This Master Thesis is devoted to the study of $n$-plectic manifolds and the Strongly Homotopy Lie algebras, also called $L_{\infty}$-algebras, that can be associated to them. Since multisymplectic geometry and $L_{\infty}$-algebras are relevant in Theoretical Physics, and in particular in String Theory, we introduce the relevant background material in order to make the exposition accessible to non-experts, perhaps interested physicists. The background material includes graded and homological algebra theory, fibre bundles, basics of group actions on manifolds and symplectic geometry. We give an introduction to $L_{\infty}$-algebras and define $L_{\infty}$-morphisms in an independent way, not yet related to multisymplectic geometry, giving explicit formulae relating $L_{\infty}[1]$-algebras and $L_{\infty}$-algebras. We give also an account of multisymplectic geometry and $n$-plectic manifolds, connecting them to $L_{\infty}$-algebras. We then introduce, closely following the work {1304.2051} of Yael Fregier, Christopher L. Rogers and Marco Zambon, the concept of homotopy moment map. The new results presented here are the following: we obtain specific conditions under which two $n$-plectic manifolds with strictly isomorphic Lie-$n$ algebras are symplectomorphic, and we study the construction of an homotopy moment map for a product manifold, assuming that the factors are $n$-plectic manifolds equipped with the corresponding homotopy moment maps.
We introduce a notion of a homotopy momentum section on a Lie algebroid over a pre-multisymplectic manifold. A homotopy momentum section is a generalization of the momentum map with a Lie group action and the momentum section on a pre-symplectic manifold, and is also regarded as a generalization of the homotopy momentum map on a multisymplectic manifold. We show that a gauged nonlinear sigma model with Wess-Zumino term with Lie algebroid gauging has the homotopy momentum section structure.
Consider a closed non-degenerate 3-form $ω$ with an infinitesimal action of a Lie algebra $\mathfrak{g}$. Motivated by the fact that the observables associated to $ω$ form a Lie 2-algebra, we introduce homotopy moment maps defined on a Lie 2-algebra rather than just on the Lie algebra $\mathfrak{g}$. We formulate existence criteria and provide a construction for such homotopy moment maps, by characterizing them in terms of cohomology.
A Lie system is a non-autonomous system of first-order ordinary differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional real Lie algebra of vector fields, a so-called Vessiot--Guldberg Lie algebra. In this work, multisymplectic structures are applied to the study of the reduction of Lie systems through their Lie symmetries. By using a momentum map, we perform a reduction and reconstruction procedure of multisymplectic Lie systems, which allows us to solve the original problem by analysing several simpler multisymplectic Lie s…
Field Theories in Physics can be formulated giving a local Lagrangian density. Locality is imposed using the infinite jet bundle. That bundle is viewed as a pro-finite dimensional smooth manifold and that point of view has been compared to different topological and Frechét structures on it. A category of local (insular) manifolds has been constructed. Noether's second theorem is reviewed and the notion of Lie pseudogroups is explored using these concepts. The L-infinity algebra of local observables is defined depending only on the cohomology of the Lagrangian (using a result in multisymplectic…
In this paper, we construct a homotopy Poisson algebra of degree 3 associated to a split Lie 2-algebroid, by which we give a new approach to characterize a split Lie 2-bialgebroid. We develop the differential calculus associated to a split Lie 2-algebroid and establish the Manin triple theory for split Lie 2-algebroids. More precisely, we give the notion of a strict Dirac structure and define a Manin triple for split Lie 2-algebroids to be a CLWX 2-algebroid with two transversal strict Dirac structures. We show that there is a one-to-one correspondence between Manin triples for split Lie 2-alg…
Weighted Lie algebroids were recently introduced as Lie algebroids equipped with an additional compatible non-negative grading, and represent a wide generalisation of the notion of a VB -algebroid. There is a close relation between two term representations up to homotopy of Lie algebroids and VB - algebroids. In this paper we show how this relation generalises to weighted Lie algebroids and in doing so we uncover new and natural examples of higher term representations up to homotopy of Lie algebroids. Moreover, we show how the van Est theorem generalises to weighted objects.
