On Transformations Preserving the Basis Conditions of a Spin Structure Group in Four-Dimensional Super String Theory in Free Fermionic Formulation

Abstract

Let $\Xi$ stand for a finite abelian spin structure group of four-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 $mod2$ entry by entry summation $\oplus$ of these vectors. Let $B=\{b_i,i=1,\cdots ,k+1\}$ be a set of spin structure vectors such that $b_i$ have only entries 0 and 1 for any $i=1,\cdots ,k$, while $b_{k+1}$ is allowed to have any rational entries belonging to $]-1,1]$ with even $N_{k+1}$, where $N_{k+1}$ stands for the least positive integer such that $N_{k+1}{b}_{k+1}=0mod2$. Let $B$ be a basis of $\Xi$, i.e., let $B$ generate $\Xi$, and let ${\Lambda }_{m,n}$ stand for the transformation of $B$ which replaces $b_n$ by $b_m\oplus b_n$ for any $m\ne k+1$, $n\ne 1$, $m\ne n$. We prove that if $B$ satisfies the axioms for a basis of spin structure group $\Xi$, then $B^{\prime }={\Lambda }_{m,n}B$ also satisfies the axioms. Since the transformations ${\Lambda }_{m,n}$ for different $m$ and $n$ generate all nondegenerate transformations of the basis $B$ that preserve the vector $b_1$ and a single vector $b_{k+1}$ with general rational entries, we conclude that the axioms are conditions for the whole group $\Xi$ 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.

10 pages, MIU-THP-94/67

References in corpus (1)

• The Search for a realistic flipped SU(5) string model