Basu - Topics in Loop Groups and Vertex Algebras

7/27/2019 Basu - Topics in Loop Groups and Vertex Algebras

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Topics in Loop Groups and Vertex Algebras

Somnath Basu

Expository Notes (version 2.1)



Index

0 Introduction

1 Basics of Lie Groups

1. Definitions and results

2. Semisimple Lie groups (details ?)

3. Homotopy groups

4. Cohomology of Lie groups

5. Cohomology of Lie algebras

2 Loop Groups

1. Some properties

2. Twisted loop groups3. Cohomology of loop groups

3 Central Extensions

1. Lie algebra extensions

2. Group extensions when G is simply connected

3. Group extensions when G is semisimple but not simply connected

4 Affine Lie Algebras

1. Generalized Cartan matrices

2. Central extensions

3. Heisenberg algebra

4. Virasoro algebra

5 Vertex Algebras

1. Definition and properties

2. The axiom of locality

3. Associativity and OPE

4. Some examples

5. Lattice algebras6. Affine vertex algebras (Borcherds ? Fuchs ? affine vs root ?)

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Contents

1 Introduction 1

2 Basics of Lie Groups 22.1 Definitions and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 Semisimple Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Homotopy groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Cohomology of Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Cohomology of Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Loop Groups 163.1 Some properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Twisted loop groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 Cohomology of loop groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Central Extensions 234.1 Lie algebra extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 G is simply connected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.3 G is semisimple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Affine Lie Algebras 355.1 Generalized Cartan matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2 Central extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.3 Heisenberg algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.4 Virasoro algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6 Vertex Algebras 466.1 Definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.2 The axiom of locality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.3 Associativity and OPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.4 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.5 Lattice Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.6 Affine vertex algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

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1 Introduction

This is an expository account written in preparation for the minor topic in my orals examination.No claim to originality is being made.

My goal was to understand affine algebras arising as central extensions of loop algebra and

relate it to vertex algebras. More specifically, we try to verify this process for SL2(C).These notes are divided into five sections. The second section briefly reviews basic Lie theory,

homotopy groups and cohomology of Lie groups. The next section introduces loop groups and someof its properties. In the fourth section we study central extensions of loop algebra and analyze whenit extends to an extension of loop groups. The fifth section deals with affine algebras and someexamples. In the last section we discuss the theory of vertex algebras, keeping affine algebras inmind and the loop algebra sl2(C)((t)) in particular.

Most of the material presented in this notes is often taken verbatim from the references listedbelow. I have also benefited from several discussions with my minor advisor Prof. A. Kirillov, Jr.Since the aim was for a detailed understanding on my part, some proofs are more detailed and certainarguments more elaborate than that necessary for an advanced reader. A few proofs are omittedand appropriate references are given. Otherwise, it is self-contained and, except for a few occasions,notationally consistent with the sources we follow at the respective sections.

References

[1] R. Borcherds, What is a vertex algebra?, http://arxiv.org/abs/q-alg/9709033

[2] E. Frenkel, Langlands Correspondence for Loop Groups, Cambridge Studies in Advanced Mathe-matics 103 (2007)

[3] E. Frenkel, Vertex Algebras and Algebraic Curves, Talk given at Seminaire Bourbaki in June 2000

[4] J. Fuchs, Affine Lie Algebras and Quantum Groups, Cambridge Monographs on Mathematical

Physics (1992)

[5] W. Greub et al, Connections, Curvature and Cohomology, vol II , Academic Press (1973)

[6] V. Kac, Vertex Algebras for Beginners, University Lecture Series 10, American Mathematical Society(1998)

[7] A. Kirillov Jr., An Introduction to Lie Groups and Lie Algebras, Cambridge Studies in AdvancedMathematics 113 (2008)

[8] D. Milicic, Lectures on Lie Groups, http://www.math.utah.edu/ milicic/lie.pdf

[9] A. Pressley, W. Segal, Loop Groups, Oxford Mathematical Monographs (1986)

[10] P. Woit, http://www.math.columbia.edu/∼woit/RepThy/, Section Old Lecture Notes

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2 Basics of Lie Groups

For this section a Lie group means a real smooth Lie group unless mentioned otherwise.

In this section we review some basic results in Lie theory. We briefly discuss the classificationof semisimple Lie groups and sketch a proof of the classical result that π2(G) = 0 for any connected

Lie group. We also study the structure of the cohomology ring of G. Finally we review group coho-mology and provide some interesting interpretations of first few cohomology groups of g.

References We will follow [7] chapter 3 and §4.8 for §2.1, ??? for §2.2, [8] chapter 3 for §2.3 and[5] chapter 4 for §2.4.

2.1 Definitions and results

Recall that a Lie group G is a smooth manifold equipped with a compatible smooth groupstructure. Some examples are the classical groups, i.e., various subgroups of the general lineargroups. The connected component G0 is a normal Lie subgroup and G/G0 is discrete. It follows

from the theory of covering spaces that any Lie group G has a universal cover G which is also a Liegroup. The tangent spaces at the identities of G and G are isomorphic and T 1G is called the Lie algebra of G, denoted by g. The covering map is a smooth homomorphism with kernel π1(G). Itturns out that π1(G) is a discrete central subgroup of G.

A Lie subgroup H of G is a subgroup which is also a submanifold. It is known that a subgroupH of G is a Lie subgroup if and only if it is closed. It then follows that a neighbourhood U of theidentity generates G0. Thus, if f : G1 → G2 is a morphism of Lie groups with G2 connected andf ∗ : g1 → g2 is surjective then so is f . Note that if H is a normal Lie subgroup then G/H has acanonical structure of a Lie group.

A natural way Lie groups arise is by the realization of a group acting on a manifold M , i.e., asa subgroup of Diff(M ). If the G-action on M is transitive, then it is called a G-homogeneous space .

Observe that the group G acts on itself by the left action Lg (left multiplication by g), the rightaction Rg (right multiplication by g−1) and the adjoint action Adg (conjugation by g). The adjointaction, when thought of as Ad : G → GL(g), induces a map ad = Ad∗ : g → gl(g).

As with manifolds, exponentiating a vector x at 1, i.e., x ∈ g, we get an element of G. Thismap exp : g → G is a local diffeomorphism at 0 ∈ g. For compact groups, this is surjective but is nottrue in general. The crucial connection that helps us translate results from groups to algebras is :

Proposition 2.1.1. Let G1, G2 be connected Lie groups. Then any morphism φ : G1 → G2 is uniquely determined by the linear map φ∗ : g1 → g2.

The Lie algebra comes equipped with a Lie bracket [, ] : g × g → g, a bilinear skew-symmetric mapdefined by

(2.1) [x, y] := ad x.y

and satisfies the Jacobi identity

(2.2) [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0.

This is equivalent to saying that ad x is a derivation of [, ]. For classical groups, this bracket is justthe commutator of matrices.

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We define a subspace h to be a subalgebra of g if it is closed under [, ]. A subspace h is an ideal in g if [x, y] ∈ h for any x ∈ g, y ∈ h. It follows that g/h has a canonical structure of a Lie algebra.We remark in passing that if H is a normal Lie subgroup then h is an ideal. Conversely, if H isconnected and h is an ideal then H is normal. Finally, the centre z(g) of g is the set of elements xsuch that ad x ≡ 0. This is an ideal and is also the Lie algebra of Z (G), the centre of G.

The fundamental theorems of Lie theory are stated for completeness :Theorem 2.1.2. If G1 is a connected, simply connected Lie group then Hom(G1, G2) = Hom(g1, g2).

Theorem 2.1.3. There is a bijection between connected immersed subgroups H ⊂ G and subalgebras h ⊂ g; the correspondence is given by H → h = T 1H .

Theorem 2.1.4. Any finite dimensional Lie algebra is isomorphic to a Lie algebra of a Lie group.

In particular, we conclude from the above results that the categories of finite dimensional Lie algebrasand connected, simply connected Lie groups are equivalent.

There is an interesting interplay between real and complex Lie algebras and groups. Given areal Lie algebra g, one can define its complexification gC to be g⊗RC with the obvious commutator.Then g is called a real form of gC. These notions extend to (connected) Lie groups as well.

Definition 2.1.5. Let G be a connected complex Lie group and let K ⊂ G be a real Lie subgroupsuch that k is a real form of g. Then K is called a real form of G

It can be shown that any real form k can be obtained from a real form K of G.

Example 1 SU (2) is a compact real form of the complex group SL2(C) : sl2(C) consists of traceless 2 × 2 matrices and su(2) consists of traceless, skew-Hermitian matrices. If A ∈ sl2(C) then

A =1

2(A − A∗) − i

2(iA − (iA)∗) ∈ su(2) ⊕ i su(2).

A basis of su(2) is given by the Pauli matrices

(2.3) σ1 = 0 1−1 0 , σ2 = 0 i

i 0 , σ3 = i 0

0 −i .

We have the commutation relations

(2.4) [σ1, σ2] = σ3, [σ2, σ3] = σ1, [σ3, σ1] = σ2.

Notice that if we identify SU (2) with S 3 and su(2) with R3 (identifying σi with the standard basisei), then the bracket is just the cross product.

For sl2(C) we take the basis

(2.5) e =

0 10 0

, f =

0 01 0

, h =

1 00

−1

.

The commutation relations are

(2.6) [e, f ] = h, [h, e] = 2e, [h, f ] = −2f.

Each of these Lie algebras has a G-invariant (non-degenerate) symmetric bilinear form defined byx, y = −tr(xy). Then

(2.7) h, h = −2, e, f = f, e = −1

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and all other inner products are zero for sl2(C). Also

(2.8) σi, σ j = 2δ ij.

One can define an isomorphism su(2)C ∼= sl2(C) by sending

(2.9) σ1 → e − f, σ2 → i(e + f ) σ3 → ih.

Example 2 SL2(R) is a real form of SL2(C) : Choose e ,f ,h as a basis of sl2(R) as well as of sl2(C).

We now review some facts from representation theory of compact groups. Recall that anycompact Lie group G has a unique bi-invariant Haar measure µ such that

G

dµ = 1. It is alsoinvariant under the inverse map. It is well known that any finite dimensional representation of Gis unitary and hence completely reducible. If we are considering representations ρ : g → gl(V ) of g instead, then ρ(x)ρ(y) is well defined although xy in general doesn’t make sense. Moreover, thecommutation relation in g imply other relations on these operators. For example, if g = sl2(C) thenρ(e)ρ(f ) − ρ(f )ρ(e) = ρ(h) for any choice of ρ. We define

Definition 2.1.6. Let g be a Lie algebra (over R or C). The universal enveloping algebra , U g, of g isthe associative algebra with unit and with generators x ∈ g modulo bilinearity and xy − yx = [x, y].

Notice thatU g = T g/(xy − yx − [x, y]), x , y ∈ g,

where T g is the tensor algebra of g.

Example 1 The universal enveloping algebra of su(2) is the associative algebra over R generatedby σi’s with the relations σ1σ2 − σ2σ1 = σ3, σ2σ3 − σ3σ2 = σ1, σ3σ1 − σ1σ3 = σ2.

Example 2 The universal enveloping algebra of sl2(C) is the associative algebra over C generatedby e ,f ,h with the relations ef

−f e = h,he

−eh = 2e,hf

−f h =

−2f .

U g is universal in the sense that any linear map ρ : g → A from g to an associative algebraA satisfying ρ(x)ρ(y) − ρ(y)ρ(x) = ρ([x, y]) extends uniquely to ρ : U g → A. Consequently, thecategories of representations of g and U g-modules are equivalent. U g has certain central elementswhich can be used to construct intertwining operators.

Example 1 Let C = 12 σ2

1 + 12 σ2

2 + 12 σ2

3. It is central in U su(2) and is called the Casimir operator for su(2).

Example 2 Define C = ef + f e + 12 h2 ∈ U sl2(C). It is central in U sl2(C). Hence, for any

representation ρ : sl2(C) → gl(V ), the element ρ(C ) : V → V commutes with the action of sl2(C)and by Schur’s lemma, C acts by a constant in every irreducible representation. The element C iscalled the Casimir operator for sl2(C).

It is possible to put a grading on U g and the Poincare-Birkhoff-Witt theorem asserts thatthis is naturally isomorphic to the symmetric algebra S g. As a consequence, the natural inclusiong → U g is injective and U g has no zero divisors.

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2.2 Semisimple Lie groups

Let g be a Lie algebra over a field k of characteristic zero. There is a largest solvable ideal r of g,called the radical of g. Recall that g is semisimple if r is 0. This is equivalent to saying that thereare no proper abelian ideals in g.compact

←→semisimple complex group

compact real formKilling form ←→ semisimple ←→ Dynkin diagrams ←→ Cartan matrixKilling form is negative definite if and only if G is compact and semisimpleroot systemsevery bilinear form on G is invariant if G is semisimplethere are finitely many homomorphisms iα : SU (2) → G which generate g; surjective etc.g = [g, g]

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2.3 Homotopy groups

For any connected abelian Lie group G of dimension n, its Lie algebra g is just Rn with thetrivial commutator. Since Rn is simply connected, the isomorphism of Lie algebras lift to a morphismof Lie groups ϕ : Rn → G, which is also a covering projection. Thus, any simply connected andconnected abelian Lie group G of dimension n is isomorphic to Rn. One can similarly show that any

connected abelian Lie group G is isomorphic to Tr ×Rn−r for some r. Thus, any connected abeliangroup has Rn as its universal cover.

We will be interested in connected compact Lie groups henceforth. Recall that for such a groupG,

(i) maximal tori exist and its Lie algebras correspond to maximal abelian Lie algebras,(ii) any two maximal tori are conjugate,(iii) the map exp : g → G is surjective,(iv) Any maximal torus T has an associated Weyl group W = N G(T )/T which is finite. This

is unique up to isomorphism for any choice of maximal torus.(v) The centre Z of G is the intersection of all maximal tori in G.

To describe the structure of the universal covering group G, we need

Lemma 2.3.1. Let G be a connected Lie group and C be a discrete central subgroup of G such that G/C is compact. Then there exists a compact neighbourhood D of identity such that int(D) · C = G.

Proof Choose an open neighbourhood U of 1 in G such that U is compact. Since the naturalprojection p : G → G/C is open, p(U ) is an open neighbourhood of 1 in G/C and its translatesby elements of G/C cover G/C . By the compactness of G/C , finitely many translates cover G/C .Choose preimages for these translates in G, say g1, . . . , gk, and set D as the union of giU . This is acompact set and p(int(D)) ⊇ G/C , whence int(D) · C = G.

As an immediate corollary, in the hypothesis above, G is compact if and only if C is finite.In fact, let D be as in the proof above. Since D2 is compact, it is contained in finitely many of translates of D by elements of C , i.e.,

D2 ⊂ Dc1 ∪ Dc2 ∪ · · · ∪ Dcm.

If Γ is the subgroup generated by c1, . . . , cm then D2 ⊂ D · Γ. An easy induction shows thatDn ⊂ D · Γ. Since G is connected, the union of powers of D exhaust G. Consequently, G = D · Γand any c ∈ C is of the form db where d ∈ D ∩ C, b ∈ Γ. Since D is compact and C is discrete, D ∩ C is finite and C is finitely generated. Using this for the covering projection π : G → G of a connectedcompact Lie group G with C = ker π ∼= π1(G), we get

Corollary 2.3.2. The fundamental group of a connected compact Lie group is finitely generated.

There exists an invariant inner product on g since G is compact. Using this, it is easy to showthat any ideal h in g has an orthogonal complementary ideal h⊥ such that g is the direct sum of

both. This fact is used to show that

(2.10) g = z ⊕ [g, g] ,

where z is the Lie algebra of Z , the centre of G.

Proposition 2.3.3. Let G be a connected compact Lie group. Then the following are equivalent :(i) Z is finite;(ii) the universal cover G is compact.

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Proof Assume (i). This implies that z is zero and g = [g, g]. It suffices to show that C = ker π(which is finitely generated) is finite. If not, then it contains Z as a subgroup, thereby providing amorphism ϕ : C → Z. It can be shown that this extends to a surjective morphism ϕ : G → R sinceG/C ∼= G is compact. Then the kernel of ϕ∗ : g → R is of codimension 1 and contains [g, g], whichcontradicts g = [g, g].

Conversely, assume that Z is infinite. Z has finitely many components since it is compact.Therefore the the identity component of Z is infinite and z is non-zero. Let q = dim z. Let K be the integral subgroup of G corresponding to [g, g] and K be the universal covering group of K .Then K × Rq is isomorphic to G since the corresponding algebras are. Hence G is not compact, acontradiction.

Since z is a maximal abelian ideal, using the decomposition g = z ⊕ [g, g] it follows that [g, g] issemisimple. Let K be the integral subgroup of G corresponding to [g, g]. Let H = G/Z 0, where Z 0is the connected component of Z . Then there is a covering map ϕ : K → H . Since H has a finitecentre, the universal covering group H is compact. This means K is compact and therefore, closedin G. So K is a semisimple Lie group.

Now consider the compact Lie group K × Z 0 and define ϕ : K × Z 0 → G by ϕ(k, z) = kz. ϕ

is a morphism of Lie groups and ϕ∗ is an isomorphism of Lie algebras. Therefore it is a coveringprojection. The kernel

ker ϕ = (k, z) | kz = 1 = (c, c−1) | c ∈ K ∩ Z 0is a finite central subgroup of K × Z 0. Thus, we have proved

Proposition 2.3.4. Let G be a connected compact Lie group. Then ϕ induces an isomorphism of (K × Z 0)/D with G, where D = (c, c−1) | c ∈ K ∩ Z 0.

This reduces the classification of connected compact Lie groups to connected compact semisimple Liegroups. Moreover, the universal cover K of K is a connected, simply connected compact semisimpleLie group and the cover of Z 0 is Rq, where q = dim Z 0. Thus, the universal of G is isomorphic to

K × R

q

. As a consequenceProposition 2.3.5. Let G be a connected compact semisimple Lie group. Then its universal covering group G is compact.

If G is connected compact semisimple then g is semisimple and is the direct sum of its minimalideals ki, i = 1, . . . , p , which are simple Lie algebras. Let K i be the integral subgroup of G corre-sponding to ki and K i be the universal covering group. Then K 1 ×· · ·×K p is isomorphic to G, whichis compact. Thus, K i and hence K i are both compact. Therefore the map ϕ : K 1 × · · · × K p → Ggiven by ϕ(k1, . . . , k p) = k1 · · · k p is a covering projection (use the fact that the exponential map issurjective for compact groups). This discussion establishes the following result.

Proposition 2.3.6. Any connected compact semisimple Lie group G is a quotient by a finite central

subgroup of a product K 1 × · · · × K p of connected simple Lie groups.

If we want to calculate the second homotopy group of any connected Lie group, then we canassume it is compact since a well known result asserts that any connected Lie group deformationretracts to its maximal compact subgroup. Since higher homotopy groups (πk for k > 1) doesn’tdepend on the cover we may as well assume that G is a connected, simply connected compact Liegroup. Then 2.3.4 implies

πk(G) ∼= πk(K × Z 0) = πk(K ),

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since quotienting by a finite (or even discrete) group doesn’t change πk and πk(Z 0) = 0. Again, wemay assume K is simply connected. It is already compact and semisimple. By 2.3.6,

πk(K ) ∼= πk(K 1) × · · · × πk(K p),

where K i’s are compact and simply connected. From the classification theorem, the possibilities for

connected compact simple Lie groups are :(i) SU (n) - simply connected,(ii) SO(n) (n = 4) - has a universal double cover Spin(n) (if n > 2).

Remark We have purposely left out the exceptional groups G2, F 4, E 6, E 7 and E 8 since it is beyondthe scope of this discussion. It was shown by A. Borel and later by R. Bott that π2 is zero and π3 isZ for these groups.

Using the homotopy exact sequence of the fibration SU (n − 1) → SU (n) → S 2n−1 we see thatπ2(SU (n − 1)) ∼= π2(SU (n)). Since π2(SU (2)) = 0, it follows that

(2.11) π2(SU (n)) = 0, n ≥ 2.

Exactly the same way, it can be shown that

(2.12) π3(SU (n)) = Z, n ≥ 2.

Similarly, using SO(n − 1) → SO(n) → S n−1 and the fact that SO(4) ∼= SO(3) × S 3, it is easilyshown that

(2.13) π2(SO(n)) = 0 n ≥ 2,

and we also get

(2.14) π3(SO(n)) = Z, n ≥ 3, n = 4.

Thus we have established :

Theorem 2.3.7. Let G be any connected Lie group G. Then π1(G) is finitely generated, π2(G) = 0and π3(G) is a finitely generated free abelian group.

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2.4 Cohomology of Lie groups

Let G be a connected compact Lie group of dimension n and let L : G×G → G denote the leftaction. We are working with real coefficients throughout this section unless specified otherwise. Anelement α ∈ C i(G) is called a left-invariant form if L∗gα = α for any g ∈ G. Let C (G) denote the spaceof all forms and let C L(G) denote the space of left-invariant forms, which is a graded subalgebra

and is closed under d - the coboundary map/the de Rham differential. Thus, the inclusion mapι : C L(G) → C (G) induces

(2.15) ι∗ : H ∗L(G;R) → H ∗(G;R),

a map of graded algebras. Observe that(1) C L(G) is isomorphic to Λg∗ as graded algebras.(2) C L(G) is the exterior algebra over the vector space C 1L(G).

We have a map ρ : C (G) → C L(G) defined by

(2.16) α →

GL∗gαdµ,

where µ is the unique bi-invariant Haar measure on G with total measure 1. Observe that ρ isidentity on C L(G), i.e., ρι = Id : C L(G) → C L(G). It is also a cochain map, i.e., commutes with d.This implies that ι∗ is injective. We claim that

Proposition 2.4.1. The map ι∗ : H ∗L(G;R) → H ∗(G;R) is an isomorphism.

Proof Suppose we construct a chain map h : C i(G) → C i−1(G) of degree −1 such that

ιρ − Id = dh + hd

on C ∗(G). Then ι∗ρ∗ = Id and ι∗ is therefore surjective. Since it is injective from the previousdiscussion, it is an isomorphism. We construct h as a composition of hGL∗ where hG : C (G

×G)

→C (G) is homogeneous of degree −1.Let π1 : G × G → G denote the projection of the trivial G-bundle (over G) to G. We have a

map G

: C (G × G) → C (G) called the fibre integral and is defined at g ∈ G by integrating it overthe fibre at g. It is a homogeneous map of degree −n and commutes with d. We now define a degree0 map

I Ω : C (G × G) → C (G)

by setting

(2.17) I Ω(ω)(g) :=

Gω ∧ π∗1Ω,

where Ω is the normalized left-invariant volume form on G. For any α∈

C ∗(G)

[(I ΩL∗)α] (g) =

GL∗α ∧ π∗1(Ω) =

G

(L∗gα)(g) dµ.

