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Renormalization and motivic Galois theory
 International Math. Research Notices
"... Abstract. We investigate the nature of divergences in quantum field theory, showing that they are organized in the structure of a certain “ motivic Galois group ” U ∗ , which is uniquely determined and universal with respect to the set of physical theories. The renormalization group can be identifie ..."
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Cited by 24 (13 self)
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Abstract. We investigate the nature of divergences in quantum field theory, showing that they are organized in the structure of a certain “ motivic Galois group ” U ∗ , which is uniquely determined and universal with respect to the set of physical theories. The renormalization group can be identified canonically with a one parameter subgroup of U ∗. The group U ∗ arises through a Riemann–Hilbert correspondence. Its representations classify equisingular flat vector bundles, where the equisingularity condition is a geometric formulation of the fact that in quantum field theory the counterterms are independent of the choice of a unit of mass. As an algebraic group scheme, U ∗ is a semidirect product by the multiplicative group Gm of a prounipotent group scheme whose Lie algebra is freely generated by one generator in each positive integer degree. There is a universal singular frame in which all divergences disappear. When computed as iterated integrals, its coefficients are certain rational numbers that appear in the local index formula of Connes–Moscovici. When working with formal Laurent series over Q, the data of equisingular flat vector bundles
Picard groups in Poisson geometry
 MOSCOW MATH. J
, 2003
"... We study isomorphism classes of symplectic dual pairs P ← S → P, where P is an integrable Poisson manifold, S is symplectic, and the two maps are complete, surjective Poisson submersions with connected and simplyconnected fibres. For fixed P, these Morita selfequivalences of P form a group Pic(P) ..."
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Cited by 12 (2 self)
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We study isomorphism classes of symplectic dual pairs P ← S → P, where P is an integrable Poisson manifold, S is symplectic, and the two maps are complete, surjective Poisson submersions with connected and simplyconnected fibres. For fixed P, these Morita selfequivalences of P form a group Pic(P) under a natural “tensor product ” operation. Variants of this construction are also studied, for rings (the origin of the notion of Picard group), Lie groupoids, and symplectic groupoids.
Quantum fields and motives
 J. Geom. Phys
, 2006
"... The main idea of renormalization is to correct the original Lagrangian of a quantum field theory by an infinite series of counterterms, labelled by the Feynman graphs that encode the combinatorics of the perturbative expansion of the theory. These counterterms have the effect of cancelling the ultra ..."
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Cited by 10 (2 self)
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The main idea of renormalization is to correct the original Lagrangian of a quantum field theory by an infinite series of counterterms, labelled by the Feynman graphs that encode the combinatorics of the perturbative expansion of the theory. These counterterms have the effect of cancelling the ultraviolet divergences. Thus, in the procedure of perturbative renormalization, one introduces a counterterm C(Γ) in the initial Lagrangian for every divergent one particle irreducible (1PI) Feynman diagram Γ. In the case of a renormalizable theory, all the necessary counterterms C(Γ) can be obtained by modifying the numerical parameters that appear in the original Lagrangian. It is possible to modify these parameters and replace them by (divergent) series, since they are not observable, unlike actual physical quantities that have to be finite. One of the fundamental difficulties with any renormalization procedure is a systematic treatment of nested and overlapping divergences in multiloop diagrams. Dimensional regularization and minimal subtraction. One of the most effective renormalization techniques in quantum field theory is
Meeting Descartes and Klein somewhere in a noncommutative space
 Highlights of Mathematical Physics
"... Abstract. We combine the coordinate method and Erlangen program in the framework of noncommutative geometry through an investigation of symmetries of noncommutative coordinate algebras. As the model we use the coherent states construction and the wavelet transform in functional spaces. New examples ..."
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Cited by 10 (8 self)
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Abstract. We combine the coordinate method and Erlangen program in the framework of noncommutative geometry through an investigation of symmetries of noncommutative coordinate algebras. As the model we use the coherent states construction and the wavelet transform in functional spaces. New examples are a three dimensional spectrum of a nonnormal matrix and a quantisation procedure from symplectomorphisms.
Category Theory: an abstract setting for analogy and comparison
 In: What is Category Theory? Advanced Studies in Mathematics and Logic, Polimetrica Publisher
, 2006
"... and comparison ..."
Brane and string field structure of elementary particles
, 2006
"... The main relevant features of quantum (field) theories are examined in order to set up the physical and mathematical foundations of the algebraic quantum theory. It then appears that the two quantizations of QFT, as well as the attempt of unifying it with general relativity, lead us to consider that ..."
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Cited by 3 (3 self)
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The main relevant features of quantum (field) theories are examined in order to set up the physical and mathematical foundations of the algebraic quantum theory. It then appears that the two quantizations of QFT, as well as the attempt of unifying it with general relativity, lead us to consider that the internal structure of an elementary fermion must be twofold and composed of three embedded internal (bi)structures which are vacuum and mass (physical) bosonic fields decomposing into packets of pairs of strings behaving like harmonic oscillators characterized by integers µ corresponding to normal modes at µ (algebraic) quanta.
C ∞Smooth Singularities Exposed: Chimeras of the Differential Spacetime Manifold
, 2004
"... The glaringly serious conflict between the principle of general covariance of General Relativity (GR) and the existence of C ∞smooth singularities assailing the differential spacetime manifold on which the classical relativistic field theory of gravity vitally depends, is resolved by using the basi ..."
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Cited by 3 (2 self)
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The glaringly serious conflict between the principle of general covariance of General Relativity (GR) and the existence of C ∞smooth singularities assailing the differential spacetime manifold on which the classical relativistic field theory of gravity vitally depends, is resolved by using the basic manifold independent and, in extenso, Calculusfree concepts, techniques and results of Abstract Differential Geometry (ADG). As a physical toy model to illustrate these ideas, the ADGtheoretic resolution of both the exterior, but more importantly, of the inner, Schwarzschild singularities of the gravitational field of a point particle is presented, with the resolution of the latter being carried out entirely by finitisticalgebraic and sheaftheoretic means, and in two different ways. First, by regarding it as a localized, ‘static’ pointsingularity, we apply Sorkin’s finitary topological poset discretization scheme in its Gel’fand dual representation in terms of ‘discrete ’ differential incidence algebras [297, 298] and the finitary spacetime sheaves thereof [289]. Then we exercise the ADG machinery on those sheaves in the manner of [249, 250, 251] to show that the vacuum Einstein equations still hold over the classically offensive locus occupied by the pointmass both at the ‘discrete’
Noncommutative geometry through monoidal categories I
"... Abstract. After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasicoherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, HopfGalois extensions) can be given geometric meaning extending ..."
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Cited by 3 (0 self)
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Abstract. After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasicoherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, HopfGalois extensions) can be given geometric meaning extending their geometric interpretations in the commutative case. On the other hand, we show that some constructions in commutative geometry (e.g. faithfully flat descent theory, principal fibrations, equivariant and infinitesimal geometry) can be interpreted as noncommutative geometric constructions applied to commutative objects. For such generalized geometry we define global invariants constructing cyclic objects from which we derive Hochschild, cyclic and periodic cyclic homology (with coefficients) in the standard way. Contents