Hitchin systems on ll-curves

Andrei Tyurin

24.02.1942 – 27.10.2002

To Prof. Friedrich Hirzebruch on his 75 th birthday

1 Introduction

Recall (see, for example [T1] and [T2]) that the Deligne-Mumford compactification \overline{M_{g}} of the moduli space M_{g} of smooth curves of genus g containes a finite configuration of points P ⊂ \overline{M_{g}} corresponding to the large limit curves and enumerated by 3-valent graphs. Given a 3-valent graph Γ, we write:

• V (Γ) = {v_{i}} for the set of its vertices;

• E(Γ) = {e} for the set of its edges;

• E⃗(Γ) = {e⃗} for the set of all orientation choices for all the edges;

• v_{s}(e⃗), v_{t}(e⃗) for the source and target vertices of an oriented edge e⃗ ∈ E⃗;

• S(v) ⊂ E(Γ) for the star of a vertex v ∈ V (Γ), i. e. for the set of all (non oriented) edges incident to v;

Topologically, each 3-valent graph Γ is equivalent to some 3-dimensional handlebody H_{Γ} whose boundary ∂H_{Γ} = Σ_{Γ} is a smooth compact Riemann surface obtained from Γ by pumping its edges to tubes and vertices to trinions, that is 2-spheres with 3 holes. It is easy to see that the cardinalities |E(Γ)| = 3g−3, |V (Γ)| = 2g−2, where g is the genus of Σ_{Γ}. By the construction, Σ_{Γ} is equipped with a trinion decomposition, i. e. it is sliced into ‘pairs of pants’ obtained by cutting some meridian C_{e} out of each tube e ∈ E(Γ). Contracting these 3g−3 meridians to points, we obtain a connected reducible algebraic curve P_{Γ} glued from 2g − 2 Riemann spheres P_{v} = ℂℙ^{1}, v ∈ V (Γ), along triples of points (p_{e_{1}}, p_{e_{2}}, p_{e_{3}}) ∈ P_{v} corresponding to the edges of the star S(v) = {e_{1}, e_{2}, e_{3}}. This reducible algebraic curve P_{Γ} is Deligne - Mumford stable and has the arithmetical genus g.

The restriction of the canonical sheaf O(K_{Γ}) to each component P_{v} coincides with the sheaf of meromorphic differentials ω with simple poles at (p_{e_{1}}, p_{e_{2}}, p_{e_{3}}) ∈ P_{v}:

O_{P_{Γ}}(K_{Γ})|_{P_{v}} = O_{P_{v}}(K_{P_{v}} + p_{e_{1}} + p_{e_{2}} + p_{e_{3}}) = O_{P_{v}}(1) .

Thus, a global holomorphic section s ∈ H^{0}(P_{Γ}, O(K_{Γ})) is a collection of meromorphic differentials {ω_{v}} on P_{v} whose residues at the poles satisfy the following compatibility relations:

res_{p_{e}}ω_{v_{s}(e)} + res_{p_{e}}ω_{v_{t}(e)} = 0 , ∀ e ∈ E(Γ)

∑_{e∈S(v)} res_{p_{e}}ω(v) = 0 , ∀ v ∈ V (Γ) (1.1)

These linear constraints lead to the right value for dim H^{0}(P_{Γ}, O(K_{Γ})) = g, the dimension of the space of global holomorphic differentials. This agrees with the Deligne - Mumford prediction that the vector bundle π : V_{1} → \overline{M_{g}}, whose fiber at a smooth C ∈ M_{g} is H^{0}(C, O(K_{C})), can be continued into each ll-curve point as a vector bundle of rank g. However, the properties of the complete canonical linear system |K_{Γ}| depend on the topology of Γ (more precisely, on the thickness of the graph, see [A]).

