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

TL;DR

This paper proves that certain basis transformations preserve the axioms of a spin structure group in four-dimensional superstring theory, establishing these transformations as generators of the model's discrete symmetry group.

Contribution

It demonstrates that basis transformations preserving axioms generate the entire symmetry group of superstring models in free fermionic formulation.

Findings

Transformations $\Lambda_{m,n}$ preserve the basis axioms.

These transformations generate all nondegenerate basis transformations.

The axioms define the symmetry group of the superstring models.

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 $\rbrack -1,\, 1\rbrack $ and the group operation is the $mod\, \, 2 $ 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 $\rbrack -1,\, 1\rbrack $ with even $N_{k+1}$, where $N_{k+1}$ stands for the least positive integer such that $N_{k+1}b_{k+1}= 0\,mod\,2$. 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 $B'=\Lambda_{m, n}B$ 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.