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Abstract: Abstract We prove that the Podleś spheres \(S_q^2\) converge in quantum Gromov–Hausdorff distance to the classical 2-sphere as the deformation parameter q tends to 1. Moreover, we construct a q-deformed analogue of the fuzzy spheres, and prove that they converge to \(S_q^2\) as their linear dimension tends to infinity, thus providing a quantum counterpart to a classical result of Rieffel. PubDate: 2022-06-01
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Abstract: Abstract New eigenvalue enclosures for the block operator problem arising in the study of stability of the Ekman boundary layer are proved. This solves an open problem in [19] on the existence of open sets of eigenvalues in domains of Fredholmness of the analyzed operator family. PubDate: 2022-06-01
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Abstract: Abstract We analyze the problem of global reconstruction of functions as accurately as possible, based on partial information in the form of a truncated power series at some point, and additional analyticity properties. This situation occurs frequently in applications. The question of the optimal procedure was open, and we formulate it as a well-posed mathematical problem. Its solution leads to a practical method which provides dramatic accuracy improvements over existing techniques. Our procedure is based on uniformization of Riemann surfaces. As an application, we show that our procedure can be implemented for solutions of a wide class of nonlinear ODEs. We find a new uniformization method, which we use to construct the uniformizing maps needed for special functions, including solution of the Painlevé equations \(P_\mathrm{I}\) – \(P_{\mathrm{V}}\) . We also introduce a new rigorous and constructive method of regularization, elimination of singularities whose position and type are known. If these are unknown, the same procedure enables a highly sensitive resonance method to determine the position and type of a singularity. In applications where less explicit information is available about the Riemann surface, our approach and techniques lead to new approximate, but still much more precise reconstruction methods than existing ones, especially in the vicinity of singularities, which are the points of greatest interest. PubDate: 2022-06-01
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Abstract: Abstract We derive a new bound on the effectiveness of the Petz map as a universal recovery channel in approximate quantum error correction using the second sandwiched Rényi relative entropy \(\widetilde{D}_{2}\) . For large Hilbert spaces, our bound implies that the Petz map performs quantum error correction with order- \(\epsilon \) accuracy whenever the data processing inequality for \(\widetilde{D}_{2}\) is saturated up to terms of order \(\epsilon ^2\) times the inverse Hilbert space dimension. Conceptually, our result is obtained by extending (Cree and Sorce in J Phys A Math Theor, 2022. http://iopscience.iop.org/article/10.1088/1751-8121/ac5648), in which we studied exact saturation of the data processing inequality using differential geometry, to the case of approximate saturation. Important roles are played by (i) the fact that the exponential of the second sandwiched Rényi relative entropy is quadratic in its first argument, and (ii) the observation that the second sandwiched Rényi relative entropy satisfies the data processing inequality even when its first argument is a non-positive Hermitian operator. PubDate: 2022-06-01
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Abstract: Abstract We prove the Hölder continuity of the integrated density of states for a class of quasi-periodic long-range operators on \(\ell ^2({\mathbb {Z}}^d)\) with large trigonometric polynomial potentials and Diophantine frequencies. Moreover, we give the Hölder exponent in terms of the cardinality of the level sets of the potentials, which improves, in the perturbative regime, the result obtained by Goldstein and Schlag (Geom. Funct. Anal. 18:755-869, 2008). Our approach is a combination of Aubry duality, generalized Thouless formula and the regularity of the Lyapunov exponents of analytic quasi-periodic \(GL(m,{\mathbb {C}})\) cocycles which is proved by quantitative almost reducibility method. PubDate: 2022-06-01
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Abstract: Abstract Generalizing Frassek et al. (Adv. Math. 401, 108283 (2022). https://doi.org/10.1016/j.aim.2022.108283), we construct a family of SO(2r), Sp(2r), \(SO(2r\!+\!1)\) rational Lax matrices \(T_D(z)\) , polynomial in the spectral parameter z, parametrized by \(\Lambda ^+\) -valued divisors D on \({\mathbb {P}}^1\) . To this end, we provide the RTT realization of the antidominantly shifted extended Drinfeld Yangians of \({\mathfrak {g}}=\mathfrak {so}_{2r}, \mathfrak {sp}_{2r}, \mathfrak {so}_{2r+1}\) , and of their coproduct homomorphisms. PubDate: 2022-06-01
