Unramified covers and branes on the Hitchin system☆
Introduction
Among the many fundamental contributions of Narasimhan and Ramanan to the study of moduli of vector bundles on curves are the Hecke correspondence [50], [51] and the study of generalized Prym varieties as fixed points [52]. In this paper we use these ideas to explore mirror symmetry for the moduli space of Higgs bundles in the spirit of the seminal work of A. Kapustin and E. Witten [45]. More precisely, we exhibit pairs of dual branes for the Langlands self dual group . The interest of our construction relies on two aspects: firstly the branes we consider are sheaves (rather than just submanifolds) and the duality is realized via an explicit Fourier–Mukai transform; secondly, we are making progress in the understanding of mirror symmetry in the singular locus of the Hitchin system, since the branes lie entirely over this locus. As far as we know, this is the first example of dual branes lying over the singular locus, where mirror symmetry is explicitly realized by a Fourier–Mukai transform. Finally, it is also important to note that among the branes we construct are the fixed loci under tensorization by an order n line bundle, central in the work of Hausel and Thaddeus [33] on topological mirror symmetry, so our construction ought to be important in a deeper understanding of the topological mirror symmetry phenomenon. In the remainder of this section, we explain our constructions and results in more detail.
N. Hitchin introduced in [37] Higgs bundles over a smooth projective complex curve X of genus as solutions to certain equations obtained by dimensional reduction of the self-dual equations on a 4-manifold. These are pairs , where E is a holomorphic vector bundle over X and φ is a holomorphic one-form with values in . The moduli space of Higgs bundles of rank n and degree d is a holomorphic symplectic manifold carrying a hyperkähler metric. Moreover, it admits the structure of an algebraically completely integrable system given by the Hitchin map . Here the Hitchin base is an affine space whose dimension is half that of , and the components of are the coefficients of the characteristic polynomial of φ. The fiber of over a generic point of the Hitchin base is a torsor for an abelian variety, namely the Jacobian of an associated spectral curve.
The concept of a G-Higgs bundle can be defined for any complex (and even real) reductive Lie group G. In these terms, the above definition becomes that of a -Higgs bundle. The Hitchin map can also be defined in this generality, and it has been shown that it is an algebraically completely integrable system for any complex reductive Lie group G [38], [19], [59], [15].
A new development arose with the discovery by T. Hausel and M. Thaddeus [33] of a close relation between Higgs bundles, mirror symmetry and the Langlands correspondence. They proved that the moduli spaces of Higgs bundles for the group and its Langlands dual group form a pair of SYZ-mirror partners [63], in the sense that the respective Hitchin maps have naturally isomorphic bases and their fibers over corresponding points are, generically, half-dimensional torsors for a pair of dual abelian varieties. This was subsequently generalized by N. Hitchin [40] for the self-dual group and then by R. Donagi and T. Pantev [16] for any pair of Langlands dual groups. The duality is reflected by a Fourier–Mukai transform between the moduli spaces interchanging fibers of the Hitchin map over corresponding points in the base. These dualities were obtained over the locus of the Hitchin base where the corresponding spectral curves are smooth.
As mentioned above, the moduli space is hyperkähler. This means that it carries three natural complex structures , and verifying the quaternionic relations and a metric which is Kähler with respect to all three holomorphic structures. In the present case, is the natural complex structure on the moduli space of Higgs bundles , while the complex structures and arise via the non-abelian Hodge Theorem, which identifies with the moduli space of projectively flat -connections (see [37], [60]).
A. Kapustin and E. Witten considered in [45] certain special subvarieties of , equipped with special sheaves. The pair composed by such a subvariety and the corresponding sheaf is called a brane. For each of the complex structures on a brane is classified as follows: it is of type A if it is a Lagrangian subvariety with respect to the corresponding Kähler form and the sheaf over it is equipped with a flat connection, and it is of type B if it is a holomorphic subvariety and the sheaf over it is also holomorphic. Thus, for instance, a (BBB)-brane is a subvariety equipped with a sheaf, holomorphic with respect to all three complex structures , and ; in other words, it is a hyperholomorphic subvariety equipped with a hyperholomorphic sheaf. A (BAA)-brane is a subvariety which is holomorphic with respect to , and Lagrangian with respect to the Kähler forms and associated to and (hence complex Lagrangian for ), and which in addition supports a flat vector bundle. There are only two other possible types of branes on , namely (ABA)- and (AAB)-branes. Again all this holds for any complex Lie group and not just .
