GAUGED FLOER HOMOLOGY FOR HAMILTONIAN ISOTOPIES I:
DEFINITION OF THE FLOER HOMOLOGY GROUPS
GUANGBO XU
Abstract. We construct the vortex Floer homology group V HF (M,µ;H) for an aspherical Hamil-
tonian G-manifold (M,ω) with moment map µ and a class of G-invariant Hamiltonian loop Ht,
following the proposal of [3]. This is a substitute for the ordinary Hamiltonian Floer homology of
the symplectic quotient of M . We achieve the transversality of the moduli space by the classical
perturbation argument instead of the virtual technique, so the homology can be defined over Z or
Z2.
Contents
1. Introduction 1
2. Basic setup and outline of the construction 7
3. Asymptotic behavior of the connecting orbits 14
4. Fredholm theory 21
5. Compactness of the moduli space 28
6. Floer homology 31
Appendix A. Transversality by perturbing the almost complex structure 37
References 46
1. Introduction
1.1. Background. Floer homology, introduced by Andreas Floer (see [8], [9]), has been a great
triumph of J-holomorphic curve technique invented by Gromov [17] in many areas of mathemat-
ics. Hamiltonian Floer homology gives new invariants of symplectic manifolds and its Lagrangian
submanifolds and has been the most important approach towards the solution to the celebrated
Arnold conjecture initiated in the theory of Hamiltonian dynamics; the Lagrangian intersection
Floer homology is the basic language in defining the Fukaya category of a symplectic manifold
and stating Kontsevich’s homological mirror symmetry conjecture; several Floer-type homology
theory, including the instanton Floer homology ([7], [4]), Heegaard-Floer theory ([29]), Seiberg-
Witten Floer homology ([22]), ECH theory ([20], [21]), has become tools of understanding lower
dimensional topology.
All these different types of Floer theory, are all certain infinite dimensional Morse theory, whose
constructions essentially apply Witten’s point of view ([34]). Basically, if f : X → R is certain
Date: December 24, 2013.
1
2 GUANGBO XU
smooth functional on manifold X (which could be infinite dimensional), then with an appropriate
choice of metric on X, we can study the equation of negative gradient flow of f , of the form
x′(t) +∇f(x(t)) = 0, t ∈ (−∞,+∞). (1.1)
If some natural energy functional defined for maps from R to X is finite for a solution to the above
equation, then x(t) will converges to a critical point of f . Assuming that all critical points of f is
nondegenerate, then usually we can define a Morse-type index (or relative indices) λf : Critf →Z. Then for a given pair of critical points a−, a+ ∈ Critf , the moduli space of solutions to the
negative gradient flow equation which are asymptotic to a± as t → ±∞, denoted by M(a−, a+),
has dimension equal to λf (a−)−λf (a+), if f and the metric are perturbed generically. If λf (a−)−λf (a+) = 1, because of the translation invariance of (1.1), we expect to have only finitely many
geometrically different solutions connecting a− and a+. In many cases (which we call the oriented
case), we can also associate a sign to each such solutions.
On the other hand, we define a chain complex over Z2 (and over Z in the oriented case), spanned
by critical points of f and graded by the index λf ; the boundary operator ∂ is defined by the
(signed) counting of geometrically different trajectories of solutions to (1.1) connecting two critical
points with adjacent indices. We expect a nontrivial fact that ∂ ∂ = 0. So a homology group is
derived.
1.2. Hamiltonian Floer homology and the transversality issue. In Hamiltonian Floer the-
ory, we have a compact symplectic manifold (X,ω) and a time-dependent Hamiltonian Ht ∈C∞(X), t ∈ [0, 1]. We can define an action functional AH on a covering space LX of the con-
tractible loop space of X. The space LX consists of pairs (x,w) where x : S1 → X is a contractible
loop and w : D→ X with w|∂D = x; the action functional is defined as
AH(x,w) = −∫Dw∗ω −
∫S1
Ht(x(t))dt. (1.2)
The Hamiltonian Floer homology is formally the Morse homology of the pair(LX,AH
).
The critical points are pairs (x,w) where x : S1 → X satisfying x′(t) = XHt(x(t)), where XHt is
the Hamiltonian vector field associated to Ht; these loops are 1-periodic orbits of the Hamiltonian
isotopy generated by Ht. Then, choosing a smooth S1-family of ω-compatible almost complex
structures Jt on X which induces an L2-metric on the loop space of X, (1.1) is written as the Floer
equation for a map u from the infinite cylinder Θ = R× S1 to X, as
∂u
∂s+ Jt
(∂u
∂t−XHt(u)
)= 0. (1.3)
Here (s, t) is the standard coordinates on Θ. This is a perturbed Cauchy-Riemann equation, so
Gromov’s theory of pseudoholomorphic curves is adopted in Floer’s theory.
There is always an issue of perturbing the equation in order to make the moduli spaces transverse,
so that the ambiguity of counting of solutions doesn’t affect the resulting homology. Floer originally
defined the Floer homology in the monotone case, which was soon extended by Hofer-Salamon
([19]) and Ono ([28]) to the semi-positive case. Finally by applying the “virtual technique”, Floer
homology is defined for general compact symplectic manifold by Fukaya-Ono ([15]) and Liu-Tian
([23]).
GAUGED FLOER HOMOLOGY 3
1.3. Hamiltonian Floer theory in gauged σ-model. In this paper, we consider a new type of
Floer homology theory proposed in [3] and motivated from Dostoglou-Salamon’s study of Atiyah-
Floer conjecture (see [6]). The main analytical object is the symplectic vortex equation, which was
also independently studied initially in [3] and by Ignasi Mundet in [25].
The symplectic vortex equation is a natural elliptic system appearing in the physics theory
“2-dimensional gauged σ-model”. Its basic setup contains the following ingredients:
(1) The target space is a triple (M,ω, µ), where (M,ω) is a symplectic manifold with a Hamil-
tonian G-action, and µ is a moment map of the action. We also choose a G-invariant,
ω-compatible almost complex structure J on M .
(2) The domain is a triple (P,Σ,Ω), where Σ is a Riemann surface, P → Σ is a smooth G-
bundle, and Ω is an area form on Σ.
(3) The “fields” are pairs (A, u), where A is a smooth G-connection on P , and u : Σ→ P ×GMis a smooth section of the associated bundle.
Then we can write the system of equation on (A, u):
∂Au = 0;
∗FA + µ(u) = 0.(1.4)
Here ∂Au is the (0, 1)-part of the covariant derivative of u with respect to A; FA is the curvature
2-form of A; ∗ is the Hodge star operator associated to the conformal metric on Σ with area form
Ω; µ(u) is the composition of µ with u, which, after choosing a biinvariant metric on g, is identified
with a section of adP → Σ. This equation contains a symmetry under gauge transformations on
P . Moreover, its solutions are minimizers of the Yang-Mills-Higgs functional:
YMH(A, u) :=1
2
(‖dAu‖2L2 + ‖FA‖2L2 + ‖µ(u)‖2L2
)(1.5)
which generalizes the Yang-Mills functional in gauge theory and the Dirichlet energy in harmonic
map theory.
Now, similar to Hamiltonian Floer theory, consider the following action functional on a covering
space of the space of contractible loops in M × g. Let H : M × S1 → R be an S1-family of G-
invariant Hamiltonians; for any contractible loop x := (x, f) : S1 → M × g with a homotopy class
of extensions of x : S1 →M , represented by w : D→M , the action functional (given first in [3]) is
AH(x, f, w) := −∫Dw∗ω +
∫S1
(µ(x(t)) · f(t)−Ht(x(t))) dt. (1.6)
The critical loops of AH corresponds to periodic orbits of the induced Hamiltonian on the symplectic
quotient M := µ−1(0)/G. The equation of negative gradient flows of AH , is just the symplectic
vortex equation on the trivial bundle G × Θ, with the standard area form Ω = ds ∧ dt, and the
connection A is in temporal gauge (i.e., A has no ds component). If choosing an S1-family of
G-invariant, ω-compatible almost complex structures Jt, then the equation is written as a system
4 GUANGBO XU
of (u,Ψ) : Θ→M × g: ∂u
∂s+ Jt
(∂u
∂t+XΨ(u)− YHt
)= 0;
∂Ψ
∂s+ µ(u) = 0.
(1.7)
Solutions with finite energy are asymptotic to loops in CritAH . Then the moduli space of such
trajectories, especially those zero-dimensional ones, gives the definition of the boundary operator in
the Floer chain complex, and hence the Floer homology group. We call these homology theory the
vortex Floer homology. We have to use certain Novikov ring Λ, which will be defined in Section
2, as the coefficient ring, and the vortex Floer homology will be denoted by V HF (M,µ;H,J ; Λ).
The main part of this paper is devoted to the analysis about (1.7) and its moduli space, in order
to define V HF (M,µ;H,J ; Λ).
1.4. Lagrange multipliers. The action functional (1.6) seems to be already complicated, not to
mention its gradient flow equation (1.7). However, the action functional (1.6) is just a Lagrange
multiplier of the action functional (1.2). Indeed there is a much simpler situation in the case of
the Morse theory of a finite-dimensional Lagrange multiplier function, which is worth mentioning
in this introduction as a model.
Suppose X is a Riemannian manifold and µ : X → R is a smooth function, with 0 a regular
value. Then consider a function f : X → R whose restriction to X = µ−1(0) is Morse. Then
critical points of f |X are the same as critical points of the Lagrange multiplier F : X × R → Rdefined by F (x, η) = f(x) + ηµ(x), and the Morse index as a critical point of f |X is one less than
the index as a critical point of F . Then instead of considering the Morse-Smale-Witten complex
of f |X , we can consider that of F . In generic situation, these two chain complexes have the same
homology (with a grading shifting), and a concrete correspondence can be constructed through the
“adiabatic limit” (for details, see [33]).
Indeed, the vortex Floer homology proposed by Cieliebak-Gaio-Salamon and studied in this
paper is an infinite-dimensional and equivariant generalization of this Lagrange multiplier technique.
Therefore, the vortex Floer homology is expected to coincide with the ordinary Hamiltonian Floer
homology of the symplectic quotient (the proof of this correspondence will be treated in separate
work).
1.5. Advantage in achieving transversality. It seems that by considering the complicatd equa-
tion (1.7) and the moduli spaces we can only recover what we have known of the Hamiltonian Floer
theory of the symplectic quotient. But the trade-off is that the most crucial and sophisticated
step–transversality of the moduli space–can be achieved more easily. The advantage of lifting to
gauged σ-model is because, in many cases, M has simpler topology than M . So the issue caused by
spheres with negative Chern numbers is ruled out by topological reason. This phenomenon allows
us to achieve transversality of the moduli space by using the traditional “concrete perturbation” to
the equation. Moreover, when using virtual technique, the Floer homology group of the symplectic
quotient can only be defined over Q but here it can be defined over Z or Z2.
GAUGED FLOER HOMOLOGY 5
1.6. Computation of the Floer homology group and adiabatic limits. The ordinary Hamil-
tonian Floer homology HF (M,H) of a compact symplectic manifold can be shown to be canon-
ically independent of the Hamiltonian H and to be isomorphic to the singular homology of M .
This correspondence plays significant role in proving the Arnold conjecture. To prove this isomor-
phism, basically two methods have been used. One is to use a time-independent Morse function
as the Hamiltonian, and try to prove that when the function is very small in C2-norm, there is no
“quantum contribution” when defining the boundary operator in the Floer chain complex; this was
also Floer’s original argument. Another is via the Piunikhin-Salamon-Schwarz (PSS) construction,
introduced in [30].
For the case of the vortex Floer homology, it has been well-expected to be isomorphic to the
singular homology of the symplectic quotient M . To prove this isomorphism we can try the similar
methods as for the ordinary Hamiltonian Floer homology (which we will discuss in Section 6.3), as
well as the adiabatic limit method, which we discuss here.
Indeed, for any λ > 0, we consider a variation of (1.7)∂u
∂s+ Jt
(∂u
∂t+XΨ(u)− YHt
)= 0;
∂Ψ
∂s+ λ2µ(u) = 0.
(1.8)
This can be viewed as the symplectic vortex equation over the cylinder R × S1 with area form
replaced by λ2dsdt. The moduli space of solutions to the above equation also defines a Floer
homology group, and by continuation method we can show that this homology is (canonically)
independent of λ.
Then we would like to let λ approach to ∞. By a simple energy estimate, solutions of (1.8) will
“sink” into the symplectic quotient M and become Floer trajectors of the induced pair (H, J); at
isolated points there will be energy blow up, and certain “affine vortices” will appear, which are
finite energy solutions to the symplectic vortex equation over the complex plane C. In the Gromov-
Witten setting, the work of Gaio-Salamon [16] shows that (in special cases), the Hamiltonian-
Gromov-Witten invariants with low degree insertions coincide with the Gromov-Witten invariants
of the symplectic quotient, via the Kirwan map κ : H∗G(M) → H∗(M). The high degree part
shall be corrected, by the contribution from the affine vortices. This leads to the definition of the
“quantum Kirwan map” (see [38], [35]).
In the case of vortex Floer homology, as long as we can carefully analyze the contribution of
affine vortices (maybe with similar restriction on M as in [16]), we could prove that V HF (M,µ;H)
is isomorphic to HF (M ;H), with appropriate changes of coefficients.
It is an interesting topic to consider the reversed limit λ→ 0, and it actually motivated the work
of the author with S. Schecter [33], where they considered the nonequivariant, finite dimensional
Morse homology. In [33] it was shown that, the Morse-Smale trajectories, as λ→ 0, will converge to
certain “fast-slow” trajectories, and the counting of such trajectories defines a new chain complex,
which also computes the same homology.
1.7. Gauged Floer theory for Lagrangian intersections. In Frauenfelder’s thesis and [12], he
used the symplectic vortex equation on the strip R× [0, 1] to define the “moment Floer homology”
6 GUANGBO XU
for certain types of pairs of Lagrangians (L0, L1) in M . The Lagrangians are not G-invariant
in general, but their intersections with µ−1(0) reduce to a pair of Lagrangians (L0, L1) in the
symplectic quotient M . Then by the calculation in the Morse-Bott case, he managed to prove the
Arnold-Givental conjecture with certain topological assumption on M .
Woodward also defined a version of gauged Floer theory in [36], where he considered a pair of
Lagrangians L0, L1 in the symplectic quotient M . They lift to a pair of G-invariant Lagrangians
L0, L1 ⊂ µ−1(0) ⊂M . Then his equation for connecting orbits is the naive limit of the symplectic
vortex equation on the strip R × [0, 1], by setting the area form to be zero. Since the strip is
contractible, the equation is just the J-holomorphic equation on the strip, and two solutions are
regarded equivalent if they differ by a constant gauge transformation. Then he applied this Floer
theory to the fibres of the toric moment map for any toric orbifold and showed the relation between
the nondisplacibility of toric fibres and the Hori-Vafa potential, which reproduces and extends the
results of Fukaya et. al. [13] [14].
Both of the above take advantage of the simpler topology of M than the symplectic quotient, as
we mentioned above, to avoid certain virtual technique. Further work are expected to relate the
Hamiltonian gauged Floer theory we studied here and the Lagrangian versions, for example, by
constructing the so-called “open-close map”.
1.8. Organization and conventions of this paper. In Section 2 we give the basic setup, includ-
ing the action functional, the definition of the Floer chain complex and the equation of connecting
orbits. In Section 3 we proved that each finite energy solution is asymptotic to critical loops of the
action functional. In Section 4 we study the Fredholm theory of the equation of connecting orbits
(modulo gauge transformations); we show that the linearized operator is a Fredholm operator whose
index is equal to the difference of Conley-Zehnder indices of the two ends of the connecting orbit. In
Section 5 we prove that our moduli space is compact up to breaking, if assuming the nonexistence
of nontrivial holomorphic spheres. In Section 6 we summarize the previous constructions and give
the definition of the vortex Floer homology (where we postpone the proof of transversality). We
also prove the invariance of the homology group by using continuation method. In the final Section
we give some discussions on our further work along this line.
In the appendices we provide detailed proof of several technical theorems. Most importantly,
we showed that by using concrete perturbation of the almost complex structure, we can achieve
transversality of the moduli space, which allows us to avoid the more sophisticated virtual technique.
We use Θ to denote the infinite cylinder R×S1, with the axial coordinate s and angular coordinate
t. We denote Θ+ = [0,+∞)× S1 and Θ− = (−∞, 0]× S1.
G is a connected compact Lie group, with Lie algebra g. Any G-bundle over Θ is trivial, and we
just consider the trivial bundle P = G×Θ. Any connection A can be written as a g-valued 1-form
on Θ. We always use Φ to denote its ds component and Ψ to denote its dt component.
