Seiberg-Witten differentials on the Hitchin base

In this note we describe explicitly, in terms of Lie theory and cameral data, the covariant (Gauss--Manin) derivative of the Seiberg--Witten differential defined on the weight-one variation of Hodge structures that exists on a Zariski open subset of the base of the Hitchin fibration. Dedicated to Tony Pantev on the occasion of his 60th birthday.


Introduction
The base of the Hitchin integrable system ( [Hit87]) supports a family of cameral curves, and, as a consequence, carries various Hodge-theoretic and differentialgeometric structures ( [DM96], [DM], [Mar]).In particular, the Zariski open subset of the base, corresponding to smooth cameral curves with generic ramification carries a weight-one variation of Hodge structures (VHS) with a Seiberg-Witten differential.Our goal in this note is to describe the covariant (Gauss-Manin) derivative of the Seiberg-Witten differential explicitly in terms of Lie theory and cameral data.
We recall now the main ingredients and constructions, starting with the Hodgetheoretic ones.
The notions of polarised Z-VHS or Q-VHS are introduced analogously, by replacing V R with appropriate locally constant sheaves V Z or V Q of Zor Q-modules, respectively.
The prototypical example is that of a geometric VHS, i.e., one arising from a family of compact Kähler (e.g., projective) manifolds.
By Griffiths Transversality, ∇ induces an O B -module homomorphism and hence, taking a direct sum over the different p, an O B -module homomorphism which satisfies θ ∧ θ = 0.
The pair E = p F p /F p+1 , θ is an example of a Higgs bundle on B. This example played an important rôle in Carlos Simpson's study of Higgs bundles on higher-dimensional varieties ( [Sim92], [Sim88]).
Consider a polarised Z-VHS (V, ∇, V Z , V • , S, . ..) of weight w = 1.An abstract Seiberg-Witten differential on it is a section λ SW ∈ H 0 (B, V 1 ), for which the O Bmodule homomorphism Given such data, we obtain a refinement of the weight-1 filtration to a weight-3 filtration For links to projective special Kähler geometry ("N = 2 supergravity") and weight-3 VHS, satisfying the Calabi-Yau condition, one can check [HHP10] Furthermore, given such data, there is an associated fibration of complex tori The polarisation S gives rise to an isomorphism Vert ≃ (V 1 ) ∨ , and hence λ SW induces, by composition with the dual of its defining isomorphism Such an isomorphism is also induced by a choice of symplectic form on J .There is unique symplectic form ω λ on J , which induces i λ and such that the 0-section is Lagrangian.
We next recall the construction of the family of cameral covers over the Hitchin base, and introduce a weight-1 VHS with a Seiberg-Witten differential on it.
First, we fix the following data: • A simple complex Lie group G of rank l, together with a choice of Borel and Cartan subgroups T ⊂ B ⊂ G.We denote by t ⊂ b ⊂ g the respective Lie algebras and by W the corresponding Weyl group.• A compact (connected) Riemann surface X of genus g ≥ 2 (or equivalently, a non-singular proper algebraic curve over C).We do not need to fix a particular projective embedding of X.Additionally, we choose: • Homogeneous generators I 1 , . . ., I l of the ring These additional choices are not necessary for the entire discussion, but are needed for the explicit calculation in Theorem A.
Two explicit examples of invariant polynomials -for SL 3 (C) and G 2 -are given in Equations ( 30) and (32), respectively.
Notice that while t/W is a priori just a cone, the choice of generators {I k } allows us to identify it with C l .Notice also that we may interpret {I k } as elements of C[g] G , via Chevalley's theorem.
The chosen simple roots determine an isomorphism t ≃ C l , v → (α 1 (v), . . ., α l (v)), using which we further identify χ : t → t/W with a finite map I : C l → C l .We may abuse the notation for these maps, e.g., write χ = (I 1 , . . ., I l ) instead of I, etc.
