Abelianisation of Logarithmic $\mathfrak{sl}_2$-Connections

We prove a functorial correspondence between a category of logarithmic $\mathfrak{sl}_2$-connections on a curve $X$ with fixed generic residues and a category of abelian logarithmic connections on an appropriate spectral double cover $\pi : \Sigma \to X$. The proof is by constructing a pair of inverse functors $\pi^{\text{ab}}, \pi_{\text{ab}}$, and the key is the construction of a certain canonical cocycle valued in the automorphisms of the direct image functor $\pi_\ast$.


§ 1. Introduction
This paper describes an approach to analysing meromorphic connections on Riemann surfaces. The technique, called abelianisation, is to introduce a decorated graph Γ on a Riemann surface X in order to establish a correspondence between meromorphic connections on vector bundles of higher rank over X and meromorphic connections on line bundles (which we call abelian connections) over a multi-sheeted ramified cover Σ → X. Namely, given a flat vector bundle E on X, an application of the standard local theory of singular differential equations near each pole allows one to extract valuable asymptotic information in the form of locally defined flat filtrations on E, first discovered by Levelt [Lev61]. These filtrations, often called Levelt filtrations, can be organised into a single flat line bundle L but only over Σ, and E can be recovered from L using the combinatorial data encoded in Γ.

Main result.
In this paper, we restrict our attention to the simplest case of sl 2 -connections with logarithmic singularities and generic residues. Our main result (Theorem 3.3) is a natural equivalence between a category of sl 2 -connections on X and a category of logarithmic abelian connections on a double cover Σ of X. More precisely, fix (X, D) a compact smooth complex curve with a finite set of marked points, fix the data of generic residues along D, and choose an appropriate meromorphic quadratic differential ϕ on X with double poles along D. Then ϕ gives rise to a double cover π : Σ → X (called the spectral curve) ramified at R ⊂ Σ, a graph Γ on X (called the Stokes graph), and a transversality condition on the Levelt filtrations extracted at nearby poles as dictated by Γ. Then there is a natural equivalence of categories: sl 2 -connections on X with logarithmic poles at D with very generic residues transverse with respect to Γ abelian connections on Σ with logarithmic poles at π −1 (D) ∪ R with fixed residues equipped with odd structure Given a flat vector bundle E on X, the abelianisation functor π ab Γ extracts Levelt filtrations along D and glues them into a flat line bundle L over Σ. In order to recover E from L, the main difficulty is that the naive guess that E is the pushforward π * L is incorrect because π * L necessarily has logarithmic singularities along the branch locus. The solution is to realise the combinatorial content of the Stokes graph Γ in cohomology: we construct a canonical cocycle V on X (called the Voros cocycle) which deforms the pushforward functor π * , as a functor, and this deformation is the nonabelianisation functor π Γ ab . The Voros cocycle is constructed in a completely standardised and combinatorial way from the Stokes graph Γ. This is significant because it means V is constructed without reference to any specific choice of E or L, thereby setting up an equivalence of categories.
Evidently, ∇ has logarithmic singularities at each point u i with residue Res u i ∇ = A i . The residue Res ∇ along D is then simply the full collection of the chosen matrices {A 1 , . . . , A d }. The eigenvalues ±λ i ∈ C of each A i are the Levelt exponents of ∇, so the residue data of ∇ is a = λ 2 1 , . . . , λ 2 d .
2.6. The central object of study in this paper is the category of logarithmic sl 2connections on (X, D) with fixed generic residue data a, for which we shall use the following shorthand notation: Conn 2 X := Conn 2 sl (X, D; a) ⊂ Conn 2 sl (X, D) .
(4) 2.7. Local diagonal decomposition. Fix a point p ∈ D, and consider a connection germ (E p , ∇ p ) at p with generic Levelt exponents ±λ p at p, where Re(λ) > 0. A coordinate trivialisation E p ∼ −→ C{z} 2 transforms ∇ p to a logarithmic sl 2 -differential system d + A(z)z −1 dz, where A(z) is some sl 2 -matrix of holomorphic function germs. By [Was76, Theorems 5.1, 5.4], there exists a holomorphic SL 2 gauge transformation which transforms the given differential system into the diagonal system d + diag(−λ p , +λ p )z −1 dz which depends only on λ p and z. This classical theorem about singular ordinary differential equations admits vast generalisations, but we do not need them here. Together with the fixed ordering on the Levelt exponents, it induces a graded decomposition of E p with respect to which ∇ p is diagonal.

