ALBERT

Beschreibung: We obtain an asymptotic formula for $$nimes n$$ n × n Toeplitz determinants as $$n ightarrow infty $$ n → ∞ , for non-negative symbols with any fixed number of Fisher–Hartwig singularities, which is uniform with respect to the location of the singularities. As an application, we prove a conjecture by Fyodorov and Keating (Philos Trans R Soc A 372: 20120503, 2014) regarding moments of averages of the characteristic polynomial of the Circular Unitary Ensemble. In addition, we obtain an asymptotic formula regarding the momentum of impenetrable bosons in one dimension with periodic boundary conditions.

Beschreibung: The Haag–Kastler net of local von Neumann algebras is constructed in the ultraviolet finite regime of the Sine–Gordon model, and its equivalence with the massive Thirring model is proved. In contrast to other authors, we do not add an auxiliary mass term, and we work completely in Lorentzian signature. The construction is based on the functional formalism for perturbative Algebraic Quantum Field Theory together with estimates originally derived within Constructive Quantum Field Theory and adapted to Lorentzian signature. The paper extends previous work by two of us.

Beschreibung: Lassalle and Nekrasov discovered in the 1990s a surprising correspondence between the rational Calogero–Moser system with a harmonic term and its trigonometric version. We present a conceptual explanation of this correspondence using the rational Cherednik algebra and establish its quasi-invariant extension. More specifically, we consider configurations $${mathcal {A}}$$ A of real hyperplanes with multiplicities admitting the rational Baker–Akhiezer function and use this to introduce a new class of non-symmetric polynomials, which we call $${mathcal {A}}$$ A -Hermite polynomials. These polynomials form a linear basis in the space of $${mathcal {A}}$$ A -quasi-invariants, which is an eigenbasis for the corresponding generalised rational Calogero–Moser operator with harmonic term. In the case of the Coxeter configuration of type $$A_N$$ A N this leads to a quasi-invariant version of the Lassalle–Nekrasov correspondence and its higher order analogues.

Beschreibung: We consider a variety of lattice spin systems (including Ising, Potts and XY models) on $$mathbb {Z}^d$$ Z d with long-range interactions of the form $$J_x = psi (x) e^{-|x|}$$ J x = ψ ( x ) e - | x | , where $$psi (x) = e^{{mathsf o}(|x|)}$$ ψ ( x ) = e o ( | x | ) and $$|cdot |$$ | · | is an arbitrary norm. We characterize explicitly the prefactors $$psi $$ ψ that give rise to a correlation length that is not analytic in the relevant external parameter(s) (inverse temperature $$ eta $$ β , magnetic field $$h$$ h , etc). Our results apply in any dimension. As an interesting particular case, we prove that, in one-dimensional systems, the correlation length is non-analytic whenever $$psi $$ ψ is summable, in sharp contrast to the well-known analytic behavior of all standard thermodynamic quantities. We also point out that this non-analyticity, when present, also manifests itself in a qualitative change of behavior of the 2-point function. In particular, we relate the lack of analyticity of the correlation length to the failure of the mass gap condition in the Ornstein–Zernike theory of correlations.

Beschreibung: Given a Lie superalgebra $${mathfrak {g}}$$ g with a subalgebra $${mathfrak {g}}_{ge 0}$$ g ≥ 0 , and a finite-dimensional irreducible $${mathfrak {g}}_{ge 0}$$ g ≥ 0 -module F, the induced $${mathfrak {g}}$$ g -module $$M(F)={mathcal {U}}({mathfrak {g}})otimes _{{mathcal {U}}({mathfrak {g}}_{ge 0})}F$$ M ( F ) = U ( g ) ⊗ U ( g ≥ 0 ) F is called a finite Verma module. In the present paper we classify the non-irreducible finite Verma modules over the largest exceptional linearly compact Lie superalgebra $${mathfrak {g}}=E(5,10)$$ g = E ( 5 , 10 ) with the subalgebra $${mathfrak {g}}_{ge 0}$$ g ≥ 0 of minimal codimension. This is done via classification of all singular vectors in the modules M(F). Besides known singular vectors of degree 1,2,3,4 and 5, we discover two new singular vectors, of degrees 7 and 11. We show that the corresponding morphisms of finite Verma modules of degree 1,4,7, and 11 can be arranged in an infinite number of bilateral infinite complexes, which may be viewed as “exceptional” de Rham complexes for E(5, 10).

