Revisiting the computation of cohomology classes of the Witt algebra using conformal field theory and aspects of conformal algebra

In this article, we revisit some aspects of the computation of the cohomology class of $H^2 ( \text{Witt}, \mathbb{C})$ using some methods in two-dimensional conformal field theory and conformal algebra to obtain the one-dimensional central extension of the Witt algebra to the Virasoro algebra. Even though this is well-known in the context of standard mathematical physics literature, the operator product expansion of the energy-momentum tensor in two-dimensional conformal field theory is presented almost axiomatically. In this paper, we attempt to reformulate it with the help of a suitable modification of conformal algebra (as developed by V. Kac), and apply it to compute the representative element of the cohomology class which gives the desired central extension. This paper was written in the scope of an undergraduate's exploration of conformal field theory and his attempt to gain insight on the subject from a mathematical perspective.


Introduction
The computation of the cohomology class H 2 (Witt, C) is well-known in the context of central extension of the Witt algebra and conformal field theory (CFT). However, we note that this computation in the opinion of these authors is unclear, especially in the mathematical physics literature dealing with CFT. In particular, in [1] we find that the form of the operator product expansion of the energy-momentum tensor is presented almost axiomatically as (1.1) In this article we compute H 2 (Witt, C) analytically using ideas from CFT and some tools from Kac's conformal algebra [2]. In section 2 we present the necessary background material on Lie algebras, their cohomology (in the finite-dimensional

Lie Algebra Cohomology
(Co)homology first arises in algebraic topology, where it involves associating a sequence of groups to a topological space in order to study various properties of the topological space. It also can be generalized to study other objects, such as Lie algebras. In this section we present the basic definitions and discuss the properties in the cohomology theory of finite-dimensional Lie algebras. However, in section 4 we discuss the infinite-dimensional Lie algebra of vector fields on C \ {0} or its restriction on S 1 known as the Witt algebra, whose cohomology can be handled similarly with appropriate modifications.

Lie algebra cohomology with complex coefficients
Let g be a finite-dimensional complex Lie algebra and let ω : g × ... × g = g k → C be a k-linear form. Such a k-linear form is called alternating if the following is true: ω(X 1 , ..., X i , ..., X j , ..., X k ) = −ω(X 1 , ..., X j , ..., X i , ..., X k ) (2.2) where X 1 , ..., X k ∈ g. The set of all alternating k-linear forms is denoted by C k (g, C) and is called the k-th cochain. Note that C 0 (g, C) := C. We recall that given η ∈ C p (g, C), θ ∈ C q (g, C), and ω ∈ C r (g, C), we can define a product ∧ with the following properties: • η ∧ θ ∈ C p+q (g, C), We call this the wedge product or exterior product. This gives C * (g, C) := ∞ k=0 C k (g, C) the structure of a ring.
Given ω ∈ C k (g, C), we define the coboundary operator ∂ k : C k (g, C) → C k+1 (g, C) for all k ≥ 1 as follows: where X 1 , . . . , X k+1 ∈ g andX n signifies that the element has been removed. If k = 0 we define ∂ 0 ω = 0. We can use the coboundary operator to construct a long sequence, known as the Chevalley-Eilenberg Complex denoted by C: Remark 2.2. For simplicity we write ∂ k = ∂ if there is no chance of confusion.
, ω ∈ C q (g, C), Proof. We prove the claim by induction on p. For the case p = 0, choose η ∈ C 0 (g, C) and ω ∈ C q (g, C), then since η is a scalar ∂(η ∧ ω) = η∂(ω) = η ∧ ∂(ω). Let us assume that the statement is true for η ∈ C p−1 (g, C), then choose θ ∈ C 1 (g, C) and let Combining, the first two terms of the previous expression we have The claim follows by linearity for any η ′ ∈ C p (g, C) Proof. We prove the claim by induction on k. If k = 1, then for any ω ∈ C 1 (g, C) By the Jacobi identity on g, we get ∂ 2 • ∂ 1 = 0. Let the induction hypothesis be true for k = q − 1. Consider η ′ = θ ∧ η where θ ∈ C 1 (g, C) and η ∈ C q−1 (g, C). Then by Once again, it follows by linearity that ∂ 2 (η ′ ) = 0 for all η ′ ∈ C q (g, C). [5] If ω ∈ Im ∂ k−1 , then ω ∈ C k (g, C) is called a k-coboundary. The set of all k-coboundaries is denoted by B k (g, C).
If ω ∈ Ker ∂ k , then ω ∈ C k (g, C) is called a k-cocycle. The set of all k-cocycles is denoted by Z k (g, C).
Given a k-coboundary ω, we know that ω = ∂ω ′ for some ω ′ ∈ C k−1 (g, C). Applying the coboundary operator yields ∂ω = ∂ 2 ω ′ . It follows that ∂ω = 0, which implies that Definition 2.2. (Singular cohomology) The k th singular cohomology with values in C, H k (g, C), is defined by . Remark 2.3. If g is an infinite-dimensional Lie algebra, we must consider continuous k-linear forms, obtained by topologizing the Lie algebra g and C. For example, let M be a smooth compact manifold and let g be the Lie algebra of all smooth vector fields on M with the C ∞ topology, then the corresponding cohomology is called the Gelfand-Fuchs cohomology. Details can be found in [6,7,8].

