Dirac masses and mixings in the (geo)SM(EFT) and beyond

We report a set of exact formulae for computing Dirac masses, mixings, and CP-violation parameter(s) from 3×3 Yukawa matrices Y valid when YY† → U†YY†U under global $$ \mathrm{U}{(3)}_{Q_L} $$ U 3 Q L flavour symmetry transformations U. The results apply to the Standard Model Effective Field Theory (SMEFT) and its ‘geometric’ realization (geoSMEFT). We thereby complete, in the Dirac flavour sector, the catalogue of geoSMEFT parameters derived at all orders in the $$ \sqrt{2\left\langle {H}^{\dagger }H\right\rangle } $$ 2 H † H /Λ expansion. The formalism is basis-independent, and can be applied to models with decoupled ultraviolet flavour dynamics, as well as to models whose infrared dynamics are not minimally flavour violating. We highlight these points with explicit examples and, as a further demonstration of the formalism’s utility, we derive expressions for the renormalization group flow of quark masses, mixings, and CP-violation at all mass dimension and perturbative loop orders in the (geo)SM(EFT) and beyond.

Computational Design of Planar Multistable Compliant Structures

This article presents a method for designing planar multistable compliant structures. Given a sequence of desired stable states and the corresponding poses of the structure, we identify the topology and geometric realization of a mechanism—consisting of bars and joints—that is able to physically reproduce the desired multistable behavior. In order to solve this problem efficiently, we build on insights from minimally rigid graph theory to identify simple but effective topologies for the mechanism. We then optimize its geometric parameters, such as joint positions and bar lengths, to obtain correct transitions between the given poses. Simultaneously, we ensure adequate stability of each pose based on an effective approximate error metric related to the elastic energy Hessian of the bars in the mechanism. As demonstrated by our results, we obtain functional multistable mechanisms of manageable complexity that can be fabricated using 3D printing. Further, we evaluated the effectiveness of our method on a large number of examples in the simulation and fabricated several physical prototypes.

Quasi-split symmetric pairs of 𝑈(𝔰𝔩_{𝔫}) and Steinberg varieties of classical type

We provide a Lagrangian construction for the fixed-point subalgebra, together with its idempotent form, in a quasi-split symmetric pair of type A n − 1 A_{n-1} . This is obtained inside the limit of a projective system of Borel-Moore homologies of the Steinberg varieties of n n -step isotropic flag varieties. Arising from the construction are a basis of homological origin for the idempotent form and a geometric realization of rational modules.

A Geometric and Combinatorial Exploration of Hochschild Lattices

Hochschild lattices are specific intervals in the dexter meet-semilattices recently introduced by Chapoton. A natural geometric realization of these lattices leads to some cell complexes introduced by Saneblidze, called the Hochschild polytopes. We obtain several geometrical properties of the Hochschild lattices, namely we give cubic realizations, establish that these lattices are EL-shellable, and show that they are constructible by interval doubling. We also prove several combinatorial properties as the enumeration of their $k$-chains and compute their degree polynomials.

Euler Transformation of Polyhedral Complexes

We propose an Euler transformation that transforms a given [Formula: see text]-dimensional cell complex [Formula: see text] for [Formula: see text] into a new [Formula: see text]-complex [Formula: see text] in which every vertex is part of the same even number of edges. Hence every vertex in the graph [Formula: see text] that is the [Formula: see text]-skeleton of [Formula: see text] has an even degree, which makes [Formula: see text] Eulerian, i.e., it is guaranteed to contain an Eulerian tour. Meshes whose edges admit Eulerian tours are crucial in coverage problems arising in several applications including 3D printing and robotics. For [Formula: see text]-complexes in [Formula: see text] ([Formula: see text]) under mild assumptions (that no two adjacent edges of a [Formula: see text]-cell in [Formula: see text] are boundary edges), we show that the Euler transformed [Formula: see text]-complex [Formula: see text] has a geometric realization in [Formula: see text], and that each vertex in its [Formula: see text]-skeleton has degree [Formula: see text]. We bound the numbers of vertices, edges, and [Formula: see text]-cells in [Formula: see text] as small scalar multiples of the corresponding numbers in [Formula: see text]. We prove corresponding results for [Formula: see text]-complexes in [Formula: see text] under an additional assumption that the degree of a vertex in each [Formula: see text]-cell containing it is [Formula: see text]. In this setting, every vertex in [Formula: see text] is shown to have a degree of [Formula: see text]. We also present bounds on parameters measuring geometric quality (aspect ratios, minimum edge length, and maximum angle of cells) of [Formula: see text] in terms of the corresponding parameters of [Formula: see text] for [Formula: see text]. Finally, we illustrate a direct application of the proposed Euler transformation in additive manufacturing.

