Mirror symmetry for Nahm branes

The Dirac--Higgs bundle is a hyperholomorphic bundle over the moduli space of stable Higgs bundles of coprime rank and degree. We provide an algebraic generalization to the case of trivial degree and the rank higher than $1$. This allow us to generalize to this case the Nahm transform defined by Frejlich and the second named author, which, out of a stable Higgs bundle, produces a vector bundle with connection over the moduli space of rank 1 Higgs bundles. By performing the higher rank Nahm transform we obtain a hyperholomorphic bundle with connection over the moduli space of stable Higgs bundles of rank $n$ and degree 0, twisted by the gerbe of liftings of the projective universal bundle. Such hyperholomorphic vector bundles over the moduli space of stable Higgs bundles can be seen, in the physicist's language, as BBB-branes twisted by the above mentioned gerbe. We refer to these objects as Nahm branes. Finally, we study the behaviour of Nahm branes under Fourier--Mukai transform over the smooth locus of the Hitchin fibration, checking that the resulting objects are supported on a Lagrangian multisection of the Hitchin fibration, so they describe partial data of BAA-branes.


Context
The Nahm transform was originally described as a geometrical correspondence between solutions of the self-duality Yang-Mills equations (also known as instantons) in R 4 which are invariant under dual groups of translations [ADHM78,Nah82,Nah84,Hit83,CG84,BvB89,Nak93,Jar02]. In [Jar04], the second-named author reviewed the Nahm transform in several situations and gave an interpretation as a nonlinear version of the Fourier transform which, given a family of self-dual connections over a spin four-manifold with non-negative scalar curvature, produces a vector bundle with connection over the parametrizing space of the family. Such bundle is constructed by considering, at each point of the parametrizing space, the cokernel of the associated Dirac operator. The connection is hyperkähler whenever both varieties, the base manifold and the parametrizing space, are hyperkähler.
The study of instantons that are invariant under translations in two directions led Hitchin to introduce Higgs bundles in [Hit87a] as solutions of the dimensional reduction to a Riemann surface of the self-dual connections in 4 dimensions. It turns out that the moduli space M of Higgs bundles has a very rich geometry; in particular, it can be constructed as a hyperkähler quotient in the context of gauge theory [Hit87a,Sim94,Sim95,Don87,Cor88] inheriting a hyperkähler structure. When the rank and the degree are coprime, all semistable Higgs bundles are stable, and Hausel [Hau98] showed that a universal bundle exists. In degree 0, however, there is no universal bundle over the stable locus M st or even over any other Zariski open set of the moduli space (see Ramanan [Ram73] and Drezet and Narasimhan [DN89] for a proof in the case of vector bundles that extends naturally to Higgs bundles). There exists nonetheless a local universal bundle in the étale topology of the moduli space of stable Higgs bundles M st , as indicated by Simpson [Sim95].
Another important feature of the moduli space M of stable Higgs bundles with coprime rank and degree is the existence of the so called Dirac-Higgs bundle, originally considered by Hitchin and studied in detail by Hausel in [Hau98]. Later, Blaavand [Bla15] extended the construction of the Dirac-Higgs bundle to the moduli space of parabolic Higgs bundles.
One way to describe the Dirac-Higgs bundle is as the hyperholomorphic bundle on M obtained via the Nahm transform, as defined in [FJ08], associated to the universal bundle. To be more precise, the Nahm transform of a Higgs bundle is defined by considering the index bundle associated to the family obtained by twisting the original Higgs bundle with the universal family of rank 1 Higgs bundles. This transform underlies the Fourier-Mukai transform for Higgs bundles defined by Bonsdorff [Bon06], which also equipped it with an autodual connection [Bon10].
Another interesting feature of the moduli space M of Higgs bundles is that it admits a fibration M Ñ B over a vector space, becoming an algebraically completely integrable system [Hit87b] which is known as the Hitchin system. The notion of Higgs bundles generalizes naturally to any structure group G. It was shown in [HT03,DG02,DP12] that Hitchin systems for Langlands dual structure groups, G and G L , are dual, satisfying thereby the requirements of being Strominger-Yau-Zaslow (SYZ) mirror partners [SYZ96], which allows for the identification of T-duality with mirror symmetry between them. Since the group G " GLpn, Cq is Langlands self-dual, we obtain a self-dual Hitchin system in this case, which is the one that we study in this paper.
The rich geometry of the moduli space of Higgs bundles M makes it an object of interest for theoretical physics. In [BJSV95,HMS95] it was shown that the dimensional reduction of an N " 4 Super Yang-Mills theory in 4 dimensions gives a 2 dimensional sigma model with hyperkähler target M, and, hence, S-duality in the former becomes T-duality (mirror symmetry) in the latter. This was the starting point for the ground-breaking article by Kapustin and Witten [KW07] (see also [Wit18]), where they relate the Geometric Langlands Conjecture and S-duality in the original N " 4 super Yang-Mills theory. Following Kapustin and Witten, a pBBBq-brane is a pair consisting of a hyperkähler submanifold and a hyperholomorphic vector bundle. Similarly, a pBAAq-brane is given by a submanifold which is complex Lagrangian with respect to the first Kähler structure, and a flat vector bundle. In String Theory, branes are geometrical objects that encode the Dirichlet boundary conditions, and mirror symmetry [KW07,Wit18] predicts a 1-1 correspondence between pBBBq-branes on the moduli space of G-Higgs bundles and pBAAq-branes on its G L counterpart.

Our constructions
In this paper we generalize the constructions of the Dirac-Higgs bundle and the Nahm transform of a stable Higgs bundle to the case of trivial degree and rank higher than 1. This provides a class of (space filling) pBBBq-branes that we transform under Fourier-Mukai, obtaining a partial description of the mirror dual pBAAq-brane.
In the case of coprime rank and degree, the universal Higgs bundle plays a central role in the construction of the Dirac-Higgs bundle. In our case, however, there is no universal Higgs bundle at hand, not even locally. To surpass this obstacle, we consider the gerbe of liftings of the projective universal bundle 1 , and we introduce the notions of sheaves twisted and shifted by such gerbe. We can then construct the Dirac-Higgs bundle as a vector bundle twisted by our gerbe, showing that it is equipped with a hyperholomorphic connection. The techniques used in the construction of the Dirac-Higgs bundle can then be applied to define the Nahm transform of a stable Higgs bundle which is, again, a bundle twisted by our gerbe and equipped with a hyperholomorphic connection. This constitutes a family of pBBBq-branes (one for each stable Higgs bundle) which we call Nahm branes.
The second step is to study the behaviour of these Nahm branes under mirror symmetry. We work over the smooth locus of the Hitchin fibration, where mirror symmetry is expected to be realized via Fourier-Mukai transform. We check that the transformed sheaf is supported on a complex Lagrangian multisection of 1 We are grateful to the anonymous referee for suggesting this approach. the Hitchin fibration. This is part of the data of a pBAAq-brane, providing evidence for the existence of a correspondence between pBBBq and pBAAq-branes conjectured by Kapustin and Witten.