We proceed by constructing from these data two rank-l vector bundles on X.The first one is t ⊗ C K X ≃ K ⊕l X , whose total space will be denoted by M : The group W acts (fibrewise, via its action on t) on M .The resulting quotient U is a priori just a cone bundle, but the choice of {I k } allows us to give it the structure of a vector bundle of rank l: (2) We can also think of U \{0} as the C × -bundle with fibre t/W , associated to the C × -bundle K X \{0}.
The morphism χ : t → t/W induces a morphism χ : M → tot U of X-varieties (not of vector bundles!): (3) We write B for the Hitchin base -the space of global sections of U : Any b ∈ B determines a W -cover p b : X b → X as the pullback of χ : M → tot U via (the evaluation map of) the section b: This W -cover is called the cameral cover of X (corresponding to b).We may occasionally write p : X → X if the point b ∈ B is fixed or understood.By construction X b is a closed subscheme of M that can be singular or nonreduced.The cameral cover X b ⊂ M inherits from M a W -action (and thus has lots of automorphisms).For a generic choice of b it is a non-singular ramified Galois W -cover with simple ramification.We write B ⊆ B for the open set of generic cameral covers.
There is a weight-1 Z-variation of Hodge structures Intrinsically, it is defined as follows.Let Λ ⊆ t be the cocharacter lattice and p : X → B the universal cameral curve.Let also p W * be the W -invariant pushforward functor.Then we set , and the Hodge filtration is induced by the naive filtration The Gauss-Manin connection can be identified with the d 1 differential of the spectral sequence, induced by the Koszul-Leray filtration on Ω • X .The polarisation pairing S is given by S b (α, β) = α ∪ β, X b .For more details, see section 3.2 and the references therein, as well as [HHP10][8.1]and [DP12].
On M there is a canonical t-valued Liouville form λ, see section 2.4.The Liouville form λ determines a Seiberg-Witten differential, λ SW ∈ Γ(B, V 1 ), via λ SW (b) = λ| X b , and, as in (1), we have that the map In [HM98][Proposition 2.11], an isomorphism with the same domain and codomain as in (5) is described as the composition of pullback on global sections (by π), contraction with ω and restriction to X b , see also Proposition 3.1.In [HHP10][Proposition 8.2] it is shown, using a hypercohomology calculation, that the isomorphism described by Hurtubise and Markman coincides with the isomorphism (4).Some of the above relations for G = SL 2 are discussed in [DH75][Proposition 1], see also [MSWW19][Eq.(3)].
The above isomorphism can also be considered from an integrable systems viewpoint.Indeed, consider the universal family of generic cameral curves p : X → B ⊆ B. The relative Prym fibration Prym X /B → B is in fact an algebraic completely integrable system.The fibre Prym X b over b ∈ B is an abelian variety, whose tangent space is Serre dual to H 0 ( X b , t ⊗ C K X b ) W , the right hand side of (5).The isomorphism (5) actually amounts to lifting a tangent vector in T B,b to a vector field along the fibre Prym X b and then pairing it with the symplectic form on the Prym fibration.This is the viewpoint, taken, e.g., by Hurtubise and Markman.
Our goal in this note is to provide an explicit and global (on X and X b ) description of (5) in terms of Lie theory and the covering p b : X b → X.
The simplest case, that of G = SL 2 , is given in Example 5.1, where we show that Equation (5) specialises to is the tautological section and λ SW is the Liouville (Seiberg-Witten) form.The expression on the right hand side can in fact also be rewritten as − p * g 2λ , and in this form it coincides (up to scaling factors) with Our main result is a general formula for ∇ GM g λ SW for the case of an arbitrary (complex, simple) group G.
Let DI be the Jacobi matrix of the adjoint quotient I = (I 1 , . . ., I l ) : and ι the natural algebra homomorphism from Sym(t ∨ ) into H 0 M, n≥0 π * K n X , introduced in Equation (22).Finally, α i = ι(α i ) and λ i = e i ⊗ α i , where {e i } is the basis of t, dual to {α i }.In this notation, the Liouville form is λ SW = i λ i .
