## Noncommutative geometry, quantum fields and motives (2008)

Venue: | Colloquium Publications, Vol.55, American Mathematical Society |

Citations: | 53 - 11 self |

### BibTeX

@INPROCEEDINGS{Connes08noncommutativegeometry,,

author = {Alain Connes and Matilde Marcolli},

title = {Noncommutative geometry, quantum fields and motives},

booktitle = {Colloquium Publications, Vol.55, American Mathematical Society},

year = {2008}

}

### OpenURL

### Abstract

### Citations

879 |
Operator Algebras and Quantum Statistical Mechanics
- Bratteli, Robinson
- 1987
(Show Context)
Citation Context ...β. This parameter is an inverse temperature β = 1/T where one sets the Boltzmann constant equal to one. The equilibrium states are defined by the KMS (Kubo–Martin–Schwinger) condition as follows (cf. =-=[33]-=-, [160], [161]). Definition 3.6. Suppose given a C ∗ -dynamical system (A, σt), that is, a C ∗ -algebra A together with a 1-parameter group of automorphisms σ : R → Aut(A). For a given 0 < β < ∞, a st... |

762 |
A Guide to Quantum Groups
- Chari, Pressley
- 1994
(Show Context)
Citation Context ..., consistently with our notion of odd bimodule in Definition 1.170. The relation to the group ring of SU(2) suggests possible connections with the theory of quantum groups at roots of unity (cf. e.g. =-=[54]-=- §11 and §9.3.C). We shall give in §18.3 a completely different conceptual explanation for taking as an input, instead of ALR, the algebra M2(H) ⊕ M4(C) which leads, by the order one condition of §13.... |

152 |
The spectral action principle
- Chamseddine, Connes
- 1997
(Show Context)
Citation Context ...s again a product geometry of a Riemannian spin four-manifold by a finite geometry F but the latter will be more subtle than the case of (MN (C), MN (C), 0). For the noncommutative geometry F used in =-=[46]-=- to obtain the Standard Model coupled to gravity, all the ingredients are finite-dimensional. The algebra AF = C ⊕ H ⊕ M3(C) (i.e. the direct sum of the algebras C of complex numbers, H of quaternions... |

113 |
Lectures on Non-Perturbative Canonical Gravity", World Scienti
- Ashtekar
- 1991
(Show Context)
Citation Context ...dom. This is similar to what happens in the form of Einstein gravity given by the Palatini action, where the basic dynamical variables are the tetrads (the vierbein) ea µ of (1.581) (see for instance =-=[7]-=- §3.2). It is nevertheless surely important to compute the full spectral action (including the terms independent of Λ) in the presence of the additional degrees of freedom. The above characterization ... |

112 |
A Lefschetz fixed point formula for elliptic complexes
- Atiyah, Bott
- 1967
(Show Context)
Citation Context ...uaternions H as the base, but endowing them with an additional structure, namely their complex structure when viewed as a right vector space over C. We will use the Atiyah-Bott Lefschetz formula (cf. =-=[9]-=-) applied to the ¯∂-complex, which generates the crucial term of theorem 2.60, that is, 1 (2.499) , |1 − u|H with the reduced norm |u|H as above, while a more naive approach without the use of an elli... |

106 |
Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory
- Bost, Connes
- 1995
(Show Context)
Citation Context ...ometric notion: the commensurability relation on Q-lattices. This formulation leads us to consider more general types of noncommutative adelic quotients and their relation to Galois theory. We follow =-=[30]-=-, [86], [88], [90], [91]. All of the cases discussed in this section are quantum statistical mechanical systems, with nontrivial phase transition phenomena, and with thermodynamical equilibrium states... |

