The Virasoro Algebra and Some Exceptional Lie and Finite Groups

We describe a number of relationships between properties of the vacuum Verma module of a Virasoro algebra and the automorphism group of certain vertex operator algebras. These groups include the Deligne exceptional series of simple Lie groups and some exceptional finite simple groups including the Monster and Baby Monster.


Introduction
This paper is based on a talk given at the Lochlainn O'Raifeartaigh Symposium on Non-Perturbative and Symmetry Methods in Field Theory, Budapest, 2006. We describe a number of relationships between the vacuum Verma module of a Virasoro algebra and the Deligne exceptional series of Lie algebras and also some exceptional finite groups such as the Monster and Baby Monster. The setting in which this is explained is the theory of Vertex Operator Algebras (VOAs) or chiral conformal field theory. In particular, we construct certain Casimir vectors that are invariant under the VOA automorphism group. If these vectors are elements of the vacuum Verma module then the VOA Lie or Griess algebra structure is constrained. This is an idea originally introduced by Matsuo for Griess algebras [11] and further developed for Lie algebras in [12]. Our approach is more general with weaker assumptions and is based on a consideration of a certain rational correlation function. The constraints on the Lie or Griess algebra structure arise from an analysis of the expansion of this correlation function in various domains. The paper contains a quite elementary description of the main ingredients where many of the detailed proofs are omitted. These will appear elsewhere [15].

The Virasoro algebra, Verma modules and the Kac determinant
We begin with a number of basic definitions and properties for Virasoro algebras e.g. [9]. The Virasoro Algebra Vir with central charge C is given by The Vacuum Verma Module V (C, 0) for Vir is defined as follows. Let 1 ∈V (C, 0) denote the vacuum vector where L n 1 = 0, n ≥ −1.
In general we may consider a Verma Module V (C, h) defined in terms of highest weight vector v ∈ V (C, h) obeying v is called a primary vector of level h. Then V (C, h) is generated by the Virasoro descendents of v. However, we are primarily interested only in the vacuum module V (C, 0) here. V (C, 0) is an irreducible module provided no descendent vector is itself a primary vector. The irreducibility of V (C, 0) can be established by considering the Kac determinant defined as follows. Define a symmetric bilinear form ·, · on V (C, 0) with for u, v ∈ V (C, 0) and normalization 1, 1 = 1. Note that u, v = 0 for u, v of different Virasoro level. Thus we consider the level n Gram matrix for vacuum descendents u, v ∈ V (n) (C, 0). Then V (C, 0) is irreducible iff the Kac determinant det M (n) = 0 (with a similar formulation for V (C, h)) [9].

Some exceptional group numerology
Let us consider the prime factors of the Kac determinant det M (n) for level n ≤ 10 for particular values of C. We observe some coincidences with properties of a number of exceptional Lie and finite groups. Later on we will explain the underlying reason for these coincidences and obtain many other relationships with the Virasoro structure.
Deligne's exceptional Lie algebras A 1 , A 2 , G 2 , D 4 , F 4 , E 6 , E 7 , E 8 . This set of simple Lie algebras has been shown recently to share a surprising number of representation theory properties in common [2]. For example, the dimension of the adjoint representation d of each of these algebras can be expressed in terms of the dual Coxeter number h ∨ for the given algebra (as originally found by Vogel): Let us compare d to the independent factors C and 5C + 22 of the level 4 Kac determinant det M (4) for particular values of C: Table 1.
The first row of Table 1 shows h ∨ whereas the second row shows the prime factorization of d for each algebra. The third row shows the prime factorization for particular values of C (where C is the rank of the algebra for the simply-laced cases A 1 , A 2 , D 4 , E 6 , E 7 , E 8 ). Notice that each prime divisor of d is a prime divisor of the numerator of either C or 5C + 22 and hence the numerator of det M (4) .
Some prime divisors of the order of a number of exceptional finite groups are also related to the Kac determinant factors. We highlight three examples.
The Monster simple group M. The classification theorem of finite simple groups states that a finite simple group is either one of several infinite families of simple groups (e.g. the alternating groups A n for n ≥ 5) or else is one of 26 sporadic finite simple groups. The largest sporadic group is the Monster group M of (prime factored) order |M| = 2 46 · 3 20 · 5 9 · 7 6 · 11 2 · 13 3 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71.
The Baby Monster simple group B. The second largest sporadic group is the Baby Monster group B of order The prime divisors 2, 3, 5, 23, 31, 47 of |B| are divisors of the numerator of the independent factors of det M (6) for C = 23 1 2 : Table 3.

