USC SEMINAR: FALL 2012
SABIN CAUTIS
Abstract. This is an introduction to vertex operator algebras and related topics.
Contents
1. Introduction
2. Lie algebras and some representation theory
2.1. The universal enveloping algebra U (sl2 )
2.2. The category U̇ (sl2 )
2.3. The category U̇ (sln )
2.4. The category U̇ (Γ)
2.5. Quantum deformation
3. Quantum affine algebras
3.1. The Kac-Moody presentation
3.2. The loop presentation
3.3. Quantum Heisenberg (sub)algebras
3.4. The loop presentation: revisited
3.5. The Fock space representation of U̇q (b
h)
b n )loop
3.6. The Basic representation of U̇q (sl
3.7. The Frenkel-Kac-Segal construction
4. Vertex algebras
4.1. Definitions
4.2. Example #1: the Heisenberg vertex algebra
References
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1. Introduction
The aim of these notes is to introduce the basic concepts behind vertex operator algebras and sketch
their relation to the following topics:
• the representation theory of (affine) Lie algebras
• conformal field theory
• 3-manifold invariants (Reshetykhin-Turaev invariants)
2. Lie algebras and some representation theory
For convenience we will work over the base field k = C.
1
2
SABIN CAUTIS
2.1. The universal enveloping algebra U (sl2 ).
matrices equipped with the Lie bracket [X, Y ] = XY
0
0 1
F :=
E :=
1
0 0
subject to relations
(1)
The Lie algebra sl2 consists of 2 × 2 traceless
− Y X. It has basis
1 0
0
H :=
0 −1
0
[E, F ] = H, [H, E] = 2E, [H, F ] = −2F.
Instead of studying sl2 (as a Lie algebra) one can instead study the representation theory of its universal enveloping algebra U (sl2 ) (as an associative algebra). This is defined as the unital, associative
C-algebra with generators E, F, H and relations (1).
Now consider aLfinite dimensional U (sl2 ) module V . We can use H to break up V into (generalized)
eigenspaces V = λ V (λ) where λ ∈ C. In other words, if v ∈ V (λ) then H(v) = λv. Note that, since
V is finite dimensional, only finitely many V (λ) are nonzero.
Lemma 2.1. If v ∈ V (λ) then E(v) ∈ V (λ + 2) and F (v) ∈ V (λ − 2).
Proof. This is an exercise using relations (1).
Exercise 1. Let v ∈ V (λ) be a highest weight vector (i.e. E(v) = 0) (such a vector always exists since
dim(V ) < ∞). Show that, for m ∈ N, we have E m F m (v) = m!λ(λ − 1) . . . (λ − m + 1)v.
This means that, unless λ ∈ N, F m (v) 6= 0 for any m ∈ N. So, if dim(V ) < ∞, we must have that
V (λ) = 0 unless λ ∈ Z. Moreover, if v ∈ V (N ) is a highest weight vector (for some N ∈ N), then
F m (v) 6= 0 for exactly m = 0, . . . , N .
Conclusion. We find that any finite dimensional U (sl2 )-module breaks up as follows
E
(2)
E
$$
V (−N )
ff
···
E
((
V (λ − 2)
hh
V (λ)
V (λ + 2)
hh
F
F
E
((
&&
V (N )
· · · cc
F
F
for some N ∈ N.
Example 1. The simplest U (sl2 )-module is C where E(C) = F (C) = H(C) = 0.
Example 2. The standard U (sl2 )-module is V = C2 = hv− , v+ i where
(3)
H(v− ) = −v− , H(v+ ) = v+ , E(v− ) = v+ , E(v+ ) = 0, F (v− ) = 0, F (v+ ) = v−
E
!!
C __
C
F
Example 3. More generally, for each N ∈ N there exists an irreducible, (N + 1)-dimensional, U (sl2 )module VN which decomposes as
E
E
!!
C __
F
C __
F
E
E
!!
C
...
!!
C __
F
!!
C __
C
F
It is spanned by v, F (v), . . . , F N (v). The action of E on F m (v) can be deduced from the [E, F ] = H
relation and the fact that E(v) = 0.
Exercise 2. In the example above, show that EF m (v) = cm F m−1 (v) and determine cm ∈ C.
Proposition 2.2. Any finite dimensional U (sl2 )-module is isomorphic to a direct sum of modules VN .
VERTEX OPERATOR ALGEBRAS AND RELATED TOPICS
3
2.2. The category U̇ (sl2 ). An additive C-linear category is a category enriched in C-vector spaces.
This means that for objects A, B, the space Hom(A, B) is a vector space over C.
Example 4. The category V of vector spaces over C is an additive C-linear category where the objects
are vector spaces and the morphisms are linear maps between them.
The discussion above suggests that we define an additive category U̇ (sl2 ) as follows:
• the objects are indexed by n ∈ N
• the morphisms are generated by E, F and 1n , where 1n denotes the projection onto n
subject to the relations E1n = 1n+2 E1n = 1n+2 E, F 1n = 1n−2 F 1n = 1n−2 F and
(4)
EF 1n = F E1n + n · 1n
for any n ∈ Z. Note that 1n 1m = δm,n 1m . Equation (4) is equivalent to the commutator relation
[E, F ] = H in the algebra U (sl2 ).
Key point. A representation of U (sl2 ) is equivalent to a functor U̇ (sl2 ) → V.
Example 5. The standard representation of U (sl2 ) corresponds to the functor which takes n 7→ 0
unless n = ±1 in which case −1 7→ C = hv− i and 1 7→ C = hv+ i. Then E, F are mapped as in (3).
2.2.1. The integral version. For various reasons, such as wanting to work over Z instead of C, one
r
r
often uses the notation E (r) := Er! and F (r) := Fr! . Throwing in these extra generators, subject to the
obvious relations
r+s
r+s
E (r) E (s) ∼
E (r+s) and F (r) F (s) ∼
F (r+s)
=
=
r
r
gives us Lusztig’s integral (because it can be defined over Z) version of U̇ (sl2 ). We will subsequently
use this notation for convenience.
Example 6. (Compare with Exercise 1) If v ∈ V (λ) is a highest weight vector then, for m ∈ N, we
have
λ
E (m) F (m) (v) =
v.
m
Exercise 3. If v ∈ V (λ) show that
EF (m) (v) = (λ − m + 1)v + F (m) E(v)
and
F E (m) (v) = −(λ + m − 1)v + F E (m) (v).
What about E (n) F (m) (v)?
2.3. The category U̇ (sln ). One can carry out a similar analysis for sln instead of sl2 . In this case,
U (sln ) is the associative algebra generated by E1 , . . . , En−1 , F1 , . . . , Fn−1 and H1 , . . . , Hn−1 subject to
the following relations:

the sl2 relations for each triple {Ei , Fi , Hi }



 if |i − j| = 1 then [H , E ] = −E , [H , F ] = F , [H , H ] = 0, [E , F ] = 0,
i
j
j
i
j
j
i
j
i
j
(5)
2
2
2
2

E
+
E
E
and
2
·
F
F
F
=
F
F
+
F
F
as
well
as
2
·
E
E
E
=
E
j i
i j i
j i
i j i

i j
i j


if |i − j| > 1 then {Ei , Fi , Hi } commute with all of {Ej , Fj , Hj }.
L
If V is a finite dimensional representation of U (sln ) then we have a decomposition V = λ V (λ)
where V (λ) is a simultaneous eigenvalue for H1 , . . . , Hn . In other words, {H1 , . . . , Hn−1 } generate a
lattice Zn−1 and λ is an element of the dual lattice such that if v ∈ V (λ) then Hi (v) = λ(Hi )v.
The lattice spanned by all such λ is called the weight lattice X. It is spanned by {Λ1 , . . . , Λn−1 }
which are called fundamental weights. The V (λ)’s are called the weight spaces of V .
4
SABIN CAUTIS
To understand how the Es and F s move you between the different V (λ)’s one introduces the root
sublattice Q ⊂ X spanned by {α1 , . . . , αn−1 }, called the roots. To describe this sublattice (somewhat
indirectly) we introduce a symmetric, nondegenerate form h·, ·i : X ⊗Z X → Q defined by


if i = j
2
hαi , αj i = −1 if |i − j| = 1


0
otherwise
and satisfying hαi , Λj i = δi,j .
Example 7. For sl2 , X ∼
= Z is spanned by Λ and Q = 2Z ⊂ Z = X is spanned by α = 2Λ. For sl3 ,
X∼
= Z2 is spanned by hΛ1 , Λ2 i with Q ⊂ X spanned by α1 = 2Λ1 − Λ2 and α2 = −Λ1 + 2Λ2 .
Exercise 4. For sln , write α1 , . . . , αn−1 as linear combinations of fundamental weights Λ1 , . . . , Λn−1 .
Lemma 2.3. If v ∈ V (λ) then Ei (v) ∈ V (λ + αi ) and Fi (v) ∈ V (λ − αi ).
Proof. This is an exercise using the relations above. Compare with Lemma 2.1.
Example 8. The standard representation V of U (sl3 ) has a weight space decomposition
V = V (Λ1 ) ⊕ V (Λ1 − α1 ) ⊕ V (Λ1 − α1 − α2 ).
If v ∈ V (Λ1 ) then F1 (v) and F2 F1 (v) form a basis of V .
Recall that in the sl2 case, any irreducible representation is generated by a highest weight vector.
The same is true here. In this case we say that v ∈ V (λ) is a highest weight vector if Ei (v) = 0 for all
1 ≤ i ≤ n − 1.
Example 9. The irreducible U (sl3 )-module below is generated by v ∈ V (α1 + α2 ). Note that the
weight space V (0) is 2-dimensional, spanned by F1 F2 (v) and F2 F1 (v). Maps E1 , E2 , which point in
the opposite direction, are omitted in order to simplify the diagram.
F1
C ❅oo
❅❅
❅❅F2
❅❅
C oo
F1
C oo
C ∋ v❊
❊❊
❊❊F2
❊❊
❊❊
""
F1
C2 ❋oo
❋❋
❋❋F2
❋❋
❋❋
##
F1
C
C
P
A weight λ ∈ X is called dominant if λ = i ai Λi where ai ≥ 0. A weight space V (λ) can contain
highest weight vectors only if λ is dominant.
Example 10. In the case of sl2 , X ∼
= Z and dominant weights were indexed by N ⊂ Z.
Theorem 2.4. For each dominant weight λ ∈ X there exists a unique, irreducible U (sln )-module Vλ
generated by a highest weight vector v ∈ Vλ (λ). Any finite dimensional U (sln )-module is a direct sum
of such Vλ ’s.
Remark 2.5. There are various algebraic and geometric ways to construct these Vλ . There are also
interesting formulas for dim(Vλ ) as well as dim(Vλ (µ)). However, for our purpose it will usually suffice
to know that such Vλ exist.
We can now define a category U̇ (sln ) as follows:
• the objects are indexed by λ ∈ X
• the morphisms are generated by Ei , Fi and 1λ , where 1λ denotes the projection onto λ,
VERTEX OPERATOR ALGEBRAS AND RELATED TOPICS
5
subject to the relations Ei 1λ = 1λ+αi Ei 1λ = 1λ+αi Ei , Fi 1λ = 1λ−αi Fi 1λ = 1λ−αi Fi ,
(
Fi Ei 1λ + hλ, αi i · 1λ if i = j
(6)
Ei Fj 1λ =
Fj Ei 1λ
if i 6= j.
(7)
(2)
(2)
Ei Ej Ei 1λ = Ei Ej 1λ + Ej Ei 1λ ,
(2)
(2)
Fi Fj Fi 1λ = Fi Fj 1λ + Fj Fi 1λ
if |i − j| = 1
and [Ei , Ej ] = 0 = [Fi , Fj ] if |i − j| > 1. This captures all the relations from (5) and generalizes the
definition of U̇ (sl2 ).
(r)
Er
(r)
Fr
Remark 2.6. In (7) above we used the familiar notation Ei := r!i , Fi := r!i . It turns out that this
relation, called the Serre relation, is redundant. So, once again, (6) is the most important relation.
Key point. A representation of U (sln ) is equivalent to a functor U̇ (sln ) → V.
2.4. The category U̇ (Γ). The key ingredient in defining U̇ (sln ) was the sublattice Q ⊂ X together
with the bilinear form h·, ·i on X. This information is captured in the Cartan matrix


2 −1 0
0 ...
0
−1 2 −1 0 . . .
0


. . . . . . . . . . . . . . . . . . 

