We study quantum geometry of Nakajima quiver varieties of two different types—framed A-type quivers and ADHM quivers. While these spaces look completely different we find a surprising connection between equivariant K-theories thereof with a nontrivial match between their equivariant parameters. In particular, we demonstrate that quantum equivariant K-theory of \(A_n\) quiver varieties in a certain

The note corrects the aforementioned paper (Bertola in Commun Math Phys 294(2):539–579, 2010). The consequences of the correction are traced and the examples updated.

We study a three-dimensional, incompressible, viscous, micropolar fluid with anisotropic microstructure on a periodic domain. Subject to a uniform microtorque, this system admits a unique nontrivial equilibrium. We prove that this equilibrium is nonlinearly unstable. Our proof relies on a nonlinear bootstrap instability argument which uses control of higher-order norms to identify the instability at

Adapting tools that we introduced in Jézéquel (J Spectr Theory 10(1):185–249, 2020) to study Anosov flows, we prove that the trace formula conjectured by Dyatlov and Zworski in (Ann. Sci. Éc. Norm. Supér. (4) 49(3):543–577, 2016) holds for Anosov flows in a certain class of regularity (smaller than \({\mathcal {C}}^\infty \) but larger than the class of Gevrey functions). The main ingredient of the

In this paper we investigate the tau-functions for the stationary sector of Gromov–Witten theory of the complex projective line and its version, relative to one point. In particular, we construct the integral representation for the points of the Sato Grassmannians, Kac–Schwarz operators, and quantum spectral curves. This allows us to derive the matrix models.

We establish global regularity and stability for the irrotational relativistic Euler equations with equation of state \(\bar{p}{}=K\bar{\rho }{}\), where \(0

In topological phases of matter, the interplay between intrinsic topological order and global symmetry is an interesting task. In the study of topological orders with discrete global symmetry, an important systematic approach is the construction of exactly soluble lattice models. However, for continuous global symmetry, in particular the electromagnetic U(1), the lattice approach has been less systematically

Uniform integer-valued Lipschitz functions on a domain of size N of the triangular lattice are shown to have variations of order \(\sqrt{\log N}\). The level lines of such functions form a loop O(2) model on the edges of the hexagonal lattice with edge-weight one. An infinite-volume Gibbs measure for the loop O(2) model is constructed as a thermodynamic limit and is shown to be unique. It contains

Pure quantum spin-s states can be represented by 2s points on the sphere, as shown by Majorana (Nuovo Cimento 9:43–50, 1932)—the description has proven particularly useful in the study of rotational symmetries of the states, and a host of other properties, as the points rotate rigidly on the sphere when the state undergoes an SU(2) transformation in Hilbert space. We present here an extension of this

We consider a particle moving in one dimension, its velocity being a reversible diffusion process, with constant diffusion coefficient, of which the invariant measure behaves like \((1+|v|)^{-\beta }\) for some \(\beta >0\). We prove that, under a suitable rescaling, the position process resembles a Brownian motion if \(\beta \ge 5\), a stable process if \(\beta \in [1,5)\) and an integrated symmetric

We propose a reformulation of the probabilistic component of the Multi-Scale Analysis adapted to the alloy-type Anderson models with a non-negligible dependence at large distances due to an infinite range of the media-particle interaction. Despite a considerable wealth of results accumulated in the spectral theory of random operators with short-range potentials, much less is known about the alloy models

It has long been suggested that the Cauchy horizon of dynamical black holes is subject to a weak null singularity, under the mass inflation scenario. We study in spherical symmetry the Einstein–Maxwell–Klein–Gordon equations and while we do not directly show mass inflation, we obtain a “mass inflation/ridigity” dichotomy. More precisely, we prove assuming (sufficiently slow) decay of the charged scalar

Perverse schobers are categorical analogs of perverse sheaves. Examples arise from varieties admitting flops, determined by diagrams of derived categories of coherent sheaves associated to the flop: in this paper we construct mirror partners to such schobers, determined by diagrams of Fukaya categories with stops, for examples in dimensions 2 and 3. Interpreting these schobers as supported on loci

