Papers

We prove that learning an unknown mixed fermionic Gaussian state on $m$ modes to trace distance $\epsilon$ requires $\Omega(m^3/\epsilon^2)$ copies when measurements act on one copy at a time, even with arbitrary POVMs, fresh ancillas and classical adaptivity, but without quantum memory between copies. The bound holds for $0<\epsilon\le1/3600$ and separates this model from the known collective rate $\Theta(m^2/\epsilon^2)$. It follows from a uniform single-copy Fisher-information budget and a dimension-independent comparison between trace distance and Gaussian parameters. On covariance matrices of operator norm at most $1-c$, we prove a Frobenius-to-trace-norm continuity bound with constant $[2c(2-c)]^{-1/2}$; matchgate shadows then attain $O(m^3/(c\epsilon^2))$ copies. This also gives explicit error certificates for thermal free-fermion states. For a class of mixed passive bosonic Gaussian states with total mean photon number at most one, a reduction to bounded-block qudit tomography gives a single-copy lower bound $\Omega(m^3/(\epsilon^2\sqrt{\log(m/\epsilon)}))$, versus collective complexity $\Theta(m^2/\epsilon^2)$. These separations answer the mixed-state measurement-resource question posed by Chen et al. A two-mode example has optimal two-copy Fisher trace $5$, versus the separable ceiling $4$. An IBM replication gave a pre-registered witness lower bound $4.12$ at one-sided $95\%$ shot-noise confidence; its interpretation assumes the prescribed preparations and a common measurement channel and does not bound preparation systematics.

We compute the effective six-wave interaction in spatial and temporal Hamiltonian Dysthe equations and compare it with the complete degree-six coefficient of the deep-water Euler Hamiltonian. On every compact subset $K$ of a generic quadratic-dispersion resonance patch, exact resonances persist under a small cubic correction to the dispersion, and \[ W_6\bigl(\Phi_{\lambda,\varepsilon}(z)\bigr) =\varepsilon\frac{4BD}{aL(z)}+O_K(\varepsilon^2) \] uniformly in the resonance point $z$ and the deformation parameter $\lambda$. Here $L$ is the spectral diameter and $B,D$ are the local cubic and mean-flow coefficients. Thus $BD\ne0$ gives nonzero scattering on relatively open resonance sets. At the Craig–Guyenne–Sulem temporal coefficients, the corresponding exact Euler resonances satisfy \[ \frac{\mu}{4}W_6^{\rm E}\bigl(1+\mu\Psi_\mu(z)\bigr) =-\frac{8}{L(z)}+O_K(\mu), \] where $\mu$ is the relative bandwidth. The leading term comes from nine contractions of quartic vertices; every other connected degree-six contribution is uniformly bounded. We classify the coefficient values for which the generic six-wave coefficient vanishes identically; in the temporal case, this also gives cancellation on every non-permutation four-wave resonance. Outside these sets, the Zakharov–Schulman condition restricts every regular $C^3$ quadratic leading symbol to $\operatorname{span}\{1,k,\omega(k)\}$, the span of the symbols of wave action, momentum, and linear energy. Consequently the physical Dysthe models cannot support an inverse-scattering hierarchy with infinitely many linearly independent regular $C^3$ quadratic leading symbols. The cancellation sets contain known nonlinear Schrödinger, mixed Chen–Lee–Liu, Calogero–Moser derivative nonlinear Schrödinger, and Hirota representatives.

We consider the FPUT $\alpha$-model with $N$ particles and initial data with small total energy $E$, independent of $N$. In particular, we prove that if the energy is initially given to Fourier modes of wave number $\pi k/N\leq \theta_*$, then it does not flow to modes of wave number larger than $\theta_*$ up to long times of order $E^{-(1-\epsilon/2)}$, $\theta_*$ and $\epsilon$ being small numbers. The proof relies on the closeness of the Toda lattice to the FPUT system. Unlike previous approaches, we do not make use of the action-angle variables of the Toda system, but we exploit the Flaschka integrals of the latter. Actually, we find a suitable recombination of the Flaschka integrals whose quadratic part is used to control the packet, the higher order part fulfilling very good estimates.

