1 Introduction
The moduli space $\operatorname {\mathrm {M}} _C(2,\mathcal {L})$ of stable rank 2 vector bundles with fixed determinant of odd degree on a curve C of genus $g\geq 2$ is an important and wellstudied object, its history going back to the 1960s [Reference Newstead44, Reference Mumford and Newstead37, Reference Narasimhan and Seshadri43]. It is a smooth projective Fano variety of dimension $3g3$ when considered as an algebrogeometric object, and a monotone symplectic manifold of dimension $6g6$ when considered as a symplectogeometric object. They also arise as moduli spaces of flat unitary $\operatorname {\mathrm {SU}}(2)$ connections on Riemann surfaces, or character varieties of conjugacy classes.
Associated to $\operatorname {\mathrm {M}} _C(2,\mathcal {L})$ , one can study its invariants, such as its Betti numbers [Reference Harder and Narasimhan22] or Chow motive [Reference del Baño1]. The idea behind the description of the Chow motive is to find a decomposition which expresses it in terms of easier objects that still possess a connection to the curve C.
In this article we will discuss 3 types of decompositions, which have (conjectural) relations to each other:

• In Conjecture A, we propose a semiorthogonal decomposition of the derived category $\operatorname {\mathrm {{\mathbf {D}}}}^{\mathrm {b}}(\operatorname {\mathrm {M}} _C(2,\mathcal {L}))$ .

• In Theorem B, we describe the critical values and critical loci of the graph potentials from [Reference Belmans, Galkin and Mukhopadhyay8, Reference Belmans, Galkin and Mukhopadhyay7]. This describes orthogonal decompositions of the Fukaya category resp. the category of matrix factorisations.
When seen as candidate Landau–Ginzburg mirrors to $\operatorname {\mathrm {M}} _C(2,\mathcal {L})$ , this description is consistent with the eigenvalue decomposition of the quantum multiplication $\mathrm {c}_1(\operatorname {\mathrm {M}} _C(2,\mathcal {L}))*_0$ on quantum cohomology $\operatorname {\mathrm {QH}}^{\bullet }(\operatorname {\mathrm {M}} _C(2,\mathcal {L}))$ following Muñoz as described in Proposition 3.2.

• In Theorem C, we describe an identity in $\mathrm {K}_{0}(\mathrm {Var}/k)$ , the Grothendieck ring of varieties, which relates to various known and expected motivic decompositions.
The decompositions discussed in this article for $\operatorname {\mathrm {M}} _C(2,\mathcal {L})$ can serve as a blueprint for other decompositions of moduli spaces of sheaves, and we will discuss related work in Sections 2.1 and 4.1.
Semiorthogonal decompositions of the derived category
Semiorthogonal decompositions encode many important geometric properties of varieties (see Kuznetsov’s ICM address [Reference Kuznetsov31] for an introduction) and currently play an important role in the geometry and construction of moduli spaces of sheaves. Hence, we propose the following conjecture.
Conjecture A. Let C be a smooth projective curve of genus $g\geq 2$ . Then there exists a semiorthogonal decomposition
This was conjectured independently by Narasimhan (as communicated in [Reference Lee34]) and the authors (see [Reference Belmans4, Conjecture 7]). We also give a further refinements of this conjecture in Sections 2.2 and 2.3. We suggest that there exists a

• a minimal Lefschetz decomposition in the sense of [Reference Kuznetsov29] with respect to the generator of the Picard group $\mathcal {O}_{\operatorname {\mathrm {M}} _C(2,\mathcal {L})}(1)$ ;

