# Calabi-Yau structures for multiplicative preprojective algebras

aa r X i v : . [ m a t h . R T ] F e b Calabi–Yau structures for multiplicative preprojectivealgebras

Tristan Bozec ∗ , Damien Calaque † , Sarah Scherotzke ‡ Abstract

In this paper we deal with Calabi–Yau structures associated with (diﬀerentialgraded versions of) deformed multiplicative preprojective algebras, of which we pro-vide concrete algebraic descriptions. Along the way, we prove a general result thatstates the existence and uniqueness of negative cyclic lifts for non-degenerate relativeHochschild classes.

Contents k [ x ± ] k [ x ± ] . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Relative Calabi–Yau structures on evaluations k [ x ± ] → k . . . . . . . . . . . 103.3 A Calabi–Yau cospan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 A quiver . . . . . . . . . . . . . . . . . 144.2 Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.3 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 k [ x ± ], derived loop stacks, and the adjoint quotient . . 195.2 Moduli of objects of k h x ± , y ± i , pair of pants, and fusion . . . . . . . . . . . 205.3 Open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 ∗ IMAG, Univ. Montpellier, CNRS, Montpellier, France [email protected] † IMAG, Univ. Montpellier, CNRS, Montpellier, France [email protected] ‡ Mathematical Institute, University of Luxembourg, Luxembourg [email protected] Introduction

Given a quiver Q , that is an oriented graph, one may consider the preprojective algebraassociated to Q over some ﬁeld k . It can be deﬁned as a quotient of the path algebra kQ of the double quiver Q , obtained by adjoining to each edge e : i → j between two vertices i and j a reverse edge e ∗ : j → i . We quotient by a single relation µ = X e ∈ Q [ e, e ∗ ] , a signed combination of 2-cycles in Q . Originally introduced by Gelfand–Ponomarev [15](see also [24]) in a strictly algebraic context, the preprojective algebra turns out bearinga strong geometric signiﬁcance. It may indeed be understood as the algebraic structureunderlying the cotangent to the moduli of representations of Q (see [5] for a fully precisestatement). Its representations correspond to the 0-ﬁber of the moment map associated tothe linear group acting by conjugation at each vertex. This fact has been extensively usedby Lusztig [19], and later Nakajima [21], to geometrically realize quantum groups and theirrepresentations, in particular through the deﬁnition of lagrangian subvarieties of symplecticquiver moduli.Multiplicative variants of these preprojective algebras have been introduced by Crawley-Boevey and Shaw [13] in the course of their study of the Deligne–Simpson problem. It isdeﬁned by performing a quotient by Y e ∈ Q (1 + ee ∗ )(1 + e ∗ e ) − of an appropriate localization of kQ . These variants turn out to naturally appear in variousareas such as character varieties [4, 26], local systems and perverse sheaves over Riemannsurfaces of nodal curves [36, 2, 3], or integrable systems [11] among others. The geometricframework in which multiplicative quiver varieties seem to be better studied is the one ofquasi-hamiltonian reduction and group-valued moment maps from [1], as shown by Van denBergh [34, 35].Multiplicative preprojective algebras ﬁt both into the quasi-hamiltonian formalism andinto its non-commutative analogue as developped by Van den Bergh in [35]. In the ﬁrstcase, group-valued moment maps and the quasi-hamiltonian formalism have a nice interpre-tation within the framework of shifted symplectic geometry of [22], in terms of lagrangianmorphisms and derived lagrangian intersections (see [8, 25]).Using the non-commutative analogue of quasi-hamiltonian formalism, one obtains thatmultiplicative preprojective algebras come equipped with double quasi-Poisson structures [34].Furthermore, Fern´andez and Herscovich have recently proved in [14] that double quasi-Poisson structures give rise to pre-Calabi–Yau structures in the sense of Iyudu–Kontsevich–Vlassopoulos [16] , extending a similar result from loc. cit. for double Poisson structures.In the same way as shifted Poisson structures in the sense of [9, 23] arise on the sourceof morphisms equipped with a lagrangian structure [20] (actually, shifted Poisson structuresare conjectured to be equivalent to lagrangian thickenings), it is expected that pre-Calabi–Yau structures in the sense of [16] often (if not always) arise on the target of Calabi–Yaumorphisms in the sense of [6].Hinging on these observations, our goal is to directly construct Calabi–Yau structureson appropriate algebraic objects, and get back the usual lagrangian morphisms associatedwith group-valued moment maps on moduli spaces. Namely: We would like to warn the reader that pre-Calabi–Yau structures in loc. cit. are diﬀerent from theones considered e.g. in [5, 32] (the latter being non-commutative pre-symplectic strctures, rather than non-commutative Poisson structures).

21) We ﬁrst study a Calabi–Yau structure on k [ x ± ], seen as the multiplicative analog of k [ x ].Using this Calabi-Yau structure, we obtain using [6] a 1-shifted symplectic structure onmoduli of perfect complexes. We show that restricting to the moduli of representationswe recover the usual 1-shifted symplectic structure on the adjoint quotient which iscrucial in the derived symplectic interpretation of the quasi-hamiltonian formalism;(2) We give a 1-Calabi-Yau on cospans which allow us to retrieve standard lagrangiancorrespondences when applying the moduli of objects functor Perf . We recover inparticular the lagrangian correspondence that was shown in [25] to underly the fusionproduct from [1];(3) We give a relative 1-Calabi–Yau structure on the algebraic counterpart of the group-valued moment map. This is done via a gluing procedure called fusion.(4) Via pushouts of Calabi-Yau cospans, we obtain a 2-Calabi-Yau structure on the diﬀer-ential graded multiplicative preprojective algebra, deﬁned in Theorem 4.11. The zerotruncation of the diﬀerential graded multiplicative preprojective algebra is the originalmultiplicative preprojective algebra.

Description of the paper

In section 2, we provide a short recollection on Calabi–Yau structures, after [6]. We alsoshow that, in the case of smooth dg-categories sitting in degree 0, the required cyclic lift ofthe non-degenerate relative Hochschild class, in the deﬁnition of a Calabi–Yau structure ona morphism, automatically exists and is unique. This extends to the relative case a resultof [29], and is of independent interest, see Theorem 2.5.Section 3 uses this result to produce 1-Calabi–Yau structures on k [ G m ] = k [ x ± ] and thecospan deﬁned by k [ x ± ] ∐ k [ y ± ] → k h x ± , y ± i ← k [ z ± ], denoted by F in this introductiononly. In each case we deﬁne explicit Hochschild classes that we prove to be non-degenerate.Thanks to section 2, these admit a unique cyclic lift. We also study evaluation morphisms k [ x ± ] → k .In section 4, using the Calabi-Yau structures of the previous section, we show in Theo-rem 4.8 that the multiplicative moment map is 1-Calabi-Yau. The quiver A = • → • servesas a building block. Again, this structure is made explicit and proven to be non-degenerate“by hand”, whereas its cyclic lift exists thanks to section 2. The cospan F studied ear-lier then serves in a gluing process ( a.k.a. fusion ) to extend our result to arbitrary quivers.Pushouts along evaluation morphisms yields a 2-Calabi-Yau structure on a dg-algebra whose0-truncation is the classical multiplicative preprojective algebra, c.f. Theorem 4.11.Finally, section 5 justiﬁes our choices of Hochschild classes deﬁning Calabi–Yau struc-tures. We prove that when taking

Perf , we retrieve standard symplectic structures. Namely,the 1-shifted symplectic structure on

Perf k [ x ± ] matches the one on the derived loop stack L Perf k . We also prove that the Calabi–Yau structure on F corresponds to a particulargluing of the boundaries of the pair-of-pants. We conjecture that the structures we geton our dg-variants of multiplicative preprojective algebras yield standard quasi-hamiltonianstructures on multiplicative quiver varieties. Related works

In [3], Bezrukavnikov and Kapranov prove that certain triangulated categories of microlocalcomplexes on nodal curves have a Calabi–Yau property, which roughly corresponds to theexistence of an almost Calabi–Yau structure according to our terminology. However, it isnot clear if it admits an actual Calabi–Yau (i.e. cyclic) lift. In loc. cit. the authors mention3 dg-version of the multiplicative preprojective algebra and expect that it is a Calabi–Yaudg-algebra. Our results show that this is indeed true. This expectation was motivated by theexistence of an equivalence of abelian categories between microlocal sheaves on nodal curveswith rational components on the one side, and modules over the multiplicative preprojectivealgebra on the other side. They could not conclude, because it is not known if a similarequivalence holds for the dg-version of the multiplicative preprojective algebra.A similar approach is considered by Shende and Takeda in their work [28] on Calabi–Yaustructures of topological Fukaya categories. It is possible that, following some suggestionsfrom [28, § § A building bloc remains to be dealt with.Yeung [37, § Acknowledgements

The ﬁrst and second author have received funding from the European Research Council(ERC) under the European Union’s Horizon 2020 research and innovation programme (GrantAgreement No. 768679).

