Abstract
These lectures were given in Session 1: “Vertex algebras, W-algebras, and applications” of INdAM Intensive research period “Perspectives in Lie Theory” at the Centro di Ricerca Matematica Ennio De Giorgi, Pisa, Italy, December 9, 2014–February 28, 2015.
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Keywords
- Hamiltonian PDE
- Lenard-Magri scheme
- Lie conformal algebra
- Poisson vertex algebra
- Pre-vertex algebra
- Quantum field
- Variational complex
- Vertex algebra
- Zhu algebra
-
U[z]: polynomials with coefficients in a vector space U.
-
U[z, z −1]: Laurent polynomials.
-
U[[z]]: formal power series.
-
U((z)): formal Laurent series.
-
U[[z, z −1]]: bilateral series.
-
\(\mathbb{Z}_{+} =\{ 0,1,2,\ldots \}\).
-
\(\mathbb{F}\): the base field, a field of characteristic 0. All vector spaces are considered over \(\mathbb{F}\).
These lecture notes were typeset by Vidas Regelskis (Lectures 1 and 6), Tamás F. Görbe (2), Xiao He (3), Biswajit Ransingh (4) and Laura Fedele (5 and 6).
The author would like to thank all the above mentioned scribes for their work, especially Laura Fedele and Vidas Regelskis for many corrections to the edited manuscript.
1 Lecture 1 (December 9, 2014)
In the first lecture we give the definition of a vertex algebra and explain calculus of formal distributions. We end the lecture by giving two examples of non-commutative vertex algebras: the free boson and the free fermion.
1.1 Definition of a Vertex Algebra
From a physicist’s point of view, a vertex algebra can be understood as an algebra of chiral fields of a 2-dimensional conformal field theory. This point of view is explained in my book [17].
From a mathematician’s point of view a vertex algebra can be understood as a natural “infinite” analogue of a unital commutative associative differential algebra. Recall that a differential algebra is an algebra V with a derivation T. A simple, but important remark is that a unital algebra V is commutative and associative if and only if
where \(\hat{a}\) denotes the operator of left multiplication by a ∈ V.
Exercise 1
Prove this remark.
A vertex algebra is roughly a unital differential algebra with a product, depending on a parameter z, satisfying a locality axiom, similar to (1). To be more precise, let me first introduce the notion of a z-algebra. (Sorry for the awkward name, but I was unable to find a better one.)
Definition 1
A z-algebra is a vector space V endowed with a bilinear (over \(\mathbb{F}\)) product, valued in V ((z)), a ⊗ b ↦ a(z)b, endowed with a derivation T of this product:
-
(i)
T(a(z)b) = (Ta)(z)b + a(z)Tb,
such that the following consistency property holds:
-
(ii)
(Ta)(z) = ∂ z a(z).
Here and further on we denote by a(z) the operator of left multiplication by a ∈ V in the z-algebra V. Using the standard notation
we can write
The bilinear (over \(\mathbb{F}\)) product a (n) b is called the nth product. Note that a(z) is an \(\mathop{\mathrm{End}}\nolimits V\) -valued distribution, i.e., an element of \((\mathop{\mathrm{End}}\nolimits V )[[z,z^{-1}]]\). Moreover, a(z) is a quantum field, i.e., a (n) b = 0 for b ∈ V and sufficiently large n (depending on b).
Remark 1
Axioms (i) and (ii) of a z-algebra imply the following translation covariance property of a(z):
Moreover, the translation covariance of a(z) and either of the axioms (i) or (ii) in Definition 1 imply the other axiom.
Next, we define a unital z-algebra.
Definition 2
A unit element of a z-algebra V is a non-zero vector 1 ∈ V, such that
Lemma 1
Let V be a vector space, let 1 ∈ V and \(T \in \mathop{\mathrm{End}}\nolimits V\) be such that T1 = 0. Then
-
(a)
For any translation covariant (with respect to T) quantum field a(z), we have a(z)1 ∈ V [[z]].
-
(b)
$$\displaystyle{ a(z)\mathbf{1} = e^{zT}a\;\;(=\sum _{ n=0}^{\infty }\frac{z^{n}} {n!} \,T^{n}(a)),\mathit{\mbox{ where }}a = a_{ (-1)}1. }$$(5)
Proof
For (a) we have to prove that a (n)1 = 0 for all \(n \in \mathbb{Z}_{+}\). Since a(z) is a quantum field, a (n)1 = 0 for \(n\geqslant N\) with some \(N \in \mathbb{Z}_{+}\). Also by translation covariance we have [T, a (n)] = −na (n−1) for all \(n \in \mathbb{Z}\). Apply both sides of the last equality to 1:
Therefore a (n)1 = 0 for n > 0 implies a (n−1)1 = 0. Hence a (n)1 = 0 for all \(n \in \mathbb{Z}_{+}\) and a(z)1 ∈ V [[z]].
Now we prove (b). By (a), the LHS of (5) lies in V [[z]]. Both sides are solutions of the differential equation
For the RHS it is obvious, and for the LHS it follows from (4) and T1 = 0:
Both sides obviously satisfy the same initial condition f(0) = a, hence they are equal. □
Since, 1(−1)1 = 1 and T is a derivation of nth products, we have in a unital z-algebra:
Note that Lemma 1(a) implies that a(z)1 ∈ V [[z]], and by Lemma 1(b) one actually has (5). Lemma 1(b) implies that
so that the derivation T is “built in” the product of a unital z-algebra.
Now we can define a vertex algebra.
Definition 3
-
(a)
A z-algebra is called local if
$$\displaystyle\begin{array}{rcl} & & (z - w)^{N_{ab} }a(z)b(w) \\ & & = (z - w)^{N_{ab} }b(w)a(z),\mbox{ for some }N_{ab} \in \mathbb{Z}_{+}(\mbox{ depending on }a,b \in V ).\qquad {}\end{array}$$(11) -
(b)
A vertex algebra is a local unital z-algebra.
A frequently asked question is: why one cannot cancel \((z - w)^{N_{ab}}\) on both sides of (11)? As we will see in a moment, the answer is: due to the existence of the delta function. In fact, the case N ab = 0 for all a, b ∈ V is not very interesting, since all such vertex algebras correspond bijectively to unital commutative associate differential algebras, as Exercise 2 below demonstrates.
Example 1
A commutative vertex algebra, i.e., [a(z), b(w)] = 0 for all a, b ∈ V, can be constructed as follows. Take V to be a unital commutative associative algebra with a derivation T. Then V is a commutative vertex algebra with the product a(z)b = (e zT a)b.
Exercise 2
Check that the above example is indeed a commutative vertex algebra. Using Lemma 1, prove that all commutative vertex algebras are of the form given in Example 1.
Remark 2
A unital z-algebra V is a vector space with unit element 1 and bilinear products a (n) b, \(n \in \mathbb{Z}\). (Recall that T is obtained by (10).) Through these bilinear products we can naturally define z-algebra homomorphisms/isomorphisms, and subalgebras/ideals. Namely, a homomorphism between two z-algebras V and V ′ is a linear map f such that f(1) = 1 and \(f(a)_{(n)}f(b) = f(a_{(n)}b),\,\forall a,b \in V\) and \(\forall n \in \mathbb{Z}\). It is an isomorphism if it is a homomorphism of z-algebras and also an isomorphism as vector spaces. A subalgebra is a subspace W of V which contains 1, such that \(a_{(n)}b \in W,\,\forall a,b \in W\) and \(\forall n \in \mathbb{Z}\). And an ideal is a subspace I such that \(a_{(n)}b,b_{(n)}a \in I,\,\forall a \in V,\,\forall b \in I\) and \(\forall n \in \mathbb{Z}\). Note that both a subalgebra and an ideal are T-invariant due to (9), and if an ideal I contains 1, then it must be the whole vertex algebra V.
Now I will give another definition of a vertex algebra, in the spirit of quantum field theory, using language closer to physics: a unit element is called a vacuum vector, element of a vector space is called a state, etc.
Definition 4
A vertex algebra is a vector space V (the space of states) with a non-zero vector | 0〉 (the vacuum vector) and a linear map from V to the space of \(\mathop{\mathrm{End}}\nolimits V\)-valued quantum fields (the state-field correspondence) a ↦ a(z), satisfying the following axioms:
-
vacuum axiom: | 0〉(z) = I V , a(z) | 0〉 = a + (Ta)z + …, where \(T \in \mathop{\mathrm{End}}\nolimits V\);
-
translation covariance axiom (4);
-
locality axiom (11).
Remark 1 demonstrates that a vertex algebra defined in the spirit of differential algebra is a vertex algebra defined in the spirit of quantum field theory. However, in order to prove the converse, one has to show that axiom (ii) of Definition 1 holds. This will follow from the proof of the Extension theorem in Lecture 3.
Definition 5
Given a vertex algebra V, the map of the space of its quantum fields to V, defined by
is called the field-state correspondence. This map is obviously surjective. If this map is also injective, then the inverse map
is called the state-field correspondence.
The first fundamental theorem, which allows one to construct non-commutative vertex algebras, is the so-called Extension theorem.
Theorem 1 (Extension Theorem)
Let V be a vector space, | 0〉 ∈ V a non-zero vector, \(T \in \mathop{\mathrm{End}}\nolimits V\) and
a collection of \(\mathop{\mathrm{End}}\nolimits V\) -valued quantum fields indexed by a set J. Suppose that the following properties hold:
-
(i)
(vacuum axiom) T | 0〉 = 0,
-
(ii)
(translation covariance) [T, a j(z)] = ∂ z a j(z) for all j ∈ J,
-
(iii)
(locality) \((z - w)^{N_{ij}}[a^{i}(z),a^{\,j}(w)] = 0\) for all i, j ∈ J with some \(N_{ij} \in \mathbb{Z}_{+}\) ,
-
(iv)
(completeness) \(\mathop{\mathrm{span}}\nolimits \{a_{(n_{1})}^{j_{1}}\cdots a_{(n_{ s})}^{j_{s}}\vert 0\rangle \mid j_{ i} \in J,\ n_{i} \in \mathbb{Z},\ s \in \mathbb{Z}_{+}\} = V\) .
Let
denote the set of all translation covariant quantum fields a(z) such that a(z), a
j(z) is a local pair for all j ∈ J. Then the map
is bijective and the inverse map
endows V with a structure of a vertex algebra (in the sense of Definition
4
) with vacuum vector
\(\left \vert 0\right>\)
and translation operator T.
Remark 3
By conditions (ii) and (iii) we have 
, hence the name Extension theorem.
Some historical remarks:
-
Vertex algebras first appeared implicitly in the paper of Belavin et al. [4] in 1984.
-
The first definition of vertex algebras was given by Borcherds [5] in 1986.
-
The Extension Theorem was proved in [6]. In [17] a weaker version was given.
-
Connection to physics (Wightman axioms of a quantum field theory [20] in the 1950s) is discussed e.g. in [17].
Remark 4 (Super Version)
A vertex superalgebra V is a local unital z-superalgebra V, cf. Definition 3. Namely V is a vector superspace
a(z)b ∈ V α+β ((z)) if a ∈ V α , b ∈ V β , and TV α ⊂ V α , \(\alpha,\beta \in \mathbb{Z}/2\mathbb{Z}\). An element a ∈ V has parity p(a) = α if a ∈ V α . Finally, the locality axiom (11) is written as \((z - w)^{N_{ab}}[a(z),b(w)] = 0\), where the commutator is understood in the “super” sense, i.e.
All the identities in the “super” case are obtained from the respective identities in the purely even case by the Koszul-Quillen rule: there is a sign change if the order of two odd elements is reversed; no change otherwise. It is a general convention to drop the adjective “super” in the case of vertex superalgebras.
1.2 Calculus of Formal Distributions
Definition 6
Let U be a vector space. A U-valued formal distribution a(z) is an element of U[[z, z −1]]:
The residue of a(z) is
Most often one uses a different indexing of coefficients:
Note that a(z) is a linear function on the space of test functions \(\mathbb{F}[z,z^{-1}]\):
and it is easy to see that one thus gets all linear functions on \(\mathbb{F}[z,z^{-1}]\).
A formal distribution in two variables z and w is an element a(z, w) ∈ U[[z, z −1, w, w −1]].
Definition 7
A formal distribution a(z, w) is called local if (z − w)N a(z, w) = 0 for some \(N \in \mathbb{Z}_{+}\).
Example 2
The formal delta function δ(z, w), defined by
is an example of an \(\mathbb{F}\)-valued formal distribution in two variables. It is local since (z − w)δ(z, w) = 0. In fact, one can write δ(z, w) as
where i z, w denotes the expansion in the domain | z | > | w | and i w, z denotes the expansion in the domain | z | < | w |, i.e.
The following formula, which is derived by differentiating (21) and (22) \(n \in \mathbb{Z}_{+}\) times, will be useful:
Let us list some properties of the formal delta function, which are straightforward by (24):
-
1.
\((z-w)^{m}\dfrac{\partial _{w}^{n}\delta (z,w)} {n!} = \left \{\begin{array}{@{}l@{\quad }l@{}} \dfrac{\partial _{w}^{n-m}\delta (z,w)} {(n - m)!} \quad &\text{if}\ n\geqslant m\geqslant 0, \\ 0, \quad &\text{if}\ m> n,\end{array} \right.\)
-
2.
δ(z, w) = δ(w, z),
-
3.
∂ z δ(z, w) = −∂ w δ(z, w),
-
4.
a(z)δ(z, w) = a(w)δ(z, w), where a(z) is any formal distribution,
-
5.
\(\mathop{\mathrm{Res}}\nolimits a(z)\delta (z,w)dz = a(w)\).
Theorem 2 (Decomposition Theorem)
Any local formal distribution a(z, w) can be uniquely decomposed as a finite sum of derivatives of the formal delta function with formal distributions in w as coefficients:
Moreover
Proof
Multiply both sides of (25) by (z − w) j and take residues. Using properties of the delta function listed above we obtain (26). To show (25) we set
with c j(w) given by (26). It is immediate that
hence
By definition b(z, w) is local, therefore (29) implies that b(z, w) = 0. □
Remark 5
If we have a local pair \(a(z),b(z) \in \mathfrak{g}[[z,z^{-1}]]\), where \(\mathfrak{g}\) is a Lie (super)algebra (i.e. [a(z), b(w)] is a local formal distribution in z and w), then, by the Decomposition theorem, we have:
where the sum over j is finite, and
Using (24) and comparing the coefficients of z m w n on both sides of (30), we find
1.3 Free Boson and Free Fermion Vertex Algebras
Example 3 (Free Boson)
Let \(B = \mathbb{F}[x_{1},x_{2},\ldots ]\), | 0〉 = 1, \(T =\sum _{j\geqslant 2}jx_{j} \frac{\partial } {\partial x_{j-1}}\) and
so that
The quantum field a(z) is called the free boson field. Since (34) is equivalent to
we have
i.e., a(z) is local with itself.
The translation covariance of the free boson field a(z), that is [T, a (n)] = −na (n−1), \(\forall n \in \mathbb{Z}\), can be verified directly. Vacuum axiom and completeness are obviously satisfied. Locality is (36). So, by the Extension theorem, B carries a vertex algebra structure.
Example 4 (Free Fermion)
Let F = Λ[ξ 1, ξ 2, …] be a Grassmann superalgebra, i.e.,
Let | 0〉 = 1 and \(T =\sum _{j\geqslant 1}j\xi _{j+1} \frac{\partial } {\partial \xi _{j}}\), where \(\frac{\partial } {\partial \xi _{j}}\) is an odd derivation of the superalgebra F (i.e. \(\frac{\partial (ab)} {\partial \xi _{j}} = \frac{\partial a} {\partial \xi _{j}} b + (-1)^{p(a)}\frac{\partial b} {\partial \xi _{j}}\)), such that
Set
then
The odd quantum field φ(z) is called the free fermion field. Since (39) is equivalent to
φ(z) is local to itself. As in Example 3, the vacuum axiom and completeness are immediate. Translation covariance follows from the exercise below.
Exercise 3
Show that the free fermion field is translation covariant, i.e.,
2 Lecture 2 (December 11, 2014)
In the first lecture we discussed the two simplest examples of non-commutative vertex algebras (see Examples 3 and 4). In this lecture we will consider further important examples, among them a generalization of those two mentioned previously. First, we need to introduce the necessary notions.
2.1 Formal Distribution Lie Algebras and Their Universal Vertex Algebras
Definition 8
A formal distribution Lie (super)algebra is a pair 
, where \(\mathfrak{g}\) is a Lie (super)algebra and 
is a collection of pairwise local \(\mathfrak{g}\)-valued formal distributions \(a^{\,j}(z) =\sum _{n\in \mathbb{Z}}a_{(n)}^{\,j}z^{-n-1}\), j ∈ J, such that the coefficients \(\{a_{(n)}^{\,j}\mid j \in J,\ n \in \mathbb{Z}\}\) span \(\mathfrak{g}\). A formal distribution Lie (super)algebra 
is called regular if:
-
(i)
the \(\mathbb{F}[\partial _{z}]\)-span of
is closed under all nth products for \(n \in \mathbb{Z}_{+}\), $$\displaystyle{ a(z)_{(n)}b(z):=\mathop{ \mathrm{Res}}\nolimits (w - z)^{n}[a(w),b(z)]dw, }$$(42)i.e., if a(z) and b(z) are elements of the form ∑ j ∈ J f j (∂ z )a j(z), where \(f_{j}(\partial _{z}) \in \mathbb{F}[\partial _{z}]\), and only finitely many f j (∂ z ) ≠ 0, then their nth product for \(n \in \mathbb{Z}_{+}\) is still an element of the same form.
