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49
Noncommutative geometry, quantum fields and motives
 Colloquium Publications, Vol.55, American Mathematical Society
, 2008
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Anatomy of a gauge theory
, 2006
"... We exhibit the role of Hochschild cohomology in quantum field theory with particular emphasis on gauge theory and Dyson–Schwinger equations, the quantum equations of motion. These equations emerge from Hopf and Lie algebra theory and free quantum field theory only. In the course of our analysis we ..."
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Cited by 19 (7 self)
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We exhibit the role of Hochschild cohomology in quantum field theory with particular emphasis on gauge theory and Dyson–Schwinger equations, the quantum equations of motion. These equations emerge from Hopf and Lie algebra theory and free quantum field theory only. In the course of our analysis we exhibit an intimate relation between the SlavnovTaylor identities for the couplings and the existence of Hopf subalgebras defined on the sum of all graphs at a given loop order, surpassing the need to work on single diagrams. 0
Mixed Hodge Structures and Renormalization in Physics
 Commun. Num. Theor. Phys
"... 1.1. This paper is a collaboration between a mathematician and a physicist. It is based on the observation that renormalization of Feynman amplitudes in physics is closely related to the theory of limiting mixed Hodge structures in mathematics. Whereas classical physical renormalization methods invo ..."
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Cited by 17 (2 self)
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1.1. This paper is a collaboration between a mathematician and a physicist. It is based on the observation that renormalization of Feynman amplitudes in physics is closely related to the theory of limiting mixed Hodge structures in mathematics. Whereas classical physical renormalization methods involve manipulations with the integrand of a divergent integral, limiting Hodge theory involves moving the chain of integration so
FEYNMAN MOTIVES OF BANANA GRAPHS
"... Abstract. We consider the infinite family of Feynman graphs known as the “banana graphs ” and compute explicitly the classes of the corresponding graph hypersurfaces in the Grothendieck ring of varieties as well as their Chern–Schwartz–MacPherson classes, using the classical Cremona transformation a ..."
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Cited by 17 (12 self)
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Abstract. We consider the infinite family of Feynman graphs known as the “banana graphs ” and compute explicitly the classes of the corresponding graph hypersurfaces in the Grothendieck ring of varieties as well as their Chern–Schwartz–MacPherson classes, using the classical Cremona transformation and the dual graph, and a blowup formula for characteristic classes. We outline the interesting similarities between these operations and we give formulae for cones obtained by simple operations on graphs. We formulate a positivity conjecture for characteristic classes of graph hypersurfaces and discuss briefly the effect of passing to noncommutative spacetime. 1.
The massless higherloop twopoint function
 Commun. Math. Phys
, 2009
"... Abstract. We introduce a new method for computing massless Feynman integrals analytically in parametric form. An analysis of the method yields a criterion for a primitive Feynman graph G to evaluate to multiple zeta values. The criterion depends only on the topology of G, and can be checked algorith ..."
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Cited by 14 (2 self)
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Abstract. We introduce a new method for computing massless Feynman integrals analytically in parametric form. An analysis of the method yields a criterion for a primitive Feynman graph G to evaluate to multiple zeta values. The criterion depends only on the topology of G, and can be checked algorithmically. As a corollary, we reprove the result, due to Bierenbaum and Weinzierl, that the massless 2loop 2point function is expressible in terms of multiple zeta values, and generalize this to the 3, 4, and 5loop cases. We find that the coefficients in the Taylor expansion of planar graphs in this range evaluate to multiple zeta values, but the nonplanar graphs with crossing number 1 may evaluate to multiple sums with 6 th roots of unity. Our method fails for the five loop graphs with crossing number 2 obtained by breaking open the bipartite graph K3,4 at one edge. 1.
ALGEBROGEOMETRIC FEYNMAN RULES
"... Abstract. We give a general procedure to construct algebrogeometric Feynman rules, that is, characters of the Connes–Kreimer Hopf algebra of Feynman graphs that factor through a Grothendieck ring of immersed conical varieties, via the class of the complement of the affine graph hypersurface. In par ..."
