Let A be a star product on a symplectic manifold (M,ω0), 1 t [ω] its Fedosov class, where ω is a deformation of ω0. We prove that for a complex polarization of ω there exists a commutative subalgebra, O, in A that is isomorphic to the algebra of functions constant along the polarization. Let F (A) consists of elements of A whose commutator with O belongs to O. Then, F (A) is a Lie algebra which...

The minimal fractal manifold (MFM) defines a space-time continuum endowed with arbitrarily small deviations from four-dimensions 4 , ( D     << 1). It was recently shown that MFM is a natural consequence of the Renormalization Group which brings up a series of unforeseen solutions to the challenges raised by the Standard Model. In this brief report we argue that MFM may be treated as asympt...

There is an equivalence relation on the set of smooth maps of a manifold into the stable unitary group, defined using a Chern-Simons type form, whose equivalence classes form an abelian group under ordinary block sum of matrices. This construction is functorial, and defines a differential extension of odd K-theory, fitting into natural commutative diagrams and exact sequences involving K-theory...

The notion of convergent star product is generally understood as the data of a one parameter family {Et}t∈I ⊂ C∞(M) of function algebras on a Poisson manifold (M, { , }). On each of them one is given an associative algebra structure ⋆t which respect to which the function space Et is closed. The family of products {⋆t} should moreover define in some sense a deformation of the commutative pointwi...

We describe all FMm1,m2,n1,n2 -natural operators D : Qproj-proj QT ∗ transforming projectable-projectable classical torsion-free linear connections ∇ on fibred-fibred manifolds Y into classical linear connections D(∇) on cotangent bundles T ∗Y of Y . We show that this problem can be reduced to finding FMm1,m2,n1,n2 -natural operators D : Qproj-proj (T ∗,⊗pT ∗⊗⊗qT ) for p = 2, q = 1 and p = 3, q...

We present some ideas for a possible Noncommutative Floer Homology. The geometric motivation comes from an attempt to build a theory which applies to practically every 3-manifold (closed, oriented and connected) and not only to homology 3-spheres. There is also a physical motivation: one would like to construct a noncommutative topological quantum field theory. The two motivations are closely r...

We introduce a higher dimensional generalization of the affine Kac-Moody algebra using language factorization algebras. In particular, on any complex manifold there is currents associated to Lie algebra. classify local cocycles these current algebras, and compare them central extensions algebras recently proposed by Faonte-Hennion-Kapranov. A goal this paper witness as symmetries class holomorp...

Let M be a closed, oriented, n-manifold, and LM its free loop space. In [4] a commutative algebra structure in homology, H∗(LM), and a Lie algebra structure in equivariant homology H 1 ∗ (LM), were defined. In this paper we prove that these structures are homotopy invariants in the following sense. Let f : M1 → M2 be a homotopy equivalence of closed, oriented n-manifolds. Then the induced equiv...

Unless specified otherwise, by “manifolds” and “varieties” we shall always mean, respectively, complex manifolds and complex algebraic varieties. Consequently, “dimension” (of a manifold or variety) always refers to the complex dimension. We also fix, once and for all, a commutative Noetherian ring k of finite global dimension as our ring of “coefficients”; the reader is welcome to take k = Z,Q...

Let M be a closed, oriented, n-manifold, and LM its free loop space. In [3] a commutative algebra structure in homology, H∗(LM), and a Lie algebra structure in equivariant homology H 1 ∗ (LM), were defined. In this paper we prove that these structures are homotopy invariants in the following sense. Let f : M1 → M2 be a homotopy equivalence of closed, oriented n-manifolds. Then the induced equiv...