-
(ii)
there exists a derivation \(T \in \mathop{\mathrm{Der}}\nolimits \,\mathfrak{g}\) such that
$$\displaystyle{ T(a^{\,j}(z)) = \partial _{ z}a^{\,j}(z),\ \text{i.e.},\ T(a_{ (n)}^{\,j}) = -na_{ (n-1)}^{\,j}\quad \forall j \in J. }$$(43)
is closed under all nth products for The annihilation subalgebra of \(\mathfrak{g}\) is \(\mathfrak{g}_{-} = \mathop{\mathrm{span}}\nolimits \{a_{(n)}^{\,j}\mid j \in J,\ n \in \mathbb{Z}_{+}\}\).
Exercise 4
Show that \(\mathfrak{g}_{-}\) is a T-invariant subalgebra of \(\mathfrak{g}\). (Hint: use the commutation formulas (32) and (43).)
The following theorem allows one to construct vertex algebras via the Extension theorem. Let \(U(\mathfrak{g})\) denote the universal enveloping algebra of \(\mathfrak{g}\).
Theorem 3
Let
be a regular formal distribution Lie algebra, and let
\(\mathfrak{g}_{-}\)
be the annihilation subalgebra. Let
\(V = U(\mathfrak{g})/U(\mathfrak{g})\mathfrak{g}_{-}\)
(also known as the induced
\(\mathfrak{g}\)
-module
\(\mathrm{Ind}_{\mathfrak{g}_{-}}^{\mathfrak{g}}(\mathbb{F})\)
) and let π be the representation of
\(\mathfrak{g}\)
in V induced via the left multiplication. Let
\(\vert 0\rangle =\bar{ 1}\)
be the image of 1 in V and
\(T \in \mathop{\mathrm{End}}\nolimits V\)
be the endomorphism of V induced by the derivation of
\(\mathfrak{g}\)
. Let
be the collection of
\(\mathop{\mathrm{End}}\nolimits V\)
-valued formal distributions
Then
consists of quantum fields and
satisfies the conditions of the Extension theorem, hence V is a vertex algebra, which we denote by
.
Proof
The only non-obvious part is to check that all \(\pi \big(a^{\,j}(z)\big)\) are quantum fields, i.e., \(\pi \big(a^{\,j}(z)\big)v \in V ((z))\) for each v ∈ V. Due to the PBW theorem, it is sufficient to check it for vectors of the following form (we use the same notation for elements in \(U(\mathfrak{g})\) and their images in V ):
We argue by induction on s. For s = 0 we have v = | 0〉, hence
The last equality follows from the fact that a (n) j | 0〉 = 0 for \(n\geqslant 0\). We proceed by proving the induction step:
By assumption of induction, the second term in the right-hand side is in V ((z)), so we only need to show that the first term is also in V ((z)). Now recall the commutation formula (32). We have
where \((a_{(k)}^{\,j}a^{j_{1}})_{(m+n_{ 1}-k)}\) is the Fourier coefficient of the formal distribution \(a^{\,j}(z)_{(k)}a^{j_{1}}(z)\). By the regularity property, we know that \(a^{\,j}(z)_{(k)}a^{j_{1}}(z)\) is contained in the \(\mathbb{F}[\partial _{z}]\)-span of 
, thus we can assume that
Since \(a^{\,j}(z),a^{j_{1}}(z)\) is a local pair, we know that there exists an integer \(N \in \mathbb{Z}_{+}\) such that \(a^{\,j}(z)_{(k)}a^{j_{1}}(z) = 0\) for \(k\geqslant N\). This allows us to rewrite formula (48) as follows,
By assumption of induction, for each k,
thus the first term in the right-hand side of (47) is also in V ((z)). □
Remark 6
Recall that by the Decomposition theorem for any local pair a(z), b(w) we have
which is equivalent to the commutator formula
where (a ( j) b)(w) = a(w)( j) b(w) is given by (31). This, along with the obvious formula
allows us to convert the commutator formula into the decomposition formula, thereby establishing locality.
Let us now discuss the next important example of a non-commutative vertex algebra.
Example 5
Let \(\mathfrak{g} = \mathrm{Vir}\) be the Virasoro algebra with commutation relations
Consider the formal distribution
so that L (n) = L n−1. Then the commutation relations (55) can be written in the equivalent form
Indeed, by (57) we have: \(L_{(0)}L = \partial L,\ L_{(1)}L = 2L,\ L_{(3)}L = \tfrac{C} {2},\) and L ( j) L = 0 for all other \(j\geqslant 0.\) Hence by (53) and (54), (57) is equivalent to (55). It follows that L(z) is local with itself, hence (Vir, {L(z), C}) is a formal distribution Lie algebra. Furthermore, it is regular. There are two conditions (1) and (2) we need to check: (1) is obvious, for (2) take T = adL −1, then [L −1, L n ] = (−1 − n)L n−1, which gives (43). The annihilation subalgebra is
So, by Theorem 3 and the Extension theorem, we get the associated vertex algebra
called the universal Virasoro vertex algebra. One can make it slightly smaller by taking \(c \in \mathbb{F}\) and factorizing by the ideal generated by (C − c). Let V c stand for the corresponding factor vertex algebra, which is called the universal Virasoro vertex algebra with central charge c.
Remark 7
V c can be non-simple for certain values of c. Namely, V c is non-simple if and only if [16]
Exercise 5
The vertex algebra V c has a unique maximal ideal J c.
Let V c = V c∕J c. Since c in (60) is symmetric in p and q we may assume that p < q. The smallest example p = 2, q = 3 gives c = 0; V 0 is the one-dimensional vertex algebra. The next example is p = 3, q = 4 when c = 1∕2; the vertex algebra \(V _{\frac{1} {2} }\) is related to the Ising model. The simple vertex algebras V c with c of the form (60) are called discrete series vertex algebras. They play a fundamental role in conformal field theory [4].
Example 6
Let \(\mathfrak{g}\) be a finite dimensional Lie algebra with a non-degenerate symmetric invariant bilinear form (. |. ). Let \(\widehat{\mathfrak{g}} = \mathfrak{g}[t,t^{-1}] + \mathbb{F}K\) be the associated Kac-Moody affinization, with commutation relations
where \(a,b \in \mathfrak{g}\), \(m,n \in \mathbb{Z}\). Let \(a(z) =\sum _{n\in \mathbb{Z}}(at^{n})z^{-n-1}\) and 
be an (infinite) collection of formal distributions. The commutation relations (61) are equivalent to
Hence 
is a local family. So 
is a formal distribution Lie algebra. The annihilation subalgebra is \(\widehat{\mathfrak{g}}_{-} = \mathfrak{g}[t]\).
Exercise 6
Show that the formal distribution Lie algebra 
defined above is regular with T = −∂
t
.
The associated vertex algebra 
is called the universal affine vertex algebra associated to
\(\big(\mathfrak{g},(.\vert.)\big)\). Again, it can be made a little smaller by taking \(k \in \mathbb{F}\) and considering
which is called the universal affine vertex algebra of level k. There are certain values of k for which \(V ^{k}(\mathfrak{g})\) is non-simple (it is a known set of rational numbers [16]).
Example 7
Let A be a finite dimensional vector superspace with a non-degenerate skewsymmetric bilinear form 〈. |. 〉:
Take the associated Clifford affinization
with commutation relations
Consider the formal distributions
and define 
to be
Then the commutation relations (66) are equivalent to
Hence 
is a local family, and 
is a formal distribution Lie superalgebra. Its annihilation subalgebra is \(\hat{A}_{-} = A[t]\). Furthermore, 
is regular with T = −∂
t
and
is the associated vertex algebra called the vertex algebra of free superfermions.
Exercise 7
-
(1)
If A is a 1-dimensional odd superspace we get the free fermion vertex algebra F = F(A).
-
(2)
If \(\mathfrak{g}\) is the 1-dimensional Lie algebra \(\mathbb{F}\), with bilinear form (a | b) = ab and level k = 1, then we get the free boson vertex algebra \(B = V ^{1}(\mathbb{F})\).
Exercise 8
Show that the vertex algebra F(A) is always simple.
2.2 Formal Cauchy Formulas and Normally Ordered Product
We proceed by proving some statements which are analogous to the Cauchy formula and are true for any formal distribution. Let U be a vector space and \(a(z) =\sum _{n\in \mathbb{Z}}a_{(n)}z^{-n-1}\) be a U-valued formal distribution. We call
the creation part or “positive” part of a(z) and
the annihilation part or “negative” part of a(z). Note that \(\partial _{z}\big(a(z)_{\pm }\big) =\big (\partial _{z}a(z)\big)_{\pm }\).
Proposition 1
Formal Cauchy formulas can be written as follows:
-
(a)
For the “positive” and “negative” parts of a(z) we have
$$\displaystyle{ a(w)_{+} =\mathop{ \mathrm{Res}}\nolimits a(z)i_{z,w} \frac{1} {z - w}dz,\qquad - a(w)_{-} =\mathop{ \mathrm{Res}}\nolimits a(z)i_{w,z} \frac{1} {z - w}dz. }$$(73) -
(b)
For the derivatives of a(z)± we have
$$\displaystyle\begin{array}{rcl} & & \frac{1} {n!}\,\partial _{w}^{n}a(w)_{ +} =\mathop{ \mathrm{Res}}\nolimits a(z)i_{z,w} \frac{1} {(z - w)^{n+1}}dz, \\ & & -\frac{1} {n!}\,\partial _{w}^{n}a(w)_{ -} =\mathop{ \mathrm{Res}}\nolimits a(z)i_{w,z} \frac{1} {(z - w)^{n+1}}dz. {}\end{array}$$(74)
Proof
Use property (5) of the delta function and (22) to get
Collect the (non-negative) powers of w on both sides to get (a). Differentiating (a) by wn times gives (b). □
Multiplying two quantum fields naïvely would lead to divergences. The next definition is introduced to circumvent this problem.
Definition 9
The normally ordered product of \(\mathop{\mathrm{End}}\nolimits V\)-valued quantum fields a(z) and b(z) is defined by
It must be proved that: a(z)b(z): is an “honest” quantum field, i.e., all the divergences are removed.
Proposition 2
If a(z) and b(z) are quantum fields then so is: a(z)b(z): .
Proof
Apply: a(z)b(z): , defined by (76), to any vector v ∈ V:
Since b(z) is assumed to be a quantum field, b(z)v in the first term of the right-hand side of (77) is a Laurent series by definition. The creation part a(z)+ has only non-negative powers of z, therefore a(z)+ b(z)v is still a Laurent series. In the second term a(z)− v consists of finitely many terms with negative powers, i.e., it is a Laurent polynomial. Now b(z)a(z)− v is a Laurent series multiplied by a Laurent polynomial which is still a Laurent series. Hence we proved that: a(z)b(z): is a sum (or a difference) of two Laurent series, thus it is a Laurent series. □
Exercise 9
Let a(z) and b(z) be quantum fields. Show that their nth product a(z)(n) b(z), \(n \in \mathbb{Z}_{+}\) and derivatives ∂ z a(z), ∂ z b(z) are also quantum fields.
On the space of quantum fields we have defined a(w)(n) b(w) for \(n\geqslant 0\). Introduce
so that a(w)(−1) b(w) = : a(w)b(w): . Thus for each \(n \in \mathbb{Z}\) we have the nth product a(w)(n) b(w). Using the formal Cauchy formulas above, we get the unified formulas for all nth products of quantum fields
Remark 8
For a local pair of quantum fields physicists write
This way of writing is useful but might be confusing, since different parts of it are expanded in different domains. Therefore it is worth giving a rigorous interpretation of (80) by writing
and
By taking the difference (81)– (82) we get
Conversely, by separating the negative (resp. non-negative) powers of z in (83) we get (81) [resp. (82)]. We still need to explain (80) for negative n. By Taylor’s formula in the domain | z − w | < | w | ([17], (2.4.3)), we have
i.e., the nth products for negative n are “contained” in the normally ordered product.
2.3 Bakalov’s Formula and Dong’s Lemma
Locality of the pair a(z), b(z) of \(\mathop{\mathrm{End}}\nolimits V\)-valued quantum fields means that
Denote either side of this equality by F(z, w). Then for each \(k \in \mathbb{Z}_{+}\) we have
The first term of the left-hand side of (86) is
while the second term of the right-hand side of (86) is
Applying the unified formula (79) the sum of (87) and (88) can be written as
Hence we obtain Bakalov’s formula
which holds for each non-negative integer k and sufficiently large positive integer N. The second equality follows from the first one by properties (3) and (5) of the formal delta function.
Remark 9
Since a(z) and b(z) are quantum fields, it follows from (85) that F(z, w)v lies in the space V [[z, w]][z −1, w −1] for each v ∈ V. Hence (90) makes sense.
Remark 10
It follows from (85) that if we replace a(z) in this equation by ∂ z k a(z) for some positive integer k, then it still holds with N replaced by N + k.
Lemma 2 (Dong)
If a(z), b(z) and c(z) are pairwise mutually local quantum fields, then a(z)(n) b(z), c(z) is a local pair for any \(n \in \mathbb{Z}\) .
Proof
[1] It suffices to prove that for N and k as in (90) we have for some \(M \in \mathbb{Z}_{+}:\)
where ± is the Koszul-Quillen sign, if (85) holds for all three pairs (a, b), (a, c) and (b, c). We let M = 2N + k. By Bakalov’s formula (90), the left-hand side of (91) is equal to
Due to (85) for the pair (b, c), we can permute c(z 3) with b(z 2) (up to the Koszul-Quillen sign), and after that similarly permute c(z 3) and the (k − i)th derivative of a(z 1), using Remark 10. We thus obtain the right-hand side of (91). □
3 Lecture 3 (December 16, 2014)
In this lecture, we will prove the Extension theorem, the Borcherds identity and the skewsymmetry. We will also introduce the concepts of conformal vector, conformal weight and Hamiltonion operators. In the end, we give some properties of the Formal Fourier Transform.
3.1 Proof of the Extension Theorem
First of all, let us give a name for the data which appeared in the Extension theorem.
Definition 10
A pre-vertex algebra is a quadruple 
, where V is a vector space with a non-zero element | 0〉, \(T \in \mathop{\mathrm{End}}\nolimits V\) and 
is a collection of quantum fields with values in \(\mathop{\mathrm{End}}\nolimits V\) satisfying the following conditions:
-
(i)
(vacuum axiom) T | 0〉 = 0,
-
(ii)
(translation covariance) [T, a j(z)] = ∂ z a j(z) for all j ∈ J,
-
(iii)
(locality) \((z - w)^{N_{ij}}[a^{i}(z),a^{\,j}(w)] = 0\) for all i, j ∈ J with some \(N_{ij} \in \mathbb{Z}_{+}\),
-
(iv)
(completeness) \(\mathop{\mathrm{span}}\nolimits \{a_{(n_{1})}^{j_{1}}\cdots a_{(n_{ s})}^{j_{s}}\vert 0\rangle \mid j_{ i} \in J,\ n_{i} \in \mathbb{Z},\ s \in \mathbb{Z}_{+}\} = V\).
Let 
be a pre-vertex algebra. Define
where I
V
is the constant field equal to the identity operator I
V
on V. Let, as in Lecture 1, 
be the set of all translation covariant quantum fields a(z), such that a(z), a
j(z) is a local pair for all j ∈ J. The following is a more precise version of the Extension theorem, stated in Lecture 1.
Theorem 4 (Extension Theorem)
For a pre-vertex algebra
, let
be defined as above, then we have,
-
(a)
, -
(b)
The map
(93)is well-defined and bijective. Denote by sf the inverse map.
-
(c)
The z-product a(z)b: = sf (a)b endows V with a vertex algebra structure, which extends the pre-vertex algebra structure.
,
Remark 11
-
(1)
The map fs is called the field-state correspondence since it sends a field to a vector in V, called a “state” in physics. Its inverse map, called the state-field correspondence, is denoted by
(94) -
(2)
Denote by
the space of translation covariant quantum fields. By Lemma 1, a(z) | 0〉 ∈ V [[z]] for 
, hence fs(a(z)) ∈ V is well-defined.
the space of translation covariant quantum fields. By Lemma 
, hence fs(a(z)) ∈ V is well-defined.Lemma 3
contains I
V
, it is ∂
z
-invariant and is closed under all nth product, i.e.,
for any
\(n \in \mathbb{Z}\)
if
.
Proof
Since [T, I
V
] = 0 = ∂
z
I
V
, we have 
. Now if a(z) is translation covariant, we need to show that [T, ∂
z
a(z)] = ∂
z
∂
z
a(z) and so ∂
z
a(z) is also translation covariant. But
and
For the last part of this lemma, let us recall the definition of the nth product,
We want to prove [T, a(w)(n) b(w)] = ∂ w (a(w)(n) b(w)). Both T and ∂ w commute with \(\mathop{\mathrm{Res}}\nolimits\), moreover, ∂ w commutes with i z, w and i w, z . So we have
Note that ∂ w (z − w)n = −∂ z (z − w)n and \(-\mathop{\mathrm{Res}}\nolimits a(z)i_{w,z}\partial _{z}(z - w)^{n}dz =\mathop{ \mathrm{Res}}\nolimits (\partial _{z}a(z))i_{w,z}(z - w)^{n}dz\). So
This shows that ∂ w is a derivation for the nth product. Now
Since both a(z), b(z) are translation covariant, we have
This completes the proof. □
We have inclusions
The first inclusion is because for any 
, we have 
. The second inclusion is by Lemma 3 and Dong’s Lemma (locality). The last inclusion is by definition.