Organization of the paper
In Sections 2.1 and 2.2 we review the properties of the theory of Higgs bundles and the Hitchin system. In Section 2.3 we survey, after restricting ourselves to the smooth locus of the Hitchin fibration, the Fourier-Mukai transform, which is expected to realize mirror symmetry in this context. In Section 2.4 we review gerbes and the notions of sheaves twisted by a gerbe, we also describe an example of gerbe over the moduli space of Higgs bundles of particular importance for us, the gerbe of liftings of the projective universal bundle. In Section 2.5 we review the Dirac-Higgs bundle in general, and we provide an algebraic construction over the moduli space of Higgs bundles with 0 degree as a bundle twisted by the gerbe of liftings. In Section 3 we study the behaviour of spectral data of Higgs bundles under tensorization, crucial for understanding our generalization of the Nahm transform of a stable Higgs bundle to rank higher than 1, which we achieve in Section 4, thereby constructing Nahm pBBBq-branes. In Section 5.1 we adapt the Fourier-Mukai transform for sheaves twisted by a gerbe, in order to study in Sections 5.2 and 5.3, respectively, the transform of the Dirac-Higgs bundle and of the Nahm brane associated to any stable Higgs bundle.

Non-abelian Hodge theory
In this section we introduce the moduli space of Higgs bundles, an object with an extremely rich geometry. In particular, it is equipped with a hyperkähler structure.
Let us consider a smooth projective curve X over C of genus g ě 2. Denote by E the unique up to isomorphism C 8 -bundle of rank n over X and consider a Hermitian metric h on it. Denote by G the Gauge group of unitary automorphism of E preserving the metric, and its complexification, G C , parametrizing all automorphisms of E. Recall that E equipped with a Dolbeault operator B E gives rise to a holomorphic vector bundle E. Out of B E and the metric h, one can construct B E and B E`BE is a unitary connection on E.
A Higgs pair of rank n on X is pair pB E , ϕq where B E is a Dolbeault operator on E fixing an integrable complex structure on it, and ϕ is an element of Ω 1,0 X pEndpEqq. Note that the space A of Higgs pairs is an affine space modeled in the infinite dimensional vector space Ω 0,1 X pEndpEqq ' Ω 1,0 X pEndpEqq. A Higgs bundle over X is a Higgs pair pB E , ϕq satisfying B E ϕ " 0 (hence B E ϕ˚" 0, where ϕ˚is the adjoint of ϕ with respect to the metric h). Equivalently, a Higgs bundle is a pair E " pE, ϕq, where E is a holomorphic vector bundle on X, and ϕ P H 0 pX, EndpEq b K X q is a holomorphic section of the endomorphisms bundle, twisted by the canonical bundle K X . Let us write B Ă A for the G C -invariant subset parametrizing Higgs bundles.
Recall  [Hit87a,Sim94,Sim95,Don87,Cor88] between M and the moduli space of flat connections of rank n. This is a consequence of the construction of these moduli spaces as a hyperkähler quotient of the space of Higgs pairs A by the gauge group G C of complex automorphisms of E. Tangent to any Higgs pair pB E , ϕq we can consider its infinitesimal deformations 9 α dz P Ω 0,1 X pEndpEqq and 9 ϕ dz P Ω 1,0 X pEndpEqq, with z being a holomorphic coordinate of the base curve. Also, we consider 9 α˚dz P Ω 1,0 X pEndpEqq and 9 ϕ˚dz P Ω 0,1 X pEndpEqq to be infinitesimal deformations of B E and ϕ˚respectively. The hyperkähler structure on A is given by the following metric on this space, (2.1) r g pp 9 α 1 , 9 ϕ 1 q, p 9 α 2 , 9 ϕ 2 qq " 1 4π ż X tr p 9 α1 9 α 2`9 α2 9 α 1`9 ϕ 1 9 ϕ2`9 ϕ 2 9 ϕ1 q dz^dz, and the complex structures r Γ 1 , r Γ 2 , and r Γ 3 " r Γ 1r Γ 2 . We denote by r ω j p¨,¨q " r gp¨, r Γ j p¨qq the associated Kähler forms and consider the symplectic forms r Λ j " r ω j`1`i r ω j´1 which are holomorphic for the corresponding r Γ j . From each of the Kähler forms r ω j one can construct a moment map µ j . The space of Higgs bundles B can be identified with pµ 2 q´1p0q X pµ 3 q´1p0q inside A. Then, after an infinite-dimensional version of Kempf-Ness theorem, M can be identified with the hyperkähler quotient The G-invariant complex structures r Γ j descend naturally to complex structures Γ j on M. Also, the 2-forms r ω j and r Λ j are G-invariant, so they provide naturally the Kähler forms ω j on M, and the holomorphic symplectic forms Λ j .
Given a Higgs bundle E " pE, ϕq, consider the complex Let pq 1 , . . . , q n q be a basis of GLpn, Cq-invariant polynomials with degpq i q " i. The Hitchin fibration is the dominant morphism h : M ÝÑ B :" À n i"1 H 0 pX, K bi X q pE, ϕq Þ ÝÑ pq 1 pϕq, . . . , q n pϕqq , and we refer to B as the Hitchin base. Consider the total space TotpK X q of the canonical bundle, and the obvious algebraic surjection p : TotpK X q Ñ X; let λ be the tautological section of the pullback bundle p˚K X Ñ TotpK X q. Given an element b " pb 1 , . . . , b n q P B we construct the associated spectral curve S b Ă TotpK X q by considering the vanishing locus of the section λ n`p˚b 1 λ n´1`¨¨¨`p˚b n´1 λ`p˚b n P H 0 pX, p˚K bn X q. Restricting p to S b yields a finite morphism p b : S b Ñ X of degree n.
One can further consider the pull-back of p˚K bn X to the product TotpK X qˆB, which is naturally equipped with a section obtained by pull-back of λ. Seeing the b i as coordinates in B, one obtains a second section of our bundle, whose vanishing locus provides naturally a family of spectral curves S Ă TotpK X qˆB for which we naturally have that S X pTotpK X qˆtbuq " S b . Restricting the projection pˆ1 B : TotpK X qˆB Ñ XˆB, we obtain a finite morphism of degree n: For every b P B, the corresponding spectral curve S b belongs to the linear system |nX|, and, by Bertini's theorem, it is generically smooth and irreducible. Furthermore, since the canonical divisor of the symplectic surface TotpK X q is zero, the genus of S b is given by Thanks to Riemann-Roch theorem, p˚O S b is a degree´pn 2´n qpg´1q vector bundle of rank n. This motivates the notation δ :" pn 2´n qpg´1q.
Following [BNR89], we consider the push-forward (2.4) E L :" p˚L of a torsion free sheaf L on S b of rank 1 and degree δ, which is a vector bundle on X of rank n and degree δ`degpπ˚O S b q " 0. We consider as well the multiplication by the restriction to S b of tautological section, Note that this induces the following twisted endomorphism of L whose push-forward returns the Higgs field Thanks to the spectral correspondence [Hit87b,BNR89,Sim95,Sch98,dCat17], each Hitchin fibre is identified with the compactified Jacobian of the corresponding spectral curve h´1 pbq -Jac Fixing a point x 0 P X in our curve, we construct a smooth sectionσ : B Ñ Jac

