12 results on '"Chern–Simons invariants"'
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2. From torus bundles to particle–hole equivariantization.
- Author
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Cui, Shawn X., Gustafson, Paul, Qiu, Yang, and Zhang, Qing
- Abstract
We continue the program of constructing (pre)modular tensor categories from 3-manifolds first initiated by Cho–Gang–Kim using M theory in physics and then mathematically studied by Cui–Qiu–Wang. An important structure involved in the construction is a collection of certain SL (2 , C) characters on a given manifold, which serve as the simple object types in the corresponding category. Chern–Simons invariants and adjoint Reidemeister torsions also play a key role, and they are related to topological twists and quantum dimensions, respectively, of simple objects. The modular S-matrix is computed from local operators and follows a trial-and-error procedure. It is currently unknown how to produce data beyond the modular S- and T-matrices. There are also a number of subtleties in the construction, which remain to be solved. In this paper, we consider an infinite family of 3-manifolds, that is, torus bundles over the circle. We show that the modular data produced by such manifolds are realized by the Z 2 -equivariantization of certain pointed premodular categories. Here the equivariantization is performed for the Z 2 -action sending a simple (invertible) object to its inverse, also called the particle–hole symmetry. It is our hope that this extensive class of examples will shed light on how to improve the program to recover the full data of a premodular category. [ABSTRACT FROM AUTHOR] more...
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- 2022
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3. Cluster Algebras from Surfaces : Lecture Notes for the CIMPA School Mar del Plata, March 2016
- Author
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Schiffler, Ralf, Hussin, Véronique, Series Editor, Assem, Ibrahim, editor, and Trepode, Sonia, editor
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- 2018
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4. Cohomology theory for biological time series.
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Pinčák, Richard, Kanjamapornkul, Kabin, and Bartoš, Erik
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CHRONOBIOLOGY , *TIME series analysis , *GENETIC code , *DIRAC operators , *AMINO acids - Abstract
We use Khovanov cohomology in biological time series data to model the curvature in the docking system of the co‐receptor CCR5 with Δ32 and V3 loop time series data. We use modified Dirac operator to measure the transition state in the gene expression. We provide the proof of the existence of genetic code as knot and link between the interaction of behavior field in the genetic code. We define 20 transition states in 20 types of amino acids as new types of Yang‐Mills field in the passive layer of DNA‐RNA transcription. [ABSTRACT FROM AUTHOR] more...
- Published
- 2020
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5. Eta invariants and the hypoelliptic Laplacian.
- Author
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Bismut, Jean-Michel
- Subjects
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ORBITAL interaction , *HYPOELLIPTIC operators , *GRAPHIC methods , *TOPOLOGICAL algebras , *MATHEMATICAL invariants - Abstract
The purpose of this paper is to give a new proof of the results of Moscovici and Stanton on orbital integrals associated with eta invariants on compact locally symmetric spaces. Moscovici and Stanton used methods of harmonic analysis on reductive groups. Here, we combine our approach to orbital integrals that uses the hypoelliptic Laplacian with the introduction of a rotation on certain Clifford algebras. Probabilistic methods play an important role in establishing key estimates. In particular, we construct a suitable Ito calculus associated with certain hypoelliptic diffusions. [ABSTRACT FROM AUTHOR] more...
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- 2019
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6. Trivializations of differential cocycles.
- Author
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Redden, Corbett
- Subjects
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COCYCLES , *CURVATURE , *COHOMOLOGY theory , *DIFFERENTIAL algebra , *SQUARE - Abstract
Associated to a differential character is an integral cohomology class, referred to as the characteristic class, and a closed differential form, referred to as the curvature. The characteristic class and curvature are equal in de Rham cohomology, and this is encoded in a commutative square. In the Hopkins-Singer model, where differential characters are equivalence classes of differential cocycles, there is a natural notion of trivializing a differential cocycle. In this paper, we extend the notion of characteristic class, curvature, and de Rham class to trivializations of differential cocycles. These structures fit into a commutative square, and this square is a torsor for the commutative square associated to characters with degree one less. Under the correspondence between degree two differential cocycles and principal circle bundles with connection, we recover familiar structures associated to global sections. [ABSTRACT FROM AUTHOR] more...
