Comparing (6.73) and (6.80) we obtain:
(−i)a+b+c
(a+b−c−1)!2πihhABCii= X
C′∈Wc
FABC′hhC′Cii(−i)2c (6.81) With the same considerations for hA[B, C]i we obtain:
(−i)a+b+c
(b+c−a−1)!2πihhABCii= X
A′∈Wa
FBCA′ hhA′Aii(−i)2a (6.82) The last two formulae allow us to find a new condition on the structure constantsFABC involving only 2-point amplitudes:
(a+b−c−1)!(−1)c X
C′∈Wc
FABC′hhC′Cii= (b+c−a−1)!(−1)a X
A′∈Wa
FBCA′ hhA′Aii (6.83) or
(a+b−c−1)!(−1)chhΓ∗(A, B)a+b−c−1, Cii= (b+c−a−1)!(−1)ahhA,Γ∗(B, C)b+c−a−1ii (6.84) There are two ways how to look at this condition: either one assumes a given quadratic form hh·,·ii, which amounts to fixing bases of the finite-dimensional reduced field spaces Va: then (6.83) is indeed an additional constraint on the structure constants FABC . Or one regards the reduced bracket (6.10) subject to the structure relations (6.7) and (6.67) as the primary structure: then (6.84) is an invariance condition on the quadratic form, in the same way as the invariance conditiong([X, Y], Z) =g(X,[Y, Z]) on a quadratic form on a Lie algebra. This invariant quadratic form on the reduced Lie algebra corresponds to the vacuum expectation functional on the original commutator algebra.
6.8 Axiomatization of chiral conformal QFT
The upshot of the previous analysis is a new axiomatization of chiral conformal quantum field theory. It consists of the three data:
• a graded reduced space of fields V =L
a∈NVa,
• a generalized Lie bracket Γ∗ =P
m≥0Γ∗m :V ×V →V
• and a quadratic form hh· ·ii:V ×V →R
These data should enjoy the features outlined before: Va are real linear spaces; the bracket is filtered: Γ∗(Va × Vb) ⊂ L
m≥0Va+b−1−m, and satisfies the graded symmetry (6.56) and generalized Jacobi identity (6.65); the quadratic form is symmetric, positive definite, respects the grading, and is invariant (6.84) with respect to the bracket.
Notice that the unitarity bound (absence of negative scaling dimensions) has been imposed through the specification of the reduced space V. Although the local intertwiner bases, and therefore also the coefficient matrices Y in the Jacobi identity do involve “intermediate” rep-resentations of negative dimensions (a+b −1−m may be < 0), these do not contribute to the present axiomatization because they multiply non-existent structure constants. Recall also that the possibly singular instances of the Jacobi identity have to be understood as explained in the end of Sect. 6.6.
One may impose further physically motivated constraints, e.g., the existence of a stress-energy tensor as a distinguished field T ∈ V2 whose structure constants FT AA take canonical values; or the generation of the entire reduced space by iterated brackets of a finite set of fields, formulated as a surjectivity property of the bracket.
As a simple example, one may consider the constraints on the structure constants for the commutator of two fieldsA, B of dimension one. The only possibility in this case is dimC = 1.
The generalized Jacobi identity just reduces to the classical Jacobi identity for the structure constants of some Lie algebra g. Likewise, the invariance property of the quadratic form becomes the classicalg-invariance of the quadratic formh(A, B) =hhABiiong. The positivity condition on the quadratic form implies that g must be compact, and that h is a multiple of the Cartan-Killing metric. In other words: one obtains precisely the Kac-Moody algebras as solutions to this part of the constraints. The quantization of the level is expected to arise by the interplay between the positivity condition with the higher generalized Jacobi identities.
Other approaches [Zamolodchikov, 1986; Bouwknegt, 1988; Blumenhagen et al., 1991] to the classification of “W-algebras” have, of course, exploited essentially the same consistency relations for a set of generating fields. Our focus here is, however, on the entire structure including all “composite” fields, and the possibility to formulate a deformation theory, to which we turn in the next chapter.
7 Cohomology and deformations of the reduced Lie algebra
In this chapter we will study the deformation theory of the reduced Lie bracket. The motivat-ing example for us was [Hollands, 2008], where deformations in the settmotivat-ing of OPE(operator product expansion) approach to quantum field theory on curved space–time were studied. We consider formal deformations, defined as a perturbative series, such that the reduced Jacobi identity is respected. Following the standard strategy from other deformation theories of alge-braic structures, we first construct a cohomology complex related to our deformation problem.
To construct this complex, which we call reduced Lie algebra cohomology complex, we will adapt the scheme used to define a Lie algebra cohomology to our case. The functions, which build the cochain spaces of our cochain complex, will possess a complicated symmetry property, a generalization of (anti-)symmetry, which we will define in the first section of this chapter and we will call it ZBε-symmetry.
We still have not calculated the cohomology groups of this complex, but we have shown that the non-trivial first order perturbations in the deformation theory belong to the second cohomology group and we have computed the obstruction operators to their integration. As our cohomology complex is obtained merely by stripping off the test functions from the field Lie algebra cohomology complex, we expect that it inherits all the nice properties of the Lie case, in particular that the obstructions to lift a perturbation to higher order lie in the third cohomology group. In such case we would be immediately able to relate the rigidity of the reduced Lie bracket to the content of the second cohomology group and the integrability of the elements of this group to the content of the third cohomology group.
7.1 Z
Bε-symmetry
The reduced bracket (6.54) obeys the symmetry rule (6.7). The reduced Jacobi identity (6.65) obeys the symmetry rule (6.66). A symmetry rule, generalizing the last two rules for structures with more arguments, will be the following:
Definition 7.1 (ZBε-symmetry). LetV be the reduced space as in Section 6.5 and let us con-sider the maps ω∗nB(·, ...,·)m1...mn−1 : V| ×...{z×V}
n
→V. Let an be the n-tuple of scaling dimen-sions ai, let Xi ∈Vai and let mn−1 will be the n-tuple (m1, ..., mn−1). Let ω∗nB(X1, ..., Xn)mn−1
be non-zero only for mi ≤Pn
s=ias−Pn−1
t=i+1mt−n+i. We will say that ω∗Bn(X1, ..., Xn)mn−1
are ZBε-symmetric if:
mn−1 as in Definition 6.4 and σin is the permutation {i1, ..., in} of the indices {1, ..., n}.
From now on we will use the notation P
in :=P
i16=...6=in
i1...in∈[1,...n]
Notation. We will be interested in those ZBε-symmetric maps for which B is the default bracket scheme as in Section 6.3. We will call such maps Zε-symmetric. From now on, whenever the label B stands for the default bracket scheme, we will just omit it.
Observation. It follows from the definition that:
ωB∗n(X1, ..., Xn)mbn−1
To show this, one uses:
ZB,aε is completely anti–symmetric in the arguments Xi(fi).
Proof. It follows directly from the definitions that:
ωn(X1(f1), ..., Xn(fn)) = X
7.2 Reduced Lie algebra cohomology