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Feynman Integrals and Mirror Symmetry

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Partition Functions and Automorphic Forms

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Abstract

In this text we describe various approaches to the computation of Feynman integrals. One approach uses toric geometry to derive differential equations satisfied by the imaginary part of the Feynman integrals. We then discuss how this can be used to obtain the full differential equation acting on the integral. In a second part of this text we explain how Calabi–Yau geometry is naturally associated to Feynman integrals and that mirror symmetry plays some role in evaluating some particular Feynman integrals.

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Notes

  1. 1.

    An ideal I of a ring R, is the subset , such that 1) 0 ∈ I, 2) for all a, b ∈ I then a + b ∈ I, 3) for a ∈ I and b ∈ R, a ⋅ b ∈ R. For P(x 1, …, x n) an homogeneous polynomial in \(R=\mathbb C[x_1,\dots ,x_n]\) the Jacobian ideal of P is the ideal generated by the first partial derivative \(\{\partial _{x_i} P(x_1,\dots ,x_n)\}\) [41]. Given a multivariate polynomial P(x 1, …, x n) its Jacobian ideal is easily evaluated using Singular command jacob( P) . The hypersurface P(x 1, ⋯ , x n) = 0 for an homogeneous polynomial, like the Symanzik polynomials, is of codimension 1 in the projective space \(\mathbb P^{n-1}\). The singularities of the hypersurface are determined by the irreducible factors of the polynomial. This determines the cohomology of the complement of the graph hypersurface and the number of independent master integrals as shown in [42].

  2. 2.

    Consider an homogeneous polynomial of degree d

    this is called a toric polynomial if it is invariant under the following actions

    for \((t_1,\dots ,t_n)\in \mathbb C^n\) and α ij and β ij integers. The second Symanzik polynomial has a natural torus action acting on the mass parameters and the kinematic variables as we will see on some examples below. We refer to the book [41] for more details.

  3. 3.

    The convergence of these series is discussed in [52, §3-2] and [50, §5.2].

  4. 4.

    This quantity is the usual Yukawa coupling of particle physics and string theory compactification. The Yukawa coupling is determined geometrically by the integral of the wedge product of differential forms over particular cycles [58]. The Yukawa couplings which depend non-trivially on the internal geometry appear naturally in the differential equations satisfied by the periods of the underlying geometry as explained for instance in these reviews [48, 59].

  5. 5.

    The Jacobi theta functions are defined by , and .

  6. 6.

    A del Pezzo surface is a two-dimensional Fano variety. A Fano variety is a complete variety whose anti-canonical bundle is ample. The anti-canonical bundle of a non-singular algebraic variety of dimension n is the line bundle defined as the nth exterior power of the inverse of the cotangent bundle. An ample line bundle is a bundle with enough global sections to set up an embedding of its base variety or manifold into projective space.

  7. 7.

    The graph polynomial (3.10) for higher loop sunset graphs defines Fano variety, which is as well a Calabi–Yau manifold.

  8. 8.

    The fan of a toric variety is defined in the standard reference [75] and the review oriented to a physicists audience in [49].

  9. 9.

    Feynman integrals are period integrals of mixed Hodge structures [28, 77]. At a singular point some cycles of integration vanish, the so-called vanishing cycles, and the limiting behaviour of the period integral is captured by the asymptotic behaviour of the cohomological Hodge theory. The asymptotic Hodge theory inherits some filtration and weight structure of the original Hodge theory.

  10. 10.

    It has been already noticed in [80] the special role played by the Mahler measure and mirror symmetry.

  11. 11.

    We would like to thank Albrecht Klemm for discussions and communication that helped clarifying the link between the work in [20] and the analysis in [21].

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Acknowledgements

It is a pleasure to thank Charles Doran and Albrecht Klemm for discussions. The research of P. Vanhove has received funding the ANR grant “Amplitudes” ANR-17- CE31-0001-01, and is partially supported by Laboratory of Mirror Symmetry NRU HSE, RF Government grant, ag. N∘ 14.641.31.0001.

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  1. CEA, DSM, Institut de Physique Théorique, IPhT, CNRS, MPPU, URA2306, Gif-sur-Yvette, France

    Pierre Vanhove

  2. National Research University, Higher School of Economics, Moscow, Russia

    Pierre Vanhove

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  1. CNRS U.M.R. 8524, Université de Lille and IUF, Villeneuve d’Ascq Cedex, France

    Valery A. Gritsenko

  2. National Research University Higher School of Economics, Laboratory of Mirror Symmetry and Automorphic Forms, Moscow, Russia

    Vyacheslav P. Spiridonov

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Vanhove, P. (2020). Feynman Integrals and Mirror Symmetry. In: Gritsenko, V.A., Spiridonov, V.P. (eds) Partition Functions and Automorphic Forms. Moscow Lectures, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-030-42400-8_7

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