2. JK residues and flags

In this Section we introduce Jeffrey–Kirwan residues of hyperplane arrangements through their characterisation in terms of flags, started in [ Reference Brion and Vergne9 ] and studied systematically in [ Reference Szenes and Vergne29 ].

Given an n-dimensional real linear space $\mathfrak{a}$ with a fixed full rank lattice $\Gamma$ , we consider the duals $V \;:\!=\; \mathfrak{a}^\ast$ , $\Gamma^\ast \subseteq V$ . Fix a finite set of generators $\mathfrak{A}$ for $\Gamma^\ast$ , and suppose it is projective (i.e., contained in a strict half-space of V). We also fix elements $f_1, \ldots, f_n \in \mathfrak{A}$ giving an ordered basis of V, and the top form on $\mathfrak{a}$ defined as $d\mu \;:\!=\; f_1 \wedge \cdots \wedge f_n$ . Given a subset $S \subseteq V$ , we will denote by $\mathfrak{B}(S)$ the set of all distinct bases of V consisting of elements of S.

Definition 2·1. Let $x \in \mathfrak{a}_\mathbb{C}$ . We say that x is regular (with respect to $\mathfrak{A}$ ) if it is not contained in the union of hyperplanes

\begin{align*} \bigcup_{f \in \mathfrak{A}\otimes_\mathbb{R}\mathbb{C}} V(f) \subset \mathfrak{a}_\mathbb{C}. \end{align*}

The set of regular points of $\mathfrak{a}_\mathbb{C}$ is a dense open cone, denoted by $\mathfrak{a}_\mathbb{C}^{reg}$ .

We also need to introduce the dual notion.

Definition 2·2. Let $S \subseteq V$ be a finite set and let $\zeta \in V$ . We say that $\zeta$ is S-regular if

\begin{align*} \zeta \notin \bigcup_{\substack{I \subseteq S\\ | I | = n-1}} \text{Span}_\mathbb{R}(I). \end{align*}

In particular, we will say that $\zeta$ is regular (and write $\zeta \in V^{reg}$ ) if it is $\mathfrak{A}$ -regular. The connected components of the dense open cone $V^{reg}$ are called ( $\mathfrak{A}$ -)chambers of V. Moreover, denoting by $\Sigma\mathfrak{A}$ the set of all sums of distinct elements of $\mathfrak{A}$ , we say that $\zeta$ is sum-regular if it is $\Sigma \mathfrak{A}$ -regular.

The classical Jeffrey–Kirwan residue associated with a hyperplane arrangement (see [ Reference Brion and Vergne9 ]), which we denote here with $\text{J-K}_{\xi}^{\mathfrak{A}}$ , is a $\mathbb{C}$ -linear functional, depending on a chamber $\xi$ , defined on the linear space of germs of meromorphic functions f which are regular in a neighbourhood of the origin in $\mathfrak{a}_{\mathbb{C}}^{reg}$ .

The original definition of $\text{J-K}_{\xi}^{\mathfrak{A}}$ , which we do not recall here, involves a formal Laplace transform. According to [ Reference Szenes and Vergne29 , Lemma 2·2], there is a cycle C in $\mathfrak{a}_\mathbb{C}^{reg}$ , depending on $\xi$ , such that

\begin{equation*} \text{J-K}_{\xi}^{\mathfrak{A}}(f) = \left(\frac{1}{2\pi {\rm i}}\right)^{n}\int_C f d\mu \end{equation*}

for all meromorphic functions f which are regular in a neighbourhood of the origin in $\mathfrak{a}_\mathbb{C}^{reg}$ (cf. (1·2)).

Definition 2·4. The (finite) set $\mathcal{FL}(\mathfrak{A})$ is given by flags of V of the form

\begin{align*} \mathfrak{F} = [\lbrace 0 \rbrace = F_0 \subset F_1 \subset \ldots \subset F_n = V] \end{align*}

such that, for all $j=0, \ldots, n$ , $\mathfrak{A}$ contains a basis of $F_j$ and $\text{dim}_\mathbb{R}F_j = j$ .

Consider a flag $\mathfrak{F} \in \mathcal{FL}(\mathfrak{A})$ . Pick a basis $\gamma^{\mathfrak{F}}$ as in the Lemma above. This basis induces an isomorphism of $\mathbb{C}$ -linear spaces $\mathfrak{a}_\mathbb{C} \cong \mathbb{C}^n$ and so an isomorphism between the germs of meromorphic functions at the origin of $\mathfrak{a}_\mathbb{C}$ and of $\mathbb{C}^n$ , which we denote by $\mathcal{M}_{\mathfrak{a}_\mathbb{C},0}$ , respectively $\mathcal{M}_{\mathbb{C}^n,0} $ .

Definition 2·6. The flag residue morphism associated with the flag $\mathfrak{F}$ is the composition

\begin{align*} \mathcal{M}_{\mathfrak{a}_\mathbb{C},0} \xrightarrow{\sim} \mathcal{M}_{\mathbb{C}^n,0} \xrightarrow{\text{IR}_0} \mathbb{C}. \end{align*}

Here $\text{IR}_0$ denotes the iterated residue operation, recalled in Appendix A. By [ Reference Szenes and Vergne29 , lemma 2·5], this does not depend on the choice of the basis $\gamma^{\mathfrak{F}}$ , so we have a well defined map $\operatorname{Res}_{(\cdot)}\; :\; \mathcal{FL}(\mathfrak{A}) \rightarrow \left( \mathcal{M}_{\mathfrak{a}_\mathbb{C},0} \right)^\ast$ .

A flag $\mathfrak{F} \in \mathcal{FL}(\mathfrak{A})$ determines a partition of $\mathfrak{A}$ into the subsets

\begin{equation*}\left \lbrace \mathfrak{P}_j \;:\!=\; \mathfrak{A} \cap F_{j} \setminus F_{j-1} \right \rbrace_{j=1}^n. \end{equation*}

We define the vectors

\begin{align*} \kappa^{\mathfrak{F}}_j \;:\!=\; \sum_{s=1}^j \sum_{\alpha \in \mathfrak{P}_s} \alpha \in \Sigma \mathfrak{A},\, j=1, \ldots, n. \end{align*}

Note that $\kappa \;:\!=\; \kappa^{\mathfrak{F}}_n$ does not depend on the flag $\mathfrak{F}$ .

Definition 2·7. A flag $\mathfrak{F} \in \mathcal{FL}(\mathfrak{A})$ is called proper if the vectors $\lbrace \kappa^{\mathfrak{F}}_j\rbrace_{j=1}^n$ form a basis of V. We introduce the function $\nu \;:\; \mathcal{FL}(\mathfrak{A}) \longrightarrow \lbrace 0, \pm 1 \rbrace$ given by

\begin{align*} \nu(\mathfrak{F}) \;:\!=\; \begin{cases} 0 & \text{if } \mathfrak{F} \text{ is not proper}, \\[5pt] 1 & \text{if } \mathfrak{F} \text{ is proper and } \lbrace \kappa_1^{\mathfrak{F}}, \ldots, \kappa_r^{\mathfrak{F}} \rbrace \text{ is positively oriented w.r.t. } d\mu,\\[5pt] -1\,\, & \text{if } \mathfrak{F} \text{ is proper and } \lbrace \kappa_1^{\mathfrak{F}}, \ldots, \kappa_r^{\mathfrak{F}} \rbrace \text{ is negatively oriented w.r.t. } d\mu. \end{cases} \end{align*}

For each flag $\mathfrak{F}$ we introduce the cone

\begin{align*} \mathfrak{c}^+(\mathfrak{F}, \mathfrak{A}) \;:\!=\; \sum_{j=1}^n \mathbb{R}_{>0} \kappa_j^{\mathfrak{F}}. \end{align*}

If $\zeta \in V$ is a sum-regular vector, we define the set of flags for which $\zeta$ is in this cone:

\begin{align*} \mathcal{FL}^+(\mathfrak{A}, \zeta) \;:\!=\; \left\lbrace \mathfrak{F} \in \mathcal{FL}(\mathfrak{A})\,: \zeta \in \mathfrak{c}^+(\mathfrak{F}, \mathfrak{U}) \right \rbrace. \end{align*}

We can now define a notion of Jeffrey–Kirwan residue for arbitrary meromorphic functions.

Definition 2·9. Let $\zeta$ be a sum-regular vector of V. We define the JK residue map $\operatorname{JK}^{\mathfrak{A}}_\zeta \;:\; \mathcal{M}_{\mathfrak{a}_\mathbb{C},0} \rightarrow \mathbb{C}$ as

\begin{align*} \operatorname{JK}^{\mathfrak{A}}_\zeta(f) = \sum_{\mathfrak{F}\in \mathcal{FL}^+(\mathfrak{A}, \zeta)} \nu(\mathfrak{F})\operatorname{Res}_{\mathfrak{F}}(f). \end{align*}

The classical Jeffrey–Kirwan residue can now be expressed in terms of the flag residues introduced above.

Theorem 2·12 ([ Reference Szenes and Vergne29 , theorem 2·6]). Let $\xi$ be a chamber of V and let $\zeta \in \xi$ be a sum-regular vector. Then, for every $f \in \mathcal{M}_{\mathfrak{a}_\mathbb{C},0}$ which is regular (locally around the origin) on $\mathfrak{a}_\mathbb{C}^{reg}$ , we have

\begin{align*} \operatorname{J-K}^{\mathfrak{A}}_\xi(f) = \operatorname{JK}^{\mathfrak{A}}_\zeta(f). \end{align*}

When $\mathfrak{A}$ is a basis of V, the JK residue takes a special form.

Lemma 2·13. Let $\mathfrak{A}$ be a basis of V and let $\zeta \in V$ be a $\Sigma \mathfrak{A}$ -regular vector. If $\zeta \notin \operatorname{Span}_{\mathbb{R}_{>0}}\{\mathfrak{A}\}$ then $\operatorname{JK}_{\zeta}^{\mathfrak{A}}=0$ as a morphism. Otherwise $\operatorname{JK}_{\zeta}^{\mathfrak{A}}$ is the composition

\begin{align*} \mathcal{M}_{\mathfrak{a}_\mathbb{C},0} \xrightarrow{\sim} \mathcal{M}_{\mathbb{C}^n,0}\xrightarrow{\operatorname{IR}_0} \mathbb{C}, \end{align*}

where the first isomorphism is given by the choice of the unique ordering of $\mathfrak{A}$ such that the components $c_1, \ldots, c_n$ of $\zeta$ with respect to this ordered basis satisfy

\begin{align*} 0 < c_n < \cdots < c_1. \end{align*}

Proof. Choose an ordering $f_1, \ldots, f_n$ for the elements of $\mathfrak{A}$ such that this ordered basis is positively oriented. The set of flags $\mathcal{FL}(\mathfrak{A})$ is in bijection with the set of permutations $S_n$ via

\begin{align*} S_n \ni \sigma \mapsto \mathfrak{F}_\sigma \in \mathcal{FL}(\mathfrak{A}), \qquad (\mathfrak{F}_\sigma)_j \;:\!=\; \text{Span}_\mathbb{R}(f_{\sigma(1)}, \ldots, f_{\sigma(j)}). \end{align*}

Every flag $\mathfrak{F}_\sigma$ is proper, since the set

(2·1) \begin{align} \big\lbrace f_{\sigma(1)}, f_{\sigma(1)} + f_{\sigma(2)}, \ldots, \sum_{i=1}^n f_{\sigma(i)} \big\rbrace \end{align}

is a basis of V. Note moreover that the orientation of this basis coincides with $(\!-\!1)^\sigma$ . Assume that $\zeta \in \mathfrak{c}^+(\mathfrak{F}_\sigma, \mathfrak{A})$ . Then its coordinates $\zeta_1, \ldots, \zeta_n$ with respect to the basis (2·1) are all positive. On the other hand its coordinates $c_1,\ldots, c_n$ with respect to the basis $f_{\sigma(1)}, \ldots, f_{\sigma(n)}$ are given by

\begin{align*} c_{j} \;:\!=\; \sum_{i=1}^j \zeta_i. \end{align*}

Thus $\zeta \in \mathfrak{c}^+(\mathfrak{F}_\sigma, \mathfrak{A})$ implies that $0 < c_n < \ldots < c_1$ . This condition cannot be satisfied if $\zeta \notin \operatorname{Span}_{\mathbb{R}_{>0}}\{\mathfrak{A}\}$ , hence we obtain the first part of the claim.

On the other hand, if $\zeta \notin \operatorname{Span}_{\mathbb{R}_{>0}}\{\mathfrak{A}\}$ , then its components with respect to $f_1, \ldots, f_n$ are all positive and distinct since $\zeta$ is sum-regular. So there exist a unique permutation $\sigma \in S_n$ such that $0 < c_n < \cdots < c_1$ , and we have

\begin{align*} \text{JK}_\zeta^{\mathfrak{A}}(\!\ast\!) = \nu(\mathfrak{F}_\sigma)\text{Res}_{\mathfrak{F}_\sigma}(\!\ast\!). \end{align*}

We have already seen that $\nu(\mathfrak{F}_\sigma) = (\!-\!1)^\sigma$ , and that the same number also gives the orientation of the basis $f_{\sigma(1)}, \ldots, f_{\sigma(n)}$ . By definition, in order to compute the flag residue, we should express the meromorphic function under the linear change of basis to $(\!-\!1)^\sigma f_{\sigma(1)}, f_{\sigma(2)}, \ldots, f_{\sigma(n)}$ . By Proposition A·2 we see that this is equivalent to multiplying by $(\!-\!1)^\sigma$ the residue obtained by considering the basis $f_{\sigma(1)}, \ldots, f_{\sigma(n)}$ . Then the result follows from $(\!-\!1)^\sigma (\!-\!1)^\sigma = 1$ .

Corollary 2·14. Let $\mathfrak{A}$ be a basis of V and let $\xi$ be a chamber. If $\xi =\sum_{u \in \mathfrak{A}}\mathbb R_{>0} u$ then the functional $\operatorname{J-K}_{\xi}^{\mathfrak{A}}$ is the composition

\begin{align*} \mathcal{M}_{\mathfrak{a}_\mathbb{C},0} \xrightarrow{\sim} \mathcal{M}_{\mathbb{C}^n,0}\xrightarrow{\operatorname{IR}_0} \mathbb{C} \end{align*}

restricted to the set of meromorphic functions that are holomorphic on $\mathfrak{a}_\mathbb{C}^{reg}$ in a neighborhood of the origin. Here the first isomorphism is induced by an arbitrary choice of an ordering for the elements of $\mathfrak{A}$ .

3. JK residues and quivers

In this Section we describe the application of JK residues to quiver invariants, following [ Reference Beaujard, Mondal and Pioline1, Reference Córdova and Shao12 ].

