CONFORMAL FIELD THEORIES
Definition 0.1 (Segal, see for example [Hen]). A full conformal field theory is a
symmetric monoidal functor
{︂
}︂
∙ 1 dimensional compact oriented smooth manifolds
−→ {Hilbert spaces} .
∙ conformal cobordisms
(Here the monoidal structure on the left is given by disjoint union, while the
monoidal structure on the right is given by the completed tensor product.)
Definition 0.2. A chiral conformal field theory consists of the following data:
∙ To each 1-manifold 𝑆 as above we associate a category 𝒞𝑆 .
Furthermore, to each object 𝜆 ∈ 𝒞, we associate a Hilbert space 𝐻𝜆 .
∙ To each complex cobordism Σ from between 1-manifolds 𝑆in and 𝑆out , we
associate a functor
𝑓Σ : 𝒞in → 𝒞out ,
depending on the complex structure of Σ in a conformal way.
Furthermore, to each object 𝜆 ∈ 𝒞in we associate a linear map 𝐻𝜆 →
𝐻𝑓Σ (𝜆) .
As a model for chiral CFT, we will use vertex algebras. Given a vertex algebra
𝑉 , the category 𝒞𝑆 1 will be given by the category Rep(𝑉 ). Given a representation
𝜆 ∈ Rep(𝑉 ), the corresponding Hilbert space will be given by the vector space
underlying the representation. And the linear maps should be constructed from
chiral blocks in some way.
Given a chiral CFT, we need to make a lot of choices in a compatible way in order
to get a full CFT. The FRS theorem says that given a chiral CFT corresponding to
a rational vertex algebra 𝑉 , making all of these choices is equivalent to choosing a
Frobenius algebra object 𝒜 ∈ Rep(𝑉 ). (When 𝑉 is rational, Rep(𝑉 ) is a modular
tensor category, so it is reasonable to talk about Frobenius algebra objects.)
We will define some of these terms in the rest of this talk, and give examples of
vertex algebras.
1. Vertex algebras
Definition 1.1. A vertex algebra (𝑉, |0⟩, 𝑇, 𝑌 (·, 𝑧)) consists of the following data:
∙ the space of states: a graded complex vector space
⨁︁
𝑉 =
𝑉𝑖
𝑖≥0
∙ the vacuum vector : |0⟩ ∈ 𝑉0
∙ the translation operator : 𝑇 : 𝑉 → 𝑉 a linear map of degree 1
Date: February 2015.
1
2
CONFORMAL FIELD THEORIES
∙ the vertex operators: 𝑌 (·, 𝑧) : 𝑉 → End 𝑉 [[𝑧, 𝑧 −1 ]] a linear map such that if
we have 𝐴 ∈ 𝑉𝑖 and write
∑︁
𝑌 (𝐴, 𝑡) =
𝐴(𝑛) 𝑧 −𝑛−1 ,
𝑛∈Z
then the (−𝑛 − 1)th coefficient 𝐴(𝑛) ∈ End 𝑉 is of degree −𝑛 + 𝑖 − 1.
These data are subject to the following conditions:
∙ The vacuum axiom:
𝑌 (|0⟩, 𝑡) = id𝑉
Furthermore, for any 𝐴 ∈ 𝑉 , 𝐴(𝑛) |0⟩ = 0 for 𝑛 ≥ 0, and 𝐴(−1) |0⟩ = 𝐴.
∙ The translation axiom:
[𝑇, 𝑌 (𝐴, 𝑧)] = 𝜕𝑧 𝑌 (𝐴, 𝑧)
∀𝐴 ∈ 𝑉
𝑇 |0⟩ = 0
∙ The locality axiom: For any 𝐴, 𝐵 ∈ 𝑉 there exists 𝑁 ∈ N such that
(𝑧 − 𝑤)𝑁 [𝑌 (𝐴, 𝑧), 𝑌 (𝐵, 𝑤)] = 0 ∈ End 𝑉 [[𝑧 ± , 𝑤± ]]
Remark 1.2. We can modify this definition in the super setting, resulting in vertex
superalgebras.
1.1. Example: Heisenberg and lattice vertex algebras.
Definition 1.3. The Heisenberg Lie algebra is the central extension ĥ of the commutative Lie algebra C((𝑡))
0 → C1 → ĥ → C((𝑡)) → 0,
with generators {𝑏𝑛 = 𝑡𝑛 } and 1 satisfying the commutation relations
[1, 𝑏𝑛 ] = 0,
[𝑏𝑛 , 𝑏𝑚 ] = 𝑛𝛿𝑛,−𝑚 1.
