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144
From polygons and symbols to polylogarithmic functions
 JHEP 1210 (2012) 075, arXiv:1110.0458 [mathph
"... Abstract: We present a review of the symbol map, a mathematical tool that can be useful in simplifying expressions among multiple polylogarithms, and recall its main properties. A recipe is given for how to obtain the symbol of a multiple polylogarithm in terms of the combinatorial properties of an ..."
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Cited by 43 (8 self)
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Abstract: We present a review of the symbol map, a mathematical tool that can be useful in simplifying expressions among multiple polylogarithms, and recall its main properties. A recipe is given for how to obtain the symbol of a multiple polylogarithm in terms of the combinatorial properties of an associated rooted decorated polygon. We also outline a systematic approach to constructing a function corresponding to a given symbol, and illustrate it in the particular case of harmonic polylogarithms up to weight four. Furthermore, part of the ambiguity of this process is highlighted by exhibiting a family of nontrivial elements in the kernel of the symbol map for arbitrary weight.
A Topos for Algebraic Quantum Theory
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2009
"... The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C ..."
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Cited by 31 (5 self)
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The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C*algebra of observables A induces a topos T (A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*algebra A. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum �(A) in T (A), which in our approach plays the role of the quantum phase space of the system. Thus we associate a locale (which is the topostheoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on �, and selfadjoint elements of A define continuous functions (more precisely, locale maps) from � to Scott’s interval domain. Noting that open subsets of �(A) correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos T (A). These results were inspired by the topostheoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.
Renormalization and effective field theory
, 2011
"... This is a preliminary version of the book Renormalization and Effective Field Theory published by the American Mathematical Society (AMS). This preliminary version is made available with the permission of the AMS and may not be changed, edited, or reposted at any other website without explicit writt ..."
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Cited by 22 (1 self)
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This is a preliminary version of the book Renormalization and Effective Field Theory published by the American Mathematical Society (AMS). This preliminary version is made available with the permission of the AMS and may not be changed, edited, or reposted at any other website without explicit written permission from the author and the AMS. Author's preliminary version made available with permission of the publisher, the American Mathematical SocietyContents
Scalar curvature for the noncommutative two torus
 J. Noncommut. Geom
"... Abstract. The scalar curvature for the noncommutative four torus T4Θ, where its flat geometry is conformally perturbed by a Weyl factor, is computed by making the use of a noncommutative residue that involves integration over the 3sphere. This method is more convenient since it does not require the ..."
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Abstract. The scalar curvature for the noncommutative four torus T4Θ, where its flat geometry is conformally perturbed by a Weyl factor, is computed by making the use of a noncommutative residue that involves integration over the 3sphere. This method is more convenient since it does not require the rearrangement lemma and it is advantageous as it explains the simplicity of the final functions of one and two variables, which describe the curvature with the help of a modular automorphism. In particular, it readily allows to write the function of two variables as the sum of a finite difference and a finite product of the one variable function. The curvature formula is simplified for dilatons of the form sp, where s is a real parameter and p ∈ C∞(T4Θ) is an arbitrary projection, and it is observed that, in contrast to the two dimensional case studied by A. Connes and H. Moscovici, unbounded functions of the parameter s appear in the final formula. An explicit formula for the gradient of the analog
Spectral triples on the superVirasoro algebra
"... We construct infinite dimensional spectral triples associated with representations of the superVirasoro algebra. In particular the irreducible, unitary positive energy representation of the Ramond algebra with central charge c and minimal lowest weight h = c/24 is graded and gives rise to a net of ..."
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We construct infinite dimensional spectral triples associated with representations of the superVirasoro algebra. In particular the irreducible, unitary positive energy representation of the Ramond algebra with central charge c and minimal lowest weight h = c/24 is graded and gives rise to a net of even θsummable spectral triples with nonzero Fredholm index. The irreducible unitary positive energy representations of the NeveuSchwarz algebra give rise to nets of even θsummable generalised spectral triples where there is no Dirac operator but only a superderivation.
