Higher rank motivic Donaldson-Thomas invariants of $\mathbb{A}^3$ via wall-crossing, and asymptotics

We compute, via motivic wall-crossing, the generating function of virtual motives of the Quot scheme of points on $\mathbb{A}^3$, generalising to higher rank a result of Behrend, Bryan and Szendr\H{o}i. We show that this motivic partition function converges to a Gaussian distribution, extending a result of Morrison.


INTRODUCTION
This paper has a two-fold goal: to compute, and to study the asymptotic behavior of the generating function of rank r motivic Donaldson-Thomas invariants of 3 , namely the series DT points r ( 3 , q ) = n≥0 Quot 3 ( ⊕r , n) vir · q n ∈ M q .
Here M is a suitable motivic ring and [ · ] vir ∈ M is the virtual motive (cf. § 1.1), induced by the critical locus structure on the Quot scheme Quot 3 ( ⊕r , n) parametrising 0-dimensional quotients of length n of the free sheaf ⊕r .
The following is our first main result. The result was first obtained in the case r = 1 by Behrend, Bryan and Szendrői [3] via an explicit motivic vanishing cycle calculation. Formula (0.2) follows by combining Formula (0.1) and Lemma 2.5. The approach of § 2.3, where we prove Formula (0.1), is based on the techniques of motivic wall-crossing for framed objects developed by Mozgovoy [23], allowing us to express the invariants for Quot 3 ( ⊕r , n), which we view as 'r -framed' Donaldson-Thomas invariants, in terms of the universal series of the invariants of unframed representations of the 3-loop quiver in a critical chamber. These ideas can be employed to compute framed motivic Donaldson-Thomas invariants of small crepant resolutions of affine toric Calabi-Yau 3-folds [8], which also exhibit similar factorisation properties.
The fact that partition functions of rank r invariants factor as r copies of partition functions of rank 1 invariants, shifted just as in Formula (0.2), has also been observed in the context of K-theoretic Donaldson-Thomas theory of 3 [13], as well as in string theory [24]. The exponential form of Formula (0.1) has been exploited in [29] to define higher rank motivic Donaldson-Thomas invariants for an arbitrary smooth quasi-projective 3-fold.
Formula (0.1) allows us to interpret the refined Donaldson-Thomas invariants of Quot 3 ( ⊕r , n) in terms of a weighted count of r -tuples of plane partitions π = (π 1 , . . ., π r ) of total size n (also known in the physics literature as r -colored plane partitions). Setting T = 1/2 , the coefficient of q n in DT points r ( 3 , q ) can be written as where S n,r is a certain explicit random variable on the space of r -tuples of plane partitions. In § 3, we describe the asymptotic behavior of (a renormalisation of ) the refined DT generating series, generalising A. Morrison's result for r = 1 [20]. We discuss the relationship with Morrison's work in § 3.1.
The following is our second main result. It will be proved in § 3.2.
1. BACKGROUND MATERIAL 1.1. Rings of motives and the motivic quantum torus. Let K 0 (St ) be the Grothendieck ring of stacks. It can be defined as the localisation of the ordinary Grothendieck ring of varieties K 0 (Var ) at the classes [GL k ] of general linear groups [4]. The invariants we want to study will live in the extended ring 1.1.1. The virtual motive of a critical locus. Let U be a smooth d -dimensional -scheme, f : U → 1 a regular function. The virtual motive of the critical locus crit f = Z (d f ) ⊂ U , depending on the pair (U , f ), is defined in [3] as the motivic class where [φ f ] ∈ Kμ 0 (Var ) is the (absolute) motivic vanishing cycle class defined by Denef and Loeser [12]. The 'μ' decoration means that we are consideringμ-equivariant motives, whereμ is the group of all roots of unity. However, the motivic invariants studied here will live in the subring M ⊂ Mμ of classes carrying the trivial action.

