Showing papers on "Symmetry (geometry) published in 2019"

Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases

Abstract: We study generalized discrete symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. In particular, we describe 't Hooft anomalies and classify gapped phases stabilized by these symmetries, including new 1+1D topological phases. The algebra of these operators is not a group but rather is described by their fusion ring and crossing relations, captured algebraically as a fusion category. Such data defines a Turaev-Viro/Levin-Wen model in 2+1D, while a 1+1D system with this fusion category acting as a global symmetry defines a boundary condition. This is akin to gauging a discrete global symmetry at the boundary of Dijkgraaf-Witten theory. We describe how to "ungauge" the fusion category symmetry in these boundary conditions and separate the symmetry-preserving phases from the symmetry-breaking ones. For Tambara-Yamagami categories and their generalizations, which are associated with Kramers-Wannier-like self-dualities under orbifolding, we develop gauge theoretic techniques which simplify the analysis. We include some examples of CFTs with fusion category symmetry derived from Kramers-Wannier-like dualities as an appetizer for the Part II companion paper.

Classification of crystalline topological insulators and superconductors with point group symmetries

Abstract: Crystalline topological phases have recently attracted a lot of experimental and theoretical attention. Key advances include the complete elementary band representation analyses of crystalline matter by symmetry indicators and the discovery of higher-order hinge and corner states. However, current classification schemes of such phases are either implicit or limited in scope. We present a scheme for the explicit classification of crystalline topological insulators and superconductors. These phases are protected by crystallographic point group symmetries and are characterized by bulk topological invariants. The classification paradigm generalizes the Clifford algebra extension process of each Altland-Zirnbauer symmetry class and utilizes algebras which incorporate the point group symmetry. Explicit results for all point group symmetries of three-dimensional crystals are presented as well as for all symmorphic layer groups of two-dimensional crystals. We discuss future extensions for treatment of magnetic crystals and defected or higher-dimensional systems as well as weak and fragile invariants.

On the positive geometry of conformal field theory

Abstract: It has long been clear that the conformal bootstrap is associated with a rich geometry. In this paper we undertake a systematic exploration of this geometric structure as an object of study in its own right. We study conformal blocks for the minimal SL(2, R) symmetry present in conformal field theories in all dimensions. Unitarity demands that the Taylor coefficients of the four-point function lie inside a polytope U determined by the operator spectrum, while crossing demands they lie on a plane X. The conformal bootstrap is then geometrically interpreted as demanding a non-empty intersection of U ∩ X. We find that the conformal blocks enjoy a surprising positive determinant property. This implies that U is an example of a famous polytope — the cyclic polytope. The face structure of cyclic polytopes is completely understood. This lets us fully characterize the intersection U∩X by a simple combinatorial rule, leading to a number of new exact statements about the spectrum and four-point function in any conformal field theory.

On gapped boundaries for SPT phases beyond group cohomology

Abstract: We discuss a strategy to construct gapped boundaries for a large class of symmetry-protected topological phases (SPT phases) beyond group cohomology. This is done by a generalization of the symmetry extension method previously used for cohomo- logical SPT phases. We find that this method allows us to construct gapped boundaries for time-reversal-invariant bosonic SPT phases and for fermionic Gu-Wen SPT phases for arbitrary finite internal symmetry groups.

A computational comparison of symmetry handling methods for mixed integer programs

Abstract: The handling of symmetries in mixed integer programs in order to speed up the solution process of branch-and-cut solvers has recently received significant attention, both in theory and practice. This paper compares different methods for handling symmetries using a common implementation framework. We start by investigating the computation of symmetries and analyze the symmetries present in the MIPLIB 2010 instances. It turns out that many instances are affected by symmetry and most symmetry groups contain full symmetric groups as factors. We then present (variants of) six symmetry handling methods from the literature. Their implementation is tested on several testsets. On very symmetric instances used previously in the literature, it is essential to use methods like isomorphism pruning, orbital fixing, or orbital branching. Moreover, tests on the MIPLIB instances show that isomorphism pruning, orbital fixing, or adding symmetry breaking inequalities allow to speed-up the solution process by about 15% and more instances can be solved within the time limit.

Non-invertible anomalies and mapping-class-group transformation of anomalous partition functions.

Abstract: Recently, it was realized that anomalies can be completely classified by topological orders, symmetry protected topological (SPT) orders, and symmetry enriched topological orders in one higher dimension. The anomalies that people used to study are invertible anomalies that correspond to invertible topological orders and/or symmetry protected topological orders in one higher dimension. In this paper, we introduce a notion of non-invertible anomaly, which describes the boundary of generic topological order. A key feature of non-invertible anomaly is that it has several partition functions. Under the mapping class group transformation of space-time, those partition functions transform in a certain way characterized by the data of the corresponding topological order in one higher dimension. In fact, the anomalous partition functions transform in the same way as the degenerate ground states of the corresponding topological order in one higher dimension. This general theory of non-invertible anomaly may have wide applications. As an example, we show that the irreducible gapless boundary of 2+1D double-semion (DS) topological order must have central charge $c=\bar c \geq \frac{25}{28}$.

