SEMISIMPLE FROBENIUS (SUPER)MANIFOLDS

AND QUANTUM COHOMOLOGY OF P^{r}

Yu. I. Manin

Max–Planck–Institut für Mathematik, Bonn, Germany

S. A. Merkulov

Department of Mathematics, University of Glasgow, UK

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Abstract. We introduce and study a superversion of Dubrovin’s notion of semisimple Frobenius manifolds. We establish a correspondence between semisimple Frobenius (super)manifolds and special solutions to the (supersymmetric) Schlesinger equations. Finally, we calculate the Schlesinger initial conditions for solutions describing quantum cohomology of projective spaces.

0. Introduction

B. Dubrovin introduced the notion of Frobenius manifold and made an extensive study of it in [D]. Roughly speaking, it is a triple (M, g, ∘) where M is a manifold, g is a flat Riemannian metric on it, and ∘ is a O_{M}–linear commutative and associative multiplication on the tangent sheaf T_{M}, with compatibility conditions (see 1.1.1 below). An important class of examples is supplied by quantum cohomology (see [KM].) Actually, quantum cohomology furnishes versions of Dubrovin’s definition in which M may be supermanifold, or even a formal supermanifold.

An important subclass of Frobenius manifolds consists of semisimple ones. This means that tangent spaces with ∘–multiplication are semisimple algebras. This is possible only if M has no odd coordinates, by purely formal reasons.

In [Ma1], p. 41, one of the authors suggested that it would be interesting to construct a natural superization of the notion of semisimple Frobenius manifolds. This is one of the goals of this paper. To avoid any misunderstanding, we must stress that semisimple Frobenius supermanifolds in the sense of this work are not Frobenius manifolds in the category of supermanifolds in the sense of [KM] or [Ma1].

Our extension is based upon Dubrovin’s theory of semisimple Frobenius manifolds reducing their classification to that of isomonodromic deformations: see [D]. This reduction exists in two versions. The first version leads to the deformation of a flat connection on a vector bundle on P^{1} having two singular points, a regular and an irregular one (see [D] and [S].) The second one deals with connections having only regular singularities. The two versions are related by the formal Laplace transform, as was explained in [KM].

In the classical paper [Sch], L. Schlesinger constructed the universal deformation space of the connections with regular singularities on P^{1} (see [Mal3] for a modern treatment.) Schlesinger’s equations govern the dependence of the universal connection on the deformation parameters. Semisimple Frobenius manifolds correspond to some solutions to Schlesinger’s equations, with the structure group reduced to the orthogonal one, and supplied with an additional piece of data. These solutions are called here “strict special” ones.

Our supeversion of the Dubrovin theory includes a superization of Schlesinger’s equations, of the notion of strict special solutions, and of the correspondence between them and semisimple Frobenius manifolds briefly described above. This is one of the arguments for the naturality of our definition. An additional detail is that the (rather mysterious) structural action of the Virasoro algebra on any semisimple Frobenius manifold is now replaced by that of the Neveu–Schwarz superalgebra. We hope that super–Schlesinger equations may present an independent interest.

Since the structure of the superversion is closely parallel to that of the pure even one, we start with a report on the theory of semisimple Frobenius manifolds and with its application to the quantum cohomology of projective spaces.

Quantum cohomology of a projective algebraic manifold V is the pair (H^{∗}(V), Φ) consisting of the usual cohomology space, say, with complex coefficients, and the potential Φ, a formal function on the cohomology space whose Taylor coefficients are numerical invariants of V counting the number of parametrized rational curves subject to certain incidence conditions: cf. e. g. [KM], [M1] and [BM]. Its third derivatives form the structure constants of the quantum cohomology algebra of V.

For projective spaces, it turns out to be semisimple, and we characterize the relevant special solutions by their initial conditions.

The paper is structured as follows. §1 contains an overview of the basic facts about Frobenius manifolds. Omitted proofs can be found in [D] and [Ma1]. In §2, we discuss the Schlesinger equations and interrelations between them and Frobenius manifolds. The version we explain here is taken from [Ma1]; for a closely related treatment see [H]. An important amelioration is the Theorem 1.14 and the related notion of strict speciality. §3 is devoted to the quantum cohomology of projective spaces; see [Ma2] and [Ma1], §III.5 for the special case of P^{2}. In §4, we supersymmetrize the notion of semisimple Frobenius manifolds. Finally, in §5 we discuss a correspondence between semisimple Frobenius supermanifolds and special solutions of supersymmetric Schlesinger equations.

§1. Frobenius manifolds

1.1. Frobenius manifolds. Throughout this paper, we work in the category of complex manifolds M. A metric on M is an even symmetric pairing g : S^{2}(T_{M}) → O_{M}, inducing an isomorphism g′ : T_{M} → T_{M}^{∗}. Here O_{M} is the structure sheaf, and T_{M} is the tangent sheaf.

An affine flat structure on M is a subsheaf T_{M}^{f} ⊂ T_{M} of linear spaces of pairwise commuting vector fields, such that T_{M} = O_{M} ⊗_{C} T_{M}^{f}. Sections of T_{M}^{f} are called flat vector fields. The metric g is compatible with the structure T_{M}^{f}, if g(X, Y) is constant for flat X, Y.

An affine flat structure can be equivalently described by a complete atlas whose transition functions are affine linear, because for a maximal commuting set of linearly independent vector fields (X_{a}) one can find local coordinates such that X_{a} = ∂/∂x^{a}, and they are defined up to a constant shift. If a metric g is compatible with an affine flat structure, it is flat in the sense of the formalism of Riemannian geometry. The parallel transport endows T_{M}^{f} with the structure of local system.

1.1.1. Definition. Let M be a manifold. Consider a triple (T_{M}^{f}, g, A) consisting of an affine flat structure, a compatible metric, and an even symmetric tensor A : S^{3}(T_{M}) → O_{M}.

Define an O_{M}–bilinear symmetric multiplication ∘ = ∘_{A,g} on T_{M}:

T_{M} ⊗ T_{M} → S^{2}(T_{M}) →^{A′} T_{M}^{∗} →^{g′} T_{M} : X ⊗ Y → X ∘ Y  (1.1)

where prime denotes a partial dualization, or equivalently,

A(X, Y, Z) = g(X ∘ Y, Z) = g(X, Y ∘ Z).  (1.2)

This means that the metric is invariant with respect to the multiplication. a). M endowed with this structure is called a pre–Frobenius manifold.

b). A local potential Φ for (T_{M}^{f}, A) is a local even function such that for any flat local tangent fields X, Y, Z

A(X, Y, Z) = (XY Z)Φ.  (1.3)

A pre–Frobenius manifold is called potential one, if A everywhere locally admits a potential.

c). A pre–Frobenius manifold is called associative, if the multiplication ∘ is associative.

d). A pre–Frobenius manifold is called Frobenius, if it is simultaneously potential and associative.

If a potential Φ exists, it is unique up to adding a quadratic polynomial in flat local coordinates.

In flat local coordinates (x^{a}) (1.3) becomes A_{abc} = ∂_{a}∂_{b}∂_{c}Φ, and (1.2) can be rewritten as

∂_{a} ∘ ∂_{b} = ∑_{c} A_{ab}^{c}∂_{c},  (1.4)

where

A_{ab}^{c} := ∑_{e} A_{abe}g^{ec}, (g^{ab}) := (g_{ab})^{−1}.

Furthermore,

(∂_{a} ∘ ∂_{b}) ∘ ∂_{c} = (∑_{e} A_{bc}^{e}∂_{e}) ∘ ∂_{c} = ∑_{ef} A_{ab}^{e}A_{ec}^{f}∂_{f},

∂_{a} ∘ (∂_{b} ∘ ∂_{c}) = ∂_{a} ∘ ∑_{e} A_{bc}^{e}∂_{e} = ∑_{ef} A_{bc}^{e}A_{ae}^{f}∂_{f} =

= ∑_{ef} A_{bc}^{e}A_{ea}^{f}∂_{f}.  (1.5)

Comparing the coefficients of ∂_{f} in (1.5), lowering the superscripts and expressing A_{abc} through a potential, we finally see that the notion of the Frobenius manifold is a geometrization of the following highly non–linear and overdetermined system of PDE:

∀a, b, c, d : ∑_{ef} Φ_{abe}g^{ef}Φ_{fcd} = ∑_{ef} Φ_{bce}g^{ef}Φ_{fad}.  (1.6)

They are called Associativity Equations, or WDVV (Witten–Dijkgraaf–Verlinde– Verlinde) equations.

Following B. Dubrovin, we will now express (1.6) as a flatness condition.

1.2. The first structure connection. Let (M, g, A) be a pre–Frobenius manifold (we omit T_{M}^{f} in the notation, since it can be reconstructed from g.) Define the following objects:

a). The connection ∇_{0} : T_{M} → Ω_{M}^{1} ⊗ T_{M} well determined by the condition that flat vector fields are ∇_{0}–horizontal.

Denote its covariant derivative along a vector field X by

∇_{0,X}(Y) = i_{X}(∇_{0}(Y)), i_{X}(df ⊗ Z) = Xf ⊗ Z.

b). A pencil of connections depending on an even parameter λ:

∇_{λ} : T_{M} → Ω_{M}^{1} ⊗ T_{M} : ∇_{λ,X}(Y) := ∇_{0,X}(Y) + λX ∘ Y.  (1.7)

We will call ∇_{λ} the first structure connection of (M, g, A).

In flat coordinates (1.7) reads:

∇_{λ,∂_{a}}(∂_{b}) = λ ∑_{c} A_{ab}^{c}∂_{c} = λ ∂_{a} ∘ ∂_{b} = (−1)^{ab}λ∂_{b} ∘ ∂_{a} = (−1)^{ab}∇_{λ,∂_{b}}(∂_{a}).

Therefore ∇_{λ} has vanishing torsion for any λ. In particular, ∇_{0} is the Levi–Civita connection for g.

1.2.1. Theorem. Let ∇_{λ} be the structure connection of the pre–Frobenius manifold (M, g, A). Put ∇_{λ}^{2} = λ^{2}R_{2} + λ R_{1} (there is no constant term since ∇_{0}^{2} = 0.) Then

a). R_{1} = 0 ⟺ (M, g, A) is potential.

b). R_{2} = 0 ⟺ (M, g, A) is associative.

Therefore (M, g, A) is Frobenius, iff ∇_{λ} is flat.

This can be proved by the direct computation.

1.3. Identity. Let (M, g, A) be a pre–Frobenius manifold. A vector field e on M is called identity, if e ∘ X = X for all X.

If e exists at all, it is uniquely defined by ∘, hence by g and A.

Conversely, given A and e, there can exist at most one metric g making (M, g, A) a pre–Frobenius manifold with this identity:

g(X, Y) = A(e, X, Y).

This follows from (1.2). If A has a potential Φ, this translates into a non–homogeneous linear differential equation for Φ supplementing the Associativity Equations (1.6):

∀ flat X, Y, eXY Φ = g(X, Y).  (1.8)

In most (although not all) important examples e itself is flat. If this is the case, one can everywhere locally find a flat coordinate system (x^{0}, …, x^{n}) such that e = ∂/∂x^{0} = ∂_{0}, and (1.8) becomes

∀ a, b, Φ_{0ab} = g_{ab}.  (1.9)

Since all g_{ab} are constants, we get the following result.

On a potential pre–Frobenius manifold with flat identity e = ∂_{0} (in a flat coordinate system) we have modulo terms of degree ≤ 2:

Φ(x^{0}, …, x^{n}) = \frac{1}{2}x^{0} ( ∑_{a,b≠0} g_{ab}x^{a}x^{b} + ∑_{a≠0} g_{0a}x^{0}x^{a} + \frac{1}{3} g_{00}(x^{0})^{2}) + Ψ(x^{1}, …, x^{n}).

(1.10)

The metric g identifies T_{M} and T_{M}^{∗}. We will call the co–identity and denote ε the 1–form which is the image of e. ε is defined by

∀X ∈ T_{M}, i_{X}(ε) = g(X, e).

If (x^{a}) is a local coordinate system,then

ε = ∑_{a} dx^{a}g(∂_{a}, e).

Finally, if e and (x^{a}) are flat, then g(∂_{a}, e) are constant, and

ε = dη, η = ∑_{a} x^{a}g(∂_{a}, e).  (1.11)

1.4. Euler field. We will say that a vector field E on a manifold with flat metric (M, g) is conformal, if Lie_{E}(g) = Dg for some constant D. In other words, for all vector fields X, Y we have

E(g(X, Y)) − g([E, X], Y) − g(X, [E, Y]) = Dg(X, Y).  (1.12)

It follows that in flat coordinates we have E = ∑_{a} E^{a}(x)∂_{a} where E^{a}(x) are polynomials of degree ≤ 1. In fact, E is a sum of infinitesimal rotation, dilation and constant shift. Hence [E, T_{M}^{f}] ⊂ T_{M}^{f}. Moreover, the operator

V : T_{M}^{f} → T_{M}^{f}, V(X) := [X, E] − \frac{D}{2} X

is skewsymmetric:

∀ flat X, Y : g(V(X), Y) + g(X, V(Y)) = 0.

1.4.1. Definition. Let E be a vector field on a pre–Frobenius manifold (M, g, A). It is called an Euler field, if it is conformal, and Lie_{E}(∘) = d_{0}∘ for some constant d_{0}, that is, for all vector fields X, Y,

[E, X ∘ Y] − [E, X] ∘ Y − X ∘ [E, Y] = d_{0}X ∘ Y.  (1.13)

Clearly, any scalar multiple of an Euler field is also an Euler field. One can use this remark in order to normalize E by requiring that some non–vanishing eigenvalue becomes one. A convenient choice is often d_{0} = 1, if we have reasons to restrict ourselves to the d_{0} ≠ 0 case.

1.4.2. Proposition. Let E be a conformal vector field on a Frobenius manifold (M, g, Φ). Then E is Euler, iff

EΦ = (d_{0} + D)Φ + a quadratic polynomial in flat coordinates.

1.4.3. Case of semisimple ad E. We will call the set of eigenvalues of −ad E on T_{M}^{f}, together with d_{0} and D, the spectrum of E. We will say that E is semi– simple, if ad E, acting on flat fields, is. For semisimple E we can construct many homogeneous elements of O_{M}(∗) and T_{M}(∗) explicitly.