Let Q be a quiver without loops or oriented cycles. We define an equivalence relation on $ Q_1$ by

\begin{align*} \alpha \sim \beta \iff (t(\alpha), h(\alpha)) = (t(\beta), h(\beta)), \end{align*}

and consider a new quiver $\bar{ Q}$ , called the reduced quiver of Q, defined by $\bar{ Q}_0 = Q_0$ and $\bar{ Q}_1 \;:\!=\; Q_1/\sim$ , with head and tail functions induced on the quotient by those of Q. For every arrow $\alpha \in\bar{ Q}_1$ we define the multiplicity $m_\alpha = | \pi^{-1}(\alpha)| = \langle h(\alpha), t(\alpha)\rangle$ , where $\langle -, - \rangle$ is the skew-symmetrised Euler form of Q.

Fix a dimension vector $d \in \mathbb Z_{\geq 0} Q_0$ . The complexified gauge group $G_d$ is a Lie group with Lie algebra $\mathfrak{g}$ , and its Cartan subalgebra is the abelian Lie algebra of diagonal matrices

\begin{align*} \mathfrak{h} = \bigoplus_{v \in Q_0} \mathbb{C}^{d_v} \leq \mathfrak{g} = \prod_{v \in Q_0} \text{Mat}_{d_v \times d_v}(\mathbb{C}). \end{align*}

The roots of $\mathfrak{g}$ are the $\mathbb{C}$ -linear functionals $\mathfrak{g} \rightarrow \mathbb{C}$ of the form

\begin{align*} r_{v,i,j} (M) \;:\!=\; (M_v)_{j,j} - (M_v)_{i,i}, \text{ for } v \in Q_0, \, i\neq j \in \lbrace 1, \ldots, d_v \rbrace. \end{align*}

The weights of the representation of Lie groups

\begin{align*} G_d \longrightarrow \text{GL}(\text{Rep}_d( Q)) \end{align*}

given by basechange are the $\mathbb{C}$ -linear functionals $\mathfrak{h} \rightarrow \mathbb{C}$ of the form

\begin{align*} \rho_{\alpha, i,j}(M) \;:\!=\; (M_{h(\alpha)})_{j,j} - (M_{t(\alpha)})_{i,i}, \end{align*}

for $\alpha \in Q_1, \, i \in \lbrace 1, \ldots, d_{t(\alpha)}\rbrace, \, j \in \lbrace 1, \ldots, d_{h(\alpha)}\rbrace$ .

We fix a vertex $\overline{v}$ of Q and choose an index $\overline{k} \in \lbrace 1, \ldots, d_{\overline{v}} \rbrace$ . Consider the hyperplane

\begin{align*} \mathfrak{a}_\mathbb{C} \;:\!=\; V({u}_{\overline{v}, \overline{k}}) \subset \mathfrak{h}. \end{align*}

There is an obvious isomorphism $\mathfrak{a}_{\mathbb C} \cong \bigoplus_{v \in Q_0} \mathbb{C}^{d_v - \delta_{v, \overline{v}}}$ , which we use implicitly in the following. Let $V_\mathbb{C}$ be the dual linear space of $\mathfrak{a}_\mathbb{C}$ . We denote by $\mathfrak{R}_\mathbb{C}$ the image of the set of roots of $G_d$ under the natural projection $\mathfrak{h}^\ast \rightarrow V_\mathbb{C}$ , and similarly by $\mathcal{W}_{\mathbb{C}}$ the image of the set of weights.

Remark 3·2. Note that the elements of $\mathfrak{R}_\mathbb{C}$ and of $\mathcal{W}_\mathbb{C}$ have real coefficients with respect to the canonical basis of $V_\mathbb{C}$ . So we have real linear spaces $\mathfrak{a}$ $(:\!=\; V(x_{\overline{v}, \overline{k}}) \subset \bigoplus_{v \in Q_0} \mathbb{R}^{d_v})$ , $V \;:\!=\; \mathfrak{a}^\ast$ , and subsets $\mathfrak{R}\subset V$ , $\mathcal{W}\subset V$ , such that

\begin{align*} \mathfrak{a}_\mathbb{C} \cong \mathfrak{a} \otimes_\mathbb{R} \mathbb{C},\, V_\mathbb{C} \cong V \otimes_\mathbb{R} \mathbb{C}, \, \mathfrak{R}_\mathbb{C} \cong \mathfrak{R} \otimes_\mathbb{R} \mathbb{C},\, \mathcal{W}_\mathbb{C} \cong \mathcal{W} \otimes_\mathbb{R} \mathbb{C}. \end{align*}

For a fixed R-charge, we consider the following affine hyperplanes of $\mathfrak{a}_\mathbb{C}$ :

\begin{align*} &H_r \;:\!=\; V(r - 1), \, r \in \mathfrak{R}_\mathbb{C},\\[5pt] &H_\rho \;:\!=\; V(\rho + R_\rho), \, \rho \in \mathcal{W}_\mathbb{C}. \end{align*}

A stability vector $\theta \in \mathbb R Q_0$ is normalised for (Q, d) if it belongs to the hyperplane

\begin{align*} \mathcal{H} \;:\!=\; \big\lbrace \sum_{v \in Q_0} d_v \theta_v = 0 \big\rbrace \subset \mathbb R Q_0. \end{align*}

With this normalisation, d is $\theta$ -coprime if, for all $0<e<d$ , we have $\theta(e)\neq 0$ .

Let $\theta \in \mathbb{R} Q_0 $ be a stability vector for Q. We extend it to an element $\tilde{\theta} \in V$ by

(3·1) \begin{align} \tilde{\theta}(e_{v,k}) \;:\!=\; \theta_v. \end{align}

Definition 3·5. Let $\zeta$ be a $\Sigma \mathfrak{A}$ -regular element belonging to the same chamber of V as $-\tilde{\theta}$ . We define the global Jeffrey–Kirwan residue

\begin{align*} \operatorname{JK}\;:\; \text{Mer}(\mathfrak{a}_\mathbb{C}) \Longrightarrow \mathbb{C} \end{align*}

given by

\begin{align*} \operatorname{JK}(f, \zeta) = \sum_{x \in \mathfrak{M}_{sing}} \text{JK}^{\mathfrak{A}_x}_{\zeta} \left(f\left(\ast + x \right)\right). \end{align*}

Remark 3·7. Note that, in general, $\operatorname{JK}(f, \zeta)$ depends on the choice of $\zeta$ . However, if $f\in \text{Mer}(\mathfrak{a}_\mathbb{C})$ is such that, for every $x \in \mathfrak{M}_{sing}$ , there is a ball $D \subset \mathfrak{a}_\mathbb{C}$ around the origin such that f is holomorphic on the $\mathfrak{A}_x$ -regular locus

\begin{equation*} (x + D)\setminus \bigcup_{l \in \mathfrak{A}_x} V(l(\ast - x)), \end{equation*}

then, by Theorem 2·12, $\operatorname{JK}(f,\zeta)$ depends only on the chamber containing $\zeta$ and not on the particular choice of sum-regular vector.

Definition 3·8. Fix $z \in \mathbb{C}\setminus \mathbb{Z}$ . The one-loop factor is the meromorphic function $Z_{1-loop} \in \text{Mer}(\mathfrak{a}_\mathbb{C})$ , depending on z, given by

\begin{align*} &Z_{1-loop}(u,z) =\\[5pt] & \left(- \frac{\pi z}{\sin(\pi z)} \right)^{\vert d \vert -1} \left( \prod_{r \in \mathfrak{R}_\mathbb{C}} \frac{\sin(\pi z r(u))}{\sin(\pi z( r (u) - 1 ))} \right) \left( \prod_{\rho \in \mathcal{W}_\mathbb{C}} \frac{\sin(\pi z( \rho(u) + R_\rho - 1))}{\sin(\pi z(\rho (u) + R_\rho))} \right)^{m_\rho}. \end{align*}

Suppose $\tilde{\theta}$ is a $\mathfrak{A}$ -regular vector and that $\mathfrak{A}_x$ is projective for every $x \in \mathfrak{M}_{sing}$ (the latter condition holds automatically if $\mathfrak{A}_x$ is a basis; this will always be the case in our applications). The quantity of physical interest (i.e., the $\tau \to {\rm i} \infty$ limit of the corresponding partition function on $T^2 = \mathbb C/\langle1, \tau\rangle$ ) is given by

\begin{equation*} \frac{1}{\prod_{v \in Q_0} d_v !} \operatorname{JK}\left(Z_{1-loop}\left(u, z \right),\zeta\right),\,\text{ for } z \in \mathbb{C}\setminus\mathbb{Z}. \end{equation*}

If the quiver moduli space $\mathcal{M}^{\theta-sst}_d$ is smooth of dimension D, then physics predicts that its Poincaré polynomial in singular cohomology P is determined by

\begin{align*} P(e^{i\pi z}) = e^{i \pi z D} \frac{1}{\prod_{v \in Q_0} d_v !} \operatorname{JK}\left(Z_{1-loop}\left(u, z \right),\zeta\right),\,\text{ for } z \in \mathbb{C}\setminus\mathbb{Z}. \end{align*}

More generally, if $\mathcal{M}^{\theta-sst}_d$ is singular, then the Poincaré polynomial should be replaced by a suitable refined (Donaldson–Thomas) quiver invariant [ Reference Joyce and Song20, Reference Kontsevich and Soibelman21, Reference Meinhardt and Reineke25 ].

Note that, for $\alpha, \beta \in \mathbb{R}$ , we have

\begin{align*} \frac{\sin(\pi z(y+ \alpha))}{\sin(\pi z(y+ \beta))} \xrightarrow{z \rightarrow 0} \frac{y+ \alpha}{y+ \beta}, \end{align*}

locally uniformly for $y \in \mathbb{C}^*$ . Thus, locally uniformly on $\mathfrak{a}_\mathbb{C}^{reg}$ , the one-loop factor $Z_{1-loop}(u, z)$ converges to the meromorphic function

(3·2) \begin{align} \operatorname{Z}_Q(u) \;:\!=\; (\!-\!1)^{\vert d \vert -1}\left( \prod_{r \in \mathfrak{R}_\mathbb{C}} \frac{ r(u)}{ r (u) - 1} \right) \prod_{\rho \in \mathcal{W}_\mathbb{C}} \left(\frac{\rho(u) + R_\rho - 1}{\rho (u) + R_\rho} \right)^{m_\rho} \in \text{Mer}(\mathfrak{a}_\mathbb{C}). \end{align}

Remark 3·9. Note that this normalisation for $\operatorname{Z}_Q$ differs from (1·1) by a sign $(\!-\!1)^D$ , with

\begin{align*} D = \sum_{\alpha \in Q_1} d_{t(\alpha)} d_{h(\alpha)} - \vert d \vert + 1. \end{align*}

In the rest of the paper we follow this sign convention.

Lemma 3·10. As $z \to 0$ , we have

\begin{equation*} \operatorname{JK}\left(Z_{1-loop}\left(u, z \right),\zeta\right) \to \operatorname{JK}(\operatorname{Z}_Q, \zeta). \end{equation*}

Proof. Let $x \in \mathfrak{M}_{sing}$ . We know that there is an open ball $D \subset \mathfrak{a}_\mathbb{C}$ around x such that $\mathfrak{a}_\mathbb{C}^{reg} \cap D$ is the complement in D of a finite number of hyperplanes (corresponding to the set of functionals $\mathfrak{A}_x$ ). This implies that there is a product of annuli $A \subset D$ , centred at x, such that $A \subset \mathfrak{a}_\mathbb{C}^{reg}$ , hence satisfying

(3·3) \begin{align} Z_{1-loop}(*, z)_{\vert A} \xrightarrow{z\rightarrow 0} Z_Q(*)_{\vert A} \end{align}

uniformly. Corollary 2·14 and Proposition A·4 now imply that

\begin{align*} \lim_{z \rightarrow 0} \operatorname{JK}_{\zeta}^{\mathfrak{A}_x}(Z_{1-loop}(* + x, z)) = \operatorname{JK}_{\zeta}^{\mathfrak{A}_x}(Z_Q(*+ x)), \end{align*}

for all $x \in \mathfrak{M}_{sing}$ , as required.

In particular, if $\mathcal{M}^{\theta-sst}_d$ is smooth, physics predicts the identity

\begin{align*} \chi\left(\mathcal{M}^{\theta-sst}_d( Q) \right) = \frac{1}{\prod_{v \in Q_0} d_v !} \operatorname{JK}(\operatorname{Z}_Q, \zeta). \end{align*}

More generally, if $\mathcal{M}^{\theta-sst}_d$ is singular, then the topological Euler characteristic should be replaced by a suitable generalised (Donaldson–Thomas) quiver invariant [ Reference Joyce and Song20, Reference Kontsevich and Soibelman21 ].

4. Abelianisation

In this Section we discuss abelianisation for JK residues and quiver invariants, following [ Reference Beaujard, Mondal and Pioline1, Reference Manschot, Pioline and Sen24, Reference Reineke, Stoppa and Weist26 ].

Fix a quiver Q without loops or oriented cycles, with dimension vector d, and let $i \in Q_0$ denote a fixed vertex. Abelianisation results are conveniently expressed in terms of a new, infinite “blown-up” quiver $\widehat{Q}^i$ , obtained by replacing the vertex i with vertices $i_{k, l}$ , $k, l \geq 1$ (all other vertices are unchanged). Similarly, a single arrow $i \to j$ ( $j \to i$ ) is replaced by l arrows $i_{k, l} \to j$ (respectively $j \to i_{k, l}$ ) for all $k, l \geq 1$ .

A multiplicity vector $m_* \vdash d_i$ , that is, a collection of integers $m_l$ , $l \geq 1$ , satisfying $\sum_l l m_l = d_i$ , defines a dimension vector $\widehat{d}$ for $\widehat{Q}^i$ by the rule

\begin{equation*} \widehat{d}_{i_{k, l}}(m_*) = \begin{cases} 1, \quad 1 \leq k \leq m_l\\[5pt] 0, \quad k > m_l. \end{cases} \end{equation*}

Similarly, a stability vector $\zeta$ for Q induces a stability vector $\widehat{\zeta}$ for $\widehat{Q}^i$ , defined by

\begin{equation*} \widehat{\zeta}_{i_{k, l}} = l \zeta_i,\, k, l\geq 1. \end{equation*}

Note that if $\zeta$ is normalised then we have $\sum_{k,l} \widehat{d}_{i_{k,l}} \widehat{\zeta}_{i_{k,l}} = \sum_l m_l l \zeta_i = d_i \zeta_i$ , hence $\sum_{w \in \widehat{Q}^i_0} \widehat{d}_w \widehat{\zeta}_w = \sum_{v \in Q_0} d_v \zeta_v = 0$ and so $\widehat{\zeta}$ is a normalised stability vector.

