Principal bundles on elliptic fibrations

arXiv:alg-geom/9702002v2 22 Dec 1997 alg-geom/9702002 Principal bundles on elliptic fibrations Ron Y. Donagi Institute for Advanced Study Princeton, NJ 08540, USA and Department of Mathematics, University of Pennsylvania Philadelphia, PA 19104, USA ABSTRACT A central role in recent investigations of the duality of F-theory and heterotic strings is played by the moduli of principal bundles, with various structure groups G, over an elliptically fibered Calabi-Yau manifold on which the heterotic theory is compactified. In this note we propose a simple algebro-geometric technique for studying the moduli spaces of principal G-bundles on an arbitrary variety X which is elliptically fibered over a base S: The moduli space itself is naturally fibered over a weighted projective base parametrizing spectral covers S̃ of S, and the fibers are identified as translates of distinguished Pryms of these covers. In nice situations, the generic Prym fiber is isogenous to the product of a finite group and an abelian subvariety of P ic(S̃). January 1997 1. Introduction Moduli spaces of principal G-bundles on K3 surfaces, and on Calabi-Yaus of other dimensions, are basic ingredients of the compactification of heterotic strings, especially for semisimple structure groups G contained in E8 × E8 or in Spin(32)/Z2. More recently, duality of the heterotic string with F -theory [1] has suggested the special importance of moduli spaces of G-bundles on elliptically fibered Calabi-Yaus. The purpose of this note is to propose a general technique for studying G-bundles on (not necessarily CY) elliptic fibrations X → S, with arbitrary semisimple structure group G. The idea is quite simple. To a principal G-bundle P over X we associate its spectral data. This consists of the cameral cover S̃ → S, a Galois cover with covering group the Weyl group W of G; of a collection of multisection maps vλ : S̃ → X, one for each character λ ∈ Λ of the maximal torus T ⊂ G, subject to the condition of W -equivariance (the spectral covers are the various images of S̃ in X); and of a T -bundle over S̃, subject to a certain twisted form of W -equivariance. When the cameral cover is reasonably nice, it determines the distinguished Prym Prym(S̃/S), an extension of an abelian variety by a finite group. The G-bundles on X with a given cameral cover S̃ (and any consistent collection of maps to X) can then be parametrized by the distinguished Prym. An open subset of the moduli of G-bundles can thus be fibered over the parameter space for nice cameral covers, the fibers being the distinguished Pryms. When the cameral cover is not-so-nice, the description becomes less precise; but the whole construction has a local character, so bad behavior can be traced to specific singular loci. We avoid much of the difficulty by considering regularized bundles, or bundles with some additional structure, specified by a reduction of the structure group, over each s ∈ S, to an abelian subgroup which is the centralizer in G of a regular element. (We call such a subgroup a regular centralizer.) Nice bundles have a unique regularization, special ones may have large families of regularizations, and others will have none, but can become regularized after some blowing up in the base S. The reason that regular centralizers are the right structure can be seen already from the behavior of G-bundles on a single elliptic curve E, and already for G = SL(2). The structure group of “most” semistable G-bundles on E can be reduced to a maximal torus T ⊂ G. For G = SL(2), there is an essentially unique exception: the non-trivial extension of O by itself (or the same tensored with one of the four spin bundles). The structure group in this case can still be reduced to an abelian subgroup, though not to T : it can be reduced to the group of upper triangular matrices with 1 or -1 on the diagonal, i.e. to the centralizer of a non-zero nilpotent element in SL(2). As an abstract group, this is Z2 times the additive group (C, +). Now the family of bundles which admit a reduction to a given torus T is parametrized by P1 , or more naturally by E modulo its involution. There are only 8 isomorphism classes of bundles with reduction to a nilpotent regular centralizer: the above four extensions of L by L, where L is a spin bundle, as well as the four trivial extensions L ⊕ L. An important point is that all regular centralizers in G fit nicely into one family, C. For SL(2), the base of this family is P2 = (P1 × P1 )/Z2 : the tori are parametrized by the complement of the diagonal P1 , while the exotics sit over P1 . 