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Local mirror symmetry and the sunset Feynman integral. (English) Zbl 1390.14123

This article comprises a comprehensive study of the sunset Feynman integral, resulting from a two loop propagator-type diagram with arbitrary masses running in the loops. An interesting approach is given in this paper using some theory of motives.
Covering all important aspects of this very exhaustive piece of work in a short review is rather impossible. So we give an overview with the author’s words on the plan of the paper: “The plan of the paper is the following. In Part II, we analyse the sunset Feynman integral (1.1). In Section 3.1 we describe the geometry of the sunset family of elliptic curve and in Section 3.2 derive the Picard-Fuchs equation following Griffiths’s approach in [P.A. Griffiths, Ann. Math. (2) 90, 460–495 (1969; Zbl 0215.08103)] for deriving the Picard-Fuchs equation from the cohomology of smooth projective hyperspace defined by rational form in \(\mathbb{P}^2\). In Section 3.3 we derive the expression (1.2) of the sunset integral in terms of elliptic dilogarithm. In Section 3.3.1 we show how to reproduce the all equal masses result of [the first and the third author, J. Number Theory 148, 328–364 (2015; Zbl 1319.81044)] and Section 3.3.2 contains numerical verification of the three different masses case. We give a proof of these results using a motivic approach in Section 4. Part III of the paper deals with the mirror symmetry construction. In Section 5 we describe the degeneration from a compact Calabi-Yau 3-fold X to the local Hori-Vafa model Y, and show in Theorem 5.3 that the third homology of Y matches the invariant part of the limiting mixed Hodge structure of \(H^3(\mathrm{X})\). In Section 6 we describe the variation of Hodge structure arising on the A-model obtained by considering the Batyrev mirror of X. By comparing the limiting mixed Hodge structures of the A-model and B-model, we prove in Theorem 6.1 a strong form of local mirror symmetry – equality of variations of \(\mathbb{Q}\)-mixed Hodge structure. The particular case of the sunset integral is discussed in Section 7. In the appendix A we recall the main properties of Jacobi theta functions, and in the appendix B we give the detailed coefficients entering the derivation of the Picard-Fuchs equation in Section 3.2.”

MSC:

14J33 Mirror symmetry (algebro-geometric aspects)
81T18 Feynman diagrams
14F42 Motivic cohomology; motivic homotopy theory
14H52 Elliptic curves
11G55 Polylogarithms and relations with \(K\)-theory
14J30 \(3\)-folds
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14D06 Fibrations, degenerations in algebraic geometry
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
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