The Moore-Tachikawa conjecture via shifted symplectic geometry
Peter Crooks Maxence Mayrand

	
	
	The Moore–Tachikawa conjecture via shifted symplectic geometry]The Moore–Tachikawa conjecture via

		shifted symplectic geometry
	
	Peter Crooks]Peter Crooks
	Maxence Mayrand]Maxence Mayrand
	[Peter Crooks]Department of Mathematics and Statistics
 Utah State University 
 3900 Old Main Hill 
 Logan, UT 84322, USA
	peter.crooks@usu.edu
	[Maxence Mayrand]Département de mathématiques 
 Université de Sherbrooke 
 2500 Bd de l’Université 
 Sherbrooke, QC, J1K 2R1, Canada
	maxence.mayrand@usherbrooke.ca
	
	53D17 (primary); 14L30, 57K16 (secondary) 
	
	
	§ ABSTRACT 
		We use shifted symplectic geometry to construct the Moore–Tachikawa topological quantum field theories (TQFTs) in a category of Hamiltonian schemes. Our new and overarching insight is an algebraic explanation for the existence of these TQFTs, i.e. that their structure comes naturally from three ingredients: Morita equivalence, as well as multiplication and identity bisections in abelian symplectic groupoids. Using this insight, we generalize the Moore–Tachikawa TQFTs in two directions. 

		The first generalization concerns a 1-shifted version of the Weinstein symplectic category . Each  abelianizable quasi-symplectic groupoid  is shown to determine a canonical 2-dimensional TQFT η_:_2⟶. We recover the open Moore–Tachikawa TQFT and its multiplicative counterpart as special cases. 
		
		Our second generalization is an affinization process for TQFTs. We first enlarge Moore and Tachikawa's category  of holomorphic symplectic varieties with Hamiltonian actions to , a category of affine Poisson schemes with Hamiltonian actions of affine symplectic groupoids.
		We then show that if X is an affine symplectic groupoid that is abelianizable when restricted to an open subset U  X statisfying Hartogs' theorem, then  determines a TQFT η_ : _2.
		In more detail, we first devise an affinization process sending 1-shifted Lagrangian correspondences in  to Hamiltonian Poisson schemes in .
		The TQFT is obtained by composing this affinization process with the TQFT η_|_U : _2 of the previous paragraph.
		Our results are also shown to yield new TQFTs outside of the Moore–Tachikawa setting.

		

		
		
	
	
	
	
	
	
	
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§ INTRODUCTION

	
	

 §.§ Context

	The Moore–Tachikawa conjecture <cit.> inspires important, topical, and ongoing research at the interface of geometric representation theory, Lie theory, low-dimensional topology, holomorphic symplectic geometry, and theoretical physics. It arises in a string-theoretic context, from a conjectural class of 6-dimensional superconformal quantum field theories (SQFTs). One may associate such a class to each connected complex simple Lie group G. The Higgs branch of this conjectural class would be a 2-dimensional topological quantum field theory (TQFT) η_G:_2⟶, i.e. a symmetric monoidal functor, valued in a category of holomorphic symplectic varieties with Hamiltonian actions. The objects of  are complex semisimple groups, and a morphism from G_1 to G_2 is a holomorphic symplectic variety with a Hamiltonian action of G_1 × G_2. The composition of M ∈(G_1, G_2) and N ∈(G_2, G_3) is the Hamiltonian reduction (M × N)  G_2. Properties of this TQFT would necessarily include η_G(S^1)=G and η_G([
		baseline=-2.5pt,
		every tqft/.append style=
			transform shape, rotate=90, tqft/circle x radius=4pt,
			tqft/circle y radius= 2pt,
			tqft/boundary separation=0.6cm, 
			tqft/view from=incoming,
		
		]
		[
		tqft/cup,
		name=d,
		every incoming lower boundary component/.style=draw,
		every outgoing lower boundary component/.style=draw,
		every incoming boundary component/.style=draw,
		every outgoing boundary component/.style=draw,
		cobordism edge/.style=draw,
		cobordism height= 1cm,
		];
	)=G×𝒮, where 𝒮 is a Kostant slice in the Lie algebra of G. Moore and Tachikawa's conjecture is that there exists a 2-dimensional TQFT _2⟶ satisfying these two conditions. While formulatable as a problem in pure mathematics, a resolution of this conjecture would clearly have implications for string theory. An affirmative answer would be some evidence that the aforementioned SQFTs actually exist. We refer the reader to Tachikawa's ICM paper <cit.> for further context.
	
	The Moore–Tachikawa conjecture was formulated less than 15 years prior to our completing this manuscript. In this short time, it has witnessed progress and been found to make deep connections to adjacent subjects <cit.>. Progress on the conjecture itself is made in unpublished work of Ginzburg–Kazhdan <cit.>. These authors first construct the open Moore–Tachikawa varieties, thereby proving a relative of the Moore–Tachikawa conjecture. If the affinizations of such varieties were known to be of finite type, then the Moore–Tachikawa conjecture would follow. The work of Braverman–Finkelberg–Nakajima <cit.> implies that the relevant affine schemes are of finite type for G of Lie type A. In particular, the Moore–Tachikawa conjecture holds in type A.
	
	Ginzburg and Kazhdan's approach to the Moore–Tachikawa conjecture involves a kind of Hamiltonian reduction by abelian group schemes. This is formalized and generalized in <cit.>, where we introduce Hamiltonian reduction by symplectic groupoids along pre-Poisson subvarieties. We thereby recover the open Moore–Tachikawa varieties constructed by Ginzburg–Kazhdan, as well as their affinizations. In very rough terms, this is achieved by replacing G with its cotangent groupoid T^*G^*. One might therefore suspect the following: our recovering the Ginzburg–Kazhdan construction is a shadow of a more general “affinization process", in which the results of <cit.> are used to affinize TQFTs constructed via symplectic groupoids. This is the first of two main themes in our manuscript.
	
	An appealing feature of the Moore–Tachikawa conjecture is its apparent amenability to a wide variety of techniques, including some not used by Ginzburg–Kazhdan or Braverman–Finkelberg–Nakajima. It is in this context that we are led to shifted symplectic geometry, as introduced by Pantev–Töen–Vaquie–Vezzosi <cit.>. In subsequent work <cit.>, Calaque suggests that shifted symplectic geometry should facilitate a rigorous construction of the Moore–Tachikawa TQFT. This is largely consistent with a result in <cit.>, where Bălibanu and the second named author develop a Hamiltonian reduction theory for 1-shifted symplectic groupoids (a.k.a. quasi-symplectic groupoids). One consequence is a multiplicative counterpart of the Ginzburg–Kazhdan construction described above. With this in mind, the second main theme of our manuscript is the relevance of shifted symplectic geometry to the Moore–Tachikawa conjecture.  
	
	

 §.§ Overarching principle

	The overarching principle of this manuscript is that the structure underlying certain TQFTs is encoded in multiplication and identity bisections in abelian symplectic groupoids. We thereby offer an algebraic explanation for the existence of the Moore–Tachikawa TQFT in this enlarged category, as well as that of other TQFTs. These ideas are made somewhat more precise in the next few subsections, and completely precise in the main body of the manuscript.   
	
	

 §.§ TQFTs in a 1-shifted Weinstein symplectic category

	For the moment, we may work in the smooth or holomorphic categories.
	A first useful idea, suggested by Calaque <cit.>, is to temporarily replace the Moore–Tachikawa category  of holomorphic symplectic varieties with a 1-shifted version  of Weinstein symplectic category <cit.>; see Section <ref>. The objects of  are quasi-symplectic groupoids <cit.>, interpreted as presentations of 1-shifted symplectic stacks.
	
	We then recall that a 2-dimensional TQFT in a symmetric monoidal category 𝐂 is equivalent to a commutative Frobenius object in 𝐂.
	We subsequently establish that any abelian symplectic groupoid gives rise to a commutative Frobenius object in .
	
	The multiplication and unit of the aforementioned commutative Frobenius object are essentially given by groupoid multiplication and the identity section, respectively.
	
	 In the interest of associating TQFTs to a wider class of groupoids, we proceed as follows. A quasi-symplectic groupoid is called abelianizable if it is Morita equivalent to abelian symplectic groupoid. A notion of Morita transfer then implies that every abelianizable quasi-symplectic groupoid determines a TQFT. This amounts to the following result, the underlying details of which are given in Section <ref>.

	
	Every abelianizable quasi-symplectic groupoid  completes to a commutative Frobenius object in  whose product _μ∈(×, ) and unit _η∈(⋆, ) are induced by groupoid multiplication and identity bisection in the abelianization, respectively. It thereby determines a TQFT η_:_2⟶.
	
	
	In this context, one naturally seeks sufficient conditions for a quasi-symplectic groupoid X to be abelianizable. This leads us to introduce the notion of an admissible global slice. We define it to be a submanifold S X that intersects every  orbit in X transversely in a singleton, such that the isotropy group _x is abelian for all x∈ S. We show X to be abelianizable if there exists an admissible global slice S X with the property that the 3-form on X pulls back to an exact 3-form on S.  
	
	

 §.§ Affinizations of TQFTs
 We now work exclusively over ℂ. It is useful to specialize Main Theorem <ref> as follows. Let G be a connected semisimple affine algebraic group with Lie algebra . Write T^*G for the pullback of T^*G^* to the regular locus ^*. One may verify that T^*G is Morita equivalent to 𝒵_G⟶𝔠Spec((Sym)^G), the universal centralizer of G. This fact implies that T^*G is abelianizable. By means of Main Theorem <ref>, T^*G turns out to determine the open Moore–Tachikawa TQFT. As mentioned above, Ginzburg and Kazhdan affinize some of the morphisms in the image of this TQFT<cit.>. These affinizations are subsequently shown to satisfy the relations of a TQFT, despite not necessarily being of finite type.
	
	Our second main result is that the aforementioned affinization process applies in far greater generality, and that the TQFT relations essentially follow from Main Theorem <ref>. To this end, we enlarge  to what we call the algebraic Moore–Tachikawa category ; it is very loosely described as a category with affine symplectic groupoids as objects, and morphisms consisting of isomorphism classes of affine Poisson schemes with Hamiltonian actions of product groupoids. Our previously developed machinery of scheme-theoretic coisotropic reduction <cit.> allows one to compose morphisms in .
	We then devise an affinization process which sends 1-shifted Lagrangian correspondences in  to Hamiltonian Poisson schemes in .
	
	In light of the above, we introduce the notion of a Hartogs abelianizable affine symplectic groupoid X. By this, we mean that X is abelianizable when restricted to an open subset U  X whose complement has codimension at least two in X.
	We then show that the composition of the TQFT associated to |_U in Main Theorem <ref> with the affinization process gives  the structure of a commutative Frobenius object in .
	A key ingredient in the proof is that U  X satisfies Hartogs' theorem, enabling us to complete the relevant structures on |_U to .
	In more detail, our second main result is as follows.
	
	
	
	
	Let X be an affine symplectic groupoid that is abelianizable over an open subset U  X whose complement has codimension at least two in X.
	Then  completes to a commutative Frobenius object in  whose product _μ∈(×, ) and unit _η∈(⋆, ) are the affinizations of (|_U)_μ and (|_U)_η in Main Theorem <ref>, respectively.
	In this way,  determines a TQFT η_:_2⟶.
	
	
	In analogy with the discussion following Main Theorem <ref>, one should seek sufficient conditions for an affine symplectic groupoid to be Hartogs abelianizable. This leads us to introduce the notion of an admissible Hartogs slice; it is defined somewhat analogously to an admissible global slice. We show that an affine symplectic groupoid admitting an admissible Hartogs slice is Hartogs abelianizable. 
	
	Let us again consider the cotangent groupoid T^*G^* of a connected complex semisimple affine algebraic group G. The universal centralizer witnesses T^*G^* as Hartogs abelianizable. Main Theorem <ref> therefore applies and yields the Moore–Tachikawa TQFT. Perhaps surprisingly, Main Theorem <ref> turns out to yield a strictly larger class of TQFTs. The following are some details in this direction. 
	
	One might seek conditions on an complex affine algebraic group G for T^*G^* to be Hartogs abelianizable. This leads us to introduce the notion of a Moore–Tachikawa group, i.e. a complex affine algebraic group G with Lie algebra  that satisfies the following properties:
	
	
  * the set ^* of regular elements has a complement of codimension at least two in ^*;
	
  * the stabilizer subgroup G_ξ is abelian for all ξ∈^*;
	
  * the pullback of the cotangent groupoid T^*G to ^* is abelianizable.
	This holds, in particular, if there exists an affine slice S ^* for the coadjoint action on the regular locus.
	
	Results of Kostant <cit.> imply that every complex reductive group is Moore–Tachikawa. On the other hand, Main Theorem <ref> implies that every Moore–Tachikawa group induces a TQFT. These considerations motivate one to find examples non-reductive Moore–Tachikawa groups. We provide one such example in Section <ref>. We also obtain examples of TQFTs arising from certain Slodowy slices, e.g. a Slodowy slice to the minimal nilpotent orbit in 𝔰𝔩_n.

	

 §.§ Organization
 Each section begins with a summary of its contents. In Section <ref>, we develop pertinent results and techniques regarding shifted Lagrangians. This leads to Section <ref>, where we define and discuss the 1-shifted Weinstein symplectic “category"  and its completion . We then prove Main Theorem 1 in Section <ref>. 
	
	Our attention subsequently turns to the matter of affinizing TQFTs. In Section <ref>, we use scheme-theoretic coisotropic reduction to define and study the category . Main Theorem <ref> is also proved in Section <ref>. Section <ref> is then devoted to the implications of Main Theorem <ref> for constructing the Moore–Tachikawa TQFT. In Sections <ref> and <ref>, we use Main Theorem <ref> to produce TQFTs different from those of Moore–Tachikawa. Section <ref> provides a TQFT for an affine symplectic groupoid integrating a Slodowy slice to the minimal nilpotent orbit in 𝔰𝔩_n. Section <ref> then provides an example of a non-reductive Moore–Tachikawa group. 
	
	
	

 §.§ Acknowledgements

	We thank Ana Bălibanu, Damien Calaque, Zoïk Dubois, and Tom Gannon for highly useful discussions. P.C. and M.M were partially supported by the Simons Foundation Grant MPS-TSM-00002292 and NSERC Discovery Grant RGPIN-2023-04587, respectively.








	

§ 1-SHIFTED LAGRANGIAN CORRESPONDENCES

	
	In this section, we review background material on quasi-symplectic groupoids, shifted Lagrangians, and Morita equivalence in a differential-geometric context, following <cit.>. We also provide proofs of certain basic, technical facts which we have not been able to find in the literature. 
	
	We work in the smooth category throughout this section. If one makes the obvious adjustments to terminology (e.g. replacing real and smooth with complex and holomorphic, respectively), all results hold in the holomorphic category as well.
	
	

 §.§ Notation and conventions on Lie groupoids

	 Given a Lie groupoid , the manifolds of arrows and objects are denoted  and , respectively. One has source and target maps (, ) :, multiplication : ⟶, inversion : ⟶, and the identity section : ⟶. The Lie algebroid of  is A__*, together with ρ__*|_A_ as its anchor map.
	
	Consider a Lie groupoid  and smooth map μ:N⟶. The pullback of  by μ is denoted μ^*; it is a groupoid with arrows (μ^*) = N μμ N and objects (μ^*) = N. A sufficient condition for μ^* to be a Lie groupoid is for ∘_ : N μ→ to be a surjective submersion, where _:  N μ⟶ is the canonical map.

	The restriction of a groupoid  to a subset S is the groupoid |_S ^-1(S) ∩^-1(S), i.e. the pullback of  by the inclusion S.

	For an action of a Lie groupoid  on a manifold M, the action groupoid is denoted ⋉ M.
	It has source map (g, p) = p and target map (g, p) = g · p.
	
	

 §.§ Quasi-symplectic groupoids

	Quasi-symplectic groupoids feature prominently in this manuscript. For the sake of completeness and self-containment, we recall their definition below.
	
	
		A quasi-symplectic groupoid is a triple (, ω, ϕ) of a Lie groupoid , 2-form ω on , and a 3-form ϕ on , such that dω = ^*ϕ - ^*ϕ, = 2, and ω_x ∩_* ∩_* = 0 for all x ∈.
		The IM-form of (, ω, ϕ) is the map 
    σ_ω : A_⟶ T^*,   a↦^*(i_aω).

We write ^- for the quasi-symplectic groupoid (, -ω, -ϕ).
A symplectic groupoid is a quasi-symplectic groupoid whose 2-form ω is non-degenerate and whose 3-form ϕ is trivial.
	

The central purpose of quasi-symplectic groupoids is that they are presentations of 1-shifted symplectic stacks <cit.>.





















	
	
	

 §.§ 1-shifted Lagrangians

	Let us recall the following crucial definition.
	
	 
		A 1-shifted Lagrangian on a quasi-symplectic groupoid (, ω, ϕ) is a Lie groupoid morphism μ : Ł⟶, together with a 2-form γ on Ł, such that the following conditions are satisfied:
		
			
  * 
			μ^*ω = ^*γ - ^*γ;
			
  * 
			μ^*ϕ = -dγ;
			
  * 
			the map
			
    A_Ł   {(v, a) ∈ TŁ⊕μ^*A_ : μ_*(v) = ρ_(a)  and  i_vγ = μ^* σ_ω(a)}
    
    				b     (ρ_Ł(b), μ_*(a))

			is an isomorphism.
		
		See <cit.> for the equivalence between this definition and the original one in <cit.>.
	


The notion of 1-shifted Lagrangians is closely related to that of Hamiltonian -spaces, as defined by Xu <cit.>. In more detail, let a quasi-symplectic groupoid  act on a manifold M. A 1-shifted Lagrangian structure on the projection ⋉ M → is precisely a 2-form on M for which M is a Hamiltonian -space.


	Given two quasi-symplectic groupoids _1 and _2, a 1-shifted Lagrangian correspondence from _1 to _2 is a 1-shifted Lagrangian Ł⟶_1 ×_2^-.
	We usually denote a 1-shifted Lagrangian correspondence by a span
	
    [row sep=1.5em,between origins,column sep=3em,between origins]
    		   Łdldr   
    _1       _2.

