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Erratum to Tropical mirror symmetry for elliptic curves (J. reine angew. Math. 732 (2017), 211–246)

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Published/Copyright: June 13, 2018
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Journal für die reine und angewandte Mathematik (Crelles Journal)
From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2020 Issue 760

We thank Elise Goujard and Martin Möller for pointing out and closing a gap in Theorem 3.2 of our work [1]:

The statement of the theorem is not accurate. The Feynman integrals in question are indeed quasimodular forms, however, they are not necessarily homogeneous. The following is the accurate formulation of the statement:

Theorem 3.2.

For all Feynman graphs Γ and orders Ω as in Definition 2.5, the function IΓ,Ω is a quasimodular form of mixed weight. The highest appearing weight is 6⁢g-6.

In our work [1], the proof of Theorem 3.2 had relied on Proposition 3.3 whose proof contains a gap in line 7 on page 235.

Theorem 3.2 in its accurate form as stated above follows from [3, Corollary 8.4]. In [3], Goujard and Möller studied quasimodularity questions for graph sums like our Feynman integrals in a more general context. They also provide an example of a Feynman integral which is a quasimodular function of mixed weight in Section 5.4.

Interestingly, if we sum all our Feynman integrals – i.e., the sum over all eligible graphs Γ and all orders Ω, ∑Γ1|AutΓ|⁢∑ΩIΓ,Ω, which, by [1, mirror symmetry Theorem 2.6] (see, for instance, [2, Theorem 3]) equals the generating function of covers of an elliptic curve – we get a function which is quasimodular of pure weight 6⁢g-6 (see [4]): the lower order contributions cancel in the sum. In [1, Example 3.5], we have computed a sum

IΓ=∑ΩIΓ,Ω

of all Feynman integrals for one fixed graph Γ. It turns out that in this example, the lower order terms already cancel and IΓ is quasimodular of pure weight 6⁢g-6. It is an interesting open question whether the cancellation always happens already at the level of a single graph, or whether in general we have to sum over all graphs to observe the cancellation. It is computationally challenging to produce more examples.

References

[1] J. Böhm, K. Bringmann, A. Buchholz and H. Markwig, Tropical mirror symmetry for elliptic curves, J. reine angew. Math. 732 (2017), 211–246. 10.1515/crelle-2014-0143Search in Google Scholar

[2] R. Dijkgraaf, Mirror symmetry and elliptic curves, The moduli space of curves, Progr. Math. 129, Birkhäuser, Boston (1995), 149–163., 10.1007/978-1-4612-4264-2_5Search in Google Scholar

[3] E. Goujard and M. Möller, Counting Feynman-like graphs: Quasimodularity and Siegel–Veech weight, preprint (2016), https://arxiv.org/abs/1609.01658. 10.4171/JEMS/924Search in Google Scholar

[4] M. Kaneko and D. Zagier, A generalized Jacobi theta function and quasimodular forms, The moduli space of curves, Progr. Math. 129, Birkhäuser, Boston (1995), 149–163. 10.1007/978-1-4612-4264-2_6Search in Google Scholar

Received: 2018-05-22
Published Online: 2018-06-13
Published in Print: 2020-03-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Probabilistic Schubert calculus
  3. On cohomological Hall algebras of quivers: Generators
  4. Intersections of two Grassmannians in ℙ9
  5. Erratum to Tropical mirror symmetry for elliptic curves (J. reine angew. Math. 732 (2017), 211–246)
  6. Erratum to A parabolic flow toward solutions of the optimal transportation problem on domains with boundary (J. reine angew. Math. 672 (2012), 127–160)
  7. Maximal time existence of unnormalized conical Kähler–Ricci flow
  8. A remark on a converse theorem of Cogdell and Piatetski-Shapiro
  9. Undistorted purely pseudo-Anosov groups
  10. Dendroidal spaces, Γ-spaces and the special Barratt--Priddy--Quillen theorem
  11. On local integration of Lie brackets
https://doi.org/10.1515/crelle-2018-0017

Articles in the same Issue

  1. Frontmatter
  2. Probabilistic Schubert calculus
  3. On cohomological Hall algebras of quivers: Generators
  4. Intersections of two Grassmannians in ℙ9
  5. Erratum to Tropical mirror symmetry for elliptic curves (J. reine angew. Math. 732 (2017), 211–246)
  6. Erratum to A parabolic flow toward solutions of the optimal transportation problem on domains with boundary (J. reine angew. Math. 672 (2012), 127–160)
  7. Maximal time existence of unnormalized conical Kähler–Ricci flow
  8. A remark on a converse theorem of Cogdell and Piatetski-Shapiro
  9. Undistorted purely pseudo-Anosov groups
  10. Dendroidal spaces, Γ-spaces and the special Barratt--Priddy--Quillen theorem
  11. On local integration of Lie brackets
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