Higher Dimensional Generalization Research Articles

Parallel transport for vector bundles on 𝑝-adic varieties

We develop a theory of étale parallel transport for vector bundles with numerically flat reduction on a p p -adic variety. This construction is compatible with natural operations on vector bundles, Galois equivariant and functorial with respect to morphisms of varieties. In particular, it provides a continuous p p -adic representation of the étale fundamental group for every vector bundle with numerically flat reduction. The results in the present paper generalize previous work by the authors on curves. They can be seen as a p p -adic analog of higher-dimensional generalizations of the classical Narasimhan-Seshadri correspondence on complex varieties. Moreover, they provide new insights into Faltings’ p p -adic Simpson correspondence between small Higgs bundles and small generalized representations by establishing a class of vector bundles with vanishing Higgs field giving rise to actual (not only generalized) representations.

The Polyanalytic Reproducing Kernels

Let $$\nu $$ be a rotation invariant Borel probability measure on the complex plane having moments of all orders. Given a positive integer q, it is proved that the space of $$\nu $$ -square integrable q-analytic functions is the closure of q-analytic polynomials, and in particular it is a Hilbert space. We establish a general formula for the corresponding polyanalytic reproducing kernel. New examples are given and all known examples, including those of the analytic case are covered. In particular, weighted Bergman and Fock type spaces of polyanalytic functions are introduced. Our results have a higher dimensional generalization for measure on $$\mathbb {C}^p$$ which are in rotation invariant with respect to each coordinate.

Holographic OPE coefficients from AdS black holes with matters

We study the OPE coefficients cΔ, J for heavy-light scalar four-point functions, which can be obtained holographically from the two-point function of a light scalar of some non-integer conformal dimension ΔL in an AdS black hole. We verify that the OPE coefficient cd,0 = 0 for pure gravity black holes, consistent with the tracelessness of the holographic energy-momentum tensor. We then study the OPE coefficients from black holes involving matter fields. We first consider general charged AdS black holes and we give some explicit low-lying examples of the OPE coefficients. We also obtain the recursion formula for the lowest-twist OPE coefficients with at most two current operators. For integer ΔL, although the OPE coefficients are not fully determined, we set up a framework to read off the coefficients γΔ,J of the log(z overline{z} ) terms that are associated with the anomalous dimensions of the exchange operators and obtain a general formula for γΔ,J. We then consider charged AdS black holes in gauged supergravity STU models in D = 5 and D = 7, and their higher-dimensional generalizations. The scalar fields in the STU models are conformally massless, dual to light operators with ΔL = d − 2. We derive the linear perturbation of such a scalar in the STU charged AdS black holes and obtain the explicit OPE coefficient cd−2,0. Finally, we analyse the asymptotic properties of scalar hairy AdS black holes and show how cd,0 can be nonzero with exchanging scalar operators in these backgrounds.

Higher-dimensional Buchdahl and Janis–Robinson–Winicour transformations in the Einstein–Maxwell system with a massless scalar field

We present higher-dimensional generalizations of the Buchdahl and Janis–Robinson–Winicour transformations which generate static solutions in the Einstein–Maxwell system with a massless scalar field. While the former adds a nontrivial scalar field to a vacuum solution, the latter generates a charged solution from a neutral one with the same scalar field. Applying these transformations to (i) a static solution with an Einstein base manifold, (ii) a multi-center solution, and (iii) a four-dimensional cylindrically symmetric solution, we construct several new exact solutions.

Perverse Coherent Sheaves on Blow-ups at Codimension 2 Loci

Abstract Let $f \colon X \to Y$ be the blow-up of a smooth projective variety $Y$ along its codimension two smooth closed subvariety. In this paper, we show that the moduli space of stable sheaves on $X$ and $Y$ are connected by a sequence of flip-like diagrams. The result is a higher dimensional generalization of the result of Nakajima and Yoshioka, which is the case of $\dim Y=2$. As an application of our general result, we study the birational geometry of the Hilbert scheme of two points.

Prediction of Multidimensional Spatial Variation Data via Bayesian Tensor Completion

This paper presents a multidimensional computational method to predict the spatial variation data inside and across multiple dies of a wafer. This technique is based on tensor computation. A tensor is a high-dimensional generalization of a matrix or a vector. By exploiting the hidden low-rank property of a high-dimensional data array, the large amount of unknown variation testing data may be predicted from a few random measurement samples. The tensor rank, which decides the complexity of a tensor representation, is decided by an available variational Bayesian approach. Our approach is validated by a practical chip testing data set, and it can be easily generalized to characterize the process variations of multiple wafers. Our approach is more efficient than the previous virtual probe techniques in terms of memory and computational cost when handling high-dimensional chip testing data.

Planar tropical cubic curves of any genus, and higher dimensional generalisations

We study the maximal values of Betti numbers of tropical subvarieties of a given dimension and degree in \mathbb TP^n . We provide a lower estimate for the maximal value of the top Betti number, which naturally depends on the dimension and degree, but also on the codimension. In particular, when the codimension is large enough, this lower estimate is larger than the maximal value of the corresponding Hodge number of complex algebraic projective varieties of the given dimension and degree. In the case of surfaces, we extend our study to all tropical homology groups. As a special case, we prove that there exist planar tropical cubic curves of genus g for any non-negative integer g .

