A pedestrian explanation of conformal blocks

Question

I would be very happy if someone could take a stab at conveying what conformal blocks are and how they are used in conformal field theory (CFT). I'm finally getting the glimmerings of understanding from reading Moore and Read's wonderful paper. But I think/hope this site has people who can explain the notions involved in a simpler and more intuitive manner.

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Edit: Here is a simple example, taken from pg 8 of the reference cited above ...

In a 2D CFT we have correlation functions of fields $ \phi_i(z,\bar z) $, (where $ z = x+\imath y$) at various points on the complex plane. The n-point correlation function can be expanded as:

$$ \left \langle \prod_{a=1}^n \phi_{i_a}(z_a,\bar z_a) \right \rangle = \sum_p | F_{p\; i_{1} \dots i_n}(z_{1} \dots z_n)|^2 $$

Here $p$ labels members of a basis of functions $ F_{p\; i_1 \dots i_n}(z_{1} \dots z_n) $ which span a vector space for each n-tuple $(z_{1} \dots z_n)$

These functions $F_p$ are known as conformal blocks, and appear to give a "fourier" decomposition of the correlation functions.

This is what I've gathered so far. If someone could elaborate with more examples that would be wonderful !

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Edit: It is proving very difficult to decide which answer is the "correct" one. I will give it a few more days. Perhaps the situation will change !

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The "correct" answer goes to (drum-roll): David Zavlasky. Well they are all great answers. I chose David's for the extra five points because his is the simplest, IMHO. He also mentions the "cross-ratio" which is a building block of CFT.

Answer 1

Now that we have a physicist's perspective, I don't feel too bad outlining conformal blocks from a mathematician's point of view. Presumably there is a dictionary connecting the two worlds, but I don't understand physics well enough to say coherent sentences about it. I apologize in advance for any confusion - this is not a very pedestrian topic.

I'll approach conformal blocks from the standpoint of conformal vertex algebras, which typically appear in mathematics as algebraic structures that you can use to prove theorems in representation theory. Vertex algebras are vector spaces $V$ equipped with a "multiplication with singularities" $V \otimes V \to V((z))$ that encodes a best effort at multiplying quantum fields (which are sometimes called "operator-valued distributions"). Left multiplication by an element $u$ yields a formal power series $\sum_{n \in \mathbb{Z}} u_n z^{-n-1}$ whose coefficients are operators. To make a vertex algebra conformal is to choose a distinguished vector $\omega$ whose corresponding operators generate an action of the Virasoro algebra, which is a central extension of the complexified Lie algebra of polynomial vector fields on the circle. You don't lose much conceptually by thinking of Virasoro as the tangent space of the group $Diff(S^1)$ at the identity, but there is a "nonzero central charge" anomaly in play that can make the central extension necessary. The circle shows up here because it is the boundary of a puncture where we will insert a field.

My understanding of the physical interpretation is the following incomplete and possibly incorrect picture: Inside a 2D conformal field theory, there is an algebra of (say, left-moving) chiral symmetries, and this is precisely the information captured by the conformal vertex algebra. The space of states in the theory decomposes into a set of "sectors" which are modules of the vertex algebra. If we choose a Riemann surface (which is a sphere in most textbooks), and attach states from various sectors to a set of distinct points, we should get a set of amplitudes, which are values of chiral correlation functions attached to these input data. I have heard that there is some way to pass from the chiral stuff to the conformal field theory proper, where the ambiguity in the correlators disappears and one gets honest correlation functions, but I haven't seen it in the math literature. In any case, conformal blocks live inside this machine - given sectors attached to points on a Riemann surface, a conformal block is a gadget that eats choices of states in those sectors, and outputs values of correlation functions in a manner consistent with the chiral symmetries.

Here is a sketch of the mathematical construction, due to Edward Frenkel (and described in more detail in his book Vertex Algebras and Algebraic Curves with David Ben-Zvi): There is a "positive half" of the Virasoro algebra, spanned by generators $-z^n\frac{d}{dz}$ for $n \geq 0$, and it generates the Lie algebra of derivations on the infinitesimal complex disk, and also acts on the conformal vertex algebra $V$. We can use this action to construct a vector bundle $\mathscr{V}$ with flat connection on our Riemann surface of choice by the Gelfand-Kazhdan "formal geometry" method (which I won't describe). Given punctures $p_1, \dots, p_n$, one constructs, from the De Rham complex of $\mathscr{V}$, a Lie algebra $L$ that acts naturally on $n$-tuples of $V$-modules. Given $V$-modules $M_i$ attached at points $p_i$, a conformal block is an $L$-module map from $\bigotimes M_i$ to the trivial module.

