A pedestrian explanation of conformal blocks
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:
- 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).
- 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.
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)
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
- Belavin A. Infinite conformal symmetry in two-dimensional quantum field theory. Nuclear Physics B. 1984;241(2):333-380. Available at: http://dx.doi.org/10.1016/0550-3213(84)90052-X.
- 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: http://projecteuclid.org/euclid.cmp/1103941860.
- Zamolodchikov AB. Conformal symmetry in two-dimensional space: Recursion representation of conformal block. Theoretical and Mathematical Physics. 1987;73(1):1088-1093. Available at: http://www.springerlink.com/content/khq7730604681676/.
and of course DiFrancesco et al's book.
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).