• a Ringel–Samokhintype decomposition (which we will introduce), which involves a symmetry under an antiinvolution.
We will explain the state of the art for these in Section 2.1.
Eigenvalue and critical value decompositions
Under mirror symmetry, $\operatorname {\mathrm {M}} _C(2,\mathcal {L})$ corresponds to a Landau–Ginzburg model $f\colon Y\to \mathbb {A}^1$ where Y is a quasiprojective variety and f is a regular function on it. In the introduction to Section 3, we recall the motivation to describe on one hand the eigenvalues of the quantum multiplication $\mathrm {c}_1(\operatorname {\mathrm {M}} _C(2,\mathcal {L}))*_0$ and on the other hand the critical values of f.
In [Reference Belmans, Galkin and Mukhopadhyay8] we have introduced graph potentials, and in [Reference Belmans, Galkin and Mukhopadhyay7] we explained how they can be seen as the first step in the construction of the mirror Landau–Ginzburg model. The following theorem summarises the situation and gives further evidence for Conjecture A and the claim that graph potentials are (partial) mirror Landau–Ginzburg mirrors to $\operatorname {\mathrm {M}} _C(2,\mathcal {L})$ .
Theorem B. The critical values of graph potential for the necklace graph in Figure 4 are given by
The dimension of the critical loci with modulus $8g88k$ is equal to k, where $k=0,\ldots ,g1$ . For trivalent graphs with a perfect matching, the critical values form a subset of (2).
The important conclusions are that graph potentials already see all the (expected) critical values, even if they are only restrictions of the full Landau–Ginzburg mirror to certain torus charts, and that the critical loci are of the expected dimension.
This result mirrors the eigenvalue decomposition of quantum cohomology $\operatorname {\mathrm {QH}}^{\bullet }(\operatorname {\mathrm {M}} _C(2,\mathcal {L}))$ described in Proposition 3.2, as obtained by Muñoz. It gives further evidence to the claim that graph potentials can be used as building blocks for the full Landau–Ginzburg mirror of $\operatorname {\mathrm {M}} _C(2,\mathcal {L})$ .
The Grothendieck ring of varieties
The third type of decomposition we consider is an identity for $\operatorname {\mathrm {M}} _C(2,\mathcal {L})$ in the Grothendieck ring of varieties. This ring encodes the cutandpaste relation, and because many invariants of varieties satisfy such a cutandpaste relation it can be seen as the universal invariant encoding this. As such, it implies various known identities for motivic invariants of $\operatorname {\mathrm {M}} _C(2,\mathcal {L})$ , and as we explain in Corollary 4.1, it is consistent with Conjecture A. The result is the following:
Theorem C. We have the equality
in $\mathrm {K}_{0}(\mathrm {Var}/k)$ , for some class T such that $(1+{\mathbb {L}}) \cdot T = 0$ .
We expect that $T=0$ , but our method of proof is not strong enough to remove this error term.
2 Semiorthogonal decompositions
In this section, we will discuss the structure of the bounded derived category of coherent sheaves $\operatorname {\mathrm {{\mathbf {D}}}}^{\mathrm {b}}(\operatorname {\mathrm {M}} _C(2,\mathcal {L}))$ . We also suggest two semiorthogonal decompositions with additional structure, namely a Lefschetz decomposition (as introduced by Kuznetsov [Reference Kuznetsov29]). We introduce the notion of a Ringel–Samokhin decomposition, related to the structure of quasihereditary algebras, and an informal analogy between moduli spaces of bundles with flag varieties.
2.1 On the conjecture
In Conjecture A, we have formulated a conjecture giving a semiorthogonal decomposition into geometrically meaningful pieces, namely copies of $\operatorname {\mathrm {{\mathbf {D}}}}^{\mathrm {b}}(\operatorname {\mathrm {Sym}}^i C)$ for $i=0,\ldots ,g1$ . This was conjectured independently by Narasimhan (as communicated in [Reference Lee34]) and the authors (see [Reference Belmans4, Conjecture 7]). The conjecture does not specify the embedding functors.
Moreover, it was a folklore conjecture that each of the pieces $\operatorname {\mathrm {{\mathbf {D}}}}^{\mathrm {b}}(\operatorname {\mathrm {Sym}}^iC)$ for $i=1,\ldots ,g1$ is indecomposable (i.e., does not admit further semiorthogonal decompositions). This was proven for $i\leq \lfloor \frac {g+3}{2}\rfloor 1$ in [Reference Bastianelli, Belmans, Okawa and Ricolfi3, Corollary D], with a weaker result being given in [Reference Biswas and Gómez11, Theorem 1.3], and finally for all $i\leq g1$ in [Reference Lin36, Theorem 1.4].
State of the art
The first steps (before the precise conjecture was phrased) were taken by Narasimhan [Reference Narasimhan41, Reference Narasimhan42], and independently, by Fonarev–Kuznetsov [Reference Fonarev and Kuznetsov17] (for C generic), who have shown that the Fourier–Mukai functor $\Phi _{\mathcal {W}}$ , where $\mathcal {W}$ is the universal vector bundle on $C\times \operatorname {\mathrm {M}} _C(2,\mathcal {L})$ , is fully faithful. Together with the exceptional objects $\mathcal {O}_{\operatorname {\mathrm {M}} _C(2,\mathcal {L})}$ and $\Theta $ , this gives rise to 3 components in (1).
The first and third authors have shown in [Reference Belmans and Mukhopadhyay10] that it is possible to twist the embedding $\Phi _{\mathcal {W}}$ by $\Theta $ to obtain the embedding of the four lowdimensional components in (1) (for $g\geq 12$ ). The embedding of a copy of $\operatorname {\mathrm {{\mathbf {D}}}}^{\mathrm {b}}(\operatorname {\mathrm {Sym}}^2C)$ was recently obtained by Narasimhan–Lee in [Reference Lee and Narasimhan35, Theorem 1.2] (for C not hyperelliptic and $g\geq 16$ ).
Most importantly, in [Reference Tevelev and Torres50] and [Reference Xu and Yau53] a semiorthogonal decomposition that contains all expected components was obtained by Tevelev–Torres and Xu–Yau. The essential ingredient in the first approach is Thaddeus’ wallcrossing picture, which is also used in Section 4 to prove Theorem C. The second approach uses Teleman’s Borel–Weil–Bott theory on the moduli stack of $\mathrm {SL}_2$  bundles. What remains to be proven is that the complement to the subcategory generated by these components is trivial.
Outlook
This conjecture forms a part of a greater program, which aims at finding natural semiorthogonal decompositions of the bounded derived categories of coherent sheaves on the moduli spaces of (stable) objects in the bounded derived categories of coherent sheaves on algebraic varieties.
For moduli of higher rank vector bundles there exist various motivic decompositions (see also Section 4.1), and these suggest semiorthogonal decompositions of $\operatorname {\mathrm {{\mathbf {D}}}}^{\mathrm {b}}(\operatorname {\mathrm {M}} _C(r,\mathcal {L}))$ with $\gcd (r,\deg \mathcal {L})=1$ . In this case, $\operatorname {\mathrm {M}} _C(r,\mathcal {L})$ is a smooth projective Fano variety of dimension $(r^21)(g1)$ and index $2$ . The motivic decompositions suggest that derived categories of products of $\operatorname {\mathrm {Sym}}^iC$ will play a role, and for $r=3$ , this was made precise in [Reference Gómez and Lee20, Conjecture 1.9].
Related to this is the indecomposability of $\operatorname {\mathrm {{\mathbf {D}}}}^{\mathrm {b}}(\operatorname {\mathrm {Sym}}^iC)$ for all $i\leq g1$ which is proven in [Reference Lin36, Theorem 1.4]. The semiorthogonal decomposition of the derived categories of $\operatorname {\mathrm {Sym}}^iC$ for $i\geq g$ was obtained by Toda in [Reference Toda52] and studied again in [Reference Jiang and Leung25, §1.4] and [Reference Belmans and Krug9, Theorem D].
Another important class of moduli spaces of coherent sheaves is punctual Hilbert schemes. The study of (partial) semiorthogonal decompositions was started in [Reference Krug and Sosna28]. It was generalized to higher dimensions in [Reference Krug, Ploog and Sosna27, 6]. Further generalizations to nested Hilbert schemes were obtained in [Reference Jiang and Leung25, §3.1.4] and [Reference Belmans and Krug9, Theorem E]. In a different direction, a semiorthogonal decomposition for the Hilbert square of a cubic hypersurface was obtained in [Reference Belmans, Fu and Raedschelders5] using the Fano variety of lines.
2.2 Lefschetz decompositions
A more structured form of Conjecture A is given by that of a Lefschetz decomposition, a notion introduced by Kuznetsov [Reference Kuznetsov29] in his theory of homological projective duality.
Definition 2.1. Let X be a smooth projective variety and $\mathcal {O}_X(1)$ be a line bundle on X. A (right) Lefschetz decomposition of $\operatorname {\mathrm {{\mathbf {D}}}}^{\mathrm {b}}(X)$ with respect to the line bundle $\mathcal {O}_X(1)$ is a semiorthogonal decomposition
such that
We say that n is the length of the decomposition, and $\mathcal {A}_0$ is the starting block.
The decomposition (4) is rectangular if A_{ n−1} = … = A_{0}.
By [Reference Kuznetsov30, Lemma 2.18(i)], a Lefschetz decomposition is completely determined by its starting block, as one can inductively define A_{ i }:=^{⊥}(A_{0} ⊗O_{ X }(−i)) ∩A_{ i−1}. This allows us to define the notion of a minimal Lefschetz collection, by considering the (partial) inclusion order on the starting blocks for Lefschetz collections of $\operatorname {\mathrm {{\mathbf {D}}}}^{\mathrm {b}}(X)$ with respect to $\mathcal {O}_X(1)$ .
In [Reference Kuznetsov and Smirnov32, Definition 1.3(iii)], the notion of residual category for a Lefschetz exceptional collection is introduced, to measure how far it is from being rectangular. One can generalize this to an arbitrary Lefschetz decomposition as in [Reference Kuznetsov and Smirnov32, Definition 2.6].
Definition 2.2. Let $\operatorname {\mathrm {{\mathbf {D}}}}^{\mathrm {b}}(X)=\langle \mathcal {A}_0,\mathcal {A}_1\otimes \mathcal {O}_X(1),\ldots ,\mathcal {A}_{n1}\otimes \mathcal {O}_X(n1)\rangle $ be a Lefschetz decomposition of length n with respect to the line bundle $\mathcal {O}_X(1)$ . The residual category $\mathcal {R}$ of this decomposition is defined as
Remark 2.3. As discussed in [Reference Kuznetsov and Smirnov33] (in the context of Fano varieties with vanishing odd cohomology), properties of the residual category are conjecturally related to the structure of fiber over zero for the quantum cohomology of X. For Fano varieties with odd cohomology, one aspect of a generalization of this conjecture is the prediction that the dimension of the Hochschild homology of the residual category is equal to the length of the eigenspace associated to the eigenvalue 0. By Proposition 3.2, this eigenspace is isomorphic to $\operatorname {\mathrm {H}}^{\bullet }(\operatorname {\mathrm {Sym}}^{g1}C)$ (of dimension $(2g+1)!/g!^2$ ), which, by the Hochschild–Kostant–Rosenberg decomposition, is isomorphic to $\operatorname {\mathrm {HH}}_{\bullet }(\operatorname {\mathrm {Sym}}^{g1}C)$ . Hence, the equality holds.
This brings us to the following more precise form of Conjecture A, incorporating a Lefschetz structure.
Conjecture 2.4. Let C be a smooth projective curve of genus g. Then there exists a minimal Lefschetz decomposition of $\operatorname {\mathrm {{\mathbf {D}}}}^{\mathrm {b}}(\operatorname {\mathrm {M}} _C(2,\mathcal {L}))$ of length 2 with respect to $\Theta $ , whose first block has a semiorthogonal decomposition
so, in particular, the residual category is $\operatorname {\mathrm {{\mathbf {D}}}}^{\mathrm {b}}(\operatorname {\mathrm {Sym}}^{g1}C)$ .
For $g=2$ , this conjecture is known by considering the semiorthogonal decomposition [Reference Bondal and Orlov14, Theorem 2.9] and mutating the object $\mathcal {O}_{\operatorname {\mathrm {M}} _C(2,\mathcal {L})}(1)$ to the right.
Proposition 2.5 (Bondal–Orlov).
Let C be a curve of genus 2. Then
is a minimal Lefschetz decomposition, whose residual category is equivalent to $\operatorname {\mathrm {{\mathbf {D}}}}^{\mathrm {b}}(C)$ .
Homological projective duality
After constructing a (minimal) Lefschetz decomposition, the next step would be to understand the homological projective dual of $\operatorname {\mathrm {M}} _C(2,\mathcal {L})$ as in [Reference Kuznetsov29]. This would give the structure of the derived categories of linear sections of $\operatorname {\mathrm {M}} _C(2,\mathcal {L})$ , and in particular, for a smooth hyperplane section. By Theorem 1.1 of op. cit., we have that Conjecture 2.4 gives a semiorthogonal decomposition
To indicate the complexity of the homological projective dual, observe that for $g=2$ , the category $\mathcal {C}_H$ consists of 7 exceptional objects (because it’s the complement to $\mathcal {O}(1)$ in the derived category of $\operatorname {\mathrm {Bl}}_5\mathbb {P}^2\cong Q\cap Q'\cap H$ in $\mathbb {P}^5$ ).
For $g=3$ , one can use the description from [Reference Desale and Ramanan16] of $\operatorname {\mathrm {M}} _C(2,\mathcal {L})$ for a hyperelliptic curve C as the zero locus of $(\operatorname {\mathrm {Sym}}^2\mathcal {U}^{\vee })^{\oplus 2}$ on $\operatorname {\mathrm {Gr}} (g1,2g+2)$ , together with the Borel–Weil–Bott theorem and the Koszul sequence, to compute that the Hodge diamond of a hyperplane section is given by
Using the Hochschild–Kostant–Rosenberg decomposition and additivity of Hochschild homology, we see that
There is no obvious modular interpretation for (a semiorthogonal decomposition of) $\mathcal {C}_H$ , and it does not look like the homological projective dual has an easy description.
2.3 Ringel–Samokhintype decompositions
There is another way in which we can impose further conditions on a semiorthogonal decomposition for $\operatorname {\mathrm {{\mathbf {D}}}}^{\mathrm {b}}(\operatorname {\mathrm {M}} _C(2,\mathcal {L}))$ . This condition is inspired by the theory of hereditary algebras and Ringel duality, and by similar semiorthogonal decompositions for $\operatorname {\mathrm {{\mathbf {D}}}}^{\mathrm {b}}(G/B)$ obtained by Samokhin, and encodes special symmetries not found in most semiorthogonal decompositions.
Let X be a smooth projective variety, and consider a semiorthogonal decomposition
of length $n+1\geq 2$ . Let $\sigma $ be an antiequivalence of $\operatorname {\mathrm {{\mathbf {D}}}}^{\mathrm {b}}(X)$ which is moreover an involution. Let $\mathcal {B}_{k}$ be the image of $\mathcal {A}_{nk}$ under this involution for $k=0,\ldots ,n$ . Then we have a semiorthogonal decomposition
However, consider the left dual decomposition in the sense of [Reference Samokhin47, Definition 3.6] denoted by
This brings us to the following definition.
Definition 2.6. A decomposition (12) is of Ringel–Samokhin type if there exists an antiequivalence $\sigma $ of $\operatorname {\mathrm {{\mathbf {D}}}}^{\mathrm {b}}(X)$ such that for all $k=0,\ldots ,n$ , the subcategory $\mathcal {C}_k$ is the image of $\mathcal {B}_k$ under $\sigma $ .
Remark 2.7. In [Reference Samokhin47], Samokhin studied such exceptional collections for the derived category $\operatorname {\mathrm {{\mathbf {D}}}}^{\mathrm {b}}(G/B)$ , where $G/B$ is the full flag variety of a group of rank 2, and the antiequivalence is given by ${{\mathbf {R}}{\mathcal {H}}om}(,\omega _{G/B}^{1/2})$ .
We can refine Conjecture A into a semiorthogonal decomposition of Ringel–Samokhintype as follows.
Conjecture 2.8. Let C be a smooth projective curve of genus $g\geq 2$ . Then there exists a Ringel–Samokhintype decomposition
with the antiequivalence given by ${{\mathbf {R}}{\mathcal {H}}om}(,\mathcal {O}_{\operatorname {\mathrm {M}} _C(2,\mathcal {L})}(1))$ .
This conjecture highlights how semiorthogonal decompositions of $\operatorname {\mathrm {M}} _C(2,\mathcal {L})$ are expected to have strong symmetry properties, much stronger than found for most varieties exhibiting semiorthogonal decompositions.
For $g=2$ , we can illustrate this conjecture using the following general result for Ringel–Samokhintype decompositions of length 3.
Proposition 2.9. Let X be a smooth projective variety. Assume that we have a semiorthogonal decomposition
where $\mathcal {L}_1,\mathcal {L}_2$ are exceptional line bundles. Then this decomposition is of Ringel–Samokhintype if and only if $\mathcal {L}_1\otimes \mathcal {L}_2^{\vee }$ is a theta characteristic of X.
Proof. A decomposition of the form (16) is determined by the line bundles $\mathcal {L}_1$ and $\mathcal {L}_2$ , as the subcategory $\mathcal {A}$ consists of the objects $E\in \operatorname {\mathrm {{\mathbf {D}}}}^{\mathrm {b}}(X)$ such that
The left dual decomposition of (16) is given by
However, we consider the antiequivalence given by dualizing and tensoring with a line bundle $\mathcal {M}$ to obtain the semiorthogonal decomposition
These two semiorthogonal decompositions agree if and only if $\mathcal {L}\cong \mathcal {L}_2^{\otimes 2}$ , so that we have $(\mathcal {L}_1^{\vee }\otimes \mathcal {L}_2)^{\otimes 2}\cong \omega _X^{\vee }$ , and hence $\mathcal {L}_1\otimes \mathcal {L}_2^{\vee }$ is a theta characteristic of X.
By Proposition 2.5, using that $\operatorname {\mathrm {M}} _C(2,\mathcal {L})$ is of index 2, we obtain the following corollary.
Corollary 2.10. Conjecture 2.8 holds for $g=2$ .
3 Eigenvalue and critical value decompositions
To understand the results in this section, we will briefly recall homological mirror symmetry for Fano varieties. If X is a (smooth projective) Fano variety, its mirror is expected to be a Landau–Ginzburg model $f\colon Y\to \mathbb {A}^1$ (where Y is a quasiprojective variety), such that we have equivalences of triangulated categories
On the first line, we have that the Fukaya category of X has an orthogonal decomposition, indexed by the eigenvalues of $\mathrm {c}_1(X)*_0$ (see [Reference Smith49]). For the matrix factorization category, we have an orthogonal decomposition, indexed by the critical values of f [Reference Orlov46, Proposition 1.14]. Hence, under the homological mirror symmetry conjecture, these sets (of eigenvalues, and critical values) are the same.
For the second line, there is no such natural (semiorthogonal) decomposition. But the philosophy behind Dubrovin’s conjecture (which a priori is only formulated in the case of semisimple quantum cohomology) predicts how an orthogonal decomposition of the triangulated category $\operatorname {\mathrm {Fuk}} (X)$ gives rise to a semiorthogonal decomposition of $\operatorname {\mathrm {{\mathbf {D}}}}^{\mathrm {b}}(X)$ . Generalizations of this philosophy to not necessarily semisimple quantum cohomology are discussed in [Reference Sanda and Shamoto48].
Hence, to understand semiorthogonal decompositions for $\operatorname {\mathrm {{\mathbf {D}}}}^{\mathrm {b}}(\operatorname {\mathrm {M}} _C(2,\mathcal {L}))$ using mirror symmetry, one can study either of the following:

1. the eigenvalues of the quantum multiplication $\mathrm {c}_1(\operatorname {\mathrm {M}} _C(2,\mathcal {L}))*_0$ on the quantum cohomology $\operatorname {\mathrm {QH}}^{\bullet }(\operatorname {\mathrm {M}} _C(2,\mathcal {L}))$ ;

2. the critical values of the potential $f\colon Y\to \mathbb {A}^1$ .
Remark 3.1. There is no worked out candidate construction for the Landau–Ginzburg mirror $(Y,f)$ of $\operatorname {\mathrm {M}} _C(2,\mathcal {L})$ yet, let alone a proof of homological mirror symmetry. But the graph potentials we have constructed and studied in [Reference Belmans, Galkin and Mukhopadhyay8, Reference Belmans, Galkin and Mukhopadhyay7] can be seen as open cluster charts of Y, and gluing these tori together along the birational transformations between them (which we can do because of the compatibilities from [Reference Belmans, Galkin and Mukhopadhyay8, Theorems 2.12 and 2.13]) is a first step one can take in the construction of the Landau–Ginzburg model mirror to $\operatorname {\mathrm {M}} _C(2,\mathcal {L})$ . We will recall this construction in Section 3.2.
A better understanding of the critical loci would be the next step in understanding if and how these graph potentials can be glued together to obtain (part of) the Landau–Ginzburg model. In Appendix A, we discuss the dimension of the different critical loci as a first step in this program.
3.1 Muñoz’s eigenvalue decomposition for quantum cohomology
The eigenvalues of quantum multiplication with $\mathrm {c}_1$ have been described, albeit via an indirect route. The following summarizes this description.
Proposition 3.2 (Muñoz).
The eigenvalue decomposition for the quantum multiplication by $\operatorname {\mathrm {c}}_1(\operatorname {\mathrm {M}} _C(2,\mathcal {L}))$ on $\operatorname {\mathrm {QH}}^{\bullet }(\operatorname {\mathrm {M}} _C(2,\mathcal {L}))$ is
where

