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AN INTERTWINING RELATION for EQUIVARIANT SEIDEL MAPS 3 Cohomology of the Classifying Space of S1

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AN INTERTWINING RELATION for EQUIVARIANT SEIDEL MAPS 3 Cohomology of the Classifying Space of S1 AN INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS TODD LIEBENSCHUTZ-JONES Abstract. The Seidel maps are two maps associated to a Hamiltonian circle ac- tion on a convex symplectic manifold, one on Floer cohomology and one on quantum cohomology. We extend their definitions to S1-equivariant Floer cohomology and S1-equivariant quantum cohomology based on a construction of Maulik and Ok- ounkov. The S1-action used to construct S1-equivariant Floer cohomology changes after applying the equivariant Seidel map (a similar phenomenon occurs for S1- equivariant quantum cohomology). We show the equivariant Seidel map on S1- equivariant quantum cohomology does not commute with the S1-equivariant quan- tum product, unlike the standard Seidel map. We prove an intertwining relation which completely describes the failure of this commutativity as a weighted version of the equivariant Seidel map. We will explore how this intertwining relationship may be interpreted using connections in an upcoming paper. We compute the equivariant Seidel map for rotation actions on the complex plane and on complex projective space, and for the action which rotates the fibres of the tautological line bundle over projective space. Through these examples, we demonstrate how equi- variant Seidel maps may be used to compute the S1-equivariant quantum product and S1-equivariant symplectic cohomology. 1. Introduction For us, equivariant will always mean S1-equivariant. The Seidel maps on the quantum and Floer cohomology of a closed symplectic manifold M are two maps associated to a Hamiltonian S1-action σ on M [Sei97]. They are compatible with each other via the PSS isomorphisms, which are maps that identify quantum cohomology and Floer cohomology [Sei97, Theorem 8.2]. McDuff and Tolman used Seidel maps to recover the Batyrev presentation of the quantum cohomology of closed toric manifolds [MT06]. Ritter extended the definition of Seidel maps to (non-closed) convex symplectic manifolds [Rit14]. He used the Seidel maps to determine the quantum cohomology and the symplectic cohomology of convex toric manifolds [Rit16]. arXiv:2010.03342v2 [math.SG] 3 Mar 2021 Let ρ be a Hamiltonian S1-action on a closed or convex symplectic manifold M. ∗ The equivariant quantum cohomology EQHρ (M) has three important compatible algebraic structures: it is a ring equipped with the equivariant quantum product ρ; it is a module over the Novikov ring Λ; and it has a geometric Z[u]-module structure∗ denoted ⌣. In this paper, we introduce an equivariant quantum Seidel map Date: March 4, 2021. Correspondence: [email protected]. Institution: University of Oxford, UK. 1 2 TODD LIEBENSCHUTZ-JONES corresponding to an additional Hamiltonian S1-action σ on M which commutes with 1 ρ. We could, for example, set ρ = IdM , or we could let ρ and σ be two S -actions which are part of a Hamiltonian torus action on M. Theorem 1.1. There is an equivariant quantum Seidel map ∗ ∗+2I(σe) EQ e : EQH (M) EQH ∗ (M) (1.1) Sσ ρ → σ ρ which is a Λ Z[u]-module homorphism. The codomain of (1.1) is the equivariant quantum cohomology⊗ of M with the pullback action σ∗ρ = σ−1ρ. (1.2) Our main theorem describes the relationship between EQ σe and equivariant quan- tum multiplication. Unlike for the (non-equivariant) quantumS Seidel map, this rela- tionship is not simply that the map commutes with the quantum product. We can nonetheless exploit the relationship to derive the equivariant quantum product, which we demonstrate for a few examples in Sections 2.3 and 8. Theorem 1.2 (Intertwining relation). The equation + − EQ σe(x α ) EQ σe(x) α = u ⌣ EQ σ,αe (x) (1.3) S ∗ρ − S σ∗ρ S ∗ + ∗ − ∗ holds for all x EQHρ (M). Here, α EHρ (M) and α EHσ∗ρ(M) are two equivariant cohomology∈ classes which are related∈ via the clutching∈ bundle, and e ± ∗ ∗+2I(σ)+ α −2 EQ σ,αe : EQH (M) EQH ∗ | | (M) (1.4) S ρ → σ ρ is a map defined in Section 7.4.1. Maulik and Okounkov gave the first definition of equivariant quantum Seidel maps as part of their work on quiver varieties [MO19, Section 8]. They also proved that the maps satisfied the intertwining relation [MO19, Proposition 8.1]. Braverman, Maulik and Okounkov used equivariant quantum Seidel maps to derive the equivariant quantum product of the Springer resolution [BMO11]. Iritani recovered Givental’s mirror theorem using a new version of the equivariant quantum Seidel map which applied to big equivariant quantum cohomology [Iri17]. Givental’s mirror theorem describes the equivariant genus-zero Gromov-Witten invariants of a toric variety. Maulik and Okounkov’s work on the equivariant quantum Seidel map applies to smooth quasi-projective varieties X with a holomorphic symplectic structure which are equipped with the action of a reductive group G (such X