99 |
Quantum field theory techniques in graphical enumeration, Hdv
- Bessis, Itzykson, et al.
- 1980
(Show Context)
Citation Context ...nes of graphs (cf. Proposition 1.30 below) as an even grading comes from the fact that it is customary in physics to think of internal lines of graphs as a pair of half-lines, see for instance [170], =-=[19]-=-. In fact we saw in §3.1 that the graphs are obtained from pairings of half-lines. Thus the grading is actually given by the number of half-lines that contribute to the internal lines of6. THE CONNES... |

96 |
Vector bundles and homogeneous spaces
- ATIYAH, HIRZEBRUCH
- 1961
(Show Context)
Citation Context ...pondences. In the topological setting, one considers smooth manifolds X and Y . Recall that the Gysin map f! : K ∗ (X) → K ∗ (Y ) in topological K-theory is defined for K-oriented maps f : X → Y (cf. =-=[10]-=-), where a K-orientation of f : X → Y is specified by an element of the form κ + 2c, with c ∈ H 2 (X, Z) and κ = w2(X) − f ∗ w2(Y ) mod 2, with w2 the Stiefel–Whitney class, satisfying f κ+2c bundle w... |

82 |
Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, Phys
- Broadhurst, Kreimer
- 1997
(Show Context)
Citation Context ...tween Feynman integrals and mixed Tate motives. We summarize a few facts, conjectures, and results that help in clarifying this general picture. • The extensive computations of Broadhurst and Kreimer =-=[35]-=- show that multiple zeta values appear as residues of Feynman graphs. • Multiple zeta values are periods of mixed Tate motives. There are different ways in which one can concretely realize multiple ze... |

70 |
The analytic continuation of generalized functions with respect to a parameter
- BERNSTEIN
- 1972
(Show Context)
Citation Context ...t of z is sufficiently large. We now discuss the existence of a meromorphic continuation to the complex plane. The mathematical treatment, cf. e.g. [128] is based on the following result of Bernstein =-=[17]-=-, applied to the polynomial det(A(t)). Lemma 1.6. Let Q(t) be a polynomial in n variables. There exists a polynomial q(D) and a polynomial differential operator L(D) in n variables, whose coefficients... |

69 |
Axial vector vertex in spinor electrodynamics
- Adler
- 1969
(Show Context)
Citation Context ..., . . . , N} ⊂ Γ(1) The3. FEYNMAN DIAGRAMS 44 and its complement Γ (1) int ⊂ Γ(1) is the set of internal lines. The geometric realization of a graph is the one-dimensional space (1.71) |Γ| = Γ (1) × =-=[0, 1]-=- ∪∂ (Γ (0) ∪ {1, 2, . . . , N}) obtained by gluing the endpoints of the lines using the maps ∂j. The graph is planar when it can be represented by a planar picture. It can always be represented by a s... |

62 |
Théorie bivariante de Kasparov et opérateurs non bornes dans les C∗-modules hilbertiens
- Baaj, Julg
(Show Context)
Citation Context ...⊗D : KK(A, D) ⊗ KK(D, B) → KK(A, B) . One also has external tensor products (4.5) KK(A, B) ⊗ KK(C, D) → KK(A ⊗ C, B ⊗ D). These admit a simpler description in terms of unbounded Kasparov modules (cf. =-=[12]-=-). One modifies Definition 4.1 in the following way. Definition 4.4. Unbounded Kasparov modules are triples (E, φ, D), with (E, φ) as in Definition 4.1, where D is an unbounded regular self-adjoint op... |

58 |
K-theory for Operator Algebras, second edition
- Blackadar
- 1998
(Show Context)
Citation Context ...ble C ∗ -algebras and with the morphisms given as follows. For A and B in Obj(KK), one has (4.1) Hom(A, B) = KK(A, B), where KK(A, B) is Kasparov’s bivariant K-theory ([178], cf. also §8 and §9.22 of =-=[20]-=-). We describe in more detail the way the morphisms (4.1) are obtained. First recall that a Hilbert module E over a C ∗ -algebra B is a right B-module endowed with a positive B-valued inner product wi... |

