The local motivic DT/PT correspondence

Abstract We show that the Quot scheme QLn=QuotA3(IL,n) parameterising length n quotients of the ideal sheaf of a line in A3 is a global critical locus, and calculate the resulting motivic partition function (varying n), in the ring of relative motives over the configuration space of points in A3. As in the work of Behrend–Bryan–Szendrői, this enables us to define a virtual motive for the Quot scheme of n points of the ideal sheaf IC⊂OY, where C⊂Y is a smooth curve embedded in a smooth 3‐fold Y, and we compute the associated motivic partition function. The result fits into a motivic wall‐crossing type formula, refining the relation between Behrend's virtual Euler characteristic of QuotY(IC,n) and of the symmetric product SymnC. Our ‘relative’ analysis leads to results and conjectures regarding the pushforward of the sheaf of vanishing cycles along the Hilbert–Chow map QLn→Symn(A3), and connections with cohomological Hall algebra representations.


Overview
Let C be a smooth curve embedded in a smooth 3-fold Y with ideal sheaf I C ⊂ O Y . For an integer n 0, the Quot scheme Q n C = Quot Y (I C , n) parameterises closed subschemes Z ⊂ Y containing C and differing from it by an effective zero-cycle of length n. The main purpose of this paper is to construct a virtual motive for this Quot scheme, that we view as a 1-dimensional analogue of the degree 0 motivic Donaldson-Thomas invariant [Hilb n Y ] vir defined by Behrend, Bryan and Szendrői [3]. The Quot scheme Q n C can be seen as a moduli space of curves and points in Y , where the curve C is fixed. This geometric situation presents a new feature that was absent in the purely 0-dimensional case: wall-crossing. More precisely, it is proved in [32,Proposition 5.1] that the generating function of the Behrend weighted Euler characteristics χ(Q n C ) satisfies the wall-crossing type formula n 0 where M (t) = m 1 (1 − t m ) −m is the MacMahon function. We show that the motivic partition function encoding the motivic classes (0.1) admits a factorisation similar to (0.2), where the point contribution, refining the factor M (−t) χ(Y ) , is precisely the motivic partition of the Hilbert schemes Hilb n Y computed in [3]. The curve contribution, on the other hand, refines the factor (1 + t) −χ (C) and is given by the (shifted) motivic zeta function of the curve C, namely n 0 Our approach to the problem is a natural extension of the approach of Behrend, Bryan and Szendrői, in that our definitions and calculations take place with respect to the natural local model L ⊂ A 3 given by fixing a line in affine space -since they consider only finite-dimensional quotients of O A 3 , their local model is simply A 3 . As in their work, for general embeddings C ⊂ Y , we build [Q n C ] vir out of the local model via power structures. We leave for another day the question of whether this virtual motive accords with the virtual motive one obtains from the machinery of (−1)-shifted symplectic stacks, and concentrate on calculating everything in sight for the local theory. Furthermore, since the key to gluing local models appears to be the direct image of the vanishing cycles sheaf to the configuration space of points on A 3 , we prove all of our results in the lambda ring of motives relative to this configuration space. We conjecture, moreover, that the wall-crossing type identity (0.2) can be categorified, that is, lifted to an isomorphism between the vanishing cycle cohomologies of the relevant moduli spaces (see Section 5.1).
Just as Behrend, Bryan and Szendrői realise the local model Hilb n A 3 as a critical locus and show that the associated motivic Donaldson-Thomas invariants [Hilb n A 3 ] vir are determined, via power structures, by the motivic weights of the punctual Hilbert schemes Hilb n (A 3 ) 0 , we show, for two convenient local models that can also be realised as critical loci, that the induced virtual motives are determined by the motivic contribution of the punctual loci. In our case, we also need to consider the contribution of points embedded on the curve C ⊂ Y , and this is what gives rise to the factor (0.3) in our motivic wall-crossing formula.
The appearance of symmetric products is pretty natural and has a neat interpretation in terms of the (C-local) DT/PT correspondence: on a Calabi-Yau 3-fold Y , the symmetric products Sym n C ⊂ P χ(OC )+n (Y, [C]) are precisely the C-local moduli spaces in the stable pair theory of Y , just as Q n C ⊂ I χ(OC )+n (Y, [C]) are the C-local moduli spaces in Donaldson-Thomas theory.
For a rigid curve C ⊂ Y in a Calabi-Yau 3-fold, one can interpret the classes (0.1) as motivic Donaldson-Thomas invariants, in the same spirit as in the 0-dimensional case.
We next give an overview of our main results. The main technical tool we use is a motivic stratification technique, that we apply to the study of the (relative) motivic Donaldson-Thomas invariants of the Quot schemes Q n C0 , where C 0 ⊂ X is the exceptional curve in the resolved conifold X = Tot(O P 1 (−1) ⊕ O P 1 (−1)).

Main results
The first step towards the construction of the motivic classes (0.1) consists in setting up a convenient local model. With respect to the local model we then prove the following as part of Theorem 2.2.
Theorem A. The Quot scheme Q n L is a global critical locus.
An analogous statement is proven in [3, Proposition 3.1] for the Hilbert scheme Hilb n (A 3 ), which is realised as the critical locus of a function on the non-commutative Hilbert scheme.
Via the theory of motivic vanishing cycles [12], Theorem A produces a relative virtual motive where the maps hc n are Hilbert-Chow morphisms. The following result, proven in Section 4.2, follows from Corollary 4.6 and the main calculation of Section 3.

Theorem B. There is an identity
where Δ n : A 3 → Sym n A 3 is the small diagonal, and Passing to absolute motives, the first factor in (0.4) becomes the (signed) motivic partition function of Hilb n (A 3 ) computed in [3] and reviewed in Section 1.7. The operation Exp ∪ in Theorem B is a lift to the lambda ring M Sym(A 3 ) of the usual plethystic exponential for power series with coefficients in the ring of absolute motives. These motivic exponentials are reviewed in Section 1.5.1. We let Q L/A 3 (t) = n 0 Q n L vir · t n be the absolute partition function. Up to a sign, it is obtained by pushing (0.4) forward to a point. The absolute version of Theorem B then reads Let P n crv ⊂ Q n L be the closed subset parameterising quotients I L F such that F is entirely supported at the origin 0 ∈ L. We assign a motivic weight P n crv vir ∈ M C to this locus. The subscript 'crv' stands for 'curve'. The punctual Hilbert scheme P n pt ⊂ Q n L , which we view as parameterising quotients supported at a single point in A 3 \ L, also inherits a motivic weight, that agrees (as shown in Proposition 4.3) with the class [Hilb n (A 3 ) 0 ] vir defined in [3] starting from the critical structure on Hilb n (A 3 ). We show in Theorem 4.7 that [Q n L ] vir is determined by these two types of 'punctual' motivic classes. They moreover allow us to define a virtual motive [Q n C ] vir ∈ M C for every smooth curve C ⊂ Y in a smooth quasi-projective 3-fold Y . In other words, the class we define satisfies χ[Q n C ] vir = χ(Q n C ). We then consider the generating function For a smooth quasi-projective variety X of dimension at most 3, let Hilb n X vir · t n be the motivic partition function of the Hilbert scheme of points. In Theorem 4.9, we prove the following explicit formula, generalising (0.5).
Theorem C. Let Y be a smooth quasi-projective 3-fold, C ⊂ Y a smooth curve. Then Hilb n−j Y vir · Sym j C vir (7) in M C . In other words, we have a factorisation that, rewritten in terms of the motivic exponential, reads In the above formulas, one has [U ] vir = L −(dim U )/2 [U ] ∈ M C for a smooth scheme U . One can view the factorisation Q C/Y = Z Y · Z C as a motivic refinement of the identity (0.2). Indeed, we have M (−t) χ(Y ) = χZ Y (t), and (1 + t) −χ(C) = n χ(Sym n C)t n = χZ C (t). The relation (0.2) says that Quot schemes and symmetric products are related by a χ-weighted wall-crossing type formula, and Theorem C upgrades this statement to the motivic level.

Calabi-Yau 3-folds
Let Y be a smooth projective Calabi-Yau 3-fold. For an integer m ∈ Z and a homology class β ∈ H 2 (Y, Z), the moduli space I m (Y, β) of ideal sheaves I Z ⊂ O Y with Chern character (1, 0, −β, −m) carries a symmetric perfect obstruction theory and the Donaldson-Thomas invariant DT m β ∈ Z is by definition the degree of the associated virtual fundamental class. These invariants are related to the stable pair invariants of Pandharipande-Thomas [25] by a well-known wall-crossing formula [6,41], and the same is true for the C-local invariants DT n C ∈ Z. The numbers DT • C represent the contribution of C to the full virtual invariants DT • [C] . The C-local wall-crossing formula [31, Theorem 1.1], written term by term, reads where DT k 0 = χ(Hilb k Y ) are the degree 0 DT invariants of Y , PT j C = n g,C · χ(Sym j C) are the C-local stable pair invariants of Y and n g,C is the BPS number of C (see [26] and Section 4.3). There is an identity χ[Q n C ] vir = DT n C when n g,C = 1 (Corollary 4.10). Indeed, in this case, equation (0.8) is equivalent to (0.2). This is especially meaningful from the point of view of motivic DT theory in the situation of the following example.
Example 0.1. Assume C ⊂ Y is a smooth rigid curve, that is, H 0 (C, N C/Y ) = 0. Then C has BPS number 1, the Quot scheme Q n C is a connected component of the Hilbert scheme I χ(OC )+n (Y, [C]), and the motivic class [Q n C ] vir is a motivic Donaldson-Thomas invariant in the sense that its Euler characteristic computes the degree of the virtual fundamental class of Q n C . In this case, the formula Q C/Y = Z Y · Z C of Theorem C can be regarded as a C-local motivic DT/PT correspondence, refining the enumerative correspondence DT C = DT 0 · PT C spelled out in (0.8).