According to [45], mirror symmetry conjecturally interchanges (BBB)-branes and (BAA)-branes, and mathematically this duality should again be realized via a Fourier–Mukai transform (in complex structure ). The support of the (BAA)-brane should depend not only on the support of the dual (BBB)-brane but also on the hyperholomorphic sheaf over it (and vice-versa). A similar story holds for pairs of (ABA)-branes and also for pairs of (AAB)-branes.
Since Kapustin and Witten's paper—and because of it—an intense study of several kinds of branes on Higgs bundle moduli spaces has been carried out. Some examples may be found in [41], [5], [10], [36], [9], [42], [23], [20], [6], [21], [11], [35] (see also [1] for a survey on this subject). Most of these works mainly focus either on the smooth locus of the Hitchin system (exceptions are [6], [21], [11]) or only deal with the support of the branes and not with the sheaves on it (exceptions are [41], [42], [23], [20], [21]).
Starting from a connected unramified cover of degree n and Galois group Γ, we introduce in this paper new types of (BBB)-branes and (BAA)-branes on , the moduli space for the self-dual group . As required in the general picture, our (BBB)-branes come equipped with flat, hence hyperholomorphic, bundles. We explicitly prove (when ) that their (fiberwise) Fourier–Mukai transform generically yields a sheaf supported exactly over the support of our (BAA)-brane. As expected, the support of the (BAA)-brane depends on the hyperholomorphic bundle over the (BBB)-brane.
These branes are supported on a subspace of the singular locus of the Hitchin system. For a dense open subset of nodal and integral spectral curves, the normalization of these curves is C itself. Since is unramified, is, by definition, contained in the so-called endoscopic locus of (cf. [34], [54]). So our construction (more precisely, its analogue for the Langlands dual groups and ) may eventually be relevant in the context of geometric endoscopy, introduced by E. Frenkel and E. Witten in [22].
In the following we outline our construction in more detail, starting with the (BBB)-branes. Fix the rank n to coincide with the degree of p and set to be the locus of Higgs bundles obtained as a pushforward under p of Higgs bundles in . Let be the image of under the Hitchin map . As a direct consequence of non-abelian Hodge theory, one concludes that is a hyperholomorphic subvariety. The pushforward by p yields an isomorphism between and the quotient of by the Galois group, acting by pullback. From this, one defines a hyperholomorphic line bundle over , naturally associated to a flat line bundle on X. We call the pair a rank 1 Narasimhan–Ramanan (BBB)-brane. We represent it by and write for its restriction to . More generally, we can construct a rank n coherent and hyperholomorphic sheaf on which is canonically associated to a flat line bundle over C, and we call the pair a rank n Narasimhan–Ramanan (BBB)-brane and represent it by .
Suppose is a Galois -cover, and let be the standard generator. Parallel transport of the lifts from X to C provides a line bundle of order n. In this case, it basically follows from [52] that the locus coincides with the subvariety of points fixed by tensorization of by , i.e. . The study of was our original motivation. So this justifies the name chosen for the (BBB)-branes appearing in this paper.
If our (BBB)-branes are intimately related to the work of Narasimhan–Ramanan in [52], the construction of our (BAA)-branes is closely linked to their work on Hecke modifications of vector bundles published in [50], [51]. Hecke modifications in the context of Higgs bundles have previously appeared in several papers; see, for example, [43], [44], [56], [64], [65]. Before describing the construction, we recall that under certain assumptions on the values of the rank and the degree, there exists a Hitchin section on the moduli space constructed out of a line bundle . The pushforward under p defines a Hitchin–type section of . We define the subvariety of those Higgs bundles over obtained as Hecke modifications of this Hitchin–type section at the divisor of singularities of the corresponding integral and nodal spectral curve (which has length δ) classified by . The notation we use for these subvarieties is chosen to recognize the pioneer work Narasimhan and Ramanan on Hecke modifications. We prove next that the subvarieties are complex Lagrangian with respect the holomorphic symplectic form on . This shows that this subvariety is the support of a (BAA)-brane on , when endowed with a flat bundle.
