Supersymmetric extensions of the standard model generically contain stable non-topological solitons, Q-balls, which carry baryon or lepton number. We show that large Q-balls can be copiously produced in the early universe, can survive until the present time, and can contribute to dark matter.
Let $\mathbb{F}_{q}^{n}$ be a vector space of dimension $n$ over the finite field $\mathbb{F}_{q}$ . A $q$ -analog of a Steiner system (also known as a $q$ -Steiner system), denoted ${\mathcal{S}}_{q}(t,\!k,\!n)$ , is a set ${\mathcal{S}}$ of $k$ -dimensional subspaces of $\mathbb{F}_{q}^{n}$ such that each $t$ -dimensional subspace of $\mathbb{F}_{q}^{n}$ is contained in exactly one element of ${\mathcal{S}}$ . Presently, $q$ -Steiner systems are known only for $t\,=\,1\!$ , and in the trivial cases $t\,=\,k$ and $k\,=\,n$ . In this paper, the first nontrivial $q$ -Steiner systems with $t\,\geqslant \,2$ are constructed. Specifically, several nonisomorphic $q$ -Steiner systems ${\mathcal{S}}_{2}(2,3,13)$ are found by requiring that their automorphism groups contain the normalizer of a Singer subgroup of $\text{GL}(13,2)$ . This approach leads to an instance of the exact cover problem, which turns out to have many solutions.
We propose the notion of q-characters for finite-dimensional representations of quantum affine algebras. It is motivated by our theory of deformed W-algebras. We show that the q-characters give rise to a homomorphism from the Grothendieck ring of representations of a quantum affine algebra to a polynomial ring. We conjecture that the image of this homomorphism is equal to the intersection of certain "screening operators". We also discuss the connection between q-characters and Bethe Ansatz.