Topological K-theory of quasi-BPS categories for Higgs bundles

Let $C$ be a smooth projective curve of genus $g$, and let the group $G$ be either $\mathrm{GL}(r),\mathrm{SL}(r)$ or $\mathrm{PGL}(r)$. We denote by

$\displaystyle\pi\colon M_{G}(\chi)\to B$

the moduli space of semistable $G$-Higgs bundles with Euler characteristic $\chi$ together with the Hitchin fibration with Hitchin base $B$. In the case of $G=\mathrm{GL}(r)$, it consists of pairs

$\displaystyle(F,\theta),\ \theta\colon F\to F\otimes\Omega_{C},$

where $F$ is a vector bundle on $C$ with $(\operatorname{rank}(F),\chi(F))=(r,\chi)$ and $(F,\theta)$ satisfies a stability condition.

In general, the stack $M_{G}(\chi)$ is singular, however it is a smooth Deligne-Mumford stack if $(r,\chi)$ are coprime. Suppose that both of $\chi$ and $w$ are coprime with $r$. In this case, Hausel–Thaddeus [HT03] proposed that the pair of smooth Deligne-Mumford stacks (together with some Brauer classes)

$\displaystyle(M_{G}(\chi),M_{G^{L}}(w))$

is a mirror pair [SYZ01]. In particular, Hausel–Thaddeus proposed the equality of the stringy Hodge numbers of $M_{G}(\chi)$ and $M_{G^{L}}(w)$, which was proved by Groechenig–Wyss–Ziegler [GWZ20]. At the categorical level, one expects a derived equivalence [DP12] (called “the Dolbeault Langlands equivalence” [BZN18]):

(1.1)

$\displaystyle D^{b}(M_{G}(\chi),\alpha^{-w})\stackrel{{\scriptstyle\sim}}{{\to}}D^{b}(M_{G^{L}}(w),\beta^{\chi}),$

where $\alpha,\beta$ are some canonical Brauer classes, see for example [Hau22, Section 2.4]. Heuristically (following Donagi–Pantev [DP12]), the conjectural equivalence (1.1) may be regarded as a classical limit of the geometric de Rham Langlands correspondence [AG15]:

(1.2)

$\displaystyle\mathrm{Ind}_{\mathcal{N}}D^{b}(\mathcal{L}\mathrm{ocSys}_{G})\simeq\text{D-mod}(\mathcal{B}\mathrm{un}_{G^{L}}).$

Here $\mathcal{L}\mathrm{ocSys}_{G}$ is the moduli stack of $G$-flat connections on $C$ and $\mathcal{B}\mathrm{un}_{G^{L}}$ is the moduli stack of $G^{L}$-bundles on $C$. The equivalence (1.1) may be regarded as an extension over the full Hitchin base of the Fourier-Mukai equivalence between dual abelian schemes, which are constructed from the Picard schemes of the smooth spectral curves. The equivalence (1.1) should match Hecke and Wilson operators on the two sides [KW07, DP12, Hau22], but we do not discuss this aspect in the paper.

The existence of an equivalence (1.1) is an open problem in most of the cases. Besides the equality of Hodge stringy numbers conjectured in [HT03], it is known that versions of (1.1) hold in cohomology (by Groechenig–Wyss–Ziegler [GWZ20], see also [LW21]), for Hodge structures (by Maulik–Shen [MS21]), for Chow and Voevodsky motives (by Hoskins–Pepin Lehalleur [HLb]), and for relative Chow motives (by Maulik–Shen–Yin [MSY]). Groechenig–Shen considered a comparison in topological K-theory [GS], and showed an equivalence of (integral) topological K-theories spectra:

(1.3)

$\displaystyle K^{\rm{top}}(M_{G}(\chi),\alpha^{-w})\stackrel{{\scriptstyle\sim}}{{\to}}K^{\rm{top}}(M_{G^{L}}(w),\beta^{\chi}).$

The first main difficulty in proving the equivalence (1.1) is the construction of a candidate Fourier-Mukai kernel, which should be an extension of the Poincaré sheaf, which is initially defined only over the locus of smooth spectral curves. Such an extension was constructed by Arinkin [Ari13] over the elliptic locus (when the spectral curve is reduced and irreducible), by Melo–Rapagnetta–Viviani [MRV19a, MRV19b] over the locus of reduced spectral curves, and by Li [Li21] for rank two meromorphic Higgs bundles. However, this is not an impediment in proving the statements in [GWZ20, LW21, MS21, HLb, GS, MSY], which follow from a good understanding of the comparison of the two sides over the elliptic locus. For example, any extension over the full Hitchin base of Arinkin’s kernel induces the isomorphism (1.3).

The purpose of this paper is to give a version of the equivalence (1.3) for integers $\chi,w$ which are not necessary coprime with $r$, using the quasi-BPS categories studied in the previous paper [PTa].

So far in the literatures, the studies of mirror symmetry of Higgs bundles have been restricted to the case when $(r,\chi)$ are coprime. In this case, the moduli space $M_{G}(\chi)$ is a smooth Deligne-Mumford stack, and its derived category has several nice properties, for example it is smooth over $\mathbb{C}$ and proper over the Hitchin base $B$.