56 |
Algebraic cycles and higher K-theory, Adv
- Bloch
- 1986
(Show Context)
Citation Context ...1(A) S ch rel n Dn/n! � HCn−1(A) B/n �� HHn(A) I chn � HCn(A) S � ch rel n−1 � HCn−2(A). In a related perspective, motivic cohomology can be described in terms of the higher Chow groups of Bloch (cf. =-=[22]-=-, [286]). It was shown in [238] that one can formulate a version of the higher Chow group complex that is modelled on the (b, B)-bicomplex of cyclic cohomology and obtain an analog of the exact sequen... |

51 | D.: Motives associated to Graph Polynomials
- Bloch, Esnault, et al.
(Show Context)
Citation Context ...o interpret the residues of Feynman integrals as periods, are very general in the sense that they generate the Grothendieck ring of motives. However, more recent results of Bloch, Esnault and Kreimer =-=[25]-=- show, for certain classes of graphs, that the Feynman integral, expressed as a period through the Schwinger parameterization, comes from a piece of the cohomology of the graph variety that is actuall... |

48 |
Connes, Universal formulas for noncommutative geometry actions, Phys
- Chamseddine, A
- 1996
(Show Context)
Citation Context ...pter 1 follows closely our joint work with Chamseddine [52], which is based on the model introduced by Connes in [73], as well as on the previous work of Chamseddine and Connes on the spectral action =-=[45]-=-, [46], [47]. Since we do not assume that the reader has much familiarity with particle physics, we begin §9 by giving a brief overview of the Standard Model. We introduce the various parameters, the ... |

47 | Gravity and the standard model with neutrino mixing
- Chamseddine, Connes, et al.
(Show Context)
Citation Context ...tary particle physics and an approach to a simple mathematical understanding of its structure via noncommutative geometry. The second part of Chapter 1 follows closely our joint work with Chamseddine =-=[52]-=-, which is based on the model introduced by Connes in [73], as well as on the previous work of Chamseddine and Connes on the spectral action [45], [46], [47]. Since we do not assume that the reader ha... |

42 |
Vortices in holomorphic line bundles over closed Kähler manifolds
- Bradlow
- 1990
(Show Context)
Citation Context ...dity to arbitrary deg(L) one can introduce an additional7. A NONCOMMUTATIVE GEOMETRY PERSPECTIVE 647 sufficiently large real parameter in the equation, which corresponds to scaling the volume, as in =-=[32]-=-. We discuss this point later. The configuration space BL(C) is set of all pairs (A, u) of connections and smooth sections of L modulo gauge, completed in suitable Sobolev norms so that the linearizat... |

32 |
Distributions sur un groupe localement compact et applications à l’étude des représentations des groupes ℘-adiques
- Bruhat
- 1961
(Show Context)
Citation Context ... the additive group K. Given such an α the associated Fourier transform is ∫ (2.95) (Fαρ)(y) = ˆϱ(y) = ϱ(x)α(xy) dx . which makes sense for any element f ∈ S(K) of the Bruhat-Schwartz space S(K) (cf. =-=[36]-=- for the definition of S(K)). By the Plancherel Theorem Fα extends to a unitary operator (2.96) 〈Fα(ξ), Fα(η)〉 = 〈ξ, η〉 , ∀ξ, η ∈ L 2 (K) , where one has normalized the additive Haar measure da on K s... |

29 |
Riemann’s zeta function: a model for quantum chaos
- Berry
- 1986
(Show Context)
Citation Context ... MECHANICS OF ZETA 325 50 40 30 20 10 20 40 60 80 100 120 140 Figure 4. The number of zeros N(E) and its approximation by 〈N(E)〉. 3.1. Spectral lines and the Riemann flow. It was observed by Berry in =-=[18]-=- that there is an interesting similarity between the expression in (2.26) and the semi-classical formula for the number of eigenvalues (2.32) N(E) = # Eigenvalues of H in [0, E] of a Hamiltonian opera... |