The double canonical system |2K_{Γ}| of an ll-curve is much more regular. Namely, a holomorphic quadratic differential Ω ∈ H^{0}(P_{Γ}, O(2K_{Γ}) is given by a collection of meromorphic quadratic differentials {Ω_{v}} on the components P_{v} such that their bi-residues satisfy the following condition:

bi res_{p_{e}}ω_{v_{1}} = bi res_{p_{e}}ω_{v_{2}} , ∀ e = (v_{1}, v_{2}) ∈ E(Γ)  (1.2)

(see [T4],[T5]). The system of nodes {p_{e}} ∈ P_{Γ}, e ∈ E(Γ), gives an identification

H^{0}(P_{Γ}, O(2K_{Γ}))^{∗} = ℂ^{E(Γ)}  (1.3)

under which the basic linear form H_{e}, which corresponds to e ∈ E(Γ), takes quadratic differential Ω to H_{e}(Ω) = bi res_{p_{e}}Ω . So, the moduli space \overline{M_{g}} has a smooth orbifold structure at each ll-point P_{Γ} ∈ \overline{M_{g}} and the fiber of the tangent bundle T_{P_{Γ}} \overline{M_{g}} at P_{Γ} equals

T_{P_{Γ}} \overline{M_{g}} = H^{0}(P_{Γ}, O_{P_{Γ}}(2K_{P_{Γ}})^{∗} = ℂ^{E(Γ)}  (1.4)

Moreover, each basic tangent direction ℂ · H_{e}, e ∈ E(Γ), can be integrated to some rational curve C_{e⊂Γ} = ψ_{e⊂Γ}(ℙ^{1}) ⊂ \overline{M_{g}} (see (3.1) – (3.12) in [T2]).

So, the Deligne - Mumford compactification \overline{M_{g}} contains the configuration of points P = {P_{Γ}} parameterized by 3-valent graphs Γ of genus g and the configuration of rational curves^{1} C = ⋃_{e⊂Γ} C_{e⊂Γ} corresponding to flags

e ⊂ Γ in such a way that three flags from the same nest (see (2.17), (2.20) in [T2]) lead to the same curve C_{e⊂Γ}. In the previous paper [T2] we described the complex gauge theory on ll-curves P ⊂ \overline{M_{g}}. Here we investigate the Hitchin systems on ll-curves.

2 Framed vector bundles on ll-curves

We write M^{ss}(P_{Γ}) for the moduli space of topologically trivial semistable holomorphic vector bundles of rank 2 on P_{Γ}. By the definition, M^{ss}(P_{Γ}) parameterizes rank 2 vector bundles E that have no positive line sub bundles L ⊂ E and are restricted to trivial holomorphic vector bundles E|_{P_{v}} over each sphere P_{v}, v ∈ V (Γ). Any topologically trivial rank 2 vector bundle E on P_{Γ} can be framed by fixing some trivialization

E|_{P_{v}} = P_{v} × ℂ^{2} = P_{v} × V_{0}

over each component of P_{v} ⊂ P_{Γ} (here V_{0} is some fixed 2-dimensional vector space). We write F(P_{Γ}) for the space of framed topologically trivial vector bundles on P_{Γ}. The framing forgetful map

f : F(Γ) → M^{ss}(P_{Γ})  (2.1)

gives a principal bundle with the ‘complex gauge’ group

Map(V (Γ), SL(2, ℂ)) ,  (2.2)

1we consider C = ∪C_{e⊂Γ} as one reducible curve

where SL(2, ℂ) = SL(V_{0}) consists of linear automorphisms of V_{0} with determinant 1.

Each framing of E can be considered as a function

a : E⃗(Γ) → SL(2, ℂ) such that a(\overleftarrow{e}) = a(\overrightarrow{e})^{−1}  (2.3)

where \overleftarrow{e}, \overrightarrow{e} are two opposite orientation choices for edge e ∈ E(Γ). In these terms, complex gauge transformation g ∈ Map(V (Γ), SL(2, ℂ)) acts on a framing a by the formula

g(a)(e⃗) = g(v_{s}) · a(e⃗) · g(v_{t})^{−1}.  (2.4)

Factorizing (2.1) through this action, we get

M^{ss}(P_{Γ}) = F(Γ)/Map(V (Γ), SL(2, ℂ)).  (2.5)