1. the eigenvalues are $8(1g),8(2g)\sqrt {1},8(3g),\ldots ,8(g3),8(g2)\sqrt {1},8(g1)$ ;

2. $H_m$ is isomorphic (as a vector space) to $\operatorname {\mathrm {H}}^{\bullet }(\operatorname {\mathrm {Sym}}^{g1m}C)$ .
Proof. This is a combination of various results of Muñoz. First we have [Reference Muñoz40, Proposition 20], describing the eigenvalues of multiplication by the generator of the Picard group on the instanton Floer homology of the 3manifold given by the product of the curve (seen as a real manifold) and $S^1$ . The (conjectural) identification as rings from [Reference Muñoz40, Theorem 1] with the quantum cohomology of $\operatorname {\mathrm {M}} _C(2,\mathcal {L})$ is in turn given by [Reference Muñoz39, Corollary 21] after an explicit description in terms of generators and relations for both rings. Hence, we can interpret any result for instanton Floer homology as a result for quantum cohomology. In [Reference Muñoz40, Conjecture 24], the conjectural decomposition was given, and in [Reference Muñoz38, Corollary 3.7], it was proved.
Property $\mathcal {O}$
An interesting symmetry property for the quantum cohomology of a Fano variety X of index r is obtained by considering the exceptional collection $\mathcal {O}_X,\ldots ,\mathcal {O}_X(r1)$ in $\operatorname {\mathrm {{\mathbf {D}}}}^{\mathrm {b}}(X)$ . The existence of this exceptional collection can be encoded in quantum cohomology using [Reference Galkin, Golyshev and Iritani19, Definition 3.1.1] as follows.
Definition 3.3. Let X be a smooth projective Fano variety, of index $r\geq 1$ . Define
Then we say that X has property $\mathcal {O}$ if

1. T is an eigenvalue of $\mathrm {c}_1(X)*_0$ , of multiplicity 1;

2. if u is another eigenvalue of $\mathrm {c}_1(X)*_0$ such that $u=T$ , then there exists a primitive rth root of unity $\zeta $ such that $u=\zeta T$ .
In [Reference Galkin, Golyshev and Iritani19, Conjecture 3.1.2], it was conjectured that this property holds for all Fano varieties. In [Reference Cheong and Li15], it was checked for all homogeneous varieties $G/P$ , and in [Reference Sanda and Shamoto48, Corollary 7.7], the case of complete intersections of index $r\geq 2$ in ${\mathbb {P}}^n$ was checked.
Hence, by Proposition 3.2, we immediately obtain the following result.
Corollary 3.4. Property $\mathcal {O}$ holds for $\operatorname {\mathrm {M}} _C(2,\mathcal {L})$ , where $T=8(g1)$ .
3.2 Graph potentials
Now we consider the other side of the mirror, and introduce graph potentials as candidate building blocks. Whilst no construction of the homological mirror exists yet as discussed in Remark 3.1 we will study the geometry of the critical values and loci in Section 3.3. Depending on the viewpoint, this gives further evidence that graph potentials are indeed building blocks for the mirror to $\operatorname {\mathrm {M}} _C(2,\mathcal {L})$ , or it gives further evidence for Conjecture A.
Graph potentials
We briefly recall the construction from [Reference Belmans, Galkin and Mukhopadhyay8, §2.1]. Let $\gamma =(V,E)$ be a connected undirected trivalent graph, of genus g. Thus, $\#V=2g2$ and $\#E=3g3$ . As explained in loc. cit., it is natural to define everything in terms of graph (co)homology. The relevant space is
in which a vertex $v\in V$ , which is adjacent to the edges $e_{v_i},e_{v_j},e_{v_k}$ , defines cochains $x_i,x_j,x_k$ by $x_i(e_{v_a})=\delta _{i,v_a}$ .
We will also decorate trivalent graphs, by assigning a color to every vertex (i.e., we consider $c\colon V\to \mathbb {F}_2$ where $c(v)=0$ (resp. $c(v)=1$ ) means the vertex v is uncolored (resp. colored)). An uncolored vertex is drawn as a circle whilst a colored vertex corresponds to a disc .
Construction 3.5. The vertex potential $\widetilde {W}_{v,c(v)}$ for a vertex $v\in V$ in a colored graph $(\gamma ,c)$ is the sum of the four monomials
where $(s_i,s_j,s_k)\in \mathbb {F}_2^{\oplus 3}$ ranges over all sign choices such that $s_i+s_j+s_k=c(v)$ . Here, $x_i$ is the coordinate variables in $\mathbb {Z}[\widetilde {N}_{\gamma }]$ corresponding to the ith edge in an enumeration $e_1,\ldots ,e_{3g3}$ of the edges.
Hence, in the variables $x,y,z$ there are precisely two cases:
The global structure of the colored graph $(\gamma ,c)$ then defines the graph potential as the sum of vertex potentials
Example 3.6. In genus $g=2$ , there are precisely two trivalent graphs: the Theta graph and the dumbbell graph [Reference Belmans, Galkin and Mukhopadhyay8, Figure 1]. In Figure 1, we have shown the Theta graph with the second vertex colored, and following Construction 3.5 we get that the graph potential is given by
More examples are given in [Reference Belmans, Galkin and Mukhopadhyay8, §2.1].
Elementary transformations
Trivalent graphs correspond to pairofpants decompositions, and for a given surface there exist many such decompositions up to isotopy. By Hatcher–Thurston [Reference Hatcher and Thurston23, Appendix], they are related via certain operations, and for our purposes, we only need operation (I) of op. cit. The operation this induces on trivalent graphs is called an elementary transformation, and the local picture is given in Figure 2
In [Reference Belmans, Galkin and Mukhopadhyay8, §2.2], we have discussed how elementary transformations transform the associated graph potentials. The main results (given as Corollary 2.9 and Theorems 2.12 and 2.13 in op. cit.) state that

• up to biregular automorphism of the torus the graph potential only depends on the parity of the coloring;

• up to rational change of coordinates the graph potential only depends on the genus of the trivalent graph.
3.3 Critical value decomposition for graph potentials
Our final goal is to prove Theorem B. In this section, we will describe the critical values and discuss the conifold point of the graph potential. The computation of the critical loci is deferred to Appendix A.
Critical values
Let $\gamma $ be a trivalent graph, and let c be a coloring. By [Reference Belmans, Galkin and Mukhopadhyay8, Corollary 2.9], we can assume that c has, at most, one colored vertex. To determine the critical values of the graph potential $\widetilde {W}_{\gamma ,c}$ we need to determine solutions to the equations
As for the description of the behavior of the graph potential under elementary transformations in the previous section, we will study the local picture for (colored) trivalent graphs. These are given in Figure 3, where Figure 3(a) (resp. Figure 3(b)) corresponds to the case where the coloring is trivial (resp. the vertex $v_2$ is the unique colored vertex).
We can write the potential in the uncolored (resp. colored) case as
resp.
where we have split off the frozen part which does not involve the variable x. We call these expressions (without the frozen part) edge potentials.
Hence, we are interested in solutions to the equations $x\frac {\partial }{\partial x}\widetilde {W}_{\gamma ,0}=0$ , resp. $x\frac {\partial }{\partial x}\widetilde {W}_{\gamma ,c}=0$ , which can be rewritten by clearing the denominators as
resp.
Perfect matchings
The graph potential is defined as a sum over the vertex potentials associated to the vertices. But for certain graphs we can alternatively write the graph potential as a sum of edge potentials. For this we need a perfect matching (i.e., a subset $P\subseteq E$ of edges such that every vertex is contained in precisely one $e\in P$ . If the genus of $\gamma $ is g, then $\#P=g1$ ).
Given a perfect matching P for $\gamma $ , we can rewrite the graph potential by summing over the edges in the matching and using the expressions (29), (30) for the edge potentials. Let e denote the edge between $v_1$ and $v_2$ , and label the variables in Figure 3 as $a_e,b_e,c_e,d_e,x_e$ . Then in the uncolored case we have
whilst in the colored case we let $e_c\in E$ denote the colored vertex and omit the subscript $e_c$ from the variables, so that
By Petersen’s theorem, a trivalent graph which is bridgeless (i.e., we cannot remove an edge to make it disconnected) has at least one perfect matching. Such a graph exists for every genus g. Using [Reference Belmans, Galkin and Mukhopadhyay8, Theorems 2.12 and 2.13], we have that the graph potentials for different graphs of the same genus are related via rational changes of coordinates, hence its critical values do not depend on the choice of graph.
Proposition 3.7. Let $\gamma $ be a trivalent graph. Let c be a coloring with, at most, one colored vertex. Then the following are critical values of the graph potential $\widetilde {W}_{\gamma ,c}$ :