Along this paper, we use the same notation and terminology as in [5]. We recall brieﬂy themost important information. Note also that in this paper k is a ﬁeld of characteristic zero.We denote by Mod k the category of cochain complexes over k . A dg-category is a Mod k -enriched category and the category of dg-categories with dg-functors is denoted by Cat k .We refer to [18, 30] for a detailed introduction to dg-categories and their homotopy theory.If M is a model category, we will write M for the corresponding ∞ -category obtainedby localizing along weak equivalences.We use the notation “Map” to distinguish the space of ∞ -categorical morphisms fromthe set of 1-categorical morphisms, for which we use the notation “Hom”. The underlinedversions designate their enriched counterparts (unless otherwise speciﬁed, the enrichmentis over complexes). If a category has a symmetric monoidal structure which is closed, weuse upper case letters for the internal enrichment, i.e. Hom and

Map , for categories and ∞ -categories, respectively. 4ecall the Hochschild chains functorHH :

Cat k −→ Mod k ; A A L ⊗ A e A op , where A e := A ⊗ A op . We write HH i ( A ) for the ( − i )-th cohomology of HH( A ). Dually,HH i ( A ) is deﬁned as the i -th cohomology of R Hom A e ( A , A ).Hochschild chains carry a mixed structure, which is given in the standard explicit modelby Connes’s B -operator. The negative cyclic complex of A , denoted by HC − ( A ), is deﬁnedas the homotopy ﬁxed points of HH( A ) with respect to the mixed structure; it comes witha natural transformation ( − ) ♮ : HC − ⇒ HH. As before, HC − i ( A ) stands for the ( − i )-thcohomology of HC − ( A ).Recall the inverse dualizing functor ( − ) ∨ : Mod A e −→ Mod op A e that is given as follows: for a right A e -module M , and an object a ∈ Ob( A op ⊗ A ), M ∨ ( a ) := R Hom

Mod ( A e )op (cid:0) M ◦ τ, A e ( a, − ) (cid:1) . where τ is the anti-involution τ : A e ˜ −→ ( A e ) op such that A ◦ τ = A op .A dg-category A is smooth if A is a perfect A e -module. For smooth dg-categories, wehave the following equivalences( − ) ♭ : HH( A ) ∼ −→ R Hom

Mod A e ( A ∨ , A )and R Hom

Mod A e ( A , A ) ≃ A ∨ L ⊗ A e A . Deﬁnition 2.1.

Let A be a smooth dg-category.(1) A class c : k [ n ] → HH( A ) such that c ♭ : A ∨ [ n ] → A is an equivalence is called non-degenerate . Such a non-degenerate Hochschild class is called an almost n -Calabi–Yau structure on A .(2) A n -Calabi–Yau structure on A is a is class c : k [ n ] → HC − ( A ) such that c ♮ is non-degenerate.We now recall (relative) Calabi–Yau structures on morphisms and cospans of dg-categories,following Brav–Dyckerhoﬀ [6] and To¨en [31, § Deﬁnition 2.2.

Let A f −→ C g ←− B be a cospan of smooth dg-categories.(1) An almost n -Calabi–Yau structure on this cospan is the data of a homotopy commutingdiagram k [ n ] c B / / c A (cid:15) (cid:15) HH( B ) (cid:15) (cid:15) HH( A ) / / HH( C )such that c A and c B are non-degenerate in the sense of Deﬁnition 2.1(1), and such thatthe homotopy HH( f )( c A ) ∼ HH( g )( c B ) is non-degenerate in the following sense: the5nduced (homotopy) commuting square C ∨ [ n ] g ∨ / / f ∨ (cid:15) (cid:15) ( B ∨ [ n ]) L ⊗ B e C e c ♭ B ⊗ id ≃ B L ⊗ B e C eg ⊗ id (cid:15) (cid:15) ( A ∨ [ n ]) L ⊗ A e C e c ♭ A ⊗ id ≃ A L ⊗ A e C e f ⊗ id / / C is cartesian.(2) An n -Calabi–Yau structure on the cospan is a homotopy commuting diagram k [ n ] c B / / c A (cid:15) (cid:15) HC − ( B ) (cid:15) (cid:15) HC − ( A ) / / HC − ( C )such that the image under ( − ) ♮ is an almost n -Calabi–Yau structure.(3) If A = ∅ then we call these (almost) n -Calabi–Yau structures on the morphism g .Recall that n -Calabi–Yau cospans do compose: after [6, Theorem 6.2], the non-degeneracyproperty is preserved under composition.We ﬁnally note that whenever A = B = ∅ , an n -Calabi–Yau structure on ∅ → C ← ∅ is the same as an ( n + 1)-Calabi–Yau structure on C . In particular, the push-out of two n -Calabi–Yau morphisms automatically inherits an ( n + 1)-Calabi–Yau structure. Proposition 2.3 ([29], Section 5) . Suppose B is a smooth dg-category. If B is almost n -Calabi–Yau, then HH i ( B ) ≃ HH n − i ( B ) for every i ∈ Z . Furthermore, if B is concentratedin degree zero then(a) HH i ( B ) = 0 for all i = 0 , , . . . , n ;(b) HC − i ( B ) = 0 for all i > n ;(c) the natural map HC − n ( B ) → HH n ( B ) is an isomorphism.In particular, every almost n -Calabi–Yau structure on B admits an n -Calabi–Yau lift.Proof. This is essentially [29, Proposition 5.5, Corollary 5.6 & Proposition 5.7]. We repro-duce the proof here for the reader’s convenience.We have an isomorphism of B e -modules c ♭ : B ∨ [ n ] ≃ B . It yieldsHH n − i ( B ) ≃ Hom Ho ( Mod B e ) ( B ∨ [ n ] , B [ i ]) c ♭ ≃ Hom Ho ( Mod B e ) ( B , B [ i ]) ≃ HH i ( B ) . If B is concentrated in degree zero then its Hochschild homology and cohomology are con-centrated in non-negative degrees, and thus, using the above identiﬁcations, HH i ( B ) = 0 forall i = 0 , , . . . , n . We then consider the negative cyclic complex, which is given by takingformal power series in a degree 2 variable u with coeﬃcients in the Hochschild complex, and6iﬀerential being given as d − uδ , where δ is the mixed diﬀerential. The ﬁrst page of thespectral sequence associated with the ﬁltration by powers of u reads as follows:0 u HH ( B ) uδ / / u HH ( B ) uδ / / · · · u n +1 HH n ( B )HH ( B ) uδ / / u HH ( B ) uδ / / · · · u n HH n ( B ) 0HH ( B ) uδ / / u HH ( B ) uδ / / · · · · · · HH n ( B ) 0 0 · · · This proves (a) and (b).

Remark 2.4.

Under the assumption of Proposition 2.3, the duality isomorphism extends toHochschild homology with values in any B -bimodule M : H i ( B , M ) ≃ H n − i ( B , M ). Moreover,if both B and M are concentrated in degree 0, then (a) still holds: H i ( B , M ) vanishes forall i = 0 , . . . , n . Theorem 2.5.

