Geometric approach to the Feynman integrals
Geometric approach to the Feynman integrals
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DissertationDate of Exam
23.12.2021Date of Publication
07.10.2022Advisor
Klemm, AlbrechtCo-Referee
Foerste, StefanMetadata
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Abstract
In this PhD thesis, the Feynman integrals are evaluated using an algebraic geometry approach known in string theory. The GKZ description of periods and certain classes of relative periods on Calabi-Yau (l - 1)-folds has been used in order to solve the l-loop banana amplitudes with their general mass dependence.
There are different ways of computing Feynman integrals. These integrals satisfy a set of differential equations whose solutions give the answer of the integral. One of the approaches to derive a system of differential equations for Feynman integrals is the integration by part (IBP) identity. There is also an alternative approach to obtain these differential equations for a Feynman integral using the geometric interpretation of the Feynman integrals.
From the algebro-geometric point of view, Feynman integrals are the periods of the mixed Hodge structure on the relative cohomology groups. Varying physical parameters leads to a variation of the Hodge structure. The geometric interpretation comes from the polar locus of the integrand. The poles of the integrand define a Calabi-Yau manifold, which is often in a toric variety. These period integrals satisfy a system of linear homogeneous differential equations, so-called Picard-Fuchs differential ideal (PFDI). With toric geometry we can derive a finite set of differential operators, so-called GKZ hypergeometric system and extract PFDI. We obtain GKZ system from the variation of the Hodge structure and we benefit from the symmetries of the graphs more efficiently. GKZ systems are generalization of hypergeometric system and use the symmetries of the integrand, i.e. symmetries in its parameter space. As examples we compute the mass dependencies of the banana amplitudes up to the four-loop case.
For the two-loop banana Feynman integral, the so-called sunset diagram, the polar locus of the integrand is a special family of elliptic curves E, i.e. a Calabi-Yau one-fold. The integral is related to the period integral of the local mirror M of the non-compact Calabi-Yau three-fold W, defined as the total space of the anti-canonical line bundle over the degree three del Pezzo surface S, which is P2 blown up in three generic points. For the three-loop case, the vanishing locus of denominator of the GKZ integral defines a K3 surface and obviously, it defines a Calabi-Yau (l - 1)-fold for the l-loop case.
Later, we show that the coefficients of the linear combination of the solutions, which leads to the Banana Feynman integral, have geometrical interpretation and can be obtained by the so-called Gb-class evaluation in the ambient space of the mirror. We explain that in the equal mass case the relevant physical subslices in series of the Calabi-Yau manifolds are complete intersections of two constrains in (P)l+1. We calculate the Gb-class for this case which happens to match perfectly with the coefficients that we obtain numerically.
There are different ways of computing Feynman integrals. These integrals satisfy a set of differential equations whose solutions give the answer of the integral. One of the approaches to derive a system of differential equations for Feynman integrals is the integration by part (IBP) identity. There is also an alternative approach to obtain these differential equations for a Feynman integral using the geometric interpretation of the Feynman integrals.
From the algebro-geometric point of view, Feynman integrals are the periods of the mixed Hodge structure on the relative cohomology groups. Varying physical parameters leads to a variation of the Hodge structure. The geometric interpretation comes from the polar locus of the integrand. The poles of the integrand define a Calabi-Yau manifold, which is often in a toric variety. These period integrals satisfy a system of linear homogeneous differential equations, so-called Picard-Fuchs differential ideal (PFDI). With toric geometry we can derive a finite set of differential operators, so-called GKZ hypergeometric system and extract PFDI. We obtain GKZ system from the variation of the Hodge structure and we benefit from the symmetries of the graphs more efficiently. GKZ systems are generalization of hypergeometric system and use the symmetries of the integrand, i.e. symmetries in its parameter space. As examples we compute the mass dependencies of the banana amplitudes up to the four-loop case.
For the two-loop banana Feynman integral, the so-called sunset diagram, the polar locus of the integrand is a special family of elliptic curves E, i.e. a Calabi-Yau one-fold. The integral is related to the period integral of the local mirror M of the non-compact Calabi-Yau three-fold W, defined as the total space of the anti-canonical line bundle over the degree three del Pezzo surface S, which is P2 blown up in three generic points. For the three-loop case, the vanishing locus of denominator of the GKZ integral defines a K3 surface and obviously, it defines a Calabi-Yau (l - 1)-fold for the l-loop case.
Later, we show that the coefficients of the linear combination of the solutions, which leads to the Banana Feynman integral, have geometrical interpretation and can be obtained by the so-called Gb-class evaluation in the ambient space of the mirror. We explain that in the equal mass case the relevant physical subslices in series of the Calabi-Yau manifolds are complete intersections of two constrains in (P)l+1. We calculate the Gb-class for this case which happens to match perfectly with the coefficients that we obtain numerically.
Classification (DDC)
530 PhysikSafari, Reza: Geometric approach to the Feynman integrals. - Bonn, 2022. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5-68243
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5-68243
@phdthesis{handle:20.500.11811/10347,
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5-68243,
author = {{Reza Safari}},
title = {Geometric approach to the Feynman integrals},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2022,
month = oct,
note = {In this PhD thesis, the Feynman integrals are evaluated using an algebraic geometry approach known in string theory. The GKZ description of periods and certain classes of relative periods on Calabi-Yau (l - 1)-folds has been used in order to solve the l-loop banana amplitudes with their general mass dependence.
