Jacobi Identity Research Articles

Higher-dimensional general Jacobi identities I

It was shown by the author [International Journal of Theoretical Physics 36 (1997), 1099-1131] in synthetic differential geometry that what is called the general Jacobi identity obtaining in microcubes underlies the Jacobi identity of vector fields. It is well known in the theory of Lie algebras that a plethora of higher-dimensional generalizations of the Jacobi identity hold, though it is usually established not as a direct derivation from the axioms of Lie algebras but by making an appeal to the so-called Poincare-Birkhoff-Witt theorem. The general Jacobi identity was rediscovered by Kirill Mackenzie in the second decade of this century [Geometric Methods in Physics, 357-366, Birkhauser/Springer 2013]. The principal objective in this paper is to investigate a four-dimensional generalization of the general Jacobi identity in detail. In a subsequent paper we will propose a uniform method for establishing a bevy of higher-dimensional generalizations of the Jacobi identity under a single umbrella.

GEMPIC: geometric electromagnetic particle-in-cell methods

We present a novel framework for finite element particle-in-cell methods based on the discretization of the underlying Hamiltonian structure of the Vlasov–Maxwell system. We derive a semi-discrete Poisson bracket, which retains the defining properties of a bracket, anti-symmetry and the Jacobi identity, as well as conservation of its Casimir invariants, implying that the semi-discrete system is still a Hamiltonian system. In order to obtain a fully discrete Poisson integrator, the semi-discrete bracket is used in conjunction with Hamiltonian splitting methods for integration in time. Techniques from finite element exterior calculus ensure conservation of the divergence of the magnetic field and Gauss’ law as well as stability of the field solver. The resulting methods are gauge invariant, feature exact charge conservation and show excellent long-time energy and momentum behaviour. Due to the generality of our framework, these conservation properties are guaranteed independently of a particular choice of the finite element basis, as long as the corresponding finite element spaces satisfy certain compatibility conditions.

Wronskian and Grammian solutions for a [formula omitted]-dimensional Date–Jimbo–Kashiwara–Miwa equation

In this paper, a (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation is investigated. Based on the Hirota method and auxiliary variable, Bäcklund transformation is obtained. Under certain conditions, the Hirota bilinear forms can be reduced to the Plücker and Jacobi identities via the Wronskian and Grammian solutions. Through the Wronskian technique and Pfaffian derivative formulae, N-soliton solutions in the Wronskian and Grammian are given. Graphically presented, soliton collisions are elastic both in the Wronskian and Grammian, and after each collision, the solitons keep their amplitudes unchanged except for the phase shifts.

Super W1+∞ n-Algebra in the Supersymmetric Landau Problem**Supported by National Natural Science Foundation of China under Grant Nos. 11375119, 11475116, and 11547101

We analyze the super n-bracket built from associative operator products. Since the super n-bracket with n even satisfies the so-called generalized super Jacobi identity, we deal with the n odd case and give the generalized super Bremner identity. For the infinite conserved operators in the supersymmetric Landau problem, we derive the super n-algebra which satisfies the generalized super Jacobi and Bremner identities for the n even and odd cases, respectively. Moreover the super sub-2n-algebra is also given.

Lie superbialgebra structures on the Lie superalgebra C3+A and deformation of related integrable Hamiltonian systems

Admissible structure constants related to the dual Lie superalgebras of particular Lie superalgebra (C3+A) are found by straightforward calculations from the matrix form of super Jacobi and mixed super Jacobi identities which are obtained from adjoint representation. Then, by making use of the automorphism supergroup of the Lie superalgebra (C3+A), the Lie superbialgebra structures on the Lie superalgebra (C3+A) are obtained and classified into inequivalent 31 families. We also determine all corresponding coboundary and bi-r-matrix Lie superbialgebras. The quantum deformations associated with some Lie superbialgebras (C3+A) are obtained, together with the corresponding deformed Casimir elements. As an application of these quantum deformations, we construct a deformed integrable Hamiltonian system from the representation of the Hopf superalgebra Uλ(Cp=12,𝜖⊕A1,1)(C3+A).

Gravity Amplitudes as Generalized Double Copies of Gauge-Theory Amplitudes.