The new pair ( $\widehat{Q}^i$ , $\widehat{d}$ ) induces new spaces $\widehat{\mathfrak{a}}^i$ , $\widehat{V}^i$ and functionals $\widehat{\mathfrak{A}}^i\;:\!=\; \widehat{\mathfrak{R}}^i \cup \widehat{\mathcal{W}}^i$ . There is an isomorphism of $\mathbb{R}$ -linear spaces induced by the identity $\sum_{v \in Q_0} d_{v} = \sum_{w \in \widehat{Q}_0} \widehat{d}_{w}$ ,

\begin{align*} \bigoplus_{v \in Q_0} \mathbb{R}^{d_v} \cong \bigoplus_{w \in \widehat{ Q}_0} \mathbb{R}^{\widehat{d}_w}. \end{align*}

The choice of a reference coordinate $\overline{u}_{s}$ induces an isomorphism between the corresponding subspaces $\mathfrak{a} \cong \widehat{\mathfrak{a}}^i$ . We will identify these two linear spaces via this isomorphism. We will work in the second space, where the coordinates associated with a vertex $w \in Q_0\setminus \lbrace i \rbrace$ will be denoted by $u_{w,1}, \ldots, u_{w, d_w}$ and the coordinate associated with the blown-up node $i_{k,l}$ will be denoted by $u_{i,k,l}$ . Let us describe the two sets of functionals $\widehat{\mathcal{W}}^i, \widehat{\mathfrak{R}}^i$ on $\mathfrak{a}^i$ .

Proof. Let us denote by $\pi$ the projection $\widehat{Q}^i_1 \rightarrow Q_1$ . By Remark 3·1 we have a projection $\widehat{\mathcal{W}} \rightarrow \overline{\widehat{ Q}}_1$ whose fibre over the arrow $\alpha$ is the family of functionals

\begin{align*} \widehat{\mathcal{W}}^i_\alpha = \left\lbrace u \mapsto u_{h(\alpha),k} - u_{t(\alpha),j} \; :\; k \in \lbrace 1, \ldots, \widehat{d}_{h(\alpha)} \rbrace, j \in \lbrace 1, \ldots, \widehat{d}_{t(\alpha)} \rbrace \right \rbrace. \end{align*}

If the arrow $\alpha$ is between two vertices in $ Q_0 \setminus \lbrace i \rbrace$ then the functionals in the corresponding fibre are precisely those in the fibre $\mathcal{W}_{\pi(\alpha)}$ over $\pi(\alpha)$ for the projection $\mathcal{W}\rightarrow \overline{ Q}_1$ . If this is not the case then one (and only one) of $h(\alpha)$ , $t(\alpha)$ is a vertex of the form $i_{k,l}$ . We consider only the case $t(\alpha) = v \in Q_0 \setminus \lbrace i\rbrace$ , $h(\alpha) = i_{k,l}$ since the other case is completely analogous. The fibre above this arrow is the family of functionals

\begin{align*} \widehat{\mathcal{W}}_\alpha = \left \lbrace u \mapsto u_{i,k,l} - u_{v,j} \; :\; j \in \lbrace 1, \ldots, {d}_{t(\alpha)} \rbrace \right \rbrace \end{align*}

which is clearly contained in the fibre $\mathcal{W}_{\pi(\alpha)}$ . It is immediate to see that in this case

\begin{align*} \mathcal{W}_{\pi(\alpha)} = \bigcup_{\beta \in \pi^{-1}(\pi (\alpha))} \widehat{\mathcal{W}}_\beta. \end{align*}

Let us now consider the elements of $\widehat{\mathfrak{R}}$ . They correspond to vertices, meaning that there is a projection $\widehat{\mathfrak{R}}\rightarrow \widehat{ Q}_0$ whose fibre above w is the family of functionals

\begin{align*} \widehat{\mathfrak{R}}_w \;:\!=\; \left\lbrace u \mapsto u_{w,k} - u_{w,j} \;:\; k \neq j \in \lbrace 1, \ldots, \widehat{N}_{w} \rbrace\right \rbrace. \end{align*}

If $w \in Q_0 \setminus \lbrace i \rbrace$ , then this is precisely $\mathfrak{R}_w$ , the fibre of $\mathfrak{R} \rightarrow Q_0$ over w. On the other hand we have that $\widehat{\mathfrak{R}}_{i_{k,l}} = \emptyset$ since $\widehat{d}_{i_{k,l}} = 1$ .

This proposition shows, in particular, that the hyperplane arrangement defined in the dual space V by $\widehat{\mathfrak{A}}^i$ is contained (strictly if $d_{i}>1$ ) in the one generated by $\mathfrak{A}$ .

A physical argument (see [ Reference Beaujard, Mondal and Pioline1 , section 3.4]) suggests a remarkable abelianisation identity for JK residues:

(4·1) \begin{equation} \operatorname{JK}(\operatorname{Z}_Q(d), \zeta) = d_i!\sum_{m_* \vdash d_i} \prod_{l\geq1} \frac{1}{m_l!}\left(\frac{(\!-\!1)^{l-1}}{l^2}\right)^{m_l} \operatorname{JK}(\operatorname{Z}_{\widehat{Q}^i}(\widehat{d}(m_*)), \widehat{\zeta}). \end{equation}

The right–hand side is well defined by the previous Corollary.

The analogue of (4·1) is known in the case of quiver invariants [ Reference Manschot, Pioline and Sen24, Reference Reineke, Stoppa and Weist26 ], namely we have

(4·2) \begin{equation} \bar{\chi}_Q(d, \zeta) = \sum_{m_* \vdash d_i} \prod_{l\geq1} \frac{1}{m_l!}\left(\frac{(\!-\!1)^{l-1}}{l^2}\right)^{m_l} \bar{\chi}_{\widehat{Q}^i}(d(m_*), \widehat{\zeta}), \end{equation}

so indeed (4·1) is compatible with the conjectural identity (1·3).

Remark 4·3. However, the physical argument for (4·1) is not based on the correspondence with quiver invariants, but rather on an application of the “Cauchy–Bose identity”

\begin{equation*} \det \frac{1}{\sinh(\mu_i - \nu_j)} = \frac{\prod_{i < j} \sinh(\mu_i - \nu_j)\sinh(\nu_j - \nu_i)}{\prod_{i, j} \sinh(\mu_i - \nu_j)} \end{equation*}

to the product

\begin{equation*} \prod^{d_i}_{s \neq s^{\prime} = 1} \frac{\sin \pi\hbar(u_{i,s^{\prime}} - u_{i, s})}{\sin \pi\hbar(u_{i, s} - u_{i,s^{\prime}}-1)}, \end{equation*}

appearing in the 1-loop factor $Z_{1-loop}(u,z)$ , which specialises to the factor

\begin{equation*} \prod^{d_i}_{s \neq s^{\prime} = 1} \frac{u_{i,s^{\prime}} - u_{i, s}}{u_{i, s} - u_{i,s^{\prime}}-1} \end{equation*}

appearing in $\operatorname{Z}_Q(d)$ .

Clearly, applying (4·1) to each vertex $i \in Q_0$ makes the gauge group completely abelian: with a straightforward extension of our notation, we have a conjectural identity

(4·3) \begin{equation} \operatorname{JK}(\operatorname{Z}_Q(d), \zeta) = \prod_{i\in Q_0}d_i!\sum_{m_* \vdash d} \prod_{l\geq1} \frac{1}{m_{i,l}!}\left(\frac{(\!-\!1)^{l-1}}{l^2}\right)^{m_{i,l}} \operatorname{JK}(\operatorname{Z}_{\widehat{Q}}(\widehat{d}(m_*)), \widehat{\zeta}). \end{equation}

In this paper, we assume that this conjectural identity holds or, equivalently for our purposes, we take the right–hand side of (4·3) as the definition of the relevant residue operation.

Definition 4·4. The abelianised JK residue of the meromorphic form $\operatorname{Z}_Q(d)$ is given by

\begin{equation*} \operatorname{JK}_{ab}(\operatorname{Z}_Q(d), \zeta) = \prod_{i\in Q_0}d_i!\sum_{m_* \vdash d} \prod_{l\geq1} \frac{1}{m_{i,l}!}\left(\frac{(\!-\!1)^{l-1}}{l^2}\right)^{m_{i,l}} \operatorname{JK}(\operatorname{Z}_{\widehat{Q}}(\widehat{d}(m_*)), \widehat{\zeta}). \end{equation*}

Each meromorphic form $\operatorname{Z}_{\widehat{Q}}(\widehat{d}(m_*))$ is defined on a torus

\begin{equation*} \mathbb{T}(\widehat{d}(m_*)) = \big(\prod_{i\in \widehat{Q}_0}(\mathbb C^*)^{\widehat{d}_i(m_*)}\big)/\mathbb C^*. \end{equation*}

We spell out the details for the complete bipartite quiver $K(\ell_1, \ell_2)$ : this is given by $\ell_1$ sources, with a single arrow connecting each source to each of $\ell_2$ sinks. We endow this with a compatible stability vector, i.e. one with constant value $\zeta_1$ ( $\zeta_2$ ) on all sources (respectively, sinks). Then, $\widehat{Q}$ above may be replaced by a quiver $\mathcal{N}$ with

\begin{align*} &\mathcal{N}_0 = \{i_{(w, m)}:(w, m) \in \mathbb N^2\} \cup \{j_{(w, m)}:(w, m) \in \mathbb N^2\},\\[5pt] &\mathcal{N}_1 = \{\alpha_1, \ldots, \alpha_{w w'}\;:\; i_{(w,m)} \to j_{(w',m')}\}. \end{align*}

A dimension vector for $K(\ell_1, \ell_2)$ is the same as a pair of ordered partitions

\begin{equation*} (P_1, P_2) = \big(\sum^{\ell_1}_{i=1} p_{1i}, \sum^{\ell_2}_{j=1} p_{2j}\big). \end{equation*}

Refinements $(k^1, k^2) \vdash (P_1, P_2)$ are defined by

\begin{equation*} p_{1i} = \sum_w w k^1_{w i},\,p_{2j} = \sum_w w k^2_{w j}. \end{equation*}

The number of entries of weight w in $k^i$ is

\begin{equation*} m_w(k^i) = \sum^{\ell_i}_{j=1} k^i_{wj},\, i=1, 2. \end{equation*}

Thus, refinements $(k^1, k^2) \vdash (P_1, P_2)$ give dimension vectors for $\mathcal{N}$ :

\begin{equation*} d(k^1, k^2)(q_{(w,m)}) = \begin{cases} 1 \quad 1 \leq m \leq m_w(k^p)\\[5pt] 0 \quad m > m_w(k^p), \end{cases} \end{equation*}

with $p = 1, 2$ according to $q = i, j$ . Finally, the stability vector $\widehat{\zeta}$ for $\mathcal{N}$ is given by

\begin{equation*} \widehat{\zeta}(i_{(w, m)}) = w \zeta_1,\,\widehat{\zeta}(j_{(w, m)}) = w \zeta_2. \end{equation*}

Then, the definition of abelianised JK residues (4·3) may be written in the form

(4·4) \begin{align} \nonumber&\frac{1}{(P_1, P_2)!}\operatorname{JK}_{ab}(\operatorname{Z}_{K(\ell_1, \ell_2)}(P_1, P_2), \zeta)\\[5pt] &= \sum_{(k^1, k^2) \vdash (P_1, P_2)} \prod^2_{i=1} \prod^{l_i}_{j=1} \prod_w \frac{(\!-\!1)^{k^i_{w,j}(w-1)}}{k^i_{w,j}!w^{2k^i_{w,j}}}\operatorname{JK}(\operatorname{Z}_{\mathcal{N}}(d(k^1,k^2)), \widehat{\zeta}). \end{align}

The meromorphic form $\operatorname{Z}_{\mathcal{N}}(d(k^1,k^2))$ is defined on the torus

\begin{equation*} \mathbb{T}(d(k^1,k^2)) = \big(\prod_{q_{(w,m)} \in \mathcal{N}_0} (\mathbb C^*)^{d(k^1,k^2)(q_{(w,m)})}\big)/\mathbb C^*. \end{equation*}

It can be written conveniently by introducing the variables

\begin{align*} \{u_{(w, m)}, \textrm{ corresponding to } i_{w,m} \in \mathcal{N}_0;\, v_{(w', m')}, \textrm{ corresponding to } j_{w',m'} \in \mathcal{N}_0\}. \end{align*}

Then, we have

\begin{align*} &\operatorname{Z}_{\mathcal{N}}(d(k^1,k^2))\\[5pt] &= (\!-\!1)^D\prod_w \prod^{m_w(k^1)}_{m=1}\prod_{w'} \prod^{m_{w'}(k^2)}_{m'=1}\left(\frac{v_{(w', m')}-u_{(w, m)}+1-R^{(w,m)}_{(w',m')}/2}{u_{(w, m)}-v_{(w', m')}+R^{(w,m)}_{(w',m')}/2}\right)^{w w'}. \end{align*}

As we already observed, the analogue of (4·4) for quiver invariants is known:

(4·5) \begin{align} \bar{\chi}_{K(\ell_1, \ell_2)}((P_1, P_2), \zeta) = \sum_{(k^1, k^2) \vdash (P_1, P_2)} \prod^2_{i=1} \prod^{\ell_i}_{j=1} \prod_w \frac{(\!-\!1)^{k^i_{w,j}(w-1)}}{k^i_{w,j}!w^{2k^i_{w,j}}}\bar{\chi}_{\mathcal{N}}(d(k^1,k^2), \widehat{\zeta}). \end{align}

5. Stable spanning trees

It is easy to see that a tree does not contain unoriented loops. Moreover, it contains either a source with a single arrow, or a sink with a single arrow. Note that if Q admits a spanning tree then it is obviously connected.

Let

\begin{equation*} \mathbb{I} = \sum_{v \in Q_0} v \in \mathbb Z_{\geq 0} Q_0 \end{equation*}

denote the full abelian dimension vector.

Note that the projection map $ Q_1 \rightarrow \bar{ Q}_1$ induces a surjective morphism

\begin{align*} N^\theta( Q) \Longrightarrow N^\theta(\bar{ Q}). \end{align*}

The fibre over $T \in N^\theta(\bar{ Q})$ has cardinality $\prod_{\alpha \in T} m_\alpha$ .

A representation of Q with dimension vector $\mathbb{I}$ is the same as an element $R \in \mathbb C Q_1$ , by associating to an arrow $\alpha$ the linear morphism $\mathbb C \xrightarrow{\times R_\alpha} \mathbb C$ . With a slight abuse of notation, we write $\mathbb{I}$ for the abelian representation that associates the identity morphism with every arrow. The following result is standard.

Recall that the space of normalised stability vectors is given by

\begin{align*} \mathcal{H} = V\big(\sum_{v \in Q_0} d_v \theta_v \big) \subset \mathbb R Q_0. \end{align*}

Spanning trees of Q describe certain natural bases of $\mathcal{H}$ . Let $e_v$ be the canonical basis of $\mathbb R Q_0 $ . For every arrow $\alpha \in Q_1$ , let $e_\alpha \;:\!=\; e_{h(\alpha)} - e_{t(\alpha)} \in \mathcal{H}$ .