1 Over this P2 lives the space parametrizing SL(2)-bundles on E together with a reduction of their structure group to some regular centralizer. This space looks like the threefold P1 × P1 × E divided by the obvious Z2 , with the singularities blown up (yielding fibers of type I0∗ over P1 ) and the components of multiplicity 2 discarded. Over this threefold there is a “universal C-bundle”, from which our G-bundles can be induced. We will see that this picture generalizes to any G, and is the basic ingredient behind the reconstruction of a G-bundle from its spectral data. Our construction is motivated by our previous work [2], and in a sense is contained in it. The objects studied in [2] may appear, at first sight, different than the ones that concern us here. There we were mostly interested in K-valued Higgs fields on S, where K is a line (or vector) bundle on S. As we review in section 4, the parametrization of these “K-valued” objects is reduced to that of abstract, or unvalued, objects, and those are precisely the ones that come up in connection with G-bundles on X → S. From this point of view, we may think of a G-bundle on X as a Higgs bundle on S whose “values” are in the elliptic curves of the fibration. The main construction, or rather its reduction to the construction in [2], is in section 4. Some basics on G-bundles on a single elliptic curve are gathered in section 2, while the behavior in families is discussed in section 3. There we also address the technical question of how to describe the family of all cameral covers. For this we need to understand the global properties of the moduli of G-bundles on a fixed elliptic curve, as well as the modular behavior seen when the curve is varied. The former were worked out by Looijenga [3] and Bernstein and Shvartsman [4], and the latter, for all simply connected groups except E8, by Wirthmuller [5]. It is a pleasure to thank Ed Witten, for asking the questions about moduli of Gbundles on elliptic fibrations which got me interested in the subject, and for drawing my attention to references [3], [4]. His joint work with Friedman and Morgan [6], just posted to hep-th, has some overlap with this note. Roughly speaking, the emphasis in [6] is on a description of the parameter space for spectral covers, while we focus on the fiber, which parametrizes bundles with a given cover. The theorem of Looijenga is recovered in full in [6], and there is also a discussion of Wirthmuller’s work. Friedman, Morgan and Witten also obtain applications to the duality between F-theory and the heterotic string. They present several beautiful descriptions of the moduli space of G-bundles on an elliptic curve. While we use deformations (of semisimple and semistable bundles, in the beginning of section 2) only to obtain a rather rough local picture of this moduli space, they base their main construction on the deformations of a “minimally unstable” bundle. This gives them a global description, as well as the proof of Looijenga’s theorem. In our approach, we take the results of Looijenga, Bernstein-Shvartsman, and Wirthmuller as given, and use them to describe the parameter space of the covers. The main novelty of our approach is the direct construction of the bundle corresponding to given spectral data, via the regular centralizers. I am also grateful to Tony Pantev, for many discussions and for showing me an advance 2 copy of another related work [7], to Eduard Looijenga who explained his works [3] to me and who told me about [5], and to Pierre Deligne for some very helpful comments on the manuscript. I have also benefitted greatly from conversations with J. Bernstein, R. Lazarsfeld, R. Livne, and V. Sadov on various aspects of the work described here. This work was partially supported by NSF grant DMS95-03249, a Lady Davis Fellowship from the Hebrew University, and grants from the University of Pennsylvania Research Foundation and the Harmon Duncombe Foundation. 