Fourier-Mukai transform and mirror symmetry for the Hitchin system
A very active field of research is the occurrence of mirror symmetry phenomena between Higgs moduli spaces of pairs of Langlands dual groups. In the so-called semi-classical limit, mirror symmetry is expected to be realised via a Fourier-Mukai transform relative to the Hitchin fibration. In this section we review this correspondence within the framework of the moduli space of Higgs bundles for GLpn, Cq, which is Langlands self-dual. Hence, the mirror of M is (conjecturally) itself. Even if the Fourier-Mukai transform extends out of the locus of smooth Hitchin fibres [Ari13, MRV19a,MRV19b], we restrict here to the original construction of Mukai over (families of) abelian varieties. We do so because the locus of smooth Hitchin fibres is dense in the Higgs moduli space, hence the study of the duality there is enough for our purposes.
Let us denote by B 1 Ă B the Zariski open subset given by those points b P B such that S b is smooth. We denote the restriction of S and M to B 1 by Note that all the points of M 1 are associated to line bundles over smooth spectral curves, which are automatically stable. Therefore, M 1 is contained in the stable locus, By the autoduality of smooth relative jacobians, we know that Jac 0 B 1 pS 1 q -Jac δ B 1 pS 1 q _ ; it thus follows from (2.7) that this is further isomorphic to Jac δ B 1 pS 1 q _ -M 1 . Then, one can consider the commuting diagram (2.8) whereσ is the constant section considered in Section 2.2, andσ is the section given by considering the structural sheaf on each Jac δ pS b q. We will study mirror symmetry in the sense of Strominger-Yau-Zaslow [SYZ96] in this context. Since the relative scheme M 1 " Jac δ B 1 pS 1 q has a sectionσ , it is well known (see [BLR90,8.2, Proposition 4] for instance) that its relative Jacobian carries a Poincaré bundle P Ñ Jac δ B 1 pS 1 qˆB1 Jac δ B 1 pS 1 q _ . With it, we can consider the relative Fourier-Mukai transforms After [Muk81], this is an equivalence of categories since where 1´1 Jac denotes the involution given by inverting elements on each Jac δ pS b q under the group structure (recall (2.6)), and d is defined in (2.3). We say that a sheaf F on Jac δ B 1 pS 1 q is -WIT (after Weak Index Theorem) if its image under RF is a complex supported in degree . Let us denote by Wit ´J ac δ B 1 pS 1 q¯the category of -WIT sheaves. It follows from (2.11) that the Fourier-Mukai transform RF induces an equivalence of categories

Flat unitary gerbes and the universal bundle
When the rank and the degree are coprime, the moduli space of stable Higgs bundles is fine and the universal bundle plays an important role in the construction of the Dirac-Higgs bundle (see Section 2.5 below). In our case, with the degree being trivial, there is no universal bundle at hand, not even over the stable locus or any other Zariski open subset of the moduli space (this was proven for vector bundles by Ramanan [Ram73] and reproved by Drezet and Narasimhan [DN89] but a similar discussion holds for Higgs bundles). The best we have at hand is a local universal bundle in the étale topology constructed by Simpson [Sim95].
Since our universal bundle only exists locally, we need to introduce gerbes. We refer to [Hit01] for a nice introduction to gerbes. Given an algebraic variety Y , denote by TorspUp1q, Y q the group of Up1q-torsors over Y , i.e. the group of flat unitary line bundles on Y . A flat unitary gerbe in the étale topology on Y is a sheaf of categories in the étale topology over Y such that its restriction to each étale open subset Y 1 Ñ Y is a torsor for the group TorspUp1q, Y 1 q. Given an étale covering tY i Ñ Y u iPI , a gerbe provides a category (a groupoid indeed) for every Y i , the natural transformations of these categories in the intersections Y ij :" Y iˆY Y j are realized via tensoring by flat unitary line bundles L ij Ñ Y ij . Therefore, a gerbe defines a set of flat unitary line bundles over the intersections tL ij Ñ Y ij u i,jPI such that L ij -L´1 ji and over the triple If β is a flat unitary gerbe in the étale topology and In this case, we also say that the cover tY i u iPI is fine enough for F. A β-twisted sheaf F is a β-twisted vector bundle of rank n if all the F i are locally free sheaves of rank n. We say that ∇ " t∇ i u iPI is a connection on the β-twisted vector bundle F if ∇ i is a connection on each of the F i satisfying the compatibility relations ζi where ∇ ij is the flat unitary connection naturally defined on L ij via the flat unitary gerbe β. Given two β-twisted sheaves F 1 " tF 1,i Ñ Y i u iPI and F 2 " tF 2,i Ñ Y i u iPI that are fine enough for the same étale covering, a morphism ψ : F 1 Ñ F 2 of β-twisted sheaves is a collection of morphisms of sheaves tψ i : F 1,i Ñ F 2,i u iPI satisfying ζi ψ i -1 L ij b ζj ψ j for any i, j P I. In a similar way, the notions of quotient of sheaves, complex of sheaves and cohomology generalize naturally to the context of β-twisted sheaves. If f : Z Ñ Y is a morphism of algebraic varieties, β a flat unitary gerbe in the étale topology over Y and F a β-twisted sheaf on Y , we define f˚F to be tfi F i Ñ Y iˆY Zu, where f i : Y iˆY Z Ñ Y i is the projection to the first factor and we note that tY iˆY Z Ñ Zu iPI is an étale covering. Similarly, for a f˚β-twisted sheaf F on Z with a fine enough étale covering of the form tY iˆY Z Ñ Zu iPI , we set f˚F :" tf i,˚Fi Ñ Y i u iPI . Thanks to the projection formula and base-change theorems, f˚F is a β-twisted sheaf as Given a β-twisted sheaf F 1 and vector bundle F 2 over Y , we define F 1 b F 2 to be the β-twisted sheaf As we said at the beginning of this section, Simpson [Sim95, Theorem 4.7 (4)] ensures the existence of local universal Higgs bundles pU i , Φ i q Ñ XˆZ i for a certain étale covering tZ i Ñ M st u iPI of the stable locus. By universality, there exists a line bundle over each intersection This defines a flat unitary gerbe in the étale topology over M st that we denote by β for the rest of the paper. After having set our gerbe β, observe that U " tU i Ñ XˆZ i u iPI is a πMβ-twisted vector bundle over M st , where π M : XˆM st Ñ M st is the obvious projection. In view of this, we refer to as the universal πMβ-twisted Higgs bundle. This object will be crucial in our description of the Dirac Higgs bundle which we shall address in the next subsection.
We finish this section adapting Kapustin-Witten's definition of a pBBBq-brane [KW07] to this context. Suppose that Y is equipped with a hyperkähler structure, we say that pF, ∇q is a space filling pBBBq-brane if F is a β-twisted vector bundle and ∇ is a connection which is hyperholomorphic, i.e. is p1, 1q with respect to the three complex structures of Y .

