aa r X i v : . [ m a t h . G T ] F e b A SYMPLECTIC FORMULA OF GENERALIZED CASSONINVARIANTS
SHAOYUN BAI
Abstract.
Suppose Y is an integer homology 3-sphere, Taubes [Tau90] provedthat the number of irreducible critical orbits of the perturbed Chern-Simonsfunctional on Y , counted with signs, is equal to the algebraic intersection num-ber of two character varieties associated with Heegaard splittings when thestructure group is SU(2). Taubes’ result established a relationship betweengauge theory and the Casson invariant. This article proves an analogous iden-tification result for SU( n ) generalized Casson invariants. As a special case, weshow that the SU(3) Casson invariant of Boden-Herald [BH98] can be equiva-lently calculated by taking an appropriate intersection number of Lagrangiansubmanifolds. Introduction
In 1985, Casson introduced an invariant for integer homology 3-spheres (see[AM90,Mar88]) by considering the SU(2)–representations of the fundamental group.Suppose Y is an integer homology 3-sphere with a Heegaard splitting Y = H ∪ Σ H ,then the inclusions of Σ in H and H induce embeddings of the SU(2)–charactervarieties of π ( H ) and π ( H ) into the SU(2)–character variety of π (Σ). Thecharacter variety of Y is given by the intersection of the character varieties of M and M as a set. After taking a sufficiently small generic perturbation onthe embeddings, one obtains a well-defined intersection number of the charactervarieties of π ( H ) and π ( H ). The intersection number is always even (see, forexample, [Sav11, Corollary 17.6]), and the Casson invariant is defined to be half ofthis intersection number.On the other hand, let P be the trivial SU(2) bundle over Y with a fixed triv-ialization, and let C ( Y ) be the affine space of smooth connections on P . Let θ bethe flat connection associated with the given trivialization of P . For B ∈ C ( Y ),suppose B = θ + b where b ∈ Ω ( Y, su (2)), then the Chern-Simons functional
CS : C ( Y ) → R is defined by CS( B ) = 12 Z Y Tr (cid:18) db ∧ b + 23 b ∧ b ∧ b (cid:19) . After fixing a Riemannian metric on Y , the formal gradient of CS at B ∈ C ( Y )is equal to ∗ F B , where ∗ is the Hodge star operator of the given metric; there-fore, the critical points of CS are given by flat connections, and the set of criticalorbits of CS has a one-to-one correspondence with the conjugacy classes of SU(2)–representations of π ( Y ). Taubes [Tau90] proved that after adding a generic per-turbation to the Chern-Simons functional, the signed count of its critical orbits isequal to two times the Casson invariant. A natural question about the Casson invariant is whether it can be extended tomore general groups and more general 3-manifolds. This question has been stud-ied intensively since the introduction of the theory, and many constructions aregiven using either intersection of character varieties arisen from Heegaard split-tings or the Chern-Simons functionals. The Heegaard-splitting approach has beenstudied by Boyer-Nicas [BN90], Walker [Wal92], Cappell-Lee-Miller [CLM90], andCurtis [Cur94]. More recently, Abouzaid-Manolescu [AM17] studied the intersec-tion of SL ( C ) character varieties arisen from Heegaard splittings and obtaineda sheaf-theoretic 3-manifold invariant. On the gauge theory side, Boden-Herald[BH98] defined an SU(3) Casson invariant for integer homology spheres by intro-ducing real-valued correction terms on the reducible critical orbits. Variations ofthe SU(3) Casson invariant have also been defined by Boden-Herald-Kirk [BHK01]and Cappell-Lee-Miller [CLM02]. Together with Zhang, the author [BZ20] gener-alized the construction of Boden-Herald and defined perturbative SU( n )–Cassoninvariants for integer homology spheres. The construction of [BZ20] is reviewed inSection 2.This article studies the relationship of the gauge-theoretic generalization of Cas-son invariants in [BZ20] with Heegaard splittings. We obtain an identification resultanalogous to Taubes’ theorem [Tau90]: the quantities entering into the definitionof the SU( n )–Casson invariant could all be expressed in terms of quantities con-structed using finite-dimensional symplectic geometry. As a special case, we showthat the SU(3) Casson invariant of Boden-Herald [BH98] is equal to an appropri-ate equivariant intersection number of character varieties associated with Heegaardsplittings.Now we describe the contents of this paper in more detail. Let Y = H ∪ Σ H be a closed, oriented 3-manifold with a Heegaard splitting, and let G be a compactLie group with Lie algebra g . In general, the G –character variety of π (Σ) havesingularities. Their appearance put forth difficulties on constructing perturbationsand studying intersection theory from the differential-geometric approach. To getaround this difficulty, we make use of the extended moduli space of flat connec-tions M g (Σ ′ ) on Σ ′ introduced by Jeffrey [Jef94] and Huebschmann [Hue95] tostudy an equivalent question using equivariant geometry, where Σ ′ be the Riemannsurface obtained from Σ be removing a disc. Roughly speaking, there is an opensubset ˆ M g (Σ ′ ) ⊂ M g (Σ ′ ) which is a smooth symplectic manifold and there is aHamiltonian G –action on ˆ M g (Σ ′ ) whose moment map reduction µ − (0) /G is nat-urally identified with the G –character variety of π (Σ). The G –character varietiesof π ( H ) and π ( H ) could be lifted to two smooth Lagrangian submanifolds L , L of ˆ M g (Σ ′ ) which are invariant under the G –action. They are also contained in µ − (0). Details of the definition and properties of the extended moduli space willbe reviewed in Section 3. Note that the extended moduli spaces were used in thework of Manolescu-Woodward [MW12] to define a symplectic version of instantonFloer homology.We are then interested in studying the G –equivariant intersection theory of L and L . In Section 4, we introduce a transversality result on Hamiltonian pertur-bations that allows us to perturb L and L equivariantly so that their intersectionis non-degenerate (see Definition 4.1). Actually, in Section 5.1, we introduce thenotion of a compatible pair of a holonomy perturbation f q on the 3–manifold Y and a G –equivariant Hamiltonian perturbation Φ H on ˆ M g (Σ ′ ) such that there is a SYMPLECTIC FORMULA OF GENERALIZED CASSON INVARIANTS 3 one-to-one correspondence between gauge orbits of f q –perturbed flat connectionson Y and G –orbits of Φ H ( L ) ∩ L . Moreover, the non-degeneracy of connection isequivalent to the non-degeneracy of Lagrangian intersection for such a compatiblepair. Therefore, we can use the equivariant transversality result on the symplecticside to construct preferred holonomy perturbation on Y to simplify the calculationof generalized Casson invariants.Let p, q be two intersection points of L and L after perturbation, and choosea disk D such that ∂D = γ ∪ γ , where γ and γ are two arcs connecting p and q on L and L respectively. Suppose H ⊂ G is a closed subgroup and forevery z ∈ D we assume that the stabilizer of z contains H as a subgroup. Forsuch data, we define an equivariant Maslov index µ H ( D ) in Section 4.2 which takesvalue in the representation ring of H . On the other hand, the disk D defines a1-parameter family B t ( D ) of (not necessarily) flat G connections on Y connectingthe perturbed flat connections on Y induced by p and q . In [BZ20], the authorsdefined the equivariant spectral flow for a family of self-adjoint operators. The mainidentification result of this paper is the following theorem, where the more precisestatement is given in Theorem 5.19. Theorem 1.1.
The equivariant Maslov index µ H ( D ) is equal to the equivariantspectral flow associated with the 1-parameter family of connections B t ( D ) on Y . The proof of Theorem 1.1 is presented in Section 5. It follows from a combi-nation of Nicolaescu’s arguments in [Nic95], which allow us to identify the H –equivariant spectral flow of B t ( D ) with certain infinite-dimensional Maslov in-dex (Theorem 5.8), Dostoglou-Salamon’s adiabatic limit result in [DS94] whichreduces the infinite-dimensional Maslov index to some finite-dimensional Maslovindex (Theorem 5.17), and some observations regarding the relation between tan-gent space of the extended moduli space ˆ M g (Σ ′ ) and twisted harmonic forms.After a further identification between the symplectic areas and Chern-Simonsinvariants coming into the definitions of SU(3)–Casson invariants on symplecticand gauge-theoretic sides, we prove Theorem 1.2.
Let Y = H ∪ Σ H be an integer homology –sphere equipped witha Heegaard splitting such that the genus of Σ is at least . Suppose Φ H is a suffi-ciently small G –equivariant Hamiltonian perturbation on ˆ M g (Σ ′ ) such that Φ H ( L ) intersect L non-degenerately. Let [˜ θ ] ∈ Φ H ( L ) ∩ L be the intersection comingfrom perturbing the product connection [ θ ] ∈ L ∩ L . Given any p ∈ Φ H ( L ) ∩ L ,let D ( p ) be a disc in ˆ M g (Σ ′ ) as above which connects p and [˜ θ ] . For any p , choosea point ˆ p in L ∩ L close to p and let D (ˆ p ) be a disc connecting ˆ p and [ θ ] which isclose to D ( p ) . Then the Boden-Herald SU(3) –Casson invariant is equal to λ SU(3) ( Y ) = X [ p ] ∈ (Φ H ( L ) ∩ L ) irr ( − µ ( D ( p )) − X [ p ] ∈ (Φ H ( L ) ∩ L ) red ( − µ t ( D ( p )) ( µ n ( D ( p )) − ω ( D (ˆ p ))2 π + 1) . (1.1) In the above formula, (Φ H ( L ) ∩ L ) irr is the set of G –orbits of (Φ H ( L ) ∩ L ) whose stabilizer is isomorphic to the center of SU(3) and (Φ H ( L ) ∩ L ) red is the SHAOYUN BAI set of G –orbits of (Φ H ( L ) ∩ L ) whose stabilizer is isomorphic to U(1) which cor-responds to perturbed
SU(2) –flat connections on Y . The index µ ( D ( p )) is the equi-variant Maslov index with respect to the trivial group, while the indices µ t ( D ( p )) and µ n ( D ( p )) are the trivial and nontrivial component of the U(1) –equivariant Maslovindex µ U(1) ( D ( p )) respectively. The symbol ω ( D (ˆ p )) is the symplectic area of thedisc D (ˆ p ) with respect to the symplectic form on ˆ M g (Σ ′ ) . The reader might have realized that this is parallel to the definition of Casson-Walker invariant [Wal92, equation (2.2)] for rational homology spheres.We hope that the symplectic approach used in this paper could help understandstructural aspects of the generalized Casson invariants. Indeed, in [Wal92], Walkerproved that the SU(2)–Casson-Walker invariant is combinatorial in nature basedon studying various isotopies of character varieties relevant to Dehn surgeries on3–manifolds. The combinatorial approach was generalized by Lescop in [Les96]to define invariants for all 3–manifolds. The symplectic geometry provides muchflexibility, though the use of Chern-Simons type invariants in our definition mightput obstructions to the understanding. However, given a knot K ⊂ Y , let Y /k ( K )be the 3–manifold obtained by doing Dehn surgery of Y along K with slope 1 /k , itis reasonable to expect that the SU( n )–Casson invariants of Y /k ( K ) might satisfythe asymptotic growing rate k n − as k → ∞ for a fixed n . When n = 2, this isthe classical surgery formula for Casson-Walker invariants, see [Wal92, Chapter 3];when n = 3, such speculation is supported by the discussion in [BHK05, Section6]. We will not discuss this topic further in this paper. Acknowledgements.
The author would like to thank his advisor John Pardonfor constant support and encouragement. Special thanks are given to Boyu Zhangfor numerous discussions and suggestions.2.
Recap of –dimensional gauge theory In this section, we recall some definitions and results from [BZ20] which will beused in this paper. Interested readers should consult loc.cit. for full details.2.1.
Some linear algebra.
Let G be a connected compact Lie group. Write therepresentation ring of G as R ( G ) and let R ( G ) ⊗ R be its natural R –extension. Definition 2.1.
Define R G to be the set of conjugation equivalence classes of thetriple ( H, V, ρ ) , where H is a closed subgroup of G , and ρ : H → Hom(
V, V ) is afinite-dimensional R –linear representation of H . We say that ( H, V, ρ ) is conjuga-tion equivalent to ( H ′ , V ′ , ρ ′ ) , if there exists g ∈ G and an isomorphism ϕ : V → V ′ ,such that H ′ = gHg − , and ρ ′ ( ghg − ) = ϕ ◦ ρ ( h ) ◦ ϕ − . Given a closed subgroup H ⊂ G , let R G ([ H ]) be the subset of R G consisting ofelements represented by representations of H . Any σ ∈ R G is called irreducible ifit is represented by an irreducible representation. Definition 2.2.
Let H be a closed subgroup of G . Define i HG : R H → R G be the tautological map by identifying subgroups of H as subgroups of G . SYMPLECTIC FORMULA OF GENERALIZED CASSON INVARIANTS 5
Given a compact Lie group H , if V is an irreducible R -linear H –representation,recall that V is said to be of type R , C or H , if hom H ( V, V ) ∼ = R , C or H respectively. Definition 2.3.
Suppose V is an orthogonal irreducible H –representation of type K ∈ { R , C , H } . Given r ∈ Z > , define d V ( r ) to be the dimension of self-adjointmaps on V r intertwining with the H action. In other words, d V ( r ) = r ( r + 1) if K = R , r if K = C , r − r if K = H . Definition 2.4.
Given σ ∈ R G with representative given by ( H, V, ρ ) such that V is an orthogonal H –representation, suppose the isotypic decomposition of V as an H –representation is given by V ∼ = V ⊕ a ⊕ · · · ⊕ V ⊕ a m m . Define the dimension function of σ , denoted by d ( σ ) , to be the following quantity: d ( σ ) = m X i =1 d V ( a i ) . This definition is independent of the choice of the representative of the equiv-alence class σ . According to Schur’s lemma, d ( σ ) is equal to the dimension of H –equivariant self-adjoint linear endomorphisms, denoted by Sym H ( V ), of the rep-resentation V . The following lemma is [BZ20, Lemma 3.18]. Lemma 2.5.