Proposition (Local diagonal decomposition).
Let (E p , ∇ p , M p ) be the germ of a logarithmic sl 2 -connection at p ∈ D with generic Levelt exponents ±λ p . Then there is a canonical ordered decomposition where (Λ ± p , ∂ ± p ) is a rank-one logarithmic connection germ at p with residue ±λ p . Moreover, M induces a flat skew-symmetric isomorphism M p : Here, "skew-symmetric" means that M p is multiplied by −1 under the switching map. The order on the Levelt exponents −λ p ≺ +λ p determines a ∇ p -invariant filtration E • p := Λ − p ⊂ E p on the vector bundle germ E p , which we will refer to as the Levelt filtration in reference to the more general such concept studied by Levelt in his thesis [Lev61].
We will refer to the ∇ p -invariant filtration E • p := Λ − p ⊂ E p , given by the order on the Levelt exponents −λ p ≺ +λ p , as the Levelt filtration on the vector bundle germ E p . Clearly, any pair of logarithmic sl 2 -connection germs (E p , ∇ p ), (E p , ∇ p ) with the same generic Levelt exponents ±λ p at p are isomorphic and any such isomorphism is necessarily diagonal with respect to the diagonal decompositions. Any morphism (E p , ∇ p ) → (E p , ∇ p ) necessarily preserves the Levelt filtration, as the following lemma explains.
2.9. Lemma. Suppose (E, ∇), (E , ∇ ) is a pair of sl 2 -connections on (X, D) with the same local exponents along D. Then any morphism φ : E → E of connections restricts to a map φ p : L p → L p between the Levelt subbundles near p for every p ∈ D.
Proof. Let ψ, ψ be (possibly multivalued) flat sections of L p , L p , respectively, and let ψ be a (possibly multivalued) flat section of E linearly independent from ψ , all defined over a punctured neighbourhood of p. Thus, (ψ , ψ ) is an ordered basis of (multivalued) flat sections of E over a punctured neighbourhood of p. The sections ψ, ψ decay to 0 as x → 0 for any local coordinate x centred at p, whilst ψ is unbounded. Being a morphism of connections, the map φ necessarily sends ψ to a constant linear combination of aψ + bψ . But since φ is holomorphic at p and ψ decays at p, the section φ(ψ) must also decay to 0 as x → 0. This forces b = 0.
Note: one may alternatively choose a sectorial neighbourhood of p to dismiss the question of multivaluedness. In this case, we need to use that fact that φ is bounded as x → 0.

Logarithmic Connections and Double Covers
Logarithmic connections can be pulled back and pushed forward along ramified covers. In this section we describe these operations, restricting ourselves to the simplest case of double covers π : Σ → X with simple ramification and which are trivial over the polar divisor D. Thus, let C := π −1 (D) and let R ⊂ Σ be the ramification divisor. Here and everywhere, we assume that R has no higher multiplicity and that the branch locus B := π(R) ⊂ X is disjoint from D. We denote by σ : Σ → Σ the canonical involution.
2.11. Odd abelian connections. Connections on line bundles are sometimes called abelian connections. The line bundle O Σ (R) carries a canonical logarithmic connection ∂ R , defined to be the connection for which the canonical map is flat. Explicitly, if z is a local coordinate on Σ vanishing at r ∈ R, then the local section z −1 ∈ O Σ (R) gives a trivialisation, in which ∂ R is given by 2.12. Definition (odd abelian connection). An odd abelian logarithmic connection on (Σ, R∪C) is the data (L, ∂, µ) consisting of an abelian logarithmic connection on (Σ, R ∪ C) equipped with a skew-symmetric isomorphism µ : intertwining ∂ ⊗ σ * ∂ and ∂ R .
Here, "skew-symmetric" means µ satisfies σ * µ = −µ. Abelian connections with a similar structure but over the punctured curve Σ \ C ∪ R have appeared in [HN16, §4.2] under the name equivariant connections. We refer to the isomorphism µ as the odd structure on (L, ∂).
2.13. Proposition (residues of odd connections). The residue of any odd abelian connection (L, ∂, µ) at a ramification point is −1/2. In particular, the monodromy of ∂ around a ramification point is −1. Furthermore, if p ∈ D and p ± ∈ C are the two preimages of p, then the residues of ∂ at p ± satisfy Proof. The residue of ∂ R at r ∈ R is −1. If λ = Res r ∂, then the residue of the connection ∂ ⊗ σ * ∂ at r is 2λ, so the odd structure on L forces λ = −1/2. Next, since σ(p − ) = p + , the residue at p − of σ * ∂ is equal to the residue of ∂ at p + . This means ∂ ⊗ σ * ∂ has residue Res p − ∂ + Res p + ∂ at p − . But the residue of ∂ R at p − is 0, so the odd structure on L forces the identity.
by the rule π * ∇(π * e) = π * (∇e) for any local section e ∈ E. Clearly, the Levelt exponents of ∇ at p ∈ D and the Levelt exponents of π * ∇ at any preimagep ∈ C of p are the same. More interesting is pushing connections forward along π. The direct image functor π * of O Σ -modules can be used to pushforward connections from Σ down to X, but the relationship between the polar divisors is more complicated (see [GLP18,proposition 2.17] for more generality).