There is a small ε > 0 such that for the ε-ball g∗ε ⊂ g∗ centered at the origin of g∗, Uε := µ−1(g∗ε )
can be identified with µ−1(0)× g∗ε . We denote by πµ : Uε → µ−1(0) the projection on the the first
component, and by πµ : Uε →M the composition with the projection µ−1(0)→M .
GAUGED FLOER HOMOLOGY 7
1.9. Acknowledgments. The author would like to thank his PhD advisor Gang Tian for introduc-
ing him to this field and the support and encouragement. He would like to thank Urs Frauenfelder,
Kenji Fukaya, Chris Woodward, and Weiwei Wu for many helpful discussions and encouragement.
2. Basic setup and outline of the construction
Let (M,ω) be a symplectic manifold. We assume that it is aspherical, i.e., for any smooth map
f : S2 → M ,
∫S2
f∗ω = 0. This implies that for any ω-compatible almost complex structure J on
M , there is no nonconstant J-holomorphic spheres.
Let G be a connected compact Lie group which acts on M smoothly. The infinitesimal action g 3ξ 7→ Xξ ∈ Γ(TM) is an anti-homomorphism of Lie algebra. We assume the action is Hamiltonian,
which means that there exists a smooth function µ : M → g∗ satisfying
µ(gx) = µ(x) Ad−1g , ∀ξ ∈ g, d (µ · ξ) = ιXξω. (2.1)
Suppose we have a G-invariant, time-dependent Hamiltonian H ∈ C∞c(M × S1
)with compact
support. For each t ∈ S1, the associated Hamiltonian vector field YHt ∈ Γ(TM) is determined by
ω(YHt , ·) = dHt ∈ Ω1(M). (2.2)
The flow of YHt is a one-parameter family of diffeomorphisms
φHt : M →M,dφHt (x)
dt= YHt
(φHt (x)
)(2.3)
which we call a Hamiltonian path.
To achieve transversality, we put the following restriction on Ht. It will be used only in the
appendix.
Hypothesis 2.1. There exists a nonempty interval I ⊂ S1 such that Ht ≡ 0 for t ∈ I.
On the other hand, we need to put several assumptions to the given structures, which are still
general enough to include the most important cases (e.g., toric manifolds as symplectic quotients
of Euclidean spaces).
Hypothesis 2.2. We assume that µ : M → g∗ is proper, 0 ∈ g∗ is a regular value of µ and the
G-action restricted to µ−1(0) is free.
With this hypothesis, µ−1(0) is a smooth submanifold of M and the symplectic quotient M :=
µ−1(0)/G is a symplectic manifold, which has a canonically induced symplectic form ω. Also, the
Hamiltonian function Ht descends to a time-dependent Hamiltonian
Ht : M → R (2.4)
by the G-invariance of Ht. It is easy to check that YHt is tangent to µ−1(0) and the projection
µ−1(0)→M pushes YHt forward to YHt. Then we assume
Hypothesis 2.3. The induced Hamiltonian Ht : M → R is nondegenerate in the usual sense.
Finally, in the case when M is noncompact, we need the convexity assumption (cf. [2, Definition
2.6]).
8 GUANGBO XU
Hypothesis 2.4. There exists a pair (f, J), where f : M → [0,+∞) is a G-invariant and proper
function, and J is a G-invariant, ω-compatible almost complex structure on M , such that there
exists a constant c0 > 0 with
f(x) ≥ c0 =⇒ 〈∇ξ∇f(x), ξ〉+ 〈∇Jξ∇f(x), Jξ〉 ≥ 0, df(x) · JXµ(x) ≥ 0, ∀ξ ∈ TxM. (2.5)
In this paper, to achieve transversality, we need to perturb J near µ−1(0) (see the appendix).
The above condition is only about the behavior “near infinity”, so such perturbations don’t break
the hypothesis.
2.1. Equivariant topology.
2.1.1. Equivariant spherical classes. Recall that the Borel construction of M acted by G is MG :=
EG ×G M , where EG → BG is a universal G-bundle over the classifying space BG. Then the
equivariant (co)homology of M is defined to be the ordinary (co)homology of MG, denoted by
HG∗ (M) for homology and H∗G(M) for cohomology.
On the other hand, for any smooth manifold N , we denote by S2(N) to be the image of the
Hurwitz map π2(N) → H2(N ;Z), and classes in S2(N) are called spherical classes. We define the
equivariant spherical homology of M to be SG2 (M) := S2(MG).
Geometrically, any generator of SG2 (M) can be represented by the following object: a smooth
principal G-bundle P → S2 and a smooth section φ : S2 → P ×GM . We denote the class of the
pair (P, φ) to be [P, φ] ∈ SG2 (M).
2.1.2. Equivariant symplectic form and equivariant Chern numbers. The equivariant cohomology
of M can also be computed using the equivariant de Rham complex(Ω∗(M)G, dG
). In Ω2(M)G,
there is a distinguished closed form ω−µ, called the equivariant symplectic form, which represents
an equivariant cohomology class.
We are interested in the pairing 〈[ω − µ], [P, u]〉 ∈ R. It can be computed in the following way.
Choose any smooth connection A on P . Then there exists an associated closed 2-form ωA on
P ×G M , called the minimal coupling form. If we trivialize P locally over a subset U ⊂ S2,
such that A = d+ α, α ∈ Ω1(U, g) with respect to this trivialization, then ωA can be written as
ωA = π∗ω − d(µ · α) ∈ Ω2(U ×M). (2.6)
Then we have
〈[ω − µ], [P, u]〉 =
∫S2
u∗ωA (2.7)
which is independent of the choice of A.
On the other hand, any G-invariant almost complex structure J on X makes TX an equivariant
complex vector bundle. So we have the equivariant first Chern class
cG1 := cG1 (TM) ∈ H2G(M ;Z). (2.8)
This is independent of the choice of J .
GAUGED FLOER HOMOLOGY 9
2.1.3. Kirwan maps. The cohomological Kirwan map is a map
κ : H∗(M ;R)→ H∗(M ;R). (2.9)
Here we take R-coefficients for simplicity. It is easy to check that
κ([ω − µ]) = [ω] ∈ H2(M ;R), κ(cG1 ) = c1(TM) ∈ H2(M ;R). (2.10)
We define
NG2 (M) = ker[ω − µ] ∩ kercG1 ⊂ SG2 (M), N2(M) = ker[ω] ∩ kerc1(TM) ⊂ S2(M), (2.11)
and
Γ := SG2 (M)/NG2 (M). (2.12)
2.2. The spaces of loops and equivalence classes. Let P be the space of smooth contractible
parametrized loops in M × g and a general element of P is denoted by
x := (x, f) : S1 →M × g. (2.13)
Let P be a covering space of P, consisting of triples x := (x, f, [w]) where x = (x, f) ∈ P and
[w] is an equivalence class of smooth extensions of x to the disk D. The equivalence relation is
described as follows. For each pair w1, w2 : D → M both bounding x : S1 → M , we have the
continuous map
w12 := w1#(−w2) : S2 →M (2.14)
by gluing them along the boundary x. We define
w1 ∼ w2 ⇐⇒ [w12] = 0 ∈ SG2 (M). (2.15)
Denote by LG := L∞G := C∞(S1, G) the smooth free loop group of G. Then for any point
x0 ∈M , we have the homomorphism
l(x0) : π1(G)→ π1(M,x0) (2.16)
which is induced by mapping a loop t 7→ γ(t) ∈ G to a loop t 7→ γ(t)x0 ∈ M . For different
x1 ∈M and a homotopy class of paths connecting x0 and x1, we have an isomorphism π1(M,x0) 'π1(M,x1); it is easy to see that l(x0) and l(x1) are compatible with this isomorphism. This means
kerl(x0) ⊂ π1(G) is independent of x0. Then we define
LMG :=γ : S1 → G | [γ] ∈ kerl(x0) ⊂ π1(G)
. (2.17)
Let L0G ⊂ LMG be the subgroup of contractible loops in G.
It is easy to see that LMG acts on P (on the right) by
P × LMG → P((x, f), h) 7→ h∗(x, f)(t) =
(h(t)−1x(t),Ad−1
h(t)(f(t)) + h(t)−1∂th(t)).
(2.18)
Here the action on the second component can be viewed as the gauge transformation on the space
of G-connections on the trivial bundle S1×G. (For short, we denote by d log h the g-valued 1-form
h−1dh, which is the pull-back by h of the left-invariant Maurer-Cartan form on G.)
10 GUANGBO XU
But LMG doesn’t act on P naturally; only the subgroup L0G does: for a contractible loop
h : S1 → G, extend h arbitrarily to h : D→ G. The homotopy class of extensions is unique because
π2(G) = 0 for any connected compact Lie group ([1]). Then the class of (h−1x, h∗f, [h−1w]) in P
is independent of the extension. It is easy to see that the covering map P→ P is equivariant with
respect to the inclusion L0G→ LMG. Hence it induces a covering
P/L0G→ P/LMG. (2.19)
2.3. The deck transformation and the action functional. We now define an action of SG2 (M)
on P/L0G. Take a class A ∈ SG2 (M) represented by a pair (P, u), where P → S2 is a principal
G-bundle and u : S2 → P ×GM is a section of the associated bundle.
Consider Un ' C∗ ∪ ∞ ⊂ S2 as the complement of the south pole 0 ∈ S2. Take an arbitrary
trivialization φ : P |Un → Un ×G, which induces a trivialization
φ : P ×GM |Un → Un ×M. (2.20)
Then φ u is a map from Un to M and there exists a loop h : S1 → G and x ∈M such that
limr→0
φ u(reiθ) = h(θ)x. (2.21)
Note that the homotopy class of h is independent of the trivialization φ and the choice of x.
Now, for any element (x, f, [w]) ∈ P, find a smooth path γ : [0, 1] → M such that γ(0) = w(0)
and γ(1) = x. Then define γh : S1 × [0, 1]→M by γ(θ, t) = h(θ)γ(t).
On the other hand, view D \ 0 ' (−∞, 0]× S1. Consider the map
wh(r, θ) = h(θ)w(r, θ)
and the “connected sum”:
u#w := (φ u) #γh#wh : D→M (2.22)
which extends the loop
xh(θ) = h(θ)x(θ). (2.23)
Denote fh := Adhf − ∂th · h−1. Then we define the action by
A#[x, f, [w]] =[xh, fh, [u#w],
], ∀A ∈ SG2 (M). (2.24)
On the other hand, there exists a morphism
SG2 (M)→ kerl(x0) ⊂ π1(G) (2.25)
which sends the homotopy class of [P, u] to the homotopy class of h : S1 → G where h is the one
in (2.21). Then it is easy to see the following.
Lemma 2.5. The action (2.24) is well-defined (i.e., independent of the representatives and choices)
and is the deck transformation of the covering P/L0G→ P/LMG.
GAUGED FLOER HOMOLOGY 11
Now by this lemma, we denote P :=(P/L0G
)/NG
2 , which is again a covering of P := P/LMG,
with the group of deck transformations isomorphic to Γ. We will use x to denote an element in P
or P/L0G and use [x] an element in P.
We can define a 1-form BH on P by
T(x,f)P 3 (ξ, h) 7→∫S1
ω (x(t) +Xf − YHt , ξ(t)) + 〈µ(x(t)), h(t)〉 dt ∈ R. (2.26)
Its pull-back to P is exact and one of the primitives is the following action functional on P:
AH(x, f, [w]) := −∫Bw∗ω +
∫S1
(µ(x(t)) · f(t)−Ht(x(t))) dt. (2.27)
The zero set of the one-form BH consists of pairs (x, f) such that
µ(x(t)) ≡ 0, x(t) +Xf(t)(x(t))− YHt(x(t)) = 0. (2.28)
The critical point set of AH are just the preimage of ZeroBH under the covering P→ P.
Lemma 2.6. AH is L0G-invariant and BH is LMG-invariant.
Proof. Take any h : S1 → G, extend it smoothly to some h : D→ G. Then we see that
(h−1w)∗ω = ω(∂x(h−1w), ∂y(h
−1w))dxdy = w∗ω + d
(µ(h−1w) · d log h
). (2.29)
Also, we see that
µ(h−1(t)x(t)) ·(
Ad−1h(t)f(t) + h(t)−1h′(t)
)= µ(x(t)) · f(t) +
(µ(h−1w) · d log h
)∣∣∂D . (2.30)
By Stokes’ theorem and the G-invariance of Ht, we obtain the invariance of AH . The LMG-
invariance of BH follows from equivariance of the involved terms in a similar way.
So we have the induced action functional AH : P/L0G→ R and it satisfies the following.
Lemma 2.7. For any [x] = [x, f, [w]] ∈ P/L0G and any A ∈ SG2 (M), we have
AH (A#[x]) = AH ([x])− 〈[ω − µ], A〉. (2.31)
Proof. Use the same notation as we define the action A#[x], we see that∫D\0
w∗hω =
∫Dw∗ω +
∫ 0
−∞ds
∫S1
dtω (h∗∂sw, h∗X∂t log h(w))
=
∫Dw∗ω −
∫ 0
−∞
∫ 1
0d (µ(w) · d log h) =
∫Dw∗ω −
∫S1
(µ(x(t))− µ(w(0)))d log h. (2.32)
Also ∫S1×[0,1]
γ∗hω = −∫S1
(µ(w(0))− µ(x)) d log h. (2.33)
In the same way we can calculate∫S2\0
(φ u)∗ ω = 〈[ω − µ], [P, u]〉 −∫S1
µ(x)d log h. (2.34)
12 GUANGBO XU
So we have
AH (A#[x]) = −∫D
(u#w
)∗ω +
∫S1
⟨µ(h(t)x(t)),Adh(t)f(t)− h′(t)h(t)−1
⟩−Ht(h(t)x(t))
dt
= −〈[ω − µ], A〉 −∫Dw∗ω +
∫S1
〈µ(x(t)), f(t)〉 −Ht(x(t))dt = −〈[ω − µ], A〉+ AH ([x]) (2.35)
This lemma implies that AH descends to a well-defined function
AH : P→ R. (2.36)
Our Floer theory will be formally a Morse theory of the pair (P,AH).
Before we move on to the chain complex, we see that AH is a Lagrange multiplier function
associated to the action functional AH of the induced Hamiltonian H on the symplectic quotient
M . Let PM be the space of contractible loops in M and let PM be pairs (x, [w]) where x ∈ PM
and w : D→M extends x; [w] = [w′] if (−w)#w′ is annihilated by both ω and c1(TM). Then for
any (x, [w]) ∈ PM , we can pull-back the principal G-bundle µ−1(0)→ M to D. Any trivialization
(or equivalently a section s) of this bundle over D induces a map ws : D→ µ−1(0) whose boundary
restriction, denoted by x : S1 → µ−1(0), lifts x. Now, if (x, [w]) ∈ CritAH , i.e.
0 = x′(t)− YHt(x(t)) = (πµ)∗
(x′(t)− YHt(x(t))
)(2.37)
there exists a smooth function, fs : S1 → g such that
x′(t) +Xfs(t)(x(t))− YHt(x(t)) = 0. (2.38)
Then this gives a map
ι : PM → P/L0G
(x, [w]) 7→ [xs, fs, [ws]](2.39)
By the correspondence of the symplectic forms and Chern classes between upstairs and downstairs,
we have
Proposition 2.8. The class [xs, fs, [ws]] is independent of the choice of the section s and only
depends on the homotopy class of w. Moreover, it induces a map
ι :(PM ,CritAH
)→ (P,CritAH) (2.40)
2.4. The Floer chain complex. For R = Z2, Z or Q, we consider the Novikov ring of formal
power series over a base ring R to be the downward completion of R[Γ]:
ΛR := Λ↓R :=
∑B∈Γ
λBqB
∣∣∣∣∣ ∀K > 0,# B ∈ Γ | 〈[ω − µ], B〉 > K, λB 6= 0 <∞
. (2.41)
We define the free ΛR-module generated by the set CritAH ⊂ P to be V CF (M,µ;H; ΛR). We
define an equivalence relation on V CF (M,µ;H; ΛR) by
[x] ∼ qB[x′]⇐⇒ B#
[x′]
= [x] ∈ P. (2.42)
Denote by V CF (M,µ;H; ΛR) the quotient ΛR-module by the above equivalence relation, which is
will be graded by a Conley-Zehnder type index which will be defined later in Section 4.
