Maurer-Cartan Equations Research Articles

Maurer-Cartan equations and black hole superpotential inN=8supergravity

We retrieve the non-BPS extremal black hole superpotential of N=8, d=4 supergravity by using the Maurer-Cartan equations of the symmetric space E_7(7)/SU(8). This superpotential was recently obtained with different 3- and 4-dimensional techniques. The present derivation is independent on the reduction to d=3.

The conformal current algebra on supergroups with applications to the spectrum and integrability

We compute the algebra of left and right currents for a principal chiral model with arbitrary Wess-Zumino term on supergroups with zero Killing form. We define primary fields for the current algebra that match the affine primaries at the Wess-Zumino-Witten points. The Maurer-Cartan equation together with current conservation tightly constrain the current-current and current-primary operator product expansions. The Hilbert space of the theory is generated by acting with the currents on primary fields. We compute the conformal dimensions of a subset of these states in the large radius limit. The current algebra is shown to be consistent with the quantum integrability of these models to several orders in perturbation theory.

A period map for generalized deformations

For every compact Kähler manifold we give a canonical extension of Griffith’s period map to generalized deformations, intended as solutions of Maurer–Cartan equation in the algebra of polyvector fields. Our construction involves the notion of Cartan homotopy and a canonical L_∞ -structure on mapping cones of morphisms of differential graded Lie algebras.

Generating higher-order Lie algebras by expanding Maurer–Cartan forms

By means of a generalization of the Maurer–Cartan expansion method, we construct a procedure to obtain expanded higher-order Lie algebras. The expanded higher-order Maurer–Cartan equations for the case G=V0⊕V1 are found. A dual formulation for the S-expansion multialgebra procedure is also considered. The expanded higher-order Maurer–Cartan equations are recovered from S-expansion formalism by choosing a special semigroup. This dual method could be useful in finding a generalization to the case of a generalized free differential algebra, which may be relevant for physical applications in, e.g., higher-spin gauge theories.

A connection between twistors and superstring sigma models on coset superspaces

We consider superstring sigma models that are based on coset superspaces G/H in which H arises as the fixed point set of an order-4 automorphism of G. We show by means of twistor theory that the corresponding first-order system, consisting of the Maurer-Cartan equations and the equations of motion, arises from a dimensional reduction of some generalised self-dual Yang-Mills equations in eight dimensions. Such a relationship might help shed light on the explicit construction of solutions to the superstring equations including their hidden symmetry structures and thus on the properties of their gauge theory duals.

β–γ systems and the deformations of the BRST operator

We describe the relation between simple logarithmic CFTs associated with closed and open strings, and their ‘infinite metric’ limits, corresponding to the β–γ systems. This relation is studied on the level of the BRST complex: we show that the consideration of metric as a perturbation leads to a certain deformation of the algebraic operations of the Lian–Zuckerman type on the vertex algebra, associated with the β–γ systems. The Maurer–Cartan equations corresponding to this deformed structure in the quasi-classical approximation lead to the nonlinear field equations. As an explicit example, we demonstrate that using this construction, Yang–Mills equations can be derived. This gives rise to a nontrivial relation between the Courant–Dorfman algebroid and homotopy algebras emerging from the gauge theory. We also discuss a possible algebraic approach to the study of beta-functions in sigma-models.

Dual formulation of the Lie algebra S-expansion procedure

The expansion of a Lie algebra entails finding a new bigger algebra G through a series of well-defined steps from an original Lie algebra g. One incarnation of the method, the so-called S-expansion, involves the use of a finite Abelian semigroup S to accomplish this task. In this paper we put forward a dual formulation of the S-expansion method, which is based on the dual picture of a Lie algebra given by the Maurer–Cartan forms. The dual version of the method is useful in finding a generalization to the case of a gauge free differential algebra, which, in turn, is relevant for physical applications in, e.g., supergravity. It also sheds new light on the puzzling relation between two Chern–Simons Lagrangians for gravity in 2+1 dimensions, namely, the Einstein–Hilbert Lagrangian and the one for the so-called “exotic gravity.”

Cartan’s Structure of Symmetry Pseudo-Group and Coverings for the r-th Modified Dispersionless Kadomtsev–Petviashvili Equation

We derive two non-equivalent coverings for the r-th dKP equation from Maurer–Cartan forms of its symmetry pseudo-group. Also we find Bäcklund transformations between the obtained covering equations.

Quantum Analogy of Poisson Geometry, Related Dendriform Algebras and Rota–Baxter Operators

We will introduce an associative (or quantum) version of Poisson structure tensors. This object is defined as an operator satisfying a “generalized” Rota–Baxter identity of weight zero. Such operators are called generalized Rota–Baxter operators. We will show that generalized Rota–Baxter operators are characterized by a cocycle condition so that Poisson structures are so. By analogy with twisted Poisson structures, we propose a new operator “twisted Rota–Baxter operators,” which is a natural generalization of generalized Rota–Baxter operators. It is known that classical Rota–Baxter operators are closely related with dendriform algebras. We will show that twisted Rota–Baxter operators induce NS-algebra, which is a twisted version of dendriform algebra. The twisted Poisson condition is considered as a Maurer–Cartan equation up to homotopy. We will show the twisted Rota–Baxter condition also is so. And we will study a Poisson-geometric reason, how the twisted Rota–Baxter condition arises.

