Gauged Floer homology and spectral invariants
We define a version of spectral invariant in the vortex Floer theory for a -Hamiltonian manifold . This defines potentially new (partial) symplectic quasi-morphism and quasi-states when is not semi-positive. We also establish a relation between vortex Hamiltonian Floer homology  and Woodward’s quasimap Floer homology  by constructing a closed-open string map between them. This yields applications to study non-displaceability problems of subsets in .
The main theme of this paper lies in the intersection of several different directions of interests in symplectic geometry. Classically, one of the fundamental questions driving the development of symplectic geometry is to understand the fixed point set of a Hamiltonian diffeomorphism, that is, the Arnold’s conjecture. A relative version of Arnold’s conjecture considers the intersection of a Lagrangian submanifold with its image under a Hamiltonian diffeomorphism. It is by now well-known that this line of problems is intertwined with Gromov-Witten invariants, in a way that the Hamiltonian Floer cohomology ring, generated by fixed points of Hamiltonians, is isomorphic to the quantum cohomology ring defined by counting holomorphic spheres . Although it is more complicated in general, but there is a rough correspondence in the open string (Lagrangian boundary) cases, see  for example.
Another important mainstream in the research of symplectic geometry owes to the discovery of a bi-invariant metric, Hofer’s metric on for arbitrary symplectic manifolds . This in turn is closely tied to an extra action filtration structure on the Floer complex, which is usually referred to as the spectral theory. In particular, Viterbo  constructed a spectral invariant for , which was later generalized by Schwarz and Oh, see for example, .
We recall briefly the construction of the chain level theory here. For a generic time dependent Hamiltonian , we can construct the Floer homology group (with some appropriate coefficient ring ). is the homology of certain chain complex of -modules with some distinguished generators; there is an action functional (which is sensitive in ) defined on those generators. is canonically isomorphic to the singular homology of the underlying manifold in the same coefficient ring. Then, any homology class is represented by a Floer chain
Then we define the spectral number formally as
One can show that is independent of various choices made in order to define the Floer chain complex, and it only depends on the Hamiltonian path . An important property is that the spectral invariant lower bounds the Hofer norm of a Hamiltonian diffeomorphism.
A remarkable connection was built between the filtered Floer theory and the displaceability problem of subsets of symplectic manifolds by Entov and Polterovich in their series of papers, see for example [10, 11, 12]. In a nutshell, they combine the ideas from dynamical systems, function theory as well as symplectic geometry to construct certain functionals on , called the (partial) symplectic quasi-states and Calabi quasimorphisms. They used these objects to study symplectic intersections and introduced the notion of heavy and superheavy subsets of symplectic manifolds. Later such constructions are extended by ,  to include the contributions of the big quantum cohomology in general symplectic manifolds.
The primary purpose of this paper is to explore a similar picture in the vortex context. The vortex Floer theory, as the last strand of the braid, is defined when there is a compact Lie group acting on the symplectic manifold . The solutions of vortex equations exhibited similar structures to those of Floer equations, thus the relation between vortex theory and Hamiltonian/Lagrangian Floer theory has attracted much interests since. In the Gromov-Witten context, a vortex equation was studied by Mundet and Cieliebak-Gaio-Salamon [22, 8, 23, 7] to define a Hamiltonian Gromov-Witten invariant. In  a version of vortex Hamiltonian Floer theory (closed string) was proposed, which was only realized very recently by the second named author [37, 35, 36]. The open string theory counterpart of symplectic vortex equations were considered first in  to obtain new non-displaceability results. Recently Chris Woodward defined a quasimap Floer cohomology using an adiabatic limit version of the equation in , and have achieved great advance in understanding the non-displaceability of the toric fibers.
In this paper, we try to understand two aspects of the vortex Floer homology theory constructed by the second named author : (1) its filtration structure, and (2) justify the fact that the vortex Hamiltonian Floer theory is an appropriate closed string theory versus Woodward’s quasi-map Floer theory. Combining these two aspects of considerations, we are able to study the non-displaceability problems and quasi-morphism(state) theory in the vortex context.
As the first step, we define the spectral invariants in vortex context. The result can be vaguely stated as follows.
Let be an aspherical, equivariantly convex, Hamiltonian -manifold with compact smooth symplectic quotient (see conditions (H1)–(H3) at the beginning of Section 2). Then there exists a function
satisfying usual axioms of the spectral invariants except for the normalization and symplectic invariance (replaced by Hamiltonian invariance).
The proof of the above theorem is not a straightforward adaption from the Hamiltonian Floer case: among others, one of the common difficulty in recovering usual properties of spectral invariants in ordinary Floer theory lies in that, a priori, we have vortex Floer trajectories which do not descend to the symplectic quotient. An adiabatic limit technique is hence developed to handle the problem. Details of the proof was provided in Section 3.4.