It is in general quite difficult to do any explicit calculations with conformal blocks, because of the amount of geometry involved. If your Riemann surface has handles, you will have to deal with a choice of complex structure, and if it has a lot of punctures, you have to deal with a complicated configuration space of points. You typically see tree-level diagrams with 4 inputs, because:

1. That is where the bare minimum of geometry appears - since the automorphism group of the complex projective line is triply transitive, the configuration space of four points is a thrice-punctured line (by which I mean a sphere).
2. Depending on the level of detail you seek, it is often all that you need - the spaces of blocks can be assembled by gluing surfaces together out of pants and taking sums over sectors where the sewing happens. In the complex algebro-geometric picture, this sewing means sticking spheres together transversely at points to get a nodal curve. One then deforms to get a smooth complex curve, and does a parallel transport along the corresponding path in the moduli space of marked curves. The four-point configuration is a situation where you have exactly one sewing operation (and the other such situation is a punctured torus, which is important for getting characters).

In fact, when the conformal field theory is suitably well-behaved (read: rational), one gets dimensions of spaces of all conformal blocks from just the dimensions of three-point genus zero blocks, also known as structure constants of the fusion algebra. One sees this in the Verlinde formula, for example.

I think examples of conformal blocks have a certain necessary complexity, but here is an overview of a reasonably simple case that is motivated by the WZW model. Pick a simple Lie group, like $SU(2)$, and a level $\ell$ (which we can view as a positive integer). One constructs the vertex algebra and its modules as level $\ell$ integrable representations of the affine Kac-Moody Lie algebra $\hat{\mathfrak{sl}_2}$, which is a central extension of the loop algebra of the complexification of the Lie algebra $\mathfrak{su}_2$. If we choose a Riemann surface (such as a sphere), and decorate points with just the vacuum module, we get a space of conformal blocks that is the space of global sections of a certain line bundle $L_G^{\otimes \ell}$ on the moduli space of $SU(2)$ bundles on the surface. Here $L_G$ is the ample generator of the Picard group of the moduli space.

Answer 2

I did a bit of reading about this, and it turns out that conformal blocks are actually quite relevant to my research! So I figured it was worth the time to investigate in some more detail. I've never studied conformal field theory formally, but I hope I'm not writing anything outright wrong here. (I lost my first draft and had to reconstruct it, which is why it's taken so long)

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In conformal field theory, it's common to represent coordinates on a two-dimensional space by using complex numbers, so $\vec{r} = (x,y)$ becomes $\rho = x + iy$. In this notation, the theory is invariant under the action of a Möbius transformation (a.k.a. conformal transformation),

$$\rho \to \frac{a\rho + b}{c\rho + d}$$

in which $a$, $b$, $c$, and $d$ are complex constants that satisfy $ad - bc \neq 0$. The transformation has three complex degrees of freedom - in other words, if you specify three initial points and three final points on the complex plane, there is a unique Möbius transformation that maps those three initial points to the three final points.

So any function of four coordinates on the plane, for example a four-point correlation function of quantum fields,

$$G_4 = \langle \phi_1(\rho_1,\rho_1^*) \phi_2(\rho_2,\rho_2^*) \phi_3(\rho_3,\rho_3^*) \phi_4(\rho_4,\rho_4^*) \rangle$$

has only one real degree of freedom, after you factor out the gauge freedoms corresponding to the Möbius transformation. In other words, you can map any three of those coordinates on to three fixed reference points (for example $0$, $1$, and $\infty$), and you're left with a function of only one variable, something like

$$x = \frac{(\rho_4 - \rho_2)(\rho_3 - \rho_1)}{(\rho_4 - \rho_1)(\rho_3 - \rho_2)}$$

This opens the door to write $G_4$ as a simple function of this one ratio (at least, simpler than a function of four independent coordinates).

The particular part of CFT in which conformal blocks are applied (as far as I can tell; I'm starting to get a little out of my depth here) has to do with Virasoro algebras. Specifically, the way the individual fields $\phi_i$ transform under a conformal transformation is described by the group defined by the Virasoro algebra. The four-point function $G_4$ can be written as a sum of contributions from different representations of the group,

$$G_4(\rho_1,\rho_2,\rho_3,\rho_4) = \sum_l G_l f(D_l, d_i, C, x) f(D_l, d_i, C, x^*)$$

Here $l$ indexes the different representations; $C$ is a constant (the "central charge" of the Virasoro algebra); and $d_i$ and $D_l$ are anomalous dimensions of the external fields and the internal field respectively. The function $f$ is called a conformal block.