However, it is important to study the (derived) moduli stacks of semistable Higgs bundles $\mathcal{M}_{G}(\chi)$ for general $\chi$. First, all such moduli stacks are used in the definition of the categorical Hall algebra of the surface $\mathrm{Tot}_{C}(\Omega_{C})$ [PS23], and needed to be studied in order to categorify theorems known for (Kontsevich–Soibelman [KS11]) cohomological Hall algebras [KK, DHSM].

Second, the stack $\mathcal{L}\mathrm{ocSys}_{G}$ degenerates to $\mathcal{M}_{G}(0)$, see [Sim97, Proposition 4.1]. Thus, when studying the limit of the de Rham Langlands equivalence (1.2) (following Donagi–Pantev), the limit of the right hand side should be a category of ind-coherent sheaves on $\mathcal{M}_{G}(0)$.

Third, for $\mathrm{G}=\mathrm{SL}(r)$, principal $\mathrm{SL}(r)$-Higgs bundles have degree zero, thus their Euler characteristic is equal to $r(1-g)$, which is divisible by $r$. Therefore, considering quasi-BPS categories for $\chi=r(1-g)$ is essential in the categorical study of principal $\mathrm{SL}$-Higgs bundles.

In [PTa], we introduced some admissible subcategories, called quasi-BPS categories:

(1.4)

$\displaystyle\mathbb{T}_{G}(\chi)_{w}\subset D^{b}(\mathcal{M}_{G}(\chi))_{w}$

for $w\in\mathbb{Z}$ corresponding to a weight with respect to the action of the center of $G$. The construction of the categories (1.4) is part of the problem of categorifying BPS invariants on Calabi-Yau 3-folds [PTc, PTe, PTd], or more generally of categorifying the constructions and theorems from (numerical or cohomological) Donaldson-Thomas theory [Tod24]. Indeed, the quasi-BPS categories (1.4) have analogous properties to the BPS cohomology (defined by Kinjo–Koseki [KK] and Davison–Hennecart–Schlegel Mejia [DHSM]) of the local Calabi-Yau threefold $X=\mathrm{Tot}_{C}(\Omega_{C})\times\mathbb{A}^{1}_{\mathbb{C}}$, see [PTa] for more details. In this paper, we make this relation precise by computing the topological K-theory of quasi-BPS categories in terms of BPS cohomology, see Proposition 4.1, Proposition 4.5, and Theorem 6.9.

If the vector $(r,\chi,w)$ is primitive, i.e. $\gcd(r,\chi,w)=1$, the category (1.4) is smooth over $\mathbb{C}$ and proper over the Hitchin base $B$. In this case, we regard it as a non-commutative analogue of the Hitchin system. Note that neither $\chi$ nor $w$ may be coprime with $r$, even if $(r,\chi,w)$ is primitive. In [PTa], we conjectured the existence of an equivalence

(1.5)

$\displaystyle\mathbb{T}_{G}(w)_{-\chi}\stackrel{{\scriptstyle\sim}}{{\to}}\mathbb{T}_{G^{L}}(\chi)_{w}$

extending the Donagi–Pantev equivalence over the locus of smooth spectral curves, which is nothing but the equivalence (1.1) if both $\chi$ and $w$ are coprime with $r$.

The purpose of this paper is to provide evidence towards the equivalence (1.5), namely to prove that extensions of the Poincaré sheaf induce an isomorphism of the (rational) topological K-theories of the two categories in (1.5). We thus obtain a generalization of the Groechenig–Shen theorem (1.3) beyond the coprime case. Note that (1.3) holds integrally, and that we also prove versions for integral topological K-theory for twisted Higgs bundles.

For a dg-category $\mathscr{D}$, Blanc [Bla16] introduced its topological K-theory spectrum

$\displaystyle K(\mathscr{D})\in\mathrm{Sp}.$

We denote by $K(\mathscr{D})_{\mathbb{Q}}:=K(\mathscr{D})\wedge H\mathbb{Q}$ its rationalization.

We use the following notations

$\displaystyle\mathbb{T}(r,\chi)_{w}:=\mathbb{T}_{\mathrm{GL}(r)}(\chi)_{w},\ \mathbb{T}_{\mathrm{SL}(r),w}:=\mathbb{T}_{\mathrm{SL}(r)}(r(1-g))_{w},\ \mathbb{T}_{\mathrm{PGL}(r)}(\chi):=\mathbb{T}_{\mathrm{PGL}(r)}(\chi)_{0}.$

We also use the notation $\mathbb{T}(r,\chi)_{w}^{\rm{red}}$ for the reduced quasi-BPS category, which is a category obtained from the usual quasi-BPS categories by removing a redundant derived structure from $\mathcal{M}_{G}(\chi)$. The following is the main theorem in this paper:

The main ingredient in the proof of the above theorem is a computation of the (rational) topological K-theory of BPS categories. For example, for $G=\mathrm{GL}(r)$, we show in Theorem 6.9 that

(1.7)

