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Izv. RAN. Ser. Mat., 2016, Volume 80, Issue 5, Pages 5–40 (Mi izv8492)  

This article is cited in 5 scientific papers (total in 5 papers)

Feynman amplitudes and limits of heights

O. Aminia, S. J. Bloch, J. I. Burgos Gilb, J. Fresánc

a CNRS--DMA, École Normale Supérieure, Paris, France
b Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, Madrid, Spain
c Eidgenössische Technische Hochschule Zürich, Zürich, Switzerland

Abstract: We investigate from a mathematical perspective how Feynman amplitudes appear in the low-energy limit of string amplitudes. In this paper, we prove the convergence of the integrands. We derive this from results describing the asymptotic behaviour of the height pairing between degree-zero divisors, as a family of curves degenerates. These are obtained by means of the nilpotent orbit theorem in Hodge theory.

Keywords: Feynman amplitudes, low-energy limit, asymptotics of the archimedean height pairing, Symanzik polynomials, nilpotent orbit theorem, biextension mixed Hodge structures, regularized Green functions.

Funding Agency Grant Number
Ministerio de Economía y Competitividad de España MTM2013--42135--P
SEV--2015--0554
Deutsche Forschungsgemeinschaft SFB 1085
Swiss National Science Foundation 200021--150099
200020--162928
J. I. Burgos Gil was partially supported by the MINECO research projects MTM2013–42135–P and ICMAT Severo Ochoa project SEV–2015–0554, and the DFG project SFB 1085 “Higher Invariants”. J. Fresán acknowledges support from the SNFS grants 200021–150099 and 200020–162928.


DOI: https://doi.org/10.4213/im8492

Full text: PDF file (757 kB)
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English version:
Izvestiya: Mathematics, 2016, 80:5, 813–848

Bibliographic databases:

UDC: 512.73, 512.72, 512.721, 512.75
MSC: 32G20, 14D05, 37P30, 81Q30, 81T30
Received: 15.12.2015
Revised: 18.04.2016

Citation: O. Amini, S. J. Bloch, J. I. Burgos Gil, J. Fresán, “Feynman amplitudes and limits of heights”, Izv. RAN. Ser. Mat., 80:5 (2016), 5–40; Izv. Math., 80:5 (2016), 813–848

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. R. de Jong, “Point-like limit of the hyperelliptic Zhang-Kawazumi invariant”, Pure Appl. Math. Q., 11:4 (2015), 633–653  crossref  mathscinet  zmath  isi  scopus
    2. S. Bloch, M. Kerr, P. Vanhove, “Local mirror symmetry and the sunset Feynman integral”, Adv. Theor. Math. Phys., 21:6 (2017), 1373–1453  crossref  mathscinet  isi  scopus
    3. J. I. Burgos Gil, D. Holmes, R. De Jong, “Singularities of the biextension metric for families of abelian varieties”, Forum Math. Sigma, 6 (2018), e12, 56 pp.  crossref  mathscinet  isi
    4. de Jong R., “Faltings Delta-Invariant and Semistable Degeneration”, J. Differ. Geom., 111:2 (2019), 241–301  crossref  mathscinet  zmath  isi
    5. Amini O., “The Exchange Graph and Variations of the Ratio of the Two Symanzik Polynomials”, Ann. Inst. Henri Poincare D, 6:2 (2019), 155–197  crossref  mathscinet  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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