18.276 SPRING 2015 LECTURE NOTES
LECTURE BY VICTOR KAC, NOTES BY DANIEL THOMPSON AND DONGKWAN KIM
Abstract. These are notes of lectures by Victor Kac in 18.276 Vertex Algebras at MIT in Spring 2015.
Lecture 9
All definitions for algebras below admit straightforward generalizations to the case of superalgebras using
the Quillen rule for parity.
9.1. The correspondence between formal distributions Lie algebras and Lie conformal algebras.
Let g be a Lie algebra and g[[w, w−1 ]] the space of g-valued formal distributions. This is an F[∂]-module via
(∂a)(w) = ∂w a(w), and it is closed with respect to all nth products a(w)(n) b(w), n ∈ Z+ .
For a = a(w), b = b(w) ∈ g[[w, w−1 ]] define the λ-bracket by
λ
[aλ b](w) = Fz,w
[a(z), b(w)]
= Res eλ(z−w) [a(z), b(w)]dz
X
λn
(z − w)n [a(z), b(w)]dz
=
Res
n!
n≥0
=
X λn
a(w)(n) b(w).
n!
n≥0
Note that if the pair (a, b) is local, then [aλ b](w) ∈ g[[w, w−1 ]][λ].
Proposition 9.1. The λ-bracket satisfies the following properties
(sesquilinearity) [(∂a)λ b] = −λ[aλ b], [aλ (∂b)] = (∂ + λ)[aλ b]
(Jacobi) [aλ [bµ c]] = [[aλ b]λ+µ c] + [bµ [aλ c]]
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(skew-commutativity) if (a, b) is a local pair then [bλ a] = −[a−λ−∂ b] = − n≥0
(−λ−∂)n
(a(n) b)
n!
λ
stated last time:
Proof. The proof uses the properties of Fz,w
λ
λ
λ
∂z a(z, w) = −λFz,w
a(z, w) = [∂w , Fz,w
]a(z, w)
(α) Fz,w
λ
−λ−∂w
(γ) Fz,w a(w, z) = Fz,w
a(z, w) if a(z, w) is local
λ
µ
λ+µ λ
(δ) Fz,w
Fx,w
= Fx,w
Fz,x .
Exercise 9.2. Derive sesquilinearity of the λ-bracket from (α).
Solution.
λ
λ
λ
[(∂a)λ b] = Fz,w
[∂z a(z), b(w)] = Fz,w
∂z [a(z), b(w)] = −λFz,w
[a(z), b(w)] = −λ[aλ b]
λ
λ
λ
[aλ (∂b)] = Fz,w
[a(z), ∂w b(w)] = Fz,w
∂w [a(z), b(w)] = (∂w + λ)Fz,w
[a(z), b(w)] = (∂ + λ)[aλ b]
Exercise 9.3. Derive skew-commutativity of the λ-bracket from (γ).
Solution. Since (a, b) is a local pair,
λ
−λ−∂w
[bλ a] = Fz,w
[b(z), a(w)] = Fz,w
[b(w), a(z)] = −[a−λ−∂ b]
Date: April 23, 2015.
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LECTURE BY VICTOR KAC, NOTES BY DANIEL THOMPSON AND DONGKWAN KIM
The Jacobi identity follows from property (δ) and the Jacobi identity for Lie algebras:
[a(x), [b(z), c(w)]] = [[a(x), b(z)], c(w)] + [b(z), [a(x), c(w)]]
Now apply
λ
µ
Fz,w
Fx,w
to both sides:
λ
µ
[aλ [bµ c]](w) = Fz,w
[a(x), Fx,w
[b(z), c(w)]]
λ+µ λ
λ
µ
= Fx,w
Fz,x [[a(x), b(z)], c(w)] + Fz,w
Fx,w
[b(z), [a(x), c(w)]]
= [[aλ b]λ+µ c] + [bµ [aλ c]].
Definition 9.4. A Lie conformal algebra is an F[∂]-module R, endowed with a λ-bracket [aλ b] : R⊗R → R[λ]
satisfying sesquilinearity, Jacobi, and skew-commutativity.
We have a map {formal dist. Lie alg.} → {Lie conformal alg.} as follows. Given a FDLA (g, F), let F be
the minimal space of g-valued formal distributions which contain F, is ∂w invariant, and is closed under all
nth products, n ∈ Z+ .
Exercise 9.5. Dong’s lemma holds: if a, b, c ∈ g[[w±1 ]] are pairwise local, then ((a(n) b), c) is a local pair for
any n ∈ Z+ .
Solution. The usual proof of Dong’s lemma is still valid here. One is referred to Lecture Note 6, Section 3.
Hence (g, F) is an FDLA, isomorphic by definition to our original FLDA (g, F). By the proposition, the
λ
λ-bracket on F given by [aλ b](w) = Fz,w
[a(z), b(w)] is a Lie conformal algebra.