At the same time, (2.3) and (2.4) show that a function a from (2.3) is nothing but a flat SL(2, ℂ)-connection on the graph Γ. Moreover, the reframing group (2.2) is really the complex gauge group acting on these connections. Thus, the quotient (2.5) coincides with the space of the gauge orbits of flat connections, which, in its own turn, is the same as the space ℜ𝔢𝔭(π_{1}(Γ), SL(2, ℂ)), of equivalence classes of representations of the fundamental group π_{1}(Γ). In other words,

M^{ss}(P_{Γ}) = ℜ𝔢𝔭(π_{1}(Γ), SL(2, ℂ)) .  (2.6)

Let us write F_{g} for the free group with g generators. Then there is an isomorphism of fundamental groups

π_{1}(Γ) = π_{1}(H_{Γ}) = F_{g}

where H_{Γ} is the handlebody whose boundary ∂H_{Γ} = Σ_{Γ} is the pumped graph. So, the space of representation classes (2.6) is described as the factor of the direct product of g copies of SL(2, ℂ) through diagonal action of SL(2, ℂ) by conjugations:

ℜ𝔢𝔭(π_{1}(Γ), SL(2, ℂ)) = (SL(2, ℂ))^{g}/ ad_{diag} SL(2, ℂ)  (2.7)

The right hand side here is called the Schottki space of genus g and denoted by S_{g}. So, M^{ss}(P_{Γ}) = S_{g}.

On the other side, we have the canonical surjection

r : π_{1}(Σ_{Γ}) → π_{1}(H_{Γ}) = π_{1}(Γ) .  (2.8)

Let us choose some standard presentation

π_{1}(Σ_{Γ}) = ⟨ a_{1}, …, a_{g}, b_{1}, …, b_{g} | ∏_{i=1}^{g}[a_{i}, b_{i}] = 1⟩  (2.9)

such that

\begin{aligned}ker(r) & = ⟨a_{1}, …, a_{g}⟩ ≃ F_{g} \\ π_{1}(Γ) = π_{1}(H_{Γ}) & = ⟨r(b_{1}), …, r(b_{g})⟩ ≃ F_{g} \end{aligned}  (2.10)

Then there are two Schottki subspaces S_{g}^{a}, S_{g}^{b} ⊂ ℜ𝔢𝔭(π_{1}(Σ_{Γ}), SL(2, ℂ)) associated with the presentation (2.9):

S_{g}^{a} = {ϱ ∈ ℜ𝔢𝔭(π_{1}(Σ_{Γ}), SL(2, ℂ)) | ϱ(a_{i}) = 1 for 1 ≤ i ≤ g}
S_{g}^{b} = {ϱ ∈ ℜ𝔢𝔭(π_{1}(Σ_{Γ}), SL(2, ℂ)) | ϱ(b_{i}) = 1 for 1 ≤ i ≤ g} (2.11)

Further, each oriented edge e⃗ ∈ E⃗(Γ) gives a homotopy class of oriented 1-cycle C⃗_{e} presented by some (oriented) meridian rounding about the tube corresponding to this edge. Joining these meridians with some fixed base point p ∈ P_{v_{0}} ⊂ Σ_{Γ}, which lives on the pumped vertex v_{0} ∈ V (Γ) say, we get elements [Ce⃗] of the pointed fundamental group π_{1}(Σ_{Γ})_{p}. These 3g − 3 cycles [Ce⃗] obviously lay in the kernel ker(r) of the projection (2.8).