Proposition
(pushforward of odd abelian connections). The direct image π * extends to a functor Moreover, for any ∂ ∈ Conn 1 odd (Σ, R ∪ C), if ±λ ∈ C are its residues at the two preimages p ± ∈ C of a point p ∈ D, then the Levelt exponents of π * ∂ at p are ±λ.
To show that the odd structure on L induces an sl 2 -structure on π * L, recall that there is a canonical isomorphism det(π * L) ∼ = det(π * O Σ ) ⊗ Nm(L), where Nm(L) is the norm of L [HP12, Cor. 3.12]. For a double cover, there is a canonical isomorphism π * Nm(L) ∼ = L ⊗ σ * L. Moreover, it is easy to see that π * det(π * O Σ ) is canonically isomorphic to O Σ (−R). The statement about the residues is obvious because π is unramified over D.
2.17. Image of π * . One can show that the monodromy of π * ∂ around the branch locus B is a quasi-permutation representation of the double cover Σ → X [Kor04]. As a result, no connection on (X, D) is the pushforward of an abelian connection on Σ. In other words, the image of the pushforward functor π * in Conn 2 sl (X, B ∪ D) does not even intersect the subcategory Conn 2 sl (X, D). Abelianisation fixes this problem: in §3.3, we will explicitly construct a deformation of the pushforward functor π * which does map into Conn 2 sl (X, D). § 2.

Spectral Curves for Quadratic Differentials
Let ϕ be a quadratic differential on (X, D), by which we mean a meromorphic quadratic differential on X with at most order-two poles along D; i.e., it is a global holomorphic section of S 2 Ω 1 X (2D). The standard reference is [Str84]; see also [BS15,§ §2,3]. By the Riemann-Roch Theorem, 2.18. Quadratic residue. In any local coordinate x centred at p ∈ D, a quadratic differential ϕ with a double pole at p is expanded as ϕ = (a p x −2 +· · · ) dx 2 . The coefficient a p ∈ C is a coordinate-independent quantity, called the (quadratic) residue of ϕ at p and denoted Res p (ϕ). The residue of ϕ along D is thus a global section a = Res(ϕ) ∈ H 0 X (O D ), same as what we called residue data in §2.1. There is a quadratic residue short exact sequence: 2.19. Lemma. For any a ∈ H 0 X (O D ), there exists a meromorphic quadratic differential ϕ on (X, D) with Res(ϕ) = a.
Proof. By the Kodaira Vanishing Theorem, H 1 X S 2 Ω 1 X (D) = 0, which implies that the residue map Res : H 0 X S 2 Ω 1 X (2D) → H 0 X O D is surjective. This means that any residue data a decorating the divisor D can be lifted to a quadratic differential ϕ.

2.20.
In view of (14), the only configuration (X, D) for which there is a unique quadratic differential ϕ with specified residues is (g X , |D|) = (0, 3) (i.e., P 1 with three marked points). In this case, the three-dimensional vector space of quadratic differentials H 0 X S 2 Ω 1 X (2D) can be parameterised by the residues α, β, γ at the three points of D. Identifying (X, D) with (P 1 , {0, 1, ∞}), one can show that the unique quadratic differential with residues α, β, γ at the double poles 0, 1, ∞ is 2.21. Generic quadratic differentials. We will say that a quadratic differential ϕ is generic if all zeroes are simple. The subspace of generic quadratic differentials in H 0 is obviously open dense given as the complement of a hypersurface. If (g X , |D|) = (0, 3), then the space of quadratic differentials is at least onedimensional; but if (g X , |D|) = (0, 3), this is a condition on the residues of ϕ. One can use (16) to calculate that the open subspace of generic quadratic differentials for (g X , |D|) = (0, 3) is the complement of the quadratic hypersurface , assume in addition that a is contained in the complement of the hypersurface (17). Then there exists a generic quadratic differential ϕ on (X, D) such that Res(ϕ) = a.
2.24. The log-cotangent bundle. Let Y be the total space of Ω 1 X (D), sometimes called the log-cotangent bundle, and let p : Y → X be the projection map. Like the usual cotangent bundle, the log-cotangent bundle Y has a canonical one-form, which can be constructed as follows. Let θ ∈ H 0 Y, p * Ω 1 X (D) be the tautological section. Then the fibre product exists in the category of vector bundles, because p : Y → X is a surjective submersion. Unravelling the definition of the fibre product, we find that A consists of all vector fields on Y that are tangent to the divisor on Y is then defined as the image of the tautological section θ under this map.

Example. Take
Then Ω 1 P 1 (D) has a trivialisation over the affine z-chart given by the logarithmic one-form z −1 (z − 1) −1 dz. With respect to this trivialisation, the canonical one-form η Y is simply yz −1 (z − 1) −1 dz where y is the linear coordinate in the fibre.