9.2. Application to a vertex algebra V. Recall that we proved that (V, F) is a FDLA with respect to
the bracket of operators, and F = F. Hence F is a Lie conformal algebra with respect to the λ-bracket.
Therefore, by the nth product identity, we get
Theorem 9.6. (V, F[T ], [aλ b]) is a Lie conformal algebra.
Theorem 9.7. Let V be a vertex algebra with |0i vacuum vector, T translation operator, and a(n) b for
n ∈ Z. Then (V, |0i, T, : ab := a(−1) b, [aλ b]) satisfies the following properties:
(a) (V, F[T ], [·λ ·]) is a Lie conformal algebra
(b) (V, |0i, T, : :) is a unital differential algebra with identity element |0i, T derivation, which satisfies
R0
(1) (quasi-commutativity) : ab : − : ba := −T [aλ b]dλ
RT
RT
(2) (quasi-associativity) :: ab : c : − : a : bc ::=: ( 0 adλ)[bλ c] : + : ( 0 bdλ)[aλ c] :
(c) : : and [·λ ·] are related by the noncommutative Wick formula
Z λ
[aλ : bc :] =: [aλ b]c : + : b[aλ c] : +
[[aλ b]µ c]dµ
0
The converse of this theorem is also true; it can be found in [2].
9.3. A detour on Lie conformal algebras. Conversely, given a Lie conformal algebra R, one constructs
the maximal FDLA (LieR, F), where
LieR = R[t, t−1 ]/(∂ + ∂t )R[t, t−1 ],
X
F = {a(z) =
a(n) z −n−1 }a∈R ,
n∈Z
where a(n) denotes the image of atn . The bracket on F is given by
X m
[a(m) , b(n) ] =
(a(j) b)(m+n−j) ,
j
j≥0
th
with a(j) b the j product in R.
Then (LieR, F) is a regular FDLA with T = ∂. Moreover, any FDLA (g, F), with F = R is obtained as
a quotient of (LieR, F) by an “irregular” ideal.
18.276 SPRING 2015 LECTURE NOTES
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Exercise 9.8. Let ĝ = g[t, t−1 ] ⊕ FK, K central, be the affine (untwisted) Lie algebra corresponding to the
finite-dimensional Lie algebra g. Recall the bracket is given by
[atm , btn ] = [a, b]tm+n + mδm,−n (a|b)K.
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This is a FDLA with F = {a(z) = n∈Z (atn )z −n−1 }a∈g ∪ {K}.
Then F = F[∂z ]F ⊕ FK with the λ-bracket
(1)
[aλ b] = [a, b] + λ(a|b)
for a, b ∈ g, [Kλ a] = 0, extended to F by sesquilinearity. The LCA F is isomorphic to the current LCA:
Cur g = (F[∂] ⊗ g) ⊕ F
with the λ-bracket as in (1). The maximal FDLA is ĝ.
Note that (ĝ/(ρ(t)), F), where ρ is a non-invertible Laurent polynomial, produces the LCA Cur g under
the above correspondence.
Solution. First note that F̄ should be defined as given since it is the minimal space which contains F and
is closed under ∂z . Meanwhile, the bracket in F is defined by
X
[a(z), b(w)] =
[atm , btn ]z −m−1 w−n−1
m,n∈Z
=
X
m,n∈Z
=
X
[a, b]tm+n z −m−1 w−n−1 +
X
m(a|b)Kz −m−1 wm−1
m∈Z
([a, b]t
m+n
w
−m−n−1
)z
−m−1
wm + ∂w
m,n∈Z
X
(a|b)Kz −m−1 wm
m∈Z
!
=
X
n
[a, b]t w
−n−1
+ (a|b)K∂w
n∈Z
!
X
z
−m−1
w
m
m∈Z
= ([a, b](w) + (a|b)K∂w ) δ(z − w).
It implies that a(0) b = [a, b], a(1) b = (a|b)K, and the other products are zero. Thus by definition of λ-bracket
we have [aλ b] = [a, b] + λ(a|b)K. But it is the same as the λ-bracket on Cur g, thus by sesquilinearity F̄
and Cur g are isomorphic. Conversely, the maximal FDLA associated with Cur g is given by Lie Cur g =
(Cur g)[t, t−1 ]/(∂ + ∂t )(Cur g)[t, t−1 ] = g[t, t−1 ] ⊕ F = ĝ as desired.
Example 9.9. The Virasoro LCA, correspoinding to the Virasoro FDLA (V ir, F = {L(z) =
3
is V ir = F[∂]L + FC with [Lλ L] = (∂ + 2λ)L + λ12 c.
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References
[1] Kac, Victor G. Vertex algebras for beginners. Vol. 10. Providence: American Mathematical Society, 1998.
[2] Bakalov, Bojko, and Victor G. Kac. ”Field algebras.” arXiv preprint math.QA/0204282 (2002).
E-mail address: dthomp@math.mit.edu
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Ln z −n−2 , C}),