Free Differential Algebras Research Articles

Construction of free quasi-idempotent differential Rota-Baxter algebras by Gröbner-Shirshov bases

Differential operators and integral operators are linked together by the first fundamental theorem of calculus. Based on this principle, the notion of a differential Rota-Baxter algebra was proposed by Guo and Keigher. Recently, the subject has attracted more attention since it is associated with many areas in mathematics, such as integro-differential algebras. This paper considers differential Rota-Baxter algebras in the quasi-idempotent operator context. We establish a Gröbner-Shirshov basis for free commutative quasi-idempotent differential algebras (resp. Rota-Baxter algebras, resp. differential Rota-Baxter algebras). This provides a linear basis of a free object in each of the three corresponding categories by the Composition-Diamond lemma. Communicated by P. Kolesnikov

Supergravities and branes from Hilbert-Poincaré series

The Molien-Weyl integral formula and the Hilbert-Poincaré series have proven to be powerful mathematical tools in relation to gauge theories, allowing to count the number of gauge invariant operators. In this paper we show that these methods can also be employed to construct Free Differential Algebras and, therefore, reproduce the associated pure supergravity spectrum and nonperturbative objects. Indeed, given a set of fields, the Hilbert-Poincaré series allows to compute all possible invariants and consequently derive the cohomology structure.

Novel Free Differential Algebras for Supergravity

We develop the theory of Free Integro-Differential Algebras (FIDA) extending the powerful technique of Free Differential Algebras constructed by D. Sullivan. We extend the analysis beyond the superforms to integral- and pseudo-forms used in supergeometry. It is shown that there are novel structures that might open the road to a deeper understanding of the geometry of supergravity. We apply the technique to some models as an illustration and we provide a complete analysis for D = 11 supergravity. There, it is shown how the Hodge star operator for supermanifolds can be used to analyze the set of cocycles and to build the corresponding FIDA. A new integral form emerges which plays the role of the truly dual to 4-form F(4) and we propose a new variational principle on supermanifolds.

Chiral higher spin gravity in (A)dS4 and secrets of Chern–Simons matter theories

Chiral Higher Spin Gravity with cosmological constant is constructed as a Free Differential Algebra, i.e. at the level of equations of motion, which is a smooth deformation of its flat space cousin [34]. Chiral Higher Spin Gravity is a unique class of local higher spin theories; its very existence implies that there is a closed and, most likely, integrable sub-sector of Chern–Simons Matter Theories, which has important consequences both for the theories themselves and for three-dimensional bosonization duality.

Minimal models of field theories: SDYM and SDGR

There exists a natural L∞-algebra or Q-manifold that can be associated to any (gauge) field theory. Perturbatively, it can be obtained by reducing the L∞-algebra behind the jet space BV-BRST formulation to its minimal model. We explicitly construct the minimal models of self-dual Yang-Mills and self-dual gravity theories, which also represents their equations of motion as Free Differential Algebras. The minimal model regains all relevant information about the field theory, e.g. actions, charges, anomalies, can be understood in terms of the corresponding Q-cohomology.

Minimal models of field theories: Chiral higher spin gravity

There exists a unique class of local higher spin gravities with propagating massless fields in $4d$---chiral higher spin gravity. Originally, it was formulated in the light-cone gauge. We construct a covariant form of this theory as a free differential algebra up to the next-to-leading order, i.e., at the level of equations of motion. It also contains the recently discovered covariant forms of the higher spin extensions of self-dual Yang-Mills and self-dual gravity, as well as self-dual Yang-Mills and self-dual gravity themselves. From the mathematical viewpoint the result is equivalent to taking the minimal model (in the sense of ${L}_{\ensuremath{\infty}}$-algebras) of the jet-space extension of the BV-BRST formulation of chiral higher spin gravity, thereby, containing also information about (presymplectic AKSZ) action, counterterms, anomalies, etc.

Differential graded algebras, Steenrod cup-one products, binomial operations, and Massey products

Motivated by the construction of Steenrod cup-i products in the singular cochain algebra of a space and in the algebra of non-commutative differential forms, we define a category of binomial cup-one differential graded algebras over the integers and over prime fields of positive characteristic. The Steenrod and Hirsch identities bind the cup-product, the cup-one product, and the differential in a package that we further enhance with a binomial ring structure arising from a ring of integer-valued rational polynomials. This structure allows us to define the free binomial cup-one differential graded algebra generated by a set and derive its basic properties. It also provides the context for defining restricted triple Massey products, which have a smaller indeterminacy than the classical ones, and hence, give stronger homotopy type invariants.