Let (∂_{a}) be a local basis of T_{M}^{f} such that

[∂_{a}, E] = d_{a}∂_{a}  (1.14)

where (d_{a}) form a part of the spectrum of E. Putting E = ∑ E^{a}(x)∂_{a}, we find from (1.14) that ∂_{a}E^{b} = δ_{a}^{b} d_{a}. Hence if ∂_{a} = ∂/∂x^{a}, we have

E = ∑_{a: d_{a}≠0}(d_{a}x^{a} + r^{a})∂_{a} + ∑_{b: d_{b}=0} r^{b}∂_{b}.

By shifting x^{a}, we can make r^{a} = 0 for d_{a} ≠ 0. Multiplying x^{b} by a constant, we can make r^{b} = 0 or 1 for d_{b} = 0. So finally we can choose local flat coordinates in such a way that

E = ∑_{a: d_{a}≠0} d_{a}x^{a}∂_{a} + ∑_{some b: d_{b}=0} ∂_{b}.  (1.15)

Clearly, E assigns definite degrees to the following local functions:

Ex^{a} = d_{a}x^{a} for d_{a} ≠ 0; E exp x^{b} = exp x^{b} or 0 for d_{b} = 0.  (1.16)

Assume now that M has an identity e. From (1.13) we get

[e, E] = d_{0}e.  (1.17)

Hence our notation for the spectrum will be consistent, if in the case of flat e we put e = ∂_{0}, and otherwise do not use 0 as one of the subscripts in (1.14).

Formula (1.12) in the basis (1.13) becomes

∀ a, b : g(d_{a}∂_{a}, ∂_{b}) + g(∂_{a}, d_{b}∂_{b}) = Dg_{ab}

that is,

(d_{a} + d_{b} − D)g_{ab} = 0.  (1.18)

In particular, g(e, e) = 0 unless D = 2d_{0}.

1.5. Extended structure connection. Let M be a pre–Frobenius manifold with a conformal vector field E. Put M̂ := M × (P_{λ}^{1} \ {0, ∞}), where P_{λ}^{1} is the completion of Spec C[λ, λ^{−1}]. Furthermore, put T̂ = pr_{M}^{∗}(T_{M}). If X is a vector field on M, it may be lifted to M̂ in two different guises: as a vector field annihilating λ, denoted again X, and as a section of T̂, then denoted X̂.

Choose a constant d_{0} and put E := E − d_{0}λ\frac{∂}{∂λ} ∈ TM̂. Clearly, X̂ for flat X span

T̂, whereas flat X and E span TM̂, provided d_{0} ≠ 0, which we will assume.

1.5.1. Definition. Let M be a pre–Frobenius manifold with a conformal field E, and d_{0} a non–zero constant. The extended structure connection for M is the connection ˆ∇ on the sheaf T̂ on M̂, defined by the following formulas for its covariant derivatives: for any local vector fields X ∈ T_{M}, Y ∈ T_{M}^{f},

ˆ∇_{X}(Ŷ) := λ{ \atop X ∘ Y} ,  (1.19)
ˆ∇_{E}(Ŷ) := { \atop [E, Y]}.  (1.20)

1.5.2. Theorem. The extended structure connection is flat iff M is Frobenius and E is Euler with Lie_{E} (∘) = d_{0}∘.

From (1.19) and (1.20) one can derive a formula for the covariant derivative in the λ–direction: if Y is flat, we have

{ \atop [E, Y]} = ˆ∇_{E−d_{0}λ∂/∂λ}(Ŷ) = ˆ∇_{E}(Ŷ) − d_{0}λ ˆ∇_{∂/∂λ}(Ŷ) = λ{ \atop E ∘ Y} − d_{0}λ ˆ∇_{∂/∂λ}(Ŷ)

so that

d_{0} ˆ∇_{∂/∂λ}(Ŷ) = { \atop E ∘ Y} − \frac{1}{λ}{ \atop [E, Y]}.  (1.21)

1.6. Semisimple Frobenius manifolds. Let (M, g, A) be an associative pre– Frobenius manifold of dimension n.

1.6.1. Definition. M is called semisimple (resp. split semisimple) if an isomorphism of the sheaves of O_{M}–algebras

(T_{M}, ∘) ˜→ (O_{M}^{n}, componentwise multiplication)  (1.22)

exists everywhere locally (resp. globally.)

This means that in a local (resp. global) basis (e_{1}, …, e_{n}) of T_{M} the multiplication takes form

(∑ f_{i}e_{i}) ∘ (∑ g_{j}e_{j}) = ∑ f_{i}g_{i}e_{i},

and in particular,

e_{i} ∘ e_{j} = δ_{ij}e_{j}.  (1.23)

Such a family of idempotents is well defined up to renumbering. Another way of saying this is that a semisimple manifold comes with the structure group of T_{M} reduced to S_{n}. Notice that e_{i} are generally not flat, so that this reduction is not compatible with that induced by T_{M}^{f}, with the structure group GL(n).

Hence if M is semisimple, there exists an unramified covering of degree ≤ n!, upon which the induced pre–Frobenius structure is split.

Denote by (ν^{i}) the basis of 1–forms dual to (e_{i}). ¿From (1.2) and (1.23) we find

g(e_{i}, e_{k}) = g(e_{i} ∘ e_{i}, e_{k}) = g(e_{i}, e_{i} ∘ e_{k}) = δ_{ik}g_{ii}.

We will denote g_{ii} by η_{i}. We see that the symmetric 2-form representing g is diagonal in the basis (ν^{i}):

g = ∑_{i} η_{i}(ν^{i})^{2}.  (1.24)

Moreover, according to (1.2), A(e_{i}, e_{j}, e_{k}) = δ_{ij}δ_{ik}η_{i}, so that the symmetric 3-form representing A, is diagonal with the same coefficients:

A = ∑_{i} η_{i}(ν^{i})^{3}.  (1.25)

Finally, e := ∑_{i} e_{i} is the identity in (T_{M}, ∘), and the co–identity, defined in 2.1.2, nicely complements (1.24) and (1.25):

ε = ∑_{i} η_{i}ν^{i}.  (1.26)

Thus the Definition 1.61 can be restated as follows:

1.6.2. Definition. The structure of the semisimple pre–Frobenius manifold on M is determined by the following data:

a). A reduction of the structure group of T_{M} to S_{n}, specified by a choice of local bases (e_{i}) and dual bases (ν^{i}).

b). A flat metric g, diagonal in (e_{i}), (ν^{i}).

c). A diagonal cubic tensor A with the same coefficients as g.

Associativity of (T_{M}, ∘) is automatic in both descriptions. However, potentiality (and the flatness of g which we postulated) are non–trivial conditions.

1.7. Theorem. The structure described in the Definition 3.2 is Frobenius iff the following conditions are satisfied:

a). [e_{i}, e_{j}] = 0, or equivalently, e_{i} = ∂/∂u^{i}, ν^{i} = du^{i} for a local coordinate system (u^{i}) called canonical one.

b). η_{i} = e_{i}η for a local function η defined up to addition of a constant. Equivalently, ε is closed.

We will call η the metric potential of this structure. (Sometimes this term refers to h such that g_{ab} = ∂_{a}∂_{b}h; our meaning is different.)

Canonical coordinates are defined up to renumbering and constant shifts.

1.8. The Darboux–Egoroff equations. The Theorem 1.7 establishes a (not very explicit) equivalence between the following functional spaces on M (modulo self–evident equivalence):

a). Flat coordinates (x^{1}, …, x^{n}), flat metric g_{ab}, function Φ(x) satisfying the Associativity Equations (1.6) and semisimplicity.

b). Canonical coordinates (u^{1}, …, u^{n}), function η(u) such that the metric g = ∑ e_{i}η(du^{i})^{2} is flat, where e_{i} = ∂/∂u^{i}.

The constraints on η, implicit in b), are called the Darboux–Egoroff equations. In order to write them down explicitly, let us introduce the rotation coefficients of the potential metric:

γ_{ij} := \frac{1}{2} \frac{η_{ij}}{\sqrt{η_{i}η_{j}}}  (1.27)

where as before, η_{i} = e_{i}η, η_{ij} = e_{i}e_{j}η.

1.8.1. Proposition. The diagonal potential metric g = ∑ e_{i}η(du^{i})^{2} is flat iff ∀k ≠ i ≠ j ≠ k :

e_{k}γ_{ij} = γ_{ik}γ_{kj}  (1.28)

and

eγ_{ij} = 0.  (1.29)

1.8.2. Proposition. Let e be the identity, and ε the co–identity of the semisimple Frobenius manifold. Then

a). ε = dη, where η is the metric potential.

b). e is flat iff for all i, eη_{i} = 0, or equivalently, eη = g(e, e) = const. This condition is satisfied in the presence of an Euler field with D ≠ 2d_{0} (see (1.12), (1.13), (1.14).)

c). If e is flat, and (x^{a}) is a flat coordinate system, then

η = ∑_{a} x^{a}g(∂_{a}, e) + const.  (1.30)

The formula (1.30) shows that in the passage from the (x^{a}, Φ)–description to the (u^{i}, η)–description the main information is encoded in the transition formulas u^{i} = u^{i}(x), at least in the presence of flat identity.

Like the identity, the Euler field is almost uniquely defined by the canonical coordinates, if it exists at all.

1.9. Theorem. Let E be a vector field on the semisimple Frobenius manifold M, d_{0} a constant.

a). We have Lie (∘) = d_{0}(∘), iff

E = d_{0} ∑_{i}(u^{i} + c^{i})e_{i},  (1.31)

where c^{i} are some constants.

b). For the field of the form (3.17) and a constant D, we have Lie_{E}(g) = Dg iff for all i, Eη_{i} = (D − 2d_{0})η_{i}, or equivalently

Eη = (D − d_{0})η + const.  (1.32)

Thus in the presence of a non–vanishing Euler field we may and will normalize the canonical coordinates so that E = d_{0} ∑ u^{i}e_{i}.

1.10. A pencil of flat metrics. Equations (1.28) are stable with respect to a semigroup of coordinate changes. Namely, let f_{i} be arbitrary functions of one variable such that ǔ^{i} := f_{i}(u^{i}) form a local coordinate system, ě_{i} = ∂/∂ǔ^{i}, η̌_{i} = ě_{i}η etc.

1.10.1. Proposition. If (e_{i}, γ_{ij}) satisfy (1.28), then (ě_{i}, γ̌_{ij}) satisfy (1.28) as well.

In order to satisfy (1.29) as well, we will have to restrict ourselves to the one– parameter family of local coordinate changes

ǔ^{i} = log (u^{i} − λ), ě_{i} = (u^{i} − λ)e_{i}, ǧ_{λ} = ∑_{i}(u^{i} − λ)^{−1}e_{i}η(du^{i})^{2}  (1.33)

which make sense on M_{λ} := {x ∈ M | ∀i, u^{i} ≠ λ}.

1.10.2. Theorem. Let M be a semisimple Frobenius manifold with canonical coordinates (u^{i}) and metric potential η. Then the following statements are equivalent.

a). For all λ, the structure (1.33) is semisimple Frobenius on M_{λ}.

b). The same for a particular value of λ.

c). For all i ≠ j,

∑_{k} u^{k}e_{k}γ_{ij} = −γ_{ij}.  (1.34)

Moreover, (1.34) is satisfied if E = ∑_{k} u^{k}e_{k} is the Euler field on M with d_{0} = 1.

Notice that generally ě = ∑ ě_{k} is not flat for ǧ_{λ} and Ě = ∑ ǔ^{k}ě_{k} is not an Euler field.

If E is Euler, the metric ǧ_{λ} in (1.33) can be written in coordinate free form:

ǧ_{λ}(X, Y) = g((E − λ)^{−1} ∘ X, Y).  (1.35)

In fact (1.35) is flat on any Frobenius manifold with semisimple Euler field on it: cf. [D].

1.11. The second structure connection. From now on, we will restrict ourselves to the case of semisimple complex Frobenius manifolds carrying an Euler field with d_{0} = 1 and admitting a global system of canonical coordinates (u^{i}). We will call the second structure connection ˇ∇_{λ} the Levi–Civita connection of the flat metric (1.35), depending on a parameter λ and defined on the open subset M_{λ} ⊂ M where u^{i} ≠ λ for all i. Put M̌ := ∪_{λ}(M_{λ} × {λ}) ⊂ M × P_{λ}^{1} and denote by Ť the restriction of pr_{M}^{∗}(T_{M}) to M̌.

We will construct a flat extension of ˇ∇ of ˇ∇_{λ} to Ť which will also be referred to as the second structure connection. Both extensions ˆ∇ and ˇ∇ will be further studied as isomonodromic deformations of their restrictions to the λ–direction parametrized by M.

More precisely, assume that T_{M}^{f} is a trivial local system (for instance, because M is simply connected.) Put T := Γ(M, T_{M}^{f}). Then ˆ∇ (resp. ˇ∇) induces an integrable family of connections with singularities on the trivial bundle on P_{λ}^{1} with the fiber T. The first connection ˆ∇ is singular only at λ = 0 and λ = ∞ but whereas 0 is a regular (Fuchsian) singularity, ∞ is irregular one, so that ˆ∇ cannot be an algebraic geometric Gauss–Manin connection, and its monodromy involves the Stokes phenomenon. To the contrary, the second connection ˇ∇ generally has only regular singularities at infinity and at λ = u^{i} whose positions thus depend on the parameters. It is determined by the conventional monodromy representation and has a chance to define a variation of Hodge structure. For more details, see the next section.

It turns out that both deformations have a common moduli space and deserve to be studied together. In fact, fiberwise they are more or less formal Laplace transforms of each other. More to the point, they form a dual pair in the sense of J. Harnad.

In our calculations the key role will be played by the O_{M}–linear skew symmetric operator V : T_{M} → T_{M} which is the unique extension of the operator defined in 1.4 on flat vector fields by the formula

V(X) = [X, E] − \frac{D}{2} X for X ∈ T_{M}^{f}.  (1.36)

1.11.1. Proposition. a). We have for arbitrary X ∈ T_{M} :

V(X) = ∇_{0,X}(E) − \frac{D}{2} X.  (1.37)

b). Let e_{j} = ∂/∂u^{j}, f_{j} = e_{j}/\sqrt{η_{j}}. Then

V(f_{i}) = ∑_{j≠i}(u^{j} − u^{i})γ_{ij}f_{j}.  (1.38)

Formula (1.37) defines an O_{M}–linear endomorphism of T_{M} which coincides with (1.36) on the flat fields, as a calculation in flat coordinates shows. To check (1.38), we can use (1.37) and the classical explicit expressions for the Levi–Civita connection, cf. below.

