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70
Sharp Inequalities, The Functional Determinant, And The Complementary Series
 TRANS. AMER. MATH. SOC
, 1995
"... Results in the spectral theory of differential operators, and recent results on conformally covariant differential operators and on sharp inequalities, are combined in a study of functional determinants of natural differential operators. The setting is that of compact Riemannian manifolds. We conce ..."
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Cited by 57 (8 self)
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Results in the spectral theory of differential operators, and recent results on conformally covariant differential operators and on sharp inequalities, are combined in a study of functional determinants of natural differential operators. The setting is that of compact Riemannian manifolds. We concentrate especially on the conformally flat case, and obtain formulas in dimensions two, four, and six for the functional determinants of operators which are well behaved under conformal change of metric. The two dimensional formulas are due to Polyakov, and the four dimensional formulas to Branson and rsted; the method is sufficiently streamlined here that we are able to present the six dimensional case for the first time. In particular, we solve the extremal problems for the functional determinants of the conformal Laplacian and of the square of the Dirac operator on S 2 , and in the standard conformal classes on S 4 and S 6 . The S 2 results are due to Onofri, and the S 4 results...
The Covariant Technique For Calculation Of OneLoop Effective Action
, 1990
"... We develop a manifestly covariant technique for heat kernel calculation in the presence of arbitrary background fields in a curved space.The four lowestorder coefficients of SchwingerDe Witt asymptotic expansion are explicitly computed.We calculate also the heat kernel asymptotic expansion up to t ..."
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Cited by 34 (16 self)
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We develop a manifestly covariant technique for heat kernel calculation in the presence of arbitrary background fields in a curved space.The four lowestorder coefficients of SchwingerDe Witt asymptotic expansion are explicitly computed.We calculate also the heat kernel asymptotic expansion up to the terms of third order in the rapidly varying background fields (curvatures). This approximate series is summed up and the covariant nonlocal expressions for heat kernel,ifunction and oneloop effective action are obtained.Other related problems are discussed. y The permanent address after 1 August, 1990: Nuclear Physics Department, Institute of Physics, Rostov State University, Stachki 194, RostovonDon 344104, USSR I. G. Avramidi: Nuclear Physics B 355 (1991) 712754 2 1. Introduction One of the most fruitful approaches in quantum field theory, especially in gauge theories and gravity, is the background field method which assumes ever greater importance in recent years [122 ]. The b...
Heat kernels and functional determinants on the generalized cone
 Commun. Math. Phys
, 1996
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Residues Of The Eta Function For An Operator Of Dirac Type
 J. FUNCT ANAL
, 1992
"... We compute the asymptotics of Tr L 2 (P e \GammatP 2 ) where P is a first order operator of Dirac type; this is equivalent to evaluating the residues of the eta function. ..."
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Cited by 24 (5 self)
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We compute the asymptotics of Tr L 2 (P e \GammatP 2 ) where P is a first order operator of Dirac type; this is equivalent to evaluating the residues of the eta function.
Quantum ergodicity of boundary values of eigenfunctions
, 2002
"... Abstract. Suppose that Ω ⊂ R n is a bounded, piecewise smooth domain. We prove that the boundary values (Cauchy data) of eigenfunctions of the Laplacian on Ω with various boundary conditions are quantum ergodic if the classical billiard map β on the ball bundle B ∗ (∂Ω) is ergodic. Our proof is base ..."
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Cited by 19 (12 self)
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Abstract. Suppose that Ω ⊂ R n is a bounded, piecewise smooth domain. We prove that the boundary values (Cauchy data) of eigenfunctions of the Laplacian on Ω with various boundary conditions are quantum ergodic if the classical billiard map β on the ball bundle B ∗ (∂Ω) is ergodic. Our proof is based on the classical observation that the boundary values of an interior eigenfunction φλ, ∆φλ = λ 2 φλ is an eigenfunction of an operator Fh on the boundary of Ω with h = λ −1. In the case of the Neumann boundary condition, Fh is the boundary integral operator induced by the double layer potential. We show that Fh is a semiclassical Fourier integral operator quantizing the billiard map plus a ‘small ’ remainder; the quantum dyanmics defined by Fh can be exploited on the boundary much as the quantum dynamics generated by the wave group were exploited in the interior of domains with corners and ergodic billiards in the work of ZelditchZworski (1996). Novelties include the facts that Fh is not unitary and (consequently) the boundary values are equidistributed by measures which are not invariant under β and which depend on the boundary conditions. Ergodicity of boundary values of eigenfunctions on domains with ergodic billiards was conjectured by S. Ozawa (1993), and was almost simultaneously proved by GerardLeichtnam (1993) in the case of convex C 1,1 domains (with continuous tangent planes) and with Dirichlet boundary conditions. Our methods seem to be quite different. Motivation to study piecewise smooth domains comes from the fact that almost all known ergodic domains are of this form. Contents
Heat kernel coefficients of the Laplace operator on the Ddimensional ball
 J. Math. Phys
, 1996
"... ball ..."