Suppose σ ∈ R G ([ H ]) is represented by the triple ( H, V, ρ ) suchthat V is an orthogonal H –representation. Let Sym
H,σ ( V ) = { s ∈ Sym H ( V ) | σ ∼ =( H, ker( s ) , ρ | ker( s ) ) } . Then Sym
H,σ ( V ) is a smooth submanifold of Sym H ( V ) ofcodimension d ( σ ) . If K ⊂ Sym H ( V ) is a linear space, let Π : V → ker( s ) be theorthogonal projection, then s + K is transverse to Sym
H,σ ( V ) at s if and only if themap taking l ∈ K to ( x Π( l ( x ))) ∈ Sym H (ker( s )) is surjective. Some –dimensional gauge theory. Let Y be a smooth, oriented, closed 3–manifold equipped with a Riemannian metric. Suppose G is a compact, connectedand simply-connected Lie group and let g be its Lie algebra. Let P = Y × G bethe product principal G –bundle over Y . We use θ to represent the trivial productconnection on P . Fix an integer k ≥
2. Let C ( Y ) be the affine space of L k –connections over P , which is an affine space modeled on L k ( T ∗ Y ⊗ g ). Let G Y be the L k +1 –gauge group of P . Then G Y acts smoothly on C ( Y ) via gauge transformation.Given B = θ + b ∈ C ( Y ), where b ∈ L k ( T ∗ Y ⊗ g ), let Stab( B ) and Orb( B ) be thestablizer group and orbit of B under the G Y –action respectively. For H ⊂ G , let C ( Y ) H be the subset of C ( Y ) consisting of connections whose stabilizer contains asubgroup of G which is conjugate to H .Recall the Chern-Simons functional CS on C ( Y ) is an R –valued smooth functionCS( θ + b ) := 12 Z Y d θ b ∧ b + 13 Z Y [ b ∧ b ] ∧ b. (2.1)Using the Riemannian metric on Y , we have(grad CS)( B ) = ∗ F B for all B ∈ C ( Y ) , where F B = d θ b + [ b, b ] is the curvature of B and ∗ is the Hodge star operator. Inparticular, critical points of CS correspond to flat connections over P . SHAOYUN BAI
Next we discuss about a slight generalization of the holonomy perturbation[Don87], [Flo88]. We follow the notation of [KM11]. This form of holonomy per-turbation will be used later in Section 5. Regard the circle S as R / Z . We identifythe open unit disc in the plane D with the open cube ( − , . Definition 2.6. A cylinder datum is a tuple ( q , · · · , q m , β, h s ) with m ∈ Z > thatsatisfies the following conditions. (1) q i : S × D ∼ = S × ( − , → Y is a smooth immersion for i = 1 , · · · , m ; (2) there exists ǫ > such that q , . . . , q m coincide on ( − ǫ, ǫ ) × D , (3) β is a non-negative, compactly supported smooth function on ( − , , suchthat Z − β = 1;(4) For any − ≤ s ≤ , h s : G m → R is a smooth function that is invariantunder the diagonal action of G by conjugations and the family { h s } − ≤ s ≤ depends smoothly on s . Given B ∈ C ( Y ) and cylinder datum q = ( q , · · · , q m , β, h s ), we can define the holonomy map of B byHol q ( B ) : D → G m z (cid:0) Hol q ,z ( B ) , · · · , Hol q m ,z ( B ) (cid:1) , where Hol q i ,z ( B ) is the element in G obtained by the holonomy of B along the loop q i ( S × { z } ) for 1 ≤ i ≤ m using the trivialization P { }×{ z } ∼ = G . Then the cylinderfunction associated to q is defined to be the map f q : C ( Y ) → R B Z − Z − β ( λ ) h s (Hol q ( B )) dλds, where we use the coordinate z = ( λ, s ) under the identification D ∼ = ( − , .When h s = χ ( s ) h such that h : G m → R is a smooth function invariant underthe diagonal action of G by conjugations and χ : ( − , → R is a compactlysupported smooth function with R − χ ( s ) ds = 1, the 2–form β ( λ ) χ ( s ) dλ ∧ ds definesa non-negative bump 2–form over D with total integration 1 and f q is the usualcylinder function as in [KM11]. These cylinder functions are smooth over C ( Y )and have well-defined formal gradients. Then one can follow the procedure in[KM11, Definition 3.6] to construct the Banach space of holonomy perturbations,which is denoted by P . Given any π ∈ P , write the associated function over C ( Y )as f π and let V π be the formal gradient of f π . Let DV π be the derivative of V π . Definition 2.7.
A connection B is called π –flat if ∗ F B + V π ( B ) = 0 . In other words, B is a critical point of the functional CS + f π . Given a pair B ∈ C ( Y ) and π ∈ P , we can define a self-adjoint Fredholm operatorwith index zero which has discrete real spectrum K B,π : L k ( g ) ⊕ L k ( T ∗ Y ⊗ g ) → L k − ( g ) ⊕ L k − ( T ∗ Y ⊗ g ) SYMPLECTIC FORMULA OF GENERALIZED CASSON INVARIANTS 7 by K B,π ( ξ, b ) := ( d ∗ B b, d B ξ + ∗ d B b + DV π ( B )( b )) . (2.2)When B is π –flat, define Hess B,π to be the operatorHess
B,π : ker d ∗ B ∩ L k ( T ∗ Y ⊗ g ) → ker d ∗ B ∩ L k − ( T ∗ Y ⊗ g )given by Hess B,π := ∗ d B + DV π ( B ) . This is a self-adjoint Fredholm operator with index zero. Note that both K B,π andHess
B,π are Stab( B )–equivariant operators. Definition 2.8. A π –flat connection B is called non-degenerate if Hess
B,π is anisomorphism. A perturbation π ∈ P is called non-degenerate if all the critical pointsof CS + f π are non-degenerate. Because the classical holonomy perturbations all arise as special cases of ourformulation of holonomy perturbation, the transversality results proved in [BZ20,Section 4] all carry over due to the abundance of perturbations. The followingproposition is [BZ20, Corollary 4.20, 4.21].
Proposition 2.9.
The set P reg ⊂ P of non-degenerate holonomy perturbations isof Baire second category.For any pair π , π ∈ P reg , one can find a generic smooth path π t : [0 , → P from π to π , such that there are only countably many t where π t is degenerate.Moreover, for every such t there is exact one degenerate critical orbit Orb( B ) of CS + f π t , and the kernel of Hess
B,π t is an irreducible representation of Stab( B ) . Next we recall definitions and results which lead to the definition of SU( n )–Casson invariants using gauge theory.Let V be a Hilbert space and D ⊂ V is a space whose image in V is dense and theinclusion map is a compact operator. Let D t : D → V , t ∈ [0 ,
1] be a 1–parameterfamily of self-adjoint operators whose spectra are real and discrete. Let H be aconnected compact Lie group and V admits a linear H –action which preserves D .Suppose that for all 0 < t <
1, the operator D t is equivariant with respect to the H –action. Given any W which is a finite-dimensional irreducible H –representation,we have an induced family of self-adjoint operators on Hom H ( W, V ) and let thespectral flow of this family be n W . We use the convention that if D or D hasnontrivial kernel, the spectral flow of the family { D t } ≤ t ≤ is calculated by thespectral flow of { D t + ǫ · id } ≤ t ≤ where ǫ > R irr ( H ) bethe set of finite-dimensional irreducible H –representations. Definition 2.10.
Define the H –equivariant spectral flow of { D t } ≤ t ≤ , denotedby Sf H ( D t ) to be the following element in the representation ring R ( H ) : X W ∈R irr ( H ) n W · [ W ] . (2.3)Let π ∈ P be a non-degenerate perturbation and suppose B is a π –flat connec-tion. Suppose H is a closed subgroup of G and the group Stab( B ) = H . Definition 2.11.
Define Sf H ( B, π ) ∈ R ( H ) to be the H –equivariant spectral flowgiven by the homotopy from K B,π to K θ, constructed from the linear homotopyfrom ( B, π ) to ( θ, . SHAOYUN BAI
Because of the homotopy invariance of the spectral flow, Sf H ( B, π ) could becalculated by any path of H –equivariant operators connecting K B,π and K θ, .Let n ≥ G is given by SU( n ) and C n is the standard representation of SU( n ). We further assume that Y is an integerhomology sphere. Let Σ n be the set of tuples of pairs of integers(( n , m ) , . . . , ( n r , m r )) , r ≥ n = P ri =1 n i m i . We also assume that n ≤ · · · ≤ n r and m i ’s are innon-decreasing order if the corresponding n i ’s are the same. By the discussion in[BZ20, Section 5], given any connection B ∈ C ( Y ), there exists σ ∈ Σ n , such thatthe vector bundle E = P × SU( n ) C n could be decomposed into E ∼ = E ( n ) ⊕ m ⊕ · · · ⊕ E ( n r ) ⊕ m r (2.4)satisfying(1) E ( n i ) is a trivial C –vector bundle over Y of rank n i for 1 ≤ i ≤ r ;(2) The connection B preserves this decomposition and is decomposed intodirect sums of irreducible unitary connections on each component E ( n i ).Such a connection B is called of type σ . Fix a σ ∈ Σ n , connections with abovedecomposition define a subset C σ ( Y ) ⊂ C ( Y ). All elements in C σ ( Y ) have stabilizersconjugate to a subgroup H σ ⊂ SU( n ). Then C σ ( Y ) ⊂ C ( Y ) H σ and the linear pathfrom B ∈ C σ ( Y ) to θ stays in C ( Y ) H σ .Suppose g : E → E is a gauge transformation that is decomposed into the directsum of gauge transformations g i : E ( n i ) → E ( n i ) using (2.4). If B ∈ C σ ( Y ), onecan find τ , . . . , τ r ∈ R ( H σ ) such that the H σ –equivariant spectral flow from K B,π to K g ( B ) ,π is given by X n i ≥ deg( g i ) · τ i which is independent of B and π ∈ P , where deg( g i ) is the degree of the map H ( Y ; Z ) → H (U( n ); Z ) induced by g i . If B is furthermore a flat connection andlet B i be the E ( n i )–component of B , defineCS σ ( B ) = X n i ≥ CS( B i )4 π n i · τ i ∈ R ( H σ ) ⊗ R . (2.5)Now we are ready to recall the definition of the SU( n )–Casson invariant. Let C F ( Y ) ⊂ C ( Y ) be the subspace consisting of flat connection on P . Let U ⊂ C ( Y )be an open subset containing C F ( Y ) such that the inclusion C F ( Y ) ∩ C ( Y ) H σ ֒ → U ∩ C ( Y ) H σ induces a one-to-one correspondence on connected components. By Uhlenbeck com-pactness, there exists r > π ∈ P satisfies k π k < r , thespace of π –flat connections is contained in U . Definition 2.12.
Suppose k π k < r is a non-degenerate perturbation and let B be a π –perturbed flat connection of type σ ∈ Σ n . Take g ∈ G Y such that g ( B ) ∈ C ( Y ) H σ SYMPLECTIC FORMULA OF GENERALIZED CASSON INVARIANTS 9 and let g ( ˆ B ) be a genuine flat connection lying in the same connected componentof U ∩ C ( Y ) H σ as g ( B ) . Define ind( B, π ) to be Sf H σ ( g ( B ) , π ) − [ker d g ( B ) ] − CS σ ( g ( ˆ B )) ∈ R ( H σ ) ⊗ R , (2.6) where [ker d g ( B ) ] is the H σ –representation associated to the Lie algebra of Stab( B ) . The following is proved in [BZ20, Section 5].
Theorem 2.13.
There exists a set e R SU( n ) , a well-defined map R ( H σ ) ⊗ R → e R SU( n ) and a Z –submodule f Bif
SU( n ) ⊂ Z e R SU( n ) with the following significance. Let [ind( B, π )] be the image of ind( B, π ) in e R SU( n ) . If k π k < r is a non-degenerateperturbation, define ind π := X Orb( B ) is π – flat [ind( B, π )] ∈ Z e R SU( n ) . Then for any pair of non-degenerate π , π ∈ P such that k π i k < r for i = 1 , ,we have ind π − ind π ∈ f Bif
SU( n ) . Therefore the image of ind π in the Z –module Z e R SU( n ) / f Bif
SU( n ) is a topologicalinvariant of the integer homology sphere Y . One can then use this quantity toconstruct Casson-type invariants using different normalizations. In particular, the R –valued SU(3)–Casson invariant of Boden-Herald [BH98] could be recovered as aspecial case of the above construction.Our goal is to present ind( B, π ) using geometric quantities constructed fromfinite-dimensional symplectic geometry when Y is equipped with a Heegaard split-ting and the perturbation π has certain adapted behavior. This is what the sym-plectic formula is referring to.3. Extended moduli spaces
In this section, we recall the construction of the extended moduli space in [Jef94]and discuss about some of its properties. Most importantly, we will see that theessential information of the tangent spaces of the extended moduli space is encodedin the corresponding space of twisted harmonic forms. We also recall the construc-tion of Lagrangian submanifolds in extended moduli spaces from [MW12] whichwill be used in this paper.Suppose Σ is a closed Riemann surface with genus h ≥ p ∈ Σ be a base point. Define Σ ′ = Σ \ D p to be the Riemannsurface with one boundary component obtained by removing a disc D p around p .Suppose G is a simply-connected compact Lie group and g is its Lie algebra. Weidentify g with its dual g ∗ using the Killing form. Let P be the product principal G –bundles over Σ. We continue to use θ to represent the trivial product connection.Fix an integer k ≥ . Let A (Σ) and A (Σ ′ ) be the affine space of L k –connectionsover P and P | Σ ′ respectively. Denote the L k +1 –gauge groups of P and P | Σ ′ by G Σ and G Σ ′ respectively. Let M (Σ) = A F (Σ) / G Σ be the moduli space of flatconnections over P , where A F (Σ) is the space of flat connections over P . Wealso need the group G c Σ ′ consisting of elements in G Σ ′ which are identity near theboundary of Σ ′ . Definition 3.1.
