Book•
Heat Kernels and Dirac Operators
01 Apr 1992-
TL;DR: In this article, the authors present a formal solution for the trace of the heat kernel on Euclidean space, and show that the trace can be used to construct a heat kernel of an equivariant vector bundle.
Abstract: 1 Background on Differential Geometry.- 1.1 Fibre Bundles and Connections.- 1.2 Riemannian Manifolds.- 1.3 Superspaces.- 1.4 Superconnections.- 1.5 Characteristic Classes.- 1.6 The Euler and Thorn Classes.- 2 Asymptotic Expansion of the Heat Kernel.- 2.1 Differential Operators.- 2.2 The Heat Kernel on Euclidean Space.- 2.3 Heat Kernels.- 2.4 Construction of the Heat Kernel.- 2.5 The Formal Solution.- 2.6 The Trace of the Heat Kernel.- 2.7 Heat Kernels Depending on a Parameter.- 3 Clifford Modules and Dirac Operators.- 3.1 The Clifford Algebra.- 3.2 Spinors.- 3.3 Dirac Operators.- 3.4 Index of Dirac Operators.- 3.5 The Lichnerowicz Formula.- 3.6 Some Examples of Clifford Modules.- 4 Index Density of Dirac Operators.- 4.1 The Local Index Theorem.- 4.2 Mehler's Formula.- 4.3 Calculation of the Index Density.- 5 The Exponential Map and the Index Density.- 5.1 Jacobian of the Exponential Map on Principal Bundles.- 5.2 The Heat Kernel of a Principal Bundle.- 5.3 Calculus with Grassmann and Clifford Variables.- 5.4 The Index of Dirac Operators.- 6 The Equivariant Index Theorem.- 6.1 The Equivariant Index of Dirac Operators.- 6.2 The Atiyah-Bott Fixed Point Formula.- 6.3 Asymptotic Expansion of the Equivariant Heat Kernel.- 6.4 The Local Equivariant Index Theorem.- 6.5 Geodesic Distance on a Principal Bundle.- 6.6 The heat kernel of an equivariant vector bundle.- 6.7 Proof of Proposition 6.13.- 7 Equivariant Differential Forms.- 7.1 Equivariant Characteristic Classes.- 7.2 The Localization Formula.- 7.3 Bott's Formulas for Characteristic Numbers.- 7.4 Exact Stationary Phase Approximation.- 7.5 The Fourier Transform of Coadjoint Orbits.- 7.6 Equivariant Cohomology and Families.- 7.7 The Bott Class.- 8 The Kirillov Formula for the Equivariant Index.- 8.1 The Kirillov Formula.- 8.2 The Weyl and Kirillov Character Formulas.- 8.3 The Heat Kernel Proof of the Kirillov Formula.- 9 The Index Bundle.- 9.1 The Index Bundle in Finite Dimensions.- 9.2 The Index Bundle of a Family of Dirac Operators.- 9.3 The Chern Character of the Index Bundle.- 9.4 The Equivariant Index and the Index Bundle.- 9.5 The Case of Varying Dimension.- 9.6 The Zeta-Function of a Laplacian.- 9.7 The Determinant Line Bundle.- 10 The Family Index Theorem.- 10.1 Riemannian Fibre Bundles.- 10.2 Clifford Modules on Fibre Bundles.- 10.3 The Bismut Superconnection.- 10.4 The Family Index Density.- 10.5 The Transgression Formula.- 10.6 The Curvature of the Determinant Line Bundle.- 10.7 The Kirillov Formula and Bismut's Index Theorem.- References.- List of Notation.
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TL;DR: Network embedding assigns nodes in a network to low-dimensional representations and effectively preserves the network structure as discussed by the authors, and a significant amount of progress has been made toward this emerging network analysis paradigm.
Abstract: Network embedding assigns nodes in a network to low-dimensional representations and effectively preserves the network structure. Recently, a significant amount of progresses have been made toward this emerging network analysis paradigm. In this survey, we focus on categorizing and then reviewing the current development on network embedding methods, and point out its future research directions. We first summarize the motivation of network embedding. We discuss the classical graph embedding algorithms and their relationship with network embedding. Afterwards and primarily, we provide a comprehensive overview of a large number of network embedding methods in a systematic manner, covering the structure- and property-preserving network embedding methods, the network embedding methods with side information, and the advanced information preserving network embedding methods. Moreover, several evaluation approaches for network embedding and some useful online resources, including the network data sets and softwares, are reviewed, too. Finally, we discuss the framework of exploiting these network embedding methods to build an effective system and point out some potential future directions.
