Ryszard Nest

Algebraic index theorem

We prove the Atiyah-Singer index theorem where the algebra of pseudodifferential operators is replaced by an arbitrary deformation quantization of the algebra of functions on a symplectic manifold.

Riemann–Roch Theorems via Deformation Quantization, I

We deduce the Riemann-Roch type formula expressing the microlocal Euler class of a perfect complex of D-modules in terms of the Chern character of the associated symbol complex and the Todd class of the manifold from the Riemann-Roch type theorem for periodic cyclic cocycles of a symplectic deformation quantization. The proof of the latter is contained in the sequel to this paper.

Triangulated Categories: Homological algebra in bivariant K-theory and other triangulated categories. I

We use homological ideals in triangulated categories to get a sufficient criterion for a pair of subcategories in a triangulated category to be complementary. We apply this criterion to construct the Baum-Connes assembly map for locally compact groups and torsion-free discrete quantum groups. Our methods are related to the abstract version of the Adams spectral sequence by Brinkmann and Christensen.

On spectral triples in quantum gravity: I

This paper establishes a link between noncommutative geometry and canonical quantum gravity. A semi-finite spectral triple over a space of connections is presented. The triple involves an algebra of holonomy loops and a Dirac-type operator, which resembles a global functional derivation operator. The commutation relation between the Dirac operator and the algebra has a structure related to the Poisson bracket of general relativity. Moreover, the associated Hilbert space corresponds, up to a certain symmetry group, to the Hilbert space of diffeomorphism-invariant states known from loop quantum gravity. Correspondingly, the square of the Dirac operator has, in terms of loop quantum gravity, the form of a global area-squared operator. Furthermore, the spectral action functional resembles a partition function of quantum gravity. The construction is background independent and is based on an inductive system of triangulations. This paper is the first of two papers on the subject.

C ∗ -ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY

We define the filtrated K-theory of a C � -algebra over a finite topo- logical space X and explain how to construct a spectral sequence that computes the bivariant Kasparov theory over X in terms of filtrated K-theory. For finite spaces with totally ordered lattice of open subsets, this spectral sequence becomes an exact sequence as in the Universal Coefficient Theorem, with the same consequences for classification. We also exhibit an example where filtrated K-theory is not yet a complete invariant. We describe two C � -algebras over a space X with four points that have isomorphic filtrated K-theory without being KK(X)-equivalent. For this space X, we enrich filtrated K-theory by another K-theory functor to a com- plete invariant up to KK(X)-equivalence that satisfies a Universal Coefficient Theorem.

Topological quantum field theories from generalized 6j symbols

Given an associative algebra with a distinguished finite set of representations that is closed under a (deformed) tensor product, and satisfies some technical assumptions, we define generalized 6j-symbols, and show that they can be associated, in a natural way, with certain labeled tetrahedra. Given a 3-dimensional compact oriented manifold M with boundary ∂M = Σ we choose an arbitrary triangulation of M and exploit the above correspondence between 6j-symbols and labeled tetrahedra to construct a vectorspace UΣ and a vector Z(M) ∈ UΣ, independent of , and fulfilling the axioms of a topological quantum field theory as formulated by Atiyah [11]. Examples covered by our approach are quantum groups corresponding to the classical simple Lie algebras as well as, expectedly, chiral algebras of 2-dimensional rational conformal field theories.

A New Spectral Triple over a Space of Connections

A new construction of a semifinite spectral triple on an algebra of holonomy loops is presented. The construction is canonically associated to quantum gravity and is an alternative version of the spectral triple presented in [1].

On Spectral Triples in Quantum Gravity II

A semifinite spectral triple for an algebra canonically associated to canonical quantum gravity is constructed. The algebra is generated by based loops in a triangulation and its barycentric subdivisions. The underlying space can be seen as a gauge fixing of the unconstrained state space of Loop Quantum Gravity. This paper is the second of two papers on the subject. email: johannes.aastrup@uni-muenster.de email: grimstrup@nbi.dk email: rnest@math.ku.dk

Noncommutative Scalar Solitons: Existence and Nonexistence

We study the variational equations for solitons in noncommutative scalar field theories in an even number of spatial dimensions. We prove the existence of spherically symmetric solutions for a sufficiently large noncommutativity parameter θ and we prove the absence of spherically symmetric solutions for small θ.

The Existence and Stability of Noncommutative Scalar Solitons

Abstract: We establish existence and stability results for solitons in noncommutative scalar field theories in even space dimension 2d. In particular, for any finite rank spectral projection P of the number operator 𝒩 of the d-dimensional harmonic oscillator and sufficiently large noncommutativity parameter θ we prove the existence of a rotationally invariant soliton which depends smoothly on θ and converges to a multiple of P as θ→∞. In the two-dimensional case we prove that these solitons are stable at large θ, if P=PN, where PN projects onto the space spanned by the N+1 lowest eigenstates of 𝒩, and otherwise they are unstable. We also discuss the generalisation of the stability results to higher dimensions. In particular, we prove stability of the soliton corresponding to P=P0 for all θ in its domain of existence. Finally, for arbitrary d and small values of θ, we prove without assuming rotational invariance that there do not exist any solitons depending smoothly on θ.