Let F : B → C be a functor between smooth dg-categories that are concen-trated in degree zero. Every almost n -Calabi–Yau structure on F admits a unique n -Calabi–Yau lift.Proof. First of all, we know from Proposition 2.3 that the almost n -Calabi–Yau structure c B ∈ HH n ( B ) on B uniquely lifts to a n -Calabi–Yau structure c − B ∈ HC − n ( B ). The other partof the almost n -Calabi–Yau structure on F is a homotopy from F ( c B ) to 0, which amountsto the choice of a relative lift c F ∈ HH n +1 ( C , B ) of c B . Indeed, HH i ( C , B ) is deﬁned as the( − i )-th cohomology of the homotopy coﬁber (or, mapping cone) of HH( B ) → HH( C ), sothat we have a long exact sequence · · · → HH n +1 ( C , B ) → HH n ( B ) → HH n ( C ) → HH n ( C , B ) → · · · The non-degeneracy of c F tells us that the nul-homotopic sequence of C e -modules C ∨ [ n ] → B ∨ [ n ] ⊗ B e C e ≃ B ⊗ B e C e → C is actually a homotopy ﬁber sequence. Applying Hom Ho ( Mod C e ) ( − , C ) yields a long exactsequence · · · → HH k ( C ) → H k ( B , C ) ≃ H n − k ( B , C ) → HH n − k ( C ) → HH k +1 ( C ) → · · · Hence, using that Hochschild homology and cohomology of C vanishes for negative indices(because C is concentrated in degree 0), together with the version from remark 2.4 of thevanishing property (a), we get that the Hochschild homology (and cohomology) of C vanishesin degrees i = 0 , . . . , n + 1. We again look at the ﬁrst page of the Hochschild-to-negativecyclic spectral sequence: 7 u HH ( C ) uδ / / u HH ( C ) uδ / / · · · u n +1 HH n +1 ( C )HH ( C ) uδ / / u HH ( C ) uδ / / · · · u n HH n +1 ( C ) 0HH ( C ) uδ / / u HH ( C ) uδ / / · · ·· · · HH n ( C ) uδ / / u HH n +1 ( C ) 0 0HH n +1 ( C ) 0 0Putting this together, we obtain the following morphism of exact sequences:0 / / HC − n +1 ( C ) ∼ (cid:15) (cid:15) / / HC − n +1 ( C , B ) (cid:15) (cid:15) / / HC − n ( B ) ∼ (cid:15) (cid:15) / / HC − n ( C ) (cid:15) (cid:15) (cid:15) (cid:15) / / HH n +1 ( C ) / / HH n +1 ( C , B ) / / HH n ( B ) / / HH n ( C )The injectivity of the rightmost arrow follows from the fact thatHC − n ( C ) ≃ ker (cid:0) HH n ( C ) → u HH n +1 ( C ) (cid:1) , and it implies that the image of c − B via HC − n ( B ) → HC − n ( C ) vanishes (because the image of c B through HH n ( B ) → HH n ( C ) does so). Therefore c − B lifts to a relative class in HC − n +1 ( C , B ).The map from the aﬃne space of relative lifts of c − B to the aﬃne space of relative lifts of c B is aﬃne and modelled on the linear map HC − n +1 ( C ) → HH n +1 ( C ), which is an isomorphism.Using that both aﬃne spaces are non-empty, we get that the map from relative lifts of c − B torelative lifts of c B is a bijection. Hence we get that a cyclic lift of c F exists and is unique. k [ x ± ] k [ x ± ] Let A = k [ x ± ] = k [ G m ]. It is the function ring of a smooth aﬃne algebraic variety; hence1-Calabi–Yau structures on A are exactly non-vanishing top degree (here, degree 1) forms.The Calabi–Yau structure we consider on A is, up to a scalar, α := d dR log( x ) = x − d dR x .In the rest of this subsection, we provide descriptions of this 1-Calabi–Yau structure thatwill be convenient for later purposes. Remark 3.1.

Notice that the inverse morphism inv : x x − allows to identify ( A , α )with ( A , − α ). We also observe that α is invariant under rescaling maps x qx , q ∈ k × (i.e. it is of zero weight for the action of G m on itself by multiplication).8 emark 3.2. In [37], Yeung also considers a Calabi–Yau structure on k [ z ± ], which is diﬀerent form ours: Yeung’s Calabi–Yau structure is given by d dR z , and is exact, as opposedto ours. On moduli of representations, the Calabi–Yau structure we consider gives back the1-shifted symplectic structure that encodes the quasi-hamiltonian formalism (see Section 5below); we expect that Yeung’s Calabi–Yau structure rather leads to a linearized versionof it. As a matter of fact, if one considers the I -adic completion ˆ A at the kernel I of theevaluation at x = 1, the morphism k [ z ] → ˆ A sending z to log( x ) is well-deﬁned and sendsthe canonical Calabi–Yau structure d dR z on k [ z ] to α . We work with the normalized Hochschild complex C n ( A ) = A ⊗ ¯ A ⊗ n where ¯ A = A /k , withHochschild diﬀerential b . On C n ( A ), the Connes boundary map is given by B ( x ⊗ · · · ⊗ x n ) = n X i =0 ( − ni ⊗ x i ⊗ · · · ⊗ x n ⊗ x ⊗ · · · ⊗ x i − . We set 2 α n = ( x − ⊗ x ) ⊗ n − ( x ⊗ x − ) ⊗ n ∈ C n − ( A ) , so that b ( α n ) = 2(1 ⊗ α n − ) and B ( α n ) = 2 n (1 ⊗ α n ) . A direct computation then shows that α = X k ≥ k ! u k α k +1 satisﬁes ( b − uB )( α ) = 0. We want to prove that α = ( x − ⊗ x − x ⊗ x − ) is non-degenerate. First observe thatthe class of x − ⊗ x equals the one of − x ⊗ x − (and thus, the one of α ) in HH ( A ) ≃ Ω A .Indeed, in (cohomological) degree − A is Ω A = k [ x ± ] d dR x .The class of a x − ⊗ x , resp. − x ⊗ x − , is computed via the Hochschild–Kostant–Rosenberg(HKR) map a ⊗ b ad dR b , and we ﬁnd x − d dR x , resp. − xd dR ( x − ) = xx − d dR x = x − d dR x . Hence it is suﬃcient to prove that the Hochschild 1-cycle x − ⊗ x is non-degenerate.The reduced Bar resolution of A is given by¯B( A ) = M n ≥ (cid:0) A ⊗ ¯ A ⊗ n ⊗ A (cid:1) [ n ]with the usual diﬀerential being given by an alternating sum of products of successiveelements. We also have a smaller resolutionR( A ) = A e [1] ⊕ A e with diﬀerential sending 1 ⊗ x ⊗ − ⊗ x . Its dual isR( A ) ∨ = A e ⊕ A e [ − A ) → ¯B( A ). In degree 0 it is the identity, and in degree − f ⊗ g f ⊗ x ⊗ g . Using this smaller resolution we obtain the “small Hochschildcomplex”: A [1] ⊕ A with zero diﬀerential. It maps inside the standard Hochschild complex as follows: in degree0 it is the identity, and in degree − f to f ⊗ x . (this map is in fact a quasi-inverse tothe HKR quasi-isomorphism). In the small Hochschild complex, the class of interest readsas x − , and one can show that as a mapR( A ) ∨ [1] −→ R( A ) (3.3)it is nothing but the product with x − ⊗ Remark 3.4.

We could have proven non-degeneracy ﬁrst, and then use [29, Proposition5.7] (see also Proposition 2.3 above) in order to obtain the existence (and unicity) of a cycliclift. k [ x ± ]For every n -Calabi–Yau category A , with Calabi–Yau structure c , one can consider thesame category with opposite Calabi–Yau structure − c , and denote it ¯ A . Then the functor A ` ¯ A → A is relative Calabi–Yau.Let k = ke be the terminal dg-category ( e denotes the identity of the single object); itis obviously 0-Calabi–Yau, with Calabi–Yau structure being e . Proposition 3.5.

There is an equivalence k [ x ± ] ≃ k a k ` ¯ k k of -Calabi–Yau dg-categories, where the Calabi–Yau structure on the left-hand-side is α ,and the one on the right-hand-side is obtained as a Calabi–Yau push-out.Proof. First of all we introduce the interval dg-category kI : it is the k -linearization of thecategory I = 1 ˜ −→ x between them. Observe that wehave a factorization k ` k → kI → k , where the ﬁrst functor is a coﬁbration (the inclusioninto kI of its subcategory of objects), and the second functor is a trivial ﬁbration. Henceour homotopy push-out can be computed as the strict push-out k ` k ` k kI ≃ k [ x ± ]. We thusget the requested equivalence of dg-categories. It remains to prove that the 1-Calabi–Yaustructures coincide. Thanks to Proposition 2.3, it is suﬃcient to prove that the underlyingHochschild classes coincide.Finally, the 0-cycle e − e is homotopic to zero in the coﬁbrant replacement kI of k : e − e = b ( α ), where α still makes sense for kI . k [ x ± ] → k The pull-back of the closed 1-form α = d dR log( x ) along any k -point q : A = k [ x ± ] −→ k G m (i.e. q ∈ k × ) obviously vanishes. This tells us that the morphism q is relative pre-Calabi–Yau in the sense of [5] Lemma 3.6.

