twenty-seven lines on a cubic surface and heterotic string space-times

Letter to the editor Twenty-seven lines on a cubic surface and heterotic string space-times Metod Saniga * Astronomical Institute, Slovak Academy of Sciences, SK-059 60 Tatransk a Lomnica, Slovak Republic Accepted 20 December 2000 Abstract It is hypothesized that the algebra of the con®guration of 27 lines lying on a general cubic surface underlies the dimensional hierarchy of heterotic string space-times. Ó 2001 Elsevier Science Ltd. All rights reserved. 1. Introduction The correct quantitative elucidation and deep qualitative understanding of the observed dimensionality and signature of the Universe represent, undoubtedly, a crucial stepping stone on our path towards un- locking the ultimate secrets of the very essence of our being. Although there have been numerous attempts of various degrees of mathematical rigorousness and a wide range of physical scrutiny to address this issue, the subject still remains one of the toughest and most challenging problems faced by contemporary physics. In this contribution, we shall approach the problem by raising a somewhat daring hypothesis that the dimensional aspect of the structure of space-time may well be reproduced by the algebra of a geometric con®guration as simple as that of the lines situated on a cubic surface in a three-dimensional projective space. 2. The set of 27 lines on a cubic surface It is a well-known fact that on a generic cubic surface, K 3 , there is a con®guration of 27 lines [1]. Al- though this con®guration is geometrically perfectly symmetric as it stands, it exhibits a remarkable non- trivial structure when intersection/incidence relations between the individual lines are taken into account. Namely, the lines are seen to form three separate groups. The ®rst two groups, each comprising six lines, are known as Schla¯i's double-six. This is indeed a remarkable subset because the lines in either group are not incident with each other, i.e., they are mutually skew, whereas a given line from one group is skew with one and incident with the remaining ®ve lines of the other group. The third group consists of 15 lines, each one being incident with four lines of the Schla¯i set and six other lines of the group in question. The basic of the algebra can simply be expressed as: www.elsevier.nl/locate/chaos Chaos, Solitons and Fractals 12 2001) 1177±1178 * Tel.: +421-969-4467-866; fax: +421-969-4467-656. E-mail address: [email protected] http://www.ta3.sk/msaniga M. Saniga). 0960-0779/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 0 0 ) 0 0 2 6 4 - 2

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Page 1: Twenty-seven lines on a cubic surface and heterotic string space-times

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