Quivers and motivic quantum torus.
A quiver Q is a finite directed graph, determined by its sets Q 0 and Q 1 of vertices and edges, respectively, along with the maps h, t : Q 1 → Q 0 specifying where an edge starts or ends. We use the notation to denote the tail and the head of an edge a ∈ Q 1 . All quivers in this paper will be assumed connected. The path algebra Q of a quiver Q is defined, as a -vector space, by using as a -basis the set of all paths in the quiver, including a trivial path ε i for each i ∈ Q 0 . The product is defined by concatenation of paths whenever the operation is possible, and is set to be 0 otherwise. The identity element is On a quiver Q one can define the Euler-Ringel form as well as the skew-symmetric form

Definition 1.3 (r -framing)
. Let Q be a quiver with a distinguished vertex 0 ∈ Q 0 , and let r be a positive integer. We define the quiver Q by adding one vertex, labelled ∞, to the original vertices in Q 0 , and r edges ∞ → 0. We refer to Q as the r -framed quiver obtained out of (Q , 0).
Let Q be a quiver. Define its motivic quantum torus (or twisted motivic algebra) as If Q is the r -framed quiver associated to (Q , 0), one has a decomposition where we have set y ∞ = y (1,0) . A generator y α ∈ T Q will be identified with its image y (0,α) ∈ T Q .
1.3. Quiver representations and their stability. Let Q be a quiver. A representation ρ of Q is the datum of a finite dimensional -vector space ρ i for every vertex i ∈ Q 0 , and a linear map ρ(a ):

Convention 1.
Let Q be a quiver, Q its r -framing. The dimension vector of a representation ρ of Q will be denoted by The space of all representations of Q with a fixed dimension vector α ∈ Q 0 is the affine space .
) a ∈Q 1 . Following [21], we recall the notion of (semi)stability of a representation.
For every α ∈ Q 0 \ 0, we denote by ϕ(α) the real number ϕ such that Z(α) = r e −1πϕ . It is called the phase of α with respect to Z.
The sets of ζ-stable and ζ-semistable representations with given dimension vector α form a chain of open subsets 1.4. Quivers with potential. Let Q be a quiver. Consider the quotient Q /[ Q , Q ] of the path algebra by the commutator ideal. A finite linear combination of cyclic paths W ∈ Q /[ Q , Q ] is called a superpotential. Given a cyclic path w and an arrow a ∈ Q 1 , one defines the noncommutative derivative This rule extends to an operator ∂ /∂ a acting on every superpotential. The Jacobi algebra J = J Q ,W of (Q , W ) is the quotient of Q by the two-sided ideal generated by ∂ W /∂ a for all edges a ∈ Q 1 . For every α ∈ Q 0 , the superpotential W = c a c c determines a regular function The points in the critical locus crit f α ⊂ R(Q , α) correspond to α-dimensional J -modules.
1.5. Framed motivic DT invariants. Let Q be a quiver, r ≥ 1 be an integer, and consider its r -framing Q with respect to a vertex 0 ∈ Q 0 (Definition 1.3). A representation ρ of Q can be uniquely written as a pair (u, ρ), where ρ is a representation of Q and u = (u 1 , . . ., u r ) is an r -tuple of linear maps u i : ρ ∞ → ρ 0 . From now on, we assume our framed representations to satisfy dim ρ ∞ = 1, so that according to Convention 1 we can write dim ρ = (1, dim ρ). We also view ρ as a subrepresentation of ρ of dimension (0, dim ρ), based at the vertex 0 ∈ Q 0 . Definition 1.9. Fix ζ ∈ Q 0 . A representation (u, ρ) of Q (resp. a J -module) with dim ρ ∞ = 1 is said to be ζ-(semi)stable if it is (ζ ∞ , ζ)-(semi)stable in the sense of Definition 1.5, where ζ ∞ = −ζ · dimρ.
We now define motivic DT invariants for moduli stacks of r -framed representations of a given quiver Q . Fix a superpotential W on Q . Let Q be the r -framing of Q at a given vertex 0 ∈ Q 0 , and let J be the Jacobi algebra J Q ,W , where W is viewed as a superpotential on Q in the obvious way. For a generic stability parameter ζ ∈ Q 0 , and an arbitrary dimension vector α ∈ Q 0 , set As in § 1.4, consider the trace map f α : R( Q , α) → 1 , induced by W , and its restriction to the framedstable locus f ζ, α : R ζ-st ( Q , α) → 1 . Define the moduli stacks Note that we are not quotienting by GL α = GL α × × , but only by GL α . Definition 1.10. We define r -framed motivic Donaldson-Thomas invariants and the associated motivic generating functions 1.6. Dimensional reduction. We say that a quiver with potential (Q , W ) admits a cut if there is a subset I ⊂ Q 1 such that every cyclic monomial appearing in W contains exactly one edge in I . If I is a cut for (Q , W ), one can define a new quiver Q I = (Q 0 ,Q 1 \ I ). Let J W ,I be the quotient of Q I by the two-sided ideal generated by the noncommutative derivatives ∂ W /∂ a for a ∈ I . Let R( J W ,I , α) ⊂ R(Q I , α) be the space of J W ,I -modules of dimension vector α. Then one has the following dimensional reduction principle.
Example 1.12. Let Q = L 3 be the 3-loop quiver (see Figure 1, and remove the framing vertex to obtain a picture of this quiver) with the potential The quiver Q I is the 2-loop quiver and J W ,I = [x , y ]. We have d I (n) = n 2 and χ Q (n, n) = −2n 2 . Therefore Proposition 1.11 yields an identity where R( J W ,I , n) is identified with the commuting variety

