Papers

Motivated by the Casas-Loday-Pirashvili maps concerning representations of $n$-Leibniz algebras, we give an intrinsic characterization of the cohomology of $n$-Lie algebras with coefficient in a representation space $V$. We investigated the cohomology theory of $n$-Lie algebras in three different ways: cohomology theory of Leibniz algebras, infinitesimal deformation and abelian extension. In the main part of this paper, we solve the extending problems for $n$-Lie algebras. A necessary and sufficient unified-product criterion is obtained. The case of crossed products, sparse non-abelian extensions, matched pairs are studied as special cases. We also investigate the factorizations, deformation maps, and complements problem for $n$-Lie algebras. An appendix removes the general non-reduced unified product with all intermediate mixed components.

We construct a weak Lie 3-group $T_2 \mathbb B^{F_1}_n$ from the 2-category of gerbes over $\mathbb R^n / \mathbb Z^n$ and the $\mathbb R^n/ \mathbb Z^n$-action on it by pullback along translations. We also construct a homotopy equivalence between $T_2 \mathbb B^{F_1}_n$ and a different Lie 3-group $T_2 \mathbb D^{F_1}_n$, which admits a dimensional reduction to the homotopy equivalence of Lie 2-groups introduced by Nikolaus and Waldorf to model half-geometric T-duality. This is a first step towards establishing a higher form of T-duality for 2-gerbes over torus fibrations, relevant to supergravity and M-theory.

We develop a finite-corner framework for convex cones of normal maps between \(\mathcal B(\mathcal H)\) and \(\mathcal B(\mathcal K)\). The basic structural assumption is stability under finite cut–pad operations. We prove that every point-ultraweakly closed cut–pad stable cone is completely determined by its finite-dimensional corner cones, and conversely that every coherent family of closed finite-dimensional corner cones admits a unique global realization of this type. For a cut–pad stable cone that is not necessarily point-ultraweakly closed, reconstruction from the norm closures of its corner cones yields exactly its point-ultraweak closure. Combining this reconstruction principle with finite-dimensional separation and the Choi representation, we show that non-membership in a global cone is always detected on a single finite-dimensional corner by a finite-dimensional witness. We illustrate the framework for completely positive and decomposable maps; in the decomposable case, the Hermitian parts of the finite-corner witnesses are PPT.

Loday's derived product is revisited in the setting of Q-manifolds, i.e., supermanifolds equipped with an odd vector field that `squares to zero'. We interpret the Grassmann odd product as a standard product upon shifting the grading, which implies the existence of a derived (noncommutative) associative $\mathbb{Z}_2$-graded algebra associated with any Q-manifold. We apply Connes' cyclic cohomology to the derived associative algebra, giving a new cohomology on Q-manifolds distinct from the standard cohomology: we refer to this as the derived cyclic cohomology of a Q-manifold. The derived cyclic cocycles are interpreted as classically BRST-invariant functionals within the BV–BFV–BRST formalism or generalised Ruelle–Sullivan currents when applied to regular foliations via their foliation Lie algebroids.

We establish Gaussian free field fluctuations for uniformly random lozenge tilings of simply connected polygonal domains with $3d$ sides whose directions cycle through the three lattice directions. More precisely, assuming that the liquid region is connected and that the boundary data do not force the height at any interior point, we prove that the fluctuations of centered height function converge to the Gaussian free field in the liquid region, confirming a prediction of Kenyon and Okounkov from 2007. We introduce a tiling action function that encodes the geometry of the limit shape through its critical points. The action function has a complex conjugate pair of critical points in the liquid region, repeated real critical points on the arctic boundary, and distinct real critical points in the frozen region. Using this tiling action function, we construct an approximation to the inverse Kasteleyn matrix in terms of explicit single-contour and double-contour integrals and prove that the approximation is uniform throughout the polygonal domain. The convergence to the Gaussian free field then follows from standard kernel computations.

