A two-category of Hamiltonian manifolds, and a (1+1+1) field theory
AA TWO-CATEGORY OF HAMILTONIAN MANIFOLDS,AND A (1+1+1) FIELD THEORY
GUILLEM CAZASSUS
Abstract.
We define an extended field theory in dimensions ` ` , that takes the form of a “quasi 2-functor” with values in a strict2-category z H am , defined as the “completion of a partial 2-category” H am , notions which we define. Our construction extends Wehrheimand Woodward’s Floer Field theory, and is inspired by Manolescu andWoodward’s construction of symplectic instanton homology. It can beseen, in dimensions ` , as a real analog of a construction by Mooreand Tachikawa.Our construction is motivated by instanton gauge theory in dimen-sions 3 and 4: we expect to promote z H am to a (sort of) 3-category viaequivariant Lagrangian Floer homology, and extend our quasi 2-functorto dimension 4, via equivariant analogues of Donaldson polynomials. Contents
1. Introduction 22. Partial 2-categories 42.1. Definition 42.2. Completion 83. Definition of H am , L ie R and L ie C L ie R and L ie C z H am a r X i v : . [ m a t h . S G ] A p r GUILLEM CAZASSUS Introduction If n and d are integers, we denote by C ob n ` `¨¨¨` the (weak) d -categorywith objects closed n -manifolds, 1-morphisms p n ` q -manifolds with bound-ary, 2-morphisms p n ` q -manifolds with corners, ... d -morphisms p n ` d q -manifolds with corners.Donaldson polynomials are invariants of smooth 4-manifolds defined bycounting solutions to the anti-self dual equation. They contain a lot ofinformation and are sensitive to the smooth structure, but are also verydifficult to compute. A major challenge is to understand how they undergocut and paste operations.Instanton homology groups are associated to some 3-manifolds, and werefirst introduced by Floer as a categorification of the Casson invariant. Theycan also be used to define relative Donaldson invariants of 4-manifolds withboundary, which behave (almost) as a (3+1)-TQFT.Ideally, one might want to be able to recast these invariants as an "ex-tended TQFT", ie build a (symmetric monoidal) 4-functor from C ob ` ` ` ` (or a variation that incorporates cohomology classes) to some 4-category,such that this 4-functor applied to a closed 4-manifold (seen as a 4-morphism)essentially corresponds to its Donaldson polynomial.In this direction, building on the Atiyah-Floer conjecture, Wehrheim andWoodward proposed such a functorial behaviour in dimensions 2+1+1. Forthis purpose they defined a 2-category that could serve as a target for a2-functor from C ob ` ` . Their 2-category is inspired by Weinstein’s sym-plectic category: objects are (some) symplectic manifolds, 1-morphisms are(equivalence classes of sequences of) Lagrangian correspondences, and 2-morphisms are "quilted Floer homology" classes. Unfortunately, as the sym-plectic manifolds and Lagrangian correspondences are singular, Floer homol-ogy in this setting is presently not defined. Nonetheless, this constructionhas been implemented in slightly different settings, using nontrivial bundles,see [WW16, WW15].The main motivation of this paper is to extend such a functorial picturedown to dimension 1, namely to build a (symmetric monoidal) 3-functor Φ : C ob ` ` ` Ñ H am that "extends down" Wehrheim and Woodward’sFloer Field theory: We want to build a (symmetric monoidal) 3-category H am with the property that, denoting e its unit object, the endomorphism2-category End p e q is similar with Weinstein-Wehrheim-Woodward’s sym-plectic category S ymp (see remark 3.19).Viewing then C ob ` ` as the endomorphism 2-category of the empty set End pHq in C ob ` ` ` ; the functor, restricted to C ob ` ` , lands in End p e q and corresponds to Wehrheim and Woodward’s Floer field theory functor,modulo the replacement of S ymp by End p e q .In this paper we build such a theory in dimension ` ` . That is, webuild a 2-category z H am , and a quasi 2-functor Φ : C ob ` ` Ñ z H am , thatshould be monoidal and symmetric for a monoidal and symmetric structureon z H am that we will construct in [Caz].For doing so we associate moduli spaces of flat connexions to surfaceswith no closed components. We use open parts of the extended moduli space of Huebschmann and Jeffrey, following a construction of Manolescuand Woodward[MW12].These moduli spaces satisfy a "gluing equals reduction" principle (seeproposition 4.5): if Σ has k boundary components, these moduli spaces N p Σ q carry an SU p q k -Hamiltonian action (corresponding to constant gaugetransformations on each boundary component), and if Σ and S (have re-specively k and l boundary components and) are glued along m components,then N p Σ Y S qz C “ p N p Σ q ˆ N p S qq{{p SU p qq m , where C is a finite union of codimension 3 submanifolds.This gluing equals reduction principle motivates the definition of the com-position of 1-morphisms (corresponding to the spaces N p Σ q ) in H am . Letus first give a naive definition of what H am could be: ‚ objects are Lie groups, ‚ G to G are p G ˆ G q -Hamiltonian manifolds, ‚ M and N are 1-morphisms form G to G , a2-morphism from M to N is a p G, G q -Lagrangian correspondence,that is a p G ˆ G q -Lagrangian in M ´ ˆ N , (see definition 3.11), ‚ (horizontal) composition of 1-morphisms is defined as the symplecticquotient of the cartesian product: if M is a 1-morphism form G to G and N a 1-morphisms form G to G , define M ˝ h N “ p M ˆ N q{{ G , where G acts diagonally on M ˆ N , with moment map µ diag definedby µ diag p m, n q “ µ M p m q ` µ N p n q . Both actions of G and G descend to this quotient, and endow M ˝ h N with a G ˆ G -Hamiltonian action. ‚ vertical composition of 2-morphisms is defined as composition of cor-respondences in the usual way. ‚ horizontal composition of 2-morphisms is defined as quotient of theproduct of correspondences.Unfortunately, this definition faces similar problems of Weinstein’s sym-plectic category: compositions are not always well-defined. For example, M ˝ h N can be singular if 0 is not a regular value of the moment map µ diag .One way to remedy this problem would be to enlarge the class of geo-metric objects we consider (by analogy with schemes or stacks in algebraicgeometry), so that these compositions are always defined. It is likely thatshifted symplectic geometry could be used in order to do that. However, itwould then be less obvious to define Floer homology inside such more com-plicated spaces. For this reason, we follow the approach of Wehrheim andWoodward[WW16]: in order to turn Weinstein’s symplectic category to ahonest category, they define a category where morphisms consist in equiva-lence classes of sequences of Lagrangian correspondences, modulo embeddedcompositions. Composition is then defined by concatenation (and agreeswith composition of correspondences, when these are embedded). We take asimilar approach at the 2-category level: we first define a “partial 2-category” GUILLEM CAZASSUS H am , where compositions are only partially defined. We will refer to themorphisms of H am as simple morphisms. And then we complete H am toa strict 2-category z H am , where 1-morphisms consist in equivalence classesof sequences of simple 2-morphisms, and 2-morphisms equivalence classesof diagrams of simple 2-morphisms. A subtlety arise when defining the 2-morphism spaces: if u, v P hom p x, y q are 1-morphisms, the set shom p u, v q should be independent in the choice of representatives of u and v . For thisreason, we introduce a “diagram axiom” that ensures this independence ofrepresentatives.A similar category has been introduced by Moore and Tachikawa [MT12]in the holomorphic setting, and could serve as a target for SL p , C q analoguesof instanton homology, as introduced in [AM17, CM18].Organization of the paper. In section 2 we set up an algebraic frameworkfor the categories we are interested in: we introduce the notion of partial 2-categories, and construct their completion. In section 3 we define the partial2-category H am , as well as two analogous categories L ie R and L ie C . In sec-tion 4 we introduce the moduli spaces that are involved in our construction.In section 5 we construct the quasi-2-functor. In section 6 we outline somefuture directions that motivates the constructions in this paper. Acknowledgments.
We thank Paul Kirk, Artem Kotelskiy, Andy Manion,Ciprian Manolescu, Catherine Meusburger, Mike Miller, Nicolas Orantin,Raphael Rouquier, Matt Stoffregen, Chris Woodward, Guangbo Xu and Wai-Kit Yeung for helpful conversations, in particular we thank Guangbo Xu forsuggesting the relation with Seiberg-Witten theory of section 6.5. We alsothank Semon Rezchikov for pointing out the work of Moore and Tachikawa. Partial 2-categories
Definition.
We now define partial two-categories, as two categorieswhere the compositions are only partially defined. These can be seen as 2-category analogs of Wehrheim’s categories with Cerf decompositions [Weh16].In the next section we associate a strict 2-category to such partial 2-categories.We denote the sets of morphisms in a partial category shom k , and we in-troduce other sets hom k of "representatives of general morphisms", of whichthe sets of (general) morphisms hom k of the completion will be a quotient.We start by defining a partial two-precategory. A partial two-categorywill satisfy an additional axiom that will be stated later in definition 2.5. Warning:
Our conventions for compositions differ from the standard onefor composition of maps: if ϕ : x Ñ y and ψ : y Ñ z are morphisms, wedenote their composition ϕ ˝ ψ : x Ñ z .We apologize for the length of the following definition. The oppositeoperations correspond to changing orientations of cobordisms, and can besafely ignored in a first reading. Definition 2.1. A partial two-precategory C consists in: ‚ A class of objects Ob C . ‚ An involution x ÞÑ x op on objects. x op is called the opposite object. ‚ For each pair of objects x, y , a class shom p x, y q of simple 1-morphisms . ‚ For each pair of objects x, y , an involutive map shom p x, y q Ñ shom p y, x q , ϕ ÞÑ ϕ T , which we call adjunction . ‚ An opposite involution shom p x, y q Ñ shom p x op , y op q , ϕ ÞÑ ϕ op . ‚ A partial horizontal composition : for each triple x, y, z of objects, asubset of composable 1-morphisms comp p x, y, z q Ă shom p x, y q ˆ shom p y, z q , and a composition map ˝ h : comp p x, y, z q Ñ shom p x, z q , that is compatible with adjunction: if p ϕ, ψ q P comp p x, y, z q , then p ψ T , ϕ T q P comp p z, y, x q and p ϕ ˝ h ψ q T “ ψ T ˝ h ϕ T .For two objects x, y we define the class of representatives of general1-morphisms hom p x, y q to be the class of finite (and possibly emptyif x “ y ) sequences ϕ “ ˆ x ϕ (cid:47) (cid:47) x ϕ (cid:47) (cid:47) ¨ ¨ ¨ ϕ k (cid:47) (cid:47) y ˙ , with ϕ i P shom p x i ´ , x i q , x “ x and x k “ y .Define adjunction hom p x, y q Ñ hom p y, x q by: ϕ T “ ˜ y ϕ Tk (cid:47) (cid:47) x k ´ ϕ Tk ´ (cid:47) (cid:47) ¨ ¨ ¨ ϕ T (cid:47) (cid:47) x ¸ . If ψ P hom p x, y q and ϕ P hom p y, z q are such sequences, wedenote ψ h ϕ P hom p x, z q their concatenation. ‚ For any ϕ, ψ P hom p x, y q , a class of simple 2-morphisms , denoted shom p ϕ, ψ q . ‚ Identification 2-morphisms for horizontal composition of elementary1-morphisms: if p ϕ, ψ q P comp p x, y, z q , I y p ϕ, ψ q P shom pp ϕ, ψ q , ϕ ˝ h ψ q . ‚ Cyclicity: for any cyclic sequence ϕ h χ h ρ T h ψ T P hom p x, x q , coher-ent identifications shom p ϕ h χ, ψ h ρ q » shom p ψ T h ϕ, ρ h χ T q .y χ (cid:30) (cid:30) yx ϕ (cid:64) (cid:64) ψ (cid:30) (cid:30) ó z » x ϕ (cid:63) (cid:63) ñ z. χ T (cid:96) (cid:96) t ρ (cid:64) (cid:64) t ρ (cid:62) (cid:62) ψ T (cid:95) (cid:95) in the sense that the composition of two such identifications corresponds to the iden-tification associated with the new reordering. GUILLEM CAZASSUS
In other words, the set shom p ϕ, ψ q only depends on the cyclicsequence ϕ h ψ T . ‚ Opposites: shom p ϕ, ψ q Ñ shom p ϕ op , ψ op q , A ÞÑ A op . ‚ Adjunctions: involution shom p ϕ, ψ q Ñ shom p ψ, ϕ q , A ÞÑ A T .We require these involutions to be compatible with the cyclicityidentifications, in the sense that the following diagrams, where hori-zontal arrows are adjunctions and vertical arrows are cyclic identifi-cations, should commute: shom p ϕ h χ, ψ h ρ q (cid:47) (cid:47) (cid:15) (cid:15) shom p ψ h ρ, ϕ h χ q (cid:15) (cid:15) shom p ψ T h ϕ, ρ h χ T q (cid:47) (cid:47) shom p ρ h χ T , ψ T h ϕ q . ‚ A partial vertical composition : for each triple ϕ, χ, ψ in hom p x, y q ,a subset of composable 2-morphisms comp p ϕ, χ, ψ q Ă shom p ϕ, χ q ˆ shom p χ, ψ q , and a composition map ˝ v : comp p ϕ, χ, ψ q Ñ shom p ϕ, ψ q , that is compatible with adjunction: if p A, B q P comp p ϕ, χ, ψ q , then p B T , A T q P comp p ψ, χ, ϕ q and p A ˝ h B q T “ B T ˝ h A T .If ϕ : x Ñ y and ψ : y Ñ z are composable 1-morphisms, we requirethat I y p ϕ, ψ q can be vertically composed to the right and left withany other adjacent morphism.We now define diagrams and concatenation of diagrams, which are in-volved in the completion. Remark . Notice that we don’t have a horizontal composition of 2-morphismsin this definition. Such a composition will be defined only after completion.