Exercise 10
Show that the constant field T is translation covariant, but is not local to any non-constant field.
Lemma 4
Let
, and a = fs(a(z)), b = fs(b(z)). Then:
-
(a)
fs(I V ) = | 0〉,
-
(b)
fs(∂ z a(z)) = Ta,
-
(c)
fs(a(z)(n) b(z)) = a (n) b. Here we write \(a(z) =\sum _{n\in \mathbb{Z}}a_{(n)}z^{-n-1}\) .
Proof
(a) is obvious. For (b), since \(a(z)\vert 0\rangle = e^{zT}a = a + (Ta)z + \frac{T^{2}a} {2} z^{2} + o(z^{2})\) we have ∂ z a(z) | 0〉 = Ta + T 2 az + o(z), so fs(∂ z a(z)) = ∂ z a(z) | 0〉 z = 0 = Ta. For (c), by definition, we have
and the right hand side, by definition of the nth product, is equal to
Now, since a(w) | 0〉 ∈ V [[w]] and i z, w (w − z)n has only non-negative powers of w, we have
For the first term, since b(z) | 0〉 ∈ V [[z]], we can let z = 0 before we calculate the residue, which gives
This completes the proof. □
Lemma 5
Let
. Then e
wT
a(z)e
−wT = i
z, w
a(z + w).
Proof
Both sides are in \((\mathop{\mathrm{End}}\nolimits V )[[z,z^{-1}]][[w]]\), and both satisfy the differential equation \(\frac{df(w)} {dw} = (\mathrm{ad}T)f(w)\) with the initial condition f(0) = a(z). □
Lemma 6 (Uniqueness Lemma)
Let
and let a(z) be some quantum field in
. Assume that
-
(i)
fs(a(z)) = 0,
-
(ii)
a(z) is local with any element in
, -
(iii)
.
,
.Then a(z) = 0.
Proof
Let 
. By the locality of a(z) and b(z), we have (z − w)N[a(z), b(w)] = 0 for some \(N \in \mathbb{Z}_{+}\). Apply both sides to | 0〉. We get
By the property (i) we have a (−1) | 0〉 = 0 and a(z) is translation covariant, hence by Lemma 1(b), a(z) | 0〉 = 0. Now, by Lemma 1(a), b(w) | 0〉 ∈ V [[w]], so we can let w = 0 and get z N a(z)b = 0, which means a (n) b = 0 for any \(n \in \mathbb{Z}\). This is true for any b ∈ V by the property (iii). So in fact, we have a(z) = 0. □
Proof of the Extension Theorem
We have the following two properties of the map fs:
-
(i)
the map
defined by \(fs(a(z)) = a(z)\vert 0\rangle \big\vert _{z=0}\) is given by $$\displaystyle{ (a^{j_{1} }(z)_{(n_{1})}(a^{j_{2} }(z)_{(n_{2})}\cdots (a^{j_{s} }(z)_{(n_{s})}I_{V })\cdots \,)\mapsto a_{(n_{1})}^{j_{1} }a_{(n_{2})}^{j_{2} }\cdots a_{(n_{s})}^{j_{s} }\vert 0\rangle, }$$(107)and it is surjective, by (a), (c) of Lemma 4 and the completeness axiom;
-
(ii)
is injective using the Uniqueness Lemma with 
.
defined by 
is injective using the Uniqueness Lemma with 
.Recall the inclusion 
. We now have that 
is surjective and 
is injective, so we can conclude that it is in fact bijective and 
. This proves (a) and (b) in the Extension Theorem. For (c), we need to show that a(z) is translation covariant \(\forall a \in V\) and that each pair \(a(z),b(w)\,\forall a,b \in V\) is a local pair. But translation covariance comes from Lemma 3 and locality comes from Dong’s lemma. □
Corollary 1 (of the Proof)
-
(a)
\(\mathit{sf }(a_{(n_{1})}^{j_{1}}a_{(n_{ 2})}^{j_{2}}\cdots a_{ (n_{s})}^{j_{s}}\vert 0\rangle ) = (a^{j_{1}}(z)_{ (n_{1})}(a^{j_{2}}(z)_{ (n_{2})}\cdots (a^{j_{s}}(z)_{ (n_{s})}I_{V })\cdots \,)\) .
-
(b)
(Ta)(z) = ∂ z a(z).
-
(c)
(a (n) b)(z) = a(z)(n) b(z), which is called the nth product identity.
Proof
(a) is by definition since sf is the inverse of fs, while fs is given by (107). Letting s = 1, n 1 = −2 in (a) we get (b). Letting s = 2, n 1 = n, n 2 = −1 in (a) we get (c). □
Remark 12
Due to Corollary 1(b) and Remark 1, the Definitions 3 and 5 of a vertex algebra are equivalent.
Remark 13 (Special Case of (a) in the Corollary)
For \(n_{1},\ldots,n_{s} \in \mathbb{Z}_{+}\), we have,
Corollary 2 (of the Proof)
\(\mathop{\mathrm{Lie}}\nolimits _{V }:= \mathop{\mathrm{span}}\nolimits \{a_{(n)}\vert \,a \in V,\ n \in \mathbb{Z}\} \subset \mathop{\mathrm{End}}\nolimits V\) is a subalgebra of the Lie superalgebra \(\mathop{\mathrm{End}}\nolimits V\) with the commutator formula
which is equivalent to each of the following two expressions
Moreover,
\(\mathop{\mathrm{Lie}}\nolimits _{V }\)
is a regular formal distribution Lie algebra with the data
.
3.2 Borcherds Identity and Some Other Properties
Proposition 3 (Borcherds Identity)
For \(n \in \mathbb{Z}\) , a, b ∈ V, where V is a vertex algebra, we have
Proof
The left hand side of (112) is a local formal distribution in z and w. Apply to it the Decomposition theorem to get that it is equal to
where
The last equality follows from the nth product formula, all other equalities are just by definition. □
Exercise 11
Prove that a unital z-algebra satisfying the Borcherds identity is a vertex algebra.
Proposition 4 (Skewsymmetry)
For a, b ∈ V, where V is a vertex algebra, we have:
Proof
By locality, we know that, there exists \(N \in \mathbb{Z}\), such that
Apply both sides to \(\left \vert 0\right>\); by Lemma 1(b) we get
Now use Lemma 5:
For N ≫ 0, this is a formal power series in (z − w), so we can set w = 0 and get
which proves the proposition. □
Proposition 5
T is a derivation for all nth products, i.e.,
Proof
It follows from Remark 12. □
In view of the nth product identity, we let: ab: = a (−1) b and call this the normally ordered product of two elements of a vertex algebra.
Proposition 6
The nth products for negative n are expressed via the normally ordered product: \(a_{(-n-1)}b =\,: \frac{T^{n}a} {n!} b:\) .
Proof
We have \((a_{(-n-1)}b)(z) = a(z)_{(-n-1)}b(z) =\,: \frac{\partial _{z}^{n}a(z)} {n!} b(z):\) , where the first equality is the nth product identity and the second equality is (78). But we also have T(a)(z) = ∂ z a(z), hence by induction we have \(: \frac{\partial _{z}^{n}a(z)} {n!} b(z):\,=\,: \frac{(T^{n}a)(z)} {n!} b(z):\,= (\frac{(T^{n}a)} {n!} _{(-1)}b)(z)\), and by the bijection of the state-field correspondence, we have \(a_{(-n-1)}b = \frac{(T^{n}a)} {n!} _{(-1)}b =\,: \frac{T^{n}a} {n!} b:\) . □
Now we take care of the nth products a (n) b for \(n \in \mathbb{Z}_{+}\). For this we define the λ-bracket
Thus we get a quadruple (V, T, : ab: , [a λ b]), which will be shown in the next lecture to have a very similar structure to a Poisson Vertex Algebra (PVA).
3.3 Conformal Vector and Conformal Weight, Hamiltonian Operator
Definition 11
A vector L of a vertex algebra V is called a conformal vector if
-
(i)
\(L(z) =\sum _{n\in \mathbb{Z}}L_{n}z^{-n-2}\), such that,
$$\displaystyle{ [L_{m},L_{n}] = (m - n)L_{m+n} +\delta _{m,-n}\frac{m^{3} - m} {12} \,cI_{V } }$$(121)for some \(c \in \mathbb{F}\), which is called the central charge,
-
(ii)
L −1 = T,
-
(iii)
L 0 acts diagonalizably on V, its eigenvalues are called conformal weights.
Since L n−1 = L (n), using the commutator formula (109), we get
which is equivalent to [cf. (111)]
So we have
Here we assume that a is an eigenvector of L 0 with the eigenvalue Δ a . We call L 0 the energy operator. It is a Hamiltonian operator by the definition below and (123) for m = 0.
Definition 12
A diagonalizable operator H is called a Hamiltonian operator if it satisfies the equation
for any eigenvector a of H with eigenvalue Δ a .
If we write \(a(z) =\sum _{n\in -\varDelta _{a}+\mathbb{Z}}a_{n}z^{-n-\varDelta _{a}}\), then due to the equality \(a_{(n)} = a_{n-\varDelta _{a}+1}\), we have:
This is an equivalent definition of a Hamiltonian operator.
Proposition 7
If H is a Hamiltonian operator, then we have:
-
(a)
Δ | 0〉 = 0,
-
(b)
Δ Ta = Δ a + 1,
-
(c)
\(\varDelta _{a_{(n)}b} =\varDelta _{a} +\varDelta _{b} - n - 1\) .
Proof
To prove (a), we just need to know that | 0〉(z) = I V , and we use (125) with a = | 0〉. Since Ta = a (−2) | 0〉, (b) follows from (a) and (c) with b = | 0〉, n = −2. For (c), we have
□
Remark 14
-
(a)
For a conformal vector L, we have \([L_{\lambda }L] = (T + 2\lambda )L + \frac{\lambda ^{3}} {2}c\vert 0\rangle\), which implies Δ L = 2. That is why we write L(z) in the form \(L(z) =\sum _{n\in \mathbb{Z}}L_{n}z^{-n-2}\).
-
(b)
Conformal weight is a good “book-keeping device”, if we let Δ λ = Δ T = 1. Then all summands in the λ-bracket \([a_{\lambda }b] =\sum _{j\geqslant 0} \frac{\lambda ^{j}} {j!}(a_{(\,j)}b)\) have the same conformal weight Δ a + Δ b − 1.
Remark 15
The translation covariance (4) of the quantum field a(z) is equivalent to the following “global” translation covariance:
Likewise, the property (125) of a(z) is equivalent to the following “global” scale covariance:
The more general property (122) is called the conformal invariance. It is the basic symmetry of conformal field theory.
3.4 Formal Fourier Transform
Definition 13
The Formal Fourier Transform is the map F z λ: U[[z, z −1]] ↦ U[[λ]] defined by
Proposition 8
-
(a)
F z λ ∂ z a(z) = −λF z λ a(z),
-
(b)
F z λ ∂ w k δ(z, w) = e λw λ k ,
-
(c)
F z λ a(−z) = −F z −λ a(z),
-
(d)
F z λ(e zT a(z)) = F z λ+T a(z), where \(T \in \mathop{\mathrm{End}}\nolimits U\) , provided that a(z) ∈ U((z)).
Proof
-
(a)
Assume \(a(z) =\sum _{n\in \mathbb{Z}}a_{(n)}z^{-n-1}\), then \(\partial _{z}a(z) =\sum _{n\in \mathbb{Z}}(-n - 1)a_{(n)}z^{-n-2}\). Now
$$\displaystyle\begin{array}{rcl} F_{z}^{\lambda }a(z)& =& \mathop{\mathrm{Res}}\nolimits e^{\lambda z}a(z)dz \\ & =& \mathop{\mathrm{Res}}\nolimits (\sum _{i\in \mathbb{Z}_{+}} \dfrac{\lambda ^{i}z^{i}} {i!} )(\sum _{n\in \mathbb{Z}}a_{(n)}z^{-n-1})dz \\ & =& \sum _{n\in \mathbb{Z}_{+}} \dfrac{\lambda ^{n}} {n!}a_{(n)}, \\ F_{z}^{\lambda }\partial _{ z}a(z)& =& \sum _{n\in \mathbb{Z}_{+}} \dfrac{\lambda ^{n}} {n!}(-n)a_{(n-1)} \\ & =& -\lambda \sum _{n\in \mathbb{Z}_{+}} \dfrac{\lambda ^{n}} {n!}a_{(n)}. {}\end{array}$$(129) -
(b)
Recall that \(\frac{\partial _{w}^{k}\delta (z,w)} {k!} =\sum _{j\in \mathbb{Z}}{j\choose k}w^{j-k}z^{-j-1}\), so
$$\displaystyle{ \begin{array}{rl} F_{z}^{\lambda }\partial _{w}^{k}\delta (z,w)& =\mathop{ \mathrm{Res}}\nolimits e^{\lambda z}k!\sum _{j\in \mathbb{Z}_{+}}{j\choose k}w^{j-k}z^{-j-1}dz \\ & =\sum _{j\in \mathbb{Z}_{+}} \dfrac{\lambda ^{j}} {j!}k! \dfrac{j!} {k!(\,j - k)!}w^{j-k} \\ & =\lambda ^{k}\sum _{j-k\in \mathbb{Z}_{+}} \dfrac{\lambda ^{j-k}} {(\,j - k)!}w^{j-k} \\ & = e^{\lambda w}\lambda ^{k}.\end{array} }$$(130) -
(c)
By definition
$$\displaystyle{ \begin{array}{rl} F_{z}^{\lambda }a(-z)& =\mathop{ \mathrm{Res}}\nolimits (\sum _{i\in \mathbb{Z}_{+}} \dfrac{\lambda ^{i}z^{i}} {i!} )(\sum _{n\in \mathbb{Z}}a_{(n)}(-z)^{-n-1})dz \\ & =\sum _{n\in \mathbb{Z}_{+}} \dfrac{\lambda ^{n}} {n!}(-1)^{n+1}a_{(n)} \\ & = -\sum _{n\in \mathbb{Z}_{+}} \dfrac{(-\lambda )^{n}} {n!} a_{(n)} \\ & = -F_{z}^{-\lambda }a(z).\end{array} }$$(131) -
(d)
Since a(z) ∈ U((z)), e zT a(z) ∈ U((z)) is well defined. Now
$$\displaystyle{ \begin{array}{rl} F_{z}^{\lambda }(e^{zT}a(z))& =\mathop{ \mathrm{Res}}\nolimits e^{\lambda z}e^{zT}a(z)dz \\ & =\mathop{ \mathrm{Res}}\nolimits e^{(\lambda +T)z}a(z)dz \\ & = F_{z}^{\lambda +T}a(z).\end{array} }$$(132)
□
Similarly, we can define the Formal Fourier Transform in two variables.
Definition 14
The Formal Fourier Transform in two variables is the map
defined by
Proposition 9
- (α):
-
F z, w λ ∂ z a(z, w) = −λF z, w λ a(z, w) = [∂ w , F z, w λ]a(z, w),
- (β):
-
F z, w λ ∂ w k δ(z, w) = λ k ,
- (γ):
-
\(F_{z,w}^{\lambda }a(w,z) = F_{z,w}^{-\lambda -\partial _{w}}a(z,w)\) provided that a(z, w) is local,
- (δ):
-
F z, w λ F x, w μ = F x, w λ+μ F z, x λ .
Proof
Since F z, w λ = e −λw F z λ, (α) and (β) follow from the properties (a) and (b) in Proposition 8. (δ) holds since
Finally, due to the Decomposition theorem, it suffices to check (γ) (interpretation as before) for a(z, w) = c(w)∂ w k δ(z, w):
using the properties (3) and (5) of the delta function. □
4 Lecture 4 (December 18, 2014)
The Formal Fourier Transform F z λ is very important for us, since the λ-bracket (120) is [a λ b] = F z λ a(z)b, i.e., the Fourier transform of the z-product is the λ-bracket. We also note that \(: ab:\, (= a_{(-1)}b) =\mathop{ \mathrm{Res}}\nolimits \dfrac{a(z)b} {z} dz\). These observations will be important for studying properties of the normally ordered product: : and the λ-bracket. For simplicity we will further consider vertex algebras V of purely even parity only. The general case follows by the Koszul-Quillen rule.
4.1 Quasicommutativity, Quasiassociativity and the Noncommutative Wick’s Formula
Lemma 7 (Newton-Leibniz (NL) Lemma)
For any a(z) ∈ U[[z]], we have
Proof
Both sides are formal power series in λ, they are equal at λ = 0, and their derivatives by λ are also equal, so they are equal. □
Proposition 10 (Quasicommutativity of: :)
The commutator for the normally ordered product and λ-bracket are related as follows
Proof
Apply F z λ to both sides of skewsymmetry, divided by z, and set λ = 0. We get
By definition
Next, using property (d) of the FFT in Proposition 8, we have
□
Next we derive the following important identity.
Proposition 11
For a, b, c in a vertex algebra V, we have the following identity in V [[λ, w, w −1]]
Proof
The following identity in V [[z ±1, w ±1]] is obvious:
Applying to both sides F z λ = e wλ F z, w λ, we get
where we have used the nth product formula \(a(w)_{(n)}b(w) = (a_{(n)}b)(w),n \in \mathbb{Z}_{+}\).
□
We have the following two important properties of a vertex algebra.