Using the language of q-opers, we propose an algebro-geometric description of the duality between Bethe Ansatz equations for quantum XXZ spin chains and many-body trigonometric Ruijsenaars-Schneider/Macdonald systems for classical groups. Upon this duality, the energy level sets of the classical B/C/D-type Hamiltonians are found to be in bijection with the set of solutions of the XXZ Bethe Ansatz equations of type A, albeit with open boundary conditions. This dictionary generalizes previous results in which a GL(N) XXZ spin chain with twisted periodic boundary conditions was dual to an N-body Ruijsenaars-Schneider system. Our construction implements a $\mathbb{Z}/2\mathbb{Z}$ folding both at the level of the (GL(N),q)-oper data as well as at the level of the N-body trigonometric Ruijsenaars-Schneider model. The nonreduced BC-type systems appear in our analysis as well.

This paper provides the instability counterpart to our recent construction of nonspherically symmetric approximating $\kappa$-self-similar naked-singularity solutions for the Einstein–scalar field system. These singular solutions contain pervasive nonspherical borderline terms, and to prove instability the delicate renormalization procedure developed in [2] does not extend to the more singular setting considered here. To overcome these difficulties, we introduce a new iteration scheme adapted to singular backgrounds whose leading-order geometry depends on the angular variables. At each step, the nonlinear coefficients are frozen using the preceding double-null geometry, and the resulting equations are solved in a triangular order. In this way, the nonspherical borderline terms are incorporated into the approximate geometry rather than treated as perturbative errors, yielding successively sharper estimates. After sufficiently many iterations, the scheme controls the singular angular structure and produces an approximate spacetime whose difference from the exact solution satisfies the required bounds. In particular, these bounds provide an existence region large enough to carry out the instability argument. We further prove that anisotropic perturbations of the outgoing data, arbitrarily small in a scale-critical norm, lead to the formation of a trapped surface. We also formulate and verify a matter-focusing condition under which a parabolic flow argument guarantees the existence of a corresponding marginally outer trapped surface (MOTS). Together, these results establish the nonlinear instability of $\kappa$-self-similar naked singularities beyond spherical symmetry in the Einstein–scalar field system and introduce a framework for applications across Einstein systems.

We study the annealed survival probability of a random walk in an inhomogeneous Poisson environment of mobile traps on $\mathbb{Z}^d$. In dimensions $d=1,2$, we determine the asymptotics for the annealed survival probability of the walker under suitable assumptions on the average trap intensity and study how inhomogeneity in the initial trap configuration affects these asymptotics. Our results extend the asymptotics proved for the homogeneous setting in \cite{DGRS2012}. We also establish a law of large numbers, central limit theorem and large deviation principle for the inhomogeneous Poisson trap environment in $d\ge 1$, generalising the corresponding results proved for the homogeneous case considered in \cite{CG1984}. We present a class of examples of inhomogeneous trap environments where the decay rate of survival probability can be identified and also instances where it does not decay with time.

In this study, we demonstrate the characteristics of the stochastic Loewner evolution (SLE) as a one-dimensional (1D) turbulence model. First, we show that the diffusion process $x(t)$ obtained by the time coordinate change in the SLE becomes the time dependent diffusion process that satisfies Richardson's law of turbulence.Subsequently, we derive the condition on the diffusivity parameter $\kappa$ such that the diffusion process $x(t)$ satisfies Kolmogorov's 4/5th law, which determines the relation between the mean energy dissipation and velocity in the particles of turbulence. Further mathematical analysis showed that this condition is seen as the condition on Loewner entropy $S_{\mathrm{Loew}}$, which includes the white Gaussian noise term $W_{s}$ in the Loewner driving function of the present model. In addition, the numerical simulations were performed to verify the scaling relations of this 1D turbulence model. These results suggest that the standard SLE becomes a beneficial model of 1D turbulence by introducing an appropriate time coordinate change and imposing the conditions on $\kappa$ and $S_{\mathrm{Loew}}$.