This proves that I ΩL∗ = ρ. Let i : G → G × G denote the map sending g to (g, 1). Then Li = Idand consequently i∗L∗ = Id. If we construct

hG : C (G × G) → C (G)

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such that I Ω − i∗ = dhG + hGd then it follows that

(2.18) ιρ − Id = I ΩL∗ − i∗L∗ = (dhG + hGd)L∗ = d(hGL∗) + (hGL∗)d,

where the last equality holds since L∗ is a cochain map.If we change the the volume form Ω to another n-form Ψ supported in a contractible local

chart U 1 of G such that G Ψ = 1, then Ω − Ψ = dη for some (n − 1)-form η. Then the maps I Ωand I Ψ are chain homotopic. The homotopy is given by

(2.19) hη(α) = (−1)i G

α ∧ π∗1η, α ∈ C i(G × G).

Choosing Ψ has the advantage that I Ψ : C (U × G) → C (G) and clearly U × G deformation retractsto G. Thus I Ψ and i∗ are chain homotopic; hence I Ω and i∗ are also chain homotopic (via some maphG).

Remark The same proof, with slight modifications, work for a G-action T on a manifold M aswell, i.e., we can prove H ∗T (M ;R) ∼= H ∗(M ;R).

We shift focus to invariant forms which are simply forms invariant under the left and rightactions Lg and Rg respectively. In particular, these are invariant under the adjoint action Adg =LgRg. These forms are invariant under d. If we define the left action I of G × G on G by

I g1,g2(g) = g1gg−12

then the algebra of differential forms that are invariant under this action is precisely the space of invariant forms, denoted C I (G).

Lemma 2.4.2. C I (G) consists of closed forms.

Proof First observe that if τ : G → G denotes the inverse map, then

dτ g = −Rg−1L−1g , g ∈ G

and τ ∗α = (−1) pα for α ∈ C pI (G). Since dα ∈ C p+1I (G),

(−1) p+1dα = τ ∗dα = dτ ∗α = (−1) pdα,

whence dα = 0.

Since C I (G) is closed, H I (G) = C I (G) and by the remark, this is isomorphic to H (G). We haveisomorphisms

(2.20) C I (G;R) ∼= H L(G;R) ∼= H (G;R).

Since G is compact, we have the Kunneth isomorphism

κ : H ∗(G × G;R)∼=−→ H ∗(G;R) ⊗ H ∗(G;R).

We have the multiplication m : G × G → G and m∗ : C (G) → C (G × G) which passes to cohomology(using κ) inducing a map

(2.21) ∆ : H ∗(G;R) → H ∗(G;R) ⊗ H ∗(G;R)

10



of degree 0. Let i1, i2 : G → G × G be the inclusion maps opposite 1. If γ ∈ H ∗(G × G) then

γ = i∗1γ ⊗ 1 + β + 1 ⊗ i∗2γ,

where β ∈ H +(G) ⊗ H +(G). Since mi1 = mi2 = Id,

(2.22) ∆(α) = α ⊗ 1 + β + 1 ⊗ α, α ∈ H +(G;R), β ∈ H +(G;R) ⊗ H +(G;R).

Definition 2.4.3. A element α ∈ H +(G;R) is called primitive if

(2.23) ∆(α) = α ⊗ 1 + 1 ⊗ α.

The primitive elements form a graded subspace, P G, of H (G). Notice that there are no evenprimitives because if α was one such then 1 ⊗ α and α ⊗ 1 would commute, both being even. Nowlet k be the least positive number such that αk = 0. Then

0 = ∆(αk) = (α ⊗ 1 + 1 ⊗ α)k =k−1

i=1

αi ⊗ αk−i.

In particular, α ⊗ αk−1 = 0, whence α = 0. Since every homogeneous element of P G is odd, it’ssquare is zero. Thus, the inclusion P G → H (G) extends to a homomorphism

(2.24) λG : ΛP G → H (G;R)

of graded algebras. It can be shown using properties of power maps and its eigenspaces that dimP G = r and λG is an isomorphism. Thus, H (G) is of dimension 2r, where r is the rank of G.

Remark It is well known that every connected Lie group deformation retracts to any of its maximalcompact subgroup. Thus, the cohomology computations of any Lie group reduces to the compactcase.

Example 1 G = SU (2) : It is isomorphic to S 3 as a Lie group. Thus, it is simply connected andH i(SU (2);Z) is Z if i = 0, 3 and zero otherwise. It is of rank 1.

Example 2 G = SL2(R) : It is of rank 1 and homeomorphic to S 1 × D2. Hence H ∗(SL2(R);Z) ∼=H ∗(S 1;Z) as D2 is contractible, i.e., H i(SL2(R);Z) is Z if i = 0, 1 and zero otherwise.

11



2.5 Cohomology of Lie algebras

Let g be a finite dimensional Lie algebra. By 2.1.4, it corresponds to a simply connectedLie group G. For each g-module M , we naturally associate a cochain complex C k(g; M ), whosecohomology is defined to be the Lie algebra cohomology of g with values in M . We define(2.25) C k(g; M ) := Hom(Λkg, M ), k = 0, 1, . . . , dim g,

the vector space of multilinear, skew maps α : g × . . . × g k

→ M . The coboundary operator

δ : C k(g; M ) → C k+1(g; M ) is defined by

(δω)(x0, . . . , xk) :=ki=0

(−1)ixi · ω(. . . , xi, . . .)(2.26)

+

0≤i<j≤k(−1)i+ jω([xi, x j] , . . . , xi, . . . , x j, . . .).

It is easily verified, using Jacobi and the properties of g-action on M , that δ δ = 0. Since thefirst few coboundary operators are frequently used, we list it :

(δα)(x) = x · α,(2.27)

(δβ )(x, y) = x · β (y) − y · β (x) − β ([x, y]),(2.28)

(δγ )(x,y,z) = x · γ (y, z) + y · γ (z, x) + z · γ (x, y)(2.29)

−γ ([x, y] , z) − γ ([y, z] , x) − γ ([z, x] , y),

where x,y,z ∈ g and α ,β ,γ are 0, 1 and 2-cochains. For small values of k, the cohomology groupshave certain interesting interpretations.

Since our main object of interest is cohomology with values in R, we set M = R with the trivialaction of g. We will also abbreviate notation and denote C k(g;R) by C k(g) and the correspondingcohomology groups H k(g;R) by H k(g). Observe that the cohomology groups so obtained are justthe the cohomology group of left-invariant forms on G and δ is exactly d. By definition, C 0(g) = R

and C 1(g) = g∗ ∼= g. The maps (2.27),(2.28),(2.29) in our case translate into

(δα)(x) = 0,(2.30)

(δβ )(x, y) = −β ([x, y]),(2.31)

(δγ )(x,y,z) = −γ ([x, y] , z) − γ ([y, z] , x) − γ ([z, x] , y).(2.32)

Then (2.30) implies that

(2.33) H 0(g) = R.

Using (2.31) we see that H 1(g) is exactly the kernel of δ : C 1(g) → C 2(g) since the map δ :C 0(g) → C 1(g) is zero. Elements α in the kernel are precisely the ones that vanish on commutators,

i.e., α([x, y]) = 0 for any x, y ∈ g. Alternatively, these can be viewed as maps from g/ [g, g] to R,whence

(2.34) H 1(g) ∼= g/ [g, g] .

In particular, the first cohomology vanishes for a semisimple Lie algebra.To interpret H 2(g) we need to understand the kernel of (2.32), i.e., 2-cochains ω such that

(2.35) ω(([x, y] , z) + ω([y, z] , x) + ω([z, x] , y) = 0.

12



The restraint above is called the cocycle condition and is equivalent to ω being closed. Any such ωdefines a central extension

(2.36) 0 → R→ g → g → 0

with the Lie bracket on g given by

(2.37) [(x, s), (y, t)] := ([x, y] , ω(x, y)).

The bracket satisfies the Jacobi identity due to (2.35) and is skew since ω is. Conversely, given acentral extension (2.36), the bracket on g is defined as in (2.37) and ω must satisfy (2.35). Thus, thecentral extensions of g by R are in bijective correspondence with the 2-cocycles.

We try to see what relations are forced on the 2-cocycles ω, ω if the corresponding centralextensions g, g are equivalent. Recall that two extensions g and g are equivalent if there existsϕ : g → g such that the following commutes :

0 / / R / /

id

g / /

ϕ

g / /

id

0

0 / / R / / g / / g / / 0,

and ϕ preserves the commutators. Since both the extensions are g ⊕ R as vector spaces,

ϕ : g ⊕ R→ g ⊕ R

and is identity on R. Observe that ϕ is an isomorphism (by the five-lemma) and ϕ(x, 0) = x + α(x)where α ∈ C 1(g). The commutator

(2.38) [ϕ(x, 0), ϕ(y, 0)] = [(x, α(x)), (y, α(y))] = ([x, y] , ω(x, y))

and it also equals

(2.39) ϕ([(x, 0), (y, 0)]) = ϕ(([x, y] , ω(x, y)) = ([x, y] , α([x, y]) + ω(x, y)).

Thus, the 2-cocycles are cohomologous via α.

Proposition 2.5.1. Equivalence classes of central extensions of g by R are in bijective correspon-dence with elements of H 2(g).

If g is semisimple, then it turns out that there are no non-trivial central extensions since H 2(g) ∼=H 2(G) = 0.

To discuss H 3(g;R), we shall restrict ourselves to algebras such that H 1(g;R) = 0 = H 2(g;R).

The Lie algebras of any connected compact semisimple Lie group G satisfies this property. It followsfrom (2.31) that the negative of the dual of δ is a map

(2.40) δ ∗ : Λ2g → g, x ∧ y → [x, y] .

Since δ : Λg∗ → Λg∗ satisfies δ 2 = 0, the map δ ∗ extends to Λg and satisfies δ ∗δ ∗ = 0. Theresulting homology groups will be called the homology groups of g and denoted by H i(g;R). By our

13



assumption that the first two cohomology groups vanish, it follows from the duality of δ and δ ∗ thatH 1(g;R) = 0 = H 2(g;R). In fact, the explicit formula of δ ∗ is

(2.41) x0 ∧ · · · ∧ x pδ∗−→i<j

(−1)i+ j+1 [xi, x j] ∧ x0 ∧ · · · xi · · · x j · · · ∧ x p.

Notice that δ ∗ may not be a derivation.Since g ∼= g∗ as g-modules, the space of (symmetric) invariant bilinear forms on g, Bil(g) =

(S 2g)g, is isomorphic to (S 2g∗)g. With this identification, define a map

ϕ : (S 2g∗)g → (Λ3g∗)g

(2.42) B → ϕ(B) : (x ∧ y ∧ z) → B([x, y] , z) = B(δ ∗(x ∧ y), z).

The 3-form ϕ(B) is anti-symmetric since B is invariant and symmetric and [, ] is skew. The invariancefollows from the Jacobi identity and the invariance of B, viz,

ϕ(B)([w, x] ∧ y ∧ z) + ϕ(B)(x ∧ [w, y] ∧ y) + ϕ(B)(x ∧ y ∧ [w, z])

= B([[w, x] , y] , z) + B([[y, w] , x] , z) + B([x, y] , [w, z])= −B([[x, y] , w] , z) + B([x, y] , [w, z]) = 0.

Let ω ∈ (Λ3g∗)g. Since ω is closed, we have

0 = ω([x0, x1] ∧ x2 ∧ x3) − ω([x0, x2] ∧ x1 ∧ x3) + ω([x0, x3] ∧ x1 ∧ x2) = 0 by invariance

+ω([x1, x2] ∧ x0 ∧ x3) − ω([x1, x3] ∧ x0 ∧ x2) + ω([x2, x3] ∧ x0 ∧ x1)

= ω([x1, x2] ∧ x0 ∧ x3) − ω([x1, x3] ∧ x0 ∧ x2) + ω([x1, x0] ∧ x3 ∧ x2) = 0 by invariance

+ω([x2, x3]

∧x0

∧x1)

−ω([x0, x1]

∧x2

∧x3)

= ω([x2, x3] ∧ x0 ∧ x1) − ω(x2 ∧ x3 ∧ [x0, x1]).

This implies

ω(u ∧ δ ∗v) = ω(δ ∗u ∧ v)(2.43)

ω(δ ∗w ∧ y) = 0(2.44)

for u, v ∈ (Λ2g)g, w ∈ (Λ3g)g.The following proposition provides the connection between Bil(g) and H 3(G;R) ∼= (Λ3g∗)g.

Proposition 2.5.2. For a semisimple Lie algebra g, ϕ : (S 2g∗)g → (Λ3g∗)g is an isomorphism.

Proof Injectivity of ϕ follows from H 1(g;R) = 0 (equivalently g = [g, g]). To prove surjectivity,let ω ∈ (Λ3g∗)g. Define B ∈ (S 2g∗)g by

B(x, y) = ω(u ∧ y), where δ ∗u = x.

This is well defined since if δ ∗v = x then δ ∗(u − v) = 0. Since H 2(g;R) = 0, there exists w ∈ (Λ3g)g

such that δ ∗w = u − v. Then ω(δ ∗w ∧ y) = 0 by (2.44). Using (2.43) and the surjectivity of δ ∗ : Λ2g → g,

B(δ ∗u, δ ∗v) = ω(u ∧ δ ∗v) = ω(v ∧ δ ∗u) = B(δ ∗v, δ ∗u),

14



the symmetry of B follows. By definition ϕ(B) = ω. Since

B([x, w] , y) = ω(x ∧ w ∧ y) = ω(w ∧ y ∧ x) = B(x, [w, y]),

B is invariant.

In view of this result and the discussion preceding it, we conclude that Bil(g) is isomorphic toH 3(G;R). If G is simple, then it is 1-dimensional since any such bilinear form is a multiple of theKilling form on g.

15



3 Loop Groups

To discuss infinite dimensional Lie groups, one needs to make sense of infinite dimensionalsmooth manifolds first. We will consider paracompact topological spaces X modeled on some topolog-ical vector space E , i.e., X is covered by an atlas of open charts U α, φα such that φα : U α → E α ⊂ E is a homeomorphism. The transition functions are assumed to be smooth. By assumption, E willbe locally convex and complete. Suppose the manifold satisfies :

(i) E has enough smooth functions, i.e., for each open set U of E there is a non-vanishing R-valued smooth function on E which vanishes outside U .

(ii) X is Lindelof, i.e., each open covering has a countable refinement.Then the theory of differentiable forms work for X since De Rham’s theorem holds for any manifoldthat admits a smooth partition of unity. One can also define vector fields and their bracket. However,vector fields on such infinite dimensional manifolds do not have trajectories in general.

We will, however, not bother much about the geometric aspects mentioned above. We beginwith a discussion of some properties of loop groups. We see a few examples of naturally occurringsubgroups of loop groups. Finally, we provide a sketchy discussion (refer [9] for details) on the co-homology of loop groups.

References We will follow most of [9], chapter 3 for §3.1, §3.2 and §4.11 for §3.3.

3.1 Some properties

For any finite dimensional compact smooth manifold X , let Map(X ; G) denote the group of smooth maps X → G, where G is a finite dimensional Lie group. This is an infinite dimensionalLie group. If X = S 1 then Map(S 1; G) is called the loop group of G and denoted by LG. It is notconnected unless G is simply connected and we denote the connected component at the identity byL0G. The Lie algebra of Map(X ; G) is clearly Map(X ; g). If X = S 1 then Map(S 1; g) is called theloop algebra and denoted by Lg. There is the exponential map

exp : Map(X ; g) → Map(X ; G),

which is well defined and a local homeomorphism near the identity. The loop group of a compactgroup behaves a lot like the compact group itself, but there a few differences.

For a connected compact Lie group G the exponential map exp : g → G is surjective and anyelement g ∈ G lies in a one-parameter subgroup of G. This doesn’t hold true for LG. However, theimage of the map exp : Lg → LG is dense in the connected component of LG.

Example 1 Consider the simply connected group G = SU 2. Then LSU (2) is connected andexp : L su(2) → LSU (2) is given by

(3.1) A = ia β

−β −ia exp−→ (cos ∆)I +

sin∆∆

A = cos ∆ + ia sin∆

∆ β sin∆∆

−β sin∆∆ cos∆ − ia sin ∆

∆

,

where ∆ =√

detA. If γ : S 1 → SU (2) is given byγ 1(t) γ 2(t)

−γ 2(t) γ 1(t)

,

16



then we proceed to find A : S 1 → su(2) such that

eA(t) =

cos∆t + ia(t) sin∆t

∆tβ (t) sin∆t

∆t

−β (t) sin∆t

∆tcos∆t − ia(t) sin∆t

∆t

,

where ∆t = √ detAt. Defining

∆t = cos−1(Re γ 1(t))

a(t) = Im γ 1(t)∆t

sin∆t

β (t) = γ 2(t)∆t

sin∆t,

we see that a(t) and β (t) are periodic functions as long as ∆t = π. Thus, if γ ∈ LSU (2) avoids −I ,then it lies the image of L su(2). For example, γ : S 1 → SU (2) defined by

tγ

−→ eit 0

0 e−it

is not in the image of L su(2).

Example 2 Consider the non-compact group SL2(R). It is non-simply connected and henceLSL2(R) is not connected. The basic map exp : sl2(R) → SL2(R) for A ∈ sl2(R) is given by

(3.2) eA =

cos

√ detA I + sin

√ detA√

detAA if detA ≥ 0

(1 + −detA2! + (−detA)2

4! + · · · )I + (1 + −detA3! + (−detA)2

5! + · · · )A if detA < 0

In either case, tr eA ≥ −2. Let

S =

B∈

SL2(R)|

tr B≥ −

2

.

Then any element B ∈ S is the exponential of some element of sl2(R). If −2 ≤ tr B ≤ 2, then B = eA

where A is a conjugate of an element of SO(2). If tr B ≥ 2 then B = eA where A is a conjugate of Diag(α, 1/α) for α > 0. Consequently, exp : L sl2(R) → LS ∩ L0SL2(R). The image, however, is aproper subset since γ : S 1 → S ⊂ SL2(R) given by

tγ −→

cos t sin t− sin t cos t

,

is not the exponential of any element of L sl2(R).

If G has a complexification GC then LG has a complexification LGC, which is a complex Lie

group. Also, if G is semisimple then G is perfect, i.e., G = [G, G] since g = [g, g]. We can’t expectLG itself to be perfect since [LG, LG] ⊆ L0G. However, L0G is perfect when G is semisimple.

Proposition 3.1.1. If G is semisimple then L0G is perfect and in fact,

G, L0G

= L0G.

Proof Let’s first assume G = SU (2). Recall that g is generated by

σ1 =

0 1

−1 0

, σ2 =

0 ii 0

, σ3 =

i 00 −i

17



and the corresponding one-parameter subgroups, all isomorphic to S 1, are given by

(3.3)

cos t sin t

− sin t cos t

∈ T 1,

cos t i sin ti sin t cos t

∈ T 2,

eit 00 e−it

∈ T 3.

Since the Ti’s are maximal tori, they are conjugates. More explicitly,

T 2 =

0 ei5π4

−e−i5π4 0

T 1

0 ei

π4

−e−iπ4 0

, T 3 =

1√ 2

i√ 2

− i√ 2

1√ 2

T 1

1√ 2

i√ 2

− i√ 2

1√ 2

.

The natural multiplication map φ : T 1 × T 2 × T 3 → G is surjective since it is locally surjective andG is connected. Thus,

Φ : LT 1 × LT 2 × LT 3 → LG

is locally surjective at the identity; hence maps onto L0G, i.e.,

Φ0 : L0T 1 × L0T 2 × L0T 3 → L0G

is surjective. It suffices to show that L0T 3 ⊂ G, L0G by the surjectivity of Φ0 and the fact that theT i’s are conjugates.

Let γ ∈ L0T 3 be given by γ (t) 0

0 γ (t)−1

,

where γ : S 1 → C∗ has winding number zero. Thus, γ 1

2 is well defined since γ lifts to the the doublecover of C∗. Then

(3.4)

γ 00 γ −1

=

0 1

−1 0

,

γ −

1

2 0

0 γ 1

2

.

When G is any other compact connected Lie group then from the discussions in §2.2 thereexists i1, . . . , in : SU (2) → G such that the product map,

ik : (SU (2))n → G, is surjective. Thus,

(LSU (2))n → LG is locally surjective at 1. Consequently, (L0SU (2))n → L0G is surjective.

We state, without proof, the following :

Proposition 3.1.2. If G is simple then the group of automorphisms of L0G is the semidirect product Diff(S 1)×LAutG.

A proof can be found in [9], pg 31. The same proof can be used to prove :

Proposition 3.1.3. If G is simple then the maximal normal subgroups of L0G are precisely the kernels of the evaluation maps L0G

→G at the points of S 1.

18



3.2 Twisted loop groups

We would work with LG for the rest of the article. However, it is useful to know a few naturallyoccurring subgroups of LG. The first is the group LanG of real-analytic loops. If G is embedded insome U n then γ ∈ LanG can be expanded in a matrix-valued Fourier series

(3.5) γ (z) = ∞k=−∞

γ kzk,

where the series converges in some annulus around S 1 ⊂ C. LanG is topologized by the direct limittopology of the Banach Lie groups Lan,rG consisting of functions holomorphic in r ≤ z ≤ r−1. Thegroup Lan,rG has the topology of uniform convergence. Thus, LanG is a Lie group with Lie algebraLang and is clearly dense in LG.

There is a slightly smaller subgroup LratG of rational loops, i.e., loops which, when regardedas matrix-valued functions, have rational functions of z with no pole at |z| = 1 as its entries. SinceLratG is dense in LanG, it follows that LratG is dense in LG too.

The smallest subgroup that is often considered is LpolG of polynomial loops, i.e., loops whose

matrix entries are finite Laurent polynomials in z and z−1

. In other words, loops of the form (3.5)where all but finitely many of the matrices γ k are zero. This group is the union of the subsets Lpol,N Gconsisting of loops for which γ k = 0 if |k| > N . Each Lpol,N G is compact and we put the direct limittopology on LpolG. It is associated with the the Lie algebra Lpolg of all finite Laurent polynomials

ξ kzk where ξ k ∈ gC and ξ −k = ξ k. This vector space is the the direct limit of its finite dimensionalsubspaces Lpol,N g. Obviously, there is no exponential map from Lpolg to LpolG as the exponentialof a finite series is usually not finite. It is not always true that LpolG is dense in LG. For example,if G = S 1 then L0

polS 1 simply consists of constant loops. However, the following holds :

Proposition 3.2.1. If G is semisimple, then LpolG is dense in LG.

Proof Let H be the closure of LpolG in LG and let V be the subset of Lg formed by the tangent

vectors ξ such that the corresponding one-parameter subgroups γ ξ ∈ H . We claim that V is a vectorspace. By the Campbell-Baker-Hausdorff formula , for a suitable neighbourhood of the origin in g,

(3.6) exp(ξ )exp(η) = exp(ξ + η +1

2[ξ, η] +

1

12([ξ, [ξ, η]] + [η, [η, ξ ]]) + · · · )

where the remaining terms are Lie polynomials of degree 4 or higher, i.e., expressions consisting of commutators of x, y, their commutators, etc. Thus

γ ξ+η(t) = limn→∞(γ ξ(t/n)γ η(t/n))n,

and V is a closed subspace. It suffices to show that V = Lg since if γ ∈ G \ H then it can beapproximated by exponential images of loops in g, i.e., γ

∈H = H , a contradiction.

