This paper develops a theory of graded manifolds in differential geometry.
problem Defining consistent global descriptions of graded manifolds with mixed graded coordinates.
method Using sheaves of graded commutative associative algebras on topological spaces.
result Resolved known issues in the definition of graded manifolds, especially those involving mixed graded coordinates.
Constructs graded jet bundles for Z-graded manifolds and vector bundles.
problem Generalizing jet manifolds to Z-graded structures for differential equations.
method Directly constructs the sheaf of sections of the k-th order jet bundle of a Z-graded vector bundle.
result Establishes a graded version of Atiyah Lie algebroid.
This is the second in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we extend the classical notion of a dg-algebra to define, in particular, the notion of a differential graded algebra in the wo…
This paper proves equivalence between derived manifolds and differential graded manifolds.
problem Characterizing derived manifolds and their relationship to differential graded manifolds.
method Proving equivalence between the infinity categories of derived manifolds and differential graded manifolds.
result The infinity category of differential graded manifolds is equivalent to that of derived manifolds.
This paper aims at setting out the basics of Z-graded manifolds theory. We introduce Z-graded manifolds from local models and give some of their properties. The requirement to work with a completed graded symmetric algebra to define functions is made clear. Moreover, we define vector fields and ex…
Determines algebra structure of complex differential forms operators.
problem Identifying the algebra structure of differential operators on complex-valued differential forms.
method Shows it is the universal enveloping algebra of a graded Lie algebra and determines its cohomology.
result Determines the cohomology of the graded Lie algebra with respect to various inner differentials.
Geometric structures on NQ-manifolds, i.e.~non-negatively graded manifolds with an homological vector field, encode non-graded geometric data on Lie algebroids and their higher analogues. A particularly relevant class of structures consists of vector bundle valued differential forms. Symplectic forms, contac…
Normal forms for Q-structures on graded manifolds explained.
problem Understanding structures of Q-manifolds on graded manifolds.
method Local and global normal forms results for Q-structures.
result Structures are concentrated along the zero-locus of curvatures.
Constructs Fedosov dg manifolds from Lie pairs.
problem Constructing differential graded manifolds from Lie pairs.
method Fedosov iteration method and homological perturbation lemma.
result Differential graded algebras of functions on the dg manifolds are homotopy equivalent.
Constructs mixed Hodge structures on Kähler manifolds.
problem Real variations of mixed Hodge structures over compact Kähler manifolds.
method Using Sullivan's 1-minimal models of differential graded algebras associated with real variations of Hodge structures.
result Constructs real variations of mixed Hodge structures.
The paper integrates DGLA to DGLG using HCPs and Hopf algebras.
problem Integrating DGLA to DGLG.
method Definition of DGLG and HCPs, use of graded Hopf algebras.
result Construction of DGLG from DGLA and vice versa.
Study graded coverings for supermanifolds, proving their universal properties.
problem Constructing obstructions for splitting supermanifolds.
method Introduce and prove properties of infinite prolongations of differential operators.
result Infinite prolongations form a covering of supermanifolds in graded manifolds.
Extends Morse-Novikov Homology to include differential graded coefficients and fibration structures.
problem Extending Morse-Novikov Homology with differential graded coefficients.
method Constructs a Morse-Novikov complex and proves the existence of a Chas-Sullivan-like product for a fibration.
result Proves the existence of a Chas-Sullivan-like product on the Novikov completion of a fibration.
Combines generalized and graded geometry to explore new structures.
problem Exploring new structures on generalized tangent bundles of graded manifolds.
method Introduces canonical brackets, Dirac structures, and generalized complex structures.
result Canonical bracket on a generalized tangent bundle of a graded manifold.
Study curvature and torsion in Courant algebroids using graded geometry.
problem Defining curvature and torsion in Courant algebroids.
method Graded geometric approach, introducing K-curvature and K-torsion.
result Natural graded geometric definition of Courant algebroid curvature and torsion.
