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We revisit anomalous phases related to large gauge transformations, such as the Witten anomaly. The latter, known to plague $d=4$ $Sp(k)$ theories, is well-understood in terms of $\pi_4(Sp(k))=\mathbb{Z}_2$, but it also has an oblique relation to the instantons, labeled by $\pi_3(G)=\mathbb{Z}$, via the fermion zero mode counting. We revisit this relation and point out how $SU(N)$ theories escape an anomalous sign of the latter type, only thanks to the perturbative anomaly cancelation condition that restricts the chiral fermion spectrum. This leads to the question of what happens if the latter, more mundane anomaly is canceled by an inflow instead. After raising an open question about fractional D3 probe theories, we explore the simplest bottom-up model of such a kind, due to Witten and Yonekura, from which we find the relevant chiral theories to be free of such a disease despite the unrestricted chiral spectra. We close with a simple but often-overlooked observation about how fermionic zero modes enter physics differently between Euclidean and Lorentzian descriptions and point out a related issue in $d=3$., Comment: 40 pages, 7 figures

Skyrmion qubits are a new highly promising logic element for quantum information processing. However, their scalability to multiple interacting qubits remains challenging. We propose a hybrid quantum setup with skyrmion qubits strongly coupled to nanomechanical cantilevers via magnetic coupling, which harnesses phonons as quantum interfaces for the manipulation of distant skyrmion qubits. A linear drive is utilized to achieve the modulation of the stiffness coefficient of the cantilever, resulting in an exponential enhancement of the coupling strength between the skyrmion qubit and the mechanical mode. We also consider the case of a topological resonator array, which allows us to study interactions between skyrmion qubits and topological phonon band structure, as well as chiral skyrmion-skyrmion interactions. The scheme suggested here offers a fascinating platform for investigating quantum information processing and quantum simulation with magnetic microstructures., Comment: To appear in PR Research, 16 pages, 9 figures

We generalize the idea of symmetry topological field theory (SymTFT) for subsystem symmetry. We propose the 2-foliated BF theory with level $N$ in $(3+1)$d as subsystem SymTFT for subsystem $\mathbb Z_N$ symmetry in $(2+1)$d. Focusing on $N=2$, we investigate various topological boundaries. The subsystem Kramers-Wannier and Jordan-Wigner dualities can be viewed as boundary transformations of the subsystem SymTFT and are included in a larger duality web from the subsystem $SL(2,\mathbb Z_2)$ symmetry of the bulk foliated BF theory. Finally, we construct the condensation defects and twist defects of $S$-transformation in the subsystem $SL(2,\mathbb Z_2)$, from which the fusion rule of subsystem non-invertible operators can be recovered., Comment: 49 pages + appendices, 9 figures; typos corrected

Given a two-dimensional bosonic theory with a non-anomalous $\mathbb{Z}_2$ symmetry, the orbifolding and fermionization can be understood holographically using three-dimensional BF theory with level $2$. From a Hamiltonian perspective, the information of dualities is encoded in a topological boundary state which is defined as an eigenstate of certain Wilson loop operators (anyons) in the bulk. We generalize this story to two-dimensional theories with non-anomalous $\mathbb{Z}_N$ symmetry, focusing on parafermionization. We find the generic operators defining different topological boundary states including orbifolding and parafermionization with $\mathbb{Z}_N$ or subgroups of $\mathbb{Z}_N$, and discuss their algebraic properties as well as the $\mathbb{Z}_N$ duality web., Comment: 39 pages, 5 figures; references added; reference added, published version

We present a hybrid quantum system composed of a magnetic vortex and a nanomechanical resonator. We show that the gyrotropic mode of the vortex can coherently couple to the quantized mechanical motion of the resonator through magnetic interaction. Benefiting from the topologically protected properties and the low damping of vortices, as well as the excellent coherent features of nanomechanical resonators, the proposed system can achieve strong coupling and even the ultrastrong coupling regime by choosing appropriate parameters. In combination with other quantum systems, such as a nitrogen-vacancy (NV) center, coherent state transfer between the vortex excitation and the spin can be realized. This setup provides a potential platform for quantum information processing and investigations into the ultrastrong coupling regimes and macroscopic quantum physics., Comment: To appear in PR Applied, 12 pages, 9 figures

