Papers

In this article we consider the Drinfeld-Sokolov hierarchy of type $A$ and its similarity reduction. Then the similarity reduction is expressed as $2$ types of Hamiltonian systems, one is with a Poisson bracket and another a Poisson bracket of canonical coordinates. We also investigate a connection between the similarity reduction and the isomonodromy deformation system.

Lesniewski and Ruskai conjectured that the contraction coefficient of every monotone Riemannian metric under a unital stochastic map equals the Hilbert–Schmidt contraction on the traceless subspace. We disprove the conjecture with an explicit entanglement-breaking qutrit channel induced by a doubly stochastic $3\times3$ matrix. A faithful diagonal state and a commuting traceless tangent give, simultaneously for every normalized monotone metric, the exact lower bound $\eta^{\mathrm{Riem}}_{\kappa}(\Phi_K)\ge 8896/20007>(62+2\sqrt{61})/225=\Lambda_2(\Phi_K^{\dagger}\Phi_K)$. The counterexample is entirely classical on a maximal abelian subalgebra. A theorem of Hiai and Ruskai establishes the conjectured identity for all unital qubit maps, so dimension three is minimal among full matrix algebras.

We compare isotypic (n-SiC/n-Si) and anisotypic (n-SiC/p-Si) heterojunctions using Kelvin probe force microscopy, scanning tunnelling spectroscopy, precision electrical measurements with a Keithley 2450 SourceMeter, and a 0.1 farad supercapacitor charge-storage test. Two-wire probing shows an enhanced low-bias current in the isotypic structures; four-wire Kelvin sensing removes the voltage-drop contribution of external leads and contacts and collapses this current to a much lower bulk and interface limited level. The high density of interface states detected by KPFM and STS, up to five times ten to the twelfth per square centimetre per electronvolt in dislocation-rich regions, provides a plausible origin for the enhanced local response. Anisotypic junctions exhibit a genuine diode-like current-voltage characteristic and an open-circuit voltage of about 1.2 volts under beta irradiation. In the capacitor test with the patented charge-pump converter, the anisotypic structure reaches 0.25 volts versus 0.05 volts for the isotypic one after 30 minutes, giving an energy ratio of about 25. We formulate a measurement-interpretation framework based on four-wire Kelvin sensing, generator polarity with cathode to LO, and high-impedance output-off state. Our results show that anisotypic SiC/Si heterojunctions are the preferred platform for betavoltaic conversion when measured correctly.

We compare the specific dissipation $Q^{-1}\left( \omega \right) $ for the Becker, Lomnitz, and Lambert models in linear viscoelasticity. Graphically, the specific dissipation of these models behaves similarly as $\omega \rightarrow 0^{+}$ and $\omega \rightarrow +\infty$. Furthermore, we obtain analytic formulas for the asymptotic behavior of $Q^{-1}\left( \omega \right) $ in the Becker and Lomnitz models as $\omega \rightarrow 0^{+}$ and $\omega \rightarrow +\infty$, where the latter limit coincides with the specific dissipation of the Maxwell model. We also derive a novel closed-form expression for the specific dissipation in the generalized Becker model, $Q_{\nu }^{-1}\left( \omega \right) $, when the parameter $\nu \in \left( 0,1\right] $ is rational. When $\nu \rightarrow 0$, we recover the specific dissipation of the Maxwell model, whereas when $\nu =1$, we recover that of the original Becker model. In addition, we derive two new closed-form expressions for the specific dissipation for the extended Jeffreys–Lomnitz model $Q_{\alpha }^{-1}\left( \omega \right) $, the first being valid for $\alpha \in \left( 0,1\right] $, and the second for $\alpha \in \left( -\infty ,1\right] $. When $\alpha \rightarrow 0$, we recover the specific dissipation of the Lomnitz model, whereas when $\alpha =1$, we recover that of the Maxwell model. Finally, we conclude that the asymptotic behavior of $Q_{\alpha }^{-1}\left( \omega \right) $ as $\omega \rightarrow +\infty $ coincides with the specific dissipation of the Maxwell model. Keywords:

Evolving crack fields in structural health monitoring and fracture assessment must often be inferred from sparse mechanical measurements rather than dense full-field observations. We develop CNN2D–ConvGRU, a convolutional-recurrent framework for measurement-conditioned reconstruction of time-dependent phase-field brittle fracture. At each load step, the model maps a fixed-length history of phase and displacement fields and sparse current-step displacement measurements to the current full-field state. New measurements are assimilated during sequential deployment, making the framework a state-inference surrogate rather than an autonomous time integrator. It reproduces crack paths, damage evolution, and bulk displacement response, with the largest errors near propagating crack tips and steep displacement gradients and some drift at late stages. Without retraining, weights learned on a $256 \times 256$ raster are evaluated on a $512 \times 512$ raster of the same physical domain and finite-element discretization using a proportionally refined measurement grid. This empirical raster-and-sensing transfer preserves the principal damage topology and global damage evolution, although fine-scale displacement errors increase near crack tips. Comparisons with alternative spatial and temporal architectures show that CNN2D–ConvGRU offers a favorable balance between reconstruction accuracy and computational cost. Relative to repeated finite-element solutions, sequential reconstruction achieves mean speedups of $175\times$ on CPU and $253\times$ on GPU. These results demonstrate efficient full-field fracture-state reconstruction from sparse observations while retaining the spatial structure and history dependence of phase-field fracture.

Colour-kinematics duality is, even at tree level, not manifest from the standard Lagrangian of Yang-Mills theory in that the usual Feynman-diagram expansion does not follow the kinematic Jacobi identities. For a Lagrangian manifesting colour-kinematics duality, the kinematic Jacobi identities of the Feynman-diagram expansion follow from the existence of a second-order differential operator $\mathsf{b}$ acting on the algebra of colour-stripped fields that forms part of the data of a BV$^\square$-algebra. We argue that the existence of differential operators of order greater than two implies weaker but nontrivial fragments of kinematic Jacobi identities. We further show that a superspace formulation of the first-order Yang-Mills action in the Batalin-Vilkovisky formalism admits a differential operator of order six in every spacetime dimension. Therefore, the Feynman-diagram expansion of tree and loop scattering amplitudes of the first-order formulation of Yang-Mills theory automatically enjoy a weak form of colour-kinematics duality.

We study the wave kinetic equation (WKE) derived by Shavit–Bühler–Shatah from the two-dimensional Boussinesq system of internal waves. The equation is anisotropic and vector-valued, with a two-branch sign-changing dispersion relation, coupled propagation branches, and a sign-indefinite pseudo-momentum invariant. The resonant manifold has angular degeneracies that obstruct the standard analytic treatment of the collision operator. We introduce a natural angular cut-off adapted to these degeneracies, prove boundedness of the cut-off collision operator, and establish local well-posedness in weighted $L^\infty$ spaces. We also show that, even with the zero-frequency cut-off retained, removing the cut-off near $|\cos\theta|=\tfrac12$ makes the collision operator unbounded on these weighted spaces. This provides, to our knowledge, the first rigorous analytic framework for an anisotropic vector-valued WKE arising from Boussinesq dynamics.

Stochastic geometric mechanics (SGM) is known for its potential utility in quantifying uncertainty in global climate modelling of the Earth's ocean and atmosphere while also preserving the fundamental advective transport properties of ideal fluid flow. This paper is a pedagogical review of the recent developments of the mathematical framework of stochastic geometric mechanics obtained from Lie group-invariant stochastic variational principles in the context of model building for upper ocean dynamics, The paper is divided into the following five parts. Part I discusses the origins of geometric mechanics applications in deterministic fluid dynamics. Part II focuses on the example of the deterministic 3D Euler Boussinesq (EB) equations. Part III adds stochastic transport to the 3D Euler Boussinesq (EB) and derives its SALT equations. (SALT is the abbreviation of Stochastic Advection by Lie Transport.) Part IV focuses on Lagrangian Averaged Stochastic Lie Transport, abbreviated as LA-SALT. LA-SALT treats atmospheric `climate' as the ensemble expectation, while the atmospheric `weather' is treated as a field of pathwise fluctuations, as discussed in Ed Lorenz's famous 1995 lecture. Part V applies SALT and LA-SALT to create stochastic Ocean–Atmosphere Models, abbreviated as SOAM.. The SOAM approach brings us back to Hasselmann's 1976 paradigm, which decomposes a general climate model into its deterministic and stochastic parts.