The Dirac-Higgs bundle
Hitchin (see [Hit02] for instance) constructed the Dirac-Higgs bundle over the moduli space of Higgs bundle with coprime rank and degree, showing that it can be equipped with a hyperholomorphic structure. The Dirac-Higgs bundle was used by Hausel [Hau98] to study the cohomology of this moduli space. Blaavand [Bla15] studied this object over the moduli space M st , showing that it exists only as a differential object. In a local sense, Hitchin's proof of the existence of a hyperholomorphic structure on the Dirac-Higgs bundle extends to this case, allowing us to produce pBBBq-branes out of it.
Our ultimate goal is to study the mirror dual of pBBBq-branes constructed out of the Dirac-Higgs bundle. For this task, we shall Fourier-Mukai transform the sheaves underlying these pBBBq-branes. As the Fourier-Mukai transform is an algebraic device, we need an algebraic construction of the Dirac-Higgs bundle, which we address in this section. To surpass the non-existence of a universal bundle, we make use of the flat unitary gerbe β defined in the étale topology of M st .
Let us fix a Hermitian metric on the rank n topologically trivial C 8 -bundle E over X. Associated to it, consider the vector space (of infinite dimension) which comes equipped with a natural metric. Given a Higgs bundle E " pE, ϕq supported on E, we write B E for the associated Dolbeault operator and B E for the p1, 0q-part of the Chern connection constructed with the metric and B E . Hitchin introduced in [Hit02] the following Dirac-Higgs operators After (2.13), the kernel ker D˚defines a C 8 vector bundle of rank 2npg´1q over B st . If ever ker Dd escends from B st to give a vector bundle on M st we call the resulting object the Dirac-Higgs bundle. When our moduli space is equipped with a universal family, the Dirac-Higgs bundle is defined as the pull-back of D under the section M st Ñ B st obtained from the universal family. The rank one case, where M st 1 " M 1 , is one of the few cases were the construction of the Dirac-Higgs bundle is possible, and it was achieved by the second named author in [FJ08].
For n ą 1 (and trivial degree) we have already seen that no universal bundle only exists, not even Zariski locally. As we have seen in Section 2.4, for general rank, the best we can obtain is the πMβ-twisted universal bundle pU, Φq " tpU i , Φ i q Ñ XˆZ i qu over the moduli space of stable Higgs bundles M st . We can now define the family of Dirac-type operators (2.14) Recalling (2.13), let us denote, for every i P I, the rank 2npg´1q algebraic sub-bundle of ΩˆZ i D i :" ker Dp U i ,Φ i q , and consider D : which we shall call the β-twisted Dirac-Higgs bundle.
Adapting the work of Hausel [Hau98], one can describe the β-twisted Dirac-Higgs bundle in terms of the universal bundle by means of (2.12). This will show that the β-twisted Dirac-Higgs bundle is, indeed, a β-twisted bundle over M st , what justifies its name.
Proof. Assuming the isomorphism (2.15), it follows from the projection formula that D is a β-twisted bundle over M st . We then focus on the proof of this isomorphism. Considering each local universal bundle pU i , Φ i q over the étale open open subset Z i Ñ M st and recall from Section 2.4 that we denoted π M,i : Globally, one gets (2.15), and the proof is completed.
One can also define a connection on the β-twisted Dirac-Higgs bundle D " tD i Ñ Z i u iPI . Consider the trivial connection d i : Consider also the embedding j i : D i ãÑ ΩˆZ i and the projection pr i : ΩˆZ i Ñ D i defined by the natural metric on ΩˆZ i . Let us consider the connection given by the composition and note that this defines a connection on the β-twisted Dirac-Higgs bundle D, that we call the Dirac-Higgs connection. The importance of this connection comes from fact that the Dirac-Higgs connection is of type p1, 1q with respect to all complex structures, see in [Bla15, Theorem 2.6.3].
One can check that the ∇ i satisfy the compatibility conditions stated in Section 2.4 and ∇ is a connection on the β-twisted Dirac-Higgs bundle D. Hence, it equips the Dirac-Higgs bundle with a hyperholomorphic structure and pD, ∇q is a space-filling pBBBq-brane on M st .