We can now state the main result of this section. In addition to (1.36), define the operator U : T_{M} → T_{M} :

U(X) := E ∘ X,  (1.39)

so that U(f_{i}) = u^{i}f_{i}.

1.12. Theorem. For X, Y ∈ pr_{M}^{−1}(T_{M}) ⊂ TM̌ (meromorphic vector fields on T_{M×P_{λ}^{1}} independent on λ) put

ˇ∇_{X}(Y) = ∇_{0,X}(Y) − (V + \frac{1}{2} Id) (U − λ)^{−1}(X ∘ Y),  (1.40)

ˇ∇_{∂/∂λ}(Y) = (V + \frac{1}{2} Id) (U − λ)^{−1}(Y).  (1.41)

Then ˇ∇ is a flat connection on Ť whose restriction on M × {λ} defined by (1.40) is the Levi–Civita connection for ǧ_{λ}.

Remark. Rewriting ˆ∇ in the same notation, we get

ˆ∇_{X}(Y) = ∇_{0,X}(Y) + λX ∘ Y,  (1.42)
ˆ∇_{∂/∂λ}(Y) = [U + \frac{1}{λ}(V + \frac{D}{2} Id)] (Y).  (1.43)

Proof. We will first calculate the Levi–Civita connection for ǧ_{λ} in coordinates ǔ^{i} = log (u^{i} − λ) :

ě_{i} = \frac{∂}{∂ǔ^{i}} = (u^{i} − λ)e_{i}, η̌_{i} = (u^{i} − λ)η_{i}, η̌_{ij} = (u^{i} − λ)(u^{j} − λ)η_{ij} + δ_{ij}(u^{i} − λ)η_{i},

γ̌_{ij} = γ_{ij}(u^{i} − λ)^{1/2}(u^{j} − λ)^{1/2}.

Then for i ≠ j

ˇ∇ě_{i}(ě_{j}) = \frac{1}{2} \frac{η̌_{ij}}{η̌_{i}} ě_{i} + \frac{1}{2} \frac{η̌_{ij}}{η̌_{j}} ě_{j} = \frac{1}{2}(u^{i} − λ)(u^{j} − λ) (\frac{η_{ij}}{η_{i}} e_{i} + \frac{η_{ij}}{η_{j}} e_{j})

so that

ˇ∇_{e_{i}}(e_{j}) = \frac{1}{2} \frac{η_{ij}}{η_{i}} e_{i} + \frac{1}{2} \frac{η_{ij}}{η_{j}} e_{j} = ∇_{0,e_{i}}(e_{j}).  (1.44)

Similarly,

ˇ∇ě_{i}(ě_{i}) = \frac{1}{2} \frac{η̌_{ii}}{η̌_{i}} ě_{i} − \frac{1}{2} ∑_{j≠i} \frac{η̌_{ij}}{η̌_{j}} ě_{j} =

\frac{1}{2}(u^{i} − λ)^{2} [\frac{η_{ii}}{η_{i}} − \frac{1}{u^{i} − λ}] e_{i} − \frac{1}{2} ∑_{j≠i}(u^{i} − λ) (u^{j} − λ) \frac{η_{ij}}{η_{j}} e_{j}

so that

ˇ∇_{e_{i}}(e_{i}) = \frac{1}{2} [\frac{η_{ii}}{η_{i}} − \frac{1}{u^{i} − λ}] e_{i} − \frac{1}{2} ∑_{j≠i} \frac{u^{j} − λ}{u^{i} − λ} \frac{η_{ij}}{η_{j}} e_{j}.  (1.45)

Subtracting from this the Levi–Civita covariant derivative, we get

( ˇ∇_{e_{i}} − ∇_{0,e_{i}})(e_{i}) = −\frac{1}{2} \frac{1}{u^{i} − λ} e_{i} − \frac{1}{2} ∑_{j: j≠i} \frac{u^{j} − u^{i}}{u^{i} − λ} \frac{η_{ij}}{η_{j}} e_{j}  (1.46)

and

( ˇ∇_{e_{i}} − ∇_{0,e_{i}})(f_{i}) = −\frac{1}{2} \frac{1}{u^{i} − λ} f_{i} − ∑_{j≠i} \frac{u^{j} − u^{i}}{u^{i} − λ} γ_{ij} f_{j}.  (1.47)

In view of (1.38), we can write (1.44) and (1.45) together as

( ˇ∇_{e_{i}} − ∇_{0,e_{i}})(f_{j}) = − (V + \frac{1}{2} Id) (U − λ)^{−1}(e_{i} ∘ f_{j})  (1.48)

because e_{i} ∘ f_{j} = δ_{ij}f_{j}. This family of formulas is equivalent to (1.40) so that (1.40) is the Levi–Civita connection for ǧ_{λ}. In particular, it is flat for each fixed λ.

Since [X, ∂/∂λ] = 0 for X ∈ pr_{M}^{−1}(T_{M}), it remains to show that the covariant derivatives (1.40) and (1.41) commute on M̌ i. e. , that for all i, j

ˇ∇_{e_{i}} ˇ∇_{∂/∂λ}(e_{j}) = ˇ∇_{∂/∂λ} ˇ∇_{e_{i}}(e_{j}).  (1.49)

First of all, from (1.41) and (1.42) we find

ˇ∇_{∂/∂λ}(e_{j}) = \frac{1}{2} \frac{1}{u^{j} − λ} e_{j} + \frac{1}{2} ∑_{k≠j} \frac{u^{k} − u^{j}}{u^{j} − λ} \frac{η_{jk}}{η_{k}} e_{k}.  (1.50)

Together with (1.44) and (1.45) this gives for i ≠ j:

ˇ∇_{∂/∂λ} ˇ∇_{e_{i}}(e_{j}) = \frac{1}{2} \frac{η_{ij}}{η_{j}} [\frac{1}{2} \frac{1}{u^{j} − λ} e_{i} + \frac{1}{2} ∑_{k≠i} \frac{u^{k} − u^{i}}{u^{i} − λ} \frac{η_{ik}}{η_{k}} e_{k}] + (i ↔ j), (1.51)

ˇ∇_{e_{i}} ˇ∇_{∂/∂λ}(e_{j}) = \frac{1}{2} \frac{1}{u^{j} − λ} (\frac{1}{2} \frac{η_{ij}}{η_{i}} e_{i} + \frac{1}{2} \frac{η_{ij}}{η_{j}} e_{j}) +

+\frac{1}{2} ∑_{k≠j} e_{i} (\frac{u^{k} − u^{j}}{u^{j} − λ} \frac{η_{jk}}{η_{k}} ) e_{k} + \frac{1}{2} ∑_{k≠j,i} \frac{u^{k} − u^{j}}{u^{j} − λ} \frac{η_{jk}}{η_{k}} (\frac{1}{2} \frac{η_{ik}}{η_{i}} e_{i} + \frac{1}{2} \frac{η_{ik}}{η_{k}} e_{k}) +

\frac{1}{2} \frac{u^{i} − u^{j}}{u^{j} − λ} \frac{η_{ik}}{η_{k}} [\frac{1}{2} (\frac{η_{ii}}{η_{i}} − \frac{1}{u^{i} − λ}) e_{i} − \frac{1}{2} ∑_{j≠i} \frac{u^{j} − λ}{u^{i} − λ} \frac{η_{ij}}{η_{j}} e_{j}] .  (1.52)

The coincidence of coefficients of e_{k} in (1.51) and (1.52) for i ≠ j ≠ k ≠ i can be checked with the help of the following identities which are equivalent to the Darboux–Egoroff equations:

η_{ijk} = \frac{1}{2} (\frac{η_{ik}η_{jk}}{η_{k}} + \frac{η_{ij}η_{ik}}{η_{i}} + \frac{η_{ij}η_{jk}}{η_{j}} ) .

The coincidence of the coefficients of e_{i} requires a little more work, and we will give some details, again for the case i ≠ j.

In (1.51) the coefficient of e_{i} is

\frac{1}{4} \frac{1}{u^{i} − λ} \frac{η_{ij}}{η_{i}} + \frac{1}{4} \frac{u^{k} − u^{j}}{u^{j} − λ} \frac{η_{ij}^{2}}{η_{i}η_{j}} ,  (1.53)

whereas in (1.52) we get

\frac{1}{4} \frac{1}{u^{j} − λ} \frac{η_{ij}}{η_{i}} + \frac{1}{2} e_{i} (\frac{u^{i} − u^{j}}{u^{j} − λ} \frac{η_{ij}}{η_{i}} ) +

+\frac{1}{4} ∑_{k≠i,j} \frac{u^{k} − u^{j}}{u^{j} − λ} \frac{η_{ik}η_{jk}}{η_{i}η_{k}} + \frac{1}{2} (\frac{η_{ii}}{η_{i}} − \frac{1}{u^{i} − λ}) .  (1.54)

To identify (1.53) and (1.54) we have to get rid of the sum ∑_{k} in (1.54). This can be done with the help of (1.28), (1.29) and (1.34):

\frac{1}{4} ∑_{k≠i,j} \frac{u^{k} − u^{j}}{u^{j} − λ} \frac{η_{ik}η_{jk}}{η_{i}η_{k}} = \frac{1}{u^{j} − λ} \frac{η_{j}^{1/2}}{η_{i}^{1/2}} [ ∑_{k≠i,j} u^{k}γ_{ik}γ_{kj} − u^{j} ∑_{k≠i,j} γ_{ik}γ_{kj}] =

= \frac{1}{u^{j} − λ} \frac{η_{j}^{1/2}}{η_{i}^{1/2}} [−γ_{ij} − u^{i}e_{i}γ_{ij} − u^{j}e_{j}γ_{ij} + u^{j}(e_{i} + e_{j})γ_{ij}] =

= \frac{1}{2} \frac{1}{u^{j} − λ} [−\frac{η_{ij}}{η_{i}} + (u^{j} − u^{i}) (\frac{η_{iij}}{η_{i}} − \frac{1}{2} \frac{η_{ij}η_{ii}}{η_{i}^{2}} − \frac{1}{2} \frac{η_{ij}^{2}}{η_{i}η_{j}} )] .

The remaining part of the calculation is straightforward, and we leave it to the reader, as well as the case i = j which is treated similarly.

1.13. Formal Laplace transform. Assume now that T_{M}^{f} is a trivial local system. This means that if we put T := Γ(M, T_{M}^{f}), there is a natural isomorphism O_{M} ⊗ T → T_{M}.

Formulas (1.41) (resp. (1.43)) define two families of connections with singularities on the trivial vector bundle on P_{λ}^{1} with fiber T, parametrized by M. Namely, denote by ∂_{λ} the covariant derivative along ∂/∂λ on this bundle for which the constant sections are horizontal. Then the two connections are

ˇ∇_{∂/∂λ} = ∂_{λ} + (V + \frac{1}{2} Id) (U − λ)^{−1},  (1.55)

ˆ∇_{∂/∂λ} = ∂_{λ} + U + \frac{1}{λ} (V + \frac{D}{2} Id) .  (1.56)

Let M, N be two C[λ, ∂_{λ}]–modules. A formal Laplace transform M → N : Y ↦ Y^{t} is a C–linear map for which

(−λY)^{t} = ∂_{λ}(Y^{t}), (∂_{λ}Y)^{t} = λY^{t}.  (1.57)

The archetypal Laplace transform is the Laplace integral

Y^{t}(μ) = ∫ e^{−λμ}Y (λ)dλ  (1.58)

taken along a contour (not necessarily closed) in P^{1}(C). In an analytical setting we have to secure the convergence of (1.58), the possibility to derivate under the integral sign and the identity

∫ ∂_{λ}(e^{−λμ}Y (λ))dλ = 0.

However, (1.58) may admit other interpretations, for instance, in terms of asymptotic series.

Let now M (resp. N) be two C[λ, ∂_{λ}]–modules of local (or formal, or distribution) sections of P_{λ}^{1} × T so that the operators ˇ∇ · (U − λ) (resp. λ ˆ∇) make sense in M (resp. N) (cf. (1.55), resp. (1.56)), and assume that we are given a formal Laplace transform M → N.

1.13.1. Proposition. We have:

[ ˇ∇_{∂/∂λ}((U − λ)Y)]^{t} = (λ ˆ∇_{∂/∂λ} + \frac{1 − D}{2} ) Y^{t} = λ^{\frac{D+1}{2}} ˆ∇_{∂/∂λ}(λ^{\frac{1−D}{2}} Y^{t}).

In particular, λ^{\frac{1−D}{2}} Y^{t} is ˆ∇–horizontal, if (U − λ)Y is ˇ∇–horizontal. Proof. Using (1.55)–(1.57), we find:

[ ˇ∇_{∂/∂λ}((U − λ)Y)]^{t} = [(∂_{λ} · (U − λ) + V + \frac{1}{2} Id)Y ]^{t} =

= [λ (U + ∂_{λ}) + V + \frac{1}{2} Id] Y^{t} =

= [λ ˆ∇_{∂/∂λ} + \frac{1 − D}{2} Id] Y^{t} = λ^{\frac{D+1}{2}} ˆ∇_{∂/∂λ}(λ^{\frac{1−D}{2}} Y^{t}).

For a more detailed discussion of the formal Laplace transform, see [S], 1.6.

For later use we note that the connection ˇ∇ defined by (1.40), (1.41) can be further deformed. Namely, for any constant s put

ˇ∇_{X}^{(s)}(Y) = ˇ∇_{X}(Y) − s(U − λ)^{−1}(X ∘ Y),  (1.59)
ˇ∇_{∂/∂λ}^{(s)}(Y) = ˇ∇_{∂/∂λ}(Y) + s(U − λ)^{−1}(Y).  (1.60)

1.14. Theorem. ˇ∇_{X}^{(s)} is a flat connection on Ť.

This can be checked by a direct calculation similar to that in the proof of the Theorem 1.12. The formal Laplace transform of ˇ∇_{∂/∂λ}^{(s)} is given by

[ ˇ∇_{∂/∂λ}^{(s)}((U − λ)Y)]^{t} = (λ ˆ∇_{∂/∂λ} + \frac{1 − D + 2s}{2}) Y^{t} = λ^{\frac{D+1−2s}{2}} ˆ∇_{∂/∂λ}(λ^{\frac{1−D+2s}{2}} Y^{t}).

