Periods of Mirrors and Multiple Zeta Values

In a recent paper, A. Libgober showed that the multiplicative sequence {Qi(c1, . . . , ci)} of Chern classes corresponding to the power series Q(z) = Γ(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 Qi can be expressed in terms of multiple zeta values. 1. The multiplicative sequence In [6], the Hirzebruch multiplicative sequence {Qi} associated to the power series Q(z) = Γ(1 + z)−1 is considered in connection with mirror symmetry. If ei denotes the ith elementary symmetric function in the variables t1, t2, . . . , then ∞ ∑ i=0 Qi(e1, . . . , ei) = ∞ ∏ i=1 1 Γ(1 + ti) . (1) As shown in [6], the polynomials Qi(c1, . . . , ci) in the Chern classes of certain Calabi-Yau manifolds X are related to the coefficients of the generalized hypergeometric series expansion of the period (holomorphic at a maximum degeneracy point) of a mirror of X . In particular, if X is a Calabi-Yau hypersurface of dimension 4 in a nonsingular toric Fano manifold, then ∫ X Q4(c1, c2, c3, c4) = 1 24 ∑ ijkl Kijkl ∂c(0, . . . , 0) ∂ρi∂ρj∂ρk∂ρl , where the c(ρ1, . . . , ρr) are coefficients in the expansion of the period and Kijkl is the (suitably normalized) 4-point function corresponding to a mirror of X . In [6] it is shown that the polynomials Qi have the form Q1(c1) = γc1 and Qi(c1, . . . , ci) = ζ(i)ci + · · · , i > 1. In this note we show that the polynomials Qi have an explicit expression in terms of multiple zeta values (called multiple harmonic series in [3, 4]), which have previously appeared in connection with Kontsevich’s invariant in knot theory [8, 5], and in quantum field theory [1]. Received by the editors November 23, 1999 and, in revised form, October 18, 2000. 2000 Mathematics Subject Classification. Primary 14J32, 11M41; Secondary 05E05.