Periods of mirrors and multiple zeta values

Hoffman, Michael

doi:10.1090/S0002-9939-01-06263-3

Periods of mirrors and multiple zeta values
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by Michael E. Hoffman PDF

Proc. Amer. Math. Soc. 130 (2002), 971-974 Request permission

Abstract:

In a recent paper, A. Libgober showed that the multiplicative sequence $\{Q_i(c_1,\dots ,c_i)\}$ of Chern classes corresponding to the power series $Q(z)=\Gamma (1+z)^{-1}$ appears in a relation between the Chern classes of certain Calabi-Yau manifolds and the periods of their mirrors. We show that the polynomials $Q_i$ can be expressed in terms of multiple zeta values.

References

D. J. Broadhurst and D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to $9$ loops, Phys. Lett. B 393 (1997), no. 3-4, 403–412. MR 1435933, DOI 10.1016/S0370-2693(96)01623-1
Ira M. Gessel, Multipartite $P$-partitions and inner products of skew Schur functions, Combinatorics and algebra (Boulder, Colo., 1983) Contemp. Math., vol. 34, Amer. Math. Soc., Providence, RI, 1984, pp. 289–317. MR 777705, DOI 10.1090/conm/034/777705
Michael E. Hoffman, Multiple harmonic series, Pacific J. Math. 152 (1992), no. 2, 275–290. MR 1141796, DOI 10.2140/pjm.1992.152.275
Michael E. Hoffman, The algebra of multiple harmonic series, J. Algebra 194 (1997), no. 2, 477–495. MR 1467164, DOI 10.1006/jabr.1997.7127
Tu Quoc Thang Le and Jun Murakami, Kontsevich’s integral for the Homfly polynomial and relations between values of multiple zeta functions, Topology Appl. 62 (1995), no. 2, 193–206. MR 1320252, DOI 10.1016/0166-8641(94)00054-7
Anatoly Libgober, Chern classes and the periods of mirrors, Math. Res. Lett. 6 (1999), no. 2, 141–149. MR 1689204, DOI 10.4310/MRL.1999.v6.n2.a2
I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
Don Zagier, Values of zeta functions and their applications, First European Congress of Mathematics, Vol. II (Paris, 1992) Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 497–512. MR 1341859