In particular, λ^{\frac{1−D+2s}{2}} Y^{t} is ˆ∇–horizontal, if (U − λ)Y is ˇ∇^{(s)}–horizontal.

§2. Schlesinger equations

2.1. Singularities of meromorphic connections. Let N be a complex manifold, D ⊂ N a closed complex submanifold of codimension one, F a locally free sheaf of finite rank on N. A meromorphic connection with singularities on D is given by a covariant differential ∇ : F → F ⊗ Ω_{N}^{1}((r + 1)D) for some r ≥ 0. It is called flat (or integrable) if it is flat outside D. We start with a list of elementary notions and constructions that will be needed later. They depend only on the local behaviour of F and ∇ in a neighborhood of D, so we will assume D irreducible.

i) Order of singularity. We will say that ∇ as above is of order ≤ r + 1 on D if ∇_{X}(F) ⊂ F(rD) for any vector field X tangent to D (i. e. satisfying XJ_{D} ⊂ J_{D} where J_{D} is the ideal of D), and ∇_{X}(F) ⊂ F((r + 1)D) in general. Locally, if (t^{0}, t^{1}, …, t^{n}) is a coordinate system on N such that t^{0} = 0 is the equation of D, the connection matrix of ∇ in a basis of F can be written as

G_{0} \frac{dt^{0}}{(t^{0})^{r+1}} + ∑_{i=1}^{n} G_{i} \frac{dt^{i}}{(t^{0})^{r}}  (2.1)

where G_{i} = G_{i}(t^{0}, t^{1}, …, t^{n}) are holomorphic matrix functions.

Note that G_{0}(0, t^{1}, …, t^{n}) ∈ H^{0}(D, EndF) is well-defined, i.e. it does not depend on the choice of local coordinates. It is called the residue of ∇ at D and is denoted by res_{D}(∇).

ii) Restriction to a transversal submanifold. Let i : N′ → N be a closed embedding of a submanifold transversal to D, D′ = N′ ∩ D, F′ = i^{∗}(F). Then the induced connection ∇′ = i^{∗}(∇) on F′ is flat and of order ≤ r + 1 on D′ if ∇ has these properties.

iii) Residual connection. Assume that ∇ is of order ≤ 1 on D. For any given local trivialization f of [D], one can define a connection without singularities ∇^{D,f} on j^{∗}(F) where j is the embedding of D in N. Namely, to define ∇_{X′}^{D,f}(s′) where s′ ∈ j^{∗}(F), X′ ∈ T_{D}, we extend locally s′ to a section s of F, X′ to a vector field X on N, res_{D}(∇) to a section res(∇) of F ⊗ F^{∗} on N, calculate (∇_{X} − \frac{Xf}{f} res(∇))(s) and restrict it to D. One checks that the result does not depend on the choices made. In the notation of (2.1), the matrix of the residual connection can be written as (r = 0):

∑_{i=1}^{n} G_{i}(0, t^{1}, …, t^{n})dt^{i}.  (2.2)

If ∇ is flat, ∇^{D,f} is flat for any local trivialization f of [D].

iv) Principal part of order r + 1. Similarly to (2.2), we can consider the matrix function on D

G_{0}(0, t^{1}, …, t^{n})  (2.3)

which we will call the principal part of order r+1 of ∇. In more invariant terms, it is the O_{D}–linear map j^{∗}(F) → j^{∗}(F) induced by F → j^{∗}(F) : s ↦ (t^{0})^{r+1}∇_{∂/∂t^{0}}(s) |_{D}. For r ≥ 1 it depends on the choice of local coordinates, and is multiplied by an invertible local function on D when this choice is changed. Hence its spectrum is well defined globally on D for r = 0, and the simplicity of the spectrum makes sense for any r.

v) Tameness and resonance. Two general position conditions are important in the study of meromorphic singularities of order ≤ r + 1.

If r ≥ 1 (irregular case), the singularity is called tame, if the spectrum of its principal part at any point of D is simple.

If r = 0 (regular case), the singularity is called non–resonant, if it is tame and moreover, the difference of any two eigenvalues never takes an integer value on D.

2.1.1. Example: the structure connections of Frobenius manifolds. As above, we will assume that T_{M}^{f} is trivial, and its fibers are identified with the space T of global flat vector fields.

Put N = M × P_{λ}^{1}, F = O_{N} ⊗ T. We can apply the previous considerations to ˆ∇ and ˇ∇.

Analysis of ˆ∇. Clearly, ˆ∇ has singularity of order 1 at λ = 0 (i. e. on D_{0} = M × {0}) and of order 2 at λ = ∞ (i. e. on D_{∞} = M × {∞}): cf. (1.42) and (1.43). Restricting ˆ∇ to {y}×P_{λ}^{1} for various y ∈ M we get a family of meromorphic connections on P_{λ}^{1} parametrized by M.

The residual connection is defined on D_{0} = M and it coincides with the Levi–

Civita connection of g. The principal part of order 1 on D_{0} is V + \frac{D}{2} Id. The eigenvalues of this operator do not depend on y ∈ D_{0}: in 1.5 they were denoted (d_{a}). In the case of quantum cohomology the principal part is always resonant.

The principal part of order 2 on D_{∞} = M is (proportional to) U (cf. (1.43), use the local equation μ = λ^{−1} = 0 for D_{∞}.) Its eigenvalues now depend on y ∈ M : they are just the canonical coordinates u^{i}(y). We will call the point y tame if u^{i}(y) ≠ u^{j}(y) for i ≠ j. We will call M tame, if all its points are tame. Every M contains the maximum tame subset which is open and dense.

Analysis of ˇ∇. According to (1.40), (1.41), ˇ∇ has singularities of order 1 at the divisors λ = u^{i} and λ = ∞. These divisors do not intersect pairwise iff M is tame.

The principal part of order 1 at λ = u^{i} is −(V + \frac{1}{2} Id) · (e_{i}∘).

The residual connection of ˇ∇ on λ = ∞ is again the Levi–Civita connection ∇_{0} of g. In fact, using (1.48) we find

ˇ∇ = dλ ˇ∇_{∂/∂λ} + ∑_{i} du^{i} ˇ∇_{e_{i}} =

= dλ ˇ∇_{∂/∂λ} + ∑_{i} du^{i}[∇_{0,e_{i}} − (V + \frac{1}{2} Id) (U − λ)^{−1}(e_{i}∘)].

Replacing λ by the local parameter μ = λ^{−1} at infinity, we have

ˇ∇ = dμ ˇ∇_{∂/∂μ} + ∑_{i} du^{i}[∇_{0,e_{i}} − μ (V + \frac{1}{2} Id) (μ U − Id)^{−1}(e_{i}∘)]

so that the expression (2.2) (with (μ, u^{1}, …, u^{m}) in lieu of (t^{0}, t^{1}, …, t^{n})) becomes ∑_{i} du^{i}∇_{0,e_{i}} = ∇_{0}.

2.2. Versal deformation. We will now review the basic results on the deformation of meromorphic connections on P_{λ}^{1}, restricting ourselves to the case of singularities of order ≤ 2. This suffices for applications to both structure connections, on the other hand, this is precisely the case treated in full detail by B. Malgrange in [Mal4], Theorem 3.1. It says that the positions of finite poles and the spectra of the principal parts of order 2 form coordinates on the coarse moduli space with tame singularities. To be more precise, one has to rigidify the data slightly.

Let ∇^{0} be a meromorphic connection on a locally free sheaf F^{0} on P_{λ}^{1} of rank p, with m + 1 ≥ 2 tame singularities (including λ = ∞) of order ≤ 2. Call the rigidity for ∇^{0} the following data:

a). A numbering of singular points: a_{0}^{1}, …, a_{0}^{m}, a^{m+1} = ∞.

b). The subset I ⊂ {1, …, m + 1} such that a_{0}^{j} is of order 2 exactly when j ∈ I.

c). For each j ∈ I, a numbering (b_{0}^{j1}, …, b_{0}^{jp}) of the eigenvalues of the principal part at a_{0}^{j}.

Construct the space B = B(m, p, S) as the universal covering of

(C^{m} \ diagonals) × ∏_{j∈I}(C^{p} \ diagonals)

with the base point (a_{0}^{i}; b_{0}^{jk}), let b_{0} ∈ B be its lift. We denote by a^{i}, b^{jk} the coordinate functions lifted to B. Let i : P_{λ}^{1} → B×P_{λ}^{1} be the embedding λ ↦ (b_{0}, λ), and D_{j} the divisor λ = a^{j} in B × P_{λ}^{1}.

2.2.1. Theorem ([Mal4], Th. 3.1). For a given (∇^{0}, F^{0}) with rigidity, there exists a locally free sheaf F of rank p on P_{λ}^{1} × B, a flat meromorphic connection ∇ on it, and an isomorphism i^{0} : i^{∗}(F, ∇) → (F^{0}, ∇^{0}) with the following properties:

D_{j}, j = 1, …, m + 1, are all the poles of ∇, of order 1 (resp. 2) if j ≠ I (resp. j ∈ I.) If j ∈ I, then (b^{j1}, …, b^{jp}) (as functions on D_{j}) form the spectrum of the principal part of order 2 of ∇ at D_{j}.

It follows that the restrictions of ∇ to the fibers {b} × P_{λ}^{1} are endowed with the induced rigidity, and i^{0} is compatible with it.

The data (F, ∇, i^{0}) are unique up to unique isomorphism.

2.2.2. Comments on the proof. a). The case when all singularities are of order 1 is easier. It is treated separately in [Mal3], Th. 2.1. Since the second structure connection satisfies this condition, we sketch Malgrange’s argument in this case.

Choose base points a ∈ U := P_{λ}^{1} \ ∪_{j=1}^{m+1}{a_{0}^{j}} and (b_{0}, a) ∈ B × P_{λ}^{1}. Notice that (b_{0}, a) belongs to V := B × P_{λ}^{1} \ ∪_{j=1}^{m}D_{j}.

The restriction of (F^{0}, ∇^{0}) to U is determined uniquely up to unique isomorphism by the monodromy action of π_{1}(U, a) on the space F, the geometric fiber F^{0}(a) at a, which can be arbitrary. Similarly, there is a bijection between flat connections (F, ∇) on V with fixed identification F^{0}(a) → F(a) = F and actions of π_{1}(V, (a, b)) on F. Hence to construct an extension (F, ∇) to V together with an isomorphism of its restriction to U with (F^{0}, ∇^{0}), it suffices to check that i induces an isomorphism π_{1}(U, a) → π_{1}(V, (a, b)), which follows from the homotopy exact sequence and the fact that B is contractible.

This argument explains the term “isomonodromic deformation.”

Next, we must extend (F, ∇) to B × P_{λ}^{1}. It suffices to do this separately in a tubular neighborhood of each D_{j} disjoint from other D_{k}. The coordinate change λ ↦ λ − a^{j} (or λ ↦ λ^{−1}) allows us to assume that the equation of D_{j} is λ = 0. Take a neighborhood W of 0 in which F^{0} can be trivialized, describe ∇^{0} by its connection matrix, lift (F^{0}, ∇^{0}) to B × W and restrict to a tubular neighborhood of D_{j}. On the complement to D_{j}, this lifting can be canonically identified with (F, ∇) through their horizontal sections. Clearly, it is of order ≤ 1 at D_{j}.

It remains to establish that any two extensions are canonically isomorphic. Outside singularities, an isomorphism exists and is unique. An additional argument which we omit shows that it extends holomorphically to B × P_{λ}^{1}.

b). When ∇ admits singularity of order 2, this argument must be completed. The extension of (F^{0}, ∇^{0}) first to V and then to the singular divisors of order ≤ 1 can be done exactly as before. But both the existence and the uniqueness of the extension to the irregular singularities requires an additional local analysis in order to show that the simple spectrum of the principal polar part determines the singularity. When formulated in terms of the asymptotic behaviour of horizontal sections, this analysis introduces the Stokes data as a version of irregular monodromy, which also proves to be deformation invariant.

2.3. The theta divisor and Schlesinger’s equations. In this subsection we will assume that F^{0} = T ⊗ O_{P_{λ}^{1}} where T is a finite dimensional vector space which can be identified with the space of global sections of F^{0}. This is the case of the two structure connections, when the local system T_{M}^{f} is trivial.

Then there exists a divisor Θ, eventually empty, such that the restriction of F to all fibers {b} × P_{λ}^{1}, b ∉ Θ, is free. This can be proved using the fact that a locally free sheaf E on P^{1} is free iff H^{0}(P^{1}, E(−1)) = H^{1}(P^{1}, E(−1)) = 0, and that the cohomology of fibers is semi–continuous. For an analytic treatment, see [Mal4], sec. 4 and 5.

Moreover, assume that λ = ∞ is a singularity of order 1 (to achieve this for the first structure connection, we must replace λ by λ^{−1}.) Then we can identify the inverse image of F on B \Θ×P_{λ}^{1} with T ⊗O_{B\Θ×P_{λ}^{1}} compatibly with the respective trivialization of F^{0}. To this end trivialize F along λ = ∞ using the residual connection (see 2.1 iii))and then take the constant extension of each residually horizontal section along P_{λ}^{1}. (If there are no poles of order 1, one can extend this argument using a different version of the residual connection, see [Mal4], p.430, Remarque 1.4.)

Using this trivialization, we can define a meromorphic integrable connection ∂ on F with the space of horizontal sections T on B \ Θ × P_{λ}^{1}. As sections of F, they develop a singularity at Θ. Therefore, the respective connection form ∇ − ∂ is a meromorphic matrix one–form with eventual pole at Θ.

The following classical result clarifies the structure of this form in the case when all poles of ∇ are of order 1.