For a compact Lie group G we consider a lattice gauge model given by the G-Hamiltonian system which consists of the cotangent bundle of a power of G with its canonical symplectic structure and standard moment map. We explicitly construct a Fedosov quantization of the underlying symplectic manifold using the Levi–Civita connection of the Killing metric on G. We then explain and refine quantized homological

Estimating ground state energies of local Hamiltonian models is a central problem in quantum physics. The question of whether a given local Hamiltonian is frustration-free, meaning the ground state is the simultaneous ground state of all local interaction terms, is known as the Quantum k-SAT (k-QSAT) problem. In analogy to its classical Boolean constraint satisfaction counterpart, the NP-complete problem

Viewing the space of cotraces in the structural coalgebra of a principal coaction as a noncommutative counterpart of the classical Cartan model, we construct the cyclic-homology Chern–Weil homomorphism. To realize the thus constructed Chern–Weil homomorphism as a Cartan model of the homomorphism tautologically induced by the classifying map on cohomology, we replace the unital subalgebra of coaction-invariants

We construct a family of infinite-dimensional positive sub-coalgebras within the Grothendieck ring of Hecke algebras, when viewed as a Hopf algebra with respect to the induction and restriction functor. These sub-coalgebras have as structure constants the 3-point genus zero Gromov–Witten invariants of Grassmannians and are spanned by what we call cylindric Hecke characters, a particular set of virtual

We consider the ideal Fermi gas of indistinguishable particles without spin but with electric charge, confined to a Euclidean plane \({{\mathbb {R}}}^2\) perpendicular to an external constant magnetic field of strength \(B>0\). We assume this (infinite) quantum gas to be in thermal equilibrium at zero temperature, that is, in its ground state with chemical potential \(\mu \ge B\) (in suitable physical

We consider smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces and show that, when the genus of the surface is two, almost every such locally Hamiltonian flow with two non-degenerate isomorphic saddles has singular spectrum. More in general, singularity of the spectrum holds for special flows over a full measure set of interval

We study (unrooted) random forests on a graph where the probability of a forest is multiplicatively weighted by a parameter \(\beta >0\) per edge. This is called the arboreal gas model, and the special case when \(\beta =1\) is the uniform forest model. The arboreal gas can equivalently be defined to be Bernoulli bond percolation with parameter \(p=\beta /(1+\beta )\) conditioned to be acyclic, or

We prove local well-posedness for the periodic derivative nonlinear Schrödinger equation, which is \(L^2\) critical, in Fourier-Lebesgue spaces which scale like \({H}^s({\mathbb {T}})\) for \(s>0\). Our result is optimal in the sense that it covers the full subcritical regime. In particular we close the existing gap in the subcritical theory by improving the result of Grünrock and Herr (SIAM J Math

Strong invariants of even-dimensional topological insulators of independent Fermions are expressed in terms of an invertible operator on the Hilbert space over the boundary. It is given by the Cayley transform of the boundary restriction of the half-space resolvent. This dimensional reduction is routed in new representation for the K-theoretic exponential map. It allows to express the invariants via

We give an explicit description, in terms of bracket, anchor, and pairing, of the standard cochain complex associated to a Courant algebroid. In this formulation, the differential satisfies a formula that is formally identical to the Cartan formula for the de Rham differential. This perspective allows us to develop the theory of Courant algebroid connections in a way that mirrors the classical theory

In a recent paper (Chen et al. in The generalized TAP free energy, to appear in Comm. Pure Appl. Math.), we developed the generalized TAP approach for mixed p-spin models with Ising spins at positive temperature. Here we extend these results in two directions. We find a simplified representation for the energy of the generalized TAP states in terms of the Parisi measure of the model and, in particular

The Gopakumar–Vafa type invariants on Calabi–Yau 4-folds (which are non-trivial only for genus zero and one) are defined by Klemm–Pandharipande from Gromov–Witten theory, and their integrality is conjectured. In a previous work of Cao–Maulik–Toda, \(\mathop {\mathrm{DT}}\nolimits _4\) invariants with primary insertions on moduli spaces of one dimensional stable sheaves are used to give a sheaf theoretical

By means of a conformal mapping and bifurcation theory, we prove the existence of large-amplitude steady stratified periodic water waves, with a density function depending linearly on the streamfunction, which may have critical layers and overhanging profiles. We also provide certain conditions for which these waves cannot overturn.

