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Conjectured enumeration of irreducible multiple zeta values, from knots and Feynman diagrams, subm
"... Abstract Multiple zeta values (MZVs) are under intense investigation in three arenas – knot theory, number theory, and quantum field theory – which unite in Kreimer’s proposal that field theory assigns MZVs to positive knots, via Feynman diagrams whose momentum flow is encoded by link diagrams. Two ..."
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Cited by 18 (3 self)
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Abstract Multiple zeta values (MZVs) are under intense investigation in three arenas – knot theory, number theory, and quantum field theory – which unite in Kreimer’s proposal that field theory assigns MZVs to positive knots, via Feynman diagrams whose momentum flow is encoded by link diagrams. Two challenging problems are posed by this nexus of knot/number/field theory: enumeration of positive knots, and enumeration of irreducible MZVs. Both were recently tackled by Broadhurst and Kreimer (BK). Here we report largescale analytical and numerical computations that test, with considerable severity, the BK conjecture that the number, Dn,k, of irreducible MZVs of weight n and depth k, is generated by � � n≥3 k≥1(1 − xnyk) Dn,k = 1 − x3y 1−x2 + x12y2 (1−y2) (1−x4)(1−x6, which is here shown to be consistent with all shuffle identities for the corresponding iterated integrals, up to
Renormalization of multiple zeta values
 J. Algebra
, 2006
"... Abstract. Multiple zeta values (MZVs) in the usual sense are the special values of multiple variable zeta functions at positive integers. Their extensive studies are important in both mathematics and physics with broad connections and applications. In contrast, very little is known about the special ..."
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Cited by 14 (10 self)
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Abstract. Multiple zeta values (MZVs) in the usual sense are the special values of multiple variable zeta functions at positive integers. Their extensive studies are important in both mathematics and physics with broad connections and applications. In contrast, very little is known about the special values of multiple zeta functions at nonpositive integers since the values are usually singular. We define and study multiple zeta functions at integer values by adapting methods of renormalization from quantum field theory, and following the Hopf algebra approach of Connes and Kreimer. This definition of renormalized MZVs agrees with the convergent MZVs and extends the work of IharaKanekoZagier on renormalization of MZVs with positive arguments. We further show that the important
Weight Systems from Feynman Diagrams, J.Knot Th.Ram.7
 J. Knot Theor. Ramifications
, 1998
"... We find that the overall UV divergences of a renormalizable field theory with trivalent vertices fulfil a fourterm relation. They thus come close to establish a weight system. This provides a first explanation of the recent successful association of renormalization theory with knot theory. 1 ..."
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Cited by 9 (4 self)
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We find that the overall UV divergences of a renormalizable field theory with trivalent vertices fulfil a fourterm relation. They thus come close to establish a weight system. This provides a first explanation of the recent successful association of renormalization theory with knot theory. 1
Globally nilpotent differential operators and the square Ising model
, 2009
"... We recall various multiple integrals with one parameter, related to the isotropic square Ising model, and corresponding, respectively, to the nparticle contributions of the magnetic susceptibility, to the (lattice) form factors, to the twopoint correlation functions and to their λextensions. The ..."
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Cited by 7 (5 self)
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We recall various multiple integrals with one parameter, related to the isotropic square Ising model, and corresponding, respectively, to the nparticle contributions of the magnetic susceptibility, to the (lattice) form factors, to the twopoint correlation functions and to their λextensions. The univariate analytic functions defined by these integrals are holonomic and even Gfunctions: they satisfy Fuchsian linear differential equations with polynomial coefficients and have some arithmetic properties. We recall the explicit forms, found in previous work, of these Fuchsian equations, as well as their Russiandoll and direct sum structures. These differential operators are selected Fuchsian linear differential operators, and their remarkable properties have a deep geometrical origin: they are all globally nilpotent, or, sometimes, even have zero pcurvature. We also display miscellaneous examples of globally nilpotent operators emerging from enumerative combinatorics problems for which no integral representation is yet known. Focusing on the factorized
On the sum formula for the qanalogue of nonstrict multiple zeta values
 Proc. Amer. Math. Soc
"... Abstract. In this article, the qanalogues of the linear relations of nonstrict multiple zeta values called “the sum formula ” and “the cyclic sum formula” are established. 1. ..."
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Cited by 6 (2 self)
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Abstract. In this article, the qanalogues of the linear relations of nonstrict multiple zeta values called “the sum formula ” and “the cyclic sum formula” are established. 1.