Definition 2.3.
Fix a partial two-precategory C , ‚ (Representatives of general 2-morphisms) Let ϕ, ψ P hom p x, y q , de-fine the set of representatives of general 2-morphisms hom p ϕ, ψ q asthe set of planar, simply connected, polygonal diagrams of simple2-morphisms from ϕ to ψ : vertices are objects, edges are simple 1-morphisms, and faces simple 2-morphisms of C . An example of sucha diagram is diagram 1 below.(1) . . .x . . y. . . Such diagrams may contain no simple 2-morphisms: if ϕ “ ψ , D “ t ϕ u is a diagram in hom p ϕ, ψ q . In particular if ϕ P hom p x, x q is the empty sequence, D “ t x u is also a diagram in hom p ϕ, ψ q . ‚ (Vertical concatenation) Let C P hom p ϕ, ψ q and D P hom p ψ, χ q ,denote C v D P hom p ϕ, χ q their vertical concatenation, obtained by gluing the two diagramsalong ψ . ‚ (Horizontal concatenation) Let C P hom p ϕ, ψ q and D P hom p ϕ , ψ q ,denote C h D P hom p ϕ h ϕ , ψ h ψ q their vertical concatenation, obtained by gluing the two diagramsalong the common target y of ϕ and ψ .One can see that the concatenations h , h and v are associative in thestrongest possible sense, and make the set of objects, hom and hom intoa strict 2-category C , which we refer to as the pre-completion of C . Definition 2.4.
Let ϕ “ p ϕ , ¨ ¨ ¨ , ϕ n q P hom p x, y q , we say that ϕ is a composition of ϕ and that ϕ is a decomposition of ϕ if for some index i , ϕ i and ϕ i ` are composable, and ϕ “ p¨ ¨ ¨ , ϕ i ´ , ϕ i ˝ h ϕ i ` , ϕ i ` , ¨ ¨ ¨ q . Definition 2.5.
A partial 2-precategory is a partial 2-category if the follow-ing axiom holds: (Diagram axiom)
Let ϕ , ϕ , ... , ϕ k be a sequence of representativesof general 1-morphisms in hom p x, y q such that for each i , ϕ i ` is eithera composition or a decomposition of ϕ i in the sense of definition 2.4. Tosuch a sequence is associated a diagram D P hom p ϕ , ϕ k q given by patchingaltogether all the identification 2-morphisms or their adjoints arising fromthe compositions/decompositions.The axiom requires that for any such sequence with ϕ k “ ϕ , the diagram D is an identity for ϕ in the following sense: for any L P shom p ψ, ϕ q ,using cyclicity of simple 2-morphisms, L can be composed successively withall the identification 2-morphisms or their adjoints, call L ˝ v D the resulting2-morphism in shom p ψ, ϕ q . The diagram D being an identity means that L ˝ v D “ L for any such ψ and L , and also D ˝ v L “ L for any L P shom p ϕ , ψ q , with D ˝ v L defined analogously. Remark . (Consequences of the definition) It follows from the diagram ax-iom that for any composable p ϕ, ψ q P comp p x, y, z q , I y p ϕ, ψ q and its adjointare inverses, in the sense that I y p ϕ, ψ q ˝ v I y p ϕ, ψ q T and I y p ϕ, ψ q T ˝ v I y p ϕ, ψ q are identities for p ϕ, ψ q (resp. for ϕ ˝ h ψ ) with respect to vertical composition.Partial associativity of simple 1-morphisms also follows from this axiom:whenever all the compositions appearing are defined, one has p ϕ ˝ h χ q ˝ h ψ “ ϕ ˝ h p χ ˝ h ψ q . GUILLEM CAZASSUS
Completion.
We now describe how to "complete" a partial 2-category C to a strict 2-category. Loosely speaking, one concatenates morphisms whenthey cannot be composed, as in Wehrheim and Woodward’s construction ofthe symplectic category S ymp [WW16]. Definition 2.7. (Completion of a partial 2-category) Let C be a partial2-category. The following construction defines a strict 2-category p C , calledcompletion of C . ‚ Objects of p C are the same as the objects of C . ‚ Given two object x and y , the set of -morphisms hom p x, y q isdefined as the quotient of hom p x, y q by the relation generated bycompositions: we identify ϕ and ϕ if ϕ is a composition of ϕ in thesense of definition 2.4. Moreover, if ϕ P shom p x, x q is an identityfor x in the sense that it can be composed to the left and right withany adjacent simple 1-morphism ψ , and ϕ ˝ h ψ “ ψ or ψ ˝ h ϕ “ ψ ;then we identify such an identity with the empty sequence. ‚ To define the spaces of 2-morphisms we first define a set hom p ϕ, ψ q for representatives ϕ, ψ P hom p x, y q , and use the diagram axiom todefine hom pr ϕ s , r ψ sq , with r ϕ s , r ψ s the equivalence classes in hom p x, y q .Let hom p ϕ, ψ q be the quotient of hom p ϕ, ψ q by the followingrelation: if D is a diagram such that two faces A and B have aconnected intersection χ P hom p x, y q , by cyclicity we can assumethat A P shom p α, χ q and B P shom p χ, β q , for some α and β . If A and B are composable, then we identify D with the diagram obtainedby removing the edge χ , and merging the two faces A and B to asingle one A ˝ v B . We also identify the empty diagrams with identitiesfor ˝ v . These identifications generate the equivalence relation.We now define hom pr ϕ s , r ψ sq : pick two representatives ϕ, ϕ P r ϕ s ,which by assumption can be joined by a sequence of compositionsand decompositions ϕ “ ϕ , ϕ , ... , ϕ k “ ϕ . Such a sequencemight not be unique, pick any other such sequence ˜ ϕ “ ϕ , ˜ ϕ , ... , ˜ ϕ l “ ϕ . To these two sequences are associated two diagrams D , ˜ D of identification 2-morphisms, and vertical concatenation defines twomaps m : hom p ϕ, ψ q Ñ hom p ϕ , ψ q , r A s ÞÑ r D v A s , ˜ m : hom p ϕ, ψ q Ñ hom p ϕ , ψ q , r A s ÞÑ r D v A s . The diagram axiom applied to the sequence ϕ , ϕ , ¨ ¨ ¨ , ϕ k , ϕ k ´ , ¨ ¨ ¨ , ϕ shows that the map m is invertible, and that its inverse is given by r A s ÞÑ r D T v A s .Applying now the diagram axiom to the sequence ˜ ϕ , ˜ ϕ , ¨ ¨ ¨ , ˜ ϕ l , ϕ k ´ , ¨ ¨ ¨ , ϕ shows that the map ˜ m ˝ m ´ is the identity. In other words, forany pair of representatives ϕ, ϕ P r ϕ s , the sets hom p ϕ, ψ q and hom p ϕ , ψ q are canonically identified. One can similarly prove thattwo equivalent choices for ψ induce canonical identifications. It fol-lows that hom p ϕ, ψ q only depends on the classes r ϕ s and r ψ sq , andcan be denoted hom pr ϕ s , r ψ sq . ‚ Identities. Let x be an object, the identity 1-morphism associated to x is defined as the class of the empty sequence.Let r ϕ s be a 1-morphism, the identity 2-morphism is defined asthe class of the diagram with no simple 2-morphisms, consisting onlyin ϕ . ‚ The three concatenations h , h , v pass to the quotient and definerespectively composition maps: ˝ h : hom p x, y q ˆ hom p y, z q Ñ hom p x, z q , ˝ h : hom pr ϕ s , r ψ sq ˆ hom pr ϕ s , r ψ sq Ñ hom pr ϕ s ˝ h r ϕ s , r ψ s ˝ h r ψ sq , ˝ v : hom pr ϕ s , r χ sq ˆ hom pr χ s , r ψ sq Ñ hom pr ϕ s , r ψ sq , that satisfy the associativity properties of a strict 2-category. Remark . (Self-criticism of the construction of completion) In the equiva-lence relation on hom p ϕ, ψ q we only consider the case where the intersectionof the two faces A and B is connected, but it can happen that faces intersectalong a disconnected set of edges. If each of the corresponding compositionsare allowable, it would be natural to also allow such identifications, howeverthat would lead to more general notions of simple two morphisms in thedefinition of a partial 2-category, as the corresponding faces might not bepolygons anymore, but non-simply connected regions of the plane. We won’tdo that in this paper, for sake of simplicity. Remark . (Completion as a solution to a universal problem) There is prob-ably a more intrinsic way of defining completion as a solution to a universalproblem. After having a suitable definition of a partial 2-functor, a comple-tion of C could consist in a pair p p C , f q of a strict 2-category p C together witha “partial 2-functor” f : C Ñ p C such that any other partial 2-functor from C to any other strict 2-category factors through f in an essentially uniqueway. It would be interesting to work out such a definition in more detail,and compare it to the definition we give, or the possibly more natural onewe just alluded in remark 2.8.3. Definition of H am , L ie R and L ie C We shall now define the partial 2-category H am . Although this categorywill be our main object of interest, we also introduce two simpler and closelyrelated categories L ie R and L ie C , which one could view as toy models for H am . The fact that these categories satisfy the diagram axiom is nontrivialand is proved in section 3.3.3.1. Definition of the partial 2-precategories L ie R and L ie C .Definition 3.1. The following defines a partial 2-category L ie R : ‚ Objects are real Lie groups, ‚ The opposite G op of a group G is the group itself endowed with theopposite multiplication g ¨ op h “ h ¨ g. ‚ The simple 1-morphisms from G to G consist in real smooth mani-folds endowed with a left action of G and a right action of G . Thesetwo actions should commute. ‚ Adjunction of simple 1-morphisms: If M P shom p G, G q , its adjoint M T P shom p G , G q consists in the same underlying manifold, withthe new action defined as g ¨ m ¨ g “ g ´ ¨ m ¨ p g q ´ , for g P G and g P G . ‚ Opposites shom p G, G q Ñ shom p G op , G op q g ¨ op m ¨ op g “ g ´ ¨ m ¨ p g q ´ , g P G, m P M, g P G ‚ Horizontal composition of simple 1-morphisms is defined as a "co-variant product". We will say that M P shom p G , G q and M P shom p G , G q are composable if the action of G on M ˆ M defined by g ¨ p m, m q “ p mg ´ , gm q is free and proper.When this is the case, we will define the composition M ˝ h M as the quotient M ˆ G M “ p M ˆ M q{ G for this action. Since they commute with the G -action, the actionsof G and G pass to this quotient. ‚ Simple 2-morphisms. Let M “ p M , M , ¨ ¨ ¨ , M p k ´ q k q , and N “ p N , N , ¨ ¨ ¨ , N p l ´ q l q be in hom p G, G q , with M i p i ` q : G i Ñ G i ` , and N j p j ` q : H j Ñ H j ` . Denote by ś M and ś N the product of all the 1-morphisms ap-pearing respectively in M and N . A simple 2-morphism from M to N is a submanifold of the product ź M ˆ ź N , which is invariant by the action of all the groups G i and H j , where G i acts on M p i ´ q i ˆ M i p i ` q by g p m, m q “ p mg ´ , gm q and acts trivially on the other factors, except for the two extremalgroups G and G that act on M ˆ N and M p k ´ q k ˆ N p l ´ q l re-spectively by g p m, m q “ p gm, gm q , and g p m, m q “ p mg ´ , m g ´ q . We will call such submanifolds multi-correspondences . ‚ Identification 2-morphisms. If p M , M q P comp p G , G , G q arecomposable 1-morphisms, the identification 2-morphism I G p M , M q P shom pp M , M q , M ˆ G M q is given by the graph of the projection M ˆ M Ñ M ˆ G M . ‚ Cyclicity and adjunction for 2-morphisms are the obvious identifica-tions. ‚ Partial vertical composition is defined as compositions of correspon-dences: If