(8)
(hαi , αj i) = 
 0
0 ...
2 −1 0 


 0
0 . . . −1 2 −1
0
0 ...
0 −1 2
More generally, consider a graph Γ, with vertex set I and suppose Γ does not contain self-loops and
that any two vertices are connected by at most one edge (such a graph is usually called simply-laced).
Such a graph gives rise to the data (Q ⊂ X, h·, ·i) as follows:
• the lattice X is generated by {Λi : i ∈ I} and Q by {αi : i ∈ I}
• the pairing satisfies hαi , Λj i = δi,j and


if i = j
2
hαi , αj i = −1 if i and j are connected by an edge in Γ


0
otherwise.
In analogy to U̇ (sln ), this gives rise to a category U̇ (Γ) as follows:
• the objects are indexed by λ ∈ X
• the morphisms are generated by Ei , Fi where i ∈ I and by 1λ ,
subject to the relations Ei 1λ = 1λ+αi Ei 1λ = 1λ+αi Ei , Fi 1λ = 1λ−αi Fi 1λ = 1λ−αi Fi together with
relations (6) and (7) above.
•
•···•
•
• where there are n − 1 vertices,
Example 11. If Γ is the graph •
then h·, ·i is given by (8) and U̇ (Γ) = U̇ (sln ).
Notation. We will usually write hi, ji instead of hαi , αj i.
2.5. Quantum deformation. The whole story above can be q-deformed. More precisely, one can
define a C(q)-algebra Uq (sln ) which is isomorphic to U (sln ) when q = 1 (Uq (sln ) is, somewhat improperly, called a quantum group). Similarly, one can define a C(q)-linear additive category U̇q (Γ) which
recovers U̇ (Γ) when we set q = 1.
The definition of U̇q (Γ) is the same as that of U̇ (Γ) except that we replace n by the quantum integer
[n] := q n−1 + q n−3 + · · · + q −n+3 + q −n+1
6
SABIN CAUTIS
in all the definitions. For example, if Γ = •
• then
(hi, ji) =
q + q −1
−1
−1
q + q −1
and relation (6) becomes
(9)
(
Fi Ei 1λ + [hλ, αi i] · 1λ
Ei Fj 1λ =
Fj Ei 1λ
if i = j
if i =
6 j.
We will try to work with U̇q (Γ) rather than U̇ (Γ) whenever possible.
3. Quantum affine algebras
3.1. The Kac-Moody presentation. In many ways, the most interesting Lie algebras are the affine
ones. Recall that graphs of finite type are distinguished by the fact that the matrix (hi, ji) is positive
definite. Similarly, graphs of affine type are those where (hi, ji) is positive semidefinite (but not
definite).
It turns out that any such graph is obtained from a graph of finite type by adding an extra vertex.
The main example to keep in mind are those of type A where the finite graph is a chain and the affine
one is obtained by adding an extra vertex and connecting it to the two ends of the chain.
Example 12. Illustrated below is the case of sl4 . On the left is the finite graph and on the right the
affine one.
b4 .
•
• /o /o /o /o /o /o /o /o /o /o /o // •
• =Γ
Γ4 = •
•
b 4 is
Notice that the matrix associated to Γ

2 −1
−1 2
(hi, ji) = 
 0 −1
−1 0
•

0 −1
−1 0 

2 −1
−1 2
which is positive semidefinite of rank 3.
b n)
b n of n vertices is denoted Uq (Γ
b n ) = Uq (sl
The quantum enveloping algebra associated to a chain Γ
b n ). In this case the fundamental weights are denoted Λ0 , . . . , Λn−1
and likewise the category is U̇q (sl
and the roots α0 , . . . , αn−1 (so, by convention, the additional vertex is labeled 0). Thus the set of roots
is indexed by Ib := {0, 1, . . . , n −P
1}. The matrix (hi, ji)0≤i,j≤n−1 is now degenerate of rank n − 1 where
the kernel is generated by δ := i αi .
b n ) is particularly interesting is that it has
3.2. The loop presentation. One of the reasons U (sl
an alternative description as a loop algebra. It is this loop presentation that we will usually use.
The quantum affine algebra in this presentation is quite complicated, so we first discuss the usual
b n )loop (to distinguish it from the Kac-Moody presentation
(q = 1) version. We denote it by U (sl
b n ) = U (sl
b n )KM ).
U (sl
b n )loop = U (sln ) ⊗C C[t, t−1 ] ⊕ C · c. To define an algebra structure we take
As a vector space U (sl
(10)
[a ⊗ tk , b ⊗ tl ] = [a, b]tk+l + kδk,−l tr(ab)c
where c is a central element. We will abbreviate Hi ⊗ tr = Hi,r , Ei ⊗ tr = Ei,r and Fi ⊗ tr = Fi,r .
VERTEX OPERATOR ALGEBRAS AND RELATED TOPICS
7
The subalgebra generated by Hi,r where i ∈ I, r ∈ Z \ {0} is called the Heisenberg algebra, which
we denote U (b
h). It is freely generated by these elements subject to the relation