The spectral invariant we define is reduced to the usual spectral invariant when and is compact aspherical. It is not clear to the authors at the present stage whether the absence of normalization and symplectic invariance properties are only technical in general. In particular, the normalization will follow from a vortex version of PSS isomorphism (in principle the PSS isomorphism can always be constructed using the virtual technique but it is not yet known if a construction without virtual technique is possible at the time of writing). However, a key observation is that this weaker version of vortex spectral invariant defines a genuine partial symplectic quasi-morphism (quasi-state) with all formal properties, hence making sense of the notion of heaviness. This will be explained in Section 4.
There are two remarks we should make at this point. It is notable that we have constrained us to the weaker notion of partial quasi-morphisms (quasi-states) instead of the genuine quasi-morphisms (quasi-states), and only heaviness instead of superheaviness. The reason is that, while the proof of properties of superheavy sets mostly follows formally from Entov and Polterovich’s original approach, we do not have a good example where vortex Floer cohomology can be computed and confirmed to have a field idempotent element other than the Fano cases, in which the vortex Floer theory is identified with the usual Hamiltonian Floer theory.
However, one should note that our partial quasi-morphisms/states alludes a relation to Borman’s reduction on quasi-morphism/states on to , see . There is no known relation between and , so it would be interesting if one could still establish instead a reduction relation on the quasi-morphism/state level, which might also shed lights on the relation between idempotents of the two rings.
The second part of the paper investigates the relation between and Woodward’s quasi-map Floer theory with focus on examples of toric manifolds. Our main result is:
For a -manifold as in Theorem 1.1, a compact -invariant Lagrangian submanifold contained in and a brane structure , there is a closed-open map
which is a -linear map sending the identity to the identity .
This seems an appropriate correspondence of the closed-open map in the usual Floer theory (see ). The reason that vortex Hamiltonian Floer theory is connected with Woodward’s quasimap Lagrangian Floer theory is, when consider the symplectic vortex equation on the half cylinder with boundary lying in a -invariant Lagrangian submanifold, the boundary bubbles are exactly “quasidisks” (disk vortices with zero area form).
A concrete consequence of the existence of the closed-open map is
Any generic Hamiltonian diffeomorphism on a toric manifold has at least one fixed point.
We would like to emphasize that the proof of the above corollary does not involve virtual techniques, and the toric manifold in the corollary can have arbitrary negative curves. This result was firstly proved as [7, Theorem 8.3] under the assumption of (H1), (H3), while allowing the symplectic quotient to be an orbifold. Their argument is similar to that of Gromov , by degenerating the symplectic vortex equation on a sphere with Hamiltonian perturbations. In the orbifold setting, it is more convenient to use the notion of “-relative 1-periodic orbits” in instead of using 1-periodic orbits in the symplectic quotient. One should note also that the expected (but unproved) PSS isomorphism for vortex Floer theory implies the above result.
As another application of the closed-open map, following ,  and , we conclude that all critical fibers of Hori-Vafa superpotential on a toric manifold are heavy with respect to the partial quasi-morphism (quasi-state) defined by the identity in in Section 4.
All Hori-Vafa critical fibers in a toric manifolds are heavy.
We also considered a vortex version of Entov-Polterovich’s theorem, which says the product of two heavy sets is again heavy, see Theorem 6.9. Note that this includes the case when one of the product factor is heavy in Entov-Polterovich’s original sense, by setting , which is a “hybrid case”. In particular this implies heavy sets are stably non-displaceable.
We would like to thank Kenji Fukaya, Yong-Geun Oh for stimulating discussions. The second author would also like to thank Chris Woodward for many detailed explanations on quasi-map Floer homology.
2. Vortex Floer Homology
In this section we review the definition of vortex Floer homology from  for the convenience of the reader. Besides fixing notations, we unify the definition of the continuation map and the pair-of-pants product under the framework of symplectic vortex equations over surfaces with cylindrical ends. The reader may refer to  for more details of the definition of the vortex Floer chain complex and to  for details of the ring structure.
We start by giving an incomplete list of our notations and conventions in this paper.
is a compact connected Lie group. is a Hamiltonian -manifold of dimension . For any , the vector field defined by
The moment map is defined as:
The following will be our standing assumptions for : (cf. the main assumptions of ).
(H1) is aspherical. Namely, for any smooth map , .
(H2) is a proper map, is a regular value of and the restriction of the -action to is free.
(H3) (cf. [7, Definition 2.6]) If is noncompact, then there exists a pair , where is a -invariant proper function, is an almost complex structure on , such that there exists a constant and
Here is the Levi-Civita connection of the metric defined by and .
is the symplectic quotient of dimension , which is always assumed to be smooth (see (H2) below). admits a canonically induced symplectic form .