2. G-bundles on elliptic curves Let G be a connected, simply-connected complex semisimple group, T its maximal torus, E an elliptic curve, i.e. a non singular curve of genus 1 with a marked point 0 ∈ E. The moduli space MET of degree-0 semistable T -bundles on E is Hom(Λ, Ě), where Λ := Hom(T, C) is the lattice of characters of G, and Ě := Pic0 E is the dual elliptic curve, which is naturally identified with E. This moduli space is an abelian variety, in fact it is (non-naturally) isomorphic to E r , where r is the rank of G. It comes with a natural action of the Weyl group W , as well as a natural polarization (cf.[3]) whose degree is the discriminant of Λ. Over E × MET there is a universal T -bundle. In case T = C∗ , MET is just the dual elliptic curve Ě, and the universal bundle is the Poincare bundle. Via the unnatural identification of the general MET with E r , the universal bundle becomes the sum of r Poincare bundles pulled back from the r factors. In this abelian situation there is actually a moduli space of all (not necessarily semistable) T -bundles: just replace Ě = Pic0 E throughout by Pic E. The moduli space MEG of semistable G-bundles on E is the quotient MET /W , where the Weyl group W acts through Λ. This moduli space parametrizes not isomorphism classes but s-equivalence classes of semistable G-bundles. There is an open set where the two notions coincide, but over smaller strata there will be a finite number of isomorphism classes represented by each point in moduli. The set of isomorphism classes maps naturally to MEG , and this map behaves nicely in families. (This classifying map is, of course, part of the definition of a coarse moduli space.) This map can be given as follows: any semistable principal G-bundle P on E has a semistable reduction of its structure group to the Borel subgroup B ⊂ G, in other words, it is associated to some semisimple B-bundle PB . The quotient map B → T then gives an associated T -bundle PT which, up to the action of W , is independent of the choice of Borel reduction. The existence of a Borel reduction can be seen, e.g., by finding a flat holomorphic connection on P ([8]) and noting that the holonomy is abelian, as image of π1 (E). (The entire discussion extends easily to non simply connected groups, as long as we restrict attention to those bundles which are liftable to the simply connected covering group.) We can develop a feel for this moduli space by considering the deformation theory of semisimple bundles, i.e. ones associated to T -bundles. Let PT be a principal T -bundle on E, and P the associated (semisimple) G-bundle. Deformations of P are unobstructed, 3 parametrized by H 0 (adP )∗ . Now ad(P ) = ad(PT ) ⊕ (⊕α Lα ), where the sum is over the roots α ∈ R of G, and Lα is the line bundle associated to PT by the root α. For generic P each Lα is a non-trivial line bundle of degree 0 on E, so H 0 (adP ) = H 0 (adPT ). This means that all deformations of such generic P come from deformations of PT . But for non-generic semisimple P , there can be other deformations. The possible types of non-generic semisimple P are indexed by subroot systems: An arbitrary PT is given by a homomorphism p : Λ → E, which determines a sub root system R′ := R ∩ ker(p), generating a sublattice Λ′ ⊂ ker(p) ⊂ Λ. The structure group of this PT then reduces to T̄ := Hom(Λ/Λ′ , C∗ ) ⊂ T. Let G′ be the centralizer ZG (T̄ ), with Lie algebra g′ . The α for which Lα is trivial are precisely the roots of g′ , so the deformations are now parametrized by the dual g′∗ , and the general deformation has structure group G′ . Since G′ is the automorphism group of P , it acts on the deformation space, and the isomorphism type is fixed along G′ -orbits. The semisimple locus looks near PT like t, and modulo G′ this becomes t/W ′ (t is the Cartan subalgebra, W ′ is the Weyl group of R′ ), while the new transversal deformations look like the nilpotent cone in g′ . Modulo G′ , this gives just a finite set of types, with a non-separated topology. In case G = SL(n) we recover Atiyah’s description [9] of vector bundles on elliptic curves as sums of indecomposables. The possible root systems R′ correspond in this case to faces of the Weyl