• purely real critical values:
(35) $$ \begin{align} 8g816k \text{ for } k=0,\ldots,g1; \end{align} $$ 
• purely imaginary critical values:
(36) $$ \begin{align} (8g1616k)\sqrt{1} \text{ for } k=0,\ldots,g2. \end{align} $$
The critical value $0$ is listed twice, depending on the parity of g as a purely real or a purely imaginary critical value.
Proof. If we assign $\pm 1$ to the variables associated to the edges in the perfect matching, then it is immediate that assigning $1$ to all variables associated to edges outside the perfect matching gives a solution to the equations (31), (32). If we choose k edges from $P\subseteq E$ to have value $1$ , evaluating (33), (34) gives the value $8g816k$ .
If we assign $\pm \sqrt {1}$ to the variables associated to the edges in the perfect matching, then it is immediate that assigning $\sqrt {1}$ to all variables associated to edges outside the perfect matching gives a solution to the equations (31), (32). If we choose k edges from $P\subseteq E$ to have value $\sqrt {1}$ , evaluating (33), (34) gives the value $(8g1616k)\sqrt {1}$ .
Conifold points
In [Reference Galkin, Golyshev and Iritani19, Remark 3.1.6], a conjecture regarding the value of T from Corollary 3.4 was suggested, relating it to critical values of the mirror Landau–Ginzburg model.
If W is a Laurent polynomial with positive coefficients such that the origin is contained in the Newton polytope (which is the convex hull of the exponents of monomials with nonzero coefficients), then it is shown in [Reference Galkin18] that there is a unique critical point with strictly positive (real) coordinates. This critical point is called the conifold point $x_{\mathrm {con}}$ . We will determine this for the graph potentials.
Define $T_{\mathrm {con}}:=\widetilde {W}_{\gamma ,c}(x_{\mathrm {con}})$ as the value of the potential $\widetilde {W}_{\gamma ,c}$ at the conifold point. The conjecture says that
From the description of the conifold point in Proposition 3.8, we can easily check this equality. Indeed, the graph potential $\widetilde {W}_{\gamma ,c}$ is the sum over the vertex potentials $\widetilde {W}_v$ for $v\in V$ , and the evaluation at the conifold point for each of these is equal to 4. There are $2(g1)$ vertices, so $T_{\mathrm {con}}=8(g1)$ . This agrees with the value T from Corollary 3.4.
Proposition 3.8. Let $\gamma $ be a trivalent graph, and let c be a coloring. The conifold point of the graph potential $\widetilde {W}_{\gamma ,c}$ is given by $(1,\ldots ,1) \in ({\mathbb {C}}^{\times })^{3g3}$ , and the value of the graph potential at the conifold point is $8g8$ .
Proof. This follows from direct computation: for every vertex v we have $\mathrm {d} W_{v,c}(1,1,1) = 0$ and $W_{v,c} (1,1,1) = 4$ . Hence $\mathrm {d} W_{\gamma ,c} = \sum _{v\in V} \mathrm {d} W_v = 0$ and thus
As discussed in the introduction of this section, the eigenvalues of quantum multiplication should correspond to the critical values of a suitable Landau–Ginzburg model. There is a priori no reason why it would suffice to consider a single Laurent polynomial, which is the restriction of the potential to a Zariskiopen torus inside Y.
But it turns out that graph potentials already see all the critical values one expects. This is immediate by comparing Proposition 3.2 to Proposition 3.7. It should be remarked that we have not proven that there are no other critical values, but we conjecture that these are all.
One necessary feature of Landau–Ginzburg mirrors in homological mirror symmetry for Fano varieties is that the critical locus of the potential is compact and that there are no critical points at infinity. Because the critical locus contains higherdimensional components, we see the need for multiple cluster charts, as a positivedimensional proper variety needs multiple affine charts (so in this case at least g, and hence this grows to $\infty $ as the genus grows).
Remark 3.9. By Proposition 3.2, we see that the $\operatorname {\mathrm {H}}^{\bullet }(\operatorname {\mathrm {Sym}}^{g1k}C)$ appears as an eigenspace of the quantum multiplication with eigenvalue of absolute value $8k$ . However, by Appendix A we see that the critical loci all have expected dimension (for the necklace graph). This is in agreement with the appearance of $\operatorname {\mathrm {{\mathbf {D}}}}^{\mathrm {b}}(\operatorname {\mathrm {Sym}}^{g1k}C)$ in the conjectural semiorthogonal decomposition of $\operatorname {\mathrm {{\mathbf {D}}}}^{\mathrm {b}}(\operatorname {\mathrm {M}} _C(2,\mathcal {L}))$ . Further agreement between the two decompositions is provided by the discussion in Appendix A.
4 Decomposition in the Grothendieck ring of varieties
In this section, we compute the class of the moduli space $\operatorname {\mathrm {M}} _C(2,\mathcal {L})$ in the Grothendieck ring of varieties $\mathrm {K}_{0}(\mathrm {Var}/k)$ , and the main result in this section is a proof of Theorem C. This identity gives further evidence for Conjecture A by considering the appropriate motivic measure on $\mathrm {K}_{0}(\mathrm {Var}/k)$ as in Corollary 4.1.
4.1 The Grothendieck ring of varieties and motivic measures
Recall that the Grothendieck ring of varieties is generated by the isomorphism classes $[X]$ of algebraic varieties over k, modulo the relations $[X]=[U]+[Z]$ for $Z\hookrightarrow X$ a closed subvariety, and $U=X\setminus Z$ its complement. The product of varieties induces the multiplicative structure, such that $[\mathrm {pt}]$ is the unit. An important element of this ring is the class of the affine line ${\mathbb {L}}=[{\mathbb {A}}^1]=[{\mathbb {P}}^1][\mathrm {pt}]$ , also called the Lefschetz class.
An alternative presentation for this ring in characteristic 0, relevant to our goal, is the Bittner presentation from [Reference Bittner12]. It says that $\mathrm {K}_{0}(\mathrm {Var}/k)$ is isomorphic to the ring generated by isomorphism classes $[X]$ of smooth and proper algebraic varieties over k, modulo the relations $[\operatorname {\mathrm {Bl}}_ZX][E]=[X][Z]$ for $Z\hookrightarrow X$ a smooth closed subvariety, and $E\to Z$ the exceptional divisor in the blowup $\operatorname {\mathrm {Bl}}_ZX\to X$ .
Via the cutandpaste relations, one can obtain the standard identities
if $X\to Y$ is a Zariskilocally trivial fibration with fibers F, and
In order to use this alternative presentation, and because we depend on other results in the literature which are only phrased for algebraically closed fields (but likely hold more generally), we will work over an arbitrary algebraically closed field k of characteristic 0.
Motivic measures
A motivic measure is a ring morphism whose domain is the Grothendieck ring of varieties. In our setting we consider the motivic measure
from [Reference Bondal, Larsen and Lunts13, §8], obtained by using the Bittner presentation and Orlov’s blowup formula, and sending $[X]$ to the class of the (unique) dg enhancement of $\mathrm {K}_{0}(\mathrm {dgCat}/k)$ . Then the identity (3) matches up with Conjecture A.
The Grothendieck ring of dg categories $\mathrm {K}_{0}(\mathrm {dgCat}_k)$ precisely encodes semiorthogonal decompositions, as it is generated by the quasiequivalence classes $[\mathcal {C}]$ of smooth and proper pretriangulated dg categories, modulo the relations
for every semiorthogonal decomposition $\mathcal {C}=\langle \mathcal {A},\mathcal {B}\rangle $ .
Hence, the image of the equality (3) is consistent with Conjecture A (up to 2torsion, as ${\mathbb {L}}\mapsto 1$ ), as explained by the following corollary.
Corollary 4.1. We have the equality
in $\mathrm {K}_{0}(\mathrm {dgCat}/k)$ , for some class $T'$ such that $2\cdot T'=0$ .
Again we expect that $T'=0$ , but our method of proof is not strong enough to remove this error term.
This is precisely the equality induced by the semiorthogonal decomposition from Conjecture A and therefore can be seen as further evidence for it.
Comparison to other results
Motivic and cohomological invariants of $\operatorname {\mathrm {M}} _C(2,\mathcal {L})$ have been an active topic of interest for a long time, and many tools are used for this. From the identity (3), it is possible to deduce known results on certain invariants of $\operatorname {\mathrm {M}} _C(2,\mathcal {L})$ by taking the appropriate motivic measures. Examples of this are given by