Proposition 12
Assume a, b, c in a vertex algebra V . Then we have
-
(a)
Quasiassociativity formula
$$\displaystyle{:: ab: c: -: a: bc::=: \left (\int _{0}^{T}d\lambda a\right )[b_{\lambda }c]: +: \left (\int _{ 0}^{T}d\lambda b\right )[a_{\lambda }c]:. }$$(144) -
(b)
Non-commutative Wick’s formula
$$\displaystyle{ [a_{\lambda }: bc:] =: [a_{\lambda }b]c: +: b[a_{\lambda }c]: +\int _{0}^{\lambda }[[a_{\lambda }b]_{\mu }c]d\mu. }$$(145)
Proof
-
(1)
Apply \(\mathop{\mathrm{Res}}\nolimits \frac{1} {z}dz\) to the -1st product identity:
$$\displaystyle{: ab: (z)c =: a(z)b(z): c = a(z)_{+}b(z)c + b(z)a(z)_{-}c, }$$(146)and use that
$$\displaystyle\begin{array}{rcl} \mathop{\mathrm{Res}}\nolimits \frac{1} {z}(: ab: (z)c)dz& =& (: ab:)_{(-1)}c =:: ab: c:\,, \\ \mathop{\mathrm{Res}}\nolimits \frac{1} {z}(a(z)_{+}b(z)c)dz& =:& a: bc:: +\sum _{j\in \mathbb{Z}_{+}}a_{(-j-2)}b_{(\,j)}c \\ & =:& a: bc:: +: (\int _{0}^{T}d\lambda a)[b_{\lambda }c]:\,, \\ \mathop{\mathrm{Res}}\nolimits \frac{1} {z}(b(z)a(z)_{-}c)dz& =& \sum _{j\in \mathbb{Z}_{+}}b_{(-j-2)}a_{(\,j)}c \\ & =:& (\int _{0}^{T}d\lambda b)[a_{\lambda }c]:\,. {}\end{array}$$(147) -
(2)
Take \(\mathop{\mathrm{Res}}\nolimits \frac{1} {w}dw\) of both sides of formula (141):
$$\displaystyle{ \mathop{\mathrm{Res}}\nolimits \frac{1} {w}[a_{\lambda }b(w)c]dw =\mathop{ \mathrm{Res}}\nolimits \frac{1} {w}(e^{w\lambda }[a_{\lambda }b](w)c + b(w)[a_{\lambda }c])dw. }$$(148)Since \(\mathop{\mathrm{Res}}\nolimits \frac{b(w)c} {w} dw = b_{(-1)}c =: bc:\), we have
$$\displaystyle{ \mathop{\mathrm{Res}}\nolimits \frac{1} {w}[a_{\lambda }b(w)c]dw = [a_{\lambda }: bc:]. }$$(149)For the second term of the right-hand side of (148),
$$\displaystyle{ \mathop{\mathrm{Res}}\nolimits \frac{1} {w}b(w)[a_{\lambda }c]dw =: b[a_{\lambda }c]:\,. }$$(150)Using the NL Lemma 7 for the first term in the right hand side of (148), we have
$$\displaystyle{ \begin{array}{rl} \mathop{\mathrm{Res}}\nolimits e^{w\lambda }\dfrac{[a_{\lambda }b](w)c} {w} dw& = F_{w}^{\lambda }\dfrac{[a_{\lambda }b](w)c} {w} \\ & =\mathop{ \mathrm{Res}}\nolimits \dfrac{[a_{\lambda }b](w)c} {w} dw +\int _{ 0}^{\lambda }F_{w}^{\mu }[a_{\lambda }b](w)cd\mu \\ & =: [a_{\lambda }b]c: +\int _{0}^{\lambda }[[a_{\lambda }b]_{\mu }c]d\mu.\end{array} }$$(151)
□
Remark 16
The expression: (∫ 0 T dλa)[b λ c]: should be understood in the following way. We know that \([b_{\lambda }c] =\sum _{j\in \mathbb{Z}_{+}}b_{(\,j)}c \dfrac{\lambda ^{j}} {j!}\), so \(: a[b_{\lambda }c]:=\sum _{j\in \mathbb{Z}_{+}}a_{(-1)}b_{(\,j)}c \dfrac{\lambda ^{j}} {j!}\). We have \(\int _{0}^{T} \dfrac{\lambda ^{j}} {j!}d\lambda = \dfrac{T^{j+1}} {(\,j + 1)!}\); letting \(\dfrac{T^{j+1}} {(\,j + 1)!}\) act just on a we get \(\left ( \dfrac{T^{j+1}a} {(\,j + 1)!}\right )_{(-1)} = a_{(-j-2)}\), so \(: (\int _{0}^{T}d\lambda a)[b_{\lambda }c]:=\sum _{j\in \mathbb{Z}_{+}}a_{(-j-2)}b_{(\,j)}c\).
4.2 Lie Conformal Algebras vs Vertex Algebras
Let \(\mathfrak{g}\) be a Lie algebra, and let \(\mathfrak{g}[[w,w^{-1}]]\) be the space of all \(\mathfrak{g}\)-valued formal distributions. This space is an \(\mathbb{F}[\partial ]\)-module by defining
It is closed under the following (formal) λ-bracket: for \(a = a(w),b = b(w) \in \mathfrak{g}[[w,w^{-1}]]\). Let
Indeed, by definition of F z, w λ and its property (β), we have:
Thus [a λ b](w) is a generating series for jth products of a(w) and b(w). It is a formal power series in λ in general, but if the pair (a(w), b(w)) is local, \([a_{\lambda }b] \in \mathfrak{g}[[w,w^{-1}]][\lambda ]\) is polynomial in λ.
Proposition 13
Assume \(a(w),b(w),c(w) \in \mathfrak{g}[[w,w^{-1}]]\) for some Lie algebra \(\mathfrak{g}\) with ∂ = ∂ w defined as above. Denote a = a(w), b = b(w), c = c(w). Then the λ-bracket defined as above satisfies the following properties:
Proof
The sesquilinearity comes from (α) and the skewsymmetry comes from (γ) in Proposition 9 about properties of formal Fourier transform in two variables. For the Jacobi identity we have:
The last equality comes from the Jacobi identity in the Lie algebra \(\mathfrak{g}\). By property (δ) of the formal Fourier transform in Proposition 9, we have:
while F z, w λ F x, w μ[b(x), [a(z), c(w)]] = [b μ [a λ c]](w) is just by definition. □
Definition 15
A Lie conformal algebra (LCA) is an \(\mathbb{F}[\partial ]\)-module R endowed with an \(\mathbb{F}\)-bilinear λ-bracket [a λ b] ∈ R[λ] for a, b ∈ R, which satisfies the axioms of sesquilinearity, skewsymmetry and the Jacobi identity.
Example 8
The Virasoro formal distribution Lie algebra from Example 5 gives rise, by Proposition 13, to the Virasoro Lie conformal algebra
with λ-bracket
Replacing L by \(L -\tfrac{1} {2}\alpha C,\) where \(\alpha \in \mathbb{F},\) we obtain a λ-bracket with a trivial cocycle added:
Example 9
The Kac-Moody formal distribution Lie algebra from Example 6 gives rise to the Kac-Moody Lie conformal algebra
with λ-bracket \((a,b \in \mathfrak{g}):\)
Fix \(s \in \mathfrak{g};\) replacing a by a − (a | s)K, we obtain a λ-bracket with a trivial cocycle added:
Of course, adding a trivial cocycle doesn’t change the Lie conformal algebra. However this will become crucial in the proof of the integrability of the associated integrable systems.
Due to the nth product identity in a vertex algebra [Corollary 1(c)], we derive from the last proposition the following.
Proposition 14
A vertex algebra V is a Lie conformal algebra with ∂ = T, the translation operator, and λ-bracket
Proof
The λ-bracket defined by (162) is the formal Fourier transform of the z-product in V. V is obviously an \(\mathbb{F}[T]\)-module. Moreover, the Fourier coefficients of the formal distributions \(\{a(w)\vert a \in V \} \subset \mathop{\mathrm{End}}\nolimits V [[w,w^{-1}]]\) span a Lie subalgebra of \(\mathop{\mathrm{Lie}}\nolimits _{V }\) of \(\mathop{\mathrm{End}}\nolimits \,V\) (Corollary 2), and they are pairwise local, hence the skewsymmetry is always satisfied. Thus, \((\mathop{\mathrm{Lie}}\nolimits _{V },\{a(w)\}_{a\in V })\) is a formal distribution Lie algebra. Hence, by Proposition 13, the formal distributions {a(w)} a ∈ V satisfy all axioms of a Lie conformal algebra. Due to the nth product identity, Proposition 14 follows. □
We thus obtain the following
Theorem 5
Let V be a vertex algebra. Then the quintuple (V, | 0〉, T, : : , [. λ . ]) satisfies the following properties of a “quantum Poisson vertex algebra”.
-
(a)
(V, T, [⋅ λ ⋅ ]) is a Lie conformal algebra.
-
(b)
(V, | 0〉, T, : : ) is a quasicommutative, quasiassociative unital differential algebra.
-
(c)
The normally order product:: and the λ-bracket [⋅ λ ⋅ ] are related by the noncommutative Wick formula (145).
Remark 17
In fact, properties (a), (b), (c) of Theorem 5 characterize a vertex algebra structure, i.e., a quintuple (V, | 0〉, T, : : , [⋅ λ ⋅ ]) satisfying the above “quantum Poisson vertex algebra” properties, is a vertex algebra. This is proved in [2].
Example 10
(Computation with the non-commutative Wick’s formula) The simplest example is a free boson. Recall Example 3 in Lecture 1. For a free boson field a(z), we have
In the language of λ-brackets this means for a = fs(a(z)):
i.e., a (1) a = 1 and a (n) a = 0 for n = 0 or \(n\geqslant 2\).
Now let \(L:= \frac{1} {2}: aa:\), then
Indeed,
Using (164), we obtain [a λ L] = λa (since [ | 0〉 λ a] = 0). By the skewsymmetry of the λ-bracket, the first equation in (165) follows.
Next we have:
proving the second equation in (165).
Of course, there is a simpler way of manipulating with free quantum fields, see Theorem 3.3 in [17]. However, exactly the same method as above works well for arbitrary quantum fields (like currents, discussed below).
The following proposition tells us how to prove that a vector L is a conformal vector, hence how to construct a Hamiltonian operator H = L 0.
Proposition 15
Let
be a pre-vertex algebra and let L ∈ V be such that for
,
-
(i)
[L λ a] = (T + △ a λ)a + o(λ)
-
(ii)
L(z) satisfies the Virasoro relation: \([L_{\lambda }L] = (T + 2\lambda )L + \frac{\lambda ^{3}} {12}c\vert 0\rangle\) .
Then L is a conformal vector of the corresponding (by the Extension theorem) vertex algebra.
Proof
L(z) is already a Virasoro field, so we only need to prove that L
−1 = T and that L
0 acts diagonalizably on V. By completeness, V is spanned by \(a_{(k_{1})}^{j_{1}}\cdots a_{(k_{ s})}^{j_{s}}\vert 0\rangle\), where 
. Furthermore, property (i) tells us
Moreover, letting a = | 0〉 in (i), we get
Remember that T also satisfies the first equation in (166), so [L −1 − T, a (k)] = 0 for all \(k \in \mathbb{Z}\). Moreover (L −1 − T) | 0〉 = 0, so L −1 − T, being a derivation of all nth products, is zero, i.e., L −1 = T. L 0 is diagonizable by (166). □
It follows from Proposition 15 and (165) that L is a conformal vector for the free boson vertex algebra, the free boson a being primary of conformal weight 1. Exactly the same method works for the affine vertex algebras.
Exercise 12
Let \(V ^{k}(\mathfrak{g})\) be the universal affine vertex algebra of level k associated to a simple Lie algebra \(\mathfrak{g}\). Let a i, b i be dual bases of \(\mathfrak{g}\), i.e., (b i | a j) = δ ij with respect to the Killing form. Assume that k ≠ − h ∨, where 2h ∨ is the eigenvalue of the Casimir element of \(U(\mathfrak{g})\) in the adjoint representation (h ∨ is called the dual Coxeter number). Let \(L = \frac{1} {2(k+h^{\vee })}\sum _{i}: a^{i}b^{i}:\) (the so called Sugawara construction). Show that L is a conformal vector with central charge \(c = \frac{k\mathrm{dim} \mathfrak{g}} {2(k+h^{\vee })}\), all \(a \in \mathfrak{g}\) being primary of conformal weight 1.
4.3 Quasiclassical Limit of Vertex Algebras
Suppose we have a family of vertex algebras, i.e. a vertex algebra (V ℏ , T ℏ , | 0〉 ℏ , : : ℏ , [⋅ λ ⋅ ] ℏ ) over \(\mathbb{F}[[\hslash ]]\), such that
-
(i)
for v ∈ V ℏ , ℏv = 0 only if v = 0 (e.g. if V ℏ is a free \(\mathbb{F}[[\hslash ]]\)-module),
-
(ii)
[a λ b] ℏ ∈ ℏV ℏ for a, b ∈ V ℏ .
Given a vertex algebra (V
ℏ
, T
ℏ
, | 0〉
ℏ
, : :
ℏ
, [
λ
]
ℏ
) over \(\mathbb{F}[[\hslash ]]\), satisfying the above two conditions, let 
. This is a vector space over \(\mathbb{F}.\) Denote by 1 the image of | 0〉
ℏ
∈ V
ℏ
in 
and by ∂ the operator on 
induced by \(T \in \mathop{\mathrm{End}}\nolimits \,V _{\hslash }\) (ℏV
ℏ
is obviously T-invariant). The subspace (over \(\mathbb{F}\)) ℏV
ℏ
is obviously an ideal for the product: v:
ℏ
, hence we have the induced product ⋅ on 
which is bilinear over \(\mathbb{F}.\) Finally, define a λ-bracket {a
λ
b} on 
as follows. Let \(\tilde{a}\) and \(\tilde{b}\) be preimages in V
ℏ
of a and b; then we have
where \([\tilde{a}_{\lambda }\tilde{b}]^{{\prime}}\) is uniquely defined due to (i) and (ii). We let
Obviously this λ-bracket is independent of the choices of the preimages of a and b.
Definition 16
The quasiclassical limit of the family of vertex algebras V
ℏ
is the quintuple 
Definition 17
A Poisson vertex algebra is a quintuple 
which satisfies the following axioms,
-
(A)
is a Lie conformal algebra, -
(B)
is a commutative associative unital differential algebra, -
(C)
{a λ bc} = {a λ b}c + b{a λ c} for all
(left Leibniz rule).
is a Lie conformal algebra,
is a commutative associative unital differential algebra,
(left Leibniz rule).Theorem 6
The quasiclassical limit
of the family of vertex algebras V
ℏ
is a Poisson vertex algebra.
Proof
Since V
ℏ
is a vertex algebra over \(\mathbb{F}[[\hslash ]]\), due to Theorem 5 we have the quasicommutativity formula, the quasiassociativity formula and the non-commutative Wick formula for representatives in V
ℏ
of elements of 
. After taking the images of these formulas in 
, the “quantum corrections” disappear, hence 
satisfies properties (B) and (C) of a PVA. Property (C) is satisfied as well since the axioms of a Lie conformal algebra are homogeneous in its elements. □
Exercise 13
Deduce from the left Leibniz rule and the skewcommutativity of the λ-bracket of a Poisson vertex algebra, the right Leibniz rule:
Given a Lie algebra \(\mathfrak{g}\), we can associate to it two structures: the universal enveloping algebra \(U(\mathfrak{g})\) and the Poisson algebra \(S(\mathfrak{g})\). The Poisson bracket on \(S(\mathfrak{g})\) is the extension of {a, b} = [a, b] for all \(a,b \in \mathfrak{g}\) by left and right Leibneiz rules. In fact, \(S(\mathfrak{g})\) is the quasiclassical limit of \(U(\mathfrak{g}_{\hslash })\), where \(\mathfrak{g}_{\hslash }\) is the Lie algebra \(\mathbb{F}[[\hslash ]] \otimes \mathfrak{g}\) over \(\mathbb{F}[[\hslash ]]\) with bracket [a, b] ℏ = ℏ[a, b] for \(a,b \in \mathfrak{g}\). Indeed it is easy to see that the ordered monomials in a basis of \(\mathfrak{g}\) form a basis of \(U(\mathfrak{g}_{\hslash })\) over \(\mathbb{F}[[\hslash ]]\). Hence \(U(\mathfrak{g}_{\hslash })/\hslash U(\mathfrak{g}_{\hslash }) = S(\mathfrak{g})\) as associative algebras, and \(\{a,b\} = \dfrac{[\tilde{a},\tilde{b}]_{\hslash }} {\hslash } \bigg\vert _{\hslash =0} = [a,b]\) for all \(a,b \in \mathfrak{g}\) defines the Poisson structure on \(S(\mathfrak{g})\).
Similar picture holds if in place of a Lie algebra \(\mathfrak{g}\) we take a Lie conformal algebra R, and in place of \(U(\mathfrak{g})\) we take V (R), its universal enveloping vertex algebra. Recall its construction. We have the “maximal” formal distribution Lie algebra \((\mathop{\mathrm{Lie}}\nolimits R,R)\), associated to R, which is regular (see [17], Chap. 2). Then \(V (R) = V (\mathop{\mathrm{Lie}}\nolimits R,R)\) (for another construction, entirely in terms of R, see [6]).