chamber, or equivalently to parabolic subgroups containing B. For other groups there may be additional possibilites; e.g. for G2 there are, in addition to the parabolic types 0, Ashort , Along , 1 1 long long short and G2 , also the two possibilities A2 and A1 × A1 , which arise when the image of p : Λ → E happens to consist of points of order two. But in any case, the nilpotent cone breaks into a finite number of orbits. The closed orbit is 0, corresponding to the semisimple bundles. At the opposite end, there is a unique dense orbit, corresponding to regular bundles. For such bundles P, H 0 (adP ) is again of the smallest possible dimension, namely r. It is the centralizer in g′ of a regular nilpotent element. There is a natural, “universal”, space UE parametrizing G-bundles on E together with a trivialization (frame) over 0 ∈ E and a reduction to a regular centralizer. To describe it, we start with the quotient G/T which parametrizes pairs T ′ ⊂ B ′ consisting of a maximal torus T ′ and a Borel subgroup containing it, or equivalently a torus with a choice of chamber. The Weyl group W acts on G/T . The quotient G/N (where N := NG (T ) is the normalizer of T in G) parametrizes tori T ′ . Equivalently, G/T and G/N can be described in terms of Cartan subalgebras t′ ⊂ g and Borels b′ ⊂ g. In [D] we introduced the parameter spaces G/N and G/T , parametrizing all regular centralizers C ⊂ G, respectively pairs C ⊂ B. By definition, G/N is an open subset of the closure of G/N in Grass(r, g). (The closure itself contains some abelian subalgebras which are not regular centralizers and therefore need to be discarded.) Over G/N there is the vector bundle c → G/N of rank r abelian subalgebras of g, and the corresponding abelian group scheme C → G/N whose fibers are the regular centralizers 4 in G. Let ̟ be the projection G/N × E → G/N . Then ̟ ∗ C is an abelian group scheme over G/N × E. The cohomology sheaf R1 ̟∗ ̟ ∗ C can be represented by an analytic or algebraic space u′ : UE′ → G/N whose fiber over C ∈ G/N is H 1 (E, C(OE )), the moduli space of C-bundles on E. We will usually restrict attention to the subfamily u : UE → G/N parametrizing semistable C-bundles. Over UE there is a universal principal C-bundle (more precisely, a u∗ C-bundle) PUCE → UE × E. Since u∗ C is a subgroup scheme of the trivial group scheme G over UE × E, we get an associated principal G-bundle PUE → UE × E. The bundles PUCE and PUE are uniquely characterized by the properties: ∗ The restriction of PUCE to UE × {0} is the trivial C-bundle. ∗ the restriction of PUCE to {x} × E (where x := (C, p) ∈ UE , C a regular centralizer, p the isomorphism class of a C-bundle over E) is a C-bundle on E in the class p, and the restriction of PUE is the associated G-bundle. We could rephrase this as saying that UE is a fine moduli space for the data it parametrizes: a G-bundle together with a trivialization over 0 ∈ E and a semistable reduction to a regular centralizer. The basic reason for existence of the universal family is that the objects parametrized have no automorphisms: we killed them by fixing the trivialization. Over any one stratum of G/N it is easy to describe UE and PUCE . For instance, over the open stratum G/N , we start with the Poincare T -bundle over MTE × E, cross this with G/T , and divide by W which acts on both G/T and MTE . A related object which we shall need is the quotient U E := (G/T × MTE )/W . There is a natural morphism f : UE → U E which, over each C ∈ G/N , maps H 1 (E, C(OE )) to its compact part H 1 (E, C ′ (OE )), where C ′ is the quotient of C by its unipotent part, C ′ := C/(C ∩ [B, B]) for a Borel B ⊃ C. Between fibers over points of the open stratum G/N , this map is surjective (in fact, an isomorphism); but over the whole base its image is the constructible set ′ U E := {(C, t) ∈ G/T × MTE | StabW (C) ⊂ StabW (t)}/W. As described in the introduction, for G = SL(2) we have G/T = P1 × P1 and G/N = P2 . ′ The fiber of either UE or U E over a point not in the diagonal is E. Over points of ′ the diagonal, the fibers in UE , U E and U E respectively are: four lines, four points, and ′ P1 . We may think of U E as parametrizing isomorphism classes of G-bundles having a semistable reduction to a regular centralizer C, together with a trivialization over 0 ∈ E. The additional data in the fiber of UE over U E chooses such a C-structure. Thus f is the forgetful map sending a C-struture to its associated G-structure. The point is that the dimension of the normalizer NG (C) can be greater than r, so there can be a non-trivial family of C-bundles whose associated G-bundles are all isomorphic. 