Whenever the integrand of a gauge-theory loop amplitude can be arranged into a form where the Bern-Carrasco-Johansson duality between color and kinematics is manifest, a corresponding gravity integrand can be obtained simply via the double-copy procedure. However, finding such gauge-theory representations can be challenging, especially at high loop orders. Here, we show that we can, instead, start from generic gauge-theory integrands, where the duality is not manifest, and apply a modified double-copy procedure to obtain gravity integrands that include contact terms generated by violations of dual Jacobi identities. We illustrate this with three-, four- and five-loop examples in N=8 supergravity.

A classification of polynomial functions satisfying the Jacobi identity over integral domains

The Jacobi identity is one of the properties that are used to define the concept of Lie algebra and in this context is closely related to associativity. In this paper we provide a complete description of all bivariate polynomials that satisfy the Jacobi identity over infinite integral domains. Although this description depends on the characteristic of the domain, it turns out that all these polynomials are of degree at most one in each indeterminate.

Background independent double field theory at order α′ : Metric versus framelike geometry

The higher derivative corrections in double field theory are revisited to first order in $\alpha'$. In first order perturbation theory around flat space, the gauge algebra is $\alpha'$ corrected, governed by two parameters $a, b$. One parameter choice corresponds to bosonic string theory, another to heterotic string theory. These results are generalized to second order in perturbation theory by a Noether procedure. Using consistency conditions derived from the Jacobi identity, it is shown that for general $a, b$ there is no generalized metric formulation. A manifestly background independent formulation is instead given by a frame formalism, with $\alpha'$-deformed frame transformations, as proposed by Marques and Nunez. Their construction is slightly generalized to an $\alpha'$-deformed $GL(D)\times GL(D)$ frame formalism. The perturbation theory around flat space is matched to second order with the results obtained by the Noether procedure. We also obtain a formulation based on the `non-symmetric' metric ${\cal E}$, the sum of metric and $b$-field. The transformations of ${\cal E}$ under $O(D,D)$ receive non-trivial $\alpha'$ corrections, unless $a=-b$, in which case there is a generalized metric formulation, in agreement with previous results in the literature.

Background Independence and Duality Invariance in String Theory.

Closed string theory exhibits an O(D,D) duality symmetry on tori, which in double field theory is manifest before compactification. I prove that to first order in α^{'} there is no manifestly background independent and duality invariant formulation of bosonic string theory in terms of a metric, b field, and dilaton. To this end I use O(D,D) invariant second order perturbation theory around flat space to show that the unique background independent candidate expression for the gauge algebra at order α^{'} is inconsistent with the Jacobi identity. A background independent formulation exists instead for frame variables subject to α^{'}-deformed frame transformations (generalized Green-Schwarz transformations). Potential applications for curved backgrounds, as in cosmology, are discussed.

Symmetry for Flavor-Kinematics Duality from an Action.

We propose a new representation of the nonlinear sigma model that exhibits a manifest duality between flavor and kinematics. The fields couple exclusively through cubic Feynman vertices which define the structure constants of an underlying kinematic algebra. The action is invariant under a combination of internal and spacetime symmetries whose conservation equations imply flavor-kinematics duality, ensuring that all Feynman diagrams satisfy kinematic Jacobi identities. Substituting flavor for kinematics, we derive a new cubic action for the special Galileon theory. In this picture, the vanishing soft behavior of amplitudes is a by-product of the Weinberg soft theorem.

Formation of a giant component in the intersection graph of a random chord diagram

We study the number of chords and the number of crossings in the largest component of a random chord diagram when the chords are sparsely crossing. This is equivalent to studying the number of vertices and the number of edges in the largest component of the random intersection graph. Denoting the number of chords by n and the number of crossings by m, when m/(nlog⁡n) tends to a limit in (0,2/π2), we show that the chord diagram chosen uniformly at random from all the diagrams with given parameters has a component containing almost all the crossings and a positive fraction of chords. On the other hand, when m≤n/14, the size of the largest component is of order O(log⁡n). One of the key analytical ingredients is an asymptotic expression for the number of chord diagrams with parameters n and m for m<(2/π2)nlog⁡n, based on the Touchard–Riordan formula and the Jacobi identity for the generating function of Euler partition function.