In this paper, we investigate higher-order symmetries and non-geometric symmetries within the framework of $k$-cosymplectic Hamiltonian field theory. We introduce the concept of generalized infinitesimal symmetry, which relaxes the standard symmetry condition, and then define two new classes of symmetries: higher-order general infinitesimal Cartan symmetries and good non-geometric symmetries. For each of these symmetry classes, we establish a Noether-type theorem that provides explicit conserved quantities for exact $k$-cosymplectic Hamiltonian systems. These results extend and generalize previous work on standard Cartan symmetries in the $k$-cosymplectic setting. Furthermore, we extend all of these results in a parallel fashion to Hamiltonian systems on $k$-symplectic manifolds, demonstrating the unifying nature of our approach. Concrete examples are provided to illustrate the existence and applicability of the proposed symmetries. Our framework offers a systematic geometric treatment of generalized symmetries and their associated conservation laws in classical field theories.

Understanding the physical origin of the piecewise-linear computation in deep rectified linear unit (ReLU) networks remains a fundamental challenge. Here we establish an exact statistical-mechanical realization of arbitrary ReLU networks. Starting from a microscopic configuration space and its state multiplicities, we construct a partition function without prescribing a neural-network activation function. The resulting system admits equivalent descriptions in terms of cascaded quantum operations with post selection, or fermionic transport model. We rigorously prove that in $\beta\to+\infty$ limit, the thermodynamic observables of this system exactly reproduce the hidden states, outputs, and loss function of an arbitrary deep ReLU network. Crucially, we demonstrate that the piecewise-linear behavior of the ReLU network emerges from a first-order phase transition in the microscopic system, where the discontinuities precisely coincide with the boundaries of linear regions of ReLU network. This work establishes an exact physical foundation for deep neural computation and provides a novel statistical-mechanical perspective on neural architecture design.

We resolve an existing question concerning dynamical and structural stability of the E {\times} B staircases in magnetically confined fusion plasma. The problem has stirred considerable interest in the literature due to a unique combination of its scientific and practical aspects. Here, we report a new approach to the problem by combining the paradigmatic Hasegawa-Wakatani model of electrostatic drift-wave turbulence with an idea of interaction renormalization, imported from quantum field theory. It is shown based on the Schrödinger-Newton equation with renormalized interaction constant that the plasma staircase is robust and stable as soon as the nonadiabaticity parameter, {\delta}, goes above a certain critical value, and is unstable otherwise. This critical value is an exact result of the interaction model and is found to be {\delta}c = {\pi}. For {\delta} < {\delta}c , the unstable system expands in the radial direction, resulting in a radial second moment that grows with time t as t^(2/3). The results obtained bear important implications with regard to the implementation of arrays of semipermeable transport barriers in tokamaks, suggesting a threshold condition for their generation.

We study finite-time singularity formation for a nonlocal Boussinesq system arising in cold-plasma dynamics. We show that smooth, symmetric initial data with strictly positive total density and a sufficiently negative velocity slope at the origin lead to finite-time blow-up. More precisely, we give an explicit upper bound for the lifespan and show that the velocity gradient becomes unbounded.

Four-dimensional conformal gravity, twistor theory, and ambitwistor theory are reviewed within the unified framework of holomorphic conformal Cartan geometry. Starting from the spinorial representation of the geometry, the normal $\mathfrak{sl}(4,\mathbb{C})$-valued Cartan connection, its curvature $2$-form, and the associated tractor bundles are systematically formulated. Evaluated on the real Lorentzian slice, the quadratic curvature action is shown to reproduce four-dimensional Weyl–Bach conformal gravity upon fixing a Weyl gauge. Local twistor and dual twistor bundles are constructed as associated spin tractor bundles, illustrating how parallel transport yields the classical incidence relations in flat space and chiral integrability constraints in curved backgrounds: anti-self-duality for $\alpha$-surfaces and self-duality for $\beta$-surfaces. Furthermore, by pairing local twistors and dual twistors via an invariant canonical pairing, ambitwistor space is shown to parameterize complex null geodesics and symmetrically accommodate both left- and right-handed Weyl curvatures. This framework establishes a direct gauge-theoretic bridge linking conformal Cartan geometry to twistor and ambitwistor theories.

