The computation of the cohomology class is well-known in the context of central extension of the Witt algebra (see  ,  for more details) and conformal field theory (CFT) (see  ,  for more details). 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  we find that the form of the operator product expansion of the energy-momentum tensor is presented almost axiomatically as
In this article we compute analytically using ideas from CFT  and some tools from Kac’s conformal algebra  . In Section 2 we present the necessary background material on Lie algebras, their cohomology (in the finite-dimensional case), and central extensions of a Lie algebra by a one-dimensional complex vector space . Section 3 briefly introduces two-dimensional CFT. One aspect discussed in detail is the so-called energy-momentum tensor, which characterizes the two-dimensional CFT. Then in Section 4, we define the Witt algebra, which is an example of an infinite-dimensional Lie algebra, and discuss its central extension by to the Virasoro algebra. In Section 5, we compute the cohomology class analytically using CFT. In Section 5.1 we adapt results related to conformal algebra from Kac  (specifically sections 2.1-2.6 in  ) to obtain the operator product expansion of two local eigenfields of conformal weight and . In Section 5.2 we apply the results of Section 5.1 to the energy-momentum tensor and use this to compute the cohomology class. Finally, in the Conclusion and Future work we summarize the key results which lead to the construction of the Virasoro algebra, and we propose to investigate the algebra that may arise in the case is a monomial of non-zero degree.
2. Review of Key Ideas on Lie Algebras and Their Cohomology
2.1. Lie Algebras
Definition 2.1. (Lie algebra) A Lie algebra is a vector space over a field along with a bilinear map such that for all :
This bilinear map is called a Lie bracket.
Remark 2.1. The subscript is added to the bracket (i.e. ) to distinguish it from other bracket operations. If there is no potential confusion, the subscript is often omitted.
Property (2) is called the Jacobi identity. Applying bilinearity and property (1) to we obtain another property:
This is called skew-symmetry. If the characteristic of the field is not 2, then skew-symmetry implies property (2) as well. We define the dimension of a Lie algebra to be its dimension as a vector space.
A first example of a Lie algebra is the space of linear transformations on a finite-dimensional vector space V along with the Lie bracket operation defined as where is a composition, denoted . For this reason, the bracket operation is often called the commutator, and if then we say X and Y commute. Any vector space V can be considered a Lie algebra with the bracket operation for all . Such a Lie algebra is called abelian. More background on Lie algebras can be found in   .
2.2. 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 or its restriction on known as the Witt algebra, whose cohomology can be handled similarly with appropriate modifications.
2.2.1. Lie Algebra Cohomology with Complex Coefficients
Let be a finite-dimensional complex Lie algebra and let be a k-linear form. Such a k-linear form is called alternating if the following is true:
where . The set of all alternating k-linear forms is denoted by and is called the k-th cochain. Note that .
We recall that given , , and , we can define a product with the following properties:
We call this the wedge product or exterior product. This gives the structure of a ring.
Given , we define the coboundary operator for all as follows:
where and signifies that the element has been removed. If we define . We can use the coboundary operator to construct a long sequence, known as the Chevalley-Eilenberg Complex denoted by :
Remark 2.2. For simplicity we write if there is no chance of confusion.
Proposition 2.1. For , ,
Proof. We prove the claim by induction on p. For the case , choose and , then since is a scalar . Let us assume that the statement is true for , then choose and let . Since (applying the case ), then
Combining, the first two terms of the previous expression we have , hence . Therefore, Equation (6) is true if . The claim follows by linearity for any
Proposition 2.2. For all , .
Proof. We prove the claim by induction on k. If , then for any
By the Jacobi identity on , we get . Let the induction hypothesis be true for . Consider where and . Then by proposition 2.2.1, and . Once again, it follows by linearity that for all  .
If , then is called a k-coboundary. The set of all k-coboundaries is denoted by .
If , then is called a k-cocycle. The set of all k-cocycles is denoted by .
Given a k-coboundary , we know that for some . Applying the coboundary operator yields . It follows that , which implies that
Definition 2.2. (Singular cohomology) The kth singular cohomology with values in , , is defined by
Remark 2.3. If is an infinite-dimensional Lie algebra, we must consider continuous k-linear forms, obtained by topologizing the Lie algebra and . For example, let M be a smooth compact manifold and let be the Lie algebra of all smooth vector fields on M with the topology, then the corresponding cohomology is called the Gelfand-Fuchs cohomology. Details can be found in    .
2.2.2. Central Extensions and
Consider two complex Lie algebras and , and let where c is contained in the center of , i.e. for all . Consider the following short sequence
This sequence is called exact if . The splitting lemma states that if there exists a map such that , then
where I is some ideal contained in the center of .
The map is called a section of . Note that this result is a generalization of the rank-nullity theorem from linear algebra. If (9) and (10) hold for , then is called a central extension of by .
