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Noncommutative geometry, quantum fields and motives
 Colloquium Publications, Vol.55, American Mathematical Society
, 2008
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Toeplitz Quantization Of Kähler Manifolds And gl(N), N → ∞ Limits
"... For general compact Kähler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a welldefined (by operator norm estimates) classical limit. This generalizes earlier results of the authors and Klimek and Lesniewski obtained for the torus and higher genus Riemann s ..."
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Cited by 134 (10 self)
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For general compact Kähler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a welldefined (by operator norm estimates) classical limit. This generalizes earlier results of the authors and Klimek and Lesniewski obtained for the torus and higher genus Riemann surfaces, respectively. We thereby arrive at an approximation of the Poisson algebra by a sequence of finitedimensional matrix algebras gl(N), N → ∞.
The Schrödinger functional  A renormalizable probe for nonAbelian gauge theories
, 1992
"... Following Symanzik we argue that the Schrödinger functional in lattice gauge theories without matter fields has a welldefined continuum limit. Due to gauge invariance no extra counter terms are required. The Schrödinger functional is, moreover, accessible to numerical simulations. It may hence be u ..."
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Cited by 96 (8 self)
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Following Symanzik we argue that the Schrödinger functional in lattice gauge theories without matter fields has a welldefined continuum limit. Due to gauge invariance no extra counter terms are required. The Schrödinger functional is, moreover, accessible to numerical simulations. It may hence be used to study the scaling properties of the theory and in particular the evolution of the renormalized gauge coupling from low to high energies. A concrete proposition along this line is made and the necessary perturbative analysis of the Schrödinger functional is carried through to 1loop order
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 88 (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 Asymptotics of the Laplacian on a Manifold With Boundary
, 1990
"... : Let P be a secondorder differential operator with leading symbol given by the metric tensor on a compact Riemannian manifold with boundary. We compute the asymptotics of the heat equation for Dirichlet, Neumann, and mixed boundary conditions. x1 Statement of results Let M m be a compact Riemann ..."
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Cited by 88 (24 self)
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: Let P be a secondorder differential operator with leading symbol given by the metric tensor on a compact Riemannian manifold with boundary. We compute the asymptotics of the heat equation for Dirichlet, Neumann, and mixed boundary conditions. x1 Statement of results Let M m be a compact Riemannian manifold with boundary @M: Let V be a smooth vector bundle over M equipped with a connection r V : Let E be an endomorphism of V: Define P = \Gamma(\Sigma i;j g ij r V i r V j +E) : C 1 (V ) ! C 1 (V ): Every second order elliptic operator on M with leading symbol given by the metric tensor can be put in this form. Let f 2 C 1 (M): If @M = ;; then as t ! 0 + ; T r L 2 (fe \GammatP ) ' t \Gammam=2 \Sigma n t n an (f; P ) where n = 0; 1; 2; ::: ranges over the nonnegative integers. If @M 6= ;; we must impose suitable boundary conditions. Let OE 2 C 1 (V ): Dirichlet boundary conditions are BOE = OEj @M = 0: Let OE ;N be the covariant derivative of OE with respect ...
Chiral gauge theories on the lattice with exact gauge invariance
, 1999
"... A recently proposed formulation of chiral lattice gauge theories is reviewed, in which the locality and gauge invariance of the theory can be preserved if the fermion representation of the gauge group is anomalyfree. ..."
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Cited by 65 (2 self)
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A recently proposed formulation of chiral lattice gauge theories is reviewed, in which the locality and gauge invariance of the theory can be preserved if the fermion representation of the gauge group is anomalyfree.
Emergent Gravity from Noncommutative Gauge Theory
, 2007
"... We show that the matrixmodel action for noncommutative U(n) gauge theory actually describes SU(n) gauge theory coupled to gravity. This is elaborated in the 4dimensional case. The SU(n) gauge fields as well as additional scalar fields couple to an effective metric Gab, which is determined by a dyn ..."
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Cited by 60 (29 self)
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We show that the matrixmodel action for noncommutative U(n) gauge theory actually describes SU(n) gauge theory coupled to gravity. This is elaborated in the 4dimensional case. The SU(n) gauge fields as well as additional scalar fields couple to an effective metric Gab, which is determined by a dynamical Poisson structure. The emergent gravity is intimately related to noncommutativity, encoding those degrees of freedom which are usually interpreted as U(1) gauge fields. This leads to a class of metrics which contains the physical degrees of freedom of gravitational waves, and allows to recover e.g. the Newtonian limit with arbitrary mass distribution. It also suggests a consistent picture of UV/IR mixing in terms of an induced gravity action. This should provide a suitable framework for quantizing gravity.