The cohomology of semi-infinite Deligne–Lusztig varieties

We prove a 1979 conjecture of Lusztig on the cohomology of semi-infinite Deligne–Lusztig varieties attached to division algebras over local fields. We also prove the two conjectures of Boyarchenko on these varieties. It is known that in this setting, the semi-infinite Deligne–Lusztig varieties are ind-schemes comprised of limits of certain finite-type schemes {X_{h}}. Boyarchenko’s two conjectures are on the maximality of {X_{h}} and on the behavior of the torus-eigenspaces of their cohomology. Both of these conjectures were known in full generality only for division algebras with Hasse invariant {1/n} in the case {h=2} (the “lowest level”) by the work of Boyarchenko–Weinstein on the cohomology of a special affinoid in the Lubin–Tate tower. We prove that the number of rational points of {X_{h}} attains its Weil–Deligne bound, so that the cohomology of {X_{h}} is pure in a very strong sense. We prove that the torus-eigenspaces of the cohomology group {H_{c}^{i}(X_{h})} are irreducible representations and are supported in exactly one cohomological degree. Finally, we give a complete description of the homology groups of the semi-infinite Deligne–Lusztig varieties attached to any division algebra, thus giving a geometric realization of a large class of supercuspidal representations of these groups. Moreover, the correspondence {\theta\mapsto H_{c}^{i}(X_{h})[\theta]} agrees with local Langlands and Jacquet–Langlands correspondences. The techniques developed in this paper should be useful in studying these constructions for p-adic groups in general.

Higher-form symmetries in 5d

We study higher-form symmetries in 5d quantum field theories, whose charged operators include extended operators such as Wilson line and ’t Hooft operators. We outline criteria for the existence of higher-form symmetries both from a field theory point of view as well as from the geometric realization in M-theory on non-compact Calabi-Yau threefolds. A geometric criterion for determining the higher-form symmetry from the intersection data of the Calabi-Yau is provided, and we test it in a multitude of examples, including toric geometries. We further check that the higher-form symmetry is consistent with dualities and is invariant under flop transitions, which relate theories with the same UV-fixed point. We explore extensions to higher-form symmetries in other compactifications of M-theory, such as G2-holonomy manifolds, which give rise to 4d $$ \mathcal{N} $$ N = 1 theories.

Multiframings of 3-manifolds

We investigate multiframings of a closed oriented 3-manifold. We show that multiframings give a geometric realization of the tensor product of the homotopy set of framings and [Formula: see text]. We prove that the Hirzebruch defect defines a bijection from the homotopy set of multiframings to [Formula: see text] for any connected closed oriented 3-manifold, and we prove that any multiframing defined near the boundary of a compact oriented 3-manifold extends to the bounded manifold.

Geometric Realization of $\gamma$-Vectors of Subdivided Cross Polytopes

For any flag simplicial complex $\Theta$ obtained by stellar subdividing the boundary of the cross polytope in edges, we define a flag simplicial complex $\Delta(\Theta)$ whose $f$-vector is the $\gamma$-vector of $\Theta$. This proves that the $\gamma$-vector of any such simplicial complex is the face vector of a flag simplicial complex, partially solving a conjecture by Nevo and Petersen. As a corollary we obtain that such simplicial complexes satisfy the Frankl-Füredi-Kalai inequalities.