2.3.1. Theorem. a). Let (a^{1}, …, a^{m}) be the functions on B describing the λ–coordinates of finite poles of ∇ (with given rigidity.) Then

∇ = ∂ + ∑_{i=1}^{m} A_{i}(a^{1}, …, a^{m}) \frac{d(λ − a^{i})}{λ − a^{i}}  (2.4)

where A_{i} are meromorphic functions B → End (T) which can be considered as multivalued meromorphic functions of a_{i}.

b). The connection (2.4) is flat iff A_{i} satisfy the Schlesinger equations

∀j,  dA_{j} = ∑_{i≠j}[A_{i}, A_{j}] \frac{d(a^{i} − a^{j})}{a^{i} − a^{j}}.  (2.5)

c). Fix a tame point a_{0} = (a_{0}^{1}, …, a_{0}^{m}). Then arbitrary initial conditions A_{i}^{0} = A_{i}(a_{0}) define a solution of (2.5) holomorphic on B \ Θ, with eventual pole at Θ of order 1.

d). For any such solution ∇ of (2.5), define the meromorphic 1-form on B:

ω_{∇} := ∑_{i<j} Tr (A_{i}A_{j}) \frac{d(a^{i} − a^{j})}{a^{i} − a^{j}}.  (2.6)

This form is closed, and for any local equation t = 0 of Θ the form ω_{∇} − \frac{dt}{t} is locally holomorphic.

2.3.2. Corollary. For any solution ∇ of (2.5), there exists a holomorphic function τ_{∇} on B such that ω_{∇} = d log τ_{∇}. It is defined uniquely up to a multiplication by a constant.

In fact, B is simply connected.

For a proof of Theorem 2.3.1, we refer to [Mal4]: a), b), and c) are proved on pp. 406–410, d) on pp. 420–425.

2.4. Special solutions. Slightly generalizing (2.5), we will call a solution to Schlesinger’s equations any data (M, (u^{i}), T, (A_{i})) where M is a complex manifold of dimension m ≥ 2; (u^{1}, …, u^{m}) a system of holomorphic functions on M such that du^{i} freely generate Ω_{M}^{1} and for any i ≠ j, x ∈ M, we have u^{i}(x) ≠ u^{j}(x); T a finite dimensional complex vector space; A_{j} : M → End T, j = 1, …, m, a family of holomorphic matrix functions such that

∀j :  dA_{j} = ∑_{i: i≠j}[A_{i}, A_{j}] \frac{d(u^{i} − u^{j})}{u^{i} − u^{j}}.  (2.7)

Let such a solution be given. Summing (3.1) over all j, we find d(∑_{j} A_{j}) = 0. Hence ∑_{j} A_{j} is a constant matrix function; denote its value by W.

2.4.1. Definition. A solution to Schlesinger’s equations as above is called special, if dim T = m = dim M; T is endowed with a complex nondegenerate quadratic

form g; W = −V − \frac{1}{2} Id, where V ∈ End T is a skew symmetric operator with respect to g, and finally

∀j :  A_{j} = −(V + \frac{1}{2} Id)P_{j}  (2.8)

where P_{j} : M → End T is a family of holomorphic matrix functions whose values at any point of M constitute a complete system of orthogonal projectors of rank one with respect to g:

P_{i}P_{k} = δ_{ik}P_{i}, ∑_{i=1}^{m} P_{i} = Id_{T} , g(Im P_{i}, Im P_{j}) = 0  (2.9)

if i ≠ j. Moreover, we require that A_{j} do not vanish at any point of M.

2.4.2. Comment. We committed a slight abuse of language: the notion of special solution involves a choice of additional data, the metric g. However, when it is chosen, the rest of the data is defined unambiguously if it exists at all.

In fact, assume that A_{j} = WP_{j} as above do not vanish anywhere. Then they have constant rank one. Hence at any point of M we have

Ker A_{j} = Ker WP_{j} = Ker P_{j} = ⊕_{i: i≠j}Im P_{i},

so that

Im P_{i} = ∩_{j: j≠i} ⊕_{k: k≠j} Im P_{k} = ∩_{j: j≠i}Ker A_{j}.

This means that P_{j} can exist for given A_{j} only if the spaces T_{j} = ∩_{j: j≠i}Ker A_{j} are one–dimensional and pairwise orthogonal at any point of M.

Conversely, assume that this condition is satisfied. Define P_{j} as the orthogonal projector onto T_{j}. Then A_{i}P_{j} = 0 for i ≠ j because T_{j} = Im P_{j} ⊂ Ker A_{i}. Hence

A_{j} = A_{j}(∑_{i=1}^{m} P_{i}) = A_{j}P_{j} = (∑_{i=1}^{m} A_{i})P_{j} = WP_{j}.  (2.10)

Notice that all A_{j} are conjugate to diag (−\frac{1}{2}, 0, …, 0) and satisfy A_{j}^{2} + \frac{1}{2} A_{j} = 0.

These conditions, as well as ∑_{j} A_{j} = −(V+ \frac{1}{2} Id), are compatible with the equations (2.8) and so must be checked at one point only.

2.4.3. Strictly special solutions. A special solution to Schlesinger’s equations as above is called strictly special if the operators

A_{j}^{(t)} := A_{j} + tP_{j}

also satisfy Schlesinger’s equations for any t ∈ C.

2.4.4. Lemma. If W is invertible, then any special solution with given W is strictly special.

Proof. Inserting A_{j}^{(t)} into (2.7) one sees that the solution is strictly special iff

∀j : dP_{j} = ∑_{i: i≠j}(P_{i}WP_{j} − P_{j}WP_{i}) \frac{d(u^{i} − u^{j})}{u^{i} − u^{j}}.

On the other hand, replacing A_{k} by WP_{k} in (2.7), one sees that after left multiplication by W this becomes a consequence of (2.7).

2.5. From Frobenius manifolds to special solutions. Given a semisimple Frobenius manifold with flat identity and an Euler field E with d_{0} = 1, we can produce a special solution to the Schlesinger equations rephrasing the results of the previous two sections.

Namely, we first pass to a covering M of the subspace of tame points of the initial manifold such that T_{M}^{f} is trivial and a global splitting can be chosen, represented by the canonical coordinates (u^{i}). Then we put T = Γ(M, T_{M}^{f}) and A_{i} = the coefficients of the second structure connection written as in (2.4).

Since this connection is flat, (M, (u^{i}), T, (A_{i})) form a solution of (2.7).

Moreover, this solution is special. In fact, T comes equipped with the metric g. The operator A_{i} is the principal part of order 1 of ˇ∇ at λ = u^{i} which is of the form (2.8), with P_{j} = e_{j}∘.

Finally, this special solution comes with one more piece of data, the identity e ∈ T. We will axiomatize its properties in the following definition.

2.5.1. Definition. Consider a special solution to Schlesinger’s equations as in the Definition 2.4.1. A vector e ∈ T is called an identity of weight D for this solution, if

a). V(e) = (1 − \frac{D}{2} ) e.

b). e_{j} := P_{j}(e) do not vanish at any point of M.

For Frobenius manifolds with d_{0} = 1, a) is satisfied in view of (1.17) and (1.36).

Theorem 1.14 shows moreover that in this way we always obtain strictly special solutions, although the operator W need not be invertible. For example, for quantum cohomology of P^{r} (which is semisimple, cf. below) the spectrum of W is

\{a − \frac{r + 1}{2} | a = 0, …, r\}. It contains 0 if r is odd.

2.6. From special solutions to Frobenius manifolds. Let (M, (u^{i}), T, g, (A_{i})) be a strictly special solution, and e ∈ T an identity of weight D for it.

2.6.1. Theorem. These data come from the unique structure of semisimple split Frobenius manifold on M, with flat identity and Euler field, as it was described in 2.5.

Proof. Proceeding as in 2.5, but in the reverse direction, we are bound to make the following choices.

Put e_{j} = P_{j}(e) ⊂ O_{M} ⊗ T, j = 1, …, m. Identify O_{M} ⊗ T with T_{M} by setting e_{j} = ∂/∂u^{j}. Transfer the metric g from T to T_{M}. Because of 2.5.1 c), it will be flat, with T as the space of flat vector fields. Define the multiplication on T_{M} for which e_{i} ∘ e_{j} = δ_{ij}e_{j}. Put η_{i} := g(e_{i}, e_{i}).

Let T_{M}^{f} be the image of T under this identification. We will first check that it is an abelian Lie subalgebra of T_{M}. It will then follow that g is flat, so that we get a structure of semisimple pre–Frobenius manifold in the sense of the Definition 1.6.2.

Choose t ∈ C in such a way that

W^{(t)} := ∑_{j} A_{j}^{(t)} = W + t Id ∈ End T

is invertible. The section X = ∑_{j} f_{j}e_{j} of O_{M} ⊗ T lands in T_{M}^{f} iff

W^{(t)}X = ∑_{j} f_{j}W^{(t)}P_{j}(e) = (∑_{j} f_{j}A_{j}^{(t)}) (e) ∈ T.

Let ∇ be the connection on T_{M} for which T_{M}^{f} is horizontal. Applying it to (∑_{j} f_{j}A_{j}^{(t)}) (e) we see that the last condition is in turn equivalent to

∀k :  ∑_{j} \frac{∂f_{j}}{∂u^{k}} A_{j}^{(t)}(e) = − ∑_{j} f_{j} \frac{∂A_{j}^{(t)}}{∂u^{k}} (e).  (∗)

We can similarly rewrite the condition Y := ∑_{j} g_{j}e_{j} ∈ T_{M}^{f}.

Commutator of vector fields induces on O_{M} ⊗ T the bracket

[X, Y] = ∑_{j,k} (f_{j} \frac{∂g_{k}}{∂u^{j}} − g_{j} \frac{∂f_{k}}{∂u^{j}} ) e_{k}.

From (*) for Y and X we find:

∑_{j,k} f_{j} \frac{∂g_{k}}{∂u^{j}} A_{k}^{(t)}(e) = − ∑_{j} f_{j} ∑_{k} g_{k} \frac{∂A_{k}^{(t)}}{∂u^{j}} (e),

∑_{j,k} g_{j} \frac{∂f_{k}}{∂u^{j}} A_{k}^{(t)}(e) = − ∑_{j} g_{j} ∑_{k} f_{k} \frac{∂A_{k}^{(t)}}{∂u^{j}} (e).

The terms j = k in the right hand sides are the same. For j ≠ k, using the strict speciality of our solution, we find

f_{j}g_{k} \frac{∂A_{k}^{(t)}}{∂u^{j}} = f_{j}g_{k} \frac{[A_{j}^{(t)}, A_{k}^{(t)}]}{u^{j} − u^{k}}

so that the (j, k)–term of the first identity cancels with the (k, j)–term of the second one.

To establish that this structure is Frobenius, it suffices to prove that e_{i}η_{j} = e_{j}η_{i} for all i, j: see Theorem 1.7.

We have η_{j} = g(e, e_{j}). Therefore

g(e, A_{j}(e)) = −g(e, (V + \frac{1}{2} Id) P_{j}e) = g(Ve, e_{j}) − \frac{1}{2} g(e, e_{j}) = \frac{1 − D}{2} η_{j} (2.11)

since V is skewsymmetric, and e is an eigenvector of V. Furthermore, let ∇ be the Levi–Civita connection of the flat metric g. Then derivating (2.11) we find for every i, j:

\frac{1 − D}{2} \frac{∂}{∂u^{i}} η_{j} = g(∇_{e_{i}}(e), A_{j}(e)) + g(e, ∇_{e_{i}}(A_{j}(e))) =

= g(e, \frac{∂A_{j}}{∂u^{i}} (e)),  (2.12)

because e ∈ T so that ∇(e) = 0. If i ≠ j, we find from (2.5)

\frac{∂A_{j}}{∂u^{i}} = \frac{[A_{i}, A_{j}]}{u^{i} − u^{j}} = \frac{∂A_{i}}{∂u^{j}} .  (2.13)

This shows that if D ≠ 1, e_{i}η_{j} = e_{j}η_{i}.

To see that D = 1 is not exceptional, one can replace in this argument A_{j} by

A_{j}^{(t)} for any t ≠ 0, so that \frac{1 − D}{2} in (2.11) will become \frac{1 − D}{2} + t.

It remains to check that E = ∑_{i} u^{i}e_{i} is the Euler field. According to the Theorem 1.9, we must prove that Eη_{j} = (D − 2)η_{j} for all j. Insert (2.13) into (2.12) and sum over i ≠ j. We obtain:

\frac{1 − D}{2} Eη_{j} = \frac{1 − D}{2} ∑_{i: i≠j} u^{i} \frac{∂η_{j}}{∂u^{i}} + \frac{1 − D}{2} u^{j} \frac{∂η_{j}}{∂u^{j}} =

= ∑_{i: i≠j} g (e, u^{i} \frac{[A_{i}, A_{j}]}{u^{i} − u^{j}} (e)) + u^{j}g(e, \frac{∂A_{j}}{∂u^{j}} (e)).  (2.14)

From (2.8) it follows that

\frac{∂A_{j}}{∂u^{j}} = − ∑_{i: i≠j} \frac{[A_{i}, A_{j}]}{u^{i} − u^{j}}.  (2.15)

On the other hand,

u^{i} \frac{[A_{i}, A_{j}]}{u^{i} − u^{j}} = [A_{i}, A_{j}] + u^{j} \frac{[A_{i}, A_{j}]}{u^{i} − u^{j}}.  (2.16)

Inserting (2.15) and (2.16) into (2.14), we find

\frac{1 − D}{2} Eη_{j} = ∑_{i: i≠j} g(e, [A_{i}, A_{j}] (e)) + u^{j} ∑_{i: i≠j} g (e, \frac{[A_{i}, A_{j}]}{u^{i} − u^{j}} (e)) +

+u^{j}g(e, \frac{∂A_{j}}{∂u^{j}} (e)) = g(e, [ ∑_{i: i≠j} A_{i}, A_{j}] (e)) =

= −g(e, [V + \frac{1}{2} Id, (V + \frac{1}{2} Id)P_{j}] (e)).  (2.17)

Using the skew symmetry of V, we see that the last expression in (2.17) equals

\frac{1 − D}{2} (D − 2)η_{j}. Hence Eη_{j} = (D − 2)η_{j} if D ≠ 1.

Again, replacing in this argument A_{j} by A_{j}^{(t)} we see that the restriction D ≠ 1 is irrelevant.

2.7. Special initial conditions. Theorem 2.3.1 shows that arbitrary initial conditions for the Schlesinger equations determine a global meromorphic solution on the universal covering B(m) of C^{m} \ {diagonals}, m ≥ 2.