More generally, let 𝐿 be a lattice (i.e. a finitely generated free abelian group),
and suppose (·, ·) is a positive definite symmetric bilinear form on 𝐿.1
Definition 1.4. The Heisenberg Lie algebra modelled on 𝐿 is the central extension
ĥ𝐿
0 → C1 → ĥ𝐿 → 𝐿 ⊗C C((𝑡)) → 0,
defined by setting [𝐴⊗𝑓 (𝑡), 𝐵⊗𝑔(𝑡)] ∑︀
= −(𝐴, 𝐵)(Res 𝑓 (𝑡)𝑔 ′ (𝑡)𝑑𝑡), where Res ℎ(𝑡)𝑑𝑡 =
Res𝑡=0 ℎ(𝑡)𝑑𝑡 = ℎ−1 for any ℎ(𝑡) = 𝑖∈Z ℎ𝑖 𝑡𝑖 ∈ C((𝑡)).
The Lie algebra ĥ𝐿 is generated by elements of the form 𝛼(𝑛) ..= 𝛼 ⊗ 𝑡𝑛 (𝛼 ∈
𝐿, 𝑛 ∈ Z), and by 1, satisfying the relations
[𝛼(𝑚), 1] = 0,
[𝛼(𝑚), 𝛽(𝑛)] = 𝑚(𝛼, 𝛽)𝛿𝑚,−𝑛 1.
We define representations of ĥ and ĥ𝐿 called Fock spaces by inducing onedimensional representations of commutative subalgebras.
1If we assume 𝐿 is even, then the construction we’re about to describe will result in a vertex
algebra; otherwise we will obtain a vertex super algebra.
CONFORMAL FIELD THEORIES
3
For example, let C · |0⟩ be the one dimensional representation of ĥ+ ⊕ C1 on
which 1 acts as the identity and ĥ+ acts trivially. (By ĥ+ we mean the subalgebra
generated by positive powers of 𝑡.) Then we define
𝜋0 ..= 𝑈 (ĥ) ⊗𝑈 (h^+ ⊕C1) C · |0⟩.
We can see that as a ĥ-representation, 𝜋0 is isomorphic to the polynomial algebra
in infinitely many variables
C[𝑏−1 , 𝑏−2 , . . .],
where a generator 𝑏𝑛 of ĥ acts by multiplication by the variable 𝑏𝑛 for 𝑛 < 0, by
differentiation 𝑛 𝜕𝑏𝜕−𝑛 for 𝑛 > 0 and by zero for 𝑛 = 0.
Similarly, for any 𝜆 in the lattice 𝐿, we have a representation 𝜋𝜆 which is generated by a single vector |𝜆⟩ and satisfies
1|𝜆⟩ = |𝜆⟩;
𝛼(𝑛)|𝜆⟩ = 0,
𝑛 > 0;
𝛼(0)|𝜆⟩ = (𝜆, 𝛼)|𝜆⟩.
Set
𝑉𝐿 ..=
⨁︁
𝜋𝜆 .
𝜆∈𝐿
Claim 1.5.
(1) The vector space 𝑉𝐿 is a vertex (super)algebra, called the lattice vertex algebra associated to (𝐿, (·, ·)).
(2) The component 𝜋0 is a vertex subalgebra of 𝑉𝐿 , called the Heisenberg vertex
algebra.
We will sketch the construction of the data required to specify the vertex algebra
structure. For the details, see Chapter 7 of [FLM88] and also Sections 4.2 and 4.4
of [FBZ01].
∙ The parity: We set the parity of 𝜋𝜆 to be equal to (𝜆, 𝜆) mod 2. (In
particular, if (·, ·) takes values in 2Z, then 𝑉𝐿 will be purely even and hence
an ordinary vertex algebra.)
∙ The grading: We define the degree of a vector of the form
(1)
𝑚|𝜆⟩ = 𝛼1 (𝑛1 )𝛼2 (𝑛1 ) · · · 𝛼𝑘 (𝑛𝑘 )|𝜆⟩
∑︀𝑘
to be equal to (𝜆, 𝜆)/2 − 𝑗=1 𝑛𝑗 .
∙ The vacuum vector: we choose the vector |0⟩, noting that it is indeed even
and of degree 0.
∙ The translation operator: we define 𝑇 : 𝑉𝐿 → 𝑉𝐿 by setting
𝑇 · |𝜆⟩ ..= 𝜆(−1)|𝜆⟩
∀𝜆 ∈ 𝐿
[𝑇, 𝛼(𝑛)] ..= −𝑛𝛼(𝑛 − 1) ∀𝛼 ∈ 𝐿, 𝑛 ∈ Z
Note that this determines the action of 𝑇 on all of 𝑉𝐿 by induction on the
length 𝑘 of the monomial 𝑚 from (1):
𝑇 · 𝛼𝑘+1 (𝑛𝑘+1 )𝑚|𝜆⟩ = 𝛼𝑘+1 (𝑛𝑘+1 )𝑇 · 𝑚|𝜆⟩ + [𝑇, 𝛼𝑘+1 (𝑛𝑘+1 )]𝑚|𝜆⟩
It is clear that 𝑇 is even and of degree one.
