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Bejleri, Dori; Marcolli, Matilde

doi:10.1016/j.geomphys.2013.03.002

Cited by 10 publications

(12 citation statements)

References 37 publications

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“…Following the line of thoughts of Manin in [86], there should be a link between F 1 -schemes and motives that explains the similar structure of their zeta functions. There are further connections of motives and F 1 -schemes to non-commutative geometry, see Connes, Consani and Marcolli's papers [25] and [26], Bejleri and Marcolli's paper [7] and Marcolli's appendix of Sujatha and Plazas' lecture notes [107].…”

Section: Results and New Directionsmentioning

confidence: 99%

A blueprinted view on $\mathbb F_1$-geometry

Lorscheid¹

2016

Absolute Arithmetic and $\Mathbb F_1$-Geometry

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This overview paper has two parts. In the first part, we review the development of F1-geometry from the first mentioning by Jacques Tits in 1956 until the present day. We explain the main ideas around F1, embedded into the historical context, and give an impression of the multiple connections of F1-geometry to other areas of mathematics.In the second part, we review (and preview) the geometry of blueprints. Beyond the basic definitions of blueprints, blue schemes and projective geometry, this includes a theory of Chevalley groups over F1 together with their action on buildings over F1; computations of the Euler characteristic in terms of F1-rational points, which involve quiver Grassmannians; K-theory of blue schemes that reproduces the formula Ki(F1) = π st i (S 0 ); models of the compactifications of Spec Z and other arithmetic curves; and explanations about the connections to other approaches towards F1 like monoidal schemes after Deitmar, B1-algebras after Lescot, Λ-schemes after Borger, relative schemes after Toën and Vaquié, log schemes after Kato and congruence schemes after Berkovich and Deitmar.

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Section: Results and New Directionsmentioning

confidence: 99%

A blueprinted view on $\mathbb F_1$-geometry

Lorscheid¹

2016

Absolute Arithmetic and $\Mathbb F_1$-Geometry

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“…r t e η e,r η e, j β r . One obtains in this way a formulation of the Feynman integral where the relevant variety replacing the graph hypersurface complement XG is the variety Λ G defined above, see [45]. In terms of motivic properties, the varieties Λ G were introduced by Esnault and Bloch in relation to Hodge structures and shown to be always mixed-Tate, [46].…”

Section: This Determines a Complete Intersection Varietymentioning

confidence: 99%

Intersection theory, characteristic classes, and algebro-geometric Feynman rules

Aluffi¹,

Marcolli²

2022

Proceedings of MathemAmplitudes 2019: Intersection Theory &Amp; Feynman Integrals — PoS(MA2019)

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We review the basic definitions in Fulton-MacPherson Intersection Theory and discuss a theory of 'characteristic classes' for arbitrary algebraic varieties, based on this intersection theory. We also discuss a class of graph invariants motivated by amplitude computations in quantum field theory. These 'abstract Feynman rules' are obtained by studying suitable invariants of hypersurfaces defined by the Kirchhoff-Tutte-Symanzik polynomials of graphs. We review a 'motivic' version of these abstract Feynman rules, and describe a counterpart obtained by intersection-theoretic techniques.

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“…Thus, if we consider the product torification on (P d ) n , the morphism π : Bl ∆ ((P d ) n ) → (P d ) n is compatible with torifications only in the weak sense: there is a decomposition (the cell decomposition) of the variety such that there are isomorphisms on the pieces of the decomposition that perform the change of torification that makes the morphism torified, but these isomorphisms do not extend globally to the variety. Thus, in a construction of geometric or constructible torifications on the compactifications P d [n] based on the iterated blowups, as in [1] the maps π : P d [n] → (P d ) n will only be weak F 1 -morphisms, that is, morphisms in the category CT w of Proposition 5.4.…”

Section: Moduli Spaces and Wonderful Compactificationsmentioning

confidence: 99%

Absolute Arithmetic and $\mathbb F_1$-Geometry

Lorscheid¹

2016

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In this paper we answer a question raised in [25], Sec. 4, by showing that the genus zero moduli operad {M 0,n+1} can be endowed with natural descent data that allow it to be considered as the lift to Spec Z of an operad over F1. The relevant descent data are based on a notion of constructible sets and constructible functions over F1, which describes suitable differences of torifications with a positivity condition on the class in the Grothendieck ring. More generally, we do the same for the operads {T d,n+1 } (whose components were) introduced in [5]. Finally, we describe a blueprint structure on {M 0,n} and we discuss from this perspective the genus zero boundary modular operad {M 0 g,n+1 }.

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Quantum field theory overF1

Cited by 10 publications

References 37 publications

A blueprinted view on $\mathbb F_1$-geometry

A blueprinted view on $\mathbb F_1$-geometry

Intersection theory, characteristic classes, and algebro-geometric Feynman rules

Absolute Arithmetic and $\mathbb F_1$-Geometry

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