if i = j
2kδk,−l c
(11)
[Hi,k , Hj,l ] = −kδk,−l c if |i − j| = 1


0
if |i − j| > 1
where c is central.
Somewhat surprisingly, it takes a lot of work to quantize this story. It was done by Drinfeld in [Dr].
b n )loop has generators Ei,r , Fi,r
We will use the notation from [N1]. The quantum affine algebra Uq (sl
H
±d
±c/2
(i ∈ I = {1, . . . , n − 1}, r ∈ Z), q (H ∈ X), and Hi,m , q , q
where i ∈ I and m ∈ Z \ {0}. The
set of relations are as follows:
(i) q ±c/2 is central
′
′
(ii) q 0 = 1, q H q H = q H+H , [q H , Hi,m ] = 0, q d q −d = 1, q c/2 q −c/2 = 1
(iii) ψi± (z)ψj± (w) = ψj± (w)ψi± (z)
(z−q −hi,ji q c w)(z−q hi,ji q −c w) +
ψ (w)ψi− (z)
(z−q hi,ji q c w)(z−q −hi,ji q −c w) j
[q d , q H ] = 0, q d Hi,m q −d = q m Hi,m and q d Ei,r q −d = q r Ei,r , q d Fi,r q −d = q r Fi,r
s
±hi,ji ±sc/2
s=±
q
z − w)x±
(q ±sc/2 z − q ±hi,ji w)ψjs (z)x±
i (w)ψj (z) where
i (w) = (q
− 2c
+
−
+ 2c
1
c
−1
c
−1
[xi (z), xj (w)] = δij q−q−1 δ(q wz )ψi (q w) − δ(q zw )ψi (q z)
±
±
±2
z − w)x±
(z − q ±2 w)x±
i (w)xi (z)
i (z)xi (w) = (q
±
±
±
±
∓1
∓1
(z − q w)xi (z)xj (w) = xj (w)xi (z)(q z − w) if hi, ji = −1
(iv) ψi− (z)ψj+ (w) =
(v)
(vi)
(vii)
(viii)
(ix)
(x) if hi, ji ≤ 0 then
X
σ∈SN
N =1−hi,ji
X
(−1)s
s=0
N ±
±
±
±
x (z
) . . . x±
i (zσ(s) )xj (w)xi (zσ(s+1) ) . . . xi (zσ(N ) ) = 0.
s i σ(1)
In the above relations we have
X
(12)
Ei,r z −r
x+
i (z) =
x−
i (z) =
(13)
=
X
ψi± (±r)z ∓r
Fi,r z −r
r∈Z
r∈Z
ψi± (z)
X
=q
±Hi
exp ± (q − q −1 )
r>0
r≥0
(14)
δ(z)
=
X
X
z
r
Hi,±r z ∓r
r∈Z
The main things to take a way from this definition is that:
(i) it is somewhat complicated,
(ii) it is expressed as relations between Laurent series in an indeterminate z.
Even though we will give a simplified presentation of this definition in section 3.4, the indeterminate z
may play an important role later since one can give it a geometric meaning (it is the local coordinate
on a curve).
3.3. Quantum Heisenberg (sub)algebras. The quantum Heisenberg algebra Uq (b
h) is the subb
algebra of Uq (sln )loop generated by Hi,r , i ∈ I, r ∈ Z \ {0}. It is freely generated by these elements
subject to the following relations
(15)
where c is central.
[Hi,k , Hj,l ] = [lhi, ji]δk,−l
[lc]
l
8
SABIN CAUTIS
It turns out that it is convenient (for instance, from the perspective of categorification) to use a
(r)
(r)
different generating set for Uq (b
h). This set consists of Pi and Qi defined by




X (r)
X (r)
X Hi,r
X Hi,−r
zr  =
zr  =
Pi z r and exp 
Qi z r
(16)
exp 
[r]
[r]
r≥0
r≥1
r≥0
r≥1
where i ∈ I and m ∈ Z≥0 .
Example 13. For the first few terms we have
(0)
(0)
= 1 = Qi ,
Pi
(1)
Pi
(2)
Pi
=
(1)
= Hi,−1 ,
1
1 2
Hi,−2 + Hi,−1
,
[2]
2
Qi
(2)
Qi
=
= Hi,1 ,
1
1 2
Hi,2 + Hi,1
,
[2]
2
etc.
The series in (16) are sometimes known as vertex generating functions. Notice the resemblance
between them and the generating functions from (13).
Since c is central, its action on any irreducible representation will be multiplication by some ℓ ∈ C.
This ℓ is called the level of the the representation. Of particular interest is when ℓ = 1.
Proposition 3.1. If ℓ = 1 then the new generating set for Uq (b
h) satisfies the following relations:
(n)
(m)
(17)
Pi
(18)
Q i Pj
(n)
Pj
(m)
(m)
(n)
(n)
(m)
Proof. First we prove that Qi Pi
A(z) :=
=
P
k≥0 [k
X Hi,−s
s≥1
(n)
(n)
(m)
(m)
(n)
= Pj Pi , Q i Q j = Q j Q i
for any i, j ∈ I,
P
(m−k) (n−k)