3. Families of moduli spaces Let π : X → S be an elliptic fibration with a section, with non-singular X, S. We want to put the basic objects from the previous section into families parametrized by S. 5 It will be convenient to work instead with a (singular) Weierstrass model X → S, given by an equation of the form y 2 = x3 + b2 x + b3 , where bi is a section of L⊗2i and L is π∗ of the relative canonical bundle Kπ . Since π has a section, we can identify Pic0 of a smooth fiber Es := π −1 (s) with Es . Globally, the relative Jacobian Pic0 (X̄/S) is the complement in X̄ of the locus of singular points of fibers. (By definition, this singular locus includes any fiber component of multiplicity > 1.) The reason for this is that a section of π : X̄ → S must have intersection number 1 with each fiber, so it cannot pass through a singular point. This leads immediately to identification of MTX/S with the r-th cartesian power of X̄ − Sing(X̄) over the base S. A satisfactory construction of MG X/S is somewhat more delicate. Looijenga shows [3] (see also [4]) that for simply connected group G and for fixed, non-singular E, MG E is a weighted projective space. The weights are the Dynkin indices of the dual Dynkin diagram of G (i.e. the coefficients of the highest short root when expressed as a sum of simple roots, plus the coefficient 1 for the affine root). In order to be able to describe the relative object MG X/S , we need a way of relating nearby fibers. A general construction of a flat connection on such families was carried out by Saito [10] but more immediately useful results are in [5]. Wirthmuller takes the base S to be a modular curve, so that Jacobi forms give sections of powers of the “theta” bundle (giving the polarization) on MTX/S , and we are looking for the W -invariants among them. These are now bi-graded, by what he calls “weight” (as modular form) and index. The “indices” correspond to the weights in Looijenga’s weighted projective space, and also to the power of the theta bundle in which the section lives. I will refer to Wirthmuller’s “weights” as degrees, to avoid confusion with Looijenga’s weights. These degrees correspond to powers of L, the Weierstrass line bundle. One of them always turns out to be 0, the others are the degrees of the basic G-invariant polynomials. It is not clear a-priori why there should be a natural way to pair these two sequences, of indices and degrees. The end result is that MG E , for simple G other than E8, ∗ r di can be identified with the quotient (⊕i=0 L )/C , where d0 = 0, {di }ri=1 are the degrees of the invariant polynomials, and C∗ acts with weights equal to the Dynkin indices for the dual Kac-Moody algebra. At least for some groups, these results are elementary. For SL(r + 1), all indices equal 1, while the degrees are 0, 2, 3, 4, ...r + 1. So MG E is the ordinary projective space, 2 r+1 projectivization of O ⊕ L ⊕ ... ⊕ L . The fact that MG E is a projective space follows directly from Abel-Jacobi: MTE is the locus of (r + 1)-tuples of points in E which add up to 0, W is the symmetric group, so the quotient is the variety of effective divisors in the linear system r + 1 times the origin, a projective space of dimension r. But we can really identify this space: for r = 1 it is the P1 with coordinate x, of which the Weierstrass equation exhibits E as a double cover. A basis of sections of O(1) is given by 1 (of degree 0) and x (of degree 2). For r = 2 we get the P2 with functions 1, x, y, etc. Similar or easier arguments show that for G either SO(2r + 1) or Sp(r), the moduli space MG E is the 1 r-th symmetric product of the basic P , so it can be identified with the projectivization 6 (all weights equal 1) of O + L2 + L4 + ... + L2r . For Sp(r), this is as it should be: the (2) Dynkin diagram of the Kac-Moody algebra (Cˇr ) = Dr+1 has all indices equal to 1. But (2) the