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Advances in Mathematics

Volume 377, 22 January 2021, 107493
Advances in Mathematics

Unramified covers and branes on the Hitchin system

https://doi.org/10.1016/j.aim.2020.107493Get rights and content

Abstract

We study the locus of the moduli space of GL(n,C)-Higgs bundles on a curve given by those Higgs bundles obtained by pushforward under a connected unramified cover. We equip these loci with a hyperholomorphic bundle so that they can be viewed as BBB-branes, and we introduce corresponding BAA-branes which can be described via Hecke modifications. We then show how these branes are naturally dual via explicit Fourier–Mukai transform (recall that GL(n,C) is Langlands self dual). It is noteworthy that these branes lie over the singular locus of the Hitchin fibration.

As a particular case, our construction describes the behavior under mirror symmetry of the fixed loci for the action of tensorization by a line bundle of order n. These loci play a key role in the work of Hausel and Thaddeus on topological mirror symmetry for Higgs moduli spaces.

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 GL(n,C). 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 g2 as solutions to certain equations obtained by dimensional reduction of the self-dual equations on a 4-manifold. These are pairs (E,φ), where E is a holomorphic vector bundle over X and φ is a holomorphic one-form with values in End(E). The moduli space MX(n,d) 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 hX,n:MX(n,d)BX,n. Here the Hitchin base BX,n is an affine space whose dimension is half that of MX(n,d), and the components of hX,n are the coefficients of the characteristic polynomial of φ. The fiber of hX,n 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 GL(n,C)-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 SL(n,C) and its Langlands dual group PSL(n,C) 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 G2 and then by R. Donagi and T. Pantev [16] for any pair (G,GL) 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 MX(n,d) is hyperkähler. This means that it carries three natural complex structures I1, I2 and I3 verifying the quaternionic relations and a metric which is Kähler with respect to all three holomorphic structures. In the present case, I1 is the natural complex structure on the moduli space of Higgs bundles MX(n,d), while the complex structures I2 and I3=I1I2 arise via the non-abelian Hodge Theorem, which identifies MX(n,d) with the moduli space of projectively flat GL(n,C)-connections (see [37], [60]).

A. Kapustin and E. Witten considered in [45] certain special subvarieties of MX(n,d), 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 MX(n,d) 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 I1, I2 and I3; 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 I1, and Lagrangian with respect to the Kähler forms ω2 and ω3 associated to I2 and I3 (hence complex Lagrangian for Ω1=ω2+iω3), and which in addition supports a flat vector bundle. There are only two other possible types of branes on MX(n,d), namely (ABA)- and (AAB)-branes. Again all this holds for any complex Lie group and not just GL(n,C).

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 I1). 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 p:CX of degree n and Galois group Γ, we introduce in this paper new types of (BBB)-branes and (BAA)-branes on MX(n,d), the moduli space for the self-dual group GL(n,C). As required in the general picture, our (BBB)-branes come equipped with flat, hence hyperholomorphic, bundles. We explicitly prove (when d=0) 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 BpBX,n of the singular locus of the Hitchin system. For a dense open subset BnipBp of nodal and integral spectral curves, the normalization of these curves is C itself. Since p:CX is unramified, Bnip is, by definition, contained in the so-called endoscopic locus of hX,n (cf. [34], [54]). So our construction (more precisely, its analogue for the Langlands dual groups SL(n,C) and PSL(n,C)) 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 MX(n,d)p to be the locus of Higgs bundles obtained as a pushforward under p of Higgs bundles in MC(1,d)TJacd(C). Let Bp be the image of MX(n,d)p under the Hitchin map hX,n:MX(n,d)BX,n. As a direct consequence of non-abelian Hodge theory, one concludes that MX(n,d)p is a hyperholomorphic subvariety. The pushforward by p yields an isomorphism between MX(n,d)p and the quotient of TJacd(C) by the Galois group, acting by pullback. From this, one defines a hyperholomorphic line bundle L over MX(n,d)p, naturally associated to a flat line bundle L on X. We call the pair (MX(n,d)p,L) a rank 1 Narasimhan–Ramanan (BBB)-brane. We represent it by (BBB)Lp and write (BBB)nip,L for its restriction to Bnip. More generally, we can construct a rank n coherent and hyperholomorphic sheaf F on MX(n,d)p which is canonically associated to a flat line bundle F over C, and we call the pair (MX(n,d)p,F) a rank n Narasimhan–Ramanan (BBB)-brane and represent it by (BBB)Fp.

Suppose p:CX is a Galois Zn-cover, and let ξZn be the standard generator. Parallel transport of the lifts from X to C provides a line bundle LξJac0(X) of order n. In this case, it basically follows from [52] that the locus MX(n,d)p coincides with the subvariety MX(n,d)ξMX(n,d) of points (E,φ) fixed by tensorization of by Lξ, i.e. (E,φ)(ELξ,φ). The study of MX(n,d)ξ 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 MC(r,d+δ) constructed out of a line bundle JJacd+δ(C). The pushforward under p defines a Hitchin–type section of MX(n,d+δ)pBp. We define the subvariety NRnip,JMX(n,d) of those Higgs bundles over Bnip 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 Bnip. 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 NRnip,J are complex Lagrangian with respect the holomorphic symplectic form Ω1=ω2+iω3 on MX(n,d). This shows that this subvariety is the support of a (BAA)-brane on MX(n,d), when endowed with a flat bundle.