• the Betti polynomial (with values in ${\mathbb {Z}}[t]$ ) [Reference Newstead45],

• the Hodge–Poincaré polynomial (with values in ${\mathbb {Z}}[x,y]$ ) [Reference del Baño1, Corollary 5.1].
It would be interesting to weaken the assumptions on the field k for (3) and to also recover pointcounting realizations [Reference del Baño1, Corollary 5.4].
With a view towards providing evidence for Conjecture A, Lee has given in [Reference Lee34, Theorem 1.2] an isomorphism similar to (3) which holds in any semisimple category of motives, such as the category of numerical motives. It is based on the isomorphism [Reference del Baño2, Theorem 2.7] due to del Baño, which also requires this semisimplicity.
This result is stronger in the sense that it is known that the error term T vanishes. But it only works for (the Grothendieck group of) a semisimple category of motives. The motivic measure
which sends the class $[X]$ of a smooth projective variety to the class $[\mathfrak {h}(X)]$ of its Chow motive, allows one to obtain a strengthening (modulo vanishing of T) of [Reference Lee34, Theorem 1.2]. See also [Reference del Baño2, Remark 2.8].
4.2 The Thaddeus picture
We break up the proof of Theorem C into several steps. The main idea of the proof is to use Thaddeus’ variation of GIT for moduli of stable pairs and compare the class of the variety when we cross a wall using a telescopic sum. This is the ‘Thaddeus picture’ alluded to in the title of this section, and it will be given in (47).
Setup
Let d be an odd integer which is greater than $2g2$ . Let $\mathcal {L}$ be a line bundle of degree d, and let $M^d_0={\mathbb {P}}(\operatorname {\mathrm {H}}^1(C,\mathcal {L}^{\vee }))$ . For each $i \in \{1,\dots , (d1)/2\}$ , Thaddeus constructs in [Reference Thaddeus51] two smooth projective varieties $M^d_i$ and $\widetilde {M}^d_i$ such that $\widetilde {M}_i^d$ is

• a blowup of $M_{i1}^d$ along a projective bundle ${\mathbb {P}}(\mathcal {W}_{i}^{})$ over $\operatorname {\mathrm {Sym}}^iC$ ;

• a blowup of $M_i^d$ along a projective bundle ${\mathbb {P}}(\mathcal {W}_{i}^{+})$ over $\operatorname {\mathrm {Sym}}^iC$ .
Here, $\mathcal {W}_i^{}$ (respectively $\mathcal {W}_i^{+}$ ) is vector bundle on $\operatorname {\mathrm {Sym}}^iC$ of rank i (respectively of rank $d+g2i1$ ).
This leads us to the following flip diagram:
Summarizing the situation from [Reference Thaddeus51] for all i we obtain the following theorem.
Theorem 4.2 (Thaddeus).
With the above notation:

1. There is a flip $M^d_{i1} \dashrightarrow M^d_{i}$ with center $\operatorname {\mathrm {Sym}}^iC$ and type $(i,d+g2i1)$ for each $i \in \{1,\dots , (d1)/2\}$ .

2. We have that $M_0^d\cong {\mathbb {P}}^{d+g2}$ and $\widetilde {M}_1\cong M_1^d$ .

3. There is a natural map
(46) $$ \begin{align} \pi: M^d_{(d1)/2}\rightarrow \operatorname {\mathrm {M}}_C(2,\mathcal{L}) \end{align} $$with fiber ${\mathbb {P}}(\operatorname {\mathrm {H}}^0(C,\mathcal {E}))$ over a stable bundle $\mathcal {E}$ in $\operatorname {\mathrm {M}} _C(2,\mathcal {L})$ . Moreover, if $d\geq 4g3$ , then $\pi $ is a projective bundle associated to a vector bundle of rank $d+2(1g)$ .
This yields the following picture, which we refer to as the ‘Thaddeus picture’:
All morphisms in the diagram, except $\pi $ , are blowups.
To minimize the dimension of the moduli spaces and the number of steps involved in the construction, we will take $d=4g3$ . Then

1. $\dim M_i=5g5$ ;

2. we are considering moduli spaces $M_0^{4g3},\ldots ,M_{2g2}^{4g3}$ ;

