In this paper, the problem of the identification of the symmetry class of a given tensor is asked. Contrary to classical approaches which are based on the spectral properties of the linear operator describing the elasticity, our setting is based on the invariants of the irreducible tensors appearing in the harmonic decomposition of the elasticity tensor [Forte-Vianello, 1996]. To that aim we first introduce a geometrical description of the space of elasticity tensors. This framework is used to derive invariant-based conditions that characterize symmetry classes. For low order symmetry classes, such conditions are given on a triplet of quadratic forms extracted from the harmonic decomposition of the elasticity tensor $C$, meanwhile for higher-order classes conditions are provided in terms of elements of $H^{4}$, the higher irreducible space in the decomposition of $C$. Proceeding in such a way some well known conditions appearing in the Mehrabadi-Cowin theorem for the existence of a symmetry plane are retrieved, and a set of algebraic relations on polynomial invariants characterizing the orthotropic, trigonal, tetragonal, transverse isotropic and cubic symmetry classes are provided. Using a genericity assumption on the elasticity tensor under study, an algorithm to identify the symmetry class of a large set of tensors is finally provided.
We complete the proof of the x-y symmetry of symplectic invariants of [EO]. We recall the main steps of the proof of [EO2], and we include the integration constants absent in [EO2].
We describe some classes of PDE that display hidden symmetry, with reduced equations having additional symmetry operators compared to the initial equations. Relations between the concepts of hidden and conditional symmetry, and between hidden symmetry and equivalence of classes of equations, is discussed. In particular, we describe equations having hidden and conditional symmetry under rotations and boosts in the Lorentz and Euclid groups.
We characterize symmetry transformations of Lagrangian extremals generating `on shell' conservation laws. We relate symmetry transformations of extremals to Jacobi fields and study symmetries of higher variations by proving that a pair given by a symmetry of the $l$-th variation of a Lagrangian and by a Jacobi field of the $s$-th variation of the same Lagrangian (with $s<l$) is associated with an `off shell' conserved current. The conserved current associated with two symmetry transformations is constructed and, as a case of study, its expression for invariant Yang--Mills connections on Minkow…
An invariant of SPT-phases with on-site finite group $G$ symmetry for two-dimensional Fermion systems was derived in [O]. This invariant is doubled compared to the conjectured one from the invertible quantum field theory. We show that if we require CRT-symmetry (which holds automatically in quantum field theory) in addition, then our invariant reduces to the conjectured one.
Charge conjugation, parity transformation and time reversal symmetry (CPT) violation and Lorentz invariance can coexist in the framework of non-local field theory. In this article we have proposed a class of Charge conjugation, parity transformation and time reversal symmetry (CPT) violating Lorentz invariant nonlocal gauge-invariant models, which can be termed as non-local Thirring models. The conserved currents in this aspect are obtained.
In this paper a hidden extra symmetry of conformally invariant Lagrangians occuring in physics is pointed out. This symmetry is most apparent in a metric independent, i.e. in a Palatini-like presentation of the variational problem. In such presentation, the usual conformal weight of fields can be encoded as local dilatation group gauge charges. The conventional conformal invariance of Lagrangians is then equivalent to dilatation gauge invariance. The claim of the paper is, that the most commonly occurring conformally invariant Lagrangians turning up in physics are not only invariant to local d…
Ordinary differential equations (ODEs) and ordinary difference systems (O$Δ$Ss) invariant under the actions of the Lie groups $\mathrm{SL}_x(2)$, $\mathrm{SL}_y(2)$ and $\mathrm{SL}_x(2)\times\mathrm{SL}_y(2)$ of projective transformations of the independent variables $x$ and dependent variables $y$ are constructed. The ODEs are continuous limits of the O$Δ$Ss, or conversely, the O$Δ$Ss are invariant discretizations of the ODEs. The invariant O$Δ$Ss are used to calculate numerical solutions of the invariant ODEs of order up to five. The solutions of the invariant numerical schemes are compared…
We show that a generating function for open $r$-spin enumerative invariants produces a universal unfolding of the polynomial $x^r$. Further, the coordinates parametrizing this universal unfolding are flat coordinates on the Frobenius manifold associated to the Landau-Ginzburg model $(\mathbb{C},x^r)$ via Saito-Givental theory. This result provides evidence for the same phenomenon to occur in higher dimension, proven in a sequel paper.