929 citations
Book•
23 Aug 2014
TL;DR: The Spectral Calculus as mentioned in this paper is a generalization of K-theory for non-commutative spaces and algebraic spaces and algebras of functions, and it is used in Projective Systems of Non-Commutative Lattices.
Abstract: Noncommutative Spaces and Algebras of Functions.- Projective Systems of Noncommutative Lattices.- Modules as Bundles.- A Few Elements of K-Theory.- The Spectral Calculus.- Noncommutative Differential Forms.- Connections on Modules.- Field Theories on Modules.- Gravity Models.- Quantum Mechanical Models on Noncommutative Lattices.
572 citations
Book•
19 Jul 2007
TL;DR: Demailly's Holomorphic Morse Inequalities on non-compact manifolds and the Bergman Kernel on noncompact manifold have been studied in this paper, where they have been shown to have similar properties to those of the Moishezon manifold.
Abstract: Demailly's Holomorphic Morse Inequalities.- Characterization of Moishezon Manifolds.- Holomorphic Morse Inequalities on Non-compact Manifolds.- Asymptotic Expansion of the Bergman Kernel.- Kodaira Map.- Bergman Kernel on Non-compact Manifolds.- Toeplitz Operators.- Bergman Kernels on Symplectic Manifolds.
555 citations
Book•
17 Jul 2001TL;DR: Symplectic Toric Manifolds as mentioned in this paper are a type of complex manifold that is composed of two or more complex manifold types, and they can be classified into three classes: symplectic, symplectomorphisms, and symmlectic reduction.
Abstract: Symplectic Manifolds.- Symplectic Forms.- Symplectic Form on the Cotangent Bundle.- Symplectomorphisms.- Lagrangian Submanifolds.- Generating Functions.- Recurrence.- Local Forms.- Preparation for the Local Theory.- Moser Theorems.- Darboux-Moser-Weinstein Theory.- Weinstein Tubular Neighborhood Theorem.- Contact Manifolds.- Contact Forms.- Contact Dynamics.- Compatible Almost Complex Structures.- Almost Complex Structures.- Compatible Triples.- Dolbeault Theory.- Kahler Manifolds.- Complex Manifolds.- Kahler Forms.- Compact Kahler Manifolds.- Hamiltonian Mechanics.- Hamiltonian Vector Fields.- Variational Principles.- Legendre Transform.- Moment Maps.- Actions.- Hamiltonian Actions.- Symplectic Reduction.- The Marsden-Weinstein-Meyer Theorem.- Reduction.- Moment Maps Revisited.- Moment Map in Gauge Theory.- Existence and Uniqueness of Moment Maps.- Convexity.- Symplectic Toric Manifolds.- Classification of Symplectic Toric Manifolds.- Delzant Construction.- Duistermaat-Heckman Theorems.
538 citations
TL;DR: In this article, the authors studied the perturbation theory for three dimensional Chern-Simons quantum field theory on a general compact three manifold without boundary, and they showed that after a simple change of variables, the action obtained by BRS gauge fixing has a superspace formulation.
Abstract: We study the perturbation theory for three dimensional Chern--Simons quantum field theory on a general compact three manifold without boundary. We show that after a simple change of variables, the action obtained by BRS gauge fixing in the Lorentz gauge has a superspace formulation. The basic properties of the propagator and the Feynman rules are written in a precise manner in the language of differential forms. Using the explicit description of the propagator singularities, we prove that the theory is finite. Finally the anomalous metric dependence of the $2$-loop partition function on the Riemannian metric (which was introduced to define the gauge fixing) can be cancelled by a local counterterm as in the $1$-loop case. In fact, the counterterm is equal to the Chern--Simons action of the metric connection, normalized precisely as one would expect based on the framing dependence of Witten's exact solution. Invited talk at the XXth Conference on Differential Geometric Methods in Physics, New York, June 1991.
465 citations
References
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01 Nov 1975
TL;DR: In this article, the authors present a generalization of Hirzebruch's signature theorem for the case of manifolds with boundary, which can be viewed as analogous to the Gauss-Bonnet theorem for manifold with boundary.
