A mirror theorem for toric stacks

Abstract

We prove a Givental-style mirror theorem for toric Deligne-Mumford stacks X . This determines the genus-zero Gromov-Witten invariants of X in terms of an explicit hypergeometric function, called the I-function, that takes values in the Chen-Ruan orbifold cohomology of X .

Orbifold quasimap theory

“…Besides the overlap with the Mirror Theorem for toric DM stacks proved in [13] (which will be explained in the next section, see Corollary 5.3.4 (3) and the discussion after it), Theorem 4.1.2 also overlaps with the work of C. Woodward, [32,33].…”

“…This is the denominator of e C * ×T (N vir F β /QG P a,1 Part (2) of the Corollary is precisely the main result of [13]. Note that the notion of "S-extended I-function" from [13] corresponds in our terminology to the Givental small I-function for a different GIT presentation of the geometric target X.…”

“…For the convenience of the reader we present its extension to the orbifold case. As a result of this calculation, the Mirror Theorem for toric DM stacks, recently proved by different methods in [13], becomes a special case of Theorem 4.2.…”

“…This gives a closed form for the "big" I-function as well. As a result, the Mirror Theorem for toric orbifolds, recently proved by different methods in [13], becomes a special case of Theorem 4.2, see Corollary 5.6. 2.…”

Pixton’s double ramification cycle relations

“…and let ι Γ : M Γ → M g,n be the gluing morphism as in (8). A basic class on M Γ is defined as an expression of the form…”