Consider the vertex algebra V (R ℏ ) over \(\mathbb{F}[[\hslash ]]\), where R ℏ = R[[ℏ]] for the Lie conformal algebra R over \(\mathbb{F}\), with λ-bracket defined by [a λ b] ℏ = ℏ[a λ b] for a, b ∈ R. In the same way as in the Lie algebra case, the quasiclassical limit is the Poisson vertex algebra, which, as a differential algebra, is S(R) (the symmetric algebra of the \(\mathbb{F}\)-vector space R) with ∂, extended as its derivation, endowed with the λ-bracket {a λ b} = [a λ b] on R, which is extended to S(R) by the left and right Leibniz rules.
4.4 Representations of Vertex Algebras and Zhu Algebra
We have the following diagram
In the diagram, AA means associative algebras, PA means Poisson algebras, VA means vertex algebras and PV A means Poisson vertex algebras; q.lim means the quasiclassical limit and Zhu means a functor from vertex algebras to associative algebras (resp. from Poisson vertex algebras to Poisson algebras), explained below.
Let V be a vertex algebra with a Hamiltonian operator H. Throughout this section we will assume (for simplicity) that all eigenvalues of H are integers. Recall the Borcherds identity from Sect. 3.2. For a and b ∈ V with eigenvalues of H equal Δ a and Δ b respectively, we write
Then, comparing the coefficients of monomials in z and w in the Borcherds identity we have, for \(m,n,k \in \mathbb{Z}\):
Definition 18
A representation of the vertex algebra V in a vector space M is a linear map
defined for eigenvectors of H and then extended linearly to V, such that,
-
(i)
a M(z) is an \(\mathop{\mathrm{End}}\nolimits M\)-valued quantum field for all a ∈ V (i.e., given m ∈ M, a (n) M m = 0 for n ≫ 0),
-
(ii)
| 0〉M(z) = I M ,
-
(iii)
Borcherds identity holds, i.e., for \(a,b \in V,\ c \in M,\ m,n,k \in \mathbb{Z}\) we have [cf. (168)]:
$$\displaystyle\begin{array}{rcl} & & \sum _{j\geqslant 0}{k\choose j}(-1)^{j}(a_{ m+k-j}^{M}b_{ n+j}^{M}c - (-1)^{n}b_{ n+k-j}^{M}a_{ m+j}^{M}c) \\ & & \quad =\mathop{\sum }\limits_{ j\geqslant 0}{m + \bigtriangleup _{a} - 1\choose j}(a_{(k+j)}b)_{m+n+k}^{M}c\,. {}\end{array}$$(170)
Remark 18
Note that (Ta) n = (−n −Δ a )a n and Ha = Δ a a, hence, ((T + H)a)0 = 0.
Now assume that our vertex algebra V contains a conformal vector L of central charge \(c \in \mathbb{F}\) (see Definition 11), so that L −1 = T and L 0 = H is a Hamiltonian operator. Then we have \(L^{M}(z) =\sum _{n\in \mathbb{Z}}L_{n}^{M}z^{-n-2}\), and \([L_{m}^{M},L_{n}^{M}] = (m - n)L_{m+n}^{M} +\delta _{m,-n}\frac{m^{3}-m} {12} cI_{M}\).
Definition 19
A positive energy representation M of V is a representation with L 0 M acting diagonalizably on M with spectrum bounded below, i.e., \(M = \oplus _{j\geqslant h}M_{j}\) for some h, where M j = {m ∈ M | L 0 M m = jm}.
By (126) (which follows from the Borcherds identity) we have
So we have a linear map with (H + T)V contained in the kernel (by Remark 18):
Taking m = 1, k = −1, n = 0 in Borcherds identity (170) for c ∈ M h , we get, by (171),
where
Thus we get a representation of the algebra (V, ∗) in the vector space M h . The multiplication ∗ on V is not associative. However, we have the following remarkable theorem.
Theorem 7 ([21])
-
(a)
J(V ): = ((T + H)V ) ∗ V is a two-sided ideal of the algebra (V, ∗).
-
(b)
ZhuV: = (V∕J(V ), ∗) is a unital associative algebra with 1 being the image of | 0〉.
-
(c)
The map M → M h induces a map from the equivalence classes of positive energy V -modules to the equivalence classes of ZhuV -modules, which is bijective on irreducible modules.
Proof
We refer for the proof to the original paper [21] or to [6] for a simpler proof of a similar result without the assumption that the eigenvalues of H are integers. □
Exercise 14
Prove the commutator formula in Zhu algebra:
Exercise 15
Let 
be a Poisson vertex algebra and let H be a diagonalizable operator on 
, such that
where Δ a is the eigenvalue of a, and
Show that 
is a unital commutative associative algebra with the well defined Poisson bracket (cf. Exercise 14)
Exercise 16
Let V (resp. 
) be a vertex algebra (resp. Poisson vertex algebra). Then J =: (TV )V: (resp. 
) is a two-sided ideal of the algebra (V, : : ) (resp. 
), and V∕J (resp. 
) is a Poisson algebra with the product, induced by: : (resp ⋅ ), and the well defined bracket, induced by the 0th product of the λ-bracket.
Of course, Zhu’s theorem is just the beginning of the representation theory of vertex algebras, which has been a rapidly developing field in the past twenty years. Some of the most remarkable results of this theory are presented in the beautiful lecture course by T. Arakawa in this school.
5 Lecture 5 (January 14, 2015)
Given a vertex algebra V, one can construct its quasiclassical limit. As a result we get a Poisson vertex algebra (PVA). This can be done both considering a filtration of the vertex algebra V or by constructing a one parameter family of vertex algebras V ℏ , as previously done in Lecture 4. This construction resembles the way a Poisson algebra arises as a quasiclassical limit of a family of associative algebras, hence the name “Poisson” vertex algebra. The reason we are interested in such structures is that the theory of Poisson vertex algebras has important relation with the theory of integrable systems of PDE’s. This relation is parallel to (but a bit different from) the relation of Poisson algebras with the theory of integrable systems of ODE’s.
5.1 From Finite-Dimensional to Infinite-Dimensional Poisson Structures
Let us start by recalling the definition of a Poisson vertex algebra:
Definition 20
A PVA is a quintuple \((\mathcal{V },\,\partial,\,1,\,\cdot \,,\{\cdot _{\lambda }\cdot \})\) such that:
-
1.
\((\mathcal{V },\,\partial,\,1,\,\cdot \,)\) is a differential algebra;
-
2.
\((\mathcal{V },\,\partial,\{\cdot _{\lambda }\cdot \})\) is a Lie conformal algebra, whose λ-bracket satisfies the following axioms:
-
(i)
(sesquilinearity) {∂a λ b} = −λ{a λ b}, {a λ ∂b} = (∂ + λ){a λ b};
-
(ii)
(skewsymmetry) {b λ a} = −{a −∂−λ b};
-
(iii)
(Jacobi identity) {a λ {b μ c}} −{ b μ {a λ c}} = {{ a λ b} λ+μ c};
-
(i)
-
3.
{ ⋅ λ ⋅ } and ⋅ are related by the following Leibniz rules:
-
(i)
(left Leibniz rule) {a λ bc} = {a λ b}c + b{a λ c};
-
(ii)
(right Leibniz rule) {ab λ c} = {a λ+∂ c}→ b +{ b λ+∂ c}→ a.
-
(i)
Remark 19
We use the following notation: if \(\{a_{\lambda }b\} =\sum \limits _{n\in \mathbb{Z}_{+}} \frac{\lambda ^{n}} {n!}a_{(n)}b\), then when a right arrow appears it means that λ + ∂ has to be moved to the right: \(\{a_{\lambda +\partial }b\}_{\rightarrow }c =\sum \limits _{n\in \mathbb{Z}_{+}} \frac{a_{(n)}b} {n!} (\lambda +\partial )^{n}c\). However, if no arrow appears we just have \(\{a_{-\partial -\lambda }b\} =\sum \limits _{n\in \mathbb{Z}_{+}} \frac{(-\lambda -\partial )^{n}} {n!} a_{(n)}b\).
In the theory of Hamiltonian ODEs the key role is played by the Poisson bracket on the space of smooth functions 
on a manifold. Choosing local coordinates u
1, …, u
ℓ
on the manifold, we can endow 
with a structure of Poisson algebra, letting
By the Leibniz rule this extends to polynomials in the variables u i as follows:
where \(\frac{\partial f} {\partial u} = \left (\begin{array}{c} \frac{\partial f} {\partial u_{1}}\\ \vdots \\ \frac{\partial f} {\partial u_{\ell}} \end{array} \right )\), \(u = \left (\begin{array}{c} u_{1}\\ \vdots \\ u_{\ell} \end{array} \right )\), \(H = (H_{ij})_{\begin{array}{c}i,j=1\end{array}}^{\ell}\) is an ℓ × ℓ matrix with coefficients in 
, and ⋅ is the usual dot product of vectors from 
with values in 
Formula (177) extends to arbitrary functions 
. This bracket obviously satisfies the Leibniz rule, but it is not necessarily skewsymmetric, neither it satisfies the Jacobi identity. If the matrix H is skewsymmetric (i.e. H
T = −H), then the bracket (177) is skewsymmetric. If, in addition, it satisfies the Jacobi identity (which happens iff [H, H] = 0, where [⋅ , ⋅ ] is the Schouten bracket), then the matrix H is called a Poisson structure on 
.
Definition 21
The Hamiltonian ODE associated with this Poisson structure is
where the second equality follows from (177). The function 
is called the Hamiltonian of this equation.
This is a special case of what is called an evolution ODE, that is
In the theory of Hamiltonian PDEs a similar role is played by PVAs. Let us now see how to construct a similar machinery.
First of all we need to define which kind of differential algebra \(\mathcal{V }\) we want for our PVA. The basic example is the algebra of differential polynomials in ℓ variables \(\mathcal{P}_{\ell} = \mathbb{F}[u_{i}^{(n)}\,\vert \,i \in I =\{ 1,\ldots,\ell\},\,n \in \mathbb{Z}_{+}]\), which is a differential algebra with derivation ∂, called the total derivative, such that ∂u i (n) = u i (n+1).
Definition 22
An algebra of differential functions in ℓ variables \(\mathcal{V }\) is a differential algebra with a derivation ∂, which is an extension of the algebra of differential polynomials \(\mathcal{P}_{\ell}\), endowed with linear maps \(\frac{\partial } {\partial u_{i}^{(n)}}:\mathcal{ V } \rightarrow \mathcal{ V }\) for all i ∈ I, \(n \in \mathbb{Z}_{+}\), which are commuting derivations of \(\mathcal{V }\), extending the usual partial derivatives in \(\mathcal{P}_{\ell}\), and satisfying the following axioms:
-
(i)
given \(f \in \mathcal{ V }\), \(\frac{\partial f} {\partial u_{i}^{(n)}} = 0\) for all but finitely many pairs \((i,n) \in I \times \mathbb{Z}_{+}\);
-
(ii)
\([ \frac{\partial } {\partial u_{i}^{(n)}},\partial ] = \frac{\partial } {\partial u_{i}^{(n-1)}}\) (where the RHS is considered to be zero if n = 0).
Which differential algebras are algebras of differential functions? The algebra of differential polynomials \(\mathcal{P}_{\ell}\) itself clearly satisfies these axioms (it suffices to check (ii) on the generators u i (n)). One can as well consider the corresponding field of fractions \(\mathcal{Q}_{\ell} = \mathbb{F}(u_{i}^{(n)}\,\vert \,i \in I,\,n \in \mathbb{Z}_{+})\), or any algebraic extension of \(\mathcal{P}_{\ell}\) or \(\mathcal{Q}_{\ell}\), obtained by adding a solution of a polynomial equation. However, if we want both axioms to hold, we can not add a solution of an arbitrary differential equation: for example, we can add e u, solution of f ′ = fu ′, but we can not add a non-zero solution of f′ = fu.
Exercise 17
Let \(\mathcal{V } =\mathcal{ P}_{1}[v]\) with the derivation ∂, extended from \(\mathcal{P}_{1}\) by ∂v = vu 1 or by ∂v = u 1. Show that the structure of an algebra of differential functions cannot be extended from \(\mathcal{P}_{1}\) to \(\mathcal{V }\).
The reasons why we want both properties (i) and (ii) to hold will soon be clear.
We also want an analogue of the bracket given by (177) and to understand what a Poisson structure is in the infinite-dimensional case. Recall the following (non-rigorous) formula which appears in any textbook on integrable Hamiltonian PDE, cf. [19], but not [12]. It defines the Poisson bracket on generators (i, j ∈ I) as
where H = (H ji ) i, j = 1 ℓ is an ℓ × ℓ matrix differential operator on \(\mathcal{V }^{\ell}\), the u i ’s are viewed as functions in x on a one-dimensional manifold, and δ(x − y) is the usual delta function.
Example 11
The first example is given by the Gardner-Faddeev-Zakharov (GFZ) bracket, for \(\mathcal{V } =\mathcal{ P}_{1}\), and it goes back to 1971:
As in the ODE case, we can extend the bracket defined in (179) by the Leibniz rule. Then, for arbitrary \(f,\,g \in \mathcal{ V }\) we have
The basic idea is to introduce the λ-bracket by application of the Fourier transform
to both sides of (181):
Thus, for arbitrary \(f,g \in \mathcal{ V }\), we get a rigorous formula, called the Master Formula:
Here, {u j ∂+λ u i } = H ij (∂ + λ), where H(∂) = (H ij (∂)) i, j ∈ I is a matrix differential operator with coefficients in \(\mathcal{V }\) for which the λ-bracket is its symbol.
Exercise 18
Note that (MF) is similar to the formula for the Poisson bracket defined by Eq. (177). In fact, to go from the former to the latter we just put λ and ∂ equal to 0.
Theorem 8 ([3])
Let \(\mathcal{V }\) be an algebra of differential functions in the variables {u i } i ∈ I . For each pair i, j ∈ I choose \(\{u{_{i}}_{\lambda }u_{j}\} = H_{ji}(\lambda ) \in \mathcal{ V }[\lambda ]\) . Then
-
1.
The Master Formula ( MF ) defines a λ-bracket on \(\mathcal{V }\) which satisfies sesquilinearity, the left and right Leibniz rules, and extends the given λ-bracket on the variables u i ’s. Consequently, any λ-bracket on the algebra of differential polynomials, satisfying these properties, is given by the Master Formula.
-
2.
This λ-bracket is skewsymmetric provided skewsymmetry holds for every pair of variables:
$$\displaystyle{ \{u{_{i}}_{\lambda }u_{j}\} = -\{u{_{j}}_{-\lambda -\partial }u_{i}\},\quad \forall \,i,j \in I. }$$(184) -
3.
If this λ-bracket is skewsymmetric, then it satisfies the Jacobi identity, provided Jacobi identity holds for every triple of variables:
$$\displaystyle{ \{u{_{i}}_{\lambda }\{u{_{j}}_{\mu }u_{k}\}\} -\{ u{_{j}}_{\mu }\{u{_{i}}_{\lambda }u_{k}\}\} =\{ \{u{_{i}}_{\lambda }u_{k}\}_{\lambda +\mu }u_{j}\},\quad \forall \,i,j,k \in I. }$$(185)
It follows from Theorem 8 that, if the corresponding conditions on the variables u i ’s hold, the λ-bracket defined by the Master Formula (MF) endows \(\mathcal{V }\) with a structure of PVA. As in the finite-dimensional case, this structure is completely defined by \(H(\lambda ) = (H_{ij}(\lambda )) \in \text{Mat}_{\ell\times \ell}\mathcal{V }[\lambda ]\).
Definition 23
We say that the matrix differential operator \(H(\partial ) \in \text{Mat}_{\ell\times \ell}\mathcal{V }[\partial ]\) with the symbol H(λ) is a Poisson structure if the corresponding λ-bracket defines a PVA structure on \(\mathcal{V }\).
Exercise 19
The λ-bracket, given by the Master Formula, is skewsymmetric if and only if the matrix differential operator H(∂) is skewadjoint.
Example 12
Let \(\mathcal{V } =\mathcal{ P}_{1} = \mathbb{F}[u,u^{{\prime}},u^{{\prime\prime}},\ldots ]\). From the GFZ bracket defined in Example 11 we get the following λ-bracket: {u λ u} = λ. The skewsymmetry and the Jacobi identity for the λ-bracket, given by the Master Formula, are immediate by Theorem 8. The associated Poisson structure is H(∂) = ∂. This PVA is the quasi-classical limit of the family of free boson vertex algebras B ℏ .
Example 13
Let \(\mathcal{V } =\mathcal{ P}_{1} = \mathbb{F}[u,u^{{\prime}},u^{{\prime\prime}},\ldots ]\). The Magri-Virasoro PVA with central charge \(c \in \mathbb{F}\) is defined by the following λ-bracket:
Of course, it is straightforward to check that the pair u, u satisfies (184) and the triple u, u, u satisfies (185), hence, by Theorem 8, we get a PVA. It is instructive, however, to give a more conceptual proof. Consider the Lie conformal algebra Vir from Example 8. Then by Theorem 8, S(Vir) is a PVA, hence its quotient 
by the ideal, generated by C − c, is a PVA, which is obviously isomorphic to the Magri-Virasoro PVA. The corresponding family of Poisson structures is
These Poisson structures were discovered by Magri; the name is due to its connection to the Virasoro algebra. Note that 
is the quasiclassical limit of the family of universal Virasoro vertex algebras V
ℏ
12c.
The following exercise shows that the discrete series vertex algebras V c with c given by (60) is a purely quantum effect.
Exercise 20
Show that the PVA 
is simple if c ≠ 0.
