## On motives associated to graph polynomials

Venue: | Commun. Math. Phys |

Citations: | 51 - 12 self |

### BibTeX

@ARTICLE{Bloch_onmotives,

author = {Spencer Bloch and Hélène Esnault and Dirk Kreimer},

title = {On motives associated to graph polynomials},

journal = {Commun. Math. Phys},

year = {}

}

### OpenURL

### Abstract

Abstract. The appearance of multiple zeta values in anomalous dimensions and β-functions of renormalizable quantum field theories has given evidence towards a motivic interpretation of these renormalization group functions. In this paper we start to hunt the motive, restricting our attention to a subclass of graphs in four dimensional scalar field theory which give scheme independent contributions to the above functions. Calculations of Feynman integrals arising in perturbative quantum field theory [4, 5] reveal interesting patterns of zeta and multiple zeta values. Clearly, these are motivic in origin, arising from the existence of Tate mixed Hodge structures with periods given by Feynman integrals.

### Citations

1007 |
Quantum Field Theory
- Itzykson, Zuber
- 1980
(Show Context)
Citation Context ...e mixed Hodge structures with periods given by Feynman integrals. We are far from a detailed understanding of this phenomenon. An analysis of the problem leads via the technique of Feynman parameters =-=[12]-=- to the study of motives associated to graph polynomials. By the seminal work of Belkale and Brosnan [3], these motives are known to be quite general, so the question becomes under what conditions on ... |

82 |
Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, Phys
- Broadhurst, Kreimer
- 1997
(Show Context)
Citation Context ...four dimensional scalar field theory which give scheme independent contributions to the above functions. 0. Introduction Calculations of Feynman integrals arising in perturbative quantum field theory =-=[4, 5]-=- reveal interesting patterns of zeta and multiple zeta values. Clearly, these are motivic in origin, arising from the existence of Tate mixed Hodge structures with periods given by Feynman integrals. ... |

64 |
Cohomology of local systems on the complement of hyperplanes
- Esnault, Schechtman, et al.
(Show Context)
Citation Context ... the natural local coordinates and to be cones over the strict transforms of the Λi. One knows that after a finite number of such blowups, the strict transforms of the Λi will meet transversally (see =-=[10]-=- for a minimal way to do it). All blowups and coordinates will be defined over K ⊂ R, and one is reduced to checking convergence for an integral of the form ∫ (5.5) U d 4n−1 x (x 2 1 + . . . + x 2 4) ... |

59 |
Groupes fondamentaux motiviques de Tate mixtes, preprint
- Deligne
- 2003
(Show Context)
Citation Context ...hen the general guideline for what we wish to understand is the following. One has now a good candidate for a triangulated category of mixed motives over Q, defined by Voevodsky, Levine and Hanamura (=-=[6]-=-, section 1 and references there for the discussion here). One further considers the triangulated subcategory spanned by Q(n), n ∈ Z. In this category, one has Hom j ⎧ ⎪⎨ Q p = j = 0 (9.4) (Q(0), Q(p)... |

44 |
Cohomologie de SLn et valeurs de fonctions zêta aux points entiers
- Borel
- 1977
(Show Context)
Citation Context ...(9.4) (Q(0), Q(p)) = K2p−1(Q) ⊗ Q p ≥ 1, j = 1 ⎪⎩ 0 else The iterated extensions of Q(n) form an abelian subcategory which is the heart of a t-structure. Borel’s work on the K-theory of number fields =-=[2]-=-, [14] tells us that K2p−1(Q) ⊗ Q ∼ = Q for p = 2n − 3, n ≥ 2, so there is a one dimensional space of motivic extensions of Q(0) by Q(p). We want to understand their periods. Let E be a nontrivial suc... |

35 |
Cohomologie étale
- Deligne
- 1977
(Show Context)
Citation Context ... by j = j0 : P2n−1 \X → P2n−1 , ji : Xi−1 \Xi → Xi−1. Over Xi, the quadric ∑ e Aeqj e is a cone over a smooth quadric ∑ e Aeq j e ⊂ Pj(n−i)−1 , thus by homotopy invariance and base change for R(πj)! (=-=[7]-=-), one obtains Proposition 10.1. R i ⎧ j!Q(−2n + 1) i = 4n − 1 ⎪⎨ (j1)!Q(−2n − 1) i = 4n + 3 (10.4) (π4)!Q = . . . . . . ⎪⎩ (ja)!Q(−2n + 1 − 2a) i = 4n + 4a e (10.5) R i ⎧ j!Q(−n + 1) i = 2n − 1 ⎪⎨ (j... |

27 |
motives, and a conjecture of
- Belkale, Brosnan, et al.
(Show Context)
Citation Context ...g of this phenomenon. An analysis of the problem leads via the technique of Feynman parameters [12] to the study of motives associated to graph polynomials. By the seminal work of Belkale and Brosnan =-=[3]-=-, these motives are known to be quite general, so the question becomes under what conditions on the graph does one find mixed Tate Hodge structures and multiple zeta values. The purpose of this paper ... |

