An intertwining relation for equivariant Seidel maps
aa r X i v : . [ m a t h . S G ] O c t AN INTERTWINING RELATION FOREQUIVARIANT 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 quantumcohomology. We extend their definitions to S -equivariant Floer cohomology and S -equivariant quantum cohomology based on a construction of Maulik and Ok-ounkov. The S -action used to construct S -equivariant Floer cohomology changesafter applying the equivariant Seidel map (a similar phenomenon occurs for S -equivariant quantum cohomology). We show the equivariant Seidel map on S -equivariant quantum cohomology does not commute with the S -equivariant quan-tum product, unlike the standard Seidel map. We prove an intertwining relationwhich completely describes the failure of this commutativity as a weighted versionof the equivariant Seidel map. We will explore how this intertwining relationshipmay be interpreted using connections in an upcoming paper. We compute theequivariant Seidel map for rotation actions on the complex plane and on complexprojective space, and for the action which rotates the fibres of the tautological linebundle over projective space. Through these examples, we demonstrate how equi-variant Seidel maps may be used to compute the S -equivariant quantum productand S -equivariant symplectic cohomology. Introduction
Equivariant quantum Seidel maps.
An important invariant of a closed sym-plectic manifold M is its quantum cohomology QH ∗ ( M ) . A useful tool to computequantum cohomology is the (quantum) Seidel map Q S e σ : QH ∗ ( M ) → QH ∗ +2 I ( e σ ) ( M ) (1.1)associated to a Hamiltonian S -action σ on M [Sei97]. For monotone toric manifolds,these maps recover the Batyrev presentation of QH ∗ ( M ) [MT06, Section 5]. Thequantum Seidel map is computed by counting sections of a clutching bundle over P with fibre M , however many properties of the Seidel map are proved easily using anequivalent construction in Floer cohomology.In recent years, equivariant quantum cohomology has attracted attention. For us, equivariant will always mean S -equivariant. The equivariant quantum cohomology EQH ∗ ρ ( M ) corresponding to the S -action ρ is a ring with equivariant quantum prod-uct ∗ ρ . Maulik and Okounkov found that the equivariant quantum product maybe nonclassical for some quiver varieties [MO19] even though the (non-equivariant) Date : October 8, 2020.
Correspondence : [email protected]. quantum product is classical for deformation reasons. This indicates the equivari-ant quantum product records pseudoholomorphic spheres which are otherwise notdetected.In addition to being a ring,
EQH ∗ ρ ( M ) is a module over the Novikov ring Λ and isequipped with a geometric Z [ u ] -module structure denoted ⌣ . We extend the quantumSeidel maps to the equivariant setting with the following theorem. Theorem 1.1.
There is an equivariant quantum Seidel map EQ S e σ : EQH ∗ ρ ( M ) → EQH ∗ +2 I ( e σ ) σ ∗ ρ ( M ) (1.2) which is a Λ ⊗ Z [ u ] -module homorphism. Here, the codomain is equipped with thepullback action σ ∗ ρ = σ − ρ. (1.3)In their paper [MO19], Maulik and Okounkov study quiver varieties, and showan intertwining relation which describes how EQ S e σ intertwines with the equivariantquantum product. Their result holds more generally for smooth quasi-projective va-rieties X with a holomorphic symplectic structure which are equipped with an actionby a reductive group G (such X are (real) symplectic manifolds with c = 0 ). Further-more, they use a Cartan model for EQH ∗ ( X ) , which combines de Rham cohomologyon X with polynomials on the Lie algebra of G .The main result of this paper is an analogous intertwining relation in the symplecticcontext. This means we work instead with (real) symplectic manifolds M which eitherare monotone or satisfy c = 0 , and we work with a Borel model for EQH ∗ ( M ) . TheBorel model is preferable in our context because it extends to Floer theory. Theorem 1.2 (Intertwining relation) . The equation EQ S e σ ( x ∗ ρ α + ) − EQ S e σ ( x ) ∗ σ ∗ ρ α − = u ⌣ EQ S e σ,α ( x ) (1.4) holds for all x ∈ EQH ∗ ρ ( M ) . Here, α + ∈ EH ∗ ρ ( M ) and α − ∈ EH ∗ σ ∗ ρ ( M ) are twoequivariant cohomology classes which are related via the clutching bundle and EQ S e σ,α : EQH ∗ ρ ( M ) → EQH ∗ +2 I ( e σ )+ | α ± | − σ ∗ ρ ( M ) (1.5) is a map defined in Section 7.4.1. Maulik and Okounkov use an algebrogeometric technique called equivariant local-isation to prove their intertwining relation. In contrast, we construct an explicit1-dimensional moduli space whose boundary gives the relation.1.2.
Equivariant Floer Seidel maps.
In the non-compact setting, many of thesetechniques still apply for the large class of convex symplectic manifolds. This classincludes cotangent bundles and Liouville manifolds, as well as non-exact symplec-tic manifolds such as O P N ( − . The symplectic cohomology SH ∗ ( M ) of a convexsymplectic manifold M is an important invariant for understanding the symplectictopology of these spaces [Sei08].The Floer Seidel map is an isomorphism F S e σ : SH ∗ ( M ) → SH ∗ +2 I ( e σ ) ( M ) (1.6) N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 3 which fits into the commutative diagram QH ∗ ( M ) QH ∗ +2 I ( e σ ) ( M ) SH ∗ ( M ) SH ∗ +2 I ( e σ ) ( M ) Q S e σ c ∗ c ∗ F S e σ (1.7)whose vertical arrows are the natural c ∗ maps. The quantum Seidel map Q S e σ isnot an isomorphism for general convex manifolds [Rit14], even though it is alwaysan isomorphism for closed manifolds. Nonetheless, the Seidel maps can be used tocompute symplectic cohomology [Rit14, Rit16].Equivariant symplectic cohomology ESH ∗ ( M ) is increasingly an invariant of in-terest for convex symplectic manifolds M . Unlike for SH ∗ ( M ) , there is mountingevidence that, by using ESH ∗ ( M ) , we can obtain information about Hamiltonian or-bits with different stablizer groups. This is useful because these different orbits havedifferent geometric significance (for certain choices of Hamiltonians). For example,the constant orbits (with stabilizer group S ) correspond to the cohomology of M ,and we can localise the ring Z [ u ] to see this: the isomorphism Q [ u , u − ] ⊗ Z [ u ] ESH ∗ ( M ) ∼ = Q [ u , u − ] ⊗ Z H ∗ ( M ) (1.8)holds when M is a completion of a Liouville domain [Zha19, Theorem 1.1]. One mightcontrast this with the vanishing of SH ∗ ( M ) when such an M is a subcritical Steinmanifold (this follows from [Cie02, Theorem 1.1, part 1] and SH ∗ ( C N ) = 0 [Oan04,Section 3]).On the other hand, the significance of the nonconstant orbits (with finite stablizergroups) is to the Reeb dynamics, and this is recovered by looking at the positive part of ESH ∗ ( M ) . Bourgeois and Oancea showed that the positive part of equivariant sym-plectic homology is isomorphic to linearized contact homology [BO17] while Gutt usedit to distinguish nonisomorphic contact structures on spheres [Gut17, Theorem 1.4].We construct a variant of ESH ∗ ( M ) which incorporates an S -action on M . Theequivariant symplectic cohomology corresponding to the S -action ρ on M is denoted ESH ∗ ρ ( M ) . Theorem 1.3.
There is an equivariant (Floer) Seidel map on equivariant symplecticcohomology EF S e σ : ESH ∗ ρ ( M ) → ESH ∗ +2 I ( e σ ) σ ∗ ρ ( M ) (1.9) which is a Λ ⊗ Z [ u ] -module isomorphism. The diagram EQH ∗ ρ ( M ) EQH ∗ +2 I ( e σ ) σ ∗ ρ ( M ) ESH ∗ ρ ( M ) ESH ∗ +2 I ( e σ ) σ ∗ ρ ( M ) EQ S e σ EF S e σ (1.10) commutes, where the vertical arrows denote equivariant c ∗ maps. TODD LIEBENSCHUTZ-JONES
Examples.
We compute the equivariant quantum Seidel map for the followingthree spaces: the complex plane, complex projective space and the total space ofthe tautological line bundle O P N ( − . Through these examples, we demonstrate howthe map may be used to compute equivariant quantum cohomology and equivariantsymplectic cohomology.For the complex plane, we deduce the equivariant quantum Seidel map from theequivariant Floer complex which was computed in [Zha19, Section 8.1]. In both otherexamples, we find the map by directly computing some coefficients and deducing therest by repeated application of the intertwining relation (1.4).We use the parameterisation S = R / Z throughout. Example 1.4 (Complex plane) . The complex plane has a Hamiltonian circle action σ θ ( z ) = e πiθ z . The origin C ∈ C is the unique fixed point of σ . Thus C is equivari-antly contractible and its symplectic form is globally exact, so for any nonnegative r ,we have EQH ∗ σ − r ( C ) ∼ = Z [ u ] . The equivariant quantum Seidel map is EQ S σ : EQH ∗ σ − r ( C ) → EQH ∗ +2 σ − ( r +1) ( C )1 ( r + 1) u . (1.11) Example 1.5 (Projective space) . The complex projective space P n with its Fubini-Study symplectic form has a Hamiltonian action σ given by θ · [ z : z : · · · : z n ] = [ z : e πiθ z : · · · : e πiθ z n ] , (1.12)for any ( z , z , · · · , z n ) ∈ S n +1 ⊂ C n +1 . The σ -invariant Morse function f P n ([ z : · · · : z n ]) = P nk =0 k | z k | has critical points e , . . . , e n , where e k has Morse index k .For any nonnegative integer r , we have EQH ∗ σ − r ( P n ) ∼ = Z [ q ± ] b ⊗ Z [ u ] h e , . . . , e n i , (1.13)where the Novikov variable q is a formal variable of degree n + 1) . The equivariantquantum Seidel map is the map EQ S e σ : EQH ∗ σ − r ( P n ) → EQH ∗ +2 nσ − ( r +1) ( P n ) (1.14)given by e n X l =0 ( r + 1) n − l u n − l e l ,e k k − X l =0 ( r + 1) k − − l q u k − − l e l , k = 1 , . . . , n. (1.15)One way to compute the non-equivariant quantum product on P n is to compute thequantum Seidel map and deduce the product from the fact that this map intertwinesthe product (this is a key idea behind the combinatorial algorithm in [MT06, Sec-tion 5]). In our computations behind Example 1.5, we demonstrate this is true alsoin the equivariant case, so we use the equivariant quantum Seidel map to recover theequivariant quantum product. The equivariant quantum product on P n is computedusing different methods in [Mih06]. N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 5
Example 1.6 (Tautological line bundle) . The total space O P n ( − of the tautologicalline bundle over projective space P n is a monotone convex symplectic manifold [Rit14,Section 7]. The fibres are symplectic submanifolds, and the circle action σ whichrotates the fibres is a linear Hamiltonian circle action. The action σ fixes the imageof the zero section Z , and, like Example 1.4, O P n ( − equivariantly contracts onto Z with the trivial circle action. We use the same Morse function on Z ∼ = P n as forExample 1.5, so we have EQH ∗ σ − r ( O P n ( − ∼ = Z [ q ± ] b ⊗ Z [ u ] h e , . . . , e n i , (1.16)where the Novikov variable q now has degree n (so Λ = Z [ q ± ] is the Novikov ring).The equivariant quantum Seidel map is the map EQ S e σ : EQH ∗ σ − r ( O P n ( − → EQH ∗ +2 σ − ( r +1) ( O P n ( − (1.17)given by e k ( − e k +1 + ( r + 1) u e k k < n,qe + ( r + 1) u e n − ( r + 1) u qe k = n. (1.18)Unlike the non-equivariant quantum Seidel map on O P n ( − , this is an injective map.From (1.18), we derive the equivariant quantum product as given by e ∗ σ − r e k = ( e k +1 k < n, − qe + r u qe k = n. (1.19)This product has a term which is not detected by either quantum cohomology orequivariant cohomology: it exists only in equivariant quantum cohomology.We can use (1.18) to find the equivariant symplectic cohomology of O P n ( − usingan argument of Ritter [Rit14, Theorem 22]. We deduce that the equivariant symplecticcohomology ESH ∗ σ − r ( O P n ( − is a Λ b ⊗ Z [ u ] -module which is not finitely generatedand which satisfies EQH ∗ σ − r ( O P n ( − ( ESH ∗ σ − r ( O P n ( − ( (cid:0) EQH ∗ σ − r ( O P n ( − (cid:1) Z [ u ] \{ } , (1.20)where the left inclusion is the equivariant c ∗ map and the module on the right isthe localisation by Z [ u ] \ { } . If we perform this localisation to all three modules,we find that the localised equivariant symplectic cohomology is isomorphic to thelocalised equivariant quantum cohomology, which is equivalent to a version of Zhao’slocalisation theorem (1.8).We express ESH ∗ σ − r ( O P n ( − as a Λ b ⊗ Z [ u ] -submodule of the localised equivariantquantum cohomology with an explicit set of generators. The generators are definedby a recurrence relation induced by (1.18).1.4. Equivariant Seidel maps as shift operators.
In this section, we briefly de-scribe some ideas from our upcoming paper [LJ]. The work is based on the shiftoperator interpretation of the equivariant quantum Seidel map [MO19, Section 1.4]and is motivated by Seidel’s q -connection on equivariant Floer cohomology [Sei18, Sec-tions 2a and 5].We change our Novikov ring so its elements are certain formal sums of mono-mials q A where A ∈ H ( M ) and q is a formal variable. We also change to using S u × S σ -equivariant objects, where the first copy of S corresponds to the rotation of TODD LIEBENSCHUTZ-JONES the clutching bundle and has the formal variable u associated to it, and the secondcorresponds to a chosen Hamiltonian action σ . We fix a lift e σ and an assignment H ( M ) → EQH ( M ) which is compatible with the lift e σ that we denote by α e α .Define the map ∇ : H ( M ) × EQH ∗ ( M ) → EQH ∗ +2 ( M ) by ∇ α ( q A x ) = α ( A ) u q A x − e α ∗ q A x. (1.21)The first term resembles an algebraic differentiation operation, whereas the second in-corporates the equivariant quantum product. This map is called a connection becauseit behaves like differentiation algebraically.We will prove in [LJ] that Theorem 1.2 is equivalent to the commutativity of S σ = EQ S e σ with ∇ α , so that we have [ S σ , ∇ α ] = 0 . Strictly speaking, the map S σ is thecomposition of the S u × S σ -equivariant quantum Seidel map with a second map whichundoes the change in S u × S σ -action. Roughly, the first term in (1.21) takes the placeof the right-hand side of the intertwining relation (1.4).The map ∇ together with the map S σ is a flat difference-differential connection ,and, in this context, the map S σ is a shift operator . The equation [ S σ , ∇ α ] = 0 formspart of the algebraic condition of being flat . This terminology is motivated by flatconnections on bundles, however these are only algebraic conditions.The main advantage of this approach is that it extends to equivariant Floer coho-mology. While there is no equivariant pair-of-pants product, there is a map ∇ Floer which is analogous to the connection ∇ defined above, and moreover it commutes withthe equivariant Floer Seidel map.A second advantage is that this algebraic construction is readily extended to S u × ( S ) k -equivariant objects, given a Hamiltonian ( S ) k -action on M . This makes theapproach particularly powerful for toric manifolds.1.5. Outline.
We give an overview of the background material for our work in thenext section (Section 2), as well as giving more information about and intuition forour constructions and results.In Section 3, we introduce the Floer Seidel maps in detail, clarifying our assump-tions and conventions for symplectic cohomology. We introduce equivariant Floertheory and its associated module structures in Section 4 before defining the equivari-ant Floer Seidel map in Section 5.In Section 6, we introduce the quantum Seidel map (Section 6.2) and equivariantquantum cohomology (Section 6.3). We define the equivariant quantum Seidel mapin Section 7 and prove the intertwining relation in Section 7.4.Section 8 contains our three example calculations.1.6.
Acknowledgements.
The author would like to thank his supervisor AlexanderRitter for his guidance, support and ideas. The author wishes to thank Paul Seidel forhis ideas, particularly in relation to Section 1.4. The author thanks Nicholas Wilkinsand Filip Živanović for many constructive discussions and their feedback. The authorgratefully acknowledges support from the EPSRC grant EP/N509711/1. This workwill form part of his DPhil thesis. The meaning of lift in Section 1.4 is different to the one in the rest of this paper because theNovikov ring has changed.
N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 7 Overview
Background.
Seidel maps on closed symplectic manifolds.
Let M be a closed monotone sym-plectic manifold and let σ be a Hamiltonian circle action on M . In [Sei97], Seideldefined a pair of automorphisms associated to σ , one on Floer cohomology and oneon quantum cohomology. To distinguish between these maps, we call the former the Floer Seidel map and the latter the quantum Seidel map . Definition 2.1 (Floer Seidel map) . Recall that the Floer cochain complex
F C ∗ ( M ; H ) associated to the time-dependent Hamiltonian function H : S × M → R is freely-generated by the 1-periodic Hamiltonian orbits of H over the Novikov ring Λ . Givena Hamiltonian orbit x : S → M of H , the pullback orbit σ ∗ x given by ( σ ∗ x )( t ) = σ − t ( x ( t )) (2.1)is a Hamiltonian orbit of the pullback Hamiltonian σ ∗ H . Moreover, the assignment x σ ∗ x is a bijection between the orbits of H and the orbits of σ ∗ H . This assignmentcan be upgraded to an isomorphism of Floer cochain complexes F S e σ : F C ∗ ( M ; H ) → F C ∗ +2 I ( e σ ) ( M ; σ ∗ H ) (2.2)by using a lift e σ of the circle action σ to keep track of the information recorded by theNovikov ring. The quantity I ( e σ ) is a Maslov index associated to e σ (see Section 3.3for details).The Floer Seidel map F S e σ is the map induced on Floer cohomology by (2.2). Itsatisfies F S e σ ( a ∗ b ) = a ∗ F S e σ ( b ) , where ∗ denotes the pair-of-pants product. Definition 2.2 (Quantum Seidel map) . Define a clutching bundle E over S withfibre M as follows. The sphere is the union of its northern hemisphere D − and itssouthern hemisphere D + . Each hemisphere is isomorphic to a closed unit disc, andthe two hemispheres are glued along the equator S = ∂ D − = ∂ D + to get the sphere.The clutching bundle is the union of the trivial bundles D − × M and D + × M , gluedalong the equator by the relation ∂ D − × M ∋ ( t, m ) ↔ ( t, σ t ( m )) ∈ ∂ D + × M. (2.3)Thus we twist the bundle by σ when passing from the northern hemisphere to thesouthern hemisphere.The quantum Seidel map is an isomorphism Q S e σ : QH ∗ ( M ) → QH ∗ +2 I ( e σ ) ( M ) which counts pseudoholomorphic sections of this clutching bundle. It satisfies Q S e σ ( a ∗ b ) = a ∗ Q S e σ ( b ) , where ∗ denotes the quantum product. It follows that the quantumSeidel map is given by quantum product multiplication by the invertible element Q S e σ (1) . This element is the Seidel element of e σ .Using the PSS isomorphism to identify QH ∗ ( M ) with Floer cohomology, the FloerSeidel map and the quantum Seidel map are identified. Roughly, each hemisphere inthe clutching bundle corresponds to a PSS map, and the twisting along the equatorcorresponds to the Floer Seidel map.More generally, Seidel maps may be defined for loops in the Hamiltonian sym-plectomorphism group Ham( M ) based at the identity Id M . (Technically, we must TODD LIEBENSCHUTZ-JONES use a cover ] Ham( M ) , though we omit details here.) The Seidel maps are homotopyinvariants, so that the assignment π ( ] Ham( M ) , Id M ) → QH ∗ ( M ) × e σ Q S e σ (1) (2.4)is a group homomorphism. The map (2.4) is the Seidel representation .Here is a selection of results whose proofs use Seidel maps. It is by no meanscomplete. • An algorithmic and combinatorial computation of quantum cohomology fromthe moment polytope of a toric symplectic manifold [MT06, Section 5]. • An isomorphism QH ∗ ( M ) ∼ = QH ∗ (( P ) n ) whenever M admits a semifree circleaction with nonempty isolated fixed point set [Gon06]. • Computation of the Gromov width and Hofer-Zehnder capacity in terms of thevalues of the Hamiltonian of σ on fixed components, under the assumption that σ is semi-free (and its maximal fixed locus is a point) [HS17].2.1.2. Convex manifolds.