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CONFORMAL FIELD THEORIES
(2)
∙ The vertex operators: For any 𝛼 ∈ 𝐿 we set
∑︁
𝛼(𝑧) = 𝑌 (𝛼(−1)|𝜆⟩, 𝑧) ..=
𝛼(𝑛)𝑧 −𝑛−1 ;
𝑛∈Z
here 𝛼(𝑛) is regarded as a linear operator on 𝑉𝐿 : note that it has degree
−𝑛 and is even.
By the Strong Reconstruction Theorem (Theorem 3.6.1, [FBZ01]) it suffices to define the vertex operators 𝑉𝜆 (𝑧) = 𝑌 (|𝜆⟩, 𝑧) for 𝜆 ∈ 𝐿; these
together with (2) uniquely determine the remaining vertex operators. We
define
𝑆𝜆 : 𝜋𝜇 → 𝜋𝜇+𝜆
by setting 𝑆𝜆 (|𝜇⟩) ..= |𝜇 + 𝜆⟩ and requiring that [𝑆𝜆 , 𝛼(𝑛)] = whenever
𝛼 ∈ 𝐿 and 𝑛 =
̸ 0. Then we set
)︃
(︃
)︃
(︃
∑︁ 𝜆𝑛
∑︁ 𝜆𝑛
−𝑛
−𝑛
𝜆
.
0
𝑧
exp −
𝑧
𝑉𝜆 (𝑧) .= 𝑆𝜆 𝑧 exp −
𝑛
𝑛
𝑛>0
𝑛<0
(Here 𝑧0𝜆 acts on 𝜋𝜇 by 𝑧 ⟨𝜆,𝜇⟩ .)
We omit the proof that this gives a vertex (super)algebra structure on 𝑉𝐿 .
Remark 1.6. We can see that the coefficients of the vertex operators 𝑌 (|𝜆⟩, 𝑧) are
endomorphisms which act on the 𝐿-graded pieces of 𝑉𝐿 as follows:
𝜋𝜇 → 𝜋𝜇+𝜆 .
In particular, the coefficients of 𝑌 (|0⟩, 𝑧) restrict to give endomorphisms of 𝜋0 . It
is for this reason that 𝜋0 is a vertex subalgebra.
The Heisenberg vertex algebra for the lattice 𝐿 = Z is used to model the theory
of a free boson. When we introduce the lattice 𝐿 we study the compactified version
of this theory.
1.2. Example: Kac-Moody vertex algebras. Let g be a simple Lie algebra of
finite dimension 𝑑 = dim g.
Definition 1.7. The loop algebra of g is the Lie algebra
𝐿g = g ⊗C C((𝑡)),
with Lie bracket given by
[𝐴 ⊗ 𝑓 (𝑡), 𝐵 ⊗ 𝑔(𝑡)] = [𝐴, 𝐵] ⊗ 𝑓 (𝑡)𝑔(𝑡),
𝐴, 𝐵 ∈ g, 𝑓 (𝑡), 𝑔(𝑡) ∈ C((𝑡)).
Definition 1.8. The affine Kac-Moody Lie algebra associated to g is the central
extension
0 → CK → ĝ → 𝐿g → 0,
with bracket given by
[𝐴 ⊗ 𝑓 (𝑡), 𝐵 ⊗ 𝑔(𝑡)] = [𝐴, 𝐵] ⊗ 𝑓 (𝑡)𝑔(𝑡) − (Res𝑡=0 𝑓 𝑑𝑔)(𝐴, 𝐵)K.
As before, we consider a subalgebra of our Lie algebra and we induce a onedimensional represention of this subalgebra to 𝑔ˆ. In this case, we note that g[[𝑡]]⊕CK
is a Lie subalgebra, and for any 𝑘 ∈ C we consider its one-dimensional representation
C𝑘 defined by letting K act by the scalar 𝑘, and g[[𝑡]] act trivially.
CONFORMAL FIELD THEORIES
5
Definition 1.9. The vacuum representation of level 𝑘 is
𝑉𝑘 (g) ..= 𝑈 (ĝ) ⊗𝑈 (g[[𝑡]]⊕CK) C𝑘 .
Claim 1.10. We have a canonical structure of vertex algebra on 𝑉𝑘 (g).
Fact 1.11. If 𝑘 ∈ Z+ , then 𝑉𝑘 (g) is irreducible as a ĝ-module, with a proper
max 𝑘+1
submodule 𝐼𝑘 generated by (𝑒𝛼
𝑣𝑘 . (Here 𝛼max is the maximal root of the
−1 )
Lie algebra g, and 𝑣𝑘 is the image of 1 ⊗ 1 in 𝑉𝑘 (g), and is the vacuum vector.)