Qi
if i = j,
 k≥0 [k + 1]Pi
(m−1) (n−1)
(m) (n)
= Pj Q i − Pj
if hi, ji = −1,
Qi

 (m) (n)
Pj Q i
if hi, ji = 0.
[s]
(m−k)
+ 1]Pi
(n−k)
Qi
z s and B(w) :=
. Denote
X Hi,s
s≥1
[s]
ws .
(m)
Then Qi Pi
= [z m wn ] exp(B) exp(A), where [z m wn ] means taking the coefficient of the polynomial
m n
z w in the subsequent power series. Now, using relation (15), it follows that
[A, B]
=
X [2s]
s≥1
=
ws z s
X (q s + q −s )
s≥1
=
s[s]
s
ws z s
log (1 − qwz)(1 − q −1 wz) .
Thus [A, B] commutes with A and B and hence that
exp(B) exp(A) = exp(−[A, B]) exp(A) exp(B)
VERTEX OPERATOR ALGEBRAS AND RELATED TOPICS
9
(see for instance Lemma 9.43 of [N4]). Thus
[z m wn ] exp(B) exp(A)
=
=
=
1
exp(A) exp(B)
(1 − qwz)(1 − q −1 wz)
X
[z m wn ]
[s + 1](wz)s exp(A) exp(B)
[z m wn ]
X
s≥0
(m−k)
[k + 1]Pi
(n−k)
Qi
k≥0
which is what we needed to prove.
(n) (m)
(m) (n)
(m−1) (n−1)
Next we prove that Qi Pj = Pj Qi + Pj
Qi
if hi, ji = −1. Denote
A(z) :=
X Hj,−s
s≥1
(n)
(m)
Then Qi Pj
[s]
z s and B(w) :=
X Hi,s
s≥1
[s]
ws .
= [z m wn ] exp(B) exp(A). However, this time
[A, B] =
X [−1]
s≥1
s
ws z s = − log(1 − wz)
and so
[z m wn ] exp(B) exp(A)
=
[z m wn ](1 − wz) exp(A) exp(B)
=
Pj
(m)
(n)
Qi
(m−1)
+ Pj
(n−1)
Qi
which is what we needed to prove. The rest of the relations are obvious.
Subsequently, we define U̇q (b
h) to be the category where
• objects are indexed by m ∈ Z,
(r)
(r)
• the morphisms are generated by Pi , Qi and 1m where r ∈ Z≥0 and m ∈ Z,
(r)
(r)
(r)
(r)
(r)
(r)
subject to the relations that Pi 1m = 1m+r Pi 1m = 1m+1 Pi , Qi 1m = 1m−r Qi 1m = 1m−r Qi
together with the relations from Proposition (3.1).
One can generalize the discussion above to the case of integral level ℓ ∈ Z≥1 as follows. First we
recall some terminology involving Z-gradedPfinite dimensional vector spaces V = ⊕i V (i). To such a V
one can associated the polynomial fV := i q i dimV (i). This gives a bijection between isomorphism
classes of finite dimensional graded vector spaces and elements f ∈ N[q, q −1 ].
From V one can construct the associated Z-graded vector spaces Symn (V ) and Λn (V ). If V has
graded dimension f ∈ N[q, q −1 ] then we denote by Symn (f ) and Λn (f ) the graded dimensions of
the Z-graded vector spaces Symn (V ) and Λn (V ). For example, if f = q + q −1 then Symn (f ) =
q n + q n−2 + · · · + q −n+2 + q −n which is just the quantum integer [n + 1]. On the other hand, Λn (f ) is
1, [2], 1 if n = 0, 1, 2, and is zero otherwise.
It turns out [CL2] that the analogue of Proposition 3.1 for a general ℓ ∈ Z≥1 replaces equation (18)
with
P
(m−k) (n−k)
k

Qi
if i = j,
Pk≥0 Sym ([2][ℓ])Pi
(n) (m)
(m−k) (n−k)
k
k
(19)
Q i Pj =
Qi
if hi, ji = −1,
k≥0 (−1) Λ ([ℓ])Pj

 (m) (n)
Pj Q i
if hi, ji = 0.
This gives rise to a category U̇q (b
h)ℓ which generalizes U̇q (b
h) discussed above (the case ℓ = 1). Notice
that from this point of view it is not clear what to do for a nonintegral level ℓ ∈ C.
10
SABIN CAUTIS
Key point. A representation of Uq (b
h)ℓ is equivalent to a functor U̇q (b
h)ℓ → V. We will denote the
image of the object k ∈ U̇q (b
h)ℓ by V (k).
L
Thus, a representation V = k V (k) of Uq (b
h)ℓ looks like
(3)
Pi
(2)
Pi
Pi
. . . dd
Qi
''
V (k − 1)
dd hh
Pi
Pi
((
V (k)
Qi
ee hh
Qi
(( %%
V (k + 1)
hh
Pi
(( $$
V (k + 2)
gg
Qi
Pi
. ## . .
Qi
(2)
Qi
(3)
Qi
which is reminiscent of the sl2 picture from (2). In section 3.5 we will see that any such “integrable”
irreducible representation is isomorphic to the Fock space.
3.3.1. Some other generators. There is a certain symmetry worth noting. Namely, instead of (16) we
(1r )
(1r )
can define Pi
and Qi
as follows:




X
X
X
X Hi,−r
r
H
i,r r 
(1 )
(1r )
zr  =
z
=
(−1)r Pi z r and exp −
(−1)r Qi z r .
(20)
exp −
[r]
[r]
r≥0
r≥1
r≥1
r≥0
Example 14. For the first few terms we have
(10 )
Pi
(10 )
= 1 = Qi
(11 )
Pi
(12 )
Pi
,
(11 )
Qi
= Hi,−1 ,
= Hi,1 ,
1 2
1
1 2
1
(12 )
,
= − Hi,−2 + Hi,−1 , Qi = − Hi,2 + Hi,1
[2]
2
[2]
2
r
etc.
r
(1 )
(1 )
Exercise 5. Show that Pi
And Qi
satisfy the same commutation relations as in (19). [Hint:
Consider the automorphism of Uq (b
h) given by Hi,r 7→ (−1)r Hi,r .]
Proposition 3.2. For any level ℓ ∈ N we have
(1n )
(m)
(21)
Pi
(22)
Q i Pj
Pj
(1m )
(n)
(1n )
and similarly for Qi
(m)
Pj
.
(m)
(1n )
(1n )
(m)
(m)
(1n )
= Pj Pi , Q i Q j = Q j Q i
for any i, j ∈ I,
P
m−k
(1
) (n−k)
k


Qi
if i = j,
 k≥0 Λ ([2][ℓ])Pi
m−k
P
(1
)
(n−k)
k
k
=
Qi
if hi, ji = −1,
k≥0 (−1) Sym ([ℓ])Pj