The above relative pre-Calabi–Yau structure is non-degenerate.Proof.

One ﬁrst observes that both (R( A ) ∨ [1]) ⊗ A e k e and R( A ) ⊗ A e k e are isomorphic to k [1] ⊕ k with zero diﬀerential. Moreover, after applying ⊗ A e k e , the isomorphism (3.3) becomes themultiplication by q − on each component. Then recall that k ∨ = k , so that the morphism k ∨ [1] = k [1] → k [1] ⊕ k = (R( A ) ∨ [1]) ⊗ A e k e , resp. R( A ) ⊗ A e k e = k [1] ⊕ k → k is the obvious inclusion, resp. projection. Hence the map k ∨ [1] −→ ﬁb (cid:18) R( A )) ⊗ A e k e → k (cid:19) identiﬁes with the map k [1] q − −→ k [1] ≃ ﬁb (cid:0) k [1] ⊕ k → k ) (cid:1) . This proves the non-degeneracy.

Our aim is to prove that the cospan k [ x ± ] a k [ y ± ] −→ k h x ± , y ± i ←− k [ z ± ] , (3.7)where the rightmost map is z xy , is relative Calabi–Yau in the sense of [5].Set β = y − ⊗ x − ⊗ xy − y ⊗ y − x − ⊗ x , which satisﬁes α ( xy ) − ( α ( x ) + α ( y )) = b ( β ) . Lemma 3.8.

The above homotopy β is non-degenerate, and thus deﬁnes an almost -Calabi–Yau structure on the cospan (3.7) . This almost -Calabi–Yau structure lifts uniquelyto a -Calabi–Yau structure.Proof. As a preliminary observation, let us recall on the one hand that B := k h x ± , y ± i also has a small resolution as a B -bimodule:R( B ) = ( B e ) ⊕ [1] ⊕ B e with diﬀerential sending (1 ⊗ ,

0) to x ⊗ − ⊗ x , and (0 , ⊗

1) to y ⊗ − ⊗ y . ThereforeR( B ) ∨ = B e ⊕ ( B e ) ⊕ [ − ⊗ x ⊗ − ⊗ x, y ⊗ − ⊗ y ).As the maps α ( xy ) and α ( x ) + α ( y ) are homotopic via β , the following diagram ishomotopy commutative B ∨ [1] (cid:15) (cid:15) / / A ∨ ⊗ A e B e [1] α ( xy ) ≃ A ⊗ A e B e (cid:15) (cid:15) ( A ⊕ ) ∨ ⊗ A e B e [1] α ( x )+ α ( y ) ≃ A ⊕ ⊗ A e B e / / B We warn again the reader that pre-Calabi–Yau in the sense of [5] (see also [32]) is the non-commutativeanalog of pre-symplectic, and diﬀers from the pre-Calabi–Yau notion from [16] that is the non-commutativeanalog of a Poisson structure. A = k [ x ± ]. Following § A ⊗ A e B e ≃ B e [1] ⊕ B e , with diﬀerential sending 1 ⊗ x ⊗ − ⊗ x . Hence, we get that the ﬁbre of the map( B e [1]) ⊕ ⊕ ( B e ) ⊕ → ( B e ) ⊕ [1] ⊕ B e induced by α ( xy ) − α ( x ) + α ( y ) is isomorphic to R( B ) ∨ [1] = B e [1] ⊕ ( B e ) ⊕ . Then, usingTheorem 2.5, we get that β lifts to a unique homotopy β between α ( xy ) and α ( x ) + α ( y ).Therefore the cospan (3.7) carries a 1-Calabi–Yau structure. Below we give an alternativepresentation of this cospan. Observe that we have the following (strict) commuting diagram in the category (Cat smk ) k/ HC − of smooth dg-categories equipped with a negative cyclic 0-cycle (in order to lighten the no-tation, we omit coproducts): k k (cid:15) (cid:15) kk (cid:15) (cid:15) ∅ B B ✆✆✆✆✆✆✆✆✆✆ (cid:30) (cid:30) ❂❂❂❂❂❂❂❂❂ k k k ¯ k ¯ k k @ @ ✁✁✁✁✁✁✁✁✁✁ ^ ^ ❂❂❂❂❂❂❂❂❂ ❆❆❆❆❆❆❆❆ ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥ k ∅ \ \ ✾✾✾✾✾✾✾✾✾✾✾ (cid:0) (cid:0) ✁✁✁✁✁✁✁✁✁ k k O O k k O O It admits a replacement by a (homotopy coherent) commuting diagram in the ∞ -category( Cat smk ) k/ HC − : kI k ¯ I (cid:15) (cid:15) kk (cid:15) (cid:15) ∅ @ @ ✂✂✂✂✂✂✂✂✂✂✂ ❅❅❅❅❅❅❅❅❅❅ kI k ¯ I k ¯ k ¯ k k @ @ ✁✁✁✁✁✁✁✁✁✁ ` ` ❅❅❅❅❅❅❅❅❅❅ ❆❆❆❆❆❆❆❆ } } ③③③③③③③③③ k ∅ \ \ ✾✾✾✾✾✾✾✾✾✾✾ (cid:0) (cid:0) ✁✁✁✁✁✁✁✁✁ kI k ¯ I O O k k O O Observe that the above diagram strictly commutes in Cat k , but that the negative cyclic0-cycles only match up to homotopy.By composing horizontal cospans we obtain a new (homotopy) commuting diagram inthe ∞ -category ( Cat smk ) k/ HC − : k [ z ± ] (cid:15) (cid:15) ∅ rrrrrrrrrrrr % % ▲▲▲▲▲▲▲▲▲▲▲ k h x ± , y ± i ∅ e e ▲▲▲▲▲▲▲▲▲▲▲▲ y y rrrrrrrrrrr k [ x ± ] ` k [ y ± ] O O c z and c x,y in HC − ( k [ z ± ]) and HC − ( k [ x ± ] ` k [ y ± ]), respectively,together with a homotopy c x,y,z between their images in HC − ( k h x ± , y ± i ). Proposition 3.9.

The triple ( c x,y , c z , c x,y,z ) deﬁnes a -Calabi–Yau structure on (3.7) , thatcoincides with the one from Lemma 3.8.Proof. As we have already seen in Proposition 3.5, the 1-Calabi–Yau structures on k [ x ± ] ` k [ y ± ]match up: c x,y ∼ α ( x ) + α ( y ).They also match on k [ z ± ], but there is a subtelty that is worth noticing. As usual,according to the unicity of cyclic lifts from Proposition 2.3, in order to prove that c z ∼ α ( z )it is suﬃcient to prove that c ♮z ∼ α ( z ) ♮ = α ( z ). Now, computed stricly, the top horizontalpush-out gives the k -linearization C of a category with two objects 1 , x : 1 ˜ → y : 2 ˜ →

1. Of course, we have an equivalence k [ z ± ] ˜ →C , sending z to xy .Following a similar calculation as in the proof of Proposition 3.5, we get on C the Hochschild1-cycle α ( x ) + α ( y ). Up to a Hochschild boundary, this matches up with the image of α ( z ) through the equivalence given by z xy . Indeed, the formula for the homotopy β still makes sense in C .It remains to prove that the homotopy c x,y,z matches with β . As the underlyingHochschild homotopy β = β ♮ is non-degenerate (thanks to Lemma 3.8), according to theunicity of cyclic lifts from Theorem 2.5, it suﬃces to prove that the underlying Hochschildhomotopies c ♮x,y,z and β ♮ = β coincide. We already proved it, as β is the homotopy thatidentiﬁes α ( x ) + α ( y ) with α ( z ) in C . Remark 3.10.

