Kac-Moody algebra and sporadic and monster groups

[Lawrence Crowell]

I want to expand further the role the monster group has, or potentially has, in physics. There is an elegant chain leading from Lie algebra theory to the Klein j-invariant with vertex functions and scattering and then leading up the Fischer-Griess group. The monster may be the smallest part of a tower of ever larger groups that in effect maintain symmetry and conservation of the vacuum. This in effect I think maintain “nothing.” 

Lie algebras are generators of groups. These groups are elements g = e^A or e^{iA}, depending on whether you are a mathematician or a physicist, where A is a matrix in an algebra. The algebra generates the group. The matrices are A = a_{ij} for i,j from 1 to n to define n×n Cartan matrices. 

a_{ii} = 2, a_{ij} ∈ ℤ and ≤ 0 and a_{ij} = a_{ji}.

There is an associated Kac-Moody algebra over ℂ with the 3 sets of Chevalley generators e_i, f_i, and h_i with the commutation [x, y] = xy - yx relationships

[e_i, f_i] = h_i, [e_i, f_j] = 0, for i \ne j

[h_i, e_i] = a_{ij}e_j, [h_i, f_j] = a_{ij}f_j

(ad e_i)^{1-a_{ij}}e_j = 0, (ad e_i)^{1-a_{ij}}f_j = 0, for i \ne j.

The Lie algebra g(A) is finite only if the matrix A is positive definite. The Kac-Moody algebra K(A). If the principal minors of these matrices are positive this defines a finite dimensional semi-simple algebra over ℂ. Negative minors define a Kac-Moddy algebras that are infinite dimensional versions of Lie algebras.

The Kac-Moody algebra has integrable highest weight representations. The set Λ = {λ_1, λ_2, …, λ_n} is the highest weight representation π_A of a Kac-Moody algebra. This is a feature shared with the Virasoro algebra. This highest weight representation exists on a vector space V(Λ) so for any v ∈ V(Λ)

π_A(e_i)v = 0, π_A(h_i)v = λ_iv.

The π_A are all irreducible finite dimensional representations of a finite dimensional algebra. The matrices A are positive definite Cartan matrices with the standard Chevalley generators E_i, F_i, and H_i with commutators above. The matrices are of the form

A^(1) = 

|2      a_{01} a_{02} ... a_{0n}|

|a_{10}                                      |

|  .                                                |

|  .                      A                       |

|  .                                                |

|a_{n0}                                      |,

which is a positive semi-definite n+1, n+1 matrix. These matrices plus affine matrices define a graded algebra with the following commutators

[x^m, x^n] = [x, y]^{m+n} + mcδ_{m,-n},

Where c is a central extension. This resembles the Kac-Moody algebra. The commutators are established for 

e_0 = E_0, f_0 = F_0, h_0 = c – H_0

e_i = E_i, f_i = F_i, h_i = H_i,

and follow the commutation of Cartan matrices above. 

This means the group g(A) = ℂ[z,z^{-1}]⊗ℂg + cℂ. The elements form a commutative associative algebra for V = S(ℌ)⊗_ℂℂ[A] which with the order of multiplication defines the vertex operators

X(γ,z) = exp(sum_j (z^j/j)γ(-j))exp(-sum_j (z^{-j}/j)γ(j))e^γe^{γ(0)}

for z ∈ C*. We may then expand this in powers of z X(γ,z) = sum_n X(γ)z^{-n-1} to obtain a sequence of operators on V. Now for g and ℌ the simple Lie algebras with Cartan matrices A, D, E such that we have the dual ℌ and ℌ * in the product form 〈⋅|⋅〉, or as mathys have it  (⋅,⋅) and Δ = {α ∈ Q: 〈α|α〉 = 2} the root system, the standard Cartan definition a_{ii} = 2. We then choose a multiplicative function k(α|α) = (-1)^{〈α|α〉/2}. For gamma in Q there exists the operator c_γ such that c_γ(f⊗e^β)= f(γ|beta)f⊗e^β. The g = g⊕sum_{α ∈ Δ}ℂE_alpha and the commutation relationships are

[h, h] = 0, [h, E_α] = (α,h)E_α, 

[E_α, E_β] = 0, if α + β ∉ Δ∪{0}

[E_α, E_-α] = -α, [E_α, E_β] = k(α,β)

The representation of g(A^(1)) is then defined on V by the formula

π(u^(n)) = u(n) for u ∈ ℌ.

This is the basic construction of the vertex operators in string theory. The explicit representation of elements of a Kac-Moody algebra with the central extension is 

L_i = sum_j g_{αβ}A^α_{i_j}A^β_{-j},

such that L_i obeys the Virasoro algebra [L_i, L_j] = (i – j)L_{i+j} + (1/12)δ_{i.-j}. We may think of the operators for a Lie algebra as having an additional index for the ladder operation for the vibration on a bosonic string. 

The bosonic string exists in 26 dimensions. The anomaly cancellation of it is in 26 dimensions. The Virasoro algebra can be represented by the elements L_n = e^{inθ}∂/∂θ. Some elementary calculus illustrates how [L_m, L_n] = (m – n)L_{m+n}. This is called the Witt algebra, and it does have a degeneracy problem. For a large number of m and n you get the same commutator.  So we impose a kernel so that 

[L_m, L_n] = (m – n)L_{m+n} + C(m)δ_{m+n}

We can use the Jacobi identity [L_n, [L_k, L_m]] + [L_k, [L_m, L_n]] + [L_m, [L_n, L_k]] = 0 that equals

(n – m)C(k) + (m – k)C(n) + (k – n)C(m) = 0.

Then for k = 1 and m+n = -1 we can find a recursion relationship where upon we find C(m) = am^3 – bm, Now if we look at the action of [L_1, L_{-1}] on the vacuum 〈0|[L_1, L_{-1}] |0〉 = 0. For [L_2, L_{-2}] and expanded in a Kac-Moody form we have 〈0|[L_1, L_{-1}] |0〉 = ½g_{αβ} g^{αβ}  = D/2 for D the dimension of the space. So now what is D, the dimension of the space? The operator L_{-2} + 3/2L_{-1}^2 on the state |0;p〉 gives 

(L_{-2} + 3/2L_{-1}^2)|0;p〉 = (D – 26)/2 

and this is zero. So, the space or spacetime is 26 dimensions. This is the size of the basic bosonic string spacetime. 

This mathematics leads into the system of special function. These include the Jacobi θ-function, the Dedekind η-function and in particular the Klein j-invariant function. This then takes us into the domain of the sporadic groups. This leads up to the strangest and largest of them all, the Fischer-Griess monster group.

LC