Related papers: Integration of differential graded manifolds

In this paper we develop the $A_\infty$-analog of the Maurer-Cartan simplicial set associated to an $L_\infty$-algebra and show how we can use this to study the deformation theory of $\infty$-morphisms of algebras over non-symmetric…

Deformations of a Courant Algebroid E and its Dirac subbundle A have been widely considered under the assumption that the pseudo-Euclidean metric is fixed. In this paper, we attack the same problem in a setting that allows the…

We define Lie and Courant algebroids on Fr\'{e}chet manifolds. Moreover, we construct a Dirac structure on the generalized tangent bundle of a Fr\'{e}chet manifold and show that it inherits a Fr\'{e}chet Lie algebroid structure. We show…

We construct completely integrable systems on the dual of the Lie algebra of any compact Lie group $K$ with respect to the standard Lie-Poisson structure. These systems generalize key properties of Gelfand-Zeitlin systems: A) the pullback…

Cartan-Lie algebroids, i.e. Lie algebroids equipped with a compatible connection, permit the definition of an adjoint representation, on the fiber as well as on the tangent of the base. We call (positive) quadratic Lie algebroids,…

Starting with minimal requirements from the physical experience with higher gauge theories, i.e. gauge theories for a tower of differential forms of different form degrees, we discover that all the structural identities governing such…

A general position map $f:K\to M$ of a $k$-dimensional simplicial complex to a $2k$-dimensional manifold (for $k=1$, of a graph to a surface) is a $\mathbb Z_2$-embedding if $|f\sigma \cap f\tau|$ is even for any non-adjacent $k$-faces…

We study the spectral convergence of a symmetrized Graph Laplacian matrix induced by a Gaussian kernel evaluated on pairs of embedded data, sampled from a manifold with boundary, a sub-manifold of $\mathbb{R}^m$. Specifically, we deduce the…

We study exact Lagrangian immersions with one double point of a closed orientable manifold K into n-complex-dimensional Euclidean space. Our main result is that if the Maslov grading of the double point does not equal 1 then K is homotopy…

We consider homotopy actions of a Lie algebroid on a graded manifold, defined as suitable $L_{\infty}$-algebra morphisms. On the "semi-direct product" we construct a homological vector field that projects to the Lie algebroid. Our main…

In this paper, we establish the convergence of Feynman graph integrals on closed real-analytic K\"ahler manifolds and uncover the structural mechanism underlying this convergence. The key insight is that, using Getzler's rescaling…

We use Chen's iterated integrals to integrate representations up to homotopy. That is, we construct an A_infty functor from the representations up to homotopy of a Lie algebroid to those of its infinity groupoid. This construction extends…

In categorified symplectic geometry, one studies the categorified algebraic and geometric structures that naturally arise on manifolds equipped with a closed nondegenerate (n+1)-form. The case relevant to classical string theory is when n=2…

Jae-Suk Park and the second-named author introduce the deformation problem of coisotropic submanifolds of a symplectic manifold as the study of Mauer-Cartan moduli problem of an $L_\infty$ algebra attached to the foliation de-Rham complex…

We prove that dg manifolds of finite positive amplitude, i.e. bundles of positively graded curved $L_\infty[1]$-algebras, form a category of fibrant objects. As a main step in the proof, we obtain a factorization theorem using path spaces.…

A symplectic integration of a Poisson manifold $(M,\Lambda)$ is a symplectic groupoid $(\Gamma,\eta)$ which realizes the given Poisson manifold, i.e. such that the space of units $\Gamma_0$ with the induced Poisson structure $\Lambda_0$ is…

Motivated by numerical integration on manifolds, we relate the algebraic properties of invariant connections to their geometric properties. Using this perspective, we generalize some classical results of Cartan and Nomizu to invariant…

Lie algebroids can not always be integrated into Lie groupoids. We introduce a new object--``Weinstein groupoid'', which is a differentiable stack with groupoid-like axioms. With it, we have solved the integration problem of Lie algebroids.…

For any positive integer $n$ and any Lie group $\mathfrak{G}$, given a definite symmetric bilinear form on $\mathbb{R}^n$ and an $\hbox{Ad}$-invariant scalar product on the Lie algebra of $\mathfrak{G}$, we construct a variational problem…

We deal with the symmetries of a (2-term) graded vector space or bundle. Our first theorem shows that they define a (strict) Lie 2-groupoid in a natural way. Our second theorem explores the construction of nerves for Lie 2-categories,…