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

摘要

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.