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The ability of surface polaritons (SPs) to enhance and manipulate light fields down to deep-subwavelength length scales enables applications in optical sensing and nonlinear optics at the nanoscale. However, the wavelength mismatch between light and SPs prevents direct optical excitation of surface-bound modes, thereby limiting the widespread development of SP-based photonics. Free electrons are a natural choice to directly excite strongly confined SPs because they can supply field components of high momentum at designated positions with subnanometer precision. Here, we theoretically explore free-electron--SP coupling mediated by small scatterers and show that low-energy electrons can efficiently excite surface modes with a maximum probability reached at an optimum surface--scatterer distance. By aligning the electron beam with a periodic array of scatterers placed near a polariton-supporting interface, in-plane Smith--Purcell emission results in the excitation of surface modes along well-defined directions. Our results support using scattering elements to excite SPs with low-energy electrons., Comment: 17 pages, 8 figures, 71 references

Gottfried Leibniz embarked on a research program to prove all the Aristotelic categorical syllogisms by diagrammatic and algebraic methods. He succeeded in proving them by means of Euler diagrams, but didn't produce a manuscript with their algebraic proofs. We demonstrate how key excerpts scattered across various Leibniz's drafts on logic contained sufficient ingredients to prove them by an algebraic method -- which we call the Leibniz-Cayley (LC) system -- without having to make use of the more expressive and complex machinery of first-order quantificational logic. In addition, we prove the classic categorical syllogisms again by a relational method -- which we call the McColl-Ladd (ML) system -- employing categorical relations studied by Hugh McColl and Christine Ladd. Finally, we show the connection of ML and LC with Boolean algebra, proving that ML is a consequence of LC, and that LC is a consequence of the Boolean lattice axioms, thus establishing Leibniz's historical priority over George Boole in characterizing and applying (a sufficient fragment of) Boolean algebra to effectively tackle categorical syllogistic., Comment: 66 pages, 9 figures (some of which include subfigures), 5 tables (one of which includes 2 subtables). A cut-down version of this article, which removes the discussion on diagrammatic logic with Euler diagrams, was submitted to the "History and Philosophy of Logic" journal with a different title

In conventional quantum mechanics (QM), time is treated as a parameter, $t$, and the evolution of the quantum state with respect to time is described by ${\hat {H}}|\psi(t)\rangle=i\hbar \frac{d}{dt}|\psi(t)\rangle$. In a recently proposed space-time-symmetric (STS) extension of QM, position becomes the parameter and a new quantum state, $|\phi(x)\rangle$, is introduced. This state describes the particle's arrival time at position $x$, and the way the arrival time changes with respect to $x$ is governed by ${\hat {P}}|\phi(x)\rangle=-i\hbar \frac{d}{dx} |\phi(x)\rangle$. In this work, we generalize the STS extension to a particle moving in three-dimensional space. By combining the conventional QM with the three-dimensional STS extension, we have a ``full'' STS QM given by the dynamic equation ${\hat { P}}^{\mu}|{\phi }^\mu(x^{\mu})\rangle=- i \hbar~\eta^{\mu\nu}\frac{d}{dx^{\nu}}|{\phi}^\mu (x^{\mu})\rangle$, where $x^{\mu}$ is the coordinate chosen as the parameter of the state. Depending on the choice of $x^\mu$, we can recover either the Schr\"odinger equation (with $x^\mu=x^0=t$) or the three-dimensional STS extension (with $x^\mu=x^i=$ either $x$, $y$, or $z$). By selecting $x^\mu=x$, we solve the dynamic equation of the STS QM for a free particle and calculate the wave function $\langle t,y,z|\phi^1(x)\rangle$. This wave function represents the probability amplitude of the particle arriving at position ($y$,$z$) at instant $t$, given that the detector occupies the entire $yz$-plane located at position $x$. Remarkably, we find that the integral of $|\langle t,y,z|\phi (x)\rangle|^2$ in $y$ and $z$ takes the form of the three-dimensional version of the axiomatic Kijowski distribution., Comment: 7 pages

