known generalizations of the $\backslash Delta$-recursion, but there exist modified $\backslash Delta$-recursions that generate series in powers of $z$.

The Alday–Gaiotto–Tachikawa relation between two-dimensional conformal field theory and supersymmetric gauge theory, more specifically, between the conformal blocks of Liouville theory and Nekrasov partition functions{{Cite journal |arxiv = hep-th/0206161|doi = 10.4310/ATMP.2003.v7.n5.a4|title = Seiberg-Witten Prepotential from Instanton Counting|year = 2004|last1 = Nekrasov|first1 = Nikita|journal = Advances in Theoretical and Mathematical Physics|volume = 7|number = 5|pages = 831–864|s2cid = 2285041}} of supersymmetric gauge theories in four dimensions, leads to combinatorial expressions for conformal blocks as sums over Young diagrams. Each diagram can be interpreted as a state in a representation of the Virasoro algebra, times an abelian affine Lie algebra.{{Cite journal |arxiv = 1012.1312|doi = 10.1007/s11005-011-0503-z|title = On Combinatorial Expansion of the Conformal Blocks Arising from AGT Conjecture|year = 2011|last1 = Alba|first1 = Vasyl A.|last2 = Fateev|first2 = Vladimir A.|last3 = Litvinov|first3 = Alexey V.|last4 = Tarnopolskiy|first4 = Grigory M.|journal = Letters in Mathematical Physics|volume = 98|issue = 1|pages = 33–64|bibcode = 2011LMaPh..98...33A|s2cid = 119143670}}

A zero-point block does not depend on field positions, but it depends on the moduli of the underlying Riemann surface. In the case of the torus

:$\backslash frac\{\backslash Complex\}\{\backslash Z+\backslash tau\backslash Z\},$

that dependence is better written through $q=e^\{2\backslash pi\; i\backslash tau\}$ and the zero-point block associated to a representation $\backslash mathcal\{R\}$ of the Virasoro algebra is

:$\backslash chi\_\backslash mathcal\{R\}(\backslash tau)\; =\; \backslash operatorname\{Tr\}\_\backslash mathcal\{R\}\; q^\{L\_0-\backslash frac\{c\}\{24\}\},$

where $L\_0$ is a generator of the Virasoro algebra. This coincides with the character of $\backslash mathcal\{R\}.$ The characters of some highest-weight representations are:

• Verma module with conformal dimension $\backslash Delta=\backslash tfrac\{c-1\}\{24\}-P^2$:

::$\backslash chi\_P(\backslash tau)\; =\; \backslash frac\{q^\{-P^2\}\}\{\backslash eta(\backslash tau)\},$

:where $\backslash eta(\backslash tau)$ is the Dedekind eta function.

• Degenerate representation with the momentum $P\_\{(r,s)\}$:

::$\backslash chi\_\{(r,s)\}(\backslash tau)\; =\; \backslash chi\_\{P\_\{(r,s)\}\}(\backslash tau)\; -\; \backslash chi\_\{P\_\{(r,-s)\}\}(\backslash tau).$

• Fully degenerate representation at rational $b^2\; =\; -\backslash tfrac\{p\}\{q\}$:

::$\backslash chi\_\{(r,s)\}(\backslash tau)\; =\; \backslash sum\_\{k\backslash in\backslash Z\}\; \backslash left(\backslash chi\_\{P\_\{(r,s)\}+ik\backslash sqrt\{pq\}\}(\backslash tau)\; -\; \backslash chi\_\{P\_\{(r,-s)\}+ik\backslash sqrt\{pq\}\}(\backslash tau)\; \backslash right).$

The characters transform linearly under the modular transformations:

:

In particular their transformation under $\backslash tau\; \backslash to\; -\backslash tfrac\{1\}\{\backslash tau\}$ is described by the modular S-matrix. 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.A. Cappelli, J-B. Zuber, "A-D-E Classification of Conformal Field Theories", [http://www.scholarpedia.org/article/A-D-E_Classification_of_Conformal_Field_Theories Scholarpedia] These are related to the modular group representations of modular tensor categories.

An arbitrary one-point block on the torus can be written in terms of a four-point block on the sphere at a different central charge. 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 $P\_\{(0,\backslash frac12)\}\; =\; \backslash tfrac\{1\}\{4b\}$:

:$$

H^\text{torus}_{P'}(P_1|q^2) = H_{P}\left(\left.\tfrac{1}{4b},P_2,\tfrac{1}{4b},\tfrac{1}{4b}\right|q\right) \quad \text{with}\quad \left\{\begin{array}{l} b=\frac{b'}{\sqrt{2}}\\ P_2=\frac{P_1}{\sqrt{2}}\\ P=\sqrt{2}P' \end{array}\right.

where

• $H\_\{P\_s\}\backslash left(\backslash left.P\_1,P\_2,P\_3,P\_4\backslash right|q\backslash right)$ is the non-trivial factor of the sphere four-point block in Zamolodchikov's recursive representation, written in terms of momentums $P\_i$ instead of dimensions $\backslash Delta\_i$.
• $H^\backslash text\{torus\}\_\{P\}(P\_1|q)$ is the non-trivial factor of the torus one-point block $\backslash mathcal\{F\}^\backslash text\{torus\}\_\{\backslash Delta\}(\backslash Delta\_1|q)\; =\; q^\{\backslash Delta-\backslash frac\{c-1\}\{24\}\}\backslash eta(q)^\{-1\}H^\backslash text\{torus\}\_\{\backslash Delta\}(\backslash Delta\_1|q)$, where $\backslash eta(q)$ is the Dedekind eta function, the modular parameter $\backslash tau$ of the torus is such that $q=e^\{2\backslash pi\; i\backslash tau\}$, and the field on the torus has the dimension $\backslash Delta\_1$.

