- Søborg, Hovedstaden, Denmark
Ryszard Nest
University of Copenhagen, Mathematics, Faculty Member
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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... more
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 ,i f 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 re- sults 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 .
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Summary: We discuss the behaviour of a superconducting weak link ring coupled to a radio frequency tank circuit. We show that with the weak link acting quantum mechanically to connect different winding number states of the ring together... more
Summary: We discuss the behaviour of a superconducting weak link ring coupled to a radio frequency tank circuit. We show that with the weak link acting quantum mechanically to connect different winding number states of the ring together two limiting regimes ofbehaviour are ...
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We prove a $\unicode[STIX]{x1D6E4}$ -equivariant version of the algebraic index theorem, where $\unicode[STIX]{x1D6E4}$ is a discrete group of automorphisms of a formal deformation of a symplectic manifold. The particular cases of this... more
We prove a $\unicode[STIX]{x1D6E4}$ -equivariant version of the algebraic index theorem, where $\unicode[STIX]{x1D6E4}$ is a discrete group of automorphisms of a formal deformation of a symplectic manifold. The particular cases of this result are the algebraic version of the transversal index theorem related to the theorem of A. Connes and H. Moscovici for hypo-elliptic operators and the index theorem for the extension of the algebra of pseudodifferential operators by a group of diffeomorphisms of the underlying manifold due to A. Savin, B. Sternin, E. Schrohe and D. Perrot.
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In this paper we introduce a notion of quantum Hamiltonian (co)action of Hopf algebras endowed with Drinfel'd twist structure (resp., 2-cocycles). First, we define a classical Hamiltonian action in the setting of Poisson Lie groups... more
In this paper we introduce a notion of quantum Hamiltonian (co)action of Hopf algebras endowed with Drinfel'd twist structure (resp., 2-cocycles). First, we define a classical Hamiltonian action in the setting of Poisson Lie groups compatible with the 2-cocycle stucture and we discuss a concrete example. This allows us to construct, out of the classical momentum map, a quantum momentum map in the setting of Hopf coactions and to quantize it by using Drinfel'd approach.
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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... more
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 [Formula: see text] 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 [Formula: see text], 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.
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... 1 tO B. Durhuus el M. / Topological Quantum Field Theory from 6j-symbols boundary components ~t,..., E. is associated a finite dimensional complex Hi!bert space VoM =Vz, '" V~., (I) where 8M denotes the oriented ... Haahr... more
... 1 tO B. Durhuus el M. / Topological Quantum Field Theory from 6j-symbols boundary components ~t,..., E. is associated a finite dimensional complex Hi!bert space VoM =Vz, '" V~., (I) where 8M denotes the oriented ... Haahr Andersen, in preparation, [21] H. Wenz, in preparation. ...
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... The Heisenberg Group and K-Theory GEORGE A. ELLIOTT Mathematics Institute, Universitetsparken 5, DK-2100 Copenhagen O, Demnark and Department of Mathematics, University of Toronto, Toronto, Canada M5S IAt ... THE HEISENBERG GROUP AND... more
... The Heisenberg Group and K-Theory GEORGE A. ELLIOTT Mathematics Institute, Universitetsparken 5, DK-2100 Copenhagen O, Demnark and Department of Mathematics, University of Toronto, Toronto, Canada M5S IAt ... THE HEISENBERG GROUP AND K-THEORY ...
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Abstract. Let Г be a discrete subgroup of PSL (2, R) of infinite covolume with infinite conjugacy classes." Ht denotes the Hilbert space consisting of analytic functions in L2 (D,(Im z)'~ 2dzdz) and, for t> 1, nt denotes the... more
Abstract. Let Г be a discrete subgroup of PSL (2, R) of infinite covolume with infinite conjugacy classes." Ht denotes the Hilbert space consisting of analytic functions in L2 (D,(Im z)'~ 2dzdz) and, for t> 1, nt denotes the corresponding projective unitary representation of P5L (2, R) on this Hubert space. Let At be the J/oo factor given by the commutant of тг ((Г) in B (Ht). Let F denote a fundamental domain for Г in D. We assume that í> 5 and give дМ= Ж> Г¡ F the topology of disjoint union of its connected components. Suppose that/is a ...
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Fof a nilpotent differential graded Lie algebra whose components vanish in degrees below -1 we construct an explicit equivalence between the nerve of the Deligne 2-groupoid and the simplicial set of differential forms with values in the... more
Fof a nilpotent differential graded Lie algebra whose components vanish in degrees below -1 we construct an explicit equivalence between the nerve of the Deligne 2-groupoid and the simplicial set of differential forms with values in the Lie algebra introduced by V.Hinich. The construction uses the theory of non-abelian multiplicative integration.
Research Interests: Mathematics and Holonomy
In the setting of several commuting operators on a Hilbert space one defines the notions of invertibility and Fredholmness in terms of the associated Koszul complex. The index problem then consists of computing the Euler characteristic of... more
In the setting of several commuting operators on a Hilbert space one defines the notions of invertibility and Fredholmness in terms of the associated Koszul complex. The index problem then consists of computing the Euler characteristic of such a special type of Fredholm complex. In this paper we investigate transformation rules for the index under the holomorphic functional calculus. We distinguish between two different types of index results: 1) A global index theorem which expresses the index in terms of the degree function of the "symbol" and the locally constant index function of the "coordinates". 2) A local index theorem which computes the Euler characteristic of a localized Koszul complex near a common zero of the "symbol". Our results apply to the example of Toeplitz operators acting on both Bergman spaces over pseudoconvex domains and the Hardy space over the polydisc. The local index theorem is fundamental for future investigations of determin...
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In this paper we study continuous bundles of C*-algebras which are non-commutative analogues of principal torus bundles. We show that all such bundles, although in general being very far away from being locally trivial bundles, are at... more
In this paper we study continuous bundles of C*-algebras which are non-commutative analogues of principal torus bundles. We show that all such bundles, although in general being very far away from being locally trivial bundles, are at least locally trivial with respect to a suitable bundle version of bivariant K-theory (denoted RKK-theory) due to Kasparov. Using earlier results of Echterhoff and Williams, we shall give a complete classification of principal non-commutative torus bundles up to equivariant Morita equivalence. We then study these bundles as topological fibrations (forgetting the group action) and give necessary and sufficient conditions for any non-commutative principal torus bundle being RKK-equivalent to a commutative one. As an application of our methods we shall also give a K-theoretic characterization of those principal torus-bundles with H-flux, as studied by Mathai and Rosenberg which possess "classical" T-duals.
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We prove a Riemann-Roch formula for deformation quantization of complex manifolds and its corollary, an index theorem for elliptic pairs conjectured by Schapira and Schneiders.