Gauge-invariant theories and higher-degree forms

A free differential algebra is generalization of a Lie algebra in which the mathematical structure is extended by including of new Maurer-Cartan equations for higher-degree differential forms. In this article, we propose a generalization of the Chern-Weil theorem for free differential algebras containing only one p-form extension. This is achieved through a generalization of the covariant derivative, leading to an extension of the standard formula for Chern-Simons and transgression forms. We also study the possible existence of anomalies originated on this kind of structure. Some properties and particular cases are analyzed.

Simplicial gravity with coordinates

We present a formulation of Regge calculus where arbitrary coordinates are associated to each vertex of the simplicial complex and the fundamental degrees of freedom are given by the metric g μν (α) on each simplex α. The lengths of the edges, which are the usual degrees of freedom of Regge calculus, are thus determined and are left invariant under arbitrary transformations of the discrete set of coordinates, provided the metric transforms accordingly. Invariance under coordinate transformations entails tensor calculus and our formulation then follows closely the usual formalism of the continuum theory. This includes a definition of partial derivative which stems from a generalization to simplicial lattices of the symmetric finite difference operator on a cubic lattice. The definitions of parallel transport, Christoffel symbol, covariant derivatives and Riemann curvature tensor follow in a rather natural way establishing a kind of dictionary between continuum and simplicial lattice quantities. In this correspondence Einstein action becomes Regge action with the deficit angle θ replaced by sin θ. The correspondence with the continuum theory can be extended to actions with higher powers of the curvature tensor, to the vielbein formalism and to the coupling of gravity with matter fields (scalars, fermionic fields including spin 3/2 fields and gauge fields) which are then determined unambiguously and discussed in the paper. An action on the simplicial lattice for N = 1 supergravity in four dimensions is derived in this context. Another relavant result is that Yang–Mills actions on a simplicial lattice consist, even in absence of gravity, of two plaquettes terms, unlike the one plaquette Wilson action on the hypercubic lattice. An attempt is also made to formulate a discrete differential calculus to include differential forms of higher order and the gauging of free differential algebras in this scheme. However this leads to form products that do not satisfy associativity and distributive law with respect to the d operator. A proper formulation of theories that contain higher order differential forms in the context of Regge calculus is then still lacking.

Construction of free differential algebras by extending Gröbner-Shirshov bases

As a fundamental notion, the free differential algebra on a set is concretely constructed as the polynomial algebra on the differential variables. Such a construction is not known for the more general notion of the free differential algebra on an algebra, from the left adjoint functor of the forgetful functor from differential algebras to algebras, instead of sets. In this paper we show that a generator-relation presentation of a base algebra can be extended to the free differential algebra on this base algebra. More precisely, a Gröbner-Shirshov basis property of the base algebra can be extended to the free differential algebra on this base algebra, allowing a Poincaré-Birkhoff-Witt type basis for these more general free differential algebras. Examples are provided as illustrations.

Lie models of simplicial sets and representability of the Quillen functor

Extending the model of the interval, we explicitly define for each n ≥ 0 a free complete differential graded Lie algebra \(\mathfrak{L}_n\) generated by the simplices of Δn, with desuspended degrees, in which the vertices are Maurer-Cartan elements and the differential extends the simplicial chain complex of the standard n-simplex. The family \(\{\mathfrak{L{_\bullet}}\}_{n\geq0}\) is endowed with a cosimplicial differential graded Lie algebra structure which we use to construct two adjoint functors $$Simpset\underset{{\left\langle \cdot \right\rangle }}{\overset{\mathfrak{L}}{\longleftrightarrow}}DGL$$ given by \(\langle{L}\rangle_\bullet={\rm{DGL}}(\mathfrak{L}_\bullet,L)\) and \(\mathfrak{L}(K)=\lim_{\rightarrow K}\mathfrak{L_\bullet}\). This new tool lets us extend the Quillen rational homotopy theory approach to any simplicial set K whose path components are not necessarily simply connected.We prove that \(\mathfrak{L}(K)\) contains a model of each component of K. When K is a 1-connected finite simplicial complex, the Quillen model of K can be extracted from \(\mathfrak{L}(K)\). When K is connected then, for a perturbed differential ϑa, \(H_0(\mathfrak{L}(K),\partial_a)\) is the Malcev Lie completion of π1(K). Analogous results are obtained for the realization 〈L〉 of any complete DGL.