	Two 1-shifted Lagrangian correspondences
	
    [row sep=1.5em,between origins,column sep=3em,between origins]
    			   Ł_1 dldr      Ł_2 dldr   
    _1       _2       _3

	are called transverse if their homotopy fibre product Ł_1 __2Ł_2 (see, e.g. <cit.> or <cit.>) is transverse, i.e. the two maps
	
    [row sep=1em,column sep=2em]
    		Ł_1×Ł_2dr      _2[swap]dl(, )
       _2×_2

	are transverse.
	In this case,
	
    [row sep=2em,between origins,column sep=4em,between origins]
    			   Ł_1 __2Ł_2 dldr   
    _1       _3.

	is a 1-shifted Lagrangian correspondence <cit.>; it is called the composition of Ł_1 and Ł_2, and denoted Ł_2 ∘Ł_1.
	More precisely, if γ_i is the 1-shifted Lagrangian structure on Ł_i for i = 1, 2, then the 1-shifted Lagrangian structure on (<ref>) is given by _Ł_1^*γ_1 + _Ł_2^*γ_2 - __2^*ω_2, where _Ł_i : Ł_1 __2Ł_2 Ł_i and
	
    __2 : (Ł_1 __2Ł_2) = Ł_1μ_1_2μ_2Ł_2_2

	are the natural projections.
	

	
	
	

 §.§ Morita equivalences

	A morphism of Lie groupoids φ : ⟶̋ is called a Morita morphism <cit.> (a.k.a. surjective equivalence <cit.> or hypercover <cit.>) if the underlying map on objects φ : ⟶ is a surjective submersion, and the induced morphism ⟶̋(φ)^* is a Lie groupoid isomorphism.
	A Morita equivalence between two Lie groupoids _1 and _2 consists of a Lie groupoid $̋ and Morita morphisms_1 ⟵⟶̋_2.
	Morita equivalence is an equivalence relation on the set of Lie groupoids.
	
	A weaker notion is that of an essential equivalence, consisting of a Lie groupoid morphismφ: such that∘_ : φ →is a surjective submersion and→̋(φ)^*is a Lie groupoid isomorphism.
	Two Lie groupoids_1and_2are Morita equivalent if and only if there is a Lie groupoid$̋ together with essential equivalences _1 ⟵⟶̋_2 (see e.g. <cit.>).
	Hence, Morita morphisms and essential equivalences induce the same equivalence relation on the set of Lie groupoids and can be used interchangeably.
	We use mainly Morita morphisms in this paper, following <cit.>, but both approaches are equivalent.

	The notion of Morita equivalence gives rise to the notion of a symplectic Morita equivalence between two quasi-symplectic groupoids (_1, ω_2, ϕ_1) and (_2, ω_2, ϕ_2), i.e. a Morita equivalence _1 ⟵⟶̋_2 that is also a 1-shifted Lagrangian correspondence.
	Two quasi-symplectic groupoids are symplectically Morita equivalent if and only if they present isomorphic 1-shifted symplectic stacks <cit.>.
	By <cit.>, the non-degeneracy condition <ref> in Definition <ref> is automatic: a Morita equivalence _1 φ_1⟵φ_2⟶_2 is symplectic if and only if there is a 2-form γ on  such that φ_1^*ω_1 - φ_2^*ω_2 = ^*γ - ^*γ and φ_1^*ϕ_1 - φ_2^*ϕ_2 = -dγ.
	The following example therefore follows automatically.

	
	For every quasi-symplectic groupoid , the identity maps ⟵⟶ and trivial 2-form constitute a symplectic Morita equivalence.
	

	
	An alternative but equivalent <cit.> approach to symplectic Morita equivalence is via Hamiltonian bibundles <cit.>, i.e. two quasi-symplectic groupoids _1 and _2 are symplectically Morita equivalent if and only if there is manifold M endowed with a 2-form and commuting Hamiltonian actions of _1 and _2^- such that M →_2 is a principal _1-bundle and M →_1 is a principal _2-bundle.
	Such a bibundle gives rise to a symplectic Morita equivalence via the action groupoid (_1 ×_2) ⋉ M together with the two projections to _1 and _2.
	
	
	A Lagrangian Morita equivalence between two 1-shifted Lagrangians (Ł_1, γ_1) ⟶ (_1, ω_1, ϕ_1) and (Ł_2, γ_2) ⟶ (_2, ω_2, ϕ_2) is a 2-commutative diagram
	
    [column sep=4em,between origins,row sep=2em,between origins]
    			   [swap]dlψ_1ddνdrψ_2   
    Ł_1 [swap]ddμ_1[Rightarrow,shorten=6pt,swap]drθ_1      Ł_2 ddμ_2[Rightarrow,shorten=6pt]dlθ_2
       dlφ_1[swap]drφ_2   
    _1       _2,

	such that (φ_1, φ_2) is a symplectic Morita equivalence with respect to some 2-form δ on , (ψ_1, ψ_2) is a Morita equivalence of Lie groupoids, and
	
    ψ_1^*γ_1 - ψ_2^*γ_2 = ν^*δ - θ_1^*ω_1 + θ_2^*ω_2.

	In this case, we say that Ł_1 and Ł_2 are Lagrangially Morita equivalent.
	This happens if and only if Ł_1 and Ł_2 present isomorphic 1-shifted Lagrangians on the 1-shifted symplectic stacks presented by _1 and _2 <cit.>.
	In the special case where (_1, ω_1, ϕ_1) = (_2, ω_2, ϕ_2), we call this a weak equivalence of 1-shifted Lagrangians. This amounts to declaring two 1-shifted Lagrangians (Ł_1, γ_1) ⟶ (, ω, ϕ) and (Ł_2, γ_2) ⟶ (, ω, ϕ) to be weakly equivalent if there a 2-commutative diagram
	
    [row sep=3em,between origins,column sep=3em,between origins]
    			   [swap]dlψ_1drψ_2   
    Ł_1 [swap]drμ_1[shorten=14pt,Rightarrow]rrθ      Ł_2 dlμ_2

	such that (ψ_1, ψ_2) is a Morita equivalence of Lie groupoids and ψ_2^*γ_2 - ψ_1^*γ_1 = θ^*ω.


Recall that the action of ×^- on  by left and right multiplications is Hamiltonian in the sense of Xu <cit.>.
It follows from Example <ref> that (×) ⋉ is a symplectic Morita equivalence from  to .
Under the equivalence between Hamiltonian bibundles and symplectic Morita equivalences (Remark <ref>), ⟵ (×) ⋉⟶ is weakly equivalent to the 1-shifted Lagrangian correspondence ⟵⟶ of Example <ref>.
An explicit essential equivalence can be obtained by the morphism (×) ⋉, g  ((g, g), _(g)).


If _1 and _2 are symplectically Morita equivalent quasi-symplectic groupoids, then for every 1-shifted Lagrangian Ł_1 → G_1 there is a 1-shifted Lagrangian Ł_2 →_2 together with a Lagrangian Morita equivalence between them <cit.>.
Moreover, Ł_2 →_2 is unique up to weak equivalences.
We call this process of transferring 1-shifted Lagrangians from _1 to _2 Morita transfer.
	
The next three results are important for the construction of the 1-shifted Weinstein symplectic category in the next section.
	
	
	
		Consider a 2-commutative diagram of Lie groupoid morphisms
		
    [row sep=1.5em,between origins,column sep=3em,between origins]
    			_1 [swap]dddr      _2 [swap]dldd
       dd   
    Ł_1 [swap]dr      Ł_2 dl
          ,

		where the vertical arrows are Morita morphisms.
		Then Ł_1 _Ł_2 is transverse if and only if _1 __2 is transverse.
		In this case, the induced map _1 __2 ⟶Ł_1 _Ł_2 is a Morita morphism.
	
	
	
		Introduce the following labels:
		
    [row sep=2em,between origins,column sep=4em,between origins]
    			_1 [swap]ddψ_1drν_1      _2 [swap]dlν_2ddψ_2
       ddφ[Rightarrow, shorten=8pt,swap]dlθ_1[Rightarrow, shorten=8pt]drθ_2   
    Ł_1 [swap]drμ_1      Ł_2 dlμ_2
          ,

		where θ_i is a natural transformation from φν_i to μ_i ψ_i, i.e.		
    θ_i = φν_i,   θ_i = μ_i ψ_i,   and  θ_i((g_i)) ·φ(ν_i(g_i)) = μ_i(ψ_i(g_i)) ·θ_i((g_i))

		for all g_i ∈_i and i=1,2.
		Choose an Ehresmann connection τ on  <cit.>; this is a right splitting of the short exact sequence
		
    [column sep=3em]
    				0 r   ^* A_rR    Tr_*   ^*Tr    0,

		where R is right translation.
		Let  be the corresponding adjoint representation up to homotopy <cit.>; see also <cit.> for an overview in the same notation as this proof. We have the quasi-action of  on T and A_ given by 
		
    _g    : T_(g) T_(g),   _g(v) = _*(τ_g(v)) 
    _g    : (A_)_(g) (A_)_(g),   _g(a) = τ̌(a^L_g)

		for all g ∈, where τ̌ is the left splitting of (<ref>) corresponding to τ.
		As in <cit.>, we denote the basic curvature of τ by K ∈Γ(^2; (^*T, ^*A_)).
		We also set
		
    θ̇_i τ̌∘θ_i* : T_iψ_2^*μ_2^*A_,

		for i=1,2, so that the θ̇_i provide chain homotopies for the maps of chain complexes from A__i⟶ T_i to A_⟶ T <cit.>.
		
		Suppose that _1 __2 is transverse.
		Let (x_1, g, x_2) ∈ (Ł_1 _Ł_2), i.e. (μ_1(x_1), μ_2(x_2)) = ((g), (g)).
		Take (u_1, u_2) ∈ T_(g)× T_(g).
		Let y_i ∈_i be such that ψ_i(y_i) = x_i.
		Then (g) = μ_1(x_1) = μ_1(ψ_1(y_1)) = (θ_1(y_1)) and (g) = μ_2(x_2) = μ_2(ψ_2(y_2)) = (θ_2(y_2)).
		Moreover, (θ_1(y_1)) = φ(ν_1(y_1)) and (θ_2(y_2)) = φ(ν_2(y_2)).
		It follows that
		
    (ν_1(y_1), θ_2(y_2)^-1 g θ_1(y_1), ν_2(y_2))
    		∈
    		(φ)^*.

		Since ≅̋(φ)^*, there is a unique h ∈$̋ such that(h) = ν_1(y_1),(h) = ν_2(y_2), andφ(h) = θ_2(y_2)^-1 g θ_1(y_1).
		Noting thatu_1 ∈T_(g)= T_(θ_1(y_1)), there exists a uniquev_1 ∈T_(θ_1(y_1))= T_(φ(h)) such that_θ_1(y_1)(v_1) = u_1.
		A similar argument reveals the existence of a uniquev_2 ∈T_(θ_2(y_2))= T_(φ(h))such that_θ_2(y_2)(v_2) = u_2.
		We also haveφ(ν_i(y_i)) = (θ_i(y_i)), and knowφto be a submersion. One therefore hasw_i ∈T_ν_i(y_i)such thatφ_*(w_i) = v_i.
		At the same time,(y_1, h, y_2) ∈(_1 __2)and(w_1, w_2) ∈T_(h)×T_(h).
		Since_1 __2is transverse, there existδ_i ∈T_y_i_iandγ∈T_hsuch that
		
    (w_1, w_2) = (ν_1*(δ_1), ν_2*(δ_2)) + (_*(γ), _*(γ)).

		Letg̃ θ_2(y_2)^-1 g θ_1(y_1).
		Using the fact thatT_g̃= R_g̃((A_)_(g̃)) ⊕τ_g̃(T_(g̃)), we can write 
		
    φ_* γ = a^R_g̃ + τ_g̃(b)

		fora ∈(A_)_θ_2(y_2)andb ∈T_θ_1(y_1).
		Applying_*to both sides of (<ref>) yields_* φ_* (γ) = b.
		It follows that
		
    φ_* (γ) = a^R_g̃ + τ_g̃_* φ_*(γ).

		Furthermore, applying_*to both sides of (<ref>) gives
		
    ρ(a) = _* φ_* (γ) - _g̃_* φ_*(γ).

		
		
		
		
		
		
		
		
		
		
		
		
		
		We claim that
		
    (u_1, u_2) = (μ_1* (δ̃_1) + _* (γ̃), μ_2* (δ̃_2) + _* (γ̃)),

		where
		
    γ̃   τ_g _θ_1(y_1)φ_*(γ) + (θ̇_1(δ_1))^L_g + (_θ_2(y_2)(a) -θ̇_2 (δ_2) - K(g, θ_1(y_1))(φ_*(γ)) + K(θ_2(y_2), g̃)(φ_* (γ)))^R_g 
    δ̃_1    ψ_1*(δ_1) 
    δ̃_2    ψ_2*(δ_2).

		To this end, <cit.> implies that
		
    μ_1*(δ̃_1) + _*(γ̃)
    			   =
    			μ_1ψ_1*(δ_1) + _θ_1(y_1)φ_*(γ) - ρθ̇_1(δ_1) 
       = _θ_1(y_1)φ_*ν_1*(δ_1) + _θ_1(y_1)φ_*(γ) 
       = _θ_1(y_1)φ_*(w_1) 
       = u_1.

		At the same time, <cit.>, (<ref>), and the equivariance ofρtell us that
		
    μ_2*(δ̃_2) + _*(γ̃) 
       =
    			μ_2 ψ_2* (δ_2) + _g _θ_1(y_1)_* φ_*(γ) + ρ(_θ_2(y_2)(a) - θ̇_2 (δ_2) - K(g, θ_1(y_1))(_* φ_* (γ)) + K(θ_2(y_2), g̃)(_* φ_* (γ))) 
       = _θ_2(y_2)φ_* ν_2* (δ_2) + _g _θ_1(y_1)_* φ_*(γ) + _θ_2(y_2) (_*φ_*(γ) - _g̃_* φ_*(γ)) 
            - ρ(K(g, θ_1(y_1))(_* φ_* (γ))) + ρ(K(θ_2(y_2), g̃)(_* φ_* (γ))) 
       = 
    			_θ_2(y_2)φ_*ν_2*(δ_2) + _θ_2(y_2)_* φ_*(γ) 
       = u_2.

		It follows thatŁ_1 _Ł_2is transverse.
		
		Suppose now thatŁ_1 _Ł_2is transverse.
		Let(y_1, h, y_2) ∈(_1 __2), i.e.(ν_1(y_1), ν_2(y_2)) = ((h), (h)).
		Let(v_1, v_2) ∈T_(h)×T_(h).
		Note that for all(u_1, u_2) ∈(φ_*)_(h) ×(φ_*)_(h), we have(u_1, u_2) = (_*(w), _*(w)), wherew = (u_1, 0, u_2) ∈T_h≅T_(h)×_T T_φ(h)×_T T_(h).
		It therefore suffices to show that
		
    φ_* (v_1) = φ_*(ν_1*(u_1) + _*(w))
    		  and  φ_* (v_2) = φ_*(ν_2*(u_2) + _*(w))

		for someu_i ∈T_y_i_iandw ∈T_h.
		In other words, it suffices to show that
		
    φ_*(v_1 - ν_1*(u_1)) = _* γ  and  φ_*(v_2 - ν_2*(u_2)) = _* γ,

		for someu_i ∈T_y_i_iandγ∈T_φ(h).
		
		Note that(φ(h)) = φ(ν_2(y_2)) = (θ_2(y_2))and(φ(h)) = φ(ν_1(y_1)) = (θ_1(y_1)), so thatg θ_2(y_2) ·φ(h) ·θ_1(y_1)^-1 ∈is well-defined.
		Letx_1 ψ_1(y_1)andx_2 ψ_2(y_2), noting that(μ_1(x_1), μ_2(x_2)) = ((g), (g)).
		We then have(φ_*(v_1), φ_*(v_2)) ∈T_θ_1(y_1)×T_θ_2(y_2). It follows that(_θ_1(y_1)φ_*(v_1), _θ_2(y_2)φ_*(v_2)) ∈T_(g)×T_(g).
		SinceŁ_1 _Ł_2is transverse, there existũ_1 ∈T_x_1Ł_1,ũ_2 ∈T_x_2Ł_2, andw ∈T_gsuch that
		
    (_θ_1(y_1)φ_*(v_1), _θ_2(y_2)φ_*(v_2))
    		=
    		(μ_1*(ũ_1) + _*(w), μ_2*(ũ_2) + _*(w)).

		Chooseu_i ∈T_y_i_isuch thatψ_i*(u_i) = ũ_i.
		As in the first part, we can write
		
    w = a^R_g + τ_g _* (w),

		wherea ∈(A_)_(g)andρ(a) = _*(w) - _g _* (w).
		
		
		
		
		Consider the element ofT_φ(h)defined by
		
    γ   τ_φ(h)_θ_1(y_1)^-1_*(w) 
            + (_θ_2(y_2)^-1(a + θ̇_2 (u_2)) + K(θ_2(y_2)^-1, g)(_* (w)) - K(φ(h), θ_1(y_1)^-1)(_*(w)))^R_φ(h)
            -K(θ_2(y_2)^-1, θ_2(y_2)(φ_*(v_2 - ν_2*(u_2))))^R_φ(h)
            + (K(θ_1(y_1)^-1, θ_1(y_1))(φ_*(v_1 - ν_1*(u_1))) - _θ_1(y_1)^-1θ̇_1(u_1))^L_φ(h).

		We have
		
    _* γ   = _θ_1(y_1)^-1_*(w) - ρ(K(θ_1(y_1)^-1, θ_1(y_1))(φ_*(v_1 - ν_1*(u_1)))) + ρ(_θ_1(y_1)^-1θ̇_1(u_1)) 
       = _θ_1(y_1)^-1_* (w) - _θ_1(y_1)^-1_θ_1(y_1)φ_*(v_1 - ν_1*(u_1)) + φ_*(v_1 - ν_1*(u_1)) + _θ_1(y_1)^-1ρθ̇_1(u_1) 
       = _θ_1(y_1)^-1_* (w) - _θ_1(y_1)^-1_θ_1(y_1)φ_*(v_1 - ν_1*(u_1)) + φ_*(v_1 - ν_1*u_1) + _θ_1(y_1)^-1(μ_1*ψ_1*u_1 - _θ_1(y_1)φ_*ν_1*u_1) 
       = _θ_1(y_1)^-1_θ_1(y_1)φ_*(v_1) - _θ_1(y_1)^-1_θ_1(y_1)φ_*(v_1 - ν_1*(u_1)) + φ_*(v_1 - ν_1*(u_1)) - _θ_1(y_1)^-1_θ_1(y_1)φ_*ν_1*(u_1) 
       = φ_*(v_1 - ν_1*(u_1))

		and
		
    _*γ   = _φ(h)_θ_1(y_1)^-1_*(w) + _θ_2(y_2)^-1(_*(w) - _g_*(w) + μ_1*ψ_1*(u_2) - _θ_2(y_2)φ_*ν_2*(u_2)) 
            +_θ_2(y_2)^-1_g _*(w)  - _θ_2(y_2)^-1g_*(w) - _φ(h)_θ_1(y_1)^-1_*(w) + _φ(h)θ_1(y_1)^-1_*(w) 
            - _θ_2(y_2)^-1_θ_2(y_2)(φ_*(v_2 - ν_2*(u_2))) + φ_*(v_2 - ν_2*(u_2)) 
       = φ_*(v_2 - ν_2*(u_2)).