We study a stochastic PDE limit of the height function of the dynamic asymmetric simple exclusion process (dynamic ASEP). Introduced in Borodin (Symmetric elliptic functions, IRF models, and dynamic exclusion processes, 2017), the dynamic ASEP has a jump parameter \(q\in (0,1)\) and a dynamical parameter \(\alpha >0\). It degenerates to the standard ASEP height function when \(\alpha \) goes to 0 or

Wilson loop expectations at weak coupling are computed to first order, for four dimensional lattice gauge theories with finite gauge groups which satisfy some mild additional conditions. This continues recent work of Chatterjee, which considered the case of gauge group \(\mathbb {Z}_2\). The main steps are (1) reducing the first order computation to a problem of Poisson approximation, and (2) using

We prove that the twisted Kähler–Einstein metrics that arise on the base of certain holomorphic fiber space with Calabi–Yau fibers have conical-type singularities along the discriminant locus. These fiber spaces arise naturally when studying the collapsing of Ricci-flat Kähler metrics on Calabi–Yau manifolds, and of the Kähler–Ricci flow on compact Kähler manifolds with semiample canonical bundle and

We present several infinite families of potential modular data motivated by examples of Drinfeld centers of quadratic categories. In each case, the input is a pair of involutive metric groups with Gauss sums differing by a sign, along with some conditions on the fixed points of the involutions and the relative sizes of the groups. From this input we construct S and T matrices which satisfy the modular

In this paper, we study the motion of the two dimensional inviscid incompressible, infinite depth water waves with point vortices in the fluid. We show that the Taylor sign condition \(-\frac{\partial P}{\partial \vec {n}}\geqslant 0\) can fail if the point vortices are sufficiently close to the free boundary, so the water waves could be subject to Taylor instability. Assuming the Taylor sign condition

We analyze the class of Generalized Double Semion (GDS) models in arbitrary dimensions from the point of view of lattice Hamiltonians. We show that on a d-dimensional spatial manifold M the dual of the GDS is equivalent, up to constant depth local quantum circuits, to a group cohomology theory tensored with lower dimensional cohomology models that depend on the manifold M. We comment on the space-time

We introduce the notion of fully simple maps, which are maps with non self-intersecting disjoint boundaries. In contrast, maps where such a restriction is not imposed are called ordinary. We study in detail the combinatorics of fully simple maps with topology of a disk or a cylinder. We show that the generating series of simple disks is given by the functional inversion of the generating series of

We study the single electron model of a semi-infinite graphene sheet interfaced with the vacuum and terminated along a zigzag edge. The model is a Schroedinger operator acting on \(L^2(\mathbb {R}^2)\): \(H^\lambda _{\mathrm{edge}}=-\Delta +\lambda ^2 V_\sharp \), with a potential \(V_\sharp \) given by a sum of translates an atomic potential well, \(V_0\), of depth \(\lambda ^2\), centered on a subset

We show that spherical Whittaker functions on an n-fold cover of the general linear group arise naturally from the quantum Fock space representation of \(U_q(\widehat{\mathfrak {sl}}(n))\) introduced by Kashiwara, Miwa and Stern (KMS). We arrive at this connection by reconsidering the solvable lattice models known as “metaplectic ice” whose partition functions are metaplectic Whittaker functions. First

We prove the linear stability of subextremal Reissner–Nordström spacetimes as solutions to the Einstein–Maxwell equation. We make use of a novel representation of gauge-invariant quantities which satisfy a symmetric system of coupled wave equations. This system is composed of two of the three equations derived in our previous works (Giorgi in Ann Henri Poincar, 21: 24852580, 2020; Giorgi in Class Quantum

