Partha Sarathi Chakraborty
Compact quantum metric spaces are order unit spaces along with a Lip norm. On the order unit space of the selfadjoint elements of the dense subalgebra of smooth elements in the quantum Heisenberg manifold we construct Lip norms.
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Every Heisenberg manifold has a natural "sub-Riemannian" metric with interesting properties. We describe the corresponding noncommutative metric structure for Rieffel's quantum Heisenberg manifolds.
The quantum Heisenberg manifolds are noncommutive manifolds constructed by M. Rieffel as strict deformation quantizations of Heisenberg manifolds and have been studied by various authors. Rieffel constructed the quantum Heisenberg manifolds as the generalized fixed-point algebras of certain crossed product $C^*$-algebras, and they also can be realized as crossed products of $C(\mathbb{T}^2)$ by Hilbert $C^*$-bimodules in the sense of Abadie et al. In this paper, we describe how the quantum Heisenberg manifolds can also be realized as twisted groupoid $C^*$-algebras.
We discuss Morita equivalence within the family of quantum Heisenberg manifolds. The main tool employed is the generalization of a result of P. Green and M. Rieffel about Morita equivalence of transformation groups to crossed products by Hilbert bimodules.
A study of Hilbert $C^*$-bimodules over commutative $C^*$-algebras is carried out and used to establish a sufficient condition for two quantum Heisenberg manifolds to be isomorphic.
In this paper, we study the Yang-Mills functional on quantum Heisenberg manifolds using the appratuses developed by A. Connes and M. Rieffel. It is discovered that a connection on a projective module over a quantum Heisenberg manifold is a minimum of Yang-Mills functional whicih is a critical point that is different with critical points found by S. Kang.
In this paper, we discuss the Yang-Mills functional and a certain family of its critical points on quantum Heisenberg manifolds using noncommutative geometrical methods developed by A. Connes and M. Rieffel. In our main result, we construct a certain family of connections on a projective module over a quantum Heisenberg manifold that give rise to critical points of the Yang-Mills functional. Moreover, we show that this set of solutions can be described as a set of solutions to Laplace's equation on quantum Heisenberg manifolds.
By means of techniques from the Morita equivalence theory, we get finitely generated and projective modules over the quantum Heisenberg manifolds. This enables us to get some information about the range of the trace of these algebras, at the level of the K_{0}-group, in terms of the Poisson bracket in whose direction the manifolds are deformed.
The C*-algebras called Quantum Heisenberg Manifolds (QHM) were introduced by Rieffel in 1989 as strict deformation quantizations of Heisenberg manifolds. In this article, we compute the pairings of K-theory and cyclic cohomology on the QHM. Combining these calculations with other results proved elsewhere, we also determine the periodic cyclic homology and cohomology of these algebras, and obtain explicit bases of the periodic cyclic cohomology of the QHM. We further isolate bases of periodic cyclic homology, expressed as Chern characters of the K-theory.
We embed the quantum Heisenberg manifold in a crossed product algebra. This enables us to show that, in the irrational case, all tracial states on $\dc$ induce the same homomorphism on the K_0-group. We conclude that two irrational quantum Heisenberg manifolds $\dc$ and $D^c_{μ' ν'}$ are isomorphic if and only if the parameters $(μ,ν)$ and $(μ',ν')$ belong to the same orbit under the usual action of $GL_2(\ZZ)$ on the torus.
In the noncommutative geometry program of Connes there are two variations of the concept of Yang-Mills action functional. We show that for the quantum Heisenberg manifolds they agree.
For the quantum Heisenberg manifolds, using the action of Heisenberg group we construct a family of spectral triples. It is shown that associated Kasparov module is same for all these spectral triples. Then we show that element is nontrivial by explicitly computing the pairing with a unitary. Space of forms as introduced by Connes are also computed. We characterize connections that are torsionless or unitary in the sense of Frohlich et.al. Then it is shown that a torsionless unitary connection cannot exist. For some unitary connections we compute Ricci curvature and scalar curvature. This prod…
As a step toward proving an index theorem for hypoelliptic operators Heisenberg manifolds, including those on CR and contact manifolds, we construct an analogue for Heisenberg manifolds of Connes' tangent groupoid of a manifold $M$. As it is well known for a Heisenberg manifold $(M,H)$ the relevant notion of tangent is rather that of Lie group bundle of graded 2-step nilpotent Lie groups $GM$. We then construct the tangent groupoid of $(M,H)$ as a differentiable groupoid $\cG_{H} M$ encoding the smooth deformation of $M\times M$ to $GM$. In this construction a crucial use is made of a refined …
We prove six-term exact sequences of Pimsner-Voiculescu type for certain subalgebras of the Cuntz-Pimsner algebras. This sequence may, in particular, be applied to smooth subalgebras of the Quantum Heisenberg Manifolds in order to compute the generators of their cyclic cohomology. Further, our results include the known results for smooth crossed products. Our proof is based on a combination of arguments from the setting of (Cuntz-)Pimsner algebras and the Toeplitz proof of Bott-periodicity.