Tensorization and spectral data
The morphism between Higgs moduli spaces that one obtains by considering the tensor product with a particular Higgs bundle will be crucial in our description of the Nahm transform of high rank that we provide in Section 4 below. In this section we explore the behaviour of the spectral data under tensorization, generalizing partial results established in [BS16] for Higgs bundles of rank 2 and 4. This is required in Section 5.3, during our study of the mirror branes dual to the ones we obtain after Nahm transform of high rank.
As it is useful for the purpose of the remaining sections, we add to the notation related to the moduli space of Higgs bundles a sub-index indicating the rank.
Let us introduce in this section the tensorization of two Higgs bundles E " pE, ϕq and F " pF, φq, It is well known that, if E and F are semistable, then E b F is semistable too. Then, fixing some E P M n , one can define a map Remark 3.1. Note that τ E m is hyperholomorphic, meaning that it is a holomorphic morphism between pM m , Γ i m q and pM nm , Γ i nm q for each of the i " 1, 2 or 3. As it is defined, τ E m is clearly holomorphic for i " 1. To see that it is also holomorphic for i " 2, consider the vector bundle with flat connection pE 1 , ∇ E 1 q corresponding to E under the Hitchin-Kobayashi correspondence, and observe that τ E m , in the complex structure Γ 2 , sends the vector bundle with flat connection pF 1 , ∇ F 1 q corresponding to F , to m is holomorphic for i " 1 and i " 2, it is also holomorphic for i " 3 since this complex structure is given by the composition of the previous two.
The map given by the sum along the fibres of the canonical line bundle, σ : K XˆX K X ÝÑ K X , will be necessary for the description of τ E m under the spectral correspondence, which we address next.
Proposition 3.2. Let E be a semistable Higgs bundle with spectral data pS E , L E q and F a (stable) Higgs bundle, Higgs bundle with spectral data pL EbF , S EbF q satisfying where q E and q F denote the projections from S EˆX S F to S E and S F .
Proof. Since S F is smooth, L F is a line bundle over S F and q F is a smooth morphism. This implies that F is stable, so the tensor product E b F is semistable as E is so. By construction, S EbF is a projective curve contained in TotpK X q, and the restriction of the projection morphism p EbF : S EbF Ñ X is an nm-cover. Being defined as the push-forward under σ , if L EbF has a sub-sheaf of dimension 0, so does qE L E b qF L F . We have that qE L E is torsion free, as q E is a smooth morphism and L E is torsion free. Since qF L F is a line bundle, qE L E b qF L F is also torsion-free, and this implies that L EbF is torsion free as well, so the pair pS EbF , L EbF q is the spectral data of some Higgs bundle. The proof would be completed if we show that this Higgs bundle is indeed E b F . In view of (2.4) and (2.5), we need to show that E b F and ϕ b 1 F`1E b φ arise as the push-forward under p EbF of L EbF and the morphism ψ EbF : L EbF Ñ L EbF b pE bF K X given by tensoring with the tautological section λ EbF of pE bF K X .
Note that p EbF fits in the commuting diagram which is commutative and its exterior is Cartesian.
Denote by λ E the tautological section of pE K X over S E and by λ F the tautological section of pF K X over S F . In view of the previous commutative diagrams, one has that σ˚λ EbF , qE λ E and qF λ F are all sections of π˚K X , and one has the equality σ˚λ EbF " qE λ E`qF λ F .
Then, considering the morphism given by tensorization under σ˚λ EbF , one has the decomposition to be the tensorization under qE λ E , and similarly for r ψ F . Denoting by the morphism obtained by tensorizarion by λ E , one has that One can define ψ F analogously, obtaining Note that we obtain a similar expression for the last line using π " q E˝pE . Also, observe that p EbF,˚ψEbF -π˚r ψp E,˚qE,˚r ψ E`pF,˚qF,˚r ψ F .
From now on, we shall focus only on the study of p F,˚qF,˚r ψ F as the description of p E,˚qE,˚r ψ E is completely analogous.
As L F is a line bundle, the projection formula further gives p EbF,˚LEbF -p F,˚p pq F,˚qE L E q b L F q and, after (3.4), Since the exterior arrows of (3.3) provide a Cartesian diagram, and p F is flat, flat base change allow us to identify As pF E is locally free, applying again the projection formula we obtain the desired equality Similarly, one obtains and the proof is concluded.
Consider the semistable Higgs bundle E " pE, ϕq associated to the spectral data pL E , S E q, where L E is a line bundle. After Proposition 3.2, the morphism τ E m : Jac

Corollary 3.3. For every semistable Higgs bundle E " pE, ϕq of rank n, the diagram
commutes.
Finally, we study the relation of τ E m and the holomorphic 2-forms Λ 1 m and Λ 1 nm .
Proof. Recall that g nm is obtained from the Gauge invariant metric (2.1). Observe that the isomorphism E nm -E n b E m induces the isomorphism of vector bundles and note that the infinitesimal deformations of the Higgs bundles contained in the image of τ E m are of the form 1 E b 9 α m P Ω 0,1 X pEndpE nm qq and 1 E b 9 ϕ m P Ω 1,0 X pEndpE nm qq. Then, one can easily check that τ E,m g nm " rkpEqg m .

Nahm transform of high rank
In [Jar04], the Nahm transform is constructed for any a family of self-dual connections over a spin fourmanifold with non-negative scalar curvature. The construction produces a vector bundle with connection over the parametrizing space of the family once we consider, at each point of the parametrizing space, the cokernel of the associated Dirac operator. It is also possible to define naturally a connection on this bundle. The Nahm transform for Higgs bundles is defined in [FJ08] considering, for each stable Higgs bundle, the family obtained by twisting with the universal family of rank 1 Higgs bundles. Hence, for each stable Higgs bundle, we obtain a Hermitian connection over M 1 of type p1, 1q with respect to the complex structures Γ 1 1 , Γ 2 1 and Γ 3 1 . Here, we generalize the previous construction to moduli spaces of stable Higgs bundles of arbitrary rank. The main difference with the rank 1 case relies in the fact that there is no universal bundle over XˆM st n for n ą 1, not even Zariski locally. Therefore, as we did in Section 2.5, we shall work with the gerbe β n in the étale topology of M n , and make use of the definition of β n -twisted bundles.
Fix a Higgs bundle E " pE, ϕq of rank n and degree 0, supported on the Hermitian C 8 vector bundle E n . For every rank m Higgs bundle F " pF, φq with degpFq " 0, supported on the Hermitian C 8 vector bundle E m , we can consider the Higgs bundle If E and F are semistable, it is well known that E b F is semistable, although, for m ą 1, such correspondence is no longer valid when we replace semistability with stability. We recall that a crucial step in the construction of the Dirac-Higgs bundle is Hausel's vanishing statement [Hau98, Corollary 5.1.4] which ensures that ker D E " 0 whenever E is stable and different from the trivial Higgs bundle of degree 0 (where by trivial Higgs bundle we mean the pair pO X , 0q given by the trivial line bundle and zero Higgs field). Note that H 0 pO X , 0q " H 2 pO X , 0q " C. Since we intend to study the Dirac operator over the locus of Higgs bundles obtained as a tensor product, we need to study first whether or not one can generalize Hausel's vanishing statement to this locus. This justifies the following definition.
If E be a semistable Higgs bundle of degree 0, let grpEq " ' j G j be the associated graded object, where pG 1 , . . . , G l q are the stable factors of its Jordan-Hölder filtration; we say that E is without trivial factors if none of these factors G j is the trivial Higgs bundle pO X , 0q. Clearly, every nontrivial stable Higgs bundle is without trivial factors. Proof. Let E is a semistable Higgs bundle of degree 0 and let be its Jordan-Hölder filtration, and let G j " E j {E j´1 be its factors. We prove the first claim by induction on the length l of the Jordan-Hölder filtration.
If l " 1, then E is stable and the claim is just the corollary due to Hausel mentioned above. For the induction step, assume that the claim holds when the l " k´1 and consider the short exact sequence of Higgs bundles If E is without trivial factors, then G l is a nontrivial stable Higgs bundle, and E l´1 is also without trivial factors, so that, by the induction hypothesis H 0 pE l´1 q " H 2 pE l´1 q " H 0 pG l q " H 2 pG l q " 0.
The associated long exact sequence in hypercohomology then yields H 0 pEq " H 2 pEq " 0 and H 1 pEq " H 1 pE l´1 q ' H 1 pG l q " l à j"1 H 1 pG j q.
In particular, we also conclude that if E is without trivial factors, then ker D E " 0.
For the converse statement, assume that one of the factors of the Jordan-Hölder filtration, say G j with j P t1, . . . , lu, is trivial. Considering the exact sequence we obtain a surjective map H 2 pE j q H 2 pG j q " C, thus ker D E j ‰ 0. The monomorphism E j ãÑ E then provides an injective map ker D E j ãÑ ker D E , proving that ker D E ‰ 0 as well.
Lemma 4.1 shows that the rank of ker DE does not jump if we remain inside the locus of Higgs bundle without trivial factors, but it will as long as we leave this locus. Below we find conditions under which E b F is without trivial factors. (1) if n ą m and E is stable, then E b F is without trivial factors; (2) if n ă m and F is stable, then E b F is without trivial factors; (3) if n " m, E and F are stable and F fl E˚, then E b F is without trivial factors.
In particular, if n ‰ m and both E and F are stable, then E b F is without trivial factors.
Proof. The (semi)stability of E implies the (semi)stability of E˚" pE˚,´ϕ t q. If E b F has a trivial factor, then there exists ψ : O X Ñ E b F such that pϕ b 1 F`1E b φqpψq " 0. Equivalently, there exists a nontrivial morphism ψ : E˚Ñ F such that pψ b 1 K X q˝p´ϕ t q " φ˝ψ. As a consequence, the image Im ψ is a φ-invariant sub-sheaf of F and its saturation Im ψ a φ-invariant sub-bundle of F. Also, the kernel ker ψ is a p´ϕ t q-invariant bundle of E. As degpIm ψq ą degpIm ψq, note that degpIm ψq " degpker ψq " 0 and Im ψ " Im ψ due to the semistability of E˚and F and the fact that both have trivial degree. Hence Im ψ is a vector sub-bundle of F. If n ‰ m, either Im ψ or ker ψ are proper sub-bundles, contradicting the stability of F or E˚, respectively. Finally, suppose that n " m, both E and F are stable and F is not isomorphic to the dual E˚. Then E b F is without trivial factor, as otherwise the last condition will be violated.