The symplectic manifold ( M, ω ) is convex (sometimes con-ical in the literature) if it is symplectomorphic to Σ × [1 , ∞ ) with symplectic form ω = d( Rα ) away from a relatively compact subset M ⊆ M . Here, Σ is a closedcontact manifold with contact form α and R is the coordinate of [1 , ∞ ) .The cotangent bundle of a closed manifold is convex, and more generally so is thecompletion of a Liouville domain. In addition, there are many examples of convexsymplectic manifolds whose symplectic forms are not globally exact, such as the totalspace of the line bundle O P N ( − .In the region Σ × [1 , ∞ ) , the convex end , the symplectic form is exact and it growsinfinitely big as R → ∞ . Moreover the formula ω = d( Rα ) = d R ∧ α + R d α decomposesthe tangent space into the two-dimensional subspace R ∂ R + R X α and the contactstructure ker α . (Here, X α is the Reeb vector field, which is characterised by α ( X α ) =1 and d α ( X α , · ) = 0 .) These properties of ω constrain the Hamiltonian dynamics ofa Hamiltonian function H to a compact region of the form M ∪ (Σ × [1 , R ]) so longas H = λR + µ is a linear function of R outside this region (some further conditionsare also required, see Section 3.2).Once the Hamiltonian dynamics are restricted to a compact region, the machineryof Floer cohomology applies, so we can define Floer cohomology for convex symplecticmanifolds (which satisfy an appropriate monotonicity assumption) and Hamiltonianfunctions which are linear outside a compact region. Floer cohomology depends onthe slope λ of the Hamiltonian, unlike for closed manifolds whose Floer cohomologyis entirely independent of the Hamiltonian. Symplectic cohomology SH ∗ ( M ) is thedirect limit of Floer cohomology as the slope λ tends to infinity. The PSS maps areisomorphisms between the quantum cohomology QH ∗ ( M ) and the Floer cohomologyof a Hamiltonian with sufficiently small (positive) slope.2.1.3. Seidel maps on convex symplectic manifolds.
In [Rit14], Ritter extended theconstruction of Seidel maps to Hamiltonian circle actions σ on convex symplecticmanifolds, so long as the Hamiltonian K σ of σ is linear, with nonnegative slope, outsidea compact region. The Floer Seidel map is an isomorphism between F H ∗ ( M ; H ) and N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 9
F H ∗ +2 I ( e σ ) ( M ; σ ∗ H ) as for closed manifolds, however the pullback Hamiltonian σ ∗ H may have a different slope to H . The limit of the Floer Seidel maps as the slope λ of H tends to infinity induces an automorphism F S e σ of symplectic cohomology SH ∗ ( M ) .The quantum Seidel map on a convex symplectic manifold is not necessarily anisomorphism, but instead is merely a module homomorphism. Ritter computed thequantum Seidel maps on the total spaces of the line bundles O P N ( − . His calculationsdemonstrate that, even in elementary examples, the quantum Seidel map may fail tobe injective and surjective.The fact that the Floer Seidel map is an isomorphism even though the quantumSeidel map may fail to be injective and surjective is explained by the following com-mutative diagram. The non-bijectivity of the quantum Seidel map corresponds to thenon-bijectivity of the continuation map. QH ∗ ( M ) QH ∗ +2 I ( e σ ) ( M ) F H ∗ ( M ; H small ) F H ∗ +2 I ( e σ ) ( M ; H small ) F H ∗ +2 I ( e σ ) ( M ; σ ∗ H small ) Q S e σ PSS map ∼ = F S e σ ∼ = PSS map ∼ = continuation map (2.5)The Hamiltonian H small has small (positive) slope, but the Hamiltonian σ ∗ H small typically has negative slope.When the slope of K σ is positive, the quantum Seidel map may be used to computesymplectic cohomology [Rit14, Theorem 22]. This approach explicitly computes sym-plectic cohomology for various line bundles of the form O P N ( − k ) [Rit14, Theorem 5]and provides an algorithm to compute the symplectic cohomology of a Fano toricnegative line bundle using its moment polytope [Rit16, Theorem 1.5].2.1.4. Equivariant Floer cohomology.
Recall that Floer cohomology is inspired by theMorse cohomology of the loop space L M = (cid:8) contractible x : S → M (cid:9) . The loopspace L M has a canonical circle action which rotates the loops. Equivariant Floercohomology is analogously inspired by the equivariant Morse cohomology of L M withthis rotation action.Viterbo introduced the first version of equivariant Floer cohomology [Vit96, Sec-tion 5], and later Seidel introduced a second version [Sei08, Section 8b]. Bourgeoisand Oancea showed these different versions are equivalent [BO17, Proposition 2.5].We use Seidel’s approach, since its analysis is far simpler. For more information onthe history, see [BO17, Section 2]. Our conventions differ from those in the literaturein three important ways: we use a different relation for the Borel homotopy quotient(Remark 4.3), we use a geometric module structure (Section 4.4.2) and we incorporateactions on the manifold M .Denote by S ∞ the limit of the inclusions S ֒ → S ֒ → S · · · , where S k − is thoughtof as the unit sphere in C k . It is a contractible space with a free circle action. Givenany space X with a circle action, its Borel homotopy quotient , denoted S ∞ × S X , is the quotient of S ∞ × X by the relation ( w, θ · x ) ∼ ( θ · w, x ) for all θ ∈ S . Equivariantcohomology is the cohomology of S ∞ × S X .Informally, the equivariant Morse cohomology of the Borel homotopy quotient maybe obtained by doing Morse theory on S ∞ and on X , and quotienting the modulispaces by the induced relation. Similarly, the equivariant Floer cohomology of M isobtained by doing Morse theory on S ∞ and Floer theory on M , and quotienting themoduli spaces by the induced relation.The equivariant Floer cochain complex is generated by ∼ -equivalence classes [( w, x )] ,where w is a critical point in S ∞ and x is a 1-periodic Hamiltonian orbit of a time-dependent Hamiltonian H eq w . The function H eq : S ∞ × S × M → R must satisfy theidentity H eq θ · w,t ( m ) = H eq w,t + θ ( m ) (2.6)in order that the relation ∼ make sense on the pairs ( w, x ) ∈ S ∞ × L M . Such ∼ -equivalence classes are equivariant Hamiltonian orbits .Similarly, consider ∼ -equivalence classes [( v, u )] where v : R → S ∞ is a Morse tra-jectory and u : R × S → M is a Floer solution of the s -dependent Hamiltonian H eq v ( s ) .The differential on the equivariant Floer cochain complex counts these equivalenceclasses modulo the free R -action of translation.The cohomology of S ∞ /S is Z [ u ] , where u is a formal variable in degree 2. Forcertain choices of Floer data, the equivariant Floer cochain complex is F C ∗ ( M ) b ⊗ Z [ u ] with differential d eq = d + o ( u ) , where d is the (non-equivariant) Floer differential.The resulting equivariant Floer cohomology EF H ∗ ( M ; H eq ) is a graded moduleover the Novikov ring Λ . Moreover, it has a Z [ u ] -module structure coming fromthe description of the cochain complex above. In fact, it has another Z [ u ] -modulestructure given by an equivariant cup product type construction, which we call the geometric module structure and denote ⌣ .Much like in the non-equivariant setup in Section 2.1.2, EF H ∗ ( M ; H eq ) only de-pends on the slope of H eq , and the limit of EF H ∗ ( M ; H eq ) as the slopes increase toinfinity is the equivariant symplectic cohomology ESH ∗ ( M ) . Equivariant PSS mapsidentify EF H ∗ ( M ; H eq ) and EQH ∗ ( M ) when H eq has sufficiently small (positive)slope. Here, EQH ∗ ( M ) is the equivariant quantum cohomology of M for the trivialidentity circle action on M .For us, we have a Hamiltonian action σ acting on M . This means that the loopspace L ( M ) has another canonical circle action given by θ · ( t x ( t ) ) = ( t σ θ ( x ( t − θ )) ) . (2.7)This action combines the rotation action on the domain of the loops with the action σ on the target space M . All of the equivariant constructions above generalise to thenew action, and we denote these versions with a subscript σ . For example, ESH ∗ σ ( M ) is the equivariant symplectic cohomology corresponding to (2.7). We may need infinitely-many powers of u so we must use some kind of completed tensor product.Some authors use Z J u K , however we use a slightly smaller cochain complex (4.14). With our approach, EF H ∗ ( M ) is graded in the conventional sense. N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 11
Equivariant Seidel maps.
In this paper, we define new variants of the FloerSeidel map and the quantum Seidel map on equivariant Floer cohomology and equi-variant quantum cohomology respectively. We prove a number of initial properties ofthese maps, which for the most part are just like the non-equivariant maps. The mainexception is that the equivariant quantum Seidel map and the equivariant quantumproduct do not commute (Theorem 1.2).The extension of the Floer Seidel map to the equivariant setting is nontrivial becausethe equivariant Floer Seidel map pulls back the action (2.7). A similar phenomenonoccurs with the equivariant quantum Seidel map. No new analysis is required, however,since our constructions only use an S ∞ -parameterised version of the analysis used todefine the Seidel maps in [Rit14].2.2.1. Definitions.
Let M be a convex symplectic manifold which is either monotoneor whose first Chern class vanishes. Let σ and ρ be two commuting Hamiltonian circleactions on M whose Hamiltonian functions are linear outside a compact subset of M .Assume the Hamiltonian K σ of σ has nonnegative slope.Recall the cochain complex for equivariant Floer cohomology EF H ∗ ρ ( M ; H eq ) isgenerated by equivariant Hamiltonian orbits, which are certain equivalence classes ofpairs ( w, x ) ∈ S ∞ × L M under the equivalence relation ( θw, x ( t )) ∼ ρ ( w, ρ θ ( x ( t − θ ))) .In order for the map [( w, x )] [( w, σ ∗ x )] to be well-defined, the equivalence class [( w, σ ∗ x )] must be considered with the relation ∼ σ ∗ ρ , which corresponds to the pull-back circle action σ ∗ ρ = σ − ρ . Once we account for the change in the action, thedefinition of the Floer Seidel map extends naturally. Definition 2.3 (Equivariant Floer Seidel map) . The equivariant Floer Seidel map isthe map EF S e σ : EF H ∗ ρ ( M ; H eq ) → EF H ∗ +2 I ( e σ ) σ ∗ ρ ( M ; σ ∗ H eq ) (2.8)given by [( w, x )] [( w, σ ∗ x )] on equivariant Hamiltonian orbits. It is a Λ -moduleisomorphism on the cochain complex which is compatible with algebraic Z [ u ] -modulestructure. On cohomology, it is compatible with the geometric Z [ u ] -module struc-ture and with continuation maps. Under the limit as the slope of the equivariantHamiltonians tends to infinity, the maps induce a well-defined isomorphism EF S e σ : ESH ∗ ρ ( M ) → ESH ∗ +2 I ( e σ ) σ ∗ ρ ( M ) . (2.9)We discuss how the equivariant Floer Seidel map is compatible with filtrations andthe Gysin sequence in Section 5.2.For the quantum Seidel map, we put an action on the clutching bundle which liftsthe natural rotation action of the sphere and which restricts to the action ρ on thefibre above the south pole. The “twisting by σ ” across the equator forces the actionon the fibre above the north pole to be σ ∗ ρ . With this action on the clutching bundle,the quantum Seidel map extends naturally to the equivariant setup. Our construction works for closed manifolds as well, but we only discuss the convex case here tosimplify the discussion. We treat both cases in the rest of the paper (see Remark 3.3).
Definition 2.4 (Equivariant quantum Seidel map) . The equivariant quantum Seidelmap is the map EQ S e σ : EQH ∗ ρ ( M ) → EQH ∗ +2 I ( e σ ) σ ∗ ρ ( M ) (2.10)which counts equivalence classes [( w, u )] , where w ∈ S ∞ and u is a pseudoholomorphicsection of the clutching bundle. It is a Λ -module homomorphism which is compatiblewith the algebraic and geometric Z [ u ] -module structures.2.2.2. Properties.
Equivariant PSS maps identify equivariant quantum cohomologywith the equivariant Floer cohomology of an equivariant Hamiltonian of sufficientlysmall (positive) slope.
Proposition 2.5 (Compatibility with PSS maps) . An analogous commutative dia-gram to (2.5) holds for the equivariant maps:
EQH ∗ ρ ( M ) EQH ∗ +2 I ( e σ ) σ ∗ ρ ( M ) EF H ∗ ρ ( M ; H eq, small ) EF H ∗ +2 I ( e σ ) σ ∗ ρ ( M ; H eq, small ) EF H ∗ +2 I ( e σ ) σ ∗ ρ ( M ; σ ∗ H eq, small ) EQ S e σ equivariantPSS map ∼ = EF S e σ ∼ = equivariantPSS map ∼ = equivariantcontinuation map (2.11)Here, H eq, small satisfies an equivariance condition analogous to (2.6) which corre-sponds to the action ρ . In contrast, σ ∗ H eq, small and H eq, small satisfy the equivariancecondition which corresponds to the action σ ∗ ρ . Both H eq, small and H eq, small havesmall positive slope, but σ ∗ H eq, small has negative slope in general.This proposition implies (1.10).The maps for different actions compose exactly like the non-equivariant case asfollows. Proposition 2.6 (Composition of multiple actions) . Let ρ , σ and σ be commutingHamiltonian circle actions whose Hamiltonians are linear outside a compact subset of M . Suppose the Hamiltonians of σ and σ have nonnegative slope. The followingdiagrams commute. (We have omitted grading, the Hamiltonians and M from thenotation.) EQH σ ∗ ρ EF H σ ∗ ρ EQH ρ EF H ρ EQH ( σ σ ) ∗ ρ EF H ( σ σ ) ∗ ρEQ S f σ EF S f σ EQ S f σ EQ S f σ f σ EF S f σ EF S f σ f σ (2.12) N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 13
The quantum Seidel map intertwines the quantum product. More precisely, given α ∈ QH ∗ ( M ) , the operations Q S e σ and ∗ α commute, where ∗ α is the operation givenby right-multiplication by α . Thus the relation [ Q S e σ , ∗ α ] = 0 holds.The first complication for extending this relation to the equivariant setup is thatthe domain and codomain of EQ S e σ are different. We solve this by using the clutchingbundle to relate the domain and codomain. Let α ∈ EH ∗ ( E ) be an equivariantcohomology class of the clutching bundle E with respect to the action we put on it.The restriction of α to the fibre over the south pole is a class α + ∈ EH ∗ ρ ( M ) , whilethe restriction to the fibre over the north pole is a class α − ∈ EH ∗ σ ∗ ρ ( M ) .Having established this, we can ask whether the analogue of the intertwining relation [ Q S e σ , ∗ α ] = 0 holds in the equivariant setup. From the non-equivariant relation,we can immediately deduce any failure to commute is o ( u ) . The following theoremcharacterises the failure precisely. Theorem 2.7 (Intertwining relation, Theorem 1.2) . The equation EQ S e σ ( x ∗ ρ α + ) − EQ S e σ ( x ) ∗ σ ∗ ρ α − = u ⌣ EQ S e σ,α ( x ) (2.13) holds for all α ∈ EH ∗ ( E ) and x ∈ EQH ∗ ρ ( M ) , where EQ S e σ,α : EQH ∗ ρ ( M ) → EQH ∗ +2 I ( e σ )+ | α |− σ ∗ ρ ( M ) (2.14) is a map defined in Section 7.4.1. The product ∗ ρ is the equivariant quantum product for the action ρ , and the symbol ⌣ denotes the action of the geometric Z [ u ] -module structure. The map EQ S e σ,α counts (equivariant) pseudoholomorphic sections of the clutching bundle which areweighted by the (equivariant) class α . This weighting is easiest to understand when α ∈ EH ( E ) is degree 2. In this case, the map EQ S e σ,α counts exactly the samepseudoholomorphic sections u as the map EQ S e σ , but with the weight α ( u ∗ ( (cid:2) S (cid:3) )) .For the full definition, see Section 7.4.1.In the realm of algebraic geometry, Maulik and Okounkov defined an equivariantquantum Seidel map in [MO19, Section 8] (they call it a shift operator for reasonsdescribed in Section 1.4). Their intertwining property [MO19, Proposition 8.2.1] isanalogous to our intertwining relation (Theorem 1.2), though our proof is new.In the realm of algebraic geometry, Maulik and Okounkov proved an analogousintertwining relation for quiver varieties [MO19, Proposition 8.2.1], and our formularesembles theirs when α has degree 2. They prove the relation using virtual local-ization [Hor03, Chapter 27]. This technique converts counting sections into countingonly the sections which are invariant under the S -action on the clutching bundle.Any invariant section must be a constant section at a fixed point of the action σ on M , however invariant sections are allowed to bubble over the poles. The result is adecomposition of EQ S into three maps: B − ◦ F ◦ B + . The map B − counts bubblesover the north pole and B + counts bubbles over the south pole, both appropriatelymodified according to the virtual localization. The map F corresponds to the constantsection at a fixed point of the action σ .The intertwining relation is then proven for each of the three maps. For B − and B + , it is a consequence of standard relations corresponding to gravitational descendant invariants (namely, the divisor equation and the topological recursion relation [Hor03,Chapter 26]). For F , the intertwining relation is a topological result which relates α − and α + when both classes are restricted to the fixed locus of σ on M .In contrast, our proof has a Floer-theoretic flavour: we construct an explicit 1-dimensional moduli space whose boundary gives (2.13) on cohomology.To motivate our proof, consider first the following proof of the intertwining ofthe non-equivariant quantum Seidel map. Define a 1-dimensional moduli space ofpseudoholomorphic sections which intersect a fixed Poincaré dual α ∨ of α along afixed line of longitude L ⊂ S . The boundary of this moduli space occurs when theintersection point is at either pole. When it is at the south pole, we recover the term Q S ( x ∗ α ) , while at the north pole we get the term − Q S ( x ) ∗ α . Summing theseboundary components gives the equation Q S ( x ∗ α ) − Q S ( x ) ∗ α = 0 as desired.In the equivariant case, we must allow the line of longitude to vary with w ∈ S ∞ .This is because the intersection condition must be preserved by the equivalence relation ∼ . We fix an equivariant assignment of lines of longitude w L w for an invariantdense open set of w ∈ S ∞ . Note that a global assignment is not possible: the set oflines of longitude is isomorphic to S , however there are no equivariant maps S ∞ → S .For the equivariant 1-moduli space, we ask that the equivariant section [( w, u )] satisfies u ( z ) ∈ α ∨ for some z ∈ L w . As per the non-equivariant case, the two polesyield the two terms on the left-hand side of (2.13). The remaining term in (2.13)comes from a limit in which w exits the dense open set.The computation behind Example 1.5 verifies that the right-hand side of (2.13) isnonzero even in straightforward cases.3. Floer theory
In Section 3.1, we clarify the assumptions we place on our symplectic manifold.We proceed in Section 3.2 by defining Floer cohomology and symplectic cohomologyusing the same conventions as [Sei97, Rit14]. A more complete explanation of theconstruction of Floer cohomology may be found in [Sal97]. Next, we introduce theHamiltonian circle actions which yield the Seidel map in Section 3.3, and define theFloer Seidel map in Section 3.4.3.1.
Symplectic manifolds.
Let M be a n -dimensional smooth manifold with asymplectic form ω . For convenience, we assume throughout that M is nonempty andconnected. There are two additional conditions that we will impose on M . The firstestablishes a relationship between the cohomology class of ω and the first Chern class,and the second controls the behaviour of the symplectic form when the manifold isopen. Definition 3.1.
Denote by c ∈ H ( M ) the first Chern class of the symplectic vectorbundle ( T M, ω ) . The symplectic manifold M is nonnegatively monotone if either: • there is λ ≥ such that ω ( A ) = λc ( A ) for all A ∈ π ( M ) ; or • c ( A ) = 0 for all A ∈ π ( M ) . N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 15
By an analogous argument to the proof of Lemma 1.1 in [HS95], a symplecticmanifold is nonnegatively monotone if and only if the implication c ( A ) < ⇒ ω ( A ) ≤ (3.1)holds for all A ∈ π ( M ) . Such a manifold has the property that, for any compatiblealmost complex structure, all pseudoholomorphic curves have nonnegative first Chernnumber. Definition 3.2. A convex symplectic manifold is a symplectic manifold M that isequipped with a closed (2 n − -dimensional manifold Σ , a contact form α on Σ anda map ψ : Σ × [1 , ∞ ) → M (3.2)such that ψ is a diffeomorphism onto its image, the set M \ ψ (Σ × [1 , ∞ )) is relativelycompact and ψ ∗ ω = d( Rα ) (3.3)holds on Σ × [1 , ∞ ) . Here, R ∈ [1 , ∞ ) is the radial coordinate and the image of ψ isthe convex end of M . Remark 3.3.