$\displaystyle\mathcal{K}_{M}^{\rm{top}}(\mathbb{T}(r,\chi)_{w}^{\rm{red}})_{\mathbb{Q}}\cong\mathcal{BPS}_{M}[\beta^{\pm 1}],$

where $\mathcal{K}_{M}^{\rm{top}}(-)$ is the relative topological K-theory [Mou19] over $M$, $\mathcal{BPS}_{M}$ is the BPS sheaf defined by Kinjo–Koseki [KK] for $M=M_{GL(r)}(\chi)$, and $\beta$ is of degree $2$. The isomorphism (1.7) explains the use of the name of BPS categories. Note that $\mathcal{BPS}_{M}$ is (a shift of) the constant sheaf $\mathbb{Q}_{M}$ if $(r,\chi)$ are coprime, and that in general $\mathcal{BPS}_{M}$ contains $\mathrm{IC}_{M}$ as a direct summand. The two sides on (1.6) are thus isomorphic because of (cohomological) $\chi$-independence [KK, Theorem 1.2]. We use the methods of [GS] to show that an extension of Arinkin’s kernel induces the isomorphism (1.6).

Note that the isomorphism (1.7) shows the weight independence of topological K-theory of BPS categories, which is a phenomenon we also discussed in [PTe, PTc]. The equivalence (1.5) thus interchanges the weight and $\chi$-independence phenomena for Higgs bundles.

The result of Theorem 1.2 is deduced from analogous results for $L$-twisted (i.e. meromorphic) Higgs bundles, where

$$L\to C$$

is a line bundle with $\deg L>2g-2$, using the method of vanishing cycles as in [KK, MS21]. We denote by $\mathcal{M}^{L}_{G}(\chi)$ the moduli stack of $L$-twisted semistable $G$-Higgs bundles. Note that $\mathcal{M}^{L}_{G}(\chi)$ is a smooth stack, whereas $\mathcal{M}_{G}(\chi)$ is, in general, singular and non-separated. In the case of $G=\mathrm{GL}(r)$, the moduli stack $\mathcal{M}^{L}_{G}(\chi)$ consists of pairs

$\displaystyle(F,\theta),\ \theta\colon F\to F\otimes L,$

where $F$ is a vector bundle and $(F,\theta)$ satisfies a stability condition. We can similarly define the quasi-BPS category

$\displaystyle\mathbb{T}_{G}^{L}(\chi)_{w}\subset D^{b}(\mathcal{M}^{L}(\chi))_{w}.$

We use the notation as in the previous subsection, e.g. $\mathbb{T}^{L}(r,\chi)_{w}:=\mathbb{T}_{\mathrm{GL}(r)}^{L}(\chi)_{w}$.

As before, we regard quasi-BPS categories as the categorical replacement of the BPS cohomology for the local Calabi-Yau threefold $\mathrm{Tot}_{C}(L\oplus L^{-1}\otimes\Omega_{C})$. Indeed, the derived category of coherent sheaves on $\mathcal{M}^{L}(\chi)$ has a semiorthogonal decomposition in Hall products of quasi-BPS categories, analogous to the decomposition of the cohomology of $\mathcal{M}^{L}(\chi)$ in terms of the BPS cohomology of $M^{L}:=M^{L}_{GL(r)}(\chi)$, which is isomorphic to its intersection cohomology [Mei]. In Propositions 4.1 and 4.5, we compute the (rational) topological K-theory of quasi-BPS categories in terms of BPS cohomology using the results and methods for quivers [PTe]. In particular, we show that, if $(r,\chi,w)$ satisfies the BPS condition, then

(1.8)

$$\mathcal{K}_{M^{L}}^{\rm{top}}(\mathbb{T}^{L}(r,\chi)_{w})_{\mathbb{Q}}\cong\mathrm{IC}_{M^{L}}[-\dim M^{L}][\beta^{\pm 1}].$$

The vector $(r,\chi,w)$ satisfies the BPS condition if the vector

$\displaystyle(r,\chi,w+1-g^{\rm{sp}})\in\mathbb{Z}^{3}$

is primitive, where $g^{\rm{sp}}$ is the genus of the spectral curve, see the formula (2.9). Note that, if $\deg L$ is even, the BPS condition is equivalent to the vector $(r,\chi,w)$ being primitive. We say that $(r,w)$ satisfies the BPS condition if $(r,0,w)$ satisfies the BPS condition. One can also formulate, for $L$-twisted Higgs bundles, a conjectural derived equivalence analogous to (1.1), (1.5), see [PTa, Conjecture 4.3]. We prove its version for topological K-theory:

As in the case $L=\Omega_{C}$, a first step in proving part (1) is to show that both sides in (1.9) have isomorphic rational topological K-theory. This follows from (1.8) and the cohomological $\chi$-independence for twisted Higgs bundles proved by Maulik–Shen [MS23]. Similarly, part (2) uses a cohomological SL/PGL-duality for twisted Higgs bundles [MS22].

Note that the equivalences of topological K-theories in Theorem 1.2 hold integrally. Indeed, we show that the topological K-groups in Theorem 1.2 are torsion-free. In Theorem 8.19, we give a slightly stronger statement of part (2) of Theorem 1.2 which also involves weight/Euler characteristics on left/hand hand sides.