First consider the case G = SU (2). Then the elements

ξ n =

0 zn

−z−n 0

, ηn =

0 izn

iz−n 0

belong to V since the one-parameter subgroups they generate,

exp(tξ n) = cos t I + sin t ξ n, exp(tηn) = cos t I + sin t ηn,

19



are in LpolG. Since V is closed and linear, it contains every element of the form

f

0 1

−1 0

+ g

0 ii 0

,

where f and g are smooth real-valued functions on S 1. Since H is normal, V is closed under the

conjugation by G, whence V = Lg.The proof for general G follows in the usual way by choosing finitely many homomorphisms

SU (2) → G such that the images of su(2) in g span g. The argument only proves that the closureof LpolG contains L0G. To complete, observe that an element γ of a connected component of LGis just a matrix with entries being smooth functions. Since real-analytic functions are dense, wemay assume that these entries are real-analytic, which can be homotoped to polynomial functions.Consequently, LpolG contains an element from each connected component of LG and LpolG is dense.

We shall quickly review maximal abelian subgroups of LG. If A is any abelian subgroup of LG, then for any θ ∈ S 1 we get an abelian subgroup A(θ) of G by evaluating A at θ. Thus, A(θ) iscontained in a maximal torus of G. The most obvious maximal abelian subgroup of LG is LT , where

T is a maximal torus of G. More generally, if λ is a smooth map from the circle, which assigns toeach θ ∈ S 1 a maximal torus T λ(θ), then the subgroup

(3.7) Aλ = γ ∈ LG | γ (θ) ∈ T λ(θ) ∀ θis maximal abelian. Since all maximal tori are conjugate, the space of maximal tori in G can beidentified with G/N , where N is the normalizer of a fixed maximal torus T . Thus, λ : S 1 → G/N issmooth map.

If λ : S 1 → G/N is nullhomotopic then by the homotopy lifting property of the fibration

N → Gπ−→ G/N, π(g) = gT g−1,

λ lifts to λ : S 1

→G such that λ(θ) = λ(θ) T λ(θ)−1 and Aλ = λ LT λ−1. If λ1, λ2 : S 1

→G/N are

homotopic then λ1λ−12 lifts to G; call it α. It is easily verified that αAλ2α−1 = Aλ1. Conversely,if α conjugates Aλ2 to Aλ1, then π(α) is contractible, whence λ1 is homotopic to λ2. Thus, theconjugacy class of Aλ depends only on the homotopy class of λ. The fundamental group for G/N is the Weyl group W = N/T , which acts freely on the simply connected space G/T . W is a finitegroup and G/N = (G/T )/W . Thus, the set of homotopy classes of maps without base points is theset of conjugacy classes of W .

Proposition 3.2.2. If λ corresponds to the conjugacy class of w ∈ W then

(3.8) Aλ∼= γ : R → T | γ (θ + 2π) = α−1

w (γ (θ)) ∀ θ,

where αw denotes the automorphism of T via conjugation by w.

We skip the proof of this. It is worthwhile to note that there are maximal abelian subgroups otherthan the subgroups Aλ. We also have :

Definition 3.2.3. If α ∈ Aut G, then the associated twisted loop group is

(3.9) LαG = γ : R→ G | γ (θ + 2π) = α(γ (θ)) ∀ θ.

The group LαG depends only on the inner-isomorphism class of α, i.e., if β (g) = cα(g)c−1 for somec ∈ G, then LβG ∼= LαG.

20



3.3 Cohomology of loop groups

It is well known (refer §2.4) that the de Rham cohomology groups H ∗(G;R) for any compactLie group is an exterior algebra on r odd-dimensional generators, where r = rank G. The generatorscorrespond to the generators of the algebra of invariant polynomial functions on g in the followingway. If P is a polynomial of degree k, regarded as a symmetric multilinear function

g × · · · × g → R,

then define a multilinear function S on 2k − 1 variables by

(3.10) S (ξ 1, . . . , ξ 2k−1) :=

σ∈Σ2k−1

(−1)σP (

ξ σ(1), ξ σ(2)

, . . . ,

ξ σ(2k−3), ξ σ(2k−2)

, ξ σ(2k−1)).

The invariance of S follows from the invariance of [, ] and P . It is skew due to the antisymmetry of [, ].

We will see later (refer §4.1) that there is map from τ : H ∗(G;R) → H ∗(ΩG;R) of degree

−1 called the transgression . It is defined by taking a class in H ∗(G;R) and pulling it back to ΩG

by the map ev : S 1 × ΩG → G and then integrating along the fibre. τ is an isomorphism if G issimply connected (refer pg 30 ???). Thus, H ∗(ΩG;R) is a polynomial algebra on even dimensionalclasses obtained by transgressing the generators of H ∗(G;R). The class so obtained from (3.10) isthe (2k − 2)-form on ΩG whose value at γ ∈ ΩG on tangent vectors ξ 1, . . . , ξ 2k−2 ∈ Ωg is given by

(3.11)1

2π

2π

0S (ξ 1(θ), . . . , ξ 2k−2(θ), γ (θ)−1γ (θ))dθ.

It is naturally defined on LG because ϕ : G × ΩG → LG defined by

ϕ(g, γ ) = g · γ,

defines a diffeomorphism, whence

H ∗(LG;R) ∼= H ∗(G;R) ⊗ H ∗(ΩG;R).

Observe that Lg ∼= g × Ωg and

H ∗(Lg;R) ∼= H ∗(g;R) ⊗ H ∗(Ωg;R).

Since H ∗(g;R) ∼= H ∗(G;R) (refer §1.4), using (3.11) we may define a map from H ∗(Ωg;R) toH ∗(ΩG;R). Consequently, there is natural map

(3.12) ϕ : H ∗(Lg;R) → H ∗(LG;R).

However, the differential form (3.11) is not left-invariant. Nevertheless we have

Proposition 3.3.1. The (2k−2)-form defined by (3.11) on LG is cohomologous to a rational multiple of the left-invariant form obtained by making skew the map

(3.13) (ξ 1, . . . , ξ 2k−2) → 1

2π

2π

0P ([ξ 1, ξ 2] , . . . , [ξ 2k−5, ξ 2k−4] , ξ 2k−3, ξ 2k−2)dθ.

21



We wouldn’t prove it here. The interested reader may look up the proof in [9], §4.11. However,observe that the skew map obtained from (3.13) is just

(ξ 1, . . . , ξ 2k−2) →

σ∈Σ2k−2

1

2π

2π

0(−1)σP (

ξ σ(1), ξ σ(2)

, . . . ,

ξ σ(2k−5), ξ σ(2k−4)

, ξ σ(2k−3), ξ σ(2k−2))dθ.

It easily seen that this is left-invariant. We shall also state without proof the following :

Proposition 3.3.2. The map ϕ is an isomorphism.

Surjectivity follows from the previous proposition. For injectivity, we refer to §14.6 of [9].

22



4 Central Extensions

For this section, G will denote a connected compact Lie group.

A fundamental property of loop group LG is the existence of interesting central extensions

(4.1) S 1

→ LG

→LG

of LG by the circle S 1. This also induces a central extension of Lie algebras

(4.2) R→ Lg → Lg.

As a topological space, LG is just a principal S 1-bundle over LG. If G is simply connected, then sois LG. In that case, any principal S 1-bundle over LG can be made into a group. Thus, the centralextensions are classified by its topological type as a bundle. In other words, except for the productextension LG × S 1, all other extensions are non-trivial, i.e., LG is not homeomorphic to LG × S 1

as spaces and there is no continuous cross-section from LG to LG. This is equivalent to saying that

LG is not isomorphic to LG × S 1 as groups.

As we will see, not every central extension of Lie algebra (4.2) gives an extension of loop

groups. We study this process for simply connected groups where, under some suitable integrabilitycondition, the transition from extension of algebras to groups and vice-versa can always be made.We study this for semisimple groups (and not simply connected) as well.

References We shall be closely following [9] parts of chapter 4 for this section.

4.1 Lie algebra extensions

At the level of Lie algebras, the extensions correspond precisely to invariant symmetric bilinearforms on g. As a vector space

Lg is Lg ⊕ R, and the bracket is given by

(4.3) [(ξ, λ), (η, µ)] = ([ξ, η] , ω(λ, µ))

for ξ, η ∈ Lg and λ, µ ∈ R. Here ω : Lg × Lg → R is the bilinear map

(4.4) ω(ξ, η) =1

2π

2π

0

ξ (θ), η(θ)

dθ

for a chosen symmetric invariant from on g. Recall that if g is semisimple then every invariantbilinear form is symmetric.

For (4.3) to define a Lie bracket, ω must be skew, which follows from integrating by parts in(4.4); it must also satisfy the Jacobi/cocycle condition

(4.5) ω([ξ, η] , ζ ) + ω([η, ζ ] , ξ ) + ω([ζ, ξ ] , η) = 0.

This can be proved using the fact that , is invariant and

[ξ, η] =

ξ , η

+

ξ, η

.

If f ∈ Diff +(S 1) then it induces a map f ∗ : Lg → Lg via precomposition. Then

(4.6) ω(f ∗ξ, f ∗η) =1

2π

2π

0

ξ (f (θ)), η(f (θ))

f (θ)dθ =

1

2π

2π

0

ξ (τ ), η(τ )

dτ = ω(ξ, η),

23



where τ = f (θ). Thus, ω is Diff +(S 1) invariant and Diff +(S 1) acts as a group of diffeomorphisms

on Lg. We will see later that it also acts on the group extension. Notice that ω is invariant underconjugation by constant loops, i.e.,

ω(gξ,gη) = ω(ξ, η)

since

,

is invariant (here gξ denotes the adjoint action).

There are essentially no other cocycles other than ω defined by (4.4). More precisely, we mayonly consider invariant cocycles since if α is a cocycle, then gα defined by gα(ξ, η) = α(g−1ξ, g−1η)defines an equivalent extension, viz,

g ⊕ Id : (Lg ⊕ R, α) → (Lg ⊕R, gα), (ξ, λ) → (gξ,λ)

is an isomorphism of Lie algebras. Similarly, the extension given by the invariant cocycle G

gαdµ

is isomorphic to the one given by α. We have :

Proposition 4.1.1. If g is semisimple then the only continuous G-invariant cocycles on Lg are given by (4.4).

Proof Any cocycle ω : Lg×Lg → R can be extended to a complex bilinear map ω : LgC×LgC → C.Since an element ξ ∈ LgC can be expanded in a Fourier series

ξ kzk with ξ k ∈ gC, by continuity

ω is determined by values on ξ kzk. Write ω p,q(ξ, η) = ω(ξz p, ηzq) for ξ, η ∈ gC; this is a G-invariantbilinear map gC × gC → C which is necessarily symmetric since gC is semisimple. Then

ω p,q(ξ, η) = −ω(ηzq, ξz p) = −ωq,p(η, ξ ) = −ωq,p(ξ, η).

The cocycle identity translates to

(4.7) ω p+q,r + ωq+r,p + ωr+ p,q = 0, ∀ p, q, r.

It can be shown (we solve for (4.7) with relabelled variables in the later parts of §5.2) that ω p,q = 0if p + q = 0 and ω p,− p = p ω1,−1. If we write ξ =

ξ pz p and η =

ηqzq, then

(4.8) ω(ξ, η) = p

p ω1,−1(ξ p, η− p).

On the other hand

i

2π

2π

0ω1,−1(ξ (θ), η(θ))dθ =

p,q−1

2π

2π

0qω1,−1(ξ p, ηq)ei( p+q)θdθ

= p

p

2π

2π

0ω1,−1(ξ p, η− p)dθ,

which equals (4.8). Thus, ω is completely determined by ω1,−1 and is of the form (4.4).

Notation Let Bil(g) denote the space of invariant bilinear forms on g and K its dual. We also

denote the central extension Lg of Lg corresponding to any B ∈ Bil(g) by gB.

24



The above result determines the universal central extension of Lg, viz,

(4.9) 0 → K → guniv → Lg → 0

and it is universal because any extension of Lg by R (corresponding to B ∈ Bil(g)) can be obtainedas the push forward

(4.10) 0 / / K / /

B

guniv/ /

Lg / /

id

0

0 / / R / / gB / / Lg / / 0

Now assume that g is a semisimple Lie algebra corresponding to a connected compact Lie groupG. Then it follows from §2.5 (2.5.2) that K ∼= H 3(G;R). If G is simple then H 3(G;R) = R, whence

only one central extension exist up to scaling. Let Lg be the extension of Lg by R associatedto a symmetric invariant bilinear form , of g. We can define a symmetric invariant bilinear formon Lg, denoted again by , , by

(4.11) ξ, η =1

2π

2π

0ξ (θ), η(θ) dθ.

Since the extension is central, the adjoint action of Lg on itself arises from an action of Lg, given by

(4.12) η · (ξ, λ) = ([η, ξ ] , ω(η, ξ )).

Here ω is the 2-cocycle associated with the chosen bilinear form.

Proposition 4.1.2. The adjoint action of Lg on Lg comes from an action of LG given by

(4.13) γ

·(ξ, λ) = (γ

·ξ, λ

− γ −1γ , ξ ).

Proof First, we verify that this is a group action.

γ 1 · (γ 2 · (ξ, λ)) = γ 1 · (γ 2 · ξ, λ − γ −12 γ 2, ξ

)

= (γ 1 · (γ 2 · ξ ), λ − γ −12 γ 2, ξ

− γ −11 γ 1, γ 2 · ξ

)

= ((γ 1 · γ 2) · ξ, λ − γ −12 γ −1

1 γ 1γ 2, ξ − γ −1

2 · (γ −11 γ 1), ξ

)

= ((γ 1 · γ 2) · ξ, λ − (γ 1γ 2)−1γ 1γ 2, ξ − (γ 1γ 2)−1γ 1γ 2, ξ

)

= ((γ 1 · γ 2) · ξ, λ − (γ 1 · γ 2)−1(γ 1 · γ 2), ξ

).

If γ : [−1, 1] → LG such that γ (0) = η ∈ Lg then

η · (ξ, λ) = ddtt=0

(γ · ξ ), − ddtt=0γ (t)−1γ (t), ξ

= ([η, ξ ] ,

γ (0)−2γ (0), ξ − γ (0)−1η, ξ

)

= ([η, ξ ] , ω(η, ξ ))

agrees with (4.12).

25



4.2 Group extensions when G is simply connected

The Lie algebra extensions that were described in §4.1 do not always correspond to Lie group,i.e., it does not arise from a group extension of loop groups. For that to be true, a certain integrabilitycondition must hold.

Theorem 4.2.1.(i) If G is simply connected then the Lie algebra extension

R → Lg → Lg

defined by a cocycle ω corresponds to a group extension

S 1 → LG → LG

if and only if the differential form ω/2π represents an integral cohomology class on LG.(ii) In that case the group extension is completely determined by ω, and there is a unique action

of Diff +(S 1) on LG which covers its action on LG.

(iii) The cocycle ω defined by (4.4) satisfies the the integrability condition if and only if hα, hαis an even integer for each coroot hα of G.

Recall that the Lie algebra cocycle ω is a skew form on Lg and therefore defines a left-invariant2-form on LG. The cocycle condition (4.5) translates to the fact that this differential form is closed.

The part of the extension of LG over G is isomorphic to G × S 1. This follows from the fact thatisomorphism classes of principal S 1-bundles over G are in bijective correspondence with H 2(G;Z).Since G is simply connected and π2(G) = 0, so is H 2(G;Z). Thus, any S 1-bundle over G is trivial.

Therefore we can think of G as a subgroup of LG.

Proof If G is simply connected then ΩG is connected and simply connected. Since LG G × ΩGas spaces, this forces LG to be connected and

π1(LG) = π1(G) ⊕ π1(ΩG) = π1(G) ⊕ π2(G) = 0

using 2.3.7. Notice that LG leaves ω invariant under the left action.(i) Suppose that ω/2π represents an integral cohomology class. We need to define LG and a

group extension. To each piecewise smooth loop : S 1 → LG, we define

(4.14) C () := exp(i

σ

ω),

where σ is surface bounding . If σ1, σ2 are two surfaces bounding then Σ = σ1 ∪ (−σ2) is 2-cycleand

σ1 ω − σ2 ω = Σ ω ∈ 2πZ

.

Thus, C () is well defined. This assignment → C () has three properties :(1) (Independence of parametrization ) C () = C (φ), if φ : S 1 → S 1 is piecewise smooth map

of degree 1.(2) (Additivity ) If = 1 ∗ 2 then C () = C (1)C(2).(3) (LG-invariance ) C (γ · ) = C (); this follows from left invariance of ω.

Notice that if = p ∗ p−1 for some path p, then C () = 1.

26



Fix 1 ∈ LG. We define the space LG to consist of equivalence classes of triples (γ ,p ,eiθ), whereγ ∈ LG and p is a path in LG that joins 1 to γ . Two triples (γ ,p ,eiθ) and (γ , p, eiθ

) are equivalent

if γ = γ and eiθ = C ( p ∗ p−1)eiθ

. The composition in LG is given by

(4.15) (γ 1, p1, eiθ1) · (γ 2, p2, eiθ2) = (γ 1γ 2, p1 ∗ γ 1 · p2, ei(θ1+θ2)).

Using (3) and the observation following it, we can show that this is indeed well defined. Note thatany assignment C satisfying the three properties defines an extension of LG by S 1. For C as definedin (4.14), we get

(4.16) S 1ι→ LG

π→ LG,

where ι : S 1 → LG is given by eiθ → (1, 1, eiθ) and π :

(γ ,p ,eiθ) → γ . It is easy to check that

π1−(γ ) S 1.We also have

(4.17) (1, 1, eiθ1) · (γ ,p ,eiθ2) = (γ, 1 ∗ p, ei(θ1+θ2)) (γ, p ∗ γ, ei(θ1+θ2)) = (γ ,p ,eiθ2) · (1, 1, eiθ1).

To see why the middle equivalence holds, check that by using (1) and (2)

C (1 ∗ p ∗ γ −1 ∗ p−1) = C (γ −1 ∗ p−1 ∗ p) = C (γ −1)C ( p ∗ p−1) = C (γ −1).

Since γ −1 : S 1 → G ⊂ LG is a trivial loop in G choose a surface σ ⊂ G that bounds γ −1. AsH 2(G;R) = 0, the form ω on G is exact. By Stokes’s theorem,

C (γ −1) = exp(i

σ

ω) = exp(i

γ −1

0) = 1,

whence the equivalence. Thus, the group extension defined above is a central extension. We need tocheck that the induced extension of Lie algebras is given by ω. We skip the verifications about the

global topology of LG as a Lie group.Let Lg denote the Lie algebra of LG. Its elements are equivalence classes of derivatives of

(γ (t), p(t), eiθ(t)) at t = 0. Here γ (t) : (−ε, ε) → LG such that γ (0) = 1 and p is a smooth family of paths joining 1 and γ (t).

Notation We denote ddt

t=0

γ (t) by γ (0) and identify it as an element of Lg. We identify ddt

t=0

p(t)also as an element of Lg and denote it by p(0).

Thus, any element of Lg is represented by (γ (0), p(0), θ(0)). Observe that given γ (t), there is acanonical choice of paths joining γ , defined by pγ (t)(s) = γ (st). Then

(γ (t), p(t), eiθ(t)) ∼ (γ (t), pγ (t), eiθ1(t)),

where eiθ1(t) is defined in terms of θ, p and pγ . The advantage of choosing this path is that thecorresponding vector is just (γ (0), γ (0), θ1(0)). This provides an identification of Lg with Lg asvector spaces.

The Lie bracket for two elements (γ (0), θ(0)) and (η(0), φ(0)) of Lg is by definition

(4.18)d

dt

t=0

d

ds

s=0

(γ (t), pγ (t), eiθ(t)) · (η(s), pη(s), eiφ(s)) · (γ (t)−1, pγ (t)−1, e−iθ(t)).

27



The above expression equals

d

dt

t=0

d

ds

s=0

(γ (t)η(s)γ (t)−1, pγ (t) ∗ (γ (t) · pη(s)) ∗ ((γ (t)η(s)) · pγ (t)−1), eiφ(s))

∼ d

dt t=0

d

ds s=0(γ (t)η(s)γ (t)−1, γ (t) pη(t)γ (t)−1, eiφ(s)C (s, t)

= ddtt=0

(γ (t)η(0)γ (t)−1, γ (t)η(0)γ (t)−1, φ(0)C (0, t) + ddss=0

C (s, t))

= (

γ (0), η(0)

,

γ (0), η(0)

,d

dt

t=0

d

ds

s=0

C (s, t)),

where

(4.19) C (s, t) = C ( pγ (t) ∗ (γ (t) · pη(s)) ∗ ((γ (t)η(s)) · pγ (t)−1) ∗ γ (t) pγ,−1(s)γ (t)−1).

The last equality follows from the fact that

C (0, t) = C ( p(t) ∗ γ (t) ∗ (γ (t) · p(t)−1)) = C ( p(t) ∗ p−1(t)) = 1,

whence the derivative of C (0, t) vanishes at t = 0. Similar considerations show that C (s, 0) = 1.

Thus, it suffices to show

(4.20)d

dt

t=0

d

ds

s=0

C (s, t) = ω(γ (0), η(0)).

For each s, t we get a loop s,t : S 1 → ΩG defined by going from 1 to γ (t) (via pγ (t)),then to η(s) (via γ (t) · pη(s)), then to γ (t)η(s)γ (t)−1 (via (γ (t)η(s)) · pγ (t)−1) and finally to 1 (viaγ (t) · pη,−1(s) · γ (t)−1). Explicitly,

(4.21) s,t(r) =

γ (4rt) if r ∈ [0, 1/4]γ (t)η((4r − 1)s) if r ∈ [1/4, 1/2]γ (t)η(s)γ ((4r − 2)t)−1 if r ∈ [1/2, 3/4]γ (t)η((4 − 4r)s)γ (t)−1 if r ∈ [3/4, 1].

The loop s,t can be thought of as the boundary of the unit square [0, 1] × [0, 1] after reparametriza-tion, by identifying [0, 1/4] with x = 0, [1/4, 1/2] with y = 1, [1/2, 3/4] with x = 1 and [3/4, 1] withy = 0. We claim that this identification can be extended to the interior as well, i.e., the unit squaremaps into a rectangle in LG whose boundary is s,t.