A symmetric distribution of charges or currents typically produces an electromagnetic field having the same symmetry group. However, under special circumstances we show that a discrete symmetry of the source distribution leads to a larger continuous symmetry for the resulting field in restricted regions. The conditions for this promotion from discrete to continuous symmetry are that the fields be affine linear and the symmetry of the source distribution act irreducibly on the ambient space. We illustrate this effect with many examples. This phenomenon is pedagogically interesting both as a source of inspiration in electro- and magnetostatics and as an accessible example of group representation theory.

Fukui and Kawakami showed that the trigonometric Haldane–Shastry (HS) spin chain can be deformed by introducing twisted boundary conditions. In this work we revisit the resulting twisted HS chains, which we call Fukui–Kawakami (FK) chains. We analyse their spectra in dependence of the twist parameter, utilising a direct connection with the (untwisted) HS chain. This allows us to explain why part of the spectrum can be described by Fukui and Kawakami's twisted Bethe equations, and can be constructed from Yangian highest weight states. These states can be labelled by combinatorial data called `motifs', as in the HS chain, which cover some of the (deformed) descendants. We furthermore show that there are other descendants which do not follow from these Bethe equations, but whose energy can be described as a sum of two single-particle energies, suggesting additional hidden symmetries. We then focus on the special case of antiperiodic boundary conditions, and show that this chain coincides with the `minimally polarised' long-range model recently introduced by Basu-Mallick, Finkel, and González-López. Remarkably, this connection implies that the antiperiodic chain has a hidden $\mathfrak{gl}(1|1)$-symmetry, which we use to relate the HS motifs to their `supersymmetric' $\mathfrak{gl}(1|1)$ counterparts.

We determine the Boltzmann–Grad limit of the Ehrenfest “wind-tree” model with a periodic configuration of polyhedral scatterers. The central result is a loss-of-memory effect in dimension $d\geq 3$ for a generic choice of polyhedron. This yields a Markovian limiting transport process on an extended state space and a generalised linear Boltzmann equation for the time evolution of the particle density.

We study edge-state transport in a two-dimensional topological insulator with a periodically deformed edge subjected to a uniform magnetic field. Zeeman coupling breaks time-reversal symmetry and enables elastic backscattering, producing oscillations of the forbidden-band widths. In the strong-field regime, the gaps can close completely at discrete field values. In the weak-field regime, we identify an important class of periodic deformations for which the dominant semiclassical scattering is controlled by complex infinity rather than by the nearest turning points. We develop a semiclassical treatment of this process and establish its agreement with perturbation theory and direct numerical calculations. The gap modulation should produce observable oscillations of the edge conductance. Unlike conventional magnetic quantum oscillations, which are periodic in inverse field, the predicted oscillations are periodic in the magnetic field itself, with a period determined by the Fermi velocity and effective g-factor

This paper is the third and final article in a series of papers constructing the canonical conformally invariant metric on the set of loops of the conformal loop ensemble (CLE) with critical parameter $\kappa=4$. The previous two articles construct, as a subsequential limit of the renormalized graph metric on the loops of CLE$_\kappa$ as $\kappa \downarrow 4$, a conformally invariant, local metric on the loops of a CLE$_4$ whose metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. In this paper, we establish that this metric is uniquely characterized by its properties, as are its geodesics, and that it is a measurable function of the CLE$_4$. In particular, we show that the renormalized CLE$_\kappa$ graph metric converges as $\kappa \downarrow 4$ without passing to a subsequence. A key step in the proof is to show that the metric is determined by the geodesics from each loop to the domain boundary, which are in turn determined by the uniform exploration; this representation will have important applications in future work.

We continue our study of the conformal loop ensemble (CLE) with parameter $\kappa=4$, the critical threshold at or below which the loops are simple and disjoint, touching neither each other nor the domain boundary. This paper is the second in a series of three establishing that the loops of a CLE$_4$ uniquely determine a conformally invariant, local, and geodesic metric such that the metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. In this second paper, we prove the existence of geodesics, showing that any geodesic between two loops is supported on the CLE$_4$ loops (off a set of Hausdorff dimension zero) and does not intersect the domain boundary. Along the way, we establish sharp quantitative estimates for the CLE$_4$ metric geometry, including exponential tail bounds for rectangle distances and multi-scale four-arm SLE$_4$ non-intersection bounds for metric balls.