Beschreibung: We define Poisson-geometric analogues of Kitaev’s lattice models. They are obtained from a Kitaev model on an embedded graph $$Gamma $$ Γ by replacing its Hopf algebraic data with Poisson data for a Poisson-Lie group G. Each edge is assigned a copy of the Heisenberg double $${mathcal {H}}(G)$$ H ( G ) . Each vertex (face) of $$Gamma $$ Γ defines a Poisson action of G (of $$G^*$$ G ∗ ) on the product of these Heisenberg doubles. The actions for a vertex and adjacent face form a Poisson action of the double Poisson-Lie group D(G). We define Poisson counterparts of vertex and face operators and relate them via the Poisson bracket to the vector fields generating the actions of D(G). We construct an isomorphism of Poisson D(G)-spaces between this Poisson-geometrical Kitaev model and Fock and Rosly’s Poisson structure for the graph $$Gamma $$ Γ and the Poisson-Lie group D(G). This decouples the latter and represents it as a product of Heisenberg doubles. It also relates the Poisson-geometrical Kitaev model to the symplectic structure on the moduli space of flat D(G)-bundles on an oriented surface with boundary constructed from $$Gamma $$ Γ .

Beschreibung: Regularized coherent-state functional integrals are derived for ensembles of identical bosons on a lattice, the regularization being a discretization of Euclidian time. Convergence of the time-continuum limit is proven for various discretized actions. The focus is on the integral representation for the partition function and expectation values in the canonical ensemble. The connection to the grand-canonical integral is exhibited and some important differences are discussed. Uniform bounds for covariances are proven, which simplify the analysis of the time-continuum limit and can also be used to analyze the thermodynamic limit. The relation to a stochastic representation by an ensemble of interacting random walks is made explicit, and its modifications in presence of a condensate are discussed.

Beschreibung: We show that a connection can be recovered up to gauge from source-to-solution type data associated with the Yang–Mills equations in Minkowski space $${mathbb {R}}^{1+3}$$ R 1 + 3 . Our proof analyzes the principal symbols of waves generated by suitable nonlinear interactions and reduces the inversion to a broken non-abelian light ray transform. The principal symbol analysis of the interaction is based on a delicate calculation that involves the structure of the Lie algebra under consideration and the final result holds for any compact Lie group.

Beschreibung: Rindler wedges are fundamental localization regions in AQFT. They are determined by the one-parameter group of boost symmetries fixing the wedge. The algebraic canonical construction of the free field provided by Brunetti–Guido–Longo (BGL) arises from the wedge-boost identification, the BW property and the PCT Theorem. In this paper we generalize this picture in the following way. Firstly, given a $$mathbb Z_2$$ Z 2 -graded Lie group we define a (twisted-)local poset of abstract wedge regions. We classify (semisimple) Lie algebras supporting abstract wedges and study special wedge configurations. This allows us to exhibit an analog of the Haag–Kastler one-particle net axioms for such general Lie groups without referring to any specific spacetime. This set of axioms supports a first quantization net obtained by generalizing the BGL construction. The construction is possible for a large family of Lie groups and provides several new models. We further comment on orthogonal wedges and extension of symmetries.

Beschreibung: This paper is dedicated to proving the complete integrability of the Benjamin–Ono (BO) equation on the line when restricted to every N-soliton manifold, denoted by $$mathcal {U}_N$$ U N . We construct generalized action–angle coordinates which establish a real analytic symplectomorphism from $$mathcal {U}_N$$ U N onto some open convex subset of $${mathbb {R}}^{2N}$$ R 2 N and allow to solve the equation by quadrature for any such initial datum. As a consequence, $$mathcal {U}_N$$ U N is the universal covering of the manifold of N-gap potentials for the BO equation on the torus as described by Gérard–Kappeler (Commun Pure Appl Math, 2020. 10.1002/cpa.21896. arXiv:1905.01849). The global well-posedness of the BO equation on $$mathcal {U}_N$$ U N is given by a polynomial characterization and a spectral characterization of the manifold $$mathcal {U}_N$$ U N . Besides the spectral analysis of the Lax operator of the BO equation and the shift semigroup acting on some Hardy spaces, the construction of such coordinates also relies on the use of a generating functional, which encodes the entire BO hierarchy. The inverse spectral formula of an N-soliton provides a spectral connection between the Lax operator and the infinitesimal generator of the very shift semigroup.

