arXiv research

This article reviews $\infty$-bundles and their applications in geometry and physics.

Inverse function theorem and homotopy description for L-infinity bundles.

This paper introduces $\infty$- and $n$-fold vector bundles as special functors from the $\infty$- and $n$-cube categories to the category of smooth manifolds. We study the cores and "n-pullbacks" of $n$-fold vector bundles and we prove that any $n$-fold vector bundle admits a non-canonical isomorphism to a decomposed …

Introduces new connections in higher geometry.

We consider the gradient flow of the Yang-Mills-Higgs functional of twist Higgs pairs on a Hermitian vector bundle $(E,H_0)$ over a Riemann surface $X$. It is already known the gradient flow with initial data $(A_0,φ_0)$ converges to a critical point $(A_\infty, φ_\infty)$ of this functional. Using a modified Chern-Wei…

Lie-Rinehart algebras over $C^\infty$-rings defined and studied.

Given a bundle of chain complexes, the algebra of functions on its shifted cotangent bundle has a natural structure of a shifted Poisson algebra. We show that if two such bundles are homotopy equivalent, the corresponding Poisson algebras are homotopy equivalent. We apply this result to $L_\infty$-algebroids to show th…

The paper bridges diffeological bundle theory with higher topos theory.

Geometrically deforms $L_\infty$ algebras to Lie algebroids, revealing new invariants.

The paper characterizes vector bundles and differential operators using Lie algebras and their symbols.

In this paper, we relate Lie algebroids to Costello's version of derived geometry. For instance, we show that each Lie algebroid $L$-and the natural generalization to dg Lie algebroids-provides an (essentially unique) $L_\infty$ space. More precisely, we construct a faithful functor from the category of Lie algebroids …

In an earlier paper [Acta Mathematica, v. 176, 1996, 145-169, alg-geom/9505024 ] the present authors and Dennis Sullivan constructed the universal direct system of the classical Teichmüller spaces of Riemann surfaces of varying genus. The direct limit, which we called the universal commensurability Teichmüller space, $…

The paper classifies symbols of differential operators on vector bundles.

Researchers classify differential operators between 3-sphere and 2-sphere bundles.

Iterative method finds Hermitian-Einstein metrics on stable bundles.

We show that conically smooth stratified spaces embed fully faithfully into $\infty$-categories. This articulates a stratified generalization of the homotopy hypothesis proposed by Grothendieck. As such, each $\infty$-category defines a stack on conically smooth stratified spaces, and we identify the descent conditions…

We equip the direct limit of tangent bundles of paracompact finite dimensional manifolds with a structure of convenient vector bundle with structural group $GL(\infty,\mathbb{R}) =\underrightarrow{\lim}GL(\mathbb{R}^{n})$.

We associate to each stable Higgs pair $(A_0,Φ_0)$ on a compact Riemann surface $X$ a singular limiting configuration $(A_\infty,Φ_\infty)$, assuming that $\det Φ$ has only simple zeroes. We then prove a desingularization theorem by constructing a family of solutions $(A_t,tΦ_t)$ to Hitchin's equations which converge t…

We continue the study of the spectral theory associated to integrable metrics, started in our previous paper arXiv:1301.1793 [math.SP]. We introduce the notion of 1-integrable metric on line-bundles on a compact Riemann surface. We extend the spectral theory of generalized Laplacians to line-bundles equipped with 1-int…

New characterization of vector bundles using Lie algebras of symbols.

To each once-punctured-torus bundle, $T_φ$, over the circle with pseudo-Anosov monodromy $φ$, there are associated two tessellations of the complex plane: one, $Δ(φ)$, is (the projection from $\infty$ of) the triangulation of a horosphere at $\infty$ induced by the canonical decomposition into ideal tetrahedra, and the…

We describe natural abelian extensions of the Lie algebra $\aut(P)$ of infinitesimal automorphisms of a principal bundle over a compact manifold $M$ and discuss their integrability to corresponding Lie group extensions. Already the case of a trivial bundle $P = M \times K$ is quite interesting. In this case, we show th…

N. Hitchin recently introduced the notion of folded hyperKähler metrics, in relation with SL(\infty,R) Higgs bundles. We provide a construction of such metrics, and prove the local existence of the Hitchin component for SL(\infty,R).

New smooth models for string groups defined in ∞-categories.

Let $X$ be a compact real analytic manifold, and let $T^*X$ be its cotangent bundle. Let $Sh(X)$ be the triangulated dg category of bounded, constructible complexes of sheaves on $X$. In this paper, we develop a Fukaya $A_\infty$-category $Fuk(T^*X)$ whose objects are exact, not necessarily compact Lagrangian branes in…

Constructs a sheaf of Bargmann-Fock modules on Kähler manifolds.