$(\!\Rightarrow\!)$ Consider the subquiver $T \subseteq Q$ obtained by taking as nodes the heads and tails of the elements of S and by considering S as the set of arrows. Assume by contradiction that there is a vertex $v \in Q_0 \setminus T_0$ . This implies that the component of $e_\alpha$ along $e_v$ vanishes for all $\alpha \in S$ . Then the vector $e_v - e_w \in \mathcal{H}$ is not contained in $\text{Span}_\mathbb{R}\{e_\alpha \;:\; \alpha \in S\}$ , which is a contradiction. Hence $T_0 = Q_0$ . By assumption, $|S| = \text{dim}_{\mathbb{R}}(\mathcal{H}) = | Q_0 | -1$ . Assume by contradiction that T is disconnected and let C be a connected component not containing the vertex w. We have the decomposition

(5·1) \begin{align} \mathbb R Q_0 = \text{Span}_{\mathbb{R}}\{ e_v \;:\; v \in C_0 \} \oplus \text{Span}_{\mathbb{R}}\{e_v \;:\; v \notin C_0\}. \end{align}

Fix $v \in C_0$ and consider the vector $e_v - e_w \in \mathcal{H}$ . We claim that this is not contained in $\text{Span}_\mathbb{R}\{e_\alpha \, : \, \alpha \in S\}$ . To see this we assume that

\begin{align*} e_v - e_w = \sum_{\alpha \in S} k_\alpha e_\alpha = \sum_{\beta \in C_1} k_\beta e_\beta + \sum_{\gamma \in S \setminus C_1} k_\gamma e_\gamma, \end{align*}

which can be rewritten as

\begin{align*} \big(\sum_{\beta \in C_1} k_\beta e_\beta - e_v \big) + \big(\sum_{\gamma \in S \setminus C_1} k_\gamma e_\gamma - e_w \big) = 0. \end{align*}

By (5·1), we know that both summands are zero. But this is impossible since, for example, we have $\sum_{\beta \in C_1} k_\beta e_\beta \in \mathcal{H}$ and $e_v \notin \mathcal{H}$ .

$(\Leftarrow)$ Since T is a spanning tree for Q, we have $| S | = | Q_0 | - 1$ , so we just have to prove that S spans $\mathcal{H}$ . Fix another vertex $v \in Q_0$ and consider $e_v - e_w \in \mathcal{H}$ . The spanning tree T certainly contains a non-oriented path $\alpha_1, \ldots, \alpha_n$ from w to v. So there are $\beta_1, \ldots, \beta_n \in \lbrace -1, 1 \rbrace$ such that ${\beta_1} \alpha_1, \ldots, {\beta_n} \alpha_n$ is an oriented path from w to v. Then $e_v - e_w = \sum_{k=1}^n {\beta_k} e_{\alpha_k} \in \text{Span}_\mathbb{R}\{e_\alpha \, : \, \alpha \in S\}$ . The claim follows since obviously $\mathcal{H} = \text{Span}_\mathbb{R}\{e_v - e_w \, : \, v \in Q_0 \}$ .

The following definition is well-posed by Proposition 5·4.

The main fact we need for our purposes is the following.

Proposition 5·6. Let T be a tree and consider a stability vector $\theta \in \mathcal{H}$ . Let $\lbrace e_\alpha\rbrace_{\alpha \in T_1}$ be the basis of $\mathcal{H}$ given by Proposition 5·4 and consider the corresponding components of $\theta$ :

\begin{align*} \theta = \sum_{\alpha \in T_1} c_\alpha e_\alpha. \end{align*}

Then T is $\theta$ -stable if and only if $c_\alpha < 0$ for every $\alpha \in T_1$ , i.e. if and only if T is regular. In particular, we have

\begin{align*} N^\theta( Q) = \left\lbrace \theta\text{-regular spanning trees of } Q \right\rbrace. \end{align*}

Proof. For the sake of the proof, we say that a subquiver $\tilde{ Q} \subseteq Q$ of a given quiver is good if, for all $\alpha \in Q_1$ , we have that $t(\alpha) \in \tilde{ Q}$ implies $h(\alpha) \in \tilde{ Q}_0$ and $\alpha \in \tilde{ Q}_1$ . Let Q be a quiver and consider the map

\begin{align*} \left\lbrace \substack{\text{subrepresentations of the}\\ \text{abelian representation } \mathbb{I}} \right\rbrace \longrightarrow \left\lbrace \text{good subquivers of } Q \right \rbrace \end{align*}

sending a subrepresentation R to the full subquiver whose nodes are those corresponding to 1-dimensional linear spaces in R. One checks that this is a well defined bijection.

Now we claim that if $\tilde{T}$ is a good subquiver of T, then

\begin{align*} \sum_{v \in \tilde{T}_0} \theta_v = \sum_{\substack{\alpha \in Q_1\\ h(\alpha) \in \tilde{T}_0,\, t(\alpha) \notin \tilde{T}_0}} c_\alpha. \end{align*}

In order to see this note that

\begin{align*} \theta_v = \sum_{\substack{\alpha \in T_1\\ h(\alpha) = v}}c_\alpha - \sum_{\substack{\beta \in T_1\\ t(\beta) = v}}c_\beta, \end{align*}

therefore

\begin{align*} \sum_{v \in \tilde{T}_0} \theta_v = \sum_{\substack{\alpha \in T_1\\ h(\alpha) \in \tilde{T}_0}}c_\alpha - \sum_{\substack{\beta \in T_1\\ t(\beta) \in \tilde{T}_0}}c_\beta. \end{align*}

Since $\tilde{T}$ is a good subquiver we see that

\begin{align*} t(\beta) \in \tilde{T}_0 \iff h(\beta), t(\beta) \in \tilde{T}_0, \end{align*}

hence the equation above becomes

\begin{align*} \sum_{v \in \tilde{T}_0} \theta_v = \sum_{\substack{\alpha \in T_1\\ h(\alpha) \in \tilde{T}_0}}c_\alpha - \sum_{\substack{\beta \in T_1\\ t(\beta) \in \tilde{T}_0,\, h(\beta) \in \tilde{T}_0}}c_\beta = \sum_{\substack{\alpha \in Q_1\\ h(\alpha) \in \tilde{T}_0,\, t(\alpha) \notin \tilde{T}_0}} c_\alpha. \end{align*}

( $\Leftarrow$ ) is now clear: if S is a subrepresentation we can consider the associated good subquiver $\tilde{T}$ and notice that

\begin{align*} \theta(S) = \sum_{v \in \tilde{T}_0} \theta_v = \sum_{\substack{\alpha \in Q_1\\ h(\alpha) \in \tilde{T}_0,\, t(\alpha) \notin \tilde{T}_0}} c_\alpha < 0. \end{align*}

( $\Rightarrow$ ) Let $\alpha \in T_1$ and consider the full subquiver $\tilde{T}$ given by the connected component containing $h(\alpha)$ after removing $\alpha$ . If we apply the formula above to this good subquiver we obtain

\begin{align*} c_\alpha = \sum_{v \in \tilde{T}_0} \theta_v < 0. \end{align*}

6. JK residues and abelian representations

In this Section we show how the JK residue formula simplifies considerably for the abelian dimension vector $\mathbb{I}$ , at least for generic R-charges.

Fix $v_0 \in Q_0$ . Then we have $\mathfrak{a} \cong \mathbb{R}\langle Q_0 \setminus \lbrace v_0 \rbrace\rangle$ . In this case, by definition, there are no roots: $\mathfrak{R} = \emptyset$ . The following result characterises the set of weights.

Proposition 6·1. Consider the map

\begin{align*} ar\;:\; \mathcal{W} \longrightarrow \bar{ Q}_1 \end{align*}

described in Remark 3·1. In the case of the abelian dimension vector $\mathbb{I}$ , ar is a bijection. Moreover, ar induces a bijection between the set of bases $\mathfrak{B}(\mathcal{W})$ and the set of spanning trees of $\bar{ Q}$ . Finally, if $B \in \mathfrak{B}(\mathcal{W})$ and T is the corresponding spanning tree of Q, write

\begin{align*} \tilde{\theta} = \sum_{\rho \in B} c_\rho \rho; \, \theta = \sum_{\alpha \in T_1} c_\alpha e_\alpha. \end{align*}

Then $c_\rho = c_{\text{ar}(\rho)}$ for every $\rho \in B$ .

Proof. By Remark 3·1 ar is surjective and the cardinality of the fibre at an arrow $\alpha \in \bar{ Q}_1$ is $d_{t(\alpha)} d_{h(\alpha)} = 1$ since $d = \mathbb{I}$ . Thus ar is a bijection. Consider the diagram

The vertical arrow on the right is the diagonal embedding $\sim$ defined in (3·1). In the abelian case this is clearly an isomorphism of linear spaces which maps $\theta$ to $\tilde{\theta}$ , by definition. The upper horizontal map sends the arrow $\alpha$ to the the vector $e_\alpha \;:\!=\; e_{h(\alpha)} - e_{t(\alpha)} \in \mathbb{R}Q_0$ , while the lower map is simply the inclusion. This diagram is clearly commutative, and this shows that ar maps a basis of V extracted from $\mathcal{W}$ to a basis of $\mathcal{H}$ induced from $\overline{Q}_1$ . The latter set of bases is precisely the set of spanning trees of $\overline{Q}$ by Proposition 5·4. The fact that the components of $\theta$ and $\tilde{\theta}$ with respect to corresponding bases are the same follows from the commutativity of the diagram.

We will often identify $\mathcal{W}$ and $\bar{ Q}_1$ via ar implicitly.

Since, in the abelian case, we have $\mathfrak{R} = \emptyset$ , the residue $\operatorname{JK}(\operatorname{Z}_{Q}(\mathbb{I}), \zeta)$ is well defined for stability vectors $\theta$ whose extensions $\tilde{\theta}$ are $\mathcal{W}$ -regular. In general, there is a simpler condition which implies regularity.

Proof. Fix a basis B of V extracted from $\mathfrak{A}$ . We need to show that the components of $\tilde{\theta}$ with respect to B are all nonzero. Construct a new quiver T having $d_v$ vertices $v_1, \ldots, v_{d_v} \in T_0$ for every vertex $v \in Q_0$ . Given $v, w \in Q_0$ and two indices $i \in \lbrace 1, \ldots, d_v \rbrace$ $j \in \lbrace 1, \ldots, d_w \rbrace$ , there is a single arrow from $v_i$ to $w_j$ in T if and only if the functional $e_{w,j} - e_{v,i}$ belongs to $\mathcal{B}$ . We equip this new quiver with the abelian dimension vector $\mathbb{I}$ . We see that the corresponding linear space is canonically isomorphic to $\mathfrak{a}_\mathbb{C}$ and the corresponding set of functionals is precisely B. Since T has no multiple arrows and it induces a basis of V, it is a tree by Proposition 6·1. Note that the vector $\theta^\prime \in \mathbb{Z}T_0$ defined by $\theta^\prime_{v_i} \;:\!=\; \theta_v$ for every $v \in Q_0$ , $i \in \lbrace 1, \ldots, d_v \rbrace$ is a normalised stability for ( $T, \mathbb{I}$ ):

\begin{align*} \sum_{w \in T_0} \theta^\prime_{w} = \sum_{v \in Q_0} d_v \theta_v = 0. \end{align*}

Moreover we have $\tilde{\theta^\prime} = \tilde{\theta}$ . By construction, for every functional $f \in B$ there is an arrow $\alpha_f$ in T and $e_{f} = e_{\alpha_f}$ . Let $\tilde{T}$ be the subquiver of T obtained by taking the connected component of $T \setminus \lbrace \alpha_f \rbrace$ containing $h(\alpha_f)$ . By the proof of Proposition 8·6, the component of $\tilde{\theta}$ with respect to the element $f \in B$ is given by

\begin{align*} c_f = \sum_{w \in \tilde{T}_0} \theta^\prime_w. \end{align*}

On the other hand, we can partition the vertices of $\tilde{T}$ into subsets coming from the same vertex of Q. Then the sum above can be rewritten as $\sum_{v \in Q_0} n_v \theta_v$ with $n_v \leq d_v$ (possibly zero) for every vertex v of Q. Then by definition of coprimality we have that the sum is nonzero.

The case of trees is especially important for our purposes.

The following result is straightforward.

Lemma 6·4. There is a dense open cone $\mathfrak{P} \subset \mathbb R\mathcal{W}$ such that, for $R \in \mathfrak{P}$ , all the isolated intersections of the hyperplanes of the form

\begin{align*} V(\rho + R_\rho),\, \rho \in \mathcal{W} \end{align*}

are non-degenerate, that is, no $|Q_0|$ hyperplanes contain such intersection.

Now, for such generic R-charges, $R \in \mathfrak{P}$ , we will apply the general result Corollary 2·14 in order to compute the (global) Jeffrey–Kirwan residue.

Let $\phi_x \;:\; \mathcal{M}_{\mathfrak{a}_\mathbb{C}, x} \xrightarrow{\sim} \mathcal{M}_{\mathbb{C}^{|Q_0|-1}, 0}$ be the isomorphism induced by the choice of an ordering of the basis $\mathfrak{A}_x$ , composed with a translation.

Proposition 6·5. Suppose $R \in \mathfrak{P}$ and let $f \in \operatorname{Mer}(\mathfrak{a}_\mathbb{C})$ be such that, for all $x \in \mathfrak{M}_{sing}$ , there is an open ball D around x with

\begin{align*} f \in \operatorname{Hol}\big(D \setminus \bigcup_{g \in \mathfrak{A}_x} V(g(\ast - x)) \big). \end{align*}

Then, we have

\begin{align*} \operatorname{JK}(f, \zeta) = \sum_{x \in \mathfrak{M}_{sing}} \tilde{\epsilon}(x) \operatorname{IR}_0 ( \phi_x(f(\ast ))), \end{align*}

where

\begin{align*} \tilde{\epsilon}(x) \;:\!=\; \begin{cases} 1 & \text{if the components of }\tilde{\theta} \text{ w.r.t. } \mathfrak{A}_x \text{ are all negative},\\[5pt] 0 & \text{otherwise}. \end{cases} \end{align*}

Using Proposition 6·1, this computation can be rephrased in terms of stable spanning trees. Let T be a spanning tree of $\bar{ Q}$ . We denote by $x_T \in \mathfrak{M}_{sing}$ the point corresponding to the element of $\mathfrak{B}(\mathcal{W})$ associated with T via Proposition 6·1.