		It follows that_1 __2is transverse.
		
		In this case, we have a Lie groupoid morphism
		
    _1 __2    Ł_1 _Ł_2 
    
    			(l_1, h, l_2)     (ψ_1(l_1), θ_2((l_2)) ·φ(h) ·θ_1((l_1))^-1, ψ_2(l_2)).

		To see that this is a Morita morphism, we first show that the map on objects is a surjective submersion. This is accomplished by establishing the existence of local sections. We first note that≅̋φ^*. It follows that(_1 __2)= _1×_φν_1,  ×_, φν_2 _2.
		The map on objects must therefore take the form
		
    φ̃ : _1×_φν_1, ×_, φν_2_2   Ł_1×_μ_1, ×_, μ_2Ł_2
    
    			(y_1, g, y_2)     (ψ_1(y_1), θ_2(y_2) · g ·θ_1(y_1)^-1, ψ_2(y_2)).

		Letσ_ibe local sections ofψ_i : _i→Ł_i.
		The map(x_1, g, x_2)  (σ_1(x_1), θ_2(σ_2(x_2))^-1 g θ_1(σ_1(x_1)), σ_2(x_2))is a local section of (<ref>).
		A straightforward computation then shows that the induced map_1 __2 ⟶φ̃^*(Ł_1 _Ł_2)is a diffeomorphism.
	
		Consider 1-shifted Lagrangian correspondences
		
    [row sep=2em,between origins,column sep=4em,between origins]
    				   Ł_1 dldr      Ł_2 dldr   
    _1       _2       _3

		and
		
    [row sep=2em,between origins,column sep=4em,between origins]
    				   Ł_1' dldr      Ł_2' dldr   
    _1'       _2'       _3'
    			,

		where _1 ⟵Ł_1 ⟶_2 is Lagrangially Morita equivalent to _1' ←Ł_1' →_2', and _2 ⟵Ł_2 ⟶_3 is Lagrangially Morita equivalent to _2' ←Ł_2' →_3'.
		Then (<ref>) is transverse if and only if (<ref>) is transverse.
		In this case, the compositions
		
    [row sep=3em,between origins,column sep=5em,between origins]
    				   Ł_1 __2Ł_2 dldr   
    _1       _3
    			    and    [row sep=3em,between origins,column sep=5em,between origins]
    				   Ł_1' __2'Ł_2' dldr   
    _1'       _3'

		are Lagrangially Morita equivalent.
	
		Lemma <ref> implies the following: (<ref>) is transverse if and only if (<ref>) is transverse, in which case the homotopy fibre products Ł_1 __2Ł_2 and Ł_1' __2'Ł_2' are Morita equivalent as Lie groupoids.
		It remains to check compatibility with the 1-shifted Lagrangian structures.
		We have a 2-commutative diagram
		
    [row sep=7em,between origins,column sep=7em,between origins]
    				(_1, ω_1, ϕ_1)     (Ł_1, γ_1) [swap]lμ_11rμ_12[Rightarrow,shorten=]dlθ_11[Rightarrow,shorten=]drθ_12    (_2, ω_2, ϕ_2)     (Ł_2, γ_2) [swap]lμ_22rμ_23[Rightarrow,shorten=]dlθ_22[Rightarrow,shorten=]drθ_23    (_3, ω_3, ϕ_3) 
    
    				(_̋1, δ_1) [swap]uφ_1dφ_1'   _1 [swap]lν_11rν_12[swap]uψ_1dψ_1'    (_̋2, δ_2) [swap]uφ_2dφ_2'   _2 [swap]lν_22rν_23[swap]uψ_2dψ_2'    (_̋3, δ_3) [swap]uφ_3dφ_3'
    
    				(_1', ω_1', ϕ_1')     (Ł_1', γ_1') lμ_11'[swap]rμ_12'[Rightarrow,shorten=,swap]ulθ_11'[Rightarrow,shorten=,swap]urθ_12'    (_2', ω_2', ϕ_2')     (Ł_2', γ_2') lμ_22'[swap]rμ_23'[Rightarrow,shorten=,swap]ulθ_22'[Rightarrow,shorten=,swap]urθ_23'    (_3', ω_3', ϕ_3')

		of Lagrangian Morita equivalences.
		In other words, the vertical maps are Morita morphisms and the following hold:
		
    φ_i^*ω_i - φ_i'^*ω_i'    = ^*δ_i - ^*δ_i   for  i = 1, 2, 3 ; 
    φ_i^*ϕ_i - φ_i'^*ϕ_i'    = -dδ_i   for  i = 1, 2, 3 ; 
    ψ_1^*γ_1 - ψ_1'^*γ_1'    = ν_11^*δ_1 - ν_12^*δ_2 - θ_11^*ω_1 + θ_12^*ω_2 + θ_11'^*ω_1' - θ_12'^*ω_2'; 
    ψ_2^*γ_2 - ψ_2'^*γ_2'    = ν_22^*δ_2 - ν_23^*δ_3 - θ_22^*ω_2 + θ_23^*ω_3 + θ_22'^*ω_2' - θ_23'^*ω_3'.

		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		Taking homotopy fibre products in (<ref>) yields the 2-commutative diagram
		
    [row sep=6em,between origins,column sep=14em,between origins]
    				(_1, ω_1, ϕ_1)     (Ł_1 __2Ł_2, _Ł_1^*γ_1 + _Ł_2^*γ_2) [swap]lμ_11_Ł_1rμ_23_Ł_2[Rightarrow,shorten=]dlθ_11__1[Rightarrow,shorten=]drθ_23__2    (_3, ω_3, ϕ_3) 
    
    				(_̋1, δ_1) [swap]uφ_1dφ_1'   _1 __̋2_2 [swap]lν_11__1rν_23__2[swap]uψdψ'    (_̋2, δ_2) [swap]uφ_3dφ_3'
    
    				(_1', ω_1', ϕ_1')     (Ł_1' __2'Ł_2', _Ł_1'γ_1' + _Ł_2'γ_2') lμ_11'_Ł_1'[swap]rμ_23'_Ł_2'[Rightarrow,shorten=,swap]ulθ_11' __1[Rightarrow,shorten=,swap]urθ_23' __2    (_3', ω_3', ϕ_3')
    			,

		where
		
    ψ(m_1, h_2, m_2)    = (ψ_1(m_1), θ_22((m_2))^-1·φ_2(h_2) ·θ_12((m_1)), ψ_2(m_2)) 
    ψ'(m_1, h_2, m_2)    = (ψ_1'(m_1), θ_22'((m_2))^-1·φ_2'(h_2) ·θ_12'((m_1)), ψ_2'(m_2))

		for all (m_1, h_2, m_2) ∈_1 __̋2_2.
		By Lemma <ref>, ψ and ψ' are Morita morphisms.
		Since _Ł_iψ = ψ_i __i and _Ł_i'ψ' = ψ_i' __i for i = 1, 2, we have
		
    ψ^*(_Ł_1^*γ_1 + _Ł_2^*γ_2)
    			-
    			ψ'^*(_Ł_1'^*γ_1' + _Ł_2'^*γ_2')
    			
       =
    			__1^*(ψ_1^*γ_1 - ψ_1'^*γ_1') + __2^*(ψ_2^*γ_2 - ψ_2'^*γ_2') 
       =
    			__1^*(ν_11^*δ_1 - ν_12^*δ_2 - θ_11^*ω_1 + θ_12^*ω_2 + θ_11'^*ω_1' - θ_12'^*ω_2') 
        +__2^*(ν_22^*δ_2 - ν_23^*δ_3 - θ_22^*ω_2 + θ_23^*ω_3 + θ_22'^*ω_2' - θ_23'^*ω_3') 
       =
    			(ν_11__1)^*δ_1 - (ν_23__2)^*δ_2
    			-(θ_11__1)^*ω_1
    			+(θ_23__2)^*ω_3
    			+(θ_11'__1)^*ω_1'
    			-(θ_23'__2)^*ω_3';

		The second equality follows from (<ref>) and (<ref>), and last equality follows from the fact that ν_12__1 = ν_22__2 on _1 __̋2_2.
		We conclude that (<ref>) is a Lagrangian Morita equivalence.
	
		Let
		
    [row sep=2em,between origins,column sep=4em,between origins]
    				   Łdldr   
    _1       _2,

		be a 1-shifted Lagrangian correspondence. Regard _1 ⟵_1 ⟶_1 as a 1-shifted Lagrangian correspondence via the trivial 2-form on _1.
		Then the pair
		
    [row sep=2em,between origins,column sep=4em,between origins]
    				   _1 dldr      Łdldr      
    _1       _1       _2

		is transverse, and the composition
		
    [row sep=3em,between origins,column sep=5em,between origins]
    				   _1 __1Łdldr      
    _1       _2

		is weakly equivalent to (<ref>).
		The analogous statement for composition on the right by _2 ⟵_2 ⟶_2 also holds.
	
		Denote the quasi-symplectic structure on _i by (ω_i, ϕ_i) for i = 1, 2, and the 1-shifted Lagrangian structure on (<ref>) by γ.
		Let us also denote also the two morphisms in (<ref>) by μ_1:Ł⟶_1 and μ_2:Ł⟶_2.
		To establish composability in (<ref>), we need to show that the maps
		
    _1×Ł[swap]dr×μ_1      _1dl(, )
       _1×_1

		are transverse.
		Let g ∈_1 be such that ((g), (g)) = (x, μ_1(y)) for some (x, y) ∈_1×Ł, and let (u, v) ∈ T_(g)_1× T_(g)_1.
		Since the target map of a Lie groupoid is a submersion, we can write v = _* (w) for some w ∈ T_g_1.
		It follows that (u, v) = (_*(u - _*(w)), μ_1*(0)) + (_*(w), _*(w)), proving transversality.
		We also get that the projection
		
    _Ł : _1 __1ŁŁ

		is a Morita morphism, fitting into a 2-commutative diagram
		
    [row sep=5em,between origins,column sep=5em,between origins]
    				[Rightarrow,shorten=42pt]ddr__1   _1 __1Ł[swap]dl__1drμ_2_Łdd_Ł   
    _1       _2 
       Łulμ_1[swap]urμ_2   ,

		where __1 : __1⇒μ_1 _Ł is the projection (_1 __1Ł) = _1_1μ_1Ł→_1.
		In the framework of weak equivalence of 1-shifted Lagrangians as in (<ref>), we can write (<ref>) as
		
    [row sep=5em,between origins,column sep=5em,between origins]
    				   _1 __1Ł[swap]dldr_Ł   
    _1 __1Ł[swap]dr__1×μ_1 _Ł[Rightarrow,shorten=20pt]rr__1×μ_2_Ł      Łdlμ_1 ×μ_2
       _1 ×_2^-.

		By the definition of composition, the 1-shifted Lagrangian structure on the composition (<ref>) is given by __1^*0 + _Ł^*γ - __1^*ω_1.
		The statement that (<ref>) is a Lagrangian Morita equivalence then amounts to the identity
		
    _Ł^*γ - ^*(__1^*0 + _Ł^*γ - __1^*ω_1) = (__1×μ_2 _Ł)^*(__1^*ω_1 - __2^*ω_2).

		This identity from the fact that ^*ω_2 = 0 for a multiplicative form ω_2.
	













































	
	

 §.§ Strong fibre products and transversality

	Consider 1-shifted Lagrangian correspondences
	
    [row sep=2em,between origins,column sep=4em,between origins]
    			   Ł_1 [swap]dldr      Ł_2 [swap]dldr   
    _1       _2       _3.

	It is sometimes useful to consider the strong fibre productŁ_1 __2 Ł_2, defined as the standard set-theoretical fibre product on both arrows and objects.
	We say that the 1-shifted Lagrangian correspondences (<ref>) are strongly transverse if the fibre product on arrows is transverse.
	As in <cit.>, this implies that the fibre product on object is also transverse, and that the vector bundle morphisms
	
    [row sep=2em,between origins,column sep=4em,between origins]
    		A_Ł_1dr       A_Ł_2dl
        A__2

	are transverse.










	It follows that the strong fibre product is a Lie groupoid endowed with the structure of a 1-shifted Lagrangian correspondence
	
    [row sep=3em,between origins,column sep=5em,between origins]
    		   Ł_1 __2Ł_2 [swap]dldr   
    _1       _3

	with respect to_Ł_1^*γ_1 + _Ł_2^*γ_2, whereγ_iare the 1-shifted Lagrangian structures onŁ_1andŁ_2, respectively <cit.>.
	The following proposition shows that the homotopy and strong fibre products are equivalent in many situations.
	
	
		Suppose that the 1-shifted Lagrangian correspondences (<ref>) are transverse and strongly transverse.
		Suppose also that the map
		
    Ł_1μ_12μ_22Ł_2   Ł_1μ_12_2μ_22Ł_2
    
    			(l_1, l_2)     ((l_1), μ_22(l_2)^-1μ_12(l_1), (l_2))

		is a surjective submersion.
		Then the homotopy fibre product Ł_1 __2Ł_2 and the strong fibre product Ł_1 __2Ł_2 are weakly equivalent as 1-shifted Lagrangian correspondences from _1 to _3.
	
		We have a commutative diagram
		
    [row sep=3em,between origins, column sep=4em,between origins]
    			Ł_1 __2Ł_2 rrψdr      Ł_1 __2Ł_2 dl
       _1 ×_3^-,

		where
		
    ψ : Ł_1μ_12μ_22Ł_2   Ł_1μ_12_2μ_22Ł_2 
    
    			(l_1, l_2)     (l_1, _μ_12((l_1)), l_2) and
    ψ : Ł_1μ_12μ_22Ł_2   Ł_1μ_12_2μ_22Ł_2
    
    			(x_1, x_2)     (x_1, _μ_12(x_1), x_2).

		Note that ψ^*(_Ł_1^*γ_1 + _Ł_2^*γ_2 - __2^*ω_2) = _Ł_1^*γ_1 + _Ł_2^*γ_2, as ^*ω_2 = 0.
		It suffices to check that ψ is an essential equivalence.
		This amounts to checking the following.
		
  * 
			The map
			
    ∘_(Ł_1 __2Ł_2) :
    				(Ł_1μ_12μ_22Ł_2) ψ 
    				(Ł_1μ_12_2μ_22Ł_2)    Ł_1μ_12_2μ_22Ł_2
    
    				((x_1, x_2), (l_1, g, l_2))     ((l_1), μ_22(l_2) · g ·μ_12(l_1)^-1, (l_2))

			is a surjective submersion.
			
  * 
			The map (Ł_1 __2Ł_2)→ (ψ)^*(Ł_1 __2Ł_2) is a diffeomorphism.
		
		To show <ref>, note that we have a map
		
    Ł_1μ_12μ_22Ł_2   
    			(Ł_1μ_12μ_22Ł_2) ψ 
    			(Ł_1μ_12_2μ_22Ł_2) 
    
    			(l_1, l_2)     (((l_1), (l_2)), (l_1, _μ_12((l_1)), l_2)),

		whose composition with (<ref>) is (<ref>).
		Since (<ref>) is a surjective submersion, so is (<ref>).
		For <ref>, we need to check that the map
		
    Ł_1μ_12μ_22Ł_2    (Ł_1μ_12μ_22Ł_2) ψ (Ł_1μ_12_2μ_22Ł_2) ψ (Ł_1μ_12μ_22Ł_2) 
    
    			(l_1, l_2)     (((l_1), (l_2)), (l_1, _μ_12((l_1)), l_2), ((l_1), (l_2)))

		is a diffeomorphism.
		But it has an explicit inverse, given by ((x_1, x_2), (l_1, g, l_2), (y_1, y_2))  (l_1, l_2).
	

§ THE 1-SHIFTED WEINSTEIN SYMPLECTIC CATEGORY

	
	We begin this section with a rough overview of the Weinstein symplectic “category" and its Wehrheim–Woodward completion. This gives context for our subsequent definition of the1-shifted Weinstein symplectic “category". Using the basic approach of Wehrheim–Woodward to the Weinstein symplectic category, we completeto a symmetric monoidal category. One may work in the smooth or holomorphic categories, as with the previous section.
	
	

 §.§ The Weinstein symplectic “category"

	Two morphisms in a category can be composed if and only if the source of one coincides with the target of the other. By weakening this to a necessary condition for composing morphisms, one obtains the definition of a “category”.[Some authors would instead call this a precategory.] Two morphisms in a “category" are called composable if their composition is defined. A prominent instance of this discussion is Weinstein's symplectic “category”<cit.>; its objects are symplectic manifolds, and its morphisms are Lagrangian correspondences. While one can compose two Lagrangian correspondences as relations between sets, the result need not be a Lagrangian correspondence. These correspondences are called composable if they satisfy transversality conditions sufficient to ensure that their set-theoretic composition is a Lagrangian correspondence.     
	While the Weinstein symplectic “category” is not a genuine category, Wehrheim–Woodward <cit.> show that it can be completed into one by defining morphisms as sequences of composable morphisms up to a certain equivalence relation. We implement a similar approach for the 1-shifted version of the Weinstein symplectic “category".
	
	

 §.§ The 1-shifted Weinstein symplectic “category”.

	We define a 1-shifted version of Weinstein's symplectic “category”, denoted, as follows.
	Quasi-symplectic groupoids constitute the objects of. A morphism from_1to_2inis a weak equivalence class of 1-shifted Lagrangian correspondences from_1to_2. 

















	We say that two morphisms
	
    [row sep=2em,between origins,column sep=4em,between origins]
    		   Ł_1 dldr      Ł_2 dldr   
    _1       _2       _3

	inare composable if their homotopy fibre productŁ_1 __2 Ł_2is transverse.
	In this case, we define the compositionŁ_2 ∘Ł_1as the weak equivalence class ofŁ_1 __2 Ł_2.
	Proposition <ref> implies that this morphism does not depend on the representatives chosen for the weak equivalence classes being composed.
	
	Letbe a quasi-symplectic groupoid. The identity morphism_:⟶is the canonical 1-shifted Lagrangian correspondence⟵⟶(Example <ref>). The content of this statement is that every morphismŁ: ⟶$̋ in  is composable with _ and _$̋, andŁ∘_≃Ł≃_∘̋Ł; see Lemma <ref>.
	
	

 §.§ Extension of  to a symmetric monoidal category

	One may extendto a symmetric monoidal categoryin two ways. The first is a manifold-theoretic extension afforded by Wehrheim–Woodward; see <cit.> and <cit.>. The second, due to Calaque <cit.>, is algebro-geometric; one replaces algebraic quasi-symplectic groupoids by their associated 1-shifted symplectic stacks, and uses derived fibre products to form a symmetric monoidal category. 

The precise way in which we extendto a categoryis irrelevant since all computations will be done on composable morphisms in.
We therefore only need to prove the existence of an extension.
We do this in the differential-geometric context by adapting the Wehrheim–Woodward approach, as we now explain.
	