On Learning Symmetric Locomotion

Abstract: Human and animal gaits are often symmetric in nature, which points to the use of motion symmetry as a potentially useful source of structure that can be exploited for learning. By encouraging symmetric motion, the learning may be faster, converge to more efficient solutions, and be more aesthetically pleasing. We describe, compare, and evaluate four practical methods for encouraging motion symmetry. These are implemented via particular choices of structure for the policy network, data duplication, or via the loss function. We experimentally evaluate the methods in terms of learning performance and achieved symmetry, and provide summary guidelines for the choice of symmetry method. We further describe some practical and conceptual issues that arise. Because similar implementation choices exist for other types of inductive biases, the insights gained may also be relevant to other learning problems with applicable symmetry abstractions.

On Serrin’s overdetermined problem in space forms

Abstract: We consider Serrin’s overdetermined problem for the equation $$\Delta v + nK v = -\,1$$ in space forms, where K is the curvature of the space, and we prove a symmetry result by using a P-function approach. Our approach generalizes the one introduced by Weinberger to space forms and, as in the Euclidean case, it provides a short proof of the symmetry result which does not make use of the method of moving planes.

Classifying local fractal subsystem symmetry-protected topological phases

Abstract: We study symmetry-protected topological (SPT) phases of matter in $2\mathrm{D}$ protected by symmetries acting on fractal subsystems of a certain type. Despite the total symmetry group of such systems being subextensively large, we show that only a small number of phases are actually realizable by local Hamiltonians. Which phases are possible depends crucially on the spatial structure of the symmetries, and we show that in many cases, no nontrivial SPT phases are possible at all. In cases where nontrivial SPT phases do exist, we give an exhaustive enumeration of them in terms of their locality.

Strongly symmetric spectral convex bodies are Jordan algebra state spaces

Abstract: We show that the strongly symmetric spectral convex compact sets are precisely the normalized state spaces of finite-dimensional simple Euclidean Jordan algebras and the simplices Spectrality is the property that every state has a convex decomposition into perfectly distinguishable pure states; strong symmetry is transitivity, for each integer N, of the affine automorphism group of the state space on lists of N perfectly distinguishable pure states Additional assumptions combine with this theorem to give simple characterizations of finite-dimensional complex quantum state space Important aspects of quantum and classical thermodynamics and of query complexity have been generalized to classes of general probabilistic theories (GPTs) satisfying natural postulates including or implying spectrality and strong symmetry; our result shows that these apply to a narrower class of theories than might have been hoped Sorkin's notion of irreducibly k-th order interference has been studied in the GPT framework and looked for in experiments Our result shows that the assumption of no higher-order (k > 2) interference, used along with spectrality and strong symmetry to characterize the same class of Jordan-algebraic convex sets by Barnum, Mueller, and Ududec in arXiv:14034147, was superfluous It also implies that Lee and Selby's extension, on the assumption that interference has fixed maximal degree k, of the important order square root of N lower bound on the quantum black-box query complexity of searching N possibilities for one having a desired property (which is achieved by Grover's quantum algorithm), to a class of theories satisfying certain postulates allowing the formulation of a generalized notion of query algorithm, actually applies in the Jordan-algebraic setting where higher-order interference is not possible

Observability, redundancy and modality for dynamical symmetry transformations

Abstract: I provide a fairly systematic analysis of when quantities that are variant under a dynamical symmetry transformation should be regarded as unobservable, or redundant, or unreal; of when models related by a dynamical symmetry transformation represent the same state of affairs; and of when mathematical structure that is variant under a dynamical symmetry transformation should be regarded as surplus. In most of these cases the answer is `it depends': depends, that is, on the details of the symmetry in question. A central feature of the analysis is that in order to draw any of these conclusions for a dynamical symmetry it needs to be understood in terms of its possible extensions to other physical systems, in particular to measurement devices.

Radial symmetry for p-harmonic functions in exterior and punctured domains

Abstract: We prove symmetry for the p-capacitary potential satisfying Δpu=0inRN\Ω¯,u=1onΓ,lim|x|→∞u(x)=0,1

Supertranslations in higher dimensions revisited

Abstract: In this paper, we revisit the question of identifying the soft graviton theorem in higher (even) dimensions with Ward identities associated with asymptotic symmetries. Building on the prior work of [Ann. Math. Sci. Appl. 2, 69 (2017)], we compute, from first principles, the (asymptotic) charges associated to supertranslation symmetry in higher even dimensions and show that (i) these charges are nontrivial and finite and (ii) the corresponding Ward identities are indeed the soft graviton theorems.