Dynkin diagram of the Kac-Moody algebra (Bˇr ) = A2r−1 has its three extreme indices equal to 1, while the others equal 2, so we expect a weighted projective space of weights (1, 1, 1, 2, ..., 2). And that is exactly what we get, if we replace SO(2r + 1) by the simply connected Spin(2r + 1): the double covering group yields a 4-sheeted covering moduli space (the elliptic curve has 4 two-torsion points), which can be identified as the unique Z/2 × Z/2 cover each of whose three Z/2 quotients is branched along two of the three hyperplanes in Pr = Symr P1 corresponding to r-tuples containing one of the three roots of the Weierstrass equation. I have not worked out the missing case of E8, nor the remaining non simply connected groups. Presumably, even if MG E does not always turn out to be a weighted projective space, it can still be described by some universal construction in terms of the Weierstrass line bundle L. In particular, there is then a natural way to extend the family of quotients MTEs /W to a locally trivial family over the entire base S (including the discriminant) of any Weierstrass family. The most familiar instance of this is when G = SL(2): there the 1 resulting MG X/S is the P -bundle over S obtained as projectivization of O ⊕ L, which is the quotient of the Weierstrass model by its natural involution fixing the 0-section. We also need to note that the spaces UE and U E parametrizing C- and G- bundles likewise extend in families to form objects UX/S and U X/S respectively. For U X/S we take (G/T × MTX/S )/W . Near s corresponding to non-singular fiber Es , this looks like a bundle over S with fibers U E . At singular s, we have already removed the singularities of MTX/S , so we get a non-singular, but somewhat smaller, object. Likewise, we restrict UX/S to the open subset which maps to U X/S . Its points then parametrize C-bundles on the Es , and there is again a universal C-bundle on UX/S ×S X. (It is possible that for those groups for which Wirthmuller-type constructions work we could extend these universal objects to some of the singular loci, but we have not pursued this, and the construction presented below seems to avoid the issue.) 4. Spectral parametrization of bundles In this section we describe the equivalence between regularized G-bundles on an elliptic fibration with a section and the corresponding spectral data. We fix an elliptic fibration π : X → S with a “zero” section σ : S → X, where X, S are smooth. Let Es := π −1 (s) denote a fiber. Given a principal G-bundle P on X, let PS := P|σ(S) be the restriction to the zero section. There are associated bundles of groups AdP, AdPS over X, S respectively. The sheaf of automorphisms along the fibers Es , AutS (P) := π∗ AdP, can be identified as a subsheaf of AdPS . There is also an associated bundle PS /N , whose fiber over s ∈ S is the family of regular centralizers associated to the fiber Ps over σ(s). A section c : S → PS /N determines an abelian group scheme C → S, a subgroup scheme of the group scheme AdPS . A π ∗ C-torser P C on X then induces a G-bundle 7 P = P C ×C AdPS . By a regularized G-bundle we mean such a triple {P, c : S → PS /N , P C }, or equivalently a reduction of the structure of P to a group scheme C of regular centralizers. This group subscheme C ⊂ AdPS is contained in AutS (P), and extends naturally to a group subscheme of AdP. A key point is that an everywhere regular, semisimple and semistable bundle P (i.e. one whose restriction Ps to each Es has these properties) has a unique regularization, with C = AutS (P). But there are other regularized bundles, whose underlying G-bundles may not be everywhere, or even anywhere, regular. Yet these bundles too can be parametrized by their spectral data. (Being regularized means that we have chosen a reduction to a regular centralizer subgroup of the automorphisms along each Es , not necessarily that those automorphisms form a regular centralizer themselves.) If P is known to be regular, semisimple, and semistable only for generic s ∈ S then a regularization is still unique if it exists; in general though, we may have to blow up the base S to find a regularization. A cameral cover of S is a W -Galois cover S̃ → S which is modelled on G/T → G/N . (“modelled