An alternative quantization rule, in which time becomes a self-adjoint operator and position is a parameter, was proposed by Dias and Parisio [Phys. Rev. A {\bf 95}, 032133 (2017)]. In this approach, the authors derive a space-time-symmetric (STS) extension of quantum mechanics (QM) where a new quantum state (intrinsic to the particle), $|{\phi}(x)\rangle$, is defined at each point in space. $|\phi(x)\rangle$ obeys a space-conditional (SC) Schr\"odinger equation and its projection on $|t\rangle$, $\langle t|\phi(x)\rangle$, represents the probability amplitude of the particle's arrival time at $x$. In this work, first we provide an interpretation of the SC Schr\"odinger equation and the eigenstates of observables in the STS extension. Analogous to the usual QM, we propose that by knowing the "initial" state $|\phi(x_0)\rangle$ -- which predicts any measurement on the particle performed by a detector localized at $x_0$ -- the SC Schr\"odinger equation provides $|\phi(x)\rangle={\hat U}(x,x_0)|\phi(x_0)\rangle$, enabling us to predict measurements when the detector is at $x \lessgtr x_0$. We also verify that for space-dependent potentials, momentum eigenstates in the STS extension, $|P_b(x)\rangle$, depend on position just as energy eigenstates in the usual QM depend on time for time-dependent potentials. In this context, whereas a particle in the momentum eigenstate in the standard QM, $|\psi(t)\rangle=|P\rangle|_t$, at time $t$, has momentum $P$ (and indefinite position), the same particle in the state $|\phi(x)\rangle=|P_b(x)\rangle$ arrives at position $x$ with momentum $P_b(x)$ (and indefinite arrival time). By investigating the fact that $|\psi(t)\rangle$ and $|{\phi}(x)\rangle$ describe experimental data of the same observables collected at $t$ and $x$, respectively, we conclude that they provide complementary information about the same particle..., Comment: 14 pages, 2 figures

Dense micron-sized electron plasmas, such as those generated upon irradiation of nanostructured metallic surfaces by intense femtosecond laser pulses, constitute a rich playground to study light-matter interactions, many-body phenomena, and out-of-equilibrium charge dynamics. Besides their fundamental interest, laser-induced plasmas hold potential for the generation of localized terahertz radiation pulses. However, the underlying mechanisms ruling the formation and evolution of these plasmas is not yet well understood. Here, we develop a comprehensive microscopic theory to predictably describe the spatiotemporal dynamics of laser-pulse-induced plasmas. Through detailed analysis of electron emission, metal screening, and plasma cloud interactions, we investigate the spatial, temporal, and spectral characteristics of the so-generated terahertz fields, which can be extensively controlled through the metal morphology and the illumination conditions. We further describe the interaction with femtosecond electron beams to explain recent ultrafast electron microscopy experiments, whereby the position and temporal dependence of the observed electron acceleration permits assessing the associated terahertz field. Besides its potential application to the design of low-frequency light sources, our work contributes with fundamental insight on the generation and dynamics of micron-scale electron plasmas and their interaction with ultrafast electron pulses., Comment: 14 pages, 8 figures, 41 references

The synergy between free electrons and light has recently been leveraged to reach an impressive degree of simultaneous spatial and spectral resolution, enabling applications in microscopy and quantum optics. However, the required combination of electron optics and light injection into the spectrally narrow modes of arbitrary specimens remains a challenge. Here, we demonstrate microelectronvolt spectral resolution in the nanoscale mapping of photonic modes with quality factors as high as 10$^4$. We rely on mode-matching of a tightly focused laser beam to whispering gallery modes to achieve a 10$^8$-fold increase in light-electron coupling efficiency. By adapting the shape and size of free-space optical beams to address specific physical questions, our approach allows us to interrogate any type of photonic structure with unprecedented spectral and spatial detail

Developing adhesives whose bond strength can be externally manipulated is a topic of considerable interest for practical and scientific purposes. In this work, we propose a method of controlling the adhesion force of a regular fluid, such as water and/or glycerol, confined between two parallel plates by applying an external electric field. Our results show the possibility of enhancing or diminishing the bond strength of the liquid sample by appropriately tuning the intensity and direction of the electric current generated by the applied electric field. Furthermore, we verify that, for a given direction of the electric current, the adhesion force can be reduced enough for the fluid to lose its adhesive properties and begin exerting a force to move apart the confining plates. In these circumstances, we obtain an analytical expression for the minimum electric current required to detach the plates without requiring the action of an external force., Comment: 9 pages, 5 figures

