Construction of two-dimensional topological field theories with non-invertible symmetries

We construct the defining data of two-dimensional topological field theories (TFTs) enriched by non-invertible symmetries/topological defect lines. Simple formulae for the three-point functions and the lasso two-point functions are derived, and crossing symmetry is proven. The key ingredients are open-to-closed maps and a boundary crossing relation, by which we show that a diagonal basis exists in the defect Hilbert spaces. We then introduce regular TFTs, provide their explicit constructions for the Fibonacci, Ising and Haagerup ℋ3 fusion categories, and match our formulae with previous bootstrap results. We end by explaining how non-regular TFTs are obtained from regular TFTs via generalized gauging.

Generation flow in field theory and strings

Nontrivial strong dynamics often leads to the appearance of chiral composites. In phenomenological applications, these can either play the role of Standard Model particles or lift chiral exotics by partnering with them in mass terms. As a consequence, the RG flow may change the effective number of chiral generations, a phenomenon we call generation flow. We provide explicit constructions of globally consistent string models exhibiting generation flow. Since such constructions were misclassified in the traditional model searches, our results imply that more care than usually appreciated has to be taken when scanning string compactifications for realistic models.

Tensor products of finitely presented functors

We provide explicit constructions for various ingredients of right exact monoidal structures on the category of finitely presented functors. As our main tool, we prove a multilinear version of the universal property of so-called Freyd categories, which in turn is used in the proof of correctness of our constructions. Furthermore, we compare our construction with the Day convolution of arbitrary additive functors. Day convolution always yields a closed monoidal structure on the category of all additive functors. In contrast, right exact monoidal structures for finitely presented functor categories are not necessarily closed. We provide a necessary criterion for being closed that relies on the underlying category having weak kernels and a so-called finitely presented prointernal hom structure. Our results are stated in a constructive way and thus serve as a unified approach for the implementation of tensor products in various contexts.

Drinfeld–Manin solutions of the Yang–Baxter equation coming from cube complexes

The most common geometric interpretation of the Yang–Baxter equation is by braids, knots and relevant Reidemeister moves. So far, cubes were used for connections with the third Reidemeister move only. We will show that there are higher-dimensional cube complexes solving the [Formula: see text]-state Yang–Baxter equation for arbitrarily large [Formula: see text]. More precisely, we introduce explicit constructions of cube complexes covered by products of [Formula: see text] trees and show that these cube complexes lead to new solutions of the Yang–Baxter equations.

New and explicit constructions of unbalanced Ramanujan bipartite graphs

The objectives of this article are threefold. Firstly, we present for the first time explicit constructions of an infinite family of unbalanced Ramanujan bigraphs. Secondly, we revisit some of the known methods for constructing Ramanujan graphs and discuss the computational work required in actually implementing the various construction methods. The third goal of this article is to address the following question: can we construct a bipartite Ramanujan graph with specified degrees, but with the restriction that the edge set of this graph must be distinct from a given set of “prohibited” edges? We provide an affirmative answer in many cases, as long as the set of prohibited edges is not too large.

Lyapunov - Кrasovskii functionals for homogeneous systems with multiple delays

In this article, explicit constructions of Lyapunov - Кrasovskii functionals are proposed for homogeneous systems with multiple constant delays and homogeneity degree of the right- hand sides strictly greater than one. The constructions are based on the Lyapunov functions suitable for the analysis of corresponding systems with all delays equal to zero. The letter systems are assumed to be asymptotically stable. It is proved that the proposed functionals satisfy the conditions of the Кrasovskii theorem, and hence it allows us to establish the asymptotic stability of the trivial solution for arbitrary values of delays. The functionals are applied to the estimation of the attraction region of the trivial solution.

Explicit constructions of cyclic and negacyclic codes of length 3ps over 𝔽pm + u𝔽pm

Let [Formula: see text] be a prime such that [Formula: see text]. The algebraic structures of all cyclic and negacyclic codes of length [Formula: see text] over the finite commutative chain ring [Formula: see text] are obtained that the conditions depend on the factorization of polynomial [Formula: see text] over [Formula: see text]. Therefore, we classify the structures of cyclic and negacyclic codes of length [Formula: see text] over [Formula: see text] into 2 cases, i.e., [Formula: see text] and [Formula: see text]. From that we obtain the number of all cyclic and negacyclic codes of length [Formula: see text] over [Formula: see text]. After that, we give some situations for such cyclic and negacyclic codes are self-dual codes.