GAUGED FLOER HOMOLOGY 13
2.5. Gradient flow and symplectic vortex equation. Now we choose an S1-family of G-
invariant, ω-compatible almost complex structures J on M , we assume that
f(x) ≥ c0 =⇒ Jt(x) = J(x) (2.43)
where (f, J) is the convex structure which we assume to exist in Hypothesis 2.4. Then ω and Jt
defines a Riemannian metric on M , which induces an L2-metric g on the loop space LM . On the
other hand, we fix a biinvariant metric on the Lie algebra g which induces a metric g2 on Lg; it
also identifies g with g∗ and we use this identification everywhere in this paper without mentioning
it. These choices induce a metric on P.
Then, it is easy to see that formally, the equation for the negative gradient flow line of AH is
the following equation for a pair (u,Ψ) : Θ→M × g∂u
∂s+ Jt
(∂u
∂t+XΨ − YHt
)= 0,
∂Ψ
∂s+ µ(u) = 0.
(2.44)
The equation is invariant under the action of LG. The action is defined by
LG×Map (Θ,M × g) → Map (Θ,M × g)
g∗(u,Ψ)(s, t) =(g(t)−1u(s, t),Ad−1
g(t)Ψ(s, t) + g(t)−1∂tg(t)) (2.45)
Definition 2.9. The energy for a flow line u (or its Yang-Mills-Higgs functional) is defined to be
E (u) := E (u,Ψ) :=1
2‖du+ (XΨ − YHt)⊗ dt‖
2L2 +
1
2
∥∥∥∥∂Ψ
∂s
∥∥∥∥2
L2
+1
2‖µ(u)‖2L2 . (2.46)
The second equation of (2.44) is actually the symplectic vortex equation on the triple (P, u,Ψdt),
where P → Θ is the trivial G-bundle, u is a section of P×GM and Ψdt corresponds to the covariant
derivative d+ Ψdt. (For a detailed introduction to the symplectic vortex equation, see [3] or [25]).
This is why we name our theory the vortex Floer homology.
The connection d+ Ψdt has already been put in the temporal gauge, i.e., it has no ds compo-
nent. A more general equation on pairs (u, α), with α = Φds+ Ψdt ∈ Ω1(Θ)⊗ g is∂u
∂s+XΦ + Jt
(∂u
∂t+XΨ − YHt
)= 0,
∂Ψ
∂s− ∂Φ
∂t+ [Φ,Ψ] + µ(u) = 0.
(2.47)
We write the object (u, α) in the form of a triple u = (u,Φ,Ψ). The above equation is invariant
under the action by GΘ := C∞ (Θ, G), which is defined by
GΘ ×Map (Θ,M × g× g) → Map (Θ,M × g× g)
g∗
u
Φ
Ψ
(s, t) =
g(s, t)−1u(s, t)
Ad−1g(s,t)Φ(s, t) + g(s, t)−1∂sg(s, t)
Ad−1g(s,t)Ψ(s, t) + g(s, t)−1∂tg(s, t)
(2.48)
14 GUANGBO XU
A solution u = (u,Φ,Ψ) to (2.47) is called a generalized flow line. The energy of a generalized
flow line is
E(u,Φ,Ψ) =1
2‖du+XΦ ⊗ ds+ (XΨ − YHt)⊗ dt‖L2 +
1
2‖µ(u)‖2L2 +
1
2‖∂sΨ− ∂tΦ + [Φ,Ψ]‖2L2 .
(2.49)
It is easy to see that any smooth generalized flow line is gauge equivalent via a gauge transformation
in GΘ to a smooth flow line in temporal gauge; and the energy is gauge independent.
2.6. Moduli space and the formal definition of the vortex Floer homology. We focus on
solutions to (2.47) with finite energy. In Section 3 we will show that, any such solution is gauge
equivalent to a solution u = (u,Φ,Ψ) such that there exists a pair x± = (x±, f±) ∈ ZeroBH with
lims→±∞
Φ(s, t) = 0, lims→±∞
(u(s, ·),Ψ(s, ·)) = x±. (2.50)
Hence for any pair [x±] ∈ CritAH , we can consider solutions which “connect” them. We denote by
M ([x−], [x+]; J,H) (2.51)
the moduli space of all such solutions, modulo gauge transformation.
In the appendix we will show that, if we assume that Ht vanishes for t in a nonempty open
subset I ⊂ S1, and J a generic, “admissible” family of almost complex structures, then the space
M ([x−], [x+]; J,H) is a smooth manifold, whose dimension is equal to the difference of the Conley-
Zehnder indices of [x±]. Moreover, assuming that the manifold M cannot have any nonconstant
pseudoholomorphic spheres, we show thatM ([x−], [x+]; J,H) is compact modulo breakings. Finally,
there exist coherent orientations on different moduli spaces, which is similar to the case of ordinary
Hamiltonian Floer theory (see [10]). Then, the signed counting of isolated gauged equivalence
classes of trajectories in our case has exactly the same nature as in finite-dimensional Morse-Smale-
Witten theory, which defines a boundary operator
δJ : V CF∗(M,µ;H; ΛZ)→ V CF∗−1(M,µ;H; ΛZ). (2.52)
The vortex Floer homology is then defined
V HF∗ (M,µ; J,H; ΛZ) = H (V CF∗(M,µ;H; ΛZ), δJ) . (2.53)
Moreover, for a different choice of the pair (J ′, H ′), we can use continuation principle to prove
that the chain complex (V CF∗(M,µ;H ′; ΛZ), δJ ′) is chain homotopic to (V CF∗(M,µ;H; ΛZ); δJ).
There is a canonical isomorphism between the homologies, and we denote the common homology
group by V HF∗(M,µ; ΛZ). The details are given in Section 6 and the appendix.
3. Asymptotic behavior of the connecting orbits
In this section we analyze the asymptotic behavior of solutions u = (u,Φ,Ψ) to (2.47) which has
finite energy and for which u(Θ) has compact closure in M . We call such a solution a bounded
solution. We denote the space of bounded solutions by MbΘ. We can also consider the equation
on the half cylinder Θ+ or Θ− and denote the spaces of bounded solutions over Θ± by MbΘ±
.
The main theorem of this section is
GAUGED FLOER HOMOLOGY 15
Theorem 3.1. (1) Any (u,Φ,Ψ) ∈ MbΘ±
is gauge equivalent (via a smooth gauge transforma-
tion g : Θ± → G) to a solution (u′,Φ′,Ψ′) ∈ MbΘ±
such that there exist x± = (x±, f±) ∈ZeroBH and
lims→±∞
(u′(s, ·),Ψ′(s, ·)
)= x±, lim
s→±∞Φ(s, ·) = 0 (3.1)
uniformly for t ∈ S1.
(2) There exists a compact subset KH ⊂M such that for any (u,Φ,Ψ) ∈ MbΘ, we have u(Θ) ⊂
KH .
We will prove (1) for u ∈ MbΘ+
in temporal gauge, i.e., Φ ≡ 0 and the case for Θ− is the same.
Then (2) follows from a maximum principle argument, given at the end of this section. The proof
is based on estimates on the energy density, which has been given by others in several different
settings (see [2], [16]). The only possibly new ingredient is that we have a nonzero Hamiltonian
here.
3.1. Covariant derivatives. The S1-family of metrics gt := ω(·, Jt·) induces a metric connection
∇ on the bundle u∗TM . Moreover, we define
∇A,sξ = ∇sξ +∇ξXΦ, ∇A,tξ = ∇tξ +∇ξXΨ. (3.2)
Also, on the trivial bundle Θ× g, define the covariant derivative
∇A,sθ = ∇sθ + [Φ, θ], ∇A,tθ = ∇tθ + [Ψ, θ]. (3.3)
We denote by ∇A the direct sum connection on u∗TM × g. Note that it is compatible with respect
to the natural metric on this bundle.
Define the g-valued 2-form ρt on M by
〈ρ(ξ1, ξ2), η〉t = 〈∇ξ1Xη, ξ2〉t = −〈∇ξ2Xη, ξ1〉t, ξi ∈ TM, η ∈ g. (3.4)
We list several useful identities of this covariant derivative. The reader may refer to [16] for
details.
Lemma 3.2. For u in temporal gauge, we have the following equalities
∇A,sXη −X∇A,sη = ∇∂suXη, ∇A,tXη −X∇A,tη = ∇∂tu+XΨXη. (3.5)
∇A,s (∂tu+XΨ)−∇A,t∂su = X∂tΨ. (3.6)
∇A,s (dµ · Jξ)− dµ · J (∇A,sξ) = ρ(∂su, ξ), ∇A,t (dµ · Jξ)− dµ · J (∇A,tξ) = ρ(∂tu+XΨ, ξ).
(3.7)
On the other hand, since our equation is perturbed by a Hamiltonian H, we consider another
covariant derivative which takes H into account. Let φtH be the Hamiltonian isotopy defined by Ht
and
JHt :=(dφtH
)−1 Jt dφtH , t ∈ R
be the 1-parameter family of almost complex structures. They are still compatible with ω and
hence defines a family of metric gHt , with induced inner product denoted by 〈·, ·〉H,t. We denote
the induced metric connection on u∗TM by ∇H .
16 GUANGBO XU
We think (u,Ψ) as a map from R× R→M × g, periodic in the second variable. Then we have
a well-defined connection D on u∗TM → R× R, given by
(Dsξ) (s, t) =(∇Hs ξ
)(s, t), (Dtξ) (s, t) = ∇Ht ξ +∇Hξ XΨ −∇Hξ YHt . (3.8)
Lemma 3.3. D is compatible with the inner product 〈·, ·〉H,t on u∗TM .
Proof. By direct calculation, we see for ξ, η ∈ Γ(u∗TM),
∂s〈ξ, η〉H,t = 〈Dsξ, η〉H,t + 〈ξ,Dsη〉H,t,
∂t〈ξ, η〉H,t =dgtdt
(ξ, η) +⟨∇gtt ξ +∇gtξ XF , η
⟩H,t
+⟨ξ,∇gtt η +∇gtη XF
⟩Ht
= −(LYHtgt
)(ξ, η) +
⟨∇gtt ξ +∇gtξ XF , η
⟩H,t
+⟨ξ,∇gtt η +∇gtη XF
⟩Ht
= 〈Dtξ, η〉H,t + 〈ξ,Dtη〉H,t .
3.2. Estimate of the energy density.
Lemma 3.4. There exist positive constants c1 and c2 depending only on (X,ω, J, µ,Ht) and the
subset Ku, such that for any flow line (u, 0,Ψ) in temporal gauge, we have
∆(|∂su|2H,t
)≥ −c1|∂su|4H,t − c2. (3.9)
Proof. First we see that
1
2∆ |∂su|2H,t =
(∂2s + ∂2
t
)|∂su|2H,t = ∂s 〈Ds∂su, ∂su〉H,t + ∂t〈Dt∂su, ∂su〉H,t
= |Ds∂su|2H,t + |Dt∂su|2H,t +⟨(D2s +D2
t
)∂su, ∂su
⟩H,t
.(3.10)
We denote vs = ∂su ∈ Γ(R2, u∗TM) and vt = ∂tu+XΨ − YHt ∈ Γ(R2, u∗TM). Hence vs = −Jtvt.Then we have the following computation
Dsvt −Dtvs
=∇Hs (∂tu+XΨ − YHt)−∇Ht ∂su+∇H∂su (−XΨ + YHt) = X∂sΨ = −Xµ(u).(3.11)
Dsvs +Dtvt
=Ds(−Jtvt) +Dt(Jtvs)
=−∇Hs (Jtvt) +∇Ht (Jtvs) +∇HJvs(XΨ − YHt)
=−∇Hs (Jtvt) +∇Ht (Jtvs) + [Jtvs, XΨ − YHt ] +∇HXΨ−YHt(Jtvs)
=−∇Hs (Jvt) +∇Hvt(Jtvs) + Jtvs + [Jtvs, XΨ − YHt ]
=− (∇HvsJt)vt − Jt∇Hs vt + (∇HvtJt)vs + Jt∇Hvtvs + Jtvs + Jt [vs, XΨ] + [YHt , Jtvs]
=− (∇HvsJt)vt + (∇HvtJt)vs − Jt[vs, vt]− JX∂sΨ + Jtvs + Jt[vs, XΨ] + Jt [YHt , vs] + (LYHtJt)vs=− (∇HvsJt)vt + (∇HvtJt)vs + (LYHtJt)vs + JtXµ + Jtvs.
(3.12)
GAUGED FLOER HOMOLOGY 17
On the other hand, for any (s, t) ∈ Σ, any ξ ∈ Tu(s,t)M , we extend ξ and vs(s, t) to be G-invariant
vector fields locally. Then for the Riemann curvature tensor associated to JHt , we have
RH(vs, XΨ)ξ = ∇Hvs∇HXΨξ −∇HXΨ
∇Hvsξ −∇H[vs,XΨ]ξ = ∇Hvs∇
Hξ XΨ −∇H∇HvsξXΨ. (3.13)
Hence
(DsDt −DtDs) ξ
= ∇Hs(∇Ht ξ +∇Hξ (XΨ − YHt)
)−∇Ht (∇Hs ξ)−∇H∇Hs ξ(XΨ − YHt)
= RH(vs, ∂tu)ξ +∇Hs ∇Hξ (XΨ − YHt)−∇H∇Hs ξ(XΨ − YHt)−(d
dt∇Hs)ξ
= RH(vs, ∂tu+XΨ)ξ −∇Hs ∇Hξ YHt +∇H∇Hs ξYHt +∇Hξ X∂sΨ −(d
dt∇Hs)ξ
= RH(vs, vt + YHt)ξ −∇Hs ∇Hξ YHt +∇H∇Hs ξYHt −∇Hξ Xµ −
(d
dt∇Hs)ξ. (3.14)
The third equality above uses (3.13).
Then we denote
D2svs +D2
t vs = Ds (Dsvs +Dtvt) + (DtDs −DsDt) vt −Dt (Dsvt −Dtvs) =: Q1 +Q2 +Q3.
By (3.12),
〈Q1, vs〉H,t =⟨∇Hs
(−(∇HvsJt
)vt +
(∇HvtJt
)vs +
(LYHtJt
)vs + JtXµ + Jtvs
), vs
⟩H,t
≥ −C1
(|vs|3H,t + |vs|2H,t + |vs|4H,t + |vs|2H,t
∣∣∇Hs vs∣∣H,t + |vs|H,t∣∣∇Hs vs∣∣H,t) (3.15)
for some C1 > 0. Here we used the fact that µ and dµ are uniformly bounded because u(Θ+) has
compact closure.
By (3.14), for some C2 > 0, we have
〈Q2, vs〉Ht =
⟨−RH(vs, vt + YH,t)vt +∇Hs ∇HvtYHt −∇
H∇Hs vt
YHt +∇HvtXµ +
(d
dt∇Hs)vt, vs
⟩H,t
≥ −C2
(|vs|4H,t + |vs|3H,t + |vs|2H,t + |vs|H,t
∣∣∇Hs vs∣∣H,t) . (3.16)
By (3.11), for some C3 > 0, we have
〈Q3, vs〉H,t= 〈DtXµ, vs〉H,t
=⟨∇Ht Xµ +∇HXµ(XΨ − YHt), vs
⟩H,t
=⟨Xdµ·∂tu +∇HvtXµ + [Xµ, XΨ − YHt ], vs
⟩=⟨Xdµ·∂tu +∇HvtXµ −X[µ,Ψ], vs
⟩=⟨Xdµ·vt −Xdµ·XΨ
+∇HvtXµ −X[µ,Ψ], vs⟩H,t
=〈Xdµ·vt +∇HvtXµ, vs〉H,t ≥ −C3|vs|2H,t.
(3.17)
18 GUANGBO XU
Hence for some C4 > 0 and c1, c2 > 0, we have
1
2∆|vs|2H = |Dsvs|2H + |Dtvt|2H + 〈Q1 +Q2 +Q3, vs〉H ≥ |∇gts vs|2H + 〈Q1 +Q2 +Q3, vs〉H
≥ |∇gts vs|2H − C4
(4∑i=2
|vs|iH +
2∑i=1
|vs|iH |∇gts vs|H
)≥ −c1|vs|4H − c2. (3.18)
Now we consider the other part of the energy density. We have the following calculation:
1
2∆ |µ(u)|2 = |∇A,sµ|2 + |∇A,tµ|2 + 〈∇A,s∇A,sµ+∇A,t∇A,tµ, µ〉. (3.19)
And (see [16, Lemma C.2])
∇A,s∇A,sµ(u) +∇A,t∇A,tµ(u)
=∇A,sdµ · vs +∇A,tdµ · vt = ∇A,t (dµ · Jvs)−∇A,s (dµ · Jvt)
=− 2ρ(vs, vt) + dµ · Jt (∇A,tvs −∇A,svt) + dµ(Jtvs
)=dµ ·
(JtXµ + Jtvs
)− 2ρ(vs, vt).