The discrete cylinder phase space offers a natural setting for twisted photons, Josephson junctions, and certain bosonic codes. In that context, we build a number-of-zeros hierarchy for the stellar distribution of states on the discrete cylinder phase space, tied to usefulness for quantum computation. The orbital angular momentum (OAM) and phase eigenstate, having no zeros, sits at the bottom of the hierarchy and share mathematical properties with the coherent state on the plane phase space, both sharing zero rank and a positive Wigner distribution. This reveals a structural peculiarity of the cylinder: its coherent state—the wrapped coherent state—is instead the most quantum, carrying infinitely many zeros, so "coherent state" need not signal classicality across different phase-space geometries. We identify the operations that preserve or change the zero count, giving explicit access to every rank of the hierarchy. We then ask whether this hierarchy also governs localization. Using the $L^r$ norm, the inverse participation ratio, and the Wehrl entropy as complementary measures of spread in the Husimi density, we find that the two hierarchies need not track one another: the wrapped coherent state can be more localized than the OAM eigenstate even though it carries infinitely many zeros.

In the late 1980s, Baulieu and Grossman demonstrated that the supersymmetric formulation of Langevin stochastic differential equations (SDEs), proposed earlier by Parisi and Sourlas, belongs to the family of Witten-type topological field theories (TFTs). From a certain angle, this finding may appear puzzling: TFTs have no local degrees of freedom, whereas SDEs do exhibit local fluctuations. In this paper, we address this apparent contradiction in the context of the supersymmetric theory of stochastic dynamics (STS), a generalization of the Parisi-Sourlas-Baulieu-Grossman approach to SDEs of arbitrary form. We further discuss how, within STS, dynamical chaos can be identified with the spontaneous breakdown of the corresponding topological supersymmetry, while 1/f noise can be understood as a consequence of the Goldstone theorem. The butterfly effect, in turn, calls for an effective field-theoretic description that may itself possess a hidden topological structure, as we speculate on the basis of the Ginzburg-Landau approach and insights from AdS/CFT duality.

We investigate integrability of Euler-Lagrange equations associated with 3D second-order Lagrangians of the form \begin{equation*} \int f(u_{xy},u_{xt},u_{yt})\ \text{d}x\text{d}y\text{d}t. \end{equation*} It is demonstrated that there are exactly four different types of such Lagrangian densities $f$: the first one is given by the formula $f=\sqrt{u_{xy}u_{xt}u_{yt}}$, the second and the third are more complicated (although still representable in elementary functions), whereas the most generic fourth one is expressible in terms of the Lobachevsky function, revealing unexpected links to spherical/hyperbolic trigonometry and Schläfly-type formulas. Dispersionless Lax pairs and integrable dispersive deformations of the corresponding Euler-Lagrange equations are also constructed.Remarkably, dispersive deformation of the first Lagrangian density coincides with the Lagrangian of the classical Darboux system arising in the theory of triply-orthogonal coordinate systems in $\mathbb{R}^3$. Dispersive deformations of the three other cases provide Lagrangian formulation of semi-discrete and fully discrete versions of the Darboux system, with one, two and three discrete variables, respectively.

Satellite gravimetry has become essential in many areas of earth science. However, the resolution of satellite gravitational models remains low at scales of a few hundreds km and no gravity recovery methods can take full advantages of unprecedented high accuracy of satellite tracking measurements. We first provide a unified theoretical framework of parameter estimation in differential equations for satellite gravimetry and then briefly review the mathematical methods to compute the gravity field of the Earth from satellite tracking. We focus on the collocation method, Kaula linear perturbations, two-point boundary value problems and orbit-energy-based methods. The numerical integration method is also included in this review, though it has been proved to be mathematically incorrect and physically not permitted. The reason is that it has become the standard method to routinely produce global gravitational models from satellite tracking data, which have been widely applied in many different areas of earth science. Because it is not clear how the incorrect foundation would affect gravity products from satellite tracking, we do not review any applications of these products. We then present a measurement-based perturbation theory to estimate the gravity field of the Earth, which can fully utilize both precise satellite orbits of arbitrary length and unprecedented high accuracy of satellite and inter-satellite tracking. The method is theoretically free of modeling errors, is capable of extracting any small forces from satellite and inter-satellite tracking data and provides a guarantee for high-precision and high-resolution global gravity models. Finally, we assume a reference gravity model and derive local solutions to the Newton's nonlinear governing differential equations of satellite motion for scattered tracking data that can still be important in some applications.