Corollary 6·7. Let $R \in \mathfrak{P}$ and let $f \in \operatorname{Mer}(\mathfrak{a}_\mathbb{C})$ be such that, for every $x \in \mathfrak{M}_{sing}$ , there is an open ball D around x such that

\begin{align*} f \in \operatorname{Hol}\big(D \setminus \bigcup_{g \in \mathfrak{A}_x} V(g(\ast - x)) \big). \end{align*}

Then,

\begin{align*} \operatorname{JK}(f, \zeta) = \sum_{T \in N^\theta(\bar{ Q})} \operatorname{IR}_0 ( \phi_{x_T}(f(\!\ast\!))). \end{align*}

7. Proof of Theorem 1·2

Let Q be a quiver without loops or oriented cycles. Note that, in the abelian case, the 1-loop factor is given by

\begin{align*} &Z_{1-loop}(u,z) \;:\!=\;\left(- \frac{\pi z}{\sin(\pi z)} \right)^{|Q_0|-1} \left( \prod_{\rho \in \mathcal{W}_\mathbb{C}} \frac{\sin(\pi z( \rho(u) + R_\rho - 1))}{\sin(\pi z(\rho (u) + R_\rho))} \right)^{m_\rho}. \end{align*}

Therefore the rational limit $\operatorname{Z}_Q(\mathbb{I})$ of the 1-loop factor is given by

(7·1) \begin{align} \operatorname{Z}_Q(\mathbb{I}) = \operatorname{Z}_{Q, R}(\mathbb{I}) = (\!-\!1)^{|Q_0|-1} \prod_{\rho \in \mathcal{W}_\mathbb{C}} \left(\frac{\rho(u) + R_\rho - 1}{\rho (u) + R_\rho} \right)^{m_\rho}. \end{align}

Fix $\bar{R} \in \mathfrak{P}$ , and a regular stability vector $\zeta$ . Then, according to Corollary 6·7, we have, for all $\lambda \in \mathbb R_{>0}$ ,

(7·2) \begin{equation} \operatorname{JK}(\operatorname{Z}_{Q, \lambda\bar{R}}(\mathbb{I}), \zeta) = \sum_{T \in N^\theta(\bar{Q})} \operatorname{IR}_0 ( \phi_{x_T}(\operatorname{Z}_{Q, \lambda\bar{R}}(\mathbb{I})(u))). \end{equation}

Let $v_1, \ldots, v_{|Q_0|-1}$ denote the linear coordinates of $\mathbb{C}^{|Q_0|-1}$ given by its canonical basis. Then, we have an equality of the form

\begin{align*} \phi_{T}(\operatorname{Z}_{Q, \lambda\bar{R}}) = \prod_{i = 1}^{|Q_0|-1} \left(1 - \frac{1}{v_i}\right)^{m_i} \prod_{j = 1}^h \left(1 - \frac{1}{L_j(v_1, \ldots, v_{|Q_0|-1}) + \lambda\gamma_j(\bar{R})}\right), \end{align*}

for certain $L_j \in (\mathbb{C}^{|Q_0|-1})^\ast$ and $\gamma_j \in (\mathbb R\mathcal{W})^\ast$ , $\gamma_j \neq 0$ . So, if we choose $\lambda$ sufficiently large, the divisor of poles of $\phi_{T}(\operatorname{Z}_{Q, \lambda\bar{R}})$ intersects the open ball of radius 1 in a subset contained in the union of the coordinate hyperplanes. And, on the product of annuli $\left(A_0\left(\frac{1}{2}, \, 1\right)\right)^{|Q_0|-1}$ , we have the uniform convergence

\begin{align*} \phi_{T}(\operatorname{Z}_{Q, \lambda\bar{R}}) \xrightarrow{\lambda \rightarrow +\infty} \prod_{i = 1}^{|Q_0|-1} \left(1 - \frac{1}{v_i}\right)^{m_i}. \end{align*}

Then, by Proposition A·4, we have

(7·3) \begin{equation} \lim_{\lambda \to +\infty}\operatorname{IR}_0 ( \phi_{x_T}(\operatorname{Z}_{Q, \lambda\bar{R}}(\mathbb{I})(u))) = \operatorname{IR}_0\left(\prod_{i = 1}^{|Q_0|-1} \left(1 - \frac{1}{v_i}\right)^{m_i}\right). \end{equation}

Proof of Theorem 1·2. By Definition 4·4 and (7·2), we have

\begin{align*} &\frac{1}{d!}\operatorname{JK}_{ab}(\operatorname{Z}_Q(d), \zeta) = \sum_{m_* \vdash d} \prod_{l\geq1} \frac{1}{m_{i,l}!}\left(\frac{(\!-\!1)^{l-1}}{l^2}\right)^{m_{i,l}} \operatorname{JK}(\operatorname{Z}_{\widehat{Q}}(\widehat{d}(m_*)), \widehat{\zeta})\\[5pt] &=\sum_{m_* \vdash d} \prod_{l\geq1} \frac{1}{m_{i,l}!}\left(\frac{(\!-\!1)^{l-1}}{l^2}\right)^{m_{i,l}} \sum_{T \in N^\theta(\bar{Q}')} \operatorname{IR}_0 ( \phi_{x_T}(\operatorname{Z}_{Q', \lambda\bar{R}}(\mathbb{I})(u))), \end{align*}

where $Q' = Q'(m_*)$ denotes the subquiver of $\widehat{Q}$ with support $\widehat{d}(m_*)$ . Passing to the limit, using (7·3), we find

\begin{align*} \frac{1}{d!}\operatorname{JK}^{\infty}_{ab}(\operatorname{Z}_Q(d), \zeta) &= \sum_{m_* \vdash d} \prod_{l\geq1} \frac{1}{m_{i,l}!}\left(\frac{(\!-\!1)^{l-1}}{l^2}\right)^{m_{i,l}} \sum_{T \in N^\theta(\bar{Q}')}\operatorname{IR}_0\left(\prod_{a \in T_1} \left(1 - \frac{1}{v_{a}}\right)^{m_a}\right)\\[5pt] &= \sum_{m_* \vdash d} \prod_{l\geq1} \frac{1}{m_{i,l}!}\left(\frac{(\!-\!1)^{l-1}}{l^2}\right)^{m_{i,l}} |N^\theta(\bar{Q}'(m_*))| \prod_{a \in Q'_1} m_a. \end{align*}

According to a result of Weist, proved in [ Reference Weist31 ] for abelian quivers using torus localisation, we have

\begin{equation*} |N^\theta(\bar{Q}'(m_*))| \prod_{a \in Q'_1} m_a = \chi(\mathcal{M}_\mathbb{I}^{\theta-st}(Q'(m_*))). \end{equation*}

Thus, we find

\begin{align*} \frac{1}{d!}\operatorname{JK}^{\infty}_{ab}(\operatorname{Z}_Q(d), \zeta) &= \sum_{m_* \vdash d} \prod_{l\geq1} \frac{1}{m_{i,l}!}\left(\frac{(\!-\!1)^{l-1}}{l^2}\right)^{m_{i,l}} \chi(\mathcal{M}_{\widehat{d}(m_*)}^{\theta-st}(\widehat{Q})). \end{align*}

By Proposition 6·3, we have

\begin{equation*} \chi(\mathcal{M}_{\widehat{d}(m_*)}^{\theta-st}(\widehat{Q})) = \chi(\mathcal{M}_{\widehat{d}(m_*)}^{\theta-sst}(\widehat{Q})) = (\!-\!1)^D \bar{\chi}_{\widehat{Q}}(\widehat{d}(m_*), \widehat{\zeta}). \end{equation*}

So,

\begin{align*} \frac{1}{d!}\operatorname{JK}^{\infty}_{ab}(\operatorname{Z}_Q(d), \zeta) &= \sum_{m_* \vdash d} \prod_{l\geq1} \frac{1}{m_{i,l}!}\left(\frac{(\!-\!1)^{l-1}}{l^2}\right)^{m_{i,l}} \bar{\chi}_{\widehat{Q}}(\widehat{d}(m_*), \widehat{\zeta})\\[5pt] &= (\!-\!1)^D \bar{\chi}_Q(d, \zeta), \end{align*}

by abelianisation for quiver invariants, (4·2). Due to our current sign convention for JK residues (see Remark 3·9), this is the claim of Theorem 1·2.

8. Recollections on the GHK family

Here we briefly summarise some key aspects of [ Reference Gross, Hacking and Keel15 , sections 1-3]. Let (Y, D) denote a Looijenga pair such that the intersection matrix $(D_i \cdot D_j)$ is not negative semidefinite. Then, the cone of numerically effective cycles $\operatorname{NE}(Y)_{\mathbb R_{\geq 0}}$ is rational polyhedral, so the (toric) monoid $P = \operatorname{NE}(Y)$ is finitely generated. The GHK mirror to $U = Y\setminus D$ , as an algebraic holomorphic symplectic surface, exists globally as a family of affine surfaces

\begin{equation*} \mathcal{X} \to S = \operatorname{Spec} \mathbb C[P] = \operatorname{Spec}\mathbb C[\operatorname{NE}(Y)]. \end{equation*}

Initially however $\mathcal{X}$ is constructed as a formal family around the torus fixed point $0 \in S$ , corresponding to the maximal monoid ideal $\mathfrak{m} = P \setminus \{0\}$ : that is, around the large complex structure limit

\begin{equation*} \mathcal{X}_0 \cong \mathbb{V}_n \;:\!=\; \mathbb{A}^2_{x_1x_2} \cup \cdots \cup \mathbb{A}^2_{x_n x_1} \subset \mathbb{A}^n. \end{equation*}

It is enough to consider the case when (Y, D) admits a toric model

\begin{equation*} p\!:(Y, D) \Longrightarrow (\bar{Y}, \bar{D}), \end{equation*}

blowing up distinct points $x_{ij}$ on $\bar{D}_i$ for $j = 0, \ldots, \ell_i$ , with exceptional divisors $E_{ij}$ . Here, the base $(\bar{Y}, \bar{D})$ is a toric surface, endowed with a fixed toric structure, with toric anticanonical divisor $\bar{D}$ . By convention, we write $\ell_i = -1$ if no points of $\bar{D}_i$ are blown up.

In the following we write M for the character lattice of $\bar{Y}$ with a fixed identification $M \cong \mathbb Z^2$ .

Fixing an ample divisor H on $\bar{Y}$ , the orthogonal complement $(p^* H)^{\perp}$ with respect to the intersection form is a face of P, generated by the classes $[E_{ij}]$ , and $G = P \setminus (p^* H)^{\perp}$ is a prime monoid ideal contained in $\mathfrak{m}$ . In order to construct the GHK family to all orders around $0 \in S$ , it is enough to construct it to all orders around the locus $\operatorname{Spec} \mathbb C[P]/G \subset S$ : indeed the restriction of the family $\mathcal{X} \to S$ to $\operatorname{Spec} \mathbb C[P]/G$ is the trivial family

\begin{equation*} \mathcal{X}_G \cong \mathbb{V}_n \times \operatorname{Spec} \mathbb C[P]/G. \end{equation*}

In fact it is enough to construct $\mathcal{X} \to S$ to all orders around the open torus orbit

\begin{equation*} T \subset \operatorname{Spec} \mathbb C[P]/G \subset S, \end{equation*}

known as the Gross–Siebert locus, so we replace P with its localisation along $P\setminus G$ and $\mathfrak{m}$ with the maximal monoid ideal in this localisation.

Recall that the family $\mathcal{X} \to S$ is constructed from the data of an integral affine surface B with fan $\Sigma$ , the tropicalisation of Y, and of the canonical scattering diagram $\mathfrak{D}^{\rm can}$ on B, defined in terms of relative Gromov–Witten theory on (toric blowups of) Y. Nearby the Gross–Siebert locus T, however, the family admits a simpler description. The toric model $(Y, D) \to (\bar{Y}, \bar{D})$ induces a canonical piecewise linear map $\nu\;:\; B \to \bar{B} = \mathbb R^2$ , identifying $\Sigma$ with the toric fan $\bar{\Sigma}$ , and a scattering diagram

\begin{equation*} \bar{\mathfrak{D}} = \nu(\mathfrak{D}^{\rm can}) \end{equation*}

on the trivial integral affine surface $\bar{B} = \mathbb R^2$ . For any ideal I which gives an infinitesimal thickening of T, i.e., such that $\sqrt{I} = \mathfrak{m}$ , we can construct the GHK family $\bar{\mathcal{X}}_{I, \nu(\mathfrak{D}^{\rm can})}$ with data $\bar{B}$ , $\nu(\mathfrak{D}^{\rm can})$ , and there is an isomorphism of dense open subsets

\begin{equation*} p\;:\; \mathcal{X}^o_{I, \mathfrak{D}^{\rm can}} \to \bar{\mathcal{X}}^o_{I, \nu(\mathfrak{D}^{\rm can})} \end{equation*}

over $\operatorname{Spec} \mathbb C[P]/I$ (morally, p is defined away from the singular fibres of the SYZ fibration). Because of consistency on both sides, that is, by the existence of the canonical regular functions $\vartheta_q(t) = \operatorname{Lift}_{t} (q)$ on $\mathcal{X}_{I, \mathfrak{D}^{\rm can}}$ and $\bar{\vartheta}_q(t) = \overline{\operatorname{Lift}}_{t} (q)$ on $\bar{\mathcal{X}}_{I,\nu(\mathfrak{D}^{\rm can})}$ , which are defined using broken lines as a key part of the GHK construction, this is enough to determine the family $\mathcal{X}_{I, \mathfrak{D}^{\rm can}} \to S$ from the data of the scattering diagram $\bar{\mathfrak{D}}$ . Finally, the latter can be computed as the consistent completion of the initial scattering diagram

\begin{align*} \bar{\mathfrak{D}}_0 &= \{\bar{\rho}_i, \prod^{\ell_i}_{j=0}(1+ b^{-1}_{ij} \bar{X}_{i}),\,i=1,\ldots, n\}\\[5pt] &= \{\mathbb R_{\geq 0} m_i, \prod^{\ell_i}_{j=0}(1 + z^{(m_i, \bar{\varphi}(m_i) - [E_{ij}])})\} \end{align*}

over the Mumford monoid

\begin{equation*} P_{\bar{\varphi}} = \{(m, \bar{\varphi}(m) + p),\,m\in M,\, p \in P\} \subset M \times P; \end{equation*}

that is, we have

\begin{equation*} \bar{\mathfrak{D}} = \nu(\mathfrak{D}^{\rm can}) = \operatorname{Scatter}(\bar{\mathfrak{D}}_0), \end{equation*}

as scattering diagrams for $P_{\bar{\varphi}}$ . Here, $m_1, \ldots, m_n$ denote the rays of the toric fan $\bar{\Sigma}$ , and $\bar{\varphi}\;:\; \bar{B} \to P \otimes \mathbb R$ is the canonical convex piecewise linear function determined by the bending parameters $\bar{\varphi}(m_i) = p^*[ \bar{D}_i]$ .

