scholarly journals Comments on the Atiyah-Patodi-Singer index theorem, domain wall, and Berry phase

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Tetsuya Onogi ◽  
Takuya Yoda

Abstract It is known that the Atiyah-Patodi-Singer index can be reformulated as the eta invariant of the Dirac operators with a domain wall mass which plays a key role in the anomaly inflow of the topological insulator with boundary. In this paper, we give a conjecture that the reformulated version of the Atiyah-Patodi-Singer index can be given simply from the Berry phase associated with domain wall Dirac operators when adiabatic approximation is valid. We explicitly confirm this conjecture for a special case in two dimensions where an analytic calculation is possible. The Berry phase is divided into the bulk and the boundary contributions, each of which gives the bulk integration of the Chern character and the eta-invariant.

Author(s):  
Moulay-Tahar Benameur ◽  
Alan L. Carey

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.


2000 ◽  
Vol 9 (7) ◽  
pp. 481-484 ◽  
Author(s):  
Jing Hui ◽  
Wu Jian-sheng

2000 ◽  
Vol 177 (1) ◽  
pp. 203-218 ◽  
Author(s):  
Tom M.W. Nye ◽  
Michael A. Singer

Author(s):  
P. Klein ◽  
R. Varga ◽  
J. Onufer ◽  
J. Ziman ◽  
G.A. Badini-Confalonieri ◽  
...  
Keyword(s):  

2002 ◽  
Vol 16 (14n15) ◽  
pp. 1943-1950 ◽  
Author(s):  
T. FUJIWARA

The spectral flows of the hermitian Wilson-Dirac operator for a continuous family of abelian gauge fields connecting different topological sectors are shown to have a characteristic structure leading to the lattice index theorem. The index of the overlap Dirac operator is shown to coincide with the topological charge for a wide class of gauge field configurations. It is also argued that in two dimensions the eigenvalue spectra for some special but nontrivial background gauge fields can be described by a set of universal polynomials and the index can be found exactly.


1998 ◽  
Vol 442 (1-4) ◽  
pp. 259-265
Author(s):  
Fuad M. Saradzhev
Keyword(s):  

2006 ◽  
Vol 21 (17) ◽  
pp. 3575-3603 ◽  
Author(s):  
D. GAL'TSOV ◽  
S. KLEVTSOV ◽  
D. ORLOV ◽  
G. CLÉMENT

Recently it was found that the complete integration of the Einstein-dilaton-antisymmetric form equations depending on one variable and describing static singly charged p-branes leads to two and only two classes of solutions: the standard asymptotically flat black p-brane and the asymptotically nonflat p-brane approaching the linear dilaton background at spatial infinity. Here we analyze this issue in more details and generalize the corresponding uniqueness argument to the case of partially delocalized branes. We also consider the special case of codimension one and find, in addition to the standard domain wall, the black wall solution. Explicit relations between our solutions and some recently found p-brane solutions "with extra parameters" are presented.


Sign in / Sign up

Export Citation Format

Share Document