MOTIVIC DT INVARIANTS OF THE QUOT SCHEME OF POINTS
2.1. Stability on the framed 3-loop quiver. The main character in this section is the framed quiver L 3 of Figure 1, which we equip with the superpotential and with linear maps induced naturally by those defined by ρ.
From now on we identify the space of stability parameters for L 3 with .

Lemma 2.2.
Let ζ ∈ be a stability parameter, and let ρ = (u, ρ) be a representation of L 3 of dimension (1, n). Set ζ = (−nζ, ζ). Then: On the other hand, if n = 0 then ρ is simple and hence ζ-stable. In the case ζ = 0 there is nothing to prove, as all representations have phase 1/2.
Consider the following regions of the space of stability parameters : By Lemma 2.2 the space of stability parameters on L 3 admits a particularly simple wall-and-chamber decomposition = Ω + ∐ Ω 0 ∐ Ω − with one wall (the origin) and two chambers.
2.2. The virtual motive of the Quot scheme of points. Let L 3 be the r -framed 3-loop quiver (Figure 1), and fix the superpotential . Fix a stability parameter ζ + ∈ Ω + = <0 . Fixing n ≥ 0 and setting ζ + = (−nζ, ζ), n = (1, n), the quotient stack is a smooth quasi-projective variety of dimension 2n 2 + r n, called the noncommutative Quot scheme in [1]. The regular function f n : R( L 3 , n) → 1 given by taking the trace of W descends to a regular function on M ζ + ( L 3 , n), still denoted f n . We have the following description of the Quot scheme of length n quotients of ⊕r 3 .

Proposition 2.3 ([1, Thm. 2.6]). There is an identity of closed subschemes
Thanks to Proposition 2.3, we can form the virtual motives of the Quot scheme, as in § 1.1.1, and define their generating function

Remark 2.4. The main result of [3] is the formula
The series DT points 1 (Y , q ) studied in [3] for an arbitrary smooth 3-fold Y also appeared in [11] as the wallcrossing factor in the motivic DT/PT correspondence based at a fixed smooth curve C ⊂ Y in Y . This correspondence refined its enumerative counterpart [27,26]. The same phenomenon occurred in [21,22] in the context of framed motivic DT invariants. See [29, § 4] for a generalisation DT points r (Y , q ) of (2.1) to an arbitrary smooth 3-fold Y . See [28] for a plethystic formula expressing the naive motives [Quot Y (F, n)] ∈ K 0 (Var ) in terms of the motives of the punctual Quot schemes.
The following consideration will constitute the final step in proving Theorem A.