γ (yt)

γ (t)

γ (t)η(xs)

γ (t)η(s)

γ (t)η(s)γ ((1− y)t)−1

γ (t)η(s)γ (t)−1γ (t)η(xs)γ (t)−1

1

t > 0

s > 0

s < 0

t < 0(0, 0)

(0, 1)(1, 1)

(1, 0)

σ

y

x

28



Since s and t can be assumed to be sufficiently small, we may assume that the image of the loopsγ (t), η(s) lie in neighbourhood of 1 in G, which is diffeomorphic to a neighbourhood of zero in g viathe log (the inverse of exp) map. One may assume

γ (t) = exp(tξ ), η(s) = exp(sζ ),

for some ξ, ζ ∈ Lg. Then γ (yt) = γ (t)γ ((1−y)t)−1

. With this in mind, we define σ : [−1, 1]× [1, 1] →LG by

(4.22) σ(x, y) = γ (t)η(xs)γ ((1 − y)t)−1.

Observe that C (s, t) is just the exponential of i times the area of Im σ under ω. Thus, dds

s=0

C (s, t)

is just ω(γ (yt), γ (t)η(0)γ (t)−1) integrated along γ (yt) (refer figure below).

γ (yt)

γ (t)

γ (t)η(xs)γ (t)η(s)

γ (t)η(s)γ (t)−1γ (t)η(xs)γ (t)−1

1

γ (t)η(0)γ (t)−1 γ (yt)

γ (t)

1

γ (t)η(0)γ (t)−1γ (0)ddt|t=0

Similarly, ddt

t=0

dds

s=0

C (s, t) is just ω evaluated at (γ (0), η(0)). This proves (4.20) and hence

Lg ∼=

Lg as Lie algebra extensions of Lg by R.Conversely, assume that the algebra extension extends to a group extension corresponding

to the cocycle ω. We digress into discussion of circle bundles, connections and curvature at thispoint. Treat LG as a principal S 1-bundle over LG. Then a connection α on LG is a prescriptionwhich decomposes the tangent at any point of LG as the direct sum of the tangent space of thefibre and of the base, called the horizontal tangent vectors. It tells us how to lift a path in LG toa horizontal path in LG. The curvature ωα associated with this connection measures the holonomyaround infinitesimally small closed paths. The curvature is an element of H 2(LG;R) which dependsonly on the topological type of the bundle. Moreover, ωα/2π is an integral 2-cocycle and comes froma well defined element of H 2(LG;R) called the Chern class of the bundle. The Chern class describesthe topological type of the bundle completely and any element of H 2(LG;Z) arises from a bundle.Since LG is simply connected, the inclusion map

ι : H 2(LG;Z) → H 2(LG;R)

is injective, and the topological type is completely determined by ωα/2π. The splitting of Lg = Lg⊕Rwith left-invariance gives a connection whose curvature is precisely ω. Thus, ω/2π is an integral class,i.e., ω/2π ∈ H 2(LG;Z).

(ii) The extension LG is uniquely determined by the cocycle ω as discussed before. The action

of Diff +(S 1) on LG is given by

φ ·

γ, pγ , eiθ

=

γ φ, pγ φ, eiθ

,

29



where φ ∈ Diff +(S 1). This covers the action of Diff +(S 1) on LG given by

φ · γ = γ φ.

(iii) The transgression homomorphism

(4.23) τ : H 3(G)

→H 2(ΩG)

(with either Z or R coefficients) is defined as the composite

H 3(G)ev∗−→ H 3(S 1 × ΩG)

S1−→ H 2(ΩG).

When G is simply connected, π3(G) ∼= H 3(G;Z) and π2(ΩG) ∼= H 2(ΩG;Z) by Hurewicz isomor-phisms. We also have

(4.24) ρ : π3(G)∼=−→ π2(ΩG).

To understand this map identify S 3 as the smash product S 2∧S 1, with marked points 1 ∈ S 1, p ∈ S 2.Then any f : S 3 → G could be thought of as f : S 2 × S 1 → G such that S 2 × 1 ∪ p × S 1 ismapped to 1 ∈ G. Define

ρ(f ) : S 2

→ ΩG, x → f (x, ·).By construction, this is an isomorphism. Then

τ (σ)(ρ(f )) = ev∗(σ)(S 1 × ρ(f )) =1

8π2σ(f ),

where σ ∈ H 3(G).

Proposition 4.2.2. Let σ denote the left-invariant 3-form on G whose value at the identity is given by

σ(ξ , η , ζ ) = [ξ, η] , ζ ,

where , is associated with the cocycle ω. Then the transgression τ (σ) is cohomologous to the invariant form ω/2π on ΩG.

Proof Let (ti, V i), i = 1, 2, 3 be elements of T θ,γ (S 1 × ΩG). The V i’s are vector fields along γ andwe denote γ (θ)−1V i(θ) by ξ i ∈ Lg. Then

ev∗(σ)((t1, V 1), (t2, V 2), (t3, V 3))= σ(ev∗(t1, V 1), ev∗(t2, V 2), ev∗(t3, V 3))

= σ d

ds

s=0

γ 1(s)(θ + st1),d

ds

s=0

γ 2(s)(θ + st2),d

ds

s=0

γ 3(s)(θ + st3)

= σ(t1γ (θ) + V 1(θ), t2γ 2(θ) + V 2(θ), t3γ 3(θ) + V 3(θ))

= σ(t1γ (θ)−1γ (θ) + ξ 1(θ), t2γ (θ)−1γ (θ) + ξ 2(θ), t3γ (θ)−1γ (θ) + ξ 3(θ)),

where the third equality follows from the computations below.γ = γ (0)

γ (s)

θ + st

θ

V

30



As seen in the figure, the derivative at s = 0 of γ (s)(θ + st) is a linear combination of V (θ) and γ (θ).More precisely,

d

ds

s=0

γ (s)(θ + st) = lims→0

γ (s)(θ + st) − γ (0)(θ)

s

= lims→0

γ (s)(θ + st)

−γ (0)(θ + st)

s + t lims→0

γ (0)(θ + s)

−γ (0)(θ)

s= V (θ) + tγ (θ).

Since vector fields along γ is equivalent to an element of Lg, given two such fields V 1, V 2 alongγ , we denote the corresponding elements of Lg by ξ 1, ξ 2. Then

τ (2πσ)(V 1, V 2) =1

4π

2π

0ev∗(σ)(γ , V 1, V 2)dθ(4.25)

=1

4π

2π

0

γ (θ)−1γ (θ), [ξ 1(θ), ξ 2(θ)]

dθ.

Consider the 1-form β on ΩG defined by

(4.26) β (V ) =1

4π

2π

0

γ (θ)−1γ (θ), γ (θ)−1V (θ)

dθ,

where V is a vector field along γ ∈ ΩG. Then

dβ (V 1, V 2) = V 1(β (V 2)) − V 2(β (V 1)) − β ([V 1, V 2])

= V 1(β (V 2)) − V 2(β (V 1)) − τ (2πσ)(V 1, V 2).

For points close to γ , β (V 2) is given by

β (V 2) = 14π 2π

0

γ t(θ)−1γ t(θ), ξ 2 dθ,

where γ t(θ) = γ (θ)exp(tξ 2(θ)). Consequently,

V 1(β (V 2)) = limt→0

1

4π

2π

0

γ −1t γ t − γ −1γ , ξ 1

dθ

=1

4π

2π

0limt→0

γ −1t γ t − γ −1γ

t, ξ 1

dθ

=1

4π

2π

0

ξ 2(θ), ξ 1(θ)

dθ

= 12 ω(ξ 1, ξ 2),

which equals ω(V 1, V 2)/2 by left-invariance. Therefore

(4.27) dβ (V 1, V 2) = ω(V 1, V 2) − τ (2πσ)(V 1, V 2)

and τ (σ) is cohomologous to ω/2π.

The proof of (iv) is now complete using 4.2.2 and the following :

31



Proposition 4.2.3. The skew form σ of 4.2.2 defines an integral cohomology class on the simply connected group G if and only if hα, hα ∈ 2Z for each coroot hα of G.

We only illustrate the proof for G = SU (2). For the general proof we refer the reader to [9] pg 49.

Example The positive root α sends any diagonal element A ∈ su(2) to α(A) = a11a−122 . This induces

a coroot hα ∈ su(2), a diagonal matrix with i and −i as its entries. This defines a homomorphismηα : S 1 → T byηα(θ) = exp(θhα).

It extends canonically to a homomorphism ια : SU (2) → G and in this case is the identity map. For−α, the map ι−α is better viewed through S 3, i.e.,

ι−α : (x1, x2, x3, x4) → (x1, −x2, x3, x4).

It is of degree −1 and together with ια it generates π3(SU (2)). If σ is pulled back to SU (2) by ιαthen

ι∗α(σ)(η,ν ,ζ ) = [η, ν ] , ζ .

Write η,ν,σ,hα in terms of the Pauli matrices, i.e., hα = σ3 and η = 3i=1 ηiσi etc. Observe thathα, hα = 2. The skew 3-form σ on G, which is also the invariant volume form on G of integral 1,

evaluated on three vectors η =i ηiσi, ν =

i ν iσi, σ =

i ζ iσi is given by

σ(η,ν ,ζ ) = 2η · (ν × ζ ).

This implies the tautology that1

2hα, hα σ = σ = ι∗α(σ).

The proof for general G works in a similar manner.Let , be as defined in §1.1, pgs 3 − 4. Then (iv) is satisfied and the corresponding cocycle ω

satisfies the integrability condition. More explicitly, if ξ 1, ξ 2∈

Lg then write

ξ j(θ) =k∈Z

a j,keikθ

σ1 +k∈Z

b j,keikθ

σ2 +k∈Z

c j,keikθ

σ3, j = 1, 2.

Using (4.4) and (2.8)

ω(ξ 1, ξ 2) = j,k∈Z

ja1,ka2,j

π

2π

0ei( j+k−1)θdθ +

j,k∈Z

jb1,kb2,j

π

2π

0ei( j+k−1)θdθ

+ j,k∈Z

jc1,kc2,j

π

2π

0ei( j+k−1)θdθ

= j∈Z

2 j(a1,1− ja2,j + b1,1− jb2,j + c1,1− jc2,j).

32



4.3 Group extensions when G is semisimple but not simply connected

For a semisimple group (more generally for groups with Z (G) = 1),

Z (LG) = LZ (G) = 1.

Using (4.1) we get

(4.28) Z (LG) = S 1.

We write G = G/Z , where G is the universal cover of G and Z = π1(G) is a finite subgroup of thecentre of G.

We have seen that an integral bilinear form , on g determines a central extension LG of LG

by S 1. Since we may identify G canonically as a subgroup of LG, we can regard Z as a subgroup of LG. In fact, Z belongs to the centre since its adjoint action on Lg is trivial, viz,

γ · (ξ, λ) = (γ · ξ, λ − γ −1γ , ξ

= (ξ, λ).

We have an extension

Z × S 1 → LG → LG.

Since LG/Z = L0G, the above extension mod Z gives

(4.29) S 1 → (LG)/Z → L0G.

But (4.29) is usually not the restriction of an extension of the whole group LG. To understand thiswe observe that the form , on g induces a pairing

(4.30) c : Z × Z → S 1

in the following way. Let T be a maximal torus of G with Lie algebra t . Given z1, z2 ∈ Z chooseζ 1, ζ 2 ∈ t such that exp(2πζ i) = zi, i = 1, 2. Then define

c(z1, z2) = exp(2πi ζ 1, ζ 2).

It is well defined by the integrality of the inner product and is independent of the choices made.

Lemma 4.3.1. The extension (4.29) is not the restriction of an extension of LG unless the pairing c is trivial.

Proof Consider the automorphism Aλ of Lg defined to be conjugation by λ ∈ LG which does notbelong to the identity component. This lifts uniquely to an automorphism Aλ of Lg, defined by

(4.31) Aλ(ξ, µ) = (Aλξ, µ − λ−1λ, ξ

.

We apply this to ξ

∈t such that exp(2πξ )

∈Z and a loop λ defined by θ

→exp(θζ ). If we choose ζ

such that exp(2πζ ) ∈ Z then λ is a non-trivial loop in G.

Aλ(ξ, 0) = (ξ, −ζ, ξ ).

If (4.29) was the restriction of an extension of LG then any automorphism Aλ must induce an

automorphism of (LG)0 via the exponential map. Since exp(2πξ, 0) = 1, Aλ induces a map if ξ, ζ ∈ Z.

In the other direction, we have

33



Lemma 4.3.2. The conjugation action of LG on LG lifts uniquely to an action on LG. Then the

action of Z on the centre Z × S 1 of LG is induced by (4.30).

We omit the proof here (refer [9] pgs 55 − 56). We are now ready to describe a class of extensions of LG.

Proposition 4.3.3. For any integral inner product , on g there is a group LG whose identity

component is LG and whose group of components is Z ∼= π1(G). It is an extension of LG by Z × S 1

and the conjugation action of the group of components on Z × S 1 is induced by (4.30).

The extension LG is not uniquely determined by , but is unique up to the addition of an arbitrary extension of Z by S 1.

We don’t provide a proof but illustrate with an example instead.

Example Let G = SO(3) with G = SU (2) and

Z := π1(G) = Z (G) ∼= Z/2Z.

Note that G is simple and Z (LG) = S 1 by (4.28). Let T denote the maximal torus of G consistingof diagonal matrices. Since any symmetric bilinear form is a scalar multiple of the Killing form, itsuffices to work with (refer pgs 3 − 4, §1.1) the Pauli matrices as a basis of g. Then

(4.32) (a1, a2, a3), (b1, b2, b3) = 2a1b1 + 2a2b2 + 2a3b3.

The Lie algebra t is the line spanned by σ3. If ζ ∈ t satisfies exp(2πζ ) ∈ Z then ζ ∈ σ3Z. Conse-quently, the pairing c is trivial. Therefore (4.29) is the restriction of an extension of LG. Combining

the fact that any extension of π1(G) by S 1 is trivial we conclude that the extension LG is uniquelydetermined.

34



5 Affine Lie Algebras

Affine algebras are certain infinite dimensional Lie algebras that can be classified in terms of their Cartan matrices. These are a specific class inside the more general class of Kac-Moody alge-bras defined using the generalized Cartan matrices. A particular realization of affine algebras ariseas central extensions of loop algebras. Along with a discussion on affine algebras we also reviewcentral extensions for generic Lie algebras and present some examples.

References We will follow [4] §2.1 for §5.1, [4] §2.2 for §5.2 and the section on the Heisenbergalgebras in [10] for §5.3.

5.1 Generalized Cartan matrices

For any finite dimensional complex semisimple Lie algebra g with Cartan subalgebra h, let(, ) denote the (non-degenerate) Killing form. This is non-degenerate when restricted to h andinduces a non-degenerate form on h∗ as well. Fix a reduced root system R ⊂ h∗ with a polarization

R = R− R+ and the corresponding system of simple roots αir

i=1. If one chooses for each rootα a non-zero element eα in the root space gα, then g is generated by 3r generators ei := eαi , f i :=e−αi , hi := hαi | i = 1, . . . , r. The subspaces n± consisting of positive and negative roots respectivelyare subalgebras of g. In fact, n+ is generated by the ei’s, n− is generated by the f i’s and hi’s form abasis of h. In other words

g = n+ ⊕ h ⊕ n−.

After suitably normalizing, the generators satisfy the Jacobi identity and the Serre relations (some-times also known as Chevalley-Serre relations )

[hi, h j] = 0(5.1)

[hi, e j] = aije j(5.2)

[hi, f j] = −aijf j(5.3)[ei, f j] = δ ijhi(5.4)

(ad ei)1−aije j = 0(5.5)

(ad f i)1−aijf j = 0(5.6)

where the entries of the Cartan matrix A = (aij) are given by

(5.7) aij = hαi, α j = 2(αi, α j)/(αi, αi).

It is an integer valued r × r (rank g = r) matrix satisfying

aii = 2(5.8)

aij = 0 ⇔ a ji = 0(5.9)

aij ≤ 0for i = j(5.10)

det A > 0.(5.11)

This implies that rank A = r. Observe that A = DS where

Dij = δ ij/(αi, αi), S ij = 2(αi, α j)

35





labels (ai)ri=1 and dual Coxeter labels (ai)ri=1 to be the left and right eigenvectors of A respectively

with eigenvalue 0, i.e.,

(5.14)ri=1

aiaij = 0 =ri=1

aija j

together with the normalization condition

minai| i = 1, . . . , r = 1 = minai| i = 1, . . . , r.

This makes sense since one can choose the eigenvector in either case to be positive, i.e., ai > 0 (resp.ai > 0).

Let A be indecomposable. If it is of finite type, then g(A) is simple, In contrast, if A is of affine type, g(A) possess a non-trivial center. For any constant ζ the element

(5.15) 1 := ζ ri=1

aihi

is a central element since [ 1 , hi] = 0 by definition and

[1 , e j] = ζ ri=1

aiaije j = 0 = −ζ ri=1

aiaijf j = [ 1 , f j ]

by (5.14). Since the kernel of A is 1-dimensional, 1 spans Z (g(A)). The existence of such a centralelement allows for a central extension of Lie algebras

0 → C1

→ g(A) → g(A)/C → 0.

37



5.2 Central extensions

For any given complex Lie algebra g of dimension l, one can construct an r-dimensional central extension g by simply adding r central generators 1 i to a set of (ordered) generators J a of g andimposing

[J a, 1 i] = 0 for i = 1, . . . , r, a = 1, . . . , l .

The Lie bracket on g then generalize to

(5.16)

J a, J bg

= γ cabJ c + γ iab 1 i,

where γ cab are the structure constants of g. Since

J a, J b

= − J b, J a

, the new structure constantssatisfy

γ iab = −γ iba.

It also has to satisfy the Jacobi identity. Thus, the dimension of the space of solutions is the numberof independent central extensions. Of course, choosing γ iab = 0 only results in a trivial centralextension. More generally, if there are constants uic such that

(5.17) γ iab = c

uicγ cab,

then changing J a to J a + uia 1 i, the new structure constants become zero, whence g is trivial as anextension.

Let’s restrict to 1-dimensional central extensions. ThenJ a, J b

g

=

J a, J b

+ γ ab 1 ,

where γ ab can be thought of as a bilinear map

Γ : g × g → C.

It is anti-symmetric and satisfies Jacobi :

(5.18) Γ(x, [y, z]) + Γ(y, [z, x]) + Γ(z, [x, y]) = 0.

This is the defining equation of a 2-cocycle of g with values in C. Γ is a coboundary if and only if there exists a linear map γ : g → C such that γ ([x, y]) = Γ(x, y) which is exactly (5.17) when r = 1.Consequently, isomorphism classes of 1-dimensional central extensions of g correspond to H 2(g;C).For semisimple Lie algebras the cohomology ring H ∗(g;C) is freely generated by s generators of odddegree with s = rank g. Thus, there exists no non-trivial central extension of semisimple Lie algebras.

The close connection between the Dynkin diagrams of simple Lie algebras and affine algebrassuggest that the latter may be obtainable as some generalization of simple algebras which admitsnon-trivial central extension. Indeed, for any simple Lie algebra g, consider the space Lg of analyticmaps from S 1 to g. Let g = J a| a = 1, . . . , l be a basis and consider S 1 as the unit circle in C withcoordinate t. Then a topological basis of the vector space Lg is

B = J an := J a ⊗ tn| a = 1, . . . , l; n ∈ Zand this space inherits a natural bracket operation from g, viz,

(5.19)

J am, J bn

:=

J a, J b

⊗ tm+n = γ cabJ cm+n.

38



This makes Lg into a Lie algebra and is called the loop algebra over g. The Jacobi identity takes theform

(5.20) γ cdaγ ecb + γ cabγ ecd + γ cbdγ eca = 0.

Observe that under the bracket, the index n is additive and therefore provides a Z-gradation. Hence,the subset generated by J a

0’s form a subalgebra, called the zero-mode subalgebra g

0, which is isomor-

phic to g.If we look for irreducible central extensions of Lg (as in the case of simple algebras),

0 → Cr → Lg → Lg → 0

where Cr is generated by the central elements 1 i’s, then the most general ansatz for the new Liebracket is

(5.21)

J am, J bn

new

= γ cabJ cm+n + (γ iab)mn 1 i.

This must be anti-symmetric, i.e., (γ iab)mn = −(γ iba)nm and (γ iaa)mm = 0. It also satisfies Jacobi,

which splits into (5.20) and

(5.22) γ cab(γ icd)m+n p + γ cbd(γ ica)n+ pm + γ cda(γ icb)m+ p n = 0.

But (5.2) induces a r-dimensional central extension g of g by pull back :

0 / / Crι / / Lg

π1 / / Lg / / 0

0 / / Crι / /

ι

O O

g0π1 / /

ι

O O

g0 / /

ι

O O

0

By the simplicity of g0, one can choose a basis

J a0

such that the corresponding structure constants

(γ iab)00 = 0. This provides a Lie algebra splitting g0 = g0 ⊕Cr. Now g0 acts by the adjoint action onLg. Since any representation of a semisimple algebra is completely reducible, g0 maps Lg to itself.Summarizing, with J am as a basis for Lg

(5.23)

J am, J bn

new

= γ cabJ cm+n + (γ iab)mn 1 i

where (γ iab)0n = 0 for any n. By abuse of notation, we denote these new structure constants (γ iab)mnnd the basis elements J an without the tilde henceforth.

We may define bilinear forms on g0 for each 3-tuple (i,m,n), i = 1, . . . , l; m, n ∈ Z via

J a0 , J b0 := (γ iab)mn

and extended by linearity. If we set p = 0 in (5.22) then

γ cbd(γ ica)nm + γ cda(γ icb)mn = 0

which is equivalent (after swapping m and n) to

γ cbd(γ ica)mn = γ cda(γ ibc)mn.

39



Consequently

(5.24)

J b0 , J d0

, J a0

= γ cbd(γ ica)mn = γ cda(γ ibc)mn =

J b0,

J d0 , J a0

,

thereby proving that the bilinear form is invariant. By the semisimplicity of g0, it is also symmetric.Therefore it must be a scalar multiple of the Killing form K = (κ

ab), i.e.,

(5.25) (γ iab)mn = κabγ (i)mn.

Since K is symmetric, this forces γ (i)mn to be anti-symmetric. Since (γ iab)0n = 0 and K is non-degenerate,

(5.26) γ (i)n0 = γ (i)0n = 0.

Observe that for each i, γ (i)mn is non-zero for some choice of m, n for otherwise Lg would bereducible (remove 1 i). Now we write (5.22) with p = 0 and use (5.25) :

γ cdaκcdγ (i)mn = γ cbdκcaγ (i)mn.

Since γ (i)mn is non-zero for some m, n we conclude, after cyclically permuting the indices, that

(5.27) γ cabκcd = γ cbdκca = γ cdaκcb.

Since g0 is semisimple, given a there exists b such that constants γ cab, c = 1, . . . , l cannot all be zerofor if it did, then J a0 would generate an ideal in g0. By the non-degeneracy of K , γ cabκcd = 0 for somed. With these choices, apply (5.27) to (5.22) :

(5.28) γ (i)m+np + γ (i)n+ pm + γ (i)m+ p n = 0

which is equivalent to the Jacobi identity. Putting p = −n − m we get

γ (i)m+n−m−n = γ (i)m−m + γ (i)n−n,

whence

(5.29) γ (i)m−m = mγ (i)1−1.