Let us put what we have done so far in a more general perspective, byﬁrst recalling from [6, § d -manifold M , the k -linearization L ( M ) := dg(Sing( M )) of the fundamental ∞ -groupoid of M carries a d -Calabi–Yau struc-ture. Moreover, in loc. cit. the authors also prove that if N is a compact oriented ( d + 1)-manifold with boundary ∂N = M , then one gets a d -Calabi–Yau structure on the naturalfunctor L ( M ) → L ( N ). We conjecture the existence of a symmetric monoidal ( ∞ , n )-category CY sn of n -iterated s -Calabi–Yau cospans, similar to the iterated category of la-grangian correspondences sketched in [8], and rigoroulsy constructed in [10]. We also conjec-ture that the functor dg(Sing( − )) leads to a fully extended oriented TFT in every dimension:i.e. it should admit an upgrade to a symmetric monoidal ( ∞ , n )-functor Bord orn −→ CY n for every n (in particular, k is n -dualizable in CY n ). For the above presentation of theCalabi–Yau cospan structure on k h x ± , y ± i , we took inspiration from a construction of thepair-of-pants as a suitable composition of 2-iterated oriented bordisms (see subsection 5.2below, where this decomposition of the pair-of-pants is made explicit), and guessed thediagram one shall write by pretending that the conjecture was known. Remark 3.11.

The Calabi–Yau push-out of this cospan with the evaluation Calabi–Yaumorphism q : k [ z ± ] → k from § k [ x ± ] → k [ y ± ] given by x q − y (see Remark 3.1). More precisely, thepush-out gives a morphism k [ x ± ] a k [ y ± ] −→ k h x ± , y ± i / ( xy = q ) , under which the image of α ( x ) + α ( y ) is identically 0. Then using that inv gives anisomorphism between the Calabi–Yau structure and its inverse on k [ x ± ] (see Remark 3.1,again), we obtain the desired Calabi–Yau cospan from k [ x ± ] to k [ y ± ].13 Multiplicative preprojective algebras

Consider a quiver Q , which consists in a vertex set V , and an oriented edge set E : to eachedge e we associate a source s ( e ) and a target t ( e ) in V . We consider its double version Q = ( V, E = E ⊔ E ∗ ), where E ∗ consists in reverse arrows e ∗ : t ( e ) → s ( e ), and extend ∗ inan involution of E by setting e ∗∗ = e for every e ∈ E . We also set ǫ ( e ) = 1 and ǫ ( e ∗ ) = − e ∈ E . As mentioned in the introduction, Crawley-Boevey and Shaw introducedin [13] the multiplicative preprojective algebra Λ q ( Q ), where q ∈ ( k ∗ ) V . It is given as thequotient of a localization of kQ by the relation Y e ∈ E (1 + ee ∗ ) ǫ ( e ) − X v ∈ V q v e v where e v denotes the length 0 idempotent path at v . It is thus required to invert all 1 + ee ∗ for e ∈ E ∗ , which actually amounts to inverting 1 + ee ∗ for all e ∈ E . We denote kQ loc thelocalization of kQ with respect to these elements.The deﬁnition of Λ q ( Q ) a priori requires an ordering on E , but the resulting quotientactually doesn’t depend on it (up to isomorphism [13, Theorem 1.4]). Remark 4.1.

We can either view Λ q ( Q ) as an algebra, or as a category (with objects thevertices of Q , that correspond to the idempotents of Λ q ( Q )). There is a Morita morphismfrom one to another, so that it doesn’t matter for what we do (see [5, Remark 5.4]). A quiver Consider the quiver A = ( V = { , } , E = { e : 1 → } ), with orthogonal idempotents e and e satisfying 1 = e + e , and write a = e + e ∗ e and a = e + ee ∗ . Note that 1 + e ∗ e invertible ⇔ a invertible ⇔ a invertible ⇔ ee ∗ invertible , in which case (1 + e ∗ e ) − = e + a − a − = e − ea − e ∗ a − = e − e ∗ a − e (1 + ee ∗ ) − = e + a − . (4.2)Thus in the A case, the product in the multiplicative preprojective realtion reads(1 + ee ∗ )(1 + e ∗ e ) − = a + a − = (1 + e ∗ e ) − (1 + ee ∗ ) . Denote by B the localization kA [ a − , a − ], and deﬁne morphisms µ i : k [ x ± i ] → B , i ∈{ , } , by setting µ ( x ) = a − and µ ( x ) = a . Equalities (4.2) further imply a − e = ea − e ∗ a − = a − e ∗ e ∗ ea − = e − a − ee ∗ a − = e − a − . (4.3)14 .1.1 The homotopy Note that µ maps α to a ⊗ a − − a − ⊗ a , and µ maps α to a − ⊗ a − a ⊗ a − wheretensor products are performed over the algebra R = ⊕ v ∈ V ke v . Thus µ ( α ) + µ ( α ) = e ∗ e ⊗ a − − a − ⊗ e ∗ e + a − ⊗ ee ∗ − ee ∗ ⊗ a − + e ⊗ a − − a − ⊗ e + a − ⊗ e − e ⊗ a − = e ∗ e ⊗ a − − a − ⊗ e ∗ e + a − ⊗ ee ∗ − ee ∗ ⊗ a − + 1 ⊗ ( a − − a − )as a − i ⊗ e i = a − i ⊗ e ∗ e ⊗ a − − a − ⊗ e ∗ e + a − ⊗ ee ∗ − ee ∗ ⊗ a − is the image under b of e ∗ ⊗ e ⊗ a − + a − ⊗ e ∗ ⊗ e − e ∗ ⊗ a − ⊗ e − a − ⊗ e ⊗ e ∗ . Also, 1 ⊗ ( a − − a − ) = 1 ⊗ ( a − − e − a − + e ) [normalization]= 1 ⊗ ( ee ∗ a − − e ∗ ea − ) [(4.3)]= − Bb ( e ∗ ⊗ ea − ) [(4.3)]= bB ( e ∗ ⊗ ea − ) . Hence µ ( α ) + µ ( α ) is the image under b of β = e ∗ ⊗ e ⊗ a − + a − ⊗ e ∗ ⊗ e − e ∗ ⊗ a − ⊗ e − a − ⊗ e ⊗ e ∗ + B ( e ∗ ⊗ ea − )= e ∗ ⊗ e ⊗ µ + µ ⊗ e ∗ ⊗ e − e ∗ ⊗ µ − ⊗ e − µ − ⊗ e ⊗ e ∗ + 1 ⊗ e ∗ ⊗ eµ − ⊗ µ − e ⊗ e ∗ (4.4)if µ = µ ( x ) + µ ( x ). The cospan µ ∐ − µ carries an almost -Calabi–Yau structure, that liftsuniquely to a -Calabi–Yau structure thanks to Theorem 2.5.Proof. Set A = k [ x ± ] ∐ k [ x ± ] and u = µ ∐ − µ . Thanks to the existence of the homotopy β given by (4.4), the following diagram homotopy commutes. B ∨ [1] u ∨ [1] / / (cid:15) (cid:15) A ∨ [1] ⊗ A e B e / / α ⊗ A e B e (cid:15) (cid:15) coﬁb( u ∨ [1]) (cid:15) (cid:15) ﬁb( u ) / / A ⊗ A e B e u / / B (4.6)To show the non-degeneracy, we need to prove that the vertical maps there are isomor-phisms. Since A is 1-Calabi–Yau, it is suﬃcient to prove that the leftmost vertical map is15n isomorphism. Set A i = k [ x ± i ], and B ei = A i ⊗ A ie B e induced by µ i . Using the resolutionsfrom § A ⊗ A e B e with the complex( B e ⊕ B e )[1] ⊕ ( B e ⊕ B e )with diﬀerential d : ( p ⊗ q , p ⊗ q ) ( p a − ⊗ q − p ⊗ a − q , p a ⊗ q − p ⊗ a q )where p i , q i ∈ B . A B -bimodule resolution of B is given byΩ ( B ) d ′ −→ B e . By [27, Therorem 10.6] (see also [5, Remark 5.4]), we can identify Ω ( B ) with B ⊗ R kE ⊗ R B and d ′ (1 ⊗ v ⊗