(3.20)
Since u(Θ) has compact closure, we may assume that supu(Θ) |ρ| ≤ cu. Hence there exists c3, c4 > 0
such that
∆ |µ(u)|2 ≥ −c3 − c4 |vs|4 . (3.21)
3.3. Decay of energy density. To proceed, we quote the following lemma (cf. [32, Page 12]).
Lemma 3.5. Let Ω ⊂ R2 be an open subset containing the origin and e is defined over Ω. Suppose
it satisfies
∆e ≥ −A−Be2,
then ∫Br(0)
e ≤ π
16B=⇒ e(0) ≤ 8
πr2
∫Br(0)
e+Ar2
4. (3.22)
Apply the above lemma to the function |vs|2H,t+|µ(u)|2 and note that different norms are actually
equivalent, we see
Proposition 3.6. If (u,Ψ) satisfies the equation (2.44) such that u(Θ) has compact closure and
E(u,Ψ) <∞, then the energy density ∣∣∣∣∂u∂s∣∣∣∣2 +
∣∣∣∣∂Ψ
∂s
∣∣∣∣2converges to 0 as s→ ±∞, uniformly in t.
GAUGED FLOER HOMOLOGY 19
3.4. Approaching to periodic orbits.
Proposition 3.7. Any (u, 0,Ψ) ∈ MbΘ+
is gauge equivalent to a solution to (2.47) (v,Φ,Ψ) on
Θ+ with the following properties
(1) (v,Ψ)|s×S1 converges in C0 to an element of ZeroBH as s→ +∞;
(2) lims→+∞
Φ(s, ·) = 0 uniformly in t.
Proof. By the above proposition, we know that finite energy implies that µ(u(s, ·))→ 0 as s→ +∞.
Let g∗ε ⊂ g∗ be the ε-open ball of the origin. Then there is an equivariant symplectic diffeomorphism
Uε := µ−1(g∗ε )→ µ−1(0)× g∗ε . (3.23)
Then µ|Uε is just the projection of the right hand side of (3.23) onto the second factor (see for
example [18]). Let πµ be the projection on to the first factor, then we define the almost complex
structure
J0,t := π∗µ(Jt∣∣µ−1(0)
)and the vector field
Y0,Ht := π∗µ(YHt
∣∣µ−1(0)
).
Then there exists K1 > 0 depending on (M,ω, J, µ,Ht) such that
‖J0,t(x)− Jt(x)‖ ≤ K1|µ(x)|, ‖Y0,Ht(x)− YHt(x)‖ ≤ K1|µ(x)|, ∀x ∈ Uε.
We denote πµ : µ−1(g∗ε ) → M the composition of πµ with the projection µ−1(0) → M . For
(u, 0,Ψ) ∈ MbΘ+
, we have proved that µ(u) converges to 0 uniformly as s → +∞. Hence for
N := N(ε) sufficiently large, u(s, ·) maps ΘN+ := [N,+∞)× S1 into Uε. So on ΘN
+ , we have
∂su+ J0,t (∂tu+XΨ(u)− Y0,Ht(u)) = (J0,t − Jt) (∂tu+XΨ(u)− Y0,Ht) + Jt (YHt − Y0,Ht) .
(3.24)
Denoting u := πµ u : ΘN+ →M and applying (πµ)∗ to the above equality, we see on ΘN
+ ,∥∥∂su+ J t(∂tu− YHt
(u))∥∥ ≤ K2ε (3.25)
for some constant K2. Here J t is the induced almost complex structure on M . Hence for s ≥ N ,
the family of loops u(s, ·) in the quotient will be close (in C0) to some 1-periodic orbits γ : S1 →M
of YHt.
We take a lift p ∈ µ−1(0) with ππµ(p) = γ(0). We see that there exists a unique gp ∈ G such
that
φ1Hp = gpp. (3.26)
Suppose gp = exp ξp, ξp ∈ g. It is easy to see that the loop
(x(t), f(t)) :=(exp(−tξp)φtH(p), ξp
)(3.27)
is an element of ZeroBH . We will construct a gauge transformation g on Θ+ and show that
g∗(u, 0,Ψ) satisfies the condition stated in this proposition.
20 GUANGBO XU
Take a local slice of the G-action near p. That is, an embedding i : B2n−2kδ → µ−1(0) where
B2n−2kδ is the δ-ball in R2n−2k such that i(0) = p and (y, g) 7→ g(i(y)) is a diffeomorphism from
B2n−2kδ ×G onto its image.
Denote u(s, t) = (v(s, t), ξ(s, t)) ∈ µ−1(0) × g∗ with respect to the decomposition (3.23). Then
for s large enough, there exists a unique g(s) ∈ G such that
g(s)v(s, 0) ∈ i(B2n−2kδ
), g(s)v(s, 0)→ p. (3.28)
Moreover, by the fact that |∂su| converges to zero, we see
lims→+∞
∣∣g(s)−1g(s)∣∣ = 0. (3.29)
Define hs(t) ∈ G by
hs(0) = 1, hs(t)−1∂hs(t)
∂t= Ψ(s, t).
Then by the fact that lims→+∞ |∂sΨ| = 0 we see that
lims→∞
|∂s log hs(t)| = 0. (3.30)
Thus we have
d(gpp, g(s)hs(1)g(s)−1p
)≤ d
(gpp, φ
1Hg(s)v(s, 0)
)+ d
(φ1Hg(s)v(s, 0), g(s)hs(1)v(s, 0)
)+ d
(g(s)hs(1)v(s, 0), g(s)hs(1)g(s)−1p
)= d
(gpp, φ
1Hg(s)v(s, 0)
)+ d
(φ1Hv(s, 0), hs(1)v(s, 0)
)+ d (g(s)v(s, 0), p)
=: d1(s) + d2(s) + d3(s).
(3.31)
Here d is the G-invariant distance function induced by an invariant Riemannian metric. By (3.26)
and (3.28), we have d1(s) + d3(s)→ 0. By the decay of energy density, i.e.,
lims→+∞
supt|∂tu+XΨ − YHt | = 0,
we have d2(s)→ 0. Hence we have
lims→+∞
d(gpp, g(s)hs(1)g(s)−1p
)= 0. (3.32)
Since the G-action on µ−1(0) is free, we have
lims→+∞
g(s)hs(1)g(s)−1 = gp (3.33)
Then by (3.33), there exists a continuous curve ξ(s) ∈ g defined for large s, such that
g(s)hs(1)g(s)−1 = exp ξ(s), lims→+∞
ξ(s) = ξp. (3.34)
Then apply the gauge transformation
g(s, t) = hs(t)−1g(s)−1 exp(tξ(s)) (3.35)
to the pair (u, 0,Ψ), we see
(g∗u) (s, 0) = g(s, 0)−1(u(s, 0)) = g(s)u(s, 0)→ p, g∗ (Ψdt) = ξ(s)dt+ η(s, t)ds (3.36)
GAUGED FLOER HOMOLOGY 21
where
η = ∂s log g = g−1∂g
∂s.
The fact that lims→+∞ ‖η‖ = 0 follows from (3.29) and (3.30).
Hence
lims→+∞
(g∗u)|s×S1 =(exp(−tξp)φtHp, ξp
)∈ ZeroBH .
Definition 3.8. Let x± := (x±, f±) ∈ ZeroBH . We denote
M (x−, x+) := M (x−, x+; J,H) :=
(u,Φ,Ψ) ∈ Mb
Θ | lims→±∞
(u,Φ,Ψ)|s×S1 = (x±, 0)
. (3.37)
For x± = (x±, [w±]) ∈ CritAH which projects to x± via CritAH → ZeroBH , we define
M (x−, x+) := M (x−, x+; J,H) :=
(u,Φ,Ψ) ∈ M(x−, x+) | [u#w−] = [w+]. (3.38)
Then it is easy to deduce the following energy identity for which we omit the proof.
Proposition 3.9. Let x± ∈ CritAH . Then for any (u,Φ,Ψ) ∈ M (x−, x+), we have
E (u,Φ,Ψ) = AH (x−)− AH (x+) . (3.39)
3.5. Convexity and uniform bound on flow lines. We will show in this subsection the following
Proposition 3.10. There exists a compact subset KH ⊂ M such that for any (u,Φ,Ψ) ∈ MbΘ,
u(Θ) ⊂ KH .
Proof. The proof is to use maximum principle as in [2, Subsection 2.5]. We claim this proposition
is true for
KH = SuppH ∪ f−1 ([0, c1])
where
c1 = max
c0, sup|µ(x)|≤1
f(x)
(3.40)
where c0 is the one in Hypothesis 2.4.
Suppose the statement is not true. Then there exists a solution u = (u,Φ,Ψ) ∈ MbΘ which
violates this condition and (s0, t0) ∈ Θ such that u(s0, t0) /∈ SuppH and f(u(s0, t0)) > c1. On
the other hand, by the previous results, we know that lims→±∞
µ(u(s, t)) = 0 so lims→±∞
f(u(s, t)) ≤ c1.
Hence f(u) achieves its maximum at some point of Θ. As in the proof of [2, Lemma 2.7], we see that
f(u) is subharmonic on u−1 (M \KH) and hence f(u) must be constant. However, this contradicts
with the fact that lims→±∞
f(u(s, t)) ≤ c1.
4. Fredholm theory
In this section we investigate the infinitesimal deformation theory of solutions to our equation
(modulo gauge). For a similar treatment of a relevant situation, the reader may refer to [4].
22 GUANGBO XU
4.1. Banach manifolds, bundles, and local slices. First we fix two loops x± ∈ ZeroBH . For
any k ≥ 1, p > 2, we consider the space of W k,ploc -maps u := (u,Φ,Ψ) : Θ → M × g × g, such that
Φ ∈ W k,p (Θ, g) and (u,Ψ) is asymptotic to x± = (x±, f±) at ±∞ in W k,p-sense. Then this is a
Banach manifold, denoted by
Bk,p := Bk,p(x−, x+). (4.1)
The tangent space at any element u ∈ Bk,p is the Sobolev space
TuBk,p = W k,p (Θ, u∗TM ⊕ g⊕ g) . (4.2)
We denote by expt the exponential map of M×g×g, where the Riemannian metric on M is ω(·, Jt·)which is t-dependent. Then the map ξ 7→ exptuξ is a local diffeomorphism from a neighborhood of
0 ∈ TuBk,p and a neighborhood of u in Bk,p.Then consider a pair x± = (x±, f±, [w±]) ∈ CritAH with x± = (x±, f±) ∈ ZeroBH . We define
Bk,p(x−, x+) :=u = (u,Φ,Ψ) ∈ Bk,p(x−, x+) | [w−#u] = [w+]
. (4.3)
Let Gk+1,p0 be the space of W k+1,p
loc -maps g : Θ → G which is asymptotic to the identity of G at
±∞. Then this is a Banach Lie group. The gauge transformation extends to a free Gk+1,p0 -action
on Bk,p(x−, x+) (resp. Bk,p(x−, x+)), because the symplectic quotient M is a free quotient. Then
this makes the quotient
Bk,p(x−, x+) := Bk,p(x−, x+)/Gk+1,p(
resp. Bk,p(x−, x+) := Bk,p(x−, x+)/Gk+1,p0
)(4.4)
a Banach manifold. Indeed, to see this we have to construct local slices of the Gk+1,p-action. For
any u ∈ Bk,p (whose image in Bk,p is denoted by [u]), consider the operator
d∗0 : TuBk,p → W k−1,p (Θ, g)
(ξ, φ, ψ) 7→ −dµ(Jtξ)− ∂sφ− [Φ, φ]− ∂tψ − [Ψ, ψ],(4.5)
which is the formal adjoint of the infinitesimal Gk+1,p-action. Then as in gauge theory, we have a
natural identification
T[u]Bk,p ' kerd∗0 (4.6)
(where the orthogonal complement is taken with respect to the L2-inner product) and the expo-
nential map expt induces a local diffeomorphism
kerd∗0 3 ξ 7→[exptuξ
]∈ Bk,p. (4.7)
If we have g± ∈ L0G, and x′± = g∗±x± ∈ CritAH , then the pair (g−, g+) extends to a smooth
gauge transformation on Θ which identifies Bk,p(x−, x+) with Bk,p(x′−, x′+). Then, with abuse of
notation, if x± ∈ CritAH/L0G, then we can denote by Bk,p(x−, x+) to be the common quotient
space. Finally, for two pairs [x±] ∈ CritAH , we define
Bk,p([x−], [x+]) :=⋃
y±∈CritAH/L0G, [y±]=[x±]
Bk,p(y−, y+), (4.8)
which is a discrete union of Banach manifolds.
GAUGED FLOER HOMOLOGY 23
Over Bk,p(x−, x+), we have the smooth Banach space bundle Ek−1,p(x−, x+), whose fibre over u is
the Sobolev space
Ek−1,pu := W k−1,p (u∗TM ⊕ g) . (4.9)
The Gk+1,p0 -action makes Ek−1,p an equivariant bundle, hence descends to a Banach space bundle
Ek−1,p(x−, x+)→ Bk,p(x−, x+)(
or Ek−1,p ([x−], [x+])→ Bk,p ([x−], [x+])). (4.10)
Moreover, the H-perturbed vortex equation (2.47) gives a section
F : Bk,p (x−, x+)→ Ek−1,p(x−, x+) (4.11)
which is Gk+1,p-equivariant. So it descends to a section
F : Bk,p ([x−], [x+])→ Ek−1,p ([x−], [x+]) .
Then we see that M (x−, x+; J,H) is the intersection of F−1(0) with smooth objects. We define
M ([x−], [x+]; J,H) = F−1(0), (4.12)
whose elements, by the standard regularity theory about symplectic vortex equation (see [2, The-
orem 3.1]), all have smooth representatives. Therefore M ([x−], [x+]; J,H) is independent of k, p.
The linearization of F at u is
Du := dFu : (ξ, φ, ψ) 7→
(∇A,sξ + (∇ξJt) (∂tu+XΨ − YHt) + Jt (∇A,tξ −∇ξYHt) +Xφ + JXψ
∂sψ + [Φ, ψ]− ∂tφ− [Ψ, φ] + dµ(ξ)
)(4.13)
Hence the linearization of F , under the isomorphism (4.6), is the restriction of Du to (kerd∗0)⊥.
We define the augmented linearized operator
Du := Du ⊕ d∗0 : TuBk,p → Ek−1,pu ⊕W k−1,p (Θ, g) . (4.14)
It is a standard result that the Fredholm property of dF[u] is equivalent to that of Du for any
representative u. Hence in the remaining of the section we will study the Fredholm property of the
augmented operator.
4.2. Asymptotic behavior of Du. Up to an L0G-action we can choose representatives x± =
(x±, f±, [w±]) such that f± are constants θ± ∈ g. Take any u = (u,Φ,Ψ) ∈ Bk,p(x−, x+).
For ξ := (ξ, ψ, φ) ∈ TuBk,p(x−, x+), define J(ξ, ψ, φ) = (Jξ,−φ, ψ). Here ψ is the variation of Ψ
and φ is the variation of Φ. Then
Du
ξ
ψ
φ
= ∇A,s
ξ
ψ
φ
+J
∇A,tξ −∇ξYH,t∇A,tψ∇A,tφ
+
0 JLu Lu
dµ 0 0
L∗u 0 0
ξ
ψ
φ
+q(s, t)
ξ
ψ
φ
=:
∇s∂s
∂s
+ J
∇t∂t∂t
+R(s, t) + q(s, t)
ξ
ψ
φ
. (4.15)
24 GUANGBO XU
Here q(s, t) is a linear operator such that lims→±∞
q(s, t) = 0; and
Rx±(t) := lims→±∞
R(s, t) =
Jt∇(Xθ± − YHt) JtLu Lu
dµ 0 −adθ±
L∗u adθ± 0
. (4.16)
Here Lu : g→ u∗TM is the infinitesimal action along the image of u and L∗u is its dual.
The following result is implied by Hypothesis 2.3.
Proposition 4.1. x := (x, θ) ∈ ZeroBH , then for all t-dependent, G-invariant, ω-compatible almost
complex structure Jt, the self-adjoint operator
L2(S1, x∗TM ⊕ g⊕ g
)→ L2
(S1, x∗TM ⊕ g⊕ g
) ξ
ψ
φ
7→
J ∇t∂t
∂t
+Rx(t)
ξ
ψ
φ
.(4.17)
has zero kernel.