Another important aspect of the GHK family is given by the periods conjecture, [ Reference Gross, Hacking and Keel15 , conjecture 0·20]. In [ Reference Gross, Hacking and Keel15 , section 0·5·4], this is claimed in our case when $(D_i \cdot D_j)$ is not negative semidefinite, and a proof also recently appeared in [ Reference Lai and Zhou22 ]. Thus, in this case, it is known that the local system over the locus of smooth fibres $S^o \subset S$ given by

\begin{equation*} S^o \ni s \longmapsto H_2(\mathcal{X}_s, \mathbb Z)/\langle \gamma_s \rangle, \end{equation*}

where $\gamma_s$ denotes the (monodromy invariant) class of a suitable real torus (the class of the fibre of a Lagrangian fibration), is trivial. Moreover, we can identify each fibre canonically with the lattice

\begin{equation*} \mathcal{Q} = H_2(U, \mathbb Z)/\langle \gamma \rangle = \langle D_1, \ldots, D_n\rangle^{\perp} \subset H_2(Y, \mathbb Z) \end{equation*}

for a suitable 2-torus class $\gamma$ . Under this identification, writing $\tilde{\beta}$ for any lift to $H_2(\mathcal{X}_s, \mathbb Z)$ of a class $\beta \in \mathcal{Q} \subset H_2(Y, \mathbb Z)$ , we have

\begin{equation*} z^{\beta} = \exp\big(2\pi {\rm i} \int_{\tilde{\beta}} \Omega\big), \end{equation*}

as functions on the structure torus $T_Y = \operatorname{Spec} \mathbb C[H_2(Y, \mathbb Z)] \subset S$ , provided the holomorphic symplectic form $\Omega$ on the fibres of $\mathcal{X} \to S^o$ is normalised by the condition $\int_{\gamma_s} \Omega = 1$ . Thus, the monomials $z^{\beta}$ , $\beta \in \mathcal{Q} \subset H_2(Y, \mathbb Z)$ are canonical coordinates on the complex moduli space of the surfaces $\mathcal{X}_s$ , nearby the large complex structure limit $s \to 0 \in S$ .

Note that mirror to this, by construction, we have

\begin{equation*} z^{\beta} = \exp\big(2\pi {\rm i} \int_{ \beta } [B + {\rm i}\omega]\big), \end{equation*}

as $[B + {\rm i}\omega]$ varies in an open subset of the complexified Kähler cone on Y (for which the mirror is smooth).

Let us return to the case of a toric model $(Y, D)\to (\bar{Y}, \bar{D})$ , with initial scattering diagram $\bar{\mathfrak{D}}_0 \subset \bar{B} = \mathbb R^2$ . In order to make direct contact with complex or Kähler parameters, we write this in the form

\begin{equation*} \bar{\mathfrak{D}}_0 = \{\mathbb R_{\geq 0} m_i, \prod^{\ell_i}_{j=0}(1 + z^{(0, - [E_{i0}])} z^{(m_i, \bar{\varphi}(m_i) + [E_{i0} - E_{ij}])})\}, \end{equation*}

where $E_{i0}$ is some fixed choice of a reference exceptional divisor over $\bar{D}_i$ . (Note that $z^{(0, - [E_{i0}])}$ is an invertible element in $P_{\bar{\varphi}}$ ). We have

\begin{equation*} [E_{i0} - E_{ij}] \cdot D_{k} = 0,\, k = 1\ldots, n, \end{equation*}

so

\begin{equation*} \beta_{ij} = [E_{i0} - E_{ij}] \in \mathcal{Q}, \end{equation*}

and the monomial $z^{\beta_{ij}}$ , as a function on $T_Y$ , is given by a period or a complexified Kähler parameter,

\begin{align*} z^{\beta_{ij}} &= \exp\big(2\pi {\rm i} \int_{\tilde{\beta}_{ij}} \Omega\big)= \exp\big(2\pi {\rm i} \int_{\beta_{ij}} [B + {\rm i}\omega]\big). \end{align*}

9. The GHK family and generalised monodromy

Recall the monoid P is given by the localisation of $\operatorname{NE}(Y)$ along the face $(p^*H)^{\perp}$ . We consider the Mumford monoid given by

\begin{equation*} P_{\bar{\varphi}} = \{(m, \bar{\varphi}(m) + p),\,m\in M,\, p \in P\} \subset M \times P, \end{equation*}

with maximal monoid ideal $\mathfrak{m} = P_{\bar{\varphi}} \setminus P^{\times}_{\bar{\varphi}}$ . We often identify M with $M \times \{0\} \subset P'_{\bar{\varphi}}$ , implicitly. We define a skew-symmetric, integral bilinear form on the lattice $P^{\rm{gp}}_{\bar{\varphi}}$ , given by

\begin{equation*} \langle (m, \bar{\varphi}(m) + p), (m', \bar{\varphi}(m') + p')\rangle = \langle m, m'\rangle, \end{equation*}

where $\langle m, m'\rangle$ denotes the bilinear form on $M \cong \mathbb Z^2$ (with our fixed identification) induced by the determinant.

The automorphisms $\theta_{\mathfrak{d}}$ attached to the rays $(\mathfrak{d}, f_{\mathfrak{d}})$ of the scattering diagram $\bar{\mathfrak{D}}$ are $\mathbb C$ -algebra automorphisms of $R_k \;:\!=\; \mathbb C[P_{\bar{\varphi}}]/\mathfrak{m}^k$ , for $k \geq 1$ , defined by

\begin{equation*} \theta_{\mathfrak{d}}(z^{(m, \bar{\varphi}(m) + p)}) = z^{(m, \bar{\varphi}(m) + p)} f^{\langle m_{\mathfrak{d}}, m\rangle}_{\mathfrak{d}}. \end{equation*}

The initial scattering diagram $\bar{\mathfrak{D}}_0$ is described by the $\mathbb C$ -algebra automorphisms of $R_k$ acting by

\begin{equation*} \theta_{\bar{\rho}_i, j}(z^{(m,\bar{\varphi}(m) + p)}) = z^{(m,\bar{\varphi}(m) + p)} (1 + z^{(m_i, \bar{\varphi}(m_i) - [E_{ij}])})^{\langle m_i, m\rangle},\, j = 0,\ldots,\ell_i. \end{equation*}

Similarly, we can consider the monoid given by

\begin{equation*} P'_{\bar{\varphi}} = \{(m, \bar{\varphi}(m) + v),\,m\in M,\, v \in \langle [E_{i0} - E_{ij}]\rangle, \, j = 0,\ldots,\ell_i\} \subset P_{\bar{\varphi}}. \end{equation*}

Then, setting

\begin{equation*} \tau_i = z^{(0, - [E_{i0}])},\,i=1,\dots,n, \end{equation*}

and, more generally, for later applications,

\begin{equation*} \tau^{(m, \bar{\varphi}(m) + \sum_{i} k_i \beta_{ij})} = \prod_i \tau^{k_i}_i, \end{equation*}

we can regard $\theta_{\bar{\rho}_i, j}$ as $\mathbb C[\![\tau_1,\ldots,\tau_n]\!]$ -algebra automorphisms of

\begin{equation*} R'_k \;:\!=\; (\mathbb C[P'_{\bar{\varphi}}]/(\mathfrak{m}^k\cap P'_{\bar{\varphi}}))[\![\tau_1,\ldots, \tau_n]\!], \end{equation*}

acting by

\begin{equation*} \theta_{\bar{\rho}_i, j}(z^{(m,\bar{\varphi}(m) + v)}) = z^{(m,\bar{\varphi}(m) + v)} (1 + \tau_i z^{(m_i, \bar{\varphi}(m_i) + [E_{i0} - E_{ij}])})^{\langle m_i, m\rangle}. \end{equation*}

We consider the problem of consistent completion for the scattering diagram

\begin{equation*} \bar{\mathfrak{D}}'_0 = \{\mathbb R_{\geq 0} m_i, \prod^{\ell_i}_{j=1}(1 + \tau_i z^{(m_i, \bar{\varphi}(m_i) + [E_{i0} - E_{ij}])})\} \end{equation*}

over $R'_k$ ; that is, we study

\begin{equation*} \bar{\mathfrak{D}}' = \operatorname{Scatter}(\bar{\mathfrak{D}}'_0) = \{(\mathfrak{d}', f'_{\mathfrak{d}'})\}. \end{equation*}

Clearly, by setting $\tau_i = z^{(0, - [E_{i0}])}$ , we recover the scattering diagrams for the toric model $(Y',D') \to (\bar{Y}, \bar{D})$ of the log Calabi–Yau surface $U' = Y'\setminus D'$ which blows up distinct points $x_{ij}$ on $\bar{D}_{i}$ for $i = 1, \ldots, n$ , $j = 1,\ldots,\ell_i$ . Deformations of (Y’, D’) can be regarded as deformations of (Y, D) for which the blowup points $x_{i0}$ , for $i = 1, \ldots, n$ , remain fixed.

In the Introduction, Section 1·3, we explained and provided ample motivation (as well as several references) for our main technique: describing the scattering process from $\bar{\mathfrak{D}}'_0$ to $\bar{\mathfrak{D}}'$ in terms of the analytic continuation of (flat sections of) a flat connection $\nabla_{\bar{\mathfrak{D}}'_0}$ , to (flat sections of) a flat connection $\nabla_{\bar{\mathfrak{D}}'}$ . Here we follow closely the reference [ Reference Filippini, Garcia–Fernandez and Stoppa13 ] (in particular, see [ Reference Filippini, Garcia–Fernandez and Stoppa13 , sections 2 and 3] for introductory material).

Thus, we first consider the problem of constructing a meromorphic connection $\nabla_{\bar{\mathfrak{D}}'_0}$ on the trivial principal $\operatorname{Aut} (R'_k)$ -bundle over $\mathbb P^1 = \mathbb C \cup \{\infty\}$ , for which the generalised monodromy should be given by the automorphisms $\theta_{\bar{\rho}_i, j} \in \operatorname{Aut}(R'_k)$ , $j=1,\ldots,\ell_i$ , appearing in the initial scattering diagram $\bar{\mathfrak{D}}'_0$ , along some corresponding rays $\bar{\rho}_{ij} \subset \mathbb C \cong M \otimes \mathbb R$ . Note that here we use our fixed identification $M \cong \mathbb Z^2$ and the standard identification between $\mathbb R^2$ and $\mathbb C$ . The simplest possible type for such a connection requires a simple pole at infinity, and a double pole at 0.

In order to apply the results of [ Reference Filippini, Garcia–Fernandez and Stoppa13 ], we choose a homomorphism $\mathcal{Z}\!:(P'_{\bar{\varphi}})^{\rm{gp}} \to \mathbb C$ (a “central charge”), factoring through a corresponding homomorphism $\mathcal{Z}\!:\mathcal{Q} \to \mathbb C$ .

The homomorphism $\mathcal{Z}$ can be regarded as a $\mathbb C[\![\tau_1,\ldots,\tau_n]\!]$ -linear derivation of $R'_k$ , by setting $\mathcal{Z}(z^{\alpha}) = \mathcal{Z}(\alpha) z^{\alpha}$ for $\alpha \in (P'_{\bar{\varphi}})^{\rm{gp}}$ , so, as in [ Reference Filippini, Garcia–Fernandez and Stoppa13 ], we have

(9·1) \begin{align} \nabla_{\bar{\mathfrak{D}}'_0} = \frac{\mathcal{Z}}{t^2} + \frac{f}{t}, \end{align}

where t denotes a variable on $\mathbb C^* \subset \mathbb P^1$ , and f is a suitable derivation of $R'_k$ .

Proposition 9·2. There exists a unique such $\nabla_{\bar{\mathfrak{D}}'_0}$ with generalised monodromy given by the automorphisms $\theta_{\bar{\rho}_i, j}$ , $j=1,\ldots,\ell_i$ along the ray

\begin{align*} \bar{\rho}_{ij}(\mathcal{Z} ) &= \mathbb R_{> 0} \mathcal{Z} ((m_i, \bar{\varphi}(m_i) + \beta_{ij})) = \mathbb R_{> 0} \mathcal{Z}_{\bar{s}}(\beta_{ij}). \end{align*}

Let us write $\hat{\vartheta}$ for the unique normalised, $\operatorname{Aut}(R'_k)$ -valued flat section of $\nabla_{\bar{\mathfrak{D}}'_0}$ . By construction, it satisfies the (generalised) monodromy condition

\begin{equation*} \hat{\vartheta}(t^+) = \theta_{\bar{\rho}_{i}, j} \circ \hat{\vartheta}(t^-), \end{equation*}

along the ray $\bar{\rho}_{ij}(\mathcal{Z})$ . According to [ Reference Filippini, Garcia–Fernandez and Stoppa13 ], for generic values of $\mathcal{Z}$ , there is an explicit formula for $\hat{\vartheta}$ in terms of a sum over connected, rooted trees T, with vertices labelled by elements of the set

\begin{equation*} \mathcal{L} = \{\mathbb Z_{>0}(m_i, \bar{\varphi}(m_i) + \beta_{ij}), i=1,\ldots,n,\,j=1,\ldots,\ell_i\} \subset P'_{\bar{\varphi}}. \end{equation*}

Given $\alpha \in \mathcal{L}$ , we write $\alpha = k \alpha'$ with $\alpha' \in \mathcal{L}$ primitive, and set

\begin{equation*} w(\alpha) = \frac{1}{k^2}. \end{equation*}

Note that we can regard a rooted tree T as a directed graph, by fixing the unique orientation of the edges which flows away from the root. Then, we define the weight of T as

\begin{equation*} w_{T} = \frac{w(\alpha_T) \alpha_T}{|\operatorname{Aut}(T)|} \in P'_{\bar{\varphi}}\otimes\mathbb Q, \end{equation*}

where $\alpha_T$ denotes the decoration at the root $i_0 \in T_0$ . Similarly, we attach to T a meromorphic function $\operatorname{W}_T(w)$ of the variables $w = \{w_i, \, i \in T_0\}$ , given by

(9·2) \begin{equation} \operatorname{W}_T(w ) = \prod_{\{i \to j\} \in T_1} \frac{w_i}{w_j} \frac{ \langle w(\alpha_i)\alpha_i, \alpha_j\rangle}{w_j - w_i}. \end{equation}

Here, we set $w_{i_0} = w_0$ for the root. We also introduce an integration kernel $\rho_T(t)$ , depending on $\mathcal{Z}$ , given by

(9·3) \begin{equation} \rho_T(t, w) = \frac{1}{2\pi {\rm i} w_0} \frac{t}{w_0 - t} \prod_{i\in T_0} \frac{1}{2\pi {\rm i}} \exp\left(-\frac{\mathcal{Z}(\alpha_i)}{w_i}\right), \end{equation}

and an integration cycle, also depending on $\mathcal{Z}$ , given by

\begin{equation*} C_T = \{ w_0 \in \mathbb R_{> 0} \mathcal{Z}(\alpha_T);\, w_{j} \in \mathbb R_{> 0} \mathcal{Z}(\alpha_j) \textrm{ for } \{i \to j\} \in T_1\}. \end{equation*}

Proposition 9·3. Suppose any two rays $\bar{\rho}_{ij}(\mathcal{Z})$ , $\bar{\rho}_{i'j'}(\mathcal{Z})$ with $i \neq i'$ are distinct. Then, for $p \in P'_{\bar{\varphi}}$ , we have

(9·4) \begin{equation} \hat{\vartheta}(t)(z^{p}) = z^p \exp\left(\sum_T \langle p, w_T \rangle \prod_{i \in T_{0}} (\tau z)^{\alpha_i} \int_{C_T} \rho(t, w) \operatorname{W}_T(w) \right), \end{equation}

where $\int_{C_T} \rho(t, w) \operatorname{W}_T(w)$ denotes the iterated integral along the cycle $C_T$ , computed according to the orientation of T.