If E is a C^\infty complex vector bundle on an oriented C^\infty manifold Σ, diffeomorphic to a circle, then the space of sections of E has a canonical polarization in the sense of Pressley and Segal and so one has its determinantal gerbe with lien C^*, the group of nonzero complex numbers. If q:Σ-->B is a smooth famil…

Introduces symplectic flatness for connections over symplectic manifolds.

Extends Serre-Swan theorem to all finitely generated modules over smooth functions.

Quillen connection links Riemann surfaces to projective structures.

In this note we introduce the notion of a smooth structure on a conical pseudomanifold $M$ in terms of $C^\infty$-rings of smooth functions on $M$. For a finitely generated smooth structure $C^\infty (M)$ we introduce the notion of the Nash tangent bundle, the Zariski tangent bundle, the tangent bundle of $M$, and the …

We derive a bound on the $L^{\infty}$-norm of the covariant derivative of Laplace eigensections on general Riemannian vector bundles depending on the diameter, the dimension, the Ricci curvature of the underlying manifold, and the curvature of the Riemannian vector bundle. Our result implies that eigensections with sma…

We describe two constructions giving rise to curved $A_{\infty}$-algebras. The first consists of deforming $A_{\infty}$-algebras, while the second involves transferring curved dg structures that are deformations of (ordinary) dg structures along chain contractions. As an application of the second construction, given a …

The infinite matrix `Schwartz' group $G^{-\infty}$ is a classifying group for odd K-theory and carries Chern classes in each odd dimension, generating the cohomology. These classes are closely related to the Fredholm determinant on $G^{-\infty}.$ We show that while the higher (even, Schwartz) loop groups of $G^{-\infty…

Characterizes Lie-Backlund vector fields in infinite jet bundles that can be exponentiated.

We explain how deformation theories of geometric objects such as complex structures, Poisson structures and holomorphic bundle structures lead to differential Gerstenhaber or Poisson algebras. We use homological perturbation theory to obtain $A_{\infty}$ algebra structures and some canonically defined deformations of s…

The study finds effective lower bounds for spectra of random surfaces and bundles.

Let $L$ be a line bundle over $M$. In this paper we associate an $L_\infty$-algebra to any $L$-Courant algebroid (contact Courant algebroid in the sense of Grabowski). This construction is similar to the work of Roytenberg and Weinstein for Courant algebroids. Next we associate a $p$-term $L_\infty$-algebra to any isot…

We define the equivariant family index of a family of elliptic operators invariant with respect to the free action of a bundle $\GR$ of Lie groups. If the fibers of $\GR \to B$ are simply-connected solvable, we then compute the Chern character of the (equivariant family) index, the result being given by an Atiyah-Singe…

We show that the category of affine bundles over a smooth manifold M is equivalent to the category of affine spaces modelled on projective finitely generated C^\infty(M)-modules. Using this equivalence of categories, we are able to give an alternate proof of the main result of [13], showing that the characterization of…

In this paper we give a Chern-Weil-type construction of characteristic classes of fiber bundles, based on homotopy theory of C-infinity algebras. Our idea is to replace a family of closed manifolds to a family of C-infinity morphisms with family of metrics.

Study on Čech-de Rham obstruction in diffeological spaces.

Categorifies Chern-Weil theory for infinite local systems.

If a sequence of Riemannian manifolds, $X_i$, converges in the pointed Gromov-Hausdorff sense to a limit space, $X_\infty$, and if $E_i$ are vector bundles over $X_i$ endowed with metrics of Sasaki-type with a uniform upper bound on rank, then a subsequence of the $E_i$ converges in the pointed Gromov-Hausdorff sense t…

We establish an explicit correspondence between two--dimensional projective structures admitting a projective vector field, and a class of solutions to the $SU(\infty)$ Toda equation. We give several examples of new, explicit solutions of the Toda equation, and construct their mini--twistor spaces. Finally we discuss t…

A singular (or Hermann) foliation on a smooth manifold $M$ can be seen as a subsheaf of the sheaf $\mathfrak{X}$ of vector fields on $M$. We show that if this singular foliation admits a resolution (in the sense of sheaves) consisting of sections of a graded vector bundle of finite type, then one can lift the Lie brack…

We develop a description of higher gauge theory with higher groupoids as gauge structure from first principles. This approach captures ordinary gauge theories and gauged sigma models as well as their categorifications on a very general class of (higher) spaces comprising presentable differentiable stacks, as e.g. orbif…

We define four versions of equivariant instanton Floer homology ($I^+, I^-, I^\infty$ and $\widetilde I$) for a class of 3-manifolds and $SO(3)$-bundles over them including all rational homology spheres. These versions are analogous to the four flavors of monopole and Heegaard Floer homology theories. This construction…