The recursive representation of one-point blocks on the torus is

:$H^\backslash text\{torus\}\_\{\backslash Delta\}(\backslash Delta\_1|q)\; =\; 1\; +\; \backslash sum\_\{m,n=1\}^\backslash infty\; \backslash frac\{q^\{mn\}\}\{\backslash Delta-\backslash Delta\_\{(m,n)\}\}\; R^\backslash text\{torus\}\_\{m,n\}\; H^\backslash text\{torus\}\_\{\backslash Delta\_\{(m,-n)\}\}(\backslash Delta\_1|q)\backslash \; ,$

where the residues are

:$$

R^\text{torus}_{m,n} = \frac{2P_{( 0,0)} P_{( m,n)}}{\prod_{r=1-m}^m \prod_{s=1-n}^n 2P_{(r,s)}}

\prod_{r\overset{2}{=}1-2m}^{2m-1} \prod_{s\overset{2}{=}1-2n}^{2n-1} \left(P_1+P_{(r,s)}\right)\ .

Under modular transformations, one-point blocks on the torus behave as

:$$

\mathcal{F}^\text{torus}_{P}\left(P_1|-\tfrac{1}{\tau}\right) = \int_{i\mathbb{R}} dP'\ S_{P,P'}(P_1)\mathcal{F}^\text{torus}_{P'}\left(P_1|\tau\right)\ ,

where the modular kernel is

:$$

S_{P,P'}(P_1) = \frac{\sqrt{2}}{S_b(\frac{Q}{2}+P_1)}

\prod_\pm \frac{\Gamma_b(Q\pm 2P)}{\Gamma_b(\pm 2P')} \frac{\Gamma_b(\frac{Q}{2}-P_1\pm 2P')}{\Gamma_b(\frac{Q}{2}-P_1\pm 2P)} \int_{i\mathbb{R}} \frac{du}{i}\ e^{4\pi iPu} \prod_{\pm,\pm} S_b\left(\tfrac{Q}{4}+\tfrac{P_1}{2} \pm u\pm P'\right)\ .

For a four-point function on the sphere

:$\backslash left\backslash langle\; V\_\{\backslash langle\; 2,1\; \backslash rangle\}(x)\backslash prod\_\{i=1\}^3\; V\_\{\backslash Delta\_i\}(z\_i)\backslash right\backslash rangle$

where one field has a null vector at level two, the second-order BPZ equation reduces to the hypergeometric equation. A basis of solutions is made of the two s-channel conformal blocks that are allowed by the fusion rules, and these blocks can be written in terms of the hypergeometric function,

:$$

\begin{align}

\mathcal{F}^{(s)}_{P_1+\epsilon\frac{b}{2}}(z)

&= z^{\frac12+\frac{b^2}{2}+b\epsilon P_1} (1-z)^{\frac12+\frac{b^2}{2}+bP_3}

\\ &\times

F\left(\tfrac12 + b(\epsilon P_1+P_2+P_3),\tfrac12 + b(\epsilon P_1-P_2+P_3),1 + 2b\epsilon P_1,z\right),

\end{align}

with $\backslash epsilon\backslash in\backslash \{+,-\backslash \}.$ Another basis is made of the two t-channel conformal blocks,

:$$

\begin{align}

\mathcal{F}^{(t)}_{P_3+\epsilon\frac{b}{2}}(z)

&= z^{\frac12+\frac{b^2}{2}+b P_1} (1-z)^{\frac12+\frac{b^2}{2}+b\epsilon P_3}

\\ &\times

F\left(\tfrac12 + b( P_1+P_2+\epsilon P_3),\tfrac12 + b(P_1-P_2+\epsilon P_3), 1 + 2b\epsilon P_3,1-z\right).