P (1m ) Q(n)
if hi, ji = 0.
j
i
Proof. The proof is along the same lines as that of Proposition 3.1.
Finally, it turns out that a third set of generators will be useful. These are defined as follows:
n
n
X
X
(n−m) (1m )
(1n−m ) (m)
[n]
[1n ]
(−q)m [m]Qi
Qi
(−q)m [m]Qi
Qi
and Qi :=
Qi :=
m=0
n
[1 ]
Pi
:=
n
X
(1
(−q)−m [m]Pi
)
(m)
Pi
and
[n]
Pi
:=
m=0
n
X
(n−m)
(−q)−m [m]Pi
(1m )
Pi
m=0
m=0
[1]
Pi
n−m
[1]
Qi
Note that
= −q −1 Pi and
= −qQi . These generators will be important in the next section
b n )loop .
where we discuss a simplified presentation of Uq (sl
VERTEX OPERATOR ALGEBRAS AND RELATED TOPICS
11
b n )loop in analogy
3.4. The loop presentation: revisited. In this section we define the category U̇q (sl
b n )KM . It will turn out that many of the relations in the presentation of Uq (sln )loop from
with U̇q (sl
section 3.2 are in fact redundant.
b n . Recall that X
b First we need to revisit the weight lattice X
b of sl
b is
3.4.1. The weight lattice X.
generated by Λi and the root lattice by αi where
P i = 0, 1, . . . , n − 1. However, since we are in the loop
presentation, instead of α0 we will use δ := i αi . Thus, we have the following pairings
hΛi , αj i = δi,j , hΛi , δi = 1, hαi , δi = 0.
b n ). The generators are Ei,r , Fi,r , P (r) , Q(r) together with the idempo3.4.2. A simplified form of Uq (sl
i
i
b The relations are as follows.
tents 1λ for λ ∈ X.
b are mutually orthogonal idempotents with
(i) {1λ : λ ∈ X}
Ei,r 1λ = 1µ Ei,r 1λ = 1µ Ei,r
Fi,−r 1µ = 1λ Fi,−r 1µ = 1λ Fi,−r
where µ = λ + αi + rcδ and with
(r)
(r)
(r)
Pi 1λ = 1λ−crδ Pi 1λ = 1λ−crδ Pi
(r)
(r)
(r)
Qi 1λ = 1λ+crδ Qi 1λ = 1λ+crδ Qi .
(ii) The P s and Qs satisfy relation (19) where ℓ = c.
(iii) We have