There are different ways of computing Feynman integrals. These integrals satisfy a set of differential equations whose solutions give the answer of the integral. One of the approaches to derive a system of differential equations for Feynman integrals is the integration by part (IBP) identity. There is also an alternative approach to obtain these differential equations for a Feynman integral using the geometric interpretation of the Feynman integrals.
From the algebro-geometric point of view, Feynman integrals are the periods of the mixed Hodge structure on the relative cohomology groups. Varying physical parameters leads to a variation of the Hodge structure. The geometric interpretation comes from the polar locus of the integrand. The poles of the integrand define a Calabi-Yau manifold, which is often in a toric variety. These period integrals satisfy a system of linear homogeneous differential equations, so-called Picard-Fuchs differential ideal (PFDI). With toric geometry we can derive a finite set of differential operators, so-called GKZ hypergeometric system and extract PFDI. We obtain GKZ system from the variation of the Hodge structure and we benefit from the symmetries of the graphs more efficiently. GKZ systems are generalization of hypergeometric system and use the symmetries of the integrand, i.e. symmetries in its parameter space. As examples we compute the mass dependencies of the banana amplitudes up to the four-loop case.
For the two-loop banana Feynman integral, the so-called sunset diagram, the polar locus of the integrand is a special family of elliptic curves E, i.e. a Calabi-Yau one-fold. The integral is related to the period integral of the local mirror M of the non-compact Calabi-Yau three-fold W, defined as the total space of the anti-canonical line bundle over the degree three del Pezzo surface S, which is P2 blown up in three generic points. For the three-loop case, the vanishing locus of denominator of the GKZ integral defines a K3 surface and obviously, it defines a Calabi-Yau (l - 1)-fold for the l-loop case.
Later, we show that the coefficients of the linear combination of the solutions, which leads to the Banana Feynman integral, have geometrical interpretation and can be obtained by the so-called Gb-class evaluation in the ambient space of the mirror. We explain that in the equal mass case the relevant physical subslices in series of the Calabi-Yau manifolds are complete intersections of two constrains in (P)l+1. We calculate the Gb-class for this case which happens to match perfectly with the coefficients that we obtain numerically.},
url = {https://hdl.handle.net/20.500.11811/10347}
}
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5-68243,
author = {{Reza Safari}},
title = {Geometric approach to the Feynman integrals},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2022,
month = oct,
note = {In this PhD thesis, the Feynman integrals are evaluated using an algebraic geometry approach known in string theory. The GKZ description of periods and certain classes of relative periods on Calabi-Yau (l - 1)-folds has been used in order to solve the l-loop banana amplitudes with their general mass dependence.
There are different ways of computing Feynman integrals. These integrals satisfy a set of differential equations whose solutions give the answer of the integral. One of the approaches to derive a system of differential equations for Feynman integrals is the integration by part (IBP) identity. There is also an alternative approach to obtain these differential equations for a Feynman integral using the geometric interpretation of the Feynman integrals.
From the algebro-geometric point of view, Feynman integrals are the periods of the mixed Hodge structure on the relative cohomology groups. Varying physical parameters leads to a variation of the Hodge structure. The geometric interpretation comes from the polar locus of the integrand. The poles of the integrand define a Calabi-Yau manifold, which is often in a toric variety. These period integrals satisfy a system of linear homogeneous differential equations, so-called Picard-Fuchs differential ideal (PFDI). With toric geometry we can derive a finite set of differential operators, so-called GKZ hypergeometric system and extract PFDI. We obtain GKZ system from the variation of the Hodge structure and we benefit from the symmetries of the graphs more efficiently. GKZ systems are generalization of hypergeometric system and use the symmetries of the integrand, i.e. symmetries in its parameter space. As examples we compute the mass dependencies of the banana amplitudes up to the four-loop case.
For the two-loop banana Feynman integral, the so-called sunset diagram, the polar locus of the integrand is a special family of elliptic curves E, i.e. a Calabi-Yau one-fold. The integral is related to the period integral of the local mirror M of the non-compact Calabi-Yau three-fold W, defined as the total space of the anti-canonical line bundle over the degree three del Pezzo surface S, which is P2 blown up in three generic points. For the three-loop case, the vanishing locus of denominator of the GKZ integral defines a K3 surface and obviously, it defines a Calabi-Yau (l - 1)-fold for the l-loop case.
Later, we show that the coefficients of the linear combination of the solutions, which leads to the Banana Feynman integral, have geometrical interpretation and can be obtained by the so-called Gb-class evaluation in the ambient space of the mirror. We explain that in the equal mass case the relevant physical subslices in series of the Calabi-Yau manifolds are complete intersections of two constrains in (P)l+1. We calculate the Gb-class for this case which happens to match perfectly with the coefficients that we obtain numerically.},
url = {https://hdl.handle.net/20.500.11811/10347}
}
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