scholarly journals A Kastler-Kalau-Walze Type Theorem and the Spectral Action for Perturbations of Dirac Operators on Manifolds with Boundary

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Yong Wang
Keyword(s):  
Type Theorem ◽  
Dirac Operators ◽  
Compact Manifolds ◽  
Manifolds With Boundary ◽  
Spectral Action ◽  
Gravitational Action

We prove a Kastler-Kalau-Walze type theorem for perturbations of Dirac operators on compact manifolds with or without boundary. As a corollary, we give two kinds of operator-theoretic explanations of the gravitational action on boundary. We also compute the spectral action for Dirac operators with two-form perturbations on 4-dimensional compact manifolds.

scholarly journals Modified Novikov Operators and the Kastler-Kalau-Walze-Type Theorem for Manifolds with Boundary

2020 ◽  
Vol 2020 ◽  
pp. 1-28
Author(s):  
Sining Wei ◽  
Yong Wang
Keyword(s):  
Type Theorem ◽  
Compact Manifolds ◽  
Manifolds With Boundary ◽  
Spectral Action ◽  
Witten Deformation

In this paper, we give two Lichnerowicz-type formulas for modified Novikov operators. We prove Kastler-Kalau-Walze-type theorems for modified Novikov operators on compact manifolds with (respectively without) a boundary. We also compute the spectral action for Witten deformation on 4-dimensional compact manifolds.

scholarly journals Dirac–Witten Operators and the Kastler–Kalau–Walze Type Theorem for Manifolds with Boundary

2021 ◽  
Author(s):  
Tong Wu ◽  
Jian Wang ◽  
Yong Wang
Keyword(s):  
Type Theorem ◽  
Compact Manifolds ◽  
Manifolds With Boundary

AbstractIn this paper, we obtain two Lichnerowicz type formulas for the Dirac–Witten operators. And we give the proof of Kastler–Kalau–Walze type theorems for the Dirac–Witten operators on 4-dimensional and 6-dimensional compact manifolds with (resp. without) boundary.

scholarly journals An Obata-type theorem on compact Einstein manifolds with boundary

2021 ◽  
Author(s):  
Kazuo Akutagawa
Keyword(s):  
Boundary Condition ◽  
Smooth Boundary ◽  
Type Theorem ◽  
Type Inequality ◽  
Einstein Metric ◽  
Manifolds With Boundary ◽  
Einstein Manifolds ◽  
Conformal Class ◽  
Totally Geodesic ◽  
Yamabe Constant

AbstractWe show a kind of Obata-type theorem on a compact Einstein n-manifold $$(W, \bar{g})$$ ( W , g ¯ ) with smooth boundary $$\partial W$$ ∂ W . Assume that the boundary $$\partial W$$ ∂ W is minimal in $$(W, \bar{g})$$ ( W , g ¯ ) . If $$(\partial W, \bar{g}|_{\partial W})$$ ( ∂ W , g ¯ | ∂ W ) is not conformally diffeomorphic to $$(S^{n-1}, g_S)$$ ( S n - 1 , g S ) , then for any Einstein metric $$\check{g} \in [\bar{g}]$$ g ˇ ∈ [ g ¯ ] with the minimal boundary condition, we have that, up to rescaling, $$\check{g} = \bar{g}$$ g ˇ = g ¯ . Here, $$g_S$$ g S and $$[\bar{g}]$$ [ g ¯ ] denote respectively the standard round metric on the $$(n-1)$$ ( n - 1 ) -sphere $$S^{n-1}$$ S n - 1 and the conformal class of $$\bar{g}$$ g ¯ . Moreover, if we assume that $$\partial W \subset (W, \bar{g})$$ ∂ W ⊂ ( W , g ¯ ) is totally geodesic, we also show a Gursky-Han type inequality for the relative Yamabe constant of $$(W, \partial W, [\bar{g}])$$ ( W , ∂ W , [ g ¯ ] ) .

Twisted dirac operators and Kastler-Kalau-Walze theorems for six-dimensional manifolds with boundary

2020 ◽  
Vol 17 (14) ◽  
pp. 2050211
Author(s):  
Sining Wei ◽  
Yong Wang
Keyword(s):  
Vector Bundle ◽  
Dirac Operators ◽  
Manifolds With Boundary ◽  
Unitary Connection ◽  
Twisted Dirac Operators

In this paper, we establish two kinds of Kastler-Kalau-Walze type theorems for Dirac operators and signature operators twisted by a vector bundle with a non-unitary connection on six-dimensional manifolds with boundary.

Black Hole Entropy from Non-Commutative Geometry and Spontaneous Localisation

2019 ◽  
Author(s):  
Tejinder P. Singh ◽  
Palemkota Maithresh
Keyword(s):  
Black Hole ◽  
Black Hole Entropy ◽  
Entangled State ◽  
Schwarzschild Black Hole ◽  
Spectral Action ◽  
Black Hole Evaporation ◽  
Gravitational Action ◽  
Fluctuation Dissipation ◽  
Fluctuation Dissipation Theorem ◽  
Gravitational Entropy

In our recently proposed theory of quantum gravity, a black hole arises from the spontaneous localisation of an entangled state of a large number of atoms of space-time-matter [STM]. Prior to localisation, the non-commutative curvature of an STM atom is described by the spectral action of non-commutative geometry. By using the techniques of statistical thermodynamics from trace dynamics, we show that the gravitational entropy of a Schwarzschild black hole results from the microstates of the entangled STM atoms and is given (subject to certain assumptions) by the classical Euclidean gravitational action. This action, in turn, equals the Bekenstein-Hawking entropy (Area/$4{L_P}^2$) of the black hole. We argue that spontaneous localisation is related to black-hole evaporation through the fluctuation-dissipation theorem.

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