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Vacuum expectation value asymptotics for second order differential operators on manifolds with boundary
 J. Math. Phys
, 1998
"... Abstract. Let M be a compact Riemannian manifold with smooth boundary. We study the vacuum expectation value of an operator Q by studying Tr L 2 Qe −tD, where D is an operator of Laplace type on M, and where Q is a second order operator with scalar leading symbol; we impose Dirichlet or modified Neu ..."
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Cited by 16 (9 self)
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Abstract. Let M be a compact Riemannian manifold with smooth boundary. We study the vacuum expectation value of an operator Q by studying Tr L 2 Qe −tD, where D is an operator of Laplace type on M, and where Q is a second order operator with scalar leading symbol; we impose Dirichlet or modified Neumann boundary conditions. Let M be a compact smooth Riemannian manifold of dimension m with smooth boundary ∂M. We say that a second order operator D on the space of smooth sections C ∞ (V) of a smooth vector bundle over M has scalar leading symbol if the leading symbol is h ij IV ξiξj for some symmetric 2tensor h. We say that D is
Asymptotic expansion for the heat kernel for orbifolds
 Michigan Math. J
"... ABSTRACT. We study the relationship between the geometry and the Laplace spectrum of a Riemannian orbifold O via its heat kernel; as in the manifold case, the timezero asymptotic expansion of the heat kernel furnishes geometric information about O. In the case of a good Riemannian orbifold (i.e., a ..."
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Cited by 15 (3 self)
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ABSTRACT. We study the relationship between the geometry and the Laplace spectrum of a Riemannian orbifold O via its heat kernel; as in the manifold case, the timezero asymptotic expansion of the heat kernel furnishes geometric information about O. In the case of a good Riemannian orbifold (i.e., an orbifold arising as the orbit space of a manifold under the action of a discrete group of isometries), H. Donnelly [10] proved the existence of the heat kernel and constructed the asymptotic expansion for the heat trace. We extend Donnelly’s work to the case of general compact orbifolds. Moreover, in both the good case and the general case, we express the heat invariants in a form that clarifies the asymptotic contribution of each part of the singular set of the orbifold. We calculate several terms in the asymptotic expansion explicitly in the case of twodimensional orbifolds; we use these terms to prove that the spectrum distinguishes elements within various classes of twodimensional orbifolds.
matrix theory and Khomology
"... In this paper, we study a new matrix theory based on nonBPS Dinstantons in type IIA string theory and Dinstanton anti Dinstanton system in type IIB string theory, which we call Kmatrix theory. The theory correctly incorporates the creation and anihilation processes of Dbranes. The configurati ..."
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Cited by 12 (3 self)
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In this paper, we study a new matrix theory based on nonBPS Dinstantons in type IIA string theory and Dinstanton anti Dinstanton system in type IIB string theory, which we call Kmatrix theory. The theory correctly incorporates the creation and anihilation processes of Dbranes. The configurations of the theory are identified with spectral triples, which is the noncommutative generalization of Riemannian geometry á la Connes, and they represent the geometry on the worldvolume of higher dimensional Dbranes. Remarkably, the configulations of Dbranes in the Kmatrix theory are naturally classified by a Ktheoretical version of homology group, called Khomology. Furthermore, we argue that the Khomology correctly classifies the Dbrane configurations from a geometrical point of view. We also construct the boundary states corresponding to the configurations of the Kmatrix theory, and explicitly show that they represent the higher dimensional Dbranes. 1
Analytic Torsion and RTorsion for Manifolds with Boundary
, 1999
"... Hao Fang We prove a formula relating the analytic torsion and Reidemeister torsion on manifolds with boundary in the general case when the metric is not necessarily a product near the boundary. The product case has been established by W. Lu¨ck and S. M. Vishik. We find that the extra term that comes ..."
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Cited by 12 (1 self)
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Hao Fang We prove a formula relating the analytic torsion and Reidemeister torsion on manifolds with boundary in the general case when the metric is not necessarily a product near the boundary. The product case has been established by W. Lu¨ck and S. M. Vishik. We find that the extra term that comes in here in the nonproduct case is the transgression of the Euler class in the even dimensional case and a slightly more mysterious term involving the second fundamental form of the boundary and the curvature tensor of the manifold in the odd dimensional case. 1