Theorem 2.1. The inequivalent central extensions of a Lie algebra by are classified by .
Proof. Let be a central extension of arising from the following short exact sequence:
where is the canonical projection and is a section of . For , let . Thus, using the Jacobi identity in . Hence, satisfies the 2-cocycle property. Suppose is another section, and note that for all , , thus or where . Given another bilinear form arising similarly from , we would like to show that belongs to the coboundary, i.e. :
where . Hence is a 2-cocycle.
Conversely, take a 2-cocycle which is a representative element of a cohomology class in , i.e. for all :
We can define a bracket on the vector space as follows
where . If is another bilinear form satisfying (11) and (12), then and define isomorphic Lie algebra structures on if and only if there exists a map such that
In the above construction, the Lie algebra is a central extension of by obtained by associating the bilinear form . This shows that corresponding to any element of we can associate an isomorphism class of a central extension of . Hence, we have shown that there is a one-to-one correspondence between the inequivalent central extensions of a Lie algebra by and  .
3. 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 , 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  ). In this paper we are interested in dimension since the Lie algebra of infinitesimal conformal transformations is infinite dimensional and has been investigated in complete detail by Belavin et al. in  . 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 , . Let . In conformal field theory, z and are considered independent complex variables. Thus the field on becomes . If
, i.e. depends only on z, then is said to be a chiral field. We thus simply write , which is holomorphic i.e. a power series in z. On the other hand, if , we call anti-chiral and write , which is anti-holomorphic i.e. a power series in .
We are interested in the infinitesimal conformal transformation ( ) with ( ) where
satisfying the Cauchy-Riemann conditions
Note that is simply notation.
Definition 3.1.  If a field transforms under any conformal transformation and as follows:
we call a primary field of conformal dimension . If not, we call a secondary field.
As an example, given a primary field and the infinitesimal conformal transformation as discussed above, we compute :
so that . Then:
Ignoring terms of order and in the above expression, we find that the primary field is transformed under the infinitesimal conformal transformation
For more details see  .
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   for details). Recall Nöether’s theorem which essentially states that for every continuous symmetry in a field theory there is an object called current ( ) which is conserved, i.e. using Einstein summation notation
where , . For more information, see   . Let denote the energy-momentum tensor. Then from  , under the infinitesimal conformal transformation the current is
Applying Nöether’s theorem yields
Since , the above expression can be rewritten as
or using Einstein summation notation,
Since this expression is true for all conformal transformations, in particular and , then which implies that the energy-momentum tensor is traceless (i.e. ).
We now wish to complexify our coordinates, & . We make the following association:
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 . We thus have that is chiral and is anti-chiral. We can write as a Laurent series as follows:
With a change of variables, we obtain the desired form of the energy-momentum tensor :
Remark 3.1. is an example of a secondary field.
4. The Witt Algebra
4.1. 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 is defined as follows:
with a basis given by
Remark 4.1. can be thought of as a vector field over .
Note that the basis of the Witt algebra can also be interpreted from a Laurent expansion of in the infinitesimal conformal transformation about   :
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
by the definition of the commutator. Similarly,
Adding these three expressions, we get
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 i.e. , the element of the basis becomes
Proof. Using the definition in 17 and the above value for restricted to , we get
4.2. Central Extension of the Witt Algebra
It can be shown that is one-dimensional, meaning that in the following exact short sequence:
the central extension is unique up to a constant. This unique central extension is known as the Virasoro algebra. It is spanned by . The vector c is called the central charge. The bracket operation on is defined by
where is some representative element of the cohomology class of . 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  ).
5. Computation of Cohomology Class Using Conformal Field Theory
This section is adapted from Victor Kac, who develops the theory in much more generality in  . For the sake of continuity in following along  , 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 , with Conformal Weights ,
Consider a formal field . Here the word “formal” indicates that we are not concerned with convergence. We also introduce the formal delta-function defined by
Given a rational function with poles only at , , and , let (resp. ) denote the power series expansion of R in the domain (resp. ). In particular
Using the above we can conclude that
Recall that the residue in z of a field is defined as
1) For any formal field ,
1) It is sufficient to check :
4) This is an application of Equation (22):
5) We again use Equation (22):
We want to know when a formal field
has an expansion of the form
It follows from Proposition 5.1 that
Let be the subspace consisting of formal -valued distributions for which the following series converges:
1) The operator is a projector, i.e. .
Remark 5.1. Recall that a complex function is holomorphic if in some neighborhood of its domain where .
3) Any formal field from is uniquely represented in the form:
where is a formal field holomorphic in z.
1) We want to show .
2) Suppose . Then
Thus all the coefficients of are zero for all . Thus is holomorphic. Conversely, if is holomorphic then clearly .
3) Since is a projector, . The claim follows.