Axioms
The observations made in the last section can be understood in the context of Vertex Operator Algebras (VOAs) [1,8,13]. The basic idea is that the groups appearing above arise as automorphism (sub)groups of particular VOAs with the given central charge C. In this section we review the relevant aspects of VOA theory required in the subsequent sections. A Vertex Operator Algebra (VOA) consists of a Z-graded vector space V = k≥0 V (k) with dim V (k) < ∞ and with the following properties: with component operators (modes) a n ∈ EndV such that z is a formal variable here (but is taken as a complex number in physics). Note also that we are employing "physics modes" in (4.1).
The vacuum vector has vertex operator so that 1 n = δ n,0 .
• Virasoro subalgebra. There exists a conformal vector ω ∈ V (2) with where the modes L n form a Virasoro algebra (2.1) of central charge C. The Z-grading is determined by L −1 acts as a translation operator with • Locality. For any pair of vertex operators we have for sufficiently large integer N These axioms lead to the following basic VOA properties: e.g. [8,13].
Translation. For any a ∈ V then for |y| < |x| (i.e. formally expanding in y/x) In particular, the zero mode b 0 is a linear operator on V (m) . Similarly, for all m and any primary vector b ∈ V (h) , we have the familiar property from conformal field theory

The Li-Zamolodchikov metric
Assume that V (0) = C1 and L 1 v = 0 for all v ∈ V (1) . Then there exists a unique invariant bilinear form ·, · where for all a, b, c ∈ V [4, 10] and with normalization 1, Choosing a ∈ V (k) , n = k and b = 1, (4.9) implies for quasi-primary c c k a = (−1) k a, c 1. ·, · is symmetric [4] and, furthermore, is non-degenerate iff V is semisimple [10]. We call such a unique non-degenerate form the Li-Zamolodchikov or Li-Z metric. In particular, considering the Li-Z metric on the vacuum Virasoro descendents V (C, 0) ⊂ V implies that the Kac determinant det M (n) = 0 for each level n.

Lie and Griess algebras
Lie algebras. Let us consider a number of relevant subalgebras of a VOA V with Li-Z metric. Suppose that dim V (1) > 0 and define for all a, b ∈ V (1) where the second equality follows from skew-symmetry (4.4). We may thus define a bracket [a, b] = ad (a)b which satisfies the Jacobi identity so that V (1) forms a Lie algebra. Furthermore, the Li-Z metric a, b on V (1) is an invariant non-degenerate symmetric bilinear form with 2 Griess algebras. Suppose that dim V (1) = 0 and consider a, b ∈ V (2) . Skew-symmetry implies a 0 b = b 0 a so that we may define to form a commutative non-associative algebra on V (2) known as a Griess algebra. The Li-Z metric a, b on V (2) is an invariant bilinear form with

The automorphism group of a VOA
The automorphism group Aut (V ) of a VOA V consists of all elements g ∈ GL(V ) for which gY (a, z)g −1 = Y (ga, z), with gω = ω, the conformal vector. Thus the L 0 grading is preserved by Aut (V ) and every Virasoro descendent of the vacuum is invariant under Aut (V ). Furthermore, the Li-Z metric is invariant with for all a, b ∈ V and g ∈ Aut (V ). For VOAs with dim V (1) > 0, then Aut (V ) contains continuous symmetries g = exp(a 0 ) generated by elements of the Lie algebra a ∈ V (1) . VOAs with dim V

Quadratic Casimirs
We now discuss the relationship between the structure of the Lie algebra V (1) and the Virasoro algebra described earlier. We initially follow a technique due to Matsuo [11,12] for constructing a Casimir vector λ (n) ∈ V (n) which is Aut (V ) invariant. The relationship with the Virasoro algebra follow provided λ (n) ∈ V (n) (C, 0), the vacuum Virasoro descendents of level n. This occurs, for example, if V (n) contains no Aut (V ) invariants apart from the elements of V (n) (C, 0). Here we describe a new general method based on weaker assumptions for obtaining various constraints on V (1) by considering the expansion in various domains of a particular correlation function [15].

Rational correlation functions and the V (1) Killing form
Let a, b ∈ V (1) and consider the following correlation function Then we have F (a, b; x, y) is a rational function where G(a, b; x, y) is bilinear in a, b and is a homogeneous, symmetric polynomial in x, y of degree 4.
Proof . The bilinearity of G in a, b is obvious. Locality (4.2) implies that F (a, b; x, y) is of the form (5.4) [4,13] where G(a, b; x, y) is clearly symmetric. The degree and homogeneity follow from (4.1) and (4.7).
It is convenient to parameterize G(a, b; x, y) in terms of 3 independent coefficients P (a, b), Q(a, b), R(a, b) as follows F (a, b; x, y)) in x−y y we find