Zimmermann type cancellation in the free Faà di Bruno algebra
 J. Funct. Anal
"... Krattenaler (BFK) in the context of noncommutative Lagrange inversion can be identified with the inverse of the incidence algebra of Ncolored interval partitions. The (BFK) antipode and its reflection determine the (generally distinct) left and right inverses of power series with noncommuting coe ..."
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Cited by 6 (0 self)
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Krattenaler (BFK) in the context of noncommutative Lagrange inversion can be identified with the inverse of the incidence algebra of Ncolored interval partitions. The (BFK) antipode and its reflection determine the (generally distinct) left and right inverses of power series with noncommuting coefficients and N noncommuting variables. As in the case of the Faà di Bruno Hopf algebra, there is an analogue of the Zimmermann cancellation formula. The summands of the (BFK) antipode can indexed by the depth first ordering of vertices on contracted planar trees, and the same applies to the interval partition antipode. Both can also be indexed by the breadth first ordering of vertices in the nonorder contractible planar trees in which precisely one nondegenerate vertex occurs on each level. 1.
Non local theories: new rules for old diagrams
 J. High Energy Phys
"... We show that a general variant of the Wick theorems can be used to reduce the time ordered products in the GellMann & Low formula for a certain class on non local quantum field theories, including the case where the interaction Lagrangian is defined in terms of twisted products. The only necessary ..."
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Cited by 5 (2 self)
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We show that a general variant of the Wick theorems can be used to reduce the time ordered products in the GellMann & Low formula for a certain class on non local quantum field theories, including the case where the interaction Lagrangian is defined in terms of twisted products. The only necessary modification is the replacement of the StueckelbergFeynman propagator by the general propagator (the “contractor ” of Denk and Schweda) D(y − y ′ ; τ − τ ′ ) = 1
DIFFERENTIAL ALGEBRAIC BIRKHOFF DECOMPOSITION AND THE RENORMALIZATION OF MULTIPLE ZETA VALUES
"... Abstract. In the Hopf algebra approach of Connes and Kreimer on renormalization of quantum field theory, the renormalization process is viewed as a special case of the Algebraic Birkhoff Decomposition. We give a differential algebra variation of this decomposition and apply this to the study of mult ..."
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Cited by 3 (2 self)
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Abstract. In the Hopf algebra approach of Connes and Kreimer on renormalization of quantum field theory, the renormalization process is viewed as a special case of the Algebraic Birkhoff Decomposition. We give a differential algebra variation of this decomposition and apply this to the study of multiple zeta values. 1.
Hopf Algebras in General and in Combinatorial Physics: a practical introduction
, 802
"... Abstract. This tutorial is intended to give an accessible introduction to Hopf algebras. The mathematical context is that of representation theory, and we also illustrate the structures with examples taken from combinatorics and quantum physics, showing that in this latter case the axioms of Hopf al ..."
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Cited by 1 (0 self)
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Abstract. This tutorial is intended to give an accessible introduction to Hopf algebras. The mathematical context is that of representation theory, and we also illustrate the structures with examples taken from combinatorics and quantum physics, showing that in this latter case the axioms of Hopf algebra arise naturally. The text contains many exercises, some taken from physics, aimed at expanding and exemplifying the concepts introduced. 1.
NUMBER THEORY IN PHYSICS
"... always been the case for the theory of differential equations. In the early twentieth century, with the advent of general relativity and quantum mechanics, topics such as differential and Riemannian geometry, operator algebras and functional analysis, or group theory also developed a close relation ..."
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always been the case for the theory of differential equations. In the early twentieth century, with the advent of general relativity and quantum mechanics, topics such as differential and Riemannian geometry, operator algebras and functional analysis, or group theory also developed a close relation to physics. In the past decade, mostly through the influence of string theory, algebraic geometry also began to play a major role in this interaction. Recent years have seen an increasing number of results suggesting that number theory also is beginning to play an essential part on the scene of contemporary theoretical and mathematical physics. Conversely, ideas from physics, mostly from quantum field theory and string theory, have started to influence work in number theory. In describing significant occurrences of number theory in physics, we will, on the one hand, restrict our attention to quantum physics, while, on the other hand, we will assume a somewhat extensive definition of number theory, that will allow us to include arithmetic algebraic geometry. The territory is vast and an extensive treatment would go beyond the size limits imposed by the encyclopaedia. The