3. the morphism $M_{2g2}$ is a ${\mathbb {P}}^{2g2}$  fibration.
Setup for the proof
We now give some easy lemmas, as a setup for the proof in Section 4.3. The following lemma follows directly from Theorem 4.2 and (39).
Lemma 4.3. Let d be an odd integer greater than $4g4$ . Then we have that
in $\mathrm {K}_{0}(\mathrm {Var}/k)$ .
Using the description from Lemma 4.3 for $d=4g3$ and $d=4g1$ and computing the difference gives the following description. One could take a less optimal choice, at the cost of obtaining a larger coefficient in Theorem C.
Lemma 4.4. We have that
in $\mathrm {K}_{0}(\mathrm {Var}/k)$ .
For every flip diagram, as in (47), one can prove the following, where only standard bookkeeping techniques are required.
Proposition 4.5. Let d be an odd integer greater than $4g4$ . Then for $i=1,\ldots ,(d1)/2$ , the difference of the classes $[M^d_{i}]$ and $[M^d_{i1}]$ satisfies
in $\mathrm {K}_{0}(\mathrm {Var}/k)$ .
Sketch of proof.
By applying the blowup formula and the projective bundle formula twice with the appropriate codimension and relative dimension, we get the equalities
and subtracting them gives the result.
Finally, let us recall the following fundamental identity encoding the behavior of Abel–Jacobi morphisms, as discussed in [Reference Heinloth24, §3].
Proposition 4.6. Let C be a curve of genus g. Let $e\geq 0$ and denote $a=(g1)+e$ . Then we have an equality
in $\mathrm {K}_{0}(\mathrm {Var}/k)$ .
Observe that under our assumptions we have isomorphisms $\operatorname {\mathrm {Pic}} ^iC\cong \operatorname {\mathrm {Jac}} C$ for all $i\in {\mathbb {Z}}$ by the existence of a rational point.
4.3 Proof of the decomposition
We will restrict ourselves to the cases where $d=4g3$ and $4g1$ (i.e. the first two degrees for which the morphism $\pi $ in (47) is an equidimensional projective fibration). We will compare different stages of the Thaddeus picture, using the following notation for the difference of classes of moduli of stable pairs, where $i=0,\ldots ,2g2$ .
For notational convenience, we also set $\delta M^{}_{1}:=0$ .
Now for $i=0,\ldots ,2g2$ , we define
We can describe the classes $X_i$ in the following way.
Proposition 4.7. For $i=0,\ldots ,2g2$ we have that
Proof. The proof follows from the definition of $X_i$ and by applying Proposition 4.5 via the following chain of equalities.
We now use Proposition 4.5 and rewrite $X_{i}$ as
Using Propositions 4.6 and 4.7, we obtain the following result, where we use the following polynomial in ${\mathbb {L}}$ :
Proposition 4.8. For $i=0,\ldots ,g2$ , we have that
Proof. By Proposition 4.6, we get
We apply Proposition 4.7 for $i\in \{0,1,\dots , g2\}$ to (60) to get
Now by Proposition 4.7 for $i\leq g1$ , we get $X_i={\mathbb {L}}^i(1+{\mathbb {L}})[\operatorname {\mathrm {Sym}}^iC].$ Thus, the proof follows.
The following proposition is an important step in the proof of Theorem C. It shows that there are no contributions of the Jacobian of C to the class of $\operatorname {\mathrm {M}} _C(2,\mathcal {L})$ .
Proposition 4.9. We have that
in $\mathrm {K}_{0}(\mathrm {Var}/k)$ .
Proof. First let us simplify the left hand side of the above equation. We have that
We will be done if we can show that the above expression multiplied by the class of $\operatorname {\mathrm {Jac}} (C)$ is equal to $[M_{2g2}^{4g1}][M_{2g1}^{4g1}]$ . First, by Proposition 4.6 where we take $e=g$ , we get
Combining this with Proposition 4.5, we get
Now we use that
Thus, from (66) we get
and we are done.
Now we are ready to complete the proof of Theorem C.
Proof of Theorem C.
First we write the class of $\operatorname {\mathrm {M}} _C(2,\mathcal {L})$ in terms of the difference of classes of the smooth projective varieties considered by Thaddeus. We apply Lemma 4.4 to get
Now we can write
where $X_i$ ’s are as used in Proposition 4.8. Thus, applying Proposition 4.8 we get
where the last step is by Proposition 4.9.
4.4 Motivic zeta functions and a Hardertype formula
As an application of the results above we can give an analogue of Harder’s point counting formula from [Reference Harder21] (see also [Reference del Baño2, Corollary 2.11]). We do this by exhibiting an identity in the Grothendieck ring of varieties. Note however that the ground field cannot be chosen to be ${\mathbb {F}}_q$ . Hence, Harder’s formula cannot be obtained by applying the motivic measure
For any variety X, we denote the nth symmetric $\operatorname {\mathrm {Sym}}^nX$ to be $X^n/\operatorname {\mathrm {Sym}}_n$ , where $\operatorname {\mathrm {Sym}}_n$ is the symmetric group of n letters. All the symmetric powers can be put together to give Kapranov’s motivic zeta function, as introduced in [Reference Kapranov26, §1.3]:
This is a universal version of the Hasse–Weil zeta function, valid for arbitrary ground fields, where the counting measure for $k={\mathbb {F}}_q$ gives the usual Hasse–Weil zeta function.
The following theorem is due to Kapranov [Reference Kapranov26, Theorem 1.1.9] and shows how the motivic zeta function has properties similar to the usual Hasse–Weil zeta function.
Theorem 4.10 (Kapranov).
Let C be a smooth curve. The motivic zeta function of C is a rational function of the following form:
where $F_{2g}(t)$ is a polynomial of degree $2g$ . Moreover, ${\mathrm {Z}_{\mathrm {Kap}}}(C,t)$ (resp. $F_{2g}(t)$ ) satisfies the functional equation
resp.
We can rearrange and reinterpret the terms in the motivic zeta function as follows.
Proposition 4.11. We have the identity
in $\mathrm {K}_{0}(\mathrm {Var}/k)$ .
Proof. Using Proposition 4.6, we have the identityFootnote ^{1}
It now suffices to see that
which follows from an immediate verification.
The following lemma is also an immediate verification.
Lemma 4.12. We have the identity
in $\mathrm {K}_{0}(\mathrm {Var}/k)$ .
Using Proposition 4.11 and Lemma 4.12, we can then obtain the following corollary to Theorem C. The righthand side is an element of $\mathrm {K}_{0}(\mathrm {Var}/k)$ (and not some completion) by Theorem 4.10, so that the first term of the righthand side is in fact $F_{2g}({\mathbb {L}})$ .
Corollary 4.13. We have the identity
in $\mathrm {K}_{0}(\mathrm {Var}/k)$ .
Proof. It suffices to rewrite $[{\mathbb {P}}^{gi2}]$ as $\frac {1{\mathbb {L}}^{gi1}}{1{\mathbb {L}}}$ and then multiply both sides of (76) with $(1{\mathbb {L}})(1{\mathbb {L}}^2)$ and apply Lemma 4.12.
This identity is then the analogue of Harder’s formula from [Reference del Baño2, Corollary 2.11] using the functional equation for the zeta function.
A Dimension of critical loci
We will now determine the dimensions of the different components of the critical loci, which are part of Theorem B. The goal is the following proposition.
Proposition A.1. Let $\Gamma _{g,g1}$ denote the necklace graph of genus g, with the last vertex colored as in Figure 4. Let $\widetilde {W}_g$ denote the associated graph potential. Then the dimension of the critical locus of $\widetilde {W}_g$ over the critical value with absolute value $8g88k$ for $k=0,\ldots ,g1$ is $g1$ .
The proof of Proposition A.1 is a lengthy computation in order to set up an inductive description of the critical loci.
Instead of the coordinates $x_i,y_i,z_i$ , we will work with the coordinates
which makes the graph potential easier to work with.
The necklace graph from Figure 4 can be decomposed in two ways, using beads and strings. In the coordinates $u_i,v_i,z_i$ , we describe the local pictures and the labelling of the variables in Figure 5.
We will moreover use the following notation for two functions on ${{\mathbb {G}}_{\mathrm {m}}}$ :
so that $x\frac {\partial }{\partial x}J^+(x)=J^(x)$ .
With this notation and the decomposition of the necklace graph, the graph potential can be written as the sum of either bead potentials or string potentials. Here the ith bead potential is given as
and the ith string potential is given as
This allows us to write
In the description of the critical locus, we therefore have the equations
for $i=1,\ldots ,g2$ , and
Likewise, we obtain the conditions
for $i=2,\ldots ,g1$ , and
We need to analyse the system of equations given by (86), (87), (88) and (89), so that we can describe its solutions and compute the value of the graph potential at these solutions.
The expected dimension of the connected component of the critical locus with critical value given by $8(g1k)\sqrt {1}^{1+(1)^{k+1}}$ (for $k=0,\ldots ,2g2)$ ) is k for $k=0,\ldots ,g1$ , and $gk+1$ for $k=g,\ldots ,2g2$ . For $k=0,2g2$ we expect isolated critical points, whilst for $k=g1$ we obtain critical value 0 and expect a critical locus of dimension $g1$ .
Under the assumptions (88) and (89), it is possible to rewrite the string potentials by removing the $z_i^{1}$ so that the graph potential can be expressed as
We will get rid of the factor 2 because we are interested in the values $4(g1k)\sqrt {1}^{1+(1)^{k+1}}$ for the resulting expression.
The case $u_i^2=v_i^2=1$
As a starting point, we consider the situation in which the conditions (86) and (87) are satisfied by ensuring that $J^(u_i)=J^(v_i)=0$ for all $i=1,\ldots ,g1$ . We need to analyse the condition on the remaining variables $z_i$ , as well as what the resulting critical loci and critical values are.
The condition on the variables $u_i$ and $v_i$ ensures that $u_i=\pm 1$ and $v_i=\pm 1$ for all $i=1,\ldots ,g1$ .
We will encode the sign choice on the variables of the ith string potential as a matrix, so that $(\begin {smallmatrix} s_{i1}^u & s_i^u \\ s_{i1}^v & s_i^v \end {smallmatrix})$ (resp. $(\begin {smallmatrix} s_{1}^u & s_{g1}^u \\ s_{1}^v & s_{g1}^v \end {smallmatrix})$ ) denotes the sign choice of the varaibles $u_i$ and $v_i$ .
Lemma A.2. Let $u_i^2=v_i^2=1$ for all $i=1,\ldots ,g1$ . If for some string the sign choice is inadmissible (i.e., the parity of the signs in the sign matrix is odd), then at least one of the equations (88) and (89) cannot be satisfied.
Assume that, for every string, the sign choice is admissible (i.e., the parity is even). Then we have that

1. the variables $z_i$ are either free or are necessarily equal to $\pm 1,\pm \sqrt {1}$ ;