Putting p = q − m − n we get

γ (i)q−m−nm+n = γ (i)q−mm + γ (i)q−nn,

whence

(5.30) γ (i)q−mm = mγ (i)q

−1 1.

This implies that γ (i)q−1 1 = 0 if q = 0. Consequently γ (i)mn = 0 if m + n = 0 and

(5.31) γ (i)mn = mγ (i)1−1δ m+n,0.

We may choose γ (i)1−1 = k. Then the dimension r of the central elements has to be one. If not, then

1 =i 1 i and Lg generate a subalgebra of Lg, which is also an 1-dimensional central extension of

40



Lg. In conclusion, there is a unique non-trivial irreducible central extension Lg of Lg characterizedby the bracket

(5.32)

J am, J bn

= γ cabJ cm+n + mδ m,−nkκab 1

among the generators J am

of Lg and the central element 1 . Thus, the complete Lie bracket reads

(5.33) [A ⊗ f (t) + ζ 1 , B ⊗ g(t) + η 1 ] = [A, B] ⊗ f (t)g(t) + kκ(A, B)Res(g df )1 ,

where Res denotes the residue of a Laurent polynomial and A, B ∈ Lg.For reasons of applicability, particularly interesting cases are the ones where g is a compact

real Lie algebra. Since g is also simple, κ is negative definite. Choose a basis such that κab = −δ ab.Then (5.32) looks like

(5.34)

J am, J bn

= γ cabJ cm+n + kmδ abδ m+n 0 1 .

The (untwisted) affine algebra g1 is obtained from Lg by adding one more generator τ , i.e.,

(5.35) g1 = C[t, t−1] ⊗C g ⊕ C1 ⊕ Cτ

with the new Lie bracket being anti-symmetric, satisfying Jacobi identity (5.34) and

[τ, J am] = mJ am, [τ, 1 ] = 0.

The centre of g1 is still 1-dimensional. The zero mode subalgebra of g1 is still the zero modesubalgebra of Lg, which is isomorphic to g. Since τ doesn’t appear on the right hand side of anybracket (and it is the only generator with this property), Lg is the derived algebra of g1, i.e.,

Lg = [g1, g1] .

Notice that on the subalgebra Lg, τ acts by t ddt

.

Example Let g = sl2(C) with compact real form SU (2). The Lie algebra su(2) is generatedby the Pauli matrices σ1, σ2, σ3. The Killing form is κ(x, y) = trace(ad xad y). Under the mapad : su(2) → gl( su(2)),

(5.36) ad σ1 =

0 0 00 0 −10 1 0

, ad σ2 =

0 0 −10 0 01 0 0

, ad σ3 =

0 1 0−1 0 00 0 0

.

It is easily verified that σi := σi/√

2 then forms an orthonormal basis for −κ under the adjoint map.Since [σi, σ j ] = σk if (i j k) = (1 2 3), the structure constants are

(5.37) γ i jk =1√

2, if (i j k) = (1 2 3).

In the above discussion, g was required to be simple because the affine Cartan matrix is requiredto be indecomposable. If this assumption is removed, then we may work with semisimple or evenreductive algebras. One then has to introduce an independent central extension for each simplesubalgebra and for each 1-dimensional abelian subalgebra (isomorphic to C). However, the central

41



extensions of C can be identified with each other since all irreducible modules of C are 1-dimensional.If we write g = s⊕ z where s is semisimple and z is abelian, then any irreducible central extension of Lg is of the form L s ⊕ L z. The Lie bracket of L s is given by (5.32) and for z is given by

(5.38)

J am, J bn

= dabmδ m+n,0 1 ,

where a, b runs through 1 to dim z and D = (dab) is a constant anti-symmetric matrix. Lie algebraswith brackets like above are called Heisenberg algebras because it is analogous to the commutationrelations in quantum mechanics.

42



5.3 Heisenberg algebra

Let us define the notion in a general setting. Let R be a commutative ring and M an R-module.

Definition 5.3.1. M is called a Heisenberg algebra if it is freely generated by elements P ii∈I , Qii∈I and 1 and is equipped with an anti-symmetric R-module morphism [, ] : M × M → M satisfying

(5.39) [1

, P i] = 0 = [1

, Qi] ,

(5.40) [P i, P j] = 0 = [Qi, Q j] ,

(5.41) [P i, Q j] = δ ij 1 .

1 is called the central element of M .

Although the rank of M is 2|I | + 1 as an R-module, it’s rank as an algebra is |I | +1. The submodulegenerated by P ii∈I , Qii∈I is free and has trivial bracket. Thus, M can be interpreted as anextension of R2|I | by R. If we work over R = R or R = C, then this is a 1-dimensional extension(R or C) of the trivial Lie algebras (R2|I | or C2|I |). In quantum mechanics we take R = C andI =

1, 2, 3

, while in vertex algebras we take R = C and I = Z.

Let hn denote the Heisenberg algebra over R, generated freely by P 1, . . . , P n, Q1, . . . , Qn and1 , i.e.,

0 → R → hn → R2n → 0.

Observe that [hn, hn] = R1 and [[hn, hn] , hn] = 0. Thus, hn is nilpotent. If we exponentiate thealgebra elements then the group it generates is called the Heisenberg group and denoted H n. Usingthe Campbell-Baker-Hausdorff formula and the fact that commutators of order at least 2 are zero,we have :

(5.42) eX · eY = eX +Y +1

2[X,Y ].

Let ∆un denote the strictly upper triangular matrices of order n. Define ϕ : hn → ∆u

n+2 by

(5.43)ni=1

piP i +ni=1

q iQi + c1 →

p1 · · · pn c

q 1...

q n

,

where the omitted entries are taken to be zero. For any two elements ( p1, . . . , pn, q 1, . . . , q n, c) and( p1, . . . , pn, q 1, . . . , q n, c) of hn, we have

(5.44)

n

i=1

piP i +n

i=1

q iQi + c1 ,n

i=1

piP i +n

i=1

q iQi + c 1

=

n

i=1

piq i − piq i

1 .

Then it easily follows that ϕ preserves brackets. Consequently, it defines an isomorphism of theHeisenberg algebra into a subalgebra of ∆u

n+2.

Example When n = 1, we identify h1 with the strictly upper triangular 3 × 3 matrices via

ϕ : pP + qQ + c1 → 0 p c

0 0 q 0 0 0

.

43



This is an isomorphism of vector spaces and ϕ preserves the brackets. The corresponding Heisenberggroup H 1 is isomorphic to the group generated by the one-parameter families e pϕ(P ), eqϕ(Q) andecϕ(

1

). A simple calculation shows that

e pϕ(P ) = 1 r 00 1 0

0 0 1 , eqϕ(Q) = 1 0 00 1 q

0 0 1 , ecϕ(1

) = 1 0 c0 1 0

0 0 1 .

We have the identity (5.42), which applied to X = pP,Y = qQ gives

(5.45) e pϕ(P )+qϕ(Q)+(c+ pq2

)ϕ(1

) = e pϕ(P ) · eqϕ(Q) · ecϕ(1

) =

1 p c0 1 q 0 0 1

.

This proves that

(5.46) H 1 ∼= upper triangular 3 × 3 matrices with 1’s on the diagonal.

Hence H 1 is homeomorphic to R3. It can be interpreted as a twisted product of R2 with R in thefollowing way. Let ( p, q, c) denote elements of R3 as p ,q,c runs over R. Then define a product

(5.47) ( p, q, c) · ( p, q , c) = ( p + p, q + q , c + c + pq ).

This makes R3 into a group, denoted R2×R.

Remark There exists representations of Heisenberg algebras that do not exponentiate to repre-sentations of Heisenberg groups.

Even more generally, for a finite dimensional complex vector space V one may choose a skew-symmetric form (., .) on V and define a Lie algebra with

(V ⊗ C [t, t−1]) ⊕ C1

,

as the underlying vector space and

[xtn, ytm] := n(x, y)δ n,−m 1

as the Lie bracket. Notice that (xtn, ytm) = n(x, y)δ n,−m is a skew-symmetric form on V ⊗C[t, t−1].We had encountered such an example at the end of §5.2.

44



5.4 Virasoro algebra

The complex Witt algebra is the Lie algebra of meromorphic vector fields defined on the Rie-mann sphere that are holomorphic except at two fixed points. It is also the complexification of the Lie algebra of polynomial vector fields on a circle, i.e., Der C((t)) = C((t)) ∂

∂t, and the Lie alge-

bra of derivations of the ring C[z, z−1]. A basis for the Witt algebra is given by the vector fields

Ln = −tn+1 ∂ ∂t , n ∈ Z. The Lie bracket of two vector fields is given by

(5.48) [Lm, Ln] = (m − n)Lm+n.

The Witt algebra is just the Lie algebra Vect (S 1) corresponding to the simple Lie groupDiff +(S 1). Thus, up to equivalence, there is a unique central extension called the Virasoro algebra .Let 1 denote the central element. Then

[Lm, Ln] = (m − n)Lm+n + cm,n 1 .

The skew-symmetry of the bracket implies cm,n = −cn,m and Jacobi identity implies

(5.49) (n − m)cn+m,k + (m − k)cm+k,n + (k − n)ck+n,m = 0.

The last condition basically means that cm,n’s define a cocycle.If we set Ln = Ln + cn,0 1 then we get

Lm, Ln

= (m − n)Lm+n + cm,n 1 ,

where cm,n = cm,n − cm+n,0. Since this is changing the original cocycle by a coboundary, we get anequivalent extension. By abuse of notation, we call these coefficients cm,n which satisfies cm,n = −cn,mand cn,0 = 0. The cocycle condition reads (m + n)cm,n = 0, which means that only possible non-zeroterms are cn,−n; call it an. Setting k = −m − n in the cocycle condition we get

(n − m)an+m − (2m + n)an + (2n + m)am = 0.

In particular, the an’s have to satisfy

(n − 1)an+1 − (n + 2)an + (2n + 1)a1 = 0.

Thus, the an’s are determined by a1 and a2. Thus, we have a 2-dimensional family of solutions andit can be guessed that an = λn is a coboundary and an = µn3 is a non-trivial cocycle for constantsλ, µ.

Set cn,−n = 112 (n3 − n) and then the Virasoro commutator is

(5.50) [Lm, Ln] = (m

−n)Lm+n +

m3 − m

12

δ m,−n 1 .

The factor of 12 appears for historic reasons. It also follows from (5.50) that L−1, L0, L1 form asubalgebra of the Virasoro algebra isomorphic to sl2(C). This can be seen by setting e = L−1, f =L1, h = 2L0.

45



6 Vertex Algebras

The notion of a vertex algebra was introduced by R. Borcherds in 1986. This is a rigorousmathematical definition of the chiral part of a 2-dimensional quantum field theory. Examples includeassociative commutative graded algebras with a derivation of degree 1. However, this notion wasknown to physicists much earlier. We introduce the definition of a vertex algebra and analyze whatthe axiom of locality means. Among the properties of vertex algebras that can be derived from theaxioms, the associative property leads to the notion of operator product expansions (OPE). Thisis useful for doing calculations that are otherwise quite taxing. We also review ways to constructvertex algebras starting from a Lie algebra.

References We will follow [2] chapter 2 for all the subsections except §6.4, §6.5. We use [6] as ageneral reference and in particular for §6.5. For all the examples in §6.4 we use [3].

6.1 Definitions and properties

Let R be aC

-algebra. An R-valued formal power series (or formal distribution) in variablesz1, z2, . . . , zn is an arbitrary infinite series

A(z1, z2, . . . , zn) =i1∈Z

· · ·in∈Z

ai1,...,inzi11 · · · zinn ,

where each ai1,...,in ∈ R. These series form a vector space denoted by R[[z±11 , . . . , z±1

n ]]. In general, aproduct of two power series in the same variables does not make sense, but the product of a formalpower series and a polynomial is always well defined.

Let V be a Z-graded vector space V = ⊕n∈ZV n with finite dimensional homogeneous compo-nents. An endomorphism T of V is called homogeneous of degree n if T (V m) ⊆ V m+n. The vectorspace of linear endomorphisms of V , denoted by End V , consists of elements which are finite linearcombinations of homogeneous endomorphisms. It is a Z-graded algebra.

A field of conformal dimension k ∈ Z+ is an End V -valued formal power series in z,

φ(z) = j∈Z

φ jz− j−k

where each φ j is a homogeneous linear endomorphism of V of degree − j. Also, for any v ∈ V ,φ j(v) = 0 for j 0. Two fields φ(z) and ψ(w) are called mutually local if there exists N ∈ Z+, suchthat

(6.1) (z − w)N [φ(z), ψ(w)] = 0

as an element of End V [[z±1, w±1]]. We can now define what a vertex algebra is :

Definition 6.1.1. A vertex algebra is a collection of data :(i) (space of states) A Z-graded vector space V = ⊕n∈ZV n with dim(V n) < ∞,(ii) (vacuum vector) A distinguished element |0 ∈ V 0,(iii) (translation operator) An operator T : V → V of degree one,(iv) (vertex operation) A linear map Y (., z) : V → End V [[z, z−1]] taking each A ∈ V m to a field

of conformal dimension m :A ∈ V → Y (A, z) =

n∈Z

A(n)z−n−1.

46



These data are subject to the following axioms :

(i) (vacuum axiom) Y (|0, z) = IdV . Furthermore, for any A ∈ V we have

Y (A, z)|0 ∈ A + zV [[z]].

(ii) (translation axiom) For any A∈

V, [T, Y (A, z)] = ∂ zY (A, z) and T |0

= 0.(iii) (locality axiom) All fields Y (A, z) are mutually local with each other.

First observe that vacuum axiom implies that the assignment A → Y (A, z) is injective. Also one candefine a vertex algebra V 1 ⊗ V 2 for two vertex algebras V 1 and V 2. It also follows from the axiomsthat for any A ∈ V , the A(n)’s annihilate |0 for n ≥ 0 since

∂ zY (A, z)|0 = A(−2)|0 +n<−2

(−n − 1)A(n)|0z−n−2 +n≥0

(−n − 1)A(n)|0z−n−2

must equal[T, Y (A, z)] |0 ∈ T (A) + zV [[z]].

This also shows that T (A) = A(−2)|0. Thus T is not an independent datum. Also, if A ∈ V m thenA(n) has degree −n + m − 1.

We also have Goddard’s uniqueness theorem that shows how rigid the condition of localityreally is.

Theorem 6.1.2. (Goddard’s Uniqueness Theorem) Let V be a vertex algebra, A(z) be a field on V . Suppose there exists A ∈ V such that

A(z)|0 = Y (A, z)|0

and A(z) is local with respect to Y (B, w) for any B ∈ V . Then A(z) = Y (A, z).

Proof Let B∈

V . Using the locality hypothesis and the locality axiom in succession, for large N (specified below) we have

(z − w)N A(z)Y (B, w)|0 = (z − w)N Y (B, w)A(z)|0= (z − w)N Y (B, w)Y (A, z)|0= (z − w)N Y (A, z)Y (B, w)|0.

The first and the last terms above are both well defined at w = 0 for N large enough (by locality).Set w = 0 and using Y (B, w)|0

w=0= B, we have

zN A(z)B = zN Y (A, z)B.

Since B was arbitrary, this completes the proof.

Essentially, the theorem states that one can reconstruct a vertex operator from the knowledge of how it acts on the vacuum vector.

Lemma 6.1.3.

(6.2) Y (A, z)|0 = ezT A.

47



Proof Let Y (A, z) =n∈Z A(n)z−n−1. Then it follows from the vacuum axiom that A(n)|0 = 0

for n ≥ 0, A(−1)|0 = A and T (A) = A(−2)|0. We need to show

A(−n−1)|0 =1

n!T nA, n > 0.

By the translation axiom,∂ zY (A, z)|0 = [T, Y (A, z)] |0 = T Y (A, z)|0.

Equating the coefficients of zn−1, n > 0 we get

nA(−n−1)|0 = T A(−n)|0.

Using A(−1)|0 = A, the result follows from induction.

Observe that in view of the lemma, we can replace the hypothesis of Goddard’s theorem by A(z)|0 =ezT A. This is equivalent to the formulae

(6.3) ∂ zA

(z)|0

= T A

(z)|0,

A(z)

|0z=0

= A.

This means that A(z) satisfies a differential equation in z determined by T and an initial condition.Thus, Goddard’s theorem states that the vertex operator Y (A, z) is uniquely characterized as thelocal field satisfying these properties. In conformal field theory this is known as the state-field correspondence .

Corollary 6.1.4. For all A ∈ V , Y (TA ,z) = ∂ zY (A, z).

Proof It follows from the axiom of locality that given Y (B, w) there exists N > 0 such that

(z − w)N +iY (A, z)Y (B, w) = (z − w)N +iY (B, w)Y (A, z)

for any i≥

0. Setting i = 1 and differentiating the above equation we get

(z − w)N ∂ zY (A, z)Y (B, w) = (z − w)N Y (B, w)∂ zY (A, z).

This means that ∂ zY (A, z) is local with respect to all the vertex operators. Now

∂ 2zY (A, z)|0 = ∂ z [T, Y (A, z)] |0 = ∂ z(T Y (A, z))|0 = T ∂ zY (A, z)|0and

∂ zY (A, z)|0z=0

= A(−2)|0 = T A.

Hence it follows from the theorem (with A(z) = ∂ zY (A, z)) that Y (TA ,z) = ∂ zY (A, z).

We state for completeness the two forms of the reconstruction theorem.

Theorem 6.1.5. (Weak Reconstruction)Let V be a vector space, |0 ∈ V and T an endomorphism of V . Let

aα(z) =n∈Z

aα(n)z−n−1,

where α runs over an ordered set I , be a collection of fields on V such that (i) [T, aα(z)] = ∂ zaα(z);

48



(ii) T |0 = 0, aα(z)|0 ∈ aα + zV [[z]];(iii) aα(z) and aβ(z) are mutually local;(iv) The lexicographically ordered monomials aα1(n1) . . . aαm(nm)|0 with ni < 0 form a basis of V .

Then the formula

(6.4) Y (a

α1

(n1) . . . a

αm

(nm)|0, z) =

1

(−n1 − 1)! · · ·1

(−nm − 1)! : ∂ −n1

−1

a

α1(z)

· · · ∂ −nm

−1

a

αm(z)

:,

where ni < 0, defines a vertex algebra structure on V such that |0 is the vacuum vector, T is the translation operator and Y (aα, z) = aα(z) for all α ∈ I .

The terms appearing in the sum above are what is called normally ordered product of fields (refer§6.2). It is some sort of normalization technique that helps to make sense of product of two or morefields as a formal series.

The theorem is true even if we assume that the vectors in (iv) span V . And it is still possibleto derive that the resulting vertex algebra structure is unique.

Theorem 6.1.6. (Strong Reconstruction)

Let V a vector space, equipped with the structures of the previous theorem satisfying conditions (i) − (iii) and the condition (iv) The vectors aα1(n1) . . . aαm(nm)|0 with ni < 0 span V .

Then these structures together with (6.4) define a vertex algebra structure on V .Moreover, this is the unique vertex algebra structure on V satisfying (i) − (iii) and (iv) such that

Y (aα, z) = aα(z) for all α ∈ I .

Proof Choose a basis among the vectors in (iv) and define Y (A, z) by formula (6.4). Recall thatthe locality of aα(z) and aβ(z) implies locality of derivatives of both. Hence, by Dong’s lemmathe locality axiom follows. The translation covariance axiom holds because T is a derivation of thenormally ordered product. The vacuum axiom holds for Y (aαn|0, z) :

Y (aα(n)|0, z)|0z=0

= m∈Z

(−m − 1) . . . (−m + n + 1)(−n − 1)!

zn−maα(m)|0z=0

= aα(n)|0.

In general, let v = aα(n)B|0 ∈ V such that the vacuum axiom holds for B|0. Then

Y (aα(n)B|0, z)z=0

=

1

(−n − 1)!: ∂ −n−1aα(z)Y (B|0, z) : |0

z=0

=1

(−n − 1)!(∂ −n−1aα(z))+Y (B|0, z)|0

z=0

+1

(−

n−

1)!Y (B

|0

, z)(∂ −n−1aα(z))−

|0

z=0

= (aα(n) + z(. . .))(B|0 + z(. . .))z=0

= aα(n)B|0.

Hence, the vacuum axiom holds and we get a vertex algebra structure on V .If we choose another basis among the monomials in (iv) we get another structure of a vertex

algebra on V . But all the fields of this new structure are mutually local with those of the old structureand satisfy (6.3). By the uniqueness theorem, these two vertex algebra structures coincide. Thus

49



(6.4) is well defined and with Y (aα, z) = aα(z), this provides V with a unique structure of a vertexalgebra.

We recall a few useful definitions related to vertex algebras. Let V, V be vertex algebras withinfinitesimal operators T, T respectively.

Definition 6.1.7. A homomorphism of V to V is a linear degree preserving map φ : V → V suchthatϕ(A(n)B) = ϕ(A)(n)ϕ(B), A , B ∈ V, n ∈ Z.

A derivation D of degree m ∈ Z is an endomorphism of the space V such that DV l ⊂ V l+mand

D(A(n)B) = (DA)(n)B + (−1)lmA(n)(DB), A ∈ V l, B ∈ V.

Note that if D is a derivation of degree 0 and eD is a convergent series, then

eD(A(n)B) =i≥0

1

i!Di(A(n)B)

= i≥0

1i! ik=0

ik(DkA)(n)(Di−kB)

=l,k∈Z

1

l!k!(DkA)(n)(DlB)

= eD(A)(n)eD(B)

is an automomorphism.

Definition 6.1.8. A subalgebra of V is a subspace U of V containing |0 and invariant under theoperators A(n) for any A ∈ U .

It is clear that U is a vertex algebra as well, its fields being Y (A, z) = n∈Z A(n)U z−n−1. All thepertinent axioms can be verified easily.

Definition 6.1.9. An ideal of V is a T -invariant subspace J not containing |0 and invariant underthe operators A(n) for any A ∈ V .

If A ∈ J, B ∈ V then 6.3.2 implies Y (A, z)B ∈ J . This means J is invariant under A(n) for A ∈ V .The quotient space V /J also inherits a vertex algebra structure.

Definition 6.1.10. The tensor product of V and V is defined to be the vertex algebra with V ⊗ V

as the space of states, |0⊗|0 as the vacuum vector and T ⊗1+ 1⊗T as the infinitesimal translationoperator. The fields are

Y (A ⊗ A, z) = Y (A, z) ⊗ Y (A, z) =m,n∈Z

A(m) ⊗ A(n)z−m−n−2.

It is easy to verify that V ⊗ V is a vertex algebra.

50



6.2 The axiom of locality

As we will see in the first example in §6.4, commutative associative unital algebra with aderivation of degree 1 gives rise to a commutative vertex algebra and vice-versa. Since the propertyof commutativity in a vertex algebra is too restrictive, we replace it with the locality axiom. Thus,the notion of vertex algebras generalize the notion of commutative algebra.

Let v ∈ V and φ : V → C be a linear functional. Given A, B ∈ V we can form two formalpower series

(6.5) φ, Y (A, z)Y (B, w)v φ, Y (B, w)Y (A, z)v .