26 |
Introduction to the Theory of Quantized
- Bogoliubov, Shirkov
- 1973
(Show Context)
Citation Context ...latter is properly handled, one obtains only local terms as coefficients of the divergences. A renormalization procedure that takes care of both problems was developed by Bogoliubov and Parasiuk [26] =-=[27]-=- and later refined by Hepp and Zimmermann [166], [305]. It is generally referred to as the BPHZ procedure. We fix a renormalizable theory T , and let J be the set of all monomials of the Lagrangian. I... |

25 |
introduction aux motifs, Panoramas et Syntheses 17, Societe Mathematique de
- Andre
- 2004
(Show Context)
Citation Context ...lt recalled in Remark 1.82 below. For the relation in the motivic context of this notion to the usual Euler characteristic and its effect on the Tannakian property of the category of pure motives see =-=[2]-=-. Recall also the following properties of functors. Definition 1.80. A functor ω : C → C ′ is faithful if, for all X, Y ∈ Obj(C), the mapping (1.341) ω : HomC(X, Y ) → HomC ′(ω(X), ω(Y )) is injective... |

24 |
Lorentzian version of the noncommutative geometry of the standard model of particle physics
- Barrett
(Show Context)
Citation Context ... fibered structure. We work in Euclidean rather than Lorentz signature, leaving as an important problem the Wick rotation back to the Minkowski signature. For a formulation in Minkowski signature see =-=[13]-=-. The idea of interpreting the Lagrangian of the Standard Model in terms of noncommutative geometry goes back a good twenty years (Connes [68], [70], see also [85], [181], [180]). The origin of this i... |

23 | Sukochev “The local index formula in semifinite von Neumann algebras II: the even case
- Carey, Phillips, et al.
(Show Context)
Citation Context ...ing, using (1.124), to (1.904) Tr(e −λD2 ) = ∫ −λ k2 e d z k ∀λ ∈ R ∗ + . Let us first explain how to perform the construction in the slightly more general framework of type II spectral triples [16], =-=[39]-=-, [40]. The only condition which is altered (with respect to Definition 1.120) in the definition of such a triple is the compactness of the resolvent of D which only holds relative to a type II subfac... |

20 |
Dimensional renormalization and the action principle
- Breitenlohner, Maison
- 1977
(Show Context)
Citation Context ...γ µ while keeping the cyclicity of the trace is not consistent and produces contradictions ([62], §13.2). There is a more sophisticated prescription of ’t Hooft-Veltman [169] and Breitenlohner-Maison =-=[34]-=- on how to make sense of the γ5 in DimReg. In general the Ward identities determine relations between the Green’s functions arising from symmetries of the Lagrangian. These influence the renormalizabi... |

19 |
Noncommutative Yang-Mills and noncommutative relativity: A bridge over trouble water
- Carminati, Iochum, et al.
- 1999
(Show Context)
Citation Context ...V. Thus, we see that this model predicts a heavy Higgs at around 168 GeV. For a similar analysis of the Higgs mass in other variants of the noncommutative geometry approach to the Standard Model, see =-=[41]-=- and [187]. 17.11. The gravitational terms. We have seen in Theorem 1.217 above how to recover the Standard Model Lagrangian LSM from the spectral action. We have, for simplicity, considered the case ... |

17 |
Type II Noncommutative Geometry. I. Dixmier Trace in von Neumann Algebras
- Benameur, Fack
- 2006
(Show Context)
Citation Context ...esponding, using (1.124), to (1.904) Tr(e −λD2 ) = ∫ −λ k2 e d z k ∀λ ∈ R ∗ + . Let us first explain how to perform the construction in the slightly more general framework of type II spectral triples =-=[16]-=-, [39], [40]. The only condition which is altered (with respect to Definition 1.120) in the definition of such a triple is the compactness of the resolvent of D which only holds relative to a type II ... |