1) = v ⊗ − ⊗ v , where R still denotes ⊕ v ∈ V ke v . Here for A the edge set E is simply { e } . Hence, u is given by the following commutative diagram B e ⊕ B e f / / d (cid:15) (cid:15) B ⊗ R kE ⊗ R B d ′ (cid:15) (cid:15) B e ⊕ B e τ / / B ⊗ R B . where f ( p ⊗ q , p ⊗ q ) = f ( p ⊗ q ) − f ( p ⊗ q ) τ ( p ⊗ q , p ⊗ q ) = p ⊗ q − p ⊗ q . Let us give a concrete description of f . We have a k -linear map ι : kE → B ⊗ R kE ⊗ R B whichsends a path p = α · · · α n , α i ∈ E , to n X i =1 α · · · α i − ⊗ α i ⊗ α i +1 · · · α n . This map has a natural B e -linear extension B → B ⊗ R kE ⊗ R B , still denoted by ι , satisfying ι ( bb ′ ) = bι ( b ′ ) + ι ( b ) b ′ . (4.7)Then it can be checked that the maps f i : B ei → B ⊗ R kE ⊗ R B is given as B e -linear maps by f (1 ⊗

1) = ι ( a − ) f (1 ⊗

1) = − ι ( a ) . We then identify ﬁb( u ) with ( f, τ ).The resolution of B ∨ as a B e -module is given by d ′∨ : B ⊗ R B → B ⊗ R kE ⊗ R B , ⊗ e i ⊗ X α ∈ E ( α ⊗ α ∗ ⊗ − ⊗ α ∗ ⊗ α ) . In the A case, this just reads d ′∨ (1 ⊗

1) = e ⊗ e ∗ ⊗ − ⊗ e ∗ ⊗ e + e ∗ ⊗ e ⊗ − ⊗ e ⊗ e ∗ . g = α ⊗ A e B e : ( B e ⊕ B e )[1] ⊕ ( B e ⊕ B e ) → ( B e ⊕ B e )[1] ⊕ ( B e ⊕ B e )is induced by the image of α under µ ∐ − µ , hence by the internal product m with( a ⊗ − ⊗ a , a − ⊗ − ⊗ a − )on both terms, thanks to § β deﬁned by (4.4) induces a zero homotopy h of the map B ∨ [1] ugu ∨ [1] / / B . With the chosen resolutions, this yields a map h : B ⊗ R E ⊗ R B → B ⊗ R E ⊗ R B such that thetriangles in the following diagram commute B ⊗ R B d ′∨ / / fmτ ∨ (cid:15) (cid:15) B ⊗ R kE ⊗ R B τmf ∨ (cid:15) (cid:15) h x x qqqqqqqqqq B ⊗ R kE ⊗ R B d ′ / / B ⊗ R B . where τ ∨ (1 ⊗

1) = (1 ⊗ , − ⊗ f mτ ∨ (1 ⊗

1) = f m (1 ⊗ , − ⊗ f ( a ⊗ − ⊗ a , ⊗ a − − a − ⊗ a ι ( a − ) − ι ( a − ) a + ι ( a ) a − − a − ι ( a )= − ι ( a ) a − + a − ι ( a ) + ι ( a ) a − − a − ι ( a )= − ( e ∗ ⊗ e ⊗ ⊗ e ∗ ⊗ e ) a − + a − ( e ∗ ⊗ e ⊗ ⊗ e ∗ ⊗ e )+ ( e ⊗ e ∗ ⊗ ⊗ e ⊗ e ∗ ) a − − a − ( e ⊗ e ∗ ⊗ ⊗ e ⊗ e ∗ )= hd ′∨ (1 ⊗ h (1 ⊗ e ⊗

1) = a − ⊗ e ⊗ − ⊗ e ⊗ a − h (1 ⊗ e ∗ ⊗

1) = 1 ⊗ e ∗ ⊗ a − − a − ⊗ e ∗ ⊗ . The homotopy h therefore induces an isomorphism B ∨ [1] ∼ −→ ﬁb( u ) as wished (it is theleftmost vertical map in (4.6)). Following [34], we use a fusion procedure to go from the A case to the case of an arbitraryquiver Q = ( V, E ). The following endows the “noncommutative group-valued” moment mapfor kQ loc = kQ [(1 + ee ∗ ) − ] e ∈ E , that deﬁnes the multiplicative preprojective algebra, witha Calabi–Yau structure. Theorem 4.8.

There is a 1-Calabi–Yau structure on the morphism µ : a v ∈ V k [ z ± v ] −→ kQ loc z v Y e ∈ E ∩ t − ( v ) (1 + ee ∗ ) × Y e ∈ E ∩ s − ( v ) (1 + e ∗ e ) − . roof. Denote by Q sep the quiver with same edge set E but vertex set E = { v e = s ( e ) , v e ∗ = t ( e ) } . It is the disjoint union of | E | copies of A that we aim to glue by “fusing” vertices. Thiswill be done using composition of Calabi–Yau structures by means of push-outs. Thanksto § a e ∈ E ( k [ x ± e ] ∐ k [ y ± e ]) → kQ sep [(1 + ee ∗ ) − ] e ∈ E (4.9)given by x e ( e s ( e ) + e ∗ e ) − and y e + e t ( e ) + ee ∗ .For each vertex v ∈ V , ﬁx a total ordering of all edges of E with target v and the samewith E ∗ . Consider e, f = e + 1 ∈ E , both with target v . We have a 1-Calabi–Yau cospan 3.7given by k [ y ± e ] a k [ y ± f ] → k h y ± e , y ± f i ← k [ z ± e,f ]with z e,f y e y f . Similarly, if e ∗ , f ∗ = e ∗ + 1 ∈ E ∗ , both with target v , we have a 1-Calabi–Yau cospan given by k [ x ± e ] a k [ x ± f ] → k h x ± e , x ± f i ← k [ z ± e,f ]with z e,f x e x f . Finally, if e = max E t − ( v ) and f ∗ = min E ∗ t − ( v ), we have a 1-Calabi–Yau cospan given by k [ y ± e ] a k [ x ± f ] → k h y ± e , x ± f i ← k [ z ± e,f ]with z e,f y e x f . Proceeding to ordered compositions of cospans, we get a 1-Calabi–Yaucospan given by C v := k D ( y ± e ) e ∈ E ∩ t − ( v ) , ( x ± e ) e ∈ E ∩ s − ( v ) E(cid:16)` e ∈ E ∩ t − ( v ) k [ y ± e ] (cid:17) ` (cid:16)` e ∈ E ∩ s − ( v ) k [ x ± e ] (cid:17) ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ k [ z ± v ] i i ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ where coproducts and variables are ordered.Now ﬁx an ordering on V , composing the above yields a cospan a e ∈ E ( k [ x ± e ] ∐ k [ y ± e ]) → a v ∈ V C v ← a v ∈ V k [ z ± v ]that can be composed with (4.9) in order to get a 1-Calabi–Yau structure on µ as expected. Remark 4.10.

Note that this proof is independent of the choice of the function ǫ : Q →{± } deﬁning the preprojective mulitplicative algebra. Consider a family of 1-Calabi–Yau morphisms q v : k [ z ± v ] → k , v ∈ V ; that is a collection q = ( q v ) v ∈ V ∈ ( k × ) V . Thanks to Lemma 3.6 and Theorem 4.8, we have a 2-Calabi–Yaustructure on the push-out of µ with ` v ∈ V q v . To compute this push-out, let us use for each v the k [ z ± v ]-coﬁbrant replacement of k given by k h z ′ v , z ± v i where z ′ v lies in degree − z v indegree 0 and the diﬀerential is given by z ′ v z v − q v . We thus get the following.18 heorem 4.11. For every q ∈ ( k × ) V ( Q ) , there is a -Calabi–Yau structure on the dg-algebra Υ q ( Q ) deﬁned as follows: • As a graded algebra, Υ q ( Q ) is freely generated over kQ loc by the bimodule (cid:0) kQ loc (cid:1) e ⊗ R e (cid:18) ⊕ v ∈ V kz ′ v (cid:19) ; • The diﬀerential sends z ′ v to Y e ∈ E ∩ t − ( v ) (1 + ee ∗ ) × Y e ∈ E ∩ s − ( v ) (1 + e ∗ e ) − − q v . Remark 4.12. • The zeroth cohomology of Υ q ( Q ) is the deformed preprojective algebraΛ q ( Q ). • The dg-algebra Υ q ( Q ) coincides with the one of [3, § • Theorem 4.11 generalizes [37, Theorem 5.52] from star-shaped quivers to arbitrary ones.