Fix a base point b_{0} ∈ B(m). Studying the special solutions, we may and will identify T with the tangent space at b_{0} thus eliminating the gauge freedom. This tangent space is already coordinatized: we have e_{i} and e.

We will call a family of matrices A_{1}^{0}, …, A_{m}^{0} ∈ End T special initial conditions if we can find a diagonal metric g and a skew symmetric operator V such that A_{j}^{0} = −(V + \frac{1}{2} Id) P_{j} where P_{j} is the projector onto C e_{j}.

We will describe explicitly the space I(m) of the special initial conditions.

2.7.1. Notation. Let R be any equivalence relation on {1, …, m}, |R| the number of its classes. Put F (m) = (End C^{m})^{m}, Furthermore, denote F_{R}(m) the subset of families (A_{1}, …, A_{m}) in F (m) such that R coincides with the minimal equivalence relation for which iRj if Tr A_{i}A_{j} ≠ 0, and put I_{R}(m) = F_{R}(m) ∩ I(m).

2.7.2. Construction. Denote by \overline{I}(m) ⊂ C^{m} × C^{m(m−1)/2} the locally closed subset defined by the equations:

∑_{i=1}^{m} η_{i} = 0, η_{i} ≠ 0 for all i;  (2.18)

v_{ij}η_{j} = −v_{ji}η_{i} for all i, j;  (2.19)
∑_{i=1}^{m} v_{ij} := 1 − \frac{D}{2} does not depend on j.  (2.20)

Each point of \overline{I}(m) determines the diagonal metric g(e_{i}, e_{i}) = η_{i} and the operator V : e_{i} ↦ ∑_{i} v_{ij}e_{j} which is skew symmetric with respect to g and for which e is an

eigenvector. Setting A_{i} = −(V + \frac{1}{2} Id) P_{i} we get a point in I(m).

This amounts to forgetting (η_{i}) which furnishes the surjective map \overline{I}(m) → I(m) because

A_{i}(e_{j}) = 0 for i ≠ j, A_{i}(e_{i}) = −\frac{1}{2} e_{i} − ∑_{j=1}^{m} v_{ij}e_{j}.

2.7.3. Theorem. a). The space \overline{I}(m) can be realized as a Zariski open dense subset in C^{m+(m−1)(m−2)/2}.

b). Inverse image in \overline{I}(m) of any point in I_{R}(m) is a manifold of dimension 1 for |R| = 1, |R| − 1 for |R| ≥ 2.

Proof. Fixing η_{i}, we can solve (2.19) and (2.20) explicitly. Put w_{ij} = v_{ij}η_{j} so that w_{ij} = −w_{ji} and (2.20) becomes

∀ j :  ∑_{i=1}^{m} w_{ij} = η_{j}(1 − \frac{D}{2} ).  (2.21)

If we choose arbitrarily the values (w_{ij}) for all 1 ≤ i < j ≤ m − 1, we can find w_{mj} from the first m − 1 equations (2.21), and then the last equations will hold automatically:

w_{mk} = η_{k}(1 − \frac{D}{2} ) − ∑_{i=1}^{m−1} w_{ik},

∑_{i=1}^{m} w_{im} = − ∑_{k=1}^{m} w_{mk} = − ∑_{k=1}^{m−1} η_{k} (1 − \frac{D}{2} ) + ∑_{i,k=1}^{m−1} w_{ik} = η_{m} (1 − \frac{D}{2} )

because of (2.18).

It remains to determine the fiber of the projection onto I(m).

We have for i ≠ j: Tr A_{i}Aj = v_{ij}v_{ji}. Hence in the generic case when all these traces do not vanish, we can reconstruct η_{i} compatible with given v_{ij} from (2.19) uniquely up to a common factor. Generally, for i, j in the same R–equivalence class, (3.12) allows us to determine the value η_{i}/η_{j} so that we have |R| overall arbitrary factors constrained by (3.11).

2.7.4. Question. If we choose a special initial condition for the Schlesinger equation, does the solution remain special at every point?

Generically, the answer is positive. If this is the case, we obtain the action of the braid group Bd_{m} as the group of deck transformations on the space I(m).

2.8. Analytic continuation of the potential. The picture described in this section gives a good grip on the analytic continuation of a germ of semisimple Frobenius manifold (M_{0}, m_{0}) in terms of its canonical coordinates. Namely, construct the universal covering M of the subset of the tame points of M_{0}, then fix at the point b_{0} = (u^{i}(m_{0})) ∈ B(m) the initial conditions of M at m_{0}. This provides an open embedding (M, m_{0}) ⊂ (B(m), b_{0}). Loosely speaking, we find in this way a maximal tame analytic continuation of the initial germ.

Now construct some global flat coordinates (x^{a}) on B(m) corresponding to a given Frobenius structure. They map B(m) to a subdomain in C^{m}. This is the natural domain of the analytic continuation of the potential Φ of this Frobenius structure, which is the most important object for Quantum Cohomology. Unfortunately, its properties are not clear from this description.

§3. Quantum cohomology of projective spaces

In this section we will apply the developed formalism to the study of the quantum cohomology of projective spaces P^{r}, r ≥ 2. Our main goal is the calculation of the initial conditions of the relevant solutions to the Schlesinger’s equations.

3.1. Notation. We start with introducing the basic notation. Put H = H^{∗}(P^{r}, C) = ∑_{a=0}^{r} CΔ_{a}, Δ_{a} = the dual class of P^{r−a} ⊂ P^{r}. Denote the dual coordinates on H by x_{0}, …, x_{r} (lowering indices for visual convenience), ∂_{a} = ∂/∂x_{a}. The Poincaré form is (g_{ab}) = (g^{ab}) = (δ_{a+b,r}). The classical (cubic) part of the Frobenius potential is

Φ_{cl}(x) := \frac{1}{6} ∑_{a_{1}+a_{2}+a_{3}=r} x_{a_{1}}x_{a_{2}}x_{a_{3}}.  (3.1)

The remaining part of the potential is the sum of physicists’ instanton corrections to the self–intersection form:

Φ_{inst}(x) := ∑_{d=1}^{∞} Φ_{d}(x_{2}, …, x_{r})e^{dx_{1}},  (3.2)

where we will write Φ_{d} as

Φ_{d}(x_{2}, …, x_{r}) = ∑_{n=2}^{∞} ∑_{{a_{1}+···+an= \atop r(d+1)+d−3+n}} I(d; a_{1}, …, a_{n}) \frac{x_{a_{1}} … x_{a_{n}}}{n!} .  (3.3)

This means that if we assign the weight a − 1 to x_{a}, a = 2, …, n, Φ_{d} becomes the weighted homogeneous polynomial of weight (r +1)d+r −3. Moreover, if we assign to e^{dx_{1}} the weight −(r + 1), Φ_{cl} and Φ become weighted homogeneous formal series of weight r − 3. (Notice that e in the expressions e^{dx_{1}} and alike is 2, 71828 …, whereas in other contexts e means the identity vector field. This cannot lead to confusion.)

The starting point of our study in this section will be the following result.

3.2. Theorem. a). For each r ≥ 2, there exists a unique formal solution of the Associativity Equations (1.6) of the form

Φ(x) = Φ_{cl}(x) + Φ_{inst}(x)  (3.4)

for which I(1; r, r) = 1.

b). This solution has a non–empty convergence domain in H on which it defines the structure of semisimple Frobenius manifold H_{quant}(P^{r}) with flat identity e = ∂_{0} and Euler field

E = ∑_{a=0}^{r}(1 − a)x^{a}∂_{a} + (r + 1)∂_{1}  (3.5)

with d_{0} = 1, D = 2 − r.

c). The coefficient I(d; a_{1}, …, a_{n}) is the number of rational curves of degree d in P^{r} intersecting n projective subspaces of codimensions a_{1}, … a_{n} ≥ 2 in general position.

Uniqueness of the formal solution can be established by showing that the Associativity Equations imply recursive relations for the coefficients of Φ which allow one to express all of them through I(1; r, r). This is an elementary exercise for r = 2. A more general result (stated in the language of Gromov–Witten invariants but of essentially combinatorial nature) is proved in [KM], Theorem 3.1, and applied to the projective spaces in [KM], Claim 5.2.2.

Existence is a subtler fact. The algebraic geometric (or symplectic) theory of the Gromov–Witten invariants provides the numbers I(d; a_{1}, … a_{n}) satisfying the necessary relations, together with their numerical interpretation: see [KM], [BM]. Another approach consists in calculating ad hoc the “special initial conditions” for the semisimple Frobenius manifold H_{quant}(P^{r}) in the sense of the previous section and identifying the appropriate special solution to the Schlesinger equations with this manifold. For r = 2, direct estimates of the coefficients showing convergence can be found in [D], p. 185. Probably, they can be generalized to all r.

Our approach in this section consists in taking Theorem 3.2 for granted and investigating the passage to the Darboux–Egoroff picture as a concrete illustration of the general theory. The net outcome are formulas (3.18) and (3.19) for the special initial conditions.

Conversely, starting with them, we can construct the Frobenius structure on the space B(r + 1) as was explained in 2.8 above. Expressing the E–homogeneous flat coordinates (x_{0}, …, x_{r}) on this space satisfying (3.17) in terms of the canonical coordinates and then calculating the multiplication table of the flat vector fields, we can reconstruct the potential which now will be a germ of holomorphic function of (x_{a}). Because of the unicity, it must have the Taylor series (3.4). So the Theorem 3.2 a),b) can be proved essentially by reading this section in the reverse order. Of course, the last statement is of different nature.

3.3. Tensor of the third derivatives. Most of our calculations in (T , ∘) will be restricted to the first infinitesimal neighborhood of the plane x_{2} = ⋯ = x_{r} = 0 in H. This just suffices for the calculation of the Schlesinger initial conditions. We denote by J the ideal (x_{2}, …, x_{r}).

Multiplication by the identity e = ∂_{0} is described by the components Φ_{0a}^{b} = δ_{ab} of the structure tensor. Of the remaining components, we will need only Φ_{1a}^{b} which allow us to calculate multiplication by ∂_{1}, and proceed inductively. This is where the Associativity Equations are implicitly used.

Obviously, Φ_{10}^{b} = δ_{1b}.

3.3.1. Claim. We have

for 1 ≤ a ≤ r − 1 :  Φ_{1a}^{b} = δ_{a+1,b} + x_{r+1−a+b}e^{x_{1}} + O(J^{2}),  (3.6)
Φ_{1r}^{b} = δ_{b0}e^{x_{1}} + x_{b+1}e^{x_{1}} + O(J^{2}).  (3.7)

(Here and below we agree that x_{c} = 0 for c > r.

Proof. The term δ_{a+1,b} in (3.6) comes from Φ_{cl}. The remaining terms are provided by the summands of total degree ≤ 3 in x_{2}, …, x_{r} in

∂_{1}Φ_{inst} = ∑_{d≥1} de^{dx_{1}} (∑ I(d; a_{1}, a_{2})\frac{x_{a_{1}}x_{a_{2}}}{2} + ∑ I(d; a_{1}, a_{2}, a_{3})\frac{x_{a_{1}}x_{a_{2}}x_{a_{3}}}{6} )+O(J^{4}).

For n = 2, the grading condition means that d = 1, a_{1} = a_{2} = r. For n = 3, it means that d = 1, a_{1} + a_{2} + a_{3} = 2r + 1. We know that I(1; r, r) = 1. Similarly, I(1; a_{1}, a_{2}, a_{3}) = 1 in this range. This can be deduced formally from the Associativity Equations. A nice exercise is to check that this agrees also with the geometric description (for instance, only one line intersects two given generic lines and passes through a given point in the three space.) So finally

∂_{1}Φ_{inst} = (\frac{x_{r}^{2}}{2} + \frac{1}{6} ∑_{a_{1}+a_{2}+a_{3}=2r+1} x_{a_{1}}x_{a_{2}}x_{a_{3}}) e^{x_{1}} + O(J^{4}).

The term δ_{b0}e^{x_{1}} in (3.7) comes from \frac{x_{r}^{2}}{2} . Furthermore,

Φ_{inst;1ab} = x_{2r+1−a−b}e^{x_{1}} + O(J^{2})

and

Φ_{inst;1a}^{b} = Φ_{inst;1,a,r−b} = x_{r+1−a+b}e^{x_{1}} + O(J^{2}).

3.4. Multiplication table. The main formula of this subsection is

∂_{1}^{∘(r+1)} = e^{x_{1}} (∂_{0} + ∑_{b=1}^{r−1}(b + 1)x_{b+1}∂_{b}) + O(J^{2}).  (3.8)

We will prove it by consecutively calculating the powers ∂_{1}^{∘a}. The intermediate results will also be used later. (Notice that O(J^{2}) in (3.8) now means O(∑_{i} J^{2}∂_{i}).)

First, we find from (3.6) and (3.7) for 1 ≤ a ≤ r − 1:

∂_{1} ∘ ∂_{a} = ∑_{b=0}^{r} Φ_{1a}^{b}∂_{b} = ∂_{a+1} + e^{x_{1}} ∑_{b=0}^{a−1} x_{r+1−a+b}∂_{b} + O(J^{2}),  (3.9)

∂_{1} ∘ ∂_{r} = ∑_{b=0}^{r} Φ_{1r}^{b}∂_{b} = e^{x_{1}} (∂_{0} + ∑_{b=1}^{r−1} x_{b+1}∂_{b}) + O(J^{2}).  (3.10)

Then using (3.9) and induction, we obtain

for 1 ≤ a ≤ r :  ∂_{1}^{∘a} = ∂_{a} + e^{x_{1}} ∑_{b=0}^{a−2}(b + 1)x_{r+2−a+b}∂_{b} + O(J^{2}).  (3.11)

Multiplying this formula for a = r by ∂_{1} and using (3.10), we finally find (3.8). ¿From (3.11) it follows that ∂_{1}^{∘a} for 0 ≤ a ≤ r freely span the tangent sheaf.

3.5. Idempotents. Formula (3.8) allows us to calculate all e_{i} mod J^{2} thus demonstrating semisimplicity. Namely, denote by q the (r + 1)−th root of the right

hand side of (3.8) congruent to e^{\frac{x_{1}}{r+1}} mod J and put ζ = exp (\frac{2πi}{r + 1}). Then

e_{i} = \frac{1}{r + 1} ∑_{j=0}^{r} ζ^{−ij}(∂_{1} ∘ q^{−1})^{∘j}  (3.12)

satisfy

e_{i} ∘ e_{j} = δ_{ij}e_{i}, ∑_{i} e_{i} = ∂_{0}

for all i = 0, … r. A straightforward check shows this.