Our construction of (for and p of degree n) was aimed at obtaining the support of a (BAA)-brane dual to the rank 1 (BBB)-brane , for an appropriate choice of the line bundle . Towards this goal, we provide an extensive study of the spectral data of the Higgs bundles appearing in and in , the support of . For a given , let be the corresponding spectral curve and the normalization. Over the Hitchin fiber associated to , the spectral data in is given by the closure of the preimage of by the pullback under . On the other hand, the spectral data contained in are those given by pushforward under . This paves the way for our main result, Theorem 6.5, which is described below.
Theorem Let be a connected unramified n-cover. Consider the moduli space . Let . The (fiberwise) dual of the rank 1 Narasimhan-Ramanan (BBB)-brane (restricted to the locus of nodal and irreducible spectral curves) is the (BAA)-brane supported on , and whose flat bundle satisfies (6.28). Let . The (fiberwise) dual of the rank n Narasimhan-Ramanan (BBB)-brane (restricted to the locus of nodal and irreducible spectral curves) is the (BAA)-brane supported on , and whose flat bundle satisfies (6.30).
It is important to note that this duality is proved by an explicit fiberwise Fourier–Mukai transform, on the fibers over , mapping the hyperholomorphic sheaf to a sheaf supported on . This Fourier–Mukai transform is carried out using the autoduality of compactified Jacobians of integral curves with planar singularities, from the general results of D. Arinkin [4]. It uses a Hitchin section (which embeds as a subvariety of ) to identify with the corresponding , and then apply Arinkin's Fourier–Mukai functor. In order to explicitly do it, we relate this functor with the classical Fourier–Mukai functor of , via the pullback and the pushforward maps induced by the normalization morphism .
It is worth noticing in this case that appears as the pushforward of the (BBB)-brane supported over the whole moduli space , where is the pullback under of the flat line bundle over associated to . Mirror symmetry conjectures that is dual to the (BAA)-brane given by the Hitchin section associated to . As we said before, can be interpreted in terms of Hecke modifications of the pushforward of this Hitchin section. This suggests a deep relation between duality of branes in , duality in and the Hecke operators appearing in geometric Langlands conjecture (see [16]). For d non-multiple of n a similar result should hold, but the duality should require a gerbe to work out properly. We also note that the results in this paper provide evidence for the dualities suggested in [21].
Remark 1.1 We actually construct the support of the (BBB)-brane (and describe its spectral data) in a wider generality, namely in the case where the unramified cover is of degree m not necessarily equal to the rank n. In such a case, one must consider polystable Higgs bundles over C of rank r, such that . It is however unclear how to endow such (BBB)-branes with hyperholomorphic bundles. Similarly, we construct in the more general setup of a degree m cover . In the absence of a Hitchin section on we make use of very stable bundles on C, which define natural complex Lagrangian multisections of the Hitchin fibration. We explore this in Section 7.
As mentioned above, when the Galois group is cyclic, the support of our (BBB)-branes is . It is interesting to notice that plays a central role in the proof by T. Hausel and M. Thaddeus [33] of topological mirror symmetry for the moduli spaces of Higgs bundles for the Langlands dual groups and for (the general case has recently been proved by M. Groechenig, D. Wyss and P. Ziegler [28], and, more recently, by D. Maulik and J. Shen [48]). One might thus hope that further study of our dual branes in this setting may provide a better geometric understanding of the calculation by Hausel and Thaddeus. We hope to come back to this question in a future article.
Here is a brief description of the organization of the paper. In Section 2 we recall some background material on the Hitchin system. In Section 3 we study the locus , including the corresponding spectral data, for p an unramified cover of degree m, with m dividing n. Section 4 deals with the construction and description of the Narasimhan–Ramanan (BBB)-branes. In Section 5 we construct the complex Lagrangian subvarieties , which support (BAA)-branes. In Section 6, after recalling some background facts on the Fourier–Mukai transform for compactified Jacobians of integral curves and describing in Section 6.2 the role of the normalization of the curve in the transform, we prove our main duality result, namely Theorem 6.5. Finally, in Section 7, we generalize parts of the previous study to the case where has degree strictly less than n and no Hitchin section exists on .