Let us realize π_{1}(Γ)_{v_{0}} as the group of oriented cycles (e⃗_{1}, …, e⃗_{d}), v_{t}(e⃗_{i}) = v_{s}(e⃗_{i+1}) based at v_{0} = v_{s}(e⃗_{1}) = v_{t}(e⃗_{d}). Then there is a map

Int : π_{1}(Γ)_{v_{0}} → π_{1}(Σ_{Γ})_{p}

defined as Int(e⃗_{1}, …, e⃗_{d}) =^{def} [Ce⃗_{1}] ∘ [Ce⃗_{2}] ∘ ⋯ ∘ [Ce⃗_{d}]. Clearly, Int(π_{1}(Γ)) = ker(r) and the exact triple of fundamental groups:

1 → π_{1}(Γ) →^{Int} π_{1}(Σ_{Γ}) →^{r} π_{1}(Γ) → 1.

induces a chain of maps between the corresponding spaces of equivalence classes of SL(2, ℂ)-representations:

ℜ𝔢𝔭(π_{1}(Γ)) →^{r^{∗}} ℜ𝔢𝔭(π_{1}(Σ_{Γ})) →^{Int^{∗}} ℜ𝔢𝔭(π_{1}(Γ))  (2.12)

which is ‘exact’ in the sense that each composition r^{∗}∘Int^{∗} is the constant map into the identity. The middle space ℜ𝔢𝔭(π_{1}(Σ_{Γ}), SL(2, ℂ)) in (2.12) parameterizes topologically trivial flat holomorphic bundles on P_{Γ}, that is the pairs (E, h), where E ∈ M^{ss}(P_{Γ}) and h is a holomorphic flat SL(2, ℂ)-connection on E (see formula (4.16) and Proposition 4.1 from [T2]). Moreover, the following identification list can be checked at once:

Proposition 2.1

(1) The coincidence ℜ𝔢𝔭(π_{1}(Γ)) = M^{ss}(P_{Γ}) described in (2.6) identifies the second map of (2.12) : ℜ𝔢𝔭(π_{1}(Σ_{Γ})) →^{Int^{∗}} ℜ𝔢𝔭(π_{1}(Γ)) with the forgetful map

ℜ𝔢𝔭(π_{1}(Σ_{Γ})) →^{f} M^{ss}(P_{Γ}) ,  (2.13)

which takes (E, h) ⟼ E. In particular, the fibers of f = Int^{∗} have the structure of affine spaces associated with the vector spaces of Higgs fields (see for example [T4], [T5]).

(2) The first map of (2.12) : M^{ss}(P_{Γ}) = ℜ𝔢𝔭(π_{1}(Γ)) →^{r^{∗}} ℜ𝔢𝔭(π_{1}(Σ_{Γ})) gives a section for the affine bundle (2.13). So, each fiber of f = Int^{∗} is naturally identified with the vector space of the Higgs fields on the corresponding vector bundle E.

There is quite simple geometrical reason for why ℜ𝔢𝔭(π_{1}(Σ_{Γ})) → M^{ss}(P_{Γ}) has a vector bundle structure: the boundary \overline{M^{ss}(P_{Γ})} \ M^{ss}(P_{Γ}) of the compactifified moduli space contains an effective theta divisor (see [T1]); so, the obstruction for lifting of affine structure to vector one vanishes over M^{ss}(P_{Γ}).

3 Framed Higgs fields and spectral curves

Let us write Ẽ_{a} for a framed vector bundle obtained from E by a framing function a as in (2.3). Then a Higgs field ϕ : Ẽ_{a} → Ẽ_{a}⊗K_{Γ} on Ẽ_{a} is given by a collection of traceless 2×2-matrices ω(v)|, v ∈ V (Γ), whose entries ω_{ij}(v) are meromorphic differentials on P_{v} with poles at p_{e_{1}}, p_{e_{2}}, p_{e_{3}}, {e_{1}, e_{2}, e_{3}} = S(v). Their residue matrices res_{p_{e}}(ω(v)) = ||res_{p_{e}}ω_{ij}(v)|| satisfy the relations:

res_{p_{e}}ω(v_{s}(e⃗)) + a(e⃗) ∘ res_{p_{e}}ω(v_{t}(e⃗)) ∘ a(e⃗)^{−1} = 0 , ∀ e⃗ ∈ E⃗(Γ)

∑_{e∈S(v)} res_{p_{e}}ω(v) = 0 , ∀ v ∈ V (Γ) . (3.1)

These constraints are gauge invariant^{2} and linear in the Higgs fields. In particular, the dimension of the space of Higgs fields equals 3g − 3.