M , N , P P hom p G, G q , A P shom p M , N q , and B P shom p N , P q , we say that A and B are composable if A ˆ ś P and ś M ˆ B intersect transversally in ś M ˆ ś N ˆ ś P , and if theprojection ź M ˆ ź N ˆ ź P Ñ ź M ˆ ź P is an embedding when restricted to this intersection. When this isthe case, the composition A ˝ v B is defined as the image of thisintersection by this projection.We will prove that L ie R satisfies the diagram axiom in section 3.3. Remark . (Relation with the category of Lie groups) The category ofLie groups embeds in L ie R in a natural way. Let f : G Ñ G be a groupmorphism, then M f “ G , endowed with the bi-action of G and G definedby g ¨ m ¨ g “ f p g q m p g q , g P G, m P M f , g P G is an elementary 1-morphism of L ie R . Moreover the composition of groupmorphisms agrees with the horizontal composition of the M f ’s. One cantherefore think of L ie R as an enlargement of the category of Lie groups.Furthermore, M id G plays the role of the identity for G .Before defining our main partial 2-category of interest, we find interestingto point out that it is already possible to produce a (sort of) (1+1)-fieldtheory with values in the underlying 1-category of L ie R .Observe first that at the level of simple morphisms, the cartesian productendows L ie R with a sort of Frobenius algebra structure. For any object G ,we can define a unit/counit, an identity, and a product/coproduct: ‚ Unit and counit. Let e G P shom p , G q consist in the point, endowedwith the trivial action. The counit p e G q T P shom p G, q is its adjoint.Composing to the right with e G corresponds to modding out by G . ‚ Identity. For id G P shom p G, G q one can take M Id G , with Id G theidentity group morphism. ‚ Product and coproduct. They can both be obtained from a simple1-morphism M P shom p G ˆ G ˆ G, q we define now. Let G , G ,and G stand for three copies of the same group G . Their product Ă M “ G ˆ G ˆ G admits three left actions, each group G i actson Ă M by left multiplication on its corresponding factor. Take then M to be the quotient of Ă M by G , where G acts by simultaneousright multiplication on each factor. The three actions descend to M .Moreover the slice G ˆ G ˆ t e u Ă Ă M identifies M with G ˆ G , and under this identification the three actions become, with g i P G i , a P A and b P B : g . p a, b q “p g a, b q ,g . p a, b q “p a, g b q ,g . p a, b q “p ag ´ , bg ´ q . By turning the G -action to a right action (i.e. acting by g ´ )one can think of M as a product, i.e. in shom p G ˆ G , G q . Do-ing likewise with G , one can think of M as a coproduct, i.e. in shom p G , G ˆ G q .One can check that these are indeed associative and coassociative, that e G and p e G q T are indeed units and counits. To any surface with boundary, onecan then associate a generalized 1-morphism in z L ie R , after taking a pair ofpants decomposition, and associating a copy of M to each pair of pants.One can for example produce a 1-morphism in shom p G, q that corre-sponds to the punctured torus, by contracting the coproduct with the iden-tity, seen as in shom p G ˆ G, q . This corresponds to G itself, endowed withits conjugation action. Interestingly, its cotangent bundle ends up being sim-ilar with the extended moduli space that we will associate to the puncturedtorus. Definition 3.3.
One can define a complex analogue L ie C of L ie R , by tak-ing complex Lie groups as objects, complex manifolds as 1-morphisms, andcomplex multi-correspondences as 2-morphisms.3.2. Definition of the partial 2-precategories.
We first recall some stan-dard facts about Hamiltonian actions that will be relevant to the constructionof H am , for the reader’s convenience and to set some notation conventions. Definition 3.4. (Hamiltonian manifold) Let G be a Lie group. A (left)Hamiltonian G -manifold p M, ω, µ q is a symplectic manifold p M, ω q endowedwith a left G -action by symplectomorphisms, induced by a moment map µ : M Ñ g ˚ . The moment map is G -equivariant with respect to this actionand the coadjoint representation on g ˚ , and satisfies the following equation: ι X η ω “ d x µ, η y , for each η P g , where X η stands for the vector field on M induced by theinfinitesimal action, i.e. X η p m q “ ddt | t “ p e tη m q . In other words, X η is the symplectic gradient of the function x µ, η y .A right action will be said Hamiltonian with moment µ if the associatedleft action is Hamiltonian with moment ´ µ . Remark . If G is connected, the moment map determines the action. If G is discrete, a Hamiltonian action is just an action by symplectomorphisms. Definition 3.6.
Weinstein observed in [Wei81] that the data of both theaction and the moment map can be conveniently packaged as a Lagrangiansubmanifold Λ G p M q Ă T ˚ G ˆ M ´ ˆ M , defined as: Λ G p M q “ tpp q, p q , m, m q : m “ q.m, R ˚ g ´ p “ µ p m qu . When M “ T ˚ X is a cotangent bundle and the action and the moment arethe ones canonically induces from a smooth action on the base X , Λ G p M q corresponds to the conormal bundle of the graph of the action Γ G p X q Ă G ˆ X ˆ X, where one identifies T ˚ X with p T ˚ X q ´ via p q, p q ÞÑ p q, ´ p q . Definition 3.7. (Symplectic quotient) If p M, ω, µ q is a Hamiltonian mani-fold, its symplectic quotient (or reduction ) is defined as M {{ G “ µ ´ p q{ G. When is a regular value for µ , and G acts freely and properly on µ ´ p q , M {{ G is also a symplectic manifold. In this case, we will say that the actionis regular . Definition 3.8. (Canonical Lagrangian correspondence between M and M {{ G )If the action is regular in the sense of definition 3.8, the image of the map ι ˆ π : µ ´ p q Ñ M ´ ˆ M {{ G is a Lagrangian correspondence, where ι and π stand respectively for the inclusion and the projection. Remark . (Induced action on the quotient by a normal subgroup) If p M, ω, µ q is a Hamiltonian G -manifold, and if H Ă G is a normal subgroup,then M is in particular a Hamiltonian H -manifold, with moment map ob-tained by composing µ with the dual of the inclusion of the Lie algebras. Iffurthermore the H -action is regular as in definition 3.7, then M {{ H carriesa residual Hamiltonian action of G { H .In particular, if M is a symplectic manifold endowed with two commutingHamiltonian actions of two groups G and G (that is, a Hamiltonian G ˆ G -action), and if the action of G is regular, then M {{ G is a Hamiltonian G -manifold. Remark . (Action on M ´ ) If p M, ω q is a G -Hamiltonian manifold withmoment map µ , then M ´ “ p M, ´ ω q endowed with the same action is alsoHamiltonian, with moment map ´ µ . Definition 3.11. ( G -Lagrangian) A G -Lagrangian of a Hamiltonian G -manifold M is a Lagrangian submanifold L Ă M that is both containedin the zero level µ ´ p q , and G -invariant. When the G -action is regular, the G -Lagrangians of M are in one-to-one correspondence with the Lagrangianson M {{ G (Though a Lagrangian in M need not be a G -Lagrangian to inducea Lagrangian on M {{ G ).We now define H am . Definition 3.12. (The partial 2-category H am ) The following constructiondefines a partial 2-category: ‚ Objects are real Lie groups, the opposite map sends G to G op (withthe opposite group structure). ‚ The simple 1-morphisms from G to G are the symplectic manifoldsendowed with commuting Hamiltonian left G -action and right G -action, with respective moment maps µ G and µ G . Equivalently, aHamiltonian left p G ˆ G q -action with moment map µ “ p µ G , ´ µ G q : M Ñ Lie p G q ˚ ˆ Lie p G q ˚ . The moment maps are part of the data. ‚ The opposite identification shom p G, H q Ñ shom p G op , H op q is givenby taking the same symplectic manifold, and acting through g ´ in-stead of g . ‚ The adjunction identification shom p G, H q Ñ shom p H, G q is givenby reversing the symplectic structure and acting through g ´ insteadof g . ‚ Horizontal composition of simple 1-morphisms is defined as a diago-nal symplectic reduction of the product. Let two simple 1-morphisms M P shom p G , G q and M P shom p G , G q : G M (cid:47) (cid:47) G M (cid:47) (cid:47) G . Endow M ˆ M with the diagonal action of G g ¨ p m, m q “p mg ´ , gm q , which is Hamiltonian with respect to the moment map µ diagG p m , m q “ ´ µ G p m q ` µ G p m q . We will say that M and M are composable if this action isregular. In this case, we define the horizontal composition of M and M as the symplectic reduction of M ˆ M for this action,and denote it M ˆ {{ G M “ p µ diagG q ´ p q{ G , or sometimes M ˆ {{ M when G is implicit.It is endowed with its reduced symplectic structure, and the twoactions of G and G (and their moment maps) descend to the quo-tient: M ˆ {{ G M is then a simple morphism from G to G .Notice that this construction depends on the moment maps, andnot just on the Hamiltonian actions. ‚ Simple 2-morphisms: with M “ ˜ G M (cid:47) (cid:47) G M (cid:47) (cid:47) ¨ ¨ ¨ M p k ´ q k (cid:47) (cid:47) G ¸ , and N “ ˜ G N (cid:47) (cid:47) H N (cid:47) (cid:47) ¨ ¨ ¨ N p l ´ q l (cid:47) (cid:47) G ¸ , we define shom p M , N q as the set of K -Lagrangians of P , where P “ ź i M ´ i p i ` q ˆ ź j N j p j ` q , and K “ G ˆ ˜ k ´ ź i “ G i ¸ ˆ G ˆ ˜ l ´ ź j “ H j ¸ , where each factor of K acts diagonally with the diagonal momentmap on the two symplectic manifolds associated to it (if the sym-plectic manifold comes with its opposite symplectic form, one takesthe opposite moment map, in accordance with remark 3.10). We willcall such submanifolds generalized Lagrangian correspondence . ‚ The cyclicity isomorphisms are the obvious ones, as for the oppositesand adjunction of 2-morphisms. ‚ The identification 2-morphisms I G p M , M q P shom pp M , M q , M ˆ {{ G M q are defined as the canonical Lagrangian correspondence between M ˆ M and its reduction, defined in definition 3.8. ‚ Vertical composition of simple 2-morphisms is defined as composi-tion of Lagrangian correspondences. Let