Organisation of contents
The paper is organised as follows. In Section 1, we recall foundational material on rings of motivic weights and we revisit the main formula of [3] expressing the virtual motive of Hilb n X for 3-folds. In Section 2, we prove Theorem A by restricting the critical structure on Q n C0 , where C 0 ∼ = P 1 is the exceptional curve in the resolved conifold X = Tot(O P 1 (−1) ⊕ O P 1 (−1)). In Section 3, we prove that the virtual motives of Q n L and Q n C0 are determined by motivic classes Ω n pt , Ω n crv expressing the contributions of 'fully punctual loci' (cf. Definition 3.16 and Theorem 3.17). By explicitly calculating these motives in Section 4.2, we finally prove Theorem B. We then use these classes to define (cf. Definition 4.8) a virtual motive of Q n C for every smooth curve C in a smooth 3-fold Y , and in Section 4.2 we also prove Theorem C.

Background material
In this section, we set up the notation and introduce the main tools that will be used in the rest of the paper.

Grothendieck rings of varieties
Definition 1.1. Let S be a locally finite type algebraic space over C.
(i) If S is a variety, the Grothendieck group of S-varieties is the free abelian group K 0 (Var S ) generated by isomorphism classes [X → S] of finite type varieties over S, modulo the scissor relations, namely the identities [p : (ii) We denote by L = [A 1 S ] ∈ K 0 (Var S ) the Lefschetz motive, the class of the affine line over S.
(iii) The Grothendieck group of S-stacks is the free abelian group K 0 (St aff S ) generated by isomorphism classes [X → S] of locally finite type Artin S-stacks X → S with affine stabilisers, modulo the scissor and locality relations, and the following additional relation: if f : X → S is an S-stack, such that f factors as g • π for g : Y → S an S-stack and π the projection from the total space of a rank r vector bundle, then (iv) Define the group K(Var S ) = Image(K 0 (Var S ) → K 0 (St aff S )), and give it the induced ring structure.
Where S = i∈I S i is a possibly infinite union of algebraic spaces, we will write By results of Kresch [18,Section 4], we have K 0 (St aff S ) = K 0 (Var S ) L −1 , (L n − 1) −1 n 1 and so we can alternatively define K(Var S ) as the quotient of K 0 (Var S ) by the ideal (1.1) For S and S two varieties, there is an external product defined on generators by In particular, K(Var S ) is a K(Var C )-module. When we are considering the action of absolute motives on relative motives, we will often abbreviate Often for a relative motive [X → S] ∈ K(Var S ) we will denote it by [X] S , retaining the subscript to at least remind the reader of which motivic ring it lives in.
Given a morphism f : S → T of varieties, there is an induced pullback map which is a ring homomorphism given by If S comes with an associative map ν : S × S → S, we define the convolution ring structure via ν = ν ! • , that is, we set (1. 2) The resulting associative product on K(Var S ) is commutative if ν commutes with the symmetrising isomorphism. The ring is called the ring of motivic weights over S. The structures f * , f ! , and ν carry over to M S without change. When f : S → Spec C is the structure morphism of S, we use the special notation S for the pushforward f ! . Definition 1.2. We define S 0 (Var S ) ⊂ M S to be the sub semigroup of effective motives, that is, the subset of sums of elements of the form Remark 1.3. By Definition 1.2, the motive −L 1/2 is effective, as opposed to L 1/2 . This is dictated by the fact that in the language of lambda rings (Section 1.3), we make definitions so that L 1/2 is not a line element, while −L 1/2 is.

Equivariant K-groups, quotient and power maps
Let G be a finite group. A G-action on a variety X is said to be good if every point of X has a G-invariant affine open neighbourhood; all actions are assumed to be good throughout. For instance, any G-action on a quasi-projective variety is good. Moreover, for a good G-action, an orbit space X/G exists as a variety. Definition 1.4. Let S be a variety with good G-action. We let K G 0 (Var S ) denote the abelian group generated by isomorphism classes [X → S] of G-equivariant S-varieties, modulo the G-scissor relations (over S). The equivariant Grothendieck group X ] in the righthand side is taken with the G-action induced by the trivial action on A r and the isomorphism There is a natural ring structure on K G 0 (Var S ) given by fibre product; if X and Y are G-equivariant S-varieties, we give X × S Y the diagonal G-action.
We shall consider the quotient rings obtained by modding out the ideal and its image J G ⊂ K G 0 (Var S ), respectively. We let be the rings of G-equivariant motivic weights.
There exists a natural 'quotient map' defined on generators by taking the orbit space: If the G-action on S is trivial, K G 0 (Var S ) becomes a K 0 (Var S )-algebra, and π G is K 0 (Var S )linear. More generally, we have the following: where q : S → S/G is the quotient map. We have and this is the same motive as π G (u · x) = π G [U × S/G X → X → S], since G does not act on U .
By Lemma 1.5, the map (1.3) sends the ideal J G onto the ideal J S/G ⊂ K 0 (Var S/G ) defined in (1.1), therefore it descends to a K(Var S/G )-linear map This map extends to a map M G S → M S/G , still denoted π G , by setting Furthermore, by [4,Lemma 3.2], if the G-action on S is free, † π G descends to a K(Var S/G )linear map which again extends to a morphism π G : M G S → M S/G . Let S n be the symmetric group on n elements.
where X a,b,c,d is the space of homomorphisms of schemes with the domain considered as a scheme with n points, a of which are sent to A, b of which are sent to B and so on. We consider this variety as a S n -equivariant variety over S n , sending s to the point (j(s(1)), . . . , j(s(n))), where j : A ∪ B ∪ C ∪ D → S is the union of the maps f, g, h, i.
The above lemma is proved in [3] in the case S = Spec C, but the proof for the general case is the same. We remark that, by definition, there is an identity (1.4) 2.2.1. The monodromic motivic ring. Let μ n = Spec C[x]/(x n − 1) be the group of nth roots of unity. We define a good action of the procyclic group as an action that factors through a good μ n -action for some n. The additive group Mμ S carries a commutative bilinear associative product ' ' called the convolution product. See [12,Section 5] or [19,Section 7] for its definition. The product ' ' provides an alternative ring structure on the group ofμ-equivariant motivic weights, and it restricts to the usual product '·' on the subring M S ⊂ Mμ S † Without freeness, the naive quotient map may fail to respect the relation identifying G-bundles with trivial G-bundles.
of classes with trivialμ-action. The main role of ' ' will be played through the motivic Thom-Sebastiani theorem (cf. Theorem 1.19).

Lambda ring structures
The lambda ring operations on K 0 (Var C )[L −1/2 ] are defined by setting σ n (A) = σ n (A) for A effective, and then taking the unique extension to a lambda ring on K 0 (Var C )[L −1/2 ], determined by the relation (1.5) Note that σ n (−L 1/2 ) = (−L 1/2 ) n . By [8, Remark 3.5 (4)], these operations induce a lambda ring structure on the localisation K 0 (St aff C )[L −1/2 ], and thus a lambda ring structure on M C .
Remark 1.7. Note that by definition, σ n (A) = σ n (A) for A effective. The logical structure of the paper is such that we will often end up proving relations involving σ n first, and then using them to prove that the motives we consider are effective, so that we can state those same relations in terms of the more well-behaved operations σ n .
If S comes with a commutative associative map ν : S × S → S, and A ∈ M S , we likewise define where we abuse notation by denoting by ν the map S n /S n → S. As above, using the analogue of the relation (1.5) there is a unique set of lambda ring operators σ n ν agreeing with σ ν on effective motives.
As a special case, we obtain operations σ n and σ n on M C t via the isomorphism for X 0 , X 1 , . . . varieties, and then extending by linearity. Here, N is a considered as a scheme by identifying each natural number with a distinct closed point, and this scheme is considered as a commutative monoid under the addition map.

Motivic measures
Ring homomorphisms with source K(Var C ) or M C are frequently called motivic measures, realisations, or generalised Euler characteristics. We recall some of them here. Let K 0 (HS) be the Grothendieck ring of the abelian category HS of Hodge structures. For a complex variety X, taking its Hodge characteristic defines a motivic measure. The E-polynomial is the specialisation As E(A 1 C ) = uv, the E-polynomial can be extended to a motivic measure /2 ] and one has W (L) = q. Finally, specialising to q 1/2 = 1 recovers the Euler characteristic χ : K(Var C ) → Z, extending to χ : M C → Z after setting See [12, Section 2] for a natural extension to a ring homomorphism Remark 1.8. Our sign conventions differ slightly from [3]. We have chosen them so that all specialisations are homomorphisms of pre λ-rings. Note that, putting all the changes together, our convention that χ(L −1/2 ) = −1 is the same as theirs.