24 | Counting Points on Varieties over Finite Fields Related to a Conjecture of Kontsevich, Ann
- Stembridge
- 1998
(Show Context)
Citation Context ...consider turns out to be mixed Tate in “most” cases, but with a computer it is not difficult to generate cases where it may not be. We give such an example with 12 edges. Note however that Stembridge =-=[13]-=- has shown that all graphs with ≤ 12 edges are mixed Tate, so the particular example we give must in fact be mixed Tate. Techniques and results in this section should be compared with [13], which pred... |

20 |
Kreimer,D.: Knots and numbers in Φ4 theory to 7 loops and beyond
- Broadhurst
- 1995
(Show Context)
Citation Context ...four dimensional scalar field theory which give scheme independent contributions to the above functions. 0. Introduction Calculations of Feynman integrals arising in perturbative quantum field theory =-=[4, 5]-=- reveal interesting patterns of zeta and multiple zeta values. Clearly, these are motivic in origin, arising from the existence of Tate mixed Hodge structures with periods given by Feynman integrals. ... |

8 |
Condensation of determinants, Proc
- Dodgson
(Show Context)
Citation Context ...position (i, i) and zeroes elsewhere, for 1 ≤ i ≤ n. Define (mij) to be the symmetric matrix associated to the quadratic form ∑2n n+1 Ai(e∨ i ) 2 . The assertion of the lemma is now clear. Lemma 8.2 (=-=[9]-=-). Let ψ = det(mij + δijAi)1≤i,j≤n, where the mij are independent of A1, . . . , An. For 1 ≤ k ≤ n write ψk := ∂ ψ and ∂Ak ψk := ψ|Ak=0. For I, J ⊂ {1, . . . , n} with #I = #J, define ψ(I, J) to □GRA... |

4 |
Théorème de finitude pour un morphisme propre; dimension cohomologique des schémas algébriques affines
- unknown authors
- 1973
(Show Context)
Citation Context ... , Bn−1, A2, . . . , An−2)− A 2 n−1Qn−2(B1, . . . , Bn−2, A1, . . . , An−3) + 2(−1) n−1 A0 · · · An−1. Proof. Straightforward. The following lemma is a direct application of Artin’s vanishing theorem =-=[1]-=-, Théorème 3.1, and homotopy invariance, and will be the key ingredient to the computation. Lemma 11.4. Let V ⊂ P N be a hypersurface which is a cone over the hypersurface W ⊂ P a . Then one has or eq... |

4 |
Multiple zeta motives and moduli spaces
- Manin
- 2004
(Show Context)
Citation Context ...h hypersurface along these linear spaces, so the combinatorics of their blowups is important. (It is curious that arithmetically interesting periods seem to arise frequently (cf. multiple zeta values =-=[11]-=- or the study of periods associated to Mahler measure in the non-expansive case [8]) in situations where the polar locus of the integrand meets the chain of integration in combinatorially interesting ... |

1 |
Deligne periods of mixed motives,K-theory, and the entropy of certain Zn-actions
- Deninger
- 1997
(Show Context)
Citation Context ...mportant. (It is curious that arithmetically interesting periods seem to arise frequently (cf. multiple zeta values [11] or the study of periods associated to Mahler measure in the non-expansive case =-=[8]-=-) in situations where the polar locus of the integrand meets the chain of integration in combinatorially interesting ways.) Section 4 is not used in the sequel. It exhibits a natural resolution of sin... |

1 |
E-mail address: esnault@uni-essen.de IHES, 91440 Bures sur Yvette, France and Boston U., Boston MA 02215. E-mail address: kreimer@ihes.fr
- No
- 1986
(Show Context)
Citation Context ... (Q(0), Q(p)) = K2p−1(Q) ⊗ Q p ≥ 1, j = 1 ⎪⎩ 0 else The iterated extensions of Q(n) form an abelian subcategory which is the heart of a t-structure. Borel’s work on the K-theory of number fields [2], =-=[14]-=- tells us that K2p−1(Q) ⊗ Q ∼ = Q for p = 2n − 3, n ≥ 2, so there is a one dimensional space of motivic extensions of Q(0) by Q(p). We want to understand their periods. Let E be a nontrivial such exte... |

1 |
Multiple zeta motives and moduli spaces M0,n
- Goncharov, Manin
(Show Context)
Citation Context ...h hypersurface along these linear spaces, so the combinatorics of their blowups is important. (It is curious that arithmetically interesting periods seem to arise frequently (cf. multiple zeta values =-=[11]-=- or the study of periods associated to Mahler measure in the non-expansive case [8]) in situations where the polar locus of the integrand meets the chain of integration in combinatorially interesting ... |