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Given a smooth potential W : T n → R on the torus, the Quantum Guerra–Morato action functional is given by I (ψ) = ∫ ( D v D v * 2 (x) − W (x)) a (x) 2 d x , where ψ is described by ψ = a e i u ℏ , u = v + v * 2 , a = e v * − v 2 ℏ , v, v* are real functions, ∫a2(x)dx = 1, and D is the derivative on x ∈ Tn. It is natural to consider the constraint div(a2Du) = 0, which means flux zero. The a and u obtained from a critical solution (under variations τ) for such action functional, fulfilling such constraints, satisfy the Hamilton-Jacobi equation with a quantum potential. Denote ′ = d d τ . We show that the expression for the second variation of a critical solution is given by ∫a2D[v′] D[(v*)′] dx. Introducing the constraint ∫a2Du dx = V, we also consider later an associated dual eigenvalue problem. From this follows a transport and a kind of eikonal equation. [ABSTRACT FROM AUTHOR]

We prove a C∞ version of Nekhoroshev theorem for time dependent Hamiltonians in R d × T d . Precisely, we prove a result showing that for all times the energy of the system is bounded by a constant times ⟨t⟩ɛ. We apply the result to the dynamics of a charged particle in T d subject to a time dependent electromagnetic field. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an abstract KAM (Kolmogorov–Arnold–Moser) theorem. As an application, we study the two-dimensional completely resonant beam system under periodic boundary conditions. Using the KAM theorem together with partial Birkhoff normal form method, we obtain a family of Whitney smooth small–amplitude quasi–periodic solutions for the equation system. [ABSTRACT FROM AUTHOR]

We correct an error in a previous publication which led to an incorrect statement and proof of a proposition. The errors result from an interesting subtlety of the differential topology of embedded tori in three-manifolds. [ABSTRACT FROM AUTHOR]

Strominger–Yau–Zaslow (SYZ) proposed a way of constructing mirror pairs as pairs of torus fibrations. We apply this SYZ construction to toric Fano surfaces as complex manifolds, and discuss the homological mirror symmetry, where we consider Morse homotopy of the moment polytope instead of the Fukaya category. [ABSTRACT FROM AUTHOR]

Field theories with combinatorial non-local interactions such as tensor invariants are interesting candidates for describing a phase transition from discrete quantum-gravitational to continuum geometry. In the so-called cyclic-melonic potential approximation of a tensorial field theory on the r-dimensional torus it was recently shown using functional renormalization group techniques that no such phase transition to a condensate phase with a tentative continuum geometric interpretation is possible. Here, keeping the same approximation, we show how to overcome this limitation amending the theory by local degrees freedom on R d . We find that the effective r − 1 dimensions of the torus part dynamically vanish along the renormalization group flow while the d local dimensions persist up to small momentum scales. Consequently, for d > 2 one can find a phase structure allowing also for phase transitions. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the Sobolev norms of solutions for the linear wave equation with unbounded perturbations of order one remain bounded for all time. The main proof is based on the KAM reducibility of the linear wave equation. To the best of our knowledge, this is the first reducibility result for the linear wave equation with general quasi-periodic unbounded perturbations on the one-dimensional torus. [ABSTRACT FROM AUTHOR]

For a finite graph, a spectral curve is constructed as the zero set of a two-variate polynomial with integer coefficients coming from p-adic diffusion on the graph. It is shown that certain spectral curves can distinguish non-isomorphic pairs of isospectral graphs, and can even reconstruct the graph. This allows the graph reconstruction from the spectrum of the associated p-adic Laplacian operator. As an application to p-adic geometry, it is shown that the reduction graph of a Mumford curve and the product reduction graph of a p-adic analytic torus can be recovered from the spectrum of such operators. [ABSTRACT FROM AUTHOR]

Topologically ordered phases in 2 + 1 dimensions are generally characterized by three mutually related features: fractionalized (anyonic) excitations, topological entanglement entropy, and robust ground state degeneracy that does not require symmetry protection or spontaneous symmetry breaking. Such a degeneracy is known as topological degeneracy and can be usually seen under the periodic boundary condition regardless of the choice of the system sizes L1 and L2 in each direction. In this work, we introduce a family of extensions of the Kitaev toric code to N level spins (N ≥ 2). The model realizes topologically ordered phases or symmetry-protected topological phases depending on the parameters in the model. The most remarkable feature of topologically ordered phases is that the ground state may be unique, depending on L1 and L2, despite that the translation symmetry of the model remains unbroken. Nonetheless, the topological entanglement entropy takes the nontrivial value. We argue that this behavior originates from the nontrivial action of translations permuting anyon species. [ABSTRACT FROM AUTHOR]

Consider a normally degenerate Hamiltonian system with the following Hamiltonian H (θ , I , x , y , ϵ) = ω , I + λ x n + 1 n + 1 + y 2 2 + ϵ P (θ , x , y) , (θ , I , x , y) ∈ T d × R d + 2 , which is associated with the standard symplectic form dθ ∧ dI ∧ dx ∧ dy, where 0 ≠ λ ∈ R and n > 1 is an integer. The existence of response tori for the degenerate Hamiltonian system has already been proved by Si and Yi [Nonlinearity 33, 6072–6098 (2020)] if [ ∂ P (θ , 0 , 0) ∂ x ] satisfies some non-zero conditions, see condition (H) in the work of Si and Yi [Nonlinearity 33, 6072–6098 (2020)], where [·] denotes the average value of a continuous function on T d . However, when [ ∂ P (θ , 0 , 0) ∂ x ] = 0 , no results were given by Si and Yi [Nonlinearity 33, 6072–6098 (2020)] for response tori of the above system. This paper attempts at carrying out this work in this direction. More precisely, with 2p < n, if P satisfies [ ∂ j P (θ , 0 , 0) ∂ x j ] = 0 for j = 1, 2, ..., p and either λ − 1 [ ∂ p + 1 P (θ , 0 , 0) ∂ x p + 1 ] < 0 as n − p is even or λ − 1 [ ∂ p + 1 P (θ , 0 , 0) ∂ x p + 1 ] ≠ 0 as n − p is odd, we obtain the following results: (1) For λ ̃ < 0 [see λ ̃ in (2.1)] and ϵ sufficiently small, response tori exist for each ω satisfying a Brjuno-type non-resonant condition. (2) For λ ̃ > 0 and ϵ * sufficiently small, there exists a Cantor set E ∈ (0 , ϵ * ) with almost full Lebesgue measure such that response tori exist for each ϵ ∈ E if ω satisfies a Diophantine condition. In the case where λ − 1 [ ∂ p + 1 P (θ , 0 , 0) ∂ x p + 1 ] > 0 and n − p is even, we prove that the system admits no response tori in most regions. The present paper is regarded as a continuation of work by Si and Yi [Nonlinearity 33, 6072–6098 (2020)]. [ABSTRACT FROM AUTHOR]

This work develops a magnetic pseudodifferential calculus for super operators OpA(F); these map operators onto operators (as opposed to Lp functions onto Lq functions). Here, F could be a tempered distribution or a Hörmander symbol. An important example is Liouville super operators L ̂ = − i o p A (h) , ⋅ defined in terms of a magnetic pseudodifferential operator opA(h). Our work combines ideas from the magnetic Weyl calculus developed by Măntoiu and Purice [J. Math. Phys. 45, 1394–1417 (2004)]; Iftimie, Măntoiu, and Purice [Publ. Res. Inst. Math. Sci. 43, 585–623 (2007)]; and Lein (Ph.D. thesis, 2011) and the pseudodifferential calculus on the non-commutative torus from the work of Ha, Lee, and Ponge [Int. J. Math. 30, 1950033 (2019)]. Thus, our calculus is inherently gauge-covariant, which means that all essential properties of OpA(F) are determined by properties of the magnetic field B = dA rather than the vector potential A. There are conceptual differences to ordinary pseudodifferential theory. For example, in addition to an analog of the (magnetic) Weyl product that emulates the composition of two magnetic pseudodifferential super operators on the level of functions, the so-called semi-super product describes the action of a pseudodifferential super operator on a pseudodifferential operator. [ABSTRACT FROM AUTHOR]

This Review details the relationship between the isomonodromic tau-function and conformal blocks on a torus with one simple pole. It is based on the author's talk at ICMP 2021. [ABSTRACT FROM AUTHOR]

We construct point vortex equilibria with strengths quantized by multiples of 2π in a fixed background vorticity field on the surface of a curved torus. The background vorticity consists of two terms: first, a term exponentially related to the stream function and a second term arising from the curvature of the torus, which leads to a Liouville-type equation for the stream function. By using a stereographic projection of the torus onto an annulus in a complex plane, the Liouville-type equation admits a class of exact solutions, given in terms of a loxodromic function on the annulus. We show that appropriate choices of the loxodromic function in the solution lead to stationary vortex patterns with 4 n ̂ point vortices of identical strengths, n ̂ ∈ N. The quantized point vortices are stationary in the sense that they are equilibria of a "one-way interaction" model where the evolution of point vortices is subject to the continuous background vorticity, while the background vorticity distribution is not affected by the velocity field induced by the point vortices. By choosing loxodromic functions continuously dependent on a parameter and taking appropriate limits with respect to this parameter, we show that there are solutions with inhomogeneous point vortex strengths, in which the exponential part of the background vorticity disappears. The point vortices are always located at the innermost and outermost rings of the torus owing to the curvature effects. The topological features of the streamlines are found to change as the modulus of the torus changes. [ABSTRACT FROM AUTHOR]

In this paper, the commutative and spectral properties of a kth-order slant Hankel operator (k ≥ 2, a fixed integer) on the Lebesgue space of n-dimensional torus, T n , where T is the unit circle, are studied. Characterizations for the commutativity and essential commutativity between higher order slant Hankel operators and slant Toeplitz operators have been obtained. The presence of an open disk in the point spectrum of a kth-order slant Hankel operator with a unimodular inducing function has also been ensured. [ABSTRACT FROM AUTHOR]

In this paper, we establish Cwikel-type estimates for noncommutative tori for any dimension n ≥ 2. We use them to derive Cwikel–Lieb–Rozenblum inequalities and Lieb–Thirring inequalities for negative eigenvalues of fractional Schrödinger operators on noncommutative tori. The latter leads to a Sobolev inequality for noncommutative tori. On the way, we establish a "borderline version" of the abstract Birman–Schwinger principle for the number of negative eigenvalues of relatively compact form perturbations of a non-negative semi-bounded operator with isolated 0-eigenvalue. [ABSTRACT FROM AUTHOR]

We construct a (1 + 1)d topological field theory (TFT) whose topological defect lines (TDLs) realize the transparent Haagerup H 3 fusion category. This TFT has six vacua, and each of the three non-invertible simple TDLs hosts three defect operators, giving rise to a total of 15 point-like operators. The TFT data, including the three-point coefficients and lasso diagrams, are determined by solving all the sphere four-point crossing equations and torus one-point modular invariance equations. We further verify that the Cardy states furnish a non-negative integer matrix representation under TDL fusion. While many of the constraints we derive are not limited to this particular TFT with six vacua, we leave open the construction of TFTs with two or four vacua. Finally, TFTs realizing the Haagerup H 1 and H 2 fusion categories can be obtained by gauging algebra objects. This article makes a modest offering in our pursuit of exotica and the quest for their eventual conformity. [ABSTRACT FROM AUTHOR]

We study a fractional quantum Hall system with maximal filling ν = 1/3 in the thin torus limit. The corresponding Hamiltonian is a truncated version of Haldane's pseudopotential, which upon a Jordan–Wigner transformation is equivalent to a one-dimensional quantum spin chain with periodic boundary conditions. Our main result is a lower bound on the spectral gap of this Hamiltonian, which is uniform in the system size and total particle number. The gap is also uniform with respect to small values of the coupling constant in the model. The proof adapts the strategy of individually estimating the gap in invariant subspaces used for the bosonic ν = 1/2 model to the present fermionic case. [ABSTRACT FROM AUTHOR]

We consider the periodic Cauchy problem for the Gurevich–Zybin (GZ) system on the n-dimensional torus. The GZ system models dark matter as a collision-free gas in an expanding universe. We establish local in-time well-posedness for classical solutions in Sobolev spaces. Moreover, the local in-time result is extended to all time under additional assumptions. In particular, we consider three cosmological eras determined by the universal expansion parameter. In each era, global in-time existence is established if the initial density contrast is small and the rate of change of the initial velocity is sufficiently large. [ABSTRACT FROM AUTHOR]

We prove that bounded Beltrami fields are symmetric if a proportionality factor depends on two variables in the cylindrical coordinate and admits a regular level set diffeomorphic to a cylinder or a torus. [ABSTRACT FROM AUTHOR]

We review the Landau problem of an electron in a constant uniform magnetic field. The magnetic translations are the invariant transformations of the free Hamiltonian. A Kähler polarization of the plane has been used for the geometric quantization. Under the assumption of quasi-periodicity of the wavefunction, the Zak's magnetic translations in the Bravais lattice generate a non-commutative quantum torus. We concentrate on the case when the magnetic flux density is a rational number. The Bloch wavefunctions form a finite-dimensional module of the noncommutative torus of magnetic translations as well as of its commutant, which is the non-commutative torus of magnetic translations in the dual Bravais lattice. The bi-module structure of the Bloch waves is shown to be the connecting link between two Morita equivalent non-commutative tori. The main focus of our review is the Kähler structure on the Hilbert space of Bloch waves and its inherent quantum toric geometry. We reveal that the metaplectic group M p (2 , R) of the automorphisms of magnetic translation algebras is represented by the quantum optics squeezing operators. [ABSTRACT FROM AUTHOR]

We study the 1D nonlinear Schrödinger equation with non-gauge invariant quadratic nonlinearity on the torus. The Cauchy problem admits trivial global dispersive solutions, which are constant with respect to space. The non-existence of global solutions has also been studied only by focusing on the behavior of the Fourier 0 mode of solutions. However, the earlier works are not sufficient to obtain the precise criteria for the global existence for the Cauchy problem. In this paper, the exact criteria for the global existence of L2 solutions are shown by studying the interaction between the Fourier 0 mode and oscillation of solutions. Namely, L2 solutions are shown a priori not to exist globally if they are different from the trivial ones. [ABSTRACT FROM AUTHOR]

We consider a random Gaussian model of Laplace eigenfunctions on the hemisphere, satisfying the Dirichlet boundary conditions along the equator. For this model, we find a precise asymptotic law for the corresponding zero density functions, in both short range (around the boundary) and long range (far away from the boundary) regimes. As a corollary, we were able to find a logarithmic negative bias for the total nodal length of this ensemble relative to the rotation invariant model of random spherical harmonics. Jean Bourgain's research, and his enthusiastic approach to the nodal geometry of Laplace eigenfunctions, has made a crucial impact in the field and the current trends within. His works on the spectral correlations {Theorem 2.2 in the work of Krishnapur et al. [Ann. Math. 177(2), 699–737 (2013)]} and Bombieri and Bourgain [Int. Math. Res. Not. (IMRN) 11, 3343–3407 (2015)] have opened a door for an active ongoing research on the nodal length of functions defined on surfaces of arithmetic flavor, such as the torus or the square. Furthermore, Bourgain's work [J. Bourgain, Isr. J. Math. 201(2), 611–630 (2014)] on toral Laplace eigenfunctions, also appealing to spectral correlations, allowed for inferring deterministic results from their random Gaussian counterparts. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of full-dimensional invariant tori for a non-autonomous, almost-periodically forced nonlinear beam equation with a periodic boundary condition via Kolmogorov–Arnold–MoserAM theory. [ABSTRACT FROM AUTHOR]

This paper demonstrates sufficient conditions for the existence of a positive measure set of invariant KAM (Kolmogrov-Arnol'd-Moser) tori in a singly thermostated, one degree-of-freedom Hamiltonian vector field. This result is applied to four important single thermostats in the literature, and it is shown that in each case, if the Hamiltonian is real-analytic and well-behaved, then the thermostated system always has a positive measure set of invariant KAM tori for sufficiently weak coupling and high temperature. This extends the results of Legoll, Luskin, and Moeckel [Arch. Ration. Mech. Anal. 184, 449 (2007)] and Legoll et al. [Nonlinearity 44, 1673 (2009)]. [ABSTRACT FROM AUTHOR]

In this article, we prove a reducibility result for the linear Schrödinger equation on a Zoll manifold with quasi-periodic in time pseudo-differential perturbation of order less than or equal to 1/2. As far as we know, this is the first reducibility result for an unbounded perturbation on a compact manifold different from the torus. [ABSTRACT FROM AUTHOR]

In this paper, we prove an infinite dimensional Kolmogorov-Arnold-Moser theorem. As an application, it is shown that there are many small-amplitude linearly-stable quasi-periodic solutions for higher dimensional wave equations with a real Fourier multiplier, which are under nonlocal and forced perturbations with a special structure in space and short range property. [ABSTRACT FROM AUTHOR]

We study the classical Rajeev-Ranken model, a Hamiltonian system with three degrees of freedom describing nonlinear continuous waves in a 1 + 1-dimensional nilpotent scalar field theory pseudodual to the SU(2) principal chiral model. While it loosely resembles the Neumann and Kirchhoff models, its equations may be viewed as the Euler equations for a centrally extended Euclidean algebra. The model has a Lax pair and r-matrix leading to four generically independent conserved quantities in involution, two of which are Casimirs. Their level sets define four-dimensional symplectic leaves on which the system is Liouville integrable. On each of these leaves, the common level sets of the remaining conserved quantities are shown, in general, to be 2-tori. The nongeneric level sets can only be horn tori, circles, and points. They correspond to measure zero subsets where the conserved quantities develop relations and solutions degenerate from elliptic to hyperbolic, circular, or constant functions. A geometric construction allows us to realize each common level set as a bundle with base determined by the roots of a cubic polynomial. A dynamics is defined on the union of each type of level set, with the corresponding phase manifolds expressed as bundles over spaces of conserved quantities. Interestingly, topological transitions in energy hypersurfaces are found to occur at energies corresponding to horn tori, which support purely homoclinic orbits. The dynamics on each horn torus is non-Hamiltonian, but expressed as a gradient flow. Finally, we discover a family of action-angle variables for the system that apply away from horn tori. [ABSTRACT FROM AUTHOR]

In this paper, we present a torus solution to the generalized Weierstrass-Enneper representation of surfaces in R 4 . The key analytical technique will be Bloch wave functions with complex wave vectors. We will also discuss some possible uses of these solutions which derive from their explicit nature, such as Dehn surgery and the physics of exotic smooth structure. [ABSTRACT FROM AUTHOR]

We prove an infinite dimensional Kolmogorov–Arnold–Moser theorem. As an application, we use the theorem to study the two-dimensional forced nonlinear Schrödinger equation with periodic boundary conditions, and we emphasize that the forced term is not small perturbation. We obtain a Whitney smooth family of small-amplitude quasiperiodic solutions which are partially hyperbolic. [ABSTRACT FROM AUTHOR]

For the Lie algebra of derivations over a rational quantum torus, we construct a class of irreducible weight modules with all finite dimensional weight spaces, which are called the modules of highest weight type, generating the notion for the case of commutative torus. [ABSTRACT FROM AUTHOR]

We consider twisted equivariant K-theory for actions of a compact Lie group G on a space X where all the isotropy subgroups are connected and of maximal rank. We show that the associated rational spectral sequence à la Segal has a simple E2-term expressible as invariants under the Weyl group of G. Specifically, if T is a maximal torus of G, they are invariants of the π1(XT)-equivariant Bredon cohomology of the universal cover of XT with suitable coefficients. In the case of the inertia stack ΛY, this term can be expressed using the cohomology of YT and algebraic invariants associated with the Lie group and the twisting. A number of calculations are provided. In particular, we recover the rational Verlinde algebra when Y = {*}. [ABSTRACT FROM AUTHOR]

In this paper, we research the existence of a 2-dimensional invariant whiskered torus for the complex Ginzburg-Landau equation in whole space by constructing a Kolmogorov-Arnold-Moser theorem for the system of continuous spectra. [ABSTRACT FROM AUTHOR]

We give a complete rigorous proof of the full asymptotic expansion of the partition function of the dimer model on a square lattice on a torus for general weights zh, z v of the dimer model and arbitrary dimensions of the lattice m, n. We assume m is even and we show that the asymptotic expansion depends on the parity of n. We review and extend the results of Ivashkevich et al. [J. Phys. A: Math. Gen. 35, 5543 (2002)] on the full asymptotic expansion of the partition function of the dimer model, and we give a rigorous estimate of the error term in the asymptotic expansion of the partition function. [ABSTRACT FROM AUTHOR]