M , N , P be in hom p G, G q , L P shom p M , N q and L P shom p N , P q . Say that L and L are composable if L ˆ ś P and ś M ˆ L intersect transversely in ś M ˆ ś N ˆ ś P , and if the projection of this intersection to ś M ˆ ś P is an embedding. If this is the case, the vertical com-position is defined as being the image of this embedding, and is anelement in shom p M , P q . Remark . (Relation between L ie R and H am ) In [Wei10, Sec. 5], We-instein shows that there is a functor from the category of manifolds andsmooth maps to his symplectic category. One can extend such a functor to L ie R : there is a "partial 2-functor" L ie R Ñ H am . In short, it is defined atthe level of simple morphisms by sending a group to itself, a 1-morphism toits cotangent bundle, and a 2-morphism to its conormal bundle: G ÞÑ G,M ÞÑ T ˚ M,C ÞÑ N ˚ C M. We will refer to it as the cotangent 2-functor.Indeed, a smooth action of G on M lifts to an action on T ˚ M defined by g ¨ p q, p q “ p gq, p ˝ p D q L g q ´ q , with g P G , p q, p q P T ˚ M .It follows from Cartan’s formula that this action is Hamiltonian, withmoment map µ : T ˚ M Ñ g ˚ defined by µ p q, p q ¨ ξ “ p p X ξ p q qq . Moreover, the horizontal compositions of 1-morphisms are compatible: iftwo simple 1-morphisms G M (cid:47) (cid:47) G M (cid:47) (cid:47) G are composable in L ie R , then the same holds in H am for G T ˚ M (cid:47) (cid:47) G T ˚ M (cid:47) (cid:47) G , one has T ˚ p M ˆ G M q “ T ˚ M ˆ {{ G T ˚ M , and the conormal bundleof the identification 2-morphism I G p M , M q is I G p T ˚ M , T ˚ M q .If C Ă M ˆ N is a smooth correspondence, then its conormal bundle is aLagangian submanifold N ˚ C M Ă T ˚ M ˆ T ˚ N . But T ˚ M is symplectomor-phic to its opposite symplectic structure via the map p q, p q ÞÑ p q, ´ p q , soapplying this identification, one can think of it as a Lagrangian correspon-dence N ˚ C Ă p T ˚ M q ´ ˆ T ˚ N .If moreover C is invariant for a G -action, then N ˚ C will be a G -Lagrangian.If two simple 2-morphisms are vertically composable in L ie R , then the sameis true for their conormal bundles, and the conormal bundle of their compo-sition agrees with the composition of the conormal bundles. Remark . Following the arborealization program of Nadler [Nad17], itis maybe possible to extend the class of 1-morphisms in L ie R to manifoldswith certain type of singularities (with extra local structure), and extend thecotangent functor by sending a singular manifold to a Weinstein manifoldwith Lagrangian skeleton being the singular manifold. Remark . We can use this functor to transport structure from L ie R to H am , such as identities, products and coproducts, units and co-units ... thatwould lead to a p ` q -theory taking values in H am . The 2-functor we willconstruct in section 5, alghough close to, will not correspond to that. Remark . (Identities) For any Lie group G , its cotangent bundle T ˚ G ,endowed with left and right pullbacks, plays the role of an identity (i.e. canbe composed with any other morphism, and the composition is the samemorphism). It is the image of the identity of G in L ie R by the cotangentfunctor.If M : G Ñ G is a simple 1-morphism, the diagonal ∆ M P shom p M, M q plays the role of an identity. However, for a general 1-morphism representa-tive M “ p M , M , ¨ ¨ ¨ , M k q P hom p G, G q , the product of the diagonal ofall its factors is not an identity, since it is not an element in shom p M , M q ,see remark 3.17.One can nevertheless find length two identity representatives as follows:let x M “ p M , T ˚ G , M , T ˚ G , ¨ ¨ ¨ , T ˚ G k ´ , M k q P hom p G, G q , then Weinstein’s correspondence (see definition 3.6) for the action of K “ G ˆ ¨ ¨ ¨ ˆ G k ´ on ś M defines a simple 2-morphism Λ p ź M q P shom p M , x M q , and ´ Λ K p ź M q , Λ K p ź M q T ¯ P hom p M , M q is an identity for M . Remark . (Horizontal composition of simple 2-morphisms) Let M , M P hom p G, G q ,N , N P hom p G , G q ,L P shom p M , M q , and L P shom p N , N q . One could be tempted to define the horizontal compositon of L and L astheir cartesian product. Unfortunately it is generally not an element in shom p M h N , M h N q . For a counter-example, consider p M, N q : G Ñ G Ñ G , the product of the diagonals ∆ M ˆ ∆ N is not in the zero level of the momentmap of the diagonal action of G . A natural candidate for the identity 2-morphism of p M, N q would rather be the subset of M ´ ˆ M ˆ N ´ ˆ N of elements p m, m , n, n q such that µ MG p m q ` µ NG p n q “ µ MG p m q ` µ NG p n q and that p m, n q and p m , n q lie in the same orbit. This is an identity ifthe diagonal action of G is free, but is singular in general. For example if G “ U p q , M is a point, and N “ C , acted on by U p q by rotations withmoment map µ p z q “ | z | , this set consists in a single point. Remark . (Relation between L ie C and H am ) Geometric invariant theorysuggests a correspondence between (a variation of) L ie C and H am . If M isa Hamiltonian G -manifold and J is a G -invariant almost complex structure,then under some conditions one can extend the action to an action by thecomplexification G C , and the Kempf-Ness theorem says that the symplecticquotient M {{ G agrees with the GIT quotient of M by G C . This suggests acorrespondence: H am ÐÑ L ie C G ÐÑ G C M ÐÑ ML ÐÑ L, that should preserve the various operations. It would be interesting to defineand study this correspondence in more detail. Remark . (Endomorphism category of the trivial group) The endomor-phism category of the trivial group in H am , End H am p e q is similar withWehrheim and Woodward’s category S ymp , more precisely one has a func-tor from S ymp to End H am p e q . However we should point out that this functoris not surjective, since there are 1-morphisms e Ñ G Ñ G Ñ ¨ ¨ ¨ Ñ e thatcannot be simplified to a length 1 sequence. For instance, the character vari-ety of a closed surface Σ is not an object of S ymp due to its singular nature,however it can be represented by an object in End H am p e q using extendedmoduli spaces, as we shall see in section 4. However, this functor is injective faithful, since if a sequence of simple 1-morphisms (resp. a diagram of sim-ple 2-morphisms) in H am can be simplified to a length 1 sequence (resp. adiagram with a single correspondence), then the resulting 1-morphism (resp.2-morphism) is uniquely determined and given by the quotient of all inter-mediate groups.3.3. Proof of the diagram axiom.
We now prove that L ie R and H am both satisfy the diagram axiom, which implies that these are partial 2-categories, and allows one to define their strictifications.The idea of proof is similar for both categories: we use the fact that eventhough compositions are only partially defined in these categories, these canalways be defined in a set theoretical way: even if a group action is not free,one can always define the quotient set, likewise one can always define thereduction set of a Hamiltonian action. We will first prove the axiom at theset level, and then we will "lift" the statement to the initial framework, usingthe correspondence between simple 2-morphisms and subsets of a (possiblysingular) quotient set.Recall the setting: let ϕ , ϕ , ... , ϕ k “ ϕ be a sequence of representativesof general 1-morphisms in hom p x, y q , such that for any i , ϕ i ` is either acomposition or a decomposition of ϕ i . To such a sequence is associated adiagram D by patching altogether all the identification 2-morphisms. Weaim at proving that the diagram D is an identity for ϕ .For any i , let ˜ ϕ i stand for the set obtained by forcing all the compositionsappearing in ϕ i : ‚ In L ie R , this is the quotient of the product of all the one morphismsappearing in ϕ i , modulo the diagonal actions of all the groups ap-pearing in the sequence, except the first and the last. ‚ In H am , this is the "symplectic quotient" of the product of all theone morphisms appearing in ϕ i , modulo the diagonal actions of allthe groups appearing in the sequence, except the first and the last,namely the quotient set of the zero level of the associated momentmap.Since we assumed that ϕ i ` is either a composition or a decomposition of ϕ i , it follows that all the ˜ ϕ i are equal to the same set, which we will denote ˜ ϕ . Let L P shom p ψ, ϕ q , for some ψ P hom p x, y q . By composing it suc-cessively with all the identification morphisms (or their adjoint), we get foreach i , a simple 2-morphism L i P shom p ψ, ϕ i q , and we would like to provethat L “ L k .For any i , Let ˜ L i be the subset of of the product of all the 1-morphisms ap-pearing in ψ and ˜ ϕ i , that corresponds to the quotient of L i (loosely speaking, ˜ L i P “ shom p ψ, ˜ ϕ i q ” ). By construction, and with the identification of the ˜ ϕ i in mind, all these correspond to the same subset, in particular ˜ L “ ˜ L k .Now, in either L ie R or H am , two simple morphisms in shom p ψ, ϕ q areequal if and only if the corresponding subsets of `ś ψ ˘ ˆ ˜ ϕ are equal.Indeed, an invariant subset is determined by its quotient in the orbit space.We therefore have L “ L k . As L k “ L ˝ D , we just proved that D is an identity for left composition. One can similarly show that for rightcomposition. This completes the proof. l Moduli spaces of connections
Throughout this section we identify SU p q with the group of unit quater-nions, and SO p q with its quotient by Z . Their common Lie algebra g “ su p q then corresponds to the space of pure quaternions (with zeroreal part) and is equipped with the standard bi-invariant inner product of H , which we use to identify g with its dual g ˚ . We will denote B g p π q Ă g the open ball of radius π , wich is sent injectively to SU p qzt´ u by theexponential map.4.1. Moduli space of a closed 1-manifold.