Casimir operators induced by the Maurer–Cartan equations

It is shown that for inhomogeneous Lie algebras with only one Casimir operator, the latter can be explicitly constructed from the Maurer–Cartan equations by means of wedge products. It is further proved that this constraint imposes sharp bounds for the dimension of the representation R defining the semidirect product. The procedure is generalized to compute also the rational invariant of some Lie algebras.

Hodge Dual Symmetry of Green–Schwarz Superstring in AdS5 ⊗ S5

The hidden symmetry and an infinite set non-local conserved currents of the Green–Schwarz superstring on AdS5 ⊗ S5 have been pointed out by Bena et al. In this paper, we show that the Hodge dual between the Maurer–Cartan equation and the equation of motion gives the hidden symmetry in the moduli space of Green–Schwarz superstring. Thus by twisty transforming the vielbeins, we can express the currents of the paper [I. Bena, J. Polchinski, and R. Roiban, Phys. Rev. D 69 (2004) 046002] as the Lax connections by a unique spectral parameter.

Pro-algebraic homotopy types

The purpose of this paper is to generalise Sullivan's rational homotopy theory to non-nilpotent spaces, providing an alternative approach to defining Toen's schematic homotopy types over any field k of characteristic zero. New features include an explicit description of homotopy groups using the Maurer-Cartan equations, convergent spectral sequences comparing schematic homotopy groups with cohomology of the universal semisimple local system, and a generalisation of the Baues-Lemaire conjecture. For compact Kaehler manifolds, the schematic homotopy groups can be described explicitly in terms of this cohomology ring, giving canonical weight decompositions. There are also notions of minimal models, unpointed homotopy types and algebraic automorphism groups. For a space with algebraically good fundamental group and higher homotopy groups of finite rank, the schematic homotopy groups are shown to be \pi_n(X)\otimes k.

Cartan’s Structure Theory of Symmetry Pseudo-Groups, Coverings and Multi-Valued Solutions for the Khokhlov–Zabolotskaya Equation

We derive two non-equivalent coverings for the modified Khokhlov–Zabolotskaya equation from Maurer–Cartan forms of its symmetry pseudo-group. Also we find Backlund transformations between the obtained covering equations. We apply these results to constructing multi-valued solutions for the Khokhlov–Zabolotskaya equation.

Formal Maurer-Cartan structures: from CFT to Classical Field Equations

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 differential is the BRST operator of String theory. We introduce the bilinear operations which are present in GMC, and study their properties, allowing us to find the symmetries of the resulting equations which will be naturally identified with the diffeomorphism and gauge symmetries of Einstein and Yang-Mills equations correspondingly.

A NEW APPROACH IN BICOVARIANT DIFFERENTIAL CALCULUS ON SUq(2)

When reviewing the work of Aschieri–Castellani concerning the bicovariant differential calculus on GLq(2), we remarked that a peculiar basis within the space of quantum one-forms over SUq(2) can lead to a new bicovariant differential calculus on SUq(2) easy to handle than those known in the literature. New results have been obtained for the exterior product, the exterior differential, the C-structure constants, the Cartan–Maurer equations, and the q-Lie algebra.

Twisted Duality Symmetry and Integrability ofHybrid Superstring on AdS2×S2

In this paper, we show that there exists a twisted dualitysymmetry between the Maurer–Cartanequations and the equations of motion in the hybrid formalism forthe type IIB superstring in an AdS2×S2 background withRamond–Ramond flux. As a result, from the twisted dualitytransformation, we construct the Lax connection with the spectralparameter, which ensures the integrability of the system.

Coverings of Differential Equations and Cartan’s Structure Theory of Lie Pseudo-Groups

We establish relations between Maurer–Cartan forms of symmetry pseudo-groups and coverings of differential equations. Examples include Liouville’s equation, the Khokhlov–Zabolotskaya equation, and the Boyer–Finley equation.

Homotopy Lie superalgebra in Yang-Mills theory

The Yang-Mills equations are formulated in the form of generalized Maurer-Cartan equations, such that the corresponding algebraic operations are shown to satisfy the defining relations of homotopy Lie superalgebra.

Solvable Lie algebras, products by generators, and some of its applications

In this work, we enlarge the definition of products by generators of Lie algebras to the class of solvable Lie algebras. We analyze the number of independent invariant functions for the coadjoint representation of these algebras by means of the Maurer-Cartan equations and give some applications to product structures on Lie algebras.

Strongly Homotopy Lie Bialgebras and Lie Quasi-bialgebras

Structures of Lie algebras, Lie coalgebras, Lie bialgebras and Lie quasibialgebras are presented as solutions of Maurer–Cartan equations on corresponding governing differential graded Lie algebras using the big bracket construction of Kosmann–Schwarzbach. This approach provides a definition of an L ∞-(quasi)bialgebra (strongly homotopy Lie (quasi)bialgebra). We recover an L ∞-algebra structure as a particular case of our construction. The formal geometry interpretation leads to a definition of an L ∞ (quasi)bialgebra structure on V as a differential operator Q on V, self-commuting with respect to the big bracket. Finally, we establish an L ∞-version of a Manin (quasi) triple and get a correspondence theorem with L ∞-(quasi)bialgebras.