The spectral curve.
We denote by π : Σ → X the restriction to Σ of the canonical projection p : Y → X.
We also denote the ramification divisor by R ⊂ Σ and the branch divisor by B ⊂ X. As a double cover, Σ is equipped with a canonical involution σ : Σ → Σ.
If ϕ is generic, then Σ is embedded in Y as a smooth divisor, and the projection π : Σ → X is a simply ramified double cover, branched exactly at the zeroes of ϕ, and trivial over the points of D. Its genus is g Σ = |D| + 4(g X − 1) + 1. (see, e.g., [BNR89, remark 3.2]). Using the Riemann-Hurwitz formula, the number of ramification points |R| of π, which is the same as the number of zeroes |B| of ϕ, is 2.27. Example. For the quadratic differential ϕ 1 from Example 2.23, the spectral curve Σ has genus 0, hence is a copy of P 1 . If we trivialise Ω 1 P 1 (D) over the affine z-chart using the differential form z −1 (z − 1) −1 dz, then Σ is given by the equation y 2 = 1 9 (z 2 − z + 1). For both quadratic differentials ϕ 2 and ϕ 3 , the spectral curve has genus 1, so it is an elliptic curve over P 1 , and it is given by y 2 = 1 9 (z 4 + 1).
Notice that, although the quadratic differential ϕ 1 is singular at the points 0, 1 in the affine z-chart, its spectral curve Σ is perfectly well-behaved above these points (see Fig. 1). This is a manifestation of the fact that our spectral curve Σ is embedded Figure 1: A real slice of the total space of Ω 1 P 1 (D) over the real line in P 1 z . In blue is the spectral curve Σ of the quadratic differential ϕ1 from Example 2.23. Figure 2: A real slice of the total space of Ω 1 P 1 over the real line in P 1 z . In blue is the curve given by the equation (y dz) 2 = ϕ1, where ϕ1 is the quadratic differential from Example 2.23.
inside the total space of the logarithmic cotangent bundle rather than the usual cotangent bundle. In contrast, constructing a spectral curve of ϕ 1 using the same equations but in the usual cotangent bundle yields a curve which escapes from the total space above the points 0, 1 (see Fig. 2).

The canonical one-form.
Pulling back the canonical one-form η Y to Σ yields a differential form η with logarithmic poles along C := π −1 (D), called the canonical one-form on Σ. It satisfies η 2 = π * ϕ and σ * η = −η, and can therefore be thought of as the 'canonical square root' of the quadratic differential ϕ. It has zeroes along the ramification locus R, and its residues at the two preimages p ± ∈ C of any point p ∈ D satisfy Res p − η = − Res p + η and Res p ± η 2 = Res p ϕ. If the residue data a = Res(ϕ) is generic, we can fix an order on the preimages of p: If p − ≺ p + , we shall call p − a sink pole and p + a source pole. The divisor C is thus decomposed equally into sinks and sources C = C − C + . § 2.

Logarithmic Connections and Spectral Curves
In general, connections do not have an invariant notion of eigenvalues or eigenvectors. However, in the presence of a spectral curve, we can make sense of these notions as follows.

2.29.
Let π : Σ → X be the spectral curve of a generic quadratic differential ϕ with generic residue data a along D. Suppose (E, ∇) ∈ Conn 2 X is a logarithmic sl 2connection on (X, D) with residue data a. If p ∈ D, let ±λ p be the Levelt exponents at p, which by construction are the residues of η at the preimages p ± ∈ C. Consider the local diagonal decomposition Let z be a local coordinate on Σ centred at p ± in which η is in normal form ±λ p dz /z. Since Σ is unramified over p, we also use z as a local coordinate on X centred at p.
If we fix a basepoint p * near p, then examining the Levelt normal form of ∇ p with respect to the coordinate z we obtain germs of (multivalued) flat sections ψ ± p which can be expressed as ψ ± p = f ± p e ± p , where e ± p is a (univalued) generator of Λ ± p , and f ± p is the germ of a (multivalued) function defined in the coordinate z by The observation is that the integrand in this expression is precisely the canonical one-form η thought of as written in the local coordinate z near p.

2.30.
To express this in a coordinate-free way, let U ⊂ X be any simply connected open neighbourhood of p disjoint from B and all other points of D. Then U has two disjoint preimages U ± on Σ where U ± contains p ± . Let η ± be the restriction of η to U ± , and we can think of η ± as being defined on U. Define (multivalued) functions on the punctured neighbourhood Note that the germ of f ± at p is precisely f ± p , and that f ± satisfies the differential equation dlog f ± = −η ± ; moreover, f ± is nowhere-vanishing on U • . Analytically continue the solutions ψ ± p to multivalued flat sections ψ ± of E over U • , and define These sections of E form a basis of holomorphic generators over U satisfying Thus, we can think of e ± as an eigensection of ∇ with eigenvalue η ± , and the line subbundles Λ ± U ⊂ E| U that they generate determine the flat eigen-decomposition of (E, ∇) over U that uniquely continues the local diagonal decomposition of E p : 2.31. More invariantly, letŨ ⊂ Σ be any simply connected neighbourhood of a pole p ∈ C = π −1 (D) ⊂ Σ which is disjoint from R and all other points of C. Let f be any (multivalued) solution of the differential equation dlog f = −η defined over the punctured neighbourhoodŨ * :=Ũ \ {p}. Then the same calculation as above shows that the pullback π * E overŨ has a section e which is an eigensection of π * ∇ with eigenvalue η:

The Stokes Graph
Fix some generic residue data a. If (g X , |D|) = (0, 3), assume in addition that a is contained in the complement of the hypersurface (17). For any generic quadratic differential ϕ on (X, D) with residues a, let Σ be its spectral curve with canonical one-form η.  (Fig. 3).

