Higgs branches of 5d rank-zero theories from geometry

We study the Higgs branches of five-dimensional $$ \mathcal{N} $$ N = 1 rank-zero theories obtained from M-theory on two classes non-toric non-compact Calabi-Yau threefolds: Reid’s pagodas, and Laufer’s examples. Our approach consists in reducing to IIA with D6-branes and O6-planes, and computing the open-string spectra giving rise to hypermultiplets. Starting with the seven-dimensional worldvolume theories, we switch on T-brane backgrounds to give rise to bound states with angles. We observe that the resulting partially Higgsed 5d theories have discrete gauge groups, from which we readily deduce the geometry of the Higgs branches as orbifolds of quaternionic varieties.

CONDITION (K) FOR BOOLEAN DYNAMICAL SYSTEMS

We generalize Condition (K) from directed graphs to Boolean dynamical systems and show that a locally finite Boolean dynamical system $({{\mathcal {B}}},{{\mathcal {L}}},\theta )$ with countable ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ satisfies Condition (K) if and only if every ideal of its $C^*$ -algebra is gauge-invariant, if and only if its $C^*$ -algebra has the (weak) ideal property, and if and only if its $C^*$ -algebra has topological dimension zero. As a corollary we prove that if the $C^*$ -algebra of a locally finite Boolean dynamical system with ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ countable either has real rank zero or is purely infinite, then $({{\mathcal {B}}}, {{\mathcal {L}}}, \theta )$ satisfies Condition (K). We also generalize the notion of maximal tails from directed graph to Boolean dynamical systems and use this to give a complete description of the primitive ideal space of the $C^*$ -algebra of a locally finite Boolean dynamical system that satisfies Condition (K) and has countable ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ .

Euler characteristics and their congruences in the positive rank setting

The notion of the truncated Euler characteristic for Iwasawa modules is an extension of the notion of the usual Euler characteristic to the case when the homology groups are not finite. This article explores congruence relations between the truncated Euler characteristics for dual Selmer groups of elliptic curves with isomorphic residual representations, over admissible p-adic Lie extensions. Our results extend earlier congruence results from the case of elliptic curves with rank zero to the case of higher rank elliptic curves. The results provide evidence for the p-adic Birch and Swinnerton-Dyer formula without assuming the main conjecture.

The classification of certain ASH C*-algebras of real rank zero

In this paper, using ordered total K-theory, we give a K-theoretic classification for the real rank zero inductive limits of direct sums of generalized dimension drop interval algebras.

Importance of the fourth-rank zero field splitting parameters for Fe2+ (S = 2) adatoms on the CuN/Cu(100) surface evidenced by their determination based on DFT and experimental data

Equations allow to determine 2nd- and 4th-rank ZFSPs (Bkq) based on spin energy levels (λi) at B = 0. This method is applied to Fe2+ (S = 2) adatoms on CuN/Cu(100) surface using DFT and experimental data. Relative importance of ZFSPs is analyzed.

Rank Growth of Elliptic Curves in Non-Abelian Extensions

Given an elliptic curve $E/\mathbb{Q}$, it is a conjecture of Goldfeld that asymptotically half of its quadratic twists will have rank zero and half will have rank one. Nevertheless, higher rank twists do occur: subject to the parity conjecture, Gouvêa and Mazur constructed $X^{1/2-\epsilon }$ twists by discriminants up to $X$ with rank at least two. For any $d\geq 3$, we build on their work to consider the rank growth of $E$ in degree $d$$S_d$-extensions of $\mathbb{Q}$ with discriminant up to $X$. We prove that there are at least $X^{c_d-\epsilon }$ such fields where the rank grows, where $c_d$ is a positive constant that tends to $1/4$ as $d\to \infty $. Moreover, subject to a suitable parity conjecture, we obtain the same result for fields for which the rank grows by at least two.