Motivated by the work of M. L. Lapidus and M. van Frankenhuisjen on fractal explicit formulas for generalized fractal strings, we develop in this paper a distributional fractal Taylor's formula with error term and an exact distributional fractal Taylor series representation for generalized fractal strings. The explicit formulas that the aforementioned authors developed provide a way of expressing a generalized fractal string $\eta$ in terms of the underlying fractal complex dimensions, under suitable growth assumptions on its geometric zeta function, $\zeta_{\eta}$, known as the languidity assumptions. This involves a sum indexed by the complex dimensions of the generalized fractal string, summing over all residues of $\zeta_{\eta}$, multiplied by the Mellin transform of a tempered or Schwartz test function $\phi$, denoted by $\widetilde{\phi}$. Under languidity assumptions, there is an error term present. Under stronger assumptions, that is, in the case of strong languidity, no error term is present and the resulting fractal explicit formula is said to be exact. In this paper, we express the residue term in the fractional distributional explicit formulas as the fractional $(-\omega)^{\mathrm{th}}$-ordered distributional derivative of the Dirac $\delta$ distribution, denoted by $Y_{\omega}$, applied to modified test functions of the form $\phi(x) \ln^{k-1}(x)$, where $k$ runs from $1$ to the multiplicity of the pole $\omega$ of $\zeta_{\eta}$. The coefficients in this sum are written in terms of the coefficients in the principal part of the Laurent expansion at $\omega$ of the geometric zeta function $\zeta_{\eta}$, multiplied by the gamma function evaluated at $\omega$. The results obtained in this paper contribute to the broader program of characterizing fractals in terms of fractal Taylor series expansions involving their underlying fractal complex dimensions.

We study the nonlinear Hamilton–Jacobi system arising as the formal semi-classical limit of the chordal Belavin–Polyakov–Zamolodchikov equations with a common eigenvalue parameter $\lambda$. We determine the number of global real-valued solutions modulo additive constants. When $\lambda=0$, the number of solutions can be derived using the connection between the solutions and rational functions, due to Eremenko. We focus on the case when $\lambda\neq 0$ in this article. For $\lambda>0$, a Schwarzian correspondence relates the algebraic equations for the gradients to quasi-exponentials with prescribed critical points. Enumeration and transversality results of Mukhin, Tarasov, and Varchenko yield smooth real branches, and a Poisson bracket identity establishes their exactness. An energy bound rules out global real-valued solutions for $\lambda<0$.

We compute the cohomology of the projective special linear Lie superalgebra $\mathfrak{psl}(n|n)$ over a field of characteristic zero in trivial and in adjoint coefficients in full detail through its associated Hochschild–Serre spectral sequences. Applications in physics exist for its real form, the projective special unitary Lie superalgebra $\mathfrak{psu}(n|n)$ — in particular, $\mathfrak{psu}(2,2|4)$ for AdS/CFT. We also find the cohomology of $\mathfrak{sl}(m|n)$ in trivial coefficients along the way and we point out a discrepancy with a previous statement in the literature. We present the explicit cohomology groups of low degree for $n\leqslant4$.

We define a natural orientation on a spanning-tree-decorated planar map whereby, roughly speaking, each directed edge in the map is oriented to match the direction of the contour exploration of the spanning tree. We study directed distances (lengths of shortest directed paths) with respect to this orientation. We construct the Busemann function which measures directed distances to $\infty$ along a natural interface in the uniform infinite spanning-tree-decorated map. We show that this Busemann function, re-scaled appropriately, converges in law to a $3/2$-stable Lévy process. We also show that in a uniform spanning-tree-decorated map with $n$ edges, directed distances are typically of order $n^{1/3}$. Using a strong coupling argument, we deduce analogous statements for directed distances in other random planar maps in the $\sqrt 2$-Liouville quantum gravity (LQG) universality class, including uniform meandric systems and mated-CRT maps for $\gamma=\sqrt 2$. These results give the scaling dimension for a hypothetical directed version of the $\sqrt 2$-LQG metric. Our proof strategy is inspired by work of Borga and Gwynne (2025) on directed distances in bipolar-oriented triangulations.