Let s be the coordinate on S ∼ = ∂ Σ ′ . Define A g F (Σ ′ ) := { A ∈ A (Σ ′ ) (cid:12)(cid:12) F A = 0 , A = θ + ξds near the boundary for some ξ ∈ g } . Then the extended moduli space M g (Σ ′ ) is defined to be the quotient M g (Σ ′ ) := A g F (Σ ′ ) / G c Σ ′ . M g (Σ ′ ) has a finite-dimensional description similar to character varieties. Letus choose { α i , β i } ≤ i ≤ h to be a set of simple loops on Σ with common base point atsome p ′ ∈ ∂ Σ ′ ⊂ Σ ′ ⊂ Σ which gives the standard presentation of π (Σ). Choose asimple loop γ in Σ ′ based at p ′ wrapping around the boundary once then we haveΠ hi =1 [ α i , β i ] = γ . Using the holonomy map, one can show that M g (Σ ′ ) = { A i , B i ∈ G, ∀ ≤ i ≤ h, ξ ∈ g (cid:12)(cid:12) Π hi =1 [ A i , B i ] = exp( ξ ) } . (3.1)We will use both the gauge-theoretic definition and the representation-theoreticdefinition in the sequel. Note that M g (Σ ′ ) admits a G –action: given g ∈ G , it actson M g (Σ ′ ) by(( A i , B i ) ≤ i ≤ h , ξ ) (( gA i g − , gB i g − ) ≤ i ≤ h , Ad g ( ξ )) , where Ad g is the adjoint action of g on g . Use the Killing form on g to define anorm on g , the next proposition is proved in [Jef94]. Proposition 3.2. (1)
The open G –invariant subset ˆ M g (Σ ′ ) ⊂ M g (Σ ′ ) de-fined by | ξ | < δ for some sufficiently small δ > is actually a smoothmanifold; (2) There exists a symplectic form ω on ˆ M g (Σ ′ ) and the G –action definesa smooth Hamiltonian action with respect to ω . The moment map µ :ˆ M g (Σ ′ ) → g ∗ of this G –action is given by (( A i , B i ) ≤ i ≤ h , ξ ) ξ ;(3) The moment map reduction µ − (0) /G is naturally identified with M (Σ) ; (4) If [ A ] ∈ M (Σ) is represented by some A ∈ A g F (Σ ′ ) , then the stabilizer groupof A under the G Σ ′ –action, written as Stab( A ) , is isomorphic to Stab([ A ]) ,the stabilizer of [ A ] ∈ M (Σ) under the G –action. Making δ smaller if necessary, we can assume that the exponential map exp : g → G is a diffeomorphism onto its image over { ξ ∈ g (cid:12)(cid:12) | ξ | < δ } . Then the followingstatement follows from the equation (3.1). Lemma 3.3.
The map ˆ M g (Σ ′ ) → G h defined by mapping (( A i , B i ) ≤ i ≤ h , ξ ) tothe G h –component is a G –equivariant embedding, where G h is equipped with thediagonal conjugation action. (cid:3) Introduce the notation A F (Σ) := { A ∈ A F (Σ) (cid:12)(cid:12) A | D p is given by the product connection } . Then A F (Σ) could be viewed as a subspace of A g F (Σ ′ ) corresponding to ξ = 0.Choose A ∈ A F (Σ). Denote by [ A ] ∈ ˆ M g (Σ ′ ) the image of A in ˆ M g (Σ ′ ), we nowgive an explicit description of the tangent space T [ A ] ˆ M g (Σ ′ ).Let Ω i (Σ ′ ) ⊗ g (resp. Ω i (Σ) ⊗ g ) be the space of g –valued i –forms on Σ ′ (resp.Σ) for i = 0 , ,
2. Also write the subspace of Ω i (Σ ′ ) ⊗ g consisting of elements withcompact support as Ω ic (Σ ′ ) ⊗ g . Recall that s is the coordinate on ∂ Σ ′ . DefineΩ , g (Σ ′ ) = { a ∈ Ω (Σ ′ ) ⊗ g (cid:12)(cid:12) a = ξds near the boundary for some ξ ∈ g } . SYMPLECTIC FORMULA OF GENERALIZED CASSON INVARIANTS 11
We have the elliptic complexΩ c (Σ ′ ) ⊗ g Ω , g (Σ ′ ) Ω c (Σ ′ ) ⊗ g . d A d A (3.2)Then the tangent space T [ A ] ˆ M g (Σ ′ ) is identified with the first cohomology groupof (3.2), written as ˜ H , g A . The bilinear form ω Σ ′ ( a , a ) = Z Σ ′ Tr( a ∧ a )over Ω , g (Σ ′ ) defines the symplectic form on ˆ M g (Σ ′ ) under this identification.On the other hand, view A ∈ A F (Σ), we have the complexΩ (Σ) ⊗ g Ω (Σ) ⊗ g Ω (Σ) ⊗ g . d A d A (3.3)The cohomolgy groups H iA for i = 0 , , i –forms. Similarly, the bilinear form ω Σ ( a , a ) = Z Σ Tr( a ∧ a )over Ω (Σ) ⊗ g is non-degenerate and descends down to a non-degenerate pairingon H A .We need to introduce some auxiliary objects to understand the relation between˜ H , g A and H A as symplectic vector spaces. Let Σ ∼ = Σ \{ p } be the Riemann surfaceobtained from Σ ′ by attaching a cylindrical end S × [0 , + ∞ ) along ∂ Σ ′ . We stilluse P to represent the product principal G –bundle over Σ. Choose ǫ > L ,ǫk (Ω i (Σ) ⊗ g ) the completion of the space of g –valued i –forms over Σ ′ under the norm k a k L ,ǫk = ( Z Σ e ǫχ ( | a | + |∇ a | + · · · + |∇ k a | )) , where χ is a smooth function on Σ which is equal to 1 over S × [0 , + ∞ ) and ∇ issome background connection. This is the norm associated to the inner product h a , a i k,ǫ = Z Σ e ǫχ ( h a , a i + h∇ a , ∇ a i + · · · + h∇ k a , ∇ k a i ) . Note that the bilinear form on L ,ǫk (Ω i (Σ) ⊗ g ) given by ω Σ ( a , a ) = Z Σ Tr( a ∧ a )is non-degenerate, which could be proved using the Hodge ∗ –operator. Given A ∈A F (Σ), it naturally defines a connection on P | Σ , namely extending A by the productconnection on the cylindrical end. Therefore, we can define the operator d A : L ,ǫk (Ω i (Σ) ⊗ g ) → L ,ǫk − (Ω i +1 (Σ) ⊗ g )using the connection A the the L –formal adjoint operator of d A d ∗ A : L ,ǫk (Ω i (Σ) ⊗ g ) → L ,ǫk − (Ω i − (Σ) ⊗ g ) . Lemma 3.4.
For ǫ > sufficiently small, the kernel of the map d ∗ A ⊕ d A : L ,ǫk (Ω (Σ) ⊗ g ) → L ,ǫk − (Ω (Σ) ⊗ g ) ⊕ L ,ǫk − (Ω (Σ) ⊗ g ) is naturally isomorphic to H A and this isomorphism respects the symplectic struc-ture. Proof.
The map ker( d A ⊕ d ∗ A ) → H A is given by the natural inclusion L ,ǫk (Ω (Σ) ⊗ g ) → L k (Ω (Σ)), after identifying Σ with Σ \{ p } . This map is the same as the com-position H A (Σ ′ , ∂ Σ ′ ) ∼ = H c (Σ ′ , d A ) → H A , which is an isomorphism by countingdimensions. This map obviously respects the symplectic structure. (cid:3) Lemma 3.5.
There exists a map P : ker( d A ⊕ d ∗ A ) → ˜ H , g A which defines a linearsymplectic embedding. The symplectic orthogonal complement of Im( P ) is iso-morphic to ( g /H A ) ⊕ ( g /H A ) ∗ , where H A is identified with the Lie algebra ofthe stabilizer group of A ∈ A Σ . Moreover, the induced symplectic structure on ( g /H A ) ⊕ ( g /H A ) ∗ from ω Σ ′ is the same as the canonical linear symplectic form on T ∗ ( g /H A ) ∼ = ( g /H A ) ⊕ ( g /H A ) ∗ .Proof. The proof is essentially the same as the proof of [Jef94, Proposition 3.9]. In loc.cit , Jeffrey constructed a family of linear symplectic embeddings P r : ker( d A ⊕ d ∗ A ) → ˜ H , g A parametrized by r ∈ [0 , + ∞ ) and the P here is the specializationat r = 0. Although [Jef94, Proposition 3.9] was stated for G = SU(2), the samearguments work for general Lie groups. (cid:3) Using these two lemmas, and note that all the isomorphisms commute with theStab( A )–action, we conclude Corollary 3.6.
Suppose [ A ] ∈ ˆ M g (Σ ′ ) is represented by some A ∈ A F (Σ) , thenthere is a Stab( A ) –equivariant symplectic isomorphism T [[ A ]] ˆ M g (Σ ′ ) ∼ = H A ⊕ T ∗ ( g /H A ) , (3.4) where T ∗ ( g /H A ) is equipped with the canonical symplectic form and trivial Stab( A ) –action. (cid:3) Remark . The above corollary could be viewed as a consequence of the symplecticslice theorem which appeared in the proof of [SL91, Theorem 2.1]. However, wecannot apply the slice theorem directly because we need to identify the tangentbundle of the normal slice explicitly as twisted harmonic 1–forms. This will beimportant in our calculations of spectral flows later.Now suppose Y ′ is a smooth, oriented, 3–manifold with boundary ∂Y ′ = Σ.Choose the base point p ∈ Σ as above then p also define base points in both Y ′ .Assume that π (Σ) → π ( Y ′ ) induced by the inclusion defines a surjective map.Denote by P the trivial G –bundle on both Y ′ and Σ. Let A g F (Σ ′ | Y ′ ) ⊂ A g F (Σ ′ )be the space consisting of elements in A g F (Σ ′ ) which extend to flat connection on P over Y ′ . Such space is preserved by the action of G c Σ ′ . Furthermore, let A F ( Y ′ )be the space of flat connections on P | Y ′ and let G ( Y ′ ) be the based gauge groupconsisting of gauge transformations g ∈ G Y ′ such that g ( p ) = id. Then the map A F ( Y ′ ) / G ( Y ′ ) → A g F (Σ ′ | Y ′ ) / G c Σ ′ is a diffeomorphism. By [MW12, Lemma 5.1], the embedding A F ( Y ′ ) / G ( Y ′ ) ∼ = A g F (Σ ′ | Y ′ ) / G c Σ ′ ֒ → A g F (Σ ′ ) / G c Σ ′ = M g (Σ ′ )defines a Lagrangian submanifold L Y ′ in M g (Σ ′ ) with respect to the symplecticform in Proposition 3.2. The manifold L Y ′ diffeomorphic to G h and it is preservedby the G –action on M g (Σ ′ ). Additionally, this Lagrangian is contained in the 0–level set of the moment map µ − (0). As a result, L Y ′ is contained in the smoothlocus ˆ M g (Σ ′ ). SYMPLECTIC FORMULA OF GENERALIZED CASSON INVARIANTS 13
One could obtain many G –invariant Lagrangians in ˆ M g (Σ ′ ) by the construc-tion above. We will study the equivariant intersection theory of a pair of suchLagrangians and define some version of equivariant intersection number which cal-culates the generalized Casson invariants.4. Equivariant transversality of Lagrangian submanifolds
This section develops an equivariant transversality result for Lagrangian sub-manifolds inside symplectic manifolds equipped with Lie group actions, and definesthe notion equivariant Maslov index which will be used in the calculation of thegeneralized Casson invariants.Suppose (
M, ω ) is a (not necessarily closed) smooth symplectic manifold of di-mension 2 n , where ω is the symplectic form . Let G be a connected, compact Liegroup that acts on ( M, ω ) such that ∀ g ∈ G , g ∗ ω = ω . Let L , L be two orientedLagrangian submanifolds in M which are invariant under the G –action.For p ∈ M , we use Orb( p ) to denote the orbit of p under the action of G . Notethat Orb( p ) could be identified with G/ Stab( p ), where Stab( p ) is the stabilizergroup of p under the G –action. In particular, Orb( p ) is a smooth manifold. Asubset X of M is called G –invariant if G ( X ) = X ; and similarly, a subbundle E of T M | X where X is G –invariant is called G –invariant if G ( E ) = E . Definition 4.1.
We say that L and L intersect non-degenerately at p, if for p ∈ L ∩ L , we have T p L ∩ T p L = T p Orb( p ) . L and L are said to intersect non-degenerately if they intersect non-degenerately at each of their intersectionpoint.Remark . Using the terminology in [Poz99], the above definition is equivalent tosaying that L and L have clean intersection along Orb( p ) for each p ∈ L ∩ L .Note that if L and L are compact and intersect non-degenerately, they onlyintersect along finitely many G –orbits.4.1. Transversality.
Suppose (
M, ω ) , L , L are as above, we prove that L and L intersect non-degenerately after applying a generic G –equivariant Hamiltonianperturbation on L . We also characterize the failure of equivariant transversalitywhen varying the Hamiltonian perturbations in generic 1–paramter families.We start by fixing some notations from symplectic geometry. Let C ∞ G ( M ) be thespace of compactly supported smooth G –invariant time-independent Hamiltonianson the symplectic manifold M equipped with the C ∞ –topology. Definition 4.3.