Our construction of NRnip,J (for d=0 and p of degree n) was aimed at obtaining the support of a (BAA)-brane dual to the rank 1 (BBB)-brane (BBB)Lp, for an appropriate choice of the line bundle J. Towards this goal, we provide an extensive study of the spectral data of the Higgs bundles appearing in NRnip,J and in MX(n,d)p, the support of (BBB)Lp. For a given bBnip, let Xb be the corresponding spectral curve and νb:CXb the normalization. Over the Hitchin fiber associated to Xb, the spectral data in NRnip,J is given by the closure of the preimage of J by the pullback under νb. On the other hand, the spectral data contained in MX(n,d)p are those given by pushforward under νb. This paves the way for our main result, Theorem 6.5, which is described below.

Theorem

Let p:CX be a connected unramified n-cover. Consider the moduli space MX(n,0).

  • (i)

    Let J=p(LKX(n1)/2). The (fiberwise) dual of the rank 1 Narasimhan-Ramanan (BBB)-brane (BBB)Lp (restricted to the locus of nodal and irreducible spectral curves) is the (BAA)-brane supported on NRnip,J, and whose flat bundle satisfies (6.28).

  • (ii)

    Let J=FpKX(n1)/2. The (fiberwise) dual of the rank n Narasimhan-Ramanan (BBB)-brane (BBB)Fp (restricted to the locus of nodal and irreducible spectral curves) is the (BAA)-brane supported on NRnip,J, 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 Bnip, mapping the hyperholomorphic sheaf to a sheaf supported on NRnip,J. 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 Bnip as a subvariety of NRnip,J) to identify Jacδ(Xb) with the corresponding Jac0(Xb), 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 Jacδ(C), via the pullback and the pushforward maps induced by the normalization morphism νb:CXb.

It is worth noticing in this case that (BBB)Fp appears as the pushforward of the (BBB)-brane (F,F)MC(1,0) supported over the whole moduli space MC(1,0), where F is the pullback under MC(1,0)Jac0(X) of the flat line bundle over Jac0(X) associated to FX. Mirror symmetry conjectures that (F,F)MC(1,0) is dual to the (BAA)-brane given by the Hitchin section associated to F. As we said before, (BAA)nip,F 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 MX(n,0), duality in MC(1,0) 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 p:CX 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 n=mr. It is however unclear how to endow such (BBB)-branes with hyperholomorphic bundles.

Similarly, we construct NRnip,J in the more general setup of a degree m cover p:CX. In the absence of a Hitchin section on MC(r,d) 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 MX(n,d)ξ. It is interesting to notice that MX(n,d)ξ 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 SL(n,C) and PSL(n,C) for n=2,3 (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 MX(n,d)p, 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 NRnip,J, 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 p:CX has degree strictly less than n and no Hitchin section exists on MC(r,d).

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 p:CX 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 KC be the canonical bundle of C, and letη:|KC|C be the corresponding projection. As p is unramifiedKCpKX and|KC||KX|×XC, 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(N,(F,F)), where:

  • NM is a hyperholomorphic subvariety, i.e. a subvariety which is holomorphic with respect to the three complex structures I1, I2 and I3.

  • (F,F) is a hyperholomorphic sheaf supported on N, i.e. a locally free sheaf F of finite rank over the ring of C-functions on N equipped with a connection F whose curvature is of type (1,1) in the complex structures I1, I2 and I3.

In this section we construct natural

Narasimhan–Ramanan dual (BAA)-branes

We have seen in Section 2.1 that MX(n,d) is a hyperkähler variety with Kähler structures ((I1,ω1),(I2,ω2),(I3,ω3)). Following [45], a (BAA)-brane on MX(n,d) is, by definition, a pair (Σ,(W,W)), where:

  • Σ is a subvariety of MX(n,d), which is a complex Lagrangian for the holomorphic symplectic form Ω1=ω2+iω3.

  • (W,W) 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 hX,nBX,n

Mirror symmetry and branes

In two preceding sections we constructed the Narasimhan–Ramanan (BBB)-branes (BBB)Fp and (BBB)Lp (cf. Definition 4.2, Definition 4.4) and the complex Lagrangian subvarieties NRnip,J (cf. Definition 5.3). Recall that the Definition 4.2, Definition 4.4 required Assumption 1, thus r=1. Note as well that, by construction, both branes (BBB)Lp and (BBB)Fp determine coherent sheaves on MX(n,d) (with respect to the complex structure I1), and their support fiber (via the Hitchin map) over the locus Bp.

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 MX(n,d)p. In Section 5 we required Assumption 2, which is weaker than Assumption 1, to define the Lagrangian subvariety NRnip,J over the locus of Hitchin base Bnip of those spectral curves whose normalization lives in BC,rsm.

A straight-forward observation is that, when Assumption 1 fails, one can always define a (BBB)-brane on MX(n,d)p 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).

    1

    On leave from: Departamento de Matemática, Universidade de Trás-os-Montes e Alto Douro, UTAD, Quinta dos Prados, 5000-911 Vila Real, Portugal.

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