Corollary 5.1. The null space of the operator of multiplication by , in is
Any element from (28) is uniquely represented in the form
Proof. Suppose . Then
Conversely, that lies in the null space of follows by Proposition (5.1) (5).
We sometimes write a formal field in the form
Proposition 5.3. If has the expansion (29) then
Definition 5.1. A field is said to be local if for some
Corollary 5.1 says that any local formal field has the expansion (29).
Definition 5.2. Two formal fields and are said to be mutually local, simply local, or a local pair if the formal field is local, i.e. if
Given a formal field , let
This is the only way to break into a sum of “positive” and “negative” parts such that We re-define the formal field using the “positive” and “negative” parts as follows,
With this new notation in hand we can show the following: The following are equivalent to 31:
1) , where
1) This is a clear result of Corollary (5.1).
2) By (1),
Using the bilinearity of the bracket operation, . Thus
The claim follows.
3) By Equation (34),
The other case is similar.
4) By (1), has the expansion 29, thus by proposition (5.3),
By bilinearity of the bracket,
Recall that . Replace n by . Then
Recall that denotes the power series expansion of in the domain . Thus assuming we can write proposition (5.5) (3) simply as
or just the singular part:
Equation (36) is called the operator product expansion (OPE) of for .
Let H denote the Hamiltonian, essentially a semi-postive definite self-adjoint operator.
Definition 5.3. A formal field is called an eigenfield for H of conformal weight if
We often write an eigenfield of conformal weight as
In this form the condition of being an eigenfield is equivalent to
Proposition 5.6. Suppose and are eigenfields of conformal weights and respectively. Then
1) is an eigenfield of conformal weight .
2) is an eigenfield of conformal weight .
1) Let . Then
We know . Then
2) Consider two eigenfields and of conformal weight and respectively. Thus
Since the Hamiltonian acts as a derivation, i.e. , then is an eigenfield of conformal weight .
Corollary 5.2. If and are mutually local eigenfield of conformal weights and , then in the OPE
all the summands have the same conformal weight .
Proof. Let and . We know
where is an eigenfield of conformal weight . Since the Hamiltonian acts as a derivation and and are eigenfields,
On the other hand,
Thus every term of is itself an eigenfield of conformal weight .
Proposition 5.7. Take to be local eigenfields of conformal weight resp., with OPE . Supposing is constant, then .
Since H is a semi-positive definite self-adjoint operator, its eigenvalues must be non-negative real numbers. Thus .
5.2. Computing Cohomology Class Using Operator Product Expansion of the Energy-Momentum Tensor
Note that the energy-momentum tensor is a local eigenfield of conformal weight  .
1) Let and be mutually local eigenfields for H both of conformal weights . Assume is constant. Then the singular part of the operator product expansion is of the form
where each summand is of conformal weight 4.
2) If we assume moreover that , , and then
1) From proposition 5.7 and the assumption, we obtain and . Then the singular part of the OPE looks like
Exchanging z and w in Equation (39) we get
Applying Taylor’s formula expanding about w, this becomes
Due to locality, Equations (40) and (39) are equal. Thus . The coefficient of in Equation (39) is , and in Equation (40) the coefficients of are . Then . Thus can be written as
Thus (up to a constant)
2) By proposition 5.5 (5),
This along with the assumptions show that .
We would now like to consider the commutator bracket operation
In conformal field theory, motivated by Equation (36), only makes sense if or . This leads us to define the radial ordering of two operators
Remark 5.2. In the physical theory, this radial ordering is related to the ordering of time. Thus
Substituting the OPE yields
To evaluate this expression, we must perform a Taylor expansion of about w:
We substitute this expansion:
We compute the residue by pairing terms that yield and finding the coefficients:
To calculate the integral, consider the following cases: if , then ; if , then . We can thus express the integral with the Kronecker delta . We finally conclude that
Thus the 2-cocycle representing the central extension of the Witt algebra can be rewritten by comparing with the Equation (4.3),
Since the OPE calculated in theorem (5.2) is unique up to a constant, we have our justification that , and thus the Virasoro algebra is the unique central extension of the Witt algebra.
6. Conclusions and Future Work
In this article we analytically computed the representative element of the cohomology class of by using the operator product expansion of the energy-momentum tensor and the commutator using integrals from standard complex variable theory. Note that in proposition (5.7) and in theorem (5.2) we made the assumption that the eigenfield is a constant in order to get the correct form of the commutator for obtaining the Virasoro algebra. In our future work we would like to investigate the case where is a monomial in w of appropriate degree and obtain the corresponding algebra. For example, if , it can be shown by reworking proposition (5.7) that ; hence the singular part of the corresponding operator product expansion is
We intend to rework proposition (5.2) and details therein along with the corresponding algebra obtained by computing the commutator in a future article.
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