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Scott Carnahan
Scott Carnahan
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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.

  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. ItDepending 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 ana 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'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 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. 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 correlation functions in a manner consistent with the chiral symmetries.

  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. 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 at points to get a nodal curve. One then deforms to get a smooth complex curve. 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, 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 an 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'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.

  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.

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Scott Carnahan
Scott Carnahan
  • 3.2k
  • 1
  • 26
  • 23

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 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. 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 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. 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 at points to get a nodal curve. One then deforms to get a smooth complex curve. 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, 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 an 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.