Exploring Bell inequalities for the device-independent certification of multipartite entanglement depth

Techniques developed for device-independent characterizations allow one to certify certain physical properties of quantum systems without assuming any knowledge of their internal workings. Such a certification, however, often relies on the employment of device-independent witnesses catered for the particular property of interest. In this work, we consider a one-parameter family of multipartite, two-setting, two-outcome Bell inequalities and demonstrate the extent to which they are suited for the device-independent certification of genuine many-body entanglement (and hence the entanglement depth) present in certain well-known multipartite quantum states, including the generalized Greenberger-Horne-Zeilinger (GHZ) states with unbalanced weights, the higher-dimensional generalizations of balanced GHZ states, and the $W$ states. As a by-product of our investigations, we have found that, in contrast with well-established results, provided trivial qubit measurements are allowed, full-correlation Bell inequalities can also be used to demonstrate the nonlocality of weakly entangled unbalanced-weight GHZ states. Besides, we also demonstrate how two-setting, two-outcome Bell inequalities can be constructed, based on the so-called GHZ paradox, to witness the entanglement depth of various graph states, including the ring graph states, the fully connected graph states, and some linear graph states, etc.

Abundance of minimal surfaces

This article is concerned with the existence theory of closed minimal hypersurfaces in closed Riemannian manifolds of dimension at least three. These hypersurfaces are critical points for the area functional, and hence their study can be seen as a high-dimensional generalization of the classical theory of closed geodesics (Birkhoff, Morse, Lusternik, Schnirel’mann,…). The best result until very recently, due to Almgren ([2], 1965), Pitts ([37], 1981), and Schoen–Simon ([43], 1981), was the existence of at least one closed minimal hypersurface in every closed Riemannian manifold. I will discuss the methods I have developed with Andre Neves, for the past few years, to approach this problem through the variational point of view. These ideas have culminated with the discovery that minimal hypersurfaces in fact abound.

Canonical forms for single-qutrit Clifford+[formula omitted] operators

We introduce canonical forms for single qutrit Clifford+T circuits and prove that every single-qutrit Clifford+T operator admits a unique such canonical form. We show that our canonical forms are T-optimal in the sense that among all the single-qutrit Clifford+T circuits implementing a given operator our canonical form uses the least number of T gates. Finally, we provide an algorithm which inputs the description of an operator (as a matrix or a circuit) and constructs the canonical form for this operator. The algorithm runs in time linear in the number of T gates. Our results provide a higher-dimensional generalization of prior work by Matsumoto and Amano who introduced similar canonical forms for single-qubit Clifford+T circuits.

A symmetrical quasi-classical windowing model for the molecular dynamics treatment of non-adiabatic processes involving many electronic states.

In the previous work of Cotton and Miller [J. Chem. Phys. 145, 144108 (2016)], an improved symmetrical quasi-classical (SQC) windowing model for the molecular dynamics treatment of electronically non-adiabatic processes was developed in order to extend the original SQC approach to the regime of weak-coupling between the electronic states. The improved SQC model-based on triangular-shaped window functions-handled the weak-coupling limit as intended and, as a bonus, was shown to be universally superior to the original square/histogram SQC windowing model over all coupling regimes, but only for treating systems of two electronic states, as no higher-dimensional generalization was evident. This paper, therefore, provides a generalized version for treating an arbitrary number of electronic states. By construction, the benefits of the two-state triangle model-seamless treatment of weak-coupling and improved accuracy in all coupling regimes-carry over to the generalized version. Far more significant, however, is that the new model provides vastly improved windowing statistics in higher dimensions, enabling the SQC simulation of electronically non-adiabatic processes involving many more relevant electronic states than was previously practical. Capabilities are demonstrated with respect to a 24 pigment trimer model of the Fenna-Matthews-Olson light-harvesting complex, as well as treating similar 48- and 96-electronic state model problems, illustrating the scaling properties of the new method.

Multi Hive Artificial Bee Colony Programming for high dimensional symbolic regression with feature selection

Feature selection is a process that provides model extraction by specifying necessary or related features and improves generalization. The Artificial Bee Colony (ABC) algorithm is one of the most popular optimization algorithms inspired on swarm intelligence developed by simulating the search behavior of honey bees. Artificial Bee Colony Programming (ABCP) is a recently proposed high level automatic programming technique for a Symbolic Regression (SR) problem based on the ABC algorithm. In this paper, a new feature selection method based on ABCP is proposed, Multi Hive ABCP (MHABCP) for high-dimensional SR problems. The learning ability and generalization performance of the proposed MHABCP is investigated using synthetic and real high-dimensional SR datasets and is compared with basic ABCP and GP automatic programming methods. Experimental results show that MHABCP has better performance choosing relevant features in high dimensional SR problems and generalization than other methods.

Fermionic Lieb-Schultz-Mattis theorems and weak symmetry-protected phases

The Lieb-Schultz-Mattis (LSM) theorem and its higher-dimensional generalizations by Oshikawa and Hastings establish that a translation-invariant lattice model of spin-$1/2$'s can not have a non-degenerate ground state preserving both spin and translation symmetries. Recently it was shown that LSM theorems can be interpreted in terms of bulk-boundary correspondence of certain weak symmetry-protected topological (SPT) phases. In this work we discuss LSM-type theorems for two-dimensional fermionic systems, which have no bosonic analogs. They follow from a general classification of weak SPT phases of fermions in three dimensions. We further derive constraints on possible gapped symmetry-enriched topological phases in such systems. In particular, we show that lattice translations must permute anyons, thus leading to "symmetry-enforced" non-Abelian dislocations, or "genons". We also discuss surface states of other weak SPT phases of fermions.

The fermion–boson map for large d

We show that the three-dimensional map between fermions and bosons at finite temperature generalises for all odd dimensions d>3. We further argue that such a map has a nontrivial large d limit. Evidence comes from studying the gap equations, the free energies and the partition functions of the U(N) Gross–Neveu and CPN−1 models for odd d≥3 in the presence of imaginary chemical potential. We find that the gap equations and the free energies can be written in terms of the Bloch–Wigner–Ramakrishnan Dd(z) functions analysed by Zagier. Since D2(z) gives the volume of ideal tetrahedra in 3d hyperbolic space our three-dimensional results are related to resent studies of complex Chern–Simons theories, while for d>3 they yield corresponding higher dimensional generalizations. As a spinoff, we observe that particular complex saddles of the partition functions correspond to the zeros and the extrema of the Clausen functions Cld(θ) with odd and even index d respectively. These saddles lie on the unit circle at positions remarkably well approximated by a sequence of rational multiples of π.

Total Positivity and the Classification of Term Structure Shapes in the Two-Factor Vasicek Model

Using methods from the theory of total positivity, we provide a full classification of attainable term structure shapes in the two-factor Vasicek model. In particular, we show that the shapes normal, inverse, humped, dipped and hump-dip are always attainable. In certain parameter regimes up to four additional shapes can be produced. Our results show that the correlation and the difference in mean-reversion speeds of the two factor processes play a key role in determining the scope of attainable shapes. The mathematical tools from total positivity can likely be applied to higher-dimensional generalizations of the Vasicek model and to other interest rate models as well.

A class of higher-dimensional solutions of Einstein\u2019s vacuum equation

A new class of higher-dimensional exact solutions of Einstein’s vacuum equation is presented. These metrics are written in terms of the exponential of a symmetric matrix and when this matrix is diagonal the solution reduces to higher-dimensional generalizations of Kasner spacetime with a cosmological constant. On the other hand, the metrics attained when such matrix is non-diagonal have more intricate algebraic structures. In the same vein, we integrate Einstein’s equation for homogeneous spacetimes, but not necessarily isotropic, in the presence of a perfect fluid and attain exact solutions in terms of the exponential of a symmetric matrix. Such solutions have not been presented in the literature yet.

A higher-dimensional generalization of Mumford's rational pullback for Weil divisors

Mumford defined a rational pullback for Weil divisors on normal surfaces, which is linear, respects effectivity, and satisfies the projection formula. In higher dimensions, the existence of small resolutions of singularities precludes such general results. We single out a higher-dimensional situation that resembles the surface case and show for it that a rational pullback for Weil divisors exists, which is also linear, respects effectivity, and satisfies the projection formula.

Characterization of Dynamics of a Class of Polynomial Automorphisms in ℂn

A kind of higher-dimensional complex polynomial mappings [Formula: see text] is considered: [Formula: see text] where [Formula: see text], [Formula: see text] are polynomials with degrees higher than one, and [Formula: see text] are nonzero complex numbers, [Formula: see text]. Assume that each [Formula: see text] is hyperbolic on its Julia set and [Formula: see text] is sufficiently small, [Formula: see text], then there exists a bounded set on which the dynamics on the forward and backwards Julia sets are described by using the inductive and the projective limits, respectively. These results are a natural higher-dimensional generalization of the work of Hubbard and Oberste-Vorth on two-dimensional complex Hénon mappings. The combination of the symbolic dynamics and the crossed mapping is also applied to study the complicated dynamics of a class of polynomial mappings in [Formula: see text].

A Tensor Version of the Quantum Wielandt Theorem

We prove boundedness results for the injectivity regions for projected entangled pair states. Our result is a higher-dimensional generalization of the quantum Wielandt inequality.

Stable Rationality of del Pezzo Fibrations of Low Degree Over Projective Spaces

Abstract The main aim of this article is to show that a very general three-dimensional del Pezzo fibration of degrees 1, 2, and 3 is not stably rational except for a del Pezzo fibration of degree 3 belonging to explicitly described two families. Higher-dimensional generalizations are also discussed and we prove that a very general del Pezzo fibration of degrees 1, 2, and 3 defined over the projective space is not stably rational provided that the anti-canonical divisor is not ample.