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In the third paragraph of page 7 of the paper arxiv:1112.3984 it mentions that we form a basis out of hypermultiplets on the charge lattice $\Gamma$. My questions are:
In what sense do we form such a basis?
What does it mean that the BPS states are required to form a positive integral basis? (in a more general way than what is explained in the text)
Let $\Gamma'$ be the subset of $\Gamma$ made of charges of BPS states. A basis of $\Gamma'$ is a collection of elements of $\Gamma'$ such that any element of $\Gamma'$ can be written in a unique way as a linear combination of these elements. (A more familiar notion is probably the notion of basis of a vector space: a basis is a family of vectors such that any non-zero vector can be written in a unique way as a linear combination of vectors in this family. Here we don't have vector spaces but it is similar. The objects we are dealing with are mathematically called monoids: we are looking for a basis of the monoid generated by $\Gamma'$). As shown in the paper, if such basis exists, it is unique up to permutations. But it does not exist in general: the existence is an assumption of the paper. Also the fact that if the basis exists then the basis elements are charges of hypermultiplets (and not other types of BPS multiplets) is a priori unclear: it is also a hypothesis of the paper that it is the case.
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