	
	The Wehrheim–Woodward approach is as follows.
	An object ofis a quasi-symplectic groupoid.
	A morphism fromto'is a sequence of quasi-symplectic groupoids_0, _1, …, _rwith_0 = and_r = ', together with 1-shifted Lagrangian correspondences_i-1 ⟵Ł_i ⟶_ifori = 1, …, r, up to the equivalence relation generated by
	
    (
    		[row sep=1.5em,between origins,column sep=2em,between origins]
    			      Ł_i ddlddr      Ł_i+1ddlddr      
    ⋯                  ⋯
       _i-1      _i      _i+1   )
    		  ∼  (
    		[row sep=1.5em,between origins,column sep=2em,between origins]
    			      Ł_i __iŁ_i+1ddlddr      
    ⋯            ⋯
       _i-1      _i+1   )

	ifŁ_i __i Ł_i+1is transverse and
	
    (
    		[row sep=1.5em,between origins,column sep=2em,between origins]
    			      Ł_i ddlddr      
    ⋯            ⋯
       _i-1      _i   )
    		  ∼  (
    		[row sep=1.5em,between origins,column sep=2em,between origins]
    			      Ł_i' ddlddr      
    ⋯            ⋯
       _i-1      _i   )

	ifŁ_iandŁ_i'are weakly equivalent.
	As in <cit.>, this forms a category with composition given by the concatenation of sequences.
	
	The categoryturns out to carry a symmetric monoidal structure 
    ⊗ : ×.

	It is given by the Cartesian product on the level of objects.
	For two morphisms
	
    [
    	[row sep=2em,between origins,column sep=2em,between origins]
    		   Ł_1 dldr      Ł_2 dldr      Ł_r dldr   
    _0       _1       ⋯      _r
    	]
    	  and  [
    	[row sep=2em,between origins,column sep=2em,between origins]
    		   _1 dldr      _2 dldr      _s dldr   
    _̋0       _̋1       ⋯      _̋s
    	],

	one takes the tensor product as follows. Augment the morphism of smallest length with identity morphisms on its right, until to its length matches that of other morphism. Proceed to take Cartesian products of quasi-symplectic groupoids_i ×_̋iand 1-shifted LagrangiansŁ_i ×_i. Using the more concise notation[Ł_1, …, Ł_r]and[_1, …, _s]for the morphisms, assumingr ≤s, and using juxtaposition to indicate Cartesian products, one has
	
	
    [Ł_1, …, Ł_r] ⊗ [_1, …, _s] = [Ł_1 _1, …, Ł_r_r, _r_r+1, …, _r_s].

	The right-hand side of (<ref>) does not depend on the representatives chosen for the two morphisms being composed.
	
	The right-hand side of (<ref>) it clearly invariant under (<ref>).
	To establish invariance under (<ref>), let 1 ≤ i < r be such that Ł_i and Ł_i+1 are transverse. It follows that
	
    [Ł_1, …, Ł_i, Ł_i+1, …, Ł_r]
    	=
    	[Ł_1, …, Ł_i ∘Ł_i+1, …, Ł_r].

	We need to check that both presentations of this morphism yield the same tensor product with [_1, …, _s], i.e. that
	
    [Ł_1_1, …, Ł_i_i, Ł_i+1_i+1, …, Ł_r_r, _r_r+1, …, _r_s] 
       = [Ł_1_1, …, (Ł_i∘Ł_i+1)_i, Ł_i+2_i+1, …, Ł_r_r-1, _r_r, …, _r_s].

	
	By inserting identities, (<ref>) is equal to
	
    [Ł_1_1, …, Ł_i_i, Ł_i+1_̋i_(Ł_i ∘Ł_i+1)_i, _i+1_i+1, Ł_i+2_̋i+1_Ł_i+2_i+1, …, _r-1_r-1, Ł_r_̋r-1_Ł_r_r-1, _r_r, …, _r_s].

	Subsequently performing composition in the other set of pairs
	
    [Ł_1_1, …, Ł_i_i, Ł_i+1_̋i, _i+1_i+1_Ł_i+1_i+1, Ł_i+2_̋i+1, …_Ł_i+2_i+2…, _r-1_r-1_Ł_r-1_r-1, Ł_r_̋r-1, _r_r_Ł_r_r, …, _r_s],

	we get back (<ref>).
	This completes the proof.
	
	
	The unit object inis a point⋆, viewed as a quasi-symplectic groupoid. Given quasi-symplectic groupoids,$̋, and , the associator α_,,̋ : × (×̋) ⟶ (×)̋× is the groupoid ××̋, regarded as a 1-shifted Lagrangian correspondence from × (×̋) to (×)̋×.
	The left identity λ_ : ⋆×⟶ is , as a 1-shifted Lagrangian correspondence. A similar description applies for the right identity ρ_ : ×⋆⟶.
	
	The preceding discussion makes it clear that (<ref>) is a monoidal structure on .
	To address the symmetric structure, let  and $̋ be quasi-symplectic groupoids. The braiding×⟶̋×̋is the groupoid×$̋ with trivial 2-form on its base and the obvious morphisms to ×$̋ and×̋.
	The axioms of a symmetric monoidal category are immediate.
	
	

§ TQFTS VALUED IN THE 1-SHIFTED WEINSTEIN SYMPLECTIC CATEGORY


	This section is largely concerned with proving Main Theorem <ref>. We begin by recalling the equivalence between2-dimensional TQFTs in a symmetric monoidal category and commutative Frobenius objects in the same category. This framework allows us to prove that every abelian symplectic groupoid determines a commutative Frobenius object in, whose product is induced by groupoid multiplication. To extend this result to a larger class of quasi-symplectic groupoids, we call a quasi-symplectic groupoid abelianizable if it Morita equivalent to an abelian symplectic groupoid. We then use a notion of Morita transfer to prove that every abelianizable quasi-symplectic groupoid determines a commutative Frobenius object in. To conclude, we prove that a quasi-symplectic groupoid admitting an admissible global slice is necessarily abelianizable. All constructions work overℝorℂ, as in the previous two sections.   
	
	

 §.§ 2-dimensional TQFTs and commutative Frobenius objects

	Let us first recall the equivalence between 2-dimensional TQFTs and commutative Frobenius objects. A standard reference for this material is the book <cit.>; see also <cit.>, <cit.>, and <cit.>.

Let_2be the category of 2-dimensional cobordisms, i.e. objects are compact 1-dimensional oriented manifolds and morphisms are cobordisms between them.
	Suppose that(𝐂, ⊗, I, B)is a symmetric monoidal category, where⊗is the monoidal product,Iis the unit object, andBis the brading.
	A 2-dimensional topological quantum field theory valued in 𝐂 is a symmetric monoidal functor
	
    _2 𝐂.

	Such a functor is more easily described via the notion of commutative Frobenius object in𝐂, as we now recall. One finds that_2is generated by the six morphisms
	
    [
    		baseline=-2.5pt,
    		every tqft/.append style=
    			transform shape, rotate=90, tqft/circle x radius=4pt,
    			tqft/circle y radius= 2pt,
    			tqft/boundary separation=0.6cm, 
    			tqft/view from=incoming,
    		
    		]
    		[
    		tqft/cap,
    		name=d,
    		every incoming lower boundary component/.style=draw,
    		every outgoing lower boundary component/.style=draw,
    		every incoming boundary component/.style=draw,
    		every outgoing boundary component/.style=draw,
    		cobordism edge/.style=draw,
    		cobordism height= 1cm,
    		];
    		 (d-outgoing boundary)+(-0.12,-1) node η;
    	    [
    		baseline=6pt,
    		every tqft/.append style=
    			transform shape, rotate=90, tqft/circle x radius=4pt,
    			tqft/circle y radius= 2pt,
    			tqft/boundary separation=0.6cm, tqft/view from=incoming,
    		
    		]
    		[
    		tqft/reverse pair of pants,
    		name=d,
    		every incoming lower boundary component/.style=draw,
    		every outgoing lower boundary component/.style=draw,
    		every incoming boundary component/.style=draw,
    		every outgoing boundary component/.style=draw,
    		cobordism edge/.style=draw,
    		cobordism height= 1cm,
    		];
    		 (d-outgoing boundary)+(-0.5,-1) node μ;
    	    [
    		baseline=-3pt,
    		every tqft/.append style=
    			transform shape, rotate=90, tqft/circle x radius=4pt,
    			tqft/circle y radius= 2pt,
    			tqft/boundary separation=0.6cm, tqft/view from=incoming,
    		
    		]
    		[
    		tqft/cylinder,
    		name=d,
    		every incoming lower boundary component/.style=draw,
    		every outgoing lower boundary component/.style=draw,
    		every incoming boundary component/.style=draw,
    		every outgoing boundary component/.style=draw,
    		cobordism edge/.style=draw,
    		cobordism height= 1cm,
    		];
    		 (d-outgoing boundary)+(-0.5,-1) node ι;
    	    [
    		baseline=-3pt,
    		every tqft/.append style=
    			transform shape, rotate=90, tqft/circle x radius=4pt,
    			tqft/circle y radius= 2pt,
    			tqft/boundary separation=0.6cm, tqft/view from=incoming,
    		
    		]
    		[
    		tqft/pair of pants,
    		name=d,
    		every incoming lower boundary component/.style=draw,
    		every outgoing lower boundary component/.style=draw,
    		every incoming boundary component/.style=draw,
    		every outgoing boundary component/.style=draw,
    		cobordism edge/.style=draw,
    		cobordism height= 1cm,
    		];
    		 (d-outgoing boundary)+(-0.5,-0.75) node δ;
    	    [
    		baseline=-2.5pt,
    		every tqft/.append style=
    			transform shape, rotate=90, tqft/circle x radius=4pt,
    			tqft/circle y radius=2pt,
    			tqft/boundary separation=0.6cm, 
    			tqft/view from=incoming,
    		
    		]
    		[
    		tqft/cup,
    		name=d,
    		every incoming lower boundary component/.style=draw,
    		every outgoing lower boundary component/.style=draw,
    		every incoming boundary component/.style=draw,
    		every outgoing boundary component/.style=draw,
    		cobordism edge/.style=draw,
    		cobordism height=1cm,
    		];
    		 (d-incoming boundary)+(0.12,-1) node ϵ;
    	    [
    		baseline=5pt,
    		every tqft/.append style=
    			transform shape, rotate=90, tqft/circle x radius=4pt,
    			tqft/circle y radius= 2pt,
    			tqft/cobordism height=1cm, tqft/view from=incoming,
    			tqft/boundary separation=1cm,
    		
    		]
    		 [
    		tqft/cylinder to next,
    		name=d,
    		every incoming lower boundary component/.style=draw,
    		every outgoing lower boundary component/.style=draw,
    		cobordism edge/.style=draw, 
    		every incoming boundary component/.style=draw,
    		every outgoing boundary component/.style=draw,
    		];
    		 [
    		tqft/cylinder to prior,
    		every incoming lower boundary component/.style=draw,
    		every outgoing lower boundary component/.style=draw,
    		every incoming boundary component/.style=draw,
    		every outgoing boundary component/.style=draw,
    		cobordism edge/.style=draw,
    		at=((d-incoming boundary)+(0, 0.5))];
    		 (d-outgoing boundary)+(-0.5,-1.25) node τ;

	read from left to right (e.g.η: ∅⟶S^1), subject to the relations
	
    [
    			every tqft/.append style=
    				transform shape, rotate=90, tqft/circle x radius=4pt,
    				tqft/circle y radius=2pt,
    				tqft/boundary separation=0.5cm, tqft/view from=incoming,
    			
    			]
    			[
    			tqft/cap,
    			name=a,
    			cobordism height=1cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			];
    			[
    			tqft/cylinder,
    			name=b,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			at=((a-outgoing boundary)+(-0.75,0.5)) ,
    			];
    			[
    			tqft/reverse pair of pants,
    			name=c,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			at=((a-outgoing boundary)+(0,0)) ,
    			];
    			 (c-outgoing boundary)+(0.5,0)  node ;
    			[
    			tqft/cylinder,
    			name=e,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			at=((c-outgoing boundary)+(1,0)) ,
    			];
    			 (c-outgoing boundary)+(2.25,0)  node ;
    			[
    			tqft/cap,
    			name=ar,
    			cobordism height=1cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			at=((4.25, 0.5)),
    			];
    			[
    			tqft/cylinder,
    			name=br,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			at=((ar-outgoing boundary)+(-0.75,-0.5)) ,
    			];
    			[
    			tqft/reverse pair of pants,
    			name=cr,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			at=((ar-outgoing boundary)+(0,-0.5)) ,
    			];
    		
       [
    			every tqft/.append style=
    				transform shape, rotate=90, tqft/circle x radius=4pt,
    				tqft/circle y radius=2pt,
    				tqft/boundary separation=0.5cm, tqft/view from=incoming,
    			
    			]
    			[
    			tqft/pair of pants,
    			name=a,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			];
    			[
    			tqft/cylinder,
    			name=b,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			at=((a-outgoing boundary)+(0,0.5)) ,
    			];
    			[
    			tqft/cup,
    			name=c,
    			cobordism height=1cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			at=((a-outgoing boundary)+(0,0)) ,
    			];
    			 (b-outgoing boundary)+(0.5,-0.25) node ;
    			[
    			tqft/cylinder,
    			name=e,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			at=((c-outgoing boundary)+(0.75,-0.25)) ,
    			];
    			 (e-outgoing boundary)+(0.5,0) node ;
    			[
    			tqft/pair of pants,
    			name=ar,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			at=((c-outgoing boundary)+(2.5,-0.25)) ,
    			];
    			[
    			tqft/cylinder,
    			name=br,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			at=((ar-outgoing boundary)+(0,0)) ,
    			];
    			[
    			tqft/cup,
    			name=cr,
    			cobordism height=1cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			at=((ar-outgoing boundary)+(0,0.5)) ,
    			];
    		
       [
    			every tqft/.append style=
    				transform shape, rotate=90, tqft/circle x radius=4pt,
    				tqft/circle y radius=2pt,
    				tqft/boundary separation=0.5cm, tqft/view from=incoming,
    			
    			]
    			[
    			tqft/cylinder to prior,
    			name=a,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			boundary separation=1cm,
    			at=((c-outgoing boundary)+(-0.75,0.75)),
    			];
    			[
    			tqft/cylinder to next,
    			name=b,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			boundary separation=1cm,
    			at=((c-outgoing boundary)+(-0.75,0.25)) ,
    			];
    			[
    			tqft/reverse pair of pants,
    			name=c,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			at=((a-outgoing boundary)+(0,0)) ,
    			];
    			 (c-outgoing boundary)+(0.5,0)  node ;
    			[
    			tqft/reverse pair of pants,
    			name=d,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			at=((c-outgoing boundary)+(1,-0.25)) ,
    			];
    		
       [
    			every tqft/.append style=
    				transform shape, rotate=90, tqft/circle x radius=4pt,
    				tqft/circle y radius=2pt,
    				tqft/boundary separation=0.5cm, tqft/view from=incoming,
    			
    			]
    			[
    			tqft/pair of pants,
    			name=a,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			at=((a-outgoing boundary)+(0,0)) ,
    			];
    			[
    			tqft/cylinder to prior,
    			name=b,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			boundary separation=1cm,
    			at=((a-outgoing boundary)+(0,0.5)),
    			];
    			[
    			tqft/cylinder to next,
    			name=c,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			boundary separation=1cm,
    			at=((a-outgoing boundary)+(0,0)) ,
    			];
    			 (b-outgoing boundary)+(0.5,0.25)  node ;
    			[
    			tqft/pair of pants,
    			name=d,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			at=((c-outgoing boundary)+(1,-0.25)) ,
    			];
    		
       [
    			every tqft/.append style=
    				transform shape, rotate=90, tqft/circle x radius=4pt,
    				tqft/circle y radius=2pt,
    				tqft/boundary separation=0.5cm, tqft/view from=incoming,
    			
    			]
    			[
    			tqft/pair of pants,
    			name=a,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			];
    			[
    			tqft/cylinder to prior,
    			name=b,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			at=((a-outgoing boundary)+(-0.75,-0.25)) ,
    			];
    			[
    			tqft/cylinder to prior,
    			name=c,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			at=((a-outgoing boundary)+(0,0.5)) ,
    			];
    			[
    			tqft/reverse pair of pants,
    			name=d,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			at=((a-outgoing boundary)+(0,-0.5)) ,
    			];
    			 (c-outgoing boundary)+(0.5,-0.25)  node ;
    			[
    			tqft/reverse pair of pants,
    			name=e,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			at=((a-outgoing boundary)+(1.75,-0.25)) ,
    			];
    			[
    			tqft/pair of pants,
    			name=f,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			at=((e-outgoing boundary)+(0,0)) ,
    			];
    			 (f-outgoing boundary)+(0.5,0.25)  node ;
    			[
    			tqft/pair of pants,
    			name=ar,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			at=((f-outgoing boundary)+(1,0)) ,
    			];
    			[
    			tqft/cylinder to next,
    			name=br,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			at=((ar-outgoing boundary)+(-0.75,0.75)) ,
    			];
    			[
    			tqft/reverse pair of pants,
    			name=cr,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			at=((br-outgoing boundary)+(0,-0.5)) ,
    			];
    			[
    			tqft/cylinder to next,
    			name=cr,
    			cobordism height=0.75cm,
    			every incoming boundary component/.style=draw,
    			every outgoing boundary component/.style=draw,
    			cobordism edge/.style=draw,
    			at=((ar-outgoing boundary)+(0,0)) ,
    			];

	and additional relations capturing the fact that the cylinderιis the identity.
	 A 2-dimensional topological quantum field theory valued𝐂is thereby equivalent to the following: a choice of objectX ∈𝐂and four morphisms
	
    X_η   : I → X       (unit),
    
    		X_μ   : X ⊗ X → X       (multiplication), 
    
    		X_δ   : X → X ⊗ X       (comultiplication), 
    
    		X_ϵ   : X → I       (counit),

	subject to the relations indicated by (<ref>)–(<ref>), where the cylinderιis mapped to the identity_X : X ⟶Xand the twistτis mapped to the braidingB_X : X ⊗X ⟶X ⊗X<cit.>.
	For example, (<ref>) translates to the requirement thatX_μ∘B_X = X_μ. 
	
	A tuple(X,X_η,X_μ,X_δ,X_ϵ)of an objectXin𝐂and morphismsX_η, X_μ, X_δ, X_ϵsatisfying the analogues of (<ref>)–(<ref>) in𝐂is called a commutative Frobenius object in 𝐂.
	
	
	

 §.§ The case of abelian symplectic groupoids

	The following notion will be instrumental to constructing commutative Frobenius objects in.
	
	
		A groupoid is abelian if = and all isotropy groups are abelian.
	