Suijlekom, The noncommutative geometry of YangMills fields
 J. Geom. Phys
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Feynman motives and deletioncontraction relations
"... We prove a deletioncontraction formula for motivic Feynman rules given by the classes of the affine graph hypersurface complement in the Grothendieck ring of varieties. We derive explicit recursions and generating series for these motivic Feynman rules under the operation of multiplying edges in ..."
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Cited by 13 (8 self)
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We prove a deletioncontraction formula for motivic Feynman rules given by the classes of the affine graph hypersurface complement in the Grothendieck ring of varieties. We derive explicit recursions and generating series for these motivic Feynman rules under the operation of multiplying edges in a graph and we compare it with similar formulae for the Tutte polynomial of graphs, both being specializations of the same universal recursive relation. We obtain similar recursions for graphs that are chains of polygons and for graphs obtained by replacing an edge by a chain of triangles. We show that the deletioncontraction relation can be lifted to the level of the category of mixed motives in the form of a distinguished triangle, similarly to what happens in categorifications of graph invariants.
The Witt construction in characteristic one and quantization
 In Noncommutative geometry and global analysis, volume 546 of Contemp. Math
, 2011
"... Abstract. We develop the analogue of the Witt construction in characteristic one. We construct a functor from pairs (R, ρ) of a perfect semiring R of characteristic one and an element ρ > 1 of R to real Banach algebras. We find that the entropy function occurs uniquely as the analogue of the Te ..."
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Cited by 13 (4 self)
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Abstract. We develop the analogue of the Witt construction in characteristic one. We construct a functor from pairs (R, ρ) of a perfect semiring R of characteristic one and an element ρ > 1 of R to real Banach algebras. We find that the entropy function occurs uniquely as the analogue of the Teichmüller polynomials in characteristic one. We then apply the construction to the semifield R max + which plays a central role in idempotent analysis and tropical geometry. Our construction gives the inverse process of the "dequantization" and provides a first hint towards an extension R un of the field of real numbers relevant both in number theory and quantum physics.
EULER’S CONSTANT: EULER’S WORK AND MODERN DEVELOPMENTS
, 2013
"... This paper has two parts. The first part surveys Euler’s work on the constant γ =0.57721 ·· · bearing his name, together with some of his related work on the gamma function, values of the zeta function, and divergent series. The second part describes various mathematical developments involving Eul ..."
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Cited by 12 (1 self)
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This paper has two parts. The first part surveys Euler’s work on the constant γ =0.57721 ·· · bearing his name, together with some of his related work on the gamma function, values of the zeta function, and divergent series. The second part describes various mathematical developments involving Euler’s constant, as well as another constant, the Euler–Gompertz constant. These developments include connections with arithmetic functions and the Riemann hypothesis, and with sieve methods, random permutations, and random matrix products. It also includes recent results on Diophantine approximation and transcendence related to Euler’s constant.
Quantum statistical mechanics, Lseries and anabelian geometry
, 2010
"... It is known that two number fields with the same Dedekind zeta function are not necessarily isomorphic. The zeta function of a number field can be interpreted as the partition function of an associated quantum statistical mechanical system, which is a C ∗algebra with a one parameter group of auto ..."
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Cited by 11 (7 self)
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It is known that two number fields with the same Dedekind zeta function are not necessarily isomorphic. The zeta function of a number field can be interpreted as the partition function of an associated quantum statistical mechanical system, which is a C ∗algebra with a one parameter group of automorphisms, built from Artin reciprocity. In the first part of this paper, we prove that isomorphism of number fields is the same as isomorphism of these associated systems. Considering the systems as noncommutative analogues of topological spaces, this result can be seen as another version of Grothendieck’s “anabelian ” program, much like the NeukirchUchida theorem characterizes isomorphism of number fields by topological isomorphism of their associated absolute Galois groups. In the second part of the paper, we use these systems to prove the following. If there is a continuous bijection ψ: ˇ G ab K → ˇ G ab L between the character groups (viz., Pontrjagin duals) of the abelianized Galois groups of the two number fields that induces an equality of all corresponding L