Denote by M E
m the open subset of M st m given by those Higgs bundles F such that E b F is without trivial factors. In virtue of Lemma 4.2, we require that E is stable when rkpEq ě m, and under these conditions Given an integer m and a semistable E " pE, ϕq (E stable if rkpEq ě m), we define, for each F " pF, φq in M E m , the following Dirac-type operators where we make use of the isomorphism (3.5), ÝÑ Ω nm , and its adjoint D E,F :" DE bF : Ω nm ÝÑ Ω 1,1 X pE n b E m q ' Ω 1,1 X pE n b E m q. After Lemma 4.2 and (2.13), one has that dim ker D E,F " dim ker DE bF " 2nmpg´1q is fixed. As we did in the definition of the β n -twisted Dirac-Higgs bundle in Section 2.5, take an étale covering tZ m,i Ñ M E m u iPI which is fine enough for the gerbe β m and recall the β m -twisted universal Higgs bundle pU m , Φ m q " tpU m,i , Φ m,i q Ñ XˆZ m,i u iPI on XˆM st m . For all i P I, consider the families of Dirac-type operators defined point-wise as we did in (2.14), and set We define the rank m Nahm transform of the Higgs bundle E as the pair It can be shown that the rank m Nahm transform is a β m -twisted bundle with connection by providing an analogous result to Proposition 2.1 obtained by adapting the work of Hausel [Hau98].
Proof. This follows immediately from the proof of Proposition 2.1.
Remark 4.4. The Nahm transform defined by Frejlich and the second named author in [FJ08] is precisely the case m " 1 of the construction above. Note that, in this case, the gerbe β 1 is trivial, so p E 1 can be defined globally as a vector bundle over M 1 . Proposition 4.6. Consider the rank m Nahm transform of a stable Higgs bundle E of rank n. One has that Proof. By the universal property of pU nm , Φ nm q, one has that the restriction to Xˆpτ E m q´1pM st nm q of the gerbe β nm is isomorphic to (the pull-back of) β m , hence πXE b pU m , Φ m q is isomorphic to the restriction of pU nm , Φ nm q. The proof follows from this fact and Propositions 2.1 and 4.3.
We can then construct a new class of β m -twisted pBBBq-branes. Proof. We have seen in Remark 3.1 that τ E m is a hyperholomorphic morphism. Since the Dirac-Higgs connection is hyperholomorphic [Bla15, Theorem 2.6.3], thanks to Proposition 4.6, one has that the higher rank Nahm transform is hyperholomorphic as well.