But 𝐿𝑘 (g) ..= 𝑉𝑘 (g)/𝐼𝑘 is irreducible as a ĝ-module, and acquires the structure of a
simple vertex algebra.
When physicists talk about the WZW-model, they are talking about the theory
corresponding to 𝐿𝑘 (g).
1.3. Example: the Virasoro vertex algebra.
Notation 1.12. We let 𝒦 = C((𝑡)), 𝒪 = C[[𝑡]]. We denote by Der 𝒦 = C((𝑡))𝜕𝑡
the Lie algebra of continuous derivations of 𝒦, and similarly Der 𝒪 = C[[𝑡]]𝜕𝑡 .
Definition 1.13. The Virasoro Lie algebra is the central extension
0 → CC → Vir → Der 𝒦 → 0,
with Lie bracket defined by
1
(Res𝑡=0 𝑓 𝑔 ′′′ 𝑑𝑡)C.
12
It has topological generators of the form 𝐿𝑛 = −𝑡𝑛+1 𝜕𝑡 and C, and these satisfy
the relations
𝑛3 − 𝑛
[C, 𝐿𝑛 ] = 0,
[𝐿𝑛 , 𝐿𝑚 ] = (𝑛 − 𝑚)𝐿𝑛+𝑚 +
(3)
𝛿𝑛,−𝑚 .
12
(These are known as the Virasoro relations.)
Fix a constant 𝑐 ∈ C and consider the action of the Lie subalgebra Der 𝒪 ⊕ CC
of Vir on C defined by letting C act by the scalar 𝑐 and Der 𝒪 act trivially. We
denote this representation by C𝑐 , and we consider the induced representation
[𝑓 (𝑡)𝜕𝑡 , 𝑔(𝑡)𝜕𝑡 ] = (𝑓 𝑔 ′ − 𝑓 ′ 𝑔)𝜕𝑡 −
𝑈 (Vir) ⊗𝑈 (Der 𝒪⊕CC) C𝑐 .
This has a structure of vertex algebra; we call it the Virasoro vertex algebra of
central charge 𝑐, and denote it by Vir𝑐 .
Fact 1.14. The representation Vir𝑐 is reducible as a Vir-module precisely when 𝑐
2
is of the form 𝑐(𝑝, 𝑞) = 1 − 6(𝑝−𝑞)
for 𝑝, 𝑞 > 1 coprime integers. In that case, we
𝑝𝑞
can form the irreducible quotient, which we denote by 𝐿𝑐(𝑝,𝑞) ; it is a simple vertex
algebra.
When physicists talk about the minimal model they are referring to a theory
corresponding to 𝐿𝑐(𝑝,𝑞) .
2. Conformal vertex algebras and other representations
Definition 2.1. A vertex algebra is conformal of central charge 𝑐 if it is equipped
with a distinguished vector 𝜔 ∈ 𝑉2 such that the modes 𝐿𝑉𝑛 of
∑︁
𝑌 (𝜔, 𝑧) =
𝐿𝑉𝑛 𝑧 −𝑛−2
𝑛∈Z
are endomorphisms of 𝑉 satisfying the Virasoro relations of central charge 𝑐.
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CONFORMAL FIELD THEORIES
Equivalently, there is a unique vertex algebra homomorphism Vir𝑐 → 𝑉 , which
sends the vacuum vector 𝑣𝑐 of Vir𝑐 to the vacuum vector |0⟩ of 𝑉 , and which also
maps 𝐿−2 𝑣𝑐 to 𝜔.
Definition 2.2. This vector 𝜔 is called the conformal vector.
2.1. Examples.
(1) The Virasoro vertex algebra of central charge 𝑐 is conformal of central
charge 𝑐, with conformal vector 𝜔 = 𝐿−2 𝑣𝑐 .
(2) The Heisenberg vertex algebra 𝜋0 has a family of conformal vectors 𝜔𝜆 ∈ 𝜋0 ,
given by
1 2
𝑏 + 𝜆𝑏−2 .
2 −1
Then 𝜔𝜆 is conformal of central charge 𝑐𝜆 = 1 − 12𝜆2 . In particular, setting
𝜆 = 0 we obtain a conformal vector of central charge 1.
(3) Let 𝑉𝑘 (g) be the Kac-Moody vertex algebra of level 𝑘, and assume that
𝑘 ̸= −ℎ∨ , where ℎ∨ is the dual Coxeter number of g. Then the Sugawara
𝑘𝑑
construction produces a conformal vector 𝜔 of central charge 𝑘+ℎ
∨ . (Recall
that 𝑑 is the dimension of g.)