Covariant Hamiltonian for gravity coupled top-forms

We review the covariant canonical formalism initiated by D'Adda, Nelson and Regge in 1985, and extend it to include a definition of form-Poisson brackets (FPB) for geometric theories coupled to $p$-forms, gauging free differential algebras. The form-Legendre transformation and the form-Hamilton equations are derived from a $d$-form Lagrangian with $p$-form dynamical fields $\phi$. Momenta are defined as derivatives of the Lagrangian with respect to the "velocities" $d\phi$ and no preferred time direction is used. Action invariance under infinitesimal form-canonical transformations can be studied in this framework, and a generalized Noether theorem is derived, both for global and local symmetries. We apply the formalism to vielbein gravity in $d=3$ and $d=4$. In the $d=3$ theory we can define form-Dirac brackets, and use an algorithmic procedure to construct the canonical generators for local Lorentz rotations and diffeomorphisms. In $d=4$ the canonical analysis is carried out using FPB, since the definition of form-Dirac brackets is problematic. Lorentz generators are constructed, while diffeomorphisms are generated by the Lie derivative. A "doubly covariant" hamiltonian formalism is presented, allowing to maintain manifest Lorentz covariance at every stage of the Legendre transformation. The idea is to take curvatures as "velocities" in the definition of momenta.

Hidden role of Maxwell superalgebras in the free differential algebras of D\xa0=\xa04 and D\xa0=\xa011 supergravity

The purpose of this paper is to show that the so-called Maxwell superalgebra in four dimensions, which naturally involves the presence of a nilpotent fermionic generator, can be interpreted as a hidden superalgebra underlying mathcal {N}=1, {hbox {D}}=4 supergravity extended to include a 2-form gauge potential associated to a 2-index antisymmetric tensor. In this scenario, the theory is appropriately discussed in the context of Free Differential Algebras (an extension of the Maurer–Cartan equations to involve higher-degree differential forms). The study is then extended to the Free Differential Algebra describing hbox {D} = 11 supergravity, showing that, also in this case, there exists a super-Maxwell algebra underlying the theory. The same extra spinors dual to the nilpotent fermionic generators whose presence is crucial for writing a supersymmetric extension of the Maxwell algebras, both in the hbox {D} = 4 and in the hbox {D} = 11 case, turn out to be fundamental ingredients also to reproduce the hbox {D} = 4 and hbox {D} = 11 Free Differential Algebras on ordinary superspace, whose basis is given by the supervielbein. The analysis of the gauge structure of the supersymmetric Free Differential Algebras is carried on taking into account the gauge transformations from the hidden supergroup-manifold associated with the Maxwell superalgebras.

Extended gauge theory and gauged free differential algebras

Recently, Antoniadis, Konitopoulos and Savvidy introduced, in the context of the so-called extended gauge theory, a procedure to construct background-free gauge invariants, using non-abelian gauge potentials described by higher degree forms.In this article it is shown that the extended invariants found by Antoniadis, Konitopoulos and Savvidy can be constructed from an algebraic structure known as free differential algebra. In other words, we show that the above mentioned non-abelian gauge theory, where the gauge fields are described by p-forms with p≥2, can be obtained by gauging free differential algebras.

More on the hidden symmetries of 11D supergravity

In this paper we clarify the relations occurring among the osp(1|32) algebra, the M-algebra and the hidden superalgebra underlying the Free Differential Algebra of D=11 supergravity (to which we will refer as DF-algebra) that was introduced in the literature by D'Auria and Frè in 1981 and is actually a (Lorentz valued) central extension of the M-algebra including a nilpotent spinor generator, Q′. We focus in particular on the 4-form cohomology in 11D superspace of the supergravity theory, strictly related to the presence in the theory of a 3-form A(3). Once formulated in terms of its hidden superalgebra of 1-forms, we find that A(3) can be decomposed into the sum of two parts having different group-theoretical meaning: One of them allows to reproduce the FDA of the 11D Supergravity due to non-trivial contributions to the 4-form cohomology in superspace, while the second one does not contribute to the 4-form cohomology, being a closed 3-form in the vacuum, defining however a one parameter family of trilinear forms invariant under a symmetry algebra related to osp(1|32) by redefining the spin connection and adding a new Maurer–Cartan equation.We further discuss about the crucial role played by the 1-form spinor η (dual to the nilpotent generator Q′) for the 4-form cohomology of the eleven dimensional theory on superspace.