Holomorphic functions that characterize states in a two-dimensional Landau level have been central to key developments such as the Laughlin state. Their origin has historically been attributed to a special property of “Schrödinger wavefunctions” of states in the “lowest Landau level.” It is shown here that they instead arise in any Landau level as a generic mathematical property of the Heisenberg description of the non-commutative geometry of guiding centers. When quasiperiodic boundary conditions are applied to compactify the system on a torus, a new formula for the overlap between holomorphic states, in the form of a discrete sum rather than an integral, is obtained. The new formula is unexpected from the previous “lowest-Landau level Schrödinger wavefunction” interpretation. [ABSTRACT FROM AUTHOR]

In this paper, we construct a series of additional flows of the C type Kadomtsev- Petviashvili (CKP) hierarchy and the multi-component CKP hierarchy and these flows constitute an N-fold direct product of the positive half of the quantum torus symmetry. Comparing to the W∞ infinite dimensional Lie symmetry, this quantum torus symmetry has a nice algebraic structure with double indices. [ABSTRACT FROM AUTHOR]

We consider a nonlinear elliptic system arising in the study of the SU(3) Chern-Simons model on a two-dimensional flat torusΩ. Solutions of this SU(3) Chern Simons system could be classified as topological, mixed-type and non-topological solutions. In this paper, we succeed to construct bubbling mixed type solutions. This is the first result for such example in the literature. The analysis for the existence of such solution provides some important insights for us to develop the asymptotic analysis of classifying all mixed-type solution. [ABSTRACT FROM AUTHOR]

There is a mirror symmetry between symplectic manifolds and complex manifolds. In this paper, we describe the super version of mirror symmetry. One side is the super symplectic torus and the other side is the noncommutative super abelian varieties.We compare the super Lagrangian in super symplectic torus with the super 2-form on super abelian varieties. [ABSTRACT FROM AUTHOR]

The monodromy of torus bundles associated with completely integrable systems can be computed using geometric techniques (constructing homology cycles) or analytic arguments (computing discontinuities of abelian integrals). In this article, we give a general approach to the computation of monodromy that resembles the analytical one, reducing the problem to the computation of residues of polar 1-forms. We apply our technique to three celebrated examples of systems with monodromy (the champagne bottle, the spherical pendulum, the hydrogen atom) and to the case of non-degenerate focus-focus singularities, re-obtaining the classical results. An advantage of this approach is that the residue-like formula can be shown to be local in a neighborhood of a singularity, hence allowing the definition of monodromy also in the case of non-compact fibers. This idea has been introduced in the literature under the name of scattering monodromy. We prove the coincidence of the two definitions with the monodromy of an appropriately chosen compactification. [ABSTRACT FROM AUTHOR]

For the algebra of skew-derivations over a torus, we classify the irreducible weight modules with some restrictions. [ABSTRACT FROM AUTHOR]

The Weyl quantization of classical observables on the torus (as phase space) without regularity assumptions is explicitly computed. The equivalence class of symbols yielding the same Weyl operator is characterized. The Heisenberg equation for the dynamics of general quantum observables is written through the Moyal brackets on the torus and the support of theWigner transform is characterized. Finally, a dequantization procedure is introduced that applies, for instance, to the Pauli matrices. As a result we obtain the corresponding classical symbols. [ABSTRACT FROM AUTHOR]

In this paper we investigate the link between the spectrum of some periodic Schrödinger type operators and the effective Hamiltonian of the weak KAM theory. We show that the extension of some local quasimodes is linked to the localization of the Schrödinger spectrum. Such a result provides additional information with respect to the well known Bohr-Sommerfeld quantization rules, here in a more general setting than the integrable or quasi-integrable ones. [ABSTRACT FROM AUTHOR]

We study restrictions on locality-preserving unitary logical gates for topological quantum codes in two spatial dimensions. A locality-preserving operation is one which maps local operators to local operators--for example, a constant-depth quantum circuit of geometrically local gates, or evolution for a constant time governed by a geometrically local bounded-strength Hamiltonian. Locality-preserving logical gates of topological codes are intrinsically fault tolerant because spatially localized errors remain localized, and hence sufficiently dilute errors remain correctable. By invoking general properties of two-dimensional topological field theories, we find that the locality-preserving logical gates are severely limited for codes which admit non-abelian anyons, in particular, there are no locality-preserving logical gates on the torus or the sphere with M punctures if the braiding of anyons is computationally universal. Furthermore, for Ising anyons on the M-punctured sphere, locality-preserving gates must be elements of the logical Pauli group. We derive these results by relating logical gates of a topological code to automorphisms of the Verlinde algebra of the corresponding anyon model, and by requiring the logical gates to be compatible with basis changes in the logical Hilbert space arising from local F-moves and the mapping class group. [ABSTRACT FROM AUTHOR]

The scalar curvature for noncommutative four tori TΘ4, where their flat geometries are conformally perturbed by a Weyl factor, is computed by making the use of a noncommutative residue that involves integration over the 3-sphere. This method is more convenient since it does not require the rearrangement lemma and it is advantageous as it explains the simplicity of the final functions of one and two variables, which describe the curvature with the help of a modular automorphism. In particular, it readily allows to write the function of two variables as the sum of a finite difference and a finite product of the one variable function. The curvature formula is simplified for dilatons of the form sp, where s is a real parameter and p ∊ C∞(TΘ4) is an arbitrary projection, and it is observed that, in contrast to the two dimensional case studied by Connes and Moscovici, J. Am. Math. Soc. 27(3), 639-684 (2014), unbounded functions of the parameter s appear in the final formula. An explicit formula for the gradient of the analog of the Einstein-Hilbert action is also calculated. [ABSTRACT FROM AUTHOR]

We consider the scattering theory for discrete Schrödinger operators on Zd with long- range potentials. We prove the existence of modified wave operators constructed in terms of solutions of a Hamilton-Jacobi equation on the torus Td. [ABSTRACT FROM AUTHOR]

The global-in-time existence of weak solutions to the viscous quantum Magnetohy-drodynamic equations in a three-dimensional torus with large data is proved. The global existence of weak solutions to the viscous quantum Magnetohydrodynamic equations is shown by using the Faedo-Galerkin method and weak compactness techniques. [ABSTRACT FROM AUTHOR]

Consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. The aim of the paper is to develop a perturbation theory for the eigenvalue with smallest modulus with respect to perturbations of the metric. Here the application of perturbation techniques is hindered by the fact that eigenvalues of the massless Dirac operator have even multiplicity, which is a consequence of this operator commuting with the antilinear operator of charge conjugation (a peculiar feature of dimension 3). We derive an asymptotic formula for the eigenvalue with smallest modulus for arbitrary perturbations of the metric and present two particular families of Riemannian metrics for which the eigenvalue with smallest modulus can be evaluated explicitly. We also establish a relation between our asymptotic formula and the eta invariant. [ABSTRACT FROM AUTHOR]

We study a Dirac operator subject to Atiayh-Patodi-Singer-like boundary conditions on the solid torus and shows that the corresponding boundary value problem is elliptic in the sense that the Dirac operator has a compact parametrix. [ABSTRACT FROM AUTHOR]

We study a variational problem involving an anisotropic bending energy for surfaces. Surfaces with boundary and closed equilibria are discussed. [ABSTRACT FROM AUTHOR]

We study the quantization of Abelian gauge theories of principal torus bundles over compact manifolds with and without boundary. It is shown that these gauge theories suffer from a Gribov ambiguity originating in the nontriviality of the bundle of connections whose geometrical structure will be analyzed in detail. Motivated by the stochastic quantization approach, we propose a modified functional integral measure on the space of connections that takes the Gribov problem into account. This functional integral measure is used to calculate the partition function, Green’s functions, and the field strength correlating functions in any dimension by using the fact that the space of inequivalent connections itself admits the structure of a bundle over a finite dimensional torus. Green’s functions are shown to be affected by the nontrivial topology, giving rise to nonvanishing vacuum expectation values for the gauge fields. [ABSTRACT FROM AUTHOR]

In this paper we derive the equations for the motion of relativistic torus in the Minkowski space R1+n (n>=3). This kind of equations also describes the three-dimensional timelike extremal submanifolds in the Minkowski space R1+n. We show that these equations can be reduced to a (1+2)-dimensional quasilinear symmetric hyperbolic system and the system possesses some interesting properties such as nonstrict hyperbolicity, constant multiplicity of eigenvalues, linear degeneracy of all characteristic fields, strong null condition, etc. We also find and prove an interesting fact that all plane wave solutions to these equations are lightlike extremal submanifolds and vice versa except for a type of special solution. [ABSTRACT FROM AUTHOR]