Lemma 2.5.
There is an identity Proof. The claimed identity follows from a simple manipulation: The identification of (2.2) with DT points r ( 3 , q ) is proven in the PhD theses of the first and third author [6,25]. Both proofs follow the technique introduced in the r = 1 case by Behrend-Bryan-Szendrői [3]. In the next subsection, we provide a new proof of Theorem A. We exploit an r -framed version of motivic wall-crossing. This technique, inspired by [23,21,22], will be applied to small crepant resolutions of affine toric Calabi-Yau 3-folds in [8].

Calculation via wall-crossing.
In this subsection we prove Theorem A. Consider the universal generating function as an element of the motivic quantum torus. To (generic) stability parameters ζ ± ∈ Ω ± we associate elements (cf. Definition 1.10) By Lemma 2.2 (3), we have an identity whereas the series Z ζ + is, essentially, a "shift" of the generating function DT points r ( 3 , y (0,1) ). More precisely, by Proposition 2.3 we have an identification of critical loci sitting inside the noncommutative Quot scheme; in particular, the associated virtual motives are the same. The shift is intended in the following sense: since 〈(0, n), (1, 0)〉 L 3 = r n, the product rule (1.1) yields the identity Since we can express y (0,n) as the n-fold product of y (0,1) with itself, we obtain The last generating function we need to analyse is whose n-th coefficient is the virtual motive of the stack of 0-dimensional [x , y , z ]-modules of length n. This was already computed in Example 1.12: The next ingredient of the proof is a particular instance of Mozgovoy's motivic wall-crossing formula [23].
Proof. Let ρ = (u, ρ) be a J -module. Consider ζ + ∈ Ω + , and let 〈u, ρ〉 ⊂ ρ be the submodule introduced in Definition 2.1. We have that 〈u, ρ〉 is ζ + -stable by Lemma 2.2 (1) and the quotient ρ/ 〈u, ρ〉 is supported at the vertex 0. From this we deduce the decomposition A U = Z ζ + · A U . Consider now ζ − ∈ Ω − . The quotient of ρ by the submodule ρ based at the vertex 0 is the simple module supported at the framing vertex ∞. By Lemma 2.2 (3) this is the unique ζ − -stable module for the current choice of ζ − , so we obtain   Note that we have an identity where we have used (2.4) for the second equality. By Formula (2.5), the left-hand term of Formula (2.7) can then be rewritten as Therefore, by Equation Setting q = − r 2 y (0,1) , and using Equation (2.6), a simple substitution yields DT points Formula (0.1) is proved. By Lemma 2.5, Theorem A is proved.

THE NORMAL LIMIT LAW AND ASYMPTOTICS
In § 3.1, we introduce a family of random variables on the space of r -colored plane partitions, and we describe the asymptotics of the members of the family after suitable normalisation in Proposition 3.1. Theorem B, the main theorem of the section, is deduced from Proposition 3.1 in subsection 3.2. Finally, subsection 3.3 is entirely devoted to the proof of Proposition 3.1.