We consider the conformal loop ensemble (CLE) with the parameter $\kappa=4$, the critical value at or below which the loops are simple and do not intersect each other or the domain boundary. We show that the loops of a CLE$_4$ uniquely determine a conformally invariant, local, and geodesic metric so that the metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. This metric was previously constructed in unpublished work of Sheffield, Watson, and Wu. Our approach differs in that we show that the metric arises as the renormalized limit of the graph metric on CLE$_\kappa$ loops as $\kappa \downarrow 4$. In this first paper in a series of three, we prove that the subsequential limits exist and define a non-trivial conformally invariant metric on CLE$_4$ which is local and such that the metric ball growth from the boundary is given by the uniform exploration of Werner and Wu. In subsequent work, we will show that the subsequential limit exists as a true limit.

Quantum thermalization explains how an isolated many-body system naturally evolves towards a thermal state, rendering information about the initial conditions inaccessible to local measurements. This is precisely the mechanism utilized in quantum error correction, where information is protected by design through a nonlocal encoding. In this work, we leverage this connection to port the rigorous framework of (approximate) quantum error correction to the study of quantum thermalization. Treating typical late-time states as codewords, we characterize the error-correcting properties of generic thermalizing dynamics. We numerically uncover a universal relationship between the encoding rate, distance, and thermal entropy density of the emergent code. At infinite temperature, this universal curve saturates the quantum Singleton bound, achieving the same optimal limit as Haar-random codes. At finite temperature, we introduce a code family based on the Scrooge ensemble, the natural thermal analogue of the Haar ensemble, and prove it saturates the entropic quantum Singleton bound, establishing this family as optimal within entropic constraints. Our extracted universal curve independently saturates this same bound, revealing that finite-temperature thermalization is itself optimal. Finally, we show how conserved quantities limit the error-correcting behavior of thermalization: codewords with differing energies, or other conserved charges, leak only classical information, and correctability persists until the difference reaches the scale of thermal fluctuations. Our results reveal a universal optimal coding structure in thermalizing dynamics, while introducing new optimal codes that achieve fundamental limits of approximate quantum error correction.

We study the SCFT/VOA correspondence for the $(A_2,D_{3n-2})$ Argyres–Douglas theories with $n\geq2$. From the matching of the central charges and the equality of the Schur index with the vacuum supercharacter to all orders, we propose that the associated VOA is the logarithmic doublet algebra $\mathcal A(4n-2)$ of Feigin, Feigin, and Tipunin. For $n=2$, this correspondence was studied in detail by Buican and Nishinaka, and our proposal extends it to $n\geq3$. Using the Nahm sum expression for the vacuum supercharacter, we further propose a family of 3d $\mathcal N=2$ abelian Chern–Simons matter theories that flow to the 3d $\mathcal N=4$ SCFTs obtained by the R-twisted circle reduction, generalizing the $n=2$ construction. We determine the monopole superpotential and show that the Coulomb branch is $\mathbb C^2/\mathbb Z_2$ for every $n$. The superconformal index provides quantitative evidence for the proposed infrared fixed points, including the expected enhancement to $\mathcal N=4$ supersymmetry and the Higgs and Coulomb branch structure.

We consider a grand-canonical reduced BCS Hamiltonian on the full fermionic Fock space with an attractive pairing interaction supported in an energy shell around the Fermi surface. For fixed chemical potential, we prove that the exact finite-volume ground-state energy differs from the minimum of the corresponding finite-volume BCS functional by a volume-independent error. As a consequence, the BCS variational principle is exact for the ground-state energy density in the thermodynamic limit.