These two formal power series, which are both elements of C[[z±1, w±1]], actually belongs to thesubspaces C((z))((w)) and C((w))((z)) respectively. The first consists of bounded powers of w butpowers of z are not uniformly bounded, whereas the second consists of bounded below powers of zbut not uniformly bounded below powers of w.

z

w

V ((z ))((w))

V ((w))((z ))

V ((z ))((z − w)) = V ((w))((z − w))

The intersection of the two spaces consists of elements which have bounded below powers of z andw, i.e.,

(6.6) C((z))((w)) ∩ C((w))((z)) = C[[z, w]][z−1, w−1].

Each of the two subspaces above are fields and their intersection is a subalgebra. So the fractionfield of C[[z, w]][z−1, w−1] lies in both the subspaces, which we denote by C((z, w)). It consists of elements f (z, w)/g(z, w) where f, g are in C[[z, w]].

However, the embeddings of C((z, w)) into C((z))((w)) and C((w))((z)) are different. Forexample, take 1

z−w ∈ C((z, w)). Assume that |w| < |z|. Then we can expand

1

z − w= z−1

n≥0

w

z

n.

51



Since this has bounded below powers of w it lies in C((z))((w)). Now assume |z| < |w|. Then

1

z − w= −z−1

n<0

w

z

nhas bounded powers of z and hence lies in C((w))((z)). The element

iz,w1

z − w− iw,z

1

z − w=n∈Z

wnz−n−1

is denoted by δ (z − w) and is called the formal delta function . Here iz,w (resp. iw,z) denotes theexpansion operator assuming |z| > |w| (resp. |w| > |z|).

The locality axiom states that

(z − w)N φ, Y (A, z)Y (B, w)v = (z − w)N φ, Y (B, w)Y (A, z)v

as elements of C[[z±, w±]]. Due to (6.6), both are elements of C[[z, w]][z−1, w−1]. Therefore, theformal power series (6.5) are representations of the same element of C[[z, w]][z−1, w−1, (z

−w)1] in

C((z))((w)) and C((w))((z)) respectively. We are imposing, in addition, that there is a universalbound on the power of (z − w) that can occur in the denominator as φ and v vary. This is anequivalent formulation of the locality axiom which is often useful.

It is known ([6] pg 20) that the null space of the operator of multiplication by (z −w)N , N ≥ 1,in C[[z±1, w±1]] is

N −1 j=0

∂ jwδ (z − w)C[[w, w−1]].

Moreover, an element a(z, w) of the kernel can be represented in the form

(6.7) a(z, w) =

N −1

j=0

c j

(w)∂ j

wδ (z − w),

where c j(w) = Resza(z, w)(z − w) j. One generally refers to (6.7) as the OPE expansion of a(z, w)and cn(w) as the OPE coefficients .

In order to state equivalent definitions of locality we need some notation. Given a formaldistribution a(z) =

n∈Z a(n)z−n−1, let

(6.8) a(z)− =n≥0

a(n)z−n−1, a(z)+ =n<0

a(n)z−n−1.

It is easily verified that this definition satisfies

(∂a(z))± = ∂ (a(z)±).

Given two formal distributions a(z) and b(z), define the following formal distribution in z and w,called the normally ordered product :

: a(z)b(w) := a(z)+b(w) + b(w)a(z)−.

52



One can inductively define the normally ordered product of three or more fields by inductivelyapplying it from the right. The normal ordering changes if we are working in a superspace V , i.e.,V = V 0 ⊕ V 1 and p : V → Z/2Z assigns to a homogeneous vector its parity. Then

: a(z)b(w) := a(z)+b(w) + (−1) p(a) p(b)b(w)a(z)−,

is extended by bilinearity. If fields are mutually local then their normally ordered product is a fieldas well. This is the content of Dong’s lemma . It is worthwhile to note the formulae

a(z)b(w) = [a(z)−, b(w)]+ : a(z)b(w) :(6.9)

(−1) p(a) p(b)b(w)a(z) = − [a(z)+, b(w)]+ : a(z)b(w) : .(6.10)

Theorem 6.2.1. Each of the following properties is equivalent to (z − w)N [a(z), b(w)] = 0 :

(i) [a(z), b(w)] =N −1 j=0 ∂ jwδ (z − w)c j(w), where c j(w) ∈ g[[w, w−1]].

(ii) [a(z)−, b(w)] =

N −1 j=0

iz,w

1(z−w)j+1

c j(w),

− [a(z)+, b(w)] = N −1 j=0 iw,z 1(z−w)j+1c j(w),

where c j(w) ∈ g[[w, w−1]].

(iii) a(z)b(w) =N −1 j=0

iz,w

1(z−w)j+1

c j(w)+ : a(z)b(w) :,

(−1) p(a) p(b)b(w)a(z) =N −1 j=0

iw,z

1(z−w)j+1

c j(w)+ : a(z)b(w) :,

with c j(w) ∈ g[[w, w−1]].

(iv)

a(m), b(n)

=N −1 j=0

m j

c j(m+n− j), m, n ∈ Z.

Proof We have discussed the equivalence of (i) and the axiom of locality. Notice that (iii) followsfrom (ii) and (6.9). Using (i) and

∂ jwδ (z − w) =k∈Z

k

j

z−k−1wk− j

we get m,n∈Z

a(m), b(n)

z−m−1w−n−1 =

k,l∈Z

k

j

z−k−1wk− j−l−1c j(l).

Comparing coefficients we get k = m, l = m + n − j and (iv) follows.Since a(z) = a(z)+ + a(z)−, we see that the LHS of the first equation in (ii) consists only of

negative powers of z. Since

N −1 j=0

∂ jwδ (z − w)c j(w) =N −1 j=0

k∈Z

k

j

z−k−1wk− jc j(w),

the positive part of the RHS of the above equality reads

N −1 j=0

k≥0

k

j

z−k−1wk− j

c j(w) =

N −1 j=0

iz,w

1

(z − w) j+1

c j(w).

53



One can similarly derive the second equation in (ii). This proves the equivalence of (i) and (ii).

Physicists, by abuse of notation, write the relation in (iv) as

(6.11) a(z)b(w) =N −1

j=0

c j(w)

(z

−w) j+1

+ : a(z)b(w) :,

or often just write the singular part :

(6.12) a(z)b(w) ∼N −1 j=0

c j(w)

(z − w) j+1.

54



6.3 Associativity and OPE

We have already seen that the locality axiom implies that the two formal power series

Y (A, z)Y (B, w)C and Y (B, w)Y (A, z)C

are expansions of the same element from V [[z, w]][z−1, w−1, (z−

w)−1] in two different domains. Oneof these expansions, V ((z))((w)) corresponds to w being ‘small’; the other V ((w))((z)) correspondsto z being ‘small’. These expansions can be thought as being done in the domains ‘w is very closeto 0’ and ‘w very close to ∞’. The third obvious choice is ‘w is very close to z’.

Algebraically, the space corresponding to the third domain stated above is V ((w))((z − w)).Equivalently, we can take the space V ((z))((z − w)) since it is identical to V ((w))((z − w)). In termsof vertex algebra, the expression that we expect to live in this space is

Y (Y (A, z − w)B, w)C.

In a commutative vertex algebra we have

Y (A, z)Y (B, w)C = Y (Y (A, z

−w)B, w)C

which expresses the associativity property of the algebra. Analogously, we should expect thatY (Y (A, z − w)B, w)C is the expansion of the same element of V [[z, w]][z−1, w−1, (z − w)−1] (asfor the other two expressions) but in the domain V ((w))((z − w)). We show that this is indeed thecase.

Lemma 6.3.1. In any vertex algebra we have

ewT Y (A, z)e−wT = Y (A, z + w)

where negative powers of (z + w) are expanded as power series assuming that w is ‘small’.

Proof Observe that [T, Y (A, z)] = ∂ zY (A, z) and [T, ∂ z] = 0 implies that

(ad T )nY (A, z) = ∂ nz Y (A, z).

We also have the following in any Lie algebra :

ewT Ge−wT =n≥0

wn

n!(ad T )nG.

Therefore, we get

ewT Y (A, z)e−wT =n≥0

wn

n!(ad T )nY (A, z)

= n≥0

wn

n! ∂ nz Y (A, z)

= ew∂ zY (A, z).

To complete the proof, we need

ew∂ zf (z) =n≥0

wn

n!∂ nz f (z) = f (z + w),

55



which holds for any f (z) ∈ V [[z±]] and |w| < |z|.

This lemma basically tells us that exponentiating the operator T give us a translation operatorz → z + w.

Proposition 6.3.2. (Skew Symmetry) In any vertex algebra we have the identity

Y (A, z)B = ezT Y (B, −z)A

in V [[z]].

Proof By locality, there is a large integer N such that

(z − w)N Y (A, z)Y (B, w)|0 = (z − w)N Y (B, w)Y (A, z)|0.

This is actually an equality in V [[z, w]] since there are no negative powers of w (resp. z) on the left(resp. on the right). Now we compute

(z − w)N Y (A, z)Y (B, w)|0 = (z − w)N Y (B, w)Y (A, z)|0⇒ (z − w)N Y (A, z)ewT B = (z − w)N Y (B, w)ezT A

⇒(z

−w)N Y (A, z)ewT B = (z

−w)N ezT Y (B, w

−z)A

⇒ zN Y (A, z)B = zN ezT Y (B, −z)A⇒ Y (A, z)B = ezT Y (B, −z)A.

In the fourth line we have set w = 0 since there are no negative powers of w in the above equalities.The fifth line follows since zN is not a zero divisor in V [[z]].

We are now ready to prove the associativity property.

Theorem 6.3.3. In vertex algebra the expressions

Y (A, z)Y (B, w)C, Y (B, w)Y (A, z)C, and Y (Y (A, z − w)B, w)C

are the expansions in

V ((z))((w)), V ((w))((z)), and V ((w))((z − w)),

respectively, of one and the same element of V [[z, w]][z−1, w−1, (z − w)−1].

Proof The equality of the first two expressions in the suitable domains was already discussed. Wecompute

Y (A, z)Y (B, w)C = Y (A, z)ewT Y (C, −w)B = ewT Y (A, z − w)Y (C, −w)B.

In this final expression we expand negative powers of (z − w) assuming w is ‘small’. This defines amap V ((z − w))((w)) → V ((z))((w)) which is easily seen to be a isomorphism and intertwines theembeddings of V [[z, w]][z−1, w−1, (z − w)−1] into the two spaces.

On the other hand, using 6.3.2,

Y (Y (A, z − w)B, w)C = Y n

A(n)B(z − w)−n−1, wC

=n

Y (A(n)B, w)C (z − w)−n−1

=n

ewT Y (C, −w)A(n)B(z − w)−n−1

= ewT Y (C, −w)Y (A, z − w)B.

56



This calculation holds in V ((w))((z − w)). By locality we know that

Y (C, −w)Y (A, z − w)B and Y (A, z − w)Y (C, −w)B

are expansions of the same element of V [[z, w]][z−1, w−1, (z − w)−1]. Thus,

ewT Y (C,

−w)Y (A, z

−w)B = ewT Y (A, z

−w)Y (C,

−w)B

= ewT Y (A, z − w)e−wT Y (B, w)C

= Y (A, z)Y (B, w)C,

where the last equality holds when w is ‘small’. This implies that Y (A, z)Y (B, w)C and Y (Y (A, z −w)B, w)C are the expressions of the same element.

It is important to think of this property as

(6.13) Y (A, z)Y (B, w)C = Y (Y (A, z − w)B, w)C =n

Y (A(n)B, w)C (z − w)−n−1.

It also gives us a way to represent the product of two vertex operators as a linear combination of vertex operators. However, the two sides are formal power series in V ((z))((w)) and V ((w))((z

−w))

respectively. They do ‘converge’ when we apply the expressions to a fixed vector C ∈ V but theexpressions are not equal. Written in the above form and with the above understanding, the equality(6.13) is known as the operator product expansion or OPE.

To convert OPE formulae to identities involving commutators, we invoke 6.2.1 to conclude that

Y (A, z)Y (B, w) =N −1 j=0

c j(w)

(z − w) j+1+ : Y (A, z)Y (B, w) :,

where by (z − w)−1 we mean its expansion in positive powers of w/z. Thus, for any C ∈ V the seriesY (A, z)Y (B, w)C ∈ V ((z))((w)) is an expansion of

N −1 j=0

c j(w)

(z − w) j+1 + : Y (A, z)Y (B, w) : C ∈ V [[z, w]][z−1, w−1, (z − w)−1].

We use the Taylor formula to obtain the expansion of this element in V ((w))((z − w)).

: Y (A, z)Y (B, w) : =n<0

A(n)z−n−1

Y (B, w) + Y (B, w)n≥0

A(n)z−n−1

= n<0,m

A(n)

−n − 1

m

(z − w)mw−n−m−1

Y (B, w)

+Y (B, w)n≥0,m

A(n)

−n − 1

m

(z − w)mw−n−m−1

=

m≥0

(z − w)m

m!n<0

A(n)(−n − 1) . . . (−n − m)w−n−m−1Y (B, w)

+m≥0

(z − w)m

m!Y (B, w)

n≥0

A(n)(−n − 1) . . . (−n − m)w−n−m−1

=m≥0

(z − w)m

m!: ∂ mw Y (A, w)Y (B, w) : .

57



Comparing coefficients for k ≥ 0 we get

Y (A(n)B, z) =1

(−n − 1)!: ∂ −n−1

z Y (A, z) · Y (B, z) :, n < 0.

In particular, setting B = |0, n = −2 and using A(−2)|0 = T A, we get back the state-field corre-

spondence.Similarly, comparing the coefficients of (z − w)k, k < 0 of this with (6.13) we find that

c j(w) = Y (A( j)B, w), j ≥ 0.

This improves (6.13) into

(6.14) Y (A, z)Y (B, w) =N −1 j=0

Y (A( j)B, w)

(z − w) j+1+ : Y (A, z)Y (B, w) :,

where this identity makes sense in End V [[z−1, w−1]] if we expand (z − w)−1 in positive powers of w/z. Thus, 6.2.1 implies :

(6.15) [Y (A, z), Y (B, w)] =N −1 j=0

1

j!Y (A( j)B, w)∂ jwδ (z − w).

Expanding both sides of (6.15) we obtain the following identity for the commutators of Fouriercoefficients of arbitrary vertex operators :

(6.16)

A(m), B(n)

=N −1 j=0

m

j

(A( j)B)(m+n− j).

This formula shows us that the collection of all Fourier coefficients of vertex operators form aLie algebra. It also shows that the commutators depend only on the singular terms in the OPE. Inactuality, what the OPE expresses are just the terms which are singular at z = w. We can do thisbecause the structure of all the commutation relations depend only on the singular terms.

58



6.4 Some examples

We discuss a few examples of vertex algebras and conclude with a discussion on conformalvertex algebra.

Example 1 (Commutative vertex algebra)

Let V be a Z-graded unital commutative associative algebra (with finite dimensional homogeneouscomponents) and a derivation T of degree 1. Then V carries a canonical structure of vertex algebra.We define the unit of V to be the vacuum vector |0, and define the vertex operation Y by

Y (A, z) =n≥0

1

n!m(T nA)zn = m(ezT A),

where for B ∈ V , m(B) denotes the operator of multiplication by B on V . The vacuum axiom isclearly satisfied. For translation, we use the fact that T is a derivation.

[T, Y (A, z)] B = n≥0

1

n!T ((T nA)B)zn

−n≥0

1

n!(T nA)(T B)zn

=n≥0

n + 1

(n + 1)!m(T n+1A)Bzn

= ∂ zY (A, z)B.

The locality axiom is satisfied in the strong sense : for any A, B ∈ V , [Y (A, z), Y (B, w)] = 0.Conversely let V be a vertex algebra, in which locality holds in the strong sense. We would

call such a vertex algebra commutative . Then

0 = [Y (A, z), Y (B, w)] |0 ∈ Y (A, z)(B + wV [[w]]) − Y (B, w)(A + zV [[z]]),

whence Y (A, z) can only have positive powers of z, i.e., Y (A, z)∈

End V [[z]]. Using A(−

1), theconstant term of Y (A, z), we define a bilinear operation on V by setting

A B := A(−1) · B

which equals Y (A, z)Y (B, w)|0z,w=0

. This preserves the Z-gradation on V and proves commuta-tivity. By strong locality, A(−1)B(−1) = B(−1)A(−1) for any A, B ∈ V . This implies associativity:

A (C B) = A (B C ) = A(−1)B(−1)(C ) = B(−1)A(−1)(C ) = B (A C ) = (A C ) B.

Set the vacuum vector |0 to be the unit. We check that

T (A B) = T (Y (A, z)B)z=0

= [T, Y (A, z)] Bz=0 + Y (A, z)T Bz=0

= ∂ zY (A, z)Bz=0

+ A(−1)T B

= ∂ zY (A, z)Y (B, w)|0z=0,w=0

+ A(−1)B(−2)|0= Y (B, 0)∂ zY (A, z)|0

z=0+ A(−1)B(−2)|0

= B(−1)A(−2)|0 + A(−1)B(−2)|0= B T A + A T B.

59



Thus, T is a derivation of degree 1. In fact, the property that the formal power series (fields) onlyhas non-negative powers of z is equivalent to the vertex algebra being commutative.

Lemma 6.4.1. A vertex algebra is commutative if and only if Y (A, z) ∈ End V [[z]] for all A ∈ V .

Proof We have already seen that if V is commutative then Y (A, z) ∈ End V [[z]]. On the other

hand, if Y (A, z) ∈ End V [[z]] for all A ∈ V , then Y (A, z)Y (B, w) ∈ End V [[z, w]]. By locality,

(z − w)N [Y (A, z), Y (B, w)] = 0.

Since (z − w)N has no zero divisors in End V [[z, w]], it follows that V is commutative.

This sets up a correspondence between Z-graded commutative vertex algebras and Z-graded unitalcommutative associative algebras with a derivation of degree 1.

Example 2 (Heisenberg vertex algebra)

We define the Heisenberg Lie algebra

Hto be the direct sum of formal Laurent power series C((t))

and a 1-dimensional space C1 with the commutation relations

(6.17) [f (t), g(t)] = −Res(f dg)1 , [1 , f (t)] = 0.

Here Res means the residue of the Laurent series. Thus, H is a 1-dimensional central extension of thecommutative Lie algebra C((t)). Since the relations above are independent of the local coordinatet, we may define H as a central extension of the space of functions on a punctured disc without anyspecific choice of formal coordinate t.

This algebra is topologically generated by bn = tn, n ∈ Z and the central element 1 with therelations

(6.18) [bn, bm] = nδ n,−m 1 , [ 1 , bn] = 0.

There are two Lie subalgebras a, b of H (both containing C1 ) defined by taking the taking the powerseries in t and t−1 respectively. Let Ck denote the 1-dimensional representation of a = C[[t]] ⊕ C1

where C[[t]] acts by 0 and 1 acts by k. Let π be the induced representation of H on C k. Then

π = IndHa C k := U (H) ⊗U (a) Ck.

But it follows from the theory of universal algebras that if g = h1 + h2 is the sum of Lie subalgebraswith i = h1 ∩ h2 then

U (g) = U (h1) ⊗U (i) U (h2).

Therefore, using the fact that U (a) is a (U (C1 ), U (a))-bimodule and U (C1 ) is a commutative ring

and the observation above we get

U (H) ⊗U (a) Ck = (U (b) ⊗U (C1

) U (a)) ⊗U (a) Ck

= U (b) ⊗U (C1

) (U (a) ⊗U (a) Ck)

= U (b) ⊗U (C1

) Ck.

Equivalently, we may describe π as the polynomial algebra C[b−1, b−2, . . .] with bn, n < 0, acting bymultiplication, b0 acting by zero, and bn, n > 0 acting by n ∂

∂b−n. The operators bn, n < 0 are known

60



as creation operators , while the operators bn, n ≥ 0 are called annihilation operators . The module πis called the Fock representation of H. To make π into a vertex algebra, we prescribe :

(i) Z+ grading : deg b j1 · · · b jk = −ki=1 ji.

(ii) Vacuum vector : |0 = 1.(iii) The translation operator : defined by T · 1 = 0 and [T, bi] = −ibi−1.

To define the fields, we need to define Y (b−1, z) since this generates all the other fields in an appro-priate sense. Set

b(z) = Y (b−1, z) =n∈Z

bnz−n−1.

The reconstruction theorem (strong form) then states that there is a unique vertex algebra structureon π. Explicitly, the fields corresponding to other elements of π are given by

(6.19) Y (b−n1b−n2 · · · b−nk , z) =1

(n1 − 1)!(n2 − 1)! · · · (nk − 1)!: ∂ n1−1

z b(z) · · · ∂ nk−1z b(z) : .

The colons in the formula above stand for normally ordered product . To define it, let : bn1 · · · bnk :be the monomial obtained from bn1 · · · bnk by moving all the creation operators to the left and all

the annihilation operators to the right. The fact that makes this definition correct is that creationoperators (resp. annihilation operators) commute with each other. Hence it doesn’t matter howwe order the creation operators (resp. annihilation operators) amongst themselves. Now define: ∂ m1

z b(z) · · · ∂ mkz b(z) : to be the power series in z obtained from ∂ m1

z b(z) · · · ∂ mkz b(z) but replacing

each term bn1 · · · bnk by : bn1 · · · bnk :.The locality axiom needs to be checked and by the reconstruction theorem we need only verify

that the generating field b(z) is local with itself.

[b(z), b(w)] =n,m∈Z

[bn, bm] z−n−1w−m−1

= n∈Z [bn, b−n] z−n−1wn−1

=n∈Z

nz−n−1wn−1

= ∂ wδ (z − w),

where we use δ (z − w) to denoten∈Z wnz−n−1. Then

(z − w)2 [b(z), b(w)] = (z2 − 2zw + w2)(n∈Z

nz−n−1wn−1)

=n∈Z

nz−(n−1)wn−1 −n∈Z

2nz−nwn +n∈Z

nz−(n+1)wn+1

= 0.

Therefore, we have

Proposition 6.4.2. The Fock representation π with the structure given above satisfies the axioms of vertex algebra.

61



Example 3 (Virasoro algebra)

The Lie bracket for the Virasoro algebra Vir is

[Lm, Ln] = (m − n)Lm+n +m3 − m

12δ m,−n 1

Consider the induced representation Virk = IndVir DerC[[t]]⊕C1Ck, where Ck is a 1-dimensional represen-

tation, on which 1 acts by k and DerC[[t]] acts by zero. By the Poincare-Birkhoff-Witt theorem,Virk has a basis consisting of monomials of the form L j1 . . . L jm|0, j1 ≤ · · · ≤ jm ≤ −2, where |0is the image of 1 ∈ Ck in the induced representation. The monomials L j1 . . . L jm |0 are of degree−( j1 + . . . + jm). The commutation relations can be written as

Lemma 6.4.3.

