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34
Induced Gauge Theory on a Noncommutative Space
, 2007
"... We consider a scalar φ4 theory on canonically deformed Euclidean space in 4 dimensions with an additional oscillator potential. This model is known to be renormalisable. An exterior gauge field is coupled in a gauge invariant manner to the scalar field. We extract the dynamics for the gauge field fr ..."
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Cited by 53 (16 self)
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We consider a scalar φ4 theory on canonically deformed Euclidean space in 4 dimensions with an additional oscillator potential. This model is known to be renormalisable. An exterior gauge field is coupled in a gauge invariant manner to the scalar field. We extract the dynamics for the gauge field from the divergent terms of the 1loop effective action using a matrix basis and propose an action for the noncommutative gauge theory, which is a candidate for a renormalisable model.
The spectral action for Moyal planes
 J. Math. Phys
"... Extending a result of D. V. Vassilevich [50], we obtain the asymptotic expansion for the trace of a spatially regularized heat operator LΘ (f)e−t△Θ, where △Θ is a generalized Laplacian defined with Moyal products and LΘ (f) is Moyal left multiplication. The Moyal planes corresponding to any skewsymm ..."
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Cited by 44 (10 self)
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Extending a result of D. V. Vassilevich [50], we obtain the asymptotic expansion for the trace of a spatially regularized heat operator LΘ (f)e−t△Θ, where △Θ is a generalized Laplacian defined with Moyal products and LΘ (f) is Moyal left multiplication. The Moyal planes corresponding to any skewsymmetric matrix Θ being spectral triples [24], the spectral action introduced in noncommutative geometry by A. Chamseddine and A. Connes [6] is computed. This result generalizes the ConnesLott action [15] previously computed by Gayral [23] for symplectic Θ.
Heatkernel approach to UV/IR mixing on isospectral deformation manifolds
"... We work out the general features of perturbative field theory on noncommutative manifolds defined by isospectral deformation. These (in general curved) ‘quantum spaces’, generalizing Moyal planes and noncommutative tori, are constructed using Rieffel’s theory of deformation quantization by actions o ..."
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Cited by 27 (3 self)
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We work out the general features of perturbative field theory on noncommutative manifolds defined by isospectral deformation. These (in general curved) ‘quantum spaces’, generalizing Moyal planes and noncommutative tori, are constructed using Rieffel’s theory of deformation quantization by actions of R l. Our framework, incorporating background field methods and tools of QFT in curved spaces, allows to deal both with compact and noncompact spaces, as well as with periodic and nonperiodic deformations, essentially in the same way. We compute the quantum effective action up to one loop for a scalar theory, showing the different UV/IR mixing phenomena for different kinds of isospectral deformations. The presence and behavior of the nonplanar parts of the Green functions is understood simply in terms of offdiagonal heat kernel contributions. For periodic deformations, a Diophantine condition on the noncommutivity parameters is found to play a role in the analytical nature of the nonplanar part of the oneloop reduced effective action. Existence of fixed points for the action may give rise to a new kind of UV/IR mixing. Keywords: noncommutative field theory, isospectral deformation, UV/IR mixing, heat kernel, Diophantine approximation.
Heat Kernel and Number Theory on NCTorus
 Commun. Math. Phys
, 2007
"... The heat trace asymptotics on the noncommutative torus, where generalized Laplacians are made out of left and right regular representations, is fully determined. It turns out that this question is very sensitive to the numbertheoretical aspect of the deformation parameters. The central condition we ..."
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Cited by 18 (11 self)
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The heat trace asymptotics on the noncommutative torus, where generalized Laplacians are made out of left and right regular representations, is fully determined. It turns out that this question is very sensitive to the numbertheoretical aspect of the deformation parameters. The central condition we use is of a Diophantine type. More generally, the importance of number theory is made explicit on a few examples. We apply the results to the spectral action computation and revisit the UV/IR mixing phenomenon for a scalar theory. Although we find nonlocal counterterms in the NC φ 4 theory on T 4, we show that this theory can be made renormalizable at least at one loop, and may be even beyond.
Spectral action on noncommutative torus
 J. Noncommut. Geom
"... Dedicated to Alain Connes on the occasion of his 60th birthday The spectral action on noncommutative torus is obtained, using a Chamseddine– Connes formula via computations of zeta functions. The importance of a Diophantine condition is outlined. Several results on holomorphic continuation of series ..."
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Cited by 18 (9 self)
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Dedicated to Alain Connes on the occasion of his 60th birthday The spectral action on noncommutative torus is obtained, using a Chamseddine– Connes formula via computations of zeta functions. The importance of a Diophantine condition is outlined. Several results on holomorphic continuation of series of holomorphic functions are obtained in this context.
Noncommutative QFT and Renormalization
, 2006
"... Field theories on deformed spaces suffer from the IR/UV mixing and renormalization is generically spoiled. In work with R. Wulkenhaar, one of us realized a way to cure this disease by adding one more marginal operator. We review these ideas, show the application to φ 3 models and use the heat kernel ..."