We emphasise three features of this definition. • The manifold Σ , the contact form α and the diffeomorphism ψ are all partof the data of a convex symplectic manifold. Consequently, we can use thecoordinates provided by ψ without worrying about whether our constructionsare independent of this choice (c.f. Remark 3.10). • A closed symplectic manifold is convex since we allow the manifold Σ to beempty. With this convention, we are able to prove all of our results for closedmanifolds and (non-trivially) convex symplectic manifolds simultaneously. • The symplectic form ω does not have to be exact on all of M , since (3.3)applies only on the convex end of M . Indeed, symplectic forms on closedmanifolds are never globally exact.Henceforth, M will be a nonnegatively monotone convex symplectic manifold.Finally, we assume that R \ R is unbounded so that symplectic cohomology is ameaningful direct limit (see Definition 3.9). Here R is the set of Reeb periods asdefined in (3.6). Remark 3.4 (Orientations) . This paper uses integral coefficients, and thus we requireorientations on all moduli spaces. For this purpose, let o be a coherent orientation,defined as in [Rit13, Appendix B]. The proof of all orientation signs in this paper isomitted.3.2. Floer cohomology. A subset of a topological space is relatively compact if its closure is compact. The set { R ≤ R } is defined to be M \ ψ (Σ × ( R , ∞ )) and is compact for all R ≥ . We adopt the convention that the empty manifold is a disconnected closed oriented manifold ofevery dimension. This means that the dimension of a manifold X is not well-defined; the statement dim X = k is to be interpreted as ‘ X is k -dimensional’. Hamiltonian dynamics.
Let S = R / Z and D = { z ∈ C : | z | ≤ } .A (time-dependent ) Hamiltonian (function) H is a smooth function S × M → R .Its (time-dependent) Hamiltonian vector field X H is the unique S -family of vectorfields by ω ( · , X H t ) = d H t . The Hamiltonian flow ϕ H is the flow along the vector field X H , so ϕ tH satisfies ∂ t (cid:0) ϕ tH ( m ) (cid:1) = X H t (cid:0) ϕ tH ( m ) (cid:1) . (3.4)A Hamiltonian orbit is a loop x : S → M which satisfies ∂ t x ( t ) = X H t ( x ( t )) forall t ∈ S . It is nondegenerate if the linear map Dϕ H : T x (0) M → T x (1) M has noeigenvalue equal to 1. Denote by P ( H ) the set of Hamiltonian orbits of H .The Hamiltonian H is linear of slope λ if the identity H t ( ψ ( y, R )) = λR + µ holdsat infinity for some constant µ . The vector field of such a Hamiltonian is a multipleof the Reeb vector field X α , so, at infinity, we have X H t = λX α ⊕ ∈ T Σ ⊕ T [1 , ∞ ) ∼ = T M. (3.5)As such, any Hamiltonian orbit in this region corresponds to a λ -periodic flow alongthe Reeb vector field.A Reeb period is a nonzero number λ ∈ R \ { } such that there exists a point y ∈ Σ such that the flow of the Reeb vector field from y is λ -periodic. If Σ is empty, set R = ∅ , and otherwise set R = { Reeb periods } ∪ { } . (3.6)Notice kλ ∈ R for any λ ∈ R and any k ∈ Z .Thus, if H is linear of slope λ with λ / ∈ R , then all Hamiltonian orbits of H lie ina compact region of M . If moreover all Hamiltonian orbits are nondegenerate, then P ( H ) is finite.3.2.2. Almost complex structures.
Let m be a point in the convex end of M . The ω -compatible almost complex structure J is convex at the point m if − d R ◦ J = Rα holds at m . This means that the almost complex structure respects the direct sumdecomposition T y M ∼ = ( R ∂ R ⊕ R X α ) ⊕ ker α (3.7)and satisfies J ( R∂ R ) = X α at m .Let B be a manifold and let J = ( J b ) b ∈ B be a smooth family of ω -compatiblealmost complex structures. The family J is convex if, at infinity, the almost complexstructure J b is convex for all b ∈ B . Definition 3.5.
A choice of
Floer data is a pair ( H, J ) , where H is a linear time-dependent Hamiltonian with slope not in R and J is a convex time-dependent ω -compatible almost complex structure. Throughout, a time-dependent object is a smooth S -family of objects, and a B -dependent objectis a smooth B -family of objects for any manifold B . In the context of convex manifolds, a condition holds at infinity if there exists R ≥ such thatthe condition holds on ψ (Σ × [ R , ∞ )) . Notice that if finitely-many conditions each hold individuallyat infinity, then their conjunction holds at infinity (i.e., they all hold in a common region at infinity).All statements hold tautologically at infinity on a closed symplectic manifold (see Remark 3.3). The
Reeb vector field X α associated to α is the vector field on Σ uniquely defined by ı X α (d α ) = 0 and ı X α α = 1 . N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 17
Pseudoholomorphic spheres.
Let j be the standard almost complex structureon the sphere P . Let Γ = π ( M )ker c ∩ ker ω . (3.8)A curve u : P → M is J -holomorphic if ( Du ) ◦ j = J ◦ ( Du ) and u represents theclass A ∈ Γ if [ u ] = A .Let B be a manifold and J be a convex B -dependent family of ω -compatible almostcomplex structures. The moduli space M ( A, J ) is the space of pairs ( b, u ) , where u isa simple J b -holomorphic curve representing A ∈ Γ .Denote by V k ( J ) the set of pairs ( b, m ) ∈ B × M such that m lies in the image ofa nonconstant J b -holomorphic sphere u with c ( u ) ≤ k .3.2.4. Floer solutions.
Denote by L M the space of contractible smooth maps S → M .Define a cover of this space g L M as the space of pairs ( x, u ) of loops x ∈ L M andsmooth maps u : D → M satisfying x ( t ) = u ( e πit ) , considered up to the equivalence ( x, u ) ∼ ( x, u ′ ) ⇐⇒ Z D u ∗ e c = Z D u ′∗ e c and Z D u ∗ ω = Z D u ′∗ ω, (3.9)where e c is a differential 2-form representing the first Chern class c . The deck trans-formation group of g L M is Γ , which acts on g L M by ‘adding A ∈ Γ to the filling u ’.This action is described explicitly in [HS95, Section 5]. Let e P ( H ) = n ( x, u ) ∈ g L M : x ∈ P ( H ) o . (3.10)Let ( H, J ) be a choice of Floer data. The action functional associated to H is themap A H : g L M → R given by A H ( x, u ) = − Z D u ∗ ω + Z t =0 H t ( x ( t )) d t. (3.11)The set of critical points of A H equals e P ( H ) . A smooth map u : R × S → M is a Floer solution if it satisfies ∂ s u + J t ( ∂ t u − X H, t ) = 0 (3.12)for all ( s, t ) ∈ R × S . The left side of (3.12) is abbreviated by ∂ H, J ( u ) . The energy E ( u ) of a map u is E ( u ) = Z R × S k ∂ s u k J t d s ∧ d t, (3.13)where k·k J is the norm associated to the J -invariant metric ω ( · , J · ) . Suppose theHamiltonian orbits of H are nondegenerate. If the energy of the Floer solution u isfinite, then there exist two Hamiltonian orbits x ± such that lim s →±∞ u ( s, t ) = x ± ( t ) lim s →±∞ ∂ s u ( s, t ) = 0 , (3.14) The holomorphic curve u : P → M is multiply-covered if it is a composition of a holomorphicbranched covering map P → P with degree strictly greater than 1 and a second holomorphic curve.The curve u is simple otherwise. where the limits denote uniform convergence in t , and moreover E ( u ) = A H ( x − , u − ) − A H ( x + , u − u ) (3.15)for any filling u − of x − .By a maximum principle [Rit13, Appendix D], there is a compact region of M suchthat all Floer solutions lie completely within the region.Let e x ± = ( x ± , u ± ) ∈ e P ( H ) . Denote by M ( e x − , e x + ) the moduli space of Floersolutions u which satisfy (3.14) and u + = u − u .3.2.5. Moduli space of Floer trajectories.
In order to ensure the moduli space M ( e x − , e x + ) is a smooth finite-dimensional manifold and has other desired behaviour, weimpose a number of regularity conditions on the Floer data. For regular Floer data,the Hamiltonian orbits are nondegenerate, the moduli space M ( A ; J ) is a smoothcanonically-oriented manifold of dimension n + 2 c ( A ) + 1 and the moduli space M ( e x − , e x + ) of Floer solutions is a smooth oriented manifold of dimension dim M ( e x − , e x + ) = µ ( e x − ) − µ ( e x + ) . (3.16)Here, we denote by µ ( e x ) the Conley-Zehnder index for e x ∈ e P ( H ) . In addition, wehave ( t, x ( t )) / ∈ V ( J ) and ( t, u ( s, t )) / ∈ V ( J ) for all Hamiltonian orbits x and Floersolutions u ∈ M ( e x − , e x + ) when dim M ( e x − , e x + ) ≤ . Remark 3.6 (Regularity) . In this paper, we will not always list all of the regularityconditions, however we will mention those that are more uncommon. The conditionsare always motivated by improving the behaviour of the moduli spaces. Here, wehave used them to ensure the moduli spaces of Floer solutions have the structureof manifolds without bubbling in dimensions 1 and 2. In our notation, regularityconditions will always be satisfied by generic data, and hence will exist.Assume ( H, J ) is regular. When e x − = e x + , the moduli space M ( e x − , e x + ) admitsa smooth free R -action given by s -translation. The quotient f M ( e x − , e x + ) is a smoothmanifold. The 0-dimensional moduli spaces f M ( e x − , e x + ) are all compact by the so-called compactification argument (for example, see [Sal97, Section 3.1]). The uniform convergence is a priori with respect to the t -dependent metric k·k J t , but theproperty holds for any t -dependent Riemannian metric. In order to orient all other moduli spaces in this paper, we have had to choose orientation data,such as coherent orientations (or orientations of the unstable manifolds in the case of Morse theory).The orientation of M ( A ; J ) is intrinsic, however it does rely on an orientation of the parameter space S , which we fix. The 1 in this formula corresponds to dim S , and will be dim B for B -dependent almost complexstructures. In order to find that moduli spaces are manifolds of a given dimension, the standard methodis to find a Fredholm operator whose kernel describes the tangent space of the moduli space. Thecorresponding regularity condition is that the operator is onto, so that some version of the implicitfunction theorem may be applied. A version of the Sard-Smale theorem shows this to be generic[FHS95]. The nondegeneracy of the orbits follows by [AD14, Remark 5.4.8], and the avoidence ofbubbling here is due to [HS95, Sei97]. In a topological space, a subset is generic or of second category if it is a countable intersection ofopen dense subsets. A condition on (Floer) data is generic if the set of data satisfying the conditionforms a generic subset of all data. N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 19
The 1-dimensional moduli spaces f M ( e x − , e x + ) may be given the structure of anoriented compact 1-manifold with boundary by attaching the endpoints [ e x ∈ e P ( H ) µ ( x − )= µ ( x )+1 f M ( e x − , e x ) × f M ( e x , e x + ) (3.17)using the so-called gluing maps (for example, see [Sal97, Section 3.3]). Any element of(3.17), and more generally any chain of Floer trajectories, is a broken Floer trajectory.3.2.6.
Floer cochain complex.
Let ( H, J ) be a regular choice of Floer data.Let q be a formal variable. A formal power series with coefficients in Z and ex-ponents in Γ is a formal sum P A ∈ Γ α A q A . The monomial q A is given the grading c ( A ) ∈ Z . Set Λ k to be the group of formal power series which are supported byonly monomials of grading k and satisfy the condition that the set { A ∈ Γ : ω ( A ) ≤ c, α A = 0 } is finite for all c ∈ R . The Novikov ring Λ is the Z -graded ring ⊕ k ∈ Z Λ k .The degree- k Floer cochains are the formal sums X e x ∈ e P ( H ) µ ( e x )= k α e x e x with integer coefficients such that the set ne x ∈ e P ( H ) : ω ( e x ) ≤ c, α e x = 0 o is finitefor all c ∈ R . Denote by F C k ( M ; H, J ) the set of degree- k Floer cochains. The
Floercochain complex
F C ∗ ( M ; H, J ) associated to J and H is the finitely-generated free Z -graded Λ -module ⊕ k ∈ Z F C k ( M ; H, J ) . The Λ -module structure is induced by the Γ -action on e P ( H ) .The Floer cochain differential d : F C ∗ ( M ; H, J ) → F C ∗ +1 ( M ; H, J ) is the degree-1 Λ -module endomorphism given by d ( e x + ) = X e x − ∈ e P ( H ) µ ( e x − ) − µ ( e x + )=1 X [ u ] ∈ f M ( e x − , e x + ) o ([ u ]) e x − , (3.18)where o ([ u ]) ∈ {± } denotes the orientation of the point [ u ] ∈ f M ( e x − , e x + ) inducedby the coherent orientation o . Lemma 3.7.
The differential d satisfies d = 0 . The
Floer cohomology
F H ∗ ( M ; H, J ) of M with Floer data ( H, J ) is the cohomologyof ( F C ∗ ( M ; H, J ) , d ) . It is a Z -graded Λ -module. Remark 3.8.
Floer cohomology only depends on the slope of the Hamiltonian in theFloer data, and is otherwise independent of the choice of Floer data. This followsfrom the fact that, given two choices of Floer data whose Hamiltonians have the sameslope, any monotone homotopy between them induces a continuation map which is A map of the form (3.18) counts the moduli spaces f M ( e x − , e x + ) . Recall an orientation of a 0-dimensional manifold is a choice of ± for each point of the manifold. an isomorphism (continuation maps are defined in Section 3.2.7). Floer cohomologyis also independent of the coherent orientation o . For closed manifolds M , there isa Λ -module isomorphism F H ∗ ( M ; H, J ) ∼ = Λ ⊗ H ∗ ( M ) (the PSS maps of (6.9) areisomorphisms).3.2.7. Continuation maps.
Given any two regular choices of Floer data ( H − , J − ) and ( H + , J + ) , a homotopy between them is a pair ( H s,t , J s,t ) , where H s,t is a R × S -dependent Hamiltonian and J s,t is a convex R × S -dependent ( ω -compatible) almostcomplex structure, such that both are each s -dependent only on a compact region of R × S and, respectively, equal H ± and J ± for ± s ≫ . The homotopy is monotone if, at infinity, H s,t = h s ( R ) and ∂ s h ′ s ( R ) ≤ . A monotone homotopy exists only if H ± have slopes λ ± that satisfy λ − ≥ λ + .Given any monotone homotopy ( H s,t , J s,t ) which is suitably regular, the contin-uation map ϕ : F H ∗ ( M ; H + , J + ) → F H ∗ ( M ; H − , J − ) is defined by ϕ ( e x + ) = X e x − ∈ e P ( H − ) µ ( e x − )= µ ( e x + ) X u ∈M ( e x − , e x + ) o ( u ) e x − , (3.19)where M ( e x − , e x + ) is the moduli space of solutions to a parameterised version of theFloer equation (3.12) for the homotopy. Continuation maps are independent of thechoice of homotopy. Moreover, the composition of two continuation maps is itself acontinuation map. Definition 3.9.
The symplectic cohomology SH ∗ ( M ) of M is the direct limit lim −→ F H ∗ ( M ; H, J ) , (3.20)where the limit is over all choices of regular Floer data ( H, J ) ordered by slope, andthe maps between the Floer cohomologies are the continuation maps. Remark 3.10.
The symplectic cohomology SH ∗ ( M ) a priori depends on the parame-terisation of the convex end of M . Perturbations of this parameterisation do not affect SH ∗ ( M ) (see [BR20, Theorem 1.9]). With our convention in Remark 3.3, symplecticcohomology is isomorphic to Floer cohomology for closed manifolds.3.3. Hamiltonian circle actions. A Hamiltonian circle action on M is a smoothcircle action σ : S × M → M which flows along the Hamiltonian vector field of someHamiltonian function K σ : M → R . Such a circle action automatically preserves thesymplectic structure. The action is linear if K σ is linear.The vector field X σ of a circle action σ is the vector field along which the actionflows; it is given by X σ = ∂ t σ t | t =0 . Thus for our Hamiltonian circle action σ , thevector field X σ equals the Hamiltonian vector field X K σ of K σ . Lemma 3.11.
Let σ be a linear Hamiltonian circle action of nonzero slope κ . Thepositive Reeb periods form a discrete subset of (0 , ∞ ) . The space of such regular monotone homotopies is nonempty whenever H ± have slopes λ ± thatsatisfy λ − ≥ λ + . N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 21
Proof.
Without loss of generality, suppose κ > by using the action σ − t if necessary.Recall that, by (3.5), the vector field of a linear Hamiltonian of slope κ is κX α atinfinity, where X α is the Reeb vector field on (Σ , α ) . The time-1 flow along this vectorfield is the identity map. Therefore every point of Σ is on a closed Reeb orbit of period κ . For each y ∈ Σ , set P y to be the minimal positive period of the flow along X α starting at y . It is immediate that P y divides κ . Set l y = κ/P y . There is no sequence y r ∈ Σ such that l y r → ∞ as r → ∞ . This follows by an application of the Arzelà-Ascoli theorem and the nonvanishing of the Reeb vector field. Thus, the positive Reebperiods form a subset of κ (max l y )! · Z > , which is a discrete set as required. (cid:3) Remark 3.12.
A contact manifold is
Besse if every point is on a closed Reeb orbit(see [CGM20]). The above proof shows that Σ is Besse for any convex symplecticmanifold which admits a linear Hamiltonian circle action of nonzero slope.A Hamiltonian circle action σ induces a diffeomorphism on the free loop space of M which is given by ( σ ( x ))( t ) = σ t · x ( t ) (3.21)for any loop x : S → M . Lemma 3.13.
Let σ be a linear Hamiltonian circle action on a convex symplecticmanifold M . The action σ has a fixed point, and hence the map (3.21) takes con-tractible loops to contractible loops.Proof. Let K σ have slope κ . If κ ≥ , then the function K σ : M → R has a minumum,so that it has at least one critical point. If κ < , then K σ has a maximum and hencea critical point. Critical points of K σ are fixed points of σ , so σ has a fixed point.Let m ∈ M be a fixed point of σ . Let x : S → M be any contractible loop. Let u : D → M be a smooth filling of x such that u ( z ) = m for all | z | < . Such a fillingexists because M is connected. Define σ · u : D → M by ( σ · u )( re πit ) = σ t · u ( re πit ) (3.22)for all t ∈ S and r ∈ [0 , . Since u is constantly a fixed point of σ in a neighbourhoodof ∈ D , the map σ · u is well-defined. The map σ · u is a filling of σ · x . (cid:3) Remark 3.14.
For convex manifolds, the linearity hypothesis is vital. The Hamilton-ian action on T ∗ S induced by rotation of S is not linear, and the induced action onloops does not preserve contractibility. Seidel proved a more general result for closedmanifolds which applies to loops in Ham( M ) based at Id M [Sei97, Lemma 2.2]. Hisproof used the Arnold conjecture for closed manifolds. For convex manifolds, Ritterobserved (in a Technical Remark [Rit16, page 14]) that if the map (3.21) did not pre-serve contractibility, then symplectic cohomology vanished as did the quantum Seidelmap of Section 6.2.2, which renders this case uninteresting from the point of view ofthe Seidel map, so it was discarded. Lemma 3.13 means (3.21) restricts to a map σ : L M → L M . This map may belifted to the cover g L M by the argument of [Sei97, Lemma 2.4]. Denote the choice ofa lift of σ by e σ . Definition 3.15 (Maslov index of e σ ) . Given a lift e σ , and a point ( x, u ) ∈ g L M , let e σ ( x, u ) = ( σx, v ) . Let τ x : ( x ∗ T M, x ∗ ω ) → ( R n , Ω) be the restriction of a trivialisa-tion of ( T M, ω ) on u and let τ σx be the restriction of a trivialisation on v . Here, Ω isthe standard symplectic bilinear form on R n . Define the loop of symplectic matrices l ( t ) by l ( t ) = τ σx ( t ) Dσ t ( x ( t )) τ x ( t ) − . (3.23)The Maslov index I ( e σ ) associated to this loop does not depend on the choice of thepoint ( x, u ) or on the choice of trivialisations, but it does depend on the choice of lift e σ of σ .3.4. Floer Seidel map.