$f$ is useful because it can be calculated (in principle or in practice, I'm not sure which) using only information about a single representation of the Virasoro group. It can be expressed as a series in $x$ of a known form, the coefficients of which depend on the structure of the group.

Further Reading

1. Belavin A. Infinite conformal symmetry in two-dimensional quantum field theory. Nuclear Physics B. 1984;241(2):333-380. Available at: https://doi.org/10.1016/0550-3213(84)90052-X.
2. Zamolodchikov AB. Conformal symmetry in two dimensions: an explicit recurrence formula for the conformal partial wave amplitude. Communications in Mathematical Physics (1965-1997). 1984;96(3):419-422. Available at: https://doi.org/10.1007/BF01214585.
3. Zamolodchikov AB. Conformal symmetry in two-dimensional space: Recursion representation of conformal block. Theoretical and Mathematical Physics. 1987;73(1):1088-1093. Available at: https://doi.org/10.1007/BF01022967.

and of course DiFrancesco et al's book.

Answer 3

There are already nice answers from both a physical and mathematical perspective, explaining the basic idea - given the algebra of holomorphic operators (or equivalently the symmetry algebra) of a CFT, we can write down a collection of equations (the Ward identities) that the partition function of the theory must satisfy on any Riemann surface. The space of solutions of these equations is the space of conformal blocks. If we indeed have a full CFT then the partition function will be a particular conformal block. But given any conformal block we can still make sense of correlation functions on the Riemann surface, so can perform much of the field theory.

There is a fair amount of mathematical work on extending a chiral algebra to a full CFT, especially in the rational case (as Scott pointed out this is a central focus of the extended oeuvre of Fuchs, Schweigert, Runkel and collaborators). This involves finding modular invariant combination of modules for the chiral algebra, and can be reduced to finding special modules (Frobenius algebra objects in the braided tensor category of modules with some conditions). In the irrational case this theory is really in its infancy -- there's a notion of what branes should be, but there isn't a full structure theory..

I think a very illuminating point of view on conformal blocks derives from the idea that a chiral CFT is more like a three-dimensional [topological] quantum field theory than like an honest CFT (and this can be made precise in the rational case, see e.g. the book by Bakalov-Kirillov). From this point of view, we have a 3d QFT which makes sense on curved backgrounds (in fact topologically invariant), so we can assign a Hilbert space of states from quantizing the theory on a Riemann surface times R. This space of states is the space of conformal blocks. More generally we can consider line operators in this three-dimensional theory, which means we can insert operators at points of the Riemann surface times R. These operators correspond to modules for the chiral algebra, and the resulting Hilbert space is the space of conformal blocks with module insertions. If we have a non-rational CFT we don't get a full 3d topological QFT but we can still assign Hilbert spaces to Riemann surfaces or surfaces with module insertions, hence conformal blocks. (In a full-fledged theory these vector spaces would be forced to e finite dimensional by the well-definition of the trace of the Hamiltonian, which is zero in a topological theory).

Answer 4

A conformal field theory is a quantum field theory which is invariant under conformal transformations. Due to this invariance, correlation functions must obey linear equations called conformal Ward identities. Conformal blocks are not just solutions of the conformal Ward identities, but actually elements of a particular basis of solutions. Let us focus on two-dimensional CFT. In two dimensions conformal transformations are described by two Virasoro algebras, called left-moving (or holomorphic) and right-moving (or antiholomorphic).

The question was formulated in terms of $n$-point conformal blocks on the complex plane, but it is technically simpler to first consider zero-point conformal blocks on the torus. These are just characters of representations of the Virasoro algebra. Indeed, assume you want to compute a torus zero-point function (partition function), $$ Z = \mathrm{Tr}_S q^{E} \bar{q}^{\bar{E}} $$ where $q$ is the (exponentiated) modulus of the torus, $E$ and $\bar{E}$ are the energy operators respectively associated to the left- and right-moving Virasoro algebras, and $S$ is the space of states of your CFT. The space of states can be decomposed into representations of the Virasoro algebras, $$ S = \bigoplus_{R, \bar{R}} m_{R,\bar{R}} R\otimes \bar{R} $$ where $R, \bar{R}$ are representations of our two Virasoro algebras, and the integers $m_{R,\bar{R}}$ are their multiplicities. Then computing the trace over $S$ reduces to summing over states in each representation $R$ or $\bar{R}$, and such a sum is by definition a character $$ \chi_R(q) = \mathrm{Tr}_R q^{E} = \sum_L q^{E(L)} $$ where $L$ labels an orthonormal basis of $R$, made of eigenvectors of $E$. So we obtain $$ Z = \sum_{R,\bar{R}} m_{R,\bar{R}} \chi_R(q) \chi_{\bar{R}}(\bar{q}) $$ This is the conformal block decomposition of $Z$: the conformal blocks $\chi_R(q)$, $\chi_{\bar{R}}(\bar{q})$ are locally holomorphic functions of $q$ and $\bar{q}$, they are completely determined by conformal symmetry, and they are parametrized by representations of the symmetry algebra. On the other hand, the multiplicities $m_{R,\bar{R}}$ are left undetermined by the symmetry.