Let C be a circle with a basepoint. One can associate to it the moduli space G p C q of framed SU p q -connexions, i.e. the quotient of the space of connexions modulo gauge trans-formations that do not act on the base point. It is identified with the rep-resentation variety of its fundamental group, namely Hom p π p C q , SU p qq .Notice that since C has an abelian fundamental group, a choice of a differentbasepoint yields a canonically isomorphic space.If C is oriented, its orientation furnishes a preferred generator of π p C q ,which permits to identify G p C q with SU p q and therefore endows it witha group structure. If one reverse the orientation, the new generator corre-sponds to the inverse of the old one, and the identification with SU p q differby the inverse map. Therefore the group structure is changed to its opposite: G p´ C q “ G p C q op . The group structure constructed can also be defined more topologically:given two elements of G p C q , with two connexions A and A representingthem, over bundles P and P . One can cut P and P along the fiber overthe base point of C , and glue them back as in figure 1 to form a bundleover the gluig of the two segments, which one can identify with C , with thebasepoint chosen as in the picture. ˚ A A Figure 1.
Group structure on G p C q . If now C “ C \¨ ¨ ¨\ C k is a disconnected closed oriented 1-manifold, picka basepoint on each component, and define the group G p C q as the productof the G p C i q ’s.4.2. Moduli space of a surface with boundary.Definition 4.1. (Extended moduli spaces)
Let Σ be a compact orientedsurface with boundary B Σ “ B Σ \ ¨ ¨ ¨ \ B k Σ , where the B i Σ stand for the connected components of B Σ . ‚ (Extended moduli space associated to a surface, [Jef94, Def. 2.1])Define the following space of flat connexions: A g F p Σ q “ t A P Ω p Σ q b su p q | F A “ , A | ν B i Σ “ θ i ds u , where ν B Σ is a non-fixed tubular neighborhood of B Σ , s the param-eter of R { Z , and θ i P g is a constant element. The group G c p Σ q “ (cid:32) u : Σ Ñ SU p q | u | ν B Σ “ ( acts by gauge transformations on A g F p Σ q .The extended moduli space is then defined as the quotient M g p Σ , p q “ A g F p Σ q{ G c p Σ q , This space carries a closed 2-form ω defined by: ω r A s pr α s , r β sq “ ż Σ x α ^ β y , with r A s P M g p Σ , p q and α, β representing tangent vectors at r A s of M g p Σ , p q , namely d A -closed su p q -valued 1-forms, of the form η i ds near B i Σ .Furthermore it has a Hamiltonian SU p q -action, whose momentmap is given by the elements θ i P su p q such that A | ν B i Σ “ θ i ds . ‚ Denote N p Σ , p q the subset of M g p Σ , p q consisting in equivalenceclasses of connections for which | θ i | ă π . The form ω is symplectic on N p Σ , p q by [Jef94, Prop. 3.1] (the proposition is stated and proved inthe case when Σ has connected boundary, but the same proof appliesto any number of components). Remark . The moduli space N p Σ , p q is noncompact. In [MW12], Manolescuand Woodward define Floer homology inside another moduli space N c p Σ , p q ,which is a compactification of N p Σ , p q by symplectic cutting. We will ignorethis compactification in this paper, as one can think of it just as a technicalstep in proving that Floer homology is well-defined inside N p Σ , p q . Remark . With the identification of SU p q with G p C q in mind, one canthink of the group action in a more topological way: Let r A s be in N p Σ q ,and g P G p C q , represented by a connection A g on the circle. Take a pair ofpants P with an embedded trivalent graph as in figure 2. Let C , C and C denote its boundary components. Choose a trivial bundle P over P , andequip it with a flat connexion A P that corresponds to A |B Σ on C , and to A g on C . Gluing P to Σ and cutting along the trivalent graph yields a newconnection on Σ that corresponds to g. r A s . P Σ A g ÓÓ Figure 2.
Group action on N p Σ q . Remark . (Explicit descrition, [Jef94, Sec. 6.2]) Holonomies provide anexplicit description of N p Σ , p q . Assume Σ is connected. Each boundarycomponent B i Σ has a basepoint z i corresponding to the image of P R { Z bythe parametrizations. Let γ , ..., γ k be disjoin arcs connecting z to z , ..., z k , and let α , β , ..., α g , β g be loops based at z that form a symplectic basisof the fundamental group of Σ zt γ , ¨ ¨ ¨ , γ k u . With A i , B i and Γ i denotingthe holonomy of a connexion along α i , β i and γ i respectively, and θ i P B g p π q the value at B i Σ , N p Σ , p q can be identified with the space of tuples p θ , ¨ ¨ ¨ , θ k , Γ , ¨ ¨ ¨ , Γ k , A , B ¨ ¨ ¨ , A g , B g q P p B g p π qq k ˆ SU p q k ´ ` g that satisfy the relation e θ ´ Γ e θ Γ ´ ¯ ¨ ¨ ¨ ´ Γ k e θ k Γ ´ k ¯ r A , B s ¨ ¨ ¨ r A g , B g s “ . And since θ P B g p π q , it is uniquely determined by the other elements andthe above relation, so N p Σ , p q can be identified with the open (and hencesmooth) subset of elements p θ , ¨ ¨ ¨ , θ k , Γ , ¨ ¨ ¨ , Γ k , A , B ¨ ¨ ¨ , A g , B g q P p B g p π qq k ´ ˆ SU p q k ´ ` g satisfying ´ Γ e θ Γ ´ ¯ ¨ ¨ ¨ ´ Γ k e θ k Γ ´ k ¯ r A , B s ¨ ¨ ¨ r A g , B g s ‰ ´ . Proposition 4.5. (Gluing almost equals reduction) Let Σ P hom p C , C q and S P hom p C , C q be such that Σ , S and Σ Y C S have no closed com-ponents. Then one has a natural symplectomorphism N p Σ Y C S qz C “ N p Σ q ˆ {{ G p C q N p S q , where C is the subset of connexions whose holonomy around C equals ´ .Moreover, C is a union of codimension 3 coisotropic submanifolds of N p Σ Y C S q . Proof. If A Σ and A S denote two connexions on Σ and S respectively thatcoincide on C they can be glued together to a connexion on Σ Y C S . Thisdefines a gluing map p µ diagG p C q q ´ p q Ă N p Σ q ˆ N p S q Ñ N p Σ Y C S q that passes to the quotient for the diagonal G p C q -action. One can seefrom the explicit description of the moduli spaces that the induced map isinjective, and that its image is the complement of C , since the holonomiesaround C correspond to the exponential of the θ i values, which live in B g p π q .Finally, this map preserves the symplectic forms, as both are defined in ananalogous way, by integrating the forms on Σ \ S and Σ Y C S . (cid:3) Construction of the functor
We now define a z H am -valued p ` ` q -field theory. If a true “gluingequals reduction” principle would hold, i.e. if there was no submanifold C appearing in proposition 4.5, we would obtain a 2-functor from C ob ` ` to z H am . Instead, we will obtain what we will call a quasi 2-functor: thesource category will instead be C ob elem ` ` (see definition 5.6), and consist incobordisms equipped with decompositions. We then define an equivalencerelation on morphism spaces of z H am , and define a quasi-functor to be afunctor from C ob elem ` ` such that a (2-)cobordism endowed with two differentdecompositions will result in two equivalent morphisms in z H am . Remark . In future work, we expect to promote z H am to a (sort of) 3-category, using equivariant Floer homology as 3-morphisms spaces. Such aconstruction should permit to define a linearization 2-functor L : z H am Ñ C that would land in a more algebraic 2-category (such as rings, A -algebraswith ring actions, A -modules). We expect that the codimension 3 subman-ifolds C of proposition 4.5 should be invisible to equivariant Floer homology.This expectation, together with the observation in remark 5.9, shouldimply that the composition of the functor with this linearization shoulddescend to C ob ` ` : C ob elem ` ` (cid:15) (cid:15) (cid:47) (cid:47) z H am L (cid:15) (cid:15) C ob ` ` (cid:47) (cid:47) C . Ultimately, one would also hope to extend such a 2-functor to a 3-functor.5.1.
Cobordisms and decompositions to elementary pieces.Definition 5.2.