Saddle-free quadratic differentials and very generic residues.
If the horizontal foliation F has no saddle trajectories, then the quadratic differential ϕ is said to be saddle-free. It follows from [BS15,Lemma 4.11] that the subset of quadratic differentials which are saddle-free is open dense. Note that "saddle-free" may be a condition on the residue data a. For example, if (g X , |D|) = (0, 3), the quadratic differential ϕ with given residues a is unique (given by (16)) and may fail to be saddle-free. In this case, there are only two ramification points r ± ∈ Σ, so a saddle trajectory occurs if and only if the canonical one-form η satisfies Im r + r − η = 0 for a path of integration in Σ \ C ∼ = P 1 \ {6 points}. If b ± ∈ B are the two branch points, then upon identifying X ∼ = P 1 and choosing a branch cut in order to write η with √ ϕ, where ϕ is given by (16), this integral can be explicitly computed in terms of logarithms and it defines a closed real-analytic subset of C 3 αβγ . It therefore determines an explicit condition on the residues a = {α, β, γ} for the unique ϕ to be saddle-free. We will say that residue data a is very generic if there exists a generic saddle-free quadratic differential ϕ with residues a.
Ultimately, however, this apparent rigidity in our construction is artificial and can be removed by using a more topological argument. We will study this as well as other non-generic situations elsewhere.  Figure 4: From left to right: plot of critical trajectories of quadratic differentials ϕ1, ϕ2, ϕ3 from Example 2.23. In plots 1 and 2, the trajectories that escape the picture frame tend to infinity.

Example.
All three quadratic differentials ϕ 1 , ϕ 2 , ϕ 3 from Example 2.23 are saddle-free. The true plots of their critical trajectories are presented in Fig. 4. 2.36. The Stokes and spectral graphs. Now we define the main combinatorial gadgets in our construction. Let ϕ be a generic and saddle-free quadratic differential. (Stokes graph, spectral graph). The Stokes graph Γ is the graph on X whose vertices are D ∪ B and whose edges are the critical leaves of F. The spectral graph → Γ is the oriented graph on Σ whose vertices are C ∪ R and whose edges are the critical leaves of The polar vertices C are further divided into sinks and sources (cf. Part 2.28):
If p ∈ D, we will always denote its preimages in C by p − , p + where p ± ∈ C ± . They satisfy the relation σ(p ± ) = p ∓ . All spectral rays incident to a sink/source are oriented into/out of the sink/source, so spectral rays → Γ 1 are divided by parity: • positive spectral rays → Γ + 1 : polar vertex is a source; • negative spectral rays → Γ − 1 : polar vertex is a sink.

2.38.
Spectral rays always occur in pairs: the involution σ maps a spectral ray to a spectral ray of opposite parity. Stokes rays have no natural notion of parity; instead, the preimage of every Stokes ray α ∈ Γ 1 is a pair of opposite spectral rays Fig. 5). The graphs Γ, → Γ are squaregraphs: every Stokes region is a quadrilateral with two branch vertices and two polar vertices, and its boundary is made up of four Stokes rays (Fig. 6). Similarly, every spectral region is a quadrilateral with two ramification vertices and two polar vertices (one of which is a source and one is a sink), and its boundary is made up of four spectral rays (two of which are positive and two are negative). We index them as described in Fig. 6: Each branch point has three incident Stokes rays and three incident Stokes regions, but each Stokes region has two branch vertices, so there are 3|B| Stokes rays and 3 2 |B| Stokes regions in total. So, using (23), Γ 1 = 6|D| + 12(g X − 1) and Γ 2 = 3|D| + 6(g X − 1) (32) → Γ 1 = 12|D| + 24(g X − 1) and → Γ 2 = 6|D| + 12(g X − 1) .

Example.
by enlarging all edges and faces as follows. For every face I ∈ Γ 2 and every edge α ∈ Γ 1 , let U I and U α be the germs of open neighbourhoods in X • of the face I and the edge α, respectively. We continue calling them Stokes regions and Stokes rays. We define spectral regions U i and spectral rays U ± α for all i ∈ → Γ 2 , α ± ∈ → Γ 1 in the same Figure 7: Right: a schematic picture of the Stokes graph Γ (orange) of the quadratic differential ϕ1 from Example 2.23. The point at infinity has been blown up to the bounding circle drawn in orange. b1, b2 are the branch points. Left: the corresponding spectral graph → Γ on the spectral curve Σ ∼ = P 1 . The preimages of the points 0, 1, ∞ carry a label according to whether the vertex is a sink or a source. r1, r2 are the ramification points above b1, b2, respectively. way. We obtain what we call Stokes open covers of X • and Σ • , respectively: If p is a vertex of U I , then intersecting U I with the infinitesimal disc U p around p can be seen as the germ of a sectorial neighbourhood of p (or a disjoint union of two). In fact, the infinitesimal punctured disc U * p centred at p is covered by such sectorial neighbourhoods whose double intersections are the Stokes rays incident to p.