a+b
]
ac hλ,ii [1

Qi
1λ
q q
−a−b
[1
]
bc
−hλ,ii
(23)
[Ei,a , Fi,b ]1λ = q q
1λ
Pi


[hλ, ii + ac]1λ
if a + b > 0
if a + b < 0
if a + b = 0
while if i 6= j then [Ei,a , Fj,b ]1λ = 0.
(iv) For any m, n ∈ Z we have Ei,n Ei,n−1 1λ = q 2 Ei,n−1 Ei,n 1λ and Fi,n−1 Fi,n 1λ = q 2 Fi,n Fi,n−1 1λ .
(v) For any m, n ∈ Z we have
Ei,1 Ej 1λ + Ej,1 Ei 1λ = q −1 (Ej Ei,1 1λ + Ei Ej,1 1λ )
Ei,m Ej,n 1λ = Ej,n Ei,m 1λ
if
if
hi, ji = −1
hi, ji = 0
and similarly
Fi,−1 Fj 1λ + Fj,−1 Fi 1λ = q −1 (Fj Fi,−1 1λ + Fi Fj,−1 1λ )
Fi,m Fj,n 1λ = Fj,n Fi,m 1λ
if
if
hi, ji = −1
hi, ji = 0.
(vi) If hi, ji = −1 then
(Ej,n )(Ei,m )2 1λ + (Ei,m )2 (Ej,n )1λ = [2](Ei,m )(Ej,n )(Ei,m )1λ .
and similarly if we replace all Es by F s.
Theorem 3.3 ([CL2]). The relations above are equivalent to Drinfeld’s realization from section 3.2.
b n )loop as the additive category where:
So, we can now define U̇q (sl
b
• the objects are indexed by λ ∈ X,
(r)
(r)
• the morphisms are generated by Ei,r , Fi,r , Pi , Qi and 1λ
subject to the relations (i)-(vi) above. Note that, as before, relation (vi) is redundant when working
with an integrable representation.
b n )loop is equivalent to a functor U̇q (sl
b n )loop → V.
Key point. A representation of Uq (sl
12
SABIN CAUTIS
(n)
(n)
Remark 3.4. Note that one does not need to include the generators Pi and Qi in the definition
b n )loop since they can be recovered as linear combinations of compositions of E’s and F ’s from
of U̇q (sl
(23). However, it is convenient to include them since these compositions are very complicated.
3.5. The Fock space representation of U̇q (b
h). To simplify notation we will deal with the level ℓ = 1
b
b
case (i.e. U̇q (h) = U̇q (h)1 ) as the general case works exactly the same.
By a Q-integrable representation of U̇q (b
h) we will mean a functor U̇q (b
h) → V such that for any
(m) r
vector v, m ∈ N and i ∈ I we have (Qi ) (v) = 0 for r ≫ 0 (this is probably not standard terminology).
(m)
We say that v ∈ V (k) is a lowest weight vector if Qi (v) = 0 for any m ≥ 1, i ∈ I. Finally, for
(n )
(n )
n
a partition n = (n1 ≤ · · · ≤ nk ) ∈ Π we denote Pi := Pi 1 . . . Pi k where Π is the set of partitions.
n
Lemma 3.5. If v is a lowest weight vector then the set {Pi (v) : i ∈ I, n ∈ Π} is linearly independent.
(2)
(2)
Proof. Let us prove that Pi (v) and Pi Pi (v) are linearly independent. Suppose aPi (v)+bPi Pi (v) = 0
(2)
for some a, b ∈ C. Applying Qi Qi and using that Qi (v) = 0 = Qi (v) we get that (a + 2b) = 0.
(2)
Similarly, applying Qi we get [3](a + b) + b = 0. These two equations imply that a = b = 0.
Exercise 6. Extend the argument in the proof of Lemma 3.5 (not so easy).
m
Inspired by the Lemma above we define the Fock space VFock
to be the representation generated
by the lowest weight vector v which lies in weight space m. In other words, as a vector space
n
m
VFock
= Span{Pi (v) : i ∈ I, n ∈ Π}
m
with v ∈ VFock
(m). Another way to define this space is using the induction
m
VFock
:= Uq (b
h) ⊗Uq− (bh) Chvi
where Uq− (b
h) is the subalgebra generated by Q’s and v lies in degree m. Notice that
Y 1
X
m
.
dimVFock
(n)tn = tm
1 − tn
n≥1
n∈Z
0
Since VFock
is particularly important we shorten it as VFock . Note that if we think of Uq (b
h) as a graded
m
m
algebra and VFock
as graded modules then VFock
differs from VFock just by a grading shift.
Lemma 3.6. If V is an integrable representation of U̇q (b
h) then it contains a lowest weight vector.
Proof. We give a sketch. For simplicity we assume I = {i}. Take any vector in v. By integrability there exists some n1 such that Qni 1 +1 (v) = 0 but Qni 1 (v) 6= 0. Next, there exists some n2 such
(2)
(2)
that (Qi )n2 +1 Qni 1 (v) = 0 but (Qi )n2 Qni 1 (v) = 0. Now we repeat. The claim is that this process terminates, at which point we have a lowest weight vector. The latter claim is clear so we just
need to explain why we cannot have an unbounded sequence n1 , n2 , n3 , . . . (with nk ≥ 1) such that
(k)
(2)
(Qi )nk . . . (Qi )n2 Qni 1 (v) 6= 0.
The reason for this is that the set
(k)
(2)
(k)
(2)
{(Pi )nk . . . (Pi )n2 Pin1 (Qi )nk . . . (Qi )n2 Qni 1 (v), k ≥ 1}
(2)
(2)
(3)
(2)
(3)
(2)
would then be linearly independent. For example, suppose aPi Qi (v) + bPi Pi Qi Qi (v) = 0
(2) (3)
(2)
for some a, b ∈ C. Applying Qi Qi to this expression and using that Qi (v) = 0 and (Qi )2 (v) = 0
we see that the first term disappears and all but one of the terms in the expansion of the second
(2)
(3)
(3)
(3)
survives, namely when Qi annihilates Pi and Qi annihilates Qi . In this case we end up with
(3) (2)
b[4][3]Qi Qi (v) = 0 and thus b = 0 and a = 0.
VERTEX OPERATOR ALGEBRAS AND RELATED TOPICS
13
On the other hand, every weight space V (n) of V is finite dimensional. This is a contradiction and
hence there cannot be such an unbounded sequence of nonzero vectors.
(r)
Exercise 7. For r ∈ N show that Qi
of Corollary 3.7).
m
m
: VFock
(n + r) → VFock
(n) is surjective (this is used in the proof
Corollary 3.7. Any integrable representation of U̇q (b
h) is a direct sum of Fock spaces.
Proof. Consider an integrable representation V and let vs ∈ V (ns ), s ∈ S be a full set of lowest weight
vectors in V . Then we have a natural short exact sequence of Uq (b
h)-modules
M
ι
ns
′
0→
VFock −
→V →V →0
s∈S
′
where V is by definition of the quotient of ι. Note that ι is injective by (a slight generalization of)
Lemma 3.5.
We now show that V ′ = 0. For simplicity suppose S = {s} and I = {i}. If V ′ 6= 0 then by Lemma
3.6 it contains a lowest weight vector v ′ . Let v ∈ V be a lift of v ′ . Then Qi (v) is in the image of ι
n
and hence must be a linear combinations of some Pi (u), where u ∈ V is a lowest weight vector. Since
ns
Qi is surjective there exists some w ∈ VFock such that Qi (w) is this linear combination. Thus we can
(2)
(3)
replace v with v − w and then Qi (v) = 0. Now we repeat this argument with Qi , Qi and so on.