We study the dependence of the spectral gap for the generator of the Ginzburg–Landau dynamics for all \(\mathcal O(n)\)-models with mean-field interaction and magnetic field, below and at the critical temperature on the number N of particles. For our analysis of the Gibbs measure, we use a one-step renormalization approach and semiclassical methods to study the eigenvalue-spacing of an auxiliary Schrödinger

An explicit formula is obtained for the generalized Macdonald functions on the N-fold Fock tensor spaces, calculating a certain matrix element of a composition of several screened vertex operators. As an application, we prove the factorization property of the arbitrary matrix elements of the multi-valent intertwining operator (or refined topological vertex operator) associated with the Ding–Iohara–Miki

We derive a sharp scaling law for deviations of edge-isoperimetric sets in the lattice \({\mathbb {Z}}^d\) from the limiting Wulff shape in arbitrary dimensions. As the number n of elements diverges, we prove that the symmetric difference to the corresponding Wulff set consists of at most \(O(n^{(d-1+2^{1-d})/d})\) lattice points and that the exponent \((d-1+2^{1-d})/d\) is optimal. This extends the

We describe the exact WKB method from the point of view of abelianization, both for Schrödinger operators and for their higher-order analogues (opers). The main new example which we consider is the “\(T_3\) equation,” an order 3 equation on the thrice-punctured sphere, with regular singularities at the punctures. In this case the exact WKB analysis leads to consideration of a new sort of Darboux coordinate

We introduce super quantum Airy structures, which provide a supersymmetric generalization of quantum Airy structures. We prove that to a given super quantum Airy structure one can assign a unique set of free energies, which satisfy a supersymmetric generalization of the topological recursion. We reveal and discuss various properties of these supersymmetric structures, in particular their gauge transformations

The second author formulated quantum unique ergodicity for Eisenstein series in the prime level aspect in “Equidistribution of Eisenstein series in the level aspect”, Commun. Math. Phys. 289(3) 1150 (2009). We point out errors and correct the proofs with partially weakened claims.

We classify framed and oriented 2-1-0-extended TQFTs with values in the bicategories of Landau-Ginzburg models, whose objects and 1-morphisms are isolated singularities and (either \(\mathbb {Z}_2\)- or \((\mathbb {Z}_2 \times \mathbb {Q})\)-graded) matrix factorisations, respectively. For this we present the relevant symmetric monoidal structures and find that every object \(W\in \mathbb {k}[x_1,\dots

Suspended graphene samples are observed to be gently rippled rather than being flat. In Friedrich et al. (Z Angew Math Phys 69:70, 2018), we have checked that this nonplanarity can be rigorously described within the classical molecular-mechanical frame of configurational-energy minimization. There, we have identified all ground-state configurations with graphene topology with respect to classes of

A two-dimensional chiral conformal field theory can be viewed mathematically as the representation theory of its chiral algebra, a vertex operator algebra. Vertex operator algebras are especially well suited for studying logarithmic conformal field theory (in which correlation functions have logarithmic singularities arising from non-semisimple modules for the chiral algebra) because of the logarithmic

We define new deformable families of vertex operator algebras \(\mathfrak {A}[\mathfrak {g}, \Psi , \sigma ]\) associated to a large set of S-duality operations in four-dimensional supersymmetric gauge theory. They are defined as algebras of protected operators for two-dimensional supersymmetric junctions which interpolate between a Dirichlet boundary condition and its S-duality image. The \(\mathfrak

The magnetic Laplacian (also called the line bundle Laplacian) on a connected weighted graph is a self-adjoint operator wherein the real-valued adjacency weights are replaced by unit complex-valued weights \(\{\omega _{xy}\}_{xy\in E}\), satisfying the condition that \(\omega _{xy}=\overline{\omega _{yx}}\) for every directed edge xy. When properly interpreted, these complex weights give rise to magnetic

We determine the \(L_\infty \)-algebra that controls deformations of a relative Rota–Baxter Lie algebra and show that it is an extension of the dg Lie algebra controlling deformations of the underlying \(\mathsf {Lie}\mathsf {Rep}\) pair by the dg Lie algebra controlling deformations of the relative Rota–Baxter operator. Consequently, we define the cohomology of relative Rota–Baxter Lie algebras and

