arXiv research

We derive Wahlquist - Estabrook forms of the covering of Plebanski's second heavenly equation from Maurer - Cartan forms of its symmetry pseudo-group.

Local generalization of frame bundles using a weakened Maurer-Cartan equation.

Generalizes Hodge correlators using quantum master equation concepts.

New geometric proofs and interpretations of scattering diagrams and theta functions.

On a cotangent bundle $T\sp*G$ of a Lie group $G$ one can describe the standard Liouville form $θ$ and the symplectic form $d θ$ in terms of the right Maurer Cartan form and the left moment mapping (of the right action of $G$ on itself), and also in terms of the left Maurer-Cartan form and the right moment mapping, and…

The paper is devoted to the study of BRST charge in perturbed two dimensional conformal field theory. The main goal is to write the operator equation expressing the conservation law of BRST charge in perturbed theory in terms of purely algebraic operations on the corresponding operator algebra, which are defined via th…

We show that a well-known result on solutions of the Maurer--Cartan equation extends to arbitrary (inhomogeneous) odd forms: any such form with values in a Lie superalgebra satisfying $dø+ø^2=0$ is gauge-equivalent to a constant, $ø=gCg^{-1}-dg\,g^{-1}\,.$ This follows from a non-Abelian version of a chain homotopy f…

We consider the equations, arising as the conformal invariance conditions of the perturbed curved beta-gamma system. These equations have the physical meaning of Einstein equations with a B-field and a dilaton on a hermitian manifold, where the B-field 2-form is imaginary and proportional to the canonical form associat…

The paper extends Cartan development to infinite dimensional Lie groups.

Let G be a compact, connected Lie group, acting smoothly on a manifold M. Goresky-Kottwitz-MacPherson described a small Cartan model for the equivariant cohomology of M, quasi-isomorphic to the standard Cartan complex of equivariant differential forms. In this paper, we construct an explicit cochain map from the small …

Integrates L-infinity algebroids to L-infinity groupoids.

Method solves Maurer-Cartan equation using Jacobi identity.

New tilings of the 2-sphere from convex polyhedra in 3-sphere.

New integrable systems are created using matrix operations and Lie algebra elements.

Constructs Poisson structures on gauge orbits of Maurer-Cartan elements.

Develops infinite-dimensional Kempf-Ness theory for complexification-free groups.

We extend to a scheme-theoretic context the notion of a combinatorial differential form, due to A.Kock in the framework of synthetic differential geometry. We show that group-valued combinatorial forms on a scheme may be identified, under very general hypotheses, with traditional Lie algebra-valued differential forms, …

The semi-classical data attached to stacks of algebroids in the sense of Kashiwara and Kontsevich are Maurer-Cartan elements on complex manifolds, which we call extended Poisson structures as they generalize holomorphic Poisson structures. A canonical Lie algebroid is associated to each Maurer-Cartan element. We study …

A Lie-infinity morphism relates derived Leibniz algebras from Maurer-Cartan elements.

We study how to generate new Lie algebras $\mathcal{G}(N_0,..., N_p,...,N_n)$ from a given one $\mathcal{G}$. The (order by order) method consists in expanding its Maurer-Cartan one-forms in powers of a real parameter $λ$ which rescales the coordinates of the Lie (super)group $G$, $g^{i_p} \to λ^p g^{i_p}$, in a way su…

Describes higher gauge theory on string 2-group principal 2-bundles.

The paper develops a deformation theory for Dolbeault cohomology classes.

This note elaborates on Th. Voronov's construction [math/0304038,math/0412202] of $L_\infty$-structures via higher derived brackets with a Maurer-Cartan element. It is shown that gauge equivalent Maurer-Cartan elements induce $L_\infty$-isomorphic structures. Applications in symplectic, Poisson and Dirac geometry are d…

Develops a new method for solving equivalence problems in pseudo-groups.