Power structures and motivic exponentials
We recall the notion of a power structure on a commutative ring R, mainly following [14,15]. Definition 1.9. A power structure on a ring R is a map satisfying the following conditions: Notation 1.10. If α is a partition of an integer n, which we indicate α n, by writing α = (1 α1 · · · i αi · · · r αr ) we mean that there are α i parts of size i, so that we recover n as the sum |α| = i iα i . The number of distinct parts of α is denoted l(α) = i α i . The automorphism group of α is the product of symmetric groups G α = i S αi .
Let us focus on R = K 0 (Var C ). If X is a variety and A(t) = 1 + n>0 A n t n is a power series in K 0 (Var C ) t , we define (1.8) In the above formula, Δ ⊂ i X αi = X l(α) is the 'big diagonal' (where at least two entries are equal), and the class is G α -equivariant due to the 'power map' of Lemma 1.6. Gusein-Zade, Luengo and Melle-Hernández have proved [14,Theorem 2] that there is a unique power structure on K 0 (Var C ) for which the restriction to the case where all A i are effective is given by the formula (1.8), for every variety X. Moreover, by [14,Theorem 1], such a power structure is determined by the relation where is the Kapranov motivic zeta function of X. Since we always consider effective exponents when taking powers, we just recall the recipe for dealing with general A(t) and effective exponent [X]. First, note that for any such A(t) there is an effective where both factors in the right-hand side are defined via (1.8).
Proof. By [9, Remark 3.7], the power structure can be extended to K 0 (St aff C ), and so the second statement follows from ( Next we consider the first statement. Assume A(t) [X] = B(t) [X] . Write A(t) = i 0 A i t and where A 0 = B 0 = 1, and assume that we have shown that A i = B i for i < n.
Let α n. Comparing the contributions from α in the t n coefficients of A(t) [X] and B(t) [X] , by assumption they agree for α = (n), since these terms only involve A i and B i for i < n. We deduce that the terms for α = (n 1 ) agree, and so [X] · A n = [X] · B n , and the result follows by injectivity of [X]·.
As noted in [3], there is an extension of the power structure to M C uniquely determined by the substitution rules 2.5.1. Motivic exponential. It is often handy to rephrase motivic identities in terms of the motivic exponential, which is a group isomorphism † Exp : Under (1.6), this can be seen as an inclusion of groups  As in the proof of Lemma 1.11, Exp and Exp are injective. Now if (S, ν : S × S → S) is a commutative monoid in the category of schemes, with a submonoid S + such that the induced map n 1 S ×n + → S is of finite type, we similarly define and for A and B effective classes, we define The principal example will be for U a variety, and S + = n 1 Sym n (U ). We define to be the morphism sending a pair of sets of unordered points with multiplicity to their union. Note that Exp ν sends effective motives to effective motives, as the same is true of σ n ν for each n.
In order to recover a formal power series from a relative motive over Sym(U ), we consider the operation In other words, we take the direct image along the 'tautological' map # : Sym(U ) → N which sends Sym n (U ) to the point n -recall that via (1.6) we consider power series in t with coefficients in M C as the same thing as elements of M N .
the map taking i sets of points (with multiplicity) to their union (with multiplicity). Let that are not sent to the big diagonal after projection to i|αi =0 Sym i (U ) αi . Then and if A is effective, one has Proof. The second statement follows directly from the definition of the power structure. The first arises from the decomposition of the right-hand side according to incidence partition in the U factor.

The virtual motive of a critical locus
Let X be a complex scheme of finite type, and let ν X : X(C) → Z be the canonical constructible function introduced by Behrend [2]. The weighted (or virtual) Euler characteristic of X is defined via ν X as When X is a proper moduli space of stable sheaves on a Calabi-Yau 3-fold, this number agrees with the Donaldson-Thomas invariant of X by the main result of [2]. The following definition is central to this paper.
Definition 1.13 [3]. A virtual motive of a scheme X is a motivic weight ξ ∈ Mμ C such that The Behrend function of a critical locus X = crit(f ) ⊂ U agrees with the Milnor function μ f , the function counting the number of vanishing cycles [27, Corollary 2.4 (iii)]. In particular, where d = dim U and [φ f ] X is the relative motivic vanishing cycle class introduced by Denef and Loeser [12]. It is a class in Kμ 0 (Var U0 ), supported on X = crit(f ), that we view as an element of Mμ X . We write [φ f ] for the pushforward of [φ f ] X to a point. We will repeatedly use the following proposition, due to Bittner [4].
Proposition 1.15. Let G be a finite group acting freely on a smooth variety U , let q : U → U/G be the quotient map and let f be a regular function on U/G. Then: (2) there is an equality of motives Since the fibrewise Euler characteristic of MF U,f equals ν X as a function on X [3, Proposition 2.16], the absolute class is a virtual motive for X in the sense of Definition 1.13.
Remark 1.17. A critical locus X = crit(f ) ⊂ U has a canonical virtual fundamental class [X] vir ∈ A 0 X, attached to the symmetric perfect obstruction theory determined by the Hessian of f . When X is proper, Behrend's theorem [2] can be phrased as Remark 1.18. If X is a smooth scheme, it can be considered as a critical locus via the zero function f = 0 ∈ Γ(O X ). The associated virtual motive is Via the stated sign conventions, we see that χ[X] vir recovers the virtual Euler characteristic of the smooth scheme X, namely χ(X) = (−1) dim X χ(X).
We end this subsection with two results that are of crucial importance in calculations involving motivic vanishing cycles. Recall from Section 1.2.1 that the groups Mμ S carry the convolution product ' ' besides the ordinary product.
Theorem 1.19 [11,19]. Let f : U → A 1 and g : V → A 1 be regular functions on smooth varieties U and V . Consider the function f ⊕ g :

inclusion of the zero fibres, and let p U and p V be the projections
The following result will be used in Propositions 2.9 and 2.11.  (ii) Let τ : X → Y be a map to an affine variety. If, in addition to the assumption in (i), the Remark 1.21. The original statement of this theorem in [3] fixed Y to be the affinisation of X -the statement above then follows from the fact that τ must factor through the affinisation, and the direct image of a monodromy-free motive is monodromy-free.

The virtual motive of the Hilbert scheme of points
Quivers with potentials provide a large class of examples of critical loci. For instance, consider the framed 3-loop quiver Q BBS (studied by Behrend-Bryan-Szendrői) depicted in Figure 1. The arrow 1 → ∞ is called a framing, and ∞ is the framing vertex. Throughout the paper, the vertices of a framed quiver are ordered so that the framing vertex is last.
The space of (n, 1)-dimensional right CQ BBS -modules is the affine space End(C n ) 3 × C n parameterising triples of n × n matrices (A, B, C) and vectors v ∈ C n . Consider the potential W = x[y, z], viewed as an element of the path algebra C x, y, z of the (unframed) 3-loop quiver. Then by [3, Proposition 3.1], one has, as schemes, where NCHilb n is the non-commutative Hilbert scheme, defined as follows. The open subscheme parameterising tuples (A, B, C, v) such that v ∈ C n generates the C x, y, z -module defined by the triple (A, B, C), carries a free GL n -action, and NCHilb n = V n / GL n is a smooth quasiprojective variety of dimension 2n 2 + n. The generating function Z A 3 (t) for which the t n coefficient is the virtual motive The result is the equation be the virtual motive of the punctual Hilbert scheme ι : Hilb n (A 3 ) 0 → Hilb n (A 3 ) (cf. Notation 1.16), the closed subscheme parameterising subschemes entirely supported at the origin 0 ∈ A 3 . Then the generating series Behrend, Bryan and Szendrői also define a virtual motive [Hilb n X] vir for arbitrary smooth 3-folds. The motivic partition function (1.14) Remark 1.23 (Related work on Quot schemes). The identity (1.14), as well as its reformulation in terms of the motivic exponential, has been generalised in [34] to the case of Quot schemes Quot Y (F, n) where F is an arbitrary locally free sheaf on a smooth 3-fold. See also [33] for the 'non-virtual' setup. In higher rank, the starting point of motivic DT theory is the observation that Quot A 3 (O ⊕r , n) is a global critical locus [1,Theorem 2.6]. This has also been exploited to prove a plethystic formula (the Awata-Kanno conjecture in String Theory) for the partition function of higher rank K-theoretic DT invariants [13].