The moduli of objects

Perf was introduced by To¨en-Vaqui´e in [33] as a functor

Perf : Cat f.t.k → dSt Artk from the ∞ -category of ﬁnite type dg-categories to the ∞ -category of derived Artin k -stacks. For a ﬁnite type dg-category A and a commutative diﬀerential graded k -algebra B , Perf A ( B ) := Map Cat k ( A , Mod perfB ) consists in perfect B -module valued A -modules. In [22], n -shifted symplectic structures for Artin stack, as well as n -shifted lagrangian morphismsand correspondences (see also [8]) have been introduced. Calabi–Yau structures on dg-categories and functors can be considered as non-commutative analogs of shifted symplecticand lagrangian structures in the following sense: by [7, Theorem 5.5] (see also [32]), themoduli stack of objects Perf sends n -Calabi–Yau structures to (2 − n )-shifted symplecticstructures, and can be extended to a functor from n -Calabi–Yau cospans to (2 − n )-shiftedlagrangian correspondences.Another way of producing new shifted symplectic and lagrangian structures from oldones was discovered in [22, Theorem 2.5]: it is shown that for an n -shifted symplectic Artinstack X , the mapping stack Map (cid:0) − , X (cid:1) in dSt Art sends (nice enough) d -oriented Artinstacks to ( n − d )-shifted symplectic stacks. By [8, Theorem 4.8] the functor Map (cid:0) − , X (cid:1) sends (nice enough) d -oriented cospans to ( n − d )-shifted lagrangian correspondences. Notethat the Betti-stack functor, denoted by ( − ) B , maps d -oriented manifolds to (suﬃcientlynice) d -oriented derived stacks. k [ x ± ] , derived loop stacks, and the adjointquotient On the one hand, the 1-Calabi–Yau structure on k [ x ± ] as constructed in Section 3 in-duces a 1-shifted symplectic structure on the derived stack Perf k [ x ± ] . On the other hand, Perf k [ x ± ] is equivalent to the derived loop stack L Perf k := Map ( B Z , Perf k ). Knowing19hat B Z ≃ S B is 1-oriented, and that Perf k is 2-shifted symplectic (because k is 0-Calabi–Yau), we obtain, thanks to [22, Theorem 2.5], a transgressed 1-shifted symplectic structureon L Perf k . Proposition 5.1.

There is an equivalence

Perf k [ x ± ] ≃ L Perf k as -shifted symplectic derived stacks.Proof. On the one hand, recall that for every n -shifted symplectic derived stack X , thederived loop stack L X is equivalent, as an ( n − X × X × X X , where X denotes the same derived stack equipped with the opposite n -shifted symplecticstructure. Indeed, the functor Map (cid:0) ( − ) B , X (cid:1) is an oriented topological ﬁeld theory (see [8,Theorem 4.8]), and as such it sends the gluing of two oriented manifolds along a commonboundary to the corresponding derived lagrangian intersection. The case of interest for usis the one of S , that is obtained by gluing two closed intervals along two points: S ≃ pt a pt ` pt pt , where pt denotes the point with its opposite orientation.On the other hand, using Proposition 3.5 and the fact that Perf sends compositionsof Calabi–Yau cospans to compositions of lagrangian correspondences (and, in particular,Calabi–Yau pushouts to lagrangian intersections), see [7] and [5, § Perf k [ x ± ] ≃ Perf k ` k ` ¯ k k ≃ Perf k × Perf k × Perf k Perf k ≃ L Perf k as 1-shifted symplectic derived stacks.Finally, by restricting ourselves to the open substack consisting of perfect modules of am-plitude 0 and ﬁxed dimension n , we get back the transgressed 1-shifted symplectic structureon L ( BGL n ) (recall that the open embedding BGL n ֒ → Perf k is a 2-shifted symplectomor-phism). According to [25], this 1-shifted symplectic structure coincides with the explicit onegiven on the adjoint quotient [ GL n /GL n ] ≃ L ( BGL n ) by the quasi-hamiltonian formalism(see [8, 25]). Remark 5.2.

The 1-Calabi–Yau evaluation morphism q : k [ x ± ] → k , q ∈ k × , induces a1-shifted lagrangian morphism Perf k → Perf k [ x ± ] . We let the reader check that, whenrestricted on the open substacks of amplitude 0 modules of dimension n , it gives back thelagrangian morphism BGL n → [ GL n /GL n ] corresponding to the group-valued moment mappt → GL n given by q Id n . k h x ± , y ± i , pair of pants, and fusion Recall that a lagrangian structures on the correspondence

Perf k [ x ± ] ` k [ y ± ] ←− Perf k h x ± ,y ± i −→ Perf k [ z ± ] , (5.3)given by applying the moduli of objects Perf to the Calabi–Yau cospan (3.7) (see [7, Theo-rem 5.5]). Using the other description from § Map (cid:0) ( − ) B , Perf (cid:1) sends pt to

Perf k , we obtain an alternative con-struction of the lagrangian correspondence (5.3). It amounts to apply Map (cid:0) ( − ) B , Perf (cid:1) tothe diagram pt pt (cid:15) (cid:15) ptpt (cid:15) (cid:15) ∅ A A ✄✄✄✄✄✄✄✄✄✄✄ (cid:31) (cid:31) ❃❃❃❃❃❃❃❃❃❃ pt pt pt ¯pt¯pt pt > > ⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤ ` ` ❇❇❇❇❇❇❇❇❇ " " ❊❊❊❊❊❊❊❊ | | ②②②②②②②② pt ∅ ] ] ❀❀❀❀❀❀❀❀❀❀❀ (cid:127) (cid:127) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) pt pt O O pt pt O O and then horizontally compose correspondences, as Perf sends push-outs to pull-backs.Here we recall that ( − ) denotes the orientation, respectively the symplectic structure, withinverted sign. A convenient replacement of the above diagram looks as follows: (cid:7)(cid:7) (cid:7)(cid:7) (cid:15) (cid:15) (cid:7)(cid:7) (cid:7)(cid:7) o o ∼ (cid:15) (cid:15) / / (cid:7) (cid:7)(cid:7) (cid:7) (cid:15) (cid:15) ∅ C C ✟✟✟✟✟✟✟✟✟✟ (cid:29) (cid:29) ✿✿✿✿✿✿✿✿✿✿✿✿ / / (cid:7)(cid:7) •• (cid:7)(cid:7) •• (cid:7)(cid:7) •• (cid:7)(cid:7) •• o o / / (cid:7) (cid:7)(cid:7) (cid:7) • •• • ∅ Z Z ✺✺✺✺✺✺✺✺✺✺ o o (cid:2) (cid:2) ✆✆✆✆✆✆✆✆✆✆✆✆ •• •• O O •• •• o o ∼ O O / / • •• • O O (5.4)Taking pushouts along the three horizontal correspondances yields the following 1-orientedcospan/cobordism: 21 (cid:7)(cid:7) (cid:7) (cid:15) (cid:15) ∅ B B ✆✆✆✆✆✆✆✆✆✆✆✆✆ (cid:31) (cid:31) ❃❃❃❃❃❃❃❃❃❃❃❃❃❃❃ / / (cid:7) (cid:7)(cid:7) (cid:7) • •• • ∅ \ \ ✾✾✾✾✾✾✾✾✾✾✾✾✾ o o (cid:127) (cid:127) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) • •• • O O Note that the manifold at the center of the diagram is the pair of pants (see Figure 1). ! !! (cid:7) (cid:7)(cid:7) (cid:7) • •• • ! (cid:7) (cid:7)(cid:7) (cid:7) • •• • (cid:7)(cid:7) •• (cid:7)(cid:7) •• Figure 1: The decomposition of the pair of pants corresponding to (5.4), in 3d and 2d.Using that

Map (cid:0) ( − ) B , Perf (cid:1) is a fully extended TFT [10], we have that the lagrangiancorrespondence (5.3) is obtained by applying

Map (cid:0) ( − ) B , Perf (cid:1) to the oriented cobordismgiven by the pair of pants, see [8, Theorem 4.8].Hence, when restricting ourselves to the substacks of amplitude zero modules of ﬁxeddimension n , we get a lagrangian correspondence[ GL n /GL n ] × [ GL n /GL n ] ←− [( GL n × GL n ) /GL n ] −→ [ GL n /GL n ]that coincides with the one given by applying Map (cid:0) ( − ) B , BGL n (cid:1) to the pair of pants (andusing [8, Theorem 4.8]). It was shown by Safronov [25] that composition with this lagrangiancorrespondence gives back the fusion procedure from [1].22 emark 5.5. Notice that