Solving the time-dependent Schrödinger equation (TDSE) via traditional numerical methods is computationally intensive. Transformer models offer a compelling alternative, but standard implementations rely on soft constraints that cannot rigorously guarantee probability conservation. Here, we introduce a Transformer architecture that enforces TDSE probability conservation as a hard constraint. The design intrinsically ensures unitarity across temporal evolution without requiring repeated retraining. Our empirical findings show that this hard-constraint approach is not only physically exact but also computationally superior to conventional soft-constraint methods.

The weighted Max-Cut problem is an NP-hard problem with application implications. It is investigated on a cubic lattice with 113 nodes and mixed-signed random edge weights. For a fixed upper bound on edge weights, it has been demonstrated that the computational difficulty increases as the lower bound on edge weights becomes more negative. The solution time for the problem using a novel error-mitigated quantum annealing approach is compared with standard D-Wave quantum annealing (QA) and BQM hybrid solvers, as well as various classical solvers. For the QPU-embeddable weighted Max-Cut instances with mixed-signed edge weights, it has been quantitatively demonstrated that the SEMO (spin-error mitigation for optimisation) error-mitigated quantum annealing achieved substantially shorter time-to solution than standard D-Wave QA, D-Wave BQM, simulated annealing and Tabu search baselines. The error-mitigated quantum annealing approach presented in this article potentially elevates the efficiency and application scope of quantum annealing and would be applicable in solving other discrete optimisation problems that can be formulated as QUBO or Ising instances. The promising solution time advantage would be particularly impactful for time-critical optimisation applications.

Periodically driven quantum systems are commonly modeled using master equations constructed in the eigenbasis of an undriven Hamiltonian, implicitly assuming that environmental dissipation couples to static energy transitions even under strong time-periodic driving. The validity of this approximation beyond weak or near-resonant driving remains poorly understood. To address the need for a more self-consistent quantum theory approach, we formulate a nonsecular Floquet–Markov generalized master equation (F-GME) in the quasienergy basis, treating interaction-induced (internal) and drive-induced (external) nonperturbative dressing on an equal footing. We subsequently investigate dissipation in two minimal driven open quantum systems—a harmonically driven two-level system and a harmonically driven coupled-two-level-system—each weakly coupled to a Markovian bath. Comparing the F-GME to a time-independent dressed-basis master equation, we show that even for a flat-bath spectral density and weak dissipation, the two approaches can yield qualitatively different steady-state populations and emission spectra. We resolve dissipation into drive-assisted sideband processes decaying via Floquet extended-space quasienergy channels, and show these channels can hybridize through nonsecular couplings into collective Floquet–Liouville modes governing observable spectral resonances. This analysis demonstrates that time-independent dissipative descriptions can incorrectly weight multiphoton Floquet transitions by collapsing quasienergy-resolved decay pathways into static energy gaps. The F-GME framework provides a systematic diagnostic for identifying regimes where Floquet-consistent dissipation is essential and clarifies the physical origin of discrepancies between commonly used master-equation approaches.

Widely used in mechanics and mathematical physics, physical components remove the inherent coordinate-dependent scaling in curvilinear coordinates, yielding components of consistent physical dimension. In orthogonal coordinates, they are constructed by normalizing the coordinate frame and coframe; in general coordinates, however, their construction requires additional, non-trivial choices. In this paper, we extract physical components of arbitrary tensors on arbitrary Riemannian manifolds by orthonormalization of the coordinate frame. We further formulate a general normalization framework distinguishing three requirements: dimensional consistency, dual frame-coframe compatibility, and unit normalization. We show that dimensional consistency alone leaves independent general linear gauge freedoms for the contravariant and covariant components. Further requiring dual compatibility locks these into a single general linear gauge; a dual-compatible frame and coframe are both of unit length if and only if they are orthonormal. Thus, the choice of orthonormal transformations emerges as the only physical-components framework that satisfies all three requirements. We apply this framework to nonlinear elasticity and inelasticity, referring broadly to constitutive responses involving internal distortions, of which we study anelasticity, viscoelasticity, and visco-anelasticity. We examine the deformation gradient, inelastic distortions, strain measures, and stress tensors. We conclude by arguing that physical components remain neither intrinsic nor unique.