As we mentioned in the beginning of Subsection 1.2, it is important to study all quasi-BPS categories. Thus it is natural to inquire whether a derived equivalence (1.5), or isomorphisms (1.6) or (1.9), may holds for all quasi-BPS categories. An analogous such comparison was discussed in [PTb, Section 1.3] for quasi-BPS categories of points on a surface, which includes points (i.e. rank zero Higgs sheaves) on $\mathrm{Tot}_{C}(\Omega_{C})$. We mention a rational analogue of the isomorphism (1.9) in Subsection 4.2.

It is also interesting to pursue an integral version of Theorem 1.1.

The methods of this paper (and of [PTe]) may be used to compute the topological K-theory of quasi-BPS categories (or of noncommutative resolutions defined by Špenko–Van den Bergh [ŠdB17]) for other smooth (or quasi-smooth) symmetric stacks, for example for the moduli stack $\mathcal{B}\mathrm{un}(r,d)^{\mathrm{ss}}$ of semistable vector bundles of rank $r$ and degree $d$ on a smooth projective curve $C$. In Subsection 4.3, we briefly discuss the computation of topological K-theory of quasi-BPS categories of $\mathcal{B}\mathrm{un}(r,d)^{\mathrm{ss}}$ in terms of the intersection cohomology of the good moduli space $\mathrm{Bun}(r,d)^{\mathrm{ss}}$. Note that in [PTa, Subsection 3.4] we explained that $D^{b}(\mathcal{B}\mathrm{un}(r,d)^{\mathrm{ss}})$ has a semiorthogonal decomposition in Hall products of quasi-BPS categories. Thus the results in loc.cit. and in Subsection 4.3 provide a K-theoretic (or categorical) version of a theorem of Mozgovoy–Reineke [MR, Theorem 1.3].

T. P. thanks MPIM Bonn and CNRS for their support during part of the preparation of this paper. This material is partially based upon work supported by the NSF under Grant No. DMS-1928930 and by the Alfred P. Sloan Foundation under grant G-2021-16778, while T. P. was in residence at SLMath in Berkeley during the Spring 2024 semester. T. P. thanks Kavli IPMU for their hospitality and excellent working conditions during a visit in May 2024.

This work started while Y. T. was visiting to Hausdorff Research Institute for Mathematics in Bonn on November 12-18, 2023. Y. T. thanks the hospitality of HIM Bonn during his visit. Y. T. is supported by World Premier International Research Center Initiative (WPI initiative), MEXT, Japan, and JSPS KAKENHI Grant Numbers JP19H01779, JP24H00180.

In this paper, all the (derived) stacks are defined over $\mathbb{C}$. For a stack $\mathcal{X}$, we use the notion of good moduli space from [Alp13]. It generalizes the notion of GIT quotient

$$R/G\to R/\!\!/G,$$

where $R$ is an affine variety on which a reductive group $G$ acts.

For a (classical) stack $\mathcal{X}$, denote by $D(\mathrm{Sh}_{\mathbb{Q}}(\mathcal{X}))$ the derived category of complexes of $\mathbb{Q}$-constructible sheaves [Ols07].

For a torus $T$, a character $\chi$, and cocharacter $\lambda$, we denote by $\langle\lambda,\chi\rangle\in\mathbb{Z}$ its natural pairing. For a $T$-representation $V$, we denote by $V^{\lambda>0}\subset V$ the subspace spanned by $T$-weights $\beta$ with $\langle\lambda,\beta\rangle>0$. We also write $\langle\lambda,V^{\lambda>0}\rangle:=\langle\lambda,\det(V^{\lambda>0})\rangle$.

For a variety or stack $\mathcal{X}$, we denote by $D^{b}(\mathcal{X})$ the bounded derived category of coherent sheaves, which is a pre-triangulated dg-category. We denote by $\mathrm{Perf}(\mathcal{X})$ the category of perfect complexes and $D_{\rm{qcoh}}(\mathcal{X})$ the unbounded derived category of quasi-coherent sheaves. For pre-triangulated dg-subcategories $\mathcal{C}_{i}\subset D^{b}(\mathcal{X}_{i})$ for $i=1,2$, we denote by $\mathcal{C}_{1}\boxtimes\mathcal{C}_{2}$ the smallest pre-triangulated dg-subcategory of $D^{b}(\mathcal{X}_{1}\times\mathcal{X}_{2})$ which contains objects $E_{1}\boxtimes E_{2}$ for $E_{i}\in\mathcal{C}_{i}$ and closed under direct summands.

For a dg-category $\mathcal{D}$ with $\mathrm{Perf}(B)$-module structure for a scheme $B$, its semiorthogonal decomposition $\mathcal{D}=\langle\mathcal{C}_{i}\,|\,i\in I\rangle$ is called $\mathrm{Perf}(B)$-linear if $\mathcal{C}_{i}\otimes\mathrm{Perf}(B)\subset\mathcal{C}_{i}$ for all $i\in I$.