We construct a spectral triple on a quantum Heisenberg manifold, which generalizes the results of Chakraborty and Shinha, and associate to it an energy functional on the set of projections, following the approach of Mathai-Rosenberg to non-linear sigma models. The spectral triples that we construct extend the We derive a lower bound for this energy functional that is linked on the topological charge of the projection which depends on the curvature of a compatible connection. A detailed study of this lower bound is given for the Kang projection in quantum Heisenberg manifolds. These results dis…
The modular Gromov-Hausdorff propinquity is a distance on classes of modules endowed with quantum metric information, in the form of a metric form of a connection and a left Hilbert module structure. This paper proves that the family of Heisenberg modules over quantum two tori, when endowed with their canonical connections, form a continuous family for the modular propinquity.
The modular Gromov-Hausdorff propinquity is a distance on classes of modules endowed with quantum metric information, in the form of a metric form of a connection and a left Hilbert module structure. This paper proves that the family of Heisenberg modules over quantum two tori, when endowed with their canonical connections, form a family of metrized quantum vector bundles, as a first step in proving that Heisenberg modules form a continuous family for the modular Gromov-Hausdorff propinquity.
This paper has four main parts. In the first part, we construct a noncommutative residue for the hypoelliptic calculus on Heisenberg manifolds, that is, for the class of Heisenberg PsiDOs introduced by Beals-Greiner and Taylor. This noncommutative residue appears as the residual trace on integer order Heisenberg PsiDOs induced by the analytic extension of the usual trace to non-integer order Heisenberg PsiDOs. Moreover, it agrees with the integral of the density defined by the logarithmic singularity of the Schwartz kernel of the corresponding Heisenberg PsiDO. In addition, we show that this n…
This article studies sharp norm estimates for the commutator of pseudo-differential operators with multiplication operators on closed Heisenberg manifolds. In particular, we obtain a Calderon commutator estimate: If $D$ is a first-order operator in the Heisenberg calculus and $f$ is Lipschitz in the Carnot-Caratheodory metric, then $[D,f]$ extends to an $L^2$-bounded operator. Using interpolation, it implies sharp weak--Schatten class properties for the commutator between zeroth order operators and Hölder continuous functions. We present applications to sub-Riemannian spectral triples on Heise…
Motivated by the search for new examples of ``noncommutative manifolds'', we study the noncommutative geometry (in the sense of Connes) of the group C*-algebra of the three dimensional discrete Heisenberg group. We present a unified treatment of the K-homology, cyclic cohomology and derivations of this algebra.
We present a new solution to the index problem for hypoelliptic operators in the Heisenberg calculus on contact manifolds, by constructing the appropriate topological K-theory cocycle for such operators. Its Chern character gives a cohomology class to which the Atiyah-Singer index formula can be applied. Such a K-cocycle has already been constructed by Boutet de Monvel for Toeplitz operators, and, more recently, by Melrose and Epstein for the class of Hermite operators. Our construction applies to general hypoelliptic pseudodifferential operators in the Heisenberg calculus. As in the Hermite I…
We investigate critical points and minimizers of the Yang-Mills functional YM on quantum Heisenberg manifolds $D^c_{μν}$, where the Yang-Mills functional is defined on the set of all compatible linear connections on finitely generated projective modules over the QHMs. A compatible linear connection which is both a critical point and minimizer of YM is called a Yang-Mills connection. In this paper, we investigate Yang-Mills connections with constant curvature. We are interested in Yang-Mills connections on the following classes of modules over the QHMs: (i) Abadie's module $Ξ$ of trace $2μ$ and…
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