15 | Expansional in Banach algebra - Araki |

15 | General RG equations for physical neutrino parameters and their phenomenological implications
- Casas, Espinosa, et al.
(Show Context)
Citation Context ...roximate form of (1.825) given by (1.841) y σ u = 2 √ 3 g, for σ = 3. We consider the Yukawa couplings (y σ · ) as depending on the energy scale through their renormalization group equation (cf. [4], =-=[43]-=-, [243]). We set (1.842) t = log( Λ MZ ) and µ = MZe t . We consider in particular the top quark case yσ u (t) for σ = 3. The running of the top quark Yukawa coupling yt = yσ u (t), with σ = 3, is gov... |

14 | Confocal multimode resonator for millimeter through optical wavelength masers - Boyd, Gordon - 1961 |

13 |
Iterated integrals and exponential homomorphisms
- Chen
- 1954
(Show Context)
Citation Context ...f the differential equation (1.261) dh(u) = h(u) α(u)du , h(a) = 1 . A mathematical definition can be given as an iterated integral. This type of formalism was developed in the topological context in =-=[55]-=-, [56] (Chen’s iterated integral) and in the operator algebra context in [3] (Araki’s expansional). We give here a formulation adapted to the context of affine group schemes. Definition 1.50. Given a ... |

12 |
Lectures on mixed motives. Algebraic geometry—Santa Cruz
- Bloch
- 1995
(Show Context)
Citation Context ...task. Recent results such as [287], [208], [162] produced good categories of mixed motives, but these are still in general very difficult objects to deal with. We refer the reader to Bloch’s lectures =-=[21]-=- for an overview of mixed motives. 8.3.1. The Grothendieck group of varieties. Given an algebraic variety X over a field of characteristic zero (where Hironaka’s resolution of singularities holds), on... |

10 |
Inner fluctuations of the spectral action
- Chamseddine, Connes
(Show Context)
Citation Context ... the notation (1.607) α(a) = D a D −1 ∀a ∈ A.11. THE SPECTRAL ACTION PRINCIPLE 213 A A A Figure 32. The triangle graph. Notice that in general α(a) /∈ A. One has We then have the following result of =-=[48]-=-. α(ab) = α(a) α(b) a, b ∈ A. Theorem 1.161. Let (A, H, D) be a spectral triple as in Theorem 1.159. Under the tadpole vanishing hypothesis (1.604) the following holds. (1) ψ = ϕ + 1 (1.608) ψ(a0, a1,... |

9 | Brosnan Matroids, motives and a conjecture of Kontsevich Duke
- Belkale, Patrick
(Show Context)
Citation Context ... over Z are exactly the elements of MZV [1/2πi], where MZV denotes the Q-vector space MZV ⊂ R generated by the multiple zeta values (which is in fact a Q-algebra). • The result of Belkale and Brosnan =-=[14]-=- seems to point to a problem in the expectation that residues of Feynman graphs should be periods of mixed Tate motives. In fact, they show that certain varieties constructed from graphs, on which it ... |

9 |
Produits Euleriens et facteurs de type
- Bost, Connes
- 1992
(Show Context)
Citation Context ... one has γ(r, ρ) = ((r −1 Z, ρ), (Z, ρ)), while |(r −1 Z)/Z| = Covol(Z) Covol((r−1 = r Z)) We now show that the quantum statistical mechanical system (A1, σt) is the same as the Bost–Connes system of =-=[29]-=-, [30]. □ 4.2. Hecke algebras. The original construction of the Bost–Connes (BC) quantum statistical mechanical system is based on Hecke algebras of quasi-normal pairs. The main conceptual steps invol... |

