36
$\begingroup$

By browsing through the Hodge data of known Calabi-Yau threefolds, I stumbled upon an observation that frequently enough a pair of Hodge numbers $(h^{11},h^{21})$ comes together with the pair $ (h^{11}-240,h^{21}+240)$. This shift, together with mirror symmetry accounts for a certain symmetry in the top portion of the Kreuzer-Skarke plot.

Has anyone observed this shift before?

Is there an empirical explanation for this shift? Is there any relation to the roots of $E_8$?

Are there other similar but smaller shifts?

$\endgroup$
  • 2
    $\begingroup$ A very interesting observation. $\endgroup$ – Atsushi Kanazawa Mar 22 '14 at 1:00
  • $\begingroup$ In the sactter plot, do the two parabola-like upper curves match up if one is translated to the other? $\endgroup$ – Mark Gross Mar 22 '14 at 7:55
  • $\begingroup$ I believe so. I zoomed in on the picture near the vertex of the parabola and it was completely symmetric. Also notice that the top points are $(11,491),(251,251),(491,11)$ which exhibits the same shift twice. $\endgroup$ – Lev Borisov Mar 22 '14 at 14:36
  • $\begingroup$ Certainly very intriguing. I didn't realise that the curve was symmetric. $\endgroup$ – Mark Gross Mar 22 '14 at 17:45
  • 1
    $\begingroup$ Look at this image: benjaminjurke.net/dynamic/cyexp/… There is almost a perfect symmetry in the top ranges, as if there was some sort of mechanism to get the new Hodge pair if the original numbers are large enough. $\endgroup$ – Lev Borisov Mar 22 '14 at 22:00
16
$\begingroup$

In the mathoverflow answer Today's world record on the Betti numbers of Calabi-Yau three-folds., the paper http://arxiv.org/abs/arXiv:1207.4792 is cited. In this paper, the shift of Hodge numbers by (-240,240) is mentionned and the symmetry is explained, at least for the Calabi-Yau threefolds obtained from reflexive polytopes by the Batyrev's construction. E_8 plays indeed an important role (via Calabi-Yau which are K3 fibrations).

EDIT: a video conference of P.Candelas on this paper : https://www.youtube.com/watch?v=4996MUz25vg

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ Thank you! This makes sense to me at least at the combinatorial level. $\endgroup$ – Lev Borisov Mar 27 '14 at 13:14

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.