2.41.
Any double intersection U I ∩ U J of Stokes regions is either a single Stokes ray or a pair of disjoint Stokes rays with the same polar vertex but necessarily different branch vertices, and there are no nonempty triple intersections. So we define the nerves of these covers bẏ We adopt the following notational convention: if U α is a Stokes ray contained in the double intersection U I ∩ U J , then U I , U J are ordered such that going from U I to U J the Stokes ray α is crossed anti-clockwise around the branch vertex of U α .

2.42.
The restriction of the projection π : Σ → X to any spectral region U i , any spectral ray U ± α , or any infinitesimal disc U ± p around a pole p ± is an isomorphism respectively onto its image Stokes region U I = U {i,i } , Stokes ray U α , or infinitesimal disc U p around the pole p; we denote these restrictions as follows: 2.43. Example. For the differential ϕ 1 from Example 2.23, the Stokes open covers of X • = P 1 \ {0, 1, ∞} and Σ • = P 1 \ {0 ± , 1 ± , ∞ ± } are illustrated in Fig. 8. Given a connection (E, ∇) ∈ Conn 2 X , consider its local diagonal decompositions Let us analytically continue the flat abelian connection germs Λ − p , Λ − p to U I using the flat structure on E: That such connections exist is not difficult to see once phrased in terms of filtered local systems on X \ D. Explicitly, for each Stokes region U I , choose a point x I ∈ U I and two loops based at x I , one around each of the two poles on the boundary of U I . Since U I is simply connected, these loops are to be chosen contractible if the corresponding pole is filled in. Choose a 'master' basepoint x 0 in X \ D, and for each I, choose a path in X \ D that connects x I to x 0 . Finally, choose generators for π 1 (X, x 0 ) represented by loops that avoid D. Assigning a matrix in SL(2, C) to each loop, subject to the obvious homotopy conditions, defines a local system on X \ D which can always be extended to a logarithmic connection on (X, D). By appropriately fixing α β p q b Figure 9: A Stokes region UI whose polar vertices coincide. The subset of X bounded by the Stokes rays α, β in the complement of UI must contain another point q ∈ D, for otherwise all Stokes rays incident to the branch point b are also incident to p. But then the complement of Γ has a connected component which is not a horizontal strip contradicting [BS15, Lemma 3.1]. Generically, the monodromy of ∇ around the pole q does not preserve the Levelt filtration coming from p.
the matrices corresponding to the poles , we can ensure that the resulting logarithmic connection has the desired residue data. Finally, Γ-transversality cuts out a complement finitely-many algebraic conditions, one for each pair of monodromy matrices assigned to the pair of loops corresponding to each basepoint x I .
In fact, the same argument shows that (with respect to an appropriate topology) the subset of Γ-transverse connections is open dense. We do not need these details here, and only mention that these and other moduli-theoretic considerations will be described in great detail in a future publication.

Proposition (Semilocal diagonal decomposition of transverse connections).
If (E, ∇, M ) ∈ Conn 2 X (Γ), then the restriction E I := E| U I to any Stokes region U I has a canonical flat decomposition where (Λ i , ∂ i ) and (Λ i , ∂ i ) are defined by (38). Moreover, the sl 2 -structure M defines a flat skew-symmetric isomorphism M I : The main construction in this paper (Theorem 3.3) is an equivalence between Conn 2 X (Γ) and a certain category of odd abelian connections on the spectral curve Σ.

Transversality over Stokes rays.
Suppose U α is a Stokes ray contained in the double intersection U I ∩U J of two adjacent Stokes regions. Then E has two diagonal decompositions over U α : Let p ∈ D be the common polar vertex of U I , U J . Then Λ i , Λ j are continuations of the same line bundle germ Λ − p ⊂ E p , so Λ i = Λ j over the Stokes ray U α . With respect to this pair of decompositions, the identity map on E has the following uppertriangular expression, which will be exploited throughout our construction in this paper: 2.49. Remark. Note that in the definition of transversality with respect to Γ, it is not required that the two polar vertices p, p of U I be different. If p = p it may seem that no connection ∇ can be transverse with respect to Γ for such a Stokes graph, but this is not the case. This is because the Stokes region U I defines two disjoint sectorial neighbourhoods of p, so the two analytic continuations Λ i , Λ i ⊂ E I of the same germ Λ − p are generically not the same, as explained in Fig. 9. § 3. Abelianisation 3.1. As before, let (X, D) be a smooth compact curve equipped with a nonempty set of marked points D such that |D| > 2 − 2g X . Suppose D is decorated with very generic residue data a in the sense of Definition 2.3 and Part 2.34. We are studying the category of logarithmic sl 2 -connections on (X, D) with residue data a: Our method is to choose a generic saddle-free quadratic differential ϕ on (X, D) with residues a. Let π : Σ → X be the spectral curve of ϕ, and let Γ be the corresponding Stokes graph on X. Consider the subcategory of connections that are transverse with respect to Γ in the sense of Definition 2.46: 3.2. The main result of this paper is that Conn 2 X (Γ) is equivalent to a category of odd abelian connections on the spectral curve Σ as follows. For every p ∈ D, let ±λ p ∈ C be the Levelt exponents of the residue data a at p (arranged such that Re(λ p ) > 0). Put C := π −1 (D), let C ± be as in Part 2.36, let R ⊂ Σ be the ramification divisor of π, and define abelian residue data along C ∪ R as follows: Consider the category of odd abelian logarithmic connections on (Σ, C ∪ R) with residues λ, for which we use the following shorthand notation:
Expressed more explicitly, this equivalence is We will prove this theorem by constructing a pair functors, called abelianisation and nonabelianisation with respect to Γ; they are constructed in §3.1 and §3.3, respectively. In Proposition 3.30, we prove that they form an equivalence of categories. § 3.