Let C ob ` ` stand for the weak 2-category whose: ‚ Objects are oriented closed one-manifolds, endowed with an orien-tation preserving parametrization by R { Z of each connected compo-nent. ‚ ‚ Σ and S two 1-morphisms from C to C ,a 2-morphism from Σ to S is represented by a compact oriented 3-manifold with boundary Y , with a diffeomorphisms between B Y and p´ Σ q Y C Y C S . And two such Y are identified if there is a diffeo-morphism between them that is compatible with the identificationsof the boundaries.All the compositions are given by glueing.We now define elementary morphisms. We use a slightly wider class thanthe usual notion of having a function with at most one critical point, as forexample in [WW16]. Definition 5.3. (Elementary morphisms of C ob ` ` ).Objects. All objects are said to be elementary, including the empty set.1-morphisms. A 1-morphism is elementary if it has no closed components.This also includes the empty set.Strictly speaking, elementary 2-morphisms will not be 2-morphisms be-tween 1-morphisms of C ob ` ` , but rather between sequences of elementary1-morphisms, i.e. 1-morphisms with a given decomposition into elementary1-morphisms.2-morphisms. let C , C be two objects, and Σ , S be 1-morphisms from C to C , endowed with decompositions Σ “ p Σ , ¨ ¨ ¨ , Σ k q and S “ p S , ¨ ¨ ¨ , S l q into elementary 1-morphisms (i.e. with no closed component). An elemen-tary 2-morphism Y from Σ to S is either: ‚ A compression body : Y is a compression body if it admits a Morsefunction f : p Y zp C Y C qq Ñ R that is minimal on Σ , maximal on S , vertical near C and C (meaning that, for an identification ofa neighborhood of C i in Y zp C Y C with C i ˆ r , s ˆ p , s , with Σ and S corresponding to C i ˆ t , u ˆ p , s , f corresponds to theprojection to r , s ), and admits (or not) critical points that all havethe same index, 1 or 2. Moreover, for some pseudo-grandient of f ,the decomposition Σ of Σ flows down to the decomposition S of S .This definition includes the empty set. ‚ A 0-handle or a 3-handle attachment, i.e. Y is a 3-ball. ‚ A circle(s) insertion/removal. We assume that Y is trivial, i.e. itadmits a Morse function f : p Y zp C Y C qq Ñ R that is minimalon Σ , maximal on S , vertical near C and C , and with no criticalpoint. Furthermore we assume that there exists a pseudo-gradientfor f that matches the two decompositions, except for one circle (i.e. Σ contains one circle more/less than S ).Elementary morphisms generate C ob ` ` in the following sense: Proposition 5.4. (Existence of decompositions) (1)
Every 1-morphism in C ob ` ` decomposes as a sequence of elemen-tary ones. Moreover, one can find such a sequence of length 2. (2) Every 2-morphism in C ob ` ` decomposes as a sequence of ele-mentary 2-morphism between sequences of elementary 1-morphisms,Moreover such a sequence can also be chosen to have length 2. Proof. (1) If a 1-morphism Σ has closed components, one can just inserta separating curve in each of them, this splits the surface into twoelementary 1-morphisms.(2) This follows from the fact that one can find self-indexed Morse func-tions (cid:3) It follows from Cerf theory that decompositions of morphisms of C ob ` ` are unique in the following sense: Proposition 5.5. (Uniqueness of decompositions) Two decompositions of amorphism of C ob ` ` into elementary morphisms can be related by a se-quence of moves of the following type.For 1-morphisms: ‚ (Circle(s) insertion/removal) Replacing a length k sequence Σ “ p Σ , ¨ ¨ ¨ , Σ i , ¨ ¨ ¨ , Σ k q by a length k ` one p Σ , ¨ ¨ ¨ , Σ i , Σ i , ¨ ¨ ¨ , Σ k q , where p Σ i , Σ i q areobtained from Σ i by inserting some separating circles. Circle removalis the opposite move.For 2-morphisms: Let Y “ p Y , ¨ ¨ ¨ , Y k q be a sequence of elementary 2-morphisms, with Y i from Σ i to Σ i ` . ‚ (Diffeomorphism equivalence) Let Y “ p Y , ¨ ¨ ¨ , Y k q be a sequenceof elementary 2-morphisms, with Y i from Σ i to Σ i ` , with a familyof diffeomorphism ϕ i : Y i Ñ Y i such that ϕ i and ϕ i ` coincide on Σ i ` and send the decomposition Σ i ` to Σ i ` . A diffeomorphismequivalence consists in replacing Y by Y . ‚ (Cylinder creation/cancellation) Let Σ “ p Σ , ¨ ¨ ¨ , Σ k q be a sequenceof elementary 1-morphisms, with total space Σ “ Σ Y ¨ ¨ ¨ Y Σ k . By acylinder from Σ to itself we mean a compression body with no criticalpoints. A cylinder creation corresponds to inserting a cylinder at Σ i ,for some i . A cylinder cancellation is the opposite move. ‚ (Circle(s) insertion/removal) A circle insertion corresponds to keep-ing the same 3-manifolds Y i , and performing a circle insertion onone of the Σ i ’s. A circle removal is the opposite move. ‚ (Imbrication of compression bodies) Assume that for some i , Y i and Y i ` are compression bodies of the same index, then their union Y i Y Σ i ` Y i ` is again a compression body. The move is to replace Y by p Y , ¨ ¨ ¨ , Y i Y Σ i ` Y i ` , ¨ ¨ ¨ , Y k q . ‚ (Critical point switches) Assume that for some i , Y i and Y i ` cor-respond to handle attachments along disjoint attaching spheres. Themove is to replace Y by p Y , ¨ ¨ ¨ , r Y i , r Y i ` , ¨ ¨ ¨ , Y k q , where r Y i corresponds to attaching the handles of Y i ` first, and r Y i ` corresponds to attaching the handles of Y i afterwards. ‚ (Index 0-1 (or 2-3) handle creation/cancellation). Assume that forsome i , Y i is a -handle attachment, with Σ i “ p Σ i , Σ i , ¨ ¨ ¨ , Σ ki q , and Σ i ` “ p D , D , Σ i , Σ i , ¨ ¨ ¨ , Σ ki q ,where D and D are two 1-discs such that their union is the 2-sphere bounding the 0-handle. Assume also that Y i ` corresponds toa 1-handle attachment connecting D and Σ i , so that Σ i ` “ p D , D Σ i , Σ i , ¨ ¨ ¨ , Σ ki q . Assume finally that Y i ` corresponds to the removal of the circle be-tween D and D , so that Σ i ` “ p D Y p D Σ i q , Σ i , ¨ ¨ ¨ , Σ ki q . The cancellation move is to replace p Y i , Y i ` , Y i ` q by the cylinder p Y i Y Y i ` Y Y i ` q . The creation move corresponds to the oppositemove. The similar moves for 2-3 handles corresponds to the reversedcobordisms p Y i ` , Y i ` , Y i q . ‚ (Index 1 and 2 handle creation/cancellation) If now for some i , Y i and Y i ` are either compression bodies of index 1 and 2 respectively,and such that their union Y i Y Σ i ` Y i ` is a cylinder. The move isto replace Y by p Y , ¨ ¨ ¨ , Y i Y Σ i ` Y i ` , ¨ ¨ ¨ , Y k q . Proof. Σ and Σ be two elementary decompositions of agiven 1-morphism Σ . On each closed component of Σ , pick any separatingcircle disjoint from the cicles of the decompositions Σ and Σ : call C thecollection of these circles. One can go from Σ to Σ by ‚ first adding the circles C , ‚ removing all the circles corresponding to Σ , ‚ adding all the circles corresponding to Σ , ‚ removing the circles C .2-morphisms: The statement about the pieces Y i , ignoring the decom-positions of the level surfaces Σ i , is standard Cerf theory, see for example[WW16, Th. 2.2.11]. To get the correct decompositions Σ i , one can theninsert cylinders and run the same method as above for 1-morphisms. (cid:3) We now define a 2-category C ob elem ` ` , where we keep track of decomposi-tions: Definition 5.6. ( C ob elem ` ` ) Let C ob elem ` ` stand for the 2-category whose: ‚ Objects are the same as in C ob ` ` , ‚ One-morphisms consist in sequences Σ “ p Σ , ¨ ¨ ¨ , Σ k q of elementary 1-morphisms (or equivalently 1-morphism of C ob ` ` endowed with a decomposition into elementary morphisms). ‚ Σ to Σ consist in sequences Y “ p Y , ¨ ¨ ¨ , Y k q , where Y i is an elementary 2-morphism from Σ i to Σ i ` , with Σ “ Σ and Σ k ` “ Σ . ‚ Compositions are given by concatenation.5.2.
An equivalence relation on z H am . Recall that the moduli space as-sociated to the gluing of two surfaces Σ and S does not exactly correspondto the composition N p Σ q ˆ {{ N p S q , due to the presence of the subset C in proposition 4.5. Therefore the construction of section 4 does not definea functor with values in z H am , but rather a “quasi 2-functor” modulo theequivalence relation of definition 5.8 below. We will mod out the 1-morphismspaces by identifying Hamiltonian manifolds up to codimension 3 subman-ifolds. Care must be taken however at the level of 2-morphisms, since theLagrangian correspondences we will be considering can always be containedin such codimension 3 submanifolds. In order to define a nontrivial func-tor, one has to take this fact into account, this is the reason for which weintroduce the weak transversality assumption. Definition 5.7.
Let A and B be two subsets of a topological space X . Wesay that A intersects B in a weakly transverse way if A “ p A z B q . Definition 5.8.
Define the following equivalence relation on z H am : ‚ Objects are equivalent if and only if they are equal. ‚ The equivalence relation on hom p G, G q is generated by the fol-lowing identifications: let M “ p M , ¨ ¨ ¨ , M k q P hom p G, G q be arepresentative of a 1-morphism, and let C Ă M i be a codimension 3submanifold. Then we identify M with p M , ¨ ¨ ¨ , M i z C, ¨ ¨ ¨ , M k q . ‚ On hom p α, β q : suppose that for some representatives M and N of α and β respectively, we have a diagram D P hom p M , N q . Let M be some 1-morphism decorating an edge in D , and C Ă M be a(coisotropic) codimension 3 submanifold such that for any face in D ,decorated by a 2-morphism L , which by cyclicity we can think of asan element in shom p P , M q . Suppose then that inside p ś P q ˆ M ,the intersection of L with p ś P qˆ C is weakly transverse, in the senseof definition 5.7. If that transversality assumption holds for any L adjacent to M , then we declare to be equivalent the 2-morphisms of hom p α, β q associated with D and D , the new diagram built from D by replacing M by M z C , any adjacent L by L z pp ś P q ˆ C q , andleaving the rest of the diagram unchanged. Remark . In the setting described above, if one is given the diagram D and M , then one can recover the diagram D : any adjacent L corresponds tothe closure of L z pp ś P q ˆ C q inside p ś P q ˆ M .5.3. Construction on elementary morphisms.
We now define the quasi2-functor C ob elem ` ` Ñ z H am. To a 0-morphism C , we associate the group G p C q defined in section 4.1.To an elementary 1-morphism Σ , we associate the moduli space N p Σ q defined in section 4.2 (4.2).If now Σ “ p Σ , ¨ ¨ ¨ , Σ k q is a sequence of elementary 1-morphisms Σ “ C (cid:47) (cid:47) C (cid:47) (cid:47) ¨ ¨ ¨ Σ k (cid:47) (cid:47) C k , we associate to it the corresponding sequence M “ p N p Σ q , ¨ ¨ ¨ , N p Σ k qq . Lemma 5.10.
As a 1-morphism of z H am , the sequence p N p Σ q , ¨ ¨ ¨ , N p Σ k qq is independent on the choice of parametrizations of the intermediate 1-manifolds C , ..., C k ´ .Proof. Let p i , p i : R { Z \ ¨ ¨ ¨ \ R { Z Ñ C i be two parametrizations of C i , with ď i ď k ´ . Insert a cylinder r , s ˆ C i between Σ i and Σ i ` , andparametrize t u ˆ C i and t u ˆ C i by p i and p i respectively, so to have asequence ¨ ¨ ¨ Σ i (cid:47) (cid:47) p C i , p i q C i ˆr , s (cid:47) (cid:47) p C i , p i q Σ i ` (cid:47) (cid:47) ¨ ¨ ¨ and an associated sequence in z H am : ¨ ¨ ¨ N p Σ i ,p i q (cid:47) (cid:47) G p C i , p i q N p C i ˆr , s ,p i ,p i q (cid:47) (cid:47) G p C i , p i q N p Σ i ` ,p i q (cid:47) (cid:47) ¨ ¨ ¨ composing respectively at G p C, p q and G p C, p q , we obtain that the twosequences ¨ ¨ ¨ N p Σ i ,p i q (cid:47) (cid:47) G p C i , p i q N p Σ i ` ,p i q (cid:47) (cid:47) ¨ ¨ ¨¨ ¨ ¨ N p Σ i ,p i q (cid:47) (cid:47) G p C i , p i q N p Σ i ` ,p i q (cid:47) (cid:47) ¨ ¨ ¨ define the same morphism of z H am . (cid:3) To an elementary 2-morphism we will associate a diagram of Lagrangiancorrespondences. Assume first that Y is a compact 3-manifold with boundary B Y and codimension 2 corner B Y “ C \ ¨ ¨ ¨ \ C k Ă B Y, and let Σ be the compact surface with boundary obtained by cutting B Y along B Y . We parametrize each circle C i in B Y , and take the inducedparametrization of B Σ . Definition 5.11.