For a smooth stack $\mathcal{X}$ and a regular function $f\colon\mathcal{X}\to\mathbb{C}$, we denote by $\mathrm{MF}(\mathcal{X},f)$ the $\mathbb{Z}/2$-graded dg-category of matrix factorizations of $f$. If there is a $\mathbb{C}^{\ast}$-action on $\mathcal{X}$ such that $f$ is of weight one, we also consider the dg-category of graded matrix factorizations $\mathrm{MF}^{\rm{gr}}(\mathcal{X},f)$. We refer to [PT24, Section 2.6] for a review of (graded) matrix factorizations.

Let $C$ be a smooth projective curve of genus $g$. Let $L$ be a line bundle on $C$ such that either

$\displaystyle l:=\deg L>2g-2\mbox{ or }L=\Omega_{C}.$

By definition, a $L$-twisted Higgs bundle is a pair $(F,\theta)$, where $F$ is a vector bundle on $C$ and $\theta$ is a morphism

$\displaystyle\theta\colon F\to F\otimes L.$

When $L=\Omega_{C}$, it is just called a Higgs bundle. The (semi)stable $L$-twisted Higgs bundle is defined using the slope $\mu(F)=\chi(F)/\operatorname{rank}(F)$ in the usual way: a $L$-twisted Higgs bundle $(F,\theta)$ is (semi)stable if we have

$\displaystyle\mu(F^{\prime})<(\leqslant)\mu(F),$

for any sub-Higgs bundle $(F^{\prime},\theta^{\prime})\subset(F,\theta)$ such that $\operatorname{rank}(F^{\prime})<\operatorname{rank}(F)$.

A $L$-twisted Higgs bundle corresponds to a compactly supported pure one-dimensional coherent sheaf on the non-compact surface

$\displaystyle p\colon S=\mathrm{Tot}_{C}(L)\to C.$

The correspondence (called the spectral construction [BNR89]) is given as follows: for a given $L$-twisted Higgs pair $(F,\theta)$, the Higgs field $\theta$ determines the $p_{\ast}\mathcal{O}_{S}$-module structure on $F$, which in turn gives a coherent sheaf on $S$. Conversely, a pure one-dimensional compactly supported sheaf $E$ on $S$ pushes forward to a vector bundle $F$ with Higgs field $\theta$ given by the $p_{\ast}\mathcal{O}_{S}$-module structure on it.

We denote by

(2.1)

$\displaystyle\mathcal{M}^{L}(r,\chi)$

the derived moduli stack of semistable $L$-twisted Higgs bundles $(F,\theta)$ with

$\displaystyle(\operatorname{rank}(F),\chi(F))=(r,\chi).$

It is smooth (in particular classical) when $\deg L>2g-2$, and quasi-smooth when $L=\Omega_{C}$. We omit $L$ in the notation when $L=\Omega_{C}$, i.e. $\mathcal{M}(r,\chi):=\mathcal{M}^{\Omega_{C}}(r,\chi)$. We also denote by $(\mathcal{F},\vartheta)$ the universal Higgs bundle

(2.2)

$\displaystyle\mathcal{F}\in\mathrm{Coh}(C\times\mathcal{M}^{L}(r,\chi)),\ \vartheta\colon\mathcal{F}\to\mathcal{F}\boxtimes L.$

The stack (2.1) is equipped with the Hitchin map

$\displaystyle h\colon\mathcal{M}^{L}(r,\chi)\to B^{L}(r,\chi):=\bigoplus_{i=1}^{r}H^{0}(C,L^{\otimes i})$

sending $(F,\theta)$ to $\mathrm{tr}(\theta^{i})$ for $1\leqslant i\leqslant r$. When $r,\chi$ are clear from the context, we write $B^{L}:=B^{L}(r,\chi)$, $B:=B^{\Omega_{C}}$. We have the factorization

$\displaystyle h\colon\mathcal{M}^{L}(r,\chi)^{\rm{cl}}\stackrel{{\scriptstyle\pi}}{{\to}}M^{L}(r,\chi)\to B^{L},$

where the first morphism is the good moduli space morphism. A closed point $y\in M^{L}(r,\chi)$ corresponds to a polystable Higgs bundle

(2.3)

$\displaystyle E=\bigoplus_{i=1}^{k}V_{i}\otimes E_{i},$

where $E_{i}$ is a stable $L$-twisted Higgs bundle such that $(\operatorname{rank}(E_{i}),\chi(E_{i}))=(r_{i},\chi_{i})$ satisfies $\chi_{i}/r_{i}=\chi/r$ and $V_{i}$ is a finite dimensional vector space. By abuse of notation, we also denote by $y\in\mathcal{M}^{L}(r,\chi)$ the closed point represented by (2.3). It is the unique closed point in the fiber of $\mathcal{M}^{L}(r,\chi)^{\rm{cl}}\to M^{L}(r,\chi)$ at $y$. We also have the Cartesian square

(2.8)

where $(-)^{\rm{st}}$ is the open locus of stable points, the horizontal arrows are open immersions, and the left vertical arrow is a good moduli space morphism which is a $\mathbb{C}^{\ast}$-gerbe.