𝜔𝜆 =
Definition 2.3. Let 𝑉 = (𝑉, |0⟩, 𝑇, 𝑌 (·, 𝑧)) be a vertex algebra. A representation
of 𝑉 (or a 𝑉 -module) consists of the following data:
⨁︀
∙ a graded space 𝑀 = 𝑛∈Z 𝑀𝑛 such that 𝑀𝑛 = 0 for 𝑛 sufficiently small;
∙ a translation operator 𝑇𝑀 : 𝑀 → 𝑀 , linear of degree 1;
∙ vertex operators 𝑌𝑀 (·, 𝑧) : 𝑉 → End(𝑀 )[[𝑧, 𝑧 −1 ]].
We require these objects to satisfy some axioms making them compatible with the
vertex algebra structure on 𝑉 .
Definition 2.4. Suppose in addition that 𝑉 is conformal of central charge 𝑐 with
conformal vector 𝜔. Then a representation 𝑀 as above is called conformal if the
mode 𝐿𝑀
0 of 𝜔, defined by setting
∑︁
−𝑛−2
𝑌𝑀 (𝜔, 𝑧) =
𝐿𝑀
,
𝑛 𝑧
𝑛∈Z
coincides with the gradation operator on 𝑀 , up to a shift.
2.2. Examples.
(1) If we have a homomorphism of vertex algebras 𝑉 → 𝑊 , then 𝑊 is a
representation of 𝑉 . Moreover, every representation of 𝑊 automatically
becomes a representation of 𝑉 .
(2) In particular, every conformal vertex algebra of central charge 𝑐 is a conformal representation of Vir𝑐 .
(3) From Remark 1.6, we can see that each Fock space 𝜋𝜆 is a representation
of the Heisenberg vertex algebra 𝜋0 .
Definition 2.5. We say that a conformal vertex algebra 𝑉 is rational if every
𝑉 -module is completely reducible.
Here are some consequences of 𝑉 being rational:
(1) Up to isomorphism, 𝑉 has only finitely many simple modules 𝑀 .
CONFORMAL FIELD THEORIES
7
(2) The graded components 𝑀𝑛 of each simple module 𝑀 are all finite dimensional.
(3) Each simple module 𝑀 is conformal.
(4) The category Rep 𝑉 is a modular tensor category.
3. Conformal blocks
Fix a rational vertex algebra 𝑉 and a collection 𝑀1 , 𝑀2 , . . . , 𝑀𝑛 of representations of 𝑉 . We have a trivial bundle on the moduli space ℳ𝑛,𝑔 of genus 𝑔 curves
with 𝑛 marked points given by
(𝑀1 ⊗ · · · ⊗ 𝑀𝑛 )* ⊗ ℳ𝑛,𝑔 → ℳ𝑛,𝑔 .
Our goal is to identify a nice subbundle of this vector bundle by imposing conditions that come from 𝑉 and depend on our choice of marked curve (𝑋, 𝑥1 , . . . , 𝑥𝑛 ) ∈
ℳ𝑛,𝑔 . That is, for each choice (𝑋, 𝑥1 , . . . , 𝑥𝑛 ), we need to specify a collection of
linear forms
𝑀1 ⊗ · · · ⊗ 𝑀𝑛 → C.
Step 1. To begin, assume than 𝑛 = 1, and fix a smooth projective curve 𝑋 together
with a single point 𝑥1 ∈ 𝑋. Let’s take 𝑀1 to be the natural representation of 𝑉 on
itself.
The first thing to do is use 𝑉 to build a vector bundle on 𝑋. We proceed as
follows. Define
{︁
}︁
ˆ𝑋,𝑥 −∼→ 𝒪 = C[[𝑡]] .
𝒜ut𝑋 ..= (𝑥, 𝑡𝑥 ) | 𝑥 ∈ 𝑋, 𝑡𝑥 : 𝒪
(We say that 𝑡𝑥 is a formal coordinate on 𝑋 at 𝑥.) This gives a principal 𝒜ut𝒪 bundle on 𝑋. Then notice that conditions of 𝑉 being conformal give in particular
an integrable action of Der 𝒪 on 𝑉 , so that we can view 𝑉 as a representation of
𝒜ut𝒪 .
We define 𝒱 to be the associated bundle on 𝑋:
𝒱 ..= 𝒜ut𝑋 ×𝒜ut𝒪 𝑉 → 𝑋.
This is a bundle with flat connection (the construction of the connection also comes
from the action of the Virasoro on 𝑉 ). Denote by 𝒱𝑥 the restriction of 𝒱 to the
formal disc 𝐷𝑥 around 𝑥. Note that a choice of formal coordinate 𝑡𝑥 at 𝑥 gives an
identification of 𝒱𝑥 with 𝑉 .
Recall that the definition of 𝑌 (·, 𝑧) means that for every choice of 𝐴, 𝑣 ∈ 𝑉 and
𝜙 ∈ 𝑉 * we obtain an element ⟨𝜙, 𝑌 (𝐴, 𝑧)𝑣⟩ of C((𝑡)).