We study metastable behavior in a discrete nonlinear Schrödinger equation from the viewpoint of Hamiltonian systems theory. When there are \(n<\infty \) sites in this equation, we consider initial conditions in which almost all the energy is concentrated in one end of the system. We are interested in understanding how energy flows through the system, so we add a dissipation of size \(\gamma \) at the

An inclusion of von Neumann factors \(M \subset \mathcal {M}\) is ergodic if it satisfies the irreducibility condition \(M'\cap \mathcal {M}=\mathbb {C}\). We investigate the relation between this and several stronger ergodicity properties, such as R-ergodicity, which requires M to admit an embedding of the hyperfinite II\(_1\) factor \(R\hookrightarrow M\) that’s ergodic in \(\mathcal {M}\). We prove

We introduce a new class of representations of the cohomological Hall algebras of Kontsevich and Soibelman, which we call cohomological Hall modules, or CoHM for short. These representations are constructed from self-dual representations of a quiver with contravariant involution \(\sigma \) and provide a mathematical model for the space of BPS states in orientifold string theory. We use the CoHM to

We study the open refined topological string amplitudes using the refined topological vertex. We determine the refinement of holonomies necessary to describe the boundary conditions of open amplitudes (which in particular satisfy the required integrality properties). We also derive the refined holonomies using the refined Chern–Simons theory.

Whenever available, refined BPS indices provide considerably more information on the spectrum of BPS states than their unrefined version. Extending earlier work on the modularity of generalized Donaldson–Thomas invariants counting D4-D2-D0 brane bound states in type IIA strings on a Calabi–Yau threefold \(\mathfrak {Y}\), we construct the modular completion of generating functions of refined BPS indices

The Hilbert space effect algebra is a fundamental mathematical structure which is used to describe unsharp quantum measurements in Ludwig’s formulation of quantum mechanics. Each effect represents a quantum (fuzzy) event. The relation of coexistence plays an important role in this theory, as it expresses when two quantum events can be measured together by applying a suitable apparatus. This paper’s

We develop a general framework for studying ergodicity of order-preserving Markov semigroups. We establish natural and in a certain sense optimal conditions for existence and uniqueness of the invariant measure and exponential convergence of transition probabilities of an order-preserving Markov process. As an application, we show exponential ergodicity and exponentially fast synchronization-by-noise

We consider the Fermi–Pasta–Ulam–Tsingou (FPUT) chain composed by \(N \gg 1\) particles and periodic boundary conditions, and endow the phase space with the Gibbs measure at small temperature \(\beta ^{-1}\). Given a fixed \({1\le m \ll N}\), we prove that the first m integrals of motion of the periodic Toda chain are adiabatic invariants of FPUT (namely they are approximately constant along the Hamiltonian

A long-standing problem in quantum gravity and cosmology is the quantization of systems in which time evolution is generated by a constraint that must vanish on solutions. Here, an algebraic formulation of this problem is presented, together with new structures and results, which show that specific conditions need to be satisfied in order for well-defined evolution to be possible.

We investigate the linear and semilinear massive Klein–Gordon equations in geometrical frameworks of type “Conformal Cyclic Cosmology” of R. Penrose, or “Singular Bouncing Scenario” as well. We give sufficient conditions on the decay of the mass to the fields be able to propagate across the Big-Bang.

In this paper, we classify non-freely acting discrete symmetries of complete intersection Calabi–Yau manifolds and their quotients by freely-acting symmetries. These non-freely acting symmetries can appear as symmetries of low-energy theories resulting from string compactifications on these Calabi–Yau manifolds, particularly in the context of the heterotic string. Hence, our results are relevant for

We ask the question whether entropy accumulates, in the sense that the operationally relevant total uncertainty about an n-partite system \(A = (A_1, \ldots A_n)\) corresponds to the sum of the entropies of its parts \(A_i\). The Asymptotic Equipartition Property implies that this is indeed the case to first order in n—under the assumption that the parts \(A_i\) are identical and independent of each