Example 14
Given a vector space U, denote by 
the algebra of differential polynomials over U. Let \(\mathfrak{g},(.\,\vert \,.)\) be as in Example 6, let \(k \in \mathbb{F},\) and fix \(s \in \mathfrak{g}.\) Then the associated affine PVA
is defined as the algebra of differential polynomials 
, endowed with the λ-brackets \((a,b \in \mathfrak{g}):\)
The two proofs from Example 13 apply to show that 
is a PVA. Of course, up to isomorphism, it is independent of s, but the trivial cocycle is important for the associated integrable system, since we get a multiparameter family of Poisson structures. Note that 
is the quasiclassical limit of \(V _{\hslash }^{k}(\mathfrak{g}).\)
Now we recall how one passes from the definition of a Hamiltonian ODE to that of a Hamiltonian PDE. The following idea goes back to the 1970s: in order to get an “honest” Lie algebra bracket, we should not consider the whole algebra of differential functions \(\mathcal{V }\), but its quotient \(\mathcal{V }/\partial \mathcal{V }\), which is not an algebra anymore, just a vector space. Denote by ∫ the quotient map \(\int:\mathcal{ V } \rightarrow \mathcal{ V }/\partial \mathcal{V }\). The corresponding bracket is defined by
where \(\frac{\delta f} {\delta u}\) is the vector of variational derivatives of f:
Elements \(\int f \in \mathcal{ V }/\partial \mathcal{V }\) are called local functionals.
Equation (189) is analogous to Eq. (177), with variational derivatives instead of partial derivatives, and a matrix differential operator H(∂) instead of a matrix of functions. It is rather difficult to prove directly that (189) is a Lie algebra bracket on \(\mathcal{V }/\partial \mathcal{V }\). The connection to the PVA theory, explained further on, makes it very easy.
The following exercise shows that (189) is well defined.
Exercise 21
The variational derivative \(\frac{\delta f} {\delta u}\) depends only on the image of \(f \in \mathcal{ V }\) in the quotient space \(\mathcal{V }/\partial \mathcal{V }\), since \(\frac{\delta }{\delta u} \circ \partial = 0\). Deduce the latter fact from axiom (ii) in the Definition 22 of an algebra of differential functions.
Given a local functional ∫ h, in analogy with (178), one defines the associated Hamiltonian PDE as the following evolution PDE:
The local functional ∫ h is called the Hamiltonian of this equation.
We shall explain further on how these classical definitions fit nicely in the framework of Poisson vertex algebras.
5.2 Basic Notions of the Theory of Integrable Equations
An evolution equation in the infinite-dimensional case is quite the same as in the finite-dimensional case, except it is a partial differential equation.
Definition 24
Let \(\mathcal{V }\) be an algebra of differential functions in ℓ variables u 1, …, u ℓ . An evolution PDE is
where \(u = \left (\begin{array}{c} u_{1}\\ \vdots \\ u_{\ell} \end{array} \right )\) and \(F = \left (\begin{array}{c} F_{1}\\ \vdots \\ F_{\ell}\end{array} \right ) \in \mathcal{ V }^{\ell}\). Here, u i = u i (x, t) is a function in one independent variable x, and the parameter t is called time.
Given an arbitrary differential function \(f \in \mathcal{ V }\), by the chain rule we have
Since, by (191), we have \(\frac{d(u_{i}^{(n)})} {dt} = \partial ^{n}F_{ i}\), the function f evolves in virtue of Eq. (191) as
where
is a derivation of the algebra \(\mathcal{V }\), called the evolutionary vector field with characteristic \(F \in \mathcal{ V }^{\ell}\). It is now clear why Axiom (i) in Definition 22 is important: otherwise, the evolutionary vector field would give a divergent sum when applied to arbitrary functions \(f \in \mathcal{ V }\).
An important notion in the theory of integrable systems is compatibility of evolution equations:
Definition 25
Equation (191) is called compatible with the evolution PDE
where, as before, \(u = \left (\begin{array}{c} u_{1}\\ \vdots \\ u_{\ell} \end{array} \right )\) and \(G = \left (\begin{array}{c} G_{1}\\ \vdots \\ G_{\ell}\end{array} \right ) \in \mathcal{ V }^{\ell}\), if the corresponding flows commute, that is if \(\frac{d} {dt} \frac{d} {d\tau }f = \frac{d} {d\tau } \frac{d} {dt}f\) holds for every function \(f \in \mathcal{ V }\).
By the above discussion, the compatibility of evolution equations (191) and (194) is equivalent to the property that the corresponding evolutionary vector fields commute: [X F , X G ] = 0, which is a purely Lie algebraic condition. In fact, we can easily see that the commutator of two evolutionary vector fields is again an evolutionary vector field. This follows from the next exercise.
Exercise 22
Prove that [X F , X G ] = X [F, G], where [F, G]: = X F G − X G F.
Thus, the bracket [F, G] = X F G − X G F endows \(\mathcal{V }^{\ell}\) with a Lie algebra structure, called the Lie algebra of evolutionary vector fields.
If two evolutionary vector fields commute, then each of them is called a symmetry of the other. So if [X F , X G ] = 0, F is a symmetry of G and G is a symmetry of F. Note that every evolutionary vector field commutes with \(\partial = X_{u^{{\prime}}} =\sum \limits _{i\in I,\,n\in \mathbb{Z}_{+}}u_{i}^{(n+1)} \frac{\partial } {\partial u_{i}^{(n)}}\).
Let us now introduce the notion of integrability for an evolution equation.
Definition 26
Equation (191) is called Lie integrable if X F is contained in an infinite-dimensional abelian subalgebra of the Lie algebra \(\mathcal{V }^{\ell}\).
Remark 20
Informally, one says that Eq. (191) is Lie integrable if it admits infinitely many commuting symmetries.
Example 15
The linear equations over \(\mathcal{P}_{1}\):
are Lie integrable. Indeed, \(X_{u^{(m)}}(u^{(n)}) = u^{(m+n)}\) is symmetric in m and n, hence the corresponding evolutionary vector fields commute.
Example 16
The dispersionless equations over \(\mathcal{P}_{1}\):
are Lie integrable, since
is symmetric in f and g, hence the corresponding evolutionary vector fields commute.
The motivation for the definition of Lie integrability of PDE’s comes from a theorem of Lie in the theory of ODE’s, saying that if the evolution ODE in ℓ variables \(\frac{du} {dt} = F(u)\) possesses ℓ commuting symmetries with a non-degenerate Jacobian, then it can be solved in quadratures. Of course, in the PDE case the number of coordinates is infinite, therefore we need to require infinitely many commuting symmetries.
There has been a lot of work trying to establish integrability of various partial differential equations. One well-known method of constructing symmetries of an evolution equation is called recursion operator; however, in all examples the recursion operator is actually a pseudodifferential operator (which is an element of \(\mathcal{V }((\partial ^{-1}))\)), hence it can not be applied to functions, as Exercise 17 demonstrates. We will discuss a different approach, the Hamiltonian approach, which is completely rigorous.
We shall deduce Lie integrability from the stronger Liouville integrability of Hamiltonian PDE, which, analogously to the definition for ODEs, requires the existence of infinitely many integrals of motion in involution.
5.3 Poisson Vertex Algebras and Hamiltonian PDE
In order to translate the traditional language of Hamiltonian PDE’s, discussed above, to the language of PVA’s, and also, to connect the two notions of integrability, the following simple lemma is crucial.
Lemma 8 (Basic Lemma)
Let \(\mathcal{V }\) be a PVA. Let \(\bar{\mathcal{V }}:=\mathcal{ V }/\partial \mathcal{V }\) and let \(\int:\mathcal{ V } \rightarrow \bar{\mathcal{ V }}\) be the corresponding quotient map. Then we have the following well-defined brackets:
-
(i)
\(\bar{\mathcal{V }} \times \bar{\mathcal{ V }}\longrightarrow \bar{\mathcal{V }},\qquad \{\int a,\int b\}:=\int \{ a_{\lambda }b\}_{\lambda =0}\) ,
-
(ii)
\(\bar{\mathcal{V }} \times \mathcal{ V }\longrightarrow \mathcal{V },\qquad \{\int a,b\}:=\{ a_{\lambda }b\}_{\lambda =0}\) .
Moreover, (i) defines a Lie algebra bracket on \(\bar{\mathcal{V }}\) , and (ii) defines a representation of the Lie algebra \(\bar{\mathcal{V }}\) on \(\mathcal{V }\) by derivations of the product and the λ-bracket of \(\mathcal{V }\) , commuting with ∂.
Proof
It all follows directly when we put λ = 0 in the axioms for the λ-bracket { ⋅ λ ⋅ } of a PVA. First, both brackets are well defined since sesquilinearity holds for { ⋅ λ ⋅ }: for every \(a,b \in \mathcal{ V }\) we have {∂a, b} = −λ{a λ b} λ = 0 = 0 and \(\{a,\partial b\} =\{ a_{\lambda }\partial b\}_{\lambda =0} = \partial \{a_{\lambda }b\} \in \partial \mathcal{V }\).
Let us now verify the Lie algebra axioms for the first bracket: note that ∫{b −λ−∂ a} λ = 0 = ∫{b λ a} λ = 0 since only the coefficients of the 0th power of −λ − ∂ and λ respectively survive in \(\bar{\mathcal{V }}\), and they obviously coincide. By skewsymmetry of { ⋅ λ ⋅ } we have
Hence, skewsymmetry holds for (i). Similarly, the Jacobi identity for { ⋅ λ ⋅ } provides that the Jacobi identity holds for this bracket as well, just putting λ = μ = 0 in the corresponding definitions:
Therefore, \(\bar{\mathcal{V }}\) is endowed with a Lie algebra structure with the Lie bracket defined by (i).
Next, we have to check that (ii) is a representation of \(\bar{\mathcal{V }}\) on \(\mathcal{V }\), i.e., that
holds for all \(\int f,\int g \in \bar{\mathcal{ V }}\), \(a \in \mathcal{ V }\). Again, this is due to the Jacobi identity. Then we have to check that \(\bar{\mathcal{V }}\) acts on \(\mathcal{V }\) as derivations of the product. For \(a,b \in \mathcal{ V }\) and \(\int h \in \bar{\mathcal{ V }}\) we have, by the left Leibniz rule:
Similarly, by the Jacobi identity, we check that it acts by derivations of the λ-bracket. Finally, we have to check that the derivations {∫h, ⋅ } commute with ∂. For every \(a \in \mathcal{ V }\) we have
due to the sesquilinearity of { ⋅ λ ⋅ }. □
Definition 27
Given a PVA \(\mathcal{V }\) and a local functional \(\int h \in \bar{\mathcal{ V }}\), the associated Hamiltonian PDE is
The local functional ∫h is called the Hamiltonian of this equation.
In the case when the PVA \(\mathcal{V }\) is an algebra of differential functions in the variables {u i } i ∈ I and the λ-bracket is given by the Master Formula (MF), we reproduce the traditional definitions:
-
(i)
Hamiltonian PDE: \(\frac{du} {dt} =\{\int h,u\} = H \frac{\delta \int h} {\delta u}\);
-
(ii)
Poisson bracket on \(\bar{\mathcal{V }}\): \(\{\int f,\int g\} =\int \frac{\delta g} {\delta u} \cdot H \frac{\delta f} {\delta u}\).
The first claim is obvious, and the second is obtained by integration by parts.
It follows that in this case \(\bar{\mathcal{V }}\) acts on \(\mathcal{V }\) by evolutionary vector fields: \(\int f\mapsto X_{H \frac{\delta f} {\delta u} }\), and that the following holds.
Corollary 3
We have a Lie algebra homomorphism \(\bar{\mathcal{V }} \rightarrow \mathcal{ V }^{\ell}\) , \(\int f\mapsto X_{H \frac{\delta f} {\delta u} }\) .
Thus, in the case when \(\mathcal{V }\) is an algebra of differential functions with the Poisson λ-bracket given by the Master formula, the Hamiltonian equation is a special case of the evolution equation with RHS \(H \frac{\delta \int h} {\delta u}\) and the corresponding evolutionary vector field is \(X_{H \frac{\delta h} {\delta u} }\).
Definition 28
A local functional \(\int f \in \bar{\mathcal{ V }}\) is called an integral of motion of the evolution equation (191) and f is called a conserved density, if \(\int \frac{df} {dt} = 0\), or, equivalently, if ∫X F f = 0. Integrating by parts, this, in turn, is equivalent to
Hence, ∫f is an integral of motion of the Hamiltonian equation (200) if and only if f and h are in involution, that is if {∫f, ∫h} = 0.
So, we have completely translated the language of Hamiltonian PDEs into the language of PVAs.
Definition 29
The Hamiltonian PDE (200) is called Liouville integrable if ∫h is contained in an infinite-dimensional abelian subalgebra of the Lie algebra \(\bar{\mathcal{V }}\). That is, if there exists an infinite sequence of linearly independent local functionals ∫h n , such that ∫h 0 = ∫h and {∫h n , ∫h m } = 0 for all \(n,m \in \mathbb{Z}_{+}\).
By Corollary 3, integrals of motion in involution go to commuting evolutionary vector fields \(X_{H \frac{\delta \int h} {\delta u} }\). Hence Liouville integrability usually implies Lie integrability (provided we make some weak assumption on H(∂), such as H(∂) is non-degenerate). In fact, in order to check that the local functionals are linearly independent, it is usually easier to check that the corresponding evolutionary vector fields are linearly independent.
Exercise 23
Show that the equation \(\frac{du} {dt} = u^{{\prime\prime}}\) is Lie integrable, but has no non-trivial integrals of motion, hence is not Hamiltonian. On the other hand the equation \(\frac{du} {dt} = u^{{\prime\prime\prime}}\) is Hamiltonian with H = ∂, \(h = -\frac{1} {2}(u^{{\prime}})^{2}\), and it is both Lie and Liouville integrable.
Remark 21
Let F, G, … be a sequence of elements of \(\mathcal{V }^{\ell}\), such that the corresponding evolutionary vector fields commute, i.e. the corresponding evolution equations are compatible. Then we have a hierarchy of evolution equations
so that the solution of this hierarchy depends now on x and on infinitely many times: u = u(x, t 0, t 1, t 2, …).
5.4 The Lenard-Magri Scheme of Integrability
There is a very simple scheme to prove integrability, called the Lenard-Magri scheme. Although it is not a theorem, it always works in practice.
Let \(\mathcal{V }\) be an algebra of differential functions in ℓ variables u 1, …, u ℓ . First of all, introduce the following symmetric bilinear forms on \(\mathcal{V }^{\ell}\):
Given a matrix differential operator \(H(\partial ) \in \text{Mat}_{\ell\times \ell}\mathcal{V }[\partial ]\)
Note that (H(∂)F | G) = (F | H ∗(∂)G), where H ∗(∂) is the adjoint differential operator of H(∂). Indeed, defining ∗ on \(\mathcal{V }[\partial ]\) as an anti-involution such that ∗ ( f) = f and ∗ (∂) = −∂, we get (∂f | g) = −( f | ∂g) because (∂f | g) + ( f | ∂g) = ∫ ∂( fg) = 0 in \(\bar{\mathcal{V }}\). Hence, if H(∂) is skewadjoint, then the bilinear form (204) is skewsymmetric.
Proof of Liouville integrability is based on the following result.
Lemma 9 (Lenard Lemma)
Let H(∂) and K(∂) be skewadjoint differential operators on \(\mathcal{V }^{\ell}\) . Suppose elements \(\xi _{0},\ldots,\xi _{N} \in \mathcal{ V }^{\ell}\) satisfy the following Lenard-Magri relation:
Then, the 〈ξ m , ξ n 〉 = 0 for all m, n = 0, …, N, whenever we consider it with respect to H or K: 〈ξ m , ξ n 〉 H, K = 0.
Proof
Proceed by induction on i = | m − n |. If i = 0, then m = n and we get 〈ξ n , ξ n 〉 H, K = −〈ξ n , ξ n 〉 H, K because the form is skewsymmetric, therefore it is equal to zero. Now let i > 0; by skewsymmetry we may assume m > n. We have
and, by the induction hypothesis, the RHS is zero, since | m − (n + 1) | < | m − n |. Similarly we have, assuming n > m:
and again, by induction hypothesis the RHS is zero since | n − 1 − m | < | n − m |. □
This lemma is important since, if we can prove that the elements \(\xi _{m} \in \mathcal{ V }^{\ell}\) are variational derivatives, i.e. \(\xi _{m} = \frac{\delta \int h_{m}} {\delta u}\) for some local functionals ∫h m , it guarantees that ∫h m and ∫h n are in involution with respect to both brackets on \(\bar{\mathcal{V }}\). Indeed, we know that the bracket on \(\bar{\mathcal{V }}\) for the Poisson structure H is given by
therefore, if ξ n , ξ m are variational derivatives, then by Lemma 9 we get
and the same holds for K. In other words, we have the following corollary of Lenard’s lemma.
Corollary 4
Let H(∂) and K(∂) be skewadjoint differential operators on \(\mathcal{V }^{\ell}\) . Suppose that the local functionals ∫ h 0, …, ∫ h N satisfy the following relation:
Then all these local functionals are in involution with respect to both brackets {. , . } H and {. , . } K on \(\bar{\mathcal{V }}\) .
In the case when (210) holds, and K, H are Poisson structures, one says that the evolution equations
form a hierarchy of bi-Hamiltonian equations. Note that if the right-hand sides of these equations span an infinite-dimensional subspace in the space of evolutionary vector fields, then all of these equations are both Lie and Liouville integrable.