Spin-mechanical hybrid systems have been widely used in quantum information processing. However, the spin-mechanical interaction is generally weak, making it a critical challenge to enhance the spin-mechanical interaction into the strong coupling or even ultra-strong coupling regime. Here, we propose a protocol that can significantly enhance the spin-mechanical coupling strength with a diamond spin vacancy and a levitated micromagnet. A driving electrical current is used to modulate the mechanical motion of the levitated micromagnet, which induces a two-phonon drive and can exponentially enhance the spin-phonon and phonon-medicated spin-spin coupling strengths. Furthermore, a high fidelity Schrodinger cat state and an unconventional 2-qubit geometric phase gate with high fidelity and faster gate speed can be achieved using this hybrid system. This protocol provides a promising platform for quantum information processing with NV spins coupled to levitated micromagnets., Comment: 13 pages, 8 figures

We study the Seberg-Witten geometry of 5d ${\cal N}=1$ pure Yang-Mills theories compactified on a circle. The concept of the holonomy saddle implies that there are multiple 4d limits of interacting Seiberg-Witten theories from a single 5d theory, and we explore this in the simplest case of pure $SU(N)$ theories. The compactification leads to $N$ copies of locally indistinguishable 4d pure $SU(N)$ Seiberg-Witten theories in the infrared, glued together in a manner dictated by the Chern-Simons level. We show how this picture naturally builds the 5d BPS quivers which agree with the D0 probe dynamics previously proposed via the geometrically engineered local Calabi-Yau. We work out various $SU(2)$ and $SU(3)$ examples through a detailed look at the respective spectral curves. We also note a special $\mathbb{Z}_{2N}$ feature of $SU(N)_N$ spectral curves and the resulting BPS quivers, with emphasis on how the 4d holonomy saddles are affected., Comment: 22 figures

We firstly simulated disease dynamics by KAN (Kolmogorov-Arnold Networks) nearly 4 years ago, but the kernel functions in the edge include the exponential number of infected and discharged people and is also in line with the Kolmogorov-Arnold representation theorem, and the shared weights in the edge are the infection rate and cure rate, and used activation function by tanh at the node of edge. And this Arxiv preprint version 1 of March 2022 is an upgraded version of KAN, considering the invariant coarse-grained which calculated by residual or gradient of MSE loss. The improved KAN is PNN (Plasticity Neural Networks) or ELKAN (Edge Learning KNN), in addition to edge learning, it also considered the trimming of the edge. We not inspired by the Kolmogorov-Arnold representation theorem but inspired by the brain science. The ELKAN to explain brain, the variables correspond to different types of neurons, the learning edge can be explained by rebalance of synaptic strength and glial cells phagocytose synapses, and the kernel function means the discharge of neurons and synapses, different neurons and edges mean brain regions. Through testing by cosine, the ELKAN or ORPNN (Optimized Range PNN) is better than the KAN or CRPNN (Constant Range PNN).The ELKAN is more general to explore brain, such as mechanism of consciousness, interactions of natural frequencies in brain regions, synaptic and neuronal discharge frequencies, and data signal frequencies; mechanism of Alzheimer's disease, the Alzheimer's patients has more high frequencies in the upstream brain regions; long short-term relatively good and inferior memory which means gradient of architecture and architecture; turbulent energy flow in different brain regions, turbulence critical conditions need to be met; heart-brain of the quantum entanglement may occur between the emotions of heartbeat and the synaptic strength of brain potentials.

We study the infrared physics of 5d $\cal N=1$ Yang-Mills theories compactified on $\mathbb{S}^1$, with a view toward 4d and 5d limits. Global structures of the simplest Coulombic moduli spaces are outlined, with an emphasis on how multiple planar 4d Seiberg-Witten geometries are embedded in the cigar geometry of a single 5d theory on $\mathbb{S}^1$. The Coulomb phase boundaries in the decompactification limit are given particular attention and related to how the wall-crossings by 5d BPS particles turn off. On the other hand, the elliptic genera of magnetic BPS strings do wall-cross and retain the memory of 4d wall-crossings, which we review with the example of dP$_2$ theory. Along the way, we also offer a general field theory proof of the odd shift of electric charge on Sp$(k)_\pi$ instanton solitons, previously observed via geometric engineering for low-rank supersymmetric theories., Comment: 77 pages, 28 figures