We investigate limits of semigroups of stochastic matrices defined by their invariant distribution. Given probability vectors $\gamma(\beta)$ depending on a parameter $\beta$, we introduce a notion of convergence as $\beta\to\infty$ for the corresponding semigroups of $\gamma(\beta)$-preserving stochastic matrices and investigate the structure of the resulting limit. In general, the limiting semigroup differs from the semigroup preserving the limiting distribution, showing that these two operations do not commute. We develop a general framework for such limiting semigroups and study in detail the case in which the invariant distributions are Gibbs vectors at the inverse temperature $\beta$. We show that the limiting semigroup consists of block-upper-triangular stochastic matrices subject to additional substochasticity constraints. We characterise and enumerate their extremal elements and determine the preorder on probability vectors induced by the action of the semigroup. The resulting notion of Block-Triangular majorisation interpolates between ordinary majorisation and upper triangular (aka unordered) majorisation. We show that it is completely characterised by a finite family of monotones and analyse the corresponding behaviour of Rényi $\alpha$-entropies as $\beta\to\infty$.

We propose a manifestly conformally invariant off-shell definition of the Weyl tensor in Galilean geometry. We also propose definitions for the electric and magnetic parts thereof. For the latter to vanish, observers with specific kinematical properties need to exist. While this is guaranteed by the Newton–Cartan equation, it might not be true for other Galilean invariant theories. Therefore, we propose that imposing the existence of observers for which the off-shell magnetic part is zero should be a necessary condition for a Galilean invariant theory to be called `Newtonian', in the spirit of the `Newtonian' condition, introduced by Trautman, in standard Newton–Cartan gravity. As a side result, we also show that there exists a unique Galilean boost-invariant connection that can be built from a Galilean structure and a choice of Coriolis field, even when the clock form is not closed. No extra structure is needed, contrasting with the standard approach used to construct boost-invariant connections which introduces a mass gauge field.

We prove that the electric field generated in three-dimensional space by finitely many point charges of one sign has only finitely many equilibrium points, thereby answering a 1969 question of Morse and Cairns (restated by Eremenko in 2008 and, as a conjecture, by Shapiro in 2015). More generally, for nonzero charges $q_i$ of possibly mixed signs at distinct sites $\mathbf a_i\in\mathbf{R}^3$, we show that the Coulomb field has at most $2^{N-4}(N-1)(9N^2+9N+10)$ equilibria in the region where $S(\mathbf x):=\sum_iq_i|\mathbf x-\mathbf a_i|^{-3}$ does not vanish. Of course, for charges of one sign, $S$ is nonzero everywhere. The proof rules out curves of equilibria using algebraic geometry and complex analysis on an associated complex curve, and then applies a Bézout count to obtain a quantitative bound. Well-known examples show that, in the mixed-sign case, the Coulomb field can vanish on curves contained in the zero set of $S$.

We identify non-Gaussian corrections to the central limit theorem for the global and local laws, i.e. for Stieltjes transform of the empirical eigenvalue distribution of a large real symmetric or complex Hermitian Wigner matrix. We find that in the real case the rate of convergence in this CLT is substantially slower than in the complex case. For the proofs, we develop new static and dynamic hierarchies of extended cumulants up to any order that can be analyzed by a refinement of the zigzag strategy.

The localized electromechanical phase transition in dielectric elastomers involves complex moving boundaries and severe electrostatic fringe fields driven by high-curvature interfaces. Traditional phenomenological models fundamentally underestimate the configurational forces by completely ignoring the in-plane electric field components and geometric singularities. Here, we present a asymptotically exact geometric framework to resolve the domain wall propagation. By mapping the highly deformed current configuration to a regular parametric strip via conformal mapping, the field singularities are algebraically eliminated. An exact integration by parts directly translates the higher-order geometric metric into a rigorous topological mass term, projecting the global conformal electrostatics into a non-linear $\sigma$-model Lagrangian density. Incorporating the Gent strain-stiffening model, the continuous translation symmetry yields a Hamiltonian first integral. Eigenvalue analysis and Bogomol'nyi-Prasad-Sommerfield (BPS) bound calculations prove that the domain wall emerges strictly as an asymmetric heteroclinic orbit connecting two saddle points. This geometric resolution provides exact analytical scalings for the localized interface energy, domain wall thickness, and asymptotic decay lengths, eliminating all phenomenological parameters and offering a deterministic paradigm for geometric instabilities in active soft matter.