Theorem A. Once the main and additional data are chosen, the isomorphism (5) In particular, for l = 2 we have that Knowledge of λ SW and ∇ GM λ SW is essential for describing various geometric structures on B. We mention only two examples as an illustration.
First, for the Hitchin integrable system, the Donagi-Markman cubic ( [DM]), which is essentially the infinitesimal period map for the family of Hitchin Pryms, is given by the Balduzzi-Pantev formula [Bal06][Theorem 1].If we consider the cubic as a global section c of Sym Here D is the discriminant (see also section 5) and L denotes Lie derivative.In our previous work [BD14][Theorem A] we have shown that the Balduzzi-Pantev formula holds along the (good) symplectic leaves of the generalised Hitchin system.
The second example which is worth mentioning is the special Kähler metric g SK on B. It is known that for the case of G = SL 2 (C), the special Kähler metric is given by We shall discuss additional applications of Theorem A to various aspects of the geometry of B in a forthcoming work.
Acknowledgements.P.D. thanks Tony Pantev for helpful discussions related to the project, the Simons Collaboration on Homological Mirror Symmetry for support and the University of Pennsylvania for its hospitality.

Preliminaries
2.1.The Embedding of the Cameral Curve.We are now going to work at a fixed point b ∈ B (generic), and hence will write mostly p : X → X for the cameral cover.To understand (5) we need to understand K X and for that we need to know more about the normal bundle N of the closed embedding X ⊆ M .This is not difficult, since X is in fact the zero locus of a section of a vector bundle on M .
Thus the cameral curve X b is the zero locus (11) i.e., is cut out by the equation(s) Having fixed basic invariant polynomials {I k }, and hence an isomorphism U ≃ l k=1 K d k X , we can express this as the system of equations ( 13) Proof: While in general one uses the Koszul complex to compute the normal bundle, here we have that both X b and M are smooth, and moreover, X b is a complete intersection.This case is handled by a standard geometric argument, given in, e.g.Similarly to the above argument, since M is the total space of a vector bundle (namely t ⊗ C K X ) on X, its tangent bundle T M is an extension of π * T X by t ⊗ C π * K X .Restricting to X and combining with the previous result, one gets the diagram (15) 0 . It determines a 1-parameter family of deformations of X b , given by the equation ( 16) that is, X b+ǫg ǫ .For ǫ in a sufficiently small disk ∆ ρ ⊆ C the section b + ǫg ∈ B remains generic -which we assume to be the case from now on.The total space of the 1-parameter family is cut out in M × ∆ ρ by the Equation (16).
The section g determines a section of Using the simple roots as a basis for t ≃ C l , we identify X U = p −1 b (U ) (via φ) with the set of solutions of I(α 1 , . . ., α l ) = β(z) for (z, α) ∈ ∆ × C l , giving a local version of Equation (13).
Next, the trivialisations of K di X (i = 1 . . .l) and the choice of roots provide an induced trivialisation T MU | XU and (18) and, consequently, a local description of the diagram (15): Here the bottom vertical map is, in more detail, (20) having rank l everywhere on X U , under the assumption that b = (b 1 , . . ., b l ) ∈ B is generic.This is the matrix of the map pr 2 • ds from Proposition 2.1.We write DI or Dχ for the Jacobi matrix of I = (I 1 , . . ., I l ) : Finally, given a tangent vector g = (g 1 , . . ., g l ) ∈ T B,b = B, with (ψ −1 ) * g i = γ i (z)dz ⊗di on U , the corresponding 1-parameter (analytic) family of deformations of X b is cut out locally (in ∆ × C l × ∆ ρ ) by I(α) = β(z) + ǫγ(z), where ∆ ρ ⊆ C is as before.
We may occasionally suppress the pullbacks by φ and ψ, except for the cases when there is a risk of confusion, as when discussing (co)roots and some associated objects.