	
	Our strategy is to first construct a commutative Frobenius object infor each abelian symplectic groupoid, and subsequently use Morita transfer to get a TQFT for every abelianizable quasi-symplectic groupoid.
	
	Let(, ω) Nbe an abelian symplectic groupoid.
	
	The multiplication morphism_μ∈_(×, )is given by multiplication in, i.e. we define_μto be the set*  of composable arrows, together with the morphisms
	
    [row sep=3em,between origins,column sep=4em,between origins]
    			    * [hook']dldr   
    ×      .

	Sinceis abelian, the set* is a Lie groupoid overN.
	The multiplicativity ofωimplies that_μis a 1-shifted Lagrangian correspondence from×to, with respect to the trivial 2-form onN.
	Similarly, we define comultiplication_δ∈_(, ×)via
	
    [row sep=3em,between origins,column sep=4em,between origins]
    		    * [swap]dl[hook]dr   
          ×.

	The unit_η∈_(⋆, )is given by the identity section of, i.e. we define_ηto be the trivial groupoidN N, together with the morphisms
	
    [row sep=3em,between origins,column sep=4em,between origins]
    		    N dl[hook]dr   
    ⋆      .

	It follows that_ηis a 1-shifted Lagrangian correspondence with respect to the trivial 2-form onN.
	Similarly, the counit_ϵ∈_(, ⋆)is the trivial groupoid overN, together with
	
    [row sep=3em,between origins,column sep=4em,between origins]
    		    N [hook',swap]dldr   
          ⋆.

	
	We now verify that_η, _μ, _δ, _ϵsatisfy the relations (<ref>)–(<ref>) defining a commutative Frobenius object in.
	Adopt the notation^*n * ⋯* (ntimes) for the set ofn-composable arrows.
	
	
		Relations (<ref>) and (<ref>) hold in  for _η, _μ, _δ, _ϵ. For the first identity in (<ref>), we get that the 1-shifted Lagrangian correspondences
		
    [row sep=3em,between origins,column sep=4em,between origins]
    				    N ×[swap]dl_[hook]dr×       * [hook']dldr   
          ×

		are transverse, and that their composition is weakly equivalent to the identity ⟵⟶.
		Similar statements hold for the other three identities.
	
		We verify the first identity in (<ref>) only; the others are handled similarly.
		Transversality in (<ref>) is the statement that the maps
		
    [row sep=0em]
    				N^2 × N r    N^2 × N^2    ^2 l
    
    				(x, y, z) [mapsto]r    (x, y, z, z)    
        ((a), (b), (a), (b))     (a, b) [mapsto]l

		are transverse.
		We show that the intersection is weakly equivalent to the strong fibre product via Proposition <ref>.
		Strong transversality amounts to the maps
		
    [row sep=0em]
    			N ×r   ^2    ^*2l
    
    			(x, a) [mapsto]r    (x, a), (b, c)     (b, c) [mapsto]l

		being transverse; this is clear.
		Moreover, the map (<ref>) is the map
		
    ^*3   ^*2
    
    			(a, b, c)     (b^-1, c^-1a)

		which is also clearly a surjective submersion.
		The composition in (<ref>) is then weakly equivalent to the strong fibre product (N ×) _^2 ( * ) ≅.
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
	
		Relations (<ref>) and (<ref>) hold in  for _η, _μ, _δ, _ϵ.
		More precisely, (<ref>) is the statement that the 1-shifted Lagrangian correspondences
		
    [row sep=3em,between origins,column sep=4em,between origins]
    				   ×[swap]dldrswap       * [hook']dldr   
    ×      ×

		are transverse, and their composition is weakly equivalent to (<ref>).
		A similar statement holds for (<ref>).
	
		We verify (<ref>) only; the case of (<ref>) is similar.
		Transversality in (<ref>) amounts to the fact that the maps in (<ref>) are transverse.
		Strong transversality is also clear.
		With a view to applying Proposition <ref>, we note that (<ref>) corresponds to the surjective submersion
		
    ^*4   ^*2
    
    			(a, b, c, d)     (c^-1b, d^-1a).

		We conclude that the composition in (<ref>) is weakly equivalent to the strong fibre product (×) _^2 ( * ) = {((a, b), (c, d)) ∈ (×) × ( * ) : (b, a) = (c, d)}.
		The latter is isomorphic to *, together with the maps ×← * →, (a, b)  (a, b) ↦ ba = ab, i.e. to _μ.
	
		Relations (<ref>) hold in  for _η, _μ, _δ, _ϵ.
		More precisely, the first relation in (<ref>) is the statement that the pairs of 1-shifted Lagrangian correspondences
		
    [row sep=3em,between origins,column sep=4em,between origins]
    				    ( * ) ×[swap]dl×[hook]dr      × ( * ) [hook']dldr×   
    ×      ××      ×

		and
		
    [row sep=3em,between origins,column sep=4em,between origins]
    				    * [hook']dldr       * [swap]dl[hook]dr   
    ×            ×

		are both transverse, and that their compositions are weakly equivalent.
		A similar statement holds for the second identity.
	
		We show that both of these compositions are weakly equivalent to ^2, 2 using Proposition <ref>. Let us begin with (<ref>).
		Transversality is the statement that the maps
		
    [row sep=0em]
    			N^2 × N^2 r    N^3 × N^3    ^3 l
    
    			((x, y), (u, v)) [mapsto]r    ((x, x, y), (u, v, v))    
        (((a), (b), (c)), ((a), (b), (c)))     (a, b, c) [mapsto]l

		are transverse.
		Strong transversality is the statement that the maps
		
    [row sep=0em]
    			^*2×r   ^3    ×^*2l
    
    			((a, b), c) [mapsto]r    (a, b, c)     (a, (b, c)) [mapsto]l

		are transverse.
		The map (<ref>) in Proposition <ref> corresponds to
		
    ^*3 * ^*3   ^*3
    
    			((a_1, b_1, c_1), (a_2, b_2, c_2))     (a_2^-1a_1, b_2^-1b_1, c_2^-1c_1),

		which is a surjective submersion.
		It follows that the composition in (<ref>) is weakly equivalent to the strong fibre product (( * ) ×) _^3 (× ( * )).
		The latter can be identified with *  *, together with the maps 
		
    [row sep=3em,between origins,column sep=4em,between origins]
    				    *  * dldr   
    ×      ×    [row sep=3em,between origins,column sep=4em,between origins]
    				    (a, b, c) [mapsto]dl[mapsto]dr   
    
    				(ab, c)        (a, bc).

		
		
		We now consider (<ref>).
		Transversality amounts to the maps
		
    N × N r    N × N    l(, )

		being transverse.
		Strong transversality amounts to the maps
		
    [row sep=0em]
    			^*2r      ^*2l
    
    			(a, b) [mapsto]r    ab    
        cd     (c, d) [mapsto]l

		being transverse.
		The map (<ref>) in Proposition <ref> corresponds to
		
    ^*4   
    
    			(a, b, c, d)     abc^-1d^-1,

		a surjective submersion.
		The composition in (<ref>) is therefore weakly equivalent to the strong fibre product ( * ) _ ( * ) = {(a_1, a_2, b_1, b_2) ∈^*4 : a_1 a_2 = b_1 b_2}.
		There is an isomorphism from (<ref>) to the latter given by (a, b, c)  (ab, c, a, bc).
	
		If  is an abelian symplectic groupoid, then (,_η,_μ,_δ,_ϵ) is a commutative Frobenius object in . It thereby determines a 2-dimensional TQFT _2 ⟶. 
	This is an immediate consequence of Lemmas <ref>, <ref>, and <ref>.
	
		By an induction argument, we see that for all (m, n)  (0, 0), the TQFT in Theorem <ref> maps the genus-0 cobordism from m circles to n circles to ^m, n{(a, b) ∈^*m * ^*n : a_1 ⋯ a_m = b_1 ⋯ b_n} together with the natural projections to ^m and ^n. 
	

 §.§ Abelianizations of quasi-symplectic groupoids

	
	We now consider the following definition.
	
	
	An abelianization of a quasi-symplectic groupoid  is an abelian symplectic groupoid , together with a symplectic Morita equivalence from  to . A quasi-symplectic groupoid is called abelianizable if it admits an abelianization.
	
	
	LetMbe a quasi-symplectic groupoid with an abelianization⟵⟶̋.
	
	
	
	
	
	
	By Morita transfer (see Subsection <ref>), it follows that_η, _μ, _δ, _ϵtransfer in a unique way to morphisms inof the form_η: ⋆⟶,_μ: ^2 ⟶,_δ: ⟶^2, and_ϵ: ⟶⋆.
	For example, we can realize them as the homotopy fibre products of_η,_μ,_δ, and_ϵwith^̋-,×̋×^̋-,×̋^̋- ×^̋-, and$̋, respectively.
	
	
	
	
	
	
	
	
	
	Note that any other choice of abelianization  of  will give 1-shifted Lagrangians weakly equivalent to _η, _μ, _δ, _ϵ.
	The corresponding morphisms in  are therefore independent of the choice of abelianization.
	
	
		If  is an abelianizable quasi-symplectic groupoid, then (,_η, _μ, _δ, _ϵ) is a commutative Frobenius object in .
		In this way,  determines a 2-dimensional TQFT _2 ⟶.
	
	
	
		This follows from Proposition <ref> and Lemmas <ref>, <ref>, and <ref>.
	
	
	

 §.§ Global slices

	
	We now discuss an application of Theorem <ref>.
	
	
	A global slice to a Lie groupoid M is a submanifold S M intersecting every orbit exactly once and transversely. The global slice is admissible if for all x ∈ S, the isotropy group _x is abelian.





	


Let S X be an admissible global slice to a symplectic groupoid X. The Lie algebra of _x is necessarily abelian for all x∈ S. In this way, asking the groups _x to be abelian is a very mild constraint; it holds in many situations.   

	
	
	Let (, ω)  (M, ϕ) be a quasi-symplectic groupoid. Suppose that S M is an admissible global slice for which i^*ϕ=dγ is exact, where i:S⟶ M is the inclusion. Then the restriction |_S=^-1(S)∩^-1(S) S is an abelianization of  with respect to j^*ω - ^*γ + ^*γ, where j : ⟶ is the inclusion map.
	
	
	
		Note that  is the pullback of  by i.
		It follows that ∘_ : S i→ M being a surjective submersion would force  to be a Lie groupoid. We would therefore like to prove that
		
    T_xS i_*_* T_g T_(g)M,    (v, w) _*(w)

		is surjective for all (x, g) ∈ S i.
		In other words, we would like to show that the nullity of this map is (S i) -  M.
		Its kernel is clearly isomorphic to {(v, w) ∈ T_xS × (A_)_x : v = ρ(w)}.
		Since T_xM = T_xS + ρ_x, it has dimension S +  (A_)_x -  M = (S i) -  M.
		It follows that  is a Lie groupoid, and that the inclusion j : ⟶ is an essential equivalence.
		
		It follows that  is a quasi-symplectic groupoid with respect to (j^*ω, i^*ϕ).
		Since i^*ϕ = dγ is exact, the gauge transformation <cit.> (, j^*ω - ^*γ + ^*γ, 0) is a quasi-symplectic groupoid with trivial background 3-form symplectically Morita equivalent to .
		Moreover, since S is a global slice, we have _ = _.
		It follows that ρ_ = 0, so that  integrates the zero Poisson structure. This implies that  is indeed a symplectic groupoid.
		It follows that (, j^*ω - ^*γ + ^*γ) is an abelianization of (, ω, ϕ).
	
	
	It follows from Theorem <ref> that  together with the slice S determine a TQFT
	
    η_ : _2 .


	Recall that the original Moore–Tachikawa TQFT associated to a complex semisimple group G is characterised by the fact that [
		baseline=-2.5pt,
		every tqft/.append style=
			transform shape, rotate=90, tqft/circle x radius=4pt,
			tqft/circle y radius= 2pt,
			tqft/boundary separation=0.6cm, 
			tqft/view from=incoming,
		
		]
		[
		tqft/cup,
		name=d,
		every incoming lower boundary component/.style=draw,
		every outgoing lower boundary component/.style=draw,
		every incoming boundary component/.style=draw,
		every outgoing boundary component/.style=draw,
		cobordism edge/.style=draw,
		cobordism height= 1cm,
		]; is sent to the Hamiltonian G-space G ×𝒮, where 𝒮 is a Kostant slice.
	 This Hamiltonian space can also be described as ^-1(𝒮), where  is the source map of the symplectic groupoid T^*G ^*.
	 Using the correspondence of Example <ref> between Hamiltonian spaces and 1-shifted Lagrangians, the TQFTs (<ref>) generalize the Moore–Tachikawa example in the following way.
	
	
		The morphism η_([
		baseline=-2.5pt,
		every tqft/.append style=
			transform shape, rotate=90, tqft/circle x radius=4pt,
			tqft/circle y radius= 2pt,
			tqft/boundary separation=0.6cm, 
			tqft/view from=incoming,
		
		]
		[
		tqft/cup,
		name=d,
		every incoming lower boundary component/.style=draw,
		every outgoing lower boundary component/.style=draw,
		every incoming boundary component/.style=draw,
		every outgoing boundary component/.style=draw,
		cobordism edge/.style=draw,
		cobordism height= 1cm,
		];
	) = _ϵ is the 1-shifted Lagrangian on  associated with the Hamiltonian -space ^-1(S), where  acts by left multiplication.
	
	

Let |_S be the abelianization of .
Recall that _ϵ is the trivial groupoid S  S viewed as a 1-shifted Lagrangian in .
A straightforward computation shows that S  S is also a 1-shifted Lagrangian in .
Since the inclusion  is an essential equivalence, it follows that _ϵ is the 1-shifted Lagrangian S.
Recall from Example <ref> that the identity morphism ⟵⟶ is weakly equivalent to the action groupoid ⟵ (×) ⋉⟶, where × acts on  by left and right multiplication.
Hence, by Lemma <ref>, the 1-shifted Lagrangian S is weakly equivalent to its composition with ⟵ (×) ⋉⟶ on the left.
This composition is easily seen to be the 1-shifted Lagrangian ⋉^-1(S).

	



§ THE AFFINIZATION PROCESS

In contrast to the preceding sections, we now work exclusively over ℂ. We first review some pertinent material from our manuscript on scheme-theoretic coisotropic reduction. This leads to a definition and study of Hartogs Morita morphisms. We next discuss a processes by which to affinize 0-shifted symplectic stacks and 1-shifted Lagrangians. We then define what it means for an affine symplectic groupoid to be Hartogs abelianizable. By Main Theorem <ref>, such a groupoid determines a TQFT in . 
On the other hand, we introduce the algebraic Moore–Tachikawa category  as well as an affinization process taking algebraic 1-shifted Lagrangian correspondences in  to morphisms in . 

The proof of Main Theorem <ref> is subsequently given. To conclude, we prove that an affine symplectic groupoid admitting an admissible Hartogs slice is necessarily Hartogs abelianizable.



 §.§ Preliminaries
 We briefly review some conventions from <cit.>. The term algebraic groupoid is used for a groupoid object in the category of complex algebraic varieties.
An algebraic Lie groupoid is an algebraic groupoid with smooth arrow and object varieties, as well smooth morphisms for its source and target maps.
By an affine algebraic Lie groupoid, we mean an algebraic Lie groupoid whose object and arrow varieties are affine. Such a groupoid  is called an affine symplectic groupoid if  carries an algebraic symplectic form for which the graph of multiplication is coisotropic in ××^-.

The affine quotient associated to an algebraic groupoid X is the affine scheme

    X [X]^,

where [X] is the algebra of morphisms f : X ⟶, and [X]^ is the subalgebra of those satisfying ^*f = ^*f. The pullback of X by a morphism μ : Y ⟶ X is the fibre product

    μ^* Y μμ Y.

It is an algebraic groupoid over Y.
If X is an algebraic Lie groupoid and μ : Y ⟶ X is a smooth morphism, then μ^* is also an algebraic Lie groupoid.



 §.§ Morita and Hartogs Morita morphisms

A Morita morphism between algebraic Lie groupoids X and Y is a morphism of algebraic groupoids (f, μ) : (Y) ⟶ ( X) for which μ : Y ⟶ X is a surjective smooth morphism, and the induced morphism

    μ^*,    h  ((h), f(h), (h))

is an isomorphism. It is also advantegeous to introduce the weaker notion of a Hartogs Morita morphism between affine algebraic Lie groupoids X and Y; we define it to be morphism of algebraic groupoids (f, μ) : (Y) ⟶ ( X) such that μ : Y ⟶ X is a smooth morphism, the open subset U μ X has a complement of codimension at least two in X, and the induced morphism

    μ^*

is an isomorphism.
The key property of Hartogs Morita morphisms is that they preserve affine quotients, as the next result shows.


	If ( X) ⟶ (Y) is a Hartogs Morita morphism, then the induced map X ⟶ Y $̋ is an isomorphism of affine schemes.

	It suffices to consider the case in which =̋μ^* for X an affine algebraic Lie groupoid and μ : Y ⟶ X a smooth morphism whose image U μ has a complement of codimension at least two in X. Composing with μ then defines an injective algebra morphism [X]^⟶[Y]^μ^*. We are reduced to showing that this morphism is surjective.
	
	Suppose that f ∈[Y]^μ^*.
	Since μ is smooth, there is an étale covering {π_i:U_i ⟶ U}_i∈ I with local sections {σ_i : U_i ⟶ Y}_i∈ I of μ.
	We will constuct a function F ∈[X]^ mapping to f by considering the functions F_i σ_i^*f : U_i ⟶, and subsequently using étale descent.
	To this end, fix i,j∈ I, let U_ij U_i ×_X U_j, and write p_1 : U_ij⟶ U_i and p_2 : U_ij⟶ U_j for the projections. We have (p_1^*F_i)(x, y) = f(σ_i(x)) = f(σ_j(y)) = (p_2^*F_j)(x, y)
	for all (x, y) ∈ U_ij; the second equality follows from μ^*-invariance of f, as ((σ_i(x), σ_j(y)), 1_π_i(x)) ∈μ^*. We conclude that p_1^* F_i = p_2^* F_j.
	Since (-, ) is a sheaf in the étale topology, there is a function F : U ⟶ satisfying π_i^*F = F_i for all i∈ I.
	Our assumption on codimension then implies that F extends uniquely to an element F ∈[X].
	
	To see that μ^*F = f, suppose that y ∈ Y.
	Since {π_i:U_i ⟶ U}_i∈ I is a cover, there exists a point x in some U_i with π_i(x) = μ(y).
	Note that (μ^*F)(y) = F(μ(y)) = F(π_i(x)) = F_i(x) = f(σ_i(x)) = f(y), where the last equality uses the μ^*-invariance of f and the fact that μ(σ_i(x)) = π_i(x) = μ(y). This establishes that μ^*F=f.
	