Beschreibung: We study fields reminiscent of vertex operators built from the Brownian loop soup in the limit as the loop soup intensity tends to infinity. More precisely, following Camia et al. (Nucl Phys B 902:483–507, 2016), we take a (massless or massive) Brownian loop soup in a planar domain and assign a random sign to each loop. We then consider random fields defined by taking, at every point of the domain, the exponential of a purely imaginary constant times the sum of the signs associated to the loops that wind around that point. For domains conformally equivalent to a disk, the sum diverges logarithmically due to the small loops, but we show that a suitable renormalization procedure allows to define the fields in an appropriate Sobolev space. Subsequently, we let the intensity of the loop soup tend to infinity and prove that these vertex-like fields tend to a conformally covariant random field which can be expressed as an explicit functional of the imaginary Gaussian multiplicative chaos with covariance kernel given by the Brownian loop measure. Besides using properties of the Brownian loop soup and the Brownian loop measure, a main tool in our analysis is an explicit Wiener–Itô chaos expansion of linear functionals of vertex-like fields. Our methods apply to other variants of the model in which, for example, Brownian loops are replaced by disks.

Beschreibung: The generation of certifiable randomness is the most fundamental information-theoretic task that meaningfully separates quantum devices from their classical counterparts. We propose a protocol for exponential certified randomness expansion using a single quantum device. The protocol calls for the device to implement a simple quantum circuit of constant depth on a 2D lattice of qubits. The output of the circuit can be verified classically in linear time, and is guaranteed to contain a polynomial number of certified random bits assuming that the device used to generate the output operated using a (classical or quantum) circuit of sub-logarithmic depth. This assumption contrasts with the locality assumption used for randomness certification based on Bell inequality violation and more recent proposals for randomness certification based on computational assumptions. Furthermore, to demonstrate randomness generation it is sufficient for a device to sample from the ideal output distribution within constant statistical distance. Our procedure is inspired by recent work of Bravyi et al. (Science 362(6412):308–311, 2018), who introduced a relational problem that can be solved by a constant-depth quantum circuit, but provably cannot be solved by any classical circuit of sub-logarithmic depth. We develop the discovery of Bravyi et al. into a framework for robust randomness expansion. Our results lead to a new proposal for a demonstrated quantum advantage that has some advantages compared to existing proposals. First, our proposal does not rest on any complexity-theoretic conjectures, but relies on the physical assumption that the adversarial device being tested implements a circuit of sub-logarithmic depth. Second, success on our task can be easily verified in classical linear time. Finally, our task is more noise-tolerant than most other existing proposals that can only tolerate multiplicative error, or require additional conjectures from complexity theory; in contrast, we are able to allow a small constant additive error in total variation distance between the sampled and ideal distributions.

Beschreibung: We consider the nonlinear cubic Wave, the Hartree and the nonlinear cubic Beam equations on $${mathbb {T}}^2$$ T 2 and we prove the existence of different types of solutions which exchange energy between Fourier modes in certain time scales. This exchange can be considered “chaotic-like” since either the choice of activated modes or the time spent in each transfer can be chosen randomly. The key point of the construction of those orbits is the existence of heteroclinic connections between invariant objects and the construction of symbolic dynamics (a Smale horseshoe) for the Birkhoff Normal Form truncation of those equations.

Beschreibung: We consider a stochastic perturbation of the classical Lorenz system in the range of parameters for which the origin is the global attractor. We show that adding noise in the last component causes a transition from a unique to exactly two ergodic invariant measures. The bifurcation threshold depends on the strength of the noise: if the noise is weak, the only invariant measure is Gaussian, while strong enough noise causes the appearance of a second ergodic invariant measure.