We develop a unified algebraic approach to spin-boson transformations based on the formalism of $f$-deformed oscillators. In the single-mode case, we show that the Holstein-Primakoff and Dyson-Maleev transformations, together with the interpolating $\alpha$-family, arise as different factorizations of the same algebraically determined object. The standard spin-boson mappings can thus be interpreted as realizations of a common structure, which makes it possible to clearly separate the exact algebraic content on the physical subspace from effects associated with non-Hermiticity, the choice of metric, and extensions beyond the physical subspace. In the two-mode case, the same approach yields both deformed versions of the Jordan-Schwinger transformation and new exact two-mode realizations. Our results provide a unified description of known spin-boson transformations and naturally lead to new bosonic representations.

Weighted Hardy inequalities on a domain $\Omega\subset\R^n$ can be deduced from lower Hausdorff-content bounds on suitable visible parts of the boundary, and it is natural to look for visible sets described by straight segments, which are easier to verify than the curve-based visual boundary of Koskela and Lehrbäck. We show that this approach is subject to an intrinsic obstruction. If a boundary point $\xi$ is seen from $x\in\Omega$ along a segment satisfying a uniform thickness condition, then $\Omega$ contains a truncated cone with vertex $\xi$; consequently the quantitatively straight visible boundary $\partial^{\mathrm{str}}_{x,\alpha}\Omega\cap B(x,C_0d_\Omega(x))$ is a finite union of Lipschitz graphs, is $(n-1)$-rectifiable, and carries no $\lambda$-Hausdorff content for any $\lambda>n-1$. Since content conditions with $\lambda\le n-1$ already imply the $(p,\beta)$-Hardy inequality for $\beta<p-n+\lambda$ without any visibility assumption, segment-based criteria are either unsatisfiable or redundant: they never reach the regime $\beta\ge p-1$ in which visibility is genuinely needed. We show that the exponent $n-1$ is attained, so that the rectifiability theorem is sharp; we exhibit a boundary point lying in the visual boundary of Koskela and Lehrbäck but in no $\partial^{\mathrm{str}}_{x,\alpha}\Omega$; and we observe that raw straight visibility, without the thickness condition, escapes the obstruction, is sufficient for $\lambda\le n-1$, and remains open for $\lambda>n-1$.

In this work, we develop a generalized Hamiltonian formalism for spatially nonlocal field theories whose Lagrangian densities depend explicitly on both the local field and its spatially reflected counterpart. Starting from a generalized variational principle, we derive generalized Euler-Lagrange equations and introduce a generalized functional derivative that consistently accounts for reflected-field contributions. The proposed formalism is applied to three spatially nonlocal nonlinear Schrödinger equations. For the Ablowitz-Musslimani equation, we construct, to the best of our knowledge, the first standard Lagrangian density formulated directly in terms of the complex fields and derive its complete Hamiltonian formulation. The same framework is subsequently applied to the two Lagrangian nonlocal nonlinear Schrödinger equations introduced by Velasco-Juan and Fujioka, yielding consistent Hamiltonian formulations that reproduce the corresponding generalized Euler-Lagrange equations. These results establish a unified Hamiltonian framework for a broad class of spatially nonlocal nonlinear field theories.

Initial Orbit Determination (IOD) from Very Short Arcs (VSAs) remains one of the most challenging open problems in asteroid surveillance and celestial mechanics. Classical methods require angular observations spanning a sufficient fraction of the orbit, and become ill-conditioned or fail outright when only a single observing night is available, as is the case for most newly discovered Near-Earth Objects (NEOs). We present a comparative analysis of two Neural Network (NN) models that attack the ranging problem directly: both ingest a triplet of time-tagged angular measurements $(t_i,\alpha_i,\delta_i)$, $i=1,2,3$, from a single VSA and predict the geocentric range and range-rate $(\rho,\dot\rho)$, thus completing the orbital state vector. The first is a Multi-Layer Perceptron trained on a purely data-driven objective; the second augments it with a physics-informed loss. Both are trained, under an object-disjoint partition, on a sample of the $568\,127$ VSAs of $39\,031$ real NEOs available in the NEODyS-2 catalogue. Each model returns an estimate on every arc by construction, so we assess instead whether that estimate is dynamically admissible: on a held-out test set of $56\,773$ arcs the physics-informed model places $85.1\%$ of its predictions inside the admissible region, against $76.5\%$ for the data-driven baseline, whereas Gauss's and Laplace's methods return a solution on only $45.0\%$ and $47.3\%$ of the same arcs and collapse onto the degenerate root in about nine of those cases out of ten. A stratified analysis over proper motion and true range shows that the physics-informed objective is not uniformly superior to the baseline: it trades a longer error tail for admissibility and for a marked advantage on the fast, nearby arcs that are operationally the most relevant.