Let e σ be a lift of a linear Hamiltonian circle action of slope κ .In [Sei97], Seidel defined a natural automorphism on Floer cohomology associated to e σ for closed symplectic manifolds, which was extended to convex symplectic manifoldsin [Rit14].Let ( H, J ) be a regular choice of Floer data. The pullback σ ∗ J of J by σ is ( σ ∗ J ) t = ( Dσ t ) − J t Dσ t (3.24)and the pullback σ ∗ H of H by σ is given by ( σ ∗ H ) t ( m ) = H t ( σ t ( m )) − K σ ( σ t ( m )) (3.25)for all m ∈ M . By the elementary calculations in [Pol01, Section 1.4], the Hamiltonianflows satisfy ϕ tσ ∗ H = σ − t ϕ tH . The pullback Floer data ( σ ∗ H, σ ∗ J ) is a regular choiceof Floer data, with σ ∗ H of slope λ − κ if H is of slope λ .The map (3.21) induces isomorphisms between the moduli spaces of Floer solutionsof the pullback Floer data ( σ ∗ H, σ ∗ J ) and of ( H, J ) . This isomorphism is orientation-preserving with respect to the orientations induced by o .These pullback constructions yield a degree- I ( e σ ) Λ -module isomorphism F S e σ onFloer cohomology. This map F S e σ : F C ∗ ( M ; J , H, ) → F C ∗ +2 I ( e σ ) ( M ; σ ∗ J , σ ∗ H ) (3.26)is given by the Λ -linear extension of e x e σ ∗ · e x , where e σ ∗ · e x is the preimage of e x underthe map e σ . By pulling back regular monotone homotopies, it is possible to establishthat F S e σ commutes with continuation maps, so that F H ∗ ( M ; J + , H + ) F H ∗ +2 I ( e σ ) ( M ; σ ∗ J + , σ ∗ H + ) F H ∗ ( M ; J − , H − ) F H ∗ +2 I ( e σ ) ( M ; σ ∗ J − , σ ∗ H − ) ϕ F S e σ e σ ∗ ϕF S e σ (3.27)commutes. Hence F S e σ induces a degree- I ( e σ ) automorphism of symplectic cohomol-ogy. This definition conforms to the conventions for pullbacks of tensor fields.
N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 23
Seidel showed that F S e σ respects the module structure induced by the pair-of-pantsproduct for closed manifolds [Sei97, Proposition 6.3], and this was extended to convexmanifolds by Ritter [Rit14, Theorem 23]. The pair-of-pants product does not extendto the equivariant setup, so there is no analogue for this result.4. Equivariant Floer theory
In this section, we define equivariant Floer cohomology. It should be considered inanalogy to the Borel homotopy-quotient model for the equivariant cohomology of atopological space (see Section 4.2). We outline in Section 4.1 our conventions for theuniversal bundle of S , and proceed to explain our definition in Section 4.3.In the literature, the S -action used in the definition of S -equivariant Floer coho-mology is the action which rotates the domains of loops. In this paper, we incorporatean additional S -action on M into the definition. Using the trivial action on M in ourdefinition recovers the usual definition, except we use a non-standard and more geo-metric construction for the Z [ u ] -module action (see Section 4.4.2). For completeness,we give the standard construction of the Z [ u ] -module action in Section 4.4.1, thoughwe do not use it in this paper.Our definition is based on the standard definition of the Borel homotopy-quotientrather than the variant which is common in the equivariant symplectic cohomologyliterature (see Remark 4.3).Aside from these three key differences, our construction strongly resembles thosealready in the literature. In the following remark, we compare our conventions tothose in other papers, however all these remaining differences are cosmetic. Remark 4.1 (Conventions) . Our definition is close to those of [BO17, Sections 2.2–2.3] and [Gut18, Section 2.3], except we use cohomological conventions and a Novikovring. Our conventions are almost identical to Zhao’s periodic symplectic cohomology in [Zha19, Zha16] and Seidel’s definition in [Sei18], except that we use a direct sumconvention for cohomology rather than a direct product (and we use a Novikov ring).Importantly, we do not use Z [ u , u − ] / ( u Z [ u ]) in our coefficient ring unlike [Sei08,Section 8b] and [MR18, Appendix B]; indeed this is not possible with our moduleoperation.4.1. Infinite sphere.
The group S acts freely on the odd-dimensional sphere S k − ,which we consider as the subset of C k of norm 1, by multiplication θ · w = e πiθ w. (4.1)With this action, the manifold S k − is a principal S -bundle over CP k − . Thesespheres are equipped with canonical inclusion maps i k : S k − ֒ −→ S k +1 . Denote by S ∞ the direct limit of the odd-dimensional spheres under these inclusion maps. Remark 4.2.
The topological space S ∞ is actually not very nice. It is not compact,it is not first countable and in particular it is not metrisable. We use the notation of S ∞ to simplify otherwise cumbersome statements which involve the limit of spheres. Recall a topological space X is first countable if, for every point x ∈ X , there is a sequence ( U k ) k ∈ N of open subsets of X which contain x such that, for every open subset U ⊆ X with x ∈ U ,there is an inclusion U k ⊆ U for some k ∈ N . All metric spaces are first countable. For example, by a smooth map f : S ∞ → R , we mean a sequence of smooth maps f k : S k − → R , which are compatible with the inclusions in the sense that f k +1 ◦ i k = f k and lim f k = f . In this way, the space S ∞ is a principal S -bundle, with projectionmap π : S ∞ → CP ∞ .Define the smooth function F : S ∞ → R by ( w , . . . ) P k k | w k | . The function F descends to a Morse-Smale function on CP ∞ whose unique critical point of index k is the standard basis vector c k = [0 : · · · : 0 : 1 : 0 : · · · ] for all k ≥ . Recall thatthe cohomology of CP ∞ is isomorphic to Z [ u ] , with u a formal variable of degree 2,where the critical point c k corresponds to u k .The space S ∞ is equipped with the round metric. For each k , fix an identification S ↔ c k ⊂ S ∞ . Extend this identification to the unstable and stable manifolds to getan equivariant map τ k : W u ( c k ) ∪ W s ( c k ) → S , (4.2)defined by where the negative gradient flowline of F converges to along c k .The right-shift map C k → C k +1 given by ( w , . . . , w k − ) (0 , w , . . . , w k − ) in-duces an injective smooth map U : S ∞ → S ∞ . The gradient vector field of F is U -invariant.4.2. Equivariant cohomology.
Let X be a topological space with a continuouscircle action ρ : S × X → X . The Borel homotopy quotient , denoted S ∞ × S X , isthe quotient of the product S ∞ × X by the relation ( θ · w, x ) ∼ ( w, θ · x ) . Equivalently,it is the quotient of the product S ∞ × X by the free circle action θ · ( w, x ) = ( θ − · w, ρ θ ( x )) . (4.3)The equivariant cohomology of the pair ( X, ρ ) is EH ∗ ρ ( X ) = H ∗ ( S ∞ × S X ) . Theprojection S ∞ × S X → S ∞ /S = CP ∞ induces a map Z [ u ] ∼ = H ∗ ( CP ∞ ) → EH ∗ ρ ( X ) .Together with the cup product, this map gives equivariant cohomology the structureof a unital, associative and graded-commutative Z [ u ] -algebra. Remark 4.3 (Diagonal action in literature) . The literature for equivariant symplecticcohomology uses an alternative convention for the Borel homotopy quotient wherebythe free diagonal action on S ∞ × X is used. The automorphism of S ∞ given by complexconjugation takes this diagonal action back to the standard action (4.3), however itis not orientation-preserving on the quotient CP ∞ , so this different convention resultsin a different Z [ u ] -module structure. To correct for this, the transformation u
7→ − u must be used when changing convention.Let ( X, ρ X ) and ( Y, ρ Y ) be two topological spaces with circle actions as above. Thecontinuous map f : X → Y is equivariant if it intertwines the two actions, that isthe identity ρ Y,θ ( f ( x )) = f ( ρ X,θ ( x )) holds. Such an equivariant map induces a well-defined map on the Borel homotopy quotients, and hence induces a natural pullbackmap f ∗ : EH ∗ ρ Y ( Y ) → EH ∗ ρ X ( X ) .4.3. Equivariant Floer cohomology.
N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 25
Equivariant Hamiltonian orbits.
Let ρ be a symplectic circle action on M , andassume that the action flows along the Hamiltonian vector field of some linear Hamil-tonian K ρ at infinity. Such actions are linear at infinity . Notice that ρ preserves theradial coordinate R at infinity. The linear Hamiltonian circle actions of Section 3.3satisfy this assumption by definition; the difference is that we do not assume the actionis Hamiltonian on the entire manifold here.The loop space L M , being a space of maps S → M , naturally inherits two circleactions, one from the rotation action on the domain S and the other from the circleaction on the codomain M . We combine these to get the circle action on L M whichis given, for all t, θ ∈ S and all x ∈ L M , by θ · ( t x ( t ) ) = ( t ρ θ ( x ( t − θ )) ) . (4.4)An equivariant Hamiltonian is a smooth function H eq : S ∞ × ( S × M ) → R whichsatisfies H eq w,t ( m ) = H eq θ − · w,t + θ ( ρ θ ( m )) (4.5)for all t, θ ∈ S , all w ∈ S ∞ and all m ∈ M . Notice that a function satisfying (4.5)is equivariant, in the sense of Section 4.2, with respect to the natural action on thedomain, according to our convention in (4.3), and the trivial action on the codomain R . The equivariant Hamiltonian H eq is linear of slope λ if there is R such that theequation H eq w,t ( ψ ( y, R )) = λR holds when R ≥ R .An equivariant Hamiltonian orbit is an equivalence class [ w, x ] ∈ S ∞ × S L M suchthat w is a critical point of F : S ∞ → R and x is a Hamiltonian orbit of H eq w,t ( · ) . Theequivariance of the Hamiltonian H eq guarantees this definition is independent of thechoice of representative ( w, x ) .The action (4.4) on L M lifts canonically to an action on g L M . The action functional A H eq : S ∞ × g L M → R given by A H eq ( w, ( x, u )) = − Z D u ∗ ω + Z t =0 H eq w,t ( x ( t )) d t (4.6)is invariant under the action combining the lift of (4.4) and (4.3), much like theequivariant Morse functions of Section 6.3.1. Let the group G act on sets X and Y on the left. Let Map(
X, Y ) denote the space of mapsfrom X to Y . The action on Y induces an action on Map(
X, Y ) by post-composition. The actionon Map(
X, Y ) given by g · f = f ◦ g is a right action, instead of a left action. The action given by g · f = f ◦ g − is still a left action, so this is the action naturally inherited by Map(
X, Y ) . Here, g ∈ G and f ∈ Map(
X, Y ) . Equivariant Floer data. An equivariant almost-complex structure J eq is a S ∞ × S -family of almost-complex structures J eq w,t which makes the diagram T m M T m MT ρ θ ( m ) M T ρ θ ( m ) M Dρ θ J eq w,t Dρ θ J eq θ − · w,t + θ (4.7)commute for all m ∈ M . Definition 4.4.
The equivariant Hamiltonian H eq extends the sequence of (non-equivariant) Hamiltonians H k if H eq w,t ( m ) = H kt + τ k ( w ) ( ρ τ k ( w ) ( m )) (4.8)in a neighbourhood of c k in W u ( c k ) ∪ W s ( c k ) for all k ≥ . Likewise, the equivariantalmost complex structure J eq extends the sequence of almost complex structures J k if J eq t,w = Dρ − τ k ( w ) J kt + τ k ( w ) Dρ τ k ( w ) (4.9)in a neighbourhood of c k in W u ( c k ) ∪ W s ( c k ) for all k ≥ .For such equivariant data, the equivariant Hamiltonian orbits are equivalence classes [ w, x ] where w ∈ c k satisfies τ k ( w ) = 0 and x ∈ P ( H k ) . We use the shorthand ( c k , x ) for such equivariant Hamiltonian orbits. Definition 4.5.
A choice of equivariant Floer data is a pair ( H eq , J eq ) consisting ofa linear equivariant Hamiltonian function H eq and a convex equivariant ω -compatiblealmost complex structure J eq which together extend a sequence of Floer data. Remark 4.6.
Typically, one chooses the equivariant Floer data so that it is the sameat the critical points, however we relax this requirement here, allowing any sequenceof Floer data. We need (4.8) and (4.9) to hold for a sequence of non-equivariantdata ( H k , J k ) k ≥ in order to apply the standard continuation map techniques thatguarantee the desired moduli space behaviour. Proposition 4.7 (Existence of data) . Let ( H k , J k ) be any sequence of Floer datawhose Hamiltonians are all linear of the same slope λ and whose at infinity conditions are all satisfied in a common region at infinity. There is equivariant Floer data whichextends this sequence, and moreover the space of such data is contractible. Proof.
We can construct an equivariant Hamiltonian by an analogous argument to[BO17, Example 2.4]. First, we fix an invariant time-independent Hamiltonian H λ on M which is linear at infinity of slope λ . To do this, simply set H λ = 0 on { R < R } and H λ ( ψ ( y, R )) = h ( R ) on { R ≥ R } for an appropriate function h . Next, use a cut-off function near each c k to interpolate between each H k (appropriately interpretedvia (4.8)) and the fixed H λ . The result is an equivariant Hamiltonian of slope λ which An almost-complex structure is a section of the bundle
Aut(
T M ) → M . Let p : S ∞ × S × M → M be the natural projection map. An equivariant almost-complex structure is a map S ∞ × S × M → p ∗ Aut(
T M ) which is equivariant in the usual sense. That is, the linearity of the Hamiltonians and the convexity of the almost complex structures.
N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 27 extends the sequence as desired. This shows the desired existence. The space of allsuch equivariant Hamiltonians is convex and hence contractible.For the equivariant almost complex structure, consider the symplectic vector bundle E = S ∞ × S × T ∗ M / S B = S ∞ × S × M / S . (4.10)The space of (compatible) almost complex structures on E is nonempty and con-tractible [MS98, Proposition 2.63]. The proof of this constructs a retraction r fromthe space of inner products on E (which is convex and hence contractible) to thespace of compatible almost complex structures on E . By restricting this map r to anappropriate subspace of inner products, we will get a retraction to the space of convexalmost complex structures which extend the given sequence J k . It is sufficient for thesubspace to be nonempty and contractible to complete the proof.To guarantee convexity, we restrict to inner products g which satisfy g ( ∂ R , R∂ R ) = 1 g ( ∂ R , X α ) = 0 g ( X α , X α ) = R ker( α ) is g -orthogonal to ∂ R and X α (4.11)at infinity. To ensure the almost complex structures will extend J k , we further restrictto inner products which take appropriate fixed values over neighbourhoods of c k × S × M/S for each k . The space of inner products which satisfy these two conditionsremains convex and nonempty, as desired. (cid:3) Equivariant Floer solutions.
The map ( v, u ) : R → S ∞ × L M satisfies theequivariant Floer equation if v is a negative gradient flowline of F : S ∞ → R and u satisfies the equation ∂ s u + J eq v ( s ) ,t (cid:16) ∂ t u − X H v ( s ) ,t ( · ) (cid:17) = 0 . (4.12)An equivariant Floer solution is an equivalence class [ v, u ] ∈ C ∞ ( R ,S ∞ ×L M ) / S . Theequivariant Floer solution [ v, u ] converges to equivariant Hamiltonian orbits ( c k ± , x ± ) if the limits lim s →±∞ v ( s ) = w ± and (3.14) hold for some choices of representatives.For regular equivariant Floer data, the moduli space of equivariant Floer solutions M (( c k − , e x − ) , ( c k + , e x + )) is a smooth oriented manifold of dimension k − − k + + µ ( e x − ) − µ ( e x − ) . The moduli space is empty unless k − − k + ≥ . It admits a smooth R -action via s -translation that is free if the equivariant Hamiltonian orbits are distinct.(The case when k − = k + is canonically identical to the non-equivariant case using themaps τ k + .) The dimension-0 moduli spaces are compact and the dimension-1 modulispaces admit a compactification via gluing broken trajectories as per Section 3.2.5.The phenomenon of bubbling is avoided using regularity conditions. Explicitly, the regularity conditions ensure (( w, t ) , x ( t )) / ∈ V ( J eq ) for all equivariant Hamilton-ian orbits ( w, x ) and (( v ( s ) , t ) , u ( s, t )) / ∈ V ( J eq ) for all equivariant Floer solutions ( v, u ) occuring inmoduli spaces of dimension 1. These conditions are the equivariant analogues of those used by [HS95]. Remark 4.8 (Weak + monotonicity) . A symplectic manifold satisfies weak + mono-tonicity if the implication − n ≤ c ( A ) < ⇒ ω ( A ) ≤ (4.13)holds for all A ∈ π ( M ) . Weak + monotonicity is insufficient to exclude bubbling bystandard arguments for the equivariant definitions, even though it is sufficient for thenon-equivariant definitions. For regular time-dependent almost complex structures J ,the moduli space of J -holomorphic spheres of class A has dimension n + 2 c ( A ) + 1 .Therefore the moduli space is empty when c ( A ) ≪ is large and negative, becausethe dimension is negative. In the equivariant setup, however, the moduli space of J eq -holomorphic spheres has ‘dimension’ n + 2 c ( A ) + 1 + dim( S ∞ /S ) . Since this‘dimension’ is positive for any c ( A ) , the same argument does not apply. Instead, werequire nonnegative monotonicity which prohibits any J eq -holomorphic spheres withnegative first Chern class via (3.1).4.3.4. Equivariant Floer cochain complex.
Let ( H eq , J eq ) be a regular choice of equi-variant Floer data. The degree- l equivariant Floer cochains are the formal sums X k ≥ X e x ∈ e P ( H k ) µ ( e x )= l − k α k, e x ( c k , e x ) (4.14)with integer coefficients such that the set ne x ∈ e P ( H k ) : ω ( e x ) ≤ c, α k, e x = 0 o is finite for all c ∈ R and k ≥ . Denote by EF C lρ ( M ; H eq ) the Z -module of suchdegree- l equivariant Floer cochains. The equivariant Floer cochain complex is the Z -graded Λ -module ⊕ k ∈ Z EF C kρ ( M ; H eq ) .The equivariant Floer cochain differential is the Λ -module degree-1 endomorphism d : EF C ∗ ρ ( M ; H eq ) → EF C ∗ +1 ρ ( M ; H eq ) given by d ( c k + , e x + ) = X k − ≥ k + e x − ∈ e P ( H k − )2 k − − k + + µ ( e x − ) − µ ( e x + )=1 X [ v,u ] ∈ f M (( c k ± , e x ± )) o ([ v, u ]) ( c k − , e x − ) . (4.15)The differential indeed satisfies d = 0 , and the equivariant Floer cohomology , denoted EF H ∗ ρ ( M ; H eq ) , is the cohomology of ( EF C ∗ ρ ( M ; H eq ) , d ) . It is a Z -graded Λ -module.Standard homotopy techniques ensure EF H ∗ ρ ( M ; H eq ) is dependent only on theslope of the Hamiltonian. Moreover, equivariant monotone homotopies induce equi-variant continuation maps. The equivariant symplectic cohomology ESH ∗ ρ ( M ) is thedirect limit of the resulting system, just as in the non-equivariant case.4.4. Module structures.
N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 29
Algebraic module structure.
This section describes the Z [ u ] -module action usedin the literature.Recall the right-shift operator on S ∞ denoted U . Assume that the maps τ k are U -compatible and that the equivariant Hamiltonian H eq and the equivariant almost-complex structure J eq are U -invariant. These assumptions yield isomorphisms M (( c k − , e x − ) , ( c k + , e x + )) ∼ = M (( c k − + r , e x − ) , ( c k + + r , e x + )) for all r ≥ . With respect to the Z [ u ] -module structure on EF C ∗ ρ ( M ; H eq ) given by u · ( c k , e x ) = ( c k +1 , e x ) , the differential d is a Λ[ u ] -module endomorphism. Hence underthis assumption, the equivariant Floer cohomology is a Λ[ u ] -module.4.4.2. Geometric module structure.
