On vanishing of cohomology attached to certain many valued meromorphic functions

Aomoto, Kazuhiko

doi:10.2969/jmsj/02720248

Cited by 79 publications

(62 citation statements)

References 6 publications

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“…We will call H k dW := H k (M, ∇ dW ) for short from now on. As in the case of (2.4), only k = m gives a non-trivial cohomology [40] and hence we have dim H m dW = (−1) m χ(M ).…”

Section: )mentioning

confidence: 83%

From infinity to four dimensions: higher residue pairings and Feynman integrals

Mizera

Pokraka

2020

J. High Energ. Phys.

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We study a surprising phenomenon in which Feynman integrals in D = 4 − 2ε space-time dimensions as ε → 0 can be fully characterized by their behavior in the opposite limit, ε → ∞. More concretely, we consider vector bundles of Feynman integrals over kinematic spaces, whose connections have a polynomial dependence on ε and are known to be governed by intersection numbers of twisted forms. They give rise to differential equations that can be obtained exactly as a truncating expansion in either ε or 1/ε. We use the latter for explicit computations, which are performed by expanding intersection numbers in terms of Saito's higher residue pairings (previously used in the context of topological Landau-Ginzburg models and mirror symmetry). These pairings localize on critical points of a certain Morse function, which correspond to regions in the loop-momentum space that were previously thought to govern only the large-D physics. The results of this work leverage recent understanding of an analogous situation for moduli spaces of curves, where the α → 0 and α → ∞ limits of intersection numbers coincide for scattering amplitudes of massless quantum field theories.

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“…We will call H k dW := H k (M, ∇ dW ) for short from now on. As in the case of (2.4), only k = m gives a non-trivial cohomology [40] and hence we have dim H m dW = (−1) m χ(M ).…”

Section: )mentioning

confidence: 83%

From infinity to four dimensions: higher residue pairings and Feynman integrals

Mizera

Pokraka

2020

J. High Energ. Phys.

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“…The essential facts about twisted cohomology groups are [22][23][24]: (i) the twisted cohomology groups H N ω are finite-dimensional and (ii) there is a a non-degenerate inner product between H N ω and the dual twisted cohomology group (H N ω ) * = H N −ω given by the intersection number. Let e 1 |, .…”

Section: Twisted Cocyclesmentioning

confidence: 99%

Correlation functions on the lattice and twisted cocycles

Weinzierl

2020

Physics Letters B

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“…We refer to [AK, Ba, E, HK, Kim] for related results of F 2 , F 3 or E 2 , E 3 , and to [Ao1,Ao2,C,MY1] for the twisted de Rham theory.…”

Section: Introductionmentioning

confidence: 99%

Nakagiri's Monodromy Representations Associated with Appell's Hypergeometric Functions F2 and F3

Mimachi

2017

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Abstract. We study monodromy representations associated with Appell's hypergeometric functions F 2 and F 3 by using integrals of multivalued functions. In particular, we derive Nakagiri's matix elements of the representations in a di¤erent manner and relax his conditions on the parameters. As an application, we give a simple and elementary derivation of su‰cient conditions that the systems of di¤erential equations satisfied by F 2 and F 3 are irreducible.

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On vanishing of cohomology attached to certain many valued meromorphic functions

Cited by 79 publications

References 6 publications

From infinity to four dimensions: higher residue pairings and Feynman integrals

From infinity to four dimensions: higher residue pairings and Feynman integrals

Correlation functions on the lattice and twisted cocycles

Nakagiri's Monodromy Representations Associated with Appell's Hypergeometric Functions <i>F</i><sub>2</sub> and <i>F</i><sub>3</sub>

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