Let C ∞ (cid:16) [0 , , C ∞ G ( M ) (cid:17) be the space of time-dependent smooth G –invariant Hamiltonians which vanish near s = 0 and s = 1 , where s is thecoordinate on [0 , . Take { H s , H s , · · · } which form a countable dense subset of C ∞ (cid:16) [0 , , C ∞ G ( M ) (cid:17) and let N m := sup {k H s k C m , · · · , k H ms k C m } . Define H to be the completion of the subspace of C ∞ (cid:16) [0 , , C ∞ G ( M ) (cid:17) consisting offunctions of form X m ≥ a m H ms such that P m ≥ N m | a m | < + ∞ under the norm k P m ≥ a m H ms k := P m ≥ N m | a m | . Given any H s ∈ C ∞ (cid:16) [0 , , C ∞ G ( M ) (cid:17) , the Hamiltonian vector field X H s of H s isdetermined by dH s = ω ( · , X H s ) . Let Φ H s be the time 1–flow generated by integrating X H s . Then for any G –invariantLagrangian L ⊂ M , Φ H s ( L ) is also a G –invariant Lagrangian.Recall that for every manifold S , there is a canonical symplectic form dλ on T ∗ S (see, for example, [MS17, p. 90]). When S admits a smooth G –action, dλ is preserved by the induced G –action on T ∗ S . We need the following equivariantlocal version of the Lagrangian neighborhood theorem: Lemma 4.4.
Suppose L is a G –invariant Lagrangian submanifold of ( M, ω ) , andsuppose Orb( p ) ⊂ L . Let E ⊂ T M | Orb( p ) be a G –invariant Lagrangian subbundlethat is transverse to T L | Orb( p ) . Then there exists a G –invariant open subset U ⊂ M with the following property: if we write S = U ∩ L , then there is a G –equivariantsymplectomorphism ϕ from an open neighborhood of the zero section of T ∗ S to U ,such that (1) ϕ equals the identity map on S , (2) the tangent map of ϕ on S sends the fibers of T ∗ S to the fibers of E on Orb( p ) .Proof. Because E and T L | Orb( p ) are transverse to each other fiberwisely as La-grangian subbundles of T M | Orb( p ) , the map β E : E → T ∗ L | Orb( p ) given by v ω ( v, · ) is an isomorphism. Using the identification T M | Orb( p ) ∼ = E ⊕ T L | Orb( p ) ∼ = T ∗ L | Orb( p ) ⊕ T L | Orb( p ) , we can construct a fiberwise G –invariant almost complex structure J p on T M | Orb( p ) which is compatible with the fiberwise symplectic form on T M | Orb( p ) such that J p ( T L | Orb( p ) ) = E .Extend J p to a G –equivariant compatible almost complex structure J on M andlet g J = ω ( · , J · ) be the compatible Riemannian metric. After choosing sufficientlysmall G –invariant neighborhood of Orb( p ) ∈ T ∗ L so that exponential map of g J is well defined, the argument of [MS17, Theorem 3.4.13] carries over. Our specialchoice of the almost complex structure guarantees the second condition holds. (cid:3) Lemma 4.5.
Let S be a G –manifold and suppose that U is a sufficiently small G –invariant neighborhood of Orb( p ) ⊂ S . Then any G –invariant Lagrangian L ⊂ T ∗ S which is transverse to the Lagrangian foliation on T ∗ U induced by cotangent fibersis locally presented as the graph of an exact G -equivariant –form over U .Proof. Because L is transverse to the Lagrangian foliation on T ∗ U induced bycotangent fibers, it is locally represented by the graph of some 1–form θ L over U .The Lagrangian condition implies that θ L is closed.Using the slice theorem, without loss of generality we can assume that U = G × Stab( p ) U ′ where U ′ is a contractile G –invariant neighborhood of the origin in T p S/T p Orb( p ). Because the quotient of U by G is contractible, the cohomologyclass represented by the equivariant form θ L is 0. Using the de Rham complex of G –equivariant differential forms on U , we conclude that θ L is exact. (cid:3) Now we are ready to establish the equivariant transversality result. Let A ⊂ M be a compact subset such that L ∩ L is included in the interior of A . Let H ⊂ H SYMPLECTIC FORMULA OF GENERALIZED CASSON INVARIANTS 15 be a sufficiently small open ball centered at zero, such that for every H s ∈ H , themap Φ H s is defined on A , and Φ H s ( L ) ∩ L is included in the interior of A .Suppose σ ∈ R G . Define M σ ⊂ H × ( A ∩ L ) to be the set of points ( H s , p )such that(1) p ∈ Φ H s ( L ) ∩ L ,(2) The linear space (cid:0) T p Φ H s ( L ) ∩ T p ( L ) (cid:1) /T p Orb( p ) as a Stab( p )–representationrepresents σ . Proposition 4.6.
The space M σ is a Banach manifold. The projection of M σ to H , written as π σ , is Fredholm with index − d ( σ ) .Proof. Let ( H s , p ) ∈ M σ . Let Q p ⊂ T M | Orb( p ) be a G –invariant Lagrangiansubbundle such that Q p is transverse to T Φ H s ( L ) | Orb( p ) and T L | Orb( p ) alongOrb( p ). Let B be a small G –invariant open neighborhood of p in L . Then byLemma 4.4, there exists be a G –invariant open neighborhood B of p in M , suchthat(1) There exists a G –equivariant symplectomorphism to a neighborhood of thezero section of T ∗ B ,(2) T ∗ p B is tangent to Q p at p under the above diffeomorphism.After further shrinking B and B if necessary, by Lemma 4.5, Φ H s ( L ) ∩ B is thegraph of the differential of a G –invariant function on B . This function is uniquelydetermined if we require that its integration over B is 0. Therefore we have a mapfrom a sufficiently small neighborhood N ( H s ) of H s to C ∞ G ( B ) π : N ( H s ) → C ∞ G ( B )such that for each H ′ s ∈ N ( H ), we have Φ H ′ s ( L ) ∩ B is the graph of dπ ( H ′ s ), and R B π ( H ′ s ) = 0. Claim:
The map π is a submersion at H s onto the subspace (cid:8) g ∈ C ∞ G ( B ) | Z B g = 0 (cid:9) . For each g ∈ C ∞ G ( B ) such that R B g = 0, let ˆ g ∈ C ∞ G ( M ) be a smooth,compactly supported function on M , such that ˆ g | B is given by the pull-back of g from B to T ∗ B . Let χ : [0 , → R ≥ be a smooth, compactly supported functionsuch that R χ = 1. For s ∈ [0 , H s ( s ) be the integration of X H s from s to 1.Define ˆ H s = χ ( s ) · (cid:0) ˆ g ◦ Φ H s ( s ) (cid:1) . Then ddv π ( H + v ˆ H ) = g. Therefore the claim is proved.Let D be a slice of B under the G –action through p . Let D be the fixed pointsubset of D under the Stab( p )–action. Then for any (Φ H ′ s , q ) ∈ M σ such that H ′ s ∈ N ( H s ) and q ∈ B , we have q ∈ D where D is viewed as a subset of the zerosection of the Weinstein neighborhood. Choose a G –invariant Riemannian metricon B and let S p be the orthogonal complement of T p Orb( p ) in T p B and let S p bethe fixed point subset of S p under the Stab( p )–action. Using parallel transports,we can identify T D with S p . Now we can define a map ψ : N ( H s ) × D → ( S p ) ∗ × Sym
Stab( p ) ( S p ) ( H ′ s , q ) ( ∇ q ( π ( H ′ s )) | D , Hess( π ( H ′ s )) | S p ) , where ∇ and Hess stand for the gradient and Hessian respectively. It is easy to seethat M σ = ψ − ( { } × Sym
Stab( p ) ,σ ( S p )) . Using the abundance result [BZ20, Lemma 2.25] and the result established in theclaim, we conclude that the differential of ψ at ( H s , p ) is surjective. Therefore M σ is a Banach manifold near ( H s , p ). Because the inclusion ψ − ( { } × Sym
Stab( p ) ,σ ( S p )) ֒ → N ( H s ) × D is a Fredholm map of index − dim( D ) − d ( σ ), the projection map from M σ to H is Fredholm of index − d ( σ ). The proposition is proved because of the separabilityof H × ( A ∩ L ). (cid:3) Corollary 4.7.
Let L and L be two G –invariant Lagrangians in ( M, ω ) suchthat their intersection is contained in a compact subset of M . Let H be a smallneighborhood of in H .Then for a generic perturbation H s ∈ H , the pair Φ H s ( L ) and L intersect non-degenerately.Proof. Using Proposition 4.6, because d ( σ ) > σ is not given by the 0–representation, H σ := π σ ( M σ ) is a C ∞ –subvariety of H of positive codimension.As a result, [ σ ∈R G H σ is a meager subset of H . Any Hamiltonian perturbation constructed from elementsin H \ S σ ∈R G H σ establishes the transversality result. (cid:3) Now we show that for a generic 1–paramter family { H s,t } ≤ t ≤ , the intersectionΦ H s,t ( L ) ∩ L has at most one degenerate intersection orbit which corresponds toan irreducible representation for each t .Choose H and A as before. Suppose σ , σ ∈ R G . Define M σ ,σ ⊂ H × ( A ∩ L ) × ( A ∩ L ) to be the set of points ( H s , p , p ) such that(1) p , p ∈ Φ H s ( L ) ∩ L ,(2) Orb( p ) ∩ Orb( p ) = ∅ ,(3) The spaces (cid:0) T p i Φ H s ( L ) ∩ T p i ( L ) (cid:1) /T p i Orb( p i ) as Stab( p i )–representationsrepresent σ i for i = 1 , H σ ,σ be the projection of M σ ,σ onto H . The following statement followsfrom the arguments in the proof of Proposition 4.6 and [BZ20, Lemma 3.23], thusthe proof is omitted. Proposition 4.8. H σ ,σ is a C ∞ –subvariety of H of codimension at least d ( σ )+ d ( σ ) . (cid:3) Note that d ( σ ) = 1 if and only if σ is irreducible. Therefore the following corol-lary follows immediately from Proposition 4.6 and Proposition 4.8 by an applicationof the Sard-Smale theorem. Corollary 4.9.
Let { H s,t } ≤ s ≤ be a generic path in H connecting H s, , H s, ∈H \ S σ ∈R G H σ which intersect all H σ and H σ ,σ transversely. Then there are atmost countably many t such that Φ H s,t ( L ) do not intersect L non-degenerately.For every such t ∈ (0 , , there is exactly one intersection orbit Orb( p t ) such that SYMPLECTIC FORMULA OF GENERALIZED CASSON INVARIANTS 17 (cid:0) T p t Φ H s,t ( L ) ∩ T p t ( L ) (cid:1) /T p t Orb( p t ) is an irreducible Stab( p t ) –representation, and Φ H s,t ( L ) intersect L non-degenerately along any other orbits. (cid:3) An equivariant Maslov index.
In this subsection, we give the definition ofequivariant Maslov index which will appear in the index formula later on.We recall the definition of the classical Maslov index. Let ( V , ω ) be a symplecticvector space. For a, b ∈ R such that a < b , let P [ a,b ] ( V ) be the space of continuousand piecewise smooth maps q : [ a, b ] → { paris of Lagrangian subspaces in V } . Then there exists a map µ V : P [ a,b ] ( V ) → Z which is called the Maslov index uniquely characterized by the following properties(see [CLM94]):(1) (Affine scale invariance) Suppose k > l ≥ a, b ] → [ ka + l, kb + l ] be the affine linear map given by t kt + l . Then for q ∈ P [ ka + l,kb + l ] ( V ), we have µ V ( q ) = µ V ( q ◦ Γ);(2) (Homotopy invariance) Given q , q ∈ P [ a,b ] ( V ) such that q ( a ) = q ( a )and q ( b ) = q ( b ), if there is a homotopy between q and q in P [ a,b ] ( V )relative to end points, then µ V ( q ) = µ V ( q );(3) (Additivity under concatenation of paths) If a < c < b and q ∈ P [ a,b ] ( V ),then the following additive property is satisfied: µ V ( q ) = µ V ( q | [ a,c ] ) + µ V ( q | [ c,b ] );(4) (Symplectic additivity) Let ( V ′ , ω ′ ) be another symplectic vector space andsuppose q ∈ P [ a,b ] ( V ) , q ′ ∈ P [ a,b ] ( V ′ ). Then µ V ⊕ V ′ ( q ⊕ q ′ ) = µ V ( q ) + µ V ′ ( q ′ );(5) (Independence of symplectic framing) Suppose φ t : [ a, b ] → Sp( V ) is a 1–parameter family of symplectic matrices. Denote by q ( t ) = ( L ( t ) , L ( t ))for some q ∈ P [ a,b ] ( V ) and define φ ∗ ( q )( t ) = ( φ t ( L ( t )) , φ t ( L ( t ))). Then µ V ( φ ∗ ( q )( t )) = µ V ( q ( t ));(6) (Normalization) Equip R x,y ∼ = C z = x + iy with the symplectic form dx ∧ dy = i dz ∧ dz . Consider the path of Lagrangians p ( t ) in P [ − π , π ] ( R ) given by t ( R { } , R { e it } ) . Then µ R ( p | [ − π , π ] ) = 1 , µ R ( p | [ − π , ) = 0 , µ R ( p | [ − π , ) = 1 . There are many ways to construct the map µ V explicitly. For our purposelater, we use the characterization of µ V as the spectral flow of certain self-adjointoperators, see [CLM94, Section 7]. Let J ∈ GL ( V ) be a complex structure whichis compatible with ω . Given L , L ⊂ V which are linear Lagrangian subspaces,let L ([0 , L , L ) be the completion of smooth maps f : [0 , → V with f (0) ∈ L , f (1) ∈ L under L –norm induced by the metric ω ( · , J · ). Let L ([0 , V ) bethe space of L –maps from [0 ,
1] to V . Then the operator D L ,L : L ([0 , L , L ) → L ([0 , V ) f
7→ − J dfdt is a real self-adjoint unbounded Fredholm operator with discrete spectrum. Thekernel of D L ,L is the same as constant functions taking value in L ∩ L .If we take a path of pairs of Lagrangian subspaces q ( t ) = ( L ( t ) , L ( t )) t ∈ [ a,b ] in V , we have a family of self-adjoint operators { D L ( t ) ,L ( t ) } a ≤ t ≤ b . Let ǫ > D L ( a ) ,L ( a ) and D L ( b ) ,L ( b ) do not have any other eigenvaluesbetween − ǫ and ǫ except for 0. Proposition 4.10.