Using our formal coordinate 𝑡𝑥 , we obtain a meromorphic End(𝒱)-valued section
of the bundle 𝒱 * on the punctured disc 𝐷𝑥× , which we’ll call 𝒴𝑥 . That is, for any
𝑥 ∈ 𝑋, 𝐴 ∈ 𝒱𝑥 , and 𝜙 ∈ 𝒱𝑥* , we obtain a section
(4)
⟨𝜙, 𝒴𝑥 · 𝐴⟩
of 𝒱 * on 𝐷𝑥× . We can show that the definition of 𝒴𝑥 is independent of the choice
of coordinate 𝑡𝑥 .
Definition 3.1. We say that 𝜙 ∈ 𝒱𝑥* is a conformal block if for every choice 𝐴 ∈ 𝒱𝑥
the section (4) extends to give a regular section of 𝒱 * on all of 𝑋 ∖ 𝑥.
We denote by 𝐶(𝑉, 𝑋, 𝑥) the space of all conformal blocks.
8
CONFORMAL FIELD THEORIES
Step 2. Now we generalise to allow for more insertion points {𝑥1 , . . . , 𝑥𝑛 } and
arbitrary representations 𝑀𝑖 ∈ Rep(𝑉 ) associated to each point 𝑥𝑖 . For simplicity,
𝑖
let’s assume that for each 𝑖, 𝐿𝑀
acts on 𝑀𝑖 with integral eigenvalues. Then as
0
above, we construct vector bundles (with flat connection)
ℳ𝑖 = 𝒜ut𝑋 ×𝒜ut𝒪 𝑀𝑖 → 𝑋,
and we define End(ℳ𝑖 )-valued sections 𝒴𝑀𝑖 ,𝑥𝑖 of 𝒱 * on 𝐷𝑥×𝑖 .
⨂︀𝑛
*
Definition 3.2. An element 𝜙 ∈ ( 𝑖=1 ℳ𝑖,𝑥𝑖 ) is called a conformal block if for
any fixed choices 𝐴𝑖 ∈ ℳ𝑖,𝑥𝑖 , all of the sections of 𝒱 * over the punctured discs
𝐷𝑥×𝑖 given by
⟨𝜙, 𝐴1 ⊗ · · · ⊗ (𝒴ℳ𝑖 ,𝑥𝑖 · 𝐴𝑖 ) ⊗ · · · ⊗ 𝐴𝑛 ⟩
can be extended to the same regular section of 𝒱 * over 𝑋 ∖ {𝑥1 , . . . , 𝑥𝑛 }.
This gives a space 𝐶𝑉 (𝑋, (𝑥𝑖 ), (𝑀𝑖 )) of conformal blocks associated to the data
𝑉, 𝑋, 𝑥1 , . . . , 𝑥𝑛 , and 𝑀1 , . . . , 𝑀𝑛 .
Step 3. Now we assemble these spaces into a sheaf of conformal blocks. Recall
that
ℳ1 · · · ℳ𝑛 → 𝑋 𝑛
is a vector bundle with flat connection. By definition, 𝐶𝑉 (𝑋, (𝑥𝑖 ), (𝑀𝑖 )) is a sub˚𝑛 . (Here by 𝑋
˚𝑛 we mean
space of the fibre of (ℳ1 · · · ℳ𝑛 )* at (𝑥1 , . . . , 𝑥𝑛 ) ∈ 𝑋
𝑛
the complement in 𝑋 to all of the diagonals.)
˚𝑛 , we obtain a subsheaf of (ℳ1 · · · ℳ𝑛 )* ,
Allowing (𝑥𝑖 ) to vary over 𝑋
preserved by the flat connection.
Step 4. Given an element 𝜙 ∈ 𝐶𝑉 (𝑋, (𝑥𝑖 ), (𝑀𝑖 )) we obtain a horizontal section
𝜙𝑦1 ,...,𝑦𝑛 of the sheaf of conformal blocks in a neighbourhood of (𝑥1 , 𝑙𝑑𝑜𝑡𝑠, 𝑥𝑛 ).
Choosing local sections 𝐴𝑖 (𝑦𝑖 ) of each ℳ𝑖 near 𝑥𝑖 , we form a function around the
point (𝑥1 , . . . , 𝑥𝑛 ):
⟨𝜙𝑦1 ,...,𝑦𝑛 𝐴1 ⊗ · · · ⊗ 𝐴𝑛 ⟩.
Definition 3.3. This function is called a chiral correlation function.
Question 3.4. How do chiral correlation functions behave near the diagonals?