	It remains only to establish that F ∈[X]^. Given g ∈|_U^-1(U)∩^-1(U), we have (g) = π_i(x) and (g) = π_j(y) for some x ∈ U_i and y ∈ U_j. This implies that (σ_i(x), σ_j(y), g) ∈μ^*. By the μ^*-invariance of f, we have F((g)) = F(π_i(x)) = f(σ_i(x)) = f(σ_j(y)) = F(π_j(y)) = F((g)). Since |_U is open in , it follows that F ∈[X]^, completing the proof.


By replacing Morita morphisms with Hartogs Morita morphisms, we also obtain notions of Hartogs Morita equivalence, Hartogs symplectic Morita equivalence, and Hartogs Lagrangian Morita equivalence.






















 §.§ Affine Hamiltonian schemes


An affine Poisson scheme is the data of an affine schemeXand a Poisson bracket onℂ[X]. Recall that a closed subscheme ofXis called coisotropic if its ideal is a Poisson subalgebra ofℂ[X].

LetXbe an affine symplectic groupoid acting algebraically on an affine Poisson schemeM, via a mapμ: M ⟶X.
We say that the action is Hamiltonian if the graph of the action morphismμ M Mis coisotropic in×M ×M^-.
We say that an affine Poisson schemeMis an affine Hamiltonian -scheme if it comes equipped with a Hamiltonian action of.
An isomorphism of affine Hamiltonian-schemes is an equivariant isomorphism of affine Poisson schemes.


















 §.§ Affinization of 0-shifted symplectic stacks


LetXbe an affine algebraic Lie groupoid.
A 0-shifted symplectic structure onXis a closed2-formωof constant rank onXsatisfying^*ω= ^*ωandω= ρ, whereρis the anchor map.
We show that in this case, the affine quotientX  is Poisson.

First note that we have a short exact sequence of vector bundles

    0 ω TX ω 0.

We define the Poisson structure explicitly as follows.
Note that for allf ∈[X]^,df ∈(ρ)^∘= ω.
Since every short exact sequence of vector bundles on an affine variety splits, we can choose a global vector fieldX_fonXsuch thati_X_fω= df, unique up toω.
The Poisson bracket on[X]^is then given by

    {f, g}ω(X_f, X_g).

	Equation (<ref>) defines an affine Poisson scheme structure on X.

	Let R be a vector subbundle of TX such that TX = ρ⊕ R.
	We then have an isomorphism ω : R ⟶ω = (ρ)^∘.
	It follows that every -invariant morphism f : X ⟶ determines a unique section X_f of R that satisfies df = ω(X_f).
	The Poisson bracket {f, g} = ω(X_f, X_g) is independent of the choice of R, as any other choice gives vector fields differing from X_f and X_g by elements of ω.
	
	We need to show that {f, g} is -invariant. To this end, fix a ∈ and let X̂_f, X̂_g ∈ T_a be mapped by  to X_f and X_g, respectively.
	Then ^*(ω(X_f)_(a)) = (^*df)_a = (^*df)_a = (^*(ω(X_f)))_a = (^*ω)_a(X̂_f) = (^*ω)_a(X̂_f) = ^*(ω(X̂_f)_(a)).
	It follows that X_f - X̂_f ∈ω.
	We therefore have ^*{f, g}(a) = ω_(a)(X_f, X_g) = ω_(a)(X̂_f, X̂_g) = (^*ω)_a(X̂_f, X̂_g) = (^*ω)_a(X̂_f, X̂_g) = ω_(a)(X_f, X_g) = ^*{f, g}(a).
	The Jacobi identity follows from the closedness of ω.


A symplectic Hartogs Morita morphism between0-shifted symplectic affine algebraic Lie groupoids(X, ω)and (X̃, ω̃)is a Hartogs Morita morphism(f, μ) : ( X̃) →(X)of the underlying affine algebraic Lie groupoids, such thatμ^*ω= ω̃.
We now observe that the affine Poisson scheme of a 0-shifted symplectic groupoid is invariant under symplectic Hartogs Morita morphisms.


	Let X and X̃ be 0-shifted affine symplectic groupoids. If there exists a symplectic Hartogs Morita morphism ⟶, then the Poisson schemes X and X̃ are isomorphic.

	Let μ : X̃⟶ X be the map on objects coming from a Hartogs Morita morphism ⟶.
	By Proposition <ref>, the map μ^* : [X]^→[X̃]^ is an algebra isomorphism. We also have μ^*(i_μ(X_μ^*f)ω) = i_X_μ^*fμ^*ω = i_X_μ^*fω̃ = dμ^*f = μ^*df
	for all f∈ℂ[X].
	Since μ is a submersion, we have i_μ(X_μ^*f)ω = df for all f∈ℂ[X]. It follows that μ(X_μ^*f) = X_f for all f∈ℂ[X], implying that μ^* is also a Poisson algebra morphism.

	Let (X, ω) be an 0-shifted symplectic affine algebraic Lie groupoid and Y  X a smooth closed subvariety such that TY^ω = TY + ρ.
	Let I [X] be the ideal corresponding to Y and set J  I ∩[X]^.
	Then J is a Poisson ideal and hence corresponds to a coisotropic subvariety of X.
	The condition that TY^ω = TY + ρ holds, in particular, if Y is isotropic, Y = 12 X, and (ω∩ TY) = 12ω.

	Let f, h ∈ J.
	We have X_f|_Y ∈ TY^ω = TY + ρ and dh|_Y ∈ TY^∘∩ (ρ)^∘, so that {f, h}|_Y = ω(X_f, X_h)|_Y = -dh(X_f)|_Y = 0.
	For the last statement, first note that TY + ρ = TY + ω TY^ω by isotropy of Y.
	We show that this inclusion is an equality by a dimension count. Indeed, we have (TY^ω) =  X -  Y + (ω∩ TY) =  Y + ω - (ω∩ TY)  = (TY + ω).


 §.§ Affinization of 1-shifted Lagrangians

The following lemma is useful.

	Let X be an affine symplectic groupoid and Y an affine algebraic Lie groupoid.
	Suppose that ×$̋ acts on a smooth affine varietyMwith moment map(μ, ν) : M ⟶ X × Y.
	Suppose also that the projection(×)̋⋉ M →has a 1-shifted Lagrangian structureω∈Ω_M^2for whichωμ_*and the infinitesimal$̋-action on M has constant rank.
	Then ω is a 0-shifted symplectic form on ⋉̋M  M, and the induced Poisson scheme M $̋ is a Hamiltonian-scheme.
	

	Let ω be the 1-shifted Lagrangian structure on (×)̋⋉ M ⟶. Note that ω is a closed 2-form on M with ^*ω - ^*ω = _^*Ω, where Ω is the symplectic form on  and (, ) : (×)̋⋉ M  M are the source and target maps. We also know that the map
	
    μ^*A_⊕ν^*A_{(v, a) ∈ TM × A_ : μ_*v = ρ a  and  i_vω = μ^*σ_Ω a}
    
    		(a, b)     (ψ_*(a, b), a)

	is an isomorphism, where ψ_* is the action map.
	Let (, ) : ⋉̋M  M be the source and target maps of the $̋-action.
	Then = ∘ iand = ∘ i, wherei : ⋉̋M → (×)̋⋉ Mis given byi(h, p) = (_μ(p), h, p).
	In particular,^*ω - ^*ω = i^*(^*ω - ^*ω) = i^*_^*Ω = 0, since_∘ i : ⋉̋M →has its image in the identity section.
	The non-degeneracy condition then shows that
	
    ω∩μ_* = ρ_.̋

	for the anchor mapρ_for$̋. 
	Since we have ωμ_* by assumption,
	
    ω = ρ_.̋

	It follows that ω is a 0-shifted symplectic form, and hence descends to a Poisson bracket on M $̋.
	It remains to check that the residual action ofonM $̋ is Hamiltonian.
	By Lemma <ref>, it suffices to show that the graph Γ× M × M^- of the action satisfies (η∩ TΓ) = 12η, where η (Ω, ω, -ω).
	But η = {(0, u, v) : u, v ∈ω} and  TΓ∩η = {(0, v, v) : v ∈ω}.















































The following generalization is natural and important.


	Let _1  M_1 and _2  M_2 be affine symplectic groupoids and
	
    [row sep=3em,between origins,column sep=4em,between origins]
    			   Ł[swap]dlf_1drf_2[shift left=2pt]d[shift right=2pt]d   
    _1 [shift left=2pt]d[shift right=2pt]d    N dlμ_1[swap]drμ_2   _2 [shift left=2pt]d[shift right=2pt]d
    
    			M_1        M_2.

	an affine 1-shifted Lagrangian correspondence.
	Then the affine quotient
	
    Ł (_1 μ_1 N μ_2_2) Ł

	is a Hamiltonian _1 ×_2^--scheme, where Ł acts via l · (g_1, n, g_2) = (g_1f_1(l)^-1, (l), f_2(l) g_2) if (l) = n.
	If _1 ⟵Ł' ⟶_2 is another affine 1-shifted Lagrangian correspondence that is Hartogs weakly equivalent to (<ref>), then Ł and (Ł') are isomorphic Hamiltonian _1 ×_2^--schemes.

	Consider the 1-shifted Lagrangian correspondences
	
    [row sep=3em,between origins,column sep=4em,between origins]
    			    (_1 ×_1) ⋉_1 dldr      Łdldr       (_2 ×_2) ⋉_2 dldr   
    _1       _1       _2       _2;

	see Example <ref>.
	
	Their composition is the action groupoid
	
    [row sep=3em,between origins,column sep=4em,between origins]
    		    (_1 ×Ł×_2) ⋉ (_1 μ_1 N μ_2_2) dldr   
    _1       _2

	together with the 1-shifted Lagrangian structure given by the 2-form
	
    ω__1^*ω_1 + 2 _N^*γ + __2^*ω_2

	on _1 μ_1 N μ_2_2, where γ is the 1-shifted Lagrangian structure on Ł.
	
	To show that the affine quotient (<ref>) is a Hamiltonian _1 ×_2^--scheme, we apply Lemma <ref>. A first step is to check that ωμ_*, where
	
    μ : _1 μ_1 N μ_2_2  M_1 × M_2,     μ(g_1, n, g_2) = ((g_1), (g_2))

	is the moment map for the action of _1 ×_2 on _1 μ_1 N μ_2_2.
	Let (v_1, u, v_2) ∈ T_(g_1, n, g_2)(_1 μ_1 N μ_2_2) be such that (v_1, u, v_2) ∈ω.
	It follows that v_1 ∈ (_*)^ω_1 = _* and v_2 ∈ (_*)^ω_2 = _*, so that μ_*(v_1, u, v_2) = 0.
	A second step is to check that the infinitesimal Ł-action on _1 μ_1 N μ_2_2 has constant rank.
	The kernel of this infinitesimal action is given by f_1*∩ρ_Ł∩ f_2*; it is trivial by the definition of 1-shifted Lagrangians, i.e. the injectivity part of Definition <ref><ref>.
	By Lemma <ref>, (<ref>) is a Hamiltonian _1 ×_2^--scheme.
	
	
	Consider another affine 1-shifted Lagrangian correspondence
	
    [row sep=3em,between origins,column sep=4em,between origins]
    			   Ł' [swap]dlf_1'drf_2'[shift left=2pt]d[shift right=2pt]d   
    _1 [shift left=2pt]d[shift right=2pt]d    N' dlμ_1'[swap]drμ_2'   _2 [shift left=2pt]d[shift right=2pt]d
    
    			M_1        M_2.

	that is Hartogs Morita equivalent to (<ref>).
	To show that our two correspondences yield isomorphic Hamiltonian schemes, it suffices to consider the following case: there is a Hartogs Morita morphism ψ : Ł' ⟶Ł fitting into a 2-commutative diagram
	
    [row sep=3em,between origins,column sep=3em,between origins]
    		   Ł' [swap]dlf_1'drf_2'ddψ   
    _1       _2 
       Łulf_1[swap]urf_2

	with respect to some natural transformations θ_1 : f_1ψ⟹ f_1' and θ_2 : f_2ψ⟹ f_2'.
	In this case, ψ induces a Hartogs Morita morphism
	
    Ł' ⋉ (_1 μ_1' N' μ_2'_2) 
    		   Ł⋉ (_1 μ_1 N μ_2_2) 
    
    		(l, (g_1, n, g_2))     (ψ(l), (g_1 θ_1(n), ψ(n), θ_2(n)^-1 g_2)).

	Proposition <ref> implies that the induced Poisson schemes are isomorphic.
	Since the isomorphism is _1 ×_2^- equivariant, this provides an isomorphism of Hamiltonian _1 ×_2^--schemes.








































































 §.§ The algebraic Moore–Tachikawa category

We now 
define a categoryin which the aforementioned affinizations of TQFTs take values. Our category turns out to enlarge Moore and Tachikawa's categoryof holomorphic symplectic varieties; see Section <ref>. These considerations explain our choosing the notation. 

We begin with precise definitions. The objects ofare affine symplectic groupoids. To define morphisms, suppose thatandare affine symplectic groupoids. Suppose also thatMandNare affine Poisson schemes, equipped with commuting Hamiltonian actions ofand^-. DeclareMandNto be isomorphic if there exists an affine Poisson scheme isomorphismM⟶ Nthat intertwines the actions ofand^-. Morphisms fromtoare then defined to be isomorphism classes of affine Poisson schemes carrying commuting Hamiltonian actions ofand^-. To define morphism composition, consider affine symplectic groupoids X, Y, and𝒦 Z. Let us also consider[M]∈Hom(,)and[N]∈Hom(,𝒦).
It follows that Yacts diagonally on the fibre productM ×_Y N.
By <cit.>, the affine quotient(M ×_Y N) is a Poisson scheme with commuting Hamiltonian actions ofand^-.
The Poisson structure is obtained by reduction, i.e. it is characterized by the relation{j^*f_1, j^*f_2} = j^*{f_1, f_2}for allf_1, f_2 ∈[M × N]such thatj^*f_1, j^*f_2 ∈[M ×_Y N]^, wherej : M ×_Y N  M × Nis inclusion.





The previous paragraph and a routine exercise reveal thatis a category. It turns out thatalso carries a symmetric monoidal structure. One defines the tensor product of affine symplectic groupoidsandto be the affine symplectic groupoid⊗×. Given affine symplectic groupoids,,𝒦, andℒ, and morphisms[M]∈Hom(,)and[N]∈Hom(𝒦,ℒ), note thatM× Nis an affine Hamiltonian(×𝒦)×(ℐ^- ×ℒ^-)-scheme. One thereby has[M]⊗[N][M× N]∈Hom(×𝒦,×ℒ)The unit object inis the singleton groupoid{e}⟶{*}. The associator, left unitor, right unitor, and braiding are then exactly as one would expect. A routine exercise reveals thatis indeed a symmetric monoidal category.



 §.§ A “functor"
 Consider the sub-“category”ofconsisting of affine symplectic groupoids and affine algebraic 1-shifted Lagrangian correspondences.
We construct a “functor”
    

as follows.
Consider a morphism

    [row sep=2em,between origins,column sep=4em,between origins]
    	   Łdldr   
    _1       _2.

in.
Theorem <ref> implies thatŁis a morphism from_1to_2in.





















The next proposition shows thatŁŁbehaves like a functor fromto.


	The association ŁŁ has the following properties.
	
  * 
		It is well-defined: if Ł_1 and Ł_2 are weakly equivalent 1-shifted Lagrangian correspondences from _1 to _2, then Ł_1 and Ł_2 are isomorphic Hamiltonian _1 ×_2^--schemes.
		
  * 
		It preserves composition: if _1 ⟵Ł_1 ⟶_2 and _2 ⟵Ł_2 ⟶_3 are composable 1-shifted Lagrangian correspondences, then (Ł_2 ∘Ł_1) and Ł_2∘Ł_1 are isomorphic Hamiltonian _1 ×_3^--schemes.
		
  * 
		It preserves identities: for every affine symplectic groupoid , the affinization of the identity ⟵⟶ in  is isomorphic to the identity of  in .
	
	Part <ref> follows from the last part of Theorem <ref>.
	For Part <ref>, let M_i and N_i be the objects of _i and Ł_i, respectively.
	Let μ_ij : N_i ⟶ M_j be the base maps, and let μ_1 = (μ_11, μ_12), μ_2 = (μ_22, μ_23).
	We have a diagram of Lie groupoid morphisms
	
    [column sep=4em,between origins, row sep=4em,between origins]
    		      Ł_2 ∘Ł_1 [equal]d      
          Ł_1 μ_12_2 μ_22Ł_2 dldr      
       Ł_1 dldr      Ł_2 dldr
    _1       _2       _3

	over the objects
	
    [column sep=4em,between origins, row sep=4em,between origins]
    		       N_1 μ_12_2 μ_22 N_2 [swap]dl_N_1dr_N_2      
        N_1 [swap]dlμ_11drμ_12       N_2 [swap]dlμ_22drμ_23
    
    		M_1        M_2        M_3.

	Consider the map
	
    Ł_2∘Ł_1[equal]rd   ((_1 μ_11 N_1 μ_12_2) Ł_1 ∘__2∘__2 (_2 μ_22 N_2 μ_23_3) Ł_2 ) _2 
    
    		(Ł_2 ∘Ł_1)[equal]r   (_1 μ_11∘_N_1 (N_1 μ_12_2 μ_22 N_2) μ_23∘_N_2_3)  (Ł_1 μ_12∘_2 μ_22∘Ł_2)

	which descends from the map
	
    (_1 μ_11 N_1 μ_12_2) ∘__2∘__2 (_2 μ_22 N_2 μ_23_3) d    (g_1, n_1, g_2, g_2', n_2, g_3) [mapsto]d
    _1 μ_11∘_N_1 (N_1 μ_12_2 μ_22 N_2) μ_23∘_N_2_3     (g_1, n_1, (g_2g_2')^-1, n_2, g_3).

	Note that this map descends from the morphism of action groupoids
	
    (Ł_1 ×_2 ×Ł_2) ⋉( (_1 μ_11 N_1 μ_12_2) ∘__2∘__2 (_2 μ_22 N_2 μ_23_3) ) d
    
    		(Ł_1 μ_12∘_2 μ_22∘Ł_2)
    		⋉(_1 μ_11∘_N_1 (N_1 μ_12_2 μ_22 N_2) μ_23∘_N_2_3)

	given by
	
    ((l_1, a, l_2), (g_1, n_1, g_2, g_2', n_2, g_3)) [mapsto]d   where (l_1) = n_1, (l_2) = n_2, and (a) = (g_2) = (g_2')
    
    		((l_1, (g_2 g_2')^-1, l_2), (g_1, n_1, (g_2 g_2')^-1, n_2, g_3))    where (l_1) = n_1 and (l_2) = n_2.