The local model as a critical locus
denote the Quot scheme of n points of I C . Given a surjection θ : I C F , we can consider its kernel I Z ⊂ I C , and thus think of [θ] ∈ Q n C as a closed 1-dimensional subscheme Z ⊂ Y containing C as its maximal purely 1-dimensional subscheme. We will switch between these two interpretations of Q n C without further comment. Note that we generally suppress the ambient 3-fold Y from the notation. As such there is a forgetful map from Q n L to the stack of n-dimensional C[x, y, z]modules, and postcomposing with the affinisation map for this stack, a 'Hilbert-to-Chow' morphism This map is a special case of [35,Corollary 7.15]. The goal of this section is to prove the following result, which is a 1-dimensional counterpart of the analogous statement for Hilb n (A 3 ), considered in [3]. Part (A) is Theorem A from the introduction.
Theorem 2.2. Consider the embedding L ⊂ A 3 from above. Then: Part (A) is proved at the end of Section 2.1, and part (B) is proved in Proposition 2.11. The main step in the proof consists of realising Q n L as a suitable open subscheme of the Quot scheme Q n C0 , where C 0 ⊂ X is the exceptional curve in the resolved conifold X. In the next subsection, we review the critical locus structure on Q n C0 , and more generally, the noncommutative Donaldson-Thomas theory of the conifold as introduced in [39].

Conifold Geometry
Given a quiver Q = (Q 0 , Q 1 ) with potential W and a field K, we define K(Q, W ) to be the associated Jacobi algebra, that is, the quotient of the free path algebra KQ by the noncommutative derivatives ∂W/∂a for a ∈ Q 1 ranging over the arrows of Q.
Given a dimension vector d = (d(i)) i∈Q0 ∈ N Q0 , we define to be the affine space of d-dimensional right CQ-modules. Given a King stability condition ζ ∈ Q Q0 , we denote by , the open subscheme of ζ-semistable CQ-modules. Both schemes are acted on by the gauge group by change of basis.
Throughout the paper, we let X denote the resolved conifold, namely the total space of the rank 2 locally free sheaf O P 1 (−1) ⊕ O P 1 (−1). We denote by C 0 ⊂ X the zero section of this vector bundle, so C 0 ∼ = P 1 . Let π : X → C 0 denote the projection onto the zero section. The local Calabi-Yau 3-fold X is the crepant resolution of the conifold singularity and C 0 is the exceptional curve, the only strictly positive dimensional proper subvariety of X. Since C 0 is a rigid smooth rational curve in X, the Hilbert scheme I n+1 (X, [C 0 ]) of 1-dimensional subschemes Z ⊂ X with homology class [C 0 ] and χ(O Z ) = n + 1 agree (as a scheme) with the Quot scheme Q n C0 . This space was realised as a global critical locus by Nagao and Nakajima [24], by studying representations of the Jacobi algebra C(Q con , W con ). Here, Q con is the (framed) conifold quiver depicted in Figure 2, and is the Klebanov-Witten potential [16]. We denote by Q con the quiver obtained by removing fromQ con the framing vertex ∞ and the arrow ι. Since we consider right modules, a KQ conmoduleρ is defined by the following data.
(1) A right KQ con -module ρ, with KQ con -action defined via the KQ con -action onρ and the inclusion KQ con → KQ con . ( Remark 2.3. The orientation ofQ con differs from [24, Figure 4], but our notion of framing is the same as loc. cit. (see (1) and (2) above) -note that we are considering right CQ con -modules throughout -see Remark 2.4.
We identify quasicoherent O X -modules with triples (F , α 1 , α 2 ) where F ∈ QCoh(P 1 ) and α 1 , α 2 ∈ Hom O P 1 (F , F (−1)) commute in the sense that α 1 (−1) • α 2 = α 2 (−1) • α 1 . Then the above noncommutative conifold is the natural enhancement of Beilinson's noncommutative P 1 : given a complex of quasicoherent sheaves F on P 1 , we set ρ 1 and ρ 2 to be the complexes of vector spaces be the maps induced by the two sections x and y of O P 1 (1). We set ρ(a i ) = α i for i = 1, 2, and it is easy to check that the commutativity conditions are given precisely by the superpotential relations for W con , so that we obtain in this way a right C(Q con , W con )-module.
This description of the non-commutative conifold makes the translation of various geometrically defined functors rather transparent. For instance, the direct image along the projection map π * : D b (QCoh(X)) → D b (QCoh(P 1 )) becomes the forgetful map from the noncommutative conifold to Beilinson's non-commutative P 1 , forgetting the action of the arrows a 1 , a 2 , and the direct image along the inclusion C 0 → X becomes the extension by zero functor. Let and let We denote by the equivalence of derived categories, where on the right-hand side of the equivalence we have the derived category of right A con -modules with finitely generated total cohomology. We denote by Remark 2.4. As one sees from the above, the general setup leads us to consider right A conmodules. On the other hand, for an arbitrary quiver Q, there is an equivalence of categories between K-linear Q-representations and left KQ-modules. Since we consider right modules (as in [24]), if one wants to think of modules over algebras such as A con as quiver representations, one should reverse the orientation of the underlying quiver.
The chamber decomposition of the space of stability parameters forQ con was worked out by Nagao and Nakajima in [24], where the DT and PT chambers were precisely characterised. For a generic stability condition in the 'DT region' for X, defined by the conditions ζ 1 < ζ 2 and ζ 1 + ζ 2 < 0, we consider the moduli space of ζ-stable framed representations of Q con , having dimension vector (n + 1, n, 1). Here the dimension vector (n + 1, n, 1) refers to the vertices ordered as (1, 2, ∞).
The work of Nagao-Nakajima then implies that Q n C0 is isomorphic to the subscheme of N n defined by the defining relations of A con . Since these are exactly the non-commutative derivatives of the Klebanov-Witten potential (2.2), it follows from [38, Proposition 3.8] that Q n C0 is identified with the critical locus of the function g n : N n → A 1 given by taking the trace of (2.2). Consider the open subset N • n ⊂ N n parameterising stable representations ρ such that the linear map ρ(b 2 ): C n → C n+1 is injective, and let f n : N • n → A 1 be the restriction of the function g n . We now prove that Proof of Theorem 2.2 (A). For skyscraper sheaves of points O x , corresponding to representations of dimension vector (1, 1) under Φ, injectivity of ρ(b 2 ) corresponds to the condition that π(x) = ∞. We identify A 3 with X \ π −1 (∞), and we have the following Cartesian diagram where the horizontal maps are closed inclusions, and vertical maps open inclusions.
Let L be the tautological C(Q con , W con ) ⊗ O crit(gn) -module. Let G be the submodule generated by Consider the exact sequence of C(Q con , W con ) ⊗ O crit(gn) -modules. Let Spec (K) → N n be a geometric point of crit(g n ), corresponding to a K(Q con , W con )-module ρ. By the above-mentioned result of Nagao-Nakajima, Ψ(ρ) corresponds to a K-point of Q n C , and so admits a unique (up to scalar) surjective map to O C0 ⊗ K, with kernel a coherent sheaf F with 0-dimensional support. It follows that ρ admits a unique (up to scalar) non-zero map to the nilpotent simple module at vertex 1, with kernel isomorphic to Φ(F ). In particular, the space spanned by the image of ρ(b 1 ) and ρ(b 2 ) is n-dimensional. It follows that dim(V K ) = 1 for all K-points, and so V is a locally free O crit(gn) -module of rank 1, and thus G is locally free of rank (n, n).
Let Spec (K) → crit(g n ) be the inclusion of a point, and let i : X × Spec (K) → X × crit(g n ) be the induced inclusion. The O X×crit(gn) -module Ψ(G ) is a coherent sheaf after pullback along i, and so is a coherent sheaf in [5,Lemma 4.3]. The coherent sheaf Ψ(L ) is equipped with a tautological map from O X×crit(gn) , surjective by stability, inducing the tautological surjective map The condition on ρ(b 2 ) defining N • n implies that each i * Ψ(G ) has support away from π −1 (∞). Since crit(f n ) = crit(g n ) ∩ N • n , the inverse image sheaf of Ψ(G ) on crit(f n ) × A 3 is a vector bundle on crit(f n ), equipped with a surjection from I L O crit(fn) . This defines the map crit(f n ) → Q n L . The inverse is defined similarly: given a family of surjections I L F , we obtain a family of surjections I C0 → u * I L → u * F . This completes the proof of Theorem A from the Introduction.

