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VICTOR G. KAC
1. The calculus of formal distributions 2 2. Formal Fourier Transform 7 A digression on Superalgebras 8 3. Lie conformal algebras 11 Relation of Lie conformal algebras to formal distribution Lie superalgebras 12 4. Normally ordered products 19 Operator Product Expansion 20 Completions 21 5. Wick formulas 23 Rules for calculating λ-bracket 29 6. Free (super)fermions 30 7. Bosonization and the Sugawara construction 34 8. Restricted representations 40 A Construction of a Restricted Representation 43 9. Boson-Fermion correspondence 44 An Example of a Vertex Algebra 44 Boson-Fermion Correspondence 50 10. Definition of vertex algebra 52 11. Uniqueness and n-product theorems 56 12. Existence theorem 63 13. Examples of vertex algebras 68 Applications of the existence Theorem 68 14. Poisson vertex algebras 71 15. Infinite-dimensional Hamiltonian Systems 75 Relation to Poisson Vertex Algebras 78 16. Bi-Hamiltonian systems 79 17. Lattice vertex algebras I - translation invariance 83 Lattice Vertex Algebras 85 18. Lattice vertex algebras II - locality 90 19. Lattice vertex algebras III - uniqueness 95 Existence and uniqueness of � 96 20. Borcherds identity 101 Representation Theory of Vertex algebras 103 21. Representations of vertex algebras 104 22. Representations of lattice vertex algebras 110 23. Rational vertex algebras 112
Date: February 2003.
2 VICTOR G. KAC
24. Integrable modules 117 25. Twisting procedure 120 References 123
The basic bibliography for this class will be . The first rigorous definition of Vertex Algebras was given by Borcherds his famous 1986 paper on the monster groupi .
1. The calculus of formal distributions
Let U be a vector space (over the complex numbers generally)
1.1. Definition. An U-valued formal distribution is an expression
a(z) = ∑ n∈Z
where an ∈ U and z is an indeterminate. Their space is denoted by U [[z, z−1]]. Note also that U [z, z−1] will denote the space of Laurent polynomials with values in U . The linear function
Resz a(z) := a−1,
is called the residue. Clearly satisfies
Resz ∂za(z) = 0
If we consider C[z, z−1] as space of tests functions then any U-valued formal distribution induces a linear map C[z, z−1] → U by
fa ( ϕ(z)
) = Resz ϕ(z)a(z).
Exercise 1.1. Show that all U-valued linear functions on the space C[z, z−1] are obtained uniquely in this way.
Proof. Let f : C[z, z−1] → U be a linear map, define the formal distribution a(z) by
(1.1.1) a(z) = ∑ n∈Z
f ( z−1−n
Hence clearly we have f(zn) = Resz zna(z) and by linearity existence follows. Uniqueness is checked similarly:
(1.1.2) a(z) = ∑
Multiplying both sides of (1.1.2) by zk and taking residues we get f(zk) = a−1−k in accordance with (1.1.1) �
Following the last exercise, it is natural to define a(n) = Resz zna(z), then the original formal distribution is expressed as
(1.1.3) a(z) = ∑ n∈Z
The vectors a(n) are called Fourier Coefficients of the formal distribution.
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Similarly one can define formal distributions in two or more variables as
a(z, w, . . . ) = ∑
nwm . . . , am,n,... ∈ U
Denote by A the algebra of rational functions of the form (z−w)−kP (z, w) where P (z, w) ∈ C[z, z−1, w, w−1] and k is an integer number. Consider the following map called expansion in the domain |z| > |w| defined when k < 0:
iz,wz mwn(z − w)k = zmwnzk
( 1− w
)k = zm+kwn
)−k = zm+kwn
( 1 +
z + (w z
)2 + . . .
And the obvious expansion when k ≥ 0, we can extend linearly to the rest of A. Similarly we define the expansion in the domain |w| > |z| and denote it by iw,z.
Exercise 1.2. These maps are homomorphisms which commute with multiplication by zm, wn, and with ∂z and ∂w.
Proof. These maps are C linear by definition. Now the following calculation( iz,w(z − w)−1
)2 = ( 1 +
z + . . .
)( 1 +
z + . . .
) = ( 1 +
z + . . .
)2 = iz,w(z − w)−2
is enough to prove that iz,w is an homomorphism. The fact that these maps commute with multiplication by zm and wm is obvious. The only non-trivial case remaining to prove in the exercise is
∂ziz,w(z − w)−j−1 =
= ∂z ∞∑
= − ∞∑
) (m+ 1)z−m−2wm−j
= − ∞∑
(m+ 1)! (j + 1)!(m+ 1− j − 1)!