On gapped boundaries for SPT phases beyond group cohomology

Abstract: We discuss a strategy to construct gapped boundaries for a large class of symmetry-protected topological phases (SPT phases) beyond group cohomology. This is done by a generalization of the symmetry extension method previously used for cohomological SPT phases. We find that this method allows us to construct gapped boundaries for time-reversal-invariant bosonic SPT phases and for fermionic Gu-Wen SPT phases for arbitrary finite internal symmetry groups.

One can hear the shape of ellipses of small eccentricity

Abstract: We show that if the eccentricity of an ellipse is sufficiently small then up to isometries it is spectrally unique among all smooth domains. We do not assume any symmetry, convexity, or closeness to the ellipse, on the class of domains.

The graded differential geometry of mixed symmetry tensors

Abstract: We show how the theory of $\mathbb{Z}_2^n$-manifolds - which are a non-trivial generalisation of supermanifolds - may be useful in a geometrical approach to mixed symmetry tensors such as the dual graviton. The geometric aspects of such tensor fields on both flat and curved space-times are discussed.

Classification of phases for mixed states via fast dissipative evolution

Abstract: We propose the following definition of topological quantum phases valid for mixed states: two states are in the same phase if there exists a time independent, fast and local Lindbladian evolution driving one state into the other. The underlying idea, motivated by Konig and Pastawski in 2013, is that it takes time to create new topological correlations, even with the use of dissipation. We show that it is a good definition in the following sense: (1) It divides the set of states into equivalent classes and it establishes a partial order between those according to their level of "topological complexity". (2) It provides a path between any two states belonging to the same phase where observables behave smoothly. We then focus on pure states to relate the new definition in this particular case with the usual definition for quantum phases of closed systems in terms of the existence of a gapped path of Hamiltonians connecting both states in the corresponding ground state path. We show first that if two pure states are in the same phase in the Hamiltonian sense, they are also in the same phase in the Lindbladian sense considered here. We then turn to analyse the reverse implication, where we point out a very different behaviour in the case of symmetry protected topological (SPT) phases in 1D. Whereas at the Hamiltonian level, phases are known to be classified with the second cohomology group of the symmetry group, we show that symmetry cannot give any protection in 1D in the Lindbladian sense: there is only one SPT phase in 1D independently of the symmetry group. We finish analysing the case of 2D topological quantum systems. There we expect that different topological phases in the Hamiltonian sense remain different in the Lindbladian sense. We show this formally only for the $\mathbb{Z}_n$ quantum double models.

Semi-simple enlargement of the b m s 3 $$ \mathfrak{b}\mathfrak{m}{\mathfrak{s}}_3 $$ algebra from a so 2 , 2 ⊕ so 2 , 1 $$ \mathfrak{so}\left(2,\ 2\right)\oplus \mathfrak{so}\left(2,\ 1\right) $$ Chern-Simons theory

Abstract: In this work we present a BMS-like ansatz for a Chern-Simons theory based on the semi-simple enlargement of the Poincare symmetry, also known as AdS-Lorentz algebra. We start by showing that this ansatz is general enough to contain all the relevant stationary solutions of this theory and provides with suitable boundary conditions for the corresponding gauge connection. We find an explicit realization of the asymptotic symmetry at null infinity, which defines a semi-simple enlargement of the $$ \mathfrak{b}\mathfrak{m}{\mathfrak{s}}_3 $$ algebra and turns out to be isomorphic to three copies of the Virasoro algebra. The flat limit of the theory is discussed at the level of the action, field equations, solutions and asymptotic symmetry.

F-theory on quotients of elliptic Calabi-Yau threefolds

Abstract: In this work we consider quotients of elliptically fibered Calabi-Yau threefolds by freely acting discrete groups and the associated physics of F-theory compactifications on such backgrounds. The process of quotienting a Calabi-Yau geometry produces not only new genus one fibered manifolds, but also new effective 6-dimensional physics. These theories can be uniquely characterized by the much simpler covering space geometry and the symmetry action on it. We use this method to construct examples of F-theory models with an array of discrete gauge groups and non-trivial monodromies, including an example with ℤ6 discrete symmetry.

Movement from the Double Object Construction Is Not Fully Symmetrical

Abstract: A movement asymmetry arises in some languages that are otherwise symmetrical for both A- and Ā-movement in the double object construction, including Norwegian, North-West British English, and a ran...

Cohomological framework for contextual quantum computations

Abstract: We describe a cohomological framework for measurement based quantum computation, in which symmetry plays a central role. Therein, the essential information about the computational output is contained in topological invariants, namely elements of two cohomology groups. One of those invariants applies to the deterministic case, and the other to the general probabilistic case. The same invariants also witness quantumness in the form of contextuality. In result, they give rise to fundamental algebraic structures underlying quantum computation.