\end{align}

The fusing matrix is the matrix of size two such that

:$\backslash mathcal\{F\}^\{(s)\}\_\{P\_1+\backslash epsilon\_1\backslash frac\{b\}\{2\}\}(x)\; =\; \backslash sum\_\{\backslash epsilon\_3=\backslash pm\}\; F\_\{\backslash epsilon\_1,\backslash epsilon\_3\}\; \backslash mathcal\{F\}^\{(t)\}\_\{P\_3+\backslash epsilon\_3\backslash frac\{b\}\{2\}\}(x),$

whose explicit expression is

:$F\_\{\backslash epsilon\_1,\backslash epsilon\_3\}\; =\; \backslash frac\{\backslash Gamma(1-2b\backslash epsilon\_1P\_1)\backslash Gamma(2b\backslash epsilon\_3P\_3)\}\{\backslash prod\_\backslash pm\; \backslash Gamma(\backslash frac12+b(-\backslash epsilon\_1P\_1\backslash pm\; P\_2+\backslash epsilon\_3P\_3))\}.$

Hypergeometric conformal blocks play an important role in the analytic bootstrap approach to two-dimensional CFT.{{cite journal |arxiv=hep-th/9507109|last1=Teschner|first1=Joerg.|title=On the Liouville three-point function|journal=Physics Letters B|year=1995|volume=363|issue=1–2|pages=65–70|doi=10.1016/0370-2693(95)01200-A|bibcode=1995PhLB..363...65T|s2cid=15910029}}{{cite journal |arxiv=1711.08916|last1=Migliaccio|first1=Santiago|last2=Ribault|first2=Sylvain|title=The analytic bootstrap equations of non-diagonal two-dimensional CFT|journal=Journal of High Energy Physics|year=2018|volume=2018|issue=5|pages=169|doi=10.1007/JHEP05(2018)169|bibcode=2018JHEP...05..169M|s2cid=119385003}}

If $c=1,$ then certain linear combinations of s-channel conformal blocks are solutions of the Painlevé VI nonlinear differential equation.{{Cite journal |arxiv = 1207.0787|doi = 10.1007/JHEP10(2012)038|title = Conformal field theory of Painlevé VI|year = 2012|last1 = Gamayun|first1 = O.|last2 = Iorgov|first2 = N.|last3 = Lisovyy|first3 = O.|journal = Journal of High Energy Physics|volume = 2012|issue = 10|pages = 038|bibcode = 2012JHEP...10..038G|s2cid = 119610935}} The relevant linear combinations involve sums over sets of momentums of the type $P\_s+i\backslash Z.$ This allows conformal blocks to be deduced from solutions of the Painlevé VI equation and vice versa. This also leads to a relatively simple formula for the fusing matrix at $c=1.${{Cite journal |arxiv = 1308.4092|doi = 10.1007/JHEP12(2013)029|title = Painlevé VI connection problem and monodromy of c = 1 conformal blocks|year = 2013|last1 = Iorgov|first1 = N.|last2 = Lisovyy|first2 = O.|last3 = Tykhyy|first3 = Yu.|journal = Journal of High Energy Physics|volume = 2013|issue = 12|pages = 029|bibcode = 2013JHEP...12..029I|s2cid = 56401903}} Curiously, the $c=\backslash infty$ limit of conformal blocks is also related to the Painlevé VI equation.{{Cite journal |arxiv = 1309.4700|doi = 10.1007/JHEP07(2014)144|title = Classical conformal blocks and Painlevé VI|year = 2014|last1 = Litvinov|first1 = Alexey|last2 = Lukyanov|first2 = Sergei|last3 = Nekrasov|first3 = Nikita|last4 = Zamolodchikov|first4 = Alexander|journal = Journal of High Energy Physics|volume = 2014|issue = 7|pages = 144|bibcode = 2014JHEP...07..144L|s2cid = 119710593}} The relation between the $c=\backslash infty$ and the $c=1$ limits, mysterious on the conformal field theory side, is explained naturally in the context of four dimensional gauge theories, using blowup equations,{{Cite journal |arxiv = 2007.03646|title = Blowups in BPS/CFT correspondence, and Painlevé VI|year = 2020|last1 = Nekrasov|first1 = Nikita| journal = Annales Henri Poincaré|doi = 10.1007/s00023-023-01301-5}}{{Cite journal |arxiv = 2007.03660

|doi = 10.1007/JHEP12(2020)006|title = Riemann-Hilbert correspondence and blown up surface defects

|year = 2020|last1 = Jeong|first1 = Saebyeok|last2 = Nekrasov|first2 = Nikita|journal = Journal of High Energy Physics|volume = 2020|issue = 12|pages = 006|bibcode = 2020JHEP...12..006J|s2cid = 220381427}} and can be generalized to more general pairs $c,\; c\text{'}$of central charges.

The Virasoro conformal blocks that are described in this article are associated to a certain type of representations of the Virasoro algebra: highest-weight representations, in other words Verma modules and their cosets. Correlation functions that involve other types of representations give rise to other types of conformal blocks. For example:

• Logarithmic conformal field theory involves representations where the Virasoro generator $L\_0$ is not diagonalizable, which give rise to blocks that depend logarithmically on field positions.
• Representations can be built from states on which some annihilation modes of the Virasoro algebra act diagonally, rather than vanishing. The corresponding conformal blocks have been called irregular conformal blocks.{{Cite journal |arxiv = 1203.1052|doi = 10.1007/JHEP12(2012)050|title = Irregular singularities in Liouville theory and Argyres-Douglas type gauge theories|year = 2012|last1 = Gaiotto|first1 = D.|last2 = Teschner|first2 = J.|journal = Journal of High Energy Physics|volume = 2012|issue = 12|pages = 50|bibcode = 2012JHEP...12..050G|s2cid = 118380071}}

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.

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Category:Conformal field theory