The ultrafast dynamics of charge carriers in solids plays a pivotal role in emerging optoelectronics, photonics, energy harvesting, and quantum technology applications. However, the investigation and direct visualization of such non-equilibrium transport phenomena remains as a long-standing challenge, owing to the nanometer-femtosecond spatio-temporal scales at which the charge carriers evolve. Here, we propose and demonstrate a novel interaction mechanism enabling nanoscale imaging of the femtosecond dynamics of charge carriers in solids. This imaging modality, which we name charge dynamics electron microscopy (CDEM), exploits the strong interaction between terahertz (THz) electromagnetic near fields produced by the moving charges and synchronized free-electron pulses in an ultrafast scanning transmission electron microscope. The measured free-electron energy at different spatio-temporal coordinates allows us to directly retrieve the THz near-field amplitude and phase, from which we reconstruct movies of the generated charges by comparison with microscopic theory. The introduced CDEM technique thus allows us to investigate previously inaccessible spatio-temporal regimes of charge dynamics in solids, for example revealing new insight into the photo-Dember effect, showing oscillations of photo-generated electron-hole distributions inside a semiconductor. Our work lays the foundation for exploring a wide range of previously inaccessible charge-transport phenomena in condensed matter using ultrafast electron microscopy.

Understanding and actively controlling the spatio-temporal dynamics of non-equilibrium electron clouds is fundamental for the design of light and electron sources, novel high-power electronic devices, and plasma-based applications. However, electron clouds evolve in a complex collective fashion on nanometer and femtosecond scales, producing electromagnetic screening that renders them inaccessible to existing optical probes. Here, we solve the long-standing challenge of characterizing the evolution of electron clouds generated upon irradiation of metallic structures using an ultrafast transmission electron microscope to record the charged plasma dynamics. Our approach to charge dynamics electron microscopy (CDEM) is based on the simultaneous detection of electron-beam acceleration and broadening with nanometer/femtosecond resolution. By combining experimental results with comprehensive microscopic theory, we provide deep understanding of this highly out-of-equilibrium regime, including previously inaccessible intricate microscopic mechanisms of electron emission, screening by the metal, and collective cloud dynamics. Beyond the present specific demonstration, the here introduced CDEM technique grants us access to a wide range of non-equilibrium electrodynamic phenomena involving the ultrafast evolution of bound and free charges on the nanoscale.

We investigate high-harmonic generation in graphene heterostructures consisting of metallic nanoribbons separated from a graphene sheet by either a few-nanometer layer of aluminum oxide or an atomic monolayer of hexagonal boron nitride. The nanoribbons amplify the near-field at the graphene layer relative to the externally applied pumping, thus allowing us to observe third- and fifth-harmonic generation in the carbon monolayer at modest pump powers in the mid-infrared. We study the dependence of the nonlinear signals on the ribbon width and spacer thickness, as well as pump power and polarization, and demonstrate enhancement factors relative to bare graphene reaching 1600 and 4100 for third- and fifth-harmonic generation, respectively. Our work supports the use of graphene heterostructures to selectively enhance specific nonlinear processes of interest, an essential capability for the design of nanoscale nonlinear devices.

We reveal a wealth of nonlinear and recoil effects in the interaction between individual low-energy electrons and samples comprising a discrete number of states. Adopting a quantum theoretical description of combined free-electron and two-level systems, we find a maximum achievable excitation probability of $100\%$, which requires specific conditions relating to the coupling strength and the transition symmetry, as we illustrate through calculations for dipolar and quadrupolar modes. Strong recoil effects are observed when the kinetic energy of the probe lies close to the transition threshold, although the associated probability remains independent of the electron wave function even when fully accounting for nonlinear interactions with arbitrarily complex multilevel samples. Our work reveals the potential of free electrons to control localized excitations and delineates the boundaries of such control., Comment: 12 pages, 47 references, 3 figures

We propose a message passing neural network architecture designed to be equivariant to column and row permutations of a matrix. We illustrate its advantages over traditional architectures like multi-layer perceptrons (MLPs), convolutional neural networks (CNNs) and even Transformers, on the combinatorial optimization task of recovering a set of deleted entries of a Hadamard matrix. We argue that this is a powerful application of the principles of Geometric Deep Learning to fundamental mathematics, and a potential stepping stone toward more insights on the Hadamard conjecture using Machine Learning techniques.