We use the Boué-Dupuis variational method to construct the Abelian Yang-Mills-Higgs measure on the two-dimensional torus with polynomial Higgs potentials, giving, to our knowledge, the first rigorous construction of an interacting gauge theory by this approach. We first use the variational method to construct the Higgs field conditioned on the gauge field, with a covariant Gaussian free field as the reference measure. Integrating out the Higgs field produces a weight on the gauge field, which we control through a second application of the variational method. We also prove new exponential integrability estimates for the joint gauge-Higgs field and establish gauge covariance. Finally, we prove convergence of the Laplace transforms of the regularised measures and establish a new large-deviation principle for gauge-invariant observables in the semiclassical limit.

We consider the non-linear cubic defocusing Schrödinger equation on the 2D Torus ($T^2$) with a potential $V(x)\in H^{2^+}(T^2)$. The initial data, $u_0$,is given by a centered gaussian variable with covariance $(-\Delta+V)^{-1}$, and thus is almost surely in $H^{0^-}(T^2)$. We show that it is almost surely well-possed for all time in the weak sense, and that $u-e^{it(\Delta -V)}u_0$ is in $H^{s}(T^2)$ for some $s>0$, and that the solution is the limit of the solutions to the truncated equation. This result is a generalization of the results first given by Bourgain in Invariant Measures for the {2D}-Defocusing Nonlinear {Schrödinger} Equation.

We study collision times of many independent simple random walks on $\mathbb Z^2$ through the joint moment generating function of their pairwise collision local times. For a fixed number of walks, these collision times are known to be asymptotically independent after a suitable logarithmic normalisation. We investigate the extent to which this independence persists when the number of walks grows. For $N$ being the walk length, our results identify a threshold $\asymp (\log N)^{\frac{1}{3}}$, on which the transition from independence to dependence happens. The proofs combine chaos expansion techniques and a correlation inequality, which is the result of a local limit theorem.

We study the simultaneous asymptotic behaviour of homogenization and the Bourgain–Brezis–Mironescu limit for quadratic fractional energies with periodic coefficients. The functionals depend on two small parameters: the period of the coefficients and the fractional exponent approaching the local limit. We prove that the $\Gamma$-limit is completely determined by the relative scaling of these parameters. In the two extreme regimes, one recovers the expected separation of scales, yielding either the homogenized Dirichlet energy or the energy with averaged coefficients. At the critical logarithmic scaling, however, the two mechanisms coexist, and the limit is a convex combination of the homogenized and averaged energies, with weights depending explicitly on the relative scaling of the two asymptotic processes. The proof combines the homogenization analysis of the short-range interactions with a blow-up argument showing that long-range interactions are simply averaged using a Riemann–Lebesgue-type argument.

We discuss the application of $L^2$-based velocity averaging to the Wigner kinetic equation governing the quantum evolution of the Wigner transform of a density operator of a particle subjected to an external electromagnetic field. This extends the $L^2$-based averaging lemma from [F. Golse, J. Möller: Commun. Math. Sci. (2026)] to electromagnetic fields. We make use of the magnetic or gauge-invariant Wigner transform due to Stratonovich [Sov. Phys. D., (1)414-418, 1956] which differs from the usual definition by a phase factor involving the circulation of the magnetic vector potential along a line segment. The magnetic Wigner transform obeys the Wigner equation in the generalized phase space variables $(x,v=\xi-A(t,x))$, which is a more natural setting for the application of velocity averaging. Since the magnetic Wigner equation contains a first order derivative in the spatial variable, we obtain compactness in $L^2$ but no gain in regularity.

