arXiv research

The isotropy action on certain symmetric spaces is shown to be equivariantly formal.

New infinite family of hyperbolic L-space knots with specific semigroups.

Several large classes of homogeneous spaces are known to be formal---in the sense of Rational Homotopy Theory. However, it seems that far fewer examples of non-formal homogeneous spaces are known. In this article we provide several construction principles and characterisations for non-formal homogeneous spaces, which w…

In this paper, we study the formal solution space of a nonlinear PDE in a fiber bundle. To this end, we start with foundational material and introduce the notion of a pfd structure to build up a new concept of profinite dimensional manifolds. We show that the infinite jet space of the fiber bundle is a profinite dimens…

The study confirms that certain symmetric spaces are formal.

We define the notion of a formal connection for a smooth family of star products with fixed underlying symplectic structure. Such a formal connection allows one to relate star products at different points in the family. This generalizes the formal Hitchin connection introduced by the first author. We establish a necess…

We provide examples of homogeneous spaces which are neither symmetric spaces nor real cohomology spheres, yet have the property that every invariant metric is geometrically formal. We also extend the known obstructions to geometric formality to some new classes of homogeneous spaces and of biquotients, and to certain s…

A knot space in a manifold M is a space of oriented immersions from a circle S^1 to M up to Diff(S^1). Brylinski has shown that a knot space of a Riemannian threefold is formally Kahler. We prove that a space of knots in a holonomy G2 manifold is formally Kahler.

We study the formality of the total space of principal SU(2) and SO(3)-bundles over a Wolf space, that is a symmetric positive quaternionic Kähker manifold. We apply this to conclude that all the 3-Sasakian homogeneous spaces are formal. We also determine the principal SU(2) and SO(3)-bundles over the Wolf spaces whose…

We prove that for a fibration of simply-connected spaces of finite type $F\hookrightarrow E\to B$ with $F$ being positively elliptic and $H^*(F,\qq)$ not possessing non-trivial derivations of negative degree, the base $B$ is formal if and only if the total space $E$ is formal. Moreover, in this case the fibration map i…

Strong formal properties for toric and homogeneous Kähler manifolds.

New examples of non-formal Sasaki-Einstein 7-manifolds and their submanifolds found.

We discuss the question of geometric formality for rationally elliptic manifolds of dimension $6$ and $7$. We prove that a geometrically formal six-dimensional biquotient with $b_{2}=3$ has the real cohomology of a symmetric space. We also show that a rationally hyperbolic six-dimensional manifold with $b_2\leq 2$ and …

An action of a compact Lie group is called equivariantly formal, if the Leray--Serre spectral sequence of its Borel fibration degenerates at the E_2-term. This term is as prominent as it is restrictive. In this article, also motivated by the lack of junction between the notion of equivariant formality and the concept o…

Explores local structure of morphisms and formal submanifolds in formal manifolds theory.

Deform moment map on symplectic connections using star product algebras.

The article confirms Joyce's examples of G2-holonomy are formal spaces.

Deform quantization recovers scalar curvature in complex structures.

Develops thermodynamic formalism for quasimorphisms on negatively curved spaces.

We consider analytic curves $\nabla^t$ of symplectic connections of Ricci type on the torus $T^{2n}$ with $\nabla^0$ the standard connection. We show, by a recursion argument, that if $\nabla^t$ is a formal curve of such connections then there exists a formal curve of symplectomorphisms $ψ_t$ such that $ψ_t\cdot\nabla^…

\emph{Scalable spaces} are simply connected compact manifolds or finite complexes whose real cohomology algebra embeds in their algebra of (flat) differential forms. This is a rational homotopy invariant property and all scalable spaces are formal; indeed, scalability can be thought of as a metric version of formality.…

Develops calculus on Wasserstein spaces for Riemannian manifolds.

Study of nonlinear PDEs using derived geometry and BV formalism.

New IRL algorithm for continuous state spaces with formal guarantees.

Survey on finite dimensional Lie groups over real numbers.

Researchers develop weighted GJMS operators for smooth metric measure spaces.

Symplectic forms from two phase spaces are proven equivalent.

Formal normal form created for real-smooth hypersurfaces.

We prove that all generalised symmetric spaces of compact simple Lie groups are formal in the sense of Sullivan. Nevertheless, many of them, including all the non-symmetric flag manifolds, do not admit Riemannian metrics for which all products of harmonic forms are harmonic.

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

New hierarchies derived from KP hierarchy using non-formal operators and Yang-Mills action.

We show that the isotropy action of a homogeneous space $G/K$, where $G$ and $K$ are compact, connected Lie groups and $K$ is defined by an automorphism on $G$, is equivariantly formal and that $(G, K)$ is a Cartan pair.

The paper proves conditions for 2-torus manifolds to be equivariantly formal.

In this work we analyze the behavior of Massey products of closed manifolds under the blow-up construction. The results obtained in the article are applied to the problem of constructing closed symplectic non-formal manifolds. The proofs use Thom spaces as an important technical tool. This application of Thom spaces is…

Two cross caps in Euclidean $3$-space are said to be formally isometric if their Taylor expansions of the first fundamental forms coincide by taking a suitable local coordinate system. For a given $C^\infty$ cross cap $f$, we give a method to find all cross caps which are formally isometric to $f$. As an application, w…

Reformulated sigma models for complex Grassmannians using Gross-Neveu formalism.

We develop a frame and dyad gauge-independent formalism for the calculus of variations of functionals involving spinorial objects. As part of this formalism we define a modified variation operator which absorbs frame and spin dyad gauge terms. This formalism is applicable to both the standard spacetime (i.e. SL(2,C)) 2…

In the Batalin-Vilkovisky formalism, gauge conditions are expressed as Lagrangian submanifolds in the space of fields and antifields. We discuss a way of patching together gauge conditions over different parts of the space of fields, and apply this method to extend the light-cone gauge for the superparticle to a conic …

Study shows how to section map between holonomic and formal solutions.

A quantum field theory for Spin(7)-instantons derived from moduli spaces.

Generalizes embedding formalism for CFTs on curved backgrounds.

New formalization of curved spaces using pointwise affine spaces.

In this paper, we consider formal series associated with events, profiles derived from events, and statistical models that make predictions about events. We prove theorems about realizations for these formal series using the language and tools of Hopf algebras.

We review the basic elements of the geometrical formalism for description of gauge fields and the theory of invariant connections, and their applications to the coset space dimensional reduction of Yang-Mills theories. We also discuss the problem of classification of principal fibre bundles, which is important for the …

This paper explores how local behavior of meromorphic connections on the projective line determines the global connection.

These lectures are an introduction to formal semiclassical quantization of classical field theory. First we develop the Hamiltonian formalism for classical field theories on space time with boundary. It does not have to be a cylinder as in the usual Hamiltonian framework. Then we outline formal semiclassical quantizati…

The differential system for minimal Lagrangian surfaces in a $2_{\mathbb{C}}$-dimensional, non-flat, complex space form is an elliptic system defined on the bundle of oriented Lagrangian planes. This is a 6-symmetric space associated with the Lie group SL(3,$\mathbb{C}$), and the minimal Lagrangian surfaces arise as th…

Using the concept of s-formality we are able to extend the bounds of a Theorem of Miller and show that a compact k-connected 4k+3- or 4k+4-manifold with b_{k+1}=1 is formal. We study k connected n-manifolds, n= 4k+3, 4k+4, with a hard Lefschetz-like property and prove that in this case if b_{k+1}=2, then the manifold i…