Treating the time of an event as a quantum variable, we derive a scheme in which superpositions in time are used to perform operations in an indefinite causal order. We use some aspects of a recently developed space-time-symmetric formalism of events. We propose a specific implementation of the scheme and recover the Quantum SWITCH, where quantum operations are performed in an order which is entangled with the state of a control qubit. Our scheme does not rely on any exotic quantum gravitational effect, but instead on phenomena which are naturally fuzzy in time, such as the decay of an excited atom., Comment: 5 pages, 2 figures, comments and emails welcome (v2: minor address amendments and corrections)

By representing an event as the joint state of a detector-timer couple that interact with a system, we recover the familiar tensor product structure, used to describe spatially separated systems, in the context of timelike events. Furthermore, with this approach, we extend the superposition principle to the moment of occurrence of events. We then outline quantum signatures of causality that manifest through coherence in the detector state and correlation functions of time operators. Finally, we expand the scope of quantum information theoretic measures of state discrimination and information content, commonly used to characterize spatially separated systems, to events in spacetime. For causally connected events, we illustrate a deterministic relationship between events (akin to spatially entangled physical systems) where observing a previous event (one subsystem), enables us to delineate a later event (the other subsystem).

Capillary refill time (CRT) is a method for evaluating peripheral perfusion by visual assessment. CRT is especially useful for quick evaluations in the absence of sophisticated equipment. However, there are repeatability and reproducibility limitations with CRT, especially for dark skin. To test the limits of CRT repeatability and skin color independence, we developed a system and method to perform simple and robust CRT measurements. The system consists of an RGB camera and an LED lamp, with crossed circular polarizers imaging to attenuate the light reflected by the superficial layer of the skin. The capillary refill time is determined using an exponential regression on the time-dependent green channel mean pixel intensity of the region of interest after the compression is released. We limited this regression up to a data-dependent cut-off time, after which we assume the exponential model is invalid, and used the confidence interval of the uncertainty to develop a criterion to flag and discard faulty measurements. We tested the system on twenty-two volunteers with skin phototypes ranging from I to VI on the Fitzpatrick scale, applying to their forearms a 7 kPa compression for 5 s. After the release of measurements flagged as inadequate (about 20\% of measurements) by our regressions, our results indicated good precision, with high repeatability for all skin phototypes. Approximately 80\% of measurements fall within $\pm 20\%$ of the individual's expected value for CRT (mean CRT value). Our results suggest CRT can be used as a quantitative measurement and encourages further developments for the implementation of a similar method on smartphone cameras for quick and robust CRT measurements in patients' triage, monitoring, and telehealth., Comment: 14 pages, 6 figures

Current advances in ultrafast electron microscopy make it possible to combine optical pumping of a nanostructure and electron beam probing with sub{\aa}ngstrom and femtosecond spatiotemporal resolution. We present a theory predicting that this technique can reveal a rich out-of-equilibrium dynamics of plasmon excitations in graphene and graphite samples. In a disruptive departure from the traditional probing of nanoscale excitations based on the identification of spectral features in the transmitted electrons, we show that measurement of angle-resolved, energy-integrated inelastic electron scattering can trace the temporal evolution of plasmons in these structures and provide momentum-resolved mode identification, thus avoiding the need for highly-monochromatic electron beams and the use of electron spectrometers. This previously unexplored approach to study the ultrafast dynamics of optical excitations can be of interest to understand and manipulate polaritons in 2D semiconductors and other materials exhibiting a strong thermo-optical response., Comment: 24 pages, 13 figures, 101 references

Surface polaritons display short wavelengths compared to propagating light, thus enabling large spatial concentration and enhancement of electromagnetic energy. However, this wavelength mismatch is generally accompanied by poor light-to-polaritons coupling that limits potential applications in areas such as optical sensing and optoelectronics. Here, we address this problem by demonstrating that a small scatterer placed at a suitable distance from a planar surface can produce complete coupling of a focused light beam to surface polaritons. We present rigorous theoretical results for light coupling to plasmons in silver films and graphene, as well as phonon polaritons in hexagonal boron nitride films. We further formulate detailed general prescriptions on the beam profile and particle response that are required to achieve maximum coupling, which we supplement by analytical calculations for dipolar scatterers and finite-size particles. Our results open a practical route to circumvent the long-standing photon-polariton wavelength mismatch problem in nanophotonics., Comment: 18 pages, 14 figures, 58 references