The Abelianisation Functor
In this subsection, given an sl 2 -connection (E, ∇, M ) ∈ Conn 2 X (Γ), we construct an abelian connection (L, ∂, µ) ∈ Conn 1 Σ , and show that this construction is functorial. The idea is to extract the diagonal decompositions of E at the poles of ∇, analytically continue them to the spectral regions on the spectral curve, and then glue them into a flat line bundle using canonical isomorphisms that arise due to transversality.

Definition at the poles.
Given p ∈ D, consider the local diagonal decomposition E p ∼ −→ Λ − p ⊕ Λ + p from Proposition 2.8. We define (L, ∂) over the infinitesimal disc U ± p around p ± to be the pullback of the connection germ (Λ ± p , ∂ ± p ): Thus, (L ± p , ∂ ± p ) is the germ of a logarithmic abelian connection at p ± with residue ±λ p . It also follows that (π ∓ p ) * Λ ± p = σ * L ± p , so the pullback of the flat skewsymmetric isomorphism M p : 3.5. Definition on spectral regions. Let U i ⊂ Σ be a spectral region, and let p − be its sink vertex. We define (L, ∂) by uniquely continuing the germ L − p using the flat structure on π * E: Evidently, (L i , ∂ i ) := π * i (Λ i , ∂ i ) for Λ i defined by (38). Furthermore, if U i = σ(U i ), then π * i Λ i = σ * L i for Λ i defined by (38). So if I = {i, i }, the pullback to U i of the sl 2 -structure M I from Proposition 2.47 defines a flat skew-symmetric isomorphism 3.6. Gluing over spectral rays. For every α ∈ Γ 1 , consider the pair of opposite spectral rays α ± ∈ → Γ ± 1 , and let p ± ∈ C ± be their respective polar vertices. Let U I = U {i,i } , U J = U {j,j } ⊂ X be the pair of adjacent Stokes regions which intersect along the Stokes ray U α as described in Fig. 10. By transversality with respect to Γ, the vector bundle E has two diagonal decompositions over the Stokes ray U α : Then Λ i , Λ j are continuations of the same line bundle germ Λ − p ⊂ E p , so Λ i = Λ j over the Stokes ray U α . The identity map on E, written with respect to this pair of Figure 10: U ± p is a pair of opposite spectral rays, r is their common ramification vertex, and p± are their respective polar vertices. Ui, Uj are a pair of oriented Stokes regions which have U + α in their intersection, arranged such that the ordered pair (Ui, Uj) respects the cyclic anticlockwise order around r.
decompositions, is the upper triangular matrix (42). We therefore define 3.9. Finally, functoriality of our construction readily follows from the fact that morphisms of connections necessarily preserve diagonal decompositions.

Proposition
(1) The degree of L is (2) For any p ∈ D, let U p be the infinitesimal disc around p.
be the local diagonal decomposition (Proposition 2.8). Then there is a canonical flat isomorphism (3) Let U ± p be the infinitesimal disc around the preimage p ± ∈ C of p. Recall the notation π ± p := π| U ± p : U ± p ∼ −→ U p . Then there are canonical flat isomorphisms (4) Let U I ⊂ X be a Stokes region with polar vertices p, p ∈ D, and let E I (5) Let U i , U i be the spectral regions above U I incident to p − , p − ∈ C, respectively, and recall the notation (6) Finally, recall that η is the canonical one-form on the spectral curve Σ. The abelian connection ∂ − η on the abelianisation line bundle L is holomorphic along C; it has logarithmic poles only along the ramification divisor R where it has residues −1/2.
The following proposition, which readily follows from the discussion in §2.4, expresses the sense in which the abelianisation of connections is the analogue of abelianisation of Higgs bundles. Figure 11: Illustration of the construction of the abelianisation line bundle L for a connection on (P 1 , {0, 1, ∞}) from Example 2.5 using the quadratic differential ϕ1 from Example 2.23.