The spectral flow of { D L ( t ) ,L ( t ) − ǫ id } a ≤ t ≤ b is independentof the choice of J and equal to µ V ( p ) . (cid:3) Now suppose V is a H –representation where H is a connected compact Liegroup, and the H –action preserves the symplectic form ω . Let V ∼ = ( V ) ⊕ a ⊕ · · · ⊕ ( V m ) ⊕ a m be the isotypic decomposition of V . Then each isotypic piece inherits a symplecticstructure and the above decomposition is compatible with the symplectic structure.If q ( t ) = ( L ( t ) , L ( t )) t ∈ [ a,b ] is a path such that both L ( t ) and L ( t ) are preservedunder the H –action, then( V ) ⊕ a ∩ L i ( t ) ⊕ · · · ⊕ ( V m ) ⊕ a m ∩ L i ( t )gives the isotypic decomposition of L i ( t ) for i = 1 , t ∈ [ a, b ]. Write q k ( t ) asthe induced path on the isotypic piece ( V k ) ⊕ a k for 1 ≤ k ≤ m . Definition 4.11.
The equivariant Maslov index µ H ( q ) of the path q ( t ) as above isthe element m X i =1 R V i · µ ( V i ) ⊕ ai ( q i ) · [ V i ] ∈ R ( H ) . (4.1) Remark . The above quantity is well-defined because the V i –isotypic piece of L ( t ) ∩ L ( t ) would change the Maslov index by a multiple of dim R V i . Alterna-tively, we can define µ H ( q ) using the equivariant spectral flow of the H –equivariantoperator − J ddt where J is an H –equivariant complex structure on V compatiblewith ω , based on Definition 2.10 and Proposition 4.10.Now we go back to discuss about global symplectic geometry. Recall that ( M, ω )is a symplectic manifold with a symplectic G –action and L and L are two G –invariant Lagrangians of M . Suppose H is a closed subgroup of G . Let D ⊂ C z = R x,y be the closed unit disc in the plane. Suppose we have a map u : D → M whichis smooth except at ± u ( ∂ D ∩ { y ≤ } ) ⊂ L and u ( ∂ D ∩ { y ≥ } ) ⊂ L ;(2) For any z ∈ D , the Lie group Stab( u ( z )) contains H as a subgroup.Choose p = u ( −
1) as the base point. Then the symplectic vector bundle u ∗ T M is isomorphic to the product bundle T p M × D respecting the fiberwise symplectic H –action under a trivialization Ψ. Using Ψ, q u = ( u ∗ | ∂ D ∩{ y ≤ } T L , u ∗ | ∂ D ∩{ y ≥ } T L ) SYMPLECTIC FORMULA OF GENERALIZED CASSON INVARIANTS 19 defines a path of pair of H –invariant Lagrangian subspaces in T p M , after identifying ∂ D ∩ { y ≤ } and ∂ D ∩ { y ≥ } with the interval [ − ,
1] by projecting to the x –coordinate. Recall that h is the Lie algebra of H . Because H ⊂ Stab( u ( z )) for all z ∈ D , the symplectic slice theorem implies that T u ( z ) M contains the symplecticsubspace T ∗ ( g / h ), i.e. the tangent space of the cotangent bundle of Orb( u ( z )).The tangent spaces of T u ( z ) L i , z ∈ ∂ D contain a copy of g / h as a result of the slicetheorem. Define the symplectic vector space T p M = T p M/T ∗ ( g / h )with the induced symplectic structure and the path of pairs of linear Lagrangiansubspaces q u = ( u ∗ | ∂ D ∩{ y ≤ } T L / ( g / h ) , u ∗ | ∂ D ∩{ y ≥ } T L / ( g / h ))which lies inside T p M . Definition 4.13.
Define the H –equivariant Maslov index µ H ( u ) ∈ R ( H ) of themap u as above to be µ H ( u ) := µ H ( q u ) . (4.2)Using the properties of the Maslov index, it is easy to see that µ H ( u ) is indepen-dent of the trivialization Ψ. It is also invariant under the homotopies of u whichsatisfy (1) and (2) above and keep both u ( −
1) and u (1) fixed. Example . Suppose V is an orthogonal representation of G , and suppose ( M, ω )is given by the cotangent bundle of V . Let f be a smooth G –invariant function on V with finitely many critical orbits. Furthermore, we assume that f is a G –Morsefunction, i.e. the Hessian of f is non-degenerate on T p V /T p Orb( p ) if p is a criticalpoint of f . Let L be the graph of the zero section of M = T ∗ V , let L be thegraph of df . Then it is easy to see that L and L intersect non-degenerately. Let p ∈ V be critical point of f such that H ⊂ Stab( p ). Let V /T
Orb( p ) = ( V ) ⊕ a ⊕ · · · ⊕ ( V m ) ⊕ a m be the isotypic decomposition of V as an H –representation. Then for each 1 ≤ i ≤ m , the Hessian of f induces an invertible self-adjoint H –equivariant linear map( V i ) ⊕ a i ( V i ) ⊕ a i whose negative eigenspace has dimension denoted by dim R V i · ind( p, V i ). Defineind H ( p ) = m X i =1 ind( p, V i ) · [ V i ] ∈ R ( H ) . Now consider that we have a pair of critical points p and p of f which do notlie on the same G –orbit and they are both fixed under the action of H . Then wecan find a path γ in V which is invariant under the H –action connecting p and p . Note that the time- t Hamiltonian flow of the function f : T ∗ V → R is given by( v, v ′ ) → ( v, v ′ + t · df ) , where we extend f : V → R to a smooth function over T ∗ V which is constant alongeach fiber. The trace of the image of γ for t ∈ [0 ,
1] defines a map u γ : D → T ∗ V which has a well-defined Maslov index. It could be shown that µ H ( u γ ) = ind H ( p ) − ind H ( p ) which is a classical computation of Lagrangian Floer homology of cotangent bun-dles.Example 4.14 shows that in this simple case, the non-degeneracy of critical pointsof a G –invariant smooth function is equivalent to the non-degeneracy of intersectionbetween a closely-related pair of Lagrangians. Moreover, the equivariant Maslov in-dex could be computed as a relative equivariant Morse index. We extend these basiccorrespondences in the next section to infinite dimensions which eventually allow usto interpret the generalized Casson invariants using finite-dimensional symplecticconstructions. 5. The gauge-symplectic correspondence
This section proves the most important technical results of this paper. Givena three manifold Y equipped with a Heegaard splitting Y = H ∪ Σ H , we showthat there is a bijection between gauge equivalence classes of perturbed flat G –connections on Y and intersection G –orbits of Lagrangians in ˆ M g (Σ ′ ) inducedby H and H after applying a compatible G –equivariant Hamiltonian isotopy.Moreover, the non-degeneracy condition from gauge-theoretic context is shown tobe equivalent to the non-degeneracy condition of Lagrangian intersections. We thenprove several technical results which eventually reduce the computation of spectralflows on Y to the computation of Maslov indices on ˆ M g (Σ ′ ).5.1. Perturbations.
Let Y be a closed, oriented, Riemannian 3-manifold with asplitting Y = H ∪ Σ (cid:0) [ − , × Σ (cid:1) ∪ Σ H , where H and H are handlebodies and the genus of Σ is h ≥
2. Assume furtherthat the metric on [ − , × Σ is cylindrical. Let { p } ∈ Σ, D p and Σ ′ be the sameas in Section 3. Throughout this section, the product principal G –bundle over anymanifold will be denoted by P . We introduce a family of cylinder functions on Y supported on [ − , × Σ, which are closely related to Hamiltonian perturbationson the extended moduli space ˆ M g (Σ ′ ).Suppose γ j : ( R / Z ) × ( − , → Σ ′ ⊂ Σ , j = 1 , · · · , m are smooth immersions, and suppose there exists ǫ > γ j ’s coincideon ( − ǫ, ǫ ) × ( − , h s : G m → R , s ∈ [ − , G m invariant under diagonal con-jugations by G , such that h s vanishes when s is near − β : ( − , → R be a smooth, compactly supported function that integrates to 1. Define the function H s : A (Σ) or A (Σ ′ ) → R by H s ( A ) = Z − β ( λ ) h s ( ρ λ ( A )) dλ (5.1)for s ∈ [ − , H s is invariant under the gauge group actions.Using the affine linear map x x − ,
1] and [ − , s on [0 ,
1] to get the family { H ˆ s } ≤ ˆ s ≤ from the family { H s } − ≤ s ≤ . SYMPLECTIC FORMULA OF GENERALIZED CASSON INVARIANTS 21
This fits in with the convention set up in Section 4. The Hamiltonian H ˆ s descendsto a G –invariant time-dependent Hamiltonian on the moduli space ˆ M g (Σ ′ ). Theinduced Hamiltonian flow preserves the set µ − (0) ⊂ ˆ M g (Σ ′ ). Lemma 5.1.
Every time-dependent smooth G –invariant Hamiltonian on ˆ M g (Σ ′ ) could be constructed in this way.Proof. Choose { α i , β i } ≤ i ≤ h to be loops with common base point in Σ ′ such thatthey generate the fundamental group of Σ ′ . They could be thickened into immer-sions of annuli of the form described above. For any [ A ] ∈ ˆ M g (Σ ′ ), it is uniquelydetermined by its holonomies around α i , β i , ≤ i ≤ h . This is the embedding de-scribed in Lemma 3.3. Thus all the G –invariant functions on ˆ M g (Σ ′ ) all come fromrestricting conjugation invariant functions on G h . Given a smooth 1–parameterfamily of conjugation invariant functions on G h , denoted by h ˆ s , we can constructthe function H ˆ s using h ˆ s , the annuli and the cut-off function β . It is easy to seethat the function on ˆ M g (Σ ′ ) induced from H ˆ s is the same as h ˆ s . This proves thelemma. (cid:3) On the other hand, the maps { γ j } ≤ j ≤ m could be used to define smooth immer-sions q j : ( R / Z ) × ( − , → Σ ′ × [ − , j = 1 , · · · , m simply by requiring that q j | Σ ′ ×{ s } = γ j for all 1 ≤ j ≤ m . Note that the tuple q = ( q , · · · , q m , β, h s )defines a cylinder datum on Σ ′ × [ − , ⊂ Σ × [ − ,
1] as in Definition 2.6. Because h s vanishes for s near ±
1, we see that q defines a cylinder datum on Y . Thereforewe have the function f q : C Y → R which defines a holonomy perturbation. Definition 5.2.
The Hamiltonian H ˆ s : ˆ M g (Σ ′ ) → R and the cylinder function f q : C Y → R constructed as above are called a compatible pair . Suppose A ∈ C ( Y ) is in temporal gauge on [ − , × Σ, i.e. ι ( ∂∂s )( A ) = 0 where s is the coordinate on [ − , − , × Σ in this case. Let A s bethe restriction of A to the slice { s } × Σ. Then A ∈ C ( Y ) is a critical point of theperturbed Chern-Simons functional CS + f q if and only if A is flat on the complement of [ − , × Σ , (5.2) F A s = 0 for s ∈ [ − , , (5.3)˙ A − X s ( A ) = 0 for s ∈ [ − , , (5.4)where X s ( A ) is a smooth map X s : A (Σ) → Ω (Σ) ⊗ g such that dH s ( A ) a = Z Σ Tr( a ∧ X s ( A )) . (5.5)Notice that X s ( A ) ∈ Ω (Σ) ⊗ g is supported in the union of Im( γ j ).Notice that H ∪ [ − , − × Σ and [1 , × H are manifolds with boundary bothgiven by Σ. Let L , L ⊂ µ − (0) be the corresponding G –invariant Lagrangiansubmanifolds of ˆ M g (Σ ′ ) as described in Section 3. Recall that Φ H ˆ s is the time H ˆ s . Now we are ready to establish thefollowing set-theoretic correspondence. Proposition 5.3.
