Strongly Homotopy Lie Algebras from Multisymplectic Geometry

TL;DR

This thesis explores the connection between multisymplectic geometry and $L_{}$-algebras, introducing new conditions for symplectomorphism and constructing homotopy moment maps for product manifolds, relevant in theoretical physics.

Contribution

It provides new criteria for isomorphism of $n$-plectic manifolds and develops methods to construct homotopy moment maps for product manifolds.

Findings

Conditions for symplectomorphism of $n$-plectic manifolds with isomorphic Lie-$n$ algebras

Construction of homotopy moment maps for product manifolds

Explicit relations between $L_{}[1]$-algebras and $L_{}$-algebras

Abstract

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.