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Cited by 15 (5 self)
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Field theories on deformed spaces suffer from the IR/UV mixing and renormalization is generically spoiled. In work with R. Wulkenhaar, one of us realized a way to cure this disease by adding one more marginal operator. We review these ideas, show the application to φ 3 models and use the heat kernel expansion methods for a scalar field theory coupled to an external gauge field on a θdeformed space and derive noncommutative gauge field actions.
Noncommutative Induced Gauge Theories on Moyal Spaces
, 2007
"... Noncommutative field theories on Moyal spaces can be conveniently handled within a framework of noncommutative geometry. Several renormalisable matter field theories that are now identified are briefly reviewed. The construction of renormalisable gauge theories on these noncommutative Moyal spaces, ..."
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Cited by 13 (4 self)
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Noncommutative field theories on Moyal spaces can be conveniently handled within a framework of noncommutative geometry. Several renormalisable matter field theories that are now identified are briefly reviewed. The construction of renormalisable gauge theories on these noncommutative Moyal spaces, which remains so far a challenging problem, is then closely examined. The computation in 4D of the oneloop effective gauge theory generated from the integration over a scalar field appearing in a renormalisable theory minimally coupled to an external gauge potential is presented. The gauge invariant effective action is found to involve, beyond the expected noncommutative version of the pure YangMills action, additional terms that may be interpreted as the gauge theory counterpart of the harmonic term, which for the noncommutative ϕ 4theory on Moyal space ensures renormalisability. A class of possible candidates for renormalisable gauge theory actions defined on Moyal space is presented and discussed.
Heat kernel coefficients for compact fuzzy spaces
 JHEP
, 2004
"... Abstract: I discuss the trace of a heat kernel Tr(e −tA) for compact fuzzy spaces. In continuum theory its asymptotic expansion for t → +0 provides geometric quantities, and therefore may be used to extract effective geometric quantities for fuzzy spaces. For compact fuzzy spaces, however, an asympt ..."
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Cited by 8 (2 self)
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Abstract: I discuss the trace of a heat kernel Tr(e −tA) for compact fuzzy spaces. In continuum theory its asymptotic expansion for t → +0 provides geometric quantities, and therefore may be used to extract effective geometric quantities for fuzzy spaces. For compact fuzzy spaces, however, an asymptotic expansion for t → +0 is not appropriate because of their finiteness. Instead a series of functions of t can be defined, which take almost constant values in a range of t, where geometric description of a fuzzy space is effective. The values of these functions in this range give its effective geometric quantities. This method is applied to some known fuzzy spaces to check its validity. Fuzzy spaces [1, 2] have a potential for replacing the classical geometric description of spacetime. They could incorporate the quantum fluctuation of spacetime which is suggested by the existence of minimal length in semiclassical quantum gravity and string theory [3, 4] and another kind of arguments [5, 6, 7, 8, 9]. The fuzziness is roughly in the order of the Planck length, and classical geometry is expected to be effective well over the scale. Since we understand rather well spacetime dynamics in terms of geometry, it would be useful to understand what is geometry of fuzzy spaces in the study of their dynamics. Since classical geometry can be obtained from trajectories of particles, the effective geometry of fuzzy spaces will be determined by low energy propagation modes of some fields well over the scale of fuzziness. This is in accordance with the spirit of [10]. A fuzzy space is often characterized by a scalar field action, which determines the propagation of the scalar field. In continuum theory, an asymptotic expansion of a heat kernel of a Laplacian has geometric quantities as its coefficients [11, 12]. For a Laplacian A = − ∆ on a space without boundaries, the asymptotic expansion for t → +0 is given by Tr(e −tA) = ∑ t j−ν/2 d ν x ε2j(x), (1) j=0 where ν is the dimensions of the space, and ε0(x) = ε2(x) = ε4(x) =
Heat Trace Asymptotics on Noncommutative Spaces
, 2007
"... This is a minireview of the heat kernel expansion for generalized Laplacians on various noncommutative spaces. Applications to the spectral action principle, renormalization of noncommutative theories and anomalies are also considered. ..."
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Cited by 6 (4 self)
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This is a minireview of the heat kernel expansion for generalized Laplacians on various noncommutative spaces. Applications to the spectral action principle, renormalization of noncommutative theories and anomalies are also considered.
Renormalization and Induced Gauge Action on a Noncommutative Space
, 2007
"... Field theories on deformed spaces suffer from the IR/UV mxing and renormalization is generically spoiled. In work with R. Wulkenhaar, one of us realized a way to cure this desease by adding one more marginal operator. We review these ideas, show the application to φ3 models and use heat kernel expan ..."
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Cited by 4 (0 self)
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Field theories on deformed spaces suffer from the IR/UV mxing and renormalization is generically spoiled. In work with R. Wulkenhaar, one of us realized a way to cure this desease by adding one more marginal operator. We review these ideas, show the application to φ3 models and use heat kernel expansion methods for a scalar field theory coupled to an external gauge field on a θdeformed space and derive noncommutative gauge actions.