3.5.1. Proposition. We have

e_{i} = \frac{1}{r + 1} ∑_{j=0}^{r} ζ^{−ij}e^{−x_{1} \frac{j}{r+1}} (e^{x_{1}} ∑_{b=0}^{j−2} \frac{(b + 1 − j)(r + 1 − j)}{r + 1} x_{r+b+2−j}∂_{b} +

+ ∂_{j} − ∑_{b=j+1}^{r} \frac{(b + 1 − j)j}{r + 1} x_{b+1−j}∂_{b}) + O(J^{2}).  (3.13)

Proof. We have

q^{−1} = e^{− \frac{x_{1}}{r+1}} (∂_{0} − ∑_{b=1}^{r−1} \frac{b + 1}{r + 1}x_{b+1}∂_{b}) + O(J^{2}).

Together with (3.9) this gives

∂_{1} ∘ q^{−1} = e^{− \frac{x_{1}}{r+1}} (∂_{1} − ∑_{b=1}^{r−1} \frac{b + 1}{r + 1}x_{b+1}∂_{b+1}) + O(J^{2}).

Hence

(∂_{1} ∘ q^{−1})^{j} = e^{− \frac{jx_{1}}{r+1}} (∂_{1}^{∘j} − j∂_{1}^{∘(j−1)} ∘ ∑_{b=1}^{r−1} \frac{b + 1}{r + 1}x_{b+1}∂_{b+1}) + O(J^{2}).

Inserting this into (3.12) and using (3.9)–(3.11) once again, we finally obtain (3.13).

3.6. Metric coefficients in canonical coordinates. The metric potential η is simply x_{r} (see (1.11).) Hence we can easily calculate η_{i} = e_{i}x_{r}. The answer is

η_{i} = \frac{ζ^{i}}{r + 1}e^{−x_{1} \frac{r}{r+1}} − ∑_{b=2}^{r} \frac{ζ^{ib}}{(r + 1)^{2}} b(r + 1 − b) e^{−x_{1} \frac{r+1−b}{r+1}} x_{b} + O(J^{2}).  (3.14)

As an exercise, the reader can check that the same answer results from the (longer) calculation of η_{i} = g(e_{i}, e_{i}).

3.7. Derivatives of the metric coefficients. We now see that the chosen precision just suffices to calculate the restriction of η_{ij}, γ_{ij} and the matrix elements of A_{j} to the plane x_{2} = ⋯ = x_{r} = 0 any point of which can be taken as initial one.

3.7.1. Claim. We have

η_{ki} = e_{k}η_{i} = −2 \frac{ζ^{i−k}}{(ζ^{i−k} − 1)^{2}} \frac{e^{−x_{1}}}{(r + 1)^{2}} + O(J).  (3.15)

Notice that (3.15) is symmetric in i, k as it should be.

This is obtained by a straightforward calculation from (3.13) and (3.14). The numerical coefficient in (3.16) comes as a combination of ∑_{j=1}^{r} jζ^{j} and ∑_{j=1}^{r} j^{2}ζ^{j} which are then summed by standard tricks.

3.8. Canonical coordinates. We find u^{i} from the formula E ∘ e_{i} = u^{i}e_{i}. To calculate E ∘ e_{i}, use (3.5), (3.13) and (3.9)–(3.11). We omit the details. The result is:

3.8.1. Claim. We have

u^{i} = x_{0} + ζ^{i}(r + 1)e^{\frac{x_{1}}{r+1}} + ∑_{a=2}^{r} ζ^{ai}e^{\frac{ax_{1}}{r+1}} x_{a} + O(J^{2}).  (3.16)

The reader can check that e_{i}u^{j} = δ_{ij} + O(j).

3.9. Schlesinger’s initial conditions. Recall that the matrix residues A_{i} of Schlesinger’s equations for Frobenius manifolds are

A_{j}(e_{i}) = 0 for i ≠ j,
A_{j}(e_{j}) = −\frac{1}{2}e_{j} − \frac{1}{2} ∑_{k}(u^{k} − u^{j})\frac{η_{jk}}{η_{k}} e_{k}  (3.17)

(cf (1.46).) Substituting here (3.14), (3.15) and (3.16), we finally get the main result of this section.

3.9.1. Theorem. The point (x_{0}, x_{1}, 0, …, 0) has canonical coordinates u^{i} = x_{0} + ζ^{i}(r + 1)e^{\frac{x_{1}}{r+1}} .

The special initial conditions at this point (in the sense of 2.7) corresponding to H_{quant}(P^{r}) are given by

v_{jk} = −\frac{ζ^{j−k}}{1 − ζ^{j−k}}  (3.18)

and

η_{i} = \frac{ζ^{i}}{r + 1} e^{−x_{1} \frac{r}{r+1}} .  (3.19)

As an exercise, the reader can check that

− ∑_{k: k≠j} \frac{ζ^{j−k}}{1 − ζ^{j−k}} = 1 − \frac{D}{2} = \frac{r}{2}.

§4. Semisimple Frobenius supermanifolds

4.1. Supermanifolds and SUSY-structures. A (smooth, analytic, etc.) supermanifold of dimension (m|n) is a locally ringed space (M, O_{M}) with the following properties [Ma3]: (i) the structure sheaf O_{M} = O_{M,0}⊕O_{M,1} is a sheaf of ℤ_{2}- graded supercommutative rings; (ii) M_{red} = (M, O_{M,red} := O_{M}/[O_{M,1} + O_{M,1}^{2}]) is a (smooth, analytic, etc.) classical manifold of dimension m; (iii) O_{M} is locally isomorphic to the exterior algebra Λ(E) of a locally free O_{M,red}-module E of rank n. If ϕ : Λ(E) → O_{M} is any such local isomorphism, x̄^{1}, …, x̄^{m} are local coordinates on M_{red} and θ̄^{1}, …, θ̄^{n} are free local generators of E, then the set of m + n sections

x^{1} = ϕ(x̄^{1}), …, x^{m} = ϕ(x̄^{m}), θ^{1} = ϕ(θ̄^{1}), …, θ^{n} = ϕ(θ̄^{n})

of the structure sheaf O_{M} form a local coordinate system on M. Any local function f on M can be expressed as a polynomial in anticommuting odd coordinates θ^{α}, α = 1, …, n,

f(x, θ) = ∑_{k=0}^{n} ∑_{α_{1},... ,α_{k}=1}^{n} f_{α_{1}...α_{k}}(x)θ^{α_{1}} ⋯ θ^{α_{k}}

whose coefficients f_{α_{1}...α_{k}}(x) are classical (smooth, analytic, etc.) functions of the commuting variables x^{a}, a = 1, …, m.

When a need arises to use odd constants in the structure sheaf of a supermanifold M, one simply replaces M by its relative version, i.e. by a submersion of supermanifolds π : M → S whose typical fibre is M. Then (odd) constants are just (odd) elements of π^{−1}(O_{S}). The necessary changes are routine, see [Ma3] and [Ma4].

4.1.1. Definition. Let M be an (m|n)-dimensional supermanifold. A SUSYstructure on M is a rank 0|n locally split subsheaf T_{1} ⊂ T M such that the associated Frobenius form

\begin{aligned}Φ : & Λ^{2}T_{1} & ⟶ & T_{0} := T M/T_{1} \\ & X ⊗ Y & ⟶ & \frac{1}{2}[X, Y] mod T_{1} \end{aligned}

is surjective.

With any SUSY-structure on M there is canonically associated an extension

0 ⟶ T_{1} ⟶^{i} T M ⟶^{p} T_{0} ⟶ 0,  (4.1)

i.e. an element 𝔱 ∈ Ext_{O_{M}}^{1}(T_{0}, T_{1}) ≃ H^{1}(M, T_{1} ⊗ T_{0}^{∗}).

4.1.2. Examples. 1). A (1|1)-dimensional supermanifold with a SUSYstructure is called a SUSY_{1}-curve [Ma4]. 2). A SUSY-structure on a (3|2)-dimensional is equivalent to a simple conformal supergravity in 3 dimensions [Ma3]. 3). A SUSYstructure on a (4|4)-dimensional supermanifold with T_{1} being a direct sum of two integrable rank (0|2) distributions T_{l} and T_{r} is the same as a simple superconformal supergravity in 4 dimensions [Ma3].

4.2. Pre-Frobenius supermanifolds. Let S be a module over a supercommutative ring R. A left odd involution on S is by definition a map

\begin{aligned}Π_{l} : & S & ⟶ & S \\ & X & ⟶ & Π_{l}X \end{aligned}

such that Π_{l}^{2} = Id and Π_{l}(aX) = (−1)^{a}aΠ_{l}(X), Π_{l}(Xa) = Π_{l}(X)a for any X ∈ S, a ∈ R.

A right odd involution

\begin{aligned}Π_{r} : & S & ⟶ & S \\ & X & ⟶ & XΠ_{r} \end{aligned}

also satisfies, by definition, Π_{r}^{2} = Id but has different linearity properties: (aX)Π_{r} = a(XΠ_{r}), (Xa)Π_{r} = (−1)^{a}(XΠ_{r})a.

4.2.1. Definition. A pre-Frobenius structure on an (n|n)-dimensional supermanifold M is a quaternary (T_{1}, s, Π_{l}, Π_{r}) consisting of a SUSY-structure (4.1), an even isomorphism s : T M ⟶ T_{1} ⊕ T_{0} and a pair of left and right odd involutions

Π_{l,r} : T_{1} ⊕ T_{0} ⟶ T_{1} ⊕ T_{0}

such that

(i) Π_{l}(T_{0}) = (T_{0})Π_{r} = T_{1}, Π_{l}(T_{1}) = (T_{1})Π_{r} = T_{0};

(ii) a product defined by

X ∘ Y = \{ \begin{matrix}Φ(X, Y), & for any X ∈ T_{1}, Y ∈ T_{1}, \\ Π_{l}Φ(Π_{l}X, Y), & for any X ∈ T_{0}, Y ∈ T_{1}, \\ Φ(X, Y Π_{r})Π_{r}, & for any X ∈ T_{1}, Y ∈ T_{0}, \\ Φ(XΠ_{r}, Π_{l}Y), & for any X ∈ T_{0}, Y ∈ T_{0} \end{matrix}

makes T := T_{1} ⊕ T_{0} a sheaf of associative algebras;

(iii) s |_{T_{1}}= i and s|_{T_{0}} is a splitting of the extension (4.1), i.e. p ∘ s|_{T_{0}} = Id_{T_{0}}.

4.2.2. Lemma. There is an even morphism of O_{M}-modules, c : T ⊗ T → T , such that the product ∘ : T × T → T factors through the map

T × T ⟶ T ⊗ T ⟶^{c} T .

Proof. The product ∘ obviously satisfies (aX) ∘ Y = a(X ∘ Y), X ∘ (aY) = (Xa) ∘ Y and X ∘ (Y a) = (X ∘ Y)a.

4.3. Semisimple pre-Frobenius supermanifolds. Let M be a pre-Frobenius supermanifold.

4.3.1. Definition. A pre-Frobenius structure is called (split) almost semisimple if there exists a local (global) basis {eα̇}, α̇ = ˙1, …, ṅ, of T_{1} called canonical, such that Φ(eα̇, eβ̇) = δ_{α̇β̇}Π_{l}(eα̇) = δ_{α̇β̇}(eα̇)Π_{r}.

Since in the almost semisimple case Π_{l} completely determines Π_{r} and vice versa, we can and will omit the subscripts l, r. Note that the definition 4.3.1 makes sense, because the extra condition Φ(eα̇, eβ̇) = δ_{α̇β̇}Π(eα̇) is consistent with 4.2.1(ii) (and in fact implies the latter). Indeed, in the basis {eα̇, e_{α} := Π(eα̇)} of T_{1} ⊕ T_{0} the multiplication ∘ takes the form

eα̇ ∘ eβ̇ = δ_{α̇β̇}e_{α},
eα̇ ∘ e_{β} = e_{α} ∘ eβ̇ = δ_{αβ}eα̇,
e_{α} ∘ e_{β} = δ_{αβ}e_{α}.

This algebra structure is evidently associative. This table together with Lemma 4.2.2 immediately imply that

e := ∑_{α=1}^{n} e_{α}

is the identity, i.e. e ∘ X = X ∘ e = X for any X ∈ T . It also follows that

ε := ∑_{α̇=1}^{n} eα̇ = Πe

satisfies ε ∘ X = ΠX, X ∘ ε = XΠ for any X ∈ T . We call ε the Π-identity.

4.3.2. Odd multiplication. If M is an almost semisimple pre-Frobenius supermanifold, then the formulae

X • Y := X ∘ (ΠY) = \{ \begin{matrix}Φ(X, ΠY Π)Π, & for any X ∈ T_{1}, Y ∈ T_{1}, \\ Φ(XΠ, Y), & for any X ∈ T_{0}, Y ∈ T_{1}, \\ Φ(X, ΠY), & for any X ∈ T_{1}, Y ∈ T_{0}, \\ ΠΦ(ΠX, ΠY), & for any X ∈ T_{0}, Y ∈ T_{0} \end{matrix}

define an odd associative multiplication in T . Indeed, in the basis {eα̇, e_{α}} one has

eα̇ • eβ̇ = δ_{α̇β̇}eα̇,
eα̇ • e_{β} = e_{α} • eβ̇ = δ_{αβ}e_{α},
e_{α} • e_{β} = δ_{αβ}eα̇.

The roles of e and ε get interchanged: ε is the identity, that is ε • X = X • ε = X, while e is the Π-identity, that is e • X = ΠX, X • e = XΠ for any X ∈ T .

4.3.3. Definition. A pre-Frobenius structure on M is called (split) semisimple if

(i) it is (split) almost semisimple;

(ii) there is a local (global) coordinate system {u^{α}, θα̇} called canonical, such that the isomorphism s : T_{1} ⊕ T_{0} ⟶ T M is given by

s(eα̇) = ∂α̇ + θα̇∂_{α}, s(e_{α}) = ∂_{α},

where (e_{α}, eα̇) is the canonical basis, ∂α̇ = ∂/∂θα̇ and ∂_{α} = ∂/∂u^{α}.