We compute explicit equations for all (universal coverings of) Godeaux surfaces with torsion group $\mathbb Z/2.$ We show that their moduli space is irreducible and unirational of dimension 8., Comment: Exposition improved. Some references added

Although time is one of our most intuitive physical concepts, its understanding at the fundamental level is still an open question in physics. For instance, time in quantum mechanics and general relativity are two distinct and incompatible entities. While relativity deals with events (points in spacetime), with time being observer-dependent and dynamical, quantum mechanics describes physical systems by treating time as an independent parameter. To resolve this conflict, in this work, we extend the classical concept of an event to the quantum domain by defining an event as a transfer of information between physical systems. Then, by describing the universe from the perspective of a certain observer, we introduce quantum states of events with space-time-symmetric wave functions that predict the joint probability distribution of a measurement (observation) at $ (t, {\vec x}) $. Under these circumstances, we propose that a well-defined instant of time, like any other observable, arises from a single event, thus being an observer-dependent property. As a result, a counterfactual asymmetry along a particular sequence of events within a stationary quantum state gives rise to the flow of time as being successive "snapshots" from the observer's perspective. In this proposal, it is the many distinguishable states in which the observer stores information that makes the existence of time possible., Comment: 17 pages

We compute equations for Coughlan's family of Godeaux surfaces with torsion $\mathbb Z/2$, which we call $\mathbb Z/2$-Godeaux surfaces, and we show that it is (at most) 7 dimensional. We classify non-rational KSBA degenerations $W$ of $\mathbb Z/2$-Godeaux surfaces with one Wahl singularity, showing that $W$ is birational to particular either Enriques surfaces, or $D_{2,n}$ elliptic surfaces, with $n=3,4$ or $6$. We present examples for all possibilities in the first case, and for $n=3,4$ in the second.

Let $S$ be an abelian surface over an algebraically closed field $k$ with characteristic different from $2$ and $3$, and $\mathcal{L}$ a symmetric ample line bundle defining a polarisation of type $(1,3)$. Then the linear system $|\mathcal{L}|$ defines a covering map $\varphi\colon S\rightarrow \mathbb{P}^2$ of degree $6$. Furthermore, if $|\mathcal{L}|$ is base point free, then $\varphi_*\mathcal{O}_S = \mathcal{O}_{\mathbb{P}^2} \oplus \Omega^1_{\mathbb{P}^2} \oplus \Omega^1_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(-3)$. Using this decomposition, in this paper we construct the graded coordinate ring of $(S,\mathcal{L},\theta)$, where $\theta\colon G(\mathcal{L})\xrightarrow{\sim} H(1,3)$ is a level structure of canonical type. As a corollary we prove that the moduli space of such triples is rational.

Polaritonic modes in two-dimensional van der Waals materials display short in-plane wavelengths compared with light in free space. As interesting as this may look from both fundamental and applied viewpoints, such large confinement is accompanied by poor in/out optical coupling, which severely limits the application of polaritons in practical devices. Here, we quantify the coupling strength between light and 2D polaritons in both homogeneous and anisotropic films using accurate rigorous analytical methods. In particular, we obtain universal expressions for the cross sections associated with photon-polariton coupling by point and line defects, as well as with polariton extinction and scattering processes. Additionally, we find closed-form constraints that limit the maximum possible values of these cross sections. Specifically, the maximum photon-to-plasmon conversion efficiency in graphene is $\sim10^{-6}$ and $\sim10^{-4}$ for point and line scatterers sitting at its surface, respectively, when the plasmon and photon energies are comparable in magnitude. We further show that Mie resonators placed at an optimum distance from the film can boost light-to-polariton coupling to order unity. Our results bear fundamental interest for the development of 2D polaritonics and the design of applications based on these excitations., Comment: 22 pages, 6 figures, 4 tables, 54 references