Integrable and superintegrable quantum mechanical systems with position dependent masses invariant with respect to one parametric Lie groups. 1. Systems with cylindric symmetry
A nonlocal Alice-Bob Kadomtsev-Petviashivili (ABKP) system with shifted-parities ($\hat{P}_s^x$ and $\hat{P}_s^y$ parities with shifts for the space variables $x$ and $y$) and delayed time reversal ($\hat{T}_d$, time reversal with a delay) symmetries is investigated. Some types of $\hat{P}_s^y\hat{P}_s^x\hat{T}_d$ invariant solutions including multiple soliton solutions, Painlevé reductions and soliton and $p$-wave interaction solutions are obtained via $\hat{P}_s^y\hat{P}_s^x\hat{T}_d$ symmetry and the solutions of the usual local KP equation. Some special $\hat{P}_s^y\hat{P}_s^x\hat{T}_d$ sy…
We study the symmetry reduction of nonlinear partial differential equations which are used for describing diffusion processes in nonhomogeneous medium. We find ansatzes reducing partial differential equations to systems of ordinary differential equations. The ansatzes are constructed by using operators of Lie-Bäcklund symmetry of third order ordinary differential equation. The method gives the possibility to find solutions which cannot be obtained by virtue of classical Lie method. Such solutions have been constructed for nonlinear diffusion equations which are invariant with respect to one-pa…
Lie symmetry method is applied to investigate symmetries of the combined KdV-nKdV equation, that is a new integrable equation by combining the KdV equation and negative order KdV equation. Symmetries which are obtained in this article, are further helpful for reducing the combined KdV-nKdV equation into ordinary differential equation. Moreover, a set of eight invariant solutions for combined KdV-nKdV equation is obtained by using proposed method. Out of the eight solutions so obtained in which two solutions generate progressive wave solutions, five are singular solutions and one multisoliton s…
Periodically driven (Floquet) crystals are described by their quasi-energy spectrum. Their topological properties are characterized by invariants attached to the gaps of this spectrum. In this article, we define such invariants in all space dimensions, both in the case where no symmetry is present and in the case where the unitary chiral symmetry is present. When no symmetry is present, a $\mathbb{Z}$-valued invariant can be defined in each gap in all even space dimensions. This invariant does not capture all the properties of a system where chiral symmetry is present. In even space dimension,…
The maximal symmetry of a quantum system with Heisenberg commutation relations is given by the projective representations of the automorphism group of the Weyl-Heisenberg algebra. The automorphism group is the central extension of the inhomogeneous symplectic group with a conformal scaling that acts on extended phase space. We determine the subgroup that also leaves invariant a degenerate orthogonal Minkowski line element. This defines noninertial relativistic symmetry transformations that have the expected classical limit as c becomes infinite.
In this paper, we propose a method, that is based on equivariant moving frames, for development of high order accurate invariant compact finite difference schemes that preserve Lie symmetries of underlying partial differential equations. In this method, variable transformations that are obtained from the extended symmetry groups of PDEs are used to transform independent and dependent variables, and derivative terms of compact finite difference schemes (constructed for these PDEs) such that the resulting schemes are invariant under the chosen symmetry groups. The unknown symmetry parameters tha…
Nonlinear PDE's having {\bf given} conditional symmetries are constructed. They are obtained starting from the invariants of the "conditional symmetry" generator and imposing the extra condition given by the characteristic of the symmetry. Several of examples starting from the Boussinesq and including non-autonomous Korteweg-De Vries like equations are given to showcase the methodology introduced.
This paper surveys results found by the authors in the previous papers (see for example, A. Duyunova, V. Lychagin, S. Tychkov, Differential invariants for spherical layer flows of a viscid fluid, Journal of Geometry and Physics, 130, 288-292 (2018)) concerning calculation and classification of symmetry Lie algebra and differential invariants for a variety of fundamental systems in fluid mechanics, including the Euler and Navier-Stokes equations on a plane, three-dimensional space, a sphere, and a spherical layer. In each case, the symmetry Lie algebra is found, followed by a list of generating…
We prove an explicit formula for the genus-one Fan-Jarvis-Ruan-Witten invariants associated to the quintic threefold, verifying the genus-one mirror conjecture of Huang, Klemm, and Quackenbush. The proof involves two steps. The first step uses localization on auxiliary moduli spaces to compare the usual Fan-Jarvis-Ruan-Witten invariants with a semisimple theory of twisted invariants. The second step uses the genus-one formula for semisimple cohomological field theories to compute the twisted invariants explicitly.