This process terminates and we end up with a new lowest weight vector in V . This is a contradiction
meaning that V ′ = 0 and we are done.
b n )loop . As a representation of Uq (sl
b n )KM , the basic repre3.6. The Basic representation of U̇q (sl
b n )loop -module,
sentation Vbasic is the highest weight module generated by v ∈ Vbasic (Λ0 ). As a Uq (sl
Vbasic is the unique irreducible representation generated by a highest weight vector v where c · v = v
and
Hi,k+1 (v) = Ei,k (v) = Fi,k (v) = 0,
for any i ∈ I, k ≥ 0.
b n )loop is isomorphic
Lemma 3.8. The restriction of Vbasic to the Heisenberg subalgebra Uq (b
h) ⊂ Uq (sl
to VFock ⊗C C[Q] where Q is the root lattice of sln .
3.7. The Frenkel-Kac-Segal construction. We now describe a construction of Vbasic due to FrenkelKac [FK] and Segal [S] when q = 1 and Frenkel-Jing [FJ] in the quantum case. In our language, this
construction yields a functor
b n )loop → U̇q (b
Ψ : U̇q (sl
h)
which, when composed with the Fock space representation ΨFock : U̇q (b
h) → V, yields the basic representation.
At the level of objects we define Ψ by
(
b
hΛ0 , µi − 12 hµ, µi if µ ∈ Q
(24)
Λ0 − µ 7→
0
otherwise.
Notice that since Vbasic has highest weight Λ0 , all of its nonzero weight spaces are of the form Λ0 − µ
b This explains why we take Λ0 − µ to zero if µ 6∈ Q.
b
for some µ ∈ Q.
At the level of morphisms we take
X
(k) (1hλ,αi i+1+r+k )
Ei,r 1λ 7→
(25)
(−q)−k Pi Qi
1Ψ(λ)
k
(26)
Fi,r 1λ
7→
X
k
(1hλ,αi i+1−r+k )
(−q)k Pi
(k)
Qi 1Ψ(λ)
14
SABIN CAUTIS
Theorem 3.9. The composition
ΨFock
Ψ
b n )loop −
−→ V
U̇q (sl
→ U̇q (b
h) −−−
b n )loop → V.
yields the basic representation Ψbasic : U̇q (sl
Proof. As it is stated, this result follows most easily from the main theorem in [CL1] where we prove
a categorified version of this result.
To understand Lemma 3.8 consider the functor
given by n 7→
L
λ:Ψ(λ)=n
λ and
(r)
Pi 1n 7→
M
b n )loop
Φ : U̇q (b
h) → U̇q (sl
(r)
Pi 1λ
and
(r)
Qi 1n 7→
λ:Ψ(λ)=n
M
(r)
Qi 1λ .
λ:Ψ(λ)=n
Ψbasic
Φ
b n )loop −
−−−→ V corresponds to the restriction of Vbasic to Uq (b
h). On
The composition U̇q (b
h) −
→ U̇q (sl
the other hand, the composition
Φ
Ψ
b n )loop −
U̇q (b
h) −
→ U̇q (sl
→ U̇q (b
h)
L
L
is given by 1n 7→ λ:Ψ(λ)=n 1λ 7→ λ:Ψ(λ)=n 1n on objects and the identity on morphisms. Finally,
we have a bijection
1
∼
Q−
→ {λ, Ψ(λ) = n}
µ 7→ Λ0 − µ − n − hµ, µi δ
2
L
which means that Ψ ◦ Φ(1n ) = λ∈Q 1n . This explains Lemma 3.8.
4. Vertex algebras
4.1. Definitions. Given a vector space V the formal power series
X
A(z) =
Ai z −i ∈ End(V )[[z ±1 ]]
i∈Z
is called a field if for any v ∈ V we have Ai · v = 0 for i ≫ 0. Two fields A(z) and B(w) are local
with respect to each other if there exists some N ∈ Z≥1 such that, inside End(V )[[z ±1 , w±1 ]], we have
(z − w)N [A(z), B(w)] = 0.
Remark 4.1. This is not the usual definition of local but it is equivalent. It serves as a more reasonable
definition for two fields commuting.
A vertex algebra consist of the following data:
• a vector space V together with a vacuum vector |0i ∈ V ,
• a linear translation operator T : V → V ,
• a linear operation
Y (·, z) : V → End(V )[[z ±1 ]]
such that for any A ∈ V is taken to a field
X
Y (A, z) =
A(i) z −i−1 .
i∈Z
This data is subject to the following conditions
VERTEX OPERATOR ALGEBRAS AND RELATED TOPICS
15
• Y (|0i, z) = idV and for any A ∈ V
Y (A, z)|0i ∈ V [[z]]
and Y (A, z)|0i|z=0 = A,
• [T, Y (A, z)] = ∂z Y (A, z) and T |0i = 0,
• all fields Y (A, z) are local with respect to each other.
A vertex algebra is Z-graded if V is a Z-graded, |0i has degree 0, T has degree 1 and if A ∈ Vm then
degA(n) = −n + m − 1.
Remark 4.2. The action of T on V is determined by T · A = A(−2) |0i. However, it is often convenient
to include T as part of the definition.
A vertex algebra homomorphism between vertex algebras (V1 , |0i1 , T1 , Y1 ) and (V2 , |0i2 , T2 , Y2 )
is a linear map ρ : V1 → V2 such that ρ(|0i1 ) = |0i2 , ρT1 = T2 ρ and
ρ(Y1 (A, z)B) = Y2 (ρ(A), z)ρ(B).
A vertex subalgebra V ′ ⊂ V is a T -invariant subspace containing |0i and satisfying Y (A, z)B ∈
V ((z)) for any A, B ∈ V ′ .
Given two vertex algebras the data
′
(V1 ⊗C V2 , |0i1 ⊗ |0i2 , T1 ⊗ 1 + 1 ⊗ T2 , Y )
where Y (A1 ⊗ A2 , z) = Y1 (A1 , z) ⊗ Y2 (A2 , z) defines a tensor product structure on vertex algebras.
The normally ordered product of the fields A(z) and B(w) is
: A(z)B(w) := A(z)+ B(w) + B(w)A(z)−
P
where for a formal power series f (z) = n∈Z fn z n we set
X
X
f (z)+ =
fn z n , f (z)− =
fn z n .
n<0
n≥0
4.2. Example #1: the Heisenberg vertex algebra. Recall that the Heisenberg algebra associated
b 2 )) is generated by Hk , k ∈ Z \ {0} subject
to the Cartan datum (2) (in other words, associated to Uq (sl
[cl]
to [Hk , Hl ] = [2l]δk,−l l .
We now consider the Heisenberg algebra associated to the datum matrix (1) and take q = 1 = c.
This gives the algebra with generators as above and relation
[Hk , Hl ] = lδk,−l .
This is essentially the simplest Heisenberg algebra one can consider. The Fock space
VFock := Uq (b
h) ⊗Uq− (bh) Chvi
is spanned by H−i1 . . . H−ik (v) where i1 ≥ · · · ≥ ik ≥ 1 and v is in degree zero. We will write
vi := H−i1 . . . H−ik (v) for short.
On can give VFock the structure of a vertex algebra as follows. We take v = |0i and define
T · vi =
k
X
ij H−i1 . . . H−ij +1 . . . H−ik (v).
j=1
This definition is equivalent to the more elegant condition that T · v = 0 and [T, H−i ] = iH−i+1 .
Next we need to define Y (vi ). We begin by defining
X
(27)
Y (H−1 (v), z) =
Hn z −n−1
n∈Z
16
SABIN CAUTIS
where on the right side the Hn are operators acting on VFock . More generally,
X
1
Y (H−i (v), z) =
Hn z −n−1 .
∂zi−1
(i − 1)!
n∈Z
Finally, for an arbitrary vi we let
Y (vi , z) =
where f (z) =
[CL1]
[CL2]
[FJ]
[FK]
[Ka]
[Dr]
[N1]
[N4]
[S]
P
n∈Z
1
: ∂ i1 −1 f (z) . . . ∂zik −1 f (z) :
(i1 − 1)! . . . (ik − 1)! z
Hn z −n−1 .
References
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