The Morse cup product counts ‘Y’-shaped flow-lines. We adapt this product to get a Z [ u ] -module structure on equivariant Floercohomology which reflects the geometric behaviour of equivariant flowlines in S ∞ .The additional conditions placed on data for this construction are generic, in contrastto the invariance assumptions of Section 4.4.1 which are not generic.Take an S -invariant s -dependent perturbation of the function F : S ∞ → R . Thisis a smooth function [0 , ∞ ) → C ∞ ( S ∞ ) , s F s .Given two equivariant Hamiltonian orbits ( c k − , e x − ) and ( c k + , e x + ) , consider themoduli space of S -equivalence classes of triples ( v, u, v ) , where [ v, u ] is an equivariantFloer solution converging to the two equivariant Hamiltonian orbits and v : [0 , ∞ ) → S ∞ is a negative gradient flowline of F s which converges to a point on c k and satisfies v (0) = v (0) .For a regular choice of the perturbed Morse function F s , this moduli space is anoriented smooth manifold of dimension k − − k + − k + µ ( e x − ) − µ ( e x − ) . (4.17)The 0-dimensional moduli spaces are compact and the 1-dimensional moduli spacesadmit a compactification by broken solutions. The map u k : EF C ∗ ρ ( M ; H eq ) → EF C ∗ +2 kρ ( M ; H eq ) (4.18)which counts the 0-dimensional moduli spaces commutes with the equivariant differ-ential. Moreover, a standard argument yields a chain homotopy between u k ◦ u k ′ and u k + k ′ . This gives equivariant Floer cohomology the structure of a Z [ u ] -module. Todistinguish this module structure from the one in Section 4.4.1, we denote this newaction by u ⌣ · . Throughout, any s -dependent perturbation is s -dependent only on a bounded interval. In order to avoid bubbling, we moreover assume that the map M ( A, J eq | F × S ) / S × PSL P × M (cid:0) ( c k − , e x − ) , ( c k + , e x + ) (cid:1) → ( F × S M ) × ( F × S M ) (4.16)given by ([( w, t, u P ) , p ] , ( v, u )) ( w, u P ( p ) , v (0) , u (0 , t )) is transversal to the diagonal for all A ∈ Γ .Here, F denotes the intersection W u ( u k − ) ∩ W s ( u k + ) ⊂ S ∞ . The domain of the map has dimension k − − k + + 2 n + 2 c ( A ) − k − + µ ( e x − ) − k + − µ ( e x + )) . When the moduli space of equivariantFloer solutions has dimension 1 or 2 and c ( A ) = 0 , the map (4.16) is transversal to the diagonalonly if the intersection of the image with the diagonal is empty. This recovers one of the conditionsfor regular equivariant data. Data will generically satisfy this condition by an argument analogousto the proof of [HS95, Theorem 3.2]. Remark 4.9 (Comparison of Z [ u ] -module structures) . Suppose the equivariant Floerdata satisfies the conditions for the module structures in both Section 4.4.1 and thissection. Let us compare the algebraic product u k · ( c l , e x ) = ( c l + k , e x ) with the geometricproduct u k ⌣ ( c l , e x ) . Suppose [( v, u, v )] is a solution to the geometric product u k ⌣ ( c l , e x ) with end point ( c k − , e x − ) . The ‘Y’-shaped flowline [ v, v ] is a solutionto the cup product on S ∞ /S , though it may not be isolated. Therefore we have k − ≥ k + l since there are no solutions to the cup product otherwise. Moreover for k − = k + l , the ‘Y’-shaped flowline is isolated, so u is a continuation map between e x and e x − (and this continuation map is an isomorphism because the Hamiltonian hasnot changed). As such, we can informally say that the two products agree on the c k + l term. That said, the geometric product may have other terms on the c k − terms with k − > k + l unlike the algebraic product. The author has not determined whetherthe two products are chain homotopic, however anticipates that the moduli spacesin [Sei18, Sections 3 and 5] can be extended to get a chain homotopy.5. Equivariant Floer Seidel map
In this section, we extend the definition of the Floer Seidel map of Section 3.4 tothe equivariant setup introduced in Section 4.Let e σ be a lift of a linear Hamiltonian circle action and let ρ be a symplectic circleaction which is linear at infinity, as per Section 4.3.1. Assume that the σ and ρ commute.5.1. Equivariant Floer Seidel map definition.
Let ( H eq , J eq ) be a regular choiceof equivariant Floer data for the action ρ . The pullback equivariant Floer data ( σ ∗ H eq ,σ ∗ J eq ) are given by the same formulae as in the non-equivariant case (equations (3.24)and (3.25)), so we have ( σ ∗ J eq ) w,t = ( Dσ t ) − J eq w,t Dσ t (5.1)and ( σ ∗ H eq ) w,t ( m ) = H eq w,t ( σ t ( m )) − K σ ( σ t ( m )) (5.2)for all w ∈ S ∞ and m ∈ M .The pullback equivariant Floer data are regular equivariant Floer data for the pull-back action σ ∗ ρ . We show this for the pullback Hamiltonian as follows. Proof of equivariance of (5.2) . Since σ and ρ commute, the Lie bracket of their vectorfields vanishes. The function ω ( X ρ , X σ ) : M → R has Hamiltonian vector field equalto this bracket [ X σ , X ρ ] [Sil01, Proposition 18.3], and is therefore constant. The valueof the constant is 0 because σ has a fixed point by Lemma 3.11. This yields dd θ ( K σ ( ρ θ ( m ))) = (d K σ ) ρ θ ( m ) (( X ρ ) ρ θ ( m ) ) = ω ( X ρ , X σ ) | ρ θ ( m ) = 0 , (5.3) N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 31 from which we deduce that K σ is constant along ρ . We use this in line (5.4b) todeduce the desired equivariance condition: ( σ ∗ H eq ) θ − · w,t + θ (cid:0) ( σ ∗ ρ ) θ ( m ) (cid:1) = H eq θ − · w,t + θ (cid:0) σ t + θ (cid:0) ( σ ∗ ρ ) θ ( m ) (cid:1)(cid:1) − K σ (cid:0) σ t + θ (cid:0) ( σ ∗ ρ ) θ ( m ) (cid:1)(cid:1) (5.4a) = H eq θ − · w,t + θ (cid:0) ρ θ ( σ t ( m )) (cid:1) − K σ (cid:0) ρ θ ( σ t ( m )) (cid:1) (5.4b) = H eq w,t (cid:0) σ t ( m ) (cid:1) − K σ (cid:0) σ t ( m ) (cid:1) (5.4c) = ( σ ∗ H eq ) w,t ( m ) . (5.4d) (cid:3) If H eq has slope λ and the Hamiltonian K σ of σ has slope κ , then the pullback σ ∗ H eq has slope λ − κ .Recall that, by definition, e σ is a lift of the automorphism σ : L M → L M , given by ( σ ( x ))( t ) = σ t · x ( t ) , to an automorphism of g L M . We have a commutative diagram ofautomorphisms on S ∞ × L M ( w, t x ( t ) ) ( w, t σ t ( x ( t )) )( θ − · w, t ( σ ∗ ρ ) θ ( x ( t − θ )) ) ( θ − · w, t ρ θ ( σ t − θ ( x ( t − θ ))) ) Id S ∞ × e σ · θ for σ ∗ ρ · θ for ρ Id S ∞ × e σ (5.5)which lifts to S ∞ × g L M . Just like the non-equivariant case, the map (Id S ∞ × e σ ) − takesequivariant Hamiltonian orbits of H eq to equivariant orbits of σ ∗ H eq , and inducessimilar isomorphisms on the moduli spaces of equivariant Floer solutions. As such,we get an isomorphism of cochain complexes EF S e σ : EF C ∗ ρ ( M ; J eq , H eq ) → EF C ∗ +2 I ( e σ ) σ ∗ ρ ( M ; σ ∗ J eq , σ ∗ H eq )( c k , e x ) ( c k , e σ ∗ e x ) . (5.6)This equivariant Floer Seidel map preserves both the geometric and algebraic mod-ule structures on cohomology because it induces isomorphisms between the relevantmoduli spaces. Remark 5.1.
The diagram (5.5) commutes if and only if σ is indeed an action, andfails to commute if σ is merely a based loop in Ham(
M, ω ) . This diagram is thereason that our equivariant construction requires this stronger assumption, unlike thenon-equivariant case.As per the non-equivariant case, we can pullback regular equivariant monotonehomotopies. Thus for any equivariant continuation map ϕ we get the following com-mutative diagram. EF C ∗ ρ ( M ; J eq , + , H eq , + ) EF C ∗ +2 I ( e σ ) σ ∗ ρ ( M ; σ ∗ J eq , + , σ ∗ H eq , + ) EF C ∗ ρ ( M ; J eq , − , H eq , − ) EF C ∗ +2 I ( e σ ) σ ∗ ρ ( M ; σ ∗ J eq , − , σ ∗ H eq , − ) ϕ EF S e σ σ ∗ ϕEF S e σ (5.7) When σ is linear of strictly positive slope κ , this offers a way to compute equivariantsymplectic cohomology (this construction is the equivariant version of [Rit14, Theo-rem 22]). Let ε > be smaller than any positive Reeb period. For each r ∈ Z ≥ ,choose equivariant Floer data ( H eq r , J eq r ) for the action σ − r ρ of slope ε . The pull-back data (( σ − r ) ∗ H eq r , ( σ − r ) ∗ J eq r ) is equivariant for the action ρ and has slope ε + rκ .Consider the following commutative diagram, where ϕ r are the continuation maps be-tween the relevant Floer data. We have omitted M , degrees and the almost complexstructure from the notation for clarity. EF H ρ ( H eq )] EF H ρ (( σ − ) ∗ H eq ) EF H ρ (( σ − ) ∗ H eq ) . . .EF H σ − ρ ( H eq ) EF H σ − ρ (( σ − ) ∗ H eq ) . . .EF H σ − ρ ( H eq ) . . .. . . ϕ ( σ − ) ∗ ϕ EF S e σ ∼ = ( σ − ) ∗ ϕ EF S e σ ∼ = ϕ EF S e σ ∼ = ( σ − ) ∗ ϕ ϕ (5.8)The direct limit of the top line of (5.8) is, by definition, the equivariant symplecticcohomology of M for the action ρ . The diagram shows this is isomorphic to thedirect limit of the dashed maps EF S e σ ◦ ϕ r . In Section 7.3, we find another way tocharacterise these dashed maps using equivariant quantum cohomology. This diagramis crucial for our calculations in Section 8.1 and Section 8.3.5.2. Equivariant Floer Seidel map properties.
The equivariant Floer Seidel mapis a very natural operation, and as such it preserves many of the structures associatedto equivariant Floer cohomology. It is a module map for both the geometric andalgebraic module structures, and is compatible with continuation maps as describedabove. As such, after taking a direct limit, we get an isomorphism of equivariantsymplectic cohomologies EF S e σ : ESH ∗ ρ ( M ) ∼ = → ESH ∗ +2 I ( e σ ) σ ∗ ρ ( M ) . In the remainder ofthis section, we describe two further structures on equivariant symplectic cohomologywhich are compatible with the equivariant Floer Seidel map.5.2.1. Gysin sequence.
Analogously to [BO17], there is a long exact sequence on equi-variant symplectic cohomology · · · →
ESH ∗ ρ ( M ) · u → ESH ∗ +2 ρ ( M ) → SH ∗ +2 ( M ) → ESH ∗ +1 ρ ( M ) → · · · . (5.9)This is an immediate algebraic consequence of our definitions because there is a shortexact sequence of cochain complexes, where · u denotes the algebraic Z [ u ] -moduleoperation. The second map ESH ∗ ρ ( M ) → SH ∗ ( M ) is the map induced by ( c , e x ) e x and ( c k , e x ) for k > on equivariant Floer cohomology. This long exact sequenceis the Gysin exact sequence of equivariant symplectic cohomology.The equivariant Floer Seidel maps are isomorphisms on the cochain complexeswhich are compatible with the maps in (5.9), and therefore fit into the following
N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 33 commutative diagram. · · ·
ESH ∗ ρ ( M ) ESH ∗ +2 ρ ( M ) SH ∗ +2 ( M ) · · ·· · · ESH ∗ +2 I ( e σ ) σ ∗ ρ ( M ) ESH ∗ +2+2 I ( e σ ) σ ∗ ρ ( M ) SH ∗ +2+2 I ( e σ ) ( M ) · · · · u EF S e σ EF S e σ F S e σ · u (5.10)5.2.2. Filtration and positive equivariant symplectic cohomology.
For exact symplec-tic manifolds, the Floer cochain complexes may be equipped with an action filtrationwhich distinguishes between different orbits [Vit99]. Roughly, the orbits of a Hamil-tonian with small positive slope correspond, via the PSS map of Section 6.2.3, tothe quantum cohomology of M . Conversely, orbits which occur only for Hamiltonianswith larger slopes correspond to Reeb orbits on Σ , with the period of the Reeb orbitcorresponding to the slope of the Hamiltonian at the radius of the orbit, as per (3.5).In our setup, however, the action functional (3.11) fails to provide a filtration fortwo reasons. First, since we have not assumed M is exact, the value of A H ( x ) de-pends on the lift of x to g L M . Second, the action functional may not decrease alongequivariant Floer trajectories that are not constant in S ∞ (see [BO17, page 3867]).These issues may be resolved by choosing special equivariant Floer data and addinga cut-off function to the action filtration, an approach taken in [MR18, Appendix D].We briefly outline this construction.Fix two numbers < R < R < ∞ . Our equivariant Hamiltonian H eq will havepositive small slope at R = R , be quadratic and increasing on R < R < R , andit will be linear on R > R ; in particular, it will only depend on R in the region R > R , modulo a small perturbation we will ignore. We write H eq w,t ( ψ ( y, R )) = h ( R ) to emphasize this. Fix a cut-off function β : [1 , ∞ ) → R ; this is a smooth increasingfunction which is 0 on [1 , R ) and has increasing gradient on the open interval ( R , R ) .Define f h : R → R to be f h ( R ) = Z R β ′ ( r ) h ′ ( r ) d r (5.11)and define the function F h : L M → R by F h ( x ) = − Z S x ∗ ( β ( R ) α ) + Z t =0 f h ( R ◦ x ) d t. (5.12)Notice how (5.12) resembles the action functional (3.11) in the exact setup if we wereto change β to Id R (of course this choice doesn’t satisfy the requirements for β ). Thefunction F h decreases along any equivariant Floer trajectory. Therefore the inclusion EF C ∗ ρ ( M ; H eq ; F h ≥ ֒ → EF C ∗ ρ ( M ; H eq ) , (5.13)of the subcomplex generated by the orbits which satisfy F h ≥ is a chain map. The slope λ of a Hamiltonian is small if it is smaller than any positive Reeb period, i.e. itsatisfies λ < min( R ∩ (0 , ∞ )) . Choose R large enough so that σ is linear on { R > R } . The cohomology of the quotient cochain complex of (5.13) is the positive equivariantFloer cohomology , denoted
EF H ∗ ρ, + ( M ; H eq ) . This construction is compatible withcontinuation maps, so we can take a direct limit of EF H ∗ ρ, + ( M ; H eq ) under contin-uation maps with the slopes increasing. This direct limit is the positive equivariantsymplectic cohomology of M , and is written ESH ∗ ρ, + ( M ) . The equivariant PSS mapsof Section 7.3 give an isomorphism between the cohomology of the subcomplex withequivariant quantum cohomology EF H ∗ ρ ( M ; H eq ; F h ≥ ∼ = EQH ∗ ρ ( M ) . Associatedto the short exact sequence induced by the inclusion (5.13), there is a long exactsequence · · · → EQH ∗ ρ ( M ) → ESH ∗ ρ ( M ) → ESH ∗ ρ, + ( M ) → EQH ∗ +1 ρ ( M ) → · · · . (5.14)We have the following compatibility result between the filtration and the equivariantFloer Seidel map. Theorem 5.2.
For any equivariant Hamiltonian orbit ( c k , x ) , the filtration satisfies F σ ∗ h ( σ ∗ x ) = F h ( x ) . (5.15) Proof.
By definition, we have ( σ ∗ h )( R ) = h ( R ) − κR , so that f σ ∗ h ( R ) = f h ( R ) − κβ ( R ) , where κ is the slope of σ . Without loss of generality, let x occur in the region { R < R < R } since otherwise both sides of (5.15) are 0. Suppose that x has period l . The period of σ ∗ x is l − κ . We have F σ ∗ h ( σ ∗ x ) = − Z S ( σ ∗ x ) ∗ ( β ( R ) α ) + Z t =0 f σ ∗ h ( R ◦ ( σ ∗ x )) d t = − β ( R ◦ ( σ ∗ x )) ( l − κ ) + Z t =0 f σ ∗ h ( R ◦ ( σ ∗ x )) d t = − β ( R ◦ x ) ( l − κ ) + Z t =0 f σ ∗ h ( R ◦ x ) d t = − β ( R ◦ x ) ( l − κ ) + Z t =0 f h ( R ◦ x ) − κβ ( R ◦ x ) d t = − lβ ( R ◦ x ) + Z t =0 f h ( R ◦ x ) d t = − Z S x ∗ ( β ( R ) α ) + Z t =0 f h ( R ◦ x ) d t = F h ( x ) . (cid:3) Unfortunately, the pullback Hamiltonian σ ∗ H eq does not satisfy the condition thatits slope at R is small; instead its slope at R will be ε − κ if H eq had slope ε at R . To rectify the situation for positive equivariant symplectic cohomology, wehave to consider the subcomplex generated by orbits with F σ ∗ h ≥ f σ ∗ h ( R ′ ) , where R ′ is the radius at which h ′ − κ = 0 . The positive equivariant Floer Seidel map is the composition of the equivariant Floer Seidel map with the map induced onthe quotient complexes corresponding to the inclusions of these subcomplexes. More N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 35 explicitly, abusing notation, we have the following diagram.
ESH ∗ ρ, + ESH ∗ ρ ( F h < ESH ∗ +2 I ( e σ ) σ ∗ ρ ( F σ ∗ h < ESH ∗ +2 I ( e σ ) σ ∗ ρ, + ESH ∗ +2 I ( e σ ) σ ∗ ρ ( F σ ∗ h < f σ ∗ h ( R ′ )) EF S e σ, + EF S e σ (5.16)This gives us the following morphism of long exact sequences. · · · EQH ∗ ρ ESH ∗ ρ ESH ∗ ρ, + EQH ∗ +1 ρ · · ·· · · EQH ∗ +2 I ( e σ ) σ ∗ ρ ESH ∗ +2 I ( e σ ) σ ∗ ρ ESH ∗ +2 I ( e σ ) σ ∗ ρ, + EQH ∗ +2 I ( e σ )+1 σ ∗ ρ · · · EQ S e σ EF S e σ ∼ = EF S e σ, + EQ S e σ (5.17)The five lemma applied to the diagram (5.17) implies that EF S e σ, + is an isomorphismif and only if EQ S e σ is an isomorphism.6. Quantum theory
Quantum cohomology.
Morse cohomology.
Fix a Riemannian metric on M . Let f : M → R be aMorse-Smale function which increases in the radial coordinate direction at infinity.Thus the inequality ∂ R ( f ( ψ ( y, R ))) > holds at infinity. Denote by Crit( f ) the finiteset of critical points of f . The Morse index ind( x ) of a critical point x is the dimensionof the maximal subspace of the tangent space at x on which the Hessian of f is negativedefinite. The Morse cohomology of M is the cohomology of the cochain complex freelygenerated by Crit( f ) whose differential counts negative gradient trajectories betweencritical points. It is isomorphic to the (singular) cohomology of M . Through thechoice of an orientation of each unstable manifold, the count of trajectories is signed,so that Morse cohomology is a Z -graded Abelian group.6.1.2. Quantum product.
Let J be a regular convex ω -compatible almost complexstructure. Fix distinct points p − , p +1 , p +2 ∈ P . The quantum product counts quadru-ples ( u, γ − , γ +1 , γ +2 ) , where u : P → M is a simple (or constant) J -holomorphicsphere and γ − : ( −∞ , → M and γ + i : [0 , ∞ ) → M are negative gradient flow-lines, with the intersection conditions γ + i (0) = u ( p + i ) and γ − (0) = u ( p − ) . Denoteby M ( x − , x +1 , x +2 ; A ) the space of such quadruples where u represents A ∈ Γ and thelimits γ + i ( s ) → x + i and γ − ( s ) → x − hold as s → ±∞ .With a regular choice of three s -dependent perturbations f +1 , f +2 , f − of the Morse-Smale function f , the moduli spaces will all be smooth oriented manifolds with dim M ( x − , x +1 , x +2 ; A ) = ind( x − ) − ind( x +1 ) − ind( x +2 ) + 2 c ( A ) . (6.1) Via standard compactification and gluing arguments, the 0-dimensional moduli spaceis compact and the 1-dimensional moduli space may be compactified to a manifoldwhose boundary is made up of the broken trajectories (the sphere will not bubble byregularity). As such, the map C ∗ ( M ; f ; Λ) ⊗ → C ∗ ( M ; f ; Λ) given by x +1 ∗ x +2 = X A ∈ Γ x − ∈ Crit( f )dim M ( x − ,x +1 ,x +2 ; A )=0 X ( u,γ ±• ) ∈M ( x − ,x +1 ,x +2 ; A ) o ( u, γ ±• ) q A x − (6.2)is a chain map, and hence it induces a product structure on the Morse cohomology of M with coefficients in the Novikov ring Λ . This product is unital, skew-commutativeand associative, and induces the structure of a Z -graded Λ -algebra on H ∗ ( M ; f ; Λ) .This is the quantum cohomology QH ∗ ( M ) of the manifold M , and the product is the quantum product . Remark 6.1.