On the Kontsevich ★-product associativity mechanism

The deformation quantization by Kontsevich [arXiv:q-alg/9709040] is a way to construct an associative noncommutative star-product $\star=\times+\hbar \{\ ,\ \}_{P}+\bar{o}(\hbar)$ in the algebra of formal power series in $\hbar$ on a given finite-dimensional affine Poisson manifold: here $\times$ is the usual multiplication, $\{\ ,\ \}_{P}\neq0$ is the Poisson bracket, and $\hbar$ is the deformation parameter. The product $\star$ is assembled at all powers $\hbar^{k\geq0}$ via summation over a certain set of weighted graphs with $k+2$ vertices; for each $k>0$, every such graph connects the two co-multiples of $\star$ using $k$ copies of $\{\ ,\ \}_{P}$. Cattaneo and Felder [ arXiv:math/9902090 [math.QA] ] interpreted these topological portraits as the genuine Feynman diagrams in the Ikeda-Izawa model [arXiv:hep-th/9312059] for quantum gravity. By expanding the star-product up to $\bar{o}(\hbar^3)$, i.e., with respect to graphs with at most five vertices but possibly containing loops, we illustrate the mechanism Assoc = Operator(Poisson) that converts the Jacobi identity for the bracket $\{\ ,\ \}_{P}$ into the associativity of $\star$. Key words: Deformation quantization, associative algebra, Poisson bracket, graph complex, star-product PACS: 02.40.Sf, 02.10.Ox, 02.40.Gh, also 04.60.-m

Lifting a weak Poisson bracket to the algebra of forms

We detail the construction of a weak Poisson bracket over a submanifold Σ of a smooth manifold M with respect to a local foliation of this submanifold. Such a bracket satisfies a weak type Jacobi identity but may be viewed as a usual Poisson bracket on the space of leaves of the foliation. We then lift this weak Poisson bracket to a weak odd Poisson bracket on the odd tangent bundle Π T M , interpreted as a weak Koszul bracket on differential forms on M . This lift is achieved by encoding the weak Poisson structure into a homotopy Poisson structure on an extended manifold, and lifting the Hamiltonian function that generates this structure. Such a construction has direct physical interpretation. For a generic gauge system, the submanifold Σ may be viewed as a stationary surface or a constraint surface, with the foliation given by the foliation of the gauge orbits. Through this interpretation, the lift of the weak Poisson structure is simply a lift of the action generating the corresponding BRST operator of the system.

Recursion operators and bi-Hamiltonian structure of the general heavenly equation

We discover two additional Lax pairs and three nonlocal recursion operators for symmetries of the general heavenly equation introduced by Doubrov and Ferapontov. Converting the equation to a two-component form, we obtain Lagrangian and Hamiltonian structures of the two-component general heavenly system. We study all point symmetries of the two-component system and, using the inverse Noether theorem in the Hamiltonian form, obtain all the integrals of motion corresponding to each variational (Noether) symmetry. We discover that in the two-component form we have only a single nonlocal recursion operator. Composing the recursion operator with the first Hamiltonian operator we obtain second Hamiltonian operator. We check the Jacobi identities for the second Hamiltonian operator and compatibility of the two Hamiltonian structures using P. Olver’s theory of functional multi-vectors. Our well-founded conjecture is that P. Olver’s method works fine for nonlocal operators. We show that the general heavenly equation in the two-component form is a bi-Hamiltonian system integrable in the sense of Magri. We demonstrate how to obtain nonlocal Hamiltonian flows generated by local Hamiltonians by using formal adjoint recursion operator.

Two-subsystem thermodynamics for the mechanics of aging amorphous solids

The effect of physical aging on the mechanics of amorphous solids as well as mechanical rejuvenation is modeled with nonequilibrium thermodynamics, using the concept of two thermal subsystems, namely a kinetic one and a configurational one. Earlier work (Semkiv and Hütter in J Non-Equilib Thermodyn 41(2):79–88, 2016) is extended to account for a fully general coupling of the two thermal subsystems. This coupling gives rise to hypoelastic-type contributions in the expression for the Cauchy stress tensor, that reduces to the more common hyperelastic case for sufficiently long aging. The general model, particularly the reversible and irreversible couplings between the thermal subsystems, is compared in detail with models in the literature (Boyce et al. in Mech Mater 7:15–33, 1988; Buckley et al. in J Mech Phys Solids 52:2355–2377, 2004; Klompen et al. in Macromolecules 38:6997–7008, 2005; Kamrin and Bouchbinder in J Mech Phys Solids 73:269–288 2014; Xiao and Nguyen in J Mech Phys Solids 82:62–81, 2015). It is found that only for the case of Kamrin and Bouchbinder (J Mech Phys Solids 73:269–288, 2014) there is a nontrivial coupling between the thermal subsystems in the reversible dynamics, for which the Jacobi identity is automatically satisfied. Moreover, in their work as well as in Boyce et al. (Mech Mater 7:15–33, 1988), viscoplastic deformation is driven by the deviatoric part of the Cauchy stress tensor, while for Buckley et al. (J Mech Phys Solids 52:2355–2377, 2004) and Xiao and Nguyen (J Mech Phys Solids 82:62–81, 2015) this is not the case.

