arXiv research

The geometry of graded principal bundles is discussed in the framework of graded manifold theory of Kostant-Berezin-Leites. In particular, we prove that a graded principal bundle is globally trivial if and only if it admits a global graded section and, further, that the sheaf of vertical derivations on such a bundle co…

Graded bundles are a particularly nice class of graded manifolds and represent a natural generalisation of vector bundles. By exploiting the formalism of supermanifolds to describe Lie algebroids we define the notion of a weighted $A$-connection on a graded bundle. In a natural sense weighted $A$-connections are adapte…

We study the graded geometric point of view of curvature and torsion of Q-manifolds (differential graded manifolds). In particular, we get a natural graded geometric definition of Courant algebroid curvature and torsion, which correctly restrict to Dirac structures. Depending on an auxiliary affine connection K, we int…

Defines formal exponentials for graded manifolds and linearizes QP-manifolds.

The pull back of a flat bundle $E\rightarrow X$ along the evaluation map $π: \mathcal{L} X \to X$ from the free loop space $\mathcal{L} X$ to $X$ comes equipped with a canonical automorphism given by the holonomies of $E$. This construction naturally generalizes to flat $\mathbb{Z}$-graded connections on $X$. Our main …

Study symplectic scalar curvature on supermanifolds.

Characterizes fundamental groups of disjointly tree-graded spaces.

Given a supervector bundle $E = E_0\oplus E_1 \to M$, we exhibit a parametrization of Quillen superconnections on $E$ by graded connections on the Cartan-Koszul supermanifold $(M;Ω(M))$. The relation between the curvatures of both kind of connections, and their associated Chern classes, is discussed in detail. In parti…

Extends Morse-Novikov Homology to include differential graded coefficients and fibration structures.

Study on hermitian Yang-Mills connections on blown-up manifolds.

The paper studies HYM connections on stable vector bundles over Kähler manifolds.

Introduces canonical connections for sub-Riemannian manifolds with constant symbol.

We study here systems of symmetries on $|1|$--graded parabolic geometries. We are interested in smooth systems of symmetries and we discuss non--flat homogeneous $|1|$--graded geometries. We show the existence of an invariant admissible affine connection under quite weak condition on the system.

We show that the link cobordism maps defined by the author are graded and satisfy a grading change formula. Using the grading change formula, we prove a new bound for $Υ_K(t)$ for knot cobordisms in negative definite 4-manifolds. As another application, we show that the link cobordism maps associated to a connected, cl…

We define the notion of the Ricci tensor for NQ symplectic manifolds of degree 2 and show that it corresponds to the standard generalized Ricci tensor on Courant algebroids. We use an appropriate notion of connections compatible with the generalized metric on the graded manifold.

Cobordisms are naturally bigraded and we show that this grading extends to Khovanov homology, making it a triply graded theory. Although the new grading does not make the homology a stronger invariant, it can be used to show that odd Khovanov homology is multiplicative with respect to disjoint unions and connected sums…

The paper studies gradings on nilpotent Lie algebras linked to smooth algebraic varieties.

We explain how to compute the Jones polynomial of a link from one of its grid diagrams and we observe a connection between Bigelow's homological definition of the Jones polynomial and Kauffman's definition of the Jones polynomial. Consequently, we prove that the Maslov grading on the Seidel-Smith symplectic link invari…

I consider the semiclassical approximation of the graded Chern-Simons field theories describing certain systems of topological A type branes in the large radius limit of Calabi-Yau compactifications. I show that the semiclassical partition function can be expressed in terms of a certain (differential) numerical invaria…

We define and give explicit construction of the universal tree-graded space with a given collection of pieces. We apply that to proving uniqueness of asymptotic cones of relatively hyperbolic groups whose peripheral subgroups have unique asymptotic cones. Modulo the Continuum Hypothesis, we show that if an asymptotic c…

In recent years, a close connection between supergravity, string effective actions and generalized geometry has been discovered that typically involves a doubling of geometric structures. We investigate this relation from the point of view of graded geometry, introducing an approach based on deformations of graded Pois…

Study connections on Lie and Courant algebroids, defining basic curvature and Atiyah cocycle.

Generative models' evaluation scores can be misleading, leading to inflated grades.

Convolutional neural networks improve KL grade prediction from Indian knee radiographs.