are (real) symplectic manifolds with c1 = 0). Iritani’s work applies to smooth toric varieties X that admit a projective morphism to an affine variety and for which the action of the algebraic torus G on X satisfies some positivity condition. The definitions of the equivariant quantum Seidel map in this algebrogeometric context use equivariant virtual fundamental classes to count stable maps. The proofs of the intertwining relation in this context make heavy use of an algebrogeometric technique called virtual localisation, which reduces counting stable maps to counting only the G-fixed stable maps. In contrast, our results apply to closed or convex symplectic manifolds M which satisfy a monotonicity condition. We use a Borel model for equivariant quantum cohomology, which combines the Morse cohomology of our manifold M with the Morse AN INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 3 cohomology of the classifying space of S1. This model is preferable in our context because it readily extends to equivariant Floer theory. In our Borel model, we perturb the data on M using the classifying space of S1 to ensure that the moduli spaces are smooth manifolds. We therefore avoid using virtual fundamental classes to count stable maps, and instead just count the 0-dimensional moduli spaces. The G-fixed stable maps used by virtual localisation are not pseudoholomorphic curves for the perturbed data, however, so virtual localisation will not work in our context. To remedy this, we provide a new proof of the intertwining relation Theorem 1.2 which has a Morse-theoretic flavour: we construct an explicit 1-dimensional moduli space whose boundary gives the relation. We also introduce an equivariant Floer Seidel map, which is an isomorphism on equivariant Floer cohomology. We use a Borel model for equivariant Floer cohomology, which combines Morse theory on the classifying space of S1 with Floer theory on M. The S1-action ρ : S1 M M induces an S1-action on Hamiltonian orbits x : S1 M, given by × → → (θ x)(t)= ρθ(x(t θ)), (1.5) · − and it is with respect to this action that we define equivariant Floer cohomology. The equivariant Floer Seidel map is the identity map on the classifying space of S1 and it maps the Hamiltonian orbit x : S1 M to the Hamiltonian orbit (σ∗x)(t) = −1 → σt (x(t)). Much like the equivariant quantum Seidel map (1.1), the codomain of the equivariant Floer Seidel map is the equivariant Floer cohomology not for the action (1.5), but instead for the corresponding action induced by the pullback action σ∗ρ. The equivariant Floer Seidel map commutes with continuation maps, which means it induces a map on equivariant symplectic cohomology. Recall that the equivariant symplectic cohomology of M is the direct limit of equivariant Floer cohomology as the slope of the Hamiltonian function increases (see Section 4.3). Theorem 1.3. There is an equivariant Floer Seidel map on equivariant symplectic cohomology ∗ ∗+2I(σe) EF e : ESH (M) ESH ∗ (M) (1.6) Sσ ρ → σ ρ which is a Λ Z[u]-module isomorphism. The diagram ⊗ ∗ EQSσe ∗+2I(σe) b EQHρ (M) EQHσ∗ρ (M) (1.7) ∗ EF Sσe ∗+2I(σe) ESHρ (M) ESHσ∗ρ (M) commutes, where the vertical arrows denote equivariant c∗ maps. Equivariant symplectic cohomology has attracted attention in recent years because, while similar to (non-equivariant) symplectic cohomology, it possesses a number of dif- ferent properties. Chiefly, it distinguishes Hamiltonian orbits with different stabilizer groups more readily than does its non-equivariant counterpart. This is useful be- cause these different orbits have different geometric significance (for certain choices of Hamiltonians). For example, the constant orbits (with stabilizer group S1) capture 4 TODD LIEBENSCHUTZ-JONES the topology of M. Zhao restricted to the constant orbits by localising the ring Z[u], and obtained the isomorphism −1 ∗ −1 ∗ Q[u, u ] u ESH (M) = Q[u, u ] Z H (M) (1.8) ⊗Z[ ] ∼ ⊗ for completions of Liouville domains [Zha19, Theorem 1.1]. This is unlike (non- equivariant) symplectic cohomology, which vanishes for subcritical Stein manifolds (the vanishing follows from [Cie02, Theorem 1.1, part 1] and SH∗(CN ) = 0 [Oan04, Section 3]). On the other hand, the nonconstant orbits (with finite stablizer groups) capture the Reeb dynamics. We can restrict to the nonconstant orbits by looking at the positive part of ESH∗(M). Bourgeois and Oancea showed that the positive part of equivari- ant symplectic homology is isomorphic to linearized contact homology [BO17] while Gutt used it to distinguish nonisomorphic contact structures on spheres [Gut17, The- orem 1.4]. In fact, McLean and Ritter used ESH∗(M) to classify orbits with different finite stablizer groups in their new proof of the McKay correspondence [MR18]. One downside of equivariant symplectic cohomology is that it lacks an interesting algebraic structure. (Non-equivariant) symplectic cohomology has the pair-of-pants product, which equips it with a graded-commutative, associative and unital ring struc- ture.
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