Proof. Suppose (ξ, ψ, φ)T is in the kernel, which means
Jt∇tξ + Jt∇ξ (Xθ − YHt) +Xφ + JtXψ = 0, (4.18)
−dφdt
+ dµ(ξ)− [θ, φ] = 0, (4.19)
dψ
dt+ dµ(Jtξ) + [θ, ψ] = 0. (4.20)
Apply dµ Jt to (4.18), we get
dµ(JtXφ) = dµ(∇tξ +∇ξ(Xθ − YHt)).
Hence for any η ∈ g,
d
dt〈dµ(ξ), η〉g =
d
dtω(Xη, ξ)
=ω([YHt −Xθ, Xη], ξ) + ω(Xη,∇tξ −∇ξ(YHt −Xθ))
=ω(X[θ,η], ξ) + 〈dµ(JXφ), η〉g=〈dµ(ξ), [θ, η]〉g + 〈dµ(JXφ), η〉g=〈dµ(JXφ)− [θ, dµ(ξ)], η〉g.
(4.21)
Therefore,
d
dtdµ(ξ) = dµ(JXφ)− [θ, dµ(ξ)]. (4.22)
Then by (4.19),
dµ(JtXφ)
=d
dtdµ(ξ) + [θ, dµ(ξ)]
=d
dt
(dφ
dt+ [θ, φ]
)+
[θ,dφ
dt+ [θ, φ]
]=φ′′ + 2
[θ, φ′
]+ [θ, [θ, φ]].
(4.23)
GAUGED FLOER HOMOLOGY 25
Suppose ‖φ‖ takes its maximum at t = t0 ∈ S1. Then for t ∈ (t0 − ε, t0 + ε), define φ(t) =
Ade(t−t0)θφ(t). Then the right hand side of (4.23) is equal to Ade(t0−t)θ φ′′(t). Hence at t = t0,
0 ≥1
2
d2
dt2‖φ‖2 =
1
2
d2
dt2
∥∥∥φ∥∥∥2=⟨φ′′, φ
⟩+∥∥∥φ′∥∥∥2
g
=⟨dµ(JtXφ(t0)), φ(t0)
⟩+∥∥∥φ′(t0)
∥∥∥2=∥∥Xφ(t0)
∥∥2+∥∥∥φ′(t0)
∥∥∥2.
(4.24)
Hence Xφ ≡ 0, which implies φ ≡ 0 and by (4.19), ξ is tangent to µ−1(0).
Now x∗Tµ−1(0) = Et⊕Lxg, where Et = (Lxg)⊥ ∩x∗Tµ−1(0) and the orthogonal complement is
taken with respect to the Riemannian metric g = ω(·, Jt·). Then with respect to this (G-invariant)
decomposition, write
ξ = ξ⊥(t) +Xη(t).
Then take the Et-component of (4.18), and use the nondegeneracy assumption on the induced
Hamiltonian Ht on the symplectic quotient, we see that ξ⊥ ≡ 0. The only equations left is ∇tXη(t) +∇Xη(t)(Xθ − YHt) +Xψ = 0,
ψ′ + [θ, ψ] + dµ(JXη) = 0.
The first equation is equivalent to
η′ + [θ, η(t)] + ψ = 0.
Hence
η′′(t) + 2[θ, η′(t)
]+ [θ, [θ, η]] = dµ(JtXη).
This can be treated similarly as (4.23), using maximum principle, which shows that η ≡ 0 and
hence ψ ≡ 0.
Corollary 4.2. Under Hypothesis 2.3, for any x± ∈ ZeroBH and any u ∈ Bk,p (x−, x+), the
augmented linearized operator Du is Fredholm for any k ≥ 1, p ≥ 2.
Corollary 4.3. There exists δ = δ(x±) > 0 such that, for any u = (u,Φ,Ψ) ∈ M (x−, x+; J,H)
and any ξ ∈ kerDu, there exists c > 0 such that∣∣∣ξ(s, t)∣∣∣ ≤ ce−δ|s|. (4.25)
In particular, if (u, 0,Ψ) ∈ MbΘ, then |∂su| and |∂sΨ| decay exponentially.
Proof. The first part is standard, see for example [32, Lemma 2.11]. For a solution u = (u, 0,Ψ)
in temporal gauge, by the translation invariance of the equation (2.44), we see that ξ = (∂su, βs =
0, βt = ∂sF ) ∈ kerdFu. Moreover,
−d∗0ξ =− d∗0(∂su, 0, ∂sΨ) = ∂t∂sΨ + L∗u(∂su) + [Ψ, ∂sΨ]
=− ∂t(µ(u)) + dµ(Jt∂su) + [Ψ, ∂sΨ]
=− dµ(∂tu) + dµ(Jt∂su) + [Ψ, ∂sΨ]
= dµ (XΨ − YHt)− [Ψ, µ] = 0.
(4.26)
This implies that ξ ∈ kerDu. Choose a smooth gauge transformation g : Θ → G such that
g∗u satisfies the asymptotic condition of Proposition 3.7. Then g∗ξ ∈ kerDg∗u, which decays
exponentially. So does ξ.
26 GUANGBO XU
4.3. The Conley-Zehnder indices. In this subsection we define a grading on the set CritAH ,
which is analogous to the Conley-Zehnder index in usual Hamiltonian Floer theory, and we will
call it the same name.
For the induced Hamiltonian system on the symplectic quotient M , we have the usual Conley-
Zehnder index
CZ : CritAH → Z. (4.27)
We prove the following theorem
Theorem 4.4. There exists a function
CZ : CritAH → Z (4.28)
satisfying the following properties
(1) For the embedding ι : CritAH → CritAH , we have
CZ ι = CZ; (4.29)
(2) For any B ∈ Γ and [x] ∈ CritAH we have
CZ (B# [x]) = CZ ([x])− 2cG1 (B). (4.30)
(3) For [x±] = [x±, f±, [w±]] ∈ CritAH and [u] ∈ Bk,p ([x−], [x+]) with [u#w−] = [w+], we have
ind(dF[u]
)= CZ ([x−])− CZ ([x+]) . (4.31)
We first review the notion of Conley-Zehnder index in Hamiltonian Floer homology. Let A :
[0, 1]→ Sp(2n) be a continuous path of symplectic matrices such that
A(0) = I2n, det (A(1)− I2n) 6= 0. (4.32)
We can associate an integer CZ(A) to A, called the Conley-Zehnder index. We list some properties
of the Conley-Zehnder index below which will be used here (see for example [31]).
(1) For any path B : [0, 1]→ Sp(2n), we have CZ(BAB−1) = CZ(A);
(2) CZ is homotopy invariant;
(3) If for t > 0, A(t) has no eigenvalue on the unit circle, then CZ(A) = 0;
(4) If Ai : [0, 1]→ Sp(2ni) for n = 1, 2, then CZ(A1 ⊕A2) = CZ(A1) + CZ(A2);
(5) If Φ : [0, 1]→ Sp(2n) is a loop with Φ(0) = Φ(1) = Id, then
CZ(ΦA) = CZ(A) + 2µM (Φ) (4.33)
where µM (Φ) is the Maslov index of the loop Φ.
With this algebraic notion, in the usual Hamiltonian Floer theory one can define the Conley-
Zehnder indices for nondegenerate Hamiltonian periodic orbits. In our case, the induced Hamil-
tonian Ht : M → R has the usual Conley-Zehnder index
CZ : CritAH → Z. (4.34)
Then, for each x = (x, f, [w]) ∈ CritAH , the homotopy class of extensions [w] induces a homotopy
class of trivializations of x∗TM over S1. With respect to this class of trivialization, the operator
(4.17) is equivalent to an operator J0∂t + A(t), which defines a symplectic path. We define the
GAUGED FLOER HOMOLOGY 27
Conley-Zehnder index of x to be the Conley-Zehnder index of this symplectic path. By the second
and fifth axioms listed above, this index induces a well-defined function
CZ : CritAH → Z (4.35)
which satisfies (2) and (3) of Theorem 4.4.
Now we prove (1). For any contractible periodic orbits x : S1 → M of YHtand any extension
w : D→M of x, we can lift the pair (x,w) to a tuple x = (x, f, [w]) ∈ CritAH .
Proposition 4.5. If ι : CritAH → CritAH is the inclusion we described in Proposition 2.8, then
CZ ι = CZ. (4.36)
Proof. Since the Conley-Zehnder index is homotopy invariant, and the space of G-invariant ω-
compatible almost complex structures is connected, we will compute the Conley-Zehnder index
using a special type of almost complex structures, and modify the Hamiltonian H.
Starting with any almost complex structure J on M and a G-connection on µ−1(0)→M , J lifts
to the horizontal distribution defined by the connection. On the other hand, the biinvariant metric
on g gives an identification g ' g∗. We denote by η∗ ∈ g∗ the metric dual of η ∈ g. Recall that we
have a symplectomorphism
µ−1 (g∗ε ) ' µ−1(0)× g∗ε . (4.37)
For η ∈ g, we define JXη = η∗ ∈ g∗, as a vector field on µ−1(0)× g∗. Then this gives a G-invariant
almost complex structure on TM |µ−1(0), compatible with ω. Then we pullback J by the projection
µ−1(0)× g∗ε → µ−1(0) and denote the pullback by J .
We also modify Ht by requiring that Ht(x, η) = Ht(x) for (x, η) ∈ µ−1(0)×g∗ε . Then the modified
Ht can be continuously deformed to the original one, and it doesn’t change H hence doesn’t change
CritAH . Moreover, it is easy to check that for the modified pair (J,H),(LYHtJ
)Xη =
[LYHt , JXη
]= 0. (4.38)
Now for any (x,w) ∈ CritAH , we lift it to (x, f, [w]) ∈ CritAH with w : D → µ−1(0) and f
being a constant θ ∈ g. Then any symplectic trivialization of x∗TM → S1 induces a symplectic
trivialization
φ : x∗TM ' S1 ×[R2n−2k ⊕ (g⊕ g)
](4.39)
such that φ(Xη, JXζ) = (0, η, ζ). Then we see, with respect to φ, the operator (4.17) restricted to
g4 is η1
ψ
η2
φ
7→ Jd
dt
η1
ψ
η2
φ
+
φ− [θ, η2]
η2 − [θ, φ]
ψ + [θ, η1]
η1 + [θ, ψ]
=:
(Jd
dt+ S
)η1
ψ
η2
φ
. (4.40)
Here we used the property (4.38) and
J :=
[0 −Idg⊕g
Idg⊕g 0
], S =
[0 Idg⊕g − adθ
Idg⊕g + adθ 0
](4.41)
28 GUANGBO XU
Moreover, the operator (4.17) respect the decomposition in (4.39). Hence by the fourth axiom of
Conley-Zehnder indices we listed above, we have
CZ (x, θ, [w]) = CZ(x,w) + CZ(eJSt
). (4.42)
As we have shown in the proof of Proposition 4.1 that for any θ the operator (4.17) is an isomor-
phism, we can deform θ to zero and compute instead CZ(eJS0t) for
S0 =
[0 Idg⊕g
Idg⊕g 0
], (4.43)
thanks to the homotopy invariance property. Then we see that
eJS0t =
[e−t 0
0 et
](4.44)
which has no eigenvalue on the unit circle for t > 0. Therefore by the third axiom, CZ(eJS0t) =
0.
5. Compactness of the moduli space
For a general Hamiltonian G-manifold, the failure of compactness of the moduli space of con-
necting orbits comes from two phenomenon. The first is the breaking of connecting orbits, which is
essentially the same thing happened in finite dimensional Morse-Smale-Witten theory. The second
is the blow-up of the energy density, which results in sphere bubbling. Since here we have assumed
that there exists no nontrivial holomorphic sphere in M , so we only have to consider the breakings.
5.1. Moduli space of stable connecting orbits and its topology. Let’s fix a pair [x±] ∈CritAH . Denote by M([x−], [x+]) := M([x−], [x+]; J,H) =M([x−], [x+]; J,H)/R the quotient of the
moduli space by the translation in the s-direction. We denote by u the R-orbit in M([x−], [x+])
of [u] ∈M([x−], [x+]; J,H) and call it a trajectory from [x−] to [x+].
Definition 5.1. A broken trajectory from [x−] to [x+] is a collection
u :=(u(α)
)α=1,...,m
:=(u(α),Φ(α),Ψ(α)
)α=1,...,m
(5.1)
where for each α,u(α)
∈ M ([xα−1] , [xα]) and E(u(α)) 6= 0. Here [xα]α=0,...,m is a sequence of
critical points of AH and
[x0] = [x−] , [xm] = [x+] .
We regard the domain of u as the disjoint union
∪mα=1Θ
and let Θ(α) ⊂ ∪mα=1Θ the α-th cylinder.
We denote by
M ([x−] , [x+]) (5.2)
the space of all broken trajectories from [x−] to [x+]. Then naturally we have inclusion
M ([x−] , [x+])→M ([x−] , [x+]) . (5.3)
GAUGED FLOER HOMOLOGY 29
Definition 5.2. We say that a sequence of trajectories ui = ui,Φi,Ψi ∈ M ([x−] , [x+]) from
[x−] to [x+] converges to a broken trajectory
u :=(u(α)
)α=1,...,m
if: for each i, there exists sequences of numbers s(1)i < s
(2)i < · · · < s
(m)i and gauge transformations
g(α)i ∈ GΘ such that for each α (
g(α)i
)∗ (s
(α)i
)∗(ui,Φi,Ψi) (5.4)
converges to(u(α),Φ(α),Ψ(α)
)on any compact subset of Θ and such that for any sequence of (si, ti)
with
limi→∞|si − s(α)
i | =∞, ∀α
we have
limi→∞
e (ui) (si, ti) = 0.
It is easy to see that this convergence is well-defined and independent of the choices of represen-
tatives of the trajectories. We can also extend this notion to sequences of broken connecting orbits.
We omit that for simplicity.
The main theorem of this section is
Theorem 5.3 (Compactness of the moduli space of stable connecting orbits). The space
M ([x−] , [x+]) is a compact Hausdorff space with respect to the topology defined in Definition 5.2.
Indeed the proof is routine and it has been carried out in many literature for general symplectic
vortex equations, for example [25], [2], [26], [37]. Since bubbling is ruled out, the proof is almost the
same as that for finite dimensional Morse theory, while the gauge symmetry is the only additional
ingredient.
5.2. Local compactness with uniform bounded energy density. For any compact subset
K ⊂ Θ, consider a sequence of solutions ui := (ui,Φi,Ψi) such that the image of ui is contained in
the compact subset KH ⊂M and such that
lim supi→∞
E(ui) <∞.
We have
Proposition 5.4. There exists a subsequence (still indexed by i), a sequence of smooth gauge
transformation gi : K → G and a solution u∞ = (u∞,Φ∞,Ψ∞) : K → M × g× g to (2.44) on K,
such that the sequence g∗i ui converges to u∞ uniformly with all derivatives on K.
Proof. By the fact that ui is contained in the compact subset KH , and the assumption that there
exists no nontrivial holomorphic spheres in M , we have
supz∈K,i
eui(z) <∞. (5.5)
Then this proposition can be proved in the standard way, as did in [2] or [26], using Uhlenbeck’s
compactness theorem.
30 GUANGBO XU
5.3. Energy quantization. To prove the compactness of the moduli space, we need the following
energy quantization property.
Proposition 5.5. There exists ε0 := ε0(J,H) > 0, such that for any connecting orbit u ∈ MbΘ, we
have E(u) ≥ ε0.
Proof. Suppose it is not true. Then there exists a sequence of connecting orbits, represented by
solutions in temporal gauge vi := (vi, 0,Ψi) ∈ MbΘ, such that
E(vi) > 0, limi→∞
E(vi) = 0. (5.6)
We first know that there is a compact subset KH ⊂ M such that for every i, the image vi(Θ) is
contained in KH . Then we must have
limi→∞
supΘ
(|∂svi|+ |µ(vi)|) = 0. (5.7)
Indeed, if the equality doesn’t hold, then we can find a subsequence which either bubbles off a
nonconstant holomorphic sphere at some point z ∈ Θ (if the above limit is∞), or (after a sequence
of proper translation in s-direction) converges to a solution (with positive energy) on compact
subsets (if the above limit is positive and finite). Either case contradicts the assumption. Therefore
we conclude that for any ε > 0, the image of vi lies in Uε := µ−1(g∗ε ) for i sufficiently large.
