It is proved that fermions can acquire the mass through the additional non-integrable exponential factor. For this propose the special vector potential associated with the spinor field was introduced. Such a vector potential has close relation with the triality property in Dirac spinors and plays crucial role in the construction of massive term. It is shown that the change in phase of a wavefun...

An account is given of the structure and representations of chiral bosonic meromorphic conformal field theories (CFT's), and, in particular, the conditions under which such a CFT may be extended by a representation to form a new theory. This general approach is illustrated by considering the untwisted and Z 2-twisted theories, H(Λ) and˜H(Λ) respectively , which may be constructed from a suitabl...

Need to do:. • Rewrite the section on triality groups. • Verify that Z ⊂ M q in proof of 6.5. ([PR] says it's true on p. 385.)

We consider a field F and positive integers n, m, such that m is not divisible by $$\mathrm {Char}(F)$$ prime to n!. The absolute Galois group $$G_F$$ acts on the $$\mathbb {U}_n(\mathbb {Z}/m)$$ of all $$(n+1)\times (n+1)$$ unipotent upper-triangular matrices over {Z}/m$$ cyclotomically. Given $$0,1\ne z\in F$$ an arbitrary list w n Kummer elements $$(z)_F$$ , $$(1-z)_F$$ in $$H^1(G_F,\mu _m)$...