Random variables on r -colored plane partitions. We introduce a multivariate function
The coefficient of z n is a polynomial on the three variables u, v and w , which we denote by Q n (u, v, w ), whose coefficients are nonnegative integers. When u = v = w = 1, we obtain the well known MacMahon function raised to the power r , which is the generating function for the r -tuples of plane partitions. Hence, Q n (1, 1, 1) is the number of r -tuples of plane partitions of total size n, i.e. the number of r -colored plane partitions (π 1 , π 2 , . . ., π r ) such that r j =1 |π j | = n, where |π j | denotes the sum of the entries of the plane partition π j (or the number of boxes, cf. Figure 3). The polynomial Q n (u, v, w ), when divided by Q n (1, 1, 1), represents the joint probability generating function of some random variables X n , Y n , and Z n on the space of r -colored plane partitions of size n, where each r -tuple is equally likely. More precisely, we have To describe these random variables, we need to define certain parameters of plane partitions. For a plane partition π, let ∆(π) denote the sum of the diagonal parts of π, ∆ + (π) denote the sum of the upper diagonal parts, and ∆ − (π) denote the sum of the lower diagonal parts. See Figure 3 for an example of a plane partition showing the values of these parameters. The parameter ∆(π) is also known as the trace of π and it has been studied in the literature, see for instance [17] and the references therein. In particular, one has Similarly, we can find in [20] that (1 − q 2k −m z m ) −1 = π q ∆(π)+∆ + (π)−∆ − (π) z |π| .
We can easily deduce from these two identities that for an r -colored plane partition π = (π 1 , π 2 , . . ., π r ) of total size n, we have (∆(π l ) + ∆ + (π l )) , When r = 1, Kamenov and Mutafchiev [17] proved that the distribution of the trace of a random plane partition of size n, when suitably normalised, is asymptotically normal. Morrison [20] also established asymptotic normality for any random variable of the form δ∆(π) + ∆ + (π) − ∆ − (π), where π is a random plane partition of size n and δ is a fixed real number. We show that for any fixed integer r ≥ 1, any nontrivial linear combination of the variables X n , Y n and Z n , when suitably normalised, converges weakly to a normal distribution. It is worth noting that the random variable Y n is non-constant only when r > 1.
Looking at the asymptotic behaviour of the random variables X n , Y n and Z n , when divided by n 2/3 , we observe from the above result that the random variable n −2/3 Z n degenerates as n → ∞. Furthermore, by the Camér-Wold device [9], the random variables n −2/3 X n , n −2/3 Y n converge jointly to a bivariate normal distribution with a diagonal covariance matrix. The appropriate normalisation of Z n is n −1/3 (log n) −1/2 Z n which is, when centred, asymptotically normal. This asymptotic normality and the asymptotic formulas for µ n and σ 2 n agree with the main result in [17] when r = 1 and (α, β , γ) = (0, 0, 1).

Convention 2.
We shall use the Vinogradov notation ≪ interchangeably with the O-notation. For instance, by f (n) ≪ g (n) (or g (n) ≫ f (n)) as n → ∞, we mean that there exists a positive constant C such that | f (n)| ≤ C g (n) for sufficiently large n.
Theorem B now follows immediately from Proposition 3.1 as we will see next. The product on the right-hand side can be expressed in terms of our auxiliary function F (u, v, w, z ) defined at the beginning of this section. First we write it as follows
3.3. Proof of Proposition 3.1. Morrison used the method of moments to prove his result in [20]. However, due to the appearance of the second variable Y n and the complication that comes with it, we decided to use a different approach. We follow the method that Hwang used in [16] to prove limit theorems for the number of parts in the so-called restricted partitions (these are one dimensional partitions with some restrictions on the parts). The first part of the proof is based on the saddle-point method to get an asymptotic formula for Q n (u, v, w ) as n → ∞, and the second is a perturbation technique to deduce the central limit theorem.