(6.20) [T (z), T (w)] =1

12∂ 3wδ (z − w) + 2T (w)∂ wδ (z − w) + ∂ wT (w)δ (z − w)

as a formal power series in z±1, w±1.

Proof We have

[T (z), T (w)] =m,n∈Z

[Ln, Lm] z−n−2w−m−2

=n,m∈Z

(n − m)Ln+mz−n−2w−m−2 + 1

n∈Z

n3 − n

12z−n−2wn−2

= j,l∈Z

2lL jw− j−2z−l−1wl−1 + j,l∈Z

(− j − 2)L jw− j−3z−l−1wl

1

12 l∈Zl(l − 1)(l − 2)z−l−1wl−3

=1

12∂ 3wδ (z − w) + 2T (w)∂ wδ (z − w) + ∂ wT (w)δ (z − w),

where we made the substitutions l = n + 1, j = n + m.

This implies locality :(z − w)4 [T (z), T (w)] = 0.

Remark We may rewrite (6.20) as

T (z)T (w) ∼ 1

12(z − w)4+

2T (w)

(z − w)2+

∂T (w)

z − w.

In general, let L(z) be any field such that it has the following OPE with itself :

(6.21) L(z)L(w) ∼12 C

(z − w)4+

2L(w)

(z − w)2+

∂ wL(w)

z − w.

Then L(z) =n∈Z Lnz−n−2 is a Virasoro field with central charge C .

62



By the reconstruction theorem we obtain a vertex algebra structure on Virk such that

Y (L−2|0, z) = T (z) :=n∈Z

Lnz−n−2

is the generating field of Virk. The translation operator is T = L−1 and acts as T L−n|0 = (n −1)L

−n

−1

|0

. Combining with 6.1.4 we have

Y (L−n|0, z) =1

n − 2∂ zY (L−n+1|0, z),

iterations of which yield

(6.22) Y (L−n|0, z) =1

(n − 2)!∂ n−2z T (z).

Again by the reconstruction theorem

(6.23) Y (L−n1L−n2 . . . L−nm|0, z) =1

(n1 − 2)! . . . (nm − 2)!: ∂ n1−2

z T (z) . . . ∂ nm−2z T (z) : .

We also have L2L−2|0 = 12 k|0 and L0 acts on degree n elements of Virk by n.

Definition 6.4.4. A conformal vector of a vertex algebra V is a vector ν such that Y (ν, z) =n∈Z Lnz−n−2 is a Virasoro field with central charge c. It must also have the following properties :(i) L−1 = T (ii) L0 is diagonalizable on V .

We call c the central charge of ν and V a conformal vertex algebra of rank c. Y (ν, z) is called theenergy-momentum field of V .

Y (A, z) is called a primary field of conformal weight ∆ if

(6.24) Y (ν, z)Y (A, w) ∼ ∂Y (A, w)

z − w+

∆Y (A, w)

(z − w)2.

In any conformal vertex algebra Y (|0, z) is a primary field of weight 0. Note that holomorphicvertex algebras do not admit a conformal structure since the Virasoro field is not holomorphic. It

follows from this definition that Virk is a conformal vertex algebra of central charge k with conformalvector L−2|0. It follows from the representation theory (?) of Kac-Moody algebras that there areno primary states in Virk. However, the following observation is useful in checking whether primaryfields exist :

Lemma 6.4.5. Y (A, z) =n∈Z Anz−n−∆ is primary of conformal weight ∆ if and only if one of

the following equivalent conditions hold :(i) LnA = δ n,0∆A, n ∈ Z+

(ii) [Lm, Y (A, z)] = zm(z∂ + ∆(m + 1))Y (A, z), m ∈ Z(iii) [Lm, An] = ((∆ − 1)m − n)Am+n, m, n ∈ Z.

Proof Since Y (A, z) is primary, it precisely means the following :

[Y (ν, z), Y (A, w)] = δ (z − w)Y (A, w) + ∆∂ wδ (z − w)Y (A, w).Comparing coefficients gives us (iii), which is equivalent to (ii). Recall that (6.14) tells us

Y (ν, z)Y (A, w) ∼n≥0

Y (Ln−1A, w)

(z − w)n+1=

Y (TA ,w)

z − w+

Y (L0A, w)

(z − w)2+ higher order terms.

Comparing this with (6.24) we get (i).

63



6.5 Lattice Algebras

Let Q be an l-dimensional positive definite integral lattice, i.e., equipped with a Z-valuedsymmetric, positive definite bilinear form. For example, when Q is the 1-dimensional lattice with(m|n) = 2mn, then this is exactly the root lattice associated to the Lie algebra sl2(C). The groupalgebra C[Q] is an algebra with basis eα, α

∈Q and multiplication eαeβ = eα+β and e0 = 1. Let

h = C⊗Z Q be the complexification of Q and extend the bilinear form to h by bilinearity. Set h = h[t, t−1] + C1

to be the affinization of h viewed as a commutative Lie algebra. Let S be the symmetric algebra overthe space h[t, t−1]/h[t]. We shall define the space of states of the vertex algebra associated to Q as

V Q = S ⊗ C[Q]

with the vacuum vector |0 = 1 ⊗ 1. This has a parity

p(s ⊗ eα) = p(α) ≡ (α|α)mod2,

where p : Q → Z/2Z is a homomorphism.We have a representation of the Lie algebra h on S defined by setting π1( 1 ) = Id, π1(htn) be

the operator of multiplication by htn if n < 0, π1(htn) be the derivation of the algebra S defined by

(htn)(ht−m) = nδ n,m(h|h)

if n > 0 (a ∈ h, s > 0) and π1(h) = 0. We also have a representation π2 of h on C[Q] :

π2( 1 ) = 0, π2(htn)eα = δ n,0(α|h)eα, h ∈ h, α ∈ Q, n ∈ Z.

Define a representation π of

h on V Q by π = π1 ⊗ 1 + 1 ⊗ π2.

Let hn = π(htn

) and consider the following End V Q-valued fields h(z) = n∈Z hnz−n−

1

. Thenhm, hn

(s ⊗ eβ) = (π1(htm)π1(htn) − π1(htn)π1(htm))s ⊗ eβ

+s ⊗ (π2(htm)π2(htn) − π2(htn)π2(htm))eβ

= (π1(htm)π1(htn) − π1(htn)π1(htm))s ⊗ eβ,

where the last equality follows from the fact that [ π2(htm), π2(htn)] = 0. Checking case by case wesee that

(6.25)

hm, hn

= mδ m,−n(h|h).

This implies that(6.26)

h(z), h(w)

= (h|h)

n∈Z

nz−n−1wn−1 = (h|h)∂ wδ (z − w).

This is equivalent to the OPE

(6.27) h(z)h(w) ∼ (h|h)(z − w)2

.

64



We also have (z − w)2 [h(z), h(w)] = 0.Let eα denote the operator of multiplication by 1 ⊗ eα. Then

[hn, eα] (s ⊗ eβ) = hn(s ⊗ eαeβ) − eα(hn(s ⊗ eβ))

= s ⊗ π2(htn)(eα+β) − s ⊗ eαπ2(htn)(eβ)

= s⊗

δ n,0

(α + β |h)eαeβ

−s⊗

eαπ2

(htn)eβ

= δ n,0(α|h)eα(s ⊗ eβ).(6.28)

To construct the state-field correspondence, we need to define fields Γα(z) := Y (1 ⊗ eα, z) foreach α ∈ Q. Using the general OPE formula (6.14) (valid in |z| > |w|)

Y (A, z)Y (B, w) =N −1 j≥0

Y (A( j)B, w)

(z − w) j+1+ : Y (A, z)Y (B, w) :

we get the OPE

(6.29) h(z)Γα(w) ∼ (α|h)Γα(w)

z − w.

Equivalently we have(6.30) [h(z), Γα(w)] = (α|h)Γα(w)δ (z − w)

which can be rewritten as

(6.31) [hn, Γα(w)] = (α|h)wnΓα(w).

An educated guess for Γα(z) would be

(6.32) Γα(z) := eαzα0e−

j<0z−j

jαje

−j>0

z−j

jαjcα.

The operators cα satisfy

(6.33) c0 = 1, cα|0

=|0, [hn, cα] = 0.

due to (6.30). Since

hn, α−nn

= 1 we get

[hn, ewn

nα−n ] =

j≥0

hn,

w jn

j!

α−nn

j= j≥0

w jn

j!jα−n

n

j−1= wne

wn

nα−n.

We use this to verify (6.31) :

[hn, Γα(w)] = δ n,0(α|h)Γα(w) + eαzα0

hn, e−

j<0w−j

jαje

−j>0

w−j

jαj

cα

=

(α|h)Γα(w) n = 0

eαzα0e−

j<0w−j

jαj

hn, e

−j>0

w−j

jαj

cα n < 0

eαzα0 hn, e−j<0 w−j

j αj e−j>0 w−j

j αjcα n > 0

=

(α|h)Γα(w)

eαzα0e−

j<0w−j

jαj−n−1 j=1 e−

w−j

jαj

hn, ewn

nα−n

j≥−n+1 e−

w−j

jαjcα

eαzα0−n+1 j=−1 e−

j<0

w−j

jαj

hn, ewn

nα−n

j≤−n−1 e−

w−j

jαje−

j>0

w−j

jαjcα

= (α|h)wnΓα(w).

65



We also need to calculate

zα0, eβ

. On the one hand we have

eβzα0(s ⊗ eγ ) = z(α|γ )eβ(s ⊗ eγ ) = z(α|γ )(s ⊗ eβ+γ ),

while on the other hand we have

zα0eβ(s ⊗ eγ ) = zα0(s ⊗ eβ+γ ) = z(α|β+γ )(s ⊗ eβ+γ ).

This implies

(6.34) zα0eβ = z(α|β)eβzα0 .

Using (6.28) and (6.34) we get

Γα(z)Γβ(w) =

eαzα0cαe−

j<0z−j

jαje

−j>0

z−j

jαj

eβwβ0cβe−

k<0w−k

kβke−

k>0

w−k

kβk

= eαcαeβcβzα0wβ0z(α|β)e−

j<0z−j

jαj

j>0

e−z−j

jαje

wj

jβ−j

e−

k>0

w−k

kβk .

Recall that if two linear operators A and B satisfy [A, [A, B]] = 0 = [B, [A, B]] and [A, B] = cI then the Campbell-Baker-Hausdorff formula (5.42) takes the useful form

eAeB = eA+Be1

2[A,B] = eceBeA.

We will apply this for A j = −z−j

j α j, B j = wj

j β − j and c j = − wj

jzj(α|β ) to get

(6.35) e− z−j

jαje

wj

jβ−j = ecje

wj

jβ−je

− z−j

jαj

whence

Γα(z)Γβ(w) = eαcαeβcβzα0wβ0z(α|β)s>0

ecse−j<0( z−

jj αj+w

−

jj βj)e−k>0( z−k

kαk+w−k

kβk).

Thus, we need to calculates>0 ecs . For 0 ≤ r < 1 we have

log(1 − r) = − j>0

r j

j.

This implies (for c ∈ R)

e−cj>0

rj

j = (1 − r)c.

Using this with r = |w|/|z| in the domain |z| > |w| and c = (α|β ) we get

(6.36) Γα(z)Γβ(w) = eαcαeβcβzα0wβ0(z − w)(α|β)e−

j<0( z−j

jαj+w−j

jβj)

e−

j>0( z−j

jαj+w−j

jβj)

.

Similarly, in the domain |z| < |w| we have(6.37)

Γβ(w)Γα(z) = (−1)(α|β)eβcβeαcαzα0wβ0(z − w)(α|β)e−

j<0( z−j

jαj+w−j

jβj)

e−

j>0( z−j

jαj+w−j

jβj)

.

66



Locality implies that

(6.38) eαcαeβcβ = (−1)(α|α)(β|β)+(α|β)eβcβeαcα.

It follows from (6.33) that cα(s ⊗ eβ) = s ⊗ cα(eβ). Let cα(eβ) = λeν . If we assume that (·|·)is non-degenerate then

0 = [γ 0, cα]

= λγ 0(s ⊗ eν ) − (β |γ )cα(s ⊗ eβ)

= (ν − β |γ )cα(s ⊗ eβ)

implies ν = β . We may rewrite this as follows :

(6.39) cα(s ⊗ eβ) = ε(α, β )(s ⊗ eβ).

Thus, looking for solutions of cα satisfying (6.33) and (6.38), we are led to consider cα’s satisfying(6.39), where ε(α, β ) ∈ C.

Lemma 6.5.1. Equations (6.33) and (6.38) are equivalent to

ε(α, 0) = ε(0, α) = 1(6.40)

ε(α, β ) = (−1)(α|α)(β|β)+(α|β)ε(β, α)(6.41)

ε(γ, β )ε(β + γ, α) = ε(γ, α + β )ε(β, α).(6.42)

Proof (6.40) is equivalent to the first two equations in (6.33). (6.41) follows from (6.38) appliedto |0 and using (6.33). (6.42) is equivalent to (6.38) applied to 1 ⊗ eγ and using (6.41).

For an integral lattice one can explicitly construct a cocycle with values in ±1 such that

ε(α, β )ε(β, α) = (

−1)(α|α)(β|β)+(α|β).

Choose an ordered basis α1, . . . , αl of Q over Z and define

ε(αi, α j) =

(−1)(αi|αi)(αj |αj)+(αi|αj) if i < j,1 if i ≥ j,

and extend to Q by bimultiplicativity.To understand the equations better, we introduce the twisted group algebra Cε[Q]. This is the

algebra with a basis eα, α ∈ Q and the ‘twisted’ multiplication :

eαeβ = ε(α, β )eα+β, α , β ∈ Q.

The vector space of states is V Q := S ⊗Cε[Q]. With this new definition, we redefine the fieldcorresponding to 1 ⊗ eα as

Γα(z) = eαzα0e−

j<0z−j

jαje−

j>0

z−j

jαj .

Now the product of two fields can be rewritten as

Γα(z)Γβ(w) = ε(α, β )eα+β(z − w)(α|β)zα0wβ0e−

j<0( z−j

jαj+w−j

jβj)

e−

j>0( z−j

jαj+w−j

jβj)

,

67



which we write asΓα(z)Γβ(w) = ε(α, β )(z − w)(α|β)Γα,β(z, w).

We would like to write down the Taylor expansion of Γα,β(z, w) in the domain |z| > |w|. This willhelp us write the OPE.

First observe that

∂ Γα(z) = eαα0zα0−Ide−j<0 z−j

j αje−j>0 z−j

j αj

+eαzα0∂ (e−

j<0z−j

jαj)e

−j>0

z−j

jαj + eαzα0e

−j<0

z−j

jαj∂ (e

−j>0

z−j

jαj )

= Γα(z)α0z−1 + j<0

α jz− j−1Γα(z) + j>0

Γα(z)α jz− j−1

= α(z)+Γα(z) + Γα(z)α−(z)

= : α(z)Γα(z) : .

One can similarly show that

(6.43) ∂ zΓα,β(z, w) =: α(z)Γα,β(z, w) :, α , β ∈ Q.

Since ∂ = ∂ z is also a derivation of normally ordered product, i.e.,

∂ : a(z)b(z) :=: ∂a(z)b(z) : + : a(z)∂b(z) :,

we can obtain higher derivatives using this. Thus,

∂ 2zΓα,β(z, w) = : ∂α(z)Γα,β(z, w) : + : α(z)2Γα,β(z, w) :,

∂ 3zΓα,β(z, w) = : ∂ 2α(z)Γα,β(z, w) : +3 : α(z)∂α(z)Γα,β(z, w) : + : α(z)3Γα,β(z, w) : .

Our ansatz for ∂ nΓα,β(z, w), in view of the previous formulae, is

(6.44) ∂ nz Γα,β(z, w) =

k1+2k2+...=nki≥0

cn(k1, k2, . . .) : α(z)k1(∂α(z))k2 . . . Γα,β(z, w) :

where(6.45) cn(k1, k2, . . .) =

n!

(1!)k1k1!(2!)k2k2! . . ..

The cn’s satisfy the cyclic condition

cn+1(k1, k2, . . .) = (k1 + 1)cn(k1 + 1, k2 − 1, k3, . . .) + (k2 + 1)cn(k1, k2 + 1, k3 − 1, k4, . . .)

+ . . . + (kn + 1)cn(k1, . . . , kn + 1, kn+1 − 1, . . .) + cn(k1 − 1, k2, . . .).

This makes the formula (6.44), for ∂ nΓα,β(z, w), amenable to a proof by induction. Since Γα,β(w, w) =Γα+β(w), the Taylor expansion reads

Γα,β(z, w) = n≥0

k1+2k2+...=n

ki≥0

cn(k1, k2, . . .)(z − w)n : α(w)k1(∂α(w))k2 . . . Γα+β(w) : .

Therefore

(6.46) Γα(z)Γβ(w) = ε(α, β )(z − w)(α|β)

k1+2k2+...=nki≥0,n≥0

cn(k1, k2, . . .)(z − w)n : α(w)k1 . . . Γα+β(w) : .

68



By construction, the fields Γα(z)’s are mutually local and satisfy (6.30).Let us collect together the most important properties of the fields h(z) and Γα(z) :

h(z)h(w) ∼ (h|h)(z − w)2

h, h ∈ h

h(z)Γα(w) ∼(α

|h)Γα(w)

z − w h ∈ h, α ∈ Q

Γα(z)Γβ(w) ∼ ε(α, β )(z − w)(α|β)Γα,β(z, w) α, β ∈ Q

We can now state the main result of this section.

Theorem 6.5.2. Let Q be an integral lattice and let V Q = S ⊗ Cε[Q]. Then there exists a simple vertex algebra structure on V Q with the vacuum vector |0 = 1 ⊗ 1 and such that

Y (ht−1 ⊗ 1, z) = h(z), h ∈ h,

if and only if the bilinear form (·|·) is non-degenerate. Such a vertex algebra structure is unique and is independent of the choice of the cocycle ε up to isomorphism.

Moreover, the lattice vertex algebra can be explicitly constructed. Let ε be the cocycle (refer previous construction) taking values in ±1. Let V Q be the space of states with the parity

(6.47) p(s ⊗ eα) ≡ (α|α) mod 2,

with the vacuum vector as before. Define the infinitesimal translation operator T as the derivation of V Q given by (n > 0, h ∈ h, α ∈ Q) :

(6.48) T (ht−n ⊗ 1) = n(ht−n−1) ⊗ 1, T (1 ⊗ eα) = (αt−1) ⊗ eα.

For α ∈ Q let

(6.49) Γα(z) = eαzα0e−

j<0z−j

jαje

−j>0

z−j

jαj ,

where eα is the operator of left multiplication by 1 ⊗ eα. Then the state-field correspondence is given by (ni ≥ 0, hi ∈ h, α ∈ Q) :

(6.50) Y ((h1t−n1−1)(h2t−n2−1) . . . ⊗ eα, z) =: ∂ (n1)h1(z)∂ (n2)h2(z) . . . Γα(z) : .

Proof If h ∈ h is in the kernel of (·|·) then

(ht−n ⊗ eα)(ht−1 ⊗ 1) = ht−nht−1(1 ⊗ eα) = 0,

whence ht1− ⊗ 1 generates an ideal in V Q, contradicting its simplicity. Conversely, if (·|·) is non-degenerate then it follows from a previous discussion that the operators cα must satisfy (6.39).

Let J ε be the subalgebra of Cε[Q] generated by e

α

’s such that ε(α, −α) = 0. Let Qε =α ∈ Q|ε(α, −α) = 0 be a subset of Q. Note that Qε is closed under additive inverses sinceε(α, −α) = ε(−α, α). If α ∈ Qε, β ∈ Q then

ε(−α, α)ε(0, β ) = ε(α, β )ε(−α, α + β ),

which implies

(6.51) ε(α, β ) = 0, α ∈ Qε, β ∈ Q.

69



For α, β ∈ Qε

(6.52) ε(α, β )ε(α + β, −α − β ) = ε(β, −α − β )ε(α, −α) = 0.

This means α + β ∈ Qε, whence Qε is a sublattice. For eα ∈ J ε, β ∈ Q we have

(1 ⊗ eβ)(1 ⊗ eα) = ε(β, α)(1 ⊗ eα+β).

By (6.52), either eα+β ∈ J ε or α + β ∈ Qε and ε(α, β ) = 0, which implies eβ(1 ⊗ eα) = 0. Inboth cases, eβ(J ε) ⊂ J ε and J ε is an ideal in Cε[Q]. Consequently, 1 ⊗ J ε generates an ideal in V Q.Choosing ε to take non-zero values (as per the construction before) ensures that this doesn’t happen.In such a case J ε = 0, Qε = Q and (6.51) holds for any α, β ∈ Q.

Let s⊗eβ ∈ J be an element of an ideal J in V Q. Then e−β(s⊗eβ) = s⊗1 ∈ J . By inductivelymaking hn’s (n > 0) act on s ⊗ 1 we get |0, whence J = V Q, a contradiction. The last part of thetheorem just follows from 6.1.6, the strong reconstruction theorem.

Finally we discuss the conformal structure of V Q.

Proposition 6.5.3. Let Q be an integral lattice of rank l and assume that the bilinear form (·|·) is non-degenerate. Choose basis ai and bi of h such that (ai|b j) = δ ij .(i) The vector

(6.53) ν =1

2

li=1

ai−1bi−1|0

is a conformal vector of the lattice vertex algebra V Q. The central charge of the corresponding Virasoro field Y (ν, z) is l.

(ii) The fields h(z), h ∈ h are primary of conformal weight 1.(iii) The fields Γα(z) = Y (1 ⊗ eα, z), α ∈ ∆ are primary of conformal weight 1

2 (α|α).

We skip the proof but the main ideas used for proving this is outlined for a specific case in 6.5.6.We note that the conformal structure is associated to the action of the diffeomorphism group of thecircle on the loop group.

Example (Root lattice vertex algebra)

Let Q be a positive definite integral lattice. The set

∆ = α ∈ Q|(α|α) = 2

is called the root system of Q. The lattice Q is called a root lattice if it is spanned by ∆ over Z. Let

us put an order on the basis of Q generated by ∆ and define ε as before. It is well known that ∆ isisomorphic to a direct sum of finite root systems of type A, D and E . For α, β ∈ ∆ we have

0 ≤ (α + β |α + β ) = 4 + 2(α|β ),

which implies the following three possibilities :

(α|β ) ≥ 0, (α|β ) = −1, or α = −β.