Abstract: 1. Introduction. The main purpose of this paper is to present a generalization of Hirzebruch's signature theorem for the case of manifolds with boundary. Our result is in the framework of Riemannian geometry and can be viewed as analogous to the Gauss–Bonnet theorem for manifolds with boundary, although there is a very significant difference between the two cases which is, in a sense, the central topic of the paper. To explain this difference let us begin by recalling that the classical Gauss–Bonnet theorem for a surface X with boundary Y asserts that the Euler characteristic E(X) is given by a formula:where K is the Gauss curvature of X and σ is the geodesic curvature of Y in X. In particular if, near the boundary, X is isometric to the product Y x R+, the boundary integral in (1.1) vanishes and the formula is the same as for closed surfaces. Similar remarks hold in higher dimensions. Now if X is a closed oriented Riemannian manifold of dimension 4, there is another formula relating cohomological invariants with curvature in addition to the Gauss–Bonnet formula. This expresses the signature of the quadratic form on H2(X, R) by an integral formulawhere p1 is the differential 4-form representing the first Pontrjagin class and is given in terms of the curvature matrix R by p1 = (2π)−2Tr R2. It is natural to ask if (1.2) continues to hold for manifolds with boundary, provided the metric is a product near the boundary. Simple examples show that this is false, so that in general
2,202 citations
TL;DR: In this paper, the authors defined the spectrum of the problem of bounded regions of R d with a piecewise smooth boundary B and showed that if 0 > γ1 ≥ γ2 ≥ ≥ ≥ β3 ≥ etc.
Abstract: A famous formula of H. Weyl [17] states that if D is a bounded region of R d with a piecewise smooth boundary B, and if 0 > γ1 ≥ γ2 ≥ γ3 ≥ etc. ↓−∞ is the spectrum of the problem
$$\displaystyle\begin{array}{rcl} \varDelta f =\big (\partial ^{2}/\partial x_{ 1}^{2} + \cdots + \partial ^{2}/\partial x_{ d}^{2}\big)f =\gamma f\quad \mbox{ in }D,& &{}\end{array}$$
(6.1.1a)
$$\displaystyle\begin{array}{rcl} f \in C^{2}(D) \cap C(\overline{D}),& &{}\end{array}$$
(6.1.1b)
$$\displaystyle\begin{array}{rcl} f = 0\quad \mbox{ on }B,& &{}\end{array}$$
(6.1.1c)
then
$$\displaystyle\begin{array}{rcl} -\gamma _{n} \sim C(d)(n/\mbox{ vol }D)^{2/d}\quad (n \uparrow \infty ),& &{}\end{array}$$
(6.1.2)
or, what is the same,
$$\displaystyle\begin{array}{rcl} Z \equiv \mathop{\mathrm{sp}}
olimits e^{t\varDelta } =\sum _{ n\geq 1}\exp \big(\gamma _{n}t\big) \sim (4\pi t)^{-d/2} \times \mathop{\mathrm{vol}}
olimits D\quad (t \downarrow 0),& &{}\end{array}$$
(6.1.3)
where \(C(d) = 2\pi [(d/2)!]^{d/2}\).
915 citations
TL;DR: In this paper, the Gilkey Theorem as formulated on p. 284 does not apply in the sense that the coefficients of the two operators A*A and AA* (associated to the signature operator A) are polynomial functions in the gij, their derivatives and (det g)-l.
Abstract: The main error occurs on page 306 where it is implicitly assumed that the coefficients of the two operators A*A and AA* (associated to the signature operator A) are polynomial functions in the gij, their derivatives and (det g)-l . As we shall show later this is not quite t r u e t h e coefficients also involve d ] f ~ and the inverses of the principal minors of the matrix gu" Thus the form m in (5.1) is not a regular invariant of the metric in the sense of w 2, and so the Gilkey Theorem as formulated on p. 284 does not apply. To correct this we shall widen the notion of regularity (so as to include, in particular, the form ~o above) and then check that our proof of Gitkey's Theorem still holds in this wider context. In w regularity was only defined for invariants of a Riemann structure g (i.e. satisfying the naturality or invariance property (2.3)). It will perhaps make for greater clarity if we introduce our new notion of regularity for any function of g, independently of the invariance property. We shall say that f(g) is a regular function of g if, in any coordinate system, we have
685 citations
631 citations
TL;DR: In this paper, a simple derivation of the Atiyah-Singer index theorem for classical complexes and its G-index generalization using elementary properties of quantum mechanical supersymmetric systems is presented.
Abstract: Using a recently introduced index for supersymmetric theories, we present a simple derivation of the Atiyah-Singer index theorem for classical complexes and itsG-index generalization using elementary properties of quantum mechanical supersymmetric systems.
517 citations
"Heat Kernels and Dirac Operators" refers background or methods in this paper
...The limit lim ch(At) = ch(PoA[1jPo) E A(B) t->oo 289 holds with respect to each Ce-norm on compact subsets of B....
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...X[2] - Yl]GZ[l] = (R[2] +μ[1]v[1]) - (μ[l]D)G(Dv[I]) = R121....
[...]
...A superconnection on 7-l is an operator acting on A(M, l) of the form A=A101+A[1] +A12>+....
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...The operator Vo defined by the formula Vo = POA[1]Po is a connection on the superbundle ker(D)....
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...Bibliography [1] L. Alvarez-Gaume....
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