Here is one result. Suppose that all 𝑥𝑖 are near the same point 𝑥; let us choose
a coordinate 𝑧 at 𝑥, and denote by 𝑧𝑖 the corresponding coordinate at 𝑥𝑖 . Then
⟨𝜙, 𝐴1 (𝑧1 ) ⊗ · · · ⊗ 𝐴𝑛 (𝑧𝑛 )⟩ = ⟨𝜙, 𝑌 (𝐴1 , 𝑧1 )𝑌 (𝐴2 , 𝑧2 ) · · · 𝑌 (𝐴𝑛 , 𝑧𝑛 )|0⟩⟩
(as an element of C[[𝑡1 , . . . , 𝑡𝑛 ]][(𝑧𝑖 − 𝑧𝑗−1 )]𝑖̸=𝑗 ).
4. Free field realisation
This is a technique for understanding the conformal blocks of something complicated in terms of the conformal blocks of something easier.
Key observation 4.1. Let 𝑉 → 𝑊 be a morphism of conformal vertex algebras.
Let 𝑋 be a smooth projective curve with 𝑛 marked points {𝑥1 , . . . , 𝑥𝑛 }, and let
𝑀1 , . . . 𝑀𝑛 be conformal representations of 𝑊 . Recall from the first example of 2.2
that each 𝑀𝑖 automatically acquires the structure of a 𝑉 -module.
CONFORMAL FIELD THEORIES
9
We have an embedding
𝐶𝑊 (𝑋, (𝑥𝑖 ), (𝑀𝑖 )) ˓→ 𝐶𝑉 (𝑋, (𝑥𝑖 ), (𝑀𝑖 ))
⨂︀𝑛
*
of the subspaces of ( 𝑖=1 𝑀𝑖 ) . Indeed, the conditions imposed by 𝑉 are weaker
than those imposed by 𝑊 , since the action of 𝑉 factors through the action of
𝑊 . Moreover, the flat connections on the corresponding sheaves are determined
only by the action of the Virasoro on the modules 𝑀𝑖 , and this action is the same
whether we view 𝑀𝑖 as a 𝑊 -module or as a 𝑉 -module. It follows this embedding
extends to an embedding of the sheaves of conformal blocks, compatible with the
flat connections.
Therefore, if we know the horizontal sections of the sheaf of conformal blocks
corresponding to 𝑊 , we obtain horizontal sections for 𝑉 as well.
So if we are interested in studying the conformal blocks of a vertex algebra 𝑉 , we
should try to find a map from 𝑉 to some vertex algebra 𝑊 that we understand better. The simplest non-trivial vertex algebras are given by the Heisenberg algebras,
and indeed our main application of will be by constructing a map
(5)
𝑉𝑘 (g) → 𝑊,
where 𝑊 is a Heisenberg vertex algebra corresponding to some Heisenberg Lie algebra ĥ. Then horizontal sections of the sheaf of conformal blocks corresponding to
𝑊 are understood in terms of the theory of free bosonic fields, which is not too bad.
On the other hand, horizontal sections of the sheaf of conformal blocks associated
to 𝑉𝑘 (g) correspond to solutions to the Knizhnik–Zamolodchikov equations, and are
important in the study of the WZW-model.
Note that this kind of argument cannot be done at the level of the Lie algebras ĝ
and ĥ, and we do need to work with the machinery of vertex algebras and conformal
blocks. However, to produce a map
𝑉𝑘 (g) → 𝑊,
it suffices to give a map of Lie algebras g → ĥ. Moreover, the Heisenberg Lie
algebra is (roughly) the Lie algebra of differential operators on an affine space, and
so to produce the desired map we just need to give an action of the Lie algebra g
on the affine space by vector fields.
We will not give the full construction, but it amounts to defining a formal loop
space on the flag variety 𝐺/𝐵, called the semi-infinite flag manifold. Roughly, we
should think of the space of maps
𝐷× = Spec C((𝑡)) → 𝐺/𝐵.
This is the space on which g will act.
5. 𝒲-algebras
In this section, we will sketch the definition of 𝒲-algebras: these form a 1parameter family of vertex algebras associated to a simple Lie algebra g.
Step 1: Form the BRST complex. First recall that given a finite dimensional
vector space 𝑈 , 𝑈 ((𝑡)) ⊕ 𝑈 * ((𝑡))𝑑𝑡 is a complete topological vector space. We
associate to it a complete topological algebra 𝒞l (𝑈 ), called the Clifford algebra.
The Clifford algebra has an irreducible Fock space representation, which we denote
10
CONFORMAL FIELD THEORIES
⋀︀
by 𝑈 . It has the structure of a vertex superalgebra, isomorphic to a tensor product
of copies of the free fermionic vertex superalgebra.
⋀︀
We introduce a second grading on the spaces 𝒞l (𝑈 ) and 𝑈 , which we call the
charge gradation.
Now given a simple Lie algebra g with Cartan decomposition g = n+ ⊕ h ⊕ 𝑓 𝑛− ,
we form a complex by taking the tensor product
𝐶𝑘∙ (g) = 𝑉𝑘 (g) ⊗
∙
⋀︁
,
n−
where the grading of the complex ∙ is given by the charge gradation. This complex
aqcuires the structure of a vertex superalgebra.