Let us describe certain holomorphic families of flat connections, specialising to $\nabla_{\bar{\mathfrak{D}}'_0}$ . For fixed $r \in \mathbb Z/n\mathbb Z$ , $\bar{\phi} \in (0,\pi/2)$ , consider the open subset $S_r(\bar{\phi}) \subset \operatorname{Hom}(\mathcal{Q}, \mathbb C)$ such that, for $\mathcal{Z} \in S_r(\bar{\phi})$ , $i \neq r, r+1$ , we have

\begin{equation*} \bar{\rho}_{ij}(\mathcal{Z}) \subset \{ e^{i \phi} \bar{\rho}_{i},\,\phi \in (\!-\bar{\phi},\bar{\phi}) \}, \end{equation*}

while

\begin{equation*} \bar{\rho}_{rj}(\mathcal{Z}), \bar{\rho}_{r+1,j}(\mathcal{Z}) \subset \{ e^{i \phi} \bar{\sigma}_{r, r+1},\,\phi \in (\!-\bar{\phi},\bar{\phi}) \}, \end{equation*}

where $\bar{\sigma}_{r, r+1}$ denotes the corresponding cone of the fan $\bar{\Sigma}$ .

We now specialise our discussion to the case $n = 3$ , $\ell_3 = -1$ . Thus, the toric base is the surface $\bar{Y} \cong \mathbb P^2$ with standard fan $\{\bar{\rho}_1, \bar{\rho}_2, \bar{\rho}_3\} = \mathbb R_{>0}\{m_1, m_2, m_3\}$ , and the toric models $(Y, D) \to (\bar{Y}, \bar{D})$ and $(Y', D') \to (\bar{Y}, \bar{D})$ do not blow up points along the toric divisor $\bar{D}_3$ . The proper transforms $D_3$ , $D'_3$ have positive self-intersection, which implies that the log Calabi–Yau surfaces $U = Y\setminus D$ , $U' = Y' \setminus D'$ are affine.

Fix central charges $\mathcal{Z}_0,\, \mathcal{Z}^* \in S_r(\bar{\phi})$ , satisfying

\begin{align*} \bar{\rho}_{1j}(\mathcal{Z}_0) = \bar{\rho}_{2},\,\bar{\rho}_{2j}(\mathcal{Z}_0) = \bar{\rho}_{1};\,\,\bar{\rho}_{1j}(\mathcal{Z}^*) = \bar{\rho}_{1},\,\bar{\rho}_{2j}(\mathcal{Z}) = \bar{\rho}_{2}. \end{align*}

Recall that the mirror family to (Y’, D’) is constructed around $\mathcal{X}_0 \cong \mathbb{V}_n$ using the consistent scattering diagram $\bar{\mathfrak{D}}' = \operatorname{Scatter}(\bar{\mathfrak{D}}'_0) = \{(\mathfrak{d}', f'_{\mathfrak{d}'})\}$ .

Let $\hat{\vartheta}(t, \mathcal{Z})$ denote the unique normalised, $\operatorname{Aut}(R'_k)$ -valued flat section of $\nabla(\mathcal{Z})$ . Then, $\hat{\vartheta}^* = \hat{\vartheta}(t, \mathcal{Z}^*)$ satisfies the monodromy condition

(9·5) \begin{equation} \hat{\vartheta}^*(t^+) = \theta_{\mathfrak{d}'} \circ \hat{\vartheta}^*(t^-), \end{equation}

along the ray $\mathfrak{d}'$ of $\bar{\mathfrak{D}}'$ . Thus, for $p \in P'_{\bar{\varphi}}$ , we have the identity of elements of $R'_k$

(9·6) \begin{equation} \hat{\vartheta}^*(t^+)(z^p) = \theta_{\mathfrak{d}'}\left(\hat{\vartheta}^*(t^-)(z^p)\right), \end{equation}

along the ray $\mathfrak{d}'$ . In particular, this holds for the functions

\begin{equation*} \hat{\vartheta}^*_q = \hat{\vartheta}^*(t)(z^{(q, \bar{\varphi}(q))}), \, q \in M = \bar{B}_0(\mathbb Z). \end{equation*}

For sufficiently small $\bar{\phi}$ , let us fix

\begin{equation*} \tilde{t} \in \mathbb C \setminus (\{ e^{i \phi} \bar{\rho}_{i},\,\phi \in (\!-\bar{\phi},\bar{\phi}) \} \cup \{ e^{i \phi} \bar{\sigma}_{r, r+1},\,\phi \in (\!-\bar{\phi},\bar{\phi}) \}). \end{equation*}

Then, Corollary 9·5 implies that

\begin{equation*} \hat{\vartheta}_q(\mathcal{Z})= \hat{\vartheta}(\tilde{t}, \mathcal{Z})(z^{(q, \bar{\varphi}(q))}), \, q \in M = \bar{B}_0(\mathbb Z), \end{equation*}

is a holomorphic function of $\mathcal{Z} \in S_1(\bar{\phi})$ , with values in $R'_k$ , thought of as a finite dimensional complex linear space. Therefore, $\hat{\vartheta}^*_q = \hat{\vartheta}_q(\mathcal{Z}^*) $ can be computed by analytic continuation along a path starting from $\hat{\vartheta}_q(\mathcal{Z}_0)$ .

Thus, we choose a path $\gamma\;:\; [0,1] \to S_1(\bar{\phi})$ , with $\gamma(0) = \mathcal{Z}_0,\,\gamma(1) = \mathcal{Z}^*$ , and such that, for $\sigma \in [0,1]$ , the rays $\bar{\rho}_{1j}(\mathcal{Z}_{\gamma(\sigma)})$ , $\bar{\rho}_{2j'}(\mathcal{Z}_{\gamma(\sigma)})$ are distinct, except for a finite, minimal number of critical times $0 < \sigma_1 < \ldots < \sigma_{a} < 1$ , for which we have

\begin{equation*} \bar{\rho}_{1j_r}(\mathcal{Z}_{\gamma(\sigma_r)}) = \bar{\rho}_{2j'_r}(\mathcal{Z}_{\gamma(\sigma_r)}), \end{equation*}

precisely for $j_r \in \{1, \ldots, \ell_1\}$ , $j'_r \in \{1,\ldots, \ell_2\}$ .

Note that by (9·3), (9·4), the analytic continuation along $\gamma|_{(0, \sigma_1)}$ is uniquely determined as

\begin{equation*} \hat{\vartheta}_q(\mathcal{Z}_{\gamma(\sigma)}) = z^{(q, \bar{\varphi}(q))} \exp\left(\sum_T \langle q, w_T \rangle \prod_{i \in T_{0}} (\tau z)^{\alpha_i} \int_{C_T(\mathcal{Z}_{\gamma(\sigma)})} \rho_T(\tilde{t}, w, \mathcal{Z}_{\gamma(\sigma)}) \operatorname{W}_T(w) \right). \end{equation*}

The problem of analytically continuing the expression above across $\sigma_1$ , and, inductively, across the subsequent critical times $\sigma_2, \ldots, \sigma_a$ , is studied in detail in [ Reference Filippini, Garcia–Fernandez and Stoppa13 , section 5].

As in Section 3, the meromorphic function $\operatorname{W}_T(w)$ , specialised at $w_{0} = 0$ , is singular along the hyperplane arrangement defined by $(T, i_0, \mathbb{I})$ . Fixing any ordering of $T_0$ , compatible with the orientation of T as a rooted tree, identifies this with a meromorphic function nearby $0 \in \mathbb C^{|T_0|-1}$ , which we denote by $\phi(\operatorname{W}_T(w))$ .

Let us write

\begin{equation*} \beta = \sum_{i \in T_0} \alpha_i \end{equation*}

for the total decoration of T. Then, the discussion in [ Reference Filippini, Garcia–Fernandez and Stoppa13 , sections 5·1-5·3] yields a representation for the continuation along $\gamma$ of the iterated integral of $ \operatorname{W}_T(w)$ ,

\begin{equation*} \langle q, w_T\rangle \int_{C_T(\mathcal{Z})} \rho_T(\tilde{t}, w,\mathcal{Z}) \operatorname{W}_T(w), \end{equation*}

which contains a distinguished term given by the iterated residue of $ \operatorname{W}_T(w)$ ,

(9·7) \begin{equation} \epsilon(T) \operatorname{IR}_{0}(\phi(\operatorname{W}_T(w))) \langle q, w_T\rangle \frac{1}{2\pi {\rm i}}\int_{\mathbb R_{>0} \mathcal{Z}^*(\beta)} \frac{\tilde{t}}{w_0 - \tilde{t}} \exp\left(-\frac{\mathcal{Z}^*(\beta)}{w_0}\right) \frac{d w_0}{w_0}, \end{equation}

for a unique $\epsilon(T) \in \mathbb Z$ .

In Proposition 10·3 below we will give an explicit expression for $\epsilon(T)$ , at least in a special case, in terms of semistable representations.

Moreover, the term (9·7) is characterised by its behaviour as a function of $\tilde{t}$ : it is precisely the contribution to the analytic continuation which can be extended to a holomorphic function of $\tilde{t} \in \mathbb C^* \setminus \mathbb R_{>0} \mathcal{Z}^*(\beta)$ , with a branch cut discontinuity along $\mathbb R_{>0} \mathcal{Z}^*(\beta)$ . The jump along the latter ray is given by

(9·8) \begin{equation} \frac{ \langle q, w(\alpha_T) \alpha_T\rangle}{|\operatorname{Aut}(T)|} \epsilon(T) \operatorname{IR}_0(\phi(\operatorname{W}_T(w))) \exp\left(-\frac{\mathcal{Z}^*(\beta)}{\tilde{t}}\right). \end{equation}

It follows in particular that analytic continuation gives an effective procedure to compute the consistent completion of the scattering diagram $\bar{\mathfrak{D}}'_0$ . Let us write

\begin{equation*} \log f'_{\mathfrak{d}'} = \sum_{\beta} c_{\beta} z^{\beta}. \end{equation*}

Proposition 9·8. The coefficient $c_{\beta}$ is determined by the identities

(9·9) \begin{equation} \langle q, \beta\rangle c_{\beta} = \sum_{T\,:\,\sum_{i}\alpha_i = \beta} \frac{ \langle q, w(\alpha_T) \alpha_T\rangle}{|\operatorname{Aut}(T)|} \epsilon(T) \operatorname{IR}_0(\phi(\operatorname{W}_T(w))), \end{equation}

for all $q \in M = M \times \{0\} \subset P'_{\bar{\varphi}}$ .

10. Application of generalised monodromy

Recall that, according to [ Reference Gross, Hacking and Keel15 , section 3], and by our discussion in Section 9, the mirror family to the log Calabi–Yau surface underlying the Looijenga pair with toric model $(Y', D') \to (\bar{Y}, \bar{D})$ is constructed from the consistent completion $\bar{\mathfrak{D}}' = \{(\mathfrak{d}', f'_{\mathfrak{d}'})\}$ of the initial scattering diagram

\begin{equation*} \bar{\mathfrak{D}}'_0 = \{\mathbb R_{\geq 0} m_i, \prod^{\ell_i}_{j=1}(1 + \tau_i z^{(m_i, \bar{\varphi}(m_i) + [E_{i0} - E_{ij}])})\} \subset \bar{B} = \mathbb R^2, \end{equation*}

upon setting $\tau_i = z^{(0, - [E_{i0}])}$ . The consistent completion can be computed in terms of the (pushforward of the) canonical scattering diagram $(\mathfrak{D}')^{\rm can} \subset B$ :

\begin{equation*} \operatorname{Scatter}(\bar{\mathfrak{D}}'_0) = \nu_*((\mathfrak{D}')^{\rm can}) \end{equation*}

(see [ Reference Gross, Hacking and Keel15 , proposition 3·26]. Namely, we have

(10·1) \begin{equation} \log f_{\mathfrak{d}'} = \nu\left( \sum_{\beta} k_{\beta} N_{\beta} (\tau z)^{\pi_{*}\beta - \varphi_{\tau_{\mathfrak{d}'}}(k_{\beta}m_{\mathfrak{d}'})}\right). \end{equation}

Here,

(10·2) \begin{equation} N_{\beta} = \int_{[\overline{\mathfrak{M}}((\tilde{Y}')^{o}/C^o,\beta)]^{\rm vir}}1 \end{equation}

is a relative genus 0 Gromov–Witten invariant computed on the toric blowup $\pi\!:\tilde{Y}' \to Y'$ corresponding to the ray $\mathfrak{d}'$ , with respect to the unique degree $\beta$ such that

\begin{equation*} \beta \cdot \tilde{D}'_i = \begin{cases} k_{\beta} \quad \tilde{D}'_i = C\\[5pt] 0 \,\,\,\quad \tilde{D}'_i \neq C, \end{cases} \end{equation*}

for the component $C \subset \tilde{D}'$ which corresponds to the ray $\mathfrak{d}'$ (see [ Reference Gross, Hacking and Keel15 , section 1·3] for such toric blowups).

