Mirror Symmetry, Zeta Functions and Mackey Functors

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 WT are dual invertible polynomials in n+2 variables. Denote by (XW, G) and (XWT ; GT) the corresponding Berglund-Hubsch dual hypersurfaces in Pn+1, where G and GT stands for their group of symmetries. The orbifold L-series of XW and XWT satisfy: (*) Lorb(XW, s) = Lorb(XWT, s)(-1)n. The above relation was proved by Ebeling-GusseinZade (2017), by other methods. 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 from 1. 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 EndCG⊕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 ιT: GT→Z, of the dual invertible polynomials. It is known that GT 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 GT. Using a formula of the orbifold zeta function from 1 in terms of the age functions, we deduce a comparison of zeta functions for the two dual invertible polynomials as given in the above.