We study an important property of shape invariant supersymmetric quantum mechanical systems. Particularly, we demonstrate that each shape invariant supersymmetric system can constitute a $Z_3$-graded topological symmetric algebra. The latter is known to provide topological invariants which are generalizations of the Witten index. In addition, we relate the $Z_3$-graded algebra to the generators of the SO(2,1) Lie algebra underlying each shape invariant system. We generalize the results to the case of sequential shape invariant systems, in which case we find a sequence of $Z_3$-graded algebras.…
In the framework of Atiyah's axioms of topological quantum field theory with unitarity, we give a direct proof of the fact that symmetry protected topological (SPT) phases without Hall effects are classified by cobordism invariants. We first show that the partition functions of those theories are cobordism invariants after a tuning of the Euler term. Conversely, for a given cobordism invariant, we construct a unitary topological field theory whose partition function is given by the cobordism invariant. Two theories having the same cobordism invariant partition functions are isomorphic.
We introduce a ${\mathbb Z}_2$-index for time reversal invariant Hamiltonians with unique gapped ground state on quantum spin chains. We show this is an invariant of a $C^1$-classification of gapped Hamiltonians.
We study the breaking of spin symmetry for the nonlinear Hartree-Fock model describing an infinite translation-invariant interacting quantum gas (fluid phase). At zero temperature and for the Coulomb interaction in three space dimensions, we can prove the existence of a unique first order transition between a pure ferromagnetic phase at low density and a paramagnetic phase at high density. Multiple first or second order transitions can happen for other interaction potentials, as we illustrate on some examples. At positive temperature $T>0$ we compute numerically the phase diagram in the Coulom…
We describe the connection between inversion symmetry breaking and criticality in free fermionic lattice models. It is shown that for translation-invariant spinless fermions, the breaking of this symmetry in the ground state implies criticality, i.e., the existence of long-range correlations and the vanishing of the spectral gap; while for models with spin, only the asymmetry of the spin-averaged covariance matrix implies a similar conclusion. Our results are proved by introducing invariants under global translation-invariant free fermion quenches. Using this result, we identify a set of model…
We develop a mathematical framework for the computation of open orbifold Gromov-Witten invariants of [C^3/Z_n], and provide extensive checks with predictions from open string mirror symmetry. To this aim we set up a computation of open string invariants in the spirit of Katz-Liu, defining them by localization. The orbifold is viewed as an open chart of a global quotient of the resolved conifold, and the Lagrangian as the fixed locus of an appropriate anti-holomorphic involution. We consider two main applications of the formalism. After warming up with the simpler example of [C^3/Z_3], where we…
In recent years, great success has been achieved on the classification of symmetry-protected topological (SPT) phases for interacting fermion systems by using generalized cohomology theory. However, the explicit calculation of generalized cohomology theory is extremely hard due to the difficulty of computing obstruction functions. In this paper, based on the physical picture of topological invariants and mathematical techniques in homotopy algebra, we develop an algorithm to resolve this hard problem. It is well known that cochains in the cohomology of the symmetry group, which are used to enu…
Conformally invariant elliptic Liouville equation and its symmetry preserving discretization
Universitá di Roma Tre（罗马第三大学）；INFN, Sezione di Roma Tre（意大利国家核物理研究所罗马三分所）；Universitá del Salento（萨莱诺大学）；INFN, Sezione di Lecce（意大利国家核物理研究所莱切分所）；Université de Montréal（蒙特利尔大学）
We introduce the notion of Wall-Crossing Structure and discuss it in several examples including complex integrable systems, Donaldson-Thomas invariants and Mirror Symmetry. For a big class of non-compact Calabi-Yau 3-folds we construct complex integrable systems of Hitchin type with the base given by the moduli space of deformations of those 3-folds. Then Donaldson-Thomas invariants of the Fukaya category of such a Calabi-Yau 3-fold can be (conjecturally) described in two more ways: in terms of the attractor flow on the base of the corresponding complex integrable system and in terms of the sk…
We consider a (mathbb{Z}_2)-equivariant flow in (mathbb{R}^{4}) with an integral of motion and a hyperbolic equilibrium with a transverse homoclinic orbit (Gamma). We provide criteria for the existence of stable and unstable invariant manifolds of (Gamma). We prove that if these manifolds intersect transversely, creating a so-called super-homoclinic, then in any neighborhood of this super-homoclinic there exist infinitely many multi-pulse homoclinic loops. An application to a system of coupled nonlinear Schrödinger equations is considered.