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- 2015
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7. Duality and cohomology in -theory with boundary
- Author
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Sati, Hisham
- Subjects
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DUALITY theory (Mathematics) , *HOMOLOGY theory , *GEOMETRIC analysis , *PARTITIONS (Mathematics) , *MANIFOLDS (Mathematics) , *HODGE theory , *BOUNDARY value problems , *TANGENT bundles - Abstract
Abstract: We consider geometric and analytical aspects of -theory on a manifold with boundary . The partition function of the -field requires summing over harmonic forms. When is closed, Hodge theory gives a unique harmonic form in each de Rham cohomology class, while in the presence of a boundary the Hodge–Morrey–Friedrichs decomposition should be used. This leads us to study the boundary conditions for the -field. The dynamics and the presence of the dual to the -field gives rise to a mixing of boundary conditions with one being Dirichlet and the other being Neumann. We describe the mixing between the corresponding absolute and relative cohomology classes via Poincaré duality angles, which we also illustrate for the M5-brane as a tubular neighborhood. Several global aspects are then considered. We provide a systematic study of the extension of the bundle and characterize obstructions. Considering as a fiber bundle, we describe how the phase looks like on the base, hence providing dimensional reduction in the boundary case via the adiabatic limit of the eta invariant. The general use of the index theorem leads to a new effect given by a gravitational Chern–Simons term on whose restriction to the boundary would be a generalized WZW model. This suggests that holographic models of -theory can be viewed as a sector within this index-theoretic approach. [Copyright &y& Elsevier] more...
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- 2012
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8. Eta invariants and the hypoelliptic Laplacian
- Author
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Jean-Michel Bismut, Département de Mathématiques-Université de Paris XI, Université Paris-Sud - Paris 11 (UP11), and European Research Council (E.R.C.)
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Mathematics - Differential Geometry ,Pure mathematics ,Spectral theory ,General Mathematics ,Hypoelliptic equations ,Diffusion processes and stochastic analysis on manifolds ,01 natural sciences ,Harmonic analysis ,Mathematics - Analysis of PDEs ,Probabilistic method ,Selberg trace formula ,FOS: Mathematics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,0101 mathematics ,Mathematics ,Applied Mathematics ,11F72, 35H10, 58J20, 58J28, 58J65 ,010102 general mathematics ,Clifford algebra ,In- dex theory and related fixed point theorems ,Probability (math.PR) ,Chern-Simons invariants ,Eta-invariants ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Differential Geometry (math.DG) ,[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] ,Hypoelliptic operator ,Laplace operator ,Rotation (mathematics) ,Mathematics - Probability ,Analysis of PDEs (math.AP) - Abstract
The purpose of this paper is to give a new proof of results of Moscovici and Stanton on the orbital integrals associated with eta invariants on compact locally symmetric spaces. Moscovici and Stanton used methods of harmonic analysis on reductive groups. Here, we combine our approach to orbital integrals using the hypoelliptic Laplacian, with the introduction of a rotation on certain Clifford algebras. Probabilistic methods play an important role in establishing key estimates. In particular, we construct the proper It{\^o} calculus associated with certain hypoelliptic diffusions. more...
- Published
- 2016
- Full Text
- View/download PDF
9. Chern-Simons invariants in KK-theory.
- Author
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Mohsen, Omar
- Subjects
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COMPACT groups , *VON Neumann algebras , *MANIFOLDS (Mathematics) - Abstract
For a unitary representation ϕ of the fundamental group of a compact smooth manifold, Atiyah, Patodi, Singer defined the so called α -invariant of ϕ using the Chern-Simons invariants. In this article using traces on C ⁎ -algebras, we give an intrinsic definition of an element in KK with real coefficients theory whose pullback by the representation ϕ is the α -invariant. [ABSTRACT FROM AUTHOR] more...
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- 2019
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10. Chern–Simons forms on associated bundles, and boundary terms
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Johnson, David L.
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- 2007
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11. Chern–Simons line bundle on Teichmüller space
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Sergiu Moroianu, Colin Guillarmou, Département de Mathématiques et Applications - ENS Paris (DMA), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), 'Simion Stoilow' Institute of Mathematics (IMAR), Romanian Academy of Sciences, École normale supérieure - Paris (ENS Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS) more...