	One sees that this morphism of action groupoids is a Morita morphism, i.e. a pullback by a surjective smooth morphism. Moreover, the multiplicativity of the symplectic form on _2 implies that the base map preserves the 0-shifted symplectic 2-forms on both sides.
	Proposition <ref> then implies that this morphism descends to an isomorphism of Poisson schemes Ł_2∘Ł_1→ (Ł_2 ∘Ł_1).
	It is also clearly _1 ×_3-equivariant, and hence an isomorphism of Hamiltonian _1 ×_3^--schemes.
	
	We now show Part <ref>. Note that the affinization of the identity of  in  is the affine quotient (), where  acts by g · (a, b) = (ag^-1, gb); its Poisson structure is induced by the 0-shifted symplectic form on ⋉ (), i.e. the 2-form _1^*ω + _2^*ω on . 
	At the same time, we have a Morita morphism from ⋉ () to the trivial groupoid ; it is given by (g, (a, b))  ab on arrows and : → on objects.
	By the multiplicativity of ω, this is a Morita morphism of 0-shifted symplectic affine algebraic Lie groupoids.
	Proposition <ref> then shows that () is isomorphic to  as a Hamiltonian ×^--scheme.


 §.§ Hartogs abelianizations of affine symplectic groupoids

The following definition is useful.


A Hartogs abelianization of an affine symplectic groupoid X is an abelian affine symplectic groupoid (, ω_)  Y, together with a Hartogs symplectic Morita equivalence from  to . We call Hartogs abelianizable if it admits a Hartogs abelianization.


The following is a more explicit statement of the definition. Let(, ω_)  Xbe an affine symplectic groupoid. A Hartogs abelianization ofconsists of an abelian affine symplectic groupoid(, ω_)  Y, an affine Lie groupoid$̋ endowed with an algebraic closed 2-form γ on , and Hartogs Morita morphisms

    [row sep=2em,between origins, column sep=3em,between origins]
    	   [swap]dlφdrψ

satisfying φ^*ω_ - ψ^*ω_ = ^*γ - ^*γ.






As in Subsection <ref>, _η, _μ, _δ, and _ϵ transfer to affine 1-shifted Lagrangian correspondences

    [row sep=2em,between origins,column sep=3em,between origins]
    		   _ηdldr   
    ⋆          [row sep=2em,between origins,column sep=3em,between origins]
    		   _μdldr   
    ^2           [row sep=2em,between origins,column sep=3em,between origins]
    		   _δdldr   
          ^2
    	    [row sep=2em,between origins,column sep=3em,between origins]
    		   _ϵdldr   
          ⋆

by taking the homotopy fibre products of _η, _μ, _δ, and _ϵ with ^̋-, ×̋×̋^̋-, ×̋^̋- ×^̋-, and $̋, respectively.
Consider the corresponding morphisms_η,_μ,_δ, and_ϵinobtained via affinization.


If an affine symplectic groupoid  is Hartogs abelianizable, then (,_η,_μ,_δ,_ϵ) is a commutative Frobenius object in . In this way,  determines a TQFT _2⟶.

	Let U  X be the image of the base map of the Hartogs Morita morphism ψ : ⟶̋. By assumption, U is an open subset with a complement of codimension at least two in X. We also know that the correspondences (<ref>) restrict to 1-shifted Lagrangian correspondences on powers of |_U^-1(U)∩^-1(U).
	Proposition <ref> and Lemma <ref>, <ref>, and <ref> now tell us that |_U is a commutative Frobenius object in  with respect to these restricted morphisms.
	The identity _|_U in  can be identified with $̋, and the braidingB_|_Uwith×̋$̋.
	Let _ι$̋, viewed as a 1-shifted Lagrangian correspondence fromto.
	We similarly let_τ×̋$̋, viewed as a 1-shifted Lagrangian correspondence from × to × via (ψ(a), ψ(b))  (a, b) ↦ (ψ(b), ψ(a)).
	It follows that _η, _μ, _ι, _δ, _ϵ, _τ satisfy the identities indicated by (<ref>)–(<ref>) viewed as 1-shifted Lagrangian correspondences on powers of .
	Proposition <ref> then implies that their affinizations satisfy the same relations in .
	It therefore suffices to show that _ι is the identity morphism from  to  in  and, similarly, that _τ is the brading on  in .
	This follows from Proposition <ref><ref> and the last part of Theorem <ref>, as we have a Hartogs weak equivalence
	
    [row sep=2em,between origins,column sep=3em,between origins]
       dldrdd   
          
       ulur

	of 1-shifted Lagrangian correspondences from  to .
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	


 §.§ Hartogs slices


In analogy with Subsection <ref>, one might expect certain slices to induce Hartogs abelianizations of affine symplectic groupoids. This turns out to be true via the following algebro-geometric counterpart to Definition <ref>.


A Hartogs slice to an affine algebraic Lie groupoid X is a smooth closed affine subvariety S  X that is a global slice to |_U for some open subset U  X whose complement has codimension at least 2.
The Hartogs slice is admissible if the isotropy group _x is abelian for all x ∈ S.







































	Let X be an affine symplectic groupoid together with an admissible Hartogs slice S  X.
	Then the restriction |_S is a Hartogs abelianization of .

The proof is essentially the same as that of Proposition <ref>.


Consider a Hartogs sliceS Xto an affine symplectic groupoid X. Note thatSis a Poisson transversal inX. Since the source map : ⟶ Xis anti-Poisson,^-1(S) is a symplectic subvariety of. The action ofon^-1(S)by left multiplication is Hamiltonian. We may thereby view^-1(S)as a morphism→⋆in.


	Let X be an affine symplectic groupoid with an admissible Hartogs slice S  X.
	Write F_ : _2 ⟶ for the TQFT induced from Theorem <ref> by the Hartogs abelianization |_S.
	Then F_ maps [
		baseline=-2.5pt,
		every tqft/.append style=
			transform shape, rotate=90, tqft/circle x radius=4pt,
			tqft/circle y radius= 2pt,
			tqft/boundary separation=0.6cm, 
			tqft/view from=incoming,
		
		]
		[
		tqft/cup,
		name=d,
		every incoming lower boundary component/.style=draw,
		every outgoing lower boundary component/.style=draw,
		every incoming boundary component/.style=draw,
		every outgoing boundary component/.style=draw,
		cobordism edge/.style=draw,
		cobordism height= 1cm,
		]; to the Hamiltonian -scheme ^-1(S).

	By the proof of Proposition <ref>, F_([
		baseline=-2.5pt,
		every tqft/.append style=
			transform shape, rotate=90, tqft/circle x radius=4pt,
			tqft/circle y radius= 2pt,
			tqft/boundary separation=0.6cm, 
			tqft/view from=incoming,
		
		]
		[
		tqft/cup,
		name=d,
		every incoming lower boundary component/.style=draw,
		every outgoing lower boundary component/.style=draw,
		every incoming boundary component/.style=draw,
		every outgoing boundary component/.style=draw,
		cobordism edge/.style=draw,
		cobordism height= 1cm,
		];
	) is the affinization of the 1-shifted Lagrangian (S  S)  ( X).
	It then follows directly from (<ref>) that this affinization is the Hamiltonian -scheme ^-1(S).






Consider a pair of integers (m, n)  (0, 0).
One can describe the images of the genus-0 cobordism from m circles to n circles under the TQFT F_ of Theorem <ref>. To this end, let |_S.
As in Remark <ref>, the corresponding morphism from ^m to ^n in  is ^m, n{(a, b) ∈^*m * ^*n : a_1 ⋯ a_m = b_1 ⋯ b_n}, together with the projections to ^m and ^n.
Since the inclusion ⟶|_U is an essential equivalence, the corresponding 1-shifted Lagrangian correspondences on powers of  are also given by ^m, n and the natural maps

    [row sep=2em,between origins,column sep=4em,between origins]
    	   ^m, ndldr   
    ^m       ^n.

Taking the affinizations of these morphisms, we get that F_ maps the genus-0 cobordism from m circles to n circles to the affine quotient 

    {(g, h) ∈^m ×^n : (g_1) = ⋯ = (g_m) = (h_1) = ⋯ = (h_n) ∈ S}^m, n,

where ^m, n acts by (a, b) · (g, h) = (ga^-1, bh).
The action of ^m × (^-)^n giving it the structure of a Hamiltonian scheme descends from the action (a, b) · (g, h) = (ag, hb^-1).




























































































	
	
	
	
	















	
	
	
	
















	
	
	
	
















	
	
	
	
	
















	
	
	
	
	








































































§ THE SPECIAL CASE OF THE MOORE–TACHIKAWA CONJECTURE

This section is devoted to the implications of Main Theorem <ref> for constructing the Moore–Tachikawa TQFT. As with the previous section, we work exclusively over. We begin by recalling Moore and Tachikawa's categoryof holomorphic symplectic varieties with Hamiltonian actions. We then state the Moore–Tachikawa conjecture regarding the existence of a TQFT in. This conjecture turns out to have a natural cousin in, i.e. a conjecture about the existence of a TQFT in. We deduce the latter conjecture as an immediate corollary of Main Theorem <ref>. 



 §.§ The Moore–Tachikawa category

Moore and Tachikawa's conjectural TQFT would take values in the so-called category of holomorphic symplectic varieties with Hamiltonian actions<cit.>. We denote this category by, and briefly recall its construction. 

The objects ofare complex semisimple affine algebraic groups. A precise description of morphisms inhinges on the exact meaning of holomorphic symplectic variety. In this context, one means an affine Poisson variety for which the Poisson structure is non-degenerate on an open dense subset of the smooth locus. A morphism infromGtoIis an isomorphism class of holomorphic symplectic varieties carrying algebraicG× I-actions, in such a way that the actions ofG=G×{e} G× I(resp.I={e}× I G× I) are Poisson (resp. anti-Poisson). To define morphism composition, suppose that[M]∈(G,I)and[N]∈(I,K)for complex semisimple affine algebraic groupsG,I, andK. One defines[N]∘[M][(M× N^-)0I]∈(G,K).By settingG⊗ I G× Iand[M]⊗[N]=[M× N], one can realizeas a symmetric monoidal category.

We define a functorℱ:⟶as follows. Given a complex semisimple affine algebraic groupG, let(G)be the cotangent groupoidT^*G^*. Now suppose that[M]∈(G,I)for complex semisimple affine algebraic groupsGandI. It follows thatMis an affine HamiltonianT^*G× (T^*I)^--scheme. This fact allows us to letℱ([M])=[M], where the right-hand side is the isomorphism class ofMas an affine HamiltonianT^*G× (T^*I)^--scheme. It follows thatℱ:⟶includesas a symmetric monoidal subcategory of.



 §.§ The Moore–Tachikawa conjecture

LetGbe a connected complex semisimple linear algebraic group with Lie algebraand rankℓ. The Killing form determines an isomorphism between the adjoint and coadjoint representations ofG; we denoted it by(·)^∨:⟶^*,x↦ x^∨. WriteG_x GandG_ξ Gfor theG-centralizers ofx∈andξ∈^*, respectively. Their respective Lie algebras are denoted_xand_ξ. It is known that_x≥ℓ(resp._ξ≥ℓ) for allx∈(resp.ξ∈^*). The regular loci inand^*are theG-invariant open subvarieties given by_reg{x∈:_x=ℓ}  and  ^*_reg{ξ∈^*:_ξ=ℓ},respectively. Using theG-equivariance of(·)^∨:⟶^*, one deduces that^*_reg=(_reg)^∨.

LetG× Gact onGvia(g_1,g_2)· h=g_1hg_2^-1. The cotangent lift of this action is a HamiltonianG× G-variety structure onT^*G. If we use the left trivialization to identifyT^*GwithG×^*, then this lifted action admits(μ_1,μ_2):T^*G⟶^*⊕^*,   (g,ξ)↦(-Ad_g^*(ξ),ξ)as a moment map.

Let(e,h,f)∈^× 3be an𝔰𝔩_2-triple withe,h,f∈_reg, i.e. a principal𝔰𝔩_2-triple. This triple determines a Slodowy slice𝒮 e+_f. One knows that𝒮^∨= (e+_f)^∨^*is a Poisson transversal in^*<cit.>. It follows thatG×𝒮^∨=μ_2^-1(S^∨) T^*Gis a symplectic subvariety. We also observe that the Hamiltonian action ofG=G×{e} G× GonT^*GpreservesG×𝒮^∨. In this way,G×𝒮^∨is a holomorphic symplectic variety with a Hamiltonian action ofG. The Moore–Tachikawa conjecture is then stated as follows.

Let G be a connected semisimple affine algebraic group with Lie algebra . Consider a principal 𝔰𝔩_2-triple (e,h,f)∈^× 3, and set 𝒮 e+_f. There exists a TQFT η_G:_2⟶ satisfying η_G(S^1)=G and η_G([
	baseline=-2.5pt,
	every tqft/.append style=
		transform shape, rotate=90, tqft/circle x radius=4pt,
		tqft/circle y radius= 2pt,
		tqft/boundary separation=0.6cm, 
		tqft/view from=incoming,
	
	]
	[
	tqft/cup,
	name=d,
	every incoming lower boundary component/.style=draw,
	every outgoing lower boundary component/.style=draw,
	every incoming boundary component/.style=draw,
	every outgoing boundary component/.style=draw,
	cobordism edge/.style=draw,
	cobordism height= 1cm,
	];
)=G×𝒮^∨

We discuss the status of this conjecture in the next subsection.



 §.§ The Moore–Tachikawa conjecture in 

LetGbe a connected complex semisimple affine algebraic group with Lie algebra. On the other hand, recall the functorℱ:⟶. Composing the conjectural TQFTη_G:_2⟶in Conjecture <ref> withℱyields the following conjecture.   


Let G be a connected semisimple affine algebraic group with Lie algebra . Consider a principal 𝔰𝔩_2-triple (e,h,f)∈^× 3, and set 𝒮 e+_f. There exists a TQFT η_T^*G:_2⟶ satisfying η_T^*G(S^1)=(T^*G^*) and η_T^*G([
	baseline=-2.5pt,
	every tqft/.append style=
		transform shape, rotate=90, tqft/circle x radius=4pt,
		tqft/circle y radius= 2pt,
		tqft/boundary separation=0.6cm, 
		tqft/view from=incoming,
	
	]
	[
	tqft/cup,
	name=d,
	every incoming lower boundary component/.style=draw,
	every outgoing lower boundary component/.style=draw,
	every incoming boundary component/.style=draw,
	every outgoing boundary component/.style=draw,
	cobordism edge/.style=draw,
	cobordism height= 1cm,
	];
)=G×𝒮^∨.


With some effort, unpublished results of Ginzburg–Kazhdan <cit.> can be understood as implying Conjecture <ref>. Conjecture <ref> then reduces to whetherη_T^*G:_2⟶takes values in the subcategory, i.e. whether certain algebras are finitely generated. In Lie typeA, results of Braverman–Finkelberg–Nakajima <cit.> imply that the relevant algebras are indeed finitely generated.



 §.§ Proof of Moore–Tachikawa conjecture in 

We now show Conjecture <ref> to be a straightforward consequence of our shifted symplecto-geometric approach. Retain the notation and objects used in the previous subsection.


The subvariety 𝒮^∨^* is an admissible Hartogs slice to T^*G^*.

Write G· Y and G· Z^* for the G-saturations of Y and Z^*, respectively. Our task is to verify the following:

  G·𝒮^∨ is open and has a complement of codimension at least two in ^*;

   the restriction of G· S^∨⟶ (G· S^∨)/G to 𝒮^∨ is a bijection;

  ^*=T_ξ(𝒮^∨)⊕ T_ξ(G·ξ) for all ξ∈𝒮^∨;

  G_ξ is abelian for all ξ∈𝒮^∨.

At the same time, recall that the isomorphism (·)^∨:⟶^* is G-equivariant. Parts (i), (ii), and (iii) may therefore be rephrased as follows:

  G·𝒮 is open and has a complement of codimension at least 2 in ;

   the restriction of G· S⟶ (G·𝒮)/G to 𝒮 is a bijection;

  =T_x𝒮⊕ T_x(G· x) for all x∈𝒮;

  G_x is abelian for all x∈𝒮.

It is known that G·𝒮=, and that ∖ has codimension three in <cit.>. Parts (ii), (iii), and (iv) are then immediate and well-known implications of Kostant's work <cit.>.

Conjecture <ref> is true.

Recall that the target morphism :T^*G⟶^* is given by (g,ξ)=ξ. It follows that ^-1(S^∨)=G×𝒮^∨. Theorem <ref> and Lemma <ref> then imply the desired result.

There is a multiplicative counterpart to Theorem <ref>. To obtain it, one replaces T^*G and 𝒮^∨^* with the quasi-Hamiltonian G-variety D(G) G× G and a Steinberg slice in G, respectively. This perspective is explored in the second named author's joint work with Bălibanu <cit.>.

























































































§ EXAMPLES INVOLVING SLODOWY SLICES

As with the previous two sections, we work exclusively overℂ. It is tempting to wonder if Main Theorem <ref> gives examples of TQFTs that are fundamentally different from those conjectured by Moore–Tachikawa. Perhaps reassuringly, such examples exist. The current section is devoted to examples of this sort that involve Slodowy slices. We begin by addressing some Poisson-geometric properties of Slodowy slices. Particular attention is paid to the degeneracy locus of a Slodowy slice, i.e. the locus on which the Poisson bivector has non-maximal rank. The discussion subsequently turns to specific Slodowy slices. We restrictT^*_n𝔰𝔩_n^*to a Slodowy slice to the minimal nilpotent orbit in𝔰𝔩_n^*. This restriction is shown to be Hartogs abelianizable. By Main Theorem <ref>, it determines a TQFT. 



 §.§ Some general comments

Recall that the rank of a smooth Poisson varietyXis the supremum of the dimensions of its symplectic leaves. WriterkXfor this quantity, andX_reg⊂ Xfor the union of therkX-dimensional symplectic leaves ofX. It turns out thatX∖ X_regis the vanishing locus ofπ^1/2rkX, whereπis the Poisson bivector field. This implies thatX_regis an open subvariety ofX. Let us call a symplectic leaf ofXregular if it is contained inX_reg.



 §.§ Slodowy slices

LetGbe a connected semisimple affine algebraic group with Lie algebraand rankℓ. Suppose that(e,h,f)∈^× 3is an𝔰𝔩_2-triple with Slodowy slice𝒮 e+_f⊂. Recall the definitions ofandfrom Subsection <ref>. Note that the definition ofcoincides with the Poisson-theoretic one, obtained by settingX=^*in Subsection <ref>.

Let us call an adjoint (resp. coadjoint) orbit ofGregular if it is contained in(resp.).


If 𝒪^* is a regular coadjoint orbit, then 𝒪∩𝒮^∨≠∅.