Saddle-point method.
The goal here is to obtain an asymptotic formula for Q n (u, v, w ) as n → ∞, where (u, v, w ) is allowed to vary in a fixed real neighborhood of (1, 1, 1). To simplify our notation, define Φ(t ) = − log(1 − e −t ), and for real numbers a , b and c , we let The function f depends on (a , b , c ) but we drop this dependence for now to ease notation. Also, for a positive number ρ, we make the substitution z = e −τ , where τ = ρ + i t . Hence, One can easily verify that if (u, v, w ) is bounded (which is the case throughout this section), then there exists a fixed positive real number R such that the product in (3.1) converges absolutely whenever |z | < R . Hence, F (u, v, w, z ), as function of z , is analytic in a complex neighborhood of 0. By Cauchy's integral formula, we have We now use the saddle-point method to estimate the above integral. We choose ρ to be the positive solution of the equation where f x denotes the partial derivative of f with respect to x . Similar notations will be used for other partial derivatives. Note that there is a unique positive solution ρ = ρ(n, a , b , c ) of Equation (3.5) since the function defined by the series is strictly decreasing as a function of ρ, provided that a , b and c are small enough (it suffices for instance to assume that |c | + |a | + r |b | < 1). Furthermore, we observe that ρ → 0 as n → ∞. The following lemma reveals the asymptotic dependence between n and ρ.
as n → ∞, uniformly for |c |+|a |+ r |b | ≤ ε, where the implied constants in the O-terms are independent of ε.
Proof. Recall from (3.5) that Under the assumption that |c | + |a | + r |b | ≤ ε, for any m ≥ 1, 1 ≤ k ≤ m and 1 ≤ l ≤ r , we have Moreover, the function Φ ′ (x ) is an increasing function. Therefore, Multiplying by m and summing over m ≥ 1, 1 ≤ k ≤ m and 1 ≤ l ≤ r , we obtain We can obtain asymptotic estimates of the lower and upper bounds as ρ → 0 + . This can be done via Mellin transform. The reader can consult [15] for a comprehensive survey on the Mellin transform method. The Mellin transform of which has simple poles at s = 3 and s = 1. The other singularities are precisely at the negative odd integers. Thus, we have Since |ζ(i t − 2)ζ(i t )Γ (i t )| decays exponentially fast as t → ±∞, the absolute value of the integral on the right hand side is bounded by an absolute constant. The same argument works for the estimate of the upper bound f x ((1 + ε)ρ, 0). This completes the proof of the lemma.
Next, we split the integral on the right-hand side of (3.4) into two parts as follows: let where C is an absolute constant in the interval (5/3, 2), and let I 2 = Q n (e −a ρ , e −b ρ , e −c ρ ) − I 1 .
Estimate of I 1 . For the rest of this section, we work under the condition of Lemma 3.2, that is |c | + |a | + r |b | ≤ ε and ε ∈ [0, 1/2]. For −ρ C ≤ t ≤ ρ C , Equation (3.5) and a Taylor approximation of f (ρ + i t , ρ) give where the implied constant in the error term is absolute. To estimate the error term, observe that For any real number θ and τ = ρ + i θ we have .
We can estimate | f x x x (ρ, ρ)| as we did for f x (ρ, ρ) in the proof of Lemma 3.2. We obtain where the implied constant depends only on r . Therefore, for |t | ≤ ρ C we have Since we chose C > 5/3, we have ρ 3C −5 = o (1). Thus It remains to estimate the integral on the right hand side as ρ → 0 + . Note that f x x (ρ, ρ) > 0 and f x x (ρ, ρ) ≫ ρ −4 (again via Mellin transform as in Lemma 3.2), so we have where A > 0 and the hidden constants in the error terms above depend only on r . Thus, since we chose C < 2, the term ρ 4−c e −Aρ 2C −4 tends to zero faster than any power of ρ as ρ → 0 + . Hence, we obtain an estimate for I 1 (3.8) This estimate holds uniformly for |c | + |a | + r |b | ≤ ε and ε ∈ [0, 1/2].
Estimate of I 2 . We will prove that |I 2 | is much smaller than |I 1 |. To this end, we assume that where C is as before. We have A lower estimate of the same term can be found in the proof of [18,Lemma 5]. By the same argument as the one given in loc. cit., but with |t | ≥ ρ C , we get where the implied constant is independent of ε.
If we now assume that (α, β ) = (0, 0) but γ = 0, then the Mellin transform of g y y (t , 0) is r γ 2 ζ(s − 1) 2 Γ (s ) whose dominant singularity is a double pole at s = 2. This leads to the asymptotic formula g y y (ρ 0 , 0) = On the other hand, for any θ = o (ρ 0 ), we have the following: Just as in the previous case, this is enough to prove the central limit theorem. The asymptotic formula for the variance in terms of n can be obtained from (3.21) using (3.20). The proof of Proposition 3.1 is complete.