We consider -invariant Hamiltonian functions where for each , . We also consider their descendants by -quotients, where for each , . The Hamiltonian vector field of is defined by .
The set of -invariant Hamiltonian functions and diffeomorphisms are denoted respectively as
and similarly for the set of non-equivariant Hamiltonian functions and diffeomorphism:
while the universal cover of the Hamiltonian diffeomorphism groups are denoted as
From time to time we impose the following constraints on :
(H4) The Hamiltonian diffeomorphism on induced from is nondegenerate.
We then denote different classes of Hamiltonian functions as:
2.1. Equivariant topology
We briefly recall notions which will appear in the paper from basic equivariant theory.
Equivariant spherical classes
For any manifold , we denote by to be the image of the Hurwitz map . Let , we define . We can view as a subset of . Geometrically, it is convenient to represent a generator of by the following object: a smooth principal -bundle and a smooth section . We denote the class of the pair to be .
Equivariant symplectic form and equivariant Chern numbers
The equivariant cohomology of can be computed using the equivariant de Rham complex . In , there is a distinguished closed form , called the equivariant symplectic form, which represents an equivariant cohomology class.
If , then the pairing can be computed in the following way. Choose any smooth connection on . Then there exists an associated closed 2-form on , called the minimal coupling form. If we trivialize locally over a subset , such that , with respect to this trivialization, then can be written as . We have
On the other hand, any -invariant almost complex structure on makes an equivariant complex vector bundle. So we have the equivariant first Chern class . We define
2.2. The space of loops and the action functional
Let be the space of smooth contractible parametrized loops in and a general element of is denoted by
Let be a covering space of , consisting of triples where and is an equivalence class of smooth extensions of to the disk , where two extensions and are equivalent if the class of is zero in .
Denote by the smooth free loop group of . Take any point , we have the homomorphism induced from the map sending a loop to a loop . Its kernel is independent of the choice of . We define
where is the subgroup of contractible loops.
The right action of on can be explicitly written as
This action doesn’t lift to naturally; only its restriction to does. To define this lift, one assigns a canonical choice of lift of capping for each element in as follows.
For a contractible loop , extend arbitrarily to . The homotopy class of extensions is unique because for any connected compact Lie group (). Then the class of in is independent of the extension.
It is easy to see that the covering map is equivariant with respect to the inclusion . Hence it induces a covering
An element of represented by is denoted by .
One may now define an action by “connected sum” of on . For the precise definition, we refer the readers to [37, Section 2.3]. One checks that
Lemma 2.1 (, Lemma 2.4).
is the group of deck transformations of the covering .
Since , by this lemma, we form , which is again a covering of and is the group of deck transformations. We use to denote an element in represented by .
The action functional
Take . We define a 1-form on by
Its pull-back to is the differential of the following action functional on :
The zero set of is
The critical point set of its preimage under the covering .
2.3. Gradient flow, symplectic vortex equation and the moduli spaces
Let be the set of pairs where and is an -family of almost complex structures on , such that
where is the convex structure in (H3). On the other hand, we fix a biinvariant metric on , which induces an identification . Let be a constant. Formally, the equation for the negative gradient flow of is the following equation for pairs
This equation is invariant under the -action given on pairs given by
The energy of a solution is defined by
Moduli spaces of flow lines
It is more convenient to consider the symplectic vortex equation over in general gauge, i.e., the following equation on triples
We showed in [37, Section 3] that, every finite energy solution (2.3) whose image in has compact closure is gauge equivalent to a solution (which in fact can be taken to be in temporal gauge, i.e., ) and there exists a pair such that
We simply write . We also have the energy identity (cf. Theorem 2.5)
For which project to (forgetting the cappings), we can consider solutions which “connect” them, which form a moduli space
Here is the space of smooth gauge transformations on which are asymptotic to the identity at the circles at infinity. If for , then one find gauge transformation which is asymptotic to at the circles at infinity, and induces an identification between and . The moduli space in common with respect to this type of identifications is denoted by , where is the orbit of . Lastly, if , then there is an obvious identification between and . Then the moduli space in common with respect to this “capping shifting” identification is denoted by
In the case that is symplectic aspherical, we can perturb the -family of almost complex structures to achieve transversality of the moduli space. We give a short account of the method to achieve transverality and refer the reader to [37, Section 6] for more details. For our convenience we introduce the following concept which was not present in .
Fix and . A pair is called a regular pair (relative to and ), if is a -invariant lift of and the linearization of (2.3) along each bounded solution is surjective. The set of regular pairs (relative to and ) is denoted by and the union of for all is denoted by .