Let us now specialise to the case $n = 3$ , $\ell_3 = -1$ . Recall in this case the toric base $(\bar{Y}, \bar{\Sigma})$ is given by $\mathbb P^2$ with its canonical fan spanned by $\{m_1, m_2, m_3\}$ , and there are no blowups along the toric divisor $\bar{D}_3$ . Then, the degree $\beta = \beta_{(P_1, P_2)}$ is determined by a pair of ordered partitions of lengths $\ell_1, \ell_2$ ,

\begin{equation*} (P_1, P_2) = \big(\sum^{\ell_1}_{i=1} p_{1i}, \sum^{\ell_2}_{j=1} p_{2j}\big), \end{equation*}

satisfying $(|P_1|, |P_2|) = k (a, b)$ for $k > 0$ and primitive $(a, b) \in \mathbb Z^2$ , through the correspondence

\begin{equation*} H_2(\tilde{Y}', \mathbb Z) \ni\beta_{(P_1, P_2)} = \pi^*\beta_k - \sum^{\ell_1}_{i = 1} p_{1i} [E_{1i}] - \sum^{\ell_2}_{j=1} p_{2j} [E_{2j}]. \end{equation*}

Here, $\beta_k$ is the pullback to Y’ of the unique class on the toric orbifold with fan $\mathbb R_{\geq 0}\{m_1, m_2, m_3, m_4 \;:\!=\; (a, b)\}$ satisfying

\begin{equation*} \beta_k \cdot \bar{D}_1 = ka,\,\beta_k \cdot \bar{D}_2 = kb,\, \beta_k \cdot \bar{D}_4 = k. \end{equation*}

The equality $(|P_1|, |P_2|) = k (a, b)$ then implies

\begin{equation*} \beta \cdot \tilde{D}'_1 = 0,\,\beta \cdot \tilde{D}'_2 = 0,\,\beta \cdot \tilde{D}'_3 = 0,\,\beta \cdot \tilde{D}'_4 = k. \end{equation*}

We will write

\begin{equation*} N[(P_1, P_2)] = N_{\beta_{(P_1, P_2)}} \end{equation*}

for the corresponding Gromov–Witten invariant (10·2), with $C = \tilde{D}'_4$ . Thus, in this case, the weight function $f'_{\mathfrak{d}'}$ appearing in (10·1) can be identified with the formal power series

\begin{equation*} f'_{(a, b)} \in \mathbb C[x,x^{-1}, y, y^{-1}][\![s_1, \ldots, s_{\ell_1}, t_1, \ldots, t_{\ell_2}]\!] \end{equation*}

given by

(10·3) \begin{equation} \log f'_{(a, b)} = \sum_{k>0} \sum_{(|P_1|, |P_2|) = k (a, b)} k N[(P_1, P_2)] (s, t)^{(P_a, P_b)} (\tau x)^{ka} (\tau y)^{kb}, \end{equation}

where

\begin{equation*} (s, t)^{(P_a, P_b)} = \prod^{\ell_1}_{i=1} s^{p_{1i}}_i \prod^{\ell_2}_{j=1} t^{p_{2j}}_j, \end{equation*}

with $s_i$ , $t_j$ given by

\begin{align*} &s_i = z^{(0, [E_{10} - E_{1i}])} = \exp\big(2\pi {\rm i} \int_{\tilde{\beta}_{1i}} \Omega\big),\,t_j = z^{(0, [E_{20} - E_{2j}])} = \exp\big(2\pi {\rm i} \int_{\tilde{\beta}_{2j}} \Omega\big), \end{align*}

and with

\begin{equation*} (\tau x)^{ka} = z^{(k a m_1, - k a [E_{10}])},\,(\tau y)^{kb} = z^{(k b m_2, - k b [E_{20}])}. \end{equation*}

Thus, the function $\log f'_{(a, b)}$ can be identified canonically with a corresponding sum over dimension vectors d for $K(\ell_1, \ell_2)$ , namely

\begin{equation*} \log \tilde{f}'_{(a, b)} = \sum_{k > 0} \sum_{|d| = k (a, b)} k c_d z^{d}, \end{equation*}

where

(10·4) \begin{align} \nonumber d &= (P_1, P_2),\, c_d = N[(P_1, P_2)],\\[5pt] \nonumber z^d &= (s, t)^{(P_a, P_b)} = z^{(0, \sum^{\ell_1}_{i = 1} p_{1i} [E_{10}-E_{1i}] + \sum^{\ell_2}_{j=1} p_{2j} [E_{20}-E_{2j}])}\\[5pt] & =\exp\big(2\pi {\rm i} \int_{\tilde{\beta}(d)} \Omega\big), \end{align}

for the class

\begin{equation*} \beta(d) = \sum^{\ell_1}_{i = 1} p_{1i} [E_{10}-E_{1i}] + \sum^{\ell_2}_{j=1} p_{2j} [E_{20}-E_{2j}]) \in\mathcal{Q}. \end{equation*}

This identification establishes Theorem 1·4 (i) as well as the identity (1·9) in Theorem 1·4 (ii).

Next we will describe a procedure which refines the correspondence between the consistent completion of $\bar{\mathfrak{D}}'_0$ and generalised monodromy explained in Section 9. This refinement is needed in order to match abelianisation for quiver invariants.

The first step is given by the degeneration formula in Gromov–Witten theory, applied to the invariants $N[(P_1, P_2)]$ . It is shown in [ Reference Reineke, Stoppa and Weist26 , section 4] that this can be written in the form

(10·5) \begin{equation} N[(P_1, P_2)] = \sum_{(k^1, k^2) \vdash (P_1, P_2)} \prod^2_{i=1} \prod^{\ell_i}_{j=1} \prod_w \frac{(\!-\!1)^{k^i_{w,j}(w-1)}}{k^i_{w,j}!w^{k^i_{w,j}}} N^{\rm rel}[(w(k^1), w(k^2))], \end{equation}

where $(w(k^1), w(k^2))$ is a pair of weight vectors (i.e., a pair of sequences of increasing, positive integers), of lengths $\sum_w m_w(k^i)$ for $i =1, 2$ , determined by

\begin{equation*} (w(k^i))_j = w, \textrm{ for } j = \sum^{w-1}_{r} m_{r=1}(k^i) +1,\ldots,\sum^{w}_{r=1} m_r(k^i). \end{equation*}

The weight vectors $(w(k^1), w(k^2))$ encode the orders of tangency of a rational curve in $\bar{Y}$ at specified points of $\bar{D}_1$ , $\bar{D}_2$ , contained in the smooth locus of $\bar{D}$ , and $N^{\rm rel}[(w(k^1), w(k^2))]$ denotes the relative Gromov–Witten invariant virtually enumerating such curves (see [ Reference Gross, Pandharipande and Siebert18 , section 5] for more details on such relative invariants and on the degeneration formula).

The main result of [ Reference Reineke, Stoppa and Weist26 , section 4] proves an identity between relative Gromov–Witten invariants and quiver invariants,

(10·6) \begin{equation} \bar{\chi}_{\mathcal{N}}(d(k^1,k^2), \widehat{\zeta}^*) = (\!-\!1)^{(P_1, P_2)} N^{\rm rel}[(w(k^1), w(k^2))]\prod^2_{i=1} \prod^{\ell_i}_{j=1} \prod_w w^{k^i_{w,j}}, \end{equation}

where the stability vector is given by

\begin{equation*} \widehat{\zeta}^*(i_{(w, m)}) = w,\,\widehat{\zeta}^*(j_{(w, m)}) = 0, \end{equation*}

and the sign by

\begin{equation*} (\!-\!1)^{(P_1, P_2)} = (\!-\!1)^{k a b - \sum_i (p_{1 i})^2 - \sum_j (p_{2 j})^2-1}. \end{equation*}

Recall that we also have a “refined GW/Kronecker” correspondence (see [ Reference Gross and Pandharipande17 ], [ Reference Reineke and Weist27 , section 9] and [ Reference Bousseau4 ])

(10·7) \begin{equation} N[(P_1, P_2)] = (\!-\!1)^{(P_1, P_2)}\bar{\chi}_{K(\ell_1, \ell_2)}((P_1, P_2), \zeta^*), \end{equation}

where

\begin{equation*} \zeta^*(i) = 1,\, \zeta^*(j) = 0, \end{equation*}

for all sources i (respectively, sinks j) in $K(\ell_1, \ell_2)_0$ . Thus, (10·6) shows, in particular, that the abelianisation identity for quiver invariants (4·5) corresponds precisely to the Gromov–Witten degeneration formula (10·5).

The crucial point for our purposes is that combining (10·5) with (10·6) yields the identity

(10·8) \begin{equation} N[(P_1, P_2)] = (\!-\!1)^{(P_1, P_2)}\sum_{(k^1, k^2) \vdash (P_1, P_2)} \prod^2_{i=1} \prod^{\ell_i}_{j=1} \prod_w \frac{(\!-\!1)^{k^i_{w,j}(w-1)}}{k^i_{w,j}!w^{2k^i_{w,j}}} \bar{\chi}_{\mathcal{N}}(d(k^1,k^2), \widehat{\zeta}^*). \end{equation}

(Equivalently, this can be obtained by combining (4·5), (10·7)). In turn, the invariant $(\!-\!1)^{(P_1, P_2)}\bar{\chi}_{\mathcal{N}}(d(k^1,k^2), \widehat{\zeta}^*)$ can be computed in terms of generalised monodromy.

As in [ Reference Reineke, Stoppa and Weist26 , section 4], we consider the Poisson algebra attached to the quiver $Q \subset \mathcal{N}$ spanned by the dimension vector $d(k^1, k^2)$ . This is the ring

\begin{equation*} R_Q = \mathbb C[\![x_{j_{(w', m')}}, y_{i_{(w, m)}} \;:\; i_{(w, m)}, j_{(w', m')} \in Q_0]\!], \end{equation*}

endowed with the Poisson bracket defined by

\begin{equation*} \{x_{j_{(w', m')}}, y_{i_{(w, m)}}\} = \langle j_{(w', m')}, i_{(w, m)} \rangle x_{j_{(w', m')}} y_{i_{(w, m)}} = w w' x_{j_{(w', m')}} y_{i_{(w, m)}} \end{equation*}

(while all the other brackets of generators vanish). We introduce the (Poisson) automorphisms

\begin{equation*} \{\theta_{i_{(w', m')}},\,\theta_{j_{(w', m')}} \;:\; i_{(w, m)}, j_{(w', m')} \in Q_0\} \subset \operatorname{Aut}(R_Q), \end{equation*}

acting by

\begin{align*} &\theta_{j_{(w', m')}}(x_{j_{(w, m)}}) = x_{j_{(w, m)}},\\[5pt] &\theta_{j_{(w', m')}}(y_{i_{(w, m)}}) = y_{i_{(w, m)}}(1+ x_{j_{(w', m')}})^{\langle j_{(w', m')},\,i_{(w, m)} \rangle}, \end{align*}

respectively

\begin{align*} &\theta_{i_{(w', m')}}(x_{j_{(w, m)}}) = x_{j_{(w, m)}} (1+ y_{i_{(w', m')}})^{\langle i_{(w', m')},\,j_{(w, m)} \rangle},\\[5pt] &\theta_{i_{(w', m')}}(y_{i_{(w, m)}}) = y_{i_{(w, m)}}. \end{align*}

Consider the stability vectors $\zeta_0$ , $\zeta^*$ for $K(\ell_1, \ell_2)$ given by

\begin{equation*} \zeta_0(i) = 0,\, \zeta_0(j) = 1;\,\zeta^*(i) = 1,\, \zeta^*(j) = 0. \end{equation*}

We upgrade these to central charges

\begin{equation*} \mathcal{Z}_0,\mathcal{Z}^* \in \operatorname{Hom}(\mathbb Z K(\ell_1, \ell_2)_0, \mathbb C), \end{equation*}

inducing the same stability conditions. Note that stability vectors $\zeta$ and central charges $\mathcal{Z}$ have canonical lifts $\widehat{\zeta}$ , $\widehat{\mathcal{Z}}$ to Q, given by

\begin{align*} &\widehat{\zeta}(i_{(w, m)}) = w \zeta(i),\,\widehat{\zeta}(j_{(w, m)}) = w \zeta(j),\\[5pt] &\widehat{\mathcal{Z}}(i_{(w, m)}) = w \mathcal{Z}(i),\,\widehat{\mathcal{Z}}(j_{(w, m)}) = w \mathcal{Z}(j). \end{align*}

In particular, we have

\begin{align*} \widehat{\mathcal{Z}}(d(k^1, k^2)) &= \sum_{i_{(w, m)} \in Q_0} \widehat{\mathcal{Z}}(i_{(w, m)}) + \sum_{j_{(w', m')} \in Q_0} \widehat{\mathcal{Z}}(j_{(w', m')})\\[5pt] &= \sum_{i_{(w, m)} \in Q_0} w \mathcal{Z} (i) + \sum_{j_{(w', m')} \in Q_0} w \mathcal{Z} (j) = \mathcal{Z}(\beta), \end{align*}

where $\beta = \sum^2_{i =1} \sum^{\ell_i}_{j=1}p_{ij} \beta_{i j}$ .

Fix a path $\gamma\!:[0, 1] \to \operatorname{Hom}(\mathbb Z K(\ell_1, \ell_2)_0, \mathbb C)$ with $\mathcal{Z}_{\gamma(0)} = \mathcal{Z}_0$ , $\mathcal{Z}_{\gamma(1)} = \mathcal{Z}^*$ . By the results of [ Reference Filippini, Garcia–Fernandez and Stoppa13 ], explained in Section 9, there exists a unique $\operatorname{Aut}(R_Q)$ -connection $\nabla_0$ , of the form

\begin{equation*} \nabla_0 = \frac{\widehat{\mathcal{Z}}_0}{t^2} + \frac{f}{t}, \end{equation*}

with generalised monodromy given by the pairs of rays and automorphisms

\begin{equation*} \{\mathbb R_{>0} \mathcal{Z}_0(i), \prod_{i_{(w, m)}\in Q_0} \theta_{i_{(w,m)}}\}, \{\mathbb R_{>0} \mathcal{Z}_0(j), \prod_{j_{(w, m)}\in Q_0} \theta_{j_{(w,m)}}\}. \end{equation*}

Moreover, there is a holomorphic family of connections $\nabla(\mathcal{Z})$ , parametrised by $\mathcal{Z}$ in an open neighbourhood of $\gamma([0, 1])$ , of the form

\begin{equation*} \nabla(\mathcal{Z}) = \frac{\widehat{\mathcal{Z}}}{t^2} + \frac{f(\mathcal{Z})}{t}, \end{equation*}

with constant generalised monodromy, and with initial value $\nabla(\mathcal{Z}_{\gamma(0)}) = \nabla_0$ . So the canonical $\operatorname{Aut}(R_Q)$ -valued flat section $\hat{\vartheta}$ of $\nabla(\mathcal{Z}_0)$ can be continued analytically along $\gamma$ to a flat section $\hat{\vartheta}^*$ of $\nabla(\mathcal{Z}^*)$ . The section $\hat{\vartheta}$ is given explicitly by (9·4), where now $z^p$ denotes an element of $\mathbb C[\mathbb Z Q_0]$ (so we have e.g. $z^{j_{(w, m)}} = x_{j_{(w,m)}}$ ), and T is a rooted tree labelled by elements $\alpha$ of the set

\begin{equation*} \mathcal{L} = \mathbb Z_{>0}\{ i_{(w, m)}, j_{(w', m')} \} \subset \mathbb Z_{>0}Q_0. \end{equation*}

In particular, when continuing $\hat{\vartheta}$ along $\gamma$ , we need to consider the analytic continuation, up to a neighbourhood of $\gamma(1) = \mathcal{Z}^*$ , of the graded component appearing in (9·4),

(10·9) \begin{equation} \sum_{T\,:\,\sum_{i \in T_0} \alpha_i = d(k^1, k^2)} \langle -, w_T \rangle \int_{C_T(\widehat{\mathcal{Z}}_0)} \rho_T(t, w, \widehat{\mathcal{Z}}_0) \operatorname{W}_T(w) \prod_{i \in T_{0}} z^{\alpha_i}. \end{equation}

The latter continuation is piecewise holomorphic in t, with branch cuts along a finite collection of rays $\rho \subset \mathbb C^*$ , and the jump along the ray $\mathbb R_{>0} \widehat{\mathcal{Z}}^*(d(k^1, k^2))$ is given by

(10·10) \begin{align} \sum_{\sum_{i \in T_0} \alpha_i = d(k^1, k^2)} \langle -, \alpha_T\rangle \epsilon(T) \operatorname{IR}_0(\phi(\operatorname{W}_T(w))) \exp\left(-\frac{\widehat{\mathcal{Z}}^*(d(k^1, k^2))}{t}\right) z^{d(k^1,k^2)}, \end{align}

for a unique $\epsilon(T) \in \mathbb Z$ , determined inductively by the procedure of [ Reference Filippini, Garcia–Fernandez and Stoppa13 , section 5·2]. We will use an alternative expression for $\epsilon(T)$ , given in Proposition 10·3 below.