Given a unital associatve graded algebra we construct the graded q-differential algebra by means of a graded q-commutator, where q is a primitive N-th root of unity. The N-th power (N>1) of the differential of this graded q-differential algebra is equal to zero. We use our approach to construct the graded q-differentia…
Develops a chain-level model for Chas-Sullivan products using Morse theory with differential graded coefficients.
problem Chas-Sullivan products on homology of loop spaces.
method Morse theory with differential graded coefficients, functorial properties, K{ü}nneth formula, Pontryagin-Thom construction.
result Chain-level description of Chas-Sullivan products.
The paper extends the formal manifold theorem to higher dimensions and characterizes A∞-minimal models for certain differential graded algebras.
problem Characterizing formal differential graded algebras and their A∞-minimal models. method Expanding the formal manifold theorem and proving properties of A∞-minimal models for specific cases. result The de Rham complex of certain differential graded algebras has A∞-minimal models with specific non-trivial terms. The abstract generalizes a construction for splitting supermanifolds and studies Lie supergroup cases.
problem Splitting supermanifolds and understanding their structure.
method Using n-fold vector bundles and graded manifolds, the abstract generalizes a construction for splitting supermanifolds. result The images of these embeddings into the category of graded manifolds satisfy universal properties of graded coverings or semicoverings for Lie supergroups and Lie superalgebras.
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…
Bordered Heegaard Floer homology is an invariant for 3-manifolds, which associates to a surface F an algebra A(F), and to a 3-manifold Y with boundary, together with an orientation-preserving diffeomorphism from F to the boundary of Y, a module over A(F). In a previous paper, we defined relative Z/2 differential gradin…
The study introduces new foliations and structures on complex manifolds.
problem Understanding transverse Kähler structures on complex manifolds.
method Introducing holomorphic foliations and developing differential graded and bigraded algebras.
result Obtains quasi-isomorphic complexes to de Rham and Dolbeault complexes, similar to compact Kähler manifolds.
We study contact structures on nonnegatively-graded manifolds equipped with homological contact vector fields. In the degree 1 case, we show that there is a one-to-one correspondence between such structures (with fixed contact form) and Jacobi manifolds. This correspondence allows us to reinterpret the Poissonization p…
Let G be a Lie group acting by diffeomorphisms on a manifold M and consider the image of T[1]G and T[1]M, of G and M respectively, in the category of differential graded manifolds. We show that the obstruction to lift the action of T[1]G on T[1]M to an action on a R[n]-bundle over T[1]M is measured by the G equivariant…
Study Lie superalgebroids from graded Poisson structures.
problem Construct Lie superalgebroids from graded Poisson structures.
method Construct Lie superalgebroids using graded Poisson structures.
result Lie superalgebroids constructed from graded Poisson structures.
We consider the problem of integration of L_\infty-algebroids (differential graded manifolds) to L_\infty-groupoids. We first construct a "big" Kan simplicial manifold (Fréchet or Banach) whose points are solutions of a (generalized) Maurer-Cartan equation. The main analytic trick in our work is an integral transformat…
The paper examines smoothness in graded skew Clifford algebras.
problem Smoothness of graded skew Clifford algebras.
method Investigation of differential smoothness.
result Results on the differential smoothness of graded skew Clifford algebras.
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.
The paper defines flows on Z-graded manifolds and proves unique maximal flows for vector fields.
problem Lack of a treatment for flows on Z-graded manifolds. method Definition and proof of maximal flows for vector fields on Z-graded manifolds. result Every vector field admits a unique maximal flow, with conditions for vector fields invariant under flows and commuting flows.
Generalizes Lie algebra of conformal Killing vector fields to conformal Killing-Yano forms.
problem Understanding the algebraic structure of conformal Killing-Yano forms.
method Proposes a new Lie bracket and shows graded Lie algebra properties in constant and Einstein manifolds.
result Conformal Killing-Yano forms form a graded Lie algebra in specific manifolds.
Simplified holonomy map for ruled submanifolds in graded manifolds.
problem Characterizing deformability of ruled submanifolds in graded manifolds.
method Introducing natural coordinates and higher dimensional holonomy map for ruled submanifolds.
result Characterization of singularities and deformability criterion for ruled submanifolds.
Study on deforming complex manifolds and Higgs bundles.
problem Deforming holomorphic-Higgs pairs on complex manifolds.
method Introduced a DGLA and derived the Maurer-Cartan equation to govern the deformation.
result Proved the local completeness of the Kuranishi family of the deformed holomorphic-Higgs pair.
