On transformations preserving the basis conditions of a spin structure group in four-dimensional superstring theory in free fermionic formulation

Abstract

Let Ξ stand for a finite Abelian spin structure group of 4-dimensional superstring theory in free fermionic formulation whose elements are 64-dimensional vectors (spin structure vectors) with rational entries belonging to ]−1,1] and the group operation is the mod 2 entry by entry summation ⊕ of these vectors. Let B={bi, i=1,...,k+1} be a set of spin structure vectors such that bi have only entries 0 and 1 for any i=1,...,k, while bk+1 is allowed to have any rational entries belonging to ]−1,1] with even Nk+1, where Nk+1 stands for the least positive integer such that Nk+1bk+1=0 mod 2. Let B be a basis of Ξ, i.e., let B generate Ξ, and let Λm,n stand for the transformation of B which replaces bn by bm⊕bn for any m ≠ k+1, n ≠ 1, m ≠ n. It is proven that if B satisfies the axioms for a basis of spin structure group Ξ, then B′=Λm,nB also satisfies the axioms. Since the transformations Λm,n for different m and n generate all nondegenerate transformations of the basis B that preserve the vector b1 and a single vector bk+1 with general rational entries, it can be concluded that the axioms are conditions for the whole group Ξ and not just conditions for a particular choice of its basis. Hence, these transformations generate the discrete symmetry group of four-dimensional superstring models in free fermionic formulation. This is the physical meaning of the obtained result. The practical impact is that in searching for a realistic model in four-dimensional superstring theory in free fermionic formulation it is enough to search only through different groups Ξ, which number dramatically less than the number of their different bases.