Higher Spectral Flow

AbstractFor a single Dirac operator on a closed manifold the cocycle introduced by Jaffe-Lesniewski-Osterwalder [19] (abbreviated here to JLO), is a representative of Connes' Chern character map from the K-theory of the algebra of smooth functions on the manifold to its entire cyclic cohomology. Given a smooth fibration of closed manifolds and a family of generalized Dirac operators along the fibers, we define in this paper an associated bivariant JLO cocycle. We then prove that, for any l ≥ 0, our bivariant JLO cocycle is entire when we endow smoooth functions on the total manifold with the Cl+1 topology and functions on the base manifold with the Cl topology. As a by-product of our theorem, we deduce that the bivariant JLO cocycle is entire for the Fréchet smooth topologies. We then prove that our JLO bivariant cocycle computes the Chern character of the Dai-Zhang higher spectral flow.

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Higher spectral flow for Dirac operators with local boundary conditions

International Journal of Mathematics ◽

10.1142/s0129167x16500683 ◽

2016 ◽

Vol 27 (08) ◽

pp. 1650068

Author(s):

Jianqing Yu

Keyword(s):

Boundary Conditions ◽

Elliptic Boundary ◽

Dirac Operators ◽

Spectral Flow ◽

Local Boundary ◽

Manifold With Boundary ◽

Index Bundle ◽

Adjoint Extension ◽

Smooth Map ◽

Higher Spectral Flow

We consider a one parameter family [Formula: see text] of families of fiberwise twisted Dirac type operators on a fibration with the typical fiber an even dimensional compact manifold with boundary, which verifies [Formula: see text] with [Formula: see text] being a smooth map from the fibration to a unitary group [Formula: see text]. For each [Formula: see text], we impose on [Formula: see text] a certain fixed local elliptic boundary condition [Formula: see text] and get a self-adjoint extension [Formula: see text]. Under the assumption that [Formula: see text] has vanishing [Formula: see text]-index bundle, we establish a formula for the higher spectral flow of [Formula: see text], [Formula: see text]. Our result generalizes a recent result of [A. Gorokhovsky and M. Lesch, On the spectral flow for Dirac operators with local boundary conditions, Int. Math. Res. Not. IMRN (2015) 8036–8051.] to the families case.

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Higher Spectral Flow

Journal of Functional Analysis ◽

10.1006/jfan.1998.3273 ◽

1998 ◽

Vol 157 (2) ◽

pp. 432-469 ◽

Cited By ~ 19

Author(s):

Xianzhe Dai ◽

Weiping Zhang

Keyword(s):

Spectral Flow ◽

Higher Spectral Flow

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Potential and challenges of wind measurements using met-masts in complex topography for bridge design: Part II – Spectral flow characteristics

Journal of Wind Engineering and Industrial Aerodynamics ◽

10.1016/j.jweia.2021.104585 ◽

2021 ◽

Vol 211 ◽

pp. 104585

Author(s):

Zakari Midjiyawa ◽

Etienne Cheynet ◽

Joachim Reuder ◽

Hálfdán Ágústsson ◽

Trond Kvamsdal

Keyword(s):

Flow Characteristics ◽

Spectral Flow ◽

Complex Topography ◽

Bridge Design ◽

Wind Measurements

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GROUND STATE CHARGE OF FERMION SOLITON SYSTEM IN 1+1 AND 3+1 DIMENSIONS

International Journal of Modern Physics A ◽

10.1142/s0217751x93000278 ◽

1993 ◽

Vol 08 (04) ◽

pp. 705-721

Author(s):

M. RAVENDRANADHAN ◽

M. SABIR

Keyword(s):

Ground State ◽

Energy State ◽

Flow Analysis ◽

Background Field ◽

Bound State ◽

Fermion Mass ◽

Spectral Flow ◽

Bound State Spectrum ◽

Spectral Asymmetry ◽

State Charge

Ground state charge of some fermion soliton system without C-invariance is calculated in 1+1 and 3+1 dimensions by a combination of adiabatic method and spectral flow analysis. Induced charge is calculated by evolving adiabatically the fields from a vacuum having a background field which has a zero energy state and spectral symmetry. The spectral flow is calculated by an analysis of the bound state spectrum. In 1+1 dimension our calculations are in agreement with the results already found in the literature. In 3+1 dimension we study the interaction of fermions with monopoles and dyons. In the case of monopoles, even though there is spectral asymmetry, ground state charge is found to be ±1/2. It is shown that ground state charge gets contribution only from the lowest angular momentum states and is discontinuous at the fermion mass.

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SPECTRAL FLOW AND FREE FIELD REALIZATIONS OF BOSONIC STRINGS ON NAPPI–WITTEN GROUP MANIFOLD

International Journal of Modern Physics A ◽

10.1142/s0217751x1105141x ◽

2011 ◽

Vol 26 (01) ◽

pp. 149-160

Author(s):

GANG CHEN

Keyword(s):

Lie Algebra ◽

Bosonic Strings ◽

Free Field ◽

Geometric Interpretation ◽

Closed String ◽

Space Time ◽

Spectral Flow ◽

Affine Lie Algebra ◽

Group Manifold ◽

Free Field Realization

In this paper we study some aspects of closed string theories in the Nappi–Witten space–time. The effects of spectral flow on the geodesics are studied in terms of an explicit parametrization of the group manifold. The worldsheets of the closed strings under the spectral flow of the geodesics can be classified into four classes, each with a geometric interpretation. We also obtain a free field realization of the Nappi–Witten affine Lie algebra in the most general conditions using a different but equivalent parametrization of the group manifold.

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