Let H ˆ s and f q be a compatible pair. Then there is a one-to-onecorrespondence between the orbits of perturbed flat connections on Y with respectto f q and the G –orbits of Φ H ˆ s ( L ) ∩ L . Moreover, a perturbed flat connection isnon-degenerate if and only if the corresponding intersection of Φ H ˆ s ( L ) and L isnon-degenerate.Proof. Throughout the proof, we write Φ H s = Φ H ˆ s and work with the s –coordinate.Suppose A s with − ≤ s ≤ A (Σ) thatsatisfies equation (5.3) and equation (5.4). Because A s is flat, without loss of gen-erality we can assume that A s | D p is given by the product connection after applyinga gauge transformation. Then the image [ A s ] ∈ ˆ M g (Σ ′ ) defines a smooth path.Recall that H s is a family of smooth Hamiltonian functions on ˆ M g (Σ ′ ). Then [ A s ]is a Hamiltonian chord of H s by equation (5.5) and the description of the symplec-tic form ω Σ ′ . Now if A is an f q –perturbed flat connection, by equation (5.2) [ A − ](resp. [ A ]) actually lies in L (resp. L ). By equivariance we construct a G –orbitin Φ H s ( L ) ∩ L .On the other hand, suppose [ A s ] is a Hamiltonian chord of H s on ˆ M g (Σ ′ ) ∩ µ − (0). Let A s be a lift of [ A s ] to A F (Σ), namely a family of flat connections onΣ which are trivial over the disc D p , then we have Z Σ Tr( ˙ A s − X s ( A s ) , a ) = 0for any s and any a ∈ Ω (Σ) ⊗ g such that d A s a = 0. Therefore there exists φ s suchthat ˙ A s − X s ( A s ) = d A s φ s . Hence A s solves the equations (5.3) and (5.4) after a 1-parameter family of gaugetransformations. The Lagrangian boundary condition lifts to (5.2). This proves thefirst part of the statement.For the second part, suppose A ∈ C ( Y ) is a degenerate perturbed flat connection,then there exists v ∈ ker d ∗ A \{ } such that K A,f q v = 0. We have [ A − ] ∈ L ,[ A ] ∈ L , and Φ H s ( ¯ A − ) = A . Restricting v to {± }× Σ defines twisted harmonic1–forms in H A ± denoted by v ± . Under the identification T [ A ± ] ˆ M g (Σ ′ ) ∼ = H A ± ⊕ T ∗ ( g /H A ± )from Corollary 3.6, it is easy to see that v − and v define tangent vectors in T [ A ] L and T [ A ] L which are orthogonal to the directions corresponding to the G –orbitsrespectively. By linearizing equation (5.4) and integrate along the s –direction,we see that (Φ H s ) ∗ ( v − ) = v . Therefore the intersection of Φ H s ( L ) and L isdegenerate at [ A ].On the other hand, suppose the intersection of Φ H s ( L ) and L is degenerate,then we obtain a Hamiltonian chord [ A s ] of H s connecting L and L and thereexists v ± ∈ T [ A ± ] L ± such that v ± / ∈ T Orb([ A ± ]) and (Φ H s ) ∗ ( v − ) = v +1 .Therefore, we can identify v ± as elements in H A ± . Lift v ± to w ± ∈ Ω (Σ) ⊗ g ,then w ± extends to w ∈ Ω ( Y ) ⊗ g such that w ∈ ker( ∗ d A + dV f q ( A )) and w / ∈ Im d A .Decompose w as w = d A u + w ′ where d ∗ A w ′ = 0, we have w ′ ∈ Hess K A,f q , therefore A is degenerate as a perturbed flat connection. (cid:3) SYMPLECTIC FORMULA OF GENERALIZED CASSON INVARIANTS 23
Using Corollary 4.7, Lemma 5.1 and Proposition 5.3, we conclude
Proposition 5.4.
For a generic choice of small Hamiltonian H ˆ s : A (Σ) → R con-structed as above, the intersection between G –equivariant Lagrangians Φ H ˆ s ( L ) and L is non-degenerate. Accordingly, the perturbation of the Chern-Simons functional CS on Y induced by f q constructed as above is non-degenerate. (cid:3) Remark . The above statement shows that we can obtain equivariant transver-sality of CS by a perturbation supported in the neck region [ − , × Σ instead ofusing loops wiggling all over the 3–manifold Y . Of course, this would not be asurprise because the inclusion Σ ֒ → Y defines a surjection between fundamentalgroups so a generic perturbation of the character variety of Y should result form ageneric perturbation of it as a subset in the character variety of Σ.5.2. Spectral flow comparison.
Let Y = H ∪ Σ (cid:0) [ − , × Σ (cid:1) ∪ Σ H and P be the same as before. Suppose H is a closed subgroup of the structural group G . We choose a generic Hamiltonian and the corresponding compatible holonomyperturbation f q as from Proposition 5.4. Suppose B , B ∈ C ( Y ) are two criticalpoints of CS + f q such that H = Stab( B ) ⊂ Stab( B ) and let A s, , A s, be thecorresponding path in A (Σ) satisfying equations (5.3) and (5.4). Let us furtherassume that we actually have a smooth 2–parameter family A s,t ∈ A (Σ) , ( s, t ) ∈ [ − , × [0 , A s, and A s, such that the following holds: F A s,t = 0 for ( s, t ) ∈ [ − , × [0 , t ∈ [0 , , A − ,t and A ,t extend to flat connections on H ∪ [ − , − × Σ and [1 , × Σ ∪ H respectively. (5.7)Using the above two conditions, we can construct a 1–parameter family of con-nections { B t } ≤ t ≤ on Y connecting B and B which are in temporal gauge on[ − , × Σ by extending A s,t . We further assume that H ⊂ Stab( B t ) for all t ∈ [0 , . (5.8)Then we have a 1–parameter family of self-adjoint operators K B t ,f q and it has awell-defined H –equivariant spectral flow Sf H ( K B t ,f q ) as in Definition 2.10. Byapplying a homotopy and reparametrization, we can assume that for all t ∈ [0 , B t is in temporal gauge on [ − , × Σ which defines the family { A s,t } − ≤ s ≤ ⊂ A Σ and this family is constant (i.e. independent of s ) over [ − , × Σ and [1 , × Σ.On the other hand, we can consider the spaceΩ (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g . It comes with a natural symplectic form˜ ω Σ : (( a , φ , ψ ) , ( a , φ , ψ )) Z Σ Tr( a ∧ a ) + Z Σ Tr( φ ∧ ψ ) − Z Σ Tr( ψ ∧ φ )and a compatible complex structure ˜ J Σ given by( a, φ, ψ ) ( ∗ a, − ∗ ψ, ∗ φ ) . Given A ∈ A F (Σ), we define S A to be the twisted de Rham operator on Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g given by( a, φ, ψ ) ( − ∗ d A φ − d A ∗ ψ, ∗ d A a, d A ∗ a ) . This is a self-adjoint operator with respect to the natural L –norm. Now considerthe space L k ([ − , (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g )which is the L k –completion of the space of smooth maps from [ − ,
1] to Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g using the cylindrical metric on [ − , × Σ. Note thatthis space could be identified with L k (Ω ([ − , × Σ) ⊗ g ) ⊕ L k (Ω ([ − , × Σ) ⊗ g )on [ − , × Σ by mapping ( a, φ, ψ ) ∈ L k ([ − , (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g )to ( φ, a + ∗ ψds ). Because the holonomy perturbation f q is supported on [ − , × Σ,for each t ∈ [0 , K B t ,f q is equal to the twisted odd signature operator (cid:18) ∗ d B t d B t d ∗ B t (cid:19) (5.9)on (Ω ( Y ) ⊕ Ω ( Y )) ⊗ g in the complement of [ − , × Σ. The restrictions of thisoperator on [ − , − × Σ and [1 , × Σ are all of the form ∗ ( ds ∧ ∂∂s + S A ∓ ,t )under the identification described above. Let Λ − ( t ) ⊂ L (Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ) be the subspace consisting of elements that extend to L / elements inker( K B t ,f q ) on H ∪ [ − , − × Σ and let Λ ( t ) ⊂ L (Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g )be the similar subspace constructed from [1 , × Σ ∪ H . Then it is well-known thatΛ ± ( t ) defines a pair of Lagrangian subspaces in L (Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g )with respect to the symplectic form ˜ ω Σ , see [Nic95, Section 2] for example. Thepaths t Λ ± ( t ) are continuously differentiable in the sense that the paths oforthogonal projection operators P Λ ± ( t ) onto Λ ± ( t ) are continuously differentiable. Definition 5.6.
Let L ([ − , × Σ; Λ − ( t ) , Λ ( t )) be the subspace of L ([ − , (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ) consisting of elements whose restriction to {± } × Σ lies in the space Λ ± ( t ) . Let A s,t ∈ A (Σ) be the family described above. Define D ( t ) : L ([ − , × Σ; Λ − ( t ) , Λ ( t )) → L ([ − , (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ) to be the operator given by D ( t ) aφψ = ∗ ˙ a + d A s,t φ + d ∗ A s,t ψ − ∗ dX s ( A s,t )( a ) − ∗ ˙ ψ + d ∗ A s,t a ∗ ˙ φ + d A s,t a , where the dot represents taking derivative along the s –direction.Remark . The operator D ( t ) could be written as ˜ J Σ ( ∂∂s + S A s,t − dX s ( A s,t )).Using the Lagrangian boundary conditions Λ ± ( t ), it is easy to see that D ( t ) de-fines a self-adjoint operator such that the inclusion map from its domain to thebackground Hilbert space is a compact operator. Meanwhile, D ( t ) is exactly therestriction of the operator K B t , q on the region [ − , × Σ ⊂ Y . SYMPLECTIC FORMULA OF GENERALIZED CASSON INVARIANTS 25
As a consequence, we obtain a 1–parameter family of self-adjoint operators { D ( t ) } ≤ t ≤ . Although this family has varying domains L ([ − , × Σ; Λ − ( t ) , Λ ( t )),its spectral flow is nonetheless well-defined in this situation, see [SW08, AppendixA]. Note that all the discussions above are compatible with the Stab( B t )–action,therefore the H –equivariant spectral flow Sf H ( D ( t )) of the family { D ( t ) } ≤ t ≤ isalso well-defined. Now we are ready to state the main result of this subsection. Theorem 5.8.
Let { K B t ,f q } ≤ t ≤ and { D ( t ) } ≤ t ≤ be as above. Then Sf H ( K B t ,f q ) = Sf H ( D ( t )) . The proof of Theorem 5.8 is based on adapting the arguments in [Nic95]. Theidea is to modify the path of self-adjoint operators to a preferred form to simplifythe calculation of the spectral flow.We start by reducing the calculation of the equivariant spectral flow to thecalculation of the ordinary spectral flow. Identifying (Ω ( Y ) ⊕ Ω ( Y )) ⊗ g with thespace of smooth sections of the vector bundle g ⊕ T ∗ Y ⊗ g , where g stands for theproduct g –bundle over Y , for any connection B ∈ C ( Y ), its stabilizer Stab( B ) ⊂ G acts on (Ω ( Y ) ⊕ Ω ( Y )) ⊗ g by the fiberwise adjoint G –action on the g factor.Choose H ⊂ G as above. Suppose the adjoint action of H on g decomposes g intoisotypic pieces g ∼ = V ⊕ a ⊕ · · · ⊕ V ⊕ a m m , we define the vector bundle˜ E i = (Ω ( Y ) ⊕ Ω ( Y )) ⊗ V ⊕ a i i and the vector bundle E i = (Ω ( Y ) ⊕ Ω ( Y )) ⊗ K a i i , where K i = Hom H ( V i , V i ). Then the global sections of E i could be identified withHom H ( V i , (Ω ( Y ) ⊕ Ω ( Y )) ⊗ g ) . Let L k ( E i ) be the L k –completion of the space of smooth sections of E i over Y .Then this encodes the information of the V i -isotypic piece of the H –Banach space L k ( g ⊕ T ∗ Y ⊗ g ). By the H –equivariance of the operator K B t ,f q , it induces anoperator K iB t ,f q = L k ( E i ) → L k − ( E i ) . Correspondingly, we haveHom H ( V i , L k (Ω ([ − , × Σ) ⊗ g ) ⊕ L k (Ω ([ − , × Σ) ⊗ g )) ∼ = L k ( E i | [ − , × Σ )and the spaces over the Riemann surface Σ given byHom H ( V i , L k (Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g )) ∼ = L k ( E i | { }× Σ ) . The symplectic form ˜ ω Σ induces a symplectic form on L ( E i | { }× Σ ). By restrictingΛ ± ( t ) to the V i –isotypic piece, we obtain Lagrangian subspacesΛ i ± ( t ) ⊂ L ( E i | { }× Σ ) . As a result, the H –equivariant operator D ( t ) induces operators D i ( t ) : L ( E i | [ − , × Σ ; Λ i − ( t ) , Λ i ( t )) → L ( E i | [ − , × Σ )for all i = 1 , . . . , m . According to Definition 2.10, we have Lemma 5.9. Sf H ( K B t ,f q ) = Sf H ( D ( t )) if and only if Sf ( K iB t ,f q ) = Sf ( D i ( t )) for any ≤ i ≤ m . (cid:3) From now on, we will focus on the operators K iB t ,f q and D i ( t ) for a fixed i ∈ Z ∩ [1 , m ]. Definition 5.10.
An operator D : L k ( E i ) → L k − ( E i ) is said to satisfy uniquecontinuation if for any v ∈ ker( D ) such that v | U = 0 where U ⊂ H ⊂ Y is anon-empty open set, we have v ≡ . Lemma 5.11.
For any t ∈ [0 , , the operator K iB t ,f q satisfies unique continuation.Proof. Let v ∈ ker( K iB t ,f q ) and v | U = 0 where U is as above. The operator K iB t ,f q | [1 , × Σ ∪ H comes from the twisted signature operator therefore is a Diracoperator. By the unique continuation property of Dirac operators, we know that v is identically 0 over [1 , × Σ ∪ H . Over the region [ − , × Σ, the element v defines a curve v ( s ) in L ( E i | { }× Σ ) satisfying the ODE˜ J Σ ( ∂∂s + S A s,t − dX s ( A s,t )) v ( s ) = 0 . Here we use the same notation to represent the induced operators on L ( E i | [ − , × Σ ).Since the solution is uniquely determined by the initial value and v | { }× Σ = 0,we know that v | [ − , × Σ = 0. The vanishing of v over H ∪ [ − , − × Σ comesagain from the unique continuation property of the twisted signature operator on H ∪ [ − , − × Σ and the fact that we have already shown that v {− }× Σ = 0.Therefore the lemma is proved. (cid:3) Lemma 5.12.