We develop a geometric framework for conservative mechanical systems modeled on conic pseudo-Finslerian manifolds, extending classical Riemannian and semi-Riemannian mechanics to anisotropic geometries. In this setting, we define a Jacobi-type pseudo-Finslerian metric and demonstrate that motions of fixed energy correspond, up to reparametrization, to its geodesics. We establish energy and momentum conservation under suitable symmetry assumptions via a generalized Noether's theorem, and provide new analytical criteria for the completeness of both the Jacobi metric and the mechanical system. Furthermore, for mechanical systems with holonomic and non-holonomic constraints, we prove the existence and uniqueness of reaction forces satisfying d'Alembert's principle. These results provide a unified geometric foundation for constrained and unconstrained anisotropic dynamics.

We classify a class of abstract Bose field models in quantum field theory up to unitary equivalence. The class includes abstract free Bose field models, abstract van Hove–Miyatake models, and infrared-renormalized van Hove–Miyatake models. Moreover, as an application of our classification, we classify quadratic interaction models. In particular, our results answer a question of A. Arai concerning the classification of infrared-renormalized van Hove–Miyatake models.

The Schrödinger-Lohe (SL) model is a coupled system of nonlinear Schrödinger equations describing the temporal-spatial evolution of the component wave functions, and it corresponds to the infinite-dimensional counterpart of the Lohe matrix model for quantum synchronization. In this paper, we study a rigorous mean-field limit of the SL model and provide a quantitative estimate on the fluctuations between one-marginal distribution and one particle distribution in 2-Wasserstein distance. For the derived kinetic SL equation, we present sufficient conditions leading to the complete and practical synchronizations.

For a general class of two-dimensional non-interacting semi-metallic electron systems on a lattice, we derive an explicit formula for the whole conductivity matrix, by using Euclidean many-body formalism. The semi-metallic phase takes place whenever two Bloch bands touch at the Fermi energy with conical intersections and generally occurs at the transition between distinct integer quantum Hall phases. Unlike the longitudinal conductivity (which depends solely on the shape of the cones at the Fermi level), the transverse conductivity is remarkably determined both by the conical structures of the bands and by the nature of the two nearby topological phases. Generically, in the semi-metallic state, neither the longitudinal nor the transverse conductivities are quantized in integer multiples of a universal conductivity quantum; however, universality of the conductivity matrix is restored under the assumption of emergent rotational symmetry of the linearized Hamiltonian at the Fermi points. Using our formula, we compute the conductivity matrix for several physically relevant models of quantum Hall fluids at the topological phase transition, exhibiting cases where the transverse conductivity is quantized in half-integer multiples of $e^2/h$ and cases where it depends continuously on an external strain parameter, thus shedding light on its universality or non-universality features.

In this article, we investigate the two-species zero-range process, a multi-species generalization of the classical zero-range process. First, we analyze its condensation regime, which directly parallels its single-species counterpart. As our main result, we establish the dynamical metastable behavior of the location of the single condensate, showing that its motion is governed by a simple Markov chain on the accelerated time scale $N^{1+\alpha}$, where $N$ denotes the total number of particles in the system and the parameter $\alpha>1$ governs the attractivity of the system. Another novelty of this work lies in the proof technique, which integrates the recently developed resolvent approach with the $H^{1}$-approximation method to rigorously characterize metastability.

We analyse magnetic Robin Laplacian in curvilinear half-plane with a smooth boundary. It is well known that the spectrum of the Robin Laplacian is unstable with respect to boundary deformations. This means that if the boundary is a straight line then the spectrum of the Robin Laplacian is purely essential. From the other hand, the perturbation of the boundary produces eigenvalues below the essential spectrum. In this paper, the Robin-Laplace operator with a compactly supported magnetic field is considered. We prove that the spectrum of the magnetic Robin Laplacian is stable under small and local deformations of the boundary.