Can the characteristic monodromy of string scattering be derived from S-matrix consistency alone, without assuming a worldsheet? We show that finite-particle consistency fixes it to a sharply defined extent. In a massless doubly ordered identity model, two exact six-point residues generate a scalar pentagon and the odd Fay identity, forcing every analytic four-point null kernel to satisfy $q''=\kappa q$ and leaving only the linear, trigonometric, and hyperbolic branches. This converts finite-multiplicity factorization into an all-orders constraint normally associated with worldsheet geometry. For a crossing-symmetric target in the opposite KLT module, positive unsubtracted dispersion with nonzero heavy spectral weight selects the trigonometric branch, and moment determinacy fixes the Veneziano germ up to scale. A two-scale counterexample proves that module entrance is an independent input. Our result therefore identifies both a finite-to-infinite rigidity mechanism for string scattering and the precise missing principle needed for a first-principles derivation.

These notes are devoted to the kinetic Fokker–Planck equation on the tangent bundle of a compact Riemannian manifold, with motivations coming from relativistic kinetic theory of gases (à la Debbasch) and geometric analysis (à la Bismut). They are developed in particular from collaboration with Fabrice Debbasch and Yann Ollivier. The presentation is mostly self-contained, intended for readers without much prior familiarity with geometric analysis. Once the kinetic equations are reviewed, I expand the basic toolbox from functional and spectral analysis. Then the main focus is on global hypoelliptic $C^\infty$ regularization and hypocoercive exponential equilibration, both of which are addressed both in $L^2$ and $L^1$ settings. Various types of interpolation inequalities, spectral gap and entropic inequalities are established and used. Several existing results are simplified and unified, along with new ones. Some directions of research are mentioned.

We consider quantum group generalizations of the action of monomial matrices on complex Hadamard matrices. This gives rise to various notions of quantum symmetries and quantum equivalences of Hadamard matrices. We show that if one acts by a certain largest monomial quantum group, then all Hadamard matrices of a given size become quantum equivalent. Taking a more restrictive quantization leads to a notion of $s$-quantum equivalence. We exhibit examples of Butson matrices of the same size and order that are not $s$-quantum equivalent for any choice of $s$. We also show that $s$-quantum equivalence is operationally modeled by a synchronous non-local “Hadamard equivalence” game. Our methods are largely graphical calculus based, using categories generated by complementary spiders. We use these same tools to study quantum affine equivalence of quantum groups, and prove that all finite quantum groups of the same size are quantum affinely equivalent. We also provide a short graphical proof of a result of Kasprzak–Sołtan–Woronowicz asserting that quantum symmetries of finite quantum groups must be classical.

We study finite-alphabet spin models on trees evolving under independent symmetric spin-flip dynamics. On the lattice the time-evolved plus measure of the low temperature Ising model in zero external field was shown to be non-quasilocal for all sufficiently large times in \cite{EnFeHoRe02}. On the tree however the time-evolved plus phase of the Ising model behaves differently, as the quasilocal Gibbs property is first lost, but then recovered at a later time \cite{EnErIaKu12}. We first extend this result by showing that large-time reentry into the quasilocal Gibbs regime on the tree happens more generally for all finite alphabet models, when the dynamics is started in $f$-stable Gibbs states. These are defined in terms of a time-independent sharp condition on the homogeneous recursion $f$ describing the model, which can be checked explicitly. As our second and opposite result, we develop a method for proving large-time persistence of non-quasilocality of time-evolved measures. We show that even all configurations can become bad at large times, when the initial Gibbs measure corresponds to an $f$-saddle and give an explicit illustration for the Potts model. In our proofs we study the influence of spatially inhomogeneous perturbations to the solutions of a time-dependent fixed point problem.

At a black-hole exceptional point (EP), two quasinormal modes coalesce and their separate residues become ill-conditioned. Rather than postulating a near-degenerate modal fit, we derive the response constructively from the complex-scaled Regge–Wheeler–Zerilli resolvent, treating the modes as one isolated rank-two Riesz cluster. Its zeroth and first contour moments determine an exact pair resolvent on both sides of, and at, the EP, without labeling the individual modes or constructing a normalized Jordan chain. At a second-order EP, these moments determine the simple- and double-pole Laurent operators. Although the modal decomposition is singular, fixed-real-frequency transmission and the greybody factor remain real-analytic through the EP, provided that the cluster remains isolated, the complementary resolvent is regular, and no pole reaches the physical axis. Source–observer matrix elements of the Laurent operators define finite, normalization-independent amplitudes and fix both the constant and linear-in-time terms in the causal ringdown. Their equality with the coefficients from the Jost double-zero expansion shows that they are operator-defined coefficients of the specified physical response, rather than fitting parameters. Thus two cluster moments provide mode-label-free data from which both scattering and driven responses follow.