	Consider the adjoint quotient π:^*⟶Spec(ℂ[^*]^G)𝔠. The closure 𝒪⊂^* is a fiber of π<cit.>. On the other hand, the restriction π|_𝒮^∨:𝒮^∨⟶𝔠 is known to be faithfully flat with irreducible fibers of dimension 𝒮-ℓ<cit.>. It follows that 𝒮^∨∩𝒪 is a non-empty, (𝒮-ℓ)-dimensional fiber of π|_𝒮^∨. We also know that 𝒪∖𝒪 is a union of finitely many coadjoint orbits <cit.>, all of dimension strictly less than 𝒪. The intersection of each such orbit with 𝒮^∨ must therefore have dimension strictly less than 𝒪+𝒮-=𝒮-ℓ. Since 𝒮^∨∩𝒪 is non-empty and (𝒮-ℓ)-dimensional, we must have 𝒮^∨∩𝒪≠∅.


Recall theG-module isomorphism(·)^∨:⟶^*discussed in Subsection <ref>. The subvariety𝒮^∨=(e+_f)^∨is a Poisson transversal in^*<cit.>. As such,𝒮^∨is a Poisson variety. We may therefore consider the numberrank𝒮^∨and locus(𝒮^∨)_reg𝒮^∨. 


	We have rank𝒮^∨=𝒮-ℓ and (𝒮^∨)_reg=𝒮^∨∩^*_reg.	

	Suppose that ξ∈𝒮^∨. Write L_ξ⊂𝒮^∨ for the symplectic leaf of 𝒮^∨ containing ξ. It follows that T_ξL_ξ= (G·ξ)+𝒮-. This implies that T_ξL_ξ≥ T_ηL_η for all η∈𝒮^∨ if and only if (G·ξ)≥(G·η) for all η∈𝒮^∨. A rephrased version is that x∈(𝒮^∨)_reg if and only if _ξ≤_η for all η∈𝒮^∨. Lemma <ref> tells us that the latter condition holds if and only if ξ∈. We conclude that (𝒮^∨)_reg=𝒮^∨∩^*_reg.
	
	Now suppose that ξ∈ (𝒮^∨)_reg=𝒮^∨∩^*_reg. We have  T_ξL_ξ= (G·ξ)+𝒮-=(-ℓ)+𝒮-ℓ=𝒮-ℓ. It follows that rank𝒮^∨=𝒮-ℓ. 

	The following statements are true.
	
   The regular symplectic leaves of 𝒮^∨ are the intersections of 𝒮^∨ with the regular coadjoint orbits.
		
   The codimension of 𝒮^∨∖(𝒮^∨)_reg in 𝒮^∨ is at least three.
	
	We begin by proving (i). Consider the adjoint quotient π:^*⟶Spec(ℂ[^*]^G)𝔠. The restriction π|_𝒮^∨:𝒮^∨⟶𝔠 is known to be faithfully flat with irreducible fibers of dimension 𝒮-ℓ<cit.>. Since each fiber of π is a finite union of coadjoint orbits <cit.>, each fiber of π|_S^∨ must be a finite union of symplectic leaves of 𝒮^∨. It also follows that each fiber of π|_S^∨ is the closure of a (𝒮-ℓ)-dimensional symplectic leaf of 𝒮^∨.
	
	Suppose that α∈𝔠. There exists a unique regular coadjoint orbit 𝒪_α⊂ with the property that 𝒪_α=π^-1(α)<cit.>. Each irreducible component of 𝒮^∨∩𝒪_α is a (𝒮-ℓ)-dimensional symplectic leaf of 𝒮^∨ contained in 𝒮^∨∩π^-1(α). Since 𝒮^∨∩π^-1(α) is the closure of a (𝒮-ℓ)-dimensional symplectic leaf of 𝒮^∨, this leaf must be 𝒮^∨∩𝒪_α. We also know that α↦𝒪_α defines a bijection from 𝔠 to the set of regular coadjoint orbits <cit.>. It follows that each regular coadjoint orbit intersects 𝒮^∨ in a symplectic leaf. It is also clear that the regular symplectic leaves of 𝒮^∨ are the irreducible components of the intersections of 𝒮^∨ with the regular coadjoint orbits. These last two sentences combine to imply (i).
	
	We now verify (ii). Recall that a coadjoint orbit 𝒪^* is called semisimple if it corresponds to a semisimple adjoint orbit under (·)^∨:⟶^*. Consider the locus 𝔠^∘{α∈𝔠:𝒪_α is semisimple} and its complement 𝔡𝔠∖𝔠^∘. One knows that 𝒪_α is closed for all α∈𝔠^∘. Note also that (𝒮^∨∖(𝒮^∨)_reg)∩π^-1(α)=𝒮^∨∩(𝒪_α∖𝒪_α) for all α∈𝔠, as follows from the second paragraph of this proof. We conclude that 
    𝒮^∨∖(𝒮^∨)_reg    =⋃_α∈𝔠((𝒮^∨∖(𝒮^∨)_reg)∩π^-1(α)) 
        =⋃_α∈𝔠(𝒮^∨∩(𝒪_α∖𝒪_α))
       =⋃_α∈𝔡(𝒮^∨∩(𝒪_α∖𝒪_α))
       =⋃_α∈𝔡((𝒮^∨∖(𝒮^∨)_reg)∩π^-1(α)).
 Let us also observe that (𝒮^∨∖(𝒮^∨)_reg)∩π^-1(α) has codimension at least 2 in 𝒮^∨∩π^-1(α) for all α∈𝔠; this is implied by the first paragraph of the proof. The desired result now follows from 𝔡 having codimension 1 in 𝔠.


 §.§ Slodowy slices to the minimal nilpotent orbit in 𝔰𝔩_n

Let us specialize to the case=𝔰𝔩_n. Consider the𝔰𝔩_2-triple(e,h,f)∈𝔰𝔩_n^× 3given bye=[ 0 1 0 ⋯ 0; 0 0 0 ⋯ 0; ⋮ ⋮ ⋮ ⋱ ⋮; 0 0 0 ⋯ 0 ],   h=[  1  0  0  ⋯  0;  0 -1  0  ⋯  0;  0  0  0  ⋯  0;  ⋮  ⋮  ⋮  ⋱  ⋮;  0  0  0  ⋯  0 ],  and   f=[ 0 0 ⋯ 0; 1 0 ⋯ 0; 0 0 ⋯ 0; ⋮ ⋮ ⋱ ⋮; 0 0 ⋯ 0 ].Let us also consider the closed subvariety𝒯{[     0     1     0     0     ⋯     0; a_n-2     0     1     0     ⋯     0;     ⋮     ⋮     ⋮     ⋱     ⋱     ⋮;   a_2     0     0     0     ⋱     0;   a_1     0     0     0     ⋯     1;   a_0     0     0     0     ⋯     0 ]:a_0,…,a_n-2∈ℂ}of𝔰𝔩_n; it consists of the transposes of the trace-freen× ncompanion matrices. A straightforward exercise reveals that𝒯⊂𝒮 e+(𝔰𝔩_n)_f. It follows that𝒯^∨𝒮^∨.


The subvariety 𝒯^∨ is a Hartogs slice to 𝒮^∨.

	One may use <cit.> to conclude that 𝒯⊂(𝔰𝔩_n)_reg. In other words, 𝒯𝒮∩(𝔰𝔩_n)_reg. It follows that 𝒯^∨𝒮^∨∩(𝔰𝔩_n^*)_reg=(𝒮^∨)_reg, where the last instance of equality comes from Lemma <ref>.
	It remains only to prove that each regular symplectic leaf L⊂𝒮^∨ intersects 𝒯^∨ in a single point. Proposition <ref>(i) makes this the task of proving that each regular coadjoint orbit intersects 𝒯^∨ in a single point. This is equivalent to each regular adjoint orbit intersecting 𝒯 in a single point. In other words, it suffices to prove that 𝒯 is a section of the adjoint quotient π:𝔰𝔩_n⟶Spec(ℂ[𝔰𝔩_n]^SL_n).
	
	Write (tI_n-x)=t^n+f_n-2(x)t^n-2+⋯+f_1(x)t+f_0(x) for the characteristic polynomial of x∈𝔰𝔩_n. The polynomials f_0,…,f_n-2 are algebraically independent generators of ℂ[𝔰𝔩_n]^SL_n. The adjoint quotient of 𝔰𝔩_n is thereby the map f=(f_0,…,f_n-2):𝔰𝔩_n⟶ℂ^n-1. It is also straightforward to check that f([     0     1     0     0     ⋯     0; a_n-2     0     1     0     ⋯     0;     ⋮     ⋮     ⋮     ⋱     ⋱     ⋮;   a_2     0     0     0     ⋱     0;   a_1     0     0     0     ⋯     1;   a_0     0     0     0     ⋯     0 ])=(-a_0,…,-a_n-2) for all a_0,…,a_n-2∈ℂ. This makes it clear that the restriction of the adjoint quotient π:𝔰𝔩_n⟶𝔠 to 𝒯 is an isomorphism of varieties. The proof is therefore complete. 


One symplectic groupoid integrating𝒮^∨is the restriction𝒮^∨ofT^*_n𝔰𝔩_n^*to𝒮^∨.  It is also clear that𝒯^∨⊂(𝔰𝔩_n)^*_reg<cit.>, so that(_n)_ξis abelian for allξ∈𝒯^∨. These considerations combine with Proposition <ref> to imply that𝒯^∨is an admissible Hartogs slice to𝒮^∨. In light of Theorem <ref> and Proposition <ref>,𝒮^∨determines a TQFT.



§ EXAMPLES ARISING FROM NON-REDUCTIVE GROUPS


We continue to work exclusively overℂ. In this section, we begin by defining the notion of a Moore–Tachikawa groupG. The definition provides sufficient conditions forT^*G^*to be Hartogs abelianizable, and includes all reductive groups as examples. In this way, Moore–Tachikawa groups give rise to TQFTs. In order to obtain TQFTs beyond those conjectured by Moore–Tachikawa, we must give examples of non-reductive Moore–Tachikawa groups. Most of this section is devoted to the construction of such groups.



 §.§ Moore–Tachikawa groups

LetGbe an affine algebraic group with Lie algebra. Givenξ∈^*, letG_ξ Gand_ξdenote the centralizers ofξunder the coadjoint representations ofGand, respectively. We define^*_reg{ξ∈^*:_ξ≤_η for all η∈^*}.A Moore–Tachikawa group is an affine algebraic group G with the following properties:

  ^* has a complement of codimension at least two in ^*;

  G_ξ is abelian for all ξ∈^*;

   the pullback of the cotangent groupoid T^*G^* to ^* is abelianizable, i.e. Morita equivalent to an abelian symplectic groupoid.


A sufficient condition for (ii) and (iii) to hold is the existence of a smooth, closed subvarietyS^*with the following properties:

  * S;

  * S intersects every coadjoint orbit in  transversely in a singleton;

  * G_ξ is abelian for all ξ∈ S. 


One shows this condition to be sufficient in a manner analogous to the proof of Proposition <ref>.

Definition <ref> provides sufficient conditions forT^*G^*to be Hartogs abelianizable. By combining this observation with Theorem <ref>, one concludes that every Moore–Tachikawa group induces a TQFT in. On the other hand, results of Kostant <cit.> imply that every reductive group is Moore–Tachikawa. The TQFTs induced by reductive groups are essentially those conjectured by Moore–Tachikawa. This motivates us to find examples of non-reductive Moore–Tachikawa groups. We devote the next two subsections to this task.



 §.§ The semidirect product of _2 and its standard representation


LetGbe an affine algebraic group andρ : G ⟶GL(V)a finite-dimensional, algebraic representation.
Consider the group semidirect productH  G ⋉_ρ V, i.e. we considerVas a group with addition, so that multiplication inHis given by

    (g_1, v_1) · (g_2, v_2) = (g_1 g_2, v_1 + ρ_g_1(v_2)).

The Lie algebra ofHis then⋉_ρ V, with bracket

    [(x_1, v_1), (x_2, v_2)] = ([x_1, x_2], ρ_x_1 v_2 - ρ_x_2(v_1)).
















If ρ is the standard representation of _2, then the non-reductive group H _2⋉_ρ^2 is Moore–Tachikawa. In particular, H induces a TQFT _2.

Let V = ^2 be the standard representation of _2.
We identify V^* with V = ^2 in an (2, )-equivariant way via the standard symplectic form, i.e.

    ^2 ≅ V^*,    (η_1, η_2) ω((η_1, η_2), ·),

where ω((η_1, η_2), (v_1, v_2)) = η_1 v_2 - η_2 v_1.
We also identify _2^* with _2 via the invariant bilinear form (x, y) (xy).
A straightfoward computation then reveals that the coadjoint representation of  on ^* ≅_2 ×^2 is

    _(x, u)^*(ξ, η) =
    (
    [ (x_2 ξ_3 - x_3 ξ_2) - 12(η_1 u_2 + η_2 u_1)             2 (x_1 ξ_2 - x_2 ξ_1) + η_1 u_1;             2 (x_3 ξ_1 - x_1 ξ_3) - η_2 u_2 (x_3 ξ_2 - x_2 ξ_3) + 12(η_1 u_2 + η_2 u_1) ],
    [ x_1 η_1 + x_2 η_2; x_3 η_1 - x_1 η_2 ]),

where x = ([  x_1  x_2;  x_3 -x_1 ]), u = ([ u_1; u_2 ]), ξ = ([  ξ_1  ξ_2;  ξ_3 -ξ_1 ]), and η = ([ η_1; η_2 ]). If η 0, then _(x, u)^*(ξ, η) = 0 if and only if there exists a constant c ∈ such that

    x = c[  η_1 η_2   -η_1^2;    η_2^2 -η_1 η_2 ]  and  
    u = 
    -2c[ η_1 ξ_1 + η_2 ξ_2; η_1 ξ_3 - η_2 ξ_1 ].

It follows that the centralizer _(ξ,η) is 1-dimensional in this case. If η = 0, then _(0, u)^*(ξ, η) = 0 for all u∈^2. The centralizer _(ξ,η) must therefore have dimension at least two in this case. We conclude that

    ^* = {(ξ, η) ∈(2, ) ×^2 : η 0}.

This locus clearly has a complement of codimension two in ^*.


It remains to verify (ii) and (iii) in Definition <ref>. We accomplish this by verifying the sufficient condition mentioned immediately after that definition. To this end, let S^* be the image of

    σ : ^*,    
    z (
    [ 0 0; z 0 ]
    ,
    [ 1; 0 ]).

Suppose that (ξ, η) ∈^*.
Since η 0, there exists g ∈_2 such that ρ_g^*(η) = (1, 0).
We may therefore assume that η = (1, 0).
Note that the stabilizer of (1, 0) is the set of g ∈_2 of the form

    g = [ 1 a; 0 1 ],   a∈ℂ.

For such g and η = (1, 0), we have

    _(g, u)^*(ξ, η) = 
    (
    [            ξ_1 + a ξ_3 - u_22 -2a ξ_1 + ξ_2 - a^2 ξ_3 + u_1;                           ξ_3          - ξ_1 - a ξ_3 + u_22 ],
    [ 1; 0 ]).

It follows that there is a unique u = (u_1, u_2) such that _(g, u)^*(ξ, η) is S.
In other words, S intersects every regular coadjoint orbit exactly once.
Moreover, the intersection is transverse.
Equation (<ref>) also shows the following: for all z ∈, the stabilizer of σ(z) ∈^* is the set of elements of H of the form (([ 1 a; 0 1 ]),([ a^2z;  2az ])), a ∈; this is abelian.
Our proof is therefore complete




 §.§ The centralizer of a minimal nilpotent element in _3

One may weaken Definition <ref> by replacingwith an arbitraryG-invariant open subsetU^*. This open subset would be required to satisfy the following properties: 

  U has a complement of codimension at least two in ^*;

  G_ξ is abelian for all ξ∈ U;

   the pullback of the cotangent groupoid T^*G^* to U is abelianizable, i.e. Morita equivalent to an abelian symplectic groupoid.


A sufficient condition for (ii) and (iii) to hold would be the existence of a smooth, closed subvarietyS^*with the following properties:

  * S U;
	
  * S intersects every coadjoint orbit in U transversely in a singleton;
	
  * G_ξ is abelian for all ξ∈ S. 


Suppose that an affine algebraic groupGadmits aG-invariant open subsetU^*satisfying (i)–(iii). The cotangent groupoidT^*G^*is then clearly Hartogs abelianizable. As such, it determines a TQFT_2⟶.


The term Moore–Tachikawa group could have been reserved for this generalization of Definition <ref>, i.e. for an affine algebraic group G admitting a G-invariant open subset U^* satisfying (i)–(iii) above. One of the reasons for preserving Definition <ref> is the bridge it provides to pure Lie theory. A Lie theorist could find meaningful examples satisfying Definition <ref> without reading our manuscript in detail.      

If G _3 is the stabilizer of

    e [ 0 0 1; 0 0 0; 0 0 0 ]∈(3, ),

then G satisfies Conditions (i)–(iii) above. It thereby determines a TQFT _2⟶.

The group G can be written explicitly as

    G = {[    r    a    c;    0 r^-2    b;    0    0    r ] : (r, a, b, c) ∈^××^3},

with Lie algebra

    = {[   t   x   z;   0 -2t   y;   0   0   t ] : (t, x, y, z) ∈^4 }.

Identify ^* with ^4 via the coordinates dual to those in (<ref>).
For g = (r, a, b, c) ∈ G and (s, u, v, w) ∈^*, we have

    _g^*(s, u, v, w) =
    (
    s + 3 (a/r u - b r^2 v + abr w),
    u/r^3 + wb/r,
    v r^3 - war^2,
    w
    ).

One then sees that we have two transverse slices, given by

    σ_1 :	×   ^*,    (v, w)  (0, 1, v, w) 
    σ_2 :	×   ^*,    (u, w)  (0, u, 1, w).

They are slices for the set of regular elements with (u, w)  (0, 0) and (v, w)  (0, 0), respectively, which both have complements of codimension two.
If w  0, the stabilizer of σ_1(v, w) = (0, 1, v, w) ∈^* is the set of g = (r, a, b, c) such that a = v/w(r - 1/r^2) and b = 1/w(r - 1/r^2), which is abelian.
If w = 0, the stabilizer is the set of g = (r, a, b, c) such that r^3 = 1 and a = bv, which is also abelian.
It follows that σ_1 is an admissible Hartogs slice.
A similar argument shows that σ_2 has the same property.

More generally, let  be a simple Lie algebra of type A or C. Consider the centralizers G_e G and _e of a nilpotent element e∈ under the adjoint representations of G and , respectively. By <cit.>, (_e)^*_reg has a complement of codimension at least two in (_e)^*. It is therefore reasonable to expect G_e to satisfy Conditions (i)–(iii) from the beginning of this subsection, and for a more general version of Proposition <ref> to hold.








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