arXiv research

Study cohomology of hemistrict Lie 2-algebras, proving isomorphic results.

Proves integrability of strict Lie 2-algebras using cohomological methods.

New Lie 2-algebra structure for multiplicative forms on quasi-Poisson groupoids.

The semidirect product of a Lie algebra and a 2-term representation up to homotopy is a Lie 2-algebra. Such Lie 2-algebras include many examples arising from the Courant algebroid appearing in generalized complex geometry. In this paper, we integrate such a Lie 2-algebra to a strict Lie 2-group in the finite dimensiona…

New cohomology theory for Lie 2-algebras extends classical theory.

New method for moment maps in multisymplectic geometry using Lie 2-algebras.

In this paper, for a Lie 2-algebra $\g$, we construct the automorphism 2-group $\Aut(\g)$, which turns out to be an integration of the derivation Lie 2-algebra $\Der(\g)$.

Introduces new cohomology theories for Lie 2-algebras and groups.

A Lie 2-group's left-invariant vector fields are isomorphic to its Lie 2-algebra.

Symmetries of bundle gerbes modeled using multiplicative vector fields.

We show that the category of vector fields on a geometric stack has the structure of a Lie 2-algebra. This proves a conjecture of R.~Hepworth. The construction uses a Lie groupoid that presents the geometric stack. We show that the category of vector fields on the Lie groupoid is equivalent to the category of vector fi…

Study shows connection-preserving vector fields are equivalent to certain algebroid structures.

This work introduces Lie 2-algebra structures for multiplicative sections of LA-groupoids.

We construct a flat (and fake-flat) 2-connection in the configuration space of $n$ indistinguishable particles in the complex plane, which categorifies the $sl(2,C)$-Knizhnik-Zamolodchikov connection obtained from the adjoint representation of $sl(2,C)$. This will be done by considering the adjoint categorical represen…

We describe an interesting relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group $\mathrm{String}(n)$. A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the "Jacobiator". Similarly, a Lie 2-group is a c…

We study the semidirect product of a Lie algebra with a representation up to homotopy and provide various examples coming from Courant algebroids, string Lie 2-algebras, and omni-Lie algebroids. In the end, we study the semidirect product of a Lie group with a representation up to homotopy and use it to give an integra…

Maps Lie 2-groups to Weil algebras, showing cohomology isomorphisms.

Abstract: Generalizes prequantization map for 2-plectic manifolds.

Affine structures on Lie groupoids are studied, showing rich algebraic properties.

We categorify the theory of Lie algebras beginning with a new notion of categorified vector space, or `2-vector space', which we define as an internal category in Vect, the category of vector spaces. We then define a `semistrict Lie 2-algebra' to be a 2-vector space equipped with a skew-symmetric bilinear functor satis…

We develop semistrict higher gauge theory from first principles. In particular, we describe the differential Deligne cohomology underlying semistrict principal 2-bundles with connective structures. Principal 2-bundles are obtained in terms of weak 2-functors from the Cech groupoid to weak Lie 2-groups. As is demonstrat…

For a strict Lie 2-group, we develop a notion of Lie 2-algebra-valued differential forms on Lie groupoids, furnishing a differential graded-commutative Lie algebra equipped with an adjoint action of the Lie 2-group and a pullback operation along Morita equivalences between Lie groupoids. Using this notion, we define co…

Constructs parallel transport in higher gauge theory using principal 2-bundles.

The paper extends Chern-Weil-Lecomte map to $L_{\infty}$-algebras.

We study the extension of a Lie algebroid by a representation up to homotopy, including semidirect products of a Lie algebroid with such representations. The extension results in a higher Lie algebroid. We give exact Courant algebroids and string Lie 2-algebras as examples of such extensions. We then apply this to obta…

In this paper, we show that associated to any coisotropic Cartan geometry there is a twisted Courant algebroid. This includes in particular parabolic geometries. Using this twisted Courant structure, we give some new results about the Cartan curvature and the Weyl structure of a parabolic geometry. As more direct appli…

Given an n-term L-infinity algebra L, we construct a Kan simplicial manifold which we think of as the 'Lie n-group' integrating L. This extends work of Getzler math.AT/0404003 . In the case of an ordinary Lie algebra, our construction gives the simplicial classifying space of the corresponding simply connect Lie group.…

In this paper, we give the categorification of Leibniz algebras, which is equivalent to 2-term sh Leibniz algebras. They reveal the algebraic structure of omni-Lie 2-algebras introduced in \cite{omniLie2} as well as twisted Courant algebroids by closed 4-forms introduced in \cite{4form}. We also prove that Dirac struct…

It is well-known that a Lie algebroid A is equivalently described by a degree 1 Q-manifold M. We study distributions on M, giving a characterization in terms of A. We show that involutive Q-invariant distributions on M correspond bijectively to IM-foliations on A (the infinitesimal version of Mackenzie's ideal systems)…

We show how to integrate a weak morphism of Lie algebra crossed-modules to a weak morphism of Lie 2-groups. To do so we develop a theory of butterflies for 2-term L_infty algebras. In particular, we obtain a new description of the bicategory of 2-term L_infty algebras. We use butterflies to give a functorial constructi…

Decomposes elements in multiplicative multivectors and relates to Lie algebroid cohomology.