We study a general class of gradient interface models with Hamiltonian $H=\beta\sum V(\nabla\phi)$, $\beta>0$, assuming essentially that the potential $V$ is even, $V'(s)\ge \alpha s$ on $[0,\infty)$ for some $\alpha>0$, and $-M\leq V''\le C$. We establish a Helffer-Sjöstrand representation for these models, and use it to prove that their scaling limits are Gaussian Free Fields (GFFs), and that their covariances decay at the same rate as the GFF. This extends results for strictly convex potentials to a large class of non-convex potentials and to arbitrary temperatures. Additionally, we prove Brascamp-Lieb and dimension-free Poincaré inequalities for the models. We obtain these results by representing the interface as a mixture of gradient interface models with strictly convex potentials, extending an idea by Biskup-Spohn who had considered mixtures of Gaussians at moderate inverse temperature $\beta=1$. The construction of such a representation is one of the key new contributions of this work.

After a historical introduction, the most important classical and quantum entropies are introduced as constructions in classical and quantum probability theory. Classical entropies are studied from large deviation theory, including theorems of Sanov, Cramér, Gärtner-Ellis, and Varadhan, and are illustrated in some applications to both Boltzmannian and Gibbsian statistical physics. Quantum entropies superficially connect to classical entropies at the formula level, but more deeply do so via the crucial role of entropy in statistical hypothesis testing. The classical (relative) Kullback-Leibler entropy, its quantum counterpart introduced by Umegaki, as well as their deformations proposed by Renyi all fit naturally in this context. Quantum entropy faces the new problem of defining and computing the relative entropy of a pair of states on a subsystem, here formalized as a von Neumann algebra. This requires modular (aka Tomita-Takesaki) theory, which provides the framework for the relative quantum entropies introduced by Araki and Uhlmann (these encompass both the Kullback-Leibler and Umegaki entropies as special cases). To (re)define and compute these entropies in terms of density operators and traces, further constructions are needed, namely Haagerup's noncommutative L^p spaces. Von Neumann algebras also provide the setting for the Connes-Stormer-Narnhofer-Thirring entropy, which is a quantum version of the Kolmogorov-Sinai entropy in dynamical systems and ergodic theory, to which we also provide an introduction. This course was originally inspired by, and should be relevant to, black hole thermodynamics, although we discuss neither this application nor the second law. The course tries to be both mathematically rigorous and interesting to theoretical physicists. Prerequisites are undergraduate probability theory, functional analysis, and quantum theory.

We construct an exact solution-generating correspondence between restricted ansatz sectors of quartic biadjoint scalar theory, complexified $SU(2)$ Yang–Mills theory and four-dimensional general relativity with conformally flat metrics and traceless energy-momentum. A factorised biadjoint field, a Corrigan–Fairlie–'t Hooft–Wilczek gauge potential and a conformal metric are generated by a common scalar seed. Their equations of motion reduce to the same equation of motion under the identification of couplings, which retains the nonlinear dynamics on all three legs of the correspondence. The scalar equation captures only the trace of the Einstein equations; deriving the trace-free part of the Einstein equation additionally yields a source relation between Gravity and Yang-Mills. From this, we construct a non-perturbative classical double copy for the restricted ansatz sectors between quartic biadjoint scalar theory, $SU(2)$ Yang–Mills theory and gravity. We illustrate the dictionary of mappable solutions with rational and constant profiles representing $AdS_4$, planar $dS_4$ and Minkowski space, and with Jacobi-elliptic profiles generating a positive-energy recollapsing branch for negative cosmological constant and a negative-energy bouncing branch for positive cosmological constant. Using these results, we provide a new example of the Kerr-Schild double copy.