We now must address two issues:
-
1.
How can we construct vectors ξ n ’s satisfying Eq. (205)?
-
2.
How can we prove that such ξ n ’s are variational derivatives?
Although the second issue has been completely solved considering some reduced de Rham complex, called the variational complex, discussed in the next lecture, the first and basic issue is far from being resolved, though there are some partial results.
We will now see how to construct a sequence of vectors ξ n ’s satisfying the Lenard-Magri relation.
Lemma 10 (Extension Lemma)
[ 3 ] Suppose that, in addition to the hypothesis of Lemma 9 , we also have the following orthogonality condition: assume to have vectors \(\xi _{0},\ldots,\xi _{N} \in \mathcal{ V }^{\ell}\) , satisfying the Lenard-Magri relation ( 205 ), such that
where Span{ξ 0, …, ξ N }⊥ is the orthogonal complement with respect to the symmetric bilinear form ( 203 ). Then we can extend the given sequence to an infinite sequence of vectors satisfying the Lenard-Magri relation ( 205 ) for any \(n \in \mathbb{Z}_{+}\) .
Proof
It suffices to construct ξ N+1 such that Eq. (205) holds for n = N. In fact, the orthogonal complement to Span{ξ 0, …, ξ N+1} is contained in the orthogonal complement to Span{ξ 0, …, ξ N }, hence the orthogonality condition would hold for the extended sequence. By Lemma 9, H(∂)ξ N ⊥ ξ n for every n = 0, …, N. Hence, by the orthogonality condition, H(∂)ξ N ⊂ Im K(∂). Therefore, H(∂)ξ N = K(∂)ξ N+1 for some element \(\xi _{N+1} \in \mathcal{ V }^{\ell}\). We can now iterate this procedure to construct an infinite sequence of vectors. □
Now, let us address the question why the ξ n ’s, satisfying Eq. (205), are variational derivatives. Note that so far we only have used the fact that H and K are skewadjoint, but none of their other properties as Poisson structures. However, we will need these properties in order to prove that the ξ n ’s are variational derivatives. Moreover, we will need the notion of compatibility of Poisson structures:
Definition 30 (Magri Compatibility)
Given two Poisson structures H and K, they (and the corresponding λ-brackets) are compatible if any their linear combination αH + βK is again a Poisson structure.
Examples 13 and 14 provide multiparameter families of compatible Poisson structures.
The importance of compatibility of Poisson structures is revealed by the following theorem.
Theorem 9 (see [18], Lemma 7.25)
Suppose that \(H,\,K \in \mathit{\text{Mat}}_{\ell\times \ell}\mathcal{V }[\partial ]\) are compatible Poisson structures, with K non-degenerate (i.e. KM = 0 implies M = 0 for any differential operator \(M \in \mathit{\text{Mat}}_{\ell\times \ell}\mathcal{V }[\partial ]\) ). Suppose, moreover, that the Lenard-Magri relation K(∂)ξ n+1 = H(∂)ξ n holds for n = 0, 1, and that ξ 0, ξ 1 are variational derivatives: \(\xi _{0} = \frac{\delta \int h_{0}} {\delta u}\) , \(\xi _{1} = \frac{\delta \int h_{1}} {\delta u}\) for some \(\int h_{0},\int h_{1} \in \bar{\mathcal{ V }}\) . Then ξ 2 is closed in the variational complex (discussed in the next lecture).
Theorem 9 allows us to construct an infinite series of integrals of motion in involution. In fact, if we are given a pair of compatible Poisson structures H, K with K non-degenerate and we know that the first two vectors ξ 0 and ξ 1, satisfying the Lenard-Magri relation, are exact in the variational complex (i.e. they are variational derivatives), it would follow that, whenever we can construct an extending sequence of ξ n ’s, then all of them would be closed, and hence exact in some extension \(\widetilde{\mathcal{V }}\) of the algebra of differential functions \(\mathcal{V }\) (i.e. \(\xi _{n} = \frac{\delta h_{n}} {\delta u}\) for some \(h_{n} \in \widetilde{\mathcal{ V }}\)). This is a consequence of the theory of the variational complex, discussed in the next lecture. Note, however, that \(\xi _{n} \in \mathcal{ V }^{\ell}\) for all n.
Remark 22
Let \(\xi _{-1} = 0 = \frac{\delta } {\delta u}1\). If K(∂)ξ 0 = 0, then for Theorem 9 to hold it suffices to have only ξ 1 such that K(∂)ξ 1 = H(∂)ξ 0, since the first step is trivial.
Proposition 16
Suppose we have two compatible Poisson structures H and K on \(\mathcal{V }\) , with K non-degenerate, and consider a basis ξ 0 1, …, ξ 0 s of \(\mathop{\mathrm{Ker}}\nolimits \,K\) (it is finite dimensional since K is non-degenerate). Suppose that each ξ 0 i can be extended to infinity so that Eq. ( 205 ) holds for all \(n \in \mathbb{Z}_{+}\) , and hence we have ξ n i for all \(n \in \mathbb{Z}_{+}\) . Assume moreover that all vectors ξ 0 i are exact: \(\xi _{0}^{i} = \frac{\delta \int h_{0}^{i}} {\delta u}\) for some local functional ∫h 0 i . Then all the h n i ′s are in involution. Hence, we have constructed canonically an abelian subalgebra of the Lie algebra \(\bar{\mathcal{V }}\) , corresponding to the pair of compatible Poisson structures.
This proposition holds by the following result.
Lemma 11 (Compatibility Lemma)
Let H(∂) and K(∂) be skewadjoint differential operators. Suppose we have vectors \(\xi _{0},\ldots,\xi _{N} \in \mathcal{ V }^{\ell}\) such that K(∂)ξ 0 = 0 and Eq. ( 205 ) holds for n = 0, …, N. Suppose moreover to have an infinite sequence of vectors ξ 0 ′, …, ξ M ′, … satisfying Eq. ( 205 ). Then all ξ i ’s are in involution with all ξ j ′ ’s with respect to both Poisson structures H and K.
Proof
Proceed by induction on i. The induction basis follows by the fact that for i = 0 we have K(∂)ξ 0 = 0:
and
Now let N > i > 0 and suppose 〈ξ h , ξ j ′〉 H, K = 0 for all \(h\leqslant i\). We want to prove that 〈ξ i+1, ξ j ′〉 H, K = 0. We have
and
where in both cases the last equality is given by the induction hypothesis. Hence 〈ξ i , ξ j ′〉 H, K = 0 for all i, j in question. □
In the next lecture we will demonstrate the Lenard-Magri method on the example of the KdV hierarchy, hence establishing its integrability.
6 Lecture 6 (January 15, 2015)
6.1 An Example: The KdV Hierarchy
We begin this lecture with an example.
Consider the PVA \(\mathcal{P}_{1} = \mathbb{F}[u,u^{{\prime}},u^{{\prime\prime}},\ldots ]\) with two compatible λ-brackets: one is the Gardner-Faddeev-Zakharov (GFZ) λ-bracket {u λ u} K = λ, and the other one is the Magri-Virasoro (MV) λ-bracket {u λ u} H = (∂ + 2λ)u + cλ 3 for some \(c \in \mathbb{F}\). The corresponding compatible Poisson structures are K(∂) = ∂ and H(∂) = u ′ + 2u∂ + c∂ 3 respectively (see Example 13). Note that \(\mathop{\mathrm{Ker}}\nolimits \,\partial = \mathbb{F}\).
We shall use the Lenard-Magri scheme discussed in the previous lecture to construct an infinite hierarchy of integrable Hamiltonian equations: we want to construct an infinite sequence of vectors \(\xi _{n} \in \mathcal{ P}_{1}\), such that K(∂)ξ n+1 = H(∂)ξ n , \(n \in \mathbb{Z}_{+}\), and \(\xi _{0} \in \mathop{\mathrm{Ker}}\nolimits \,K(\partial )\). We also want to compute the conserved densities h n , such that \(\xi _{n} = \frac{\delta h_{n}} {\delta u}\).
We can take ξ 0 = 1 and, consequently, h 0 = u. Taking ξ −1 = 0, h −1 = 0, we can apply Theorem 9 to establish by induction on n that all the ξ n ’s, satisfying the Lenard-Magri relation, are closed in the variational complex. Since, by Corollary 5 from the next section, every closed 1-form is exact over the algebra of differential polynomials, we conclude that there exist \(h_{n} \in \mathcal{ P}_{1}\), such that \(\xi _{n} = \frac{\delta h_{n}} {\delta u}\).
The first step of the Lenard-Magri scheme:
The second step of the Lenard-Magri scheme:
And so on.
Remark 23
All ξ n ’s are defined up to adding an element of \(\mathop{\mathrm{Ker}}\nolimits \,K(\partial )\), hence, in this case, up to adding a constant.
The corresponding KdV hierarchy of Hamiltonian equations is given by \(\dfrac{du} {dt_{n}} = K(\partial )\xi _{n+1} = \partial \xi _{n+1}\), namely:
Note that for n = 1 we get the classical KdV equation, which is the simplest dispersive equation (cf. Example 16).
The hierarchy can be extended to infinity because the orthogonality condition (ξ 0)⊥ ⊂ Im K(∂) holds (see the Extension Lemma 10): since ξ 0 = 1 and \(1^{\perp } = \partial \mathcal{P}_{1} =\mathrm{ Im}\,K(\partial )\) (equivalently, if P ∈ (ξ 0)⊥ then \(\int 1 \cdot P = 0 \Leftrightarrow P \in \partial \mathcal{P}_{1}\), therefore P ∈ Im K(∂)). It is easy to show by induction that the differential order of K(∂)ξ n is 2n + 1 (if c ≠ 0), hence they are linearly independent, and we consequently have Lie integrability. Then automatically all the ∫ h n ’s are linearly independent, and we have Liouville integrability as well. Therefore the KdV equation is integrable, as are all the other equations \(\dfrac{du} {dt_{n}} = K(\partial )\xi _{n+1} = H(\partial )\xi _{n}\).
Exercise 24
Show that the next equation of the KdV hierarchy is
and the next conserved density is
6.2 The Variational Complex
As professor S. S. Chern used to say, “In life both men and women are important; likewise, in geometry both vector fields and differential forms are important”. In our theory vector fields are evolutionary vector fields over an algebra of differential functions \(\mathcal{V }\):
and, as we have already seen in Lecture 5, they commute with \(\partial = X_{u^{{\prime}}}\). Differential forms in our theory are “variational differential forms” which are obtained by the reduction of the de Rham complex over \(\mathcal{V }\) by the image of ∂.
Let J = {1, …, N}, where N can be infinite. Given a unital commutative associative algebra A, containing the algebra of polynomials \(\mathbb{F}[x_{j}\vert \,j \in J]\) and endowed with N commuting derivations \(\dfrac{\partial } {\partial x_{j}}\), extending those on the subalgebra of polynomials, the de Rham complex \(\widetilde{\varOmega }(A)\) over A consists of finite linear combinations of the form
so that we have the decomposition
Moreover, \(\widetilde{\varOmega }(A)\) is a \(\mathbb{Z}_{+}\)-graded associative commutative superalgebra with parity given by \(p(A) =\bar{ 0}\) and \(p(dx_{i}) =\bar{ 1}\). This is a complex with the usual de Rham differential, namely an odd derivation \(d:\widetilde{\varOmega } ^{k}(A)\longrightarrow \widetilde{\varOmega }^{k+1}(A)\) of \(\widetilde{\varOmega }(A)\) such that
It is easily checked that d is a differential, namely that d 2 = 0. We will denote this complex by \((\widetilde{\varOmega }(A),d)\).
Let us define now an increasing filtration on A by subalgebras:
We call A 1 the subalgebra of quasiconstants. If \(\dfrac{\partial } {\partial x_{j}}A_{j} = A_{j}\) for all j ∈ J, we call A normal. Obviously, the algebra of polynomials in any (including infinite) number of variables is normal.
Lemma 12 (Algebraic Poincaré Lemma)
Let A be a normal commutative associative algebra as above, and let \((\widetilde{\varOmega }(A),d)\) be its de Rham complex. Then
Proof
Extend the filtration of A to \(\widetilde{\varOmega }(A)\) by letting \(\widetilde{\varOmega }_{j}(A)\) be the subalgebra, generated by A j and dx 1, …, dx j . Introduce “local” homotopy operators \(K_{m}:\widetilde{\varOmega }_{ m}^{k}(A) \rightarrow \widetilde{\varOmega }_{m}^{k-1}(A)\) by
where i 1 < … < i s < m. Here the integral ∫f dx m is a preimage in A m of f ∈ A m under the map \(\frac{\partial } {\partial x_{m}}\), which exists by normality of A.
Let \(\omega \in \widetilde{\varOmega }_{m}^{k}(A)\). Then it is straightforward to check that
Hence, if \(\omega \in \widetilde{\varOmega }_{m}^{k}(A)\) is closed, then
Equivalently \(\omega \in \widetilde{\varOmega }_{m-1}^{k}(A) + d\widetilde{\varOmega }(A)\), i.e. we may assume that \(\omega \in \widetilde{\varOmega }_{m-1}^{k}(A)\) modulo adding an exact tail. Repeating the same argument we proceed downward in the filtration, and after a finite number of steps we get 0, hence \(\omega \in d\widetilde{\varOmega }(A)\). □
Let \(\mathcal{V }\) be an algebra of differential functions. Consider the lexicographic order on pairs \((m,i) \in \mathbb{Z}_{+} \times I\), and consider the corresponding filtration by subalgebras as above:
Hence we can define normality of \(\mathcal{V }\) as above.
Example 17
The algebra of differential polynomials in ℓ variables \(\mathcal{P}_{\ell}\) is normal.
The derivation ∂ of \(\mathcal{V }\) extends to an even derivation of the superalgebra \(\widetilde{\varOmega }(\mathcal{V })\) by letting ∂(du i (n)) = du i (n+1).
Exercise 25
Show that d∂ = ∂d. (Hint: use Axiom (ii) in Definition 22.)
Due to this exercise, we may consider the reduced complex
called the variational complex over the algebra of differential functions \(\mathcal{V }\).
Exercise 26
Show that ∂ is injective on \(\widetilde{\varOmega }^{k}(\mathcal{V })\) for \(k\geqslant 1\).
Theorem 10 ([3])
Let \(\mathcal{V }\) be a normal algebra of differential functions. Then
where
is the subalgebra of quasiconstants.
Proof
We have a short exact sequence of complexes
which induces a long exact sequence in cohomology:
Since \(\mathcal{V }\) is normal, by Lemma 12 we get
Now note that \(\text{H}^{k}(\partial \widetilde{\varOmega }(\mathcal{V }),d) = 0\) for k > 0. Indeed, take \(\tilde{\omega }\in \widetilde{\varOmega }^{k}(\mathcal{V })\) with \(k\geqslant 0\). If \(d(\partial \tilde{\omega }) = 0\), then \(\partial (d\tilde{\omega }) = 0\) since d and ∂ commute. So, thanks to Exercise 26, we have \(d\tilde{\omega } = 0\). By Lemma 12, since \(\tilde{\omega }\) is closed, it is exact: \(\tilde{\omega }= d\tilde{\eta }\) for some \(\tilde{\eta }\in \widetilde{\varOmega } (\mathcal{V })\), hence \(\partial \tilde{\omega } = \partial (d\tilde{\eta }) = d(\partial \tilde{\eta })\), and
Therefore the long cohomology exact sequence (229) becomes
so \(H^{k}(\varOmega (\mathcal{V }),d) = 0\) for k > 0. When k = 0 we obviously get 
. □
Let us study the variational complex more closely. We can write down explicitly the first terms of the complex \(\varOmega (\mathcal{V })\):
-
\(\varOmega ^{0}(\mathcal{V }) = \mathcal{V }/\partial \mathcal{V }\);
-
\(\varOmega ^{1}(\mathcal{V }) =\mathcal{ V }^{\ell}\);
-
\(\varOmega ^{2}(\mathcal{V }) =\{ \text{skewadjoint }\ell \times \ell\text{ matrix differential operators over }\mathcal{V }\}\).
The corresponding maps are
where \(f \in \mathcal{ V },F \in \mathcal{ V }^{\ell}\), and \((D_{F})_{ij} =\sum \limits _{n\in \mathbb{Z}_{+}} \frac{\partial F_{j}} {\partial u_{i}^{(n)}} \partial ^{n},i,j \in I\).
The first identification is clear since \(\widetilde{\varOmega }(\mathcal{V }) =\mathcal{ V }\). Let us explain how to obtain the identification \(\varOmega ^{1}(\mathcal{V }) =\mathcal{ V }^{\ell}\). We have
where last equality is due to the fact that d and ∂ commute. Integrating by parts, we get
Thus the identification \(\varOmega ^{1}(\mathcal{V })\mathop{\longrightarrow }\limits^{ \sim }\mathcal{ V }^{\ell}\) is given by
and this is an isomorphism of vector spaces. In particular, we conclude that if we take \(df =\sum \limits _{i\in I,n\in \mathbb{Z}_{+}} \frac{\partial f} {\partial u_{i}^{(n)}} du_{i}^{(n)}\) (in this case \(f_{i,n} = \frac{\partial f} {\partial u_{i}^{(n)}}\)), then the RHS of (234) becomes exactly the vector of variational derivative of f. It also explains the action of the first differential \(d:\varOmega ^{0}(\mathcal{V })\longrightarrow \varOmega ^{1}(\mathcal{V })\). Moreover, it is clear that a 1-form \(\xi \in \mathcal{ V }^{\ell}\) is exact iff \(\xi = \frac{\delta f} {\delta u}\), and it is closed iff D ξ is self-adjoint.
