Impact of Gravity on Vacuum Stability

In a pioneering paper on the role of gravity on false vacuum decay, Coleman and De Luccia showed that a strong gravitational field can stabilize the false vacuum, suppressing the formation of true vacuum bubbles. This result is obtained for the case when the energy density difference between the two vacua is small, the so called thin wall regime, but is considered of more general validity. Here we show that when this condition does not hold, however, {\it a strong gravitational field (Planckian physics) does not necessarily induce a total suppression of true vacuum bubble nucleation}. Contrary to common expectations then, gravitational physics at the Planck scale {\it does not stabilize the false vacuum}. These results are of crucial importance for the stability analysis of the electroweak vacuum and for searches of new physics beyond the Standard Model.Comment: 6 pages, 4 figure

Continuous cohomology of topological quandles

A continuous cohomology theory for topological quandles is introduced, and compared to the algebraic theories. Extensions of topological quandles are studied with respect to continuous 2-cocycles, and used to show the differences in second cohomology groups for specific topological quandles. A method of computing the cohomology groups of the inverse limit is applied to quandles.Comment: 17 page

QUANTUM INVARIANTS OF FRAMED LINKS FROM TERNARY SELF-DISTRIBUTIVE COHOMOLOGY

The ribbon cocycle invariant is defined by means of a partition function using ternary cohomology of self-distributive structures (TSD) and colorings of ribbon diagrams of a framed link, following the same paradigm introduced by Carter, Jelsovsky, Kamada, Langford and Saito in Transactions of the American Mathematical Society 2003;355(10):3947-89, for the quandle cocycle invariant. In this article we show that the ribbon cocycle invariant is a quantum invariant. We do so by constructing a ribbon category from a TSD set whose twisting and braiding morphisms entail a given TSD 2-cocycle. Then we show that the quantum invariant naturally associated to this braided category coincides with the cocycle invariant. We generalize this construction to symmetric monoidal categories and provide classes of examples obtained from Hopf monoids and Lie algebras. We further introduce examples from Hopf-Frobenius algebras, objects studied in quantum computing

The exact evaluation of hexagonal spin-networks and topological quantum neural networks

The physical scalar product between spin-networks has been shown to be a fundamental tool in the theory of topological quantum neural networks (TQNN), which are quantum neural networks previously introduced by the authors in the context of quantum machine learning. However, the effective evaluation of the scalar product remains a bottleneck for the applicability of the theory. We introduce an algorithm for the evaluation of the physical scalar product defined by Noui and Perez between spin-network with hexagonal shape. By means of recoupling theory and the properties of the Haar integration we obtain an efficient algorithm, and provide several proofs regarding the main steps. We investigate the behavior of the TQNN evaluations on certain classes of spin-networks with the classical and quantum recoupling. All results can be independently reproduced through the "idea.deploy" framework~\href{https://github.com/lullimat/idea.deploy}{\nolinkurl{https://github.com/lullimat/idea.deploy}}Comment: 15 pages (2 columns, 12+3), 16 figures. Comments are welcome

Continuous Spatiotemporal Transformers

Modeling spatiotemporal dynamical systems is a fundamental challenge in machine learning. Transformer models have been very successful in NLP and computer vision where they provide interpretable representations of data. However, a limitation of transformers in modeling continuous dynamical systems is that they are fundamentally discrete time and space models and thus have no guarantees regarding continuous sampling. To address this challenge, we present the Continuous Spatiotemporal Transformer (CST), a new transformer architecture that is designed for the modeling of continuous systems. This new framework guarantees a continuous and smooth output via optimization in Sobolev space. We benchmark CST against traditional transformers as well as other spatiotemporal dynamics modeling methods and achieve superior performance in a number of tasks on synthetic and real systems, including learning brain dynamics from calcium imaging data.Comment: Updated version, after review

AMPNet: Attention as Message Passing for Graph Neural Networks

Graph Neural Networks (GNNs) have emerged as a powerful representation learning framework for graph-structured data. A key limitation of conventional GNNs is their representation of each node with a singular feature vector, potentially overlooking intricate details about individual node features. Here, we propose an Attention-based Message-Passing layer for GNNs (AMPNet) that encodes individual features per node and models feature-level interactions through cross-node attention during message-passing steps. We demonstrate the abilities of AMPNet through extensive benchmarking on real-world biological systems such as fMRI brain activity recordings and spatial genomic data, improving over existing baselines by 20% on fMRI signal reconstruction, and further improving another 8% with positional embedding added. Finally, we validate the ability of AMPNet to uncover meaningful feature-level interactions through case studies on biological systems. We anticipate that our architecture will be highly applicable to graph-structured data where node entities encompass rich feature-level information.Comment: 16 pages (12 + 4 pages appendix). 5 figures and 7 table