How many Slater determinants does an accurate variational description of interacting fermions require? Exact expansions in a finite basis need combinatorially many, and state-of-the-art fermionic neural quantum states stack growing numbers of them. We prove that, in the norms that govern variational calculations, at most two are needed, independently of the number of particles and of the target accuracy. A single universal Slater determinant-specified in advance, independent of both the system and the state-multiplied by a smooth bosonic wave function approximates any fermionic wave function in up to three spatial dimensions in the first-order Sobolev norm, which controls the variational energy. Reaching the second-order Sobolev norm-for Coulomb interactions, the domain of the Hamiltonian, which bounds the variance of the local energy at the core of variational Monte Carlo-requires at most one additional fixed determinant, and only in three dimensions. Antisymmetry therefore costs at most two universal determinants and no expressiveness: generalized Slater-Jastrow neural quantum states are variationally complete.

We initiate the study of four point functions in the symplectic bi-Grassmannian framework for four dimensional conformal field theories. We derive factorization formulae analogous to those for scattering amplitudes and bootstrap several examples of AdS$_5$ exchange correlation functions involving scalars, photons, fermions, gluons and gravitons. These correlators are rational functions of the minors of the Grassmannian matrices in contrast to their momentum space counterparts which are much more complicated. We then develop the $GL(2,\mathbb{R})$ twistor space formalism, deriving the Penrose transform and the twistorial Grassmannian where we find remarkably simple expressions. We also find an elegant and natural extension to $\mathcal{N}=1$ supersymmetry, deriving the supersymmetric Penrose transform and the super-twistor Grassmannian. Performing a half-Fourier transform from twistor space, we derive the supersymmetric symplectic bi-Grassmannian corresponding to spinor-helicity variables. We test our formalism in the context of AdS$_5$ $\mathcal{N}=1$ super-Yang Mills theory where we impose consistent factorization and input the bootstrapped four-fermion correlator to obtain those involving gluons, scalars and fermions. Finally, we discuss the application to supergravity theories.

It is established that the reduced Hartree-Fock grand potential at positive temperature converges in the semiclassical limit to a Thomas-Fermi grand potential. In particular, a density-potential dual representation of the limiting functional is proposed for a general class of fermionic entropy functions. The corresponding dual problem has no duality gap and admits a maximizer. The resulting grand potential converges to the established zero-temperature grand potential as the temperature tends to zero and, for the Fermi-Dirac entropy, agrees with the standard positive-temperature Thomas-Fermi free-energy functional. The proof also establishes a semiclassical trace formula that is uniform over locally precompact families of potentials subject to a common confinement condition.

We introduce a procedure to obtain lump solutions via the formation of a kink-antikink pair, consisting of the superposition of kinks whose distance from the origin is controlled by a single parameter $a$. For large values of $a$, a wide plateau appears in the solution, which we call a large lump. The procedure involves the use of a first-order equation that allows the construction of the potential associated with the lump solution. We then investigate several known scalar field models where the parent kinks are capable of giving rise to novel lumps. The lump inherits the tails of the parent kink, allowing for either short-range exponential profiles or long-range profiles characterized by distinct power-law decays. We also show how to verify if an arbitrary lump solution can be obtained via our method and illustrate this possibility with a novel vacuumless lump.

We consider the notion of shattering introduced by Ben Arous and Jagannath for spherical pure $p$-spin glasses with overlap $q$. For every $p\geq 3$ and $0<\beta\leq\beta_{\mathrm{sh}}(p)$, we rule out shattering whenever $q\leq2^{-1/2}$ or $q>\sqrt{(p-2)/(p-1)}$. The proof combines a deterministic $N+1$ bound for disjoint bands in the first range with a general-$p$ sign law showing that their total marked weight has subdominant free energy in the second. A spherical-code bound and Hölder's inequality give an additional $q$-dependent obstruction; in particular, they rule out every fixed overlap for $0<\beta\leq\sqrt{\log2}$. For $p=3$, the first two ranges already exhaust every fixed $q\in(0,1)$, so the landscape is not shattered at any $T\geq T_{\mathrm{sh}}$. For $p\geq4$, the cases not covered by our criteria are confined to $2^{-1/2}<q\leq\sqrt{(p-2)/(p-1)}$ and $\sqrt{\log2}<\beta\leq\beta_{\mathrm{sh}}(p)$. In particular, this paper partially resolves Conjecture 1 of the paper above and also suggests new methods to show non-shattering.

This work provides explicit expressions, in all sectors of the Fock space: soliton, antisoliton and breather, for the fundamental building blocks -the form factors- of the generalised integral kernels which represent the exponential of the field operator in the 1+1 dimensional Sine-Gordon quantum field theory. The obtained form factors reproduce the various results obtained earlier in the literature using, in particular, the free field approach.