8 |
Über die Multiplikation der Kausalfunktionen in der Quantentheorie der Felder
- Bogoliubow, Parasiuk
- 1957
(Show Context)
Citation Context ... the latter is properly handled, one obtains only local terms as coefficients of the divergences. A renormalization procedure that takes care of both problems was developed by Bogoliubov and Parasiuk =-=[26]-=- [27] and later refined by Hepp and Zimmermann [166], [305]. It is generally referred to as the BPHZ procedure. We fix a renormalizable theory T , and let J be the set of all monomials of the Lagrangi... |

7 |
units of the SI, fundamental constants and modern quantum physics
- Bordé, Base
- 2005
(Show Context)
Citation Context ...f relativity, c does not depend upon the frequency of the light, a property that is the object of crucial experimental probing. The choice of Cesium is of course rather arbitrary and might eventually =-=[28]-=- be replaced by Hydrogen, which is more canonical and more abundant in the universe. It is natural to adapt the basic paradigm of geometry to the new standard of length. We explain briefly below that ... |

6 |
Function theory of polylogarithms, Structural properties of polylogarithms
- Bloch
- 1991
(Show Context)
Citation Context .... An example is given by the families Mk(z) (variations of mixed Hodge structures) associated to the polylogarithms Lik(z) = ∞∑ n=1 z n n k8. MOTIVES IN A NUTSHELL 154 with matrices of the form (cf. =-=[23]-=-) ⎛ 1 0 0 0 0 · · · −Li1(z) 2πi 0 0 0 · · · (1.405) −Li2(z) 2πi log z (2πi) ⎜ ⎝ 2 0 0 · · · 2πi (log z) −Li3(z) 2 2! (2πi) 2 log z (2πi) 3 ⎞ 0 · · · ⎟ ⎠ . . . . . · · · One can similarly define real m... |

6 | Scale invariance in the spectral action
- Chamseddine, Connes
(Show Context)
Citation Context ...ws closely our joint work with Chamseddine [52], which is based on the model introduced by Connes in [73], as well as on the previous work of Chamseddine and Connes on the spectral action [45], [46], =-=[47]-=-. Since we do not assume that the reader has much familiarity with particle physics, we begin §9 by giving a brief overview of the Standard Model. We introduce the various parameters, the particles an... |

6 |
Quantum gravity boundary terms from the spectral action on noncommutative space
- Chamseddine, Connes
(Show Context)
Citation Context ...orrespondence described above in §8.4, there is an alternative which is to relate two 3-geometries by cobordisms using 4-geometries with boundary. The case of geometries with boundary is developed in =-=[51]-=- where it is shown that manifolds with boundary naturally give rise, in the even dimensional case, to a spectral triple. Moreover the spectral action delivers the York–Gibbons– Hawking boundary term w... |

4 | The explicit formula and the conductor operator
- Burnol
(Show Context)
Citation Context ...analyze here the relative position of the three projections PΛ, ̂ PΛ, and NE, using identities involving the quantized calculus, as proved in [72]. The method used here is based on the idea of Burnol =-=[37]-=- which simplifies the original argument of [71]. First recall from §IV of [68] that the main idea of quantized calculus is to give an operator-theoretic version of the calculus rules, based on the ope... |

3 |
Galois theory. Edited and with a supplemental chapter by Arthur N. Milgram. Reprint of the 1944 second edition
- Artin
- 1998
(Show Context)
Citation Context ...n a group H of automorphisms of a field E one lets E H be the subfield E H = {x ∈ E ; θ(x) = x ∀θ ∈ H} . The remaining step is immediate using the following well known general result due to E. Artin (=-=[5]-=-) Proposition 4.212. Let E be a field and H a finite group of automorphisms of E. (1) One has [E : E H ] = #H. (2) Aut E H E = H. Proof. As before, we assume that the fields have characteristic zero. ... |