The authors thank D. Arinkin, B. Collier, O. Garcia-Prada, T. Hausel, N. Hitchin, C. Pauly and R. Wentworth for their interest and useful discussions, and also thank the referee for helpful remarks and corrections.
Section snippets
Higgs bundles and the Hitchin system
The purpose of this section is to recall the basics on Higgs bundle moduli spaces which will be used in the remaining part of the paper.
Unramified covers and hyperholomorphic subvarieties in the moduli space
Let be a connected unramified cover of degree m and Galois group Γ. In this section, we study the subvarieties that arise in the moduli space of Higgs bundles out of this geometrical setting. Some of the following results have been already obtained in [34].
Let be the canonical bundle of C, and let be the corresponding projection. As p is unramified and hence we have a Cartesian diagram q being the obvious projection. In particular, q is an unramified Γ-cover
Narasimhan–Ramanan (BBB)-branes for covers of maximal degree
By definition (cf. [45]), a (BBB)-brane on a hyperkähler manifold M is a pair where:
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is a hyperholomorphic subvariety, i.e. a subvariety which is holomorphic with respect to the three complex structures , and .
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is a hyperholomorphic sheaf supported on N, i.e. a locally free sheaf of finite rank over the ring of -functions on N equipped with a connection whose curvature is of type in the complex structures , and .
In this section we construct natural
Narasimhan–Ramanan dual (BAA)-branes
We have seen in Section 2.1 that is a hyperkähler variety with Kähler structures . Following [45], a (BAA)-brane on is, by definition, a pair , where:
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Σ is a subvariety of , which is a complex Lagrangian for the holomorphic symplectic form .
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is a flat bundle supported on Σ.
The purpose of the present section is to construct a natural collection of complex Lagrangian subvarieties, whose image by the Hitchin fibration
Mirror symmetry and branes
In two preceding sections we constructed the Narasimhan–Ramanan (BBB)-branes and (cf. Definition 4.2, Definition 4.4) and the complex Lagrangian subvarieties (cf. Definition 5.3). Recall that the Definition 4.2, Definition 4.4 required Assumption 1, thus . Note as well that, by construction, both branes and determine coherent sheaves on (with respect to the complex structure ), and their support fiber (via the Hitchin map) over the locus .
Branes in the absence of a Hitchin section
In Section 4 we worked under Assumption 1 to construct a family of (BBB)-branes supported on . In Section 5 we required Assumption 2, which is weaker than Assumption 1, to define the Lagrangian subvariety over the locus of Hitchin base of those spectral curves whose normalization lives in .
A straight-forward observation is that, when Assumption 1 fails, one can always define a (BBB)-brane on by considering the trivial bundle on it.
Without Assumption 2 we face
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First, second and third authors partially supported by CMUP (UIDB/00144/2020) and the project PTDC/MAT-GEO/2823/2014 funded by FCT (Portugal) with national funds. First author supported as well by FCT (Portugal) in the framework of the Investigador FCT program, fellowship reference CEECIND/04153/2017. Third author also partially supported by the Post-Doctoral fellowship SFRH/BPD/100996/2014, also funded by FCT (Portugal) with national funds. Fourth author is currently supported by the scheme H2020-MSCA-IF-2019, Agreement n. 897722 (GoH). She was formerly funded through a Beatriu de Pinós grant n. 2018 BP 332 (H2020-MSCA-COFUND-2017 Agreement n. 801370), a postdoctoral grant associated to the project FP7 - PEOPLE - 2013 - CIG - GEOMODULI number: 618471 and the Swiss National Science Foundation project NCCR SwissMAP. The authors acknowledge support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network).
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On leave from: Departamento de Matemática, Universidade de Trás-os-Montes e Alto Douro, UTAD, Quinta dos Prados, 5000-911 Vila Real, Portugal.