Perf sends the conjectural fully dualizable object k in CY n fromRemark 3.10 to the fully dualizable object Perf k in Lag n . As a consequence Perf shallintertwine the conjectural fully extended TFT from Remark 3.10 with the fully extendedTFT

Map (cid:0) ( − ) B , Perf k (cid:1) from [10] (see also [8] for a heuristic). What we have done above isfollowing this guiding idea and applying it in an ad hoc way to the case of the pair-of-pants. Before applying reduction, we have a 1-shifted lagrangian structure on the morphism

Perf kQ loc −→ Perf V ( Q ) k [ x ± ] . Fixing a dimension vector ~n ∈ ( Z ≥ ) V ( Q ) , one can consider the open substacks of dimension ~n amplitude 0 modules. This leads to a 1-shifted lagrangian structure on the morphism (cid:2) Rep ( kQ loc , ~n ) /GL ~n (cid:3) −→ (cid:2) GL ~n /GL ~n (cid:3) (note that Rep ( kQ loc , ~n ) ≃ DRep ( kQ loc , ~n )). Knowing from § Rep ( kQ loc , ~n ) is a quasi-hamiltonian GL ~n -space. We conjecture that it coincides with thequasi-hamiltonian structure on the very same space from [34, 35, 36, 4].Observe that it suﬃces to prove the conjecture for the simplest case Q = A . Indeed, inthe above references the general case is obtained from the A one by the fusion process of [1].We proceeded in the same way in § § A , one could try to prove a similar statementdirectly at the noncommutative level. To achieve this, one would ﬁrst have to rigorouslyprove that Van den Bergh’s noncommutative quasi-hamiltonian structures [35] naturallylead to relative Calabi–Yau structures (as was argued in the introduction).Finally, we believe that our 1-Calabi–Yau structure on the dg-version of the multiplica-tive preprojective algebra shall give back the non-degenerate pairing appearing in [3] (in thecase of a nodal curve with rational components) on the one hand, and could probably berecovered from [28] on the other hand. In both cases, it is very likely that the proof of thecomparison will again go through a reduction to the A quiver, using fusion. References [1] A. Alekseev, A. Malkin & E. Meinrenken, Lie group valued moment maps, J. Diﬀer-ential Geom. (1998), no. 3, 445–495.[2] A. A. Beilinson, How to glue perverse sheaves, K-theory, arithmetic and geometry,Lecture Notes in Math., , Springer, Berlin, 1987, 42–51.[3] R. Bezrukavnikov & M. Kapranov, Microlocal sheaves and quiver varieties, Annalesde la Facult´e des Sciences de Toulouse (2016), no. 2–3, 473–516.[4] P. Boalch, Global Weyl groups and a new theory of multiplicative quiver varieties,Geom. Topol. (2015), no. 6, 3467–3536.[5] T. Bozec, D. Calaque & S. Scherotzke, Relative critical loci and quiver moduli,preprint available at https://arxiv.org/abs/2006.01069 .236] C. Brav & T. Dyckerhoﬀ, Relative Calabi–Yau structures, Compositio Math. (2019), 372–412.[7] C. Brav & T. Dyckerhoﬀ, Relative Calabi–Yau structures II: Shifted Lagrangians inthe moduli of objects, preprint available at https://arxiv.org/abs/1812.11913 .[8] D. Calaque, Lagrangian structures on mapping stacks and semi-classical TFTs, Con-temporary Mathematics (2015).[9] D. Calaque, T. Pantev, B. To¨en, M. Vaqui´e & G. Vezzosi, Shifted Poisson structuresand deformation quantization, Journal of Topology (2017), no. 2, 483–584.[10] D. Calaque, R. Haugseng & C. Scheimbauer, On the AKSZ construction in derivedalgebraic geometry as an extended topological ﬁeld theory, in preparation.[11] O. Chalykh & M. Fairon, Multiplicative quiver varieties and generalised Ruijsenaars-Schneider models, J. Geom. Phys. , (2017), 413–437.[12] W. Crawley-Boevey & M. P. Holland, Noncommutative deformations of Kleiniansingularities, Duke Math. J. , no. 3, (1998), 605–635.[13] W. Crawley-Boevey & P. Shaw, Multiplicative preprojective algebras, middle convo-lution and the Deligne–Simpson problem, Adv. Math. (2006), 180–208.[14] D. Fern´andez & E. Herscovich, Double quasi-Poisson algebras are pre-Calabi–Yau,preprint available at https://arxiv.org/abs/2002.10495 .[15] I. M. Gel’fand & V. A. Ponomarev, Model algebras and representations of graphs,Funktsional. Anal. i Prilozhen. no. 3, (1979), 1–12.[16] N. Iyudu, M. Kontsevich & Y. Vlassopoulos, Pre-Calabi-Yau algebras as noncom-mutative Poisson structures, Journal of Algebra (2021), no. 1, 63–90.[17] D. Kaplan & T. Schedler, Multiplicative preprojective algebras are 2-Calabi–Yau,available at https://arxiv.org/abs/1905.12025 .[18] B. Keller, On diﬀerential graded categories, Proc. ICM Vol. II, 151–190, Eur. Math.Soc., Z¨urich, 2006.[19] G. Lusztig, G, Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer.Math. Soc. no. 2, (1991), 365–421.[20] V. Melani & P. Safronov, Derived coisotropic structures II: stacks and quantization,Selecta Math. (2018), 3119–3173.[21] H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras,Duke Math. J. no. 2, (1994), 365–416.[22] T. Pantev, B. To¨en, M. Vaqui´e & G. Vezzosi, Shifted symplectic structures, Publi-cations math´ematiques de l’IH´ES (2013), no. 1, 271–328.[23] J.P. Pridham, Shifted Poisson and symplectic structures on derived N -stacks, Jouralof Topology (2017), no. 1, 178–210.[24] C. M. Ringel, The preprojective algebra of a quiver, CMS Conf. Proc., , Amer.Math. Soc., Providence, RI (1998), 467–480.2425] P. Safronov, Quasi-Hamiltonian reduction via classical Chern-Simons theory, Ad-vances in Mathematics (2016), 733–773.[26] T. Schedler & A. Tirelli, Symplectic resolutions for multiplicative quiver va-rieties and character varieties for punctured surfaces, preprint available at https://arxiv.org/abs/1812.07687 .[27] A. H. Schoﬁeld, Representations of rings over skew ﬁelds, London Math. Soc. Lec.Note Ser. , Cambridge Univ. Press, 1985.[28] V. Shende & A. Takeda, Calabi–Yau structures on topological Fukaya categories,preprint available at https://arxiv.org/abs/1605.02721 .[29] L. de Thanhoﬀer de V¨olcsey & M. Van den Bergh, Calabi–Yau Deformations andNegative Cyclic Homology, Journal of noncommutative geometry (2018), no. 4,1255–1291.[30] B. To¨en, Lectures on dg-categories, in Topics in Algebraic and Topological K -Theory ,Lecture Notes in Mathematics , Springer, Berlin, Heidelberg.[31] B. To¨en, Derived algebraic geometry, EMS Surveys in Mathematical Sciences (2014), no. 2, 153–240.[32] B. To¨en, Structures symplectiques et de Poisson sur les champs en cat´egories,preprint available at https://arxiv.org/abs/1804.10444 .[33] B. To¨en & M. Vaqui´e, Moduli of objects in dg-categories, Annales de l’ENS (2007), 387–444.[34] M. Van den Bergh, Double Poisson algebras, Trans. Amer. Math. Soc. (2008),no. 11, 5711–5769.[35] M. Van den Bergh, Non-commutative quasi-Hamiltonian spaces, in Poisson geometryin mathematics and physics , Contemp. Math. (2008), 273–300.[36] D. Yamakawa, Geometry of multiplicative preprojective algebra, Int. Math. Res. Pap. (2008).[37] W.-K. Yeung, Relative Calabi–Yau completions, preprint available at https://arxiv.org/abs/1612.06352https://arxiv.org/abs/1612.06352