A point $b\in B^{L}$ corresponds to a support $\mathcal{C}_{b}\subset S$ of the sheaf on $S$, called the spectral curve. We denote by $g^{\rm{sp}}$ the arithmetic genus of the spectral curve, which is given by

(2.9)

$\displaystyle g^{\rm{sp}}:=1+(g-1)r-\frac{rl}{2}+\frac{r^{2}l}{2}.$

Let $\mathcal{C}\to B^{L}$ be the universal spectral curve, which is a closed subscheme of $S\times B^{L}$. By the spectral construction, the universal Higgs bundle corresponds to a universal sheaf

(2.10)

$\displaystyle\mathcal{E}\in\operatorname{Coh}(\mathcal{C}\times_{B^{L}}\mathcal{M}^{L}(r,\chi))$

which is also regarded as a coherent sheaf on $S\times\mathcal{M}^{L}(r,\chi)$ by the closed immersion $\mathcal{C}\times_{B^{L}}\mathcal{M}^{L}(r,\chi)\hookrightarrow S\times\mathcal{M}^{L}(r,\chi)$.

The good moduli space

(2.11)

$\displaystyle\pi\colon\mathcal{M}^{L}(r,\chi)^{\rm{cl}}\to M^{L}(r,\chi)$

is, locally near a point $y\in M^{L}(r,\chi)$, described in terms of the representations of the Ext-quiver of $y$. In what follows, we write

(2.12)

$\displaystyle(r,\chi)=d(r_{0},\chi_{0}),$

where $d\in\mathbb{Z}_{>0}$ and $(r_{0},\chi_{0})$ are coprime.

Let $y\in M^{L}(r,\chi)$ be a closed point corresponding to the polystable object (2.3). The associated Ext-quiver $Q_{y}$ consists of vertices $\{1,\ldots,k\}$ with the number of arrows given by

$\displaystyle\sharp(i\to j)=\dim\operatorname{Ext}_{S}^{1}(E_{i},E_{j})=\begin{cases}r_{i}r_{j}l+\delta_{ij},&l>2g-2,\\ r_{i}r_{j}(2g-2)+2\delta_{ij},&L=\Omega_{C}.\end{cases}$

Here, by the spectral construction, we regard $E_{i}$ as a coherent sheaf on $S$. The representation space of $Q_{y}$-representations of dimension vector $\bm{d}=(d_{i})_{i=1}^{k}$ for $d_{i}=\dim V_{i}$ is given by

$\displaystyle R_{Q_{y}}(\bm{d}):=\bigoplus_{(i\to j)\in Q_{y}}\operatorname{Hom}(V_{i},V_{j})=\operatorname{Ext}_{S}^{1}(E,E).$

Let $G(\bm{d})$ be the algebraic group

$\displaystyle G(\bm{d}):=\prod_{i=1}^{k}GL(V_{i})=\mathrm{Aut}(E)$

which acts on $R_{Q_{y}}(\bm{d})$ by the conjugation.

If $l>2g-2$, by the Luna étale slice theorem, étale locally at $y$, the map

$$\pi\colon\mathcal{M}^{L}(r,\chi)\to M^{L}(r,\chi)$$

is isomorphic to

(2.13)

$\displaystyle\mathcal{X}_{Q_{y}}(\bm{d}):=R_{Q_{y}}(\bm{d})/G(\bm{d})\to X_{Q_{y}}(\bm{d}):=R_{Q_{y}}(\bm{d})/\!\!/G(\bm{d}).$

Note that the above quotient stack is the moduli stack of $Q_{y}$-representations of dimension $\bm{d}$.

If $L=\Omega_{C}$, we can write $R_{Q_{y}}(\bm{d})=R(\bm{d})\oplus R(\bm{d})^{\vee}$ for some $G(\bm{d})$-representation $R(\bm{d})$. Let $\mu$ be the moment map

$\displaystyle\mu\colon R_{Q_{y}}(\bm{d})\to\mathfrak{g}(\bm{d})^{\vee},$

where $\mathfrak{g}(\bm{d})$ is the Lie algebra of $G(\bm{d})$. Then $\pi\colon\mathcal{M}(r,\chi)^{\rm{cl}}\to M(r,\chi)$ is étale locally at $y$ isomorphic to ([Sac19], [Dav, Section 5], [HLa, Section 4.2]):

(2.14)

$\displaystyle\mathscr{P}(d)^{\rm{cl}}:=\mu^{-1}(0)^{\rm{cl}}/G(\bm{d})\to P(d):=\mu^{-1}(0)^{\rm{cl}}/\!\!/G(\bm{d}),$

see [PTc, Section 4.3] for more details.

The moduli space $M^{L}(r,\chi)$ is stratified with strata indexed by the data $(d_{i},r_{i},\chi_{i})_{i=1}^{k}$ of the polystable object (2.3). The deepest stratum corresponds to $k=1$ with $(d_{1},r_{1},\chi_{1})=(d,r_{0},\chi_{0})$, which consist of polystable objects $V\otimes E_{0}$ where $\dim V=d$ and $E_{0}$ is stable with

$\displaystyle(\operatorname{rank}(E_{0}),\chi(E_{0}))=(r_{0},d_{0}).$

The associated Ext-quiver at the deepest stratum has one vertex and $(1+lr_{0}^{2})$-loops. We have the following lemma, also see [PTc, Lemma 4.3] for an analogous statement for K3 surfaces.

