Odd Dimensions Research Articles

On self-dual double circulant codes

Self-dual double circulant codes of odd dimension are shown to be dihedral in even characteristic and consta-dihedral in odd characteristic. Exact counting formulae are derived for them, generalizing some old results of MacWilliams on the enumeration of circulant orthogonal matrices. These formulae, in turn, are instrumental in deriving a Varshamov–Gilbert bound on the relative minimum distance of this family of codes.

Cocycle deformations and Galois objects for semisimple Hopf algebras of dimension p3 and pq2

Let p and q be distinct prime numbers. We study the Galois objects and cocycle deformations of the noncommutative, noncocommutative, semisimple Hopf algebras of odd dimension p3 and of dimension pq2. We obtain that the p+1 non-isomorphic self-dual semisimple Hopf algebras of dimension p3 classified by Masuoka have no non-trivial cocycle deformations, extending his previous results for the 8-dimensional Kac–Paljutkin Hopf algebra. This is done as a consequence of the classification of categorical Morita equivalence classes among semisimple Hopf algebras of odd dimension p3, established by the third-named author in an appendix.

Discrete Wigner Function Derivation of the Aaronson–Gottesman Tableau Algorithm

The Gottesman–Knill theorem established that stabilizer states and Clifford operations can be efficiently simulated classically. For qudits with odd dimension three and greater, stabilizer states and Clifford operations have been found to correspond to positive discrete Wigner functions and dynamics. We present a discrete Wigner function-based simulation algorithm for odd-d qudits that has the same time and space complexity as the Aaronson–Gottesman algorithm for qubits. We show that the efficiency of both algorithms is due to harmonic evolution in the symplectic structure of discrete phase space. The differences between the Wigner function algorithm for odd-d and the Aaronson–Gottesman algorithm for qubits are likely due only to the fact that the Weyl–Heisenberg group is not in S U ( d ) for d = 2 and that qubits exhibit state-independent contextuality. This may provide a guide for extending the discrete Wigner function approach to qubits.

Spekkens’ toy model in all dimensions and its relationship with stabiliser quantum mechanics

Spekkens’ toy model is a non-contextual hidden variable model with an epistemic restriction, a constraint on what an observer can know about reality. The aim of the model, developed for continuous and discrete prime degrees of freedom, is to advocate the epistemic view of quantum theory, where quantum states are states of incomplete knowledge about a deeper underlying reality. Many aspects of quantum mechanics and protocols from quantum information can be reproduced in the model. In spite of its significance, a number of aspects of Spekkens’ model remained incomplete. Formal rules for the update of states after measurement had not been written down, and the theory had only been constructed for prime-dimensional and infinite dimensional systems. In this work, we remedy this, by deriving measurement update rules and extending the framework to derive models in all dimensions, both prime and non-prime. Stabiliser quantum mechanics (SQM) is a sub-theory of quantum mechanics with restricted states, transformations and measurements. First derived for the purpose of constructing error correcting codes, it now plays a role in many areas of quantum information theory. Previously, it had been shown that Spekkens’ model was operationally equivalent to SQM in the case of odd prime dimensions. Here, exploiting known results on Wigner functions, we extend this to show that Spekkens’ model is equivalent to SQM in all odd dimensions, prime and non-prime. This equivalence provides new technical tools for the study of technically difficult compound-dimensional SQM.

On blowup of co-rotational wave maps in odd space dimensions

We consider co-rotational wave maps from the (1+d)-dimensional Minkowski space into the d-sphere for d≥3 odd. This is an energy-supercritical model which is known to exhibit finite-time blowup via self-similar solutions. Based on a method developed by the second author and Schörkhuber, we prove the asymptotic nonlinear stability of the “ground-state” self-similar solution.

Constant curvature black holes in Einstein-AdS gravity: Conserved quantities

We study physical properties of constant curvature black holes (CCBHs) in Einstein anti-de Sitter (AdS) gravity. These objects, which are locally AdS throughout the space, are constructed from identifications of global AdS spacetime, in a similar fashion as Banados-Teitelboim-Zanelli (BTZ) black hole in three dimensions. We find that, in dimensions equal or greater than four, CCBHs have zero mass and angular momentum. Only in odd dimensions we are able to associate a nonvanishing conserved quantity to these solutions, which corresponds to the vacuum (Casimir) energy of the spacetime.

Scattering for radial energy-subcritical wave equations in dimensions 4 and 5

ABSTRACTIn this paper, we consider the focusing and defocusing energy-subcritical, nonlinear wave equation in ℝ1+d with radial initial data for d = 4,5. We prove that if a solution remains bounded in the critical space on its interval of existence, then the solution exists globally and scatters at ±∞. The proof follows the concentration compactness/rigidity method initiated by Kenig and Merle, and the main obstacle is to show the nonexistence of nonzero solutions with a certain compactness property. A main novelty of this work is the use of a simple virial argument to rule out the existence of nonzero solutions with this compactness property rather than channels of energy arguments that have been proven to be most useful in odd dimensions.

Huygens’ principle and Dirac-Weyl equation

We investigate the validity of Huygens' principle for forward propagation in the massless Dirac-Weyl equation. The principle holds for odd space dimension n, while it is invalid for even n. We explicitly solve the cases n=1,2 and 3 and discuss generic $n$. We compare with the massless Klein-Gordon equation and comment on possible generalizations and applications.

Minimal complexity of Kochen-Specker sets does not scale with dimension

A Kochen-Specker (KS) set is a specific set of projectors and measurement contexts that prove the Bell-Kochen-Specker contextuality theorem. The simplest known KS sets in Hilbert space dimensions $d=3,4,5,6,8$ are reproduced, and several methods by which a new KS set can be constructed using one or more known KS sets in lower dimensions are reviewed and improved. These KS sets and improved methods enable the construction of explicitly critical new KS sets in all dimensions, where critical refers to the irreducibility of the set of contexts. The simplest known critical KS sets are derived in all even dimensions $d\geq10$ with at most 9 contexts and 30 projectors, and in all odd dimensions $d\geq 7$ with at most 13 contexts and 39 projectors. These results show that neither the number of contexts nor the number of projectors in a minimal KS set scales with dimension $d$.

Particle-vortex duality in topological insulators and superconductors

We investigate the origins and implications of the duality between topological insulators and topological superconductors in three and four spacetime dimensions. In the latter, the duality transformation can be made at the level of the path integral in the standard way, while in three dimensions, it takes the form of “self-duality in odd dimensions”. In this sense, it is closely related to the particle-vortex duality of planar systems. In particular, we use this to elaborate on Son’s conjecture that a three dimensional Dirac fermion that can be thought of as the surface mode of a four dimensional topological insulator is dual to a composite fermion.

Time machines and AdS solitons with negative mass

We show that in D=2n+1 dimensions, when mass is negative and all angular momenta are non-vanishing, Kerr and Kerr-AdS metrics describe smooth time machines, with no curvature singularity. Turning off the angular momenta appropriately can lead to static AdS solitons with negative mass. Setting zero the cosmological constant yields a class of Ricci-flat K\"ahler metrics in D=2n dimensions. We also show that Euclidean-signatured AdS solitons with negative mass can also arise in odd dimensions. We then construct time machines in D=5 minimal gauged supergravity that carry only magnetic dipole charges. Turning off the cosmological constant, the time machine becomes massless and asymptotically flat. It can be described as a constant time bundle over the Eguchi-Hanson instanton.

