Shift operators and toric mirror theorem

Abstract

Abstract. We give a new proof of Givental's mirror theorem for toric manifolds using shift operators of equivariant parameters. The proof is almost tautological: it gives an A-model construction of the I-function and the mirror map. It also works for non-compact or non-semipositive toric manifolds. IntroductionIn 1995, Seidel [Sei97] introduced an invertible element of quantum cohomology associated to a Hamiltonian circle action. This has had many applications in symplectic topology. Seidel himself used it to construct non-trivial elements of π 1 of the group of Hamiltonian diffeomorphisms. calculated Seidel's elements in a more general setting and obtained Batyrev's ring presentation of quantum cohomology of toric manifolds. Their method, however, does not yield explicit structure constants of quantum cohomology, i.e. genus-zero Gromov-Witten invariants.Recently, Braverman, Maulik, Okounkov and Pandharipande [OP10, BMO11, MO12] introduced a shift operator of equivariant parameters for equivariant quantum cohomology. Their shift operators reduce to Seidel's invertible elements under the nonequivariant limit. In this paper, we show that equivariant genus-zero Gromov-Witten invariants of toric manifolds are reconstructed only from formal properties of shift operators. This means that the equivariant quantum topology of toric manifolds is determined by its classical counterpart.More specifically, we give a new proof of Givental's mirror theorem for toric manifolds, which is stated as follows: Giv98b,LLY99,Iri08,Bro09], see §4.2 for more details). Let X Σ be a semi-projective toric manifold having a torus fixed point. Let I(y, z) be the cohomologyvalued hypergeometric series defined bywhere u i , i = 1, . . . , m is the class of a prime toric divisor. Then I(y, −z) lies in Givental's Lagrangian cone L X Σ associated to X Σ .We prove this theorem in the following way. Recall that equivariant genus-zero Gromov-Witten invariants of a T -variety X can be encoded by an infinite-dimensional Lagrangian submanifold L X of the symplectic vector space H X [Giv04]:The space H X is called the Givental space and L X is called the Givental cone. By the general theory, each C × -subgroup k : C × → T defines a shift operator S k acting on the Givental space H X and induces a vector field on L X :The operator S k is determined by T -fixed loci in X and their normal bundles (see Definition 3.13). For toric manifolds, we have a shift operator S i for each torus-invariant prime divisor. Then we identify the I-function I(y, z) with an integral curve of the commuting vector fields f → z −1 S i f. Theorem 1.2. Givental's I-function I(y, z) is a unique integral curve which satisfies the differential equation:and is of the form I(y, z) = ze, where we setThe I-function defines a mirror map y → τ (y) ∈ H * T (X) via Birkhoff factorization [CG07, Iri08]. As a corollary to our proof, we obtain the following relationship between the equivariant Seidel elements S i (τ ) and the mirror map. This generalizes a previous result [GI12] in the semipo...

A mirror construction for the big equivariant quantum cohomology of toric manifolds

“…Therefore the claim follows. The lemma follows from this claim and the recursive construction of τ ℓ , Υ ℓ (z) in [38,Proposition 4.6] and in Proposition 3.6.…”

“…Note that S k (τ ), S k (τ ) are defined without inverting equivariant parameters, which again follows from the fact that E k is semi-projective, see [38,Remark 3.10]…”

“…Proof. The existence and uniqueness of τ (y), Υ(y, z) along the locus {y k = 0, ∀k ∈ G} is proved in [38,Proposition 4.7]. Since the vector fields V k commute each other, we have a unique solution (τ (y), Υ(y, z)) to (3.5) which takes the form (3.4) along {y k = 0, ∀k ∈ G} (see Theorem A.2).…”

“…A Seidel element is an invertible element of quantum cohomology associated to a Hamiltonian circle action on a symplectic manifold, introduced by Seidel [56]. This can be "lifted" to the equivariant setting and yields a shift operator for equivariant parameters [47,5,42,38]. The mirror map and the mirror isomorphism in the above theorem can be described as follows: 2 This is equivalent to X Σ being a GIT quotient of a vector space.…”

Enumerative geometry and geometric representation theory

“…It is constructed geometrically counting sections in certain twisted X-bundles over P 1 . Operators of this kind, often called shift operators, find many other applications in enumerative geometry, see for example [60].…”