Relative virtual motives
Via the coherent sheaf Ψ(G ) of O X×crit(gn) -modules constructed in the proof, we obtain the map hc X : Q n C0 → Sym n (X) (2.4) extending the map (2.1). It is again a special case of the Quot-to-Chow map [35,Corollary 7.15].
In particular, we can write via the map hc X and the inclusion u : A 3 → X. Via projection to the Sym n (A 3 )-factor, we recover the map . Where it is clear which of the two Hilbert-Chow maps we mean, we will drop the subscript.
Then taking the union of the supports of the subquotients in the filtration gives the map (2.4) to Sym n (X).
Theorem 2.2 (A) has the following immediate consequence.
Corollary 2.6. The function f n : N • n → A 1 induces a relative virtual motive that is the pullback under the inclusion Q n L → Q n C0 of the relative virtual motive Proof. The space N n is a smooth scheme of dimension 2n 2 + 3n, so the machinery recalled in Section 1.6 gives the relative motive According to Notation 1.16, we can write down the absolute motives (2.6) We end this subsection with two examples of the geometric and motivic behaviour of Q n L for low n.
Example 2.7. The scaling action of the torus G 3 m on A 3 lifts to an action on Q n L . Let us consider the Quot scheme Q 2 L . We will exhibit a singular point belonging to the torus fixed locus. First of all, we have dim Q 2 L = 6. Consider the point p ∈ Q 2 L corresponding to This is depicted in Figure 3. It is easy to check that Hom A 3 (I Z , I L /I Z ), the tangent space of Q 2 L at p, is 10-dimensional, so that p is a singular point.
Example 2.8. The Hilbert scheme Hilb n A 3 is non-singular if n 3 and singular otherwise, whereas Q n L is already singular if n = 2, by Example 2.7. Let us fix n = 1, so that both trace functions (the ones giving rise to Hilb 1 A 3 and to Q 1 L , respectively) vanish, and the virtual motives are a shift of the naive motives by L −3/2 . On the Hilbert scheme side we have, using Remark 1.18, while on the Quot scheme side we find, using that Q 1 This is the first instance of formula (0.7).

Equivariance of the family
Let R n = rMod (n+1,n,1) (CQ con ) be the affine space parameterising framedQ con -modules of dimension vector (n + 1, n, 1). Let us set G = GL n+1 × GL n and let S ⊂ Γ(O Rn ) be the subalgebra of functions scaling, under the G-action, as a positive power of the given character realising framed stability. Then by general GIT we have N n = Proj S, and the natural inclusion where the affine scheme Y 0 can be viewed as the GIT quotient R n / / 0 G at the trivial character.
Proposition 2.9. There is an identity Proof. The 4-dimensional torus T = G 4 m acts on N n by t · (A 1 , A 2 Moreover, the trace function g n : N n → A 1 is T -equivariant with respect to the primitive character χ(t) = t 1 t 2 t 3 t 4 . This means that for all P ∈ N n , we have g n (t · P ) = χ(t)g n (P ). We claim that the induced action on N n by the diagonal torus G m ⊂ T is circle compact, that is, it has compact fixed locus and the limits lim t→0 t · P exist in N n for all P ∈ N n . Following the proof of [3, Lemma 3.4], we note that (2.7) is a projective G m -equivariant map, where Y 0 has a unique G m -fixed point, and moreover all orbits have this point in their closure. In other words, limits exist in Y 0 . Therefore, by properness of p n , we conclude that the G m -fixed locus in N n is compact and limits exist. This proves the claim.
Then the equation involving g n follows by part (i) of Theorem 1.20, proved by Behrend, Bryan and Szendrői. In particular, the absolute virtual motive of Q n C0 carries no monodromy,

A direct critical locus description
There is a way of writing down the above critical locus description of Q n L that does not involve pulling back from a moduli space of representations for the non-commutative conifold. † † Although, to be precise, for the proof that the description really does recover Q n L , the only method we offer will rely on the geometry of the non-commutative conifold.
Then for s = 1, 2, we set where π denotes the projection maps. Then for W r the potential

Proposition 2.10. There is an isomorphism of schemes
The result follows directly from our analysis of the non-commutative conifold. From that description, we see that in the stable locus, we have ρ(b 1 ) = 0. Then the superpotential relations become a 2 a 1 = a 1 a 2 a 1 a 2 = a 2 a 1 and the stable locus is identified with the moduli space of triples (v x , v y , N) where N is a C[x, y, z]-module, v x , v y ∈ N generate under the action of C[x, y, z], and xv y = yv x . This is the same data as a C[x, y, z]-linear surjection θ : (x, y) N , by setting v x = θ(x) and v y = θ(y). The next result contains the statement of Theorem 2.2 (B).

Proposition 2.11. There is an identity
Proof. The first statement is proved the same way as Proposition 2.9, noting that by construction, [φ fn ] = [φ hn ], and by (2.8), h n is a potential on a GIT moduli space of quiver representations admitting a contracting C * -action for which Tr(W r ) has weight three.
Next, we claim that the hypersurface h −1 n (0) ⊂ N •• n is reduced. This will follow from the claim that the variety cut out by the function Tr(W con ) on rMod ζ (n+1,n,1) (Q con ) is reduced, which is in turn weaker than the claim that the variety cut out by Tr(W con ) on rMod (n+1,n,1) (Q con ) is reduced. This variety is affine, and so it is enough to show that the function Tr(W con ) is reduced. With respect to the T-grading this function has degree (1, 1, 1, 1), and so it cannot be factorised with a repeated factor.
The statement regarding [φ fn ] Sym n (A 3 ) then follows from part (ii) of Theorem 1.20.
The proof of Theorem 2.2 is complete.

Relative DT theory of the conifold
The goal of this section is to express the motive [Q n C0 ] vir defined in (2.6) in terms of motivic contributions coming from the 'punctual loci' inside Q n C0 . For n ∈ N, we define: (i) P n pt ⊂ Q n C0 , the subvariety parameterising quotients I C0 F such that F is settheoretically supported at a fixed single point, away from C 0 ; and (ii) P n crv ⊂ Q n C0 , the subvariety parameterising quotients I C0 F such that F is settheoretically supported at a fixed single point on the curve C 0 . Items (i) and (ii), respectively, will lead (cf. Definition 3.16) to the definition of motivic weights Ω n pt , Ω n crv ∈ M C , which will be the basic building blocks for the construction of a virtual motive of the Quot scheme Q n C for an arbitrary curve C ⊂ Y in a 3-fold.

Motivic stratifications I: partially punctual strata
In what follows, we streamline proofs by deducing results for the embedded curve C 0 ⊂ X from the embedded curve L ⊂ A 3 . In order to achieve this, we introduce two automorphisms of N n (see (2.3)): Note that both of these automorphisms preserve Tr(W con ) and the ζ-stable locus, and so they induce automorphisms of Q n C0 preserving [φ gn ] Q n C 0 . The two automorphisms of C 4 defined by σ λ (x, y, z, w) = (x, y, z + λw, w) and ξ λ (x, y, z, w) = (x, y, z, w + λz) induce automorphisms of X commuting with hc X in the sense that Sym n (σ λ ) • hc X = hc X •σ λ and Sym n (ξ λ ) • hc X = hc X •ξ λ (see Remark 2.5).
Lemma 3.1. Let α ∈ Mμ Sym n (X) satisfy: Proof. The space Sym n (X) has an open cover by open subsets of the form U t = Sym n (X \ π −1 (t)) for t ∈ P 1 : if γ ∈ Sym n (X) is in the complement of ∪ t∈T U t for |T | > n, then the support of γ is spread across more than n fibres of the projection π : X → P 1 . By our assumptions, the restriction of α to U t is zero. By the scissor relations, it follows that α = 0. The next lemma is an incarnation of the fact that taking box sum with a quadratic function (locally) does not change the vanishing cycle complex, while for global triviality one has to be mindful of monodromy. The implication is that we can pass to a 'minimal' potential at the expense of keeping track of some extra monodromy data, which in the Kontsevich-Soibelman framework, and then elsewhere, is called orientation data. In the language of potentials on 3-Calabi-Yau categories, one can think of the proof of part (2) of Lemma 3.7 as working by proceeding to a 'partially minimised' potential. Lemma 3.3. Let π : Tot(V ) → X be the projection from the total space of a vector bundle on a smooth connected variety X, and let f : Tot(V ) → C be a function that is quadratic in the fibres, that is, f (z · v) = z 2 f (v) for z ∈ C, where we have given Tot(V ) the scaling action of C * . Assume X = crit(f ), where we have identified X with the zero section of Tot(V ).
(1) For x ∈ X, there is an equality Then there is an equality