A Fourier-Mukai transform for bundles twisted by gerbes
The goal of this section is to describe the mirror partners of the Nahm pBBBq-branes we have constructed in previous sections. We shall do that by Fourier-Mukai transforming the Dirac-Higgs bundle and the bundles we obtain from the high rank Nahm transform. Before that, we first need to adapt the Fourier-Mukai transform to the setting of vector bundles twisted with a gerbe, a task that we address in this subsection. This requires the introduction of some notation.
Let Y Ñ V be a smooth V -variety equipped with a sectionσ : V Ñ Y . In that case (see [BLR90,8.2, Proposition 4]), there exists a relative Jacobian Jac 0 V pY q and let us consider Y _ to be a torsor for Jac 0 V pY q. Consider an étale covering tV i Ñ V u iPI of V and observe that tζ i : are étale coverings of, respectively, Y and Y _ . Let β be a flat unitary gerbe on the étale topology over Y giving the set of flat unitary line bundles tL ij Ñ YˆV V ij u i,jPI over the intersections pYˆV V i qˆV pYˆV V j q -YˆpV iˆV V j q " YˆV V ij . These flat line bundles define naturally sectionš σ ij : V ij Ñ Jac 0 V ij pY q which can be understood as V ij -automorphisms of Y _ . By abuse of notation, we still denote them byσ In this context, if Z Ñ V is another V -variety, we say that f : Z Ñ Y _ is a β-shifted morphism of V -varieties if it is a collection of morphisms tf i : for every i, j P I. When all the f i satisfy a certain property of morphisms of varieties, we say that the β-shifted morphism f has this property. For instance, if Y _ is equipped with a symplectic form Λ, we say that the image of f : Z Ñ Y _ is Lagrangian, if f i pZˆV V i q is a Lagrangian subvariety of Y _ˆV V i with respect to pull-back of the symplectic form pζ _ i q˚Λ. Also, we say that a β-shifted coherent sheaf G is a collection of coherent sheaves tG i P Coh pY _ˆV V i qu iPI such that for every i, j P I one has Observe that f˚G is naturally a β-shifted coherent sheaf as for every i, j P I, one has thanks to the base-change theorems and the definition of β-shifted morphism.
Let us now adapt Kapustin-Witten's definition of a pBAAq-brane to this context. Suppose Y _ is equipped with hyperkähler structure and denote by Λ the holomorphic symplectic 2-form associated to the first Kähler structure. We say that a β-shifted sheaf G " tG i u iPI admits a β-shifted pBAAq-brane structure if the support of each of the G i is a Lagrangian sub-variety of Y _ˆV V i with respect to pζ _ i q˚Λ. We will now describe a certain β-shifted morphism that will be crucial for our purposes. Recall that we denoted by B 1 Ă B the locus of smooth spectral curves, and by S 1 , the restriction of S and M to B 1 . Also, we denote by M 1 the restriction of M to B 1 , and bỳ U 1 , Φ 1˘: " pU, Φq| M 1 , the restriction of the universal bundle to M 1 Ă M st . Thanks to the spectral correspondence outlined in Section 2.2, the existence of pU 1 , Φ 1 q Ñ M 1 implies that, locally in the étale topology, there exists a πMβ-twisted universal line bundle P 1 Ñ SˆB1 Jac δ B 1 pS 1 q, satisfying U 1 " p1 Jacˆp q˚P 1 , where p is the projection (2.2). We can provide a specific, fine enough covering for it.
Proposition 5.1. There exists an étale covering tV i Ñ B 1 u iPI of B 1 such that is an étale covering of M 1 which is fine enough for P 1 .
Proof. Recall that two spectral curves intersect on a divisor of length 2npg´1q. Let us pick a smooth spectral curve S i and set B 1 i to be the open subset of B 1 given by those curves S b that intersect S i in 2npg´1q different points, i.e. those curves giving a reduced intersection divisor S i X S b . Chose a collection I of spectral curves in such a way that the union of all B 1 i covers B 1 . One can easily see that for every smooth spectral curve S b there exist another one S i such that S b X S i is a reduced divisor; this guarantees the existence of a covering with the desired properties.
Inside TotpK X qˆB 1 i , we consider the intersection Observe that, by construction, this provides an étale morphism V i Ñ B 1 with image B 1 i . This gives the étale covering tV i Ñ B 1 u iPI . Denote (5.2) S 1 i :" S 1ˆB 1 V i and observe that S 1 i Ñ V i has naturally a section since V i embeds into S| B 1 i . It then follows that Jac δ V i pS 1 i q exists and it is equipped with a universal line bundle P 1 i Ñ S 1 iˆVi Jac δ V i pS 1 i q.
Recalling the Poincaré bundle P Ñ Jac δ B 1 pS 1 qˆB1 Jac δ B 1 pS 1 q _ , we observe that the πMβ-twisted universal line bundle P 1 defines naturally a πMβ-shifted closed immersion For each open étale subset of Proposition 5.1, we can consider a diagram analogous to (2.8) and a Poincaré bundle Proceeding as in (2.9) and (2.10), we define the Fourier-Mukai transforms RF i and RF i replacing P by P i . Given a complex of β-twisted coherent sheaves F ‚ , define its β-twisted Fourier-Mukai transform RF β pF ‚ q as tRF i pF ‚ i q Ñ Z i u iPI and consider an analogous definition in the case of RF β . We say that a β-twisted coherent sheaf F " tF i Ñ Z i u iPI is -WIT if each of the F i is -WIT with respect to RF i . Similarly, a β-shifted coherent sheaf G " tG i Ñ Z i u iPI is -WIT if each of the G i is -WIT with respect to RF i . We denote by Wit β pJac δ B 1 pS 1 qq the category of β-twisted -WIT sheaves on Jac δ B 1 pS 1 q and by Wit β pJac δ B 1 pS 1 q _ q the category of β-shifted -WIT sheaves on Jac δ B 1 pS 1 q _ . In the following theorem we see that the β-twisted Fourier-Mukai transform relates both categories.

Theorem 5.2. The β-twisted Fourier-Mukai transform RF β induces an equivalence of categories
Proof. We first prove that the Fourier-Mukai transform of a -WIT β-twisted coherent sheaf is a pd´ q-WIT β-shifted coherent sheaf. We observe that the following diagram 1 Jac _ˆB1 ζ i / / Z i commutes, beingπ ij the projection to the second factor. The diagram is also Cartesian as we naturally have Z iˆV i Z ij -Z ijˆV ij Z ij . Using the commutativity of the previous diagram and base change theorems, one can show that where P ij is the Poincaré bundle over Z ijˆV ij Z ij -Jac δ V ij pS ij qˆV ij Jac δ V ij pS ij q and RF ij the corresponding Fourier-Mukai transform.
Let us recall that β-twisted sheaves must satisfy that Now we recall that one of the classical properties [Muk81] of the Fourier-Mukai transform states that RF ij`Lij b p1 Jac _ˆB1 ζ j q˚F j˘-σij,˚RFij`p 1 Jac _ˆB1 ζ j q˚F j˘. Then, so we see that the RF β pFq is indeed a β-twisted sheaf. Finally, since all the RF i are equivalence of categories of WIT sheaves with inverse RF i , the same holds between β-twisted and β-shifted WIT sheaves.

The Dirac-Higgs bundle under Fourier-Mukai
Recall that M 1 Ă M st denotes the locus of the moduli space given by those Higgs bundles whose spectral curves are smooth. Let us consider the restriction to M 1 of the β-twisted Dirac-Higgs bundle In the remaining of the section we provide a description of D 1 in terms of a Fourier-Mukai transform. We shall consider the intersection of our family of smooth spectral curves S 1 with the zero section of the bundle K X Ñ X, which is identified with Xˆt0u, Note that for each b " pb 1 , . . . , b n q P B 1 one has that S b X Xˆt0u is the locus where b n " 0, being b n P H 0 pK bn X q. Recall that deg K bn X " 2npg´1q. Then Ξ 0 is a finite cover over B 1 of degree 2npg´1q, Take an étale open subset V i Ñ B 1 and the family of curves S 1 i Ñ V i as defined in (5.2) and consider the obvious projections occurring in the following commutative diagram, Let us denote by r p i : S 1 i Ñ X the composition of p i with the obvious projection r i onto the second factor. Observe that the bundle r pi K X Ñ S 1 i has a tautological section that we denote by λ i . We observe that λ i vanishes at Ξ 0 i :" Ξ 0ˆB 1 V i . Using (2.15), we give a description of D 1 which generalizes the fibrewise description given in [Hit16,Section 7]. Recall the β-shifted closed embedding i : S 1 ãÑ M 1 defined in Section 5.1. Proposition 5.3. Consider the β-shifted push-forward under i, Then,Ď 1 is a β-shifted 0-WIT sheaf on Jac δ B 1 pSq _ and Proof. We work locally over the étale open subset Z i " V iˆB 1 M 1 . Starting from (2.15) and the isomorphism (2.7), note that Next, recalling the relation between the universal bundle from Section 2.4 and the Poincaré bundle described in Section 5.1, and making use of the projection formula and base change theorems for the various morphisms in the diagram (5.5), we obtain where for the previous to last equality, we have used the (vertical) spectral sequence and the fact that λ i is an embedding.
Recall that, by the definition of i " tı i : S 1 i ãÑ Jac δ V i pS 1 i q _ u, one has that the restriction of the Poincaré bundle Pi Ñ Jac δ V i pS 1 i qˆV i Jac δ V i pS 1 i q _ to the image of the embedding 1 Jacˆıi , coincides with P 1 i . Using this, the projection formula, and base change theorems on the diagram (5.5), we have that Since ı i is a closed embedding, one has that R 0 ı i,˚p r qq is a complex supported in degree 0. with supp´Ď 1¯" i`Ξ 0˘.