Exercise 27
Show that the algebra of differential functions \(\mathcal{P}_{1}[u^{-1},\log u]\) is normal. Show that any algebra of differential functions \(\mathcal{V }\) can be included in a normal one.
Since the algebra of differential polynomials \(\mathcal{P}_{\ell}\) is normal, we obtain the following corollary of Theorem 10.
Corollary 5
Let \(\mathcal{V } =\mathcal{ P}_{\ell}\) be an algebra of differential polynomials. Then
-
(a)
\(\mathop{\mathrm{Ker}}\nolimits \frac{\delta } {\delta u} = \mathbb{F} +\mathrm{ Im}\,\partial \,.\)
-
(b)
-
(c)
is closed if and only if it is exact.
is closed if and only if it is exact.
Claim (a) is usually attributed to a paper by Gelfand-Manin-Shubin from the 1970s, though it is certainly much older. Claim (b) is called the Helmholtz criterion, and apparently, it was first proved by Volterra in the first half of the twentieth century.
If we know that \(\xi \in \mathcal{V }^{\ell}\) is a variational derivative: \(\xi = \frac{\delta h} {\delta u}\) for some \(h \in \mathcal{V }\) (which is not unique since we can add to h elements from \(\partial \,\mathcal{V }\)), there is a simple formula to find one of such h:
Exercise 28
Let
be the degree evolutionary vector field, and suppose that \(\xi \in \mathcal{V }^{\ell}\) is such that Δ(u ⋅ ξ) ≠ 0. Let h ∈ Δ −1(u ⋅ ξ). Show that
Consequently, if \(\mathop{\mathrm{Ker}}\nolimits (\varDelta +1) = 0,\) then \(\frac{\delta h} {\delta u} =\xi\).
6.3 Homogeneous Drinfeld-Sokolov Hierarchy and the Classical Affine Hamiltonian Reduction
The method of constructing solutions of the Lenard-Magri relation, described in Sect. 5.4, uses Theorem 9, which assumes that K is non-degenerate. In this section I will describe the direct method of Drinfeld and Sokolov on the example of the so called homogeneous hierarchy, which avoids the use of Theorem 9.
Consider the affine PVA \(\mathcal{V } = \mathcal{V }^{1}(\mathfrak{g},s),\) where \(\mathfrak{g}\) is a reductive Lie algebra with a non-degenerate invariant symmetric bilinear form (. | . ) and s is a semisimple element of \(\mathfrak{g},\) with compatible Poisson λ-brackets \((a,b \in \mathfrak{g}):\)
see Example 14. Note that the Poisson structure K is degenerate, as s is a central element of the corresponding λ-bracket.
The Lenard-Magri relation (210) for infinite N can be rewritten as follows:
The Drinfeld-Sokolov method of constructing solutions to this equation is as follows, see [14] and [8]. Choosing dual bases {u i } i ∈ I and {u i} i ∈ I of \(\mathfrak{g}\), let
The first step consists of finding a solution \(F(z) =\sum _{n\geqslant 0}F_{n}z^{-n} \in (\mathfrak{g} \otimes \mathcal{V })[[z^{-1}]]\) of the following equations in \(\mathbb{F}\partial \ltimes (\mathfrak{g} \otimes \mathcal{V })((z^{-1})):\)
Theorem 11
Assume that the element s is semisimple, and let \(\mathfrak{h}\) be the centralizer of s in \(\mathfrak{g}\) , so that \(\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{h}^{\perp }\) . Then
-
(a)
There exist unique \(U(z) \in z^{-1}(\mathfrak{h}^{\perp }\otimes \mathcal{V })[[z^{-1}]]\) and \(f(z) \in (\mathfrak{h} \otimes \mathcal{V })[[z^{-1}]]\) , such that
$$\displaystyle{e^{ad\,U(z)}L(z) = \partial + f(z) - z(s \otimes 1).}$$The coefficients of U(z) and f(z) can be recursively computed.
-
(b)
Let a be a central element of \(\mathfrak{h}\) . Then F a(z) = e −ad U(z)(a ⊗ 1) satisfies Eq. (237).
Define the variational derivative of \(\int f \in \mathcal{V }/\partial \mathcal{V }\) in invariant form:
The second step is given by the following.
Theorem 12
Let f(z), a and F a(z) be as in Theorem 11 . Let h a(z) = (a ⊗ 1 | f(z)). Then
-
(a)
\(F^{a}(z) = \frac{\delta \int h^{a}(z)} {\delta u} \,.\)
-
(b)
The coefficients of \(h^{a}(z) =\sum _{n\geqslant 0} \int h_{n}^{a}z^{-n}\) satisfy the Lenard-Magri relation ( 236 ).
-
(c)
All the elements \(\int h_{n}^{a} \in \mathcal{V }/\partial \mathcal{V }\) , where \(n \in \mathbb{Z}_{+}\) and a is a central element of \(\mathfrak{h}\) , are in involution with respect to both Poisson structures H and K.
For proofs of these theorems we refer to [8]. Note that the claim (c) of Theorem 12 follows from claim (b) and Lemma 11.
Example 18
Let s be a regular semisimple element of \(\mathfrak{g},\) so that \(\mathfrak{h}\) is a Cartan subalgebra, and let \(a \in \mathfrak{h}\). Then the above procedure gives the following sequence of densities of local functionals in involution, satisfying the Lenard-Magri relation:
where Δ is the set of roots of \(\mathfrak{g}\) and the root vectors e α are chosen such that (e α | e −α ) = 1.
The corresponding Hamiltonian equations are:
The next equation is more complicated, so we give it only for \(\mathfrak{g} = s\ell_{2},\,a = s,\,\alpha (s) = 1:\)
Note that the elements of \(\mathfrak{h}\) do not evolve since they are central for the Poisson structure K.
In order to construct new PVAs from existing ones, we can use the classical affine Hamiltonian reduction of a PVA \(\mathcal{V }\).
The classical affine Hamiltonian reduction of a PVA \(\mathcal{V }\), associated to a triple \((\mathcal{V }_{0},I_{0},\varphi )\), where \(\mathcal{V }_{0}\) is a PVA, \(I_{0} \subset \mathcal{V }_{0}\) is a PVA ideal and \(\varphi: \mathcal{V }_{0} \rightarrow \mathcal{V }\) is a PVA homomorphism, is
where \(\text{ad}_{\lambda }\varphi (\mathcal{V }_{0})\) means that we are taking the adjoint action of \(\mathcal{V }\varphi (I_{0})\) on \(\mathcal{V }\) with respect to the λ-bracket.
Remark 24
\(\mathcal{V }/\mathcal{V }\varphi (I_{0})\) is a differential algebra, but the λ-bracket is not well defined on this quotient. However, the λ-bracket is well defined on the subspace of invariants \((\mathcal{V }/\mathcal{V }\varphi (I_{0}))^{\text{ad}_{\lambda }\varphi (\mathcal{V }_{0})}\).
Theorem 13
The λ-bracket on \(\mathcal{W}\) given by
is well defined and it endows the differential algebra \(\mathcal{W}\) with a structure of a PVA.
Proof
Let \(\widetilde{\mathcal{W}} =\{\, f \in \mathcal{V }\,\vert \,\{\varphi (\mathcal{V }_{0})_{\lambda }f\} \subset \mathcal{V }[\lambda ]\varphi (I_{0})\}\), so that \(\mathcal{W} =\widetilde{\mathcal{ W}}/\mathcal{V }\varphi (I_{0})\). It is a subalgebra of the differential algebra \(\mathcal{V }\), and \(\mathcal{V }\varphi (I_{0})\) is its differential ideal.
Check that \(\widetilde{\mathcal{W}}\) is closed under the λ-bracket of \(\mathcal{V }\) (i.e. \(\widetilde{\mathcal{W}}\) is a PVA subalgebra): let h ∈ I 0, \(f,g \in \widetilde{\mathcal{ W}}\), then by the Jacobi identity
Finally, by the right Leibniz rule, \(\mathcal{V }\varphi (I_{0})\) is a Poisson ideal of \(\widetilde{\mathcal{W}}\): for \(f \in \widetilde{\mathcal{ W}}\) we have
□
The main example of this construction is the classical affine W-algebra.
Example 19
Consider the affine PVA \(\mathcal{V } = \mathcal{V }^{1}(\mathfrak{g},s)\) with compatible λ-brackets (235). Let f be a nilpotent element of \(\mathfrak{g}\) and let {f, h, e} be an sℓ 2-triple, containing f. Let
be the \(\frac{1} {2}ad\,h\) eigenspace decomposition (so that \(f \in \mathfrak{g}_{-1}\)). Assume that \(s \in \mathfrak{g}_{d},\) where d = max\(\{j\vert \mathfrak{g}_{j}\neq 0\}.\) Let 
let \(\varphi: \mathcal{V }_{0} \rightarrow \mathcal{V }\) be the inclusion homomorphism, and let \(I_{0} \subset \mathcal{V }_{0}\) be the differential ideal, generated by the set
It is easily checked that I 0 is a PVA ideal of \(\mathcal{V }_{0}\) with respect to both λ-brackets (235). Then the classical affine W-algebra is the corresponding classical Hamiltonian reduction for both λ-brackets:
Remark 25
The same construction does not work for vertex algebras because of the presence of quantum corrections. However, for the usual associative algebras it actually works. The quantum (finite) Hamiltonian reduction of an associative algebra A is W = W(A, A 0, I 0, φ) constructed as above, where A 0 is an associative algebra, φ: A 0 ↪ A is a homomorphism of associative algebras and I 0 ⊂ A 0 is a two-sided ideal. Thus, taking \(A = U(\mathfrak{g})\), \(A_{0} = U(\mathfrak{g}_{>0})\) and I 0 the two-sided ideal generated by the above set M, we get the quantum finite W-algebra
Theorem 14 ([8])
As a differential algebra, the W-algebra \(\mathcal{W}(\mathfrak{g},f,s)\) is isomorphic to the algebra of differential polynomials on \(\mathfrak{g}^{f}\) , the centralizer of f in \(\mathfrak{g}\) .
In particular, for f principal nilpotent we get the classical Drinfeld-Sokolov reduction.
The problem is, for which nilpotent elements f can one construct the associated with \(\mathcal{W}(\mathfrak{g},f,s)\) integrable hierarchy of Hamiltonian PDE’s? Drinfeld and Sokolov constructed such hierarchy in [14] for the principal nilpotent f. (For \(\mathfrak{g} = s\ell_{n}\) it coincides with the Gelfand-Dickey nth KdV hierarchy, n = 2 being the KdV hierarchy.) Their method is similar to that in the homogeneous case. The same method can be extended, but unfortunately not for all nilpotent elements.
Definition 31
A nilpotent element \(f \in \mathfrak{g}\) is called of semisimple type if f + s is a semisimple element of \(\mathfrak{g}\) for some \(s \in \mathfrak{g}_{d}\).
These elements are classified for all simple Lie algebras \(\mathfrak{g}\) [15]. For example, principal, subprincipal and minimal nilpotent elements are of semisimple type. In exceptional Lie algebras about one third of the nilpotent elements are of the semisimple type. In \(\mathfrak{s}\ell_{N}\) only those elements corresponding to partitions (n, …, n, 1, …, 1) of N are of semisimple type.
Theorem 15 ([8])
Let \(\mathfrak{g}\) be a simple Lie algebra. If \(f \in \mathfrak{g}\) is a nilpotent element, such that f + s is semisimple for \(s \in \mathfrak{g}_{d}\) , then there exists a bi-Hamiltonian hierarchy associated to \(\mathcal{W}(\mathfrak{g},f,s)\) , which is both Lie and Liouville is integrable.
Remark 26
In the recent paper [11] for any nilpotent element f of gℓ N and non-zero \(s \in \mathfrak{g}_{d}\) an integrable hierarchy associated to W(gℓ N , f, s) is constructed.
6.4 Non-local Poisson Structures and the Dirac Reduction
Unfortunately in many important examples the PVA structure is not enough to deal with integrable systems, as it is in the case of the KdV equation, since in practice most of the Poisson structures are non-local. Thus we need to consider non-local PVAs, for which the λ-bracket takes value in \(\mathcal{V }((\lambda ^{-1}))\). Equivalently, the associated operator \(H(\partial ) \in \text{Mat}_{\ell\times \ell}\mathcal{V }((\partial ^{-1}))\) is now a matrix pseudodifferential operator.
Still, we can work with these structures, but we have to check that the axioms for a PVA bracket still make sense when the λ-bracket is a map \(\{\cdot _{\lambda }\cdot \}:\mathcal{ V } \otimes \mathcal{ V } \rightarrow \mathcal{ V }((\lambda ^{-1}))\). Sesquilinearity and the left and right Leibniz rules are clear. For skewsymmetry we have to make sense of (λ + ∂)−1: write \((\lambda +\partial )^{-1} =\lambda ^{-1}(1 + \frac{\partial } {\lambda } )^{-1}\) and then expand in the geometric progression, so we get a Laurent series in λ. More generally, for an \(n \in \mathbb{Z}\) we let
We only have problems with the Jacobi identity, and in order for it to make sense we need the λ-bracket to satisfy an additional property, called admissibility:
The fact is that when we consider a term like {a λ {b μ c}} we have to take Laurent series in λ and then Laurent series in μ and these can not be interchanged, since what we get are completely different spaces. So, two different terms of the Jacobi identity cannot a priori be compared, and we need this admissibility property in order to do so. If H(∂) = A(∂) ∘ B(∂)−1 is a rational matrix pseudodifferential operator (that is, both A(∂), B(∂) are ℓ × ℓ matrix differential operators and B(∂) is non-degenerate), then the λ-bracket defined by the Master Formula (MF) is admissible.
Example 20
For \(\mathcal{V } =\mathcal{ P}_{1} = \mathbb{F}[u,u^{{\prime}},u^{{\prime\prime}},\ldots ]\) examples of non-local Poisson structures are:
-
H(∂) = ∂ −1 (Toda)
-
H(∂) = u ′ ∂ −1 u ′ (Sokolov).
More information about non-local PVA can be found in [7]. In particular, it is shown there that the Lenard-Magri scheme can be applied if both K(∂) and H(∂) are rational pseudodifferential operators. One of the most important examples is the following pair of compatible non-local Poisson structures on the algebra of differential polynomials in u and v, where \(\kappa \in \mathbb{F}\):
which produces the non-linear Schrödinger (NLS) equation:
An important construction, leading to non-local PVA’s, is the Dirac reduction for PVA’s, introduced in [9], which generalizes the classical Dirac reduction for Poisson algebras [13].
Theorem 16 ([9])
Let \((\mathcal{V },\{._{\lambda }.\},\cdot )\) be a (possibly non-local) PVA. Let \(\theta _{1},\ldots,\theta _{m} \in \mathcal{V }\) be some elements (constraints) such that
is a non-degenerate matrix pseudodifferential operator. For \(f,g \in \mathcal{V }\) let
Then this modified λ-bracket provides \(\mathcal{V }\) with a structure of a non-local PVA, such that all elements θ α are central. Consequently, the differential ideal of the PVA \(\mathcal{V }^{D} = (\mathcal{V },\{._{\lambda }.\}^{D},\cdot ),\) generated by the θ α ’s is a PVA ideal, so that the quotient of \(\mathcal{V }^{D}\) by this ideal is a non-local PVA.
Proof
Formula (250) defines the only λ-bracket, which satisfies sesquilinearity and skewsymmetry, and for which all the θ i are central. The proof of Jacobi identity is a long, but straightforward, calculation. □
Example 21
Consider the affine PVA \(\mathcal{V } = \mathcal{V }^{1}(s\ell_{2},s)\) with the two compatible Poisson λ-brackets {. λ . } H and {. λ . } K , given by (235). As in Example 18, choose a basis e α , e −α , s of sℓ 2, such that
and the invariant bilinear form, such that (e α | e −α ) = 1, (α | α) = −κ.
Consider the constraint θ = s (=a multiple of α). This constraint is central with respect to the λ-bracket {. λ . } K . The quotient of \(\mathcal{V }\) by the differential ideal, generated by θ, is the algebra of differential polynomials \(\mathcal{P}_{2}\) in the indeterminates u = e α , v = e −α . The induced on \(\mathcal{P}_{2}\) λ-bracket by {. λ . } K is given by the matrix K in (248), and the Dirac reduced λ-bracket {. λ . } H on \(\mathcal{P}_{2}\) is given by the matrix H in (248). The reduced by the constraint θ evolution equation (240) is the NLS equation (249).
This approach establishes integrability of the NLS equation, see [10] for details. For other approaches see [19] and [7].
Exercise 29
Dirac reduction of the affine PVA \(\mathcal{V }^{1}(\mathfrak{g},s)\) by a basis of \(\mathfrak{h}\), applied to Eq. (239), gives an integrable Hamiltonian equation on root vectors of the reductive Lie algebra \(\mathfrak{g}\):
where a and s are some fixed elements of \(\mathfrak{h}\), s being regular. Find its Poisson structures.
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Kac, V. (2017). Introduction to Vertex Algebras, Poisson Vertex Algebras, and Integrable Hamiltonian PDE. In: Callegaro, F., Carnovale, G., Caselli, F., De Concini, C., De Sole, A. (eds) Perspectives in Lie Theory. Springer INdAM Series, vol 19. Springer, Cham. https://doi.org/10.1007/978-3-319-58971-8_1
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