The operator K iB ,f q − K iB ,f q is a bounded operator on L k ( E i ) .Proof. By definition, K iB ,f q − K iB ,f q is the sum of a 0–th order differential operatordepending on B , B and the induced operator on L k ( E i ) given by the difference DV f q ( B ) − DV f q ( B ) . Use the definition and [KM11, Proposition 3.5], these 2 terms are both bounded by C ( k B k L k + k B k L k ) where C > Y , so the lemmais proved. (cid:3) Recall that for a family of self-adjoint operators { D t } ≤ t ≤ , its resonance set Z ( D t ) consists of t ∈ [0 ,
1] such that ker( D t ) = { } . Suppose t ∈ Z ( D t ) andlet P t be the orthogonal projection onto ker( D t ) . Then the resonance matrix R ( D t ) at t is defined to be the map P t ˙ D t : ker( D t ) → ker( D t ). Definition 5.13.
A family of self-adjoint operators { D t } ≤ t ≤ is said to be positive if the resonance matrices are all positive definite. It is said to be negative if thefamily {− D t } ≤ t ≤ is positive. Using Kato’s selection theorem, if the family { D t } ≤ t ≤ is positive or negative,the set Z ( D t ) is discrete. Lemma 5.14.
The –parameter family { K iB t ,f q } ≤ t ≤ defined over L ( E i ) ⊂ L ( E i ) is homotopic relative end points to the concatenation of a positive –parameter fam-ily and a negative –parameter family such that each operator in these –familiessatisfies unique continuation. SYMPLECTIC FORMULA OF GENERALIZED CASSON INVARIANTS 27
Proof.
Using Lemma 5.12, we can choose some
C > K iB ,f q − K iB ,f q over L ( E i ) is bounded above by C −
1. Consider the families K i + ( t ) = K iB ,f q + tC · id K i − ( t ) = K iB ,f q + C · id + t ( K iB ,f q − K iB ,f q − C · id)for 0 ≤ t ≤
1. It is easy to see that K i + ( t ) is positive and K i − ( t ) is negative. Theaffine homotopy between { K iB t ,f q } ≤ t ≤ and the concatenation of these 2 familiesgives the desired homotopy. For the unique continuation property, one proceedsexactly the same as in the proof of Lemma 5.11 by noticing that K i ± ( t ) could bedescribed by Dirac operators on H ∪ [ − , − × Σ ∪ [1 , × Σ ∪ H and the kernelof K i ± ( t ) consists of solutions to an ODE on [ − , × Σ. (cid:3) Given a positive or negative family { D t } ≤ t ≤ , we wish to perturb it further sothat for any t ∈ Z ( D t ), the kernel of D t becomes 1–dimensional. This is thetransversality result established in [Nic95, Proposition 3.6]. The unique continua-tion condition stated above guarantees that the proof of [Nic95, Proposition 3.6]works without change to give the following statement. Lemma 5.15.
For the two families { K i ± ( t ) } ≤ t ≤ constructed in Lemma 5.14, onecan find α ± : [0 , → C ∞ (Hom( E i , E i )) which are arbitrarily small such that the following holds: (1) α ± ( t ) is supported away from [ − , × Σ ; (2) The family K i + ( t )+ α + ( t ) (resp. K i − ( t )+ α − ( t ) ) remains to be positive (resp.negative) and for any t ∈ Z ( K i + ( t ) + α + ( t )) (resp. t ∈ Z ( K i − ( t ) + α − ( t )) ), the kernel of K i + ( t ) + α + ( t ) (resp. K i − ( t ) + α − ( t ) ) is actually –dimensional. (cid:3) Proof of Theorem 5.8.
Let us fix α ± ( t ) as in Lemma 5.15 and denote ˜ K i ± ( t ) = K i ± ( t ) + α ± ( t ). By construction, { ˜ K i ± ( t ) } ≤ t ≤ defines operators˜ K i ± ( t ) : L ( E i ) → L ( E i )for all 0 ≤ t ≤ H ∪ [ − , − × Σ ∪ [1 , × Σ ∪ H and they are cylindrical over [ − , − × Σ ∪ [1 , × Σ. Therefore,they define Lagrangian subspaces˜Λ i − ( t ) ± ⊂ L ( E i | { }× Σ )consisting of restriction of elements in L / ( E i | H ∪ [ − , − × Σ ) lying inker( ˜ K i ± ( t ) | H ∪ [ − , − × Σ ) . Similarly one can construct ˜Λ i ( t ) ± ⊂ L ( E i | { }× Σ )by considering the restriction to [1 , × Σ ∪ H . We can then construct the operators˜ D i ± ( t ) : L ( E i | [ − , × Σ ; ˜Λ i − ( t ) ± , ˜Λ i ( t ) ± ) → L ( E i | [ − , × Σ )similar to the construction of D ( t ), which define self-adjoint operators using theLagrangian boundary conditions.Making α ± ( t ) smaller if necessary, the spectral flow of the family { K B t ,f q } ≤ t ≤ is the same as the sum of the spectral flows of { ˜ K i + ( t ) } ≤ t ≤ and { ˜ K i − ( t ) } ≤ t ≤ . Similarly, by concatenating the Lagrangian boundary conditions, it is easy to seethat the spectral flow of { D i ( t ) } ≤ t ≤ is equal to the sum of the spectral flows of { ˜ D i + ( t ) } ≤ t ≤ and { ˜ D i − ( t ) } ≤ t ≤ . Using Lemma 5.9, it suffices to show that Sf ( ˜ K i + ( t )) = Sf ( ˜ D i + ( t )) and Sf ( ˜ K i − ( t )) = Sf ( ˜ D i − ( t )) . We prove the plus version of the above statement and the negative version holdsusing the same argument. From the definition, we see that Z ( ˜ K i + ( t )) = Z ( ˜ D i + ( t ))therefore it suffices to show that the resonance matrices of { ˜ K i + ( t ) } ≤ t ≤ and { ˜ D i + ( t ) } ≤ t ≤ have the same sign at any t ∈ Z ( ˜ K i + ( t )) = Z ( ˜ D i + ( t )). In otherwords, we just need to show that the resonance matrix of ˜ D i + ( t ) at t = t is pos-itive definite. Let v ∈ ker( ˜ D i + ( t )) − { } . Note that v could also be viewed asan element of ker( ˜ K i + ( t )) . Let χ : [0 , → R be a non-negative smooth functionsupported in (0 ,
1) which is strictly positive over [ , ]. Then χ defines a functionon [0 , × Σ ⊂ Y by composing it with the projection to the s –coordinate. Thisfurther extends to a smooth function over Y in the obvious way. Then for ǫ > { ˜ K i + ( t ) + ( t − t ) χ · id } t − ǫ ≤ t ≤ t + ǫ (5.10)is C –close to { ˜ K i + ( t ) } t − ǫ ≤ t ≤ t + ǫ and the affine homotopy between them preservesthe spectral flow. This is because h v, χv i L > h v, χv i L = 0, the restriction of v to the time slice { } × Σ is 0 thus it vanisheson [ − , × Σ by uniqueness of solutions to ODE therefore vanishes on whole Y byunique continuation of Dirac operators. Accordingly, we can construct the inducedaffine homotopy from { ˜ D i + ( t ) } t − ǫ ≤ t ≤ t + ǫ to { ˜ D i + ( t ) + ( t − t ) χ · id } t − ǫ ≤ t ≤ t + ǫ (5.11)preserving the spectral flow. Now notice that because χ is supported in the region[0 , × Σ and the kernels of the new families at time t are all spanned by v , wesee that the resonance matrices of these 2 families (5.10) and (5.11) have the samevalue at time t . This finishes the proof. (cid:3) An adiabatic limit result.
Let { A s,t } − ≤ s ≤ , ≤ t ≤ ⊂ A F (Σ) be a smoothfamily of flat connections on Σ. Viewing Σ as the boundary of H ∪ [ − , − × Σand [1 , × Σ ∪ H , we further suppose that A − ,t extends to a flat connection B − ,t on H ∪ [ − , − × Σ and A ,t extends to a flat connection B ,t on [1 , × Σ ∪ H for all t ∈ [0 , { Λ ± ( t ) } ≤ t ≤ be the same as in theprevious subsection. Note that we do not require that A s, or A s, comes from therestriction of a f q –perturbed flat connection on Y as in the previous subsection.This setup gives us more flexibility for later calculations. Furthermore, we assumethat Stab( A s,t ) ∼ = H ⊂ G remains to be the same for all ( s, t ) ∈ [ − , × [0 , A s,t ∈ A F (Σ), let H A s,t be the kernel of the twisted de Rham operator S A s,t : L (Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ) → L (Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ) . In other words, H A s,t is given by the total space of twisted harmonic forms on Σwith respect to the flat connection A s,t . This is a symplectic vector space using therestriction of the symplectic form ˜ ω Σ . Let π A s,t : L (Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ) → H A s,t SYMPLECTIC FORMULA OF GENERALIZED CASSON INVARIANTS 29 be the L –orthogonal projection. Define L ± ( t ) := H A ± ,t ∩ Λ ± ( t ) , for 0 ≤ t ≤ . Lemma 5.16. L i ( t ) is a Lagrangian subspace of H A i,t for i = ± .Proof. By definition, L i ( t ) is isotropic. So we just need to prove that dim L i ( t ) ≥ dim H A i,t . Notice that (0 , ker d A i,t , ⊂ L i because this comes from the Liealgebra of the stabilizer group and Stab( B i,t ) = Stab( A i,t ).Let m = dim ker d B − ,t . Recall that h is the genus of the Riemann surface Σ.Fix a base point p on Σ ∼ = ∂H , let γ , · · · , γ h be h closed curved based on p thatgenerates the fundamental group of H . Let ˆ C be the space of smooth connections B on H ∪ [ − , − × Σ such that B ( ∂ s ) = 0 on the boundary. DefineHol : ˆ C → G h to be the map given by holonomies along γ i . Let U = ( − ǫ, ǫ ) h , let Φ : U → ˆ C be amap such that Φ(0) equals B − ,t , and that Im d (Hol ◦ Φ) is surjective at zero. Byconstruction, Im d Φ(0) ⊂ Ω ( H ∪ [ − , − × Σ) ⊗ g is a h · dim G dimensional linearspace. For every u ∈ Im d Φ(0), there is a unique v such that v ∈ Im d B − ,t , ∗ v = 0on {− } × Σ, and u ′ := u − d B − ,t v ∈ ker( d ∗ B − ,t ). The kernel of Im d Φ(0) under themap u u ′ has dimension at most m by considering the holonomies, therefore theimage of Im d Φ(0) under the map u u ′ has dimension at least ( h − · dim G + m .Therefore L − ( t ) ∩ (ker( d B − ,t + d ∗ B − ,t ) , , ⊂ H A − ,t has dimension at least ( h − · dim G + m , which proved that L − ( t ) is Lagrangian.The same argument works to prove L i ( t ) is Lagrangian. (cid:3) By definition, dim H A s,t remains to be the same for all ( s, t ) ∈ [ − , × [0 , t ∈ [0 , L k ([ − , H A s,t ) ⊂ L k ([ − , (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ) L k ([ − , H A s,t ; L − ( t ) , L ( t )) ⊂ L k ([ − , × Σ; Λ − ( t ) , Λ ( t ))consisting of elements ( a, φ, ψ ) whose value at each s ∈ [ − ,
1] lies in H A s,t . Thenwe can define a self-adjoint operator D ( t ) : L ([ − , H A s,t ; L − ( t ) , L ( t )) → L ([ − , H A s,t ) (5.12)given by D ( t ) aφψ = π A s,t ∗ ˙ a − ∗ dX s ( A s,t )( a ) − ∗ ˙ ψ ∗ ˙ φ . By construction, the family { D ( t ) } ≤ t ≤ has a well-defined H –equivariant spectralflow Sf H ( D ( t )). The following is another comparison result of spectral flows. Theorem 5.17.
Let { D ( t ) } ≤ t ≤ be the family constructed in Section 5.2 and let { D ( t ) } ≤ t ≤ be the family constructed above. Then Sf H ( D ( t )) = Sf H ( D ( t )) . The proof of Theorem 5.17 is exactly the same as the proof of [DS94, Theorem6.1]. We will sketch a proof for completeness. Because we have used connectionsin temporal gauge, the covariant differentiation ∇ s in [DS94] is replaced by theordinary differentiation ∂ s . Note that the calculation of equivariant spectral flowcould be reduced to the calculation of ordinary spectral flow as in Section 5.2 sothe H –equivariance will be suppressed in this subsection.The idea is to use the 1–parameter family of operators D ǫ ( t ) : L ([ − , × Σ; Λ − ( t ) , Λ ( t )) → L ([ − , (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g )given by the formula D ǫ ( t ) aφψ = ∗ ˙ a + d A s,t φ + d ∗ A s,t ψ − ∗ dX s ( A s,t )( a ) − ∗ ˙ ψ + ǫ d ∗ A s,t a ∗ ˙ φ + ǫ d A s,t a depending on the parameter ǫ >
0, so that the spectral flow of { D ( t ) } ≤ t ≤ = { D ( t ) } ≤ t ≤ could be reduced to the spectral flow of { D ( t ) } ≤ t ≤ by passing tothe adiabatic limit ǫ → t ∈ [0 , k ( a, φ, ψ ) k ,ǫ = k a k + ǫ k φ k + ǫ k ψ k on the space L ([ − , (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ) and k ( a, φ, ψ ) k ,ǫ = k a k + k d A s,t a k + k d A s,t ∗ a k + ǫ k ˙ a k + ǫ k d A s,t φ k + ǫ k ˙ φ k + ǫ k d ∗ A s,t ψ k + ǫ k ˙ ψ k on L ([ − , (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ) using the product metric on[ − , × Σ. Sketch of proof of Theorem 5.17.
The proof breaks into five steps.