Pure spinors, intrinsic torsion and curvature in odd dimensions

We study the geometric properties of a (2m+1)-dimensional complex manifold M admitting a holomorphic reduction of the frame bundle to the structure group P⊂Spin(2m+1,C), the stabiliser of the line spanned by a pure spinor at a point. Geometrically, M is endowed with a holomorphic metric g, a holomorphic volume form, a spin structure compatible with g, and a holomorphic pure spinor field ξ up to scale. The defining property of ξ is that it determines an almost null structure, i.e. an m-plane distribution Nξ along which g is totally degenerate.We develop a spinor calculus, by means of which we encode the geometric properties of Nξ and of its rank-(m+1) orthogonal complement Nξ⊥ corresponding to the algebraic properties of the intrinsic torsion of the P-structure. This is the failure of the Levi-Civita connection ∇ of g to be compatible with the P-structure. In a similar way, we examine the algebraic properties of the curvature of ∇.Applications to spinorial differential equations are given. Notably, we relate the integrability properties of Nξ and Nξ⊥ to the existence of solutions of odd-dimensional versions of the zero-rest-mass field equation. We give necessary and sufficient conditions for the almost null structure associated to a pure conformal Killing spinor to be integrable. Finally, we conjecture a Goldberg–Sachs-type theorem on the existence of a certain class of almost null structures when (M,g) has prescribed curvature.We discuss applications of this work to the study of real pseudo-Riemannian manifolds.

Holographic calculation of boundary terms in conformal anomaly

In the presence of boundaries the integrated conformal anomaly is modified by the boundary terms so that the anomaly is non-vanishing in any (even or odd) dimension. The boundary terms are due to extrinsic curvature whose exact structure in d=3 and d=4 has recently been identified. In this note we present a holographic calculation of those terms in two different prescriptions for the holographic description of the boundary CFT. We stress the role of supersymmetry when discussing the holographic description of N=4 SYM on a 4-manifold with boundaries.

General conditions for maximal violation of non-contextuality in discrete and continuous variables

The contextuality of quantum mechanics can be shown by the violation of inequalities based on measurements of well chosen observables. An important property of such observables is that their expectation value can be expressed in terms of probabilities for obtaining two exclusive outcomes. Examples of such inequalities have been constructed using either observables with a dichotomic spectrum or using periodic functions obtained from displacement operators in phase space. Here we identify the general conditions on the spectral decomposition of observables demonstrating state independent contextuality of quantum mechanics. Our results not only unify existing strategies for maximal violation of state independent non-contextuality inequalities but also lead to new scenarios enabling such violations. Among the consequences of our results is the impossibility of having a state independent maximal violation of non-contextuality in the Peres–Mermin scenario with discrete observables of odd dimensions.

Mapping the surgery exact sequence for topological manifolds to analysis

In this paper we prove the existence of a natural mapping from the surgery exact sequence for topological manifolds to the analytic surgery exact sequence of Higson and Roe. This generalizes the fundamental result of Higson and Roe, but in the treatment given by Piazza and Schick, from smooth manifolds to topological manifolds. Crucial to our treatment is the Lipschitz signature operator of Teleman. We also give a generalization to the equivariant setting of the product defined by Siegel in his Ph.D. thesis. Geometric applications are given to stability results for rho classes. We also obtain a proof of the APS delocalized index theorem on odd dimensional manifolds, both for the spin Dirac operator and the signature operator, thus extending to odd dimensions the results of Piazza and Schick. Consequently, we are able to discuss the mapping of the surgery sequence in all dimensions.

Restriction of Averaging Operators to Algebraic Varieties over Finite Fields

We study $L^p \to L^r$ estimates for restricted averaging operators related to algebraic varieties $V$ of $d$-dimensional vector spaces over finite fields $\mathbb{F}_q$ with $q$ elements. We observe properties of both the Fourier restriction operator and the averaging operator over $V \subset \mathbb{F}_q^d$. As a consequence, we obtain optimal results on the restricted averaging problems for spheres and paraboloids in dimensions $d \geq 2$, and cones in odd dimensions $d \geq 3$. In addition, when the variety $V$ is a cone lying in an even dimensional vector space over $\mathbb{F}_q$ and $-1$ is a square number in $\mathbb{F}_q$, we also obtain sharp estimates except for two endpoints.

Thermal phase transitions of dimensionally continued AdS black holes

We study the thermal phase transitions of charged black holes in dimensionally continued gravity in anti--de Sitter space. We find the van der Waals--like phase transition in the temperature-entropy plane of the black holes with spherical horizons in even dimensions, and there is no such phase transition of the black holes with flat and hyperbolic geometries. Near the critical inflection point, the critical exponent is computed and its value does not depend on the dimension. The Maxwell equal area law is checked to be fulfilled in the temperature-entropy diagram for the first order phase transition. In odd dimensions, there are no thermal phase transitions for the black hole with any geometry of the horizon.