on” means “obtained locally as pullback via maps of S to the base”.) Recall that G/N parametrizes regular centralizers in G, while G/T parametrizes pairs of a regular centralizer and a Borel containing it. A regularized bundle {P, c, P C } determines the following data: (1) A cameral cover S̃ → S. (2) A W -equivariant morphism v : S̃ → MTX/S . (Equivalently, a W -invariant morphism v ′ : Λ × S̃ → Pic0 (X/S). By way of terminology, we refer to the image of v as the universal spectral cover. The various other spectral covers are the images under v ′ of λ × S̃, for λ ∈ Λ.) (3) A homomorphism ℓ : Λ → Pic S̃ (or equivalently, a T -bundle on S̃) satisfying the twisted W -equivariance property of [2]. These are obtained as follows. (1) The cameral cover S̃ is the cover of S induced via c from the cover PS /T → PS /N , which indeed looks locally like G/T → G/N . (2) A point s̃ ∈ S̃ above s ∈ S corresponds to a choice of Borel in Ps containing the regular centralizer Cs ⊂ Auts (Ps ). This extends uniquely to a subbundle of Borels in P|Es . Via the natural quotient map B → B/[B, B] = T , there is an associated T -bundle, identified with a point of MTEs . When S̃ is reduced, we get the map v by letting s and s̃ vary, and the W -equivariance holds, since it does so fiber-by-fiber. In the general case, the C-bundle P C on X together with the inclusion of C into the universal Borel bundle B over S̃ induce on S̃ ×S X a B-bundle, hence a quotient T -bundle. Our morphism v is the classifying map for this bundle. (3) Above S̃ we have a bundle of Borels; the T -bundle is associated to it as above. We will discuss the shifted W -equivariance below. The main point of this note is that a regularized G-bundle can be reconstructed from its spectral data. To see this, it will be convenient to introduce an intermediate object, the principal G-Higgs bundle, cf. [2]. This is simply a pair (PS , C) consisting of a principal 8 G-bundle PS on S together with a family C of regular centralizers in Ad PS . (To avoid worrying about how the different types of centralizers fit together, we can instead consider the vector subbundle c of regular centralizers in the bundle ad PS of Lie algebras. This , of course, is equivalent data.) Now as above, to a principal G-Higgs bundle (PS , C) corresponds an abstract cameral cover S̃ → S and a homomorphism ℓ : Λ → Pic S̃. (The constructions of items (1) and (3) above used only the available data on S.) As W acts on both sides, we can consider the subgroup of W -equivariant homomorphisms. In [2] this was called the distinguished Prym of S̃/S. In case S, S̃ are non-singular, this is a finite group times an abelian subvariety of Hom(Λ, Pic(S̃)). The main observation in [2] is that the family of principal Higgs bundles with a given cameral cover S̃ is, if non-empty, parametrized by a translate of the distinguished Prym. The exact point by which we need to translate will not be crucial for us here. It is described in sections 5.2 and 5.3 of [2] as the sum of a cohomological shift term depending only on the group G, and further twist terms coming from the fixed divisors for the action of W on S̃. An analogous and more familiar situation applies in case G = SL(n) when we replace the cameral cover by the degree-n spectral cover π : S̄ → S (S̄ is the image under v ′ of λ1 × S̃, where λ1 is the first fundamental weight.) The twisting along the ramification then corresponds to the relative Todd class (=half the ramification) which enters into the Grothendieck-Riemann-Roch formula: In order for π∗ (L) to have determinant 0, c1 (L) needs to be the class of half the ramification. (The original purpose of [2] was to describe the fibers of the Hitchin map, from moduli of K-valued Higgs bundles on a variety S, to K-valued spectral data. Here K can be the canonical bundle of S, as in Hitchin’s original work [11], but it can also be an arbitrary line bundle, or (with some additional symmetry conditions imposed) even a vector bundle on S as in Simpson’s works. It turned out to be convenient to separate the problem into considerations of “eigenvectors” and “eigenvalues”: we introduce the somewhat abstract “principal G-Higgs bundles” and show that they correspond to abstract spectral data (this is the eigenvector