On the hidden maxwell superalgebra underlying supergravity

AbstractIn this work, we expand the hidden ‐Lorentz superalgebra underlying supergravity, reaching a (hidden) Maxwell superalgebra. The latter can be viewed as an extension involving cosmological constant of the superalgebra underlying supergravity in flat spacetime. We write the Maurer‐Cartan equations in this context and we find some interesting extensions of the antisymmetric 3‐form A(3) appearing in the Free Differential Algebra in Minkowski space. The structure of Free Differential Algebras is obtained by considering the zero curvature equations. We write the parametrization of A(3) in terms of 1‐forms and we rend the topological features of its extensions manifest. We interestingly find out that the structure of these extensions, and consequently the structure of the corresponding boundary contribution , strongly depends on the form of the extra fermionic generator appearing in the hidden Maxwell superalgebra. The model we develop in this work is defined in an enlarged superspace with respect to the ordinary one, and the extra bosonic and fermionic 1‐forms required for the closure of the hidden Maxwell superalgebra must be considered as physical fields in this enlarged superspace.

Hidden gauge structure of supersymmetric free differential algebras

The aim of this paper is to clarify the role of the nilpotent fermionic generator Q' introduced in Ref. [3] and appearing in the hidden supergroup underlying the free differential algebra (FDA) of D=11 supergravity. We give a physical explanation of its role by looking at the gauge properties of the theory. We find that its presence is necessary, in order that the extra 1-forms of the hidden supergroup give rise to the correct gauge transformations of the p-forms of the FDA. This interpretation is actually valid for any supergravity containing antisymmetric tensor fields, and any supersymmetric FDA can always be traded for a hidden Lie superalgebra containing extra fermionic nilpotent generators. As an interesting example we construct the hidden superalgebra associated with the FDA of N=2, D=7 supergravity. In this case we are able to parametrize the mutually non local 2- and 3-form B^(2) and B^(3) in terms of hidden 1-forms and find that supersymmetry and gauge invariance require in general the presence of two nilpotent fermionic generators in the hidden algebra. We propose that our approach, where all the invariances of the FDA are expressed as Lie derivatives of the $p$-forms in the hidden supergroup manifold, could be an appropriate framework to discuss theories defined in enlarged versions of superspace recently considered in the literature, such us double field theory and its generalizations.

Minimal D = 7 supergravity and the supersymmetry of Arnold-Beltrami flux branes

In this paper we study some properties of the newly found Arnold-Beltrami flux-brane solutions to the minimal $D=7$ supergravity. To this end we first single out the appropriate Free Differential Algebra containing both a gauge $3$-form $\mathbf{B}^{[3]}$ and a gauge $2$-form $\mathbf{B}^{[2]}$: then we present the complete rheonomic parametrization of all the generalized curvatures. This allows us to identify two-brane configurations with Arnold-Beltrami fluxes in the transverse space with exact solutions of supergravity and to analyze the Killing spinor equation in their background. We find that there is no preserved supersymmetry if there are no additional translational Killing vectors. Guided by this principle we explicitly construct Arnold-Beltrami flux two-branes that preserve $0$, $1/8$ and $1/4$ of the original supersymmetry. Two-branes without fluxes are instead BPS states and preserve $1/2$ supersymmetry. For each two-brane solution we carefully study its discrete symmetry that is always given by some appropriate crystallographic group $\Gamma$. Such symmetry groups $\Gamma$ are transmitted to the $D=3$ gauge theories on the brane world--volume that occur in the gauge/gravity correspondence. Furthermore we illustrate the intriguing relation between gauge fluxes in two-brane solutions and hyperinstantons in $D=4$ topological sigma-models.

Free integro-differential algebras and Gröbner–Shirshov bases

The notion of commutative integro-differential algebra was introduced for the algebraic study of boundary problems for linear ordinary differential equations. Its noncommutative analog achieves a similar purpose for linear systems of such equations. In both cases, free objects are crucial for analyzing the underlying algebraic structures, e.g. of the (matrix) functions.In this paper we apply the method of Gröbner–Shirshov bases to construct the free (noncommutative) integro-differential algebra on a set. The construction is from the free Rota–Baxter algebra on the free differential algebra on the set modulo the differential Rota–Baxter ideal generated by the noncommutative integration by parts formula. In order to obtain a canonical basis for this quotient, we first reduce to the case when the set is finite. Then in order to obtain the monomial order needed for the Composition-Diamond lemma, we consider the free Rota–Baxter algebra on the truncated free differential algebra. A Composition-Diamond lemma is proved in this context, and a Gröbner–Shirshov basis is found for the corresponding differential Rota–Baxter ideal.

Higher form gauge fields and their nonassociative symmetry algebras

We show that geometric theories with $p$-form gauge fields have a nonassociative symmetry structure, extending an underlying Lie algebra. This nonassociativity is controlled by the same Chevalley-Eilenberg cohomology that classifies free differential algebras, $p$-form generalizations of Cartan-Maurer equations. A possible relation with flux backgrounds of closed string theory is pointed out.