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Abstract: Abstract The Bagger–Witten line bundle is a line bundle over moduli spaces of two-dimensional SCFTs, related to the Hodge line bundle of holomorphic top-forms on Calabi–Yau manifolds. It has recently been a subject of a number of conjectures, but concrete examples have proven elusive. In this paper we propose a new, intrinsically geometric definition of the Bagger–Witten line bundle, whose restriction to the moduli spaces of complex structures of Calabi–Yau manifolds we explicitly compute in some concrete examples. We also conjecture a new criterion for UV completion of four-dimensional supergravity theories in terms of properties of the Bagger–Witten line bundle. PubDate: 2022-06-01
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Abstract: Abstract We recall Borcherds’s approach to vertex algebras via “singular commutative rings”, and introduce new examples of his constructions which we compare to vertex algebras, chiral algebras, and factorization algebras. We show that all vertex algebras (resp. chiral algebras or equivalently factorization algebras) can be realized in these new categories \(\text {VA}(A,H,S)\) . PubDate: 2022-06-01
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Abstract: Abstract This paper develops a concept of relative Cauchy evolution for the class of homotopy algebraic quantum field theories (AQFTs) that are obtained by canonical commutation relation quantization of Poisson chain complexes. The key element of the construction is a rectification theorem proving that the homotopy time-slice axiom, which is a higher categorical relaxation of the time-slice axiom of AQFT, can be strictified for theories in this class. The general concept is illustrated through a detailed study of the relative Cauchy evolution for the homotopy AQFT associated with linear Yang-Mills theory, for which the usual stress-energy tensor is recovered. PubDate: 2022-06-01
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Abstract: Abstract We introduce a notion of upper Green regular solutions to the Lax-Oleinik semi-group that is defined on the set of \(C^0\) functions of a closed manifold via a Tonelli Lagrangian. Then we prove some weak \(C^2\) convergence results to such a solution for a large class of approximated solutions as (1) the discounted solution (see [DFIZ16]); (2) the image of a \(C^0\) function by the Lax-Oleinik semi-group; (3) the weak K.A.M. solutions for perturbed cohomology class. This kind of convergence implies the convergence in measure of the second derivatives. Moreover, we provide an example that is not upper Green regular and to which we have \(C^1\) convergence but not convergence in measure of the second derivatives. PubDate: 2022-06-01
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Abstract: Abstract We consider sequential hypothesis testing between two quantum states using adaptive and non-adaptive strategies. In this setting, samples of an unknown state are requested sequentially and a decision to either continue or to accept one of the two hypotheses is made after each test. Under the constraint that the number of samples is bounded, either in expectation or with high probability, we exhibit adaptive strategies that minimize both types of misidentification errors. Namely, we show that these errors decrease exponentially (in the stopping time) with decay rates given by the measured relative entropies between the two states. Moreover, if we allow joint measurements on multiple samples, the rates are increased to the respective quantum relative entropies. We also fully characterize the achievable error exponents for non-adaptive strategies and provide numerical evidence showing that adaptive measurements are necessary to achieve our bounds. PubDate: 2022-06-01
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Abstract: Abstract We study the tree scattering amplitudes of Yang–Mills and General Relativity as functions of complex momenta, using a homological and geometrical approach. This approach uses differential graded Lie algebras, one for YM and one for GR, whose Maurer Cartan equations are the classical field equations. The tree amplitudes are obtained as the \(L_\infty \) minimal model brackets, given by a trivalent Feynman tree expansion. We show that they are sections of a sheaf on the complex variety of momenta, and that their residues factor in a characteristic way. This requires classifying the irreducible codimension one subvarieties where poles occur; constructing dedicated gauges that make the factorization manifest; and proving a flexible version of gauge independence to be able to work with different gauges. The residue factorization yields a simple recursive characterization of the tree amplitudes of YM and GR, by exploiting Hartogs’ phenomenon for singular varieties. This is similar to and inspired by Britto–Cachazo–Feng–Witten recursion, but avoids BCFW’s trick of shifting momenta, hence avoids conditions at infinity under such shifts. PubDate: 2022-06-01