Let Y and Σ be as above, ‚ (Moduli space associated to Y ) Let N p Y q “ A F p Y q{ G c p Y q , where A F p Y q “ (cid:32) A P Ω p Y q b g | F A “ , A | νC i “ θ i ds ( , with θ i P B g p π q , νC i a non-fixed neighborhood of C i , and s P R { Z the parameter of C i ; and G c p Y q “ (cid:32) u : Y Ñ SU p q | u | νC i “ ( . ‚ (Correspondence associated to Y ) Let L p Y q Ă N p Σ q denote theimage of N p Y q by the restriction map to Σ . For arbitrary Y , L p Y q might not be smooth. However, when this is thecase, Stokes formula implies that the map N p Y q Ñ N p Σ q is Lagrangian.Indeed, tangent vectors at r A s of N p Y q can be represented by classes of g -valued 1-forms that are d A -closed, and locally constant near B Y . For twosuch forms α, β , ω pr α s , r β sq “ ż Σ x α ^ β y “ ż Y d x α ^ β y “ . We will see that L p Y q is a smooth embedded Lagrangian for 0-handle at-tachments, and for some compression bodies and circle insertions.0-handle attachments. In this case, Y is a 3-ball, and Σ consists in a disjointunion of punctured spheres Σ i . For each component, by the explicit descrip-tion in remark 4.4, N p Σ i q is identified with an open subset of B g p π q k i ´ ˆ SU p q k i ´ , where k i stands for the number of boundary components. Sinceall the boundary circles bound discs in Y , under the previous identification, L p Y q corresponds to the subset ź i t u ˆ SU p q k i ´ Ă ź i B g p π q k i ´ ˆ SU p q k i ´ , as the holonomy along the γ -curves joining the boundaries can take arbitraryvalues. It follows that L p Y q is smooth, and therefore Lagrangian.Circle insertion. Assume now that Y : Σ Ñ S corresponds to a circle inser-tion as defined in definition 5.6, and assume moreover that Σ has length two,and S length one, i.e. Σ “ C (cid:47) (cid:47) C (cid:47) (cid:47) C , and S “ C S (cid:47) (cid:47) C , with S » Σ Y C Σ . From proposition 4.5, we have the identification N p S qz C “ N p Σ q ˆ {{ G p C q N p Σ q , under which L p Y q corresponds to the identification 2-morphism I G p C q p N p Σ q , N p Σ qq , modulo the codimension 3 subset C . Indeed, if A and A are flat connexionson Σ and Σ that extend flatly to Y , then they must coincide on C , whichcorresponds to the condition of being in the zero level of the moment mapfor the diagonal G p C q -action. And since A is flat and Y z p C Y C q is atrivial cobordims, the restriction A | S must be gauge equivalent to the gluingof A and A on Σ Y C Σ . It follows from the explicit description of themoduli spaces that is a regular value of the moment maps (and thereforethe diagonal moment map), and that the G p C q -action is free. This implythat I G p C q p N p Σ q , N p Σ qq , and therefore L p Y q , are smooth.2-handle attachmnents. Assume that Y : Σ Ñ S corresponds to 2-handleattachments on Σ which we assume to be connected (but S may be discon-nected, yet with no closed components). We will show that L p Y q is inducedby a fibered coisotropic submanifold on N p Σ q , in the following sense: Definition 5.12.
A coisotropic submanifold C Ă M of a symplectic man-ifold p M, ω q is said to be fibered if its characteristic foliation T C K ω Ă T C corresponds to the vertical foliation ker dp of a fibration p : C Ñ B . In thiscase, ω induces a symplectic form on B , and L “ p i ˆ p qp C q Ă M ´ ˆ B is aLagrangian correspondence. We will say that a Lagrangian correspondence Λ Ă M ´ ˆ M is induced by C if for some symplectomorphism B » M , Λ corresponds to L .From the following proposition, it is enough to consider the case when Y is a single handle attachment. Proposition 5.13.
Let M , M , M be three symplectic manifolds, and L Ă M ´ ˆ M , L Ă M ´ ˆ M be Lagrangian correspondences inducedby fibered coisotropics C Ă M , and C Ă M . Then the composition of L with L is embedded, and is induced by the fibered coisotropic C “ p ´ p C q , where p : C Ñ M denotes the fibration.Proof. Let p x , x , x q P L ˆ M X M ˆ L , we have x P C , x “ p p x q ,and x “ p p x q , with p “ p ˝ p : C Ñ M . It follow by differentiating that p D x p i ˆ p qp T x C q ˆ T x M q X p T x M ˆ D x p i ˆ p qp T x C qq“ D x p i ˆ p ˆ p qp T x C q , and L ˝ L “ p i ˆ p qp C q . (cid:3) Assume now that Σ is connected, and Y : Σ ñ S corresponds to a single2-handle attachment. Assume first that the attaching circle is separatingin Σ , and cuts it in two surfaces Σ and Σ . Denote C , ..., C k and C ,..., C k the components of B Σ contained respectively in Σ and Σ . Byassumption, k , k ě , otherwise S would have a closed component. Eachof these circles have a basepoint, corresponding to the image of r s P R { Z .Pick disjoint embedded paths γ , ..., γ k and γ , ..., γ k joining the basepointof C (resp. C ) to the other boundary components. Connect also C and C by a path δ disjoint from the other curves, and meeting the attachingcircle at one point. Fix finally α i , β i , ..., α ig i , β ig i a symplectic basis of thefundamental group of Σ i z ` δ Y γ Y ¨ ¨ ¨ Y γ k ˘ based at (the basepoint of) C i . See figure 3. These curves flow down to analogous curves on S . Σ Σ Figure 3.
Handle attachment along a separating curve.
Let θ ij P B g p π q denoting the values of a connexion at C ij , and A ij , B ij , Γ ij the holonomies along the corresponding curves, ∆ the holonomy along δ ,and let Π “ g ź i “ r A i , B i s k ź i “ ad Γ i e θ i , Π “ g ź i “ r A i , B i s k ź i “ ad Γ i e θ i , corresponding to the holonomies along a loop going aroung the attachingcircle. The spaces N p Σ q and N p S q admit the following description: N p Σ q “ (cid:32)` A , B , ¨ ¨ ¨ , A g , B g , Γ , ¨ ¨ ¨ , Γ k , θ , ¨ ¨ ¨ , θ k ,A , B , ¨ ¨ ¨ , A g , B g , Γ , ¨ ¨ ¨ , Γ k , θ , θ , ¨ ¨ ¨ , θ k , ∆ ˘ | Π ∆Π e θ ∆ ´ ‰ ´ ) , N p S q “ (cid:32)` A , B , ¨ ¨ ¨ , A g , B g , Γ , ¨ ¨ ¨ , Γ k , θ , ¨ ¨ ¨ , θ k ,A , B , ¨ ¨ ¨ , A g , B g , Γ , ¨ ¨ ¨ , Γ k , θ , ¨ ¨ ¨ , θ k ˘ | Π ‰ ´ , Π ‰ ´ u , (we have dropped θ and ∆ in N p S q ). With C “ t Π “ u Ă N p Σ q , C isfibered over N p S q , where p : C Ñ N p S q is given by forgetting ∆ and θ (indeed, C » SU p q ˆ N p S q , where ∆ corresponds to the SU p q -factor),and L p Y q is the correspondence induced by C (which is coisotropic, since L p Y q is Lagrangian).The case when the attaching circle is nonseparating is similar, and can bedescribed in an analogous way by assuming that the attaching circle is oneof the α -curve, see [Caz16, sec. 5.2.1].If now Y : Σ ñ S is an arbitrary handle attachment, one can decompose itin single handle attachments. By proposition 5.13 and the above discussion, L p Y q is again induced by a fibered coisotropic, which corresponds to the setof connexions having trivial holonomies along the attaching circles. Remark . If Y is a more general circle insertion or handle attachment,i.e. the sequences Σ and S can be longer, we have not checked whether or not L p Y q is smooth, but a priori there can be issues such as in remark 3.17. Thisis the reason why in these cases we associate more complicated diagrams,containing only correspondences as above.General circle insertions. Assume that Y : Σ ñ S , with Σ “ p Σ , ¨ ¨ ¨ , Σ i , Σ i ` , ¨ ¨ ¨ , Σ k q , and S “ p Σ , ¨ ¨ ¨ , Σ i Y Σ i ` , ¨ ¨ ¨ , Σ k q , corresponds to removing the circle between Σ i and Σ i ` . To Y we associatethe diagram D p Y q : .. ¨ ¨ ¨ . . ¨ ¨ ¨ . L N p Σ i ` q N p Σ q N p Σ i q N p Σ i Y Σ i ` q N p Σ k q , where L is the circle removal correspondence defined above (and correspond-ing to the identification 2-morphism).General 2-handle attachmnents. Assume now that Y : Σ ñ S , is a compres-sion body, and let M “ G M (cid:47) (cid:47) G M (cid:47) (cid:47) ¨ ¨ ¨ M k (cid:47) (cid:47) G k .N “ G N (cid:47) (cid:47) G N (cid:47) (cid:47) ¨ ¨ ¨ N k (cid:47) (cid:47) G k . be the sequences corresponding respectively to Σ and S . Associate then to Y the following diagram D p Y q : G M (cid:40) (cid:40) N (cid:54) (cid:54) (cid:11) (cid:19) L G M (cid:40) (cid:40) N (cid:54) (cid:54) (cid:11) (cid:19) L G (cid:39) (cid:39) (cid:54) (cid:54) (cid:11) (cid:19) ¨ ¨ ¨ (cid:43) (cid:43) (cid:51) (cid:51) (cid:11) (cid:19) G k ´ M k (cid:40) (cid:40) N k (cid:54) (cid:54) (cid:11) (cid:19) L k G k , where L i “ L p r Y i q stands for the correspondence of definition 5.11, with Y i Ă Y the piece flowing from Σ i to S i , and r Y i obtained by modding out thevertical tubes (i.e. the tubes flowing between the circles of the decomposi-tions Σ and S ) by the gradient flow of the Morse function, i.e. collapsingthem to circles. Lemma 5.15. (Independence of functions/pseudo-gradient) The diagram D p Y q , as a 2-morphism of z H am , is independent on the choice of Morsefunctions and pseudo-gradients.Proof. First, the choice of a different pseudo-gradient can have the effectof twisting the vertical tubes, which might have the effect of changing thecorrespondences L i , but not the whole diagram, as can be shown by a similarargument as in lemma 5.10.Observe then that since they all are of the same index, the number ofcritical points on each piece Y i is determined by the decomposition of Σ and S . Notice finally that the Lagrangian correspondence only depend on theattaching circles. And they change just by isotopies or handleslides, whichhas no effect on L i . (cid:3) Cerf moves invariance.