2with respect to the complex reframing group Map(V (Γ), SL(2, ℂ)) from (2.2)

Now, following the Hitchin program (see [H]) and sending a Higgs field

ϕ : Ẽ_{a} → Ẽ_{a} ⊗ K_{Γ}  (3.2)

to the corresponding quadratic differential^{3} det ω(v) = −ω_{11}^{2}(v) − ω_{12}(v) · ω_{21}(v), we get a map

π : H^{0}(P_{Γ}, ad Ẽ_{a} ⊗ K_{Γ}) → H^{0}(P_{Γ}, O_{P_{Γ}}(2 K_{Γ}))  (3.3)

The restriction of the projective line bundle ℙ(E) onto each P_{v} is the quadric ℙ(E)|_{P_{v}} = ℙ_{1} × P_{v}, which admits two projections

ℙ_{1} ⟵^{p_{f}} ℙ(E)|_{P_{v}} ⟶^{p_{v}} P_{v} .

We write H = p_{f}^{∗}O_{ℙ_{1}}(1) for the Grothendieck generator such that R^{0}p_{v}_{∗}H = E|_{P_{v}}. Then any Higgs field (3.2) is obtained as the direct image R^{0}p_{v}_{∗} of some homomorphism

Φ : H → H ⊗ p_{v}^{∗}O_{P_{v}}(1) .

Since ϕ is traceless, its restriction ϕ_{p} : ℙ(E_{p}) → ℙ(E_{p}) onto a fiber over p ∈ P_{v} is either a linear automorphism ℙ(E_{p}) →^{∼} ℙ(E_{p}), which has two distinct fixed points p_{1}, p_{2} ∈ ℙ(E_{p}), or a degenerated map ϕ_{p}, which contracts ℙ(E_{p}) to one point p_{0} ∈ ℙ(E_{p}). Thus, we get a ramified double covering

w : P̃_{v} → P_{v}  (3.4)

which depends on and encodes the initial Higgs field ϕ.

Let us suppose for simplicity that for each component P_{v}, v ∈ V (Γ), the quadratic differential det ω(v) has two distinct zero points z_{1}(v), z_{2}(v) ∈ P_{v} \ {p_{e_{1}}, p_{e_{1}}, p_{e_{1}}}, where {e_{1}, e_{2}, e_{3}} = S(v). Then the ramification divisor of the double covering (3.4) is

(det ω(v))_{0} = z_{1}(v) + z_{2}(v) ⊂ P_{v}  (3.5)

and the intersection number P̃_{v} · p_{v}^{−1}(p_{e}) = 2 on the quadric ℙ_{1} × P_{v}.

Definition 3.1 The pair (P̃_{v}, L_{ϕ}), where L_{ϕ} = H|P̃_{v} ∈ Pic P̃_{v}, is called a spectral data of the Higgs field (3.2).

3note that the constrains (3.1) automatically imply the constrains (1.2)

The Higgs field ϕ is reconstructed from its spectral data as follows. Twisting the standard adjoint sequence of sheaves

0 → O(−P̃_{v}) → O → OP̃_{v} → 0

on the quadric ℙ_{1} × P_{v} by the Grothendieck line bundle H, we get an exact triple

0 → O(−P̃_{v}) ⊗ H → H → OP̃_{v}(H) → 0 .  (3.6)

Restricting the first term to the fiber, we get

O(−P̃_{v})(H)|_{ℙ(E_{p})} = O_{ℙ_{1}}(−2) ⊗ O_{ℙ_{1}}(1) = O_{ℙ_{1}}(−1) .

Hence, applying Rf_{v}_{∗} to (3.6), we get an isomorphism

0 → E → R^{0}f_{v}_{∗}(OP̃_{v}(H)) → 0

which coincides with (3.2).