The same ideas apply to the sphere four-point function. A four-point function can be decomposed into products of three-point functions by inserting an identity operator, and we schematically obtain $$ \left< \prod_{i=1}^4 V_i(z_i,\bar{z}_i) \right> = \sum_{R,\bar{R}} m_{R,\bar{R}} \sum_{L,\bar{L}} \left< V_1V_2 \middle| (R,L),(\bar{R},\bar{L}) \right> \left< (R,L),(\bar{R},\bar{L}) \middle| V_3V_4\right> $$ Now it turns out that a three-point function $\left< V_1 V_2 \middle| (R,L),(\bar{R},\bar{L}) \right>$, is determined by conformal symmetry up to a factor $C_{1,2,(R,\bar{R})}$, which depends neither on $z_i,\bar{z}_i$ nor on $L,\bar{L}$, and we have $$ \left< \prod_{i=1}^4 V_i(z_i,\bar{z}_i) \right> = \sum_{R,\bar{R}} m_{R,\bar{R}} C_{1,2,(R,\bar{R})} C_{(R, \bar{R}), 3,4} F_R(z_i) F_{\bar{R}}(\bar{z}_i) $$ The four-point conformal block $F_R(z_i)= \sum_L \cdots$ is completely determined by conformal symmetry. It depends on all the left-moving parameters: the positions $z_i$, the $s$-channel representation $R$, and the left-moving representations which correspond to the fields $V_i$. Up to trivial factors, a four-point conformal block is actually a function of the cross-ratio $z=\frac{(z_1-z_2)(z_3-z_4)}{(z_1-z_3)(z_2-z_4)}$: this is a simple consequence of Ward identities, which holds whether you have local or global conformal symmetry. A conformal block generally does not obey any differential equation in $z$. It obeys a Belavin-Polyakov-Zamolodchikov equation only if at least one of the fields $V_i$ is a so-called degenerate field.

Conformal blocks are useful because they are universal quantities, in the sense that they are determined by conformal symmetry. In order to determine correlation functions in a specific model, all that is left to do is to compute model-dependent quantities such as the multiplicities $m_{R,\bar{R}}$ and the factors $C_{1,2,(R,\bar{R})}$. These model-dependent quantities are simpler than the correlation functions: in particular, they typically depend on fewer parameters.

For more details along these lines, see my review article.

Answer 5

Conformal field theory is the theory of scale invariance (or large-order behavior) in two dimensions. Scaling means dependence on angles only. In 2d, group of angle-preserving (conformal) transformations is infinite-dimensional, and in fact there are only a finite number of degrees of freedom in a 2d metric after conformal transformations and diffeomorphisms. (The degrees of freedom are the moduli space of Riemann surfaces.)

Fields in a theory with conformal symmetry must give representations of this symmetry algebra, and such representations are labeled by a quantum number called conformal dimension or weight. The transformations themselves are holomorphic changes of coordinates ($z \rightarrow f(z)$ and they are generated by the Lie algebra of holomorphic vector fields $L_n := -z^{n+1}\partial_z$ and their complex conjugates. You can calculate this algebra: $[L_n,L_m] = (n-m)L_{m+n}$ which is called the Virasoro algebra. (There are two of these, one with z and one with z-bar.) Quantum mechanically, this algebra can be corrected by the conformal anomaly parametrized by the central charge ("central" because the extra term commutes with all others).

Now just as in a rotation-invariant theory, if you want to know how a solution looks after a rotation you only need to know which representation the state lies in, in a conformal theory if you want to change coordinates infinitesimally you only need to know the conformal weights of the fields. But such transformations are infinitesimal coordinate changes, so this gives a differential equation that the correlator must obey. Everything in the theory can be written in terms of solutions to these differential equations -- these are called conformal blocks. (There are solutions in $\bar{z}$, too.)

This method is detailed in the classic work of Belavin, Polyakov and Zamolodchikov (NPB 241 (1988) p. 333) (another pioneer is Knizhnik).

p.s. String theory is all about 2d field theories and their dependence on the moduli of Riemann surfaces. The condition that the conformal theory be anomaly-free is the most common way of deriving dimension formulas in string theory.