2) (3) Under the conditions of (2), let G be a finite group acting freely on X, let V ∼ = V − ⊕ V + be a direct sum decomposition of G-equivariant vector bundles, and assume that f is G-invariant.
Then In part (3), we include the assumption that G acts freely so that we may apply Proposition 1.15.   The ultimate goal of this section is to show that as a relative motive over Sym(X), the virtual motive is generated under Exp ∪ by motives that are supported on the punctual locus, and constant away from C 0 , as well as constant on C 0 . To get to this point will require some work, and we break the proof up by showing first that The map ι n is the inclusion of the subspace of configurations of points which all have zcoordinate 0, which explains the notation (•, •, nilp) -the scheme-theoretic version of this condition is that the operator corresponding to the action of z is nilpotent. So, ordering the operators corresponding to x, y, z alphabetically, the first two are unconstrained, and the third is nilpotent.
There is an action of A 1 on Sym n (A 3 ) via simultaneous addition on the z coordinate of all points in a configuration, and we let be the restriction of this action. It is again an embedding, this time of the subvariety of n-tuples of points which all have the same z-coordinate (not necessarily zero). It is this locus that we call 'partially punctual'. Finally, consider the morphism obtained by projecting onto the z coordinate. We define hc z = q z • hc. More generally, for a 1 , . . . , a r distinct elements of {x, y, z} and T ⊂ Q n L , we denote by hc a1···ar : T → Sym n (A r ) the map given by composing the restriction of hc to T with the projection Sym(A 3 ) → Sym(A r ) induced by the projection A 3 → A r defined by forgetting the coordinates not contained in a 1 , . . . , a r .
The space Sym n (A 1 ) is stratified according to partitions of n, and for α n we denote by Q •,•,α L ⊂ Q n L the corresponding stratum of the stratification of Q n L given by pulling back along hc z : Q n L → Sym n (A 1 ). So, for instance, is the fibre product ofι n and hc.
Lemma 3.7. (1) There is an equalitỹ In other words, the motive on the left-hand side is constant in the A 1 -factor.
(2) Moreover, for α n there is an equality where j α is the G α -equivariant inclusion of the complement of the pullback of the big diagonal under the G α -equivariant projection onto z-coordinates: Here G α is the automorphism group for the partition type α, and ∪ is the union of points map on Sym(A 3 ).
Before we begin the proof, we give some guidance for how to read the right-hand side of (3.7). Firstly, recall from Section 1.5 that the infinite union of algebraic varieties Sym(A 3 ) has a symmetric monoidal structure ∪, given by taking unions of unordered points with multiplicity. We consider Sym i (A 2 ) × A 1 as a subvariety of Sym i (A 3 ) viaι i . We abuse notation by writing ∪ again for the map Sym(A 3 ) m → Sym(A 3 ) taking an m-tuple of sets of unordered points with multiplicity to their union. The term in big round brackets on the right is a G α -equivariant motive via Lemma 1.6.
Proof. Consider again the space N •• n ⊂ N n from Section 2.4. We define the subspace T n ⊂ N •• n by the condition that Tr(ρ(b 1 )) = 0. Then there is an isomorphism given by (ρ(a 1 ), ρ(a 2 ), ρ(b 1 ), ρ(a 1 ), ρ(a 2 ), ρ(b 1 ), t) → (ρ(a 1 ), ρ(a 2 ), ρ(b 1 ) + t · Id n×n , ρ(a 1 ), ρ(a 2 ), ρ(b 1 )), and the function h n = f n | N •• n (cf. (2.9) and Proposition 2.10) is pulled back from a function h n on T n . The stratification by partition type of ρ(b 1 ) for Q n L ⊂ N •• n is induced by a stratification of T n : for α a partition of n, define T α ⊂ T n to be the locally closed subvariety whose C-points correspond to Q r -representations for which the partition type of the generalised eigenvalues of ρ(b 1 ) are given by α.
Let Sym n 0 (A 1 ) ⊂ Sym n (A 1 ) be the closed subvariety of n-tuples summing to zero. Let be the morphism taking a module ρ to the eigenvalues (with multiplicity) of ρ(b 1 ). Note that λ| Q n L = hc z . Then the first equality follows from the commutativity of and the fact that, pulling back along the top isomorphism, the function h n becomes h n ⊕ 0. Let α = (1 α1 · · · r αr ) n be a partition. We write l(α) = i r α i for the total number of parts of α. For the proof of the second part of the lemma, we adapt the proof of [3, The closed subset V ⊂Ũ is defined by the condition that in each of the l(α) factors in the product decomposition there is only one eigenvalue. It is in fact the preimage of Q α We form the quiver Q α ( Figure 5) and dimension vector d α as follows. Set We set We give Q α the dimension vector d α = (1 α1 , . . . , r αr , 1), that is, we always assign the vertex ∞ dimension 1, and then the rest of the dimension vector is given by the ordered set of numbers α. We give this quiver the stability condition . . . , −1, n).
A (ζ α -stable) CQ α -module is the same data as a (ζ -stable) CQ r -module ρ, along with an ordered vector space decomposition of the vector space ρ 2 that is preserved by the operation ρ(b 1 ). As such, there is a forgetful functor from CQ α -modules to CQ r -modules, inducing a generically finite morphism which at the level of points amounts to setting The scheme is G α -equivariant, via the obvious G α -action on the quiver Q α .
There is a unique potential W α on Q α such that Tr(W α ) is the pullback of Tr(W r ) under this forgetful map. Precisely, we define We define the open subschemeS ⊂ N α by the condition that for every i, j l(α) the Let ρ ∈S ∩ crit(Tr(W α )). Then s(ρ)(b 1 ) = 0 by stability, and so from the superpotential relations we deduce that a 1,i,j b 1,i = b 1,j a 1,i,j and a 2,i,j b 1,i = b 1,j a 2,i,j for all i, j and so from our condition on the eigenvalues of the b 1,i we deduce that for all 1 i = j l(α) we have ρ(a 1,i,j ) = ρ(a 2,i,j ) = 0. As such, in calculating the relative vanishing cycle [φ Tr(Wα) ]S we can restrict to the setS • ⊂S defined by the condition that ρ remains stable after setting all ρ(a 1,i,j ) = ρ(a 2,i,j ) = 0 for i = j. So both the inclusionsŨ →S andŨ →S • are the inclusions of the critical locus of the function Tr(W α ) on the respective targets.
The spaceS • is a vector bundle over The map π is a Galois cover with Galois group G α , and the mapS • → S is a Galois cover in a formal neighbourhood of the morphism π.
The spaceP carries the free G α -action inherited from N α , and furthermore the vector bundlẽ S • has a direct sum decomposition V − ⊕ V + , where V − keeps track of the entries of ρ(a 1,i,j ) for i = j and V + keeps track of the entries of ρ(a 2,i,j ) for i = j, and so the decomposition is preserved by the G α -action.
Let m = rk(V + ) = rk(V − ). If we let G 1 = C * act onS • by scaling V + with weight 1 and V − with weight −1, then Tr(W α ) is C * -invariant. Let G 2 = C * act by scaling both V + and V − with weight one. Then since each term in the potential contains at most two instances of a k,i,j , for k = 1, 2, for every ρ ∈S • , there is a fixed constant C for which | Tr(W α )(z · 2 ρ)| C|z| 2 where the action is via the G 2 -action. We deduce that onS • we can write where g 0 is a function pulled back from the projection toP and g 1 is a G 1 -invariant function onS • , quadratic in the fibres. After passing to a G α -invariant Zariski open subset, we can trivialise the vector bundleS • and write where t i and s j are coordinates on the fibre of Tot(V + ) and Tot(V − ), respectively. From the equality crit(Tr(W α )) = crit(g 0 ), arguing as in the proof of the second part of Lemma 3.3, the matrix {g ij } 1 i,j m is invertible, and after a change of coordinates on Tot(V + ) we may assume that g ij = δ i−j . By the Thom-Sebastiani isomorphism, and the third part of Lemma 3.
Proof. By Lemma 3.7 (2) and (1.4), we deduce that for α n The result then follows from Proposition 1.12 and

Motivic stratifications II: fully punctual strata
Corollary 3.8 says that, considered as relative motives over Sym(A 1 ), via projection to the z coordinate, the DT invariants are generated by classes on the small diagonals, which are moreover constant as relative motives over A 1 . In other words, there are relative motives over Sym n (A 2 ) that generate the motivic DT partition function under taking the external product with the constant motive on A 1 and taking exponentials. We next show that these relative motives are themselves supported on the small diagonal (although they are non-constant, with a 'jump' at the intersection A 2 ∩ L) so that considered as a relative motive over Sym(A 3 ), the virtual motive of Q n L is generated on the small diagonal. For n n and α a partition of n − n , we define where A n ,α is the subvariety of Sym n (A 1 ) defined by the condition that 0 occurs n times, and the partition defined by the (n − n )-tuple of points away from 0 is α. We denote by z 1 , 1), . . . , (z n , 1)) (z 1 , . . . , z n ) → ((z 1 , 0), . . . , (z n , 0)) the inclusions analogous to ι n andι n defined in (3.4) and (3.6). Let l n : Sym n (A 1 ) × G m → Sym n (A 1 ) × A 1 be the inclusion. (2) More generally, there is an equality where j α is the inclusion of the complement to the preimage of the big diagonal in See the remarks following the statement of Lemma 3.7, as well as equation (1.2), for some guidance on how to read the right-hand side of the second equation.
Then Tr(W r ) is invariant under the G m -action. We define the space U n ⊂ N •• n by the condition that det(ρ(a 2 )) = 1. Consider the morphism given by restricting the G m -action. This is a Galois cover of the open subscheme Y ⊂ N •• n defined by the condition that ρ(a 2 ) is invertible. Let V n ⊂ U n be the subvariety defined by the condition that ρ(a 2 ) has only one eigenvalue. Then as a μ n -equivariant variety, there is an isomorphism where the action on the first factor is trivial, and is the action of group multiplication on the second. Here W n ⊂ U n is the subvariety defined by the condition that ρ(a 2 ) is unipotent.
Recall from Section 2.4 the function h n on N •• n . The function h n • J factors through a functionh on U n , since h is G m -invariant. By Proposition 1.15, It follows that implying the triviality along the G m -factor required for part (1). The second equality follows from the fact that only the part of the motive over V n can contribute to the pullback alongκ n , since by definition this is the pullback along the locus where ρ(a 2 ) has only one eigenvalue. For part (2), consider the quiver Q α defined as follows. Set We give this quiver the dimension vector d = (n , 1 α1 , . . . , r αr , 1) and the stability condition (−1, . . . , −1, n). As in the proof of Lemma 3.7, a (stable) Q α -representation corresponds to a (ζ -stable) Q r -representation ρ along with a direct sum decomposition of ρ 2 that is respected by the operator ρ(a 2 ). A point in Q n ,α gives rise to a d-dimensional Q α -representation, in a formal neighbourhood of which the forgetful map is a Galois covering -from this point, the proof proceeds exactly as in the proof of Lemma 3.7.
The following is proved in the same way as Corollary 3.8.
We are now two-thirds of the way toward showing that the relative DT invariants are fully punctual over Sym(A 3 ). Since the proof of Lemma 3.11 is almost identical to that of Lemma 3.9, while the proof of Lemma 3.12 is strictly simpler, we omit them. Before stating them, we introduce one last stratification; for n n, and α n − n , let Lemma 3.12.
(1) There is an equality By the same argument as Corollary 3.8, we deduce the following.
Lemma 3.15. In the notation of Corollaries 3.13 and 3.14, there is an equality Λ uni n = Λ n for all n.
Proof. There is an automorphism S n of N •• n , for all n, defined by which leaves Tr(W r ) invariant, and so preserves [Q n L → Q n L ] vir . Now    (3.11) and an equality in M Sym(X) : (3.12) Proof. By Corollary 3.8, we have n 0 Then by Corollary 3.10 and Proposition 1.12, we deduce that Then by Corollaries 3.13 and 3.14 and Proposition 1.12 again, we deduce that n 0 and the first statement follows. For the second statement, we observe that the right-hand side of (3.12) satisfies conditions (2) and (3) of Lemma 3.1, and is obviously equal to the right-hand side of (3.11) after restriction to Sym(A 3 ). Then the second statement follows from Corollary 3.2.