We define an element 𝑄 of 𝐶𝑘∙ (g) (it is given by an explicit formula, but we
haven’t introduced any of the notation necessary, so we will not give the formula
here); then the standard differential 𝑑𝑠𝑡 of charge 1 is given by the coefficient 𝑄( 0)
of the vertex operator 𝑌 (𝑄, 𝑧).
The resulting complex (𝐶𝑘∙ (g), 𝑑𝑠𝑡 ) is the standard complex of semi-infinite cohomology of the Lie algebra n+ ((𝑡)) with coefficients in 𝑉𝑘 (g).
Step 2: Twist by the Drinfeld–Sokolov character. The Drinfeld–Sokolov
𝛼
𝑛
character 𝜒 is defined on 𝑒𝛼
𝑛 = 𝑒 ⊗ 𝑡 (𝛼 a root of g, 𝑛 ∈ Z) by
{︂
1 if 𝛼 is simple and 𝑛 = −1,
𝜒(𝑒𝛼
𝑛) =
0 otherwise.
This definition extends to give a linear functional on n+ ((𝑡)), and it is a character:
𝜒([𝑥, 𝑦]) = 0.
We denote also by 𝜒 the corresponding element of n*+ ((𝑡))𝑑𝑡 ⊂ 𝒞l (n+ ).
We have that 𝜒2 = 0, and [𝑑𝑠𝑡 , 𝜒] = 0, and so it follows that 𝑑 ..= 𝑑𝑠𝑡 + 𝜒 is a
differential on 𝐶𝑘∙ (g).
Definition 5.1. The resulting complex (𝐶𝑘∙ (g), 𝑑) is called the BRST complex of
the quantum Drinfeld–Sokolov reduction.
Step 3: Take cohomology. We denote the cohomology of (𝐶𝑘∙ (g), 𝑑) by 𝐻𝑘∙ (g).
It is a vertex superalgebra, and in particular 𝐻𝑘0 (g) is a vertex algebra.
Definition 5.2. We set 𝒲𝑘 (g) ..= 𝐻𝑘0 (g); it is called the 𝒲-algebra associated to ĝ
at level 𝑘.
Example 5.3. When g = sl2 , the vertex algebra 𝒲𝑘 (sl2 ) is generated by a single
vector 𝑊1 , which can be chosen to be the conformal vector if 𝑘 ̸= −ℎ∨ = 2. It
follows that 𝒲𝑘 (sl2 ) is isomorphic to the Virasoro vertex algebra Vir𝑐 (𝑘). (Here
2
the central charge is given by 𝑐(𝑘) = 1 − 6(𝑘+1)
𝑘+2 .
When 𝑘 = −2, 𝒲−2 (sl2 ) is a commutative vertex algebra; it is isomorphic to
C[𝑆𝑛 ]𝑛≤−2 .
We have a free field realisation for 𝒲-algebras as well. Recall from Remark 1.6
that the vertex operators of the vertex algebra corresponding to a lattice 𝐿 give rise
to maps 𝜋𝜆 → 𝜋𝜆+𝜇 for
√ 𝜆, 𝜇 ∈ 𝐿. In our case, we let 𝛼1 , → 𝛼𝑑 denote the simple
roots of g; we let 𝜈 = 𝑘 + ℎ∨ , and we denote the corresponding maps by
𝑉−𝛼𝑖 /𝜈 (𝑧) : 𝜋0 → 𝜋−𝛼𝑖 /𝜈 .
CONFORMAL FIELD THEORIES
11
Definition 5.4. The operators
∫︁
𝑉−𝛼𝑖 /𝜈 (𝑧)𝑑𝑧 : 𝜋0 → 𝜋−𝛼𝑖 /𝜈
are known as the screening operators.
Recall that 𝜋0 is the Heisenberg vertex algebra, in this case associated to the
Heisenberg Lie algebra modelled on the Cartan h of g.
Theorem 5.5. For 𝑘 ̸= ℎ∨ , 𝒲𝑘 (g) is the subalgebra of 𝜋0 given by the intersection
of the kernels of the screening operators.
References
[FBZ01] Edward Frenkel and David Ben-Zvi. Vertex algebras and algebraic curves, volume 88 of
Mathematical Surveys and Monographs. American Mathematical Society, Providence,
RI, 2001.
[FLM88] I. Frenkel, J. Lepowsky, and A. Meurman. Vertex operator algebras and the Monster,
volume 134 of Pure and Applied Mathematics. Academic Press, Inc., Boston, MA, 1988.
[Hen]
André Henriques. Three-tier CFTs from Frobenius algebras. arXiv:math/1304.7328v2.