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Abstract: Abstract We establish quantitative upper and lower bounds for Schrödinger operators with complex potentials that satisfy some weak form of sparsity. Our first result is a quantitative version of an example, due to S. Bögli (Commun Math Phys 352:629–639, 2017), of a Schrödinger operator with eigenvalues accumulating to every point of the essential spectrum. The second result shows that the eigenvalue bounds of Frank (Bull Lond Math Soc 43:745–750, 2011 and Trans Am Math Soc 370:219–240, 2018) can be improved for sparse potentials. The third result generalizes a theorem of Klaus (Ann Inst H Poincaré Sect A (N.S.) 38:7–13, 1983) on the characterization of the essential spectrum to the multidimensional non-selfadjoint case. The fourth result shows that, in one dimension, the purely imaginary (non-sparse) step potential has unexpectedly many eigenvalues, comparable to the number of resonances. Our examples show that several known upper bounds are sharp. PubDate: 2022-06-01
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Abstract: Abstract On an annulus \({{\mathbb {A}}}_q :=\{z \in {{\mathbb {C}}}: q< z < 1\}\) with a fixed \(q \in (0, 1)\) , we study a Gaussian analytic function (GAF) and its zero set which defines a point process on \({{\mathbb {A}}}_q\) called the zero point process of the GAF. The GAF is defined by the i.i.d. Gaussian Laurent series such that the covariance kernel parameterized by \(r >0\) is identified with the weighted Szegő kernel of \({{\mathbb {A}}}_q\) with the weight parameter r studied by McCullough and Shen. The GAF and the zero point process are rotationally invariant and have a symmetry associated with the q-inversion of coordinate \(z \leftrightarrow q/z\) and the parameter change \(r \leftrightarrow q^2/r\) . When \(r=q\) they are invariant under conformal transformations which preserve \({{\mathbb {A}}}_q\) . Conditioning the GAF by adding zeros, new GAFs are induced such that the covariance kernels are also given by the weighted Szegő kernel of McCullough and Shen but the weight parameter r is changed depending on the added zeros. We also prove that the zero point process of the GAF provides a permanental-determinantal point process (PDPP) in which each correlation function is expressed by a permanent multiplied by a determinant. Dependence on r of the unfolded 2-correlation function of the PDPP is studied. If we take the limit \(q \rightarrow 0\) , a simpler but still non-trivial PDPP is obtained on the unit disk \({\mathbb {D}}\) . We observe that the limit PDPP indexed by \(r \in (0, \infty )\) can be regarded as an interpolation between the determinantal point process (DPP) on \({{\mathbb {D}}}\) studied by Peres and Virág ( \(r \rightarrow 0\) ) and that DPP of Peres and Virág with a deterministic zero added at the origin ( \(r \rightarrow \infty \) ). PubDate: 2022-06-01
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Abstract: Abstract We establish a surface order large deviation estimate for the magnetisation of low temperature \(\phi ^4_3\) . As a byproduct, we obtain a decay of spectral gap for its Glauber dynamics given by the \(\phi ^4_3\) singular stochastic PDE. Our main technical contributions are contour bounds for \(\phi ^4_3\) , which extends 2D results by Glimm et al. (Commun Math Phys 45(3):203–216, 1975). We adapt an argument by Bodineau et al. (J Math Phys 41(3):1033–1098, 2000) to use these contour bounds to study phase segregation. The main challenge to obtain the contour bounds is to handle the ultraviolet divergences of \(\phi ^4_3\) whilst preserving the structure of the low temperature potential. To do this, we build on the variational approach to ultraviolet stability for \(\phi ^4_3\) developed recently by Barashkov and Gubinelli (Duke Math. J. 169(17):3339–3415, 2020). PubDate: 2022-06-01
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Abstract: Abstract We refine the Lyapunov–Schmidt analysis from our recent paper (Eichmair and Koerber in Large area-constrained Willmore surfaces in asymptotically Schwarzschild 3-manifolds. arXiv preprint arXiv:2101.12665, 2021) to study the geometric center of mass of the asymptotic foliation by area-constrained Willmore surfaces of initial data for the Einstein field equations. If the scalar curvature of the initial data vanishes at infinity, we show that this geometric center of mass agrees with the Hamiltonian center of mass. By contrast, we show that the positioning of large area-constrained Willmore surfaces is sensitive to the distribution of the energy density. In particular, the geometric center of mass may differ from the Hamiltonian center of mass if the scalar curvature does not satisfy additional asymptotic symmetry assumptions. PubDate: 2022-06-01