We now prove that two decompositions of agiven cobordism yield equivalent morphisms of z H am , in the sense of defini-tion 5.8. Before starting, we point out that the weak transversality assump-tion will always be satisfied in our case. Indeed, L and p ś P qˆ C will alwaysbe irreducible real algebraic affine varieties, therefore the only way that L could fail to intersect p ś P q ˆ C weakly transversely would be to be entirelycontained in it, which is never the case, since in the holonomy descriptions, L is defined by equations such as Hol γ A “ for some curves γ , while C is defined by equations such as Hol γ A “ ´ : the trivial representation isalways contained in L , but not in C .At the level of 1-morphisms, the circle insertion/removal invariance followsfrom proposition 4.5 and the definition of the equivalence relation.We now check the moves for 2-morphisms.Diffeomorphism equivalence. Follows from the fact that a diffeomorphism ofsurfaces induces a symplectomorphism on moduli spaces, that preserves theHamiltonian actions, and a diffeomorphism of 3-manifolds maps Lagrangiancorrespondences to Lagrangian correspondences.Cylinder creation/cancellation. Follows from the fact that the diagram asso-ciated to a cylinder is an identity in z H am .Circle insertion/removal. Follows from the definition of the equivalence rela-tion in z H am (definition 5.8), and the fact that the diagram associated to acircle removal is an identification 2-morphism, modulo this relation.Imbrication of compression bodies. Follows immediately from proposition 5.13.Critical point switches. There are several cases to be distinguished, depend-ing on the index of the critical points.When both index are equal to either 1 or 2, this is a special case ofimbrication of compression bodies.When one of the critical points has index either 0 or 3, its attaching sphere(or attaching belt) is a sphere (with some circles in it) and in particular awhole connected component of the total space of Σ . It follows that thediagrams associated to the handle attachments correspond to attaching 2-cells to the sequence M associated to S in a way that does not overlap.Therefore the order of attachment doesn’t matter.It remains to check the case when the indexes are 1 and 2.Let S , S Ă Σ denote respectively a 0-shpere and a 1-sphere, disjointfrom each other. If S and S lie in different components of the sequence Σ , invariance follows from the same reasons as in the previous case. If thisis not the case, one can insert circles separating them (by circle insertioninvariance), so that they lie on distinct components.Index 0-1 (and 2-3) handle cancellation. Let p Y i , Y i ` , Y i ` q be as in propo-sition 5.5. First, N p D q » N p D q » pt , and the diagram associated to Y i consists in inserting the correspondence pt Ă N p D q ˆ N p D q . Observethen that N p Σ i q and N p D Y p D Σ i qq are identified, and both correspondto the symplectic quotient of N p D Σ i q , by the SU p q -action on B D . Thediagrams associated to Y i ` and Y i ` both correspond (modulo the equiv-alence relation of definition 5.8) to inserting the identification 2-morphismfor this reduction. It follows that the diagram associated to p Y i , Y i ` , Y i ` q is an identity, modulo the equivalence relation.Reversing the diagram, we obtain the index 2-3 cancellation move. Index 1-2 handle cancellation. Assume that Y i and Y i ` correspond to a han-dle cancellation pair, i.e. (the opposite of) Y i and Y i ` correspond to 2-handleattachments along two closed curves that intersect transversely at a singlepoint. We can assume that these curves correspond respectively to α and β , with α , β , ..., α g , β g a symplectic basis of the fundamental group ofthe intermediate surface Σ (with the γ curves removed).These two curves define two coisotropic submanifolds of N p Σ q C “ t A “ u , and C “ t B “ u that induce respectively L p Y q T and L p Y q . Since they intersect transversely,it follows that their composition is embedded, and corresponds to L p Y i Y Y i ` q , i.e. the diagonal correspondence.6. Future directions
We now outline some directions we plan to take in the future.6.1.
Existence of a (non-quasi) 2-functor, quasi-Hamiltonian ana-logue.
By using the spaces N p Σ q , we were able to construct a quasi 2-functor. A natural question follows: Question . Is it possible to replace the spaces N p Σ q in our construction bysuitable Hamiltonian manifolds, satisfying a gluing equals reduction principlein a strict sense (i.e. without C in proposition 4.5), so to obtain a 2-functor C ob ` ` Ñ z H am that still assigns to a closed surface a sequence whosecomposition is the SU p q -character variety?We believe this question can be answered partially positively, at leastin two ways. First, by using the quasi-Hamiltonian spaces constructed in[AMM98]. These spaces satisfy a gluing equals reduction principle, but theirmoment map takes values in the group, rather than its Lie algebra. Thiswould lead to the definition of an analogous partial 2-category q H am . How-ever, it seems not obvious to define Floer homology in this setting, sincethese are not symplectic manifolds.Another possible solution would be to use the infinite dimensional modulispaces M p Σ q introduced by Donaldson [Don92]: these are moduli spacesof flat connexions on Σ , but the restriction to the boundary may not beconstant, and defines a map A |B Σ : B Σ Ñ g . This moduli space is acted onby the gauge group of B Σ , which identifies with k copies of the loop group LSU p q .6.2. Extension to dimension zero.
Since the former spaces M p Σ q areinfinite dimensional, Floer homology seems also difficult to define in thissetting. However, it seems possible to use these spaces in order to define atheory extended to dimension zero: to a closed interval I one can associatethe path group G p I q “ M ap p I, SU p qq , which comes with an evaluation mapto the boundary of I : ev : G p I q Ñ SU p q . One can use these evaluationmaps to glue the groups: if S “ I Y J is a decomposition of the circle intotwo intervals, one has G p S q “ G p I q ˆ ev G p J q . This suggests the definition of a partial 3-category, where objects wouldbe finite dimensional Lie groups, 1-morphisms Banach Lie groups togetherwith morphisms similar with these evaluation maps, 2-morphisms BanachHamiltonian manifolds, and 3-morphisms G -Lagrangian correspondences.6.3. Extension to dimension four.
The aim of this project is to pro-mote z H am , or at least the pre-completion H am , to a 3-category, and extendthe quasi 2-functor defined here to a quasi 3-functor from C ob ` ` ` (or aversion enriched with cohomology classes).If G and G are objects of H am , M , M P hom p G, G q , and D , D P hom p M , M q , one would like to define a 3-morphism space hom p D , D q using equivariant Floer homology (or the chain complex defining it). Indeed,in such a situation, let M “ ź M M ´ ˆ M , L “ ź M ∆ M , L “ ź L L, G “ ź G G, where M , L and G run respectively in the set of symplectic manifolds, La-grangian multi-correspondences and Lie groups appearing in the diagram D M M D . One would then take hom p D , D q “ CF G p L , L q . Several constructions of such chain complexes appeared in the literature, forexample [Fra04, HLS16], or a construction outlined in [DF17]. We plan todefine another version relying on Wehrheim and Woodward’s quilt theory,that should be well suited for our purposes. The resulting algebraic structureshould be a 3-category analogue of Bottman and Carmeli’s p A , q -categories[BC18].6.4. Invariants for knots and sutured manifolds.
The framework de-velopped here should be well-suited for defining invariants of knots, and moregenerally sutured manifolds, similar with the ones in Heegaard-Floer theory,since a sutured manifold can be viewed as a 2-morphism in C ob ` ` . Afterapplying the functor, one gets a 2-morphism in z H am , which can be horizon-tally composed with the coadjoint orbit (cid:32) θ P g | | θ | “ π ( . That preciselycorresponds to putting a traceless condition on the holonomy of a connexionaround a meridian of a knot, as in for example [KM11].6.5. Relation with Seiberg-Witten theory.
The ideas in this sectionemerged during a conversation with Guangbo Xu. Denote F Don : C ob ` ` (cid:57)(cid:57)(cid:75) z H am the quasi 2-functor defined in this paper. One should be able to define asimilar quasi 2-functor F SW : C ob ` ` (cid:57)(cid:57)(cid:75) z H am that would correspond to Seiberg-Witten theory, using extended modulispaces of vortices analogous to the spaces N p Σ q .Following Witten’s conjecture, these two theories should be related: onecan expect that there is a natural transformation T : F Don Ñ F SW , i.e. to any k -morphism W : X Ñ Y in C ob ` ` , should correspond a p k ` q -morphism in z H am relating F Don p W q and F SW p W q as in the followingdiagram: F Don p X q T p X q (cid:15) (cid:15) F Don p W q (cid:47) (cid:47) F Don p Y q T p Y q (cid:15) (cid:15) ó T p W q F SW p X q F SW p W q (cid:47) (cid:47) F SW p Y q . One can think of T p W q as being associated with the p k ` q -morphism W ˆ r , , where the r , coordinate could correspond to a wavelength.We plan to build F SW and T in a future joint work, building on the work ofFeehan and Leness [FL01a, FL01b]. References [AM17] Mohammed Abouzaid and Ciprian Manolescu. A sheaf-theoretic model for sl (2,c) floer homology. arXiv preprint arXiv:1708.00289 , 2017.[AMM98] Anton Alekseev, Anton Malkin, and Eckhard Meinrenken. Lie group valuedmoment maps.
J. Differential Geom. , 48(3):445–495, 1998.[BC18] Nathaniel Bottman and Shachar Carmeli. p A , q -categories and relative 2-operads. arXiv e-prints , page arXiv:1811.05442, November 2018.[Caz] Guillem Cazassus. A monoidal structure on the Hamiltonian 2-category. in prepa-ration .[Caz16] G. Cazassus. Symplectic Instanton Homology: twisting, connected sums, andDehn surgery. to appear in Journal of Symplectic Geometry , June 2016.[CM18] Laurent Côté and Ciprian Manolescu. A sheaf-theoretic SL(2,C) Floer homologyfor knots. arXiv e-prints , page arXiv:1811.07000, Nov 2018.[DF17] Aliakbar Daemi and Kenji Fukaya. Atiyah-Floer Conjecture: a Formulation, aStrategy to Prove and Generalizations. arXiv e-prints , page arXiv:1707.03924,Jul 2017.[Don92] S. K. Donaldson. Boundary value problems for Yang-Mills fields. J. Geom. Phys. ,8(1-4):89–122, 1992.[FL01a] Paul M. N. Feehan and Thomas G. Leness. PU p q monopoles and links of top-level Seiberg-Witten moduli spaces. J. Reine Angew. Math. , 538:57–133, 2001.[FL01b] Paul M. N. Feehan and Thomas G. Leness. PU p q monopoles. II. Top-levelSeiberg-Witten moduli spaces and Witten’s conjecture in low degrees. J. ReineAngew. Math. , 538:135–212, 2001.[Fra04] Urs Frauenfelder. The Arnold-Givental conjecture and moment Floer homology.
Int. Math. Res. Not. , (42):2179–2269, 2004. [HLS16] K. Hendricks, R. Lipshitz, and S. Sarkar. A simplicial construction of G-equivariant Floer homology.
ArXiv e-prints , September 2016.[Jef94] Lisa C. Jeffrey. Extended moduli spaces of flat connections on Riemann surfaces.
Math. Ann. , 298(4):667–692, 1994.[KM11] P. B. Kronheimer and T. S. Mrowka. Knot homology groups from instantons.
J.Topol. , 4(4):835–918, 2011.[MT12] Gregory W Moore and Yuji Tachikawa. On 2d tqfts whose values are holomorphicsymplectic varieties. In
Proc. Symp. Pure Math , volume 85, page 191, 2012.[MW12] Ciprian Manolescu and Christopher Woodward. Floer homology on the extendedmoduli space. In
Perspectives in analysis, geometry, and topology , volume 296 of
Progr. Math. , pages 283–329. Birkhäuser/Springer, New York, 2012.[Nad17] David Nadler. Arboreal singularities.
Geom. Topol. , 21(2):1231–1274, 2017.[Weh16] K. Wehrheim. Floer Field Philosophy.
ArXiv e-prints , February 2016.[Wei81] Alan Weinstein. Symplectic geometry.
Bull. Amer. Math. Soc. (N.S.) , 5(1):1–13,1981.[Wei10] Alan Weinstein. Symplectic categories.
Port. Math. , 67(2):261–278, 2010.[WW15] Katrin Wehrheim and Chris Woodward. Floer field theory for tangles. arXivpreprint arXiv:1503.07615 , 2015.[WW16] K. Wehrheim and C. Woodward. Floer field theory for coprime rank and degree.
ArXiv e-prints , January 2016.
Department of Mathematics, Indiana University, Bloomington, IN 47405
E-mail address ::