(iii) there is an odd metric g on T M such that T_{1} ⊂ T M is isotropic and

g(∂_{α}, ∂_{β}) = −δ_{αβ}η_{β}, g(e_{α}, s(eβ̇)) = δ_{αβ̇}ηα̇,  (4.2)

where η_{α} = ∂_{α}Ψ, ηα̇ = (∂α̇ + θα̇∂_{α})Ψ and Ψ is an odd function. Such a metric is called an Egoroff metric.

Since the restriction of s : T_{1} ⊕ T_{0} ⟶ T M to T_{1} coincides with i and hence is rigidly fixed by the choice of a SUSY structure on M, we identify from now on X and s(X) for any X ∈ T_{1}. In particular, whenever the pre-Frobenius structure is semisimple, we always assume that eα̇ = ∂α̇ + θα̇∂_{α}.

Note that canonical coordinates are defined up to a transformation

θα̇ ⟶ θ̂α̇ = θα̇ + cα̇
u^{α} ⟶ û^{α} = u^{α} + c^{α} + θα̇cα̇

which satisfy ˆ∂_{α} = ∂_{α}, êα̇ = eα̇ and hence leave all the defining relations invariant.

4.3.4. Example. If M is a SUSY_{1}-curve, then any odd isomorphism T_{1} → T_{0} together with a global nowhere vanishing section of T_{1} (if any) and an odd function Ψ equips M with a semisimple pre-Frobenius structure.

4.3.5. Algebra structure in T M. The isomorphism s : T → T M translates the product ∘ from T to T M which we denote by the same symbol. The element ē:= s(e) = ∑_{α} ∂_{α} is the identity in (T M, ∘).

It is easy to check that in the basis {∂α̇, ∂_{α}} the induced multiplication ∘ takes the form

∂α̇ ∘ ∂β̇ = δ_{α̇β̇}∂_{α},
∂α̇ ∘ ∂_{β} = ∂_{α} ∘ ∂β̇ = δ_{αβ}∂α̇,
∂_{α} ∘ ∂_{β} = δ_{αβ}∂_{α},

implying that the element ε̄ = ∑_{α̇=1}^{n} ∂α̇ is the Π̄-identity in (T M, ∘), where the odd automorphism Π̄ : T M → T M is defined by

Π̄(∂α̇) = ∂_{α}, Π̄(∂_{α}) = ∂α̇.

4.3.6. Representation of the Neveu-Schwarz Lie superalgebra in T M. Let M be a semisimple pre-Frobenius supermanifold. Consider vector fields

E = ∑_{α=1}^{n}(u^{α}∂_{α} + \frac{1}{2}θα̇eα̇)

F = ∑_{α=1}^{n} u^{α}eα̇

on M. A direct calculation shows that the vector fields

𝔢_{a} := E^{∘(a+1)} = ∑_{α=1}^{n} [(u^{α})^{a+1}∂_{α} + \frac{a + 1}{2}(u^{α})^{a}θα̇eα̇] , a = 0, 1, 2, …

𝔣_{i+1/2} := Π^{i}F^{∘(i+1)} = ∑_{α=1}^{n}(u^{α})^{i+1}eα̇, i = 0, 1, 2, …

satisfy the following commutation relations

[𝔢_{a}, 𝔢_{b}] = (b − a)𝔢_{a+b},
[𝔢_{a}, 𝔣_{i}] = (i − \frac{a}{2})𝔣_{i+a},
[𝔣_{i}, 𝔣_{j}] = 2𝔢_{i+j},

where a, b = 0, 1, 2, … and i, j = \frac{1}{2}, \frac{3}{2}, \frac{5}{2}, . . ..

4.4. Flat connections on pre-Frobenius supermanifolds. On a supermanifold M with a SUSY-structure one may define the differential operator δ : O_{M} → T_{1}^{∗} as the composition

δ : O_{M} ⟶ Ω^{1}M ⟶^{i^{∗}} T_{1}^{∗}.

Let M be a semisimple pre-Frobenius supermanifold with the associated commutative diagram

0

↓

T_{0}^{∗}

p^{∗}↓
0 ⟶ O_{M} ⟶^{d} Ω^{1}M ⟶^{d} Ω^{2}M
s↑  ↓^{γ}
T_{1}^{∗} ⟶^{δ′} Λ^{2}T_{1}^{∗}
↑  ↓
0  0

and let Φ^{∗} : T_{0}^{∗} ⟶^{γ∘d∘p^{∗}} Λ^{2}T_{1}^{∗} be the (dual) Frobenius form of the SUSY-structure. Then one has

4.4.1. Proposition. There is a one-to-one correspondence between locally flat connections in a locally free sheaf F on M and covariant differentials

∇ : F ⟶ T_{1}^{∗} ⊗ F

such that

∇(fv) = δ(f)v + f∇(v), ∀f ∈ O_{M}, v ∈ F,

and the composition

C(∇^{2}) : F ⟶^{∇} T_{1}^{∗} ⊗ F ⟶^{∇} Λ^{2}T_{1}^{∗} ⊗ F ⟶ Λ^{2}T_{1}^{∗} ⊗ F/Φ^{∗}(T_{0}) ⊗ F

(which is O_{M}-linear) is zero, where the action of ∇ on T_{1}^{∗} ⊗ F is defined as follows

∇(t ⊗ v) = (−1)t̃γ ⊗ Id_{F}(t ⊗ ∇(v)) + δ′(t) ⊗ v, t ∈ T_{1}^{∗}, v ∈ F.

Proof. Given a linear connection D : F → Ω^{1}M ⊗ F, one defines ∇ as the composition

∇ : F ⟶^{D} Ω^{1}M ⊗ F ⟶^{i^{∗}⊗Id} T_{1}^{∗} ⊗ F.

Since the curvature tensor, F ∈ Ω^{2}M ⊗ F ⊗ F^{∗}, of D satisfies

F (X, Y)v = [D_{X}, D_{Y}]v − D_{[X,Y]}v

for any X, Y ∈ T M and v ∈ F, the composition ∇^{2} := ∇∘∇, viewed as a morphism Λ^{2}T_{1} ⊗ F → F, can be written explicitly as

\begin{aligned}∇^{2} : & Λ^{2}T_{1} ⊗ F & ⟶ & F \\ & X ⊗ Y ⊗ v & ⟶ & D_{2Φ(X,Y)}v + F (X, Y)v \end{aligned}

implying that C(∇^{2}) is essentially γ ⊗ Id_{F⊗F^{∗}}(F). Hence C(∇^{2}) is always O_{M}- linear and vanishes when D is flat.

In the other direction, let ∇ : F → T_{1}^{∗} ⊗ F be a covariant differential such that C(∇^{2}) = 0. Then ∇^{2} factors through the composition

∇^{2} : Λ^{2}T_{1} ⊗ F ⟶^{Φ⊗Id} T_{0} ⊗ F ⟶^{∇^{0}} F,

for some covariant differential operator ∇^{0} : F → T_{0}^{∗} ⊗ F. Define

D : F → Ω^{1}M ⊗ F ≃ T_{1}^{∗} ⊗ F ⊕ T_{0}^{∗} ⊗ F

as ∇⊕∇^{0}. A simple calculation in the canonical coordinates (which we omit) shows that D is flat. This completes the proof.

Note that in the presence of a SUSY-structure with invertible Frobenius form any covariant differential ∇ : F → T_{1}^{∗} ⊗ F can be canonically extended to a linear connection D : F → Ω^{1}M ⊗ F as ∇ ⊕ ∇^{0}, where

∇^{0} : F ⟶^{∇^{2}} Λ^{2}T_{1}^{∗} ⊗ F ⟶^{Φ^{∗}^{−1}} T_{0}^{∗} ⊗ F.

More generally, such an extension is possible whenever M comes equipped with a monomorphism Θ : T_{0} → Λ^{2}T_{1} satisfying Θ ∘ Φ = Id. For example, if M is almost semisimple pre-Frobenius, then

\begin{aligned}Θ : & T_{0} & ⟶ & Λ^{2}T_{1} \\ & e_{α} & ⟶ & eα̇ ⊗ eα̇ \end{aligned}

does have this property. When we call a covariant differential ∇ : F → T_{1}^{∗} ⊗ F a connection on F → M, we mean precisely this SUSY extension.

4.5. Semisimple Frobenius structures. Let (M, T_{1}, s, Π, g) be a semisimple pre-Frobenius supermanifold.

4.5.1. Definition. M is called semisimple Frobenius if g is flat.

4.5.2. Levi-Civita connection. If g is an odd metric on a supermanifold M, g_{AB} := (−1)Ãg(e_{A}, e_{B}) are the components of g in a basis e_{A} of T M, then the Christoffel symbols, ∇_{e_{A}}e_{B} = ∑_{C} Γ_{AB}^{C}e_{C}, of the associated Levi-Civita connection ∇ are given by

Γ_{AB}^{C} = \frac{1}{2} ∑_{D} [e_{A}g_{BD} + (−1)^{AB}e_{B}g_{AD} − (−1)^{D(A+B+1)+B}e_{D}g_{AB}

+ ∑_{M}(C_{AB}^{M}g_{MD} − (−1)^{BD+B+D}C_{AD}^{M}g_{MB} − (−1)^{A(B+D+1)+D}C_{BD}^{M}g_{MA})] g^{DC},

where C_{AB}^{M} are defined by

[e_{A}, e_{B}] = ∑_{M} C_{AB}^{M}e_{M}.

Consider now a special case when g is an Egoroff metric on a semisimple pre-Frobenius supermanifold M.

4.5.3. Proposition-definition (Darboux-Egoroff equations). The metric g is flat if and only if Ψ satisfies the equations

eμ̇γ_{α̇β̇} = γ_{μ̇α̇}γ_{μ̇β̇} for all μ̇ ≠ α̇ ≠ β̇ ≠ μ̇,  (4.3)
ēγ_{α̇β̇} = 0 for all α̇ ≠ β̇,  (4.4)

where

γ_{α̇β̇} = \frac{eα̇ηβ̇}{2\sqrt{ηα̇ηβ̇}} .

These equations are called Darboux-Egoroff equations.

A sketch of the proof. The only non-trivial commutator of basis vector fields (eα̇, ∂_{α}) is [eα̇, eβ̇] = 2δ_{α̇β̇}∂_{α} so that the only non-vanishing components of C_{AB}^{M} are C_{α̇β̇}^{γ} = 2δ_{α̇β̇}δ_{α̇γ}. Then, using the above formulae for Γ_{AB}^{C}, one obtains the following Cristoffel symbols of the Levi-Civita connection of g:

∇_{eμ̇}eα̇ = δ_{μ̇α̇}∂_{α} + \frac{eμ̇ηα̇}{2ηα̇} eα̇ − \frac{eα̇ημ̇}{2ημ̇} eμ̇,  (4.5)

∇_{eμ̇}∂_{α} = eμ̇ (\frac{η_{α}}{2ηα̇} ) eα̇ + \frac{e_{α}ημ̇}{2ημ̇} eμ̇ − δ_{μα} ∑_{β̇} eβ̇ (\frac{eβ̇ημ̇}{2ηβ̇} ) eβ̇

+ \frac{eμ̇ηα̇}{2ηα̇} ∂_{α} − δ_{μα} \frac{η_{α}}{ηα̇} ∂_{α} + δ_{μα} ∑_{β̇} \frac{eβ̇ηα̇}{2ηβ̇} ∂_{β}.  (4.6)

By Proposition 4.4.1, the connection ∇ is flat if and only if

[∇_{eμ̇}, ∇_{eν̇}] = 0 for all μ̇ ≠ ν̇.

A straightforward but very tedious calculation shows that the latter equations are equivalent to the Darboux-Egoroff equations (4.3) and (4.4).

4.6. Flat identity. We say that the ∘-identity e on a semisimple pre-Frobenius supermanifold M is flat if ∇ē = 0, where ∇ is the Levi-Civita covariant differential (4.5)-(4.6).

4.6.1. Proposition. The identity e is flat if and only if the potential Ψ satisfies the equations

ēηα̇ = 0,  (4.7)

or, equivalently,

∑_{α} η_{α} = const.  (4.8)

Proof. It follows from (4.6) that

∇_{eμ̇} ē = ∑_{β} ∇_{eμ̇}e_{β}

= ∑_{β̇} eμ̇ (\frac{η_{β}}{2ηβ̇} ) eβ̇ + ∑_{β} \frac{e_{β}ημ̇}{2ημ̇} eμ̇ − ∑_{β̇} eβ̇ (\frac{eβ̇ημ̇}{2ηβ̇} ) eβ̇

+ ∑_{β} \frac{eμ̇ηβ̇}{2ηβ̇} ∂_{β} − \frac{η_{μ}}{ημ̇} ∂_{μ} + ∑_{β} \frac{eβ̇ημ̇}{2ηβ̇} ∂_{β}

= \frac{∑_{β} e_{β}ημ̇}{2ημ̇} eμ̇,

where we used the fact that eμ̇ηα̇ + eα̇ημ̇ = 2δ_{μ̇α̇}η_{α}.

4.6.2. Proposition. Let M be a semisimple pre-Frobenius supermanifold. The Π̄-identity ε̄ is flat, i.e. ∇ε̄ = 0 for ∇ being the Levi-Civita covariant differential, if and only if the potential Ψ satisfies the equations

(θμ̇ − θα̇)eμ̇ηα̇ = 0,  (4.9)
ε̄ηα̇ = η_{α}.  (4.10)

Proof is a straightforward calculation.

4.7 Euler field. We want to introduce the notion of homogeneity of a Frobenius structure by assigning the scaling degrees 1 and 1/2 to the canonical coordinates u^{α} and θα̇ respectively (reflecting the fact that ∂_{α} = eα̇eα̇). With this motivation, we define a scaling field on a Frobenius supermanifold M as an even vector field E satisfying

[E, ∂_{α}] = −∂_{α}, [E, eα̇] = −\frac{1}{2}eα̇.

4.7.1. Proposition. If E is a scaling field, then

E = ∑_{α} [(u^{α} + c^{α} + θα̇cα̇)∂_{α} + \frac{1}{2}(θα̇ + cα̇)eα̇]

for some even constants c^{α} and odd constants cα̇.