×

\(K\)-rational d-brane crystals. (English) Zbl 1258.81071

Summary: In this paper the problem of constructing space-time from string theory is addressed in the context of D-brane physics. It is suggested that the knowledge of discrete configurations of D-branes is sufficient to reconstruct the motivic building blocks of certain Calabi-Yau varieties. The collections of D-branes involved have algebraic base points, leading to the notion of \(K\)-arithmetic D-crystals for algebraic number fields K. This idea can be tested for D0-branes in the framework of toroidal compactifications via the conjectures of Birch and Swinnerton-Dyer. For the special class of D0-crystals of Heegner type these conjectures can be interpreted as formulae that relate the canonical Néron-Tate height of the base points of the D-crystals to special values of the motivic \(L\)-function at the central point. In simple cases the knowledge of the D-crystals of Heegner type suffices to uniquely determine the geometry.

MSC:

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
83E15 Kaluza-Klein and other higher-dimensional theories
83E30 String and superstring theories in gravitational theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] DOI: 10.1016/S0393-0440(03)00030-5 · Zbl 1033.11030 · doi:10.1016/S0393-0440(03)00030-5
[2] DOI: 10.1016/j.nuclphysb.2007.01.027 · Zbl 1117.81119 · doi:10.1016/j.nuclphysb.2007.01.027
[3] DOI: 10.1007/s00220-010-1179-4 · Zbl 1213.14074 · doi:10.1007/s00220-010-1179-4
[4] DOI: 10.1016/j.geomphys.2011.04.010 · Zbl 1243.32017 · doi:10.1016/j.geomphys.2011.04.010
[5] DOI: 10.1016/S0393-0440(02)00079-7 · Zbl 1092.11028 · doi:10.1016/S0393-0440(02)00079-7
[6] Birch B. J., J. Reine Angew. Math. 212 pp 7–
[7] Birch B. J., J. Reine Angew. Math. 218 pp 79–
[8] DOI: 10.1007/BF01388809 · Zbl 0608.14019 · doi:10.1007/BF01388809
[9] Mordell L. J., Proc. Camb. Philos. Soc. 21 pp 179–
[10] DOI: 10.1007/BF02684339 · Zbl 0394.14008 · doi:10.1007/BF02684339
[11] DOI: 10.1090/S0002-9904-1975-13623-8 · Zbl 0316.14012 · doi:10.1090/S0002-9904-1975-13623-8
[12] DOI: 10.2307/2118559 · Zbl 0823.11029 · doi:10.2307/2118559
[13] DOI: 10.2307/2118560 · Zbl 0823.11030 · doi:10.2307/2118560
[14] DOI: 10.1090/S0894-0347-01-00370-8 · Zbl 0982.11033 · doi:10.1090/S0894-0347-01-00370-8
[15] Wiles A., The Millennium Prize Problems (2006) · Zbl 1155.00001
[16] Cassels J. W. S., J. Reine Angew. Math. 214 pp 65–
[17] DOI: 10.1142/S0217751X06034343 · Zbl 1111.81133 · doi:10.1142/S0217751X06034343
[18] DOI: 10.1007/BF01388984 · Zbl 0628.14018 · doi:10.1007/BF01388984
[19] DOI: 10.1007/BF01402975 · Zbl 0359.14009 · doi:10.1007/BF01402975
[20] Kolyvagin V. A., Izv. Akad, Nauk. SSSR Ser. Mat. 52 pp 522–
[21] Kolyvagin V. A., Izv. Akad, Nauk. SSSR Ser. Mat. 52 pp 1154–
[22] Kolyvagin V. A., The Grothendieck Festschrift (1990)
[23] DOI: 10.1090/S0273-0979-1989-15771-6 · Zbl 0699.10038 · doi:10.1090/S0273-0979-1989-15771-6
[24] DOI: 10.2307/1971508 · Zbl 0699.10039 · doi:10.2307/1971508
[25] DOI: 10.2307/2944316 · Zbl 0745.11032 · doi:10.2307/2944316
[26] DOI: 10.1007/978-0-387-09494-6 · Zbl 1194.11005 · doi:10.1007/978-0-387-09494-6
[27] DOI: 10.1007/978-1-4612-0851-8 · doi:10.1007/978-1-4612-0851-8
[28] Tate J., Séminaire Bourbaki 9 pp 306–
[29] DOI: 10.1007/BF01174749 · Zbl 0049.16202 · doi:10.1007/BF01174749
[30] Birch B. J., Modular Forms (1984)
[31] Waldspurger J.-L., Compos. Math. 54 pp 173–
[32] DOI: 10.1007/BFb0062705 · doi:10.1007/BFb0062705
[33] Birch B. J., Heegner Points and Rankin L-Series (2004)
[34] DOI: 10.1007/978-1-4612-0457-2_3 · doi:10.1007/978-1-4612-0457-2_3
[35] DOI: 10.1017/CBO9780511756375.008 · doi:10.1017/CBO9780511756375.008
[36] DOI: 10.2307/1970831 · Zbl 0266.10022 · doi:10.2307/1970831
[37] DOI: 10.1007/BF01455989 · Zbl 0542.10018 · doi:10.1007/BF01455989
[38] DOI: 10.1007/BF01388432 · Zbl 0588.14026 · doi:10.1007/BF01388432
[39] DOI: 10.2140/pjm.1997.181.251 · doi:10.2140/pjm.1997.181.251
[40] DOI: 10.1007/s002220050189 · Zbl 0905.11024 · doi:10.1007/s002220050189
[41] DOI: 10.1007/BFb0084592 · doi:10.1007/BFb0084592
[42] Cremona J. E., Algorithms for Elliptic Curves (1997) · Zbl 0872.14041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.