The virtual motive of the Quot scheme
In this section, we define a virtual motive Q n C vir ∈ M C for an arbitrary smooth curve C ⊂ Y in a smooth 3-fold Y . Before getting to this point, we give an explicit formula for the generating series encoding the local absolute virtual motives attached to L ⊂ A 3 and C 0 ⊂ X. We shall also prove Theorems B and C from the introduction.

The local absolute virtual motives
Let X be, as usual, the resolved conifold. Then Theorem 3.17 implies that for ι : x → X the inclusion of a point, the absolute motive only depends on whether x is in C 0 or not. We define: Explicitly, these motives are determined by the identities n 0 (−1) n P n # vir · t n = Exp n>0 Ω n # · t n for # ∈ {pt, crv}. We define the generating functions Via the morphisms defining power structures for varieties, Lemmas 3.7, 3.9, 3.11, 3.12 and 3.15 can be neatly summed up in the following corollary.
Corollary 4.1. There are equalities in Remark 4.2. Note that the exponents in the statement of the corollary are effective (despite the minus signs). Indeed, L 3 − L is the class of A 3 \ L, and L 3 + L 2 − L − 1 is the class of X \ C 0 . The ' ' subscripts are in the statement because we are yet to prove that the classes (−1) n [P n pt ] vir and (−1) n [P n crv ] vir are effective. As a result of Corollaries 4.4 and 4.6, we will be able to remove them and the statements remain true, now expressed properly in terms of the power structures on the Grothendieck rings of varieties.
The rest of this subsection is devoted to removing the decorations ' ' from the formulas in Corollary 4.1. We start with # = pt. Proof. By definition where j n : p → Sym n (A 3 ) is the inclusion of the point p = ((1, 1, 1), . . . , (1, 1, 1)). Consider the open subvariety U n ⊂ rMod (n,1) (Q r ) defined by the condition that ρ(a 1 ) and ρ(a 2 ) are invertible, and the image of ρ(a 1 ) generates ρ 2 under the action of ρ(a 1 ), ρ(a 2 ) and ρ(b 1 ). Note that this is a stronger notion of stability than ζ -stability. However, after restricting to the critical locus of Tr(W r ), using our invertibility assumptions, we have that for ρ ∈ U n the relation holds, and so we deduce that ρ satisfies our stronger notion of stability if it is ζ -stable, lies in crit(Tr(W r )), and satisfies the above invertibility assumptions. It follows that, after setting U n = U n / GL n , one has Let Q BBS denote the quiver obtained from Q r by removing a 2 and b 1 , in other words the framed 3-loop quiver of Figure 1. Then NCHilb n := rMod ζ (n,1) (Q BBS )/ GL n is the noncommutative Hilbert scheme considered by Behrend, Bryan and Szendrői in [3] (cf. Section 1.7), and U n = U n / GL n is a vector bundle over NCHilb n • , the open subscheme of NCHilb n defined by invertibility of ρ(a 1 ) and ρ(a 2 ).
We introduce a new set of matrix coordinates on U n by considering the matrices With respect to these coordinates, we can write is a function on NCHilb n considered as a function on U n by composition with the vector bundle projection U n → NCHilb n . It follows from Lemma 3.3 (2) and the motivic Thom-Sebastiani theorem that In particular, Ω n pt is effective.
Proof. By definition, we have The result follows since Exp is injective, and agrees with Exp on effective motives.
The corollary implies that in the right-hand side of the formulas in Corollary 4.1, we can remove the decoration ' ' from the first factor. In other words, for all smooth quasi-projective 3-folds U , one has Next we deal with # = crv.
In [20,Proposition 4.3], the full motivic DT and PT theories of the resolved conifold X are computed. The sign conventions in loc. cit. are different from ours, but the discrepancy amounts to the substitution After this change is done, the motivic partition function of the stable pair theory of X reads On the other hand, the DT partition Z DT function satisfies where s is the point class and T is the curve class. Again, we have adjusted the sign in the point contribution Z X . Extracting the coefficient of T (which corresponds to picking out the contribution of the reduced curve class) and multiplying by −s −1 (so that the (−s) n coefficient is I n+1 (X, [C 0 ]) = Q n C0 on the DT side) yields an identity where On the other hand, and the latter equals by Corollary 4.1 and Proposition 4.3. It follows that We shall need the following: There is an identity In particular, F crv (−s) is effective.
Since L + 1 is invertible in K(St aff C ), by Lemma 1.11 and (4.3) we deduce the lemma.
Corollary 4.6. There is an equality of motives In particular, Ω n crv is effective.
Proof. From Lemma 4.5 and the equations Since P is effective, Exp(P ) = Exp(P ). On the other hand, by definition we have the equality F crv (−s) = Exp( Ω n crv s n ). The result then follows by injectivity of Exp.
Corollary 4.1 can now be restated as follows: Theorem 4.7. The absolute virtual motives of L ⊂ A 3 and C 0 ⊂ X are given by Proof. Combining (4.1) with Lemma 4.5, we get F crv (−t) L+1 = F crv (−t) L+1 .

Proof of Theorems B and C
Proof of Theorem B. By (3.11), there is an equality Plugging in the results of Corollaries 4.6 and 4.4, we deduce n 0 where the removal of the ' ' comes from the effectiveness statements in Corollaries 4.6 and 4.4.
Guided by Theorem 4.7, we can now define a virtual motive for an arbitrary Quot scheme Q n C .
Definition 4.8. Let Y be a smooth 3-fold. For a smooth curve C ⊂ Y , we define classes [Q n C ] vir ∈ M C by the identity We also define At this point, it is not yet clear that [Q n C ] vir is a virtual motive for Q n C , but we incorporate this in the proof of Theorem C from the introduction: Hilb n−j Y vir · Sym j C vir .
In other words, we have a product decomposition where for a smooth variety X of dimension at most 3, Z X (t) denotes the motivic partition function of the Hilbert scheme of points of X.
Proof. Combining the power structure with Lemma 4.5, we find In particular, the classes [Q n C ] vir are virtual motives, because and this equals n χ(Q n C )t n by [32, Proposition 5.1].
The proof of Theorem C is complete.

Equivalent formulations.
Using the power structure on M C and the explicit formulas for Z X (t) available from [3, Section 4], we find an identity .
Another equivalent way to express the same identity is via motivic exponentials. If t is the variable used in the definition of motivic exponential (cf. Section 1.5.1), then

Local Donaldson-Thomas invariants
Let Y be a projective Calabi-Yau 3-fold, and let C ⊂ Y be a smooth curve of genus g. Recall the C-local DT invariants Proof. The main result of [31] proves that DT n C = n g,C · χ(Q n C ). By the proof of Theorem 4.9, we know that [Q n C ] vir is a virtual motive, so the result follows.
When C is infinitesimally rigid in Y , that is, H 0 (C, N C/Y ) = 0, the integer DT n C is the degree of the virtual fundamental class Remark 4.12. The formula Q L/A 3 = Z A 3 · Z L was conjectured in the second author's PhD thesis [30], where the problem was reduced to proving a motivic identity (with no 'virtualness' left to deal with) involving essentially only the stack of finite length coherent sheaves over A 2 . As it turned out, such an identity could in principle be checked by hand after performing a complete classification of finite length modules over the polynomial ring C[x, y], which is known to be a wild problem. This classification was accomplished by Moschetti and the second author in [22] for modules of length n 4. As a consequence, the motivic wall-crossing formula Q L/A 3 = Z A 3 · Z L could be proven by hand up to order 4 using this classification.