Nahm branes under Fourier-Mukai
We finished Section 4 showing that the Nahm transform is a β m -twisted pBBBq-brane on the moduli space of stable Higgs bundles M st m (or on a dense open subset in the case n " m) which called Nahm brane associated to E. Using the formalism that we presented in Section 2.3, it is then natural to ask how the Nahm branes transform under mirror symmetry, which we address in this section.
Assume for simplicity that n ‰ m and that E is a stable Higgs bundle of rank n and 0 degree with reduced spectral curve S E . As n ‰ m, the rank m Nahm transform p E m of E is a β m -twisted pBBBq-brane over M E m " M st m . Also, recall that the smooth locus of the Hitchin fibration is contained in the stable locus, Define also the family of curves inside TotpK X q parametrized by B 1 m , where σ denotes here the fibrewise sum in rmK X Ñ XˆB 1 m , where r m : B 1 mˆX Ñ X is the projection onto the second factor. After Proposition 3.2, one has the following commutative diagram for families given by the obvious projections m is a nm-cover, r p E " pp Eˆ1B 1 m q and q m are n-covers, and p m and r q E are m-covers. Noting that S E Ă TotpK X q, consider p´1q to be the (additive) inversion along the fibres of K X and consider the family of curves´S E . Define one more family of curves over B 1 m , Using a (vertical) spectral sequence similar to the one that appears in Proposition 5.3, one can show since 1 b λ m,i`r pm ,i ϕ b 1 is again injective. Next, using the description of the Poincaré bundle P m in terms of P m given in Section 2.4, the projection formula and base change theorems for the various morphisms of diagram (5.5), one obtains Finally, as in the last part of the proof of Proposition 5. We study now its support.
Theorem 5.7. The support of q E 1 m is a β m -shifted 2nmpg´1q-section of the Hitchin fibration and it is Lagrangian with respect to Λ 1 m .
Proof. Recall the étale covering tV m,i Ñ B 1 m u iPI of the locus of smooth spectral curves which is fine enough for the gerbe β m . This induces an étale covering tz i : M 1 mˆB 1 V m,i Ñ M 1 m u iPI . Since i m is a β m -shifted embedding and Ξ E m is a 2nmpg´1q-cover of B 1 m , we have that i m p´Ξ E m q is a β m -shifted 2nmpg´1q-section of M 1 m Ñ B 1 m . It remains to proof that it is Lagrangian with respect to Λ 1 m,i :" zi Λ 1 m . Recalling Corollary 5.6, we now address the proof that i m p´Ξ E m q is Lagragian with respect to Λ 1 m . Recall from Section 2.1, that Λ 1 m is defined as the exterior derivative dθ of a certain 1-form θ. Then, we see that ı m,i p´Ξ E m,i q is Lagrangian with respect to Λ 1 m,i if and only if θ i :" zi θ is a constant 1-form along ı m,i p´Ξ E m,i q for all i P I. Since ı m,i is an embedding, it suffices to prove that ım ,i θ i |´ΞE m,i is constant.
We recall the definition of θ. We recall that M 1 m Ă M st m so all the points are smooth and represented by the stable Higgs bundle E " pE, ϕq, the tangent space is T E M 1 m " H 1 pC ‚ E q, which comes naturally equipped with the map t : H 1 pC ‚ E q Ñ H 1 pX, EndpEqq. By Serre duality, the Higgs field ϕ P H 0 pEndpEq b K X q is an element of the dual space of H 1 pEndpEqq and recall that we defined θpvq " xϕ, tpvqy, for each v P H 1 pC ‚ E q.
We now study the description of θ i over Jac δ m V m,i pS 1 m,i q. By the spectral correspondence, given the spectral data L Ñ S m,b , one has that E " pp b q˚L and ϕ " pp b q˚λ b where λ b : L Ñ L b pb K X is given by tensoring by the restriction to S m,b of the tautological section λ : TotpK X q Ñ p˚K X . Note that pb K X is a sub-sheaf of the canonical bundle K S m,b , then λ b gives naturally an element of H 0 pS m,b , K S m,b q. The isomorphism M 1 m,i -Jac δ m V m,i pS 1 m,i q _ -Jac δ m V m,i pS 1 m,i q, given by the push-forward under p b and autoduality, provides as well the isomorphism between H 1 pC ‚ E q and Ext 1 TotpK X q pL, Lq and between H 1 pX, EndpEqq and Ext 1 S m,b pL, Lq -H 1 pS m,b , O S m,b q. Then, we can express θ i pvq to be xλ b , t 1 pvqy given by Serre duality, where now v P T L Jac δ m V m,i pS 1 m,i q -Ext 1 TotpK X q pL, Lq, the section λ b P H 0 pS m,b , K S m,b q is defined by the restriction of the tautological section λ : TotpK X q Ñ p˚K X to S m,b Ă TotpK X q, and t 1 : Ext 1 TotpK X q pL, Lq Ñ Ext 1 S m,b pL, Lq -H 1 pS m,b , O S m,b q is the projection to those deformations that preserve the support.
Note that, for every L 1 , L 2 P JacpS m,b q, one has naturally that Ext 1 S m,b pL 1 , L 1 q -H 1 pS m,b , O S m,b q -Ext 1 S m,b pL 2 , L 2 q. We observe that θ i is a 1-form which is constant along the fibres Jac δ m pS m,b q. On the other hand, the 1-form ım ,i θ i in S 1 m,i depends on the embedding dı m,i : T s S m,b ãÑ T Ops´s i q Jac δ m pS m,b q -H 1 pS m,b , O S m,b q. Recall that ı m,i sends the point s P S m,b to the line bundle whose meromorphic sections have pole at s P S m,b and a zero at s i . Since Serre duality x¨,¨y : H 0 pK S m,b qˆH 1 pO S m,b q Ñ C sends xλ, ξy to the sum of residues of the meromorphic differential λξ, one has that ım ,i θ i | s -ım ,i xλ b ,¨y| s -λ b psq.
So, ım ,i θ i is the one form defined by the tautological section λ : TotpK X q Ñ p˚K X .
Obviously, the tautological section λ restricted to Xˆt0u Ă TotpK X q, is the zero section. Recall that we have defined´Ξ E m,i as the intersection of S E,i and S 1 m,i inside TotpK X q. But this is equivalent to the intersection of Σ Em ,i with Xˆt0u. Therefore, λ is constantly 0 along´Ξ E m,i , that is ım ,i θ i | Ξ E m,i " 0, and this concludes the proof.
We finish placing this statement in the context of mirror symmetry.