Mirror Symmetry, Zeta Functions and Mackey Functors

Abstract

We present two alternative and new proofs for the duality between orbifold zeta functions of Berglund-Hubsch dual invertible polynomials. We re-prove the following theorem; Assume W and W^T are dual invertible polynomials in n + 2 variables. Denote by (X_W, G) and (X_W^T ; G^T) the corresponding Berglund-Hubsch dual hypersurfaces in Pⁿ⁺¹, where G and G^T stands for their group of symmetries. The orbifold L-series of X_W and X_W^T satisfy: (*) L_orb(X_W, s) = L_orb(X_W^T, s)^(-1)n. We present two proofs of the above identity (*). Our methods of proof are different. The first proof uses cohomological Mackey functors on Mackey systems. The second proof is independent and uses a formula for the orbifold zeta functions. For an orbifold (X, G) we consider a Mackey system of subgroups of G and cohomological Mackey functors on this Mackey system. We investigate the relation between the above L-series of orbifolds and the Mackey functors. We show the orbifold cohomology H^*_orb(X, C) is an End_CG[⊕_gCG/C(g)]-module, that means; the orbifold cohomology defines a cohomological Mackey functor on the Mackey system of conjugacy classes in G. This leads one to split the zeta function according to properties of G-cohomological Mackey functors. This method allows obtaining identities on orbifold zeta functions from identities in a Grothendick group associated with subgroup quotients of G. In this context, the relation (1) is a consequence of cohomological mirror symmetry and Mackey structure. In other words, we obtain the identity (1) from Mackey type of identities in a Grothendieck group followed by a multiplicative homomorphism constructed from zeta functions of a Galois representation. The second proof uses a duality between age functions ι: G→Z, and ι^T: G^T→Z, of the dual invertible polynomials. It is known that G^T is the character group of G. We show that these age functions are Fourier transforms of each other with respect to a unitary representation obtained from the natural pairing between G and G^T. Using a formula of the orbifold zeta function in terms of the age functions, we deduce a comparison of zeta functions for the two dual invertible polynomials as given in the above.