The effective gravitation theory at large scale with Lorentz violation

The Doopler effect of CMB (cosmic microwave background) may indicates the Lorentz boost violation at the cosmological scale. It is revealed that there are some anomalies in the low-l multipole expansion of the CMB spectrum in the obeservation of Planck, which are not consistent with LCDM model, the standard model of cosmology. A famous one is the existence of the axis of evil. Combining the the anomalies exhibited by the low-l multipole expansion of the CMB observation, we propose that the Lorentz symmetry may start to be violated partly from the scale of galaxy and the boost invariance is violated eventrully at the cosmological scale. We take the symmetry groups of very special relativity as the examples to illustrate the Lorentz violation effect on the large scale gravity by constructing the corresponding gauge theories as the effective large scale gravitation theories. The estimated symmetry groups for very special relativity are Sim(2), Hom(2) and E(2) which are all constructed by adding generators to the T(2) group, a 2-parameter subgroup of Lorentz group. We first review the gauge principle approach to the construction of gravitation theory based on the equivalence principle which indicates the local Lorentz invariance. The choice of dynamics of the spacetime gauge theory based on the gauge principle and the Newtonian limit at the weak coupling limit reveals that the local Lorentz invariant theory is a theory with torsion in general case. However in the case of scalar matter source, the theory is exactly the theory of general relativity with Leci-Civita connection and zero torsion. We investigate the gauge approaches for all the very special relativity cases, Sim(2), Hom(2) and E(2) as well as the case of T(2). We find that these gravitation theories share a common feather, the non-trivial distribution of space-time torsion and the contortion, which contribute an effective enegy momentum dirtribution addition to the ordinary matter souce distribution even for the case of scalar matter source, while general relativity predicts the zero torsion and contortion and space-time connection as the Levi-Civita connection in this case. To restrict the theory with a local Lorentz subgroup, one need to postpone constrain conditions on the Lorentz gauge potentials which is similar to the gauge fixing condition in the ordinary internal gauge theory. In the Sim(2) case, there needs 8 constrain conditions. The equations of motion for the gauge fields finally reduce the independent Kontortion components from 24 to 8, which is consistent with the 8 constrain equations in addition to the equation of motion for the gravitation field. The numbers of independent Kontortion components is 12 for the cases of Hom(2) and E(2) and 16 for the case of T(2), while there are 12 and 16 constrain equations in addition to the gravitation field equations respectively. The non-zero distribution of contortion may be regarded as the dark matter effect in the static case at the galaxy or the galaxy cluster scale and both the dark matter and dark energy effect in the case of expanding Robertron-Walker like solution at the cosmoligical sacle. We investigate the self consistence of the gauge theories in the Sim(2) case by examining the closure of Maurer-Cartan equation within the Sim(2) algebra and the first and second Bianchi identities induced by Jacobi identity of the Sim(2) gauge theory.

Colour-kinematics duality and the Drinfeld double of the Lie algebra of diffeomorphisms

Colour-kinematics duality suggests that Yang-Mills (YM) theory possesses some hidden Lie algebraic structure. So far this structure has resisted understanding, apart from some progress in the self-dual sector. We show that there is indeed a Lie algebra behind the YM Feynman rules. The Lie algebra we uncover is the Drinfeld double of the Lie algebra of vector fields. More specifically, we show that the kinematic numerators following from the YM Feynman rules satisfy a version of the Jacobi identity, in that the Jacobiator of the bracket defined by the YM cubic vertex is cancelled by the contribution of the YM quartic vertex. We then show that this Jacobi-like identity is in fact the Jacobi identity of the Drinfeld double. All our considerations are off-shell. Our construction explains why numerators computed using the Feynman rules satisfy the colour-kinematics at four but not at higher numbers of points. It also suggests a way of modifying the Feynman rules so that the duality can continue to hold for an arbitrary number of gluons. Our construction stops short of producing explicit higher point numerators because of an absence of a certain property at four points. We comment on possible ways of correcting this, but leave the next word in the story to future work.

Brackets in the Pontryagin Algebras of Manifolds

Given a smooth oriented manifold $M$ with non-empty boundary, we study the Pontryagin algebra $A=H_*(\Omega)$ where $\Omega$ is the space of loops in $M$ based at a distinguished point of $\partial M$. Using the ideas of string topology of Chas-Sullivan, we define a linear map $\{\{-,-\}\} : A \otimes A \to A \otimes A$ which is a double bracket in the sense of Van den Bergh satisfying a version of the Jacobi identity. For $\dim(M)>2$, the double bracket $\{\{-,-\}\}$ induces Gerstenhaber brackets in the representation algebras associated with $A$. This extends our previous work on the case $\dim(M)=2$ where $A=H_0(\Omega)$ is the group algebra of $\pi_1(M)$ and $\{\{-,-\}\}$ induces the classical Poisson brackets on the moduli spaces of representations of $\pi_1(M)$.

E(2) gauge theory model of effective gravitational theory at large scale

At the cosmological scale, there exist many anisotropic anomalies in the low-l multipoles of the CMB angular power spectrum. Especially, the normals to the octopole and quadrupole planes are aligned with the direction of the cosmological dipole at a level inconsistent with Gaussian random. The inconsistency indicates that the anomalies may not be boost effect from the CMB rest frame to the peculiar frame. It hints us that the boost invariance might be violated on a cosmological scale. There are some discrepancies between the astronomical and cosmological observations, and the predictions are solely based on general relativity and the standard model for elementary particle physics. The solutions are the introduction of dark matter and dark energy. However, all the experiments aiming at finding dark matter particles give negative result and it is still a mystery:what the dark energy is comprised of. We suppose that the Lorentz symmetry begins to be violated partly from the scale of galaxy and utilize the very special relativity symmetry group E(2) as an example to illustrate the Lorentz violation effect on the large-scale effective gravity. A local E(2) but Lorentz invariant gauge theory can be constructed based on the equivalence principle and the gauge principle. To realize the E(2) symmetry, the closure requirement of Maurer-Cartan eqnarray on E(2) algebra needs to be satisfied by postulating constraint conditions among the components of the Lorentz connection. The local Lorentz invariant gauge theory with a Hilbert-Einstein action is a theory with torsion in general case. However in the case of scalar matter source, the theory is exactly the theory of general relativity with Levi-Civita connection and zero torsion. In the E(2) gauge theory case, the closure requirement of Maurer-Cartan eqnarray for E(2) algebra postulates 12 constraint eqnarrays among the components of the Lorentz connection and the eqnarrays of motion for connection reduce the number of independent components of connection to 12. The eqnarrays of motion for the tetrad field do not contain only the involved tetrad field components nor these relevant independent components. So the whole number of variables needed to be solved is 12 more than that in general relativity while there are 12 more eqnarrays in the meantime. The torsion or the contortion field of the E(2) gauge theory is non-trivial even in the case of scalar matter source distribution. Decompose the connection into Levi-Civita one and the contortion part and rewrite the eqnarrays for tetrad field in the formalism of general relativity, then there will appear an effective energy-momentum tensor contributed by the contortion distribution, in addition to the ordinary matter source distribution even for the case of scalar matter source. We expect it to contribute at least part of the dark matter effect. We also examine the holding of the first and second Bianchi identities induced by Jacobi identity of the E(2) gauge theory. The approach of our modified gravity is different from other approach of modified gravity in the sense that we construct the modified gravity by modifying the spacetime symmetry on a large scale and the emergence of effective energy-momentum tensor caused by Lorentz violation effect is due to a purely large scale effect.

Special and exceptional mock-Lie algebras

We observe several facts and make conjectures about commutative algebras satisfying the Jacobi identity. The central question is which of those algebras admit a faithful representation (i.e., in Lie parlance, satisfy the Ado theorem, or, in Jordan parlance, are special).