This section discusses a conjecture that the extraordinary symmetries of F4 might lead to ultraviolet finiteness of the 8-dimensional F4 model. This section is only descriptive and conjectural, but is included because ultraviolet finiteness in the 8-dimensional F4 model might solve the cosmological constant problem that is observed in our 4-dimensional spacetime. It is important to describe avenues of future research that seem promising.
In 8 dimensions, the F4 model is an octonionic generalization of the supersymmetric s-models for N=2 supersymmetry (Kähler base manifold) and for N=4 supersymmetry (hyperkähler base manifold). Supersymmetric s-models, for N=2 supersymmetry with Kähler base manifold and for N=4 supersymmetry with hyperkähler base manifold, are on-shell ultraviolet finite to all orders in perturbation theory.14 The supersymmetry requirement is used to place restrictions on the fermionic structure of the theories.
In particular, N=2 supersymmetry restricts the fermionic structure to have complex structure with U(n) holonomy, and N=4 supersymmetry restricts the fermionic structure to have quaternionic structure with Sp(n) holonomy.
For an irreducible base manifold, the requirement of complex structure means that there can be at most 2 conserved spinor charges and the requirement of quaternionic structure means that there can be at most 4 conserved spinor charges.
The extension of the supersymmetric s-model structure to octonions was suggested by Kugo and Townsend15 and verified by Sudbery16 using ideas from Ramond17.
The F4 model is based on the fibrations
Spin(8) -> Spin(9) -> S8 (holonomy Spin(8)) and
Spin(9) -> F4 -> OP2 (holonomy Spin(9)) ,
so the F4 Lie algebra has the splitting
F4 = L(S8) + Spin(8) + L(OP2), where
L(S8) = O, the linear tangent space to S8, is spacetime and
L(OP2) = O + O, the linear tangent space to OP2, is the fermion spinor space.
By the fibrations G2 -> Spin(7) -> S7 and Spin(7) -> Spin(8) -> S7, the gauge boson Lie algebra Spin(8) has the splitting
Spin(8) = L(S7) + G2 + L(S7), where
L(S7) = O0, the linear tangent space to S8, is the imaginary octonions.
The F4 model is not directly supersymmetric, but it does have a symmetry between gauge bosons and fermion particles and antiparticles that is clear in its 8-dimensional formulation.
The 28 gauge bosons are O0 + G2 + O0 . Since the gauge boson term of the F4 model 8-dimensional Lagrangian is F/\*F µ F/\F/\F/\F, the dimension of F is dimF = 8/4 = 2. Since F = ¶A and dim¶ = 1, the dimensionality of A is dimA = 1.
The 16 fermion particles and antiparticles are O + O , but each particle-antiparticle pair is only one type of particle, so there are only 8 types of fermions, and the octonionic structure of the F4 model gives 8 conserved spinor charges (neutrino, red up quark, blue up quark, green up quark, red down quark, blue down quark, green down quark, and electron). Since the spinor fermion term of the F4 model 8-dimensional Lagrangian is `S8± g¶ S8± , the dimensionality of each spinor fermion (particle + antiparticle) in terms of mass is (8 - 1)/2 = 7/2 . That dimensionality, 7/2, is the ratio of the dimension (28) of the adjoint representation of Spin(8) to the dimension (8) of the vector space of Spin(8) connections A over V8.
The same formula results in 4-dimensional spinor fermion fields having dimension in terms of mass of (dim(Spin(4))/4) = 3/2 .
Therefore the 28 gauge bosons O + G2 + O , each with dimension 1, have 28 dimensions, and the 8 spinor fermions O , each with dimension 7/2, also have 28 dimensions.
The triality automorphism of F4 gives a symmetry between the
8 spinor fermions O and the 8-dimensional spacetime O.
The triality automorphism of F4 establishes an effective supersymmetry between the 28 dimensions of gauge boson fields and the 28 dimensions of spinor fermions fields.
Conjecture: The 8-dimensional F4 model is a continuum model (continuous spacetime base manifold) of massless gauge bosons and fermions. It is ultraviolet finite due to effective boson-fermion supersymmetry, with the quantum theory defined by canonical quantization. (Path integral sum over histories quantum theory is not well defined for continuum models.)