Brane tilings are bipartite graphs on a 2-torus that encode the Lagrangians of 4d N=1 supersymmetric gauge theories arising on D3-branes probing toric Calabi-Yau 3-folds. Among these bipartite graphs, only those that satisfy geometric consistency conditions correspond to well-behaved quantum field theories. We train a convolutional neural network (CNN) to distinguish geometrically consistent from inconsistent brane tilings directly from their Kasteleyn matrices. We study a family of 4d N=1 theories obtained by adding diagonal edges to the hexagonal faces of the brane tiling for the abelian orbifold C^3/Z_3 x Z_3, and find that the CNN identifies geometric inconsistency with high accuracy. For inconsistent brane tilings that can be rendered consistent by Higgsing a single bifundamental chiral field, we show that gradient-based saliency analysis can be used to localize the responsible chiral fields with accuracy well above a matched random baseline. These results demonstrate that explainable AI (XAI) can be used to identify local defects responsible for inconsistencies in supersymmetric gauge theories.

We study macroscopic thermalization in the ferromagnetic Ising model with a weak generic (highly nonlocal and many-body) random quantum perturbation. We prove that a single typical perturbation makes every initial state $|\Phi(0)\rangle$ in a specified energy shell thermalize: a measurement of a large class of macroscopic observables in the time-evolved state $|\Phi(t)\rangle=e^{-i\widetilde H_Lt}|\Phi(0)\rangle$ yields their thermal equilibrium values, within a prescribed precision and with overwhelming probability, at sufficiently large, typical times $t$. The result covers the full finite-temperature range, including the ordered phase, where we use plus boundary conditions. With periodic boundary conditions in the ordered phase, every energy eigenstate in the shell is shown to be an almost balanced macroscopic Schrödinger-cat state. Moreover, every pure initial state becomes such a cat at sufficiently large, typical times, with each branch in the corresponding macroscopic thermal equilibrium. The proof combines the theory of Roos, Sugimoto, Teufel, Tumulka, and Vogel with rigorous Ising large-deviation estimates.

The main purpose of this paper is to define and study the notion of restricted Rota-Baxter operator on restricted Lie algebras and restricted Lie triple systems. We provide the relevant properties and the usual connections with pre-Lie structures. We show that the operad of pre-Lie triple systems is the splitting of the operad of Lie triple systems. Moreover, we prove that Jacobson identities hold in pre-Lie triple systems in positive characteristic and then introduce the notion of restricted pre-Lie triple system.

Fractional partial differential equations couple a memory-dependent time derivative with a nonlocal fractional Laplacian, and their repeated solution under varying fractional orders, initial data, or diffusivity fields is computationally expensive. Neural operators offer a fast surrogate, but standard architectures must relearn the dominant linear fractional evolution from data. We propose the Fractional Fourier Neural Operator (FrFNO), a resolution-robust conditional operator. It injects the analytic Mittag–Leffler response of the frozen linear part as a once-precomputed, resolution-independent propagator table, and trains a spectral convolutional network only on the residual induced by variable coefficients and nonlinear advection, conditioned continuously on both fractional orders. The central theoretical result is that under zero-shot super-resolution the injected propagator fills the out-of-band modes that a standard Fourier neural operator sets to zero, replacing the out-of-band full-field tail by the smaller residual tail (a $K$-independent constant-factor reduction under weak perturbation). The residual and the full solution carry the same Sobolev order, so the advantage is amplitude reduction rather than a steeper tail. Mesh refinement drives the error to a residual-controlled floor. On a nonlinear two-dimensional space–time fractional Burgers problem FrFNO achieves the lowest relative $L^2$ error among six baselines (FNO, PINO, PDNO, CNO, DeepONet, U-Net) at the training resolution and under zero-shot super-resolution, yields the smallest spectral phase error, and remains the best at the integer-order limit and for long integration windows. The same construction extends to coupled fractional systems such as fractional Allen-Cahn and Navier–Stokes equations. Code, training scripts, and reference outputs are available at https://github.com/Derek2021Pang/FrFNO.