Now let us switch on the gluing procedure prescribed by the framing data a : E⃗(Γ) → SL(2, ℂ) to combine the spectral curves P̃_{v} together. Consider each P̃_{v} as projective line equipped with an involution i_{v} : P̃_{v} → P̃_{v} such that P̃_{v} →^{w} P_{v} = P̃_{v}/i_{v} is the double covering (3.4). Then the ramification points (3.5) become the fixed points of i_{v} and 3 picked points p_{e_{i}} ∈ P_{v} turn to 6 points p_{e_{i}}^{±} ∈ P̃_{v} forming a triple of i_{v}-conjugated pairs. Gluing P̃_{v_{1}} with P̃_{v_{2}} in 2 points p_{e}^{±} for each e = (v_{1}, v_{2}) ∈ S(v_{1}) ∩ S(v_{2}) ⊂ E(Γ) and all v_{1}, v_{2} ∈ V (Γ), we get a reducible curve P̃_{Γ}(ϕ) of arithmetical genus 4g − 3 with the involution

i_{ϕ} : P̃_{Γ}(ϕ) → P̃_{Γ}(ϕ)

such that the quotient P̃_{Γ}(ϕ)/i_{ϕ} = P_{Γ} is the original ll-curve we have started with. Under this procedure, the local line bundles L_{ϕ} on P̃_{v} are glued to the global line bundle

L_{ϕ} ∈ Pic P̃_{Γ}(ϕ) ,

which restricts onto each P̃_{v} as L_{ϕ}|P̃_{v} = OP̃_{v}(1).

Now, using the geometric interpretation of the Hitchin systems via Prym varieties developed in [T4], [T5] one can easily prove the following

Proposition 3.1 The initial Higgs field (3.2) on P_{Γ} is uniquely recovered from the triple (P̃_{Γ}(ϕ), i_{ϕ}, L_{ϕ}); and the fiber of the Hitchin map (3.3):

π : H^{0}(P_{Γ}, ad Ẽ_{a} ⊗ K_{Γ}) → H^{0}(P_{Γ}, O_{P_{Γ}}(2 K_{Γ}))

over a regular quadratic differential Ω ∈ H^{0}(P_{Γ}, O_{P_{Γ}}(2K_{Γ})) coincides with the Prym variety:

π^{−1}(Ω) = Prym_{w} = Pic_{1,...,1}(P̃_{Γ}(ϕ)/w^{∗}(Pic(P_{Γ})) .

Andrey Tyurin died in Bonn at 27th of October going by bus to the Institute. The death was instantaneous and unexpected. The text looked unfinished but it was decided that really it contains the result and the argument to prove it; one just needs to remember two classical papers listed below and combine these ones with [T1]. Thus it should be strongly recommended to use these sources for understanding of the present text, just slightly edited by Alexei Gorodentsev and Nikolai Tyurin.

References

[A] I. V. Artamkin. Algebraic geometry of ll-curves. in preparation

[H] N. Hitchin. Stable bundles and integral systems. Duke Math. J. v.54 n.1 (1987), 91–114.

[HT] A. Hatcher, W. Thurston. A presentation for the mapping class group of closed oriented surface. Topology 19 (3) (1980), 221–237.

[JW] L. C. Jeffrey, J. Weitsman. Bohr - Sommerfeld orbits in the moduli space of flat connections and the Verlinde dimension formula. Commun. Math. Phys. 150 (1992), 593–630.

[T1] A. Tyurin. Quantization, Classical and Quantum Field Theory and Theta-Functions. Les publication du CRM, a paraitre 2002; arXiv: math.AG/0210466

[T2] A. Tyurin. On the Mumford – Narasimhan problem.

Preprint MPI n.129 (2002), available at math.AG/0210135, 1–27.

[T3] A. Tyurin. Three mathematical facets of SU(2)-spin networksm. Preprint available at math.DG/0011035, 1–20.

[T4] A. Tyurin. Periods and principal parts on a Riemann surface with a flat structure. Izv. Akad. Nauk SSSR, Ser. Mat. 41:6 (1977), 1425–1443; translated in: Math. USSR-Izv. 11 (1979).

[T5] A. Tyurin. On periods of quadratic differentials. Uspekhi Mat. Nauk 33:6 (1978), 59–88; translated in: Russian Math. Surveys, 33:6 (1979), 169–221.