(j + 1)z−(m+1)−1wm+1−j−1
= −(j + 1) ∞∑
j + 1
= −(j + 1)iw,z(z − w)−j−2
= iz,w∂z(z − w)−j−1
and similarly for ∂w. �
4 VICTOR G. KAC
1.2. Definition. The formal δ-function is defined by
(1.1.4a) δ(z, w) = iz,w 1
z − w − iw,z
1 z − w
And substituting the definition for the corresponding expansions we get
(1.1.4b) δ(z, w) = z−1 ∑ n∈Z
)n Differentiating equations (1.1.4a) and (1.1.4b) we get
1 n! ∂nwδ(z, w) = iz,w
1 (z − w)n+1
− iw,z 1
(z − w)n+1 (1.2.5a)
= ∑ j∈Z
Exercise 1.3. Show there is a unique formal distribution, δ(z, w) ∈ C[z±1, w±1] such that Resz δ(z, w)ϕ(z) = ϕ(w) for any test function ϕ(z) ∈ C[z, z−1].
Proof. Existence is proposition 1.3(6) below. Conversely, if δ = ∑ δn,mz
nwm and we have a similar decomposition for δ′. Then we can compute
Resz δ(z, w)zk = ∑
δ−1−k,mw m = wk =
Now comparing coefficients we have δ−1−k,m = δ′−1−k,m ∀k,m ∈ Z as we wanted. �
1.3. Proposition. The formal distribution δ(z, w) satisfies the following properties
(1) (locality) (z − w)m∂nwδ(z, w) = 0 whenever m > n. (2) (z − w) 1n!∂
n wδ(z, w) =
n−1 w δ(z, w) if n ≥ 1.
(3) δ(z, w) = δ(w, z). (4) ∂zδ(z, w) = −∂wδ(w, z). (5) a(z)δ(z, w) = a(w)δ(z, w) where a(z) is any formal distribution. (6) Resz a(z)δ(z, w) = a(w) (7) exp(λ(z − w))∂nwδ(z, w) = (λ+ ∂w)nδ(z, w)
Proof. Properties (1)-(4) follows easily from the definitions and equations (1.2.5a). Perhaps properties (3) and (4) justifies the usual notation δ(z − w) for δ(z, w). Now because of (1) we have (z − w)δ(z, w) = 0 hence znδ(z, w) = wnδ(z, w) and (5) follows by linearity. Now taking residues in property (5) we get
Resz a(z)δ(z, w) = a(w)Resz δ(z, w) = a(w)
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so (6) follows. Now in order to prove (7) we expand the exponential as
eλ(z−w)∂nwδ(z, w) = ∞∑
k! n!(z − w)k ∂
n wδ(z, w) n!
= ∑ k=0
n λkn! k!
∂n−kw δ(z, w) (n− k)!
) δ(z, w)
= (λ+ ∂w) n δ(z, w)
1.4. Definition. An U-valued formal distribution is called local if
(z − w)na(z, w) = 0 n >> 0
1.5. Example. δ(z, w) is local and so are its derivatives. Also a(z, w) local implies that a(w, z) is local.
Exercise 1.4. Show that if a(z, w) is local so are ∂wa(z, w) and ∂za(z, w).
Proof. Of course by symmetry it is enough to prove the result for one derivative. If (z − w)n−1a(z, w) = 0 then we have:
(z − w)n∂za(z, w) = ∂z(z − w)na(z, w)− a(z, w)∂z(z − w)n
= −na(z, w)(z − w)n−1
proving the exercise. �
1.6. Theorem (Decomposition). Let a(z, w) be a U-valued local formal distribution. Then a(z, w) can be uniquely decomposed as in the following finite sum
(1.5.1a) a(z, w) = ∑
cj(w) ∂jwδ(z, w)
where cj(w) ∈ U [[w,w−1]] are formal distributions given by
(1.5.1b) cj(w) = Resz(z − w)ja(z, w)
Proof. Let b(z, w) = a(z, w) − ∑ cj(w)∂
j! δ(z, w). Clearly b(z, w) is local being a finite linear combination of local formal distributions. Note that
Resz(z − w)nb(z, w) =
= Resz(z − w)na(z, w)− Resz(z − w)n ∑
cj(w)∂ j wδ(z,w)
j! δ(z, w)
= cn(w)− ∑
(j − n)! Resz δ(z, w)
= cn(w)− cn(w) = 0
6 VICTOR G. KAC
Now write b(z, w) = ∑ znbn(w). Then we have Resz b(z, w) = b−1(w) = 0. By
the above calculation we have 0 = Resz(z − w)b(z, w)
= Resz zb(z, w)− wResz b(z, w) = Resz zb(z, w)
= b−2(z, w) = 0
and iterating we have b(z, w) = ∑
n∈Z+ bnz n. Now since b is local we have
(z − w)nb(z, w) = 0 ⇒ b(z, w) = 0
as we wanted. To show uniqueness we take residues on both sides of (1.5.1a), obtaining:
Resz(z − w)ka(z, w) = Resz(z − w)k ∑
cj(w) ∂jwδ(z, w)
j! = ck(w)
In accordance with (1.5.1b) . �
Given a(z, w) ∈ U [[z±1, w±1]]] we may define a linear operator