We consider the bounded derived category of coherent sheaves $D^{b}(\mathcal{M}^{L}(r,\chi))$. Note that there is an orthogonal decomposition

$$D^{b}(\mathcal{M}^{L}(r,\chi))=\bigoplus_{w\in\mathbb{Z}}D^{b}(\mathcal{M}^{L}(r,\chi))_{w},$$

where $D^{b}(\mathcal{M}^{L}(r,\chi))_{w}$ is the subcategory of $D^{b}(\mathcal{M}^{L}(r,\chi))$ of weight $w$ complexes with respect to the action of the scalar automorphisms $\mathbb{C}^{*}$ at each point of $\mathcal{M}^{L}(r,\chi)$. In this subsection, we define the quasi-BPS categories, which are subcategories of $D^{b}(\mathcal{M}^{L}(r,\chi))_{w}$.

We first construct a line bundle on $\mathcal{M}^{L}(r,\chi)$. We write $(r,\chi)=d(r_{0},\chi_{0})$ as in (2.12), and take $(a,b)\in\mathbb{Z}^{2}$ such that

(2.15)

$\displaystyle a\chi_{0}+br_{0}+(1-g)ar_{0}=1$

which is possible as $(r_{0},\chi_{0})$ are coprime. Let $u\in K(C)$ be such that $(\operatorname{rank}(u),\chi(u))=(a,b)$. Define the following line bundle on $\mathcal{M}^{L}(r,\chi)$

(2.16)

$\displaystyle\delta:=\det(Rp_{\mathcal{M}\ast}(u\boxtimes\mathcal{F}))\in\mathrm{Pic}(\mathcal{M}^{L}(r,\chi)),$

where $\mathcal{F}$ is the universal Higgs bundle (2.2) and $p_{\mathcal{M}}$ is the projection onto $\mathcal{M}$. It has diagonal $\mathbb{C}^{\ast}$-weight $d$ because of the condition (2.15) and the Riemann-Roch theorem.

Before defining quasi-BPS categories, we need to introduce some more notations. An object $A\in D^{b}(B\mathbb{C}^{\ast})$ decomposes into a direct sum

$$A=\bigoplus_{w\in\mathbb{Z}}A_{w},$$

where $A_{w}$ is of $\mathbb{C}^{\ast}$-weight $w$. Denote by $\mathrm{wt}(A)$ the set of $w\in\mathbb{Z}$ such that $A_{w}\neq 0$. In the case that $A$ is a line bundle on $B\mathbb{C}^{\ast}$, then $\mathrm{wt}(A)$ consists of one element $\mathrm{wt}(A)\in\mathbb{Z}$. We also write $A^{>0}:=\bigoplus_{w>0}A_{w}$.

We next recall the definition of (reduced or not) quasi-BPS categories in the case of $L=\Omega_{C}$. Below we fix $p\in C$. There is a closed embedding

(2.20)

$\displaystyle j\colon\mathcal{M}(r,\chi)\hookrightarrow\mathcal{M}^{\Omega_{C}(p)}(r,\chi)$

sending $(F,\theta)$ for $\theta\colon F\to F\otimes\Omega_{C}$ to $(F,\theta^{\prime})$, where $\theta^{\prime}$ is the composition

$\displaystyle\theta^{\prime}\colon F\stackrel{{\scriptstyle\theta}}{{\to}}F\otimes\Omega_{C}\hookrightarrow F\otimes\Omega_{C}(p).$

Given $(F,\theta^{\prime})$ in $\mathcal{M}^{\Omega_{C}(p)}(r,\chi)$, it comes from the image of (2.20) if and only if $\theta^{\prime}|_{p}\colon F|_{p}\to F|_{p}\otimes\Omega_{C}(p)|_{p}$ is zero. Globally, let $(\mathcal{F},\vartheta)$ be the universal Higgs bundle (2.2) for $L=\Omega_{C}(p)$, and set

$\displaystyle\mathcal{F}_{p}:=\mathcal{F}|_{p\times\mathcal{M}^{\Omega_{C}(p)}(r,\chi)}\in\mathrm{Coh}(\mathcal{M}^{\Omega_{C}(p)}(r,\chi)).$

By fixing an isomorphism $\Omega_{C}(p)|_{p}\cong\mathbb{C}$, the correspondence $(F,\theta^{\prime})\mapsto(F|_{p},\theta^{\prime}|_{p})$ gives a section $s$ of the vector bundle

(2.23)

Then we have an equivalence of derived stacks

(2.24)

$\displaystyle\mathcal{M}(r,\chi)\stackrel{{\scriptstyle\sim}}{{\to}}s^{-1}(0),$

where the right hand side is the derived zero locus of the section $s$.