Step 1-Elliptic estimate:
There is an ǫ > < ǫ < ǫ , thereexists a constant C > ǫ such that k u − π A s,t ( u ) k ,ǫ ≤ Cǫ ( k D ǫ ( t ) u k ,ǫ + k π A s,t ( u ) k L ) (5.13)for u ∈ L ([ − , × Σ; Λ − ( t ) , Λ ( t )).This is [DS94, Lemma 7.3]. Note that although our 3–manifold [ − , × Σ hasboundary, the Lagrangian boundary condition guarantees that the integration byparts formula entering the proof works without change. Meanwhile, our projectionmap π A s,t involves the H and H part so the reducible connections would notaffect the estimate. Step 2-Convergence of resolvent set:
For every C > ǫ > C > t ∈ [0 ,
1] and | λ | ≤ C . If k u k L ≤ C k D ( t ) u − λu k L for all u ∈ L ([ − , H A s,t ; L − ( t ) , L ( t )) , then k u k ,ǫ ≤ C k D ǫ ( t ) u − λu k ,ǫ for 0 < ǫ < ǫ and u ∈ L ([ − , × Σ; Λ − ( t ) , Λ ( t )).This is [DS94, Lemma 7.4]. Note that the proof is based on a direct computationand an application of equation (5.13) therefore it works in our situation withoutchange. SYMPLECTIC FORMULA OF GENERALIZED CASSON INVARIANTS 31
Step 3-Refined convergence of operators:
Define R = { ( t, λ ) ∈ [0 , × C (cid:12)(cid:12) λ is not an eigenvalue of D ( t ) } to be the resolvent set of the family { D ( t ) } ≤ t ≤ . Denote by R ǫ is resolvent setof the family { D ǫ ( t ) } ≤ t ≤ . Then for every compact subset K ⊂ R , there exists aconstant ǫ > K ⊂ R ǫ for 0 < ǫ < ǫ and k π A s,t (( λ · id − D ǫ ( t )) − v ) − ( λ · id − D ( t )) − π A s,t k L ≤ Cǫ k v k ,ǫ (5.14)for ( t, λ ) ∈ K and v ∈ L ([ − , (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ⊕ Ω (Σ) ⊗ g ).This is [DS94, Lemma 7.5]. Step 4-Multiplicity estimate:
For all C > k > t ∈ [0 ,
1] and 0 < ǫ + δ < k . If λ is an eigenvalueof D ( t ) of multiplicity m and u ⊥ ker( λ · id − D ( t )) = ⇒ k u k L ≤ C k λ u − D ( t ) u k L for all u ∈ L ([ − , H A s,t ; L − ( t ) , L ( t )), then the multiplicity of all eigenvaluesof λ of D ǫ ( t ) with | λ − λ | ≤ δ does not exceed m .This follows from [DS94, Lemma 7.6]. Step 5-Concluding the proof.
By Kato’s selection theorem, we can find δ ∈ R which is arbitrarily close to 0 such that the resonance set of the family of theoperators { D ( t ) − δ · id } ≤ t ≤ , written as Z ( D ( t ) − δ · id) is finite and containedin (0 , σ ( D ) to represent the spectrum of D . Let Z ( D ( t ) − δ · id) = { t , . . . , t N } . Recall that R ( D ( t i ) − δ · id) for i = 1 , . . . , N is the resonance matrix.Define m i := sign( R ( D ( t i ) − δ · id)) , i = 1 , . . . , N to be the signatures of the resonance matrices. Then Sf ( { D ( t ) } ≤ t ≤ ) = N X i =1 m i and m i could be computed as follows. Choose κ > λ = δ is the onlyeigenvalue of D ( t i ) in the interval [ δ − κ, δ + κ ]. Now choose τ > δ ± κ / ∈ σ ( D ( t )) for all t ∈ [ t i − τ, t i + τ ]. Then m i = { λ ∈ σ ( D ( t i − τ )) (cid:12)(cid:12) δ − κ < λ < δ }− { λ ∈ σ ( D ( t i + τ )) (cid:12)(cid:12) δ − κ < λ < δ } . By Step 2 , there exists a constant ǫ > δ / ∈ σ ( D ǫ ( t )) for t = t i − τ or t i + τ ) and δ ± κ / ∈ σ ( D ǫ ( t )) for all t ∈ ( t i − τ, t i + τ ) as long as 0 < ǫ < ǫ . Using Step3 , Step 4 and the spectral projection operators, one can show that m i = { λ ∈ σ ( D ǫ ( t i − τ )) (cid:12)(cid:12) δ − κ < λ < δ }− { λ ∈ σ ( D ǫ ( t i + τ )) (cid:12)(cid:12) δ − κ < λ < δ } . Therefore we conclude that Sf ( D ǫ ( t )) = N X i =1 m i = Sf ( D ( t ))for ǫ > Sf ( D ǫ ( t )) is independent of ǫ >
0. This concludes the proof. For the full details of the proof, we refer thereaders to [DS94, Section 7]. (cid:3)
From Maslov index to spectral flow and back.
Throughout this subsec-tion, the structure group G is equal to SU( n ) and Y is an integer homology 3–sphere.Furthermore, suppose that the genus of the Heegaard surface Σ is h ≥
3. Let H ˆ s and f q be a compatible pair of small non-degenerate perturbations. Suppose B ∈ C ( Y )is a f q –perturbed flat connection of type σ = (( n , m ) , . . . , ( n r , m r )) ∈ Σ n (see theend of Section 2). Without loss of generality, we can assume that B is in temporalgauge over [ − , × Σ and it is represented by a path A s : [ − , → A F (Σ) which isconstant for s ∈ [ − , − ∪ [1 , B is given by the direct sum of irreducible SU( n i )-connections B ( i ) on E ( n i ) where P × SU( n ) C n ∼ = E ( n ) ⊕ m ⊕ · · · ⊕ E ( n r ) ⊕ m r . (5.15)Let A ( i ) s be the induced path on the E ( n i )–component. Then A ( i ) − extends to a flatconnection B ( i ) − on H ∪ [ − , − × Σ and A ( i )1 extends to a flat connection B ( i )1 on[1 , × Σ ∪ H . Note that because Y is an integer homology sphere, the productconnection θ is a non-degenerate critical point of the unperturbed functional CS.Because f q is a small perturbation, there exists a f q –perturbed flat connection ˜ θ in temporal gauge over [ − , × Σ with stabilizer G which lies in a contractibleneighborhood of the product connection θ ∈ C ( Y ). Denote by ˜ θ ( i ) the E ( n i )–component of ˜ θ under the decomposition (5.15) and ˜ θ s is the induced path of flatconnections in A F (Σ), which can be further assumed to lie in A F (Σ). Then we canfind a path of connections A ( i ) − ,t : [0 , → A F (Σ) with A ( i ) − , = ˜ θ ( i ) − and A ( i ) − , = A ( i ) − such that A ( i ) − ,t extends to a flat connection B ( i ) − ,t on E ( n i ) over H ∪ [ − , − × Σ forall t ∈ [0 ,
1] which is irreducible expect for t = 0. Similarly, we can find the family A ( i )1 ,t on [1 , × Σ ∪ H and B ( i )1 ,t with the same properties. Then the union of paths { A ( i ) − ,t } ≤ t ≤ , { ˜ θ ( i ) s } − ≤ s ≤ , { A ( i )1 ,t } ≤ t ≤ and { A ( i ) s } − ≤ s ≤ defines a loop inside thespace of flat connections on E ( n i ), denoted by A ( i ) F (Σ). By our assumption thatthe genus of Σ is at least 3, [DU95, Corollary 2.7] shows that the subset of A ( i ) F (Σ)consisting of irreducible connections is simply-connected. Consequently, the loopconstructed above could be extended to a smooth 2–parameter family A ( i ) s,t : [ − , × [0 , → A ( i ) F (Σ)such that A ( i ) s,t is irreducible except for ( s, t ) ∈ [ − , × { } . Take the direct sum ofthese families for 1 ≤ i ≤ m and apply further gauge transformations if necessary,we obtain a 2–parameter family A s,t : [ − , × [0 , → A F (Σ) ⊂ A F (Σ) . Let ˜ θ = B and B = B , we can use A s,t to construct a family { B t } ≤ t ≤ satisfyingthe assumptions in the beginning of Subsection 5.2. According to this construction,Stab( A s,t ) remains invariant for ( s, t ) ∈ [ − , × (0 , Lemma 5.18.
Suppose the perturbations H ˆ s and f q are . Then for each ≤ i ≤ r , the symplectic area of A ( i ) s,t : [ − , × [0 , → A ( i ) F (Σ) under the Atiyah-Bottsymplectic form ω Σ is one-half of the Chern-Simons invariant of B ( i ) . SYMPLECTIC FORMULA OF GENERALIZED CASSON INVARIANTS 33
Proof.
According to our convention of Chern-Simons functional (2.1),CS( B ( i ) ) = CS( B ( i ) | H ∪ [ − , − × Σ ) + CS( B ( i ) | [ − , × Σ ) + CS( B ( i ) | [1 , × Σ ∪ H )= Z [0 , × [ − , × Σ Tr( F B t ∧ F B t )= Z [0 , × [ − , × Σ Tr(( ds ∧ ∂ s A ( i ) s,t + dt ∧ ∂ t A ( i ) s,t + F A ( i ) s,t ) )= 2 Z [0 , × [ − , × Σ Tr( ∂ s A ( i ) s,t ∧ ∂ t A ( i ) s,t ) dsdt. The second equality is an application of the Stokes’ formula over the 4–manifolds[0 , × [ − , × Σ , [0 , × ( H ∪ [ − , − × Σ) , [0 , × ([1 , × Σ ∪ H ) and notice that B ( i ) | H ∪ [ − , − × Σ , B ( i ) | [1 , × Σ ∪ H are connected to the product connection throughflat connections. The above calculation proves the lemma. (cid:3) Recall that L and L are Lagrangians in the extended moduli space ˆ M g (Σ ′ )constructed from handlebodies. The f q –perturbed flat connection B is associatedwith the G –orbit [ A ] ∈ Φ H ˆ s ( L ) ∩ L in ˆ M g (Σ ′ ) in Proposition 5.3. Let Φ H ˆ s ( η )be the time- η –flow of the Hamiltonian vector field X H ˆ s . Then the composition u ( A s,t ) := Φ H ˆ s (1 − η )([ A η − ,t ]) : ( η, t ) ∈ [0 , × [0 , → ˆ M g (Σ ′ )maps { } × [0 ,
1] to Φ H ˆ s ( L ) and maps { } × [0 ,
1] to L and the stabilizer of everypoint in the image contains H = Stab( B ) ∼ = Stab([ A ]) as a subgroup. UsingDefinition 4.13, u ( A s,t ) has a well-defined H –equivariant Maslov index. Here is theprecise statement of Theorem 1.1. Theorem 5.19. Sf H ( B, f q ) − [ker d B ] = µ H ( u ( A s,t )) .Proof. By definition, the equivariant Maslov index µ H ( u ( A s,t )) could be computedusing the equivariant spectral flow of the operator − J ∂∂s on the space of L –sectionsof the symplectic vector bundle u ( A s,t ) ∗ ( T ˆ M g (Σ ′ ))with Lagrangian boundary conditions induced by T Φ H ˆ s ( L ) and T L . Apply thefiberwise symplectomorphism induced by linearizing Φ H ˆ s (1 − η ), use the identifica-tion (3.6) and use the fact that the intersection [˜ θ ] is non-degenerate, we can seethat µ H ( u ( A s,t )) is given by the H –equivariant spectral flow of the family − D ( t ) : L ([ − , H A s,t ∩ H ; L − ( t ) , L ( t )) → L ([ − , H A s,t ∩ H )from (5.12) where t ∈ [ ǫ,
1] such that ǫ > ∩ H means that we require that the 0-form and 2–form components are 0. This is exactlywhere the shifting [ker d B ] comes in. By Theorem 5.17, we see that µ H ( u ( A s,t ))is equal to the H –equivariant spectral flow of the family {− K B t ,f q } ǫ ≤ t ≤ up to ashifting by [ker d B ]. By concatenating { K B t ,f q } ≤ t ≤ with a linear path between K ˜ θ,f q and K θ, , using Theorem 5.8, we see that µ H ( u ( A s,t )) exactly computes Sf H ( B, f q ) (recall that this is defined using the linear path from ( B, f q ) to ( θ, (cid:3) Therefore, we have the following immediate corollary from Definition 2.12, Lemma5.18 and Theorem 5.19:
Corollary 5.20. ind(
B, f q ) = µ H ( u ( A s,t )) − P n i ≥ h ω,u ( ˆ A ( i ) s,t ) i π n i · τ i , where h ω, u ( ˆ A ( i ) s,t ) i is the symplectic area of the family u ( ˆ A ( i ) s,t ) induced from the i –th component of thedecomposition of a genuine flat ˆ B near B . (cid:3) Proof of Theorem 1.2.
Let H ˆ s and f q be a compatible pair. For any G –orbits ofΦ H ˆ s ( L ) ∩ L , without loss of generality we can assume that the disc D ( p ) comesfrom a family A s,t ( p ) ⊂ A F (Σ) as in this section. If p corresponds to an irreducibleSU(3)–connection B ( p ), by Theorem 5.19 the spectral flow from K B ( p ) ,f q is equalto µ ( D ( p )) by noticing that the group acts trivially here and [ker d B ] = 0. If p corresponds to a reducible SU(3)–connection B ( p ), it has stabilizer U(1) and B ( p ) is gauge-equivalent to an SU(2)–connection. The equivariant Maslov index µ H ( u ( A s,t )) is the linear combination of the trivial representation and the weight( − µ t ( D ( p )) and µ n ( D ( p )) respectively. Then thetheorem follows from checking the formula (1.2) term-wisely with the formula in[BH98, Theorem 1]. Note that our definition of the equivariant spectral flow cancelsout the factor in loc.cit. and our convention of spectral flow adds the term[ker d B ]. Although the formula of Boden-Herald is written down using a holonomyperturbation which is not of the preferred form, the identification between (1.2)and λ SU(3) results from the independence of λ SU(3) on holonomy perturbations,established in [BZ20] and Section 2. Therefore the theorem is proved. (cid:3)
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