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SUMMARY:Computation of holonomic systems for Feynman amplitudes associated
with some simple diagrams
DTSTART;VALUE=DATE-TIME:20190319T083000Z
DTEND;VALUE=DATE-TIME:20190319T093000Z
DTSTAMP;VALUE=DATE-TIME:20210416T020715Z
UID:indico-contribution-2824@indico.mitp.uni-mainz.de
DESCRIPTION:Speakers: Toshinori Oaku (Tokyo Woman's Christian University)\
nHolonomic systems are a class of systems of linear (partial or ordinary)
differential equations. One of the most fundamental properties of a holono
mic system is that its solution space is finite-dimensional. \n\nA Feynman
amplitude is the integral of a rational function\, or more generally\, th
e product of complex powers of polynomials. \nHence we can\, in principle\
, apply the following two facts in (computational) $D$-module theory: \n\n
1.\nFor (multivariate) polynomials $f_1\,\\dots\,f_d$ and complex numbers
$\\lambda_1\,\\dots\,\\lambda_d$\, the multi-valued analytic function $f_1
^{\\lambda_1}\\cdots f_d^{\\lambda_d}$ satisfies a holonomic system\, whic
h can be computed algorithmically. \n\n2.\nIf a function satisfies a holon
omic system\, its integral with respect to some of its variables also sati
sfies a holonomic system\, which can be computed algorithmically. \n\nIn t
he integration\, it would also be natural to ragard the integrand as a loc
al cohomology class associated with $f_1\,\\dots\,f_d$\, which roughly cor
responds to the 'residue' of \n$f_1^{\\lambda_1}\\cdots f_d^{\\lambda_d}$
at $\\lambda_1 = \\cdots = \\lambda_d = -1$ (at least in positive mass cas
e for external diagrams). \nThere are also algorithms for computing a holo
nomic system for such a cohomology class. \nHowever\, actual computation\,
especially of integration\, is hard in general because of the complexity.
\nI shall present some worked out examples together with an interpretatio
n based on microlocal analysis.\n\nhttps://indico.mitp.uni-mainz.de/event/
179/contributions/2824/
LOCATION:Mainz Institute for Theoretical Physics\, Johannes Gutenberg Univ
ersity 02.430
URL:https://indico.mitp.uni-mainz.de/event/179/contributions/2824/
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