Recall that we have projections πµ : Uε → µ−1(0) and πµ : Uε → M . Then for all large i and
any s, πµ(vi(s, ·)) is C0-close to a periodic orbit of H in M . Since those orbits are discrete (in
C0-topology, for example), we may fix one such orbit γ ∈ ZeroBH and choose a subsequence (still
indexed by i) such that
lim supi→∞
sup(s,t)∈Θ
d (πµ(vi(s, t)), γ(t)) = 0. (5.8)
Then, use a fixed Riemannian metric on M with its exponential map exp, we can write
πµ(vi(s, t)) = expγ(t)ξi(s, t)
where ξi ∈ Γ(S1, γ∗TM
). Let Bε(γ
∗TM) be the ε-disk of γ∗TM . Then expγ pulls back µ−1(0)→M to a G-bundle Q→ Bε(γ
∗TM), together with a bundle map γ : Q→ µ−1(0). We can trivialize
Q by some
φ : Q→ G×Bε(γ∗TM).
Now we take a lift x := (x, f) ∈ ZeroBH of γ. Then we can write
φ(γ−1(x(t))
)= (g0(t), γ(t)). (5.9)
On the other hand, we write
φ(γ−1πµ(vi(s, t))
)= (gi(s, t), ξi(s, t)). (5.10)
Take the gauge transformation gi(s, t) = gi(s, t)g0(t)−1. Then write
v′i := (v′i,Φ′i,Ψ′i) := g∗i vi. (5.11)
GAUGED FLOER HOMOLOGY 31
Then by the exponential convergence of vi as s→ ±∞, we see that ∂sgi(s, t) decays exponentially
and hence Φ′i converges to zero as s→ ±∞. On the other hand, we see that
φ(γ−1πµ(v′i(s, t))
)= (g0(t), ξi(s, t)). (5.12)
Therefore
v′i ∈ M(x, x). (5.13)
But it is also easy to see that the homotopy class of v′i is trivial. Because the energy of connecting
orbits only depends on its homotopy class, we see that the energy of v′i, and hence the energy of vi
is actually equal to zero, which contradicts with the hypothesis.
5.4. Proof of the compactness theorem. It suffices to prove, without essential loss of gener-
ality, that for any sequence [ui] ∈ M ([x−], [x+]; J,H) represented by unbroken connecting orbits
(ui,Φi,Ψi) ∈ M (x−, x+), there exists a convergent subsequence. By the assumption that there
exists no nontrivial holomorphic sphere in M , we have
supi,Θ|∂sui +XΦi(ui)| < +∞. (5.14)
Then the limit (broken) connecting orbits can be constructed by induction and the energy quan-
tization property (Proposition 5.5) guarantees that the induction stops at finite time. The details
are standard and left to the reader.
6. Floer homology
In this section we use the moduli spacesM ([x−], [x+]; J,H) to define the vortex Floer homology
group V HF∗ (M,µ;H). We also discuss further works and related problems in the last three
subsections.
By Corollary A.12, we can choose a generic S1-family of “admissible” almost complex structures
J ∈ J regH which is regular with respect to H. Such an object is a smooth t-dependent family of
almost complex structures Jt, such that for each t, Jt is G-invariant, ω-compatible, and outside a
neighborhood U of µ−1(0), Jt ≡ J; inside U , Jt preserves a distribution gCU . Being regular implies
that for all pairs [x±] ∈ CritAH , the moduli space M ([x−], [x+]; J,H) is a smooth manifold with
dimM ([x−], [x+]; J,H) = CZ([x−])− CZ([x+]). (6.1)
Moreover, there is free R-action on M ([x−], [x+]; J,H) by time translation, whose orbit space is
denoted by M([x−], [x+]; J,H). Combining the compactness theorem, we have
Proposition 6.1. If J ∈ J regH , then
CZ([x−])− CZ([x+]) ≤ 0 =⇒M ([x−], [x+]; J,H) = ∅; (6.2)
CZ([x−])− CZ([x+]) = 1 =⇒ #M ([x−], [x+]; J,H) <∞. (6.3)
32 GUANGBO XU
6.1. The gluing map and coherent orientation. The boundary operator of the Floer chain
complex is defined by the (signed) counting of M([x−], [x+]; J,H). If we want to define the Floer
homology over Z2, then we don’t need to orient the moduli space; otherwise, the orientation of
M([x−], [x+]; J,H) can be treated in the same way as the usual Hamiltonian Floer theory, since
the augmented linearized operator Du (whose determinant is canonically isomorphic to the deter-
minant of the actual linearization dF[u]), is of the same type of Fredholm operators considered in
the abstract setting of [10]. We first give the gluing construction and then discuss the coherent
orientations of the moduli spaces.
In this subsection we construct the gluing map for broken trajectories with only one breaking.
The general case is similar. This construction is, in principle, the same as the standard construction
in various types of Morse-Floer theory (see [32] [10]), with a gauge-theoretic flavor. The gauge
symmetry makes the construction a bit more complex, since we always glue representatives, and
we want to show that the gluing map is independent of the choice of the representatives.
In this subsection, we fix the choice of the admissible family J ∈ J regH and omit the dependence
of the moduli spaces on J and H.
For any pair x± ∈ CritAH , we say that a solution u ∈ M (x−, x+) is in r-temporal gauge, if its
restrictions to [r,+∞)× S1 and (−∞, r]× S1 are in temporal gauge, for some r > 0. Now we fix a
number r = r0 and only consider solutions in r0-temporal gauge.
Now we take three elements x, y, z ∈ CritAH with
CZ(x)− 1 = CZ(y) = CZ(z) + 1. (6.4)
Assume y = (y, η) : S1 →M × g. We would like to construct, for a large R0 > 0, the gluing map
glue : M ([x], [y])× (R0,+∞)× M ([y], [z])→ M ([x], [z]) . (6.5)
Now consider two trajectories [u±] = [u±,Φ±,Ψ±], [u−] ∈ M ([x], [y]), [u+] ∈ M ([y], [z]) with
their representatives both in r0-temporal gauge and u± are asymptotic to y as s → ∓∞. Then
there exists R1 > 0 such that
±s ≥ R1 =⇒ u∓(s, t) = expy(t) ξ∓(s, t). (6.6)
Here Θ+R1
= [R1,+∞)× S1 and Θ−R1= (−∞,−R1]× S1 and ξ± ∈W k,p
(Θ∓R1
, y∗TM)
.
Next, we take a cut-off function ρ such that s ≥ 1 =⇒ ρ(s) = 1, s ≤ 0 =⇒ ρ(s) = 0. For each
R >> r0, denote ρR(s) = ρ(s−R). We construct the “connected sum”
uR(s, t) =
u−(s+R, t), s ≤ −R
2 − 1
expy(t)
(ρR
2(−s)ξ−(s+R, t) + ρR
2(s)ξ+(s−R, t)
)s ∈
[−R
2 − 1, R2 + 1]
u+(s−R, t), s ≥ R2 + 1
(6.7)
(ΦR,ΨR) (s, t) =
(Φ−(s+R, t),Ψ−(s+R, t)) , s ≤ −R
2 − 1(0, ρR
2(−s)Ψ−(s+R, t) + ρR
2(s)Ψ+(s−R, t)
), s ∈
[−R
2 − 1, R2 + 1]
(Φ+(s−R, t),Ψ+(s−R, t)) . s ≥ R2 + 1
(6.8)
Now it is easy to see that, if we change the choice of representatives u± which are also in r0-temporal
gauge, the connected sum uR := (uR,ΦR,ΨR) doesn’t change for s ∈ [−R2 − 1, R2 + 1] and hence
GAUGED FLOER HOMOLOGY 33
we obtain a gauge equivalent connected sum. Moreover, if we change y by y′ which represent the
same [y] ∈ CritAH , then we can obtain
u′− ∈ M(x, y′
), u′+ ∈ M
(y′, z
)(6.9)
which are also in r0-temporal gauge, and we obtain a connected sum u′R which is gauge equivalent
to uR.
Now we consider the augmented linearized operator DR := DuR .
Lemma 6.2. There exists c > 0 and R0 > 0 such that for every R ≥ R0 and η ∈ E2,puR⊕W 2,p (Θ, g),
we have
‖D∗Rη‖W 1,p ≤ c ‖DRD∗Rη‖Lp . (6.10)
Proof. Same as the proof of [32, Proposition 3.9]
Hence we can construct a right inverse
QR := D∗R (DRD∗R)−1 : E0,p
uR⊕ Lp (Θ, g)→ TuRB
1,p (6.11)
with
‖QR‖ ≤ c. (6.12)
Now we write QR := (QR,AR) with QR : E0,puR→ TuRB1,p (x, z). Then actually the image of QR
lies in the kernel of d∗0 and therefore QR is a right inverse to dFuR |kerd∗0. Because our construction
is natural with respect to gauge transformations, we see that QR induces an injection
QR : E0,p[uR] → T[uR]B1,p (6.13)
which is a right inverse to the linearized operator dF[uR] and which is bounded by c. By the implicit
function theorem, we have
Proposition 6.3. There exists R1 > 0, δ1 > 0 such that for each R ≥ R1, there exists a unique
ξ ∈ ImQR = kerd∗0 ⊂ TuRB1,p,∥∥∥ξ∥∥∥
W 1,p≤ δ1 such that
F(
expuR ξ)
= 0,∥∥∥ξ∥∥∥
W 1,p≤ 2c
∥∥∥F(uR)∥∥∥Lp. (6.14)
Therefore, the gluing map can be defined as
glue ([u−], R, [u+]) =[expuR ξ
]∈ M ([x], [z]; J,H) . (6.15)
On the other hand, it is easy to see that the augmented linearized operators Du for all con-
necting orbits u is of “class Σ” considered in [10]. Therefore, by the main theorem of [10], there
exists a “coherent orientation” with respect to the gluing construction. Choosing such a coher-
ent orientation, then to each zero-dimensional moduli space M([x], [y]; J,H), we can associate the
counting χJ([x], [y]) ∈ Z, where each trajectory [u] contributes to 1 (resp. -1) if the orientation of
[u] coincides (resp. differ from) the “flow orientation” of the solution. Then we define
δJ : V CFk (M,µ;H; ΛZ) → V CFk−1 (M,µ;H; ΛZ)
[x] 7→∑
[y]∈CritAH
χJ([x], [y])[y](6.16)
34 GUANGBO XU
As in the usual Floer theory, we have
Theorem 6.4. For each choice of the coherent orientation on the moduli spaces M([x], [y]; J,H),
the operator δJ in (6.16) defines a morphism of ΛZ-modules satisfying δJ δJ = 0.
This makes (V CF∗(M,µ;H; ΛZ), δJ) a chain complex of ΛZ-modules, to which will be generally
referred as the vortex Floer chain complex. Therefore the vortex Floer homology group is defined
as the ΛZ-module
V HFk (M,µ; J,H; ΛZ) :=ker (δJ : V CFk (M,µ;H; ΛZ)→ V CFk−1 (M,µ;H; ΛZ))
im (δJ : V CFk+1 (M,µ;H; ΛZ)→ V CFk (M,µ;H; ΛZ))(6.17)
6.2. The continuation map. Now we prove that the vortex Floer homology group defined above is
independent of the choice of admissible family of almost complex structures and the time-dependent
Hamiltonians, and, if we use the moduli space of (1.8) instead of (1.7) to define the Floer homol-
ogy, independent of the parameter λ > 0. So far the argument has been standard, by using the
continuation principle.
Let (Jα, Hα, λα) and(Jβ, Hβ, λβ
)be two triples where λα, λβ > 0, Hα, Hβ are G-invariant
Hamiltonians satisfy Hypothesis 2.1 and 2.3, and Jα ∈ J regHα,λα , Jβ ∈ J regHβ ,λβ
(for notations refer to
the appendix).
We choose a cut-off function ρ : R → [0, 1] with s ≤ −1 =⇒ ρ(s) = 1 and s ≥ 1 =⇒ ρ(s) = 0.
Then we define
Hs,t = ρ(s)Hαt + (1− ρ(s))Hβ
t , λs = ρ(s)λα + (1− ρ(s))λβ. (6.18)
We denote this family of Hamiltonians by H . Now, as in the appendix, we consider the space of
families of admissible almost complex structures J(Jα, Jβ
)consisting of smooth families of almost
complex structures J = (Js,t)(s,t)∈Θ, such that for each l ≥ 1,∣∣∣e|s|Js,t − Jαt ∣∣∣Cl(Θ−×M)
<∞,∣∣∣e|s|Js,t − Jβt ∣∣∣
Cl(Θ+×M)<∞. (6.19)
This is a Frechet manifold. For any J ∈ J(Jα, Jβ
), we consider the following equation on
u = (u,Φ,Ψ) ∂u
∂s+XΦ(u) + Js,t
(∂u
∂t+XΨ(u)− YHs,t(u)
)= 0;
∂Ψ
∂s− ∂Φ
∂t+ [Φ,Ψ] + λ2
sµ(u) = 0.
(6.20)
For the same reason as in Section 3, any finite energy solution whose image in M has compact
closure is gauge equivalent to a solution which is asymptotic to xα ∈ CritAHα (resp. xβ ∈ CritAHβ )
as s → −∞ (resp. s → +∞). Hence for any [xα] ∈ CritAHα and [xβ] ∈ CritAHβ , we can consider
the moduli space of solutions
N(
[xα], [xβ]; J ,H , λs
). (6.21)
GAUGED FLOER HOMOLOGY 35
One thing to check in defining the continuation map is the energy bound of solutions, which
implies the compactness of the moduli space. We define the energy to be
E(u) = E(u,Φ,Ψ)
=1
2
(|∂su+XΦ(u)|2L2 +
∣∣∂tu+XΨ(u)− YHs,t(u)∣∣2L2 +
∣∣λ−1s (∂sΨ− ∂tΦ + [Φ,Ψ])
∣∣2L2 + |λsµ(u)|2L2
)= |∂su+XΦ(u)|2L2 + |λsµ(u)|L2 (6.22)
where, the last equality holds only for u a solution to (6.20). Then we have
Proposition 6.5. For any solution u to (6.20) whose energy is finite and whose image in M has
compact closure, we have
E (u) = AHα([xα])−AHβ ([xβ])−∫
Θ
∂Hs,t
∂s(u)dsdt. (6.23)
Proof. We can transform the solution u into temporal gauge. Then the energy density is
|∂su|2 + |λsµ(u)|2 =ω(∂su, ∂tu+XΨ − YHs,t(u)
)− µ(u) · ∂Ψ
∂s
=ω(∂su, ∂tu)− ∂
∂s(µ(u) ·Ψ) +
∂
∂s(Hs,t(u))− ∂Hs,t
∂s(u).
(6.24)
Then integrating over Θ, we obtain (6.23).
Theorem 6.6. There exists a Baire subset J regH ,λs
(Jα, Jβ
)⊂ J
(Jα, Jβ
), such that for any J ∈
J regH ,λs
(Jα, Jβ
), the moduli space N ([xα], [xβ]; J ,H , λs) is a smooth oriented manifold with
dimN(
[xα], [xβ]; J ,H , λs
)= CZ([xα])− CZ([xβ]). (6.25)
So in particular, when CZ(xα) = CZ(xβ), N is of zero dimension. The algebraic count of N gives
an integer χ([xα], [xβ]
). Then we define the continuation map
contβα : V CF∗ (M,µ; Jα, Hα, λα; ΛZ) → V CF∗(M,µ; Jβ, Hβ, λβ; ΛZ
)[xα] 7→
∑[xβ ]∈CritA
Hβ
χ(
[xα], [xβ])
[xβ]. (6.26)
Now we have the similar results as in ordinary Hamiltonian Floer theory.
Theorem 6.7. The map contβα is a chain map. The induced map on the vortex Floer homology
groups is independent of the choice of the homotopy H , the family J of almost complex structures,
the cut-off function ρ. In particular, contβα is a chain homotopy equivalence. If (Jγ , Hγ) is another
admissible pair and λγ > 0, then in the level of homology
contγβ contβα = contγα.
Proof. The proof is essentially based on the construction of various gluing maps and the compact-
ness results about N([xα], [xβ]; J ,H , λs
)when CZ([xα]) − CZ([xβ]) = 1. As in the gluing map
constructed in proving the property δ2J = 0, we need to specify a gauge to construct the approxi-
mate solutions. We can still use solutions in r-temporal gauge, which is a notion independent of
the equation. We omit the details.
36 GUANGBO XU
6.3. Computation and Morse homology. In this subsection we discuss the computation of
the vortex Floer homology group. Before we proceed let us recall how to show that the ordinary
Hamiltonian Floer homology is isomorphic to the Morse homology.