Expanding the rational function
whereas expanding in −y x−y we find From the corresponding VOA expansions, P , Q, R may be computed in terms of the Li-Z metric a, b and the Lie algebra Killing form as follows: Proof . Associativity (4.5) implies we may expand in (x − y)/y to obtain The leading term of (5.7) is determined by λ (0) 0 = −d so that comparing with (5.5) we find P (a, b) = −d a, b . Note that the next to leading term automatically vanishes since λ (1) 0 = 0. We may alternatively expand F (a, b; x, y) in −y/(x − y) to find (respectively using skew-symmetry (4.4), translation (4.3), invariance of the Li-Z metric (4.9) and that a is primary (2.2)). Comparing to the leading term of (5.6) we find using (4.10) that Comparing the next to leading terms of (5.6) and (5.8) and using (4.11) and (4.12) we find We next show that if λ (2) is a vacuum Virasoro descendent then the Killing form is proportional to the Li-Z metric: Proof . If λ (2) ∈ V (2) (C, 0) then (5.2) implies λ (2) 0 = − 2d C L 0 . The n = 2 term in (5.7) is thus − 2d C a, b . Comparing to (5.5) we find and hence the result follows from Proposition 2.
Since the Li-Z metric is non-degenerate, it immediately follows from Cartan's condition that The Killing form (5.9) has previously arisen in the literature from considerations of modular invariance in the classification of V (1) for holomorphic self-dual VOAs in the work of Schellekens [14] for central charge C = 24 and Dong and Mason [3] for C = 8, 16, 24. A similar result also appears in [12] also based on a Casimir invariant approach.
We can repeat results of [14,3] concerning the decomposition of V (1) into simple components for d = C. Corollary 1 implies V (1) = g 1,k 1 ⊕ g 2,k 2 ⊕ · · · ⊕ g r,kr , where α i is a long root of g i,k i (so that (a, b) i = − a, b /k i defines a non-degenerate form on g i,k i with normalization (α i , α i ) i = 2). The dual Coxeter number h ∨ i for g i,k i is then found from the Killing form K(h α i , h α i ) = 4h ∨ i . Hence (5.9) implies that for each simple component This implies that a finite number of solutions for (5.10) exist for any pair (C, d).

Deligne's exceptional Lie algebras
We next consider the expansion (5.8) of F (a, b; x, y) to the next leading term assuming that λ (4) is also a vacuum Virasoro descendent so that (5.3) holds. This results in one further constraint on P , Q, R leading to [15] Proposition 4. Suppose that λ (n) ∈ V (n) (C, 0) for n ≤ 4. Then Note that d(C)necessarily contains the independent factors of the level 4 Kac determinant det M (4) since it must vanish for C = 0, −22/5 for which the construction of λ (4) is impossible.
For positive rational central charge C then d(C) is a positive integer for only 21 different values of C. The remarks subsequent to Corollary 1 imply that the possible values are further restricted and that V (1) must be one of the Deligne exceptional simple Lie algebras [15]:  Table 1 are now obvious since if a prime p divides d(C) then p must divide one of the factors C or 22 + 5C of det M (4) . We note that results similar to Propositions 4 and 5 also appear in [12] but are based on a number of further technical assumptions.
We conclude this section with a result concerning the constraints arising from higher level Casimirs being vacuum descendents [15]:

Griess algebras and Virasoro descendents
We can now repeat the approach taken in the last section for a VOA with a Li-Z metric with dim V (1) = 0 so that V (2) defines a Griess algebra. This was the original case considered by Matsuo [11] but was based on stronger assumptions (e.g. assuming the existence of a proper idempotent or that Aut (V ) is finite). We give a brief description of the constraints arising from quadratic Casimirs being Virasoro descendents using a similar approach to the last section. Detailed proofs will appear elsewhere [15].
Consider the correlation function for a, b ∈V (2) Then we have where G(a, b; x, y) is bilinear in a, b and is a homogeneous, symmetric polynomial in x, y of degree 8.
In this case G(a, b; x, y) is determined by 5 independent coefficients. We use associativity (4.5) as in (5.7) to expand in (x − y)/y so that Similarly to (5.8), we expand in −y/(x − y) to obtain F (a, b; x, y) = y −4 a, b + 0 + T rV (2) Analogously to Propositions 2 and 3 and employing methods of [3] we find Proposition 8. Suppose that µ (n) ∈ V (n) (C, 0) for n ≤ 4. Then G(a, b; x, y) is given by
d 2 and C are related by the next Virasoro constraint in analogy to Proposition 4: Proposition 9. Suppose that µ (n) ∈ V (n) (C, 0) for n ≤ 6. Then The formula for d 2 (C) + 1 previously appeared in [11] subject to further assumptions. The appearance of the independent factors of det M (6) in d 2 (C) follows as before. (The absence of a C factor follows from the assumed decomposition V (2) = Cω ⊕V (2) ). If a prime p divides d 2 (C) then p must divide (at least) one of the independent factors 5C + 22, 2C − 1 or 7C + 68 of det M (6) . This explains many of the prime divisor properties (but not all) for Tables 2, 3 and 4. There are exactly 36 positive rational values for C for which d 2 (C) is an integer. The simplicity of the Griess algebra can be expected to further restrict these possibilities. A full description of these "exceptional Griess algebras" and their possible realization in terms of VOAs would be of obvious interest [15].
All the factors of det M (10) appear in p(C) explaining the remarks concerning prime divisors in Table 2 where d 3 (24) = 21296876 = 2 2 · 31 · 41 · 59 · 71 as obtains for the Moonshine Module. Finally, we also find