2.3.Objects, Associated with Roots.Any linear map α ∈ t ∨ = Hom(t, C) determines, by extension of scalars, a vector bundle homomorphism t⊗ C K X → K X , denoted by the same letter.Hence, just as χ in Eq (10), such an α determines a tautological section α ∈ H 0 (M, π * K X ), which on (closed) points maps m ∈ M to α(m) = (m, α(m)) ∈ M × X tot K X .Furthermore, restricting α to X ⊂ M gives a section α X ∈ H 0 ( X, p * K X ).Occasionally, we suppress the subscript X, i.e., the restriction.
The section α vanishes along a "hyperplane divisor" tot (ker α ⊗ C K X ) ⊆ M , a rank-(l − 1) subbundle of t ⊗ C K X .The respective restrictions α i X (of sections arising from roots) vanish along divisors D αi in X, which are the ramification divisors of p : X → X.
If we choose a local chart (U, ψ) and φ : M U ≃ ∆ × t, as in (17), α is represented by (z, u) → α(u)dz, where α(u) = α, u is the natural pairing between t and t ∨ .If we further identify the preimage of π −1 (U ) in tot π * K X → M with ∆ × t × C, via φ and a trivialisation of K X , then the evaluation map of α is represented by The linear functional α ∈ t ∨ determines a function on ∆ × t, that we may denote pr * 2 α if the distinction from α is important.Furthermore, given the choice of φ, we may consider α (or rather, pr * 2 α) a function φ * α ∈ O MU (M U ) on M U .Consequently, upon restriction to X U , we get a local function φ * α ∈ O XU ( X U ) on the cameral curve.Of course, one should really write φ * pr * 2 α| XU here.The distinction between the various objects associated to a root α i becomes important when one considers their differentials.Since Naturally, we are going to write dα i for the penultimate expression, so the distinction between dα i and dα i is essential.Finally, we keep in mind that dα i = α i ∈ Hom(t, C), as with any linear map.
The assignment α i → α i determines an (injective) C-algebra homomorphism and, consequently, a homomorphism denoted by ι as well.Given a Sym(t ∨ )-valued endomorphism A with non-zero determinant det A ∈ Sym(t ∨ ), we write ι(A) −1 for the inverse of ι(A) in the ring of l × l matrices with coefficients in the field of fractions Frac H 0 M, n≥0 π * K n X , and in fact, in End(C l ) ⊗ H 0 M, n≥0 π * K n X 1 det ι(A) .We can, more generally, rewrite the global equations for X b as (23) . Probably the simplest way to introduce it is by setting ω = −dλ, ) is a tautological section, the "t-valued Liouville form".
We recall some explicit expressions for λ -although, as usual in symplectic geometry, there are various sign ambiguities in the possible definitions.
The chosen simple roots {α 1 , . . ., α l } form a basis of t ∨ , and we let {e 1 , . . ., e l } stand for the corresponding dual basis of t (consisting of fundamental coweights).
One can then set M , and write the Liouville form and the 2-form as Finally, if we choose local coordinates as in eq (17), we obtain for the pullback of λ and ω to ∆ × t 3. Background: Two Results

A Result of Hurtubise and Markman.
We begin with the special case of a result of Hurtubise and Markman [HM98][Proposition 2.11] mentioned in the introduction.We spell out some of the details of their argument for this special case.
Proposition 3.1.For each generic b ∈ B, the pullback of global sections via p b , followed by the isomorphism (14) and contraction with ω induces an isomorphism Thus, the isomorphism β (25) is a composition of two maps.The first one is pullback (adjunction) g −→ p * b g, for g ∈ B = H 0 (X, U ).The second one is the map on global sections, induced by the map of bundles (26) where s is a lift of s to a section of T M .One may denote this map simply by ω (contraction with ω), but should keep in mind the restriction to X.The proof of Proposition 3.1 relies on a dimension count, combined with good understanding of the bundle map (26) and the induced map on fibres at m ∈ X.For that, the cases when m is not a ramification point and when it is one should be considered separately.Notice that if m is not a ramification point, then T X,m So let us choose a point m ∈ M and consider the fibre of π : M → X, passing through m.We set L := π −1 (π(m)) = t ⊗ C K X,π(m) ⊆ M , and write N L for the normal bundle of the vector space L ⊆ M .