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Cited by 13 (9 self)
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Abstract. We give a general procedure to construct algebrogeometric Feynman rules, that is, characters of the Connes–Kreimer Hopf algebra of Feynman graphs that factor through a Grothendieck ring of immersed conical varieties, via the class of the complement of the affine graph hypersurface. In particular, this maps to the usual Grothendieck ring of varieties, defining motivic Feynman rules. We also construct an algebrogeometric Feynman rule with values in a polynomial ring, which does not factor through the usual Grothendieck ring, and which is defined in terms of characteristic classes of singular varieties. This invariant recovers, as a special value, the Euler characteristic of the projective graph hypersurface complement. The main result underlying the construction of this invariant is a formula for the characteristic classes of the join of two projective varieties. We discuss the BPHZ renormalization procedure in this algebrogeometric context and some motivic zeta functions arising from the partition functions associated to motivic Feynman rules. 1.
A Lie theoretic approach to renormalization
 Comm. Math. Phys
"... Abstract. Motivated by recent work of Connes and Marcolli, based on the Connes– Kreimer approach to renormalization, we augment the latter by a combinatorial, Lie algebraic point of view. Our results rely both on the properties of the Dynkin idempotent, one of the fundamental Lie idempotents in the ..."
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Cited by 12 (6 self)
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Abstract. Motivated by recent work of Connes and Marcolli, based on the Connes– Kreimer approach to renormalization, we augment the latter by a combinatorial, Lie algebraic point of view. Our results rely both on the properties of the Dynkin idempotent, one of the fundamental Lie idempotents in the theory of free Lie algebras, and on the fine properties of Hopf algebras and their associated descent algebras. Besides leading very directly to proofs of the main combinatorial properties of the renormalization procedures, the new techniques do not depend on the geometry underlying the particular case of dimensional regularization and the Riemann–Hilbert correspondence. This is illustrated with a discussion of the BPHZ renormalization scheme.
Motives associated to graphs
 Jpn. J. Math
"... Abstract. A report on recent results and outstanding problems concerning motives associated to graphs. 1. ..."
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Cited by 11 (0 self)
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Abstract. A report on recent results and outstanding problems concerning motives associated to graphs. 1.
Dyson–Schwinger Equations: from Hopf algebras to number theory
, 2006
"... We consider the structure of renormalizable quantum field theories from the viewpoint of their underlying Hopf algebra structure. We review how to use this Hopf algebra and the ensuing Hochschild cohomology to derive nonperturbative results for the shortdistance singular sector of a renormalizable ..."
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Cited by 10 (2 self)
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We consider the structure of renormalizable quantum field theories from the viewpoint of their underlying Hopf algebra structure. We review how to use this Hopf algebra and the ensuing Hochschild cohomology to derive nonperturbative results for the shortdistance singular sector of a renormalizable quantum field theory. We focus on the shortdistance behaviour and thus discuss renormalized Green functions GR(α, L) which depend on a single scale L = lnq²/µ².
Dyson Schwinger equations: From Hopf algebras to number theory
 Universality and Renormalization, volume 50 of Fields Inst. Comm
, 2007
"... Abstract. We consider the structure of renormalizable quantum field theories from the viewpoint of their underlying Hopf algebra structure. We review how to use this Hopf algebra and the ensuing Hochschild cohomology to derive nonperturbative results for the shortdistance singular sector of a reno ..."
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Cited by 9 (6 self)
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Abstract. We consider the structure of renormalizable quantum field theories from the viewpoint of their underlying Hopf algebra structure. We review how to use this Hopf algebra and the ensuing Hochschild cohomology to derive nonperturbative results for the shortdistance singular sector of a renormalizable quantum field theory. We focus on the shortdistance behaviour and thus discuss renormalized Green functions GR(α, L) which depend on a single scale L = lnq 2 /µ 2. 1