We construct membrane homology groups $\h(M)$ associated with each compact connected oriented smooth manifold, and show that $\h(M)$ is matrix graded algebra.

Develops a chain-level model for Chas-Sullivan products using Morse theory with differential graded coefficients.

This thesis generalizes structures on $\mathcal{Q}$-manifolds and Lie $n$-algebroids.

We reformulate notions from the theory of quasi-Poisson g-manifolds in terms of graded Poisson geometry and graded Poisson-Lie groups and prove that quasi-Poisson g-manifolds integrate to quasi-Hamiltonian g-groupoids. We then interpret this result within the theory of Dirac morphisms and multiplicative Manin pairs, to…

Given a graded $E_1$-module over an $E_2$-algebra in spaces, we construct an augmented semi-simplicial space up to higher coherent homotopy over it, called its canonical resolution, whose graded connectivity yields homological stability for the graded pieces of the module with respect to constant and abelian coefficien…

We introduce the notions of Atiyah class and Todd class of a differential graded vector bundle with respect to a differential graded Lie algebroid. We prove that the space of vector fields on a dg-manifold with homological vector field $Q$ admits a structure of L-infinity algebra with the Lie derivative $L_Q$ as unary …

We construct geometric examples of N-differential graded algebras such as the algebra of differential forms of depth $N$ on an affine manifold, and $N$-flat covariant derivatives.

Recently, for a family of ungraded Dirac operators over some space $B$ J. Lott constructed an index gerbe. In the present paper we show (in analogy to the holonomy formula for the determinant bundle in the graded case) that the holonomy of the index gerbe (a priori a hermitean line bundle with connection over the free …

We discuss a possible solution to an unintended consequence of having grades, certificates, rankings and other diversions in the act of transferring knowledge; and zoom in specifically to the topic of having grades, on a curve. We conduct a thought experiment, taking a chapter (and some more?) from the financial market…

New connection between dynamics and Heegaard Floer homology.

The paper connects quantum invariants to intersections of Lagrangians in symmetric power spaces.

The article explores causal structures in symmetric spaces and their relation to AQFT.

The singular instanton Floer homology was defined by Kronheimer and Mrowka in connection with their proof that the Khovanov homology is an unknot detector. We study this theory for knots and two-component links using equivariant gauge theory on their double branched covers. We show that the special generator in the sin…

Transport functions for principal bundles and Morse homology with differential graded coefficients

This work explores symplectic structures on graded manifolds and higher Lie groupoids.

A filtered manifold is a smooth manifold $M$ together with a filtration of the tangent bundle by smooth subbundles which is compatible with the Lie bracket of vector fields in a certain sense. The Lie bracket of vector fields then induces a bilinear operation on the associated graded of each tangent space of $M$ making…

We define a deformation of the triply graded Khovanov-Rozansky homology of a link $L$ depending on a choice of parameters $y_c$ for each component of $L$, which satisfies link-splitting properties similar to the Batson-Seed invariant. Keeping the $y_c$ as formal variables yields a link homology valued in triply graded …

Isomorphic algebra connects Toeplitz to Heisenberg group.

Let \gh = \gh_{-k}\oplus \cdots \oplus \gh_{l} (k >0, l \geq 0) be a finite dimensional real graded Lie algebra, with a Euclidian metric \langle \cdot , \cdot \rangle adapted to the gradation. The metric \langle\cdot , \cdot \rangle is called admissible if the codifferentials \partial^{*} : C^{k+1}(\gh_{-}, \gh ) \ra C…

New invariant connects knot homology and BPS series for plumbed knot complements.

Constructive approach to Lie algebra gradings, computing maximal and enumerating all gradings.

The geometry of an admissible Bäcklund transformation for an exterior differential system is described by an admissible Cartan connection for a geometric structure on a tower with infinite--dimensional skeleton which is the universal prolongation of a $|1|$--graded semi-simple Lie algebra.

Three definitions of graded vector bundles are shown to be equivalent.

This is the first of a series of papers where we relate tangent cones of Hermitian-Yang-Mills connections at an isolated singularity to the complex algebraic geometry of the underlying reflexive sheaf, when the sheaf is locally modelled on the pull-back of a holomorphic vector bundle from the projective space. In this …