aspect); the K-valued versions are then recovered by adding a K-valued “Higgs field” φ, a section of ad(PS ) ⊗ K, on the one side, and an “eigenvalue map” v : S̃ → K on the other. This latter map is of course analogous to our datum (2). We are thus led to think of a G-bundle on an elliptic fibration as a sort of Higgs bundle on the base taking values in the fibration instead of in a line (or vector) bundle.) Returning to our situation, from the spectral data (1)-(3) we thus retrieve the principal G-Higgs bundle (PS , C) together with a morphism v from S̃ (which is determined by (PS , C)) to MTX/S , commuting with the projections to S. It remains to recover the original regularized G-bundle {P, c, P C } (on X) from this data. This goes as follows: the situation is essentially rigid, so we can reduce to the case that S is a point, i.e. to bundles on one elliptic curve. For a given regular centralizer C, this then reduces to the straightforward verification that the canonical map: C T MC E → MorW ((G/B) , ME ) 9 is an isomorphism. Here (G/B)C is the subscheme of G/B parametrizing Borels through C (it is finite of length equal to the cardinality of W , and is not reduced except when C is a torus), and MorW is the (group of) W -equivariant morphisms. Working globally over S, the principal G-Higgs bundle (PS , C) determines an (“eigenvector”) map c : S → PS /N and its lift c̃ : S̃ → PS /T , which together with the “eigenvalue” map v : S̃ → MTX/S sends S̃ to the fiber product PS /T ×S MTX/S . This map is PS W -equivariant, so it descends to give a section ᾱ : S → U X/S := (PS /T ×S MTX/S )/W . This last object is a version of our previous U X/S which is twisted by the bundle PS : it is isomorphic to U X/S over any open subset of S over which PS is trivial. Likewise, we have PS the twisted version UX/S of UX/S . Now, the data needed to lift our section ᾱ to a section PS α of UX/S is precisely given by v : S̃ → MTX/S . (In particular, existence of v implies ′ that the image of ᾱ is contained in U X/S .) Now on UX/S ×S X we have the universal PS C-bundle trivialized along S. Twisting by PS , we get a universal C-bundle on UX/S ×S X , but instead of a trivialization we now get an identification of its restriction to S with the universal C-subbundle of PS . Pulling back via the section α to X = S ×S X gives a C-bundle P C , whose associated G-bundle is the original P . Remark1 We have emphasized the cameral covers, since they are, in our view, the most basic objects in the picture. But in order to parametrize entire components of the moduli space of G-bundles on X, it is necessary to allow the cover to vary. For this purpose, it is more convenient to consider instead the universal spectral covers. These are all obtained by pulling back one object, the W -cover MTX/S → MG X/S , by arbitrary G G sections S → MX/S . As we saw in the previous section, MX/S is, for most groups, a bundle of weighted projective spaces over S. So in a connected component (obtained by fixing the numerical invariants of the map), these maps are specified by the (weighted projectivization of) the space of sections of an appropriate vector bundle over S. In “nice” situations, a generic map of this kind will determine a non-singular universal spectral cover, and therefore also a unique cameral cover. Remark2 An alternative approach to reconstruction of a G-bundle on X from spectral data might be based on application of the equivalence of [2] directly to the principal GHiggs bundle (P, π ∗ C) on X. One then obtains a cameral cover X̃ → X and a point ℓX in a translate of its distinguished Prym. It is interesting to compare this to the spectral data (S̃, v, ℓS ) which we used above. The cover X̃ → X is clearly the pullback via π of S̃ → S, but π ∗ ℓS is not the same as ℓX : the former is trivial along fibers of π, while the behavior of the latter along fibers is equivalent to the additional datum v : S̃ → MTX/S . Our original approach has a clear advantage over this alternative in situations where X is interesting (say a K3 or CY3 ) while S is a much simpler object (P1 , Fn ). 10 References [1] C. Vafa, ‘Evidence for F-theory’, Nucl. Phys. B469(1996),403, HepTh 9602022; D. Morrison and C. 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