This paper studies differential graded modules and representations up to homotopy of Lie $n$-algebroids, for general $n\in\mathbb{N}$. The adjoint and coadjoint modules are described, and the corresponding split versions of the adjoint and coadjoint representations up to homotopy are explained. In particular, the case of Lie 2-algebroids is analysed in detail. The compatibility of a Poisson bracket with the homological vector field of a Lie $n$-algebroid is shown to be equivalent to a morphism from the coadjoint module to the adjoint module, leading to an alternative characterisation of non-de…
This paper considers the Pontryagin characters of graded vector bundles of finite rank, in the cohomology vector spaces of a Lie algebroid over the same base. These Pontryagin characters vanish if the graded vector bundle carries a representation up to homotopy of the Lie algebroid. As a consequence, this gives a strong obstruction to the existence of a representation up to homotopy on a graded vector bundle of finite rank. In particular, if a graded vector bundle $E[0]\oplus F[1]\to M$ carries a $2$-term representation up to homotopy of a Lie algebroid $A\to M$, then all the (classical) $A$-P…
The subject of this paper is strongly homotopy (SH) Lie algebras, also known as $L_\infty$-algebras. We extract an intrinsic character, the Atiyah class, which measures the nontriviality of an (SH) Lie algebra $A$ when it is extended to $L$. In fact, given such an SH Lie pair $(L, A)$, and any $A$-module $E$, there associates a canonical cohomology class, the Atiyah class $[α^E]$, which generalizes earlier known Atiyah classes out of Lie algebra pairs. We show that the Atiyah class $[α^{L/A}]$ induces a graded Lie algebra structure on $\operatorname{H}^\bullet_{\mathrm{CE}}(A,L/A[-2])$, and th…
For an element $Ψ$ in the graded vector space $Ω^*(M, TM)$ of tangent bundle valued forms on a smooth manifold $M$, a $Ψ$-submanifold is defined as a submanifold $N$ of $M$ such that $Ψ_{|N} \in Ω^*(N, TN)$. The class of $Ψ$-submanifolds encompasses calibrated submanifolds, complex submanifolds and all Lie subgroups in compact Lie groups. The graded vector space $Ω^*(M, TM)$ carries a natural graded Lie algebra structure, given by the Frölicher-Nijenhuis bracket $[-,- ]^{FN}$. When $Ψ$ is an odd degree element with $[ Ψ, Ψ]^{FN} =0$, we associate to a $Ψ$-submanifold $N$ a strongly homotopy Li…
Lie $\infty$-groupoids are simplicial Banach manifolds that satisfy an analog of the Kan condition for simplicial sets. An explicit construction of Henriques produces certain Lie $\infty$-groupoids called `Lie $\infty$-groups' by integrating finite type Lie $n$-algebras. In order to study the compatibility between this integration procedure and the homotopy theory of Lie $n$-algebras introduced in the companion paper arXiv:1809.05999, we present a homotopy theory for Lie $\infty$-groupoids. Unlike Kan simplicial sets and the higher geometric groupoids of Behrend and Getzler, Lie $\infty$-group…
This paper is devoted to the study of the relation between `formal exponential maps,' the Atiyah class, and Kapranov $L_\infty[1]$ algebras associated with dg manifolds in the $C^\infty$ context. Given a dg manifold, we prove that a `formal exponential map' exists if and only if the Atiyah class vanishes. Inspired by Kapranov's construction of a homotopy Lie algebra associated with the holomorphic tangent bundle of a complex manifold, we prove that the space of vector fields on a dg manifold admits an $L_\infty[1]$ algebra structure, unique up to isomorphism, whose unary bracket is the Lie der…