On a compact symplectic manifold M we take the Hamiltonian to be the t-independent function
εf , where ε is small and f is a Morse function on the manifold M . Then periodic orbits of εf
corresponds to critical points of f (which are denoted by z1, . . . , zk), and the Conley-Zehnder index
and the Morse index are related by
CZ(zi) = n− Ind(zi)
where 2n = dimM . Then the Floer chain complex is generated by (zi, wi), where wi is a spherical
class. Then we want to show that when
CZ(zi, wi)− CZ(zj , wj) = 0, 1 (6.27)
by taking a generic t-independent almost complex structure J on M , all Floer trajectories connect-
ing (zi, wi) and (zj , wj) are t-independent, i.e., corresponds to Morse-Smale trajectories of εf with
respect to the metric ω(·, J ·). Hence the boundary operators in the Floer chain complex and the
Morse-Smale-Witten chain complex coincide. So that
HF∗(M, εf ; Λ) ' HM2n−∗(M, εf ;Z)⊗ Λ (6.28)
The main difficulty to carry out the above argument is, when J is t-independent, one cannot
easily achieve the transversality of the moduli space of Floer trajectories. Here the cylinder Θ may
have a finite cover over itself
πk(s, t) = (ks, kt) (6.29)
and there might exist Floer trajectories which are multiple covers of other trajectories (when J
is allowed to vary with t, such objects don’t exist generically). The multiple covers might have
higher dimensional moduli than expected, which is similar to the problem caused by the negatively
covered spheres in Gromov-Witten theory. Hence to overcome this difficulty, one has to either put
topological restrictions (such as semi-positivity in [28], [19]), or to use virtual technique, to say
that, though the negative multiple covers have higher dimensional moduli, they contributes to zero
in defining the boundary operator (see [15], [23])
Now back to the case of vortex Floer homology. We choose a Morse function f : M → R so that
the induced Hamiltonian Ht := εf has its periodic orbits corresponding to the critical points of f .
Then we lift f to a function f : M → R and consider the vortex Floer homology defined for the
Hamiltonian εf . To show that the resulting homology group is isomorphic to H∗(M ; ΛQ), we have
to show that for ε small enough, the counting of Floer trajectories corresponds to the counting of
negative gradient flow trajectories of a Lagrange multiplier function associated to εf .
If we choose to put topological restrictions to avoid virtual technique, then one difficulty is the
following. To achieve transversality, we assumed that Ht vanishes for certain t ∈ S1 in the appendix.
This condition doesn’t hold for εf , which is independent of t. And this time the transversality
much be achieved by only using a t-independent almost complex structure J . So virtual technique
is probably unavoidable in this approach.
GAUGED FLOER HOMOLOGY 37
In ordinary Hamiltonian Floer homology, another way to prove the isomorphism is the so-called
Piunikhin-Salamon-Schwarz (PSS) construction, introduced in [30]. It is to consider the moduli
space of “spiked disks”, which is an object interpolating between Floer trajectories and Morse tra-
jectories. The counting of spiked disks defines a pair of chain maps between the Floer chain complex
and the Morse-Smale-Witten chain complex, and using various gluing/stretching constructions one
can prove that the two chain maps are homotopy inverses to each other. In the second paper of this
series, we will give a PSS type construction to prove the isomorphism between V HF∗(M,µ; ΛZ)
and the Morse homology of the symplectic quotient M .
Appendix A. Transversality by perturbing the almost complex structure
In this appendix, we treat the transversality of our moduli space of connecting orbits. For general
symplectic manifold, connecting orbits may develop sphere bubbles, while the expected dimension
of the moduli space of such sphere bubbles may be even larger than the expected dimension of the
moduli space of connecting orbits. In this case one must use the virtual technique to say something
of the structure of the compactified moduli space of connecting orbits. The boundary operator is
defined by the virtual count of the number of trajectories, therefore the Floer homology is only
defined over Q in general. Instead, in this section we restrict to the case where the virtual technique
is not necessary. We remark that this special case covers most interesting examples to which we
will apply our results (for example, toric manifolds as symplectic quotient of vector spaces).
First we recall the important assumption on the Hamiltonian Ht.
Hypothesis A.1. There exists a nonempty open subset I ⊂ S1 such that Ht(x) = 0 for t ∈ I.
Indeed, for any G-invariant Hamiltonian diffeomorphism of M , if it is given by the time-1 map
of some Hamiltonian path, then we can reparametrize the Hamiltonian path to make it vanish
for t lying in a small interval, while the time-1 map of the reparametrized path is the original
Hamiltonian diffeomorphism. Hence this hypothesis is not an essential restriction.
A.1. Admissible family of almost complex structures. We know that for any ε > 0 small
enough, there exists a symplectomorphism U = Uε := µ−1 (g∗ε ) ' µ−1(0) × g∗ε . Hence we have a
natural projection πµ : U → M . There is a natural foliation on U , whose leaves are Gx× g∗ε with
x ∈ µ−1(0), with dimension equal to 2dimG. The tangent planes of this foliation is a G-invariant
distribution on U , denoted by gCU .
Recall that we are also given an almost complex structure J in Hypothesis 2.4. Now we will
perturb J in a specific way. This approach was similar to that in Woodward’s erratum for [36], but
here we don’t have a Lagrangian submanifold.
Definition A.2. An admissible almost complex structure on M is a G-invariant, ω-compatible
almost complex structure J which preserves the distribution gCU on U , and coincides with J outside
U . The set of all admissible almost complex structures is denoted by J (M,U, J). We denote
by J (M,U, J) the space of smooth S1-families of admissible almost complex structures, and define
J l(M,U, J) and J l(M,U, J) the corresponding objects in the C l-category, for l ≥ 1. We abbreviate
them by J l, J l because M,U, J are all fixed.
38 GUANGBO XU
ω and any J ∈ J l induces a G-invariant Riemannian metric gJ on M . We denote by T JM
the
orthogonal complement of gCU with respect to the metric gJ . Then T JM
is isomorphic to π∗µTM , and
we have the orthogonal splitting:
TU ' π∗µTM ⊕ gCU =: T JM⊕ gCU . (A.1)
Now by the integrability of gCU , we see that for any a, b ∈ g and any J ∈ J l, we have
[JXa, JXb] ∈ gCU . (A.2)
Lemma A.3. For l ≥ 1, the space J l is a smooth Banach manifold. For any J = Jtt∈S1 ∈ J l,the tangent space TJ J l is naturally identified with the space of G-invariant sections E : S1×M →EndRTM (of class C l), supported in the closure of U , and for each t ∈ S1 satisfying
i. JtEt + EtJt = 0;
ii. ω(·, Et·) is a symmetric tensor;
iii. gCU is invariant under Et.
Now we consider the following equation for u := (u,Φ,Ψ) ∈ W k,ploc (Θ,M × g× g) with Ht satis-
fying Hypothesis (A.1) and J ∈ J l: ∂su+XΦ(u) + Jt (∂tu+XΨ(u)− YHt(u)) = 0;
∂sΨ− ∂tΦ + [Φ,Ψ] + µ(u) = 0.(A.3)
We consider only finite energy solutions and for any pair x± ∈ CritAH , denote by M (x±; J,H) the
space of all solutions which are asymptotic to x±.
We can also identify any solution u with an object v := (v,Φ,Ψ) ∈ W k,ploc
(R2,M × g× g
)by
v(s, t) = ϕHt (u(s, t)), and Φ, Ψ lifts periodically in t ∈ R. It satisfies ∂sv +XΦ(v) + JHt (∂tv +XΨ(v)) = 0;
∂sΨ− ∂tΦ + [Φ,Ψ] + µ(u) = 0;(A.4)
Here JHt =(φHt)∗ Jt for t ∈ R.
For C l almost complex structures, we have the following regularity theorem:
Theorem A.4. [2, Theorem 3.1] For any l ≥ 1 and any solution u ∈W 1,ploc (Θ,M × g× g) to (A.3)
with J ∈ J l, there exists a gauge transformation g ∈ G2,ploc (Θ, G) such that g∗u ∈ W l+1,p
loc (Θ,M ×g× g).
A.2. Existence of injective points. In this subsection we prove an important technical result,
showing that for any admissible family of complex structures J ∈ J l and any nontrivial connecting
orbit, there exist “injective” points and they form a rather large subset of Θ. We fix l ≥ 1 and J
in this subsection.
We first generalize the notion of injective points in [11]. For any C1-map u : Θ → X with
lims→±∞ u(s, t) = x±(t), we define
ΘU (u) := u−1(U), ΘUI (u) := ΘU (u) ∩ (R× I); (A.5)
CI (u) =
(s, t) ∈ ΘUI (u) | ∂su(s, t) ∈ gCU
; (A.6)
GAUGED FLOER HOMOLOGY 39
RI (u) =
(s, t) ∈ ΘUI (u) \ C (u) | u(s, t) /∈ Gu ((R \ s)× t) , u(s, t) /∈ Gx±(t)
. (A.7)
Note that the above sets are unchanged if we apply to u a gauge transformation.
Lemma A.5. For any nontrivial connecting orbit u = (u,Φ,Ψ) ∈ M (x±; J,H), the set CI(u) is
discrete in ΘUI (u).
Proof. We can assume that u is in temporal gauge, i.e., Φ ≡ 0. Denote ξ = ∂su. The vector
ξ = (ξ, 0, ∂sΨ) lies in the kernel of the linearized operator. So in particular, over R× I,
∇sξ + (∇ξJt) (∂tu+XΨ) + Jt (∇tξ +∇ξXΨ) = −JtX∂sΨ. (A.8)
The family Jt induces a family of metrics on M , and hence a decomposition
u∗TM ' u∗T JtM⊕ gCU (A.9)
over ΘU (u). Suppose ξ = ξ + Xa + JtXb for two functions a, b : ΘU (u) → g and ξ(t) ∈Γ(
ΘU (u), u∗T JtM
). Denote σ = a+
√−1b ∈ gC and Xa + JtXb = Xσ. Then over ΘU
I (u),
∇sξ =∇sξ +X∂sσ +∇ξXσ +∇XσXσ ≡ ∇sξ +∇ξXσ +∇XσXσ mod gCU ;
(∇ξJt) (Jtξ) =(∇ξJt)(Jtξ) + (∇XσJt)(Jtξ) + (∇XσJt)(JtXσ);
Jt (∇tξ +∇ξXΨ) =Jt
((∂tJt)Xb +∇tξ +X∂tσ +∇Jtξ+JtXσ−XΨ
Xσ +∇ξXΨ +∇XσXΨ
)≡Jt∇tξ + Jt∇JtXσXσ + Jt∇JξXσ + Jt∇ξXΨ mod gCU .
(A.10)
(Note that (∂tJt)Xb ∈ gCU .) Therefore we have
0 ≡ ∇sξ + Jt∇tξ +∇XσXσ + Jt∇JtXσXσ + (∇XσJt)(JtXσ)
+(∇ξXσ + (∇ξJt)(Jtξ) + (∇XσJt)Jtξ + Jt∇JtξXσ + Jt∇ξXΨ
)= ∇0,1
ξ + C(z)ξ + Jt[JtXσ, Xσ]
≡ ∇0,1ξ + C(z)ξ mod gCU .
(A.11)
Here ∇ is the connection on u∗T JtM
induced from the Levi-Civita connection for gt = ω(·, Jt·) by
orthogonal projection, and C(z) is a zero-order operator and the last congruence follows from the
integrability of gCU . Hence the section ξ satisfies the Cauchy-Riemann equation
∇0,1ξ + C(z)ξ = 0. (A.12)
Here by ∇ is the connection on u∗T JtM
induced from ∇ by the splitting (A.1).
Now we apply the Carleman similarity principle of [11]. We see that the vanishing set of ξ is
discrete or ξ ≡ 0 on ΘUI (u). If the latter happens, then it is easy to see that u is a trivial connecting
orbit.
Lemma A.6. Suppose we have (ui,Φi,Ψi) ∈W 1,ploc (Θ,M × g× g), i = 1, 2, satisfying
∂sui +XΦi(ui) + Jt (∂tui +XΨi(ui)) = 0 (A.13)
on an open subset V ⊂ Θ with ui(V ) ⊂ U , such that there exists g ∈ W 2,ploc (V,G) with g(z)u2(z) =
u1(z) for z ∈ V . Then g∗(u1,Φ1,Ψ1) = (u2,Φ2,Ψ2) on V .
40 GUANGBO XU
Proof. We assume that g ≡ 1. Then u1 ≡ u2 ≡ u. By the equation
∂su+XΦi(u) + Jt (∂tu+XΨi − YHt) = 0, (A.14)
we see that XΦ1(u) = XΦ2(u), XΨ1(u) = XΨ2(u). Since u(V ) ⊂ U we see that Φ1 ≡ Φ2, Ψ1 ≡Ψ2.
Proposition A.7. Let ui = (ui,Φi,Ψi) ∈ M1,p (x±; J,H), i = 1, 2 be solutions to (A.3) on Θ
for some J ∈ J 1, such that they coincide on a small disk Bε ⊂ Θ. Then there exists a gauge
transformation g ∈ G2,p such that g∗u2 = u1.
Proof. We apply the approach in [5, Section 4.3.4] using the unique continuation results of [27].
I. We identify ui with two solutions vi := (vi,Φi,Ψi) to (A.4) over R2. By definition of vi and
the hypothesis, v1 and v2 coincide on a small disk in R2. We will first prove that for all z0 ∈ Θ,
there exists an open disk Bρ(z0) centered at z0 on which v1 and v2 are gauge equivalent. In fact,
all such points in R2 form a nonempty open subset Ω. If Ω 6= R2, choose the largest ρ0 > 0 such
that Bρ0 = Bρ0(0) ⊂ Ω. Then we can patch gauge transformations together to obtain a global
gauge transformation g : R2 → G such that g∗v1 = v2 over the closure of Bρ0 (since Bρ0 is simply
connected). So without loss of generality, we assume that v1 and v2 coincide over the closure of
Bρ0 .
Then we want to show that indeed every z1 ∈ ∂Bρ0 also lies in Ω. Take a small disk Bρ1(z1) ⊂ R2
centered at z1 such that both v1 and v2 map Bρ1(z1) into a coordinate chart V ⊂ M , and with
respect this coordinate chart, the almost complex structure Jz(vi(z)) : Cn → Cn satisfies
‖Id + Jz(vi(z))I0‖ ≤1
2(A.15)
where I0 is the standard complex structure on Cn.
Then we can find z2 ∈ Bρ0 which is on the line segment between 0 and z1 satisfying the following
conditions.
(1) There exists ρ2 > 0 such that z1 ∈ Bρ2(z2) ⊂ Bρ1(z1);
(2) The gauge transformations gi : Bρ2(z2)→ G defined by parallel transport along the radial
direction with respect to the connection Ai satisfies
gi(w′)−1vi(w
′) ∈ V. (A.16)
(This is because ρ2 is so small so gi is very close to the identity of G.
Now we will show that v1 and v2 are gauge equivalent over Bρ2(z2). We regard v1 and v2 are
maps into V ⊂ Cn and by (A.16), we may assume that the connections are in radial gauge, i.e.,
the connection forms are fidθ, i = 1, 2. Then Xfi is a vector field on V . Then we have
∂vi∂r
+ Jz(vi(z))
(∂vi∂θ
+Xfi(z)(vi(z))
)= 0. (A.17)
GAUGED FLOER HOMOLOGY 41
Subtract one from another (using the linear structure of V ), denoting ξ(z) = v2(z) − v1(z) and
h(z) = f2(z)− f1(z), we obtain
∂ξ
∂r+ Jz(v1(z))
∂ξ
∂θ
=− (Jz(v2)− Jz(v1))∂v2
∂θ− Jz(v2)Xf2(v2) + Jz(v1)Xf1(v1)
= (Jz(v1)− Jz(v2))∂v2
∂θ− Jz(v1)Xh(v1)− (Jz(v2)− Jz(v1))Xf2 − Jz(v2)(Xf2(v2)−Xf2(v1))
= : R1(v1, v2, f1, f2).
(A.18)
Similarly, the difference between the vortex equations in the radial gauge gives
∂h
∂r= −r(µ(v2)− µ(v1)) =: R2(v1, v2, f1, f2). (A.19)
It is easy to see that there exists a constant K > 0 such that
|Rj(v1, v2, f1, f2)(r, θ)| ≤ K (|ξ|+ |h|) , j = 1, 2. (A.20)
Hence we have a differential inequality∣∣∣∣∣ ∂∂r(ξ
h
)+
(Jz(v1)∂ξ∂θ
0
)∣∣∣∣∣ ≤ K |(ξ, h)| . (A.21)
We replace ξ by ζ(z) := (Id− I0Jz(v1(z)))ξ(z) and obtain
∂ζ
∂r+ I0
∂ζ
∂θ
=−(I0∂Jz(v1)
∂r− ∂Jz(v1)
∂θ
)ξ + (Id− Jz(v1)I0)
∂ξ
∂r+ (I0 + Jz(v1))
∂ξ
∂θ
=−(I0∂Jz(v1)
∂r− ∂Jz(v1)
∂θ
)ξ + (Id− Jz(v1)I0)
(∂ξ
∂r+ Jz(v1)
∂ξ
∂θ
).