In [37, Section 6], the second named author used the following scheme to find -regular pairs. Among all smooth -families of almost complex structures and all -invariant lifts of , the second named author defined the notion of admissible families of almost complex structures ([37, Definition 6.2]) and admissible lifts ([37, Definition 6.5]) (both notions are defined relative to ). Let be the space of all smooth admissible families (relative to ). Moreover, the following was proved.
[37, Lemma 6.6, Theorem 6.8] Let and .
There exists an admissible lift .
For each such lift and each , there exists a subset of second category such that for any , , .
For any , any pair , is a smooth manifold of dimension .
The statement of the above theorem differs slightly from the original ones in . This is because when we consider vortex Floer theory for product manifolds in Subsection 6.3, we will take regular pairs of product type, which may not fall into the class of “admissible” ones in . Nevertheless, only the property of regular pairs are relevant to our construction.
On the other hand, since the target is aspherical (Hypothesis (H1)), one can compatify the moduli spaces only by adding broken flow lines. Meanwhile, one can orient the moduli spaces consistently (in the sense of ) as the case of ordinary Morse or Hamiltonian Floer theory.
2.4. The vortex Floer homology
Let be either or . The universal downward Novikov ring is
The free -module generated by is denoted by . We define an equivalence relation on by
Denote by the quotient -module by the above equivalence relation.
In the notations for moduli spaces we temporarily omit the dependence on , and . Let be the quotient of by the -action of translation. Then for , and one can define the function by
Here the number of is the algebraic count. The boundary operator is defined by the linear extension of
By a compactness argument the above sum is well-defined in and by a gluing argument proved in , . The vortex Floer homology of the Hamiltonian -manifold and the data is defined as
2.5. Symplectic vortex equation on general Riemann surfaces
To study further properties of vortex Floer cohomology, we must turn to the vortex equation on more general Riemann surfaces. Indeed (2.3) fits into this general framework when specialized to . In this section we briefly recall this general theory together with a calculation of the Yang-Mills-Higgs energy with cylindrical ends, which is a mild generalization of [37, Section 3] and [7, Proposition 2.2] but critical to our applications.
Symplectic vortex equation
Let be a Riemann surface and be a smooth -bundle. Let be the associated fibre bundle. Choose a family of -invariant, -compatible almost complex structures on . Then lifts to a complex structure on the vertical tangent bundle , which is still denoted by . On the other hand, the moment map lifts to a map , where is the co-adjoint action. We choose an -invariant metric on inducing an identification ; hence is viewed as a map from to .
Consider the space of smooth -connections on and the space of smooth sections of . For each pair , we have the covariant derivative ; is the -part of with respect to the complex structure on and on . On the other hand, is the curvature 2-form of . Choose a volume form and denote by the contraction of against . The symplectic vortex equation is the following system on pairs
The group of gauge transformations consists of smooth maps satisfying for and . It induces a left-action on by and a right action on pairs by reparametrization.
In ordinary Floer theory, one uses Hamiltonian functions to perturb the -holomorphic curve equation (see [21, Section 8] as a standard reference). In our situation, we perturb similarly, by -invariant Hamiltonians. A -connection on is a 1-form . If is a local coordinate on , then can be written as
Its curvature is
Let and be the Hamiltonian vector fields of and , both depending on . By -invariance of and , and lift to sections of . Then deforms the operator to , which is
The -perturbed symplectic vortex equation reads
where is the -part of . If we trivialize , then where and are -valued functions on , and . Then (2.9) takes the form
The energy of pairs with respect to the perturbed equation (2.2) can be generalized to the case of general perturbed symplectic vortex equation (2.8), which is the Yang-Mills-Higgs functional, given by
Here all the norms are taken with respect to the domain metric determined by and , the target metric determined by and , and the metric on the Lie algebra.
To get better control in actual applications, one needs to impose additional constraints in the case of non-compact Riemann surfaces.
Suppose is obtained by puncturing a compact Riemann surface at finitely many points . Choose mutually disjoint cylindrical ends around for each puncture with identification . Denote . Specify a trivialization of , which induces an extension of . With respect to the specified trivialization, the restriction of any connection to is written as and a section is identified with a map . If converges to a smooth contractible loop as , and is a capping to , then the “connected sum”
gives well-defined homotopy class of continuous sections of , and hence represents a homology class in .
We consider the case of (2.9) where all the auxiliary structures are of cylindrical type. Choose a volume form such that . Choose regular pairs and consider a -connection over , which is flat over and in a gauge such that
Choose a family of almost complex structures which is exponentially close to on , i.e.,
Then we have the following theorem on bounded solutions to (2.9).
Let be a solution to (2.9) on a surface with cylindrical ends (), such that and . Then the following holds.
There is a gauge transformation and , such that over , with respect to the specified trivialization of , and
We have a two form , called the minimal coupling form, so that for any choice of capping to each ,