In this paper we analyze the effects of nonlocality on the optical properties of a system consisting of a thin metallic film separated from a graphene sheet by a hexagonal boron nitride (hBN) layer. We show that nonlocal effects in the metal have a strong impact on the spectrum of the surface plasmon-polaritons on graphene. If the graphene sheet is shaped into a grating, we show that the extinction curves can be used to shed light on the importance of nonlocal effects in metals. Therefore, graphene surface plasmons emerge as a tool for probing nonlocal effects in metallic nanostructures, including thin metallic films. As a byproduct of our study, we show that nonlocal effects lead to smaller losses for the graphene plasmons than what is predicted by a local calculation. We show that these effects can be very well mimicked using a local theory with an effective spacer thickness larger than its actual value., Comment: 16 pages

The ability to confine light into tiny spatial dimensions is important for applications such as microscopy, sensing and nanoscale lasers. While plasmons offer an appealing avenue to confine light, Landau damping in metals imposes a trade-off between optical field confinement and losses. We show that a graphene-insulator-metal heterostructure can overcome that trade-off, and demonstrate plasmon confinement down to the ultimate limit of the lengthscale of one atom. This is achieved by far-field excitation of plasmon modes squeezed into an atomically thin hexagonal boron nitride dielectric h-BN spacer between graphene and metal rods. A theoretical model which takes into account the non-local optical response of both graphene and metal is used to describe the results. These ultra-confined plasmonic modes, addressed with far-field light excitation, enables a route to new regimes of ultra-strong light-matter interactions., Comment: 17 pages, 4 figures

It is well known that nonrelativistic quantum mechanics presents a clear asymmetry between space and time. Much of this asymmetry is attributed to the lack of Lorentz invariance of the theory. Nonetheless, a recent work [Phys. Rev. A \textbf{95}, 032133 (2017)] showed that even though this is partially true, there is a broader physical scenario in which space and time can be handled in nonrelativistic quantum theory in a more symmetric way. In this space-time-symmetric formalism, an additional Hilbert space is defined so that time is raised to the status of operator and position becomes a parameter. As a consequence, the Hilbert space now requires a space-conditional quantum state governed by a new quantum dynamics. In this manuscript, we reveal some physical and mathematical properties of the space-time-symmetric formalism such as: symmetries between the Hamilton-Jacobi and the space-conditional equation; the general solution for a time-independent potential; and a new Lagrangian for a spinless particle in one dimensional. Finally, we present the space-conditional equation for a particle under the effect of an electromagnetic field, and the gauge invariance of this equation is proved., Comment: 9 pages

The question of how long a particle takes to pass through a potential barrier is still a controversial topic in quantum mechanics. Arguably, the main theoretical problem in obtaining estimates for measurable times is the fact that previously defined time operators, that remained within the borders of standard quantum mechanics, present some kind of pathology. Recently, a time operator acting on an additional Hilbert space has been shown to support both Hermiticity and canonical relation with an energy observable. The theory is built in a framework which treats space and time as symmetrically as possible, in the nonrelativistic regime. In this work, we use this formalism to derive a closed analytic expression for the traversal time of a quantum particle impinging on a constant-potential barrier. We test our theory in the specific experimental scenario of a realization by Rafagni et al [Appl. Phys. Lett. {\bf 58}, 774 (1991)]. The proposed approach displays a much better performance in comparison with the B\"uttiker-Landauer and the phase-time approximations., Comment: 6 pages, 1 figure

Let $Y$ be an algebraic variety, $\mathcal{F}$ a locally free sheaf of $\mathcal{O}_Y$-modules, and $\mathcal{R}(\mathcal{F})$ the $\mathcal{O}_Y$-algebra $\operatorname{Sym}^\bullet \mathcal{F}$. In this paper we study local properties of sheaves of $\mathcal{O}_{\mathcal{R}(\mathcal{F})}$-ideals $\mathcal{I}$ such that $\mathcal{R}(\mathcal{F}))/\mathcal{I}$ is an algebraic cover of $Y$. Following the work of Miranda for triple covers, for $\mathcal{Q}$ a direct summand of $\mathcal{R}(\mathcal{F})$, we say that a morphism $\Phi\colon \mathcal{Q}\rightarrow\mathcal{R}(\mathcal{F})/\langle\mathcal{Q}\rangle$ is a covering homomorphism if it induces such an ideal. As an application we study in detail the case of Gorenstein covering maps of degree $6$ for which the direct image of $\varphi_*\mathcal{O}_X$ admits an orthogonal decomposition. These are deformation of $S_3$-Galois branch covers.