The reduction of biharmonic maps equation in terms of the Maurer-Cartan form for all smooth map of any compact Riemannian manifolds into a compact Lie group with bi-invariant Riemannian metric is obtained. By this formula, all the biharmonic curves into a compact Lie group and all biharmonic maps from a 2-dimensional o…

Higher homotopy generalizations of Lie-Rinehart algebras, Gerstenhaber-, and Batalin-Vilkovisky algebras are explored. These are defined in terms of various antisymmetric bilinear operations satisfying weakened versions of the Jacobi identity, as well as in terms of operations involving more than two variables of the L…

Relates field theory algebras to sigma model geometric structures.

We consider the local equivalence problem for the class of linear second order hyperbolic equations in two independent variables under an action of the pseudo-group of contact transformations. E. Cartan's method is used for finding the Maurer - Cartan forms for symmetry groups of equations from the class and computing …

Characterizes superforms for a sigma model and describes an integrable system.

Homotopy equivalence between formalities with different covariant derivatives.

The techniques used in this paper are based on the exterior calculus of Maurer-Cartan forms, and Weingarten surfaces are used to illustrate the methods that apply to quadratic exterior equations with constant coefficients. Isothermic {\it surfaces of constant astigmatism} (non-linear Weingarten surfaces whose differenc…

In this paper, the description of biharmonic map equation in terms of the Maurer-Cartan form for all smooth map of a compact Riemannian manifold into a Riemannian symmetric space $(G/K,h)$ induced from the bi-invariant Riemannian metric $h$ on $G$ is obtained. By this formula, all biharmonic curves into symmetric space…

This preliminary report studies immersed surfaces of constant mean curvature in $H^3$ through their {\it adjusted Gauss maps} (as harmonic maps in $S^2$) and their {\it adjusted frames} in SU(2). Lawson's correspondence between Euclidean CMC surfaces and their hyperbolic cousins is interpreted here under a different pe…

Study on deforming complex manifolds and Higgs bundles.

New geometric properties discovered in a specific Frobenius manifold.

Defines and characterizes operators on Lie ∞-algebras with respect to actions.

We show how the well-known classical field equations as Einstein and Yang-Mills ones, which arise as the conformal invariance conditions of certain two-dimensional theories, expanded up to the second order in the formal parameter, can be reformulated as Generalized/formal Maurer-Cartan equations (GMC), where the differ…

Study finite deformations from heterotic superpotential, leading to new complex effective action.

Develops Morse theory for commuting gradient-like vector fields.

Kock [Bull. Austral. Math. Soc., 25 (1982), 357-386] has considered differential forms with values in a group in a context where neighborhood relations are available. By doing so, he has made it clear where the so-called Maurer-Cartan formula should come from. In this paper, while we retain the classical definition of …

The paper studies holomorphic Poisson manifolds and their deformations under specific cohomology conditions.

We show that a flat principal bundle with compact connected structure group and its adjoint bundles of Lie groups have the same cohomology as the trivial bundle, which is done by proving they satisfy the condition for the Leray-Hirsch theorem. This information has been used to construct a cohomology class of the adjoin…

New algebraic structures for Lie 2-algebroids and their connections.

We extend the classical characterization of a finite-dimensional Lie algebra g in terms of its Maurer-Cartan algebra-the familiar differential graded algebra of alternating forms on g with values in the ground field, endowed with the standard Lie algebra cohomology operator-to sh Lie-Rinehart algebras. To this end, we …

The paper defines left-symmetric bialgebroids and their Manin triples.

Let $ω_\mathfrak{g}$ be a Lie algebra valued differential $1$-form on a manifold $M$ satisfying the structure equations $d ω_\mathfrak{g} + \frac{1}{2} ω_\mathfrak{g}\wedge ω_\mathfrak{g}=0$ where $\mathfrak{g}$ is solvable. We show that the problem of finding a smooth map $ρ:M\to G$, where $G$ is an $n$-dimensional so…

We investigate a class of Leibniz algebroids which are invariant under diffeomorphisms and symmetries involving collections of closed forms. Under appropriate assumptions we arrive at a classification which in particular gives a construction starting from graded Lie algebras. In this case the Leibniz bracket is a deriv…

The paper explores flat extensions of connections and their relation to Chern-Simons invariants.