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Abstract: Abstract We present four infinite families of mutually commuting difference operators which include the deformed elliptic Ruijsenaars operators. The trigonometric limit of this kind of operators was previously introduced by Feigin and Silantyev. They provide a quantum mechanical description of two kinds of relativistic quantum mechanical particles which can be identified with particles and anti-particles in an underlying quantum field theory. We give direct proofs of the commutativity of our operators and of some other fundamental properties such as kernel function identities. In particular, we give a rigorous proof of the quantum integrability of the deformed Ruijsenaars model. PubDate: 2022-06-01
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Abstract: Abstract The split property of a pure state for a certain cut of a quantum spin system can be understood as the entanglement between the two subsystems being weak. From this point of view, we may say that if it is not possible to transform a state \(\omega \) via sufficiently local automorphisms (in a sense that we will make precise) into a state satisfying the split property, then the state \(\omega \) has a long-range entanglement. It is well known that in 1D, gapped ground states have the split property with respect to cutting the system into left and right half-chains. In 2D, however, the split property fails to hold for interesting models such as Kitaev’s toric code. Here we show that this failure is the reason that anyons can exist in that model. There is a folklore saying that the existence of anyons, like in the toric code model, implies long-range entanglement of the state. In this paper, we prove this folklore in an infinite dimensional setting. More precisely, we show that long-range entanglement, in a way that we will define precisely, is a necessary condition to have non-trivial superselection sectors. Anyons in particular give rise to such non-trivial sectors. States with the split property for cones, on the other hand, do not admit non-trivial sectors. A key technical ingredient of our proof is that under suitable assumptions on locality, the automorphisms generated by local interactions can be “approximately factorized.” That is, they can be written as the tensor product of automorphisms localized in a cone and its complement respectively, followed by an automorphism acting near the “boundary” of \(\Lambda \) , and conjugation with a unitary. This result may be of independent interest. This technique also allows us to prove that the approximate split property, a weaker version of the split property that is satisfied in e.g. the toric code, is stable under applying such automorphisms. PubDate: 2022-06-01
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Abstract: Abstract The Van Vleck formula is a semiclassical approximation to the integral kernel of the propagator associated to a time-dependent Schrödinger equation. Under suitable hypotheses, we present a rigorous treatment of this approximation which is valid on Ehrenfest time scales, i.e. \(\hbar \) -dependent time intervals which most commonly take the form \( t \le c \log \hbar \) . Our derivation is based on an approximation to the integral kernel often called the Herman–Kluk approximation, which realizes the kernel as an integral superposition of Gaussians parameterized by points in phase space. As was shown by Robert (Rev Math Phys 22(10):1123-1145, 2010) , this yields effective approximations over Ehrenfest time intervals. In order to derive the Van Vleck approximation from the Herman–Kluk approximation, we are led to develop stationary phase asymptotics where the phase functions depend on the frequency parameter in a nontrivial way, a result which may be of independent interest. PubDate: 2022-06-01
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Abstract: Abstract The lace expansion for the Ising two-point function was successfully derived in (Sakai in Commun Math Phys 272:283–344, 2007, Proposition 1.1). It is an identity that involves an alternating series of lace-expansion coefficients. In the same paper, we claimed that the expansion coefficients obey certain diagrammatic bounds which imply faster x-space decay (as the two-point function cubed) above the critical dimension \(d_\mathrm {c}\) ( \(=4\) for finite-variance models) if the spin-spin coupling is ferromagnetic, translation-invariant, summable and symmetric with respect to the underlying lattice symmetries. However, we recently found a flaw in the proof of (Sakai in Commun Math Phys 272:283–344, 2007, Lemma 4.2), a key lemma to the aforementioned diagrammatic bounds. In this paper, we no longer use the problematic (Sakai 2007, Lemma 4.2), and prove new diagrammatic bounds on the expansion coefficients that are slightly more complicated than those in (Sakai 2007, Proposition 4.1) but nonetheless obey the same fast decay above the critical dimension \(d_\mathrm {c}\) . Consequently, the lace-expansion results for the Ising and \(\varphi ^4\) models in the literature are all saved. The proof is based on the random-current representation and its source-switching technique of Griffiths, Hurst and Sherman, combined with a double expansion: a lace expansion for the lace-expansion coefficients. PubDate: 2022-05-10
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