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2.1. The F4 Model - Spin(8) Gauge Group and 8 Dimensional Spacetime.

 To be physically realistic, the F4 model must contain spacetime, gauge bosons, and fermions. The physics should be described by a Lagrangian action of the form

ēM [ GB + SpF + GF + FP ],

where M is a spacetime base manifold, GB is a gauge boson term, and SpF is a term for spinor fermion interactions.

GF is the gauge-fixing term in the quantum Lagrangian, fixing a section of the principal fibre bundle of the gauge field over spacetime such that the covariant derivative on the principal fibre bundle splits into a horizontal part d and a vertical part s that is the nilpotent BRS transformation of the gauge group cohomology. The Fadeev-Popov ghost term FP comes from splitting the connection into a horizontal part that is the gauge bosons and a vertical part that is the ghost spin-1 fermion 1-form. A ghost is geometrically equivalent to a Maurer-Cartan 1-form.

The full quantum action is not invariant under the gauge group Spin(8), but is invariant under the nilpotent BRS transformations that define the cohomology Hopf algebra for the Lie algebra Spin(8). The classical part of the Lagrangian action,

ēM [ GB + SpF ] ,

is invariant under the gauge group Spin(8) and is now discussed in detail.

To start building the F4 model, begin by considering the Spin(9) subgroup of F4. The Spin(9) part is made up of Spin(8) (the adjoint representation) and the vector representation V8 of Spin(8). Since V8 is the base manifold, denoted by M8, and Spin(8) has 28 independent 8-curvature 2-forms, denoted by F8, it is natural to use the Spin(9) part of F4 to construct the first stage of the F4 model, a pure Spin(8) gauge theory with 8-dimensional base manifold.

The gauge bosons of a Spin(8) gauge theory are the Lie algebra infinitesimal generators of Spin(8). As discussed in Ch. V, §5 of Bröcker and tom Dieck8, simply connected compact Lie groups such as Spin(8) are in one-to-one correspondence with compact semisimple Lie algebras, which are the real Lie algebras with negative definite Killing form. By tensoring with C, there is a one-to one correspondence between compact semisimple Lie algebras and complex semisimple Lie algebras. The complex semisimple Lie algebras are classified by their root system (the reflection groups of which are Weyl groups), which in turn are classified by their Dynkin diagrams. As the root systems can be considered the most fundamental underlying structure of a Lie group, and as root systems most directly classify complex semisimple Lie algebras, the Spin(8) infinitesimal generators should fundamentally be considered to be complex. Therefore, the Spin(8) gauge bosons should be complex and their propagator amplitudes have a complex U(1) phase.

The most obvious choice for a pure gauge action is ēS8 - F8/\*F8 , with V8 = S8 = OP1 = Spin(9)/Spin(8), a compact manifold with local Spin(8) symmetry and octonionic structure.

Since F8 is a 2-form and *F8 is a 6-form, the 4-dimensional self-duality

F = *F cannot apply.

Working with spinor connections, Grossman, Kephart, and Stasheff9 and Landi10 show that the correct 8-dimensional self-duality relation is F8/\F8 = +*(F8/\F8) (self-duality) for half-spinor curvatures based on the S8+ half-spinor Spin(8) representation, and F8/\F8 = -*(F8/\F8) (antiself- duality) for half-spinor curvatures based on the mirror image S8- half-spinor Spin(8) representation. The self-duality can also be written as

*F8 [proportional to] + F8/\F8/\F8 , and the antiself-duality as *F8 [proportional to] -F8/\F8/\F8 . Since they are working with full spinor connections, their connections F8 can have both self-dual components F8+ based on S8+ half-spinors and antiself-dual components F8- based on S8- half-spinors.

Landi10 shows that, for full spinor connections F8 = F8+ + F8- on the sphere S8 (or any other sphere of dimension > 2), the field equations for the Yang-Mills Euclidean action -(1/2) ēS8 Tr F8/\*F8 are satisfied. The equations are d*F8 + A8/\*F8 - *F8/\A8 = 0 , where A8 is the spinor connection for the curvature F8 and * is the Hodge operator for the stereographic projection metric

g = (2/(1+x2))2 hmn dxmƒdxn for S8 in R9 with x2 = hmn xmxn . Landi10 also shows that for R8 with Euclidean flat metric, and for all other dimensions except 4, the Yang-Mills equations are not satisfied by the connections. That is to be expected, since Yang-Mills equations are conformally invariant only in 4 dimensions.

They also note that for V8 = S8, the half-spinor connection configurations are topological invariants with Euler number +1 for F8+ and -1 for F8-. The Euler number for the tangent bundle of S8 is 2.

The gauge boson part of the F4 model uses the V8 vector Spin(8) representation whose curvature is F8 for its connection, rather than the half-spinors S8+ and S8-. By Spin(8) triality automorphisms,

constructions for either S8+ or S8- are equivalent to those for V8.

Therefore, for the F4 model based on V8 the curvature F4 must be either purely self-dual or purely antiself-dual, and should not have both types of components. The F4 model action ēS8 - F8/\*F8 differs from the spinor action -(1/2)ēS8 Tr F8/\*F8 only by a factor of 2, so that the gauge boson part of the F4 model action is seen to be the Yang-Mills Euclidean action over S8 with self-duality relation F8/\F8 = *(F8/\F8) , or, equivalently,

*F8 [proportional to] + F8/\F8/\F8. The factor of 2 takes into account the relative magnitude of the spinor Euler numbers +1 and -1 with respect to the vector Euler number 2.


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