The paper explores Poisson structures on differentiable stacks, developing new mathematical tools.

We prove a 2-categorical analogue of a classical result of Drinfeld: there is a one-to-one correspondence between connected, simply-connected Poisson Lie 2-groups and Lie 2-bialgebras. In fact, we also prove that there is a one-to-one correspondence between connected, simply connected quasi-Poisson 2-groups and quasi-L…

In this paper, we show that the Jacobiator $J$ of a pre-Courant algebroid is closed naturally. The corresponding equivalence class $[J^\flat]$ is defined as the Pontryagin class, which is the obstruction of a pre-Courant algebroid to be deformed into a Courant algebroid. We construct a Leibniz 2-algebra and a Lie 2-alg…

Given a representation up to homotopy of a Lie algebroid on a 2-term complex of vector bundles, we define the corresponding holonomy as a strict 2-functor from a Weinstein path 2-groupoid to the gauge 2-groupoid of the underlying 2-term complex. We construct a corresponding transformation 2-groupoid and we prove that t…

These are notes for four lectures on higher structures in M-theory as presented at workshops at the Erwin Schroedinger Institute and Tohoku University. The first lecture gives an overview of systems of multiple M5-branes and introduces the relevant mathematical structures underlying a local description of higher gauge …

Leaves of Lie algebroids are Lie groupoids under certain conditions.

We consider generalizations of symplectic manifolds called n-plectic manifolds. A manifold is n-plectic if it is equipped with a closed, nondegenerate form of degree n+1. We show that higher structures arise on these manifolds which can be understood as the categorified or homotopy analogues of important structures stu…

In the first chapter, we give a precise and general description of gerbes valued in arbitrary crossed module and over an arbitrary differential stack. We do it using only Lie groupoids, hence ordinary differential geometry, by considering differential stacks as being Lie groupoids up to Morita equivalence. We prove the…

We categorify the notion of an infinitesimal braiding in a linear strict symmetric monoidal category, leading to the notion of a (strict) infinitesimal 2-braiding in a linear symmetric strict monoidal 2-category. We describe the associated categorification of the 4-term relation, leading to six categorified relations. …

In this paper, we study the algebraic properties of the higher analogues of Courant algebroid structures on the direct sum bundle $TM\oplus\wedge^nT^*M$ for an $m$-dimensional manifold. As an application, we revisit Nambu-Poisson structures and multisymplectic structures. We prove that the graph of an $(n+1)$-vector fi…

We give a complete and explicit description of the kinematical data of higher gauge theory on principal 2-bundles with the string 2-group model of Schommer-Pries as structure 2-group. We start with a self-contained review of the weak 2-category Bibun of Lie groupoids, bibundles and bibundle morphisms. We then construct…

We formulate a 4-dimensional higher gauge theoretic Chern-Simons theory. Its symmetry is encoded in a semistrict Lie 2-algebra equipped with an invariant non singular bilinear form. We analyze the gauge invariance of the theory and show that action is invariant under a higher gauge transformation up to a higher winding…

New result on finite gerbes simplifies classification of certain bundles.

In the first part of this paper, we work out a perturbative Lagrangian formulation of semistrict higher gauge theory, that avoids the subtleties of the relationship between Lie 2-groups and algebras by relying exclusively on the structure semistrict Lie 2-algebra v and its automorphism 2-group Aut(v). Gauge transformat…

We study Maurer-Cartan elements on homotopy Poisson manifolds of degree $n$. They unify many twisted or homotopy structures in Poisson geometry and mathematical physics, such as twisted Poisson manifolds, quasi-Poisson $\g$-manifolds, and twisted Courant algebroids. Using the fact that the dual of an $n$-term $L_\infty…

Defines Lie and Courant algebroids over Lie groupoids using homological vector fields.

In this paper, first we modify the definition of a Hom-Lie algebroid introduced by Laurent-Gengoux and Teles and give its equivalent dual description. Many results that parallel to Lie algebroids are given. In particular, we give the notion of a Hom-Poisson manifold and show that there is a Hom-Lie algebroid structure …