(A.22)
If we denote by P (ζ, h) =(I0∂ζ∂θ , 0
), then P is an self-adjoint differential operator on the circle S1.
Then we have the differential inequality for some K ′ > 0∥∥∥∥ ddr (ζ, h) + P (ξ, h)
∥∥∥∥L2(S1)
≤ K ′ ‖(ζ, h)‖L2(S1) . (A.23)
This is in the form considered in [27]. Since (ζ, h) vanishes for r small, (ζ, h) ≡ 0 for r ≤ ρ2.
This implies that in radial gauge, the two solutions coincide on Bρ2(z2), which contains z1. Hence
∂Bρ0 ⊂ Ω. By the compactness of ∂Bρ0 , we obtain a disk strictly larger than Bρ0 which is also
contained in Ω. This contradicts with the definition of ρ0. Therefore Ω = R2.
II. Now we prove that there exists a global gauge transformation g : Θ→ G such that g∗u2 = u1.
Indeed, there exists S > 0 such that
(−∞,−S]× S1 ⊂ ΘU (u1) ∩ΘU (u2). (A.24)
By the property of U , we see that the local gauge transformations around each point z ∈ (−∞,−S]×R appeared in I. are unique. Hence we can patch them to obtain a global gauge transformation
g : (−∞,−S]×S1 → G such that g∗u2 = u1. But it is easy to see that we can extend g to a longer
42 GUANGBO XU
cylinder (−∞,−S + ε0]× S1 with ε0 independent of S (we omit the details). Hence there exists a
global gauge transformation which identifies u1 with u2.
Lemma A.8. Suppose we have two solutions (ui,Φi,Ψi) ∈ W 2,p(Bri ,M × g× g) on Bri ⊂ R× I,
i = 1, 2 to (A.3), for some J ∈ J l, l ≥ 2, satisfying the following conditions:
(1) ui(Bri) ⊂ U, u1(0) = u2(0);
(2) ui(Bri) is an embedded surface which intersects each leaf of gCU cleanly at at most one point;
(3) For each (s, t) ∈ Br1, there exists g ∈ G and (s′, t) ∈ Br2 such that u1(s, t) = gu2(s′, t).
Then r1 ≤ r2 and there exists a gauge transformation g : Br1 → G such that g∗(u1,Φ1,Ψ1) =
(u2,Φ2,Ψ2) over Br1.
Proof. We can find an embedded submanifold S ⊂ U of codimension dimG, transverse to G-orbits,
such that u2(Br2) ⊂ S. Then we can construct a smooth map π : G ·S → Br2 such that π v = Id.
The hypothesis implies that u1(Br1) ⊂ Gu2(Br2). Then the map π u1 must take the form
(s, t) 7→ (φ(s, t), t), and there exists a unique g(s, t) ∈ G such that
u1(s, t) = g(s, t)u2(φ(s, t), t). (A.25)
Now since ui is a solution to the vortex equation, we see on Br1 ,
0 = ∂su1 +XΦ1(u1) + Jt(∂tu1 +XΨ1(u1)) ≡ (∂su2) ∂sφ+ Jt (∂tu2 + (∂su1) ∂tφ)
≡ (∂su2) (∂sφ− 1) + (∂tu2) ∂tφ(
mod gCU
).
(A.26)
But since u2 intersects with leaves of gCU cleanly, we see that the above congruence holds if and
only if ∂sφ ≡ 1 and ∂tφ ≡ 0. Hence there exists s0 ∈ R such that u1(s, t) = g(s, t)u2(s + s0, t) for
each (s, t) ∈ Br1 . Since u1(0, 0) = u2(0, 0) and the second hypothesis, s0 = 0. Hence r1 ≤ r2 and
by Lemma A.6, the restriction of u2 to Br1 is gauge equivalent to u1.
Proposition A.9. For any J ∈ J l (l ≥ 2), and any u = (u,Φ,Ψ) ∈ M (x±; J,H) a nontrivial
connecting orbit, the set RI (u) is open and dense in ΘUI (u).
Proof. I. We first prove that RI(u) is open. If it is not the case, then there exists (s0, t0) ∈ RI(u)
and a sequence (si, ti) ∈ ΘUI (u) \ RI(u) such that limi→+∞(si, ti) = (s0, t0). By the definition of
RI(u), we must have that for each i large enough, there exists gi ∈ G and s′i ∈ R \ si such that
u(si, ti) = giu(s′i, ti). (A.27)
If s′i is unbounded, then we have that (for a subsequence) u(s′i, ti) → x±(t0) as i → +∞. This
contradicts with the condition that u(s0, t0) /∈ Gx±(t0). Hence s′i must be bounded. So we may
assume that s′i → s′0 ∈ R. This implies that u(s0, t0) = g0u(s′0, t0) for some g0 ∈ G. By the
definition of RI(u), we must have s′0 = s0. Hence the two different sequences (si, ti) and (s′i, ti)
both converges to (s0, t0). Then (A.27) implies that ∂su(s0, t0) ∈ gU ⊂ gCU , which contradicts with
the fact that (s0, t0) ∈ RI(u).
II. Now we prove that RI(u) is dense. Since we know that the subset CI(u) is discrete in ΘUI (u),
we only have to show that every point in ΘUI (u) \ CI(u) can be approximated by points in RI(u).
We take a point (s0, t0) ∈ ΘUI (u) \CI(u). We may also assume that u(s0, t0) /∈ Gx±(t0); otherwise,
for any other s close to s0, (s, t0) satisfies this condition.
GAUGED FLOER HOMOLOGY 43
Now we assume that (s0, t0) is not in the closure of RI(u). Then there exists ε0 > 0 and a
nonempty open subset I0 ⊂ I with
Bε0 (s0, t0) ⊂ ΘUI0(u) \RI0(u). (A.28)
We may choose ε0 small enough and S large enough so that
(1) For any |s| ≥ S, |t− t0| ≤ ε0, u(s, t) /∈ Gu (Bε0(s0, t0));
(2) For |t− t0| ≤ ε0, u ([s0 − ε0, s0 + ε0]× t) is embedded and intersects with each G-orbit at
at most one point.
(3) Gu (Bε0(s0, t0)) ∩G(CI(u) ∩ [−S, S]× I0
)= ∅.
Now the condition (A.28) implies that for all (s, t) ∈ Bε0(s0, t0), there exists s′ ∈ R \ s such
that u(s, t) ∈ Gu(s′, t).
II.a) We claim that for each (s, t), there exists only finitely many such s′. Indeed, the first condi-
tion above implies that, if there are infinitely many such s′, then they must have an accumulation
point in [−S, S], and at the accumulation point ∂su lies in gCU . This contradicts with the third
condition. So the claim is true.
II. b) For any (s, t) ∈ Bε0(s0, t0), define
KS((s, t)) :=
(s′, t) ∈ [−S, S]× I | s′ 6= s, u(s, t) ∈ Gu(s′, t)
(A.29)
and
N := min(s,t)∈Bε0 (s0,t0)
#KS((s, t)) ≥ 1. (A.30)
We claim that there exists (s∗, t∗) ∈ Bε0(s0, t0) and ε1 > 0 such that Bε1(s∗, t∗) ⊂ Bε0(s0, t0), and
(s, t) ∈ Bε1(s∗, t∗) =⇒ #KS((s, t)) = N. (A.31)
Moreover, for any (sν , tν)→ (s∗, t∗), we have that the sequenceKS((sν , tν)) converges toKS((s∗, t∗))
as subsets.
Indeed, find any (s∗, t∗) ∈ Bε0(s0, t0) which realizes the lower bound N . If not the case, then
there exists a sequence (sν , tν) converging to (s∗, t∗) but #KS(sν , tν) > N . If the sequence of points
KS(sν , tν) have accumulations, then it will contradict with (3); if they don’t accumulate, then by
choosing a subsequence, we see that #KS(s∗, t∗) ≥ N + 1 which contradicts with the choice of
(s∗, t∗).
II. c) Hence we can actually assume that (s0, t0) = (s∗, t∗) and ε1 = ε0. In particular, we take
s1, s2, . . . , sN ∈ [−T, T ] be all numbers such that
u(s0, t0) ∈ Gu(si, t0), i = 1, . . . , N.
We claim that (which is similar to that in [11, Page 262]) there exists δ > 0 and r > 0 such that
u (B2δ(s0, t0)) ⊂ G(∪Nj=1u(Br(sj , t0))
), j 6= j′ =⇒ Br(sj , t0) ∩Br(sj′ , t0) = ∅. (A.32)
Then we define
Σj :=
(s, t) ∈ Bδ(s0, t0) | u(s, t) ∈ Gcl(u(Br(sj , t))). (A.33)
These are closed sets and Bδ(s0, t0) = Σ1∪· · ·∪ΣN . Then there exists j0 such that (s0, t0) ∈ IntΣj0 .
Then we take a small ρ > 0 such that Bρ(s0, t0) ⊂ IntΣj0 and Bρ(s0, t0) ∩Br(s1, t0) = ∅.
44 GUANGBO XU
II. d) For every (s, t) ∈ Bρ(s0, t0), we see that KS((s, t)) ∩ Br(sj0 , t0) contains a unique element
(s′, t). By Lemma A.8 we see that for ρ ≤ r and the two objects u(s0 + ·, t0 + ·) and u(sj0 + ·, t0 + ·)are gauge equivalent over Bρ(0). Then by Proposition A.7, there exists a gauge transformation
g : Θ→ G such that
g∗u = u(sj0 − s0 + ·, ·).
Let δs = sj0 − s0 6= 0. Then we see that u and u(kδs + ·, ·) are gauge equivalent for all k ∈ Z.
This implies that u(s0, t) ∈ Gx±(t) by the asymptotic behavior of finite energy solutions. This
contradicts with our choice of (s0, t0), which means that RI(u) is indeed dense in ΘUI (u).
A.3. The universal moduli space over the space of admissible almost complex struc-
tures. For l ≥ k ≥ 2, p > 2, we denote
Mk,p(
[x±]; J l, H)
:=
([u], J) | J ∈ J l, [u] = [u,Φ,Ψ] ∈Mk,p ([x±]; J,H). (A.34)
Proposition A.10. For any l ≥ k, k ≥ 1, the universal moduli space is a C l−k-Banach submanifold
of Bk,p × J l.
Proof. We just need to prove that the linearized operator
D : T[u]Bk,p × TJ J l → Ek−1,p[u] (A.35)
is surjective. It is equivalent to consider the augmented one, for any representative u ∈ M (x±; J,H),
which is
D : TuBk,p × TJ J l → Ek−1,pu ⊕W k−1,p (Θ, g) . (A.36)
Now, because the restriction of D to the first component, which is the augmented linearized
operator Du, is already Fredholm, we only need to prove that D has dense range. If not the
case, then there exists a nonzero vector η = (η, ϑ1, ϑ2) ∈ Ek−1,pu ⊕W k−1,p (Θ, g) which lies in the
L2-orthogonal complement of the image of D; therefore η also lies in the orthogonal complement
of the image of Du. Hence η ∈ kerD∗u. By elliptic regularity associated to D∗u, we see that η is
continuous. We will show that η vanishes on an nonempty open subset of RI(u), which, by the
unique continuation property of D∗u, contradicts with the fact that η 6= 0.
Indeed, on RI(u), we decompose η = η′ + η′′ with respect to the splitting (A.1) (which also
depends on t). However, it is easy to find E′, E′′ ∈ TJ J l, supported near the G-orbit of u(z) for
any z ∈ RI(u), such that E′(s, t) is zero on gCU , E′′(s, t) is zero on T JtM
, and such that D(0, E′)
(resp. D(0, E′′)) has positive L2-pairing with η′ (resp. η′′) (the concrete way of constructing E′
and E′′ are similar to that in [24]). Then this shows that η vanishes on RI(u).
Then, looking at the first component of 0 = D∗u (η, ϑ1, ϑ2), we see on RI(u),
JtXϑ1 −Xϑ2 = 0. (A.37)
By the definition of RI(u), we see that ϑ1|RI(u) = ϑ2|RI(u) = 0. This finishes our proof.
Now the projection
M(
[x±]; J l, H)→ J l (A.38)
is a Fredholm map of class C l−k. Then by Sard-Smale theorem, we obtain
GAUGED FLOER HOMOLOGY 45
Theorem A.11. For any pair [x±] ∈ CritAH , there exists l0 = l0(k, [x±]) ∈ Z+ such that for any
l ≥ l0, there exists a Baire subset J l;regH ⊂ J l, such that for any J ∈ J l;regH and any [u] = [u,Φ,Ψ] ∈M ([x±]; J,H), the linearized operator
DJ,H[u] : T[u]Bk,p → Ek−1,p[u] (A.39)
is surjective.
Using Taubes’ trick (see [24, Page 52]), we obtain
Corollary A.12. There exists a Baire subset J regH ⊂ J such that for any J ∈ J regH , the moduli
space M ([x±]; J,H) is a smooth submanifold of Bk,p([x±]) for any k ≥ 1 and p > 2.
A.4. Transversality for continuation map. It is easier to achieve transversality for the moduli
space defining the continuation map, because we are allowed to have objects which depend both
on s and t. We briefly sketch the method. Suppose we have a given homotopy Hs,t of compactly
supported Hamiltonians for (s, t) ∈ Θ such that Hs,t = Hαt for s << 0 and Hs,t = Hβ
t for s >> 0.
Now we are given two regular families of almost complex structures
Jα ∈ JHα(M,U, J)reg, Jβ ∈ JHβ (M,U, J)reg. (A.40)
Moreover, suppose we have two possibly different constants λα, λβ > 0, with a homotopy λs with
λs = λα for s << 0 and λs = λβ for s >> 0. We want to show that, for a “generic” homotopy
from Jα to Jβ (denoted by a family of almost complex structures Js,t, with Js,· ∈ J (M,U, J)), for
each pair [xα] ∈ CritAHα , [xβ] ∈ CritAHβ , the moduli space of solutions to the following equation∂u
∂s+XΦ(u) + Js,t
(∂u
∂t+XΨ(u)− YHs,t(u)
)= 0;
∂Ψ
∂s− ∂Φ
∂t+ [Φ,Ψ] + λ2
sµ(u) = 0
(A.41)
satisfying the conditions that u = (u,Φ,Ψ) is asymptotic to xα (resp. xβ) as s → −∞ (resp.
s→ +∞), is transverse.
Now take the function V (s, t) = e|s| which diverges exponentially as s→ ±∞. For l ∈ Z+∪∞,consider the space
J l(Jα, Jβ
)(A.42)
consisting of C l-family of admissible almost complex structures Js,t (with respect to the same U
and J as before) parametrized by (s, t) ∈ Θ, such that
|V (s, t)(Js,t − Jαt )|Cl(Θ−×M) <∞, |V (s, t)(Js,t − Jβt )|Cl(Θ+×M) <∞. (A.43)
Those J ’s are homotopies that are asymptotic to Jα (resp. Jβ) exponentially with their derivatives
up to order l.
Then, for each J ∈ J l(Jα, Jβ), consider the moduli space
N(
[xα], [xβ]; J ,H , λs
)⊂ Bk,p
([xα], [xβ]
)(A.44)
of solutions to (A.41). Since all element [u] ∈ M([xα], [xβ]; J ,H , λs
)will go into U as |s| →
∞, this implies that every such [u] is “irreducible” in the sense of [2]. Hence we can prove the
46 GUANGBO XU
transversality of the universal moduli space over J l(Jα, Jβ
)in the same way as in [2], because
now the perturbation could be dependent on both s and t. Then using the implicit function theorem
and Taubes’ trick, it is easy to prove the following proposition.
Proposition A.13. There exists a Baire subset J regH ,λs
(Jα, Jβ
)⊂ J
(Jα, Jβ
)such that for ev-
ery J ∈ J regH ,λs
(Jα, Jβ
)and every pair [xα] ∈ CritAHα, [xβ] ∈ CritAHβ , the moduli space
N([xα], [xβ]; J ,H , λs
)is a smooth manifold of dimension CZ([xα])− CZ([xβ]).
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Department of Mathematics, 410N Rowland Hall, University of California, Irvine, Irvine, CA 92697
USA
E-mail address: [email protected]