On completeness and generalized symmetries in quantum field theory

We review a notion of completeness in QFT arising from the analysis of basic properties of the set of operator algebras attached to regions. In words, this completeness asserts that the physical observable algebras produced by local degrees of freedom are the maximal ones compatible with causality. We elaborate on equivalent statements to this completeness principle such as the non-existence of generalized symmetries and the uniqueness of the net of algebras. We clarify that for non-complete theories, the existence of generalized symmetries is unavoidable and further that they always come in dual pairs with precisely the same “size”. Moreover, the dual symmetries are always broken together, be it explicitly or effectively. Finally, we comment on several issues raised in recent literature, such as the relationship between completeness and modular invariance, dense sets of charges, and absence of generalized symmetries in the bulk of holographic theories.

Higher-dimensional Calabi–Yau varieties with dense sets of rational points

We construct higher-dimensional Calabi–Yau varieties defined over a given number field with Zariski dense sets of rational points. We give two elementary constructions in arbitrary dimensions as well as another construction in dimension three which involves certain Calabi–Yau threefolds containing an Enriques surface. The constructions also show that potential density holds for (sufficiently) general members of the families.

Soft ωp-Open Sets and Soft ωp-Continuity in Soft Topological Spaces

We define soft ωp-openness as a strong form of soft pre-openness. We prove that the class of soft ωp-open sets is closed under soft union and do not form a soft topology, in general. We prove that soft ωp-open sets which are countable are soft open sets, and we prove that soft pre-open sets which are soft ω-open sets are soft ωp-open sets. In addition, we give a decomposition of soft ωp-open sets in terms of soft open sets and soft ω-dense sets. Moreover, we study the correspondence between the soft topology soft ωp-open sets in a soft topological space and its generated topological spaces, and vice versa. In addition to these, we define soft ωp-continuous functions as a new class of soft mappings which lies strictly between the classes of soft continuous functions and soft pre-continuous functions. We introduce several characterizations for soft pre-continuity and soft ωp-continuity. Finally, we study several relationships related to soft ωp-continuity.

On projections of products of spaces

We consider dense sets of products of topological spaces. We prove that in the product $Z^c=\prod\limits_{\alpha\in 2^\omega} Z_{\alpha},$ where $Z_\alpha=Z$ $(\alpha\in 2^\omega),$ there are dense sets such that their countable subsets have projections with additional properties. These properties entail that these dense sets contain no convergent sequences. By these properties we prove that the character of closed sets of the product is uncountable.

On some relations between ideals of nowhere dense sets in topologies on positive integers

We examine the ideals of nowhere dense sets in three topologies on the set of positive integers, namely Furstenberg’s, Rizza’s and the common division topology. We mainly concentrate on inclusions between these ideals, we present a diagram showing these and we explore all possible inclusions between them. We present a formula for the closure of a set in the common division topology. We answer a question posed by Kwela and Nowik (Topol Appl. 248:149–163, 2018) by constructing a set in $${{\mathcal {I}}}_G {\setminus } ({{\mathcal {I}}}_K \cup {{\mathcal {I}}}_F)$$ I G \ ( I K ∪ I F ) . Therefore, the main diagram of comparison between the ideals of nowhere dense sets in various topologies from the article by M. Kwela and A. Nowik is completed.