2. the critical loci have the expected dimension.
Proof. Observe that $J^+(\pm 1)=\pm 2$ , so that the coefficients of the $z_i^{\pm 1}$ in (88), (89) are $4,0,4$ . If the sign choice is inadmissible for the ith string, then in (88) exactly one of the coefficients of $z_i^{\pm 1}$ vanishes, but the system then has no solution in $(\mathbb {C}^{\times })^{3g3}$ .
If the sign choice is admissible, we can analyse the system as in Table 1. We describe $z_i\frac {\partial }{\partial z_i}\widetilde {W}_g$ and deduce the condition it imposes on $z_i$ . This proves the first part.
The dimension of the critical loci remains to be understood. For this, we consider the expression of the necklace graph potential as the sum of string potentials. The expression in (88), (89) is equal to (84) up to a sign.
Observe that the parity condition for consecutive strings force the choices of $z_i$ to be either all real or all imaginary. It follows that the critical loci have the expected dimension by the evaluation of the string potentials as given in Table 1.
Inductive description
We now consider the case where there is at least one $u_i^2\neq 1$ or $v_i^2\neq 1$ . The goal is to relate the equations of the critical locus for the necklace graph of genus g to that of the necklace graph of genus $g2$ and $g1$ .
Hence, we first need to describe what happens for $g=2$ and $g=3$ .
Lemma A.3. Proposition A.1 holds for $g=2,3$ .
Proof. For $g=2$ , we have $\widetilde {W}_2=J^+(z)(J^+(u)+J^+(v))$ and thus the system of partial derivatives
so that a casebycase analysis yields

• $J^+(z)=0$ and $J^+(u)+J^+(v)=0$ corresponds to 1dimensional critical loci with critical value 0;

• $J^(z)=0$ and $J^(u)=J^(v)=0$ corresponds to 0dimensional critical loci with critical value $\pm 8$ .
The outcome for $g=3$ is similar, using the reduction methods from the next lemmas, except for the case where $u_1^2\neq 1$ and $z_1+z_2^{1}=0$ where the reduction methods would yield $g=1$ . If $u_1^2\neq 1$ , then $z_1=z_2$ and $z_1=z_2^{1}$ by assumption. Then $z_2=\pm 1$ and $z_1=\mp 1$ . Now cancelling $z_1$ and $z_2$ gives
which implies $J(u_1)=J(v_1)$ . Substituting this, we get $J^+(u_2)=J^+(v_2)$ , and thus this critical locus has dimension 2 and critical value 0.
We can assume that $u_1^2\neq 1$ by the rational change of coordinates in the change of coloring [Reference Belmans, Galkin and Mukhopadhyay8, §2.2]. From the condition (86) for $u_1$ , we get
and thus condition (86) for $v_1$ is always satisfied.
We will first analyse the case where $z_2+z_3=0$ or $z_1+z_{g1}^{1}=0$ . Then we will be done once we’ve dealt with the case $z_1+z_{g1}^{1}\neq 0$ and $z_2+z_3\neq 0$ .
Lemma A.4. Let $u_1^2\neq 1$ . Assume that $z_2+z_3=0$ . Then the critical loci have the expected dimension.
Proof. By (93) and the assumption, we have that $z_1=z_3$ . The equations (86) for $i=1,2$ are automatically satisfied and can be ignored.
We can rewrite (88) for $i=2$ as
and (89) as
By summing these equations with (88) for $i=3$ , we obtain
Finally, observe that the first few terms in the expression of the graph potential from (90) read
so that by cancellations from the equalities $z_1=z_2=z_3$ , it reduces to that of genus $g2$ , and the soughtafter critical values are of the form $4((g2)1(k2))\sqrt {1}^{1+(1)^{k+1}}$ for $k=1,\ldots ,2g3$ .
Hence, we have reduced the system of equations and the graph potential for the necklace graph of genus g to that of genus $g2$ . By the base case from Lemma A.3, the fact that the critical loci for the necklace graph of genus $g2$ are of the expected dimension, and that we used the two equations $z_1+z_2=0$ and $z_2+z_3=0$ , we obtain that the critical loci for $k=1,\ldots ,2g3$ are also of the expected dimension $k+2$ .
A similar analysis shows the following.
Lemma A.5. Let $u_1^2\neq 1$ . Assume that $z_1+z_{g1}^{1}=0$ . Then the critical loci have the expected dimension.
Finally, we prove the following result.
Lemma A.6. Let $u_1^2\neq 1$ . Assume that $z_1+z_{g1}^{1}\neq 0$ and $z_2+z_3\neq 0$ . Then the critical loci have the expected dimension.
Proof. By (86) for $i=2$ and (87), we obtain that $u_2^2=v_2^2=u_{g1}^2=v_{g1}^2=1$ . This brings us into a situation reminiscent to that of Lemma A.2, but we will only consider the sign of $(\begin {smallmatrix} u_2 & u_{g1} \\ v_2 & v_{g1} \end {smallmatrix})$ .
We can rewrite (88) for $i=2$ as
and (89) as
so that
In case the parity is odd, then either $J^+(u_2)=J^+(v_{g1})$ and $J^+(v_2)\neq J^+(u_{g1})$ , or vice versa. But the equality of (100) forces both equalities in case one equality holds, so this situation does not arise.
In case the parity is even, we have listed the values of $J^+$ in Table 2. In the first four cases, we observe that $J^+(v_2)=J^+(v_{g1})=\pm 2$ and $J^+(v_2)=J^+(u_{g1})=\pm 2$ for the appropriate choice of signs. This implies that
hence the choice of $u_1$ and $v_1$ determine $z_1$ and $z_2$ up to a sign. The equations become those for genus $g2$ , with the additional constraint that $u_{g1}^2=v_{g1}^2=1$ . Now projecting along $u_1$ and $v_1$ reduces the equations, and we get that the dimension of the critical locus indexed by k (and genus g) is two less than that of the critical locus indexed by $k2$ and genus $g2$ , hence of expected dimension.
The last four cases remain to be studied. Now the induction will not go down by two, but rather by one. Hence, we need to ensure the factor $\sqrt {1}$ comes into play.
We claim that for $(\begin {smallmatrix} + & + \\  &  \end {smallmatrix})$ and $(\begin {smallmatrix}  & + \\  & + \end {smallmatrix})$ , resp. $(\begin {smallmatrix}  &  \\ + & + \end {smallmatrix})$ and $(\begin {smallmatrix} + &  \\ + &  \end {smallmatrix})$ , we have that

1. $J^+(u_1)=J^+(v_1)$ and $z_1^2=z_2^2=1$ ;

2. $J^+(u_1)=J^+(v_1)$ and $z_1^2=z_2^2=1$ .
We explain this in the case of $(\begin {smallmatrix} + & + \\  &  \end {smallmatrix})$ , the other cases being analogous. From Table 2 and (100), we obtain
This can be rewritten as
This is only possible if $J^+(u_1)=J^+(v_1)$ . But we have that $J^+(u_1)\neq \pm 2$ , so (89) implies that $z_1^2=1$ . Similarly, (88) for $i=2$ implies that $z_2^2=1$ . We will assume $z_1=z_2^{1}=\sqrt {1}$ , and the other cases are similar.
By the sign choices, we have $J^+(u_2)=J^+(u_{g1})=2$ , $J^+(u_{g1})=J^+(v_2)=2$ . We will now use the expression of the graph potential as a sum of bead potentials, together with the partial evaluation we can do using the equalities from above. Omitting a tedious manipulation, the presence of $z_1=z_2=\sqrt {1}$ will ensure that the resulting Laurent polynomial is the graph potential for the necklace graph of genus $g1$ , except that it is multiplied with $\sqrt {1}$ , so that all critical values are rotated by 90 degrees.
Acknowledgements
We want to thank Emanuele Macrì, M.S. Narasimhan, and Maxim Smirnov for interesting discussions.
This collaboration started in Bonn in January–March 2018 during the second author’s visit to the “Periods in Number Theory, Algebraic Geometry and Physics” Trimester Program at the Hausdorff Center for Mathematics (HIM) and the first and third author’s stay at the Max Planck Institute for Mathematics (MPIM). The remaining work was done at the Tata Institute for Fundamental Research (TIFR) during the second author’s visit in December 2019–March 2020 and the first author’s visit in February 2020. We would like to thank HIM, MPIM and TIFR for the very pleasant working conditions.
Conflict of Interest
The authors have no conflict of interest to declare.
Financial support
The first author was partially supported by the FWO (Research Foundation–Flanders). The third author was partially supported by the Department of Atomic Energy, India, under project no. 12R&DTFR5.010500 and also by the Science and Engineering Research Board, India (SRG/2019/000513).