70



Hence the complete list of OPE between the generating fields h(z) =n∈Z hnz−n−1 and Γα(z) =

n∈Z eαnz−n−1 is as follows :

h(z)h(w) ∼ (h|h)(z − w)2

if h, h ∈ h,(6.54)

h(z)Γα(w) ∼(α

|h)Γα(w)

z − w if h ∈ h, α ∈ ∆,(6.55)

Γα(z)Γβ(w) ∼ 0 if α, β ∈ ∆, (α|β ) ≥ 0,(6.56)

Γα(z)Γβ(w) ∼ ε(α, β )Γα+β(w)

z − wif α, β ∈ ∆, (α|β ) = −1,(6.57)

Γα(z)Γ−α(w) ∼ 1

(z − w)2+

α(w)

z − wif α ∈ ∆.(6.58)

These OPE are equivalent to the following commutation relations for m, n ∈ Z :hm, hn

= mδ m,−n(h|h) if h, h ∈ h,(6.59)

[hm, eαn] = (h

|α)eαm+n if h

∈h, α

∈∆,(6.60) eαm, eβn

= 0 if α, β ∈ ∆, (α|β ) ≥ 0,(6.61) eαm, eβn

= ε(α, β )eα+β

m+n if α, β ∈ ∆, (α|β ) = −1,(6.62) eαm, e−αn

= αm+n + mδ m,−n if α ∈ ∆.(6.63)

These commutation relations lead us to consider the vector space

(6.64) g = h ⊕ (⊕α∈∆Ceα)

with the Lie bracket defined by

h, h

= 0 if h, h ∈ h,(6.65)

[h, eα] = (h|α)eα if h ∈ h, α ∈ ∆,(6.66)[eα, eβ] = 0 if α, β ∈ ∆, (α|β ) ≥ 0,(6.67)

[eα, eβ] = ε(α, β )eα+β if α, β ∈ ∆, (α|β ) = −1,(6.68)

[eα, e−α] = α if α ∈ ∆,(6.69)

and with the C-valued symmetric bilinear form (·|·) : g × g → C which extends that on h by letting

(6.70) (eα|e−α) = ε(α, −α) = 1, (eα|eβ) = 0 if α = −β, (h|eα) = 0.

We arrive at the following theorem, which is usually referred to as the Frenkel-Kac construction .

Theorem 6.5.4. (Frenkel-Kac)

(1) The space g

with the bracket defined by (6.65)-(6.69) is a semisimple Lie algebra with a Cartan subalgebra h and the root space decomposition (6.64). The form (·|·) is the non-degenerate symmetric invariant bilinear form on g normalized by the condition (α|α) = 2 for α ∈ ∆.

(2) Formulae (6.59)-(6.63) define an irreducible representation of the affinization g = g[t, t−1]+C1

of the pair (g, (·|·)) with central charge k = 1 and highest weight vector |0 such that

(6.71) g[t]|0 = 0.

(3) The simple vertex algebra V Q is isomorphic to the affine vertex algebra V 1(g).

71



Proof (1) The fact that g forms a Lie algebra follows from (6.59)-(6.63) with m = n = 0. Formulae(6.59)-(6.63) also define a representation of the affinization of (g, (·|·)) with k = 1 in the space V Q(refer (5.32)). The invariance of the form (·|·) also follows by a case by case analysis. It is clearlysymmetric and non-degenerate. To show that g is semisimple we compare (6.59)-(6.63) with theSerre relations (5.1)-(5.6) and establish an equivalence. Note that (6.65) and (5.1) are the the same.

Observe that (6.66) is equivalent to (5.2)-(5.3). Equations (6.67) and (6.69) imply (5.4) while (6.67)and (6.68) are equivalent to (5.5)-(5.6).(2) Let W ⊂ V Q be any non-trivial invariant subspace of V Q. Choose s⊗eβ ∈ W . Then s⊗1 ∈ W

since eβ−1W ⊂ W . Then by inductively making hn’s (n > 0) act on s⊗1 we get |0, whence W = V Q.Thus, the representation is irreducible with central charge k = 1.

(3) The simplicity of V Q follows from the non-degeneracy of (·|·). We skip the proof of theisomorphism at this point. (?)

Example (Root lattice vertex algebra for sl2(C))

Let Q = Z with the bilinear form (m|n) = 2mn. It is a 1-dimensional root lattice with ∆ = 1, −1.

We set α = 1 to be a basis of Q over Z. The complexification is h = C with the extended bilinearform (z|w) = 2zw. The affinization C = C[t, t−1] + C1

can be thought as a commutative Lie algebra. Set S to be the symmetric algebra over C[t, t−1]/C[t].Endow C[Q] = C[eα, e−α] with the trivial cocycle. Then Cε[Q] = C[Q] and the vector space of statesis

V Z = S ⊗C[eα, e−α]

with the trivial parity and the vacuum vector |0 = 1⊗1. Recall the representation π = π1⊗1+1⊗π2

of C on V Z. Define the fields α(z) =n∈Z αnz−n−1 and

Γα(z) = e

α

z

α0

e−

j<0

z−j

jαj

e−

j>0

z−j

jαj

.For elements α, β ∈ ∆, the possibilities for (α|β ) are 2 or −2. The discussion on root lattice

vertex algebra leads us to consider the vector space

(6.72) g = C⊕ Ceα ⊕Ce−α

with the commutation relations

[α, eα] = 2eα(6.73)

[α, e−α] = −2e−α(6.74)

[eα, e−α] = α.(6.75)

This Lie algebra is isomorphic to sl2(C) and has the bilinear form (·|·) : g × g → C defined by

(λ1α + µ1eα + ν 1e−α, λ2α + µ2eα + ν 2e−α) := 2λ1λ2 + µ1ν 2 + µ2ν 1.

Observe that up to scaling by a factor of 4, this is the negative of the usual Killing form on sl2(C)with the generators e ,f ,h.

We now discuss the existence of a conformal structure on V Z and the proof uses ideas that canbe used to prove 6.5.3 in full generality. In preparation we need

72



Theorem 6.5.5. (Wick)Let a1(z), . . . , aM (z) and b1(z), . . . , bN (z) be two collection of fields such that the following hold :

(i)

ai(z)−, b j(w)

, ck(z)±

= 0 for all i,j,k, and c = a or b,(ii)

ai(z)±, b j(w)±

= 0 for all i and j.

Let

aib j

=

ai(z)−, b j(w)

denote the contraction of ai(z) and b j(w). Then

: a1(z) · · · aM (z) :: b1(w) · · · bN (w) :=(6.76)min(M,N )

s=0

i1<···<is j1=···= js

± ai1b j1 · · · aisb js

: a1(z) · · · aM (z)b1(w) · · · bN (w) :(i1,...,is,j1,...,js)

,

where the subscript (i1, . . . , is, j1, . . . , js) means the fields ai1(z), . . . , ais(z), b j1(w), . . . , b js(w) are re-moved, and the sign ± is obtained by the usual super rule, i.e., each permutation of the adjacent odd fields changes sign.

The idea behind the proof is that terms on the left hand side of (6.76) looks like a sequence of a+(z)’sfollowed by a−(z)’s and then b+(z)’s followed by b−(z)’s. One can move the a(z)−’s across b(z)+’sand use condition (i). Then the resulting contractions commute with all the fields (due to (i)) and

can be moved to the left. The indexes work out as is given in the theorem.

Theorem 6.5.6. Let Q = Z with the bilinear form (m|n) = 2mn. Let α ∈ ∆ = ±1 ⊂ Q.(i) The vector

(6.77) ν =1

4α−1α−1|0

is a conformal vector of V Q with central charge 1.(ii) The field α(z) is primary of conformal weight 1.(iii) The fields Γα(z), Γ−α(z) are primary of conformal weight 1.

Proof We apply Wick’s theorem with a1(z) = a2(z) = b1(z) = b2(z) = α(z). Recall that Y (ν, z) =14 : α(z)2 :. Therefore,

Y (ν, z)Y (ν, w) =1

16: α(z)2α(w)2 : +

1

4[α(z)−, α(w)] : α(z)α(w) : +

1

8[α(z)−, α(w)]2

∼12

(z − w)4+

1

2(z − w)2

: α(w)2 : + : α(w)∂α(w) : (z − w)

∼

12

(z − w)4+

2Y (ν, w)

(z − w)2+

∂ wY (ν, w)

z − w.

Thus, in view of the remark following 6.4.3, Y (ν, z) is a Virasoro field of central charge 1.Applying Wick’s theorem again,

Y (ν, z)α(w) =1

4: α(z)2α(w) : +

1

2[α(z)−, α(w)] α(z)

∼ α(w)(z − w)2

+∂α(w)z − w

.

Thus, α(z) is primary of conformal weight 1.We need to calculate the conformal field. Since

Y (ν, z) =n∈Z

1

4

j<0

α jαn− j + j≥0

αn− jα j

z−n−2,

73



comparing coefficients and using

[Lm, Ln] = (m − n)Lm+n +m3 − m

2,

we arrive at the following :

L0 = 12 j<0

α jα− j + 14

α0α0,(6.78)

Ln =1

4

j∈Z

αn− jα j, n = 0.(6.79)

This implies

L0(αt−n1 . . . α t−nk ⊗ eβ v

) = L0α−n1 . . . α−nkeβ|0

= n1v + α−n1L0α−n2 . . . α−nkeβ|0

= . . .= (n1 + . . . + nk)v + αt−n1 . . . α t−nkL0eβ|0= (n1 + . . . + nk +

1

2(β |β ))v,

is a diagonalizable operator.However, one cannot use Wick’s theorem to calculate Y (ν, z)Γβ(w). Instead we use the alter-

native version 6.4.5. For n > 0, we have

Ln(1 ⊗ eβ) =1

4

j≤0

αn− jα j(1 ⊗ eβ)

=

1

4 (β |α)αn(1 ⊗ eβ

) +

1

4 j<0

αn− j(αt j

⊗ eβ

)

= 0.

Since Ln’s annihilate 1 ⊗ eβ, Γ±α(z) are primary fields of conformal weight 1.It is left to show that L−1 = T , the translation operator. First, observe that L−1|0 = 0. We

check how L−1 acts on the generators 1 ⊗ eβ, αt−n (β ∈ Q, n > 0).

L−1(αt−n ⊗ 1) =1

4

j>0

α j−1(αt− jαt−n ⊗ 1) +1

2nα−1−n|0

=1

2∂ j−1,nn(αt− j ⊗ 1) +

1

2n(αt−1−n ⊗ 1)

= n(αt−1−n ⊗ 1).

L−1(1 ⊗ eβ) =1

4(β |α)α−1(1 ⊗ eβ) +

1

4α0α−1(1 ⊗ eβ)

= βt−1 ⊗ eβ.

It can be checked that L−1 acts as a derivation and hence L−1 = T .

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6.6 Affine vertex algebras

We have already seen how to associate a vertex algebra to an affine Kac-Moody algebra. Herewe study a more general presentation of the same. As before, let g denote a finite dimensionalcomplex simple Lie algebra with an ordered basis J a where a = 1, . . . , l = dim g. Recall thatthe affine Kac-Moody algebra gk has a basis consisting of J an, a = 1, . . . , l , n

∈Z and 1 with the

commutation relations (equivalent to (5.33))

[A ⊗ f (t), B ⊗ g(t)] = [A, B] ⊗ f (t)g(t) − kκ(A, B)Res(f dg)1 .

where κ is the inner product with respect to which the square length of the maximal root is 2.We group the elements associated to J a into a formal power series

J a(z) =n∈Z

J anz−n−1.

This provides a hint as to what some of the fields in this vertex algebra should be. To describe thevector space V of the vertex algebra we need a special vector |0 ∈ V . If J a(z)’s are indeed vertexoperators then J an should be linear operators on V and J an |0 = 0 for n ≥ 0. Thus the set of Lie

algebra elements which are supposed to annihilate |0 form the Lie subalgebra g[[t]] of gk. Thus,C|0 is the trivial 1-dimensional representation of g[[t]]. We define an action of 1 on |0 by setting1 |0 = 1. Let us denote the resulting representation of g[[t]] ⊕ C1 by Ck. We can now define a gk-module by using the induced representation :

V k(g) = Ind gkg[[t]]⊕C1

Ck := U ( gk) ⊗U (g[[t]]⊕C1

) Ck.

We call it the vacuum Verma module of level k.The structure of V k(g) is easy to describe. As seen in example 2, §6.4

V k(g) ∼= U (g ⊗ t−1C[[t−1]])|0.

Therefore it has a basis of lexicographically ordered monomials of the form

(6.80) J a1n1 . . . J amnm |0,

where n1 ≤ n2 ≤ . . . ≤ nm < 0 and if ni = ni+1 then ai ≤ ai+1. We define a Z-grading on gkand V k(g) by the formula deg J an = −n, deg |0 = 0. The homogeneous graded components of V k(g)are finite dimensional and are non-zero only in non-negative degrees. The picture of the first fewcomponents follows :

|0

J a

−1|0

J a−2

|0

J a−1

J b−1

|0

J a

−3|0

J a−2

J b−1

|0

J a−1

J b−1

J c−1

|0

75



The action of J an for n < 0 is just the obvious action of multiplication on the left. To apply J an forn ≥ 0 we use the commutation relations in gk to move this term through to the vacuum vector,which it annihilates.

To define the translation operator T we interpret it as the vector field −∂ t. This naturally actson g((t)) and preserves g[[t]]. Therefore it acts on V k(g) as a derivation. More explicitly, this means

[T, J an ] J a1n1 . . . J amnm |0 = T (J anJ a1n1 . . . J amnm |0) − J anT (J a1n1 . . . J amnm |0)(6.81)

= T (J an)J a1n1 . . . J amnm |0= −nJ an−1J a1n1 . . . J amnm |0.

We also have T |0 = 0 as required by the translation axiom. These two conditions uniquely specifythe action of T on V k(g).

We start defining the state-field correspondence by setting Y (|0, z) = Id as prescribed by thevacuum axiom. The elements of degree 1 are of the form J a−1|0. We set

Y (J a−1|0, z) =

n∈ZJ anz−n−1 = J a(z).

These vertex operators satisfy

Y (J a−1|0, z)|0 = J a−1|0 + zV k(g)[[z]]T, Y (J a−1|0, z)

= ∂ zY (J a−1|0, z),

where the last equality follows from (6.81).To check locality we use the commutation relations between J an ’s :

J a(z), J b(w)

=n,m∈Z

J an, J bm

z−n−1w−m−1

= n,m∈Z J

a

, J bn+m z−

n−

1

w−m−

1

+ n,m∈Znkκ(J

a

, J b

)δ n,−m 1

z−n−

1

w−m−

1

=l,n∈Z

J a, J b

l

w−l−1z−n−1wn + kκ(J a, J b)1

n∈Z

nz−n−1wn−1

=

J a, J b

(w)δ (z − w) + kκ(J a, J b)1 ∂ wδ (z − w).

This is equivalent to the OPE

(6.82) J a(z)J b(w) ∼

J a, J b

(w)

z − w+

kκ(J a, J b)1

(z − w)2.

Thus (z − w)2 annihilates J a(z), J b(w) and we have the required locality.Since the vertex operators must satisfy 6.1.4, we get

Y (J an |0, z) =1

(−n − 1)Y (T J an+1|0, z)

=1

(−n − 1)∂ zY (J an+1|0, z),

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iterations of which yield

(6.83) Y (J an |0, z) =1

(−n − 1)!∂ −n−1z J a(z).

Using the normal ordering by induction, we define a general vertex operator in V k(g)

(6.84) Y (J a1n1 . . . J amnm |0, z) =1

(−n1 − 1)!. . .

1

(−nm − 1)!: ∂ −n1−1

z J a1(z) . . . ∂ −nm−1z J am(z) : .

Theorem 6.6.1. The above formulae define a Z+-graded vertex algebra structure on V k(g).

Proof We have Y (|0, z) = Id by definition. We check that

Y (A, z)|0 = A + z(. . .)

holds when A = |0. Assume it to be true for Y (B, z) for all B ∈ V k(g)≤i. Then for any A ∈ g andn < 0

Y (AnB, z)|0 = 1(−n − 1)! : ∂ −n−1z A(z) · Y (B, z) : |0

=1

(−n − 1)!

∂ −n−1z A(z)

+

Y (B, z)|0 + Y (B, z)

∂ −n−1z A(z)

−|0

=

1

(−n − 1)

k≤n

−k − 1

−n − 2

A(k)zn−k(B + z(. . .))

+1

(−n − 1)

k≥0

−k − 1

−n − 2

Y (B, z)A(k)|0zn−k

= AnB + z(. . .).

This proves the vacuum axiom.The translation axiom is tantamount to proving

[T, : A(z)B(z) :] = ∂ z : A(z)B(z) :,

assuming that the translation axiom holds for the fields A(z), B(z), i.e., [T, A(z)±] = ∂ zA(z)± etc.Now

[T, : A(z)B(z) :] = [T, A(z)+B(z) + B(z)A(z)−]

= [T, A(z)+B(z)] + [T, B(z)A(z)−]

= T A(z)+B(z) − A(z)+B(z)T + T B(z)A(z)− − B(z)A(z)−T

= ∂ zA(z)+B(z) + A(z)+T B(z)−

A(z)+B(z)T

+T B(z)A(z)− − B(z)T A(z)− + B(z)∂ zA(z)−= ∂ z : A(z)B(z) : .

We proved the locality of the generating fields. Since derivatives of local fields are still local, we just need to verify that normally ordered products of local fields are still local, which is the statementof Dong’s lemma (refer [2] pgs 51 − 52 or [6] pgs 84 − 85). The proof is then complete by the (weak)reconstruction theorem.

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An alternative viewpoint

It is worthwhile to note at this point that the parameter z in J a(z) is just a formal variable.However, in applications we may require the parameter to take values in the unit circle. Then, interms of z, J a(z) is interpreted as the Laurent series obtained by expanding the operator-valuedquantity J a(z) around zero, where it is assumed that the series converges on S 1. The coefficients J an

can be recovered as the contour integral

J an =1

2πi

0

J a(z)zn dz.

By a relabeling of variables, we can write the commutator asJ a(z), J b(w)

=

n∈Z

z

w

n+1l∈Z

z−l−2 (γ cabJ cl + nδ l,0kκab 1 )(6.85)

=n∈Z

z

w

n+1 γ cabz

−1J c(z) + nz−2kκab 1

,

which does not converge for z, w ∈ S 1

or for any other complex values of z, w.The problem can be resolved by observing that the summation in (6.85) can be split into twoparts, one of which converges for |z| > |w| and the other converges for |z| < |w|. This motivates usto prescribe special integration contours in the relation

J am, J bn

=

1

2πi

0

1

2πi

J a(z), J b(w)

zmwn dw

dz.

More explicitly, by

J a(z), J b(w)

= J a(z)J b(w) − J b(w)J a(z) we mean the first term defined in theregion |w| < |z| and the second term defined on |w| > |z|. Then

J am, J bn

=1

2πi 01

2πi C1J a(z)J b(w)zmwn dw − 1

2πi C2J b(w)J a(z)zmwn dw

dz

=1

2πi

0

1

2πi

C1∪C2

R(J a(z)J b(w))zmwn dw

dz,

where for fixed z the w-integration contours are depicted below. The contour C1 ∪C2 can be deformedinto a contour encircling z.

0

z

C 1

C 2z

0

The function R is called the radially ordered product or operator product and is defined as

(6.86) R(J a(z)J b(w)) :=

J a(z)J b(w) if |z| > |w|,J b(w)J a(z) if |z| < |w|.

This ordering is forced on us since we aim to get a convergent power series for any complex numberz. In physical applications, the radial direction corresponds to the time direction and the radial

78



ordering reproduces the familiar time ordering of quantum field theory.Using the radial ordering we see that

R(J a(z)J b(w)) = (z − w)−2kκ(J a, J b) 1 − (z − w)−1γ cabJ c(w) + O((z − w)0).

The w-integration can be performed on the deformed contour using Cauchy’s theorem.J am, J bn

=

1

2πi

0

1

2πi

z

R(J a(z)J b(w))zmwn dw

dz

=1

2πi

0

zm1

2πi

z

(z − w)−2kκab 1 − (z − w)−1γ cabJ c(w)

wn dwdz

+1

2πi

0

zm1

2πi

z

O((z − w)0)wn dwdz

=1

2πi

0

−nzn−1kκab 1 + znγ cabJ c(z) + 0

zm dz

= γ cabJ cm+n + mδ m,−nkκab 1 .

Recall that this is exactly the commutation relations (5.32) of elements in Lg. Thus, we arrive atthe following relation :

J a(z), J b(w)

=

J a, J b

(w)δ (z − w) + kκ(J a, J b)1 ∂ wδ (z − w).

A relation of this type was encountered in physics as commutation relation of local fields in quantumfield theory.

We need to define the field T (z) of the infinitesimal operator T . Observe that the normallyordered product of fields a(z) and b(z), as defined previously in §6.2, is equivalent to the following :

(6.87) : a(z)b(z) :=1

2πi

z

R(a(w)b(z))(w − z)−1dw.

This integral can be interpreted as the difference of two contours around zero - C1 for |w| < |z| andC2 for |w| > |z| - and expanding (w − z)−1 in the respective regions in a Taylor series. This yields asplitting of : J a(z)J b(z) : into an antisymmetric part that depends on the structure constants and asymmetric part that depends on the (scaled) Killing form.

: J a(z)J b(z) : =m,n∈Z

p≥0

1

2πi

C2

w−1 z

w

pw−m−1z−n−1J amJ bn dw

− 1

2πi

C1

(−z−1)w

z

pz−n−1w−m−1J bnJ am dw

=

m,n∈Zδ −m−1≥0J amJ bn + δ m≥0J bnJ am

z−m−n−2

= m,n<0

J amJ bnz−m−n−2 + m,n≥0

J bnJ amz−m−n−2

first part I 1

ab(z)

+

m<0,n≥0

J amJ bnz−m−n−2 +

m≥0,n<0

J bnJ amz−m−n−2

second part I 2

ab(z)

.

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We analyze the the two parts :

I 1ab(z) =m,n<0

γ cabJ cm+n + J bnJ am

z−m−n−2 +

m,n≥0

−γ cabJ cm+n + J amJ bn

z−m−n−2

= I 1ba(z) +

k∈Z(−k − 1)γ cabJ ckz−k−2

= I 1ba(z) + γ cab∂ zJ c(z).

The part I 2ab(z) is clearly symmetric. Hence

: J a(z)J b(z) :=1

2γ cab∂ zJ c(z)

antisymmetric part

+ I 1ab(z) + I 2ab(z) − 1

2γ cab∂ zJ c(z)

symmetric part

=1

2γ cab∂ zJ c(z) +

κabl

T (z).

Consequently, define the operator T (z) to be the contraction of such fields with the form κ of g, i.e.,

(6.88) T (z) := κab : J a(z)J b(z) : .

Note that by the definition of normally ordered product, there is no ambiguity in the definition of T (z) and is independent of the basis chosen. This particular construction (6.88) of T (z) in termsof J a(z) is called the Sugawara construction . Observe that (6.88) is a generalization of the Casimirelement.

Remark In physical applications this field appears as the energy-momentum tensor of a two-dimensional conformal field theory.

Since g is simple, one may assume that J a forms an orthonormal basis with respect to thebilinear form κ. Then (6.84) implies

T (z) =: J a

(z)J a

(z) :=

1

2

l

a=1

Y (J a

−1J a

−1|0, z).

This field is generated by the element

T :=1

2

la=1

J a−1J a−1|0 ∈ V k(g).

We write T (z) =n∈Z T (n)z−n−1.

fff

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