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Teichmüller space ,Pure mathematics ,hyperbolic manifolds ,Mathematical analysis ,Boundary (topology) ,Hyperbolic manifold ,Orthonormal frame ,Mathematics::Geometric Topology ,Mapping class group ,Manifold ,Chern–Simons invariants ,High Energy Physics::Theory ,Line bundle ,[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] ,[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] ,32G15 ,58J28 ,Mathematics::Differential Geometry ,Compact Riemann surface ,58J28, 32G15 ,Geometry and Topology ,Mathematics::Symplectic Geometry ,renormalized volume ,Mathematics - Abstract
36 pages. Minor modifications in the introduction.; International audience; Let $X$ be a non-compact geometrically finite hyperbolic $3$-manifold without cusps of rank $1$. The deformation space $\mc{H}$ of $X$ can be identified with the Teichmüller space $\mc{T}$ of the conformal boundary of $X$ as the graph of a section in $T^*\mc{T}$. We construct a Hermitian holomorphic line bundle $\mc{L}$ on $\mc{T}$, with curvature equal to a multiple of the Weil-Petersson symplectic form. This bundle has a canonical holomorphic section defined by $e^{\frac{1}{\pi}{\rm Vol}_R(X)+2\pi i\CS(X)}$ where ${\rm Vol}_R(X)$ is the renormalized volume of $X$ and $\CS(X)$ is the Chern-Simons invariant of $X$. This section is parallel on $\mc{H}$ for the Hermitian connection modified by the $(1,0)$ component of the Liouville form on $T^*\mc{T}$. As applications, we deduce that $\mc{H}$ is Lagrangian in $T^*\mc{T}$, and that ${\rm Vol}_R(X)$ is a Kähler potential for the Weil-Petersson metric on $\mc{T}$ and on its quotient by a certain subgroup of the mapping class group. For the Schottky uniformisation, we use a formula of Zograf to construct an explicit isomorphism of holomorphic Hermitian line bundles between $\mc{L}^{-1}$ and the sixth power of the determinant line bundle. more...
- Published
- 2014
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12. Chern--Simons theory, surface separability, and volumes of 3-manifolds
- Author
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Yi Liu, Shicheng Wang, Pierre Derbez, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Peking University [Beijing], California Institute of Technology (CALTECH), and Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU) more...
- Subjects
Mathematics - Differential Geometry ,surface separabilities ,volume of representations ,010102 general mathematics ,Chern–Simons theory ,Geometric Topology (math.GT) ,01 natural sciences ,Mathematics::Geometric Topology ,Graph ,Chern-Simons invariants ,Combinatorics ,Mathematics - Geometric Topology ,Differential Geometry (math.DG) ,[MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT] ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,[MATH]Mathematics [math] ,Mathematics::Symplectic Geometry ,1991 MSC: 57M50, 51H20 ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
We study the set ${\rm vol}\left(M,G\right)$ of volumes of all representations $\rho\co\pi_1M\to G$, where $M$ is a closed oriented $3$-manifold and $G$ is either ${\rm Iso}_+{\Hi}^3$ or ${\rm Iso}_e\t{\rm SL_2(\R)}$. By various methods, including relations between the volume of representations and the Chern--Simons invariants of flat connections, and recent results of surfaces in 3-manifolds, we prove that any 3-manifold $M$ with positive Gromov simplicial volume has a finite cover $\t M$ with ${\rm vol}(\t M,{\rm Iso}_+{\Hi}^3)\ne \{0\}$, and that any non-geometric 3-manifold $M$ containing at least one Seifert piece has a finite cover $\t M$ with ${\rm vol}(\t M,{\rm Iso}_e\t{\rm SL_2(\R)}) \ne \{0\}$. We also find 3-manifolds $M$ with positive simplicial volume but ${\rm vol}(M,{\rm Iso}_+{\Hi}^3)=\{0\}$, and non-trivial graph manifolds $M$ with ${\rm vol}(M,{\rm Iso}_e\t{\rm SL_2(\R)})=\{0\}$, proving that it is in general necessary to pass to some finite covering to guarantee that ${\rm vol}(M,G)\not=\{0\}$. Besides we determine ${\rm vol}\left(M, G \right)$ when $M$ supports the Seifert geometry., Comment: 43 pages. arXiv admin note: substantial text overlap with arXiv:1111.6153 more...
- Published
- 2014
- Full Text
- View/download PDF
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