Beschreibung: We study open and closed string amplitudes at tree-level in string perturbation theory using the methods of single-valued integration which were developed in the prequel to this paper (Brown and Dupont in Single-valued integration and double copy, 2020). Using dihedral coordinates on the moduli spaces of curves of genus zero with marked points, we define a canonical regularisation of both open and closed string perturbation amplitudes at tree level, and deduce that they admit a Laurent expansion in Mandelstam variables whose coefficients are multiple zeta values (resp. single-valued multiple zeta values). Furthermore, we prove the existence of a motivic Laurent expansion whose image under the period map is the open string expansion, and whose image under the single-valued period map is the closed string expansion. This proves the recent conjecture of Stieberger that closed string amplitudes are the single-valued projections of (motivic lifts of) open string amplitudes. Finally, applying a variant of the single-valued formalism for cohomology with coefficients yields the KLT formula expressing closed string amplitudes as quadratic expressions in open string amplitudes.

Beschreibung: We are interested in studying the stationary solutions and phase transitions of aggregation equations with degenerate diffusion of porous medium-type, with exponent $$1〈 m 〈 infty $$ 1

Beschreibung: Various quantum analogues of the central limit theorem, which is one of the cornerstones of probability theory, are known in the literature. One such analogue, due to Cushen and Hudson, is of particular relevance for quantum optics. It implies that the state in any single output arm of an n-splitter, which is fed with n copies of a centred state $$ ho $$ ρ with finite second moments, converges to the Gaussian state with the same first and second moments as $$ ho $$ ρ . Here we exploit the phase space formalism to carry out a refined analysis of the rate of convergence in this quantum central limit theorem. For instance, we prove that the convergence takes place at a rate $$mathcal {O}left( n^{-1/2} ight) $$ O n - 1 / 2 in the Hilbert–Schmidt norm whenever the third moments of $$ ho $$ ρ are finite. Trace norm or relative entropy bounds can be obtained by leveraging the energy boundedness of the state. Via analytical and numerical examples we show that our results are tight in many respects. An extension of our proof techniques to the non-i.i.d. setting is used to analyse a new model of a lossy optical fibre, where a given m-mode state enters a cascade of n beam splitters of equal transmissivities $$lambda ^{1/n}$$ λ 1 / n fed with an arbitrary (but fixed) environment state. Assuming that the latter has finite third moments, and ignoring unitaries, we show that the effective channel converges in diamond norm to a simple thermal attenuator, with a rate $$mathcal {O}Big (n^{-frac{1}{2(m+1)}}Big )$$ O ( n - 1 2 ( m + 1 ) ) . This allows us to establish bounds on the classical and quantum capacities of the cascade channel. Along the way, we derive several results that may be of independent interest. For example, we prove that any quantum characteristic function $$chi _ ho $$ χ ρ is uniformly bounded by some $$eta _ ho

Beschreibung: We study the spectra of quantum trees of finite cone type. These are quantum graphs whose geometry has a certain homogeneity, and which carry a finite set of edge lengths, coupling constants and potentials on the edges. We show the spectrum consists of bands of purely absolutely continuous spectrum, along with a discrete set of eigenvalues. Afterwards, we study random perturbations of such trees, at the level of edge length and coupling, and prove the stability of pure AC spectrum, along with resolvent estimates.

Beschreibung: The gauge theories underlying gauged supergravity and exceptional field theory are based on tensor hierarchies: generalizations of Yang-Mills theory utilizing algebraic structures that generalize Lie algebras and, as a consequence, require higher-form gauge fields. Recently, we proposed that the algebraic structure allowing for consistent tensor hierarchies is axiomatized by ‘infinity-enhanced Leibniz algebras’ defined on graded vector spaces generalizing Leibniz algebras. It was subsequently shown that, upon appending additional vector spaces, this structure can be reinterpreted as a differential graded Lie algebra. We use this observation to streamline the construction of general tensor hierarchies, and we formulate dynamics in terms of a hierarchy of first-order duality relations, including scalar fields with a potential.