The paper extends deformation theory for curves of fixed degree in graded manifolds.
problem Computing the first variation of length functionals for curves of fixed degree.
method Analyzes curves in graded manifolds with Riemannian metrics and uses differential equations.
result Provides a sufficient condition for deforming curves of fixed degree.
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 LQ as unary …
This thesis generalizes structures on Q-manifolds and Lie n-algebroids.
problem Representation theory and linear structures of Q-manifolds and Lie n-algebroids. method Introduces differential graded modules and representations up to homotopy, defines Weil algebra, and studies VB-Lie n-algebroids. result Establishes an equivalence between VB-Lie n-algebroids and (n+1)-term representations up to homotopy of Lie n-algebroids. Develops theory of differential graded schemes for derived stacks.
problem Creating a theory for derived stacks using dg schemes.
method Formulates dg schemes as homotopy sites, equates to stacks on dg algebras.
result Infinity category of stacks represented by dg schemes is derived schemes.
We interpret tensors on a smooth manifold M as differential forms over a graded commutative algebra called the algebra of iterated differential forms over M. This allows us to put standard tensor calculus in a new differentially closed context and, in particular, enriches it with new natural operations. Applications wi…
We define analytic torsion of Z_2-graded elliptic complexes as an element in the graded determinant line of the cohomology of the complex, generalizing most of the variants of Ray-Singer analytic torsion in the literature. It applies to a myriad of new examples, including flat superconnection complexes, twisted analyti…
New Morse theory for path homology with coefficients.
problem Defining operations on path homology with differential graded coefficients.
method Using tools from Morse theory and string topology.
result Morse-theoretic description of a product on path homology.
Study symplectic scalar curvature on supermanifolds.
problem Define and analyze symplectic scalar curvature on supermanifolds.
method Introduced two families of odd super-Fedosov structures using graded symmetric and non-symmetric connections.
result Found non-trivial odd symplectic scalar curvature for the second family.
The paper studies how geometric transformations affect semi-classical operators on specific Lie groups.
problem Analyzing the effects of diffeomorphisms on semi-classical pseudodifferential operators.
method Examined the pull-back of semi-classical pseudodifferential operators by diffeomorphisms preserving the filtration.
result The pull-back of a semi-classical pseudodifferential operator by a Pansu differentiable diffeomorphism has a semi-classical symbol that is expressed in terms of the Pansu differential.
Expands on graded Poisson algebras, their properties, and applications.
problem None explicitly stated; focuses on overview and properties.
method Overview and discussion of properties and applications.
result Provides detailed overview of graded Poisson algebras and their contexts.
Researchers find a Poisson bracket and symplectic structure for field theories.
problem Exploring the Poisson bracket and symplectic structure in the covariant canonical formalism of fields.
method Identifying the phase space as a ringed space with a graded algebra of differential forms, they found a natural Poisson bracket and symplectic structure.
result The Poisson and symplectic structures can be even or odd depending on the manifold's dimension.
We define a differential graded algebra associated to Legendrian knots in Seifert fibered spaces with transverse contact structures. This construction is distinguished from other combinatorial realizations of contact homology invariants by the existence of orbifold points in the Reeb orbit space of the contact manifold…
Introduces Q-structures for mechanics using advanced geometry.
problem Challenges in classical differential geometric constructions in mechanics.
method Explains the use of Q-structures and differential graded manifolds.
result Q-structure preserving integrators can be useful in mechanics.
Classifies homogeneous Pfaffian forms on graded manifolds.
problem Local classification of homogeneous Pfaffian forms on graded manifolds.
method Darboux coordinates and characteristic distribution analysis.
result Darboux-type normal forms for homogeneous Pfaffian forms.
Constructs a new graded variety from algebraic data.
problem Creating a Z-graded extension of differential varieties. method Algorithm using homotopy retract data of Koszul-Tate resolution.
result Significantly reduced number of homological computations.
We introduce the concept of N-differential graded algebras (N-dga), and study the moduli space of deformations of the differential of a N-dga. We prove that it is controlled by what we call the N-Maurer-Cartan equation.