The quantum cohomology a priori depends on the Riemannian metric,the Morse-Smale function f and its three perturbations, the chosen orientations of theunstable manifolds and the almost-complex structure. However the dependence on allof this data may be removed up to canonical isomorphism via standard homotopyarguments. Moreover, quantum cohomology is independent of the parameterisationof the convex end because this information is used only to constrain all flowlines andspheres to a compact region.6.2. Quantum Seidel map.
Let e σ be a lifted linear Hamiltonian circle action on M with nonnegative slope.6.2.1. Clutching construction.
In this section, we define a symplectic M -bundle E overthe sphere associated to the action σ . Base space . The sphere S is the union of its upper hemisphere D − and lower hemi-sphere D + . Each hemisphere is a copy of the closed unit disc in the complex plane.The equator of the sphere is the circle S ∼ = R / Z . We identify the boundaries of thehemispheres with the equator via t ∈ S ↔ e πit ∈ ∂ D − ↔ e − πit ∈ ∂ D + . (6.3)The poles of the sphere are the points z ± = 0 ∈ D ± . The complement of the poles inthe sphere is isomorphic to a cylinder via the map R × S ∋ ( s, t ) (cid:26) e π ( s + it ) ∈ D − if s ≤ , e − π ( s + it ) ∈ D + if s ≥ . (6.4) Remark 6.2.
Our notation is opposite to that of Seidel [Sei97] and Ritter [Rit14], soour D ± correspond to their D ∓ . In the region at infinity where J is convex, any J -holomorphic sphere cannot achieve a maximalvalue of R by a maximum principle. As such, all the holomorphic spheres lie in a compact regionand standard compactification results apply. N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 37
Total space . The smooth manifold E is the union of the manifolds D ± × M gluedalong the boundaries via ∂ D − × M ∋ ( e πit , m ) ↔ ( e − πit , σ t m ) ∈ ∂ D + × M. (6.5)The projection map π : E → S is the union of the projection maps D ± × M → D ± .Let ι ± : M → { z ± } × M ⊂ E be the fibre inclusion maps over the poles. Symplectic bilinear form . Denote by T vert E the kernel of Dπ . With π ± M : D ± × M → M the projection map, the vector space T vert ( w,m ) E is equipped with a symplecticbilinear form Ω ( w,m ) = ( π ± M ) ∗ ( w,m ) ω m . Since the circle action σ is symplectic, thesesymplectic bilinear forms agree along the equator, so that T vert E → E is a symplecticvector bundle with symplectic bilinear form Ω . Global 2-form . There is a closed 2-form b Ω on E which restricts to Ω on T vert E . Theconstruction of b Ω for convex symplectic manifolds, due to Ritter [Rit14, Section 5],uses a special pair of Hamiltonians H E, ± : D ± × M → R to modify the fibrewisesymplectic form ω so that it becomes a well-defined closed 2-form. Almost complex structures . The sphere has an (almost) complex structure j givenby ∓ i on D ± . Denote by J ( E ) the space of almost complex structures b J on E whichsatisfy the following properties: • Dπ is ( b J , j ) -holomorphic. • b J | T vert E is a convex Ω -compatible almost complex structure on T vert E . • At infinity, b J has the form b J ( z,m ) = (cid:18) j s ⊗ X H Ez , ± − d t ⊗ J ± z X H Ez , ± J ± z (cid:19) (6.6)with respect to the decomposition T ( z,m ) E = T z D ± ⊕ T m M and the coordinates ( s, t ) on the sphere from (6.4), denoting by J ± z the fibrewise restriction b J | T vert E on each hemisphere.Given any b J ∈ J ( E ) , the 2-form b Ω + cπ ∗ ω S is symplectic and b J is ( b Ω + cπ ∗ ω S ) -compatible for large enough c > . Here, we denote by ω S the standard symplecticform on S . Remark 6.3.
The motivation for (6.6) is that any b J -holomorphic section is locallya Floer solution for ( H E , J ) whenever (6.6) applies, and hence a maximum principleforbids any (non-fixed) b J -holomorphic sections outside a compact region. The functions H E, ± : D ± × M → R must vanish near the poles, be independent of the s -coordinate near the equator, and glue according to H Et , + = σ ∗ H Et , − . The gluing condition ensuresthe Hamiltonian vector field in T vert E is well-defined along the equator. Moreover, H E, ± must bothbe monotone, by which we mean that in a region at infinity, the functions are dependent only on theradial coordinate R and the s -coordinate of (6.4), and satisfy ∂ s H E, ± ≤ . We assume σ is linear ofnonnegative slope precisely so that such Hamiltonian functions exist. Sections . Two sections s , s : S → E are Γ -equivalent if the conditions b Ω( s ) = b Ω( s ) and c ( T vert E )( s ) = c ( T vert E )( s ) hold, where c ( T vert E ) is the first Chernclass of the symplectic vector bundle T vert E → E . The property of Γ -equivalenceis independent of the choice of global 2-form b Ω which restricts to Ω . Moreover, thegroup Γ acts freely and transitively on Γ -equivalence classes of sections.Given any lift e σ and any e x ∈ g L M , we can produce a section by setting z ( z, e x ( z )) on D − and z ( z, e σ ( e x )( z )) on D + . The Γ -equivalence class of this sec-tion is independent of the choice of e x . We denote it by S e σ . It satisfies I ( e σ ) = − c ( T vert E )( S e σ ) . Every Γ -equivalence class is S e σ + A for a unique A ∈ Γ . Fixed sections . For every fixed point m ∈ M of the circle action σ , there is a constantsection s m : z ( z, m ) . The section s m is the fixed section at m . For any fixed section s m , we have that − c ( T vert E )( s m ) equals the sum of the weights of the action around m [MT06, Lemma 2.2]. Remark 6.4 (Minimal fixed sections) . A minimal fixed section is a fixed section s m at a point m in the minimal locus of the Hamiltonian K σ , i.e. K σ ( m ) = min( K σ ) .Minimal fixed sections are ( j, b J ) -holomorphic for a restricted class of almost complexstructures b J (see [MT06, Definition 2.3] and the preceding text). In this setting,we moreover have b Ω( u ) > b Ω( s m ) for any minimal fixed section s m and any ( j, b J ) -holomorphic section u which is not a minimal fixed section [MT06, Lemma 3.1].6.2.2. Quantum Seidel map definition.
The objects of focus are ( j, b J ) -holomorphicsections of E , for a suitably regular b J ∈ J ( E ) which we use throughout this section.The moduli space M ( S ) of ( j, b J ) -holomorphic sections which are in Γ -equivalenceclass S is a smooth manifold of dimension n + 2 c ( T vert E )( S ) . A sequence of suchsections u r ∈ M ( S ) with b Ω( u r ) bounded will have a subsequence which converges toa section with bubbles in the fibres.For critical points x ± ∈ Crit( f ) , denote by M ( x − , x + ; S ) the moduli space oftriples ( γ − , γ + , u ) where γ − : ( −∞ , → M and γ + : [0 , ∞ ) → M are negativegradient trajectories of f ± and u ∈ M ( S ) is a section which satisfies u ( z ± ) = γ ± (0) at the poles. Here, the functions f ± : R × M → R are s -dependent perturbationsof f , chosen such that the data ( f ± , b J ) satisfies some regularity conditions. Theseregularity conditions ensure that M ( x − , x + ; S ) is a smooth manifold of dimension ind( x − ) − ind( x + ) + 2 c ( T vert E )( S ) .Define a degree- I ( e σ ) chain map Q S e σ : C ∗ ( M ; f ; Λ) → C ∗ +2 I ( e σ ) ( M ; f ; Λ) by x + X x − ∈ Crit( f ) A ∈ Γdim M ( x − ,x + ; S e σ + A )=0 X ( γ ± ,u ) ∈M ( x − ,x + ; S e σ + A ) o (( γ ± , u )) q A x − . (6.7)The regularity conditions we impose ensure that Q S is a chain map so it induces amap on quantum cohomology. Moreover, Q S intertwines quantum multiplication in QH ∗ ( M ) , giving Q S e σ ( x ∗ y ) = x ∗ Q S e σ ( y ) (6.8) More precisely, by e x ( z ) , we mean u ( z ) for a choice of filling u of x , and similarly for e σ e x ( z ) . Theresulting Γ -equivalence class is independent of the choice. N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 39 for all x, y ∈ H ∗ ( M ; f ; Λ) . Remark 6.5.
There are two more-or-less equivalent methods to proving this inter-twining relation (6.8). One approach is to prove the intertwining of the Floer Seidelmap with the pair-of-pants product [Sei97, Proposition 6.3], and apply the ring iso-morphisms PSS ± from (6.9) to deduce the desired result. This is the approach takenby Seidel. This method will not extend to the equivariant setup for two reasons: thepair-of-pants product has no equivariant version and the homotopy Seidel constructsto prove the intertwining with the pair-of-pants product involves reparameterisingthe path σ : S → Ham( M ) , which cannot be done while maintaining (4.4) (seeRemark 5.1). The second approach directly constructs a chain homotopy either sideof (6.8). While a standard argument, I believe it has not appeared in the literature.It is this second approach we extend in Section 7.4. The non-equivariant argumentmay be derived from the equivariant argument by removing the flowlines in S ∞ .6.2.3. Gluing construction.
The PSS isomorphism is a ring isomorphism between theFloer cohomology and the quantum cohomology of a weakly monotone closed symplec-tic manifold constructed in [PSS96]. As a map, the PSS isomorphism counts spikeddiscs . These are maps from the disc to M which near the boundary act like Floersolutions and which near the centre act like a pseudoholomorphic sphere, togetherwith half-flowlines between a critical point and the centre of the disc, the spikes .To extend the definition of these maps to convex symplectic manifolds, we make thefollowing definition. Definition 6.6.
The (time-dependent) Hamiltonian H : S → C ∞ ( M ) has slopezero if H is C -bounded and, at infinity, H t ( ψ ( y, R )) = h ( R ) for a smooth function h : ( R , ∞ ) → R which satisfies < h ′ < T min , h ′′ < and h ′ → , where T min ∈ (0 , ∞ ] is the minimal Reeb period.While such a Hamiltonian is not linear, it still satisfies (3.5) and a maximum prin-ciple at infinity. Thus we can define Floer cohomology for a regular choice of Floerdata ( H , J ) . The PSS construction yields a pair of ring isomorphisms between Floercohomology and quantum cohomologyPSS − : F H ∗ ( M ; H ) → QH ∗ ( M ) , PSS + : QH ∗ ( M ) → F H ∗ ( M ; H ) (6.9)which are mutual inverses [Rit14, Theorem 37].Seidel’s gluing argument [Sei97, Section 8] proves the following diagram is commu-tative. QH ∗ ( M ) QH ∗ +2 I ( e σ ) ( M ) F H ∗ ( M ; H ) F H ∗ +2 I ( e σ ) ( M ; H ) F H ∗ +2 I ( e σ ) ( M ; σ ∗ H ) Q S e σ PSS + ∼ = F S e σ ∼ = PSS − ∼ = continuation map (6.10) i.e. a flowline with domain either [0 , ∞ ) or ( −∞ , . In [Rit14], Ritter used a non-equivariant version of (5.8) together with (6.10) to showthat if σ has positive slope, then the direct limit of the direct system QH ∗ ( M ) Q S e σ −−−→ QH ∗ +2 I ( e σ ) ( M ) Q S e σ −−−→ QH ∗ +4 I ( e σ ) ( M ) Q S e σ −−−→ · · · (6.11)is isomorphic to symplectic cohomology. This offers a method to calculate symplecticcohomology because the quantum Seidel map is quantum multiplication by the element Q S e σ (1) .6.3. Equivariant Quantum cohomology.
Equivariant Morse cohomology.
Let ρ be a smooth circle action on M , and fixa ρ -invariant Riemannian metric on M . An equivariant Morse function is a function f eq : S ∞ × M → R which is invariant under the free action (4.3), and which extendsMorse functions f k : M → R analogously to Definition 4.4. We assume that, atinfinity, the function f eq w ( · ) is increasing in the radial coordinate direction for all w ∈ S ∞ .Analogously to the equivariant Floer cohomology construction of Section 4, an equi-variant critical point is an equivalence class [ w, x ] ∈ S ∞ × S M such that w is a crit-ical point of F and x is a critical point of f eq w ( · ) . The index of such a critical point is ind( w ; F ) + ind( x ; f eq ) . We use the notation ( c k , x ) ∈ Crit( f eq ) for equivariant criticalpoints and | c k , x | for their indices.For a suitably regular equivariant Morse function, the moduli spaces of equivariantnegative gradient trajectories are smooth oriented manifolds. Equivariant (Morse)cohomology EH ∗ ρ ( M ) is the cohomology of the cochain complex generated over Z by the equivariant critical points whose differential counts the equivariant negativegradient trajectories. As in Section 6.1.1, an orientation of the unstable manifoldsmust be chosen in order that the count is signed. We omit the details. The Z [ u ] -module actions of Section 4.4.1 and Section 4.4.2 may be defined on equivariant Morsecohomology, and we opt for the geometric action u ⌣ which counts ‘Y’-shaped graphs.6.3.2. Equivariant quantum product.
Let ρ be a symplectic circle action which is lin-ear at infinity. We use a ‘Y’-shaped flowline in S ∞ for the equivariant quantumproduct so that it resembles a deformed equivariant cup product. In order to con-struct the moduli space, we need three s -dependent perturbations F − , F +1 , F +2 of thestandard function on S ∞ and three s -dependent perturbations f eq , − , f eq , + , f eq , + ofthe equivariant Morse function on M . We also need a regular convex S ∞ -dependent ω -compatible almost complex structure J eq , which is equivariant in the sense that J eq w = ( Dσ θ ) − J eq θ − · w Dσ θ for all θ ∈ S and w ∈ S ∞ .We consider septuples ( v − , v +1 , v +2 , γ − , γ +1 , γ +2 , u ) , where v ±∗ are negative gradientflowlines of F ±∗ satisfying v +1 (0) = v +2 (0) = v − (0) and γ ±∗ are negative gradientflowlines of ( f eq ∗ , ± ) v ±∗ ( · ) which intersect at 0 with a simple (or constant) J eq v − (0) -holomorphic sphere u at the points p ±∗ ∈ P (see Figure 1). When the data is suf-ficiently regular, the moduli space of ( S -equivalence classes of) such septuples withlimits ( c k ±∗ , x ±∗ ) ∈ Crit( f eq ) and with u representing A ∈ Γ is a smooth oriented N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 41 v − v +1 v +2 γ − γ +1 γ +2 p − p +1 p +2 c − c +1 c +2 x − x +1 x +2 uA Figure 1.
The equivariant quantum product counts equivalenceclasses of septuples ( v − , v +1 , v +2 , γ − , γ +1 , γ +2 , u ) . The ‘Y’-shaped graphabove maps to S ∞ while the configuration below the graph maps to M .manifold of dimension (cid:12)(cid:12) c k − , x − (cid:12)(cid:12) + 2 c ( A ) − (cid:12)(cid:12)(cid:12) c k +1 , x +1 (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) c k +2 , x +2 (cid:12)(cid:12)(cid:12) . (6.12)The equivariant quantum product ∗ ρ counts the 0-dimensional moduli spaces.The equivariant quantum cohomology EQH ∗ ρ ( M ) is the cohomology of the cochaincomplex EQC l ( M ) = ∞ Y k =0 M ( c k ,x ) ∈ Crit( f eq ) Λ l −| c k ,x | h ( c k , x ) i (6.13)with the equivariant Morse differential. It is a graded Λ[ u ] -module for both the alge-braic and the geometric u -actions. The product ∗ ρ is a unital, graded-commutative,associative product on EQH ∗ ρ ( M ) compatible with the Λ -module structure and thegeometric Z [ u ] -module structure. All of these properties follow from standard ho-motopy proofs, as does the independence from all the data chosen.Via the algebraic Z [ u ] -module structure, the equivariant quantum cochain complex(6.13) is the graded completed tensor product QC ∗ ( M ) b ⊗ Z [ u ] . By the graded com-pleted tensor product A b ⊗ Z [ u ] , where A is a graded Z -module, we mean the graded Z -module whose grading- l subgroup is (cid:0) A b ⊗ Z [ u ] (cid:1) l = ∞ Y k =0 A l − k ⊗ Z · u k . (6.14) It is difficult to see any relationship between the algebraic Z [ u ] -module structure and the equi-variant product, hence our decision to use the geometric module structure in this paper. vuγ − γ + σ D − D + z + z − c − c + x − x + Figure 2.
The equivariant quantum Seidel map counts equivalenceclasses of quadruples ( v, γ − , γ + , u ) . The map u is a section of theclutching bundle which twists the fibres by σ when passing from theupper hemisphere D − to the lower hemisphere D + . The flowlines γ ± map to the manifold M , which is identified with the fibres of the clutch-ing bundle over the poles z ± .This is a variant of the completed tensor product used by Zhao [Zha19, Section 2],but which is a graded module in the conventional sense.Since M is nonnegatively monotone, the equivariant quantum product ∗ ρ wouldbe well-defined with the graded completed tensor product replaced by an ordinarytensor product, however the resulting equivariant quantum cohomology would fail tobe isomorphic with our equivariant Floer cohomology (c.f. (4.14)).7. Equivariant quantum Seidel map
Let e σ be a lift of a linear Hamiltonian circle action of nonnegative slope and ρ asymplectic circle action linear at infinity. Assume σ and ρ commute.7.1. Clutching bundle action.
The sphere S has a natural rotation action givenby θ · ( s, t ) = ( s, t + θ ) away from the poles, using the parameterisation (6.4).The clutching bundle E from Section 6.2.1 admits a smooth circle action given by (cid:26) D − × M ∋ ( z, m ) ( e πiθ z, ( σ ∗ ρ ) θ ( m )) ∈ D − × M D + × M ∋ ( z, m ) ( e − πiθ z, ρ θ ( m )) ∈ D + × M , (7.1)which glues correctly along the equator. We denote this action by ρ E . The projectionmap is equivariant with respect to the action ρ E on E and to the rotation action on S . The poles z ± are fixed points of the rotation of S , so ρ E restricts to a circleaction in each of the fibres E z ± : the action on E z + is the action ρ and the actionon E z − is the action σ ∗ ρ . Here, we have identified these fibres E z ± with M via theinclusion maps ι ± .7.2. Equivariant quantum Seidel map definition.
The equivariant quantum Sei-del map is a version of the quantum Seidel map Q S which is equivariant with respectto the circle action ρ E of Section 7.1. Since the action ρ E restricts to different actions N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 43 on the two fibres over the poles, the equivariant quantum Seidel map will map betweenthe equivariant quantum cohomology for these two different circle actions.Let f eq , ± be equivariant Morse-Smale functions for the circle actions in the corre-sponding to the fibre E z ± , so f eq , + is equivariant with respect to the action ρ and f eq , − is equivariant with respect to the action σ ∗ ρ .We require s -dependent perturbations f eq s , ± of the equivariant Morse data andan S ∞ -dependent almost complex structure b J eq which is equivariant in the sense ofSection 6.3.2 with respect to ρ E . This almost complex structure should have similarproperties to the non-equivariant b J , so that Dπ is holomorphic and the fibrewiserestriction b J eq | T vert E is a Ω -compatible almost complex structure for all w ∈ S ∞ , and b J eq | T vert E is convex and b J eq has the form (6.6) at all points in a region at infinity.The regularity conditions will guarantee the moduli spaces below are smooth man-ifolds which in dimensions 0 and 1 compactify without bubbling.The equivariant quantum Seidel map counts ( S -equivalence classes of) quadru-ples ( v, γ − , γ + , u ) , where v is a flowline in S ∞ , the curves γ + : [0 , ∞ ) → M and γ − : ( −∞ , → M are equivariant f eq , ± s ( v ( s ) , · ) -flowlines and u is a b J eq v (0) -holomorphicsection satisfying u ( z ± ) = γ ± (0) (see Figure 2). It is a degree- I ( e σ ) Λ -module homo-morphism EQ S e σ : EQH ∗ ρ ( M ) → EQH ∗ +2 I ( e σ ) σ ∗ ρ ( M ) . A standard homotopy argumentshows that EQ S e σ commutes with the geometric Z [ u ] -module structure.7.3. Equivariant gluing.