Categorification
In this final section, we outline a programme for future work on a categorified DT/PT correspondence. Just as, since any smooth threefold analytically locally looks like A 3 , one may patch together the full cohomological understanding of the DT theory of degree 0 DT theory on A 3 obtained in [7] to try to understand the degree 0 cohomological DT theory of any Calabi-Yau 3-fold (see [42,Section 3.1]), analytically locally, the inclusion of a smooth curve inside a smooth Calabi-Yau 3-fold can be modelled by L ⊂ A 3 , and so the intention is that by proving the conjectures below we can approach a cohomological version of the DT/PT correspondence.

Further directions I: categorification
By our Theorem A, the moduli space Q n L arises as the critical locus of the function Tr(W r ) on the moduli space N •• n . Shifting the associated virtual motives, we define the generating function Then by Theorem B, we can write where Ω sh n = L −2 (1 + L −1 + · · · + L 1−n ) and Δ n : A 3 → Sym n (A 3 ) is, as ever, the inclusion of the small diagonal.
For the rest of this section, we will make free use of the language and foundational results concerning monodromic mixed Hodge modules, see [36,37] for background on mixed Hodge modules, [17] for an introduction to monodromic mixed Hodge structures, and [10] for the theory of monodromic mixed Hodge modules in cohomological Donaldson-Thomas theory. In particular, we will use the same functor In what follows, for a space X, we write Q X for the constant complex of mixed Hodge modules on X, which we think of as a complex of monodromic mixed Hodge modules by endowing it with the trivial monodromy operator. Where the choice of space X is clear, we may drop the subscript.
Given an element [Y f − → Z] ∈ Mμ Z , where Z is a variety, pick n ∈ N such that theμ action on Y factors through the projectionμ → μ n . Then we form the mapping torus where μ n acts via z · (y, z ) = (z · y, z −1 z ). We definẽ f : Y × μn G m → Z × A 1 (y, z ) → (f (y), z n ).
Set F =f ! Q Y ×μ n Gm ∈ D b (MHM(Z × A 1 )). By construction, the cohomology sheaves of this direct image are locally constant along the fibres of the projection Z × A 1 → Z, away from the zero fibre. In particular, F is a bounded complex of monodromic mixed Hodge modules, and so [F] ∈ K 0 (MMHM(Z)). Sending [Y → Z] → [F] defines a group homomorphism Ψ : Mμ Z → K 0 (MMHM(Z)), which is a λ-ring homomorphism in the event that Z is also a commutative monoid. Now consider Ψ(Q sh L/A 3 ). This is the class of the complex of monodromic mixed Hodge modules where T 1/2 is a tensor square root of the complex of mixed Hodge structures H c (A 1 , Q) -that is, a half Tate twist, concentrated in cohomological degree 1. From (5.1), we deduce that It follows as in [10, Proposition 3.5] from finiteness of the map ∪ : Sym(A 3 ) × Sym(A 3 ) → Sym(A 3 ) that the object in the square brackets on the right-hand side is pure, in the sense that its ith cohomology is pure of weight i. From semisimplicity of the category of pure mixed Hodge modules (proved by Saito, see above references), the following two statements are equivalent: • the complex of monodromic mixed Hodge modules F L/A 3 is pure; • there is an isomorphism with n keeping track of the degree on both sides, and where V is a 1-dimensional pure weight zero Hodge structure placed in degree 1. The Z-grading on the right-hand side comes from the grading on the object we are taking the symmetric algebra of (that is, we ignore the fact that Sym of any object acquires an extra Z-grading).
The isomorphism in (5.4) would categorify our Theorem B, and would lift it from a formula to an isomorphism (that is, categorify it). Since it would take quite some time to even fill in the requisite definitions for the above discussion, we leave this conjecture to future work.

Further directions II: A new CoHA module
We recall some very general theory regarding critical cohomological Hall algebras and their representations.
First, we fix a quiver Q and a potential W . The Euler form for Q is defined by A stability condition ζ ∈ Q Q0 is called generic if for any two non-zero dimension vectors of the same slope, we have γ, γ = 0. Recall that a quiver is called symmetric if for all vertices i, j ∈ Q 0 , we have # a | s(a) = i, t(a) = j = # a | s(a) = j, t(a) = i .
Note that the genericity condition is vacuous for symmetric quivers such as the three loop quiver obtained by removing the framing from the quiver Q BBS from Section 1.7, and the quiver Q con from Section 2.1.
Fix a generic stability condition ζ ∈ Q Q0 and a slope θ ∈ (0, ∞). We set Λ ζ θ ⊂ N Q0 to be the submonoid of dimension vectors of slope θ. Then as in [17] we endow the graded monodromic mixed Hodge structure with the Hall algebra product arising from the stack of short exact sequences of right CQmodules. The multiplication respects the monodromic mixed Hodge structure on (5.5)in other words (5.5) is made into an algebra object in the tensor category of Λ ζ θ -graded monodromic mixed Hodge structures.
In fact we will only consider the cohomological Hall algebra for which Q is symmetric, and ζ = (0, . . . , 0) is the degenerate stability condition, with θ = 0. So we will drop ζ and θ from the notation and just write H Q,W for the cohomological Hall algebra associated to Q and W .
Let Q ⊂ Q fr be an inclusion of quivers, with Q a full subquiver. We do not assume that Q fr is symmetric. Let I ⊂ CQ fr be the two-sided ideal generated by all paths in Q fr not contained in Q. Then CQ ∼ = CQ fr /I, and we let q : CQ fr → CQ be the induced surjection.
Let W fr be a potential extending W , in the sense that qW fr = W . Let ζ ∈ Q Q fr,0 be a stability condition extending the degenerate stability condition, that is, such that ζ| Q0 = 0. Let f ∈ N Q fr,0 \Q0 be a framing dimension vector. Define Via the usual correspondences, it is standard to check that N ζ Q fr ,f is a module for H Q,W . Now we make this setup more specific. Let Q be the three loop quiver, obtained by removing ∞ and all arrows containing it from Q r . Considering the inclusion Q ⊂ Q BBS , and setting ζ ∞ = −1 and f = 1, the module N ζ Q fr ,1 is precisely the vanishing cycle cohomology of Hilb(A 3 ). It was shown in version one of [10] that in fact this module is cyclic over the CoHA H Q,W . For a more recent example of a geometrically motivated class of modules for this CoHA, that falls under the general construction above, the reader may consult [28], where an action on the space of spiked instantons is considered.
The purpose of this subsection is to add one more geometrically motivated example to the list, namely, we consider the inclusion Q ⊂ Q r , the framing vector f = 1, and the stability condition ζ ∞ = −1. Then by Proposition 2.10, we obtain that is, the module constructed for the above data is precisely the vanishing cycle cohomologies of all of the quot schemes Q L . Keeping track of the various Tate twists, (5.7) is the hypercohomology of the Verdier dual of the monodromic mixed Hodge module φ mon Tr(Wr) Q N •• n ⊗ T −2n−n 2 considered in (5.2). According to the conjecture of the previous subsection, this module should itself be isomorphic to the underlying graded mixed Hodge module of a symmetric algebra. In fact by utilising the factorization sheaf structure implicit in our calculations in Section 3, it is not hard to find a cocommutative coproduct on (5.7), and a candidate for a compatible product, leading to our final conjecture: Conjecture 5.2. The graded mixed Hodge structure N ζ Q fr ,f is a universal enveloping algebra.
The connection with Conjecture 5.1 is that by proving a version of Conjecture 5.2 over the base Sym(A 3 ), one would deduce the purity conjecture as in [7, Theorem A].
Remark 5.3. The above module N ζ Q fr ,1 is in fact the third in a natural sequence, the first two elements of which will be well known to the reader. Firstly, we can remove the arrow a 2 from Q r to obtain a new quiver Q r . The pullback of Tr(W r ) along the extension by zero morphism rMod ζ (n,1) (Q r )/ GL n → rMod ζ (n,1) (Q r )/ GL n is induced by the potential W r = a 1 b 1 a 2 − a 2 b 1 a 1 − a 2 b 1 a 1 , and it is easy to check that there is an isomorphism H rMod ζ (n,1) (Q r )/ GL n , φ mon Tr(W r ) Q ⊗ T −n−n 2 ∼ = H(Hilb n A 2 , Q) ⊗ T −n with the usual cohomology of the Hilbert scheme for A 2 . Going further, we can remove the arrow b 1 , recovering the framed BBS quiver, with its usual potential, and the MacMahon module provided by the (vanishing cycle) cohomology of Hilb n A 3 . That the cohomology of Hilb n A 2 finds itself sandwiched between the vanishing cycle cohomology of the Hilbert scheme of A 3 and the quot scheme Q n L in this way is a mystery that we leave to future research to understand properly.