arXiv research

The goal of this note is to give a brief overview of the BV-BFV formalism developed by the first two authors and Reshetikhin in [arXiv:1201.0290], [arXiv:1507.01221] in order to perform perturbative quantisation of Lagrangian field theories on manifolds with boundary, and present a special case of Chern-Simons theory a…

Extended equivariant BV formalism to manifolds with boundaries.

In this thesis, we study the deformation problem of coisotropic submanifolds in Jacobi manifolds. In particular we attach two algebraic invariants to any coisotropic submanifold $S$ in a Jacobi manifold, namely the $L_\infty[1]$-algebra and the BFV-complex of $S$. Our construction generalizes and unifies analogous cons…

Generalizes momentum map to Courant algebroid for constrained mechanics.

This paper introduces a general perturbative quantization scheme for gauge theories on manifolds with boundary, compatible with cutting and gluing, in the cohomological symplectic (BV-BFV) formalism. Explicit examples, like abelian BF theory and its perturbations, including nontopological ones, are presented.

A geometric multisymplectic formulation of the classical BRST symmetry of constrained first-order classical field theories is described. To effect this we introduce graded analogues of the bundles and manifolds of the multisymplectic formulation of first-order field theories. The Lagrange-d'Alembert formalism is also d…

We extend the construction of the BFV-complex of a coisotropic submanifold from the Poisson setting to the Jacobi setting. In particular, our construction applies in the contact and l.c.s. settings. The BFV-complex of a coisotropic submanifold $S$ controls the coisotropic deformation problem of $S$ under both Hamiltoni…

We consider the local deformation problem of coisotropic submanifolds inside Poisson manifolds. To this end the groupoid of coisotropic sections (with respect to some tubular neighbourhood) is introduced. Although the geometric content of this groupoid is evident, it is usually a very intricate object. We provide a des…

We observe that a system of irreducible, fiber-linear, first class constraints on T*M is equivalent to the definition of a foliation Lie algebroid over M. The BFV formulation of the constrained system is given by the Hamiltonian lift of the Vaintrob description (E[1],Q) of the Lie algebroid to its cotangent bundle T*E[…

Abstract: Homotopy Poisson algebra models for reduced spaces derived from Poisson structures.

Constructs Lie-Rinehart algebra for Einstein's equations.

The ``classical BRST construction'' as developed by Batalin-Fradkin-Vilkovisky is a homological construction for the reduction of the Poisson algebra $P = C^\infty (W)$ of smooth functions on a Poisson manifold $W$ by the ideal $I$ of functions which vanish on a constraint locus. This ideal is called first class if $I$…

The relationship is established between the Fedosov deformation quantization of a general symplectic manifold and the BFV-BRST quantization of constrained dynamical systems. The original symplectic manifold $\mathcal M$ is presented as a second class constrained surface in the fibre bundle ${{\mathcal T}^*_ρ}{\mathcal …

We propose an explicit construction of the deformation quantization of the general second-class constrained system, which is covariant with respect to local coordinates on the phase space. The approach is based on constructing the effective first-class constraint (gauge) system equivalent to the original second-class o…

A $Q$-manifold $M$ is a supermanifold endowed with an odd vector field $Q$ squaring to zero. The Lie derivative $L_Q$ along $Q$ makes the algebra of smooth tensor fields on $M$ into a differential algebra. In this paper, we define and study the invariants of $Q$-manifolds called characteristic classes. These take value…

Establishes Poincaré's lemma for formal manifolds.

We define and study invariants which can be uniformly constructed for any gauge system. By a gauge system we understand an (anti-)Poisson supermanifold provided with an odd Hamiltonian self-commuting vector field called a homological vector field. This definition encompasses all the cases usually included into the noti…

Foundations laid for formal manifolds in differential geometry.

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

Study non-formal pseudo-differential operators over formal ones.

Strong formal properties for toric and homogeneous Kähler manifolds.

A metric is formal if all products of harmonic forms are again harmonic. The existence of a formal metric implies Sullivan formality of the manifold, and hence formal metrics can exist only in presence of a very restricted topology. We show that a warped product metric is formal if and only if the warping function is c…

The study shows strong formality in certain complex manifolds.

Formal manifolds with non-negative Ricci curvature have formal covers.

Compact symmetric spaces are probably one of the most prominent class of formal spaces, i.e. of spaces where the rational homotopy type is a formal consequence of the rational cohomology algebra. As a generalisation, it is even known that their isotropy action is equivariantly formal. In this article we show that $(\ma…

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

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…

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

Formality of Dolbeault DGAs on complex nilmanifolds restricted to tori.

Study on geometrically formal metrics on complex manifolds.

Research on formality problem for special holonomy manifolds.

Defines formal vertex laws related to Lie conformal algebras.

Deform quantization recovers scalar curvature in complex structures.

Formal methods verify continuous auctions at exchanges.

We describe a canonical form for linear differential operators that are formally self-adjoint or formally skew-adjoint.

There are solved standard problems related to Formal (Holomorphic) Segre preserving Mappings of non-trivial Real-Formal Hypersurfaces in $\mathbb{C}^{2}$.

Derives localization formulas in Batalin-Vilkovisky formalism.

Proves formal self-adjointness of certain differential operators.

New findings on complex manifold properties under deformations.

A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to …

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…

This paper formalizes manifolds in positive characteristic varieties.

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…

This paper explores formal verification for autonomous systems, identifying limitations and proposing improvements.

We investigate harmonic forms of geometrically formal metrics, which are defined as those having the exterior product of any two harmonic forms still harmonic. We prove that a formal Sasakian metric can exist only on a real cohomology sphere and that holomorphic forms of a formal Kähler metric are parallel w.r.t. the L…

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 …

We recall the construction of non-formal deformation quantization of the Poincare Group ISO(1,1) on its coadjoint orbit and exhibit the associated non-formal star-exponentials.

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…