The Maxwell group in 2+1 dimensions and its infinite-dimensional enhancements

Abstract: The Maxwell group in 2+1 dimensions is given by a particular extension of a semi-direct product. This mathematical structure provides a sound framework to study different generalizations of the Maxwell symmetry in three space-time dimensions. By giving a general definition of extended semi-direct products, we construct infinite-dimensional enhancements of the Maxwell group that enlarge the ISL(2, ℝ) Kac-Moody group and the $$ {\hat{\mathrm{BMS}}}_3 $$ group by including non-commutative supertranslations. The coadjoint representation in each case is defined, and the corresponding geometric actions on coadjoint orbits are presented. These actions lead to novel Wess-Zumino terms that naturally realize the aforementioned infinite-dimensional symmetries. We briefly elaborate on potential applications in the contexts of three-dimensional gravity, higher-spin symmetries, and quantum Hall systems.

Quantum walks: Schur functions meet symmetry protected topological phases

Abstract: This paper uncovers and exploits a link between a central object in harmonic analysis, the so-called Schur functions, and the very hot topic of symmetry protected topological phases of quantum matter. This connection is found in the setting of quantum walks, i.e. quantum analogs of classical random walks. We prove that topological indices classifying symmetry protected topological phases of quantum walks are encoded by matrix Schur functions built out of the walk. This main result of the paper reduces the calculation of these topological indices to a linear algebra problem: calculating symmetry indices of finite-dimensional unitaries obtained by evaluating such matrix Schur functions at the symmetry protected points $\pm1$. The Schur representation fully covers the complete set of symmetry indices for 1D quantum walks with a group of symmetries realizing any of the symmetry types of the tenfold way. The main advantage of the Schur approach is its validity in the absence of translation invariance, which allows us to go beyond standard Fourier methods, leading to the complete classification of non-translation invariant phases for typical examples.

Symmetry and Measuring: Ways to Teach the Foundations of Mathematics Inspired by Yupiaq Elders

Abstract: Evident in human prehistory and across immense cultural variation in human activities, symmetry has been perceived and utilized as an integrative and guiding principle. In our long-term collaborative work with Indigenous Knowledge holders, particularly Yupiaq Eskimos of Alaska and Carolinian Islanders in Micronesia, we were struck by the centrality of symmetry and measuring as a comparison-of-quantities, and the practical and conceptual role of qukaq [center] and ayagneq [a place to begin]. They applied fundamental mathematical principles associated with symmetry and measuring in their everyday activities and in making artifacts. Inspired by their example, this paper explores the question: Could symmetry and measuring provide a systematic and integrative way to teach the foundations of mathematical thinking? We illustrate how the fundamental structures of symmetry, measuring, and comparison-of-quantities, starting with the embodied orthogonal axes, form a basis for properties of equality, aspects of numbers and operations (including place value), geometry and number line representations, functions, algebraic reasoning, and measurement. We conclude by embedding the earlier geometric constructions of triangles and squares within the unit circle and making explicit connections to trigonometric functions.

Asymmetric truncated toeplitz operators and conjugations

Abstract: Truncated Toeplitz operators in a model space are C--symmetric with respect to a natural conjugation in that space. We show that this and another conjugation associated to an orthogonal decomposition possess unique properties and we study their relations with asymmetric truncated Toeplitz operators in terms of C--symmetry. New connections with Hankel operators are established through this approach.

New symmetries for the $U_q(sl_N)$ 6-j symbols from the Eigenvalue conjecture

Abstract: In the present paper we discuss the eigenvalue conjecture, suggested in 2012, in the particular case of $U_q(sl_2)$. The eigenvalue conjecture provides a certain symmetry for Racah coefficients and we prove that \textbf{the eigenvalue conjecture is provided by the Regge symmetry} for $U_q(sl_2)$, when three representations coincide. This in perspective provides us a kind of generalization of the Regge symmetry to arbitrary $U_q(sl_N)$.

Note on the space group selection rule for closed strings on orbifolds

Abstract: It is well-known that the space group selection rule constrains the interactions of closed strings on orbifolds. For some examples, this rule has been described by an effective Abelian symmetry that combines with a permutation symmetry to a non-Abelian flavor symmetry like D4 or Δ(54). However, the general case of the effective Abelian symmetries was not yet fully understood. In this work, we formalize the computation of the Abelian symmetry that results from the space group selection rule by imposing two conditions only: (i) well-defined discrete charges and (ii) their conservation. The resulting symmetry, which we call the space group flavor symmetry DS, is uniquely specified by the Abelianization of the space group. For all Abelian orbifolds with $$ \mathcal{N}=1 $$ supersymmetry we compute DS and identify new cases, for example, where DS contains a ℤ2 dark matter-parity with charges 0 and 1 for massless and massive strings, respectively.