We investigate the emergence of rigid polycrystalline structures in atomistic ionic particle systems as low-energy configurations. The interaction between particles of opposite charge is modeled by hard spheres that interact when they are tangential. The interaction between particles of same charge is modeled as a hard repulsion that forces a minimal distance between them. The atomistic energy is frame invariant, and no underlying reference lattice is assumed on the ionic configurations. The asymptotic behavior of configurations with finite surface energy scaling is identified by means of $\Gamma$-convergence. The related continuum theory is described by piecewise constant fields that encode the local orientation of the configuration. The limiting energy is local and concentrates at grain boundaries, which correspond to the boundaries of the regions where the underlying configuration has a constant orientation. The limiting energy density is anisotropic and depends on the relative misorientation of the two grains, their translation misfit, and the normal to their interface. Furthermore, we perform a fine analysis of surface energies for solid-solid and solid-vacuum phase transitions and determine energetically favorable orientation mismatches. This relies on a structure result for our grain boundaries, which shows that, due to the rigid setup, interpolating layers near the grain interface are energetically not favorable.

We introduce a new model for a system of interacting bosons placed in an array of sites. At its core is a nonlinear, nonlocal evolution equation, which we have dubbed the Schrödinger-Dirichlet equation. The construction is closely related to the Bose-Hubbard model and to a specific type of generalized bosons. In contrast to conventional mean-field closures, the resulting nonlinear dynamics need not preserve product structure and can generate entanglement from initially separable states. The relevant methods of analysis are based on harmonic analysis for the multiplicative group of positive rationals.

For a class of growth processes of Laplacian type in the plane, we suggest an interpretation as a “gradient descent” in the space of smooth closed curves. More precisely, we show that boundary of a growing domain moves along a gradient of a certain functional in the space of curves. In the simplest cases this functional is $\log (1/r)$, where $r$ is the external conformal radius of the growing domain.

Recent work has constructed Abelian Chern-Simons theories as categorical TQFTs, allowing us to naturally incorporate categorical defects and construct defect extensions of Abelian Chern-Simons TQFTs. We first identify the Turaev-Viro realizations of Abelian Chern-Simons theory in the center and doubled pointed modular cases, clarifying the distinction between single bulk realizations and canonical doubled ones. Then we obtain Turaev-Viro partition functions in terms of Reidemeister-torsion densities. Alternatively, the Alterfold construction supplies the associated topological boundaries, domain walls, and condensation sectors, establishing an explicit Alterfold/Chern-Simons dictionary. We show that the finite quadratic module is the invariant controlling the bulk theory, its topological symmetries, orientation-reversal invariance, and defects. We further show that multicomponent Abelian BF theory arises as the extended TQFT of an off-diagonal Abelian Chern-Simons theory, placing it naturally within the same extended framework. Finally, we demonstrate that recently proposed Abelian Chern-Simons dualities do not define a genuine TQFT duality. These results provide a concrete model for defects in Abelian topological orders and suggest a route toward the non-Abelian case.

In this paper, we completely resolve the two total positivity problems raised by Li [J. High Energy Phys., 2023(7):Paper No. 167, 44 pp., 2023], thereby determining the precise scope of this positivity structure in the one-dimensional conformal bootstrap. First, the Gauss hypergeometric kernel $\mathcal{F}(\Delta,z) = {}_2F_1(\Delta,\Delta;2\Delta;z)$ is proved to be strictly totally positive of order infinity for $\Delta > 0$ and $z \in (0,1)$. Second, the sharp lower $\Delta$-parameter threshold for the associated one-dimensional $SL(2,\mathbb{R})$ conformal block $G_{\Delta}(z) = z^{\Delta}\mathcal{F}(\Delta,z)$ is shown to be $1/2$: the conformal-block kernel $G_{\Delta}(z)$ is strictly totally positive of order infinity for $\Delta \geq 1/2$. For every $\tau \in (0,1/2)$, there exists a strictly negative odd-order minor of $G_{\Delta}(z)$ such that all its $\Delta$-values are in $(\tau,1/2)$ and all its $z$-values can be chosen arbitrarily close to one. Consequently, no restriction $z > z_0$ with $z_0 < 1$ can restore total positivity of order infinity over all $\Delta > 0$.