3 |
A PCAC puzzle, Nuovo Cimento, Vol.60A
- Bell, Jackiw
- 1969
(Show Context)
Citation Context ...singles out a large class of noncommutative manifolds. 19. Dimensional regularization and noncommutative geometry The aim of this section is to construct tools for the computation of chiral anomalies =-=[15]-=- in the general framework of noncommutative geometry. We show that a careful investigation of the meaning of the Breitenlohner-Maison prescription for the compatibility of the chiral symmetry with the... |

2 |
E.J.Piard, P.Ramond, B.D.Wright, Renormalization-group study of the standard model and its extensions: the standard model, Phys
- Arason, Kesthlyi
- 1992
(Show Context)
Citation Context ...tion group equation that we discussed in the first part of this chapter (see §6.6). Considering only 1-loop corrections, the beta function (cf. Definition 1.46 and Corollary 1.49) is given by ([187], =-=[4]-=-) (1.789) βgi = (4π)−2 bi g 3 i , with b = ( 41 , −19, −7), 6 6 so that ([255]) the renormalization group equation is then of the form (1.790) α −1 1 (Λ) = α−1 1 (MZ) − 41 12π log α −1 2 (Λ) = α−1 2 (... |

2 |
Avramidi Covariant methods for the calculation of the effective action in quantum field theory and investigation of higher-derivative quantum gravity
- G
- 1986
(Show Context)
Citation Context ...tion is analyzed. The running of the coefficients of the Euclidean higher derivative terms in (1.463), determined by the renormalization group equation, is gauge independent and is given by (see e.g. =-=[11]-=-, equations (4.49) and (4.71). See also [122], [58]) βη = − 1 (4π) 2 βω = − 1 (4π) 2 βθ = 1 (4π) 2 133 10 η2 25 + 1098 ω + 200 ω2 η 60 7(56 − 171 θ) η. 9017. THE STANDARD MODEL LAGRANGIAN FROM THE SP... |

2 | Connes, A dress for SM the beggar, hep-th 0706.3690 - Chamseddine, A |

2 |
Experimental tests of new SO(10) grand unification, Phys
- Chang, Mohapatra, et al.
- 1985
(Show Context)
Citation Context ... Λ, i.e. where the merging g2 3 = g2 2 = 5g2 1 /3 of the coupling constants supposedly takes place. The relation g2 3 = g2 2 = 5g2 1 /3 coincides with that obtained in grand unification theories (cf. =-=[53]-=- and [233] §9). We recall briefly in §17.2 below how the unification scale is computed. Corollary 1.219. The change of notations (1.781) for the Higgs fields can be reformulated in the form (1.786) H ... |

1 |
Motives associated to graphs, Japanese
- Bloch
(Show Context)
Citation Context ... suggest that, although these varieties can give rise to motives that are not mixed Tate, the motivic piece that corresponds to the period relevant for Feynman integrals is still mixed Tate. Recently =-=[24]-=-, Bloch proposed a lifting at the motivic level of the ConnesKreimer Hopf algebra. • Kreimer showed in [190] that the period matrix (1.405) of the mixed Hodge–Tate structures associated to the polylog... |

1 |
The strong CP problem in a compact Robertson-Walker universe
- Cahill
- 1985
(Show Context)
Citation Context ...scribed in §17.10. The remaining parameter of the Standard Model described in §9 above, namely the QCD vacuum angle, does not matter in our model, since we are considering a compact 4-manifold M, see =-=[38]-=-. 18. Functional integral While renormalization works remarkably well for the quantization of the classical fields involved in the Standard Model, this latter perturbative technique fails dramatically... |

1 |
Grothendieck et les motifs, preprint IHES/M/00/75
- Cartier
(Show Context)
Citation Context ...orie Quantique des Champs n’est sans doute que la première manifestation d’un groupe de symétrie des constantes fondamentales de la physique, une espèce de groupe de Galois cosmique!” Pierre Cartier, =-=[42]-=- In this passage, written in 2000, Cartier conjectured the existence of a hidden group of symmetries, closely related in structure to certain arithmetic Galois groups, and acting on the constants of p... |