Jacobi Identity Research Articles

The seeds of EFT double copy

We explore the double copy of effective field theories (EFTs), in the recently proposed generalized color-kinematics and Kawai-Lewellen-Tye (KLT) approaches. In the former, we systematically construct scalar numerators satisfying the Jacobi identities from simpler numerator seeds with trace-like permutation properties. This construction has the advantage of being easily applicable to any multiplicity, which we exemplify up to 6-point. It employs the linear map between color factors formed by single traces of generators and by products of the structure constants, which also relates the generalized KLT and color-kinematics formalisms, allowing to produce KLT kernels at arbitrary order in the EFT expansion. At 4-point, we show that all EFT kernels are generated and that they only yield double-copy amplitudes which can also be obtained from the traditional KLT kernel. We perform initial checks suggesting that the same conclusions also hold at 5-point. We focus on single-trace massless scalar EFTs which however also control the higher-derivative corrections to gauge and gravity theories.

Hamiltonian Structures for Integrable Nonabelian Difference Equations

In this paper we extensively study the notion of Hamiltonian structure for nonabelian differential-difference systems, exploring the link between the different algebraic (in terms of double Poisson algebras and vertex algebras) and geometric (in terms of nonabelian Poisson bivectors) definitions. We introduce multiplicative double Poisson vertex algebras (PVAs) as the suitable noncommutative counterpart to multiplicative PVAs, used to describe Hamiltonian differential-difference equations in the commutative setting, and prove that these algebras are in one-to-one correspondence with the Poisson structures defined by difference operators, providing a sufficient condition for the fulfilment of the Jacobi identity. Moreover, we define nonabelian polyvector fields and their Schouten brackets, for both finitely generated noncommutative algebras and infinitely generated difference ones: this allows us to provide a unified characterisation of Poisson bivectors and double quasi-Poisson algebra structures. Finally, as an application we obtain some results towards the classification of local scalar Hamiltonian difference structures and construct the Hamiltonian structures for the nonabelian Kaup, Ablowitz-Ladik and Chen-Lee-Liu integrable lattices.

Jacobi identity in polyhedral products

We show that a relation among minimal non-faces of a fillable complex K yields an identity of iterated (higher) Whitehead products in a polyhedral product over K. In particular, for the (n−1)-skeleton of a simplicial n-sphere, we always have such an identity, and for the (n−1)-skeleton of a (n+1)-simplex, the identity is the Jacobi identity of Whitehead products (n=1) and Hardie's identity for higher Whitehead products (n≥2).

The gauge-natural bilinear brackets on couples of linear vector fields and linear p-forms

We give complete description of all gauge-natural bilinear operators A transforming pairs of couples of linear vector fields and linear p-forms on a vector bundle E into couples of linear vector fields and linear p-forms on E and satisfying the Jacobi identity in Leibniz form.

Fermionic construction in the supersymmetric coset model

It is known previously that the operator product expansion (OPE) between the first [Formula: see text] multiplet and itself contains the second [Formula: see text] multiplet in the supersymmetric coset model. In this paper, by using their realizations in terms of various fermions, we compute the four kinds of OPEs between the first and the second [Formula: see text] multiplets for fixed [Formula: see text] and [Formula: see text], where the group of the coset contains [Formula: see text]. By supersymmetrizing the above OPEs in [Formula: see text] superspace and using the various Jacobi identities between the currents, we determine the [Formula: see text] supersymmetric OPE between the first and the second [Formula: see text] multiplets completely. The right-hand side of this OPE contains the various [Formula: see text] multiplets: the [Formula: see text] singlet [Formula: see text] multiplets of superspin-[Formula: see text] and the [Formula: see text] triplet [Formula: see text] multiplets of superspin-[Formula: see text]. The [Formula: see text] superspace description and the decoupling of the spin-[Formula: see text] current of the [Formula: see text] superconformal algebra are also described.

Weakly nonlocal Poisson brackets: Tools, examples, computations

We implement an algorithm for the computation of Schouten bracket of weakly nonlocal Hamiltonian operators in three different computer algebra systems: Maple, Reduce and Mathematica. This class of Hamiltonian operators encompass almost all the examples coming from the theory of (1+1)-integrable evolutionary PDEs. Program summaryProgram Title:Jacobi (Maple), CDE module cde_weaklynl.red (Reduce, official distribution), nlPVA (Mathematica)CPC Library link to program files:https://doi.org/10.17632/synmrvr74g.1Developer's repository link:https://gdeq.org/Weakly_nonlocal_Poisson_bracketsLicensing provisions: BSD 2-clauseProgramming language: Maple, Reduce (Rlisp), MathematicaSupplementary material: Example program files in the three languages (Maple, Reduce, Mathematica)Nature of problem: Calculating the Jacobi identity for weakly nonlocal Poisson bracketsSolution method: Bringing the Jacobi identity to a canonical formAdditional comments including restrictions and unusual features: Use a 2020 Maple or Mathematica version, or a 2021 Reduce snapshot

Jacobi's Triple-Product Identity and an Associated Family of Theta-Function Identities

The main object of this paper is to present a family of $q$-series identities which involve some of the theta functions of Jacobi and Ramanujan. Each of these (presumably new) $q$-series identities reveals interesting relationships among three of the theta-type functions which stem from the celebrated Jacobi's triple-product identity in a remarkably simple way. The results presented in this paper are motivated essentially by a number of recent works dealing with the subject matter which is investigated herein.

Off-shell extended graphic rule and the expansion of Berends-Giele currents in Yang-Mills theory

Tree-level color-ordered Yang-Mills (YM) amplitudes can be decomposed in terms of (n − 2)! bi-scalar (BS) amplitudes, whose expansion coefficients form a basis of Bern-Carrasco-Johansson (BCJ) numerators. By the help of the recursive expansion of Einstein-Yang-Mills (EYM) amplitudes, the BCJ numerators are given by polynomial functions of Lorentz contractions which are conveniently described by graphic rule. In this work, we extend the expansion of YM amplitudes to off-shell level. We define different types of off-shell extended numerators that can be generated by graphs. By the use of these extended numerators, we propose a general decomposition formula of off-shell Berends-Giele currents in YM. This formula consists of three terms: (i). an effective current which is expanded as a combination of the Berends-Giele currents in BS theory (The expansion coefficients are one type of off-shell extended numerators) (ii). a term proportional to the total momentum of on-shell lines and (iii). a term expressed by the sum of lower point Berends-Giele currents in which some polarizations and momenta are replaced by vectors proportional to off-shell momenta appropriately. In the on-shell limit, the last two terms vanish while the decomposition of effective current precisely reproduces the decomposition of on-shell YM amplitudes with the expected coefficients (BCJ numerators in DDM basis). We further symmetrize these coefficients such that the Lie symmetries are satisfied. These symmetric BCJ numerators simultaneously satisfy the relabeling property of external lines and the algebraic properties (antisymmetry and Jacobi identity).

10D super-Yang-Mills scattering amplitudes from its pure spinor action

Using the pure spinor master action for 10D super-Yang-Mills in the gauge b0V = QΞ, tree-level scattering amplitudes are calculated through the perturbiner method, and shown to match those obtained from pure spinor CFT techniques. We find kinematic numerators made of nested b-ghost operators, and show that the Siegel gauge condition b0V = 0 gives rise to color-kinematics duality satisfying numerators whose Jacobi identity follows from the Jacobi identity of a kinematic algebra.

The replicator dynamics of zero-sum games arise from a novel poisson algebra

We show that the replicator dynamics for zero-sum games arises as a result of a non-canonical bracket that is a hybrid between a Poisson Bracket and a Nambu Bracket. The resulting non-canonical bracket is parameterized both the by the skew-symmetric payoff matrix and a mediating function. The mediating function is only sometimes a conserved quantity, but plays a critical role in the determination of the dynamics. As a by-product, we show that for the replicator dynamics this function arises in the definition of a natural metric on which phase flow-volume is preserved. Additionally, we show that the non-canonical bracket satisfies all the same identities as the Poisson bracket except for the Jacobi identity (JI), which is satisfied for special cases of the mediating function. In particular, the mediating function that gives rise to the replicator dynamics yields a bracket that satisfies JI. This neatly explains why the mediating function allows us to derive a metric on which phase flow is conserved and suggests a natural geometry for zero-sum games that extends the Symplectic geometry of the Poisson bracket and potentially an alternate approach to quantizing evolutionary games.

Hamiltonian structure of the guiding-center Vlasov–Maxwell equations

The Hamiltonian structure of the guiding-center Vlasov-Maxwell equations is presented in terms of a Hamiltonian functional and a guiding-center Vlasov-Maxwell bracket. The bracket, which is shown to satisfy the Jacobi identity exactly, is used to show that the guiding-center momentum and angular-momentum conservation laws can also be expressed in Hamiltonian form.

Next-to-MHV Yang-Mills kinematic algebra

Kinematic numerators of Yang-Mills scattering amplitudes possess a rich Lie algebraic structure that suggest the existence of a hidden infinite-dimensional kinematic algebra. Explicitly realizing such a kinematic algebra is a longstanding open problem that only has had partial success for simple helicity sectors. In past work, we introduced a framework using tensor currents and fusion rules to generate BCJ numerators of a special subsector of NMHV amplitudes in Yang-Mills theory. Here we enlarge the scope and explicitly realize a kinematic algebra for all NMHV amplitudes. Master numerators are obtained directly from the algebraic rules and through commutators and kinematic Jacobi identities other numerators can be generated. Inspecting the output of the algebra, we conjecture a closed-form expression for the master BCJ numerator up to any multiplicity. We also introduce a new method, based on group algebra of the permutation group, to solve for the generalized gauge freedom of BCJ numerators. It uses the recently introduced binary BCJ relations to provide a complete set of NMHV kinematic numerators that consist of pure gauge.

Quantum-classical dynamical brackets

We study the problem of constructing a general hybrid quantum-classical bracket from a partial classical limit of a full quantum bracket. Introducing a hybrid composition product, we show that such a bracket is the commutator of that product. From this we see that the hybrid bracket will obey the Jacobi identity and the Leibniz rule provided the composition product is associative. This suggests that the set of hybrid variables belonging to an associative subalgebra with the composition product will have consistent quantum-classical dynamics. This restricts the class of allowed quantum-classical interaction Hamiltonians. Furthermore, we show that pure quantum or classical variables can interact in a consistent framework, unaffected by no-go theorems in the literature or the restrictions for hybrid variables. In the proposed scheme, quantum backreaction appears as quantum-dependent terms in the classical equations of motion.

Hamiltonian fluid reduction of the 1.5D Vlasov–Maxwell equations

We consider the Vlasov–Maxwell equations with one spatial direction and two momenta. By solving the Jacobi identity, we derive reduced Hamiltonian fluid models for the density, the fluid momenta, and the second order moments, related to the pressure tensor. We also provide the Casimir invariants of the reduced Poisson bracket. We show that the linearization of the equations of motion around homogeneous equilibria reproduces some essential features of the kinetic model namely, the Weibel instability.

Kinematic numerators from the worldsheet: cubic trees from labelled trees

In this note we revisit the problem of explicitly computing tree-level scattering amplitudes in various theories in any dimension from worldsheet formulas. The latter are known to produce cubic-tree expansion of tree amplitudes with kinematic numerators automatically satisfying Jacobi-identities, once any half-integrand on the worldsheet is reduced to logarithmic functions. We review a natural class of worldsheet functions called “Cayley functions”, which are in one-to-one correspondence with labelled trees, and natural expansions of known half-integrands onto them with coefficients that are particularly compact building blocks of kinematic numerators. We present a general formula expressing kinematic numerators of all cubic trees as linear combinations of coefficients of labelled trees, which satisfy Jacobi identities by construction and include the usual combinations in terms of master numerators as a special case. Our results provide an efficient algorithm, which is implemented in a Mathematica package, for computing all tree amplitudes in theories including non-linear sigma model, special Galileon, Yang-Mills-scalar, Einstein-Yang-Mills and Dirac-Born-Infeld.

Stability and Hamiltonian BRST-invariant deformations in Podolsky's generalized electrodynamics

We study the problem of stability in Podolsky's generalized electrodynamics by constructing a series of 2-parametric bounded conserved quantities. In this way, we show that the 00-component of the energy-momentum tensors could be positive definite and therefore the higher derivative system is considered to be stable. Afterwards, we derive the consistent interactions in Podolsky's theory within the framework of Hamiltonian BRST-invariant deformation procedure. The key ingredients in our analysis are the local BRST-cohomology which plays a crucial role in the determination of the first-order deformation as well as the Jacobi identity that will greatly simplify the calculations for us. We assert that in our discussions, the second-order deformation and the other higher order deformations of the BRST charge naturally turn out to be zero while the third-order as well as the corresponding higher order BRST-invariant Hamiltonian deformations also vanish completely. Moreover, we evaluate the path integral of the higher derivative constrained system before and after deformation process following the standard BRST quantization method with appropriate gauge-fixing fermions.

Nambu dynamics and its noncanonical Hamiltonian representation in many degrees of freedom systems

Abstract Nambu dynamics is a generalized Hamiltonian dynamics of more than two variables, whose time evolutions are given by the Nambu bracket, a generalization of the canonical Poisson bracket. Nambu dynamics can always be represented in the form of noncanonical Hamiltonian dynamics by defining the noncanonical Poisson bracket by means of the Nambu bracket. For the time evolution to be consistent, the Nambu bracket must satisfy the fundamental identity, while the noncanonical Poisson bracket must satisfy the Jacobi identity. However, in many degrees of freedom systems, it is well known that the fundamental identity does not hold. In the present paper we show that, even if the fundamental identity is violated, the Jacobi identity for the corresponding noncanonical Hamiltonian dynamics could hold. As an example we evaluate these identities for a semiclassical system of two coupled oscillators.

D>2 stress-tensor operator product expansion near a line

We study the $TT$ OPE in $d>2$ CFTs whose bulk dual is Einstein gravity. Directly from the $TT$ OPE, we obtain, in a certain null-like limit, an algebraic structure consistent with the Jacobi identity: $[{\cal L}_m, {\cal L}_n]= (m-n) {\cal L}_{m+n}+ C m (m^2-1) \delta_{m+n,0}$. The dimensionless constant $C$ is proportional to the central charge $C_T$. Transverse integrals in the definition of ${\cal L}_m$ play a crucial role. We comment on the corresponding limiting procedure and point out a curiosity related to the central term. A connection between the $d>2$ near-lightcone stress-tensor conformal block and the $d=2$ $\cal W$-algebra is observed. This note is motivated by the search for a field-theoretic derivation of $d>2$ correlators in strong coupling critical phenomena.

Classification of minimal Z2×Z2-graded Lie (super)algebras and some applications

This paper presents the classification over the fields of real and complex numbers, of the minimal Z2×Z2-graded Lie algebras and Lie superalgebras spanned by four generators and with no empty graded sector. The inequivalent graded Lie (super)algebras are obtained by solving the constraints imposed by the respective graded Jacobi identities. A motivation for this mathematical result is to systematically investigate the properties of dynamical systems invariant under graded (super)algebras. Recent works only paid attention to the special case of the one-dimensional Z2×Z2-graded Poincaré superalgebra. As applications, we are able to extend certain constructions originally introduced for this special superalgebra to other listed Z2×Z2-graded (super)algebras. We mention, in particular, the notion of Z2×Z2-graded superspace and of invariant dynamical systems (both classical worldline sigma models and quantum Hamiltonians). As a further by-product, we point out that, contrary to Z2×Z2-graded superalgebras, a theory invariant under a Z2×Z2-graded algebra implies the presence of ordinary bosons and three different types of exotic bosons, with exotic bosons of different types anticommuting among themselves.

Generalization of Hamiltonian mechanics to a three-dimensional phase space

Abstract Classical Hamiltonian mechanics is realized by the action of a Poisson bracket on a Hamiltonian function. The Hamiltonian function is a constant of motion (the energy) of the system. The properties of the Poisson bracket are encapsulated in the symplectic $2$-form, a closed second-order differential form. Due to closure, the symplectic $2$-form is preserved by the Hamiltonian flow, and it assigns an invariant (Liouville) measure on the phase space through the Lie–Darboux theorem. In this paper we propose a generalization of classical Hamiltonian mechanics to a three-dimensional phase space: the classical Poisson bracket is replaced with a generalized Poisson bracket acting on a pair of Hamiltonian functions, while the symplectic $2$-form is replaced by a symplectic $3$-form. We show that, using the closure of the symplectic $3$-form, a result analogous to the classical Lie–Darboux theorem holds: locally, there exist smooth coordinates such that the components of the symplectic $3$-form are constants, and the phase space is endowed with a preserved volume element. Furthermore, as in the classical theory, the Jacobi identity for the generalized Poisson bracket mathematically expresses the closure of the associated symplectic form. As a consequence, constant skew-symmetric third-order contravariant tensors always define generalized Poisson brackets. This is in contrast with generalizations of Hamiltonian mechanics postulating the fundamental identity as replacement for the Jacobi identity. In particular, we find that the fundamental identity represents a stronger requirement than the closure of the symplectic $3$-form.