The section $s$ is indeed a section $s_{0}$ of the subbundle $\mathcal{V}_{0}\subset\mathcal{V}$ consisting of traceless endomorphisms. The reduced stack is the derived zero locus of $s_{0}$

(2.27)

$\displaystyle\mathcal{M}(r,\chi)^{\rm{red}}:=s_{0}^{-1}(0)\subset\mathcal{M}(r,\chi).$

Note that the classical truncations of $\mathcal{M}(r,\chi)^{\rm{red}}$ and $\mathcal{M}(r,\chi)$ are the same. The reduced quasi-BPS category

(2.28)

$\displaystyle\mathbb{T}(r,\chi)_{w}^{\rm{red}}\subset D^{b}(\mathcal{M}(r,\chi)^{\rm{red}})$

is also similarly defined using the closed immersion $\mathcal{M}(r,\chi)^{\rm{red}}\hookrightarrow\mathcal{M}^{\Omega_{C}(p)}(r,\chi)$.

We say that the tuple $(r,\chi,w)$ satisfies the BPS condition if the tuple

$\displaystyle(r,\chi,w+1-g^{\rm{sp}})\in\mathbb{Z}^{3}$

is primitive, i.e. if $\gcd(r,\chi,w+1-g^{\rm{sp}})=1$. If $(r,\chi,w)$ satisfies the BPS condition, we say that the category (2.25), (2.28) is a (reduced or not) BPS category. We now recall the basic properties of BPS categories.

We now recall the main conjecture about the symmetry of BPS categories for $G=\mathrm{GL}(r)$ proposed in [PTa].

We denote by $(B^{L})^{\rm{sm}}\subset B^{L}$ the open subset corresponding to smooth and irreducible spectral curves, and let $\mathcal{E}^{\rm{sm}}$ be the restriction of $\mathcal{E}$ to $(B^{L})^{\rm{sm}}$. We denote by

$\displaystyle\mathcal{M}^{L}(r,\chi)^{\rm{sm}}:=\mathcal{M}^{L}(r,\chi)\times_{B^{L}}(B^{L})^{\rm{sm}},\ M^{L}(r,\chi)^{\rm{sm}}:=M^{L}(r,\chi)\times_{B^{L}}(B^{L})^{\rm{sm}}.$

Note that $\mathcal{M}^{L}(r,\chi)^{\rm{sm}}$ is the relative Picard stack of the family of smooth spectral curves $\mathcal{C}^{\mathrm{sm}}\to(B^{L})^{\rm{sm}}$. Let $\mathcal{P}^{\rm{sm}}$ be the Poincaré line bundle

$\displaystyle\mathcal{P}^{\rm{sm}}\to\mathcal{M}^{L}(r,w+1-g^{\rm{sp}})^{\rm{sm}}\times_{(B^{L})^{\rm{sm}}}\mathcal{M}^{L}(r,\chi)^{\rm{sm}}$

defined by

(2.29)		$\displaystyle\mathcal{P}^{\rm{sm}}:=\det Rp_{13\ast}(p_{12}^{\ast}\mathcal{E}^{\prime}\otimes p_{23}^{\ast}\mathcal{E})$	$\displaystyle\otimes\det Rp_{13\ast}(p_{12}^{\ast}\mathcal{E}^{\prime})^{-1}$
		$\displaystyle\otimes\det Rp_{13\ast}(p_{23}^{\ast}\mathcal{E})^{-1}\otimes\det Rp_{13\ast}\mathcal{O},$

where $p_{ij}$ are the projections from

$\displaystyle\mathcal{M}^{L}(r,w+1-g^{\rm{sp}})^{\rm{sm}}\times_{(B^{L})^{\rm{sm}}}\times\mathcal{C}^{\rm{sm}}\times_{(B^{L})^{\rm{sm}}}\times\mathcal{M}^{L}(r,\chi)^{\rm{sm}}$

onto the corresponding factors, and $\mathcal{E}^{\prime}$ is the universal sheaf on the product $\mathcal{C}\times_{B^{L}}\mathcal{M}^{L}(r,w+1-g^{\rm{sp}})$. It determines the Fourier-Mukai equivalence [Muk81, DP12]:

(2.30)

$\displaystyle\Phi_{\mathcal{P}^{\rm{sm}}}\colon D^{b}(\mathcal{M}^{L}(r,w+1-g^{\rm{sp}})^{\rm{sm}})_{-\chi+1-g^{\rm{sp}}}\stackrel{{\scriptstyle\sim}}{{\to}}D^{b}(\mathcal{M}^{L}(r,\chi)^{\rm{sm}})_{w}.$

For $l>2g-2$, we rewrite the equivalence above as the following equivalence:

(2.31)

$\displaystyle\mathbb{T}^{L}(r,w+1-g^{\rm{sp}})_{-\chi+1-g^{\rm{sp}}}|_{(B^{L})^{\rm{sm}}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathbb{T}^{L}(r,\chi)_{w}|_{(B^{L})^{\rm{sm}}}.$

The following is the main conjecture in [PTa], which says that the above equivalence extends over the full Hitchin base $B^{L}$: