We report on the Lagrangian statistics of acceleration of small (sub-Kolmogorov) bubbles and trac... more We report on the Lagrangian statistics of acceleration of small (sub-Kolmogorov) bubbles and tracer particles with Stokes number St≪1 in turbulent flow. At a decreasing Reynolds number, the bubble accelerations show deviations from that of tracer particles; i.e., they deviate from the Heisenberg-Yaglom prediction and show a quicker decorrelation despite their small size and minute St. Using direct numerical simulations, we show that these effects arise due the drift of these particles through the turbulent flow. We theoretically predict this gravity-driven effect for developed isotropic turbulence, with the ratio of Stokes to Froude number or equivalently the particle drift velocity governing the enhancement of acceleration variance and the reductions in correlation time and intermittency. Our predictions are in good agreement with experimental and numerical results. The present findings are relevant to a range of scenarios encompassing tiny bubbles and droplets that drift through the turbulent oceans and the atmosphere. They also question the common usage of microbubbles and microdroplets as tracers in turbulence research.
In this paper, we initiate the study of nonassociative strict deformation quantization of C*-alge... more In this paper, we initiate the study of nonassociative strict deformation quantization of C*-algebras with a torus action. We shall also present a definition of nonassociative principal torus bundles, and give a classification of these as nonassociative strict deformation quantization of ordinary principal torus bundles. We then relate this to T-duality of principal torus bundles with H-flux. In particular, the Octonions fit nicely into our theory.
In this paper, we initiate the study of a parametrised version of Rieffel's strict deformation qu... more In this paper, we initiate the study of a parametrised version of Rieffel's strict deformation quantization. We apply it to give a classification of noncommutative principal torus bundles, in terms of parametrised strict deformation quantization of ordinary principal torus bundles. The paper also contains a putative definition of noncommutative non-principal torus bundles.
In this paper, we use the parametrised strict deformation quantization of C*-bundles obtained in ... more In this paper, we use the parametrised strict deformation quantization of C*-bundles obtained in a previous paper, and give more examples and applications of this theory. In particular, it is used here to classify H_3-twisted noncommutative torus bundles over a locally compact space. This is extended to the case of general torus bundles and their parametrised strict deformation quantization. Rieffel's basic construction of an algebra deformation can be mimicked to deform a monoidal category, which deforms not only algebras but also modules. As a special case, we consider the parametrised strict deformation quantization of Hilbert C*-modules over C*-bundles with fibrewise torus action.
Recently, Diaconescu, Moore and Witten provided a nontrivial link between K-theory and M-theory, ... more Recently, Diaconescu, Moore and Witten provided a nontrivial link between K-theory and M-theory, by deriving the partition function of the Ramond-Ramond fields of Type IIA string theory from an E8 gauge theory in eleven dimensions. We give some relations between twisted K-theory and M-theory by adapting the method of Diaconescu-Moore-Witten and Moore-Saulina. In particular, we construct the twisted K-theory torus which defines the partition function, and also discuss the problem from the E8 loop group picture, in which the Dixmier-Douady class is the Neveu-Schwarz field. In the process of doing this, we encounter some mathematics that is new to the physics literature. In particular, the eta differential form, which is the generalization of the eta invariant, arises naturally in this context. We conclude with several open problems in mathematics and string theory.
In this paper we show that the T-duality transform of Bouwknegt, Evslin and Mathai applies to det... more In this paper we show that the T-duality transform of Bouwknegt, Evslin and Mathai applies to determine isomorphisms of certain current algebras and their associated vertex algebras on topologically distinct T-dual spacetimes compactified to circle bundles with $H$-flux.
A central result here is the computation of the entire cyclic homol- ogy of canonical smooth suba... more A central result here is the computation of the entire cyclic homol- ogy of canonical smooth subalgebras of stable continuous trace C�-algebras having smooth manifolds M as their spectrum. More precisely, the entire cyclic homology is shown to be canonically isomorphic to the continuous pe- riodic cyclic homology for these algebras. By an earlier result of the authors, one concludes
In this paper, we quantize universal gauge groups such as SU(\infty), in the sigma-C*-algebra set... more In this paper, we quantize universal gauge groups such as SU(\infty), in the sigma-C*-algebra setting. More precisely, we propose a concise definition of sigma-C*-quantum groups and explain the concept here. At the same time, we put this definition in the mathematical context of countably compactly generated groups as well as C*-compact quantum groups.
We study the geometry and topology of (filtered) algebra-bundles ${\bf\Psi}^{\mathbb Z}$ over a s... more We study the geometry and topology of (filtered) algebra-bundles ${\bf\Psi}^{\mathbb Z}$ over a smooth manifold $X$ with typical fibre $\Psi^{\mathbb Z}(Z; V)$, the algebra of classical pseudodifferential operators of integral order on the compact manifold $Z$ acting on smooth sections of a vector bundle $V$. First a theorem of Duistermaat and Singer is generalized to the assertion that the group of projective invertible Fourier integral operators ${\rm PGL}({\mathcal F}^\bullet(Z; V))$, is precisely the automorphism group, ${\rm Aut}(\Psi^{\mathbb Z}(Z; V)),$ of the filtered algebra of pseudodifferential operators. We replace some of the arguments in their paper by microlocal ones, thereby removing the topological assumption well as extending their result to sections of a vector bundle. We define a natural class of connections and B-fields the principal bundle to which ${\bf\Psi}^{\mathbb Z}$ is associated and obtain a de Rham representative of the Dixmier-Douady class, in terms of the outer derivation on the Lie algebra and the residue trace of Guillemin and Wodzicki; the resulting formula only depends on the formal symbol algebra ${\bf\Psi}^{\mathbb Z}/{\bf\Psi}^{-\infty}.$ Examples of pseudodifferential algebra bundles are given that are not associated to a finite dimensional fibre bundle over $X.$
We study both the continuous model and the discrete model of the integer quantum Hall effect on t... more We study both the continuous model and the discrete model of the integer quantum Hall effect on the hyperbolic plane in the presence of disorder, extending the results of an earlier paper [CHMM]. Here we model impurities, that is we consider the effect of a random or almost periodic potential as opposed to just periodic potentials. The Hall conductance is identified as a geometric invariant associated to an algebra of observables, which has plateaus at gaps in extended states of the Hamiltonian. We use the Fredholm modules defined in [CHMM] to prove the integrality of the Hall conductance in this case. We also prove that there are always only a finite number of gaps in extended states of any random discrete Hamiltonian. [CHMM] A. Carey, K. Hannabuss, V. Mathai and P. McCann, Quantum Hall Effect on the Hyperbolic Plane, Communications in Mathematical Physics, 190 vol. 3, (1998) 629-673.
Proceedings of the American Mathematical Society, Apr 26, 2010
Inspired by the work of Boris Vertman on refined analytic torsion for manifolds with boundary, in... more Inspired by the work of Boris Vertman on refined analytic torsion for manifolds with boundary, in this paper we extend the construction of the Cappell-Miller analytic torsion to manifolds with boundary. We also compare it with the refined analytic torsion on manifolds with boundary. As a byproduct of the gluing formula for refined analytic torsion and the comparison theorem for the Cappell-Miller analytic torsion and the refined analytic torsion, we establish the gluing formula for the Cappell-Miller analytic torsion in the case that the Hermitian metric is flat.
In this note we introduce the notion of bundle gerbe K-theory and investigate the relation to twi... more In this note we introduce the notion of bundle gerbe K-theory and investigate the relation to twisted K-theory. We provide some examples. Possible applications of bundle gerbe K-theory to the classification of D-brane charges in non-trivial backgrounds are discussed.
We introduce spherical T-duality, which relates pairs of the form $(P,H)$ consisting of a princip... more We introduce spherical T-duality, which relates pairs of the form $(P,H)$ consisting of a principal $SU(2)$-bundle $P\rightarrow M$ and a 7-cocycle $H$ on $P$. Intuitively spherical T-duality exchanges $H$ with the second Chern class $c_2(P)$. Unless $dim(M)\leq 4$, not all pairs admit spherical T-duals and the spherical T-duals are not always unique. Nonetheless, we prove that all spherical T-dualities induce a degree-shifting isomorphism on the 7-twisted cohomologies of the bundles and, when $dim(M)\leq 7$, also their integral twisted cohomologies and, when $dim(M)\leq 4$, even their 7-twisted K-theories. While spherical T-duality does not appear to relate equivalent string theories, it does provide an identification between conserved charges in certain distinct IIB supergravity and string compactifications.
We had previously defined the rho invariant $\rho_{spin}(Y,E,H, g)$ for the twisted Dirac operato... more We had previously defined the rho invariant $\rho_{spin}(Y,E,H, g)$ for the twisted Dirac operator $\not\partial^E_H$ on a closed odd dimensional Riemannian spin manifold $(Y, g)$, acting on sections of a flat hermitian vector bundle $E$ over $Y$, where $H = \sum i^{j+1} H_{2j+1} $ is an odd-degree differential form on $Y$ and $H_{2j+1}$ is a real-valued differential form of degree ${2j+1}$. Here we show that it is a conformal invariant of the pair $(H, g)$. In this paper we express the defect integer $\rho_{spin}(Y,E,H, g) - \rho_{spin}(Y,E, g)$ in terms of spectral flows and prove that $\rho_{spin}(Y,E,H, g)\in \mathbb Q$, whenever $g$ is a Riemannian metric of positive scalar curvature. In addition, if the maximal Baum-Connes conjecture holds for $\pi_1(Y)$ (which is assumed to be torsion-free), then we show that $\rho_{spin}(Y,E,H, rg) =0$ for all $r\gg 0$, significantly generalizing our earlier results. These results are proved using the Bismut-Weitzenb\"ock formula, a scaling trick, the technique of noncommutative spectral sections, and the Higson-Roe approach.
In this paper, we study both the continuous model and the discrete model of the Quantum Hall Effe... more In this paper, we study both the continuous model and the discrete model of the Quantum Hall Effect (QHE) on the hyperbolic plane. The Hall conductivity is identified as a geometric invariant associated to an imprimitivity algebra of observables. We define a twisted analogue of the Kasparov map, which enables us to use the pairing between K-theory and cyclic cohomology theory, to identify this geometric invariant with a topological index, thereby proving the integrality of the Hall conductivity in this case.
We report on the Lagrangian statistics of acceleration of small (sub-Kolmogorov) bubbles and trac... more We report on the Lagrangian statistics of acceleration of small (sub-Kolmogorov) bubbles and tracer particles with Stokes number St≪1 in turbulent flow. At a decreasing Reynolds number, the bubble accelerations show deviations from that of tracer particles; i.e., they deviate from the Heisenberg-Yaglom prediction and show a quicker decorrelation despite their small size and minute St. Using direct numerical simulations, we show that these effects arise due the drift of these particles through the turbulent flow. We theoretically predict this gravity-driven effect for developed isotropic turbulence, with the ratio of Stokes to Froude number or equivalently the particle drift velocity governing the enhancement of acceleration variance and the reductions in correlation time and intermittency. Our predictions are in good agreement with experimental and numerical results. The present findings are relevant to a range of scenarios encompassing tiny bubbles and droplets that drift through the turbulent oceans and the atmosphere. They also question the common usage of microbubbles and microdroplets as tracers in turbulence research.
In this paper, we initiate the study of nonassociative strict deformation quantization of C*-alge... more In this paper, we initiate the study of nonassociative strict deformation quantization of C*-algebras with a torus action. We shall also present a definition of nonassociative principal torus bundles, and give a classification of these as nonassociative strict deformation quantization of ordinary principal torus bundles. We then relate this to T-duality of principal torus bundles with H-flux. In particular, the Octonions fit nicely into our theory.
In this paper, we initiate the study of a parametrised version of Rieffel's strict deformation qu... more In this paper, we initiate the study of a parametrised version of Rieffel's strict deformation quantization. We apply it to give a classification of noncommutative principal torus bundles, in terms of parametrised strict deformation quantization of ordinary principal torus bundles. The paper also contains a putative definition of noncommutative non-principal torus bundles.
In this paper, we use the parametrised strict deformation quantization of C*-bundles obtained in ... more In this paper, we use the parametrised strict deformation quantization of C*-bundles obtained in a previous paper, and give more examples and applications of this theory. In particular, it is used here to classify H_3-twisted noncommutative torus bundles over a locally compact space. This is extended to the case of general torus bundles and their parametrised strict deformation quantization. Rieffel's basic construction of an algebra deformation can be mimicked to deform a monoidal category, which deforms not only algebras but also modules. As a special case, we consider the parametrised strict deformation quantization of Hilbert C*-modules over C*-bundles with fibrewise torus action.
Recently, Diaconescu, Moore and Witten provided a nontrivial link between K-theory and M-theory, ... more Recently, Diaconescu, Moore and Witten provided a nontrivial link between K-theory and M-theory, by deriving the partition function of the Ramond-Ramond fields of Type IIA string theory from an E8 gauge theory in eleven dimensions. We give some relations between twisted K-theory and M-theory by adapting the method of Diaconescu-Moore-Witten and Moore-Saulina. In particular, we construct the twisted K-theory torus which defines the partition function, and also discuss the problem from the E8 loop group picture, in which the Dixmier-Douady class is the Neveu-Schwarz field. In the process of doing this, we encounter some mathematics that is new to the physics literature. In particular, the eta differential form, which is the generalization of the eta invariant, arises naturally in this context. We conclude with several open problems in mathematics and string theory.
In this paper we show that the T-duality transform of Bouwknegt, Evslin and Mathai applies to det... more In this paper we show that the T-duality transform of Bouwknegt, Evslin and Mathai applies to determine isomorphisms of certain current algebras and their associated vertex algebras on topologically distinct T-dual spacetimes compactified to circle bundles with $H$-flux.
A central result here is the computation of the entire cyclic homol- ogy of canonical smooth suba... more A central result here is the computation of the entire cyclic homol- ogy of canonical smooth subalgebras of stable continuous trace C�-algebras having smooth manifolds M as their spectrum. More precisely, the entire cyclic homology is shown to be canonically isomorphic to the continuous pe- riodic cyclic homology for these algebras. By an earlier result of the authors, one concludes
In this paper, we quantize universal gauge groups such as SU(\infty), in the sigma-C*-algebra set... more In this paper, we quantize universal gauge groups such as SU(\infty), in the sigma-C*-algebra setting. More precisely, we propose a concise definition of sigma-C*-quantum groups and explain the concept here. At the same time, we put this definition in the mathematical context of countably compactly generated groups as well as C*-compact quantum groups.
We study the geometry and topology of (filtered) algebra-bundles ${\bf\Psi}^{\mathbb Z}$ over a s... more We study the geometry and topology of (filtered) algebra-bundles ${\bf\Psi}^{\mathbb Z}$ over a smooth manifold $X$ with typical fibre $\Psi^{\mathbb Z}(Z; V)$, the algebra of classical pseudodifferential operators of integral order on the compact manifold $Z$ acting on smooth sections of a vector bundle $V$. First a theorem of Duistermaat and Singer is generalized to the assertion that the group of projective invertible Fourier integral operators ${\rm PGL}({\mathcal F}^\bullet(Z; V))$, is precisely the automorphism group, ${\rm Aut}(\Psi^{\mathbb Z}(Z; V)),$ of the filtered algebra of pseudodifferential operators. We replace some of the arguments in their paper by microlocal ones, thereby removing the topological assumption well as extending their result to sections of a vector bundle. We define a natural class of connections and B-fields the principal bundle to which ${\bf\Psi}^{\mathbb Z}$ is associated and obtain a de Rham representative of the Dixmier-Douady class, in terms of the outer derivation on the Lie algebra and the residue trace of Guillemin and Wodzicki; the resulting formula only depends on the formal symbol algebra ${\bf\Psi}^{\mathbb Z}/{\bf\Psi}^{-\infty}.$ Examples of pseudodifferential algebra bundles are given that are not associated to a finite dimensional fibre bundle over $X.$
We study both the continuous model and the discrete model of the integer quantum Hall effect on t... more We study both the continuous model and the discrete model of the integer quantum Hall effect on the hyperbolic plane in the presence of disorder, extending the results of an earlier paper [CHMM]. Here we model impurities, that is we consider the effect of a random or almost periodic potential as opposed to just periodic potentials. The Hall conductance is identified as a geometric invariant associated to an algebra of observables, which has plateaus at gaps in extended states of the Hamiltonian. We use the Fredholm modules defined in [CHMM] to prove the integrality of the Hall conductance in this case. We also prove that there are always only a finite number of gaps in extended states of any random discrete Hamiltonian. [CHMM] A. Carey, K. Hannabuss, V. Mathai and P. McCann, Quantum Hall Effect on the Hyperbolic Plane, Communications in Mathematical Physics, 190 vol. 3, (1998) 629-673.
Proceedings of the American Mathematical Society, Apr 26, 2010
Inspired by the work of Boris Vertman on refined analytic torsion for manifolds with boundary, in... more Inspired by the work of Boris Vertman on refined analytic torsion for manifolds with boundary, in this paper we extend the construction of the Cappell-Miller analytic torsion to manifolds with boundary. We also compare it with the refined analytic torsion on manifolds with boundary. As a byproduct of the gluing formula for refined analytic torsion and the comparison theorem for the Cappell-Miller analytic torsion and the refined analytic torsion, we establish the gluing formula for the Cappell-Miller analytic torsion in the case that the Hermitian metric is flat.
In this note we introduce the notion of bundle gerbe K-theory and investigate the relation to twi... more In this note we introduce the notion of bundle gerbe K-theory and investigate the relation to twisted K-theory. We provide some examples. Possible applications of bundle gerbe K-theory to the classification of D-brane charges in non-trivial backgrounds are discussed.
We introduce spherical T-duality, which relates pairs of the form $(P,H)$ consisting of a princip... more We introduce spherical T-duality, which relates pairs of the form $(P,H)$ consisting of a principal $SU(2)$-bundle $P\rightarrow M$ and a 7-cocycle $H$ on $P$. Intuitively spherical T-duality exchanges $H$ with the second Chern class $c_2(P)$. Unless $dim(M)\leq 4$, not all pairs admit spherical T-duals and the spherical T-duals are not always unique. Nonetheless, we prove that all spherical T-dualities induce a degree-shifting isomorphism on the 7-twisted cohomologies of the bundles and, when $dim(M)\leq 7$, also their integral twisted cohomologies and, when $dim(M)\leq 4$, even their 7-twisted K-theories. While spherical T-duality does not appear to relate equivalent string theories, it does provide an identification between conserved charges in certain distinct IIB supergravity and string compactifications.
We had previously defined the rho invariant $\rho_{spin}(Y,E,H, g)$ for the twisted Dirac operato... more We had previously defined the rho invariant $\rho_{spin}(Y,E,H, g)$ for the twisted Dirac operator $\not\partial^E_H$ on a closed odd dimensional Riemannian spin manifold $(Y, g)$, acting on sections of a flat hermitian vector bundle $E$ over $Y$, where $H = \sum i^{j+1} H_{2j+1} $ is an odd-degree differential form on $Y$ and $H_{2j+1}$ is a real-valued differential form of degree ${2j+1}$. Here we show that it is a conformal invariant of the pair $(H, g)$. In this paper we express the defect integer $\rho_{spin}(Y,E,H, g) - \rho_{spin}(Y,E, g)$ in terms of spectral flows and prove that $\rho_{spin}(Y,E,H, g)\in \mathbb Q$, whenever $g$ is a Riemannian metric of positive scalar curvature. In addition, if the maximal Baum-Connes conjecture holds for $\pi_1(Y)$ (which is assumed to be torsion-free), then we show that $\rho_{spin}(Y,E,H, rg) =0$ for all $r\gg 0$, significantly generalizing our earlier results. These results are proved using the Bismut-Weitzenb\"ock formula, a scaling trick, the technique of noncommutative spectral sections, and the Higson-Roe approach.
In this paper, we study both the continuous model and the discrete model of the Quantum Hall Effe... more In this paper, we study both the continuous model and the discrete model of the Quantum Hall Effect (QHE) on the hyperbolic plane. The Hall conductivity is identified as a geometric invariant associated to an imprimitivity algebra of observables. We define a twisted analogue of the Kasparov map, which enables us to use the pairing between K-theory and cyclic cohomology theory, to identify this geometric invariant with a topological index, thereby proving the integrality of the Hall conductivity in this case.
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