Using the local description of ω, we obtain that ω fits in the following diagram: Using the normal sequence for L ⊆ M , one obtains: coincides up to sign with the canonical trivialisation.That is, 8 8 q q q q q q q q q q .Thus, in particular, ω induces, for any m ∈ L, a W -equivariant isomorphism Lemma 3.2.Consider a point m ∈ X that is not a ramification point of p : X → X.The map on fibres, induced by the bundle map (26) is an isomorphism This is again a local calculation, using the explicit form of ω.Notice that since m is not a ramification point, the composition is an isomorphism.However, at ramification points the behaviour of ω is different.
In fact, at such points the map (26) is not an isomorphism of bundles if l > 1, as is clear from the next Lemma.
This result is shown by a local calculation, which in turn boils down to a linearalgebraic result, using the explicit form of ω.It is also stated in [HM98][Lemma 2.10].
Proof of Proposition 3.1: , or, after restriction to X b , as sections of p * b K dj−1 X .Using Cramer's formula and the fact that ι is an algebra homomorphism, we can rewrite the right side of (6) as a linear combination of (restrictions of) λ i with coefficients of the kind det [ι∂ 1 I, . . ., p * b g, . . ., ι∂ l I] i.e., global meromorphic functions on X b , since both the numerator and the denominator belong to ) and denote by s ∈ H 0 ( X b , t ⊗ C K X b ) the image of g under the isomorphism (5).Let us also denote by s the section from the right hand side of Equation (6), i.e., . This is a meromorphic section of t⊗ C K X b with poles at most along the ramification of p b : X b → X.We are going to prove that s = s.We use Theorem 3.1 and the representation of the isomorphism β from Equation ( 25) is a composition of two maps.
As a first step, we show that We choose (an analytic) local coordinate z on U and use z (i.e., its pullback p * b z) as a coordinate on X U ⊆ X b \Ram(p b ).
Then, setting γ for the coordinate vector of p * b g, i.e., we obtain a lift γ of γ Note that the expression for γ is well-defined on X U : away from ramification, we can solve locally-analytically for α i in terms of z, so (DI) −1 , when restricted to a connected component of φ( X U ), is actually a section of End(C l ) ⊗ C O an ∆ .
Then, using the lift g from Equation (28), we obtain We write [dz] rather than dz since the cotangent sheaf of X U is a quotient of Ω 1 MU XU .This is precisely the expression for s from Equation (6), written locally.
Having shown (27), we now note that the sheaf of meromorphic sections of a holomorphic vector bundle on a smooth curve is trivial (see e.g.[Gun67], p.76; see also [Var], Lemma 31.25.3).As two meromorphic functions that coincide away from a finite set of points are equal, equation (27) shows that s = s.Since the two sections s and s are equal, and s is known to be W -invariant, so is s.
We refer the reader to the beautiful papers [KP94], [Hit07] for additional details on the G 2 -Hitchin system, including Langlands duality and the description of Hitchin fibres.
[EH16][Proposition 6.15].The isomorphism N X b /M ≃ p * b U is induced by the (vertical component of the) differential ds b : T M → s * b T π * U of the section s b : M → tot π * U .
(27) s| X b \Ram(p b ) = s| X b \Ram(p b ) For that we restrict the cameral cover to the complements of the ramification and branch divisors p b : X b \Ram(p b ) −→ X\Bra(p b ) and choose U ⊆ X\Bra(p b ), biholomorphic to an open disk (via ψ : U → ∆).In this case, X U ⊆ X ∩ (det ιDI = 0) has |W | (analytic) connected components, each isomorphic to U , labelled by the different Weyl chambers fixed.These are "global" equations and no choice of local trivialisation is used here: the k-th equation takes values in (the total space of) K d k X .Another global description is given in Equation (23).