Proposition (spectral properties of abelianisation
Moreover, over any spectral region U i ⊂ Σ, there is a canonical flat inclusion L → π * E with respect to which this section e is an eigensection for π * ∇ with eigenvalue η: 3.13. Example. Let us illustrate the above construction in the simplest possible explicit example. Consider a logarithmic sl 2 -connection (E, ∇) from Example 2.5 with d = 3. Namely, X = P 1 , E = O ⊕2 P 1 , and D := {0, 1, ∞}. Let A 1 , A 2 be any pair of sl(2, C)-matrices, both with eigenvalues ±1/3, and let ∇ be given by the formula (3). Then the ∇ has Levelt exponents ±1/3 at each pole.
To abelianise ∇, we must choose a generic saddle-free quadratic differential on (X, D) with residues 1/9 at each point of D. One such choice is the quadratic differential ϕ 1 from Example 2.23. Its spectral curve Σ was described in Example 2.27, its Stokes and spectral graphs were detailed in Fig. 7, and the relevant Stokes open cover was presented in Fig. 8. Finally, in Fig. 11, we illustrate the abelianisation construction by displaying which Levelt line subbundle is considered on which Stokes and spectral region. § 3.
3.14. The canonical nonabelian cocycle V . Let U α ∈ Γ 1 be a Stokes ray on X with polar vertex p ∈ D and branch vertex b ∈ B. It is a component of the intersection of exactly two Stokes regions U I , U J (see Fig. 12). Consider the pair of canonical identifications given by Proposition 3.11(4): Over the Stokes ray U α , their ratio yields a flat automorphism of (π * L, π * ∂): where π * L denotes the associated local system ker (π * ∂) on X • . The nerve of the cover U Γ of X • consists of Stokes rays, so we obtain aČech 1-cocycle V with values in the local system Aut(π * L): 3.15. Lemma. If (E, ∇, M ) ∈ Conn 2 X (Γ), let (L, ∂, µ) be its abelianisation, and consider the pushforward π * L = π * π ab Γ E. If V is the cocycle (60), then there is a canonical isomorphism Proof. The action of the cocycle V on the pushforward bundle π * L is a new bundle E := V · π * L. Explicitly, the local piece E I over a Stokes region U I is defined to be π * L| U I , and the gluing data over a Stokes ray U α ⊂ U I ∩ U J is given by V α : But this commutative square together with (58) and (59) imply that E and E are canonically isomorphic.

Transposition paths.
Let us explicitly compute each automorphism V α with respect to a pair of canonical decompositions of π * L over the Stokes ray U α . Through the isomorphisms π * L| U I Notice that Λ i = Λ j because they are continuations of the same line bundle germ at p, so using (42) we find: Now, we can decompose the map ∆ α : Λ i → Λ j through canonical inclusions, projections, and the upper-triangular expressions (42) for the identity on E as follows: Figure 12: UI , UJ , UK ⊂ X are the Stokes regions with I = {i, i }, J = {j, j }, K = {k, k }. The stokes rays Uα, Uγ, Uγ are indicated by α, β, γ (same for the spectral rays). b ∈ B is the branch point and r ∈ R is the ramification point above b.
We interpret the first and second upper-triangular expressions as the identity maps on E over U γ and U β , respectively. Since all these bundle maps are ∇-flat, the map ∆ α can be interpreted as the endomorphism of the fibre of E over a point in U α obtained as the composition of ∇-parallel transports P I , P K , P J along paths δ I contained in U I from U α to U γ , followed by δ K contained in U K from U γ to U β , followed by δ J contained in U J from U β back to U α (see Fig. 12). Explicitly: The key idea, which goes back to Gaiotto-Moore-Neitzke [GMN13a], is to notice that this expression has an interpretation as a parallel transport for the abelian connection ∂ on the spectral curve. Indeed, if we fix points p, p , p in U α , U γ , U β as shown in Fig. 12, then through the canonical identification of fibres using Proposition 3.11(4), we have: Here, ∆ + α is defined by the diagram; we used (53), and p i , p k , p j are ∂-parallel transports along the paths δ i , δ k , δ j which are the lifts of δ I , δ K , δ J as shown in Fig. 12. Figure 14: A short path ℘ on X intersecting the Stokes ray α and its lifts ℘ , ℘ to Σ. Figure 15: The concatenated path It follows immediately from the construction of E that if ℘ is a path on X • contained in a Stokes region, then P (℘) = π * p(℘). Explicitly, let ℘ , ℘ be the two lifts of ℘ to Σ. Let x, y be the startpoint and the endpoint of ℘, and similarly for ℘ , ℘ . Then, for example, the fibre E x = E| x is the direct sum of fibres L x ⊕ L x of L. With respect to these decompositions, the parallel transport P (℘) : E x −→ E y is expressed as We say that a path ℘ on X • (or Σ • ) is a short path if its endpoints do not belong to the Stokes graph Γ (or to the spectral graph → Γ) and it intersects at most one Stokes ray (or spectral ray). If ℘ is a short path on X • that intersects a Stokes ray α ∈ Γ 1 , then ℘ is divided into two segments ℘ − , ℘ + (Fig. 14). Each ℘ ± is contained in a Stokes region, so P (℘ ± ) = π * p(℘ ± ). On the other hand, the vector bundle E is constructed by gluing π * L to itself over U α by the automorphism V α , so we obtain the following formula for P (℘): P (℘) = π * p(℘ + ) · V α · π * p(℘ − ) .