COMPLETE INTERSECTIONS WITH S 1-ACTION

We give the diffeomorphism classification of complete intersections with S^1-symmetry in dimension less than or equal to 6. In particular, we show that a 6-dimensional complete intersection admits a smooth non-trivial S^1-action if and only if it is diffeomorphic to the complex projective space or the quadric. We also prove that in any odd complex dimension only finitely many complete intersections can carry a smooth effective action by a torus of rank $>1$.

Infinite loop spaces and positive scalar curvature

We study the homotopy type of the space of metrics of positive scalar curvature on high-dimensional compact spin manifolds. Hitchin used the fact that there are no harmonic spinors on a manifold with positive scalar curvature to construct a secondary index map from the space of positive scalar metrics to a suitable space from the real $K$-theory spectrum. Our main results concern the nontriviality of this map. We prove that for $2n \geq 6$, the natural $KO$-orientation from the infinite loop space of the Madsen--Tillmann--Weiss spectrum factors (up to homotopy) through the space of metrics of positive scalar curvature on any $2n$-dimensional spin manifold. For manifolds of odd dimension $2n+1 \geq 7$, we prove the existence of a similar factorisation. When combined with computational methods from homotopy theory, these results have strong implications. For example, the secondary index map is surjective on all rational homotopy groups. We also present more refined calculations concerning integral homotopy groups. To prove our results we use three major sets of technical tools and results. The first set of tools comes from Riemannian geometry: we use a parameterised version of the Gromov--Lawson surgery technique which allows us to apply homotopy-theoretic techniques to spaces of metrics of positive scalar curvature. Secondly, we relate Hitchin's secondary index to several other index-theoretical results, such as the Atiyah--Singer family index theorem, the additivity theorem for indices on noncompact manifolds and the spectral-flow index theorem. Finally, we use the results and tools developed recently in the study of moduli spaces of manifolds and cobordism categories. The key new ingredient we use in this paper is the high-dimensional analogue of the Madsen--Weiss theorem, proven by Galatius and the third named author.

Topological electromagnetic responses of bosonic quantum Hall, topological insulator, and chiral semimetal phases in all dimensions

We calculate the topological part of the electromagnetic response of Bosonic Integer Quantum Hall (BIQH) phases in odd (spacetime) dimensions, and Bosonic Topological Insulator (BTI) and Bosonic chiral semi-metal (BCSM) phases in even dimensions. To do this we use the Nonlinear Sigma Model (NLSM) description of bosonic symmetry-protected topological (SPT) phases, and the method of gauged Wess-Zumino (WZ) actions. We find the surprising result that for BIQH states in dimension $2m-1$ ($m=1,2,\dots$), the bulk response to an electromagnetic field $A_{\mu}$ is characterized by a Chern-Simons term for $A_{\mu}$ with a level quantized in integer multiples of $m!$ (factorial). We also show that BTI states (which have an extra $\mathbb{Z}_2$ symmetry) can exhibit a $\mathbb{Z}_2$ breaking Quantum Hall effect on their boundaries, with this boundary Quantum Hall effect described by a Chern-Simons term at level $\frac{m!}{2}$. We show that the factor of $m!$ can be understood by requiring gauge invariance of the exponential of the Chern-Simons term on a general Euclidean manifold, and we also use this argument to characterize the electromagnetic and gravitational responses of fermionic SPT phases with $U(1)$ symmetry in all odd dimensions. We then use our gauged boundary actions for the BIQH and BTI states to (i) construct a bosonic analogue of a chiral semi-metal (BCSM) in even dimensions, (ii) show that the boundary of the BTI state exhibits a bosonic analogue of the parity anomaly of Dirac fermions in odd dimensions, and (iii) study anomaly inflow at domain walls on the boundary of BTI states. In a series of Appendices we derive important formulas and additional results. In particular, in Appendix A we use the connection between equivariant cohomology and gauged WZ actions to give a mathematical interpretation of the actions for the BIQH and BTI boundaries constructed in this paper.