Beschreibung: We establish rapid mixing of the random-cluster Glauber dynamics on random $$varDelta $$ Δ -regular graphs for all $$qge 1$$ q ≥ 1 and $$p 2 the Glauber dynamics on random $$varDelta $$ Δ -regular graphs undergoes an exponential slowdown at $$p_u(q,varDelta )$$ p u ( q , Δ ) . More precisely, we show that for every $$qge 1$$ q ≥ 1 , $$varDelta ge 3$$ Δ ≥ 3 , and $$p

Beschreibung: The Rarita–Schwinger operator is the twisted Dirac operator restricted to $$ icefrac 32$$ 3 2 -spinors. Rarita–Schwinger fields are solutions of this operator which are in addition divergence-free. This is an overdetermined problem and solutions are rare; it is even more unexpected for there to be large dimensional spaces of solutions. In this paper we prove the existence of a sequence of compact manifolds in any given dimension greater than or equal to 4 for which the dimension of the space of Rarita–Schwinger fields tends to infinity. These manifolds are either simply connected Kähler–Einstein spin with negative Einstein constant, or products of such spaces with flat tori. Moreover, we construct Calabi–Yau manifolds of even complex dimension with more linearly independent Rarita–Schwinger fields than flat tori of the same dimension.

Beschreibung: We use coarse index methods to prove that the Landau Hamiltonian on the hyperbolic half-plane, and even on much more general imperfect half-spaces, has no spectral gaps. Thus the edge states of hyperbolic quantum Hall Hamiltonians completely fill up the gaps between Landau levels, just like those of the Euclidean counterparts.

Beschreibung: We present a new construction of the Euclidean $$Phi ^4$$ Φ 4 quantum field theory on $${mathbb {R}}^3$$ R 3 based on PDE arguments. More precisely, we consider an approximation of the stochastic quantization equation on $${mathbb {R}}^3$$ R 3 defined on a periodic lattice of mesh size $$varepsilon $$ ε and side length M. We introduce a new renormalized energy method in weighted spaces and prove tightness of the corresponding Gibbs measures as $$varepsilon ightarrow 0$$ ε → 0 , $$M ightarrow infty $$ M → ∞ . Every limit point is non-Gaussian and satisfies reflection positivity, translation invariance and stretched exponential integrability. These properties allow to verify the Osterwalder–Schrader axioms for a Euclidean QFT apart from rotation invariance and clustering. Our argument applies to arbitrary positive coupling constant, to multicomponent models with O(N) symmetry and to some long-range variants. Moreover, we establish an integration by parts formula leading to the hierarchy of Dyson–Schwinger equations for the Euclidean correlation functions. To this end, we identify the renormalized cubic term as a distribution on the space of Euclidean fields.

Beschreibung: We introduce a new approach to find the Tomita–Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo–Martin–Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann–Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered.

Beschreibung: We study geometric presentations of braid groups for particles that are constrained to move on a graph, i.e. a network consisting of nodes and edges. Our proposed set of generators consists of exchanges of pairs of particles on junctions of the graph and of certain circular moves where one particle travels around a simple cycle of the graph. We point out that so defined generators often do not satisfy the braiding relation known from 2D physics. We accomplish a full description of relations between the generators for star graphs where we derive certain quasi-braiding relations. We also describe how graph braid groups depend on the (graph-theoretic) connectivity of the graph. This is done in terms of quotients of graph braid groups where one-particle moves are put to identity. In particular, we show that for 3-connected planar graphs such a quotient reconstructs the well-known planar braid group. For 2-connected graphs this approach leads to generalisations of the Yang–Baxter equation. Our results are of particular relevance for the study of non-abelian anyons on networks showing new possibilities for non-abelian quantum statistics on graphs.

Beschreibung: The purpose of this paper is to propose a sheaf theoretic approach to the theory of quantum principal bundles over non affine bases. We study noncommutative principal bundles corresponding to $$G ightarrow G/P$$ G → G / P , where G is a semisimple group and P a parabolic subgroup.

Beschreibung: In this paper, we first construct the controlling algebras of embedding tensors and Lie–Leibniz triples, which turn out to be a graded Lie algebra and an $$L_infty $$ L ∞ -algebra respectively. Then we introduce representations and cohomologies of embedding tensors and Lie–Leibniz triples, and show that there is a long exact sequence connecting various cohomologies. As applications, we classify infinitesimal deformations and central extensions using the second cohomology groups. Finally, we introduce the notion of a homotopy embedding tensor which will induce a Leibniz$$_infty $$ ∞ -algebra. We realize Kotov and Strobl’s construction of an $$L_infty $$ L ∞ -algebra from an embedding tensor, as a functor from the category of homotopy embedding tensors to that of Leibniz$$_infty $$ ∞ -algebras, and a functor further to that of $$L_infty $$ L ∞ -algebras.