Central extensions and H 2 (g, C)
Consider two complex Lie algebras g andĝ, and let Cc := span{c} where c is contained in the center ofĝ, i.e. [X, c] = 0 for all X ∈ĝ. Consider the following short sequence This sequence is called exact if Im η = Ker π. The splitting lemma states that if there exists a map σ : g →ĝ such that σ • π = id g , then g ≃ g ⊕ Cc. (2.7) or equivalently g ≃ĝ/Cc (2.8)

Moreover
Cc ≃ I (2.9) where I is some ideal contained in the center ofĝ. The map σ is called a section of g. Note that this result is a generalization of the rank-nullity theorem from linear algebra. If (2.8) and (2.9) hold forĝ, thenĝ is called a central extension of g by Cc.
Theorem 2.1. The inequivalent central extensions of a Lie algebra g by Cc are classified by H 2 (g, C).
Given another bilinear form ω ′ arising similarly from σ ′ , we would like to show that ω − ω ′ belongs to the coboundary, i.e. ω − ω ′ = ∂ 1 (σ − σ ′ ): Conversely, take a 2-cocycle ω which is a representative element of a cohomology class in H 2 (g, C), i.e. for all X, Y, Z ∈ g: We can define a bracket on the vector spaceĝ = g ⊕ Cc as follows where α, β ∈ C. If ω ′ is another bilinear form satisfying (2.10) and (2.11), then ω and ω ′ define isomorphic Lie algebra structures on g ⊕ Cc if and only if there exists a map µ : g → C such that In the above construction, the Lie algebraĝ is a central extension of g by Cc obtained by associating the bilinear form ω. This shows that corresponding to any element of H 2 (g, C) we can associate an isomorphism class of a central extension of g. Hence, we have shown that there is a one-to-one correspondence between the inequivalent central extensions of a Lie algebra g by Cc and H 2 (g, C). [9]

A brief introduction to conformal field theory
A conformal field theory is a quantum field theory that is invariant under conformal transformations, which are transformations that preserve the angle between two lines. In a flat space-time with dimension D ≥ 3, the conformal algebra is the Lie algebra corresponding to the conformal group generated by globally-defined invertible finite transformations, which are translations, rotations, dilations, and special conformal transformations (for more details see [1] ). In this paper we are interested in dimension D = 2 since the Lie algebra of infinitesimal conformal transformations is infinite dimensional and has been investigated in complete detail by Belavin et. al. in [10]. Conformal field theory can be used to understand certain natural phenomena, and arises in string theory as well. It has long served as a meeting point between physics and mathematics, spurring progress in both fields. Consider the complexification of coordinates in In conformal field theory, z andz are considered independent complex variables. Thus the field φ(x 0 , x 1 ) on R 2 becomes φ(z,z). If ∂φ ∂z = 0, i.e. φ depends only on z, then φ is said to be a chiral field. We thus simply write φ(z), which is holomorphic i.e. a power series in z. On the other hand, if ∂φ ∂z = 0, we call φ anti-chiral and write φ(z), which is anti-holomorphic i.e. a power series inz.
We are interested in the infinitesimal conformal transformation Note thatf is simply notation.
we call φ(z,z) a primary field of conformal dimension (h,h). If not, we call φ(z,z) a secondary field.
As an example, given a primary field φ(z,z) and the infinitesimal conformal transformation as discussed above, we compute : . Then: Ignoring terms of order ǫ 2 andǭ 2 in the above expression, we find that the primary field φ(z,z) is transformed under the infinitesimal conformal transformation For more details see [1]. In our current approach, in order to study the central extension of the Witt algebra as discussed in section 4, we need to discuss the energy-momentum tensor, which is derived as follows (see [1,10] for details). Recall Nöether's theorem which essentially states that for every continuous symmetry in a field theory there is an object called current j µ (µ = 0, 1) which is conserved, i.e. using Einstein summation notation For more information, see [1,11]. Let T = T 00 T 01 T 10 T 11 denote the energy-momentum tensor. Then from [1], under the infinitesimal conformal transformation Applying Nöether's theorem yields Since ∂ µ T µν = 0, the above expression can be rewritten as T 00 ∂ 0 ǫ 0 + T 01 ∂ 0 ǫ 1 + T 10 ∂ 1 ǫ 0 + T 11 ∂ 1 ǫ 1 = 0 or using Einstein summation notation, Since this expression is true for all conformal transformations, in particular ǫ 0 = ǫx 0 and ǫ 1 = ǫx 1 , then (T 00 +T 11 )ǫ = 0 which implies that the energy-momentum tensor is traceless (i.e. T 00 + T 11 = 0). We now wish to complexify our coordinates, We make the following association: Revisiting the comp. of cohom. classes using CFT 9 From above, since the energy-momentum tensor is traceless, we have We now investigate the chirality of the energy-momentum operator: It can be similarly shown that ∂ z Tzz = 0. We thus have that T (z) is chiral and T (z) is anti-chiral. We can write T (z) as a Laurent series as follows: With a change of variables, we obtain the desired form of the energy-momentum tensor : Remark 3.1. T (z) is an example of a secondary field.

Construction of the Witt algebra
We now begin our application of the topics previously discussed with a specific Lie algebra: Definition 4.1. (Witt algebra) The Witt algebra over C * := C \ {0} is defined as follows: with a basis given by Remark 4.1. L j can be thought of as a vector field over C * .
Note that the basis of the Witt algebra can also be interpreted from a Laurent expansion of ǫ(z) in the infinitesimal conformal transformation f (z) = z + ǫ(z) about z = 0 [1,12]: We define the following commutator over the Witt algebra Proposition 4.1. The commutator defined above is a Lie bracket Proof. In order to be a Lie bracket, the commutator must be skew-symmetric and satisfy the Jacobi identity. Skew-symmetry is relatively easy to show: The Jacobi identity, on the other hand, is not difficult per se, but rather tedious. We wish to show the following: Examining the first term yields d dz A careful glance shows that this vanishes to zero, meaning Because [, ] is skew-symmetric and satisfies the Jacobi identity, it is a Lie bracket and therefore the Witt algebra is a Lie algebra.
Restricting the vector field to S 1 i.e. z = e iθ , the element of the basis L n = −z n+1 d dz becomes Proof. Using the definition in 4.1 and the above value for L n restricted to S 1 , we get

Central extension of the Witt algebra
It can be shown that H 2 (Witt, C) is one-dimensional, meaning that in the following exact short sequence: where ω is some representative element of the cohomology class of H 2 (Witt, C).
In the next section we will compute this cohomology class using standard results from Kac, which in the mind of these authors fill up the gap that seems to exist in physics literature (see for example [1]).

Computation of cohomology class using conformal field theory
This section is adapted from Victor Kac, who develops the theory in much more generality in [2]. For the sake of continuity in following along [2], we use much of the same notation. However, we introduce the term eigenfield for the Hamiltonian H of conformal weight ∆ (see definition 5.3) in our discussion.

5.1.
Operator product expansion of two eigenfields a(z), b(w) with conformal weights ∆, ∆ ′ Consider a formal field a(z, w) = m,n∈Z a m,n z m w n ∈ C[z, z −1 , w, w −1 ]. Here the word "formal" indicates that we are not concerned with convergence. We also introduce the formal delta-function δ(z − w) defined by Given a rational function R(z, w) with poles only at z = 0, w = 0, and |z| = |w|, let i z,w R (resp. i w,z R) denote the power series expansion of R in the domain |z| > |w| (resp. |w| > |z|). In particular Using the above we can conclude that Recall that the residue in z of a field f (z) = n∈Z f n z n is defined as Res z a(z) = f −1 Proposition 5.1.
(1) For any formal field f (z) ∈ C[[z, z −1 ]], (1) It is sufficient to check f (z) = n∈Z az n : We want to know when a formal field has an expansion of the form It follows from Proposition 5.1 that c n (w) = Res z a(z, w)(z − w) n (5.6) Let C[[z, z −1 , w, w −1 ]] 0 be the subspace consisting of formal C-valued distributions a(z, w) for which the following series converges: Proposition 5.2.
Remark 5.1. Recall that a complex function f (z) is holomorphic if in some neighborhood of its domain f (z) = ∞ n=0 a n z n where a i ∈ C. (3) Any formal field a(z, w) from C[[z, z −1 , w, w −1 ]] 0 is uniquely represented in the form: where b(z, w) is a formal field holomorphic in z. Proof.
Any element a(z, w) from (5.9) is uniquely represented in the form ] lies in the null space of (z − w) N follows by Proposition (5.1) (5).
We sometimes write a formal field in the form a(z) = n∈Z a n z −n−1 , a(z, w) = m,n∈Z a m,n z −m−1 w −n−1 (5.11) Here a n = Res z a(z)z n . Given a formal field a(z) = n∈Z , let a(z) − = n≥0 a n z −n−1 , a(z) + = n<0 a n z −n−1 .
This is the only way to break a(z) into a sum of "positive" and "negative" parts such that (∂a(z) ± ) = ∂(a(z) ± ) We re-define the formal field a(z)b(w) using the "positive" and "negative" parts as follows, With this new notation in hand we can show the following: Proposition 5.5. The following are equivalent to 5.12: (2) By (1), Using the bilinearity of the bracket operation, The claim follows. Recall that c j (w) = n∈Z c j n w −n−1 . Replace n by k + j − m. Then Recall that i z,w 1 (z−w) j+1 denotes the power series expansion of 1 (z−w) j+1 in the domain |z| > |w|. Thus assuming |z| > |w| we can write proposition (5.5) (3) simply as or just the singular part: (1) ∂ z a is an eigenfield of conformal weight ∆ + 1.