The results of Section 6.2.3 extend to the equivariantsetup. The equivariant PSS maps are the Λ[ u ] -module isomorphisms E PSS − : EF H ∗ ρ ( M ; H eq , ) → EQH ∗ ρ ( M ) E PSS + : EQH ∗ ρ ( M ) → EF H ∗ ρ ( M ; H eq , ) (7.2)which count equivariant spiked discs. Here, H eq , is an equivariant Hamiltonian ofslope zero with respect to the action ρ . The equivariant version of (6.10) is thefollowing commutative diagram. EQH ∗ ρ ( M ) EQH ∗ +2 I ( e σ ) σ ∗ ρ ( M ) EF H ∗ ρ ( M ; H eq , ) EF H ∗ +2 I ( e σ ) σ ∗ ρ ( M ; H eq , ) EF H ∗ +2 I ( e σ ) σ ∗ ρ ( M ; σ ∗ H eq , ) EQ S e σ E PSS + ∼ = EF S e σ ∼ = E PSS − ∼ = continuation map (7.3)If σ has positive slope, then Ritter’s argument (5.8) combined with (7.3) implies thatthe direct limit of the direct system EQH ∗ ρ ( M ) EQ S e σ −−−−→ EQH ∗ +2 I ( e σ ) σ ∗ ρ ( M ) EQ S e σ −−−−→ EQH ∗ +4 I ( e σ )( σ ) ∗ ρ ( M ) EQ S e σ −−−−→ · · · (7.4)is isomorphic to equivariant symplectic cohomology ESH ∗ ρ ( M ) . v v α γ + γ − γ α z α uc k α x α c k + x + c k − x − Figure 3.
The weighted equivariant quantum Seidel map countsequivalence classes of septuples ( v, γ + , γ − , u ; v α , γ α , z α ) . The section u of the clutching bundle intersects the flowline γ α over the point z α ∈ S .7.4. Intertwining relation.
The intertwining result (6.8) does not hold in our equi-variant setup. Instead, we have the formula EQ S e σ ( x ∗ ρ (( ι + ) ∗ α )) − EQ S e σ ( x ) ∗ σ ∗ ρ (( ι − ) ∗ α ) = u ⌣ EQ S e σ, α ( x ) . (7.5)In this equation, α ∈ EH ∗ ρ E ( E ) is an equivariant cohomology class of the clutchingbundle, and the map EQ S e σ, α is the α -weighted equivariant quantum Seidel mapdefined in the following section. The proof of (7.5) takes up the rest of this section.7.4.1. Weighted equivariant quantum Seidel map.
Fix an invariant Riemannian metricon the clutching bundle E for the action ρ E , and let f eq E be an equivariant Morse-Smalefunction for this action.Take a regular choice of the data f eq , ± s and b J eq from Section 7.2. We use an s -dependent perturbation f eq ,αE,s of f eq E on [0 , ∞ ) . We will consider septuples ( v, γ + ,γ − , u ; v α , γ α , z α ) where ( v, γ + , γ − , u ) is a quadruple from Section 7.2, z α ∈ S is apoint in the sphere and ( v α , γ α ) : [0 , ∞ ) → S ∞ × E is an equivariant f eq ,αE,s flowlinesatisfying v α (0) = v (0) and γ α (0) = u ( z α ) (see Figure 3). Given any equivariantcritical point α = ( c k α , x α ) of f eq E , together with equivariant critical points x ± =( c k ± , x ± ) ∈ Crit( f eq ) and a class A ∈ Γ , denote by M ( x − , x + , A ; α ) the moduli spaceof S -equivalence classes of septuples as above with the obvious limits and with u of class S e σ + A . For regular perturbations, these moduli spaces are smooth oriented N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 45 manifolds of dimension dim M ( x − , x + , A ; α ) = (cid:12)(cid:12) x − (cid:12)(cid:12) − (cid:12)(cid:12) x + (cid:12)(cid:12) − | α | + 2 c ( A ) + 2 . (7.6)The +2 comes from the -dimensional freedom of the point z α ∈ S . Moreover, wecan assume z α / ∈ { z ± } for moduli spaces of dimension 0 and 1 by imposing furtherregularity conditions on f eq ,αE,s .Counting this moduli space yields a map EQ S e σ, α : QC ∗ ρ ( M ) → QC ∗ +2 I ( e σ )+ | α |− σ ∗ ρ ( M ) . (7.7)Our definition may be immediately extended linearly to any α ∈ EC ∗ ρ E ( E ) . The1-dimensional moduli spaces are compactified by any of the flowlines breaking sincethere is no bubbling by regularity. This yields the equation d EQ S e σ, α ( x ) = EQ S e σ, α d ( x ) + ( − | x | EQ S e σ,d α ( x ) . (7.8)Thus for closed Morse cochains α , the map EQ S e σ, α is a chain map. Remark 7.1 (Interpretation) . Let k α = 0 and | x α | = 2 . For any quadruple ( v,γ + , γ − , u ) from Section 7.2, the flowline v α generically flows to the minimum c = c k α . Therefore, the count of flowlines γ α with γ α (0) = u ( z α ) for some z α ∈ S and γ α ( ∞ ) = x α recovers the number [ x α ] ( u ∗ (cid:2) S (cid:3) ) . This is the evaluation of thedegree-2 cohomology class [ x α ] on the degree-2 homology class ( u ∗ (cid:2) S (cid:3) ) . As such, themap EQ S e σ, α is a weighted version of EQ S e σ under which any section u has weight [ x α ] ( u ∗ (cid:2) S (cid:3) ) .7.4.2. Fix the line of longitude L = R > ∩ D ± ⊂ S ,which does not include the poles. One way to derive the commutativity of the quantumproduct ∗ and the quantum Seidel map Q S e σ is to take a non-equivariant version of themoduli space from Section 7.4.1 in which z α ∈ L . The 1-dimensional moduli space iscompactified by breaking one of the flowlines, or allowing a bubble over the pole when z α → z ± . If x α is closed, this yields a chain homotopy between ( ι − ) ∗ x α ∗ Q S e σ ( · ) and Q S e σ (( ι + ) ∗ x α ∗ · ) .The intersection condition z α ∈ L does not transform correctly under the S -action on the equivariant moduli space, however. To rectify this, take a further s -dependent perturbation F s of the Morse function F on S ∞ . We consider octu-ples ( v, γ + , γ − , u ; v α , γ α , z α ; v ) which extend the septuples in the construction ofSection 7.4.1 (see Figure 4). The map v : [0 , ∞ ) → S ∞ is a F s -flowline whichsatisfies the intersection v (0) = v (0) and the limit v ( ∞ ) ∈ c . We moreover im-pose z α ∈ τ ( v ( ∞ )) − · L , where the action · on S is the rotation action defined inSection 7.1. Notice that this condition is preserved for all ( v ( ∞ ) , z α ) ∈ c × S by thenatural circle action on c × S because c ⊂ S ∞ has the inverse action in accordancewith (4.3). The effect of this construction is that the 2-dimensional freedom of thepoint z α has been reduced to a 1-dimensional freedom ‘along L ’; the second dimensionof freedom has been absorbed into the S -action. Remark 7.2 (Regularity) . We impose regularity conditions on the data so that themoduli spaces of the above octuples are smooth manifolds, as well as the modulispaces of the above octuples without the condition z α ∈ τ ( v ( ∞ )) − · L and with v v α γ + γ − γ α z α uc k α x α c k + x + c k − x − c τ − Lv Figure 4.
The map K α counts equivalence classes of tuples likethe weighted equivariant quantum Seidel map in Figure 3, howeverwith an additional flowline v in S ∞ . Moreover, we restrict to tupleswhich satisfy z α ∈ τ − L , where τ = τ ( v ( ∞ )) is determined by theadditional flowline in S ∞ . v ( ∞ ) ∈ c k for any k . Moreover, we use regularity conditions to avoid unnecessaryintersections over the poles by asking that the projection [ v (0) , z α ] : M → S ∞ × S S intersects CP ∞ × { z ± } transversally for all the above moduli spaces. We imposefurther regularity conditions to ensure that we control the behaviour of configurationswith bubbles just as in Seidel’s argument [Sei97, Section 7].We quotient by the free S -action to get the moduli space of S -equivalence classesof above octuples with the obvious constraints, which we denote by M τ ( x − , x + , A ; α ) .It is a smooth oriented manifold of dimension dim M τ ( x − , x + , A ; α ) = (cid:12)(cid:12) x − (cid:12)(cid:12) − (cid:12)(cid:12) x + (cid:12)(cid:12) − | α | + 2 c ( A ) + 1 . (7.9)Denote by K α the map EQC ∗ ρ ( M ) → EQC ∗ +2 I ( e σ )+ | α |− σ ∗ ρ ( M ) which counts these mod-uli spaces.7.4.3. Boundary of the moduli space.
By standard compactification and gluing ar-guments, the 1-dimensional moduli spaces M τ ( x − , x + , A ; α ) will have a boundarycomposed of broken flowlines and bubbled spheres. The sum of these boundary com-ponents will be zero. Subject to further homotopies, this sum yields the desiredrelation (7.5). We list the various components of the boundary below, and detail howthey contribute to the sum. N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 47 c k α x α c k + x − c k − x + c c homotopy argument c k α x α c k + x + c k − x − c u ⌣EQ S e σ, α Figure 5.
When the flowline v breaks, we can use a homotopyargument to split the moduli space into multiplication by u and theweighted equivariant Seidel map. The flowline from c to c containsredundant information so we omit it. ( v | ( −∞ , , γ − ) breaking:: We get a contribution d K α ( x + ) . ( v | [0 , ∞ ) , γ + ) breaking:: We get a contribution K α ( d x + ) . ( v α , γ α ) breaking:: We get a contribution ( − | x + | K d α ( x + ) . If α is a closed cochain,then this contribution vanishes. v breaking:: As a consequence of the regularity conditions, the only possible break-ing of the flowline v will be to the critical point c ∈ Crit( F ) . Consider such abreaking, but before we have taken the quotient by the S -action. The brokenflowline consists of a flowline v whose limit is v ( ∞ ) = y ∈ c and a secondunparameterised F -flowline from y to τ ∈ S ∼ = c , the identification S ∼ = c given by the map τ . In such a configuration, the point z α cannot be eitherof the poles because of the regularity conditions. As a consequence, the point z α uniquely determines τ ∈ S via z α ∈ τ − · L . It may be explicitly shownthat there is a unique F -flowline whose limits are any specified points of c and c . It follows that the flowline from y to τ may be omitted without lossof generality.A standard homotopy argument will separate the flowlines v and v α in S ∞ (see Figure 5). Therefore when v breaks, we get a contribution of − u ⌣EQ S e σ, α ( x + ) to the sum, up to chain homotopy. z α → z ± with bubbling:: Due to the regularity conditions, the only possible bub-bling configuration is a single bubble in the fibre over one of the two poles z ± with the flowline to x α starting within the fibre. In this configuration, we get z α ∈ { z ± } , so the condition z α ∈ τ ( v ( ∞ )) − · L is automatically satisfied(for any value of v ( ∞ )) . As such, the flowline v may be omitted withoutlosing any information because it no longer constrains the point z α .We will treat the case of a bubble over the pole z + . If the bubble is of class B ∈ Γ , then the section u is of class S e σ + A − B . We use a standard homotopyargument to insert a broken flowline between the section and the bubble, andalso to extend the flowline to α so it breaks into a half-flowline to ( ι + ) ∗ α and c k α x α c k + x − c k − x + c homotopy argument c k α x α c k + x + c k − x − fibre EQ S e σ ∗ ρ ( ι + ) ∗ α Figure 6.
When the section bubbles over the south pole, we can usea homotopy argument to get an equivariant quantum product in thefibre above the south pole followed by the map EQ S e σ . The flowline to c is redundant, so we remove it.an equivariant functorial flowline from there to α (see Figure 6). The resultis a contribution of EQ S e σ ( x + ∗ ρ (( ι + ) ∗ α )) to the sum, up to chain homotopy.Analogously, a bubble over the pole z − yields a contribution − EQ S e σ ( x + ) ∗ σ ∗ ρ (( ι − ) ∗ α ) (7.10)to the sum, up to chain homotopy.Summing these contributions, we get the equation − u ⌣ EQ S e σ, α ( x ) + EQ S e σ ( x ∗ ρ (( ι + ) ∗ α )) − EQ S e σ ( x ) ∗ σ ∗ ρ (( ι − ) ∗ α ) = 0 (7.11)for x ∈ EQH ∗ ρ ( M ) , which rearranges to give (7.5) as desired.8. Examples
Complex plane.
The complex plane C is an exact open symplectic manifoldwhose symplectic form is given by ω = d x ∧ d y at points z = x + iy ∈ C . Thecontact form α = π d t on S and the isomorphism ψ : S × [1 , ∞ ) → C given by ( t, R )
7→ √ Re πit gives C the structure of a convex symplectic manifold. The setof Reeb periods is R = π Z . The smooth circle action σ : S × C → C given by Our construction of the pullback maps is a variant of [RV14, Section 1.3], in which we allow s -dependent perturbations of the Morse data instead of perturbing the function. Let M ± be manifoldswith Morse-Smale functions f ± : M ± → R (and metrics and orientation data). Given any ϕ : M − → M + , a functorial flowline is a pair of half-flowlines ( γ − : ( −∞ , → M − , γ + : [0 , ∞ ) → M + ) of s -dependent perturbations of the functions f ± which satisfy ϕ ( γ − (0)) = γ + (0) . When ϕ is animmersion or a submersion, it follows from standard arguments that the moduli spaces of functorialflowlines between critical points x ± is a smooth manifold of dimension ind( x − ; f − ) − ind( x + ; f + ) . Themap which counts these moduli spaces is a chain map and is denoted ϕ ∗ . The equivariant pullbackmaps are defined analogously. N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 49 θ · z = e πiθ z has Hamiltonian K σ ( z ) = π | z | and is linear of slope π . The action hasa unique fixed point C ∈ C . The lift of σ to f L C is unique because C is contractible.This lift e σ fixes the point ( S C , D C ) and has Maslov index I ( e σ ) = 2 .Let H λ linear : C → R be the autonomous linear Hamiltonian z λ | z | . If λ / ∈ R ,then the unique Hamiltonian orbit of H λ linear is the constant loop at C and has Conley-Zehnder index − (cid:4) λπ (cid:5) [Oan04, Section 3.2]. Therefore, for λ / ∈ R , the Floer cochaincomplex is F C ∗ ( C ; H λ linear ) ∼ = (cid:26) Z if ∗ = − (cid:4) λπ (cid:5) , otherwise. (8.1)The symplectic cohomology of C thus vanishes. In contrast, the equivariant cohomol-ogy ESH ∗ Id C ( C ) is isomorphic to Q [ u , u − ] as a Z [ u ] -module for the geometric and thealgebraic module structures. We derive this isomorphism in the course of our discus-sion below (see (8.7)). With the equivariant Seidel map, which is an isomorphism, weget further Z [ u ] -module isomorphisms ESH ∗ σ r ( C ) ∼ = Q [ u , u − ] for r ∈ Z .Give C the standard Riemannian metric, which is σ -invariant. The Hamiltonianfunction K σ is Morse-Smale with respect to this metric and has exactly one criticalpoint: the fixed point C of Morse index 0. Since K σ is invariant, the function K eq σ : S ∞ × C → R given by ( w, z ) K σ ( z ) is a regular equivariant Morse functionon C . Thus the Morse cohomology of C is H ∗ ( C ; K σ ) ∼ = Z h C i and the equivariantMorse cohomology is EH ∗ σ r ( C ; K eq σ ) ∼ = Z [ u ] h C i for any r ∈ Z . Since the Novikovring is Λ = Z , the quantum cohomology and the equivariant quantum cohomology aresimply the Morse cohomology and the equivariant Morse cohomology respectively.We compute the equivariant quantum Seidel map for the lifted action e σ . We cando this for the underlying actions ρ = ( σ r ) ∗ Id C = σ − r , where r is nonnegative and Id C is the identity action on C . Theorem 8.1.
Let r ≥ . The equivariant quantum Seidel map EQ S e σ : EH ∗ σ − r ( C ; K eq σ ) → EH ∗ +2 σ − ( r +1) ( C ; K eq σ ) (8.2) is the Z [ u ] -linear extension of C ( r + 1) u C . Rather than finding the equivariant quantum Seidel map directly, we will appealto (7.3) and opt to find the continuation map instead. We will use the sequence ofHamiltonians defined by Zhao in [Zha19, Section 8.1], which we outline below.For each nonnegative integer s ∈ Z ≥ , let H sπ +1quadratic : S × C → R be a time-dependent Hamiltonian function such that: • On | z | < , the function is negative, achieves its minimum at C and is Morsewith exactly one critical point at C ; • On < | z | < sπ + 2 , the function H sπ +1quadratic equals ( | z | − plus a smalltime-dependent perturbation around | z | = jπ + 1 for j = 1 , . . . , s ; and • On | z | > sπ + 2 , the function H sπ +1quadratic is linear of slope sπ + 1 . Give W u (0 C ) = { C } the orientation +1 . We slightly change H sπ +1quadratic near | z | = sπ + 2 so that the function is smooth. The Hamiltonian orbits of H sπ +1quadratic occur only at C and at R = | z | = jπ + 1 .Modulo the perturbations, the slope of H sπ +1quadratic when R = jπ + 1 is d (cid:16) H sπ +1quadratic (cid:17) d R = dd R
12 ( | z | − = R − jπ. (8.3)Thus the slope at R = jπ + 1 is the Reeb period jπ ∈ R . If we hadn’t perturbed H sπ +1quadratic in this region, we would get a S -family of Hamiltonian orbits correspondingto the Reeb orbit of period jπ , as per (3.5). Instead, as a result of Zhao’s perturbation,we get two Hamiltonian orbits, which we denote by x j − and x j . Denote by x theconstant Hamiltonian orbit at C . The orbit x l has degree − l for all ≤ l ≤ s .With this choice of Hamiltonian, the equivariant Floer cochain complex (with re-spect to the trivial action Id C ) is given by EF C ∗ Id C ( C ; H sπ +1quadratic ) = M k ≥ s M j =0 Z h ( c k , x j ) i . (8.4) Lemma 8.2 ( [Zha19, Section 8.1]) . There exist choices of all remaining data suchthat the differential of (8.4) is given by d ( c k , x j − ) = ( c k , x j − ) − j ( c k +1 , x j ) , d ( c k , x j ) = 0 (8.5) and the continuation map κ s : EF C ∗ Id C ( C ; H sπ +1quadratic ) → EF C ∗ Id C ( C ; H ( s +1) π +1quadratic ) is theinclusion map on the cochain complex. Using this explicit cochain complex, we deduce that the inclusion map Z [ u ] h x s i ∼ = L k Z h ( c k , x s ) i ֒ → EF C ∗ Id C ( C ; H sπ +1quadratic ) induces an isomorphism on cohomology.Moreover, with respect to this isomorphism, the continuation map κ s is the map x s ( s + 1) u x s +1) . Explicitly, we have EF H ∗ Id C ( C ; H sπ +1quadratic ) EF H ∗ Id C ( C ; H ( s +1) π +1quadratic ) Z [ u ] h x s i Z [ u ] h x s +1) i κ s ∼ = x s ( s +1) u x s +1) ∼ = (8.6)so that the map κ s is really multiplication by ( s + 1) u .To compute the equivariant symplectic cohomology of C , it is enough to consider thedirect limits of the continuation maps κ s because the slopes of H sπ +1quadratic are arbitrarilylarge. The map κ s is multiplication by ( s + 1) u , so it contributes to “allowing division”by ( s + 1) u in the direct limit. Thus we get the isomorphism ESH ∗ Id C ( C ) ∼ = Q [ u , u − ] . (8.7) Zhao derived the equation d ( c k , x j − ) = ( c k , x j − ) + j ( c k +1 , x j ) , which has different signsto (8.5). Her result changes to (8.5) once we apply the rule u
7→ − u to account for our differentconventions, as in Remark 4.3. The equation u ⌣ ( c k , x s ) = ( c k +1 , x s ) holds in EF C ∗ Id C ( C ; H sπ +1quadratic ) for degree reasons, sothe inclusion is a Z [ u ] -module map for the geometric module structure. N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 51
Proof of Theorem 8.1.
Fix r ≥ . Consider the following diagram, which is a combi-nation of (5.7) and (7.3). The top square commutes on the cochain complexes whereasthe bottom square commutes on cohomology. EF C ∗− rσ ( C ; H rπ +1quadratic ) EF C ∗− rσ ( C ; H ( r +1) π +1quadratic ) EF C ∗ +2 σ − ( r +1) ( C ; ( σ r +1 ) ∗ H rπ +1quadratic ) EF C ∗ +2 σ − ( r +1) ( C ; ( σ r +1 ) ∗ H ( r +1) π +1quadratic ) EF C ∗ σ − r ( C ; ( σ r ) ∗ H rπ +1quadratic ) EQC ∗ σ − r ( C ) EQC ∗ +2 σ − ( r +1) ( C ) ∼ = EF S e σr +1 κ r ∼ = EF S e σr +1 ( σ r +1 ) ∗ κ r E PSS − σ − ( r +1) EF S e σ ∼ = EQ S e σ E PSS + σ − r (8.8)Here, we use the subscript on E PSS ± to record the underlying circle action so wecan distinguish the different maps. Moreover, we use the identity ( σ r ) ∗ Id C = σ − r tosimplify notation.An inspection of Zhao’s explicit perturbation finds that there is ε ∈ {± } suchthat E PSS + σ − s (0 C ) = ε ( σ s ) ∗ x s for all s ≥ because the asymptotic behaviour of aspiked disc is the same for all these maps (see [Rit13, Appendix B] for a characterisa-tion of the coherent orientation).Thus, in cohomology, the element C is mapped in (8.8) as below. ε x r ε ( r + 1) u x r +1) ε ( σ r +1 ) ∗ x r ε ( r + 1) u ( σ r +1 ) ∗ x r +1) ε ( σ r ) ∗ x r C ε ( r + 1) u C (8.9)Thus we have C ( r + 1) u C as desired. (cid:3) This result generalises to C n and the action θ · ( z , . . . , z n ) = ( e πiθ z , . . . , e πiθ z n ) .This action σ also has exactly one fixed point, C n . There exist analogous data todescribe the equivariant cohomology with C n the unique minimal critical point. Since C n is exact and equivariantly contractible, the equivariant quantum product is trivial. Theorem 8.3.
Let r ≥ . The equivariant quantum Seidel map EQ S e σ : EH ∗ σ − r ( C n ) → EH ∗ +2 nσ − ( r +1) ( C n ) is the Z [ u ] -linear extension of the assignment C n (( r + 1) u ) n C n . This ε will depend on the choice of coherent orientation that was made in Lemma 8.2. Proof.
A similar approach to the calculation for C , using the spectral sequences from[MR18, Corollary 7.2] to deduce the differential, yields the desired formula, howeveronly up to sign. Instead, we will derive the C n case directly from Theorem 8.1.By Remark 6.4, the only holomorphic section of the clutching bundle is the minimalfixed section at C n . It follows that EQ S e σ (0 C n ) may be characterised by intersectingequivariant flowlines with the fixed point C n . Thus EQ S e σ (0 C n ) = A C n σ − ( r +1) (0 C n ) ,where we define the map A below. The theorem immediately follows from (8.11),which allows us to express A C n σ − ( r +1) (0 C n ) as the n -th power of A C σ − ( r +1) (0 C ) , and thecomputation in C from Theorem 8.1.Let X be a manifold with a smooth circle action ϕ and fixed point p ∈ X . Equip X with an equivariant Morse function. The map A pϕ : EH ∗ ϕ ( X ) → EH ∗ +dim( X ) ϕ ( X ) counts equivariant s -dependent negative gradient trajectories that intersect p at s = 0 .Notice that, since p is a fixed point, this intersection condition is independent of therepresentative of the equivariant trajectory.Given two such manifolds X and Y with circle actions ϕ X and ϕ Y and fixed points p X and p Y respectively, consider the following diagram. S ∞ × S ( X × Y ) S ∞ × S XS ∞ × S Y S ∞ /S π Y π X (8.10)Standard homotopies yield the equation A ( p X ,p Y )( ϕ X ,ϕ Y ) ◦ ( π ∗ X ⌣ π ∗ Y ) = ( π ∗ X A p X ϕ X ) ⌣ ( π ∗ Y A p Y ϕ Y ) . (8.11) (cid:3) Projective space.
The complex projective space P n is a closed monotone Kählermanifold with the Fubini-Study symplectic form ω FS and the Fubini-Study metric. ItsNovikov ring is Λ = Z [ q, q − ] , where q is a formal variable of degree n + 1) . Thestandard Morse function on P n is the function f P n ([ z : · · · : z n ]) = P nk =0 k | z k | . Thecritical points of f P n are the unit vectors e k , each with Morse index ind( e k ; f P n ) = 2 k .The Hamiltonian circle action σ given by θ · [ z : z : · · · : z n ] = [ z : e πiθ z : · · · : e πiθ z n ] preserves the metric and the function f P n . As per Section 8.1, this means wecan form a canonical equivariant Morse function f eq P n from f P n .We use the Morse functions f P n and f eq P n to describe a basis for the various coho-mologies below. In (8.12), we give module isomorphisms to each of the cohomologies,and describe the corresponding products (which are determined by the given informa-tion). The global minimum e is the unit for all products. In the following, r is any Give W u ( e k ) the orientation that comes naturally from the complex structure. As for C , we have u ⌣ ( c l , e k ) = ( c l +1 , e k ) , so we use the shorthand u l e k for the equivariantcritical point ( c l , e k ) . N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 53 nonnegative integer. H ∗ ( P n ) ∼ = Z h e k i nk =0 , e ⌣ e k = (cid:26) e k +1 < k < n,k = n.EH ∗ σ − r ( P n ) ∼ = Z [ u ] h e k i nk =0 , e ⌣ − r e k = (cid:26) e k +1 − r u e k − r u e n < k < n,k = n.QH ∗ ( P n ) ∼ = Λ h e k i nk =0 , e ∗ e k = (cid:26) e k +1 qe < k < n,k = n.EQH ∗ σ − r ( P n ) ∼ = Λ b ⊗ Z [ u ] h e k i nk =0 , e ∗ − r e k = (cid:26) e k +1 − r u e k qe − r u e n < k < n,k = n. (8.12)Let e σ be the unique lift of σ to g L P n which fixes the point ( S e , D e ) .It has Maslov index n . For r ≥ , the equivariant quantum Seidel map EQ S e σ : EQH ∗ σ − r ( P n ) → EQH ∗ +2 nσ − ( r +1) ( P n ) is given by e n X l =0 ( r + 1) n − l u n − l e l e k k − X l =0 ( r + 1) k − − l q u k − − l e l . (8.13) Proof.
Our method of determining the equivariant coefficients in (8.12) and (8.13) is tofind certain coefficients directly and to deduce the remaining coefficients by repeatedapplication of the intertwining formula (7.5). We demonstrate our method for P .The non-equivariant products and non-equivariant quantum Seidel maps are knownfor P (for example, by [MT06, (5.13)]), hence we can write the equivariant quantumproducts, using unknown integer coefficients, as e ∗ − r e = e + α r u e + β r u e (8.14) e ∗ − r e = qe + γ r u e + δ r u e + ε r u e (8.15)and the equivariant quantum Seidel map EQ S e σ : EQH ∗ σ − r ( P ) → EQH ∗ +4 σ − ( r +1) ( P ) as e e + A r u e + B r u e e qe + C r u e + D r u e + E r u e e qe + F r q u e + G r u e + H r u e + I r u e . (8.16)By Remark 6.4, the only holomorphic section which contributes a q term in (8.16)is the minimal fixed section at e . By using only small perturbations of the invariantMorse function f P , we immediately deduce β r , δ r , ε r = 0 and C r , D r , E r , G r , H r , I r =0 (hence fading these terms above) because otherwise the function would increasealong its negative gradient trajectories. Moreover, the coefficient B r is a local countof equivariant trajectories that intersect the fixed point e , and is therefore ( r + 1) since this is the same count as in Theorem 8.3. Finally, since the Borel space S ∞ × S P decomposes as CP ∞ × P in the r = 0 case, we have α , γ = 0 . We apply the intertwining relation (7.4) with α = e and x = e k . By Remark 7.1and an algebraic topology calculation, the α -weighted equivariant quantum Seidelmap counts the fixed section at e with weight 0 and any section corresponding to q with weight 1. With x = e , we get EQ S e σ ( e ∗ − r e ) − EQ S e σ ( e ) ∗ − ( r +1) e − u EQ S e σ,e ( e )= EQ S e σ ( e ) − ( e + A r u e + ( r + 1) u e ) ∗ − ( r +1) e − qe − ( qe + γ r +1 u e ) − A r u ( e + α r +1 u e ) − ( r + 1) u e = − ( γ r +1 + A r ) u e − ( A r α r +1 + ( r + 1) ) u e , (8.17)with x = e , we get EQ S e σ ( e ∗ − r e ) − EQ S e σ ( e ) ∗ − ( r +1) e − u EQ S e σ,e ( e )= EQ S e σ ( e + α r u e ) − qe ∗ − ( r +1) e − u ( qe )= ( qe + F r q u e ) + α r u ( qe ) − qe − q u e = ( F r + α r − q u e , (8.18)and with x = e , we get EQ S e σ ( e ∗ − r e ) − EQ S e σ ( e ) ∗ − ( r +1) e − u EQ S e σ,e ( e )= EQ S e σ ( qe + γ r u e ) − ( qe + F r q u e ) ∗ − ( r +1) e − u ( qe + F r q u e )= q ( e + A r u e + ( r + 1) u e ) + γ r u ( qe + F r u qe ) − q ( e + α r +1 u e ) − F r q u e − q u e − F r q u e = ( A r + γ r − α r +1 − F r − q u e + (( r + 1) + γ r F r − F r ) q u e . (8.19)The coefficients of (8.17), (8.18) and (8.19) yield the following silmultaneous equations. γ r +1 = − A r (8.20) A r α r +1 = − ( r + 1) (8.21) F r = − α r + 1 (8.22) α r +1 − A r = γ r − − F r (8.23) ( γ r − F r = − ( r + 1) (8.24)By induction, set α r , γ r = − r . Either (8.22) or (8.24) yields F r = r + 1 . The uniquesolution to (8.21) and (8.23) is α r +1 = − ( r + 1) and A r = r + 1 . Finally, (8.20) yields γ r +1 = − ( r + 1) . This proves the induction hypothesis α r +1 , γ r +1 = − ( r + 1) andhence completes the proof. (cid:3) Remark 8.4 (Inverse action) . The inverse circle action σ − is another Hamiltoniancircle action for which we can compute the equivariant quantum Seidel map. We have There is a degree-2 equivariant cohomology class α in the clutching bundle which restricts to e at both poles. N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 55 EQ S e σ EQ S e σ − = EQ S e σ e σ − = Id , so we can deduce the map EQ S e σ − by inverting(8.13). Thus the map EQ S e σ − : EQH ∗ σ − ( r +1) ( P n ) → EQH ∗− nσ − r ( P n ) is given by e q − e ,e k
7→ − ( r + 1) u q − e k + q − e k +1 , ≤ k < ne n
7→ − ( r + 1) u q − e n + e . (8.25)The assignment e q − e may be deduced directly via minimal fixed sections usingRemark 6.4. Here, e is the Poincaré dual of the subset { z = 0 } , which is the minimallocus of − K σ . No sections other than these minimal fixed sections can contribute to EQ S e σ − ( e ) for degree reasons.8.3. Tautological line bundle on projective space.
The total space O P n ( − ofthe tautological line bundle over projective space P n is one of the negative line bundlesstudied by Ritter [Rit14]. The elements of O P n ( − are of the form ([ z ] , v ) = ([ z : . . . : z n ] , ( v , . . . , v n )) ∈ P n × C n +1 , where z and v are linearly dependent. Denote by Z the image of the zero section, giving Z = { v = 0 } ∼ = P n .There is a symplectic form ω on O P n ( − such that ( O P n ( − , ω ) is a monotoneconvex symplectic manifold whose fibres and whose submanifold Z are symplecticsubmanifolds [Rit14, Section 7]. Its Novikov ring is Λ = Z [ q, q − ] , where q is a formalvariable of degree n .Let σ be the linear Hamiltonian circle action given by θ · ([ z ] , v ) = ([ z ] , e πiθ v ) . Itrotates the fibres and its fixed point set is Z . Set e σ to be the unique lift of σ whichfixes the points ( S ([ z ] , , D ([ z ] , . The Maslov index of e σ is 2.The invariant Morse function given by f (([ z ] , v )) = P nk =0 k | z k | + | v k | has criticalpoints at each of the unit vectors in Z . Denote the k -th such critical point by e k .It has Morse index ind( e k ; f ) = 2 k . Give O P n ( − the metric which is the restrictionof the standard metric on C n +1 × C n +1 , so that we recover the Fubini-Study metricalong the zero section. Thus the Morse cohomology of O P n ( − is H ∗ ( O P n ( − ∼ = Z h e k i nk =0 , e ⌣ e k = (cid:26) e k +1 k < n,k = n. (8.26)Since O P n ( − equivariantly retracts onto Z (with the trivial circle action), its equi-variant cohomology EH ∗ σ r ( O P n ( − is ring isomorphic to Z [ u ] ⊗ Z H ∗ ( O P n ( − .The quantum cohomology is QH ∗ ( O P n ( − ∼ = Λ h e k i nk =0 , e ∗ e k = (cid:26) e k +1 − qe k < n,k = n. (8.27)The quantum product is deduced via (6.8) from the the quantum Seidel map whichRitter computed [Rit14, Lemma 60]: Q S e σ ( e k ) = ( − e k +1 k < n,qe k = n. (8.28)The equivariant quantum cohomology is Λ b ⊗ Z [ u ] h e k i nk =0 . The equivariant quantumproduct and equivariant quantum Seidel maps are given by the following theorems. Orient W u ( e k ) according to the natural complex structure. Theorem 8.5.
For r ≥ , the equivariant quantum product on EQH ∗ σ − r ( O P n ( − isgiven by e ∗ − r e k = ( e k +1 k < n, − qe + r u qe k = n. (8.29) Theorem 8.6.
For r ≥ , the equivariant quantum Seidel map EQ S e σ : EQH ∗ σ − r ( O P n ( − → EQH ∗ +2 σ − ( r +1) ( O P n ( − (8.30) is given by e k ( − e k +1 + ( r + 1) u e k k < n,qe + ( r + 1) u e n − ( r + 1) u qe k = n. (8.31) Proof of Theorem 8.5 and Theorem 8.6.
We use exactly the same method as that weused in Section 8.2. Write the equivariant quantum products and equivariant quan-tum Seidel maps using variables for the unknown coefficients. By using only a smallperturbation of the Morse function, deduce that all coefficients are zero apart fromthose which occur above. The coefficients of the u e k terms and the u e n term in (8.31)are all ( r + 1) because the fibre locally resembles C . Apply the intertwining formula(7.5) to the elements e n − and e n to deduce the two remaining coefficients. (cid:3) Deducing equivariant symplectic cohomology.
The symplectic cohomology is thelimit of the quantum Seidel map (8.28). We have Q S e σ ( e n + qe ) = 0 , and Q S e σ is anisomorphism after quotienting by e n + qe . Thus we get Λ -algebra and Λ -moduleisomorphisms SH ∗ ( O P n ( − ∼ = alg. Z [ q, q − ] h e k i nk =0 ( e n + qe ) ∼ = mod. n − M k =0 Z [ q, q − ] h e k i . (8.32)In particular, in every even degree it has one copy of Z .The equivariant quantum Seidel map EQ S e σ from (8.30) is injective, and its deter-minant is det EQ S e σ = ( r + 1) n +1 u n +1 . To find the direct limit of the maps EQ S e σ ,we must use a different strategy.We localise the ring Λ b ⊗ Z [ u ] so the determinants are invertible. We denote thislocalisation by Λ b ⊗ Z [ u ] local . The degree- l elements in Λ b ⊗ Z [ u ] local are given as perthe graded completed tensor product (6.14), except finitely-many negative powers of u are permitted, and we tensor with Q . Denote by EQH ∗ σ ( O P n ( − local the tensorproduct EQH ∗ σ ( O P n ( − ⊗ (cid:0) Λ b ⊗ Z [ u ] local (cid:1) .Consider the following commutative diagram. EQH ∗ σ ( O P n ( − EQH ∗ +2 σ − ( O P n ( − EQH ∗ +2 σ − ( O P n ( − EQH ∗ σ ( O P n ( − local EQ S e σ EQ S − e σ EQ S e σ EQ S − e σ ◦ EQ S − e σ (8.33) N INTERTWINING RELATION FOR EQUIVARIANT SEIDEL MAPS 57
The direct limit of the maps EQ S e σ is the image of the (injective) dashed maps above.This gives ESH ∗ σ ( O P n ( − ∼ = ∞ [ p =0 n image (cid:16) ( EQ S − e σ ) p : EQH ∗ +2 pσ − p → EQH ∗ σ , local (cid:17)o . (8.34)We immediately deduce that ESH ∗ σ ( O P n ( − is not a finitely-generated Λ b ⊗ Z [ u ] -module because this is a strictly increasing chain of submodules. Moreover, we canfollow the element e n under the maps EQ S e σ to get e n qe + o ( u )
7→ ± qe + o ( u )
7→ · · · 7→ ± qe n + o ( u )
7→ ± q e + o ( u )
7→ · · · , (8.35)which implies that e n is not divisible by u in the direct limit (for none of the images aredivisible by u ). Therefore ESH ∗ σ ( O P n ( − is a proper submodule of the localisation.8.3.2. Finding generators.
Recall the adjugate (or adjoint ) of a nonsingular matrix A is the unique matrix A ∗ such that A ∗ A = AA ∗ = det( A )Id , so that if the inverse of A exists, it is A − = A ) A ∗ . Therefore, to find the image of the inverses ( EQ S − e σ ) p , wefind the image of the adjugates (which is a submodule of EQH ∗ σ ( O P n ( − withoutlocalisation), and then divide these elements by the determinants.We have det( EQ S p e σ ) = Q p − r =0 (( r + 1) n +1 u n +1 ) . We characterise the image of theadjugate below.Using (8.31), we have EQ S e σ ( e n + qe ) = ( r + 1) u e n . Thus, with respect to theordered basis h e n + qe , e , e , . . . , e n − , e n i , the map EQ S e σ is given by the followingmatrix. A r = · · · − ( r + 1) u r + 1) u · · · q − r + 1) u ... . . . . . . . . . ... . . . ( r + 1) u r + 1) u − r + 1) u (8.36)Note that if we permute the first and last column, the matrix (8.36) becomes lowertriangular. Set x pk = A ∗ · · · A ∗ p − e k for k = 1 , . . . , n , and set x p = A ∗ · · · A ∗ p − ( e n + qe ) .Thus the image of ( EQ S ∗ e σ ) p above is the span of { x p , . . . , x pn } . Using (8.36), we canset up a recursive formula for the x pk , which starts like x p +1 n = ( p + 1) n u n x p x p +1 n − = ( p + 1) n u n x pn − + ( p + 1) n − u n − x p . . . (8.37) Example 8.7 ( n = 1 ) . For O P ( − , the system (8.37) becomes (cid:26) x p +11 = ( p + 1) u x p x p +10 = ( q + 2( p + 1) u ) x p − ( p + 1) u x p (8.38) and has solution x p = q p − (( q + p ( p +1) u ) e − u e )+ o ( u ) . Since x p +11 = ( p +1) u x p , onecopy of ( p +1) u cancels upon division by the determinant. This yields the presentation ESH ∗ σ ( O P ( − ∼ = Λ b ⊗ Z [ u ] (cid:28) x p · · · p ( p + 1) u p +1 (cid:29) ∞ p =1 . (8.39) Remark 8.8 (Possible nicer presentation) . The author was unable to establish whetherthere is an element X ∈ ESH ∗ σ ( O P n ( − which is divisible by every power of u , orindeed by every determinant. If so, this would produce an isomorphism ESH ∗ σ ( O P ( − ∼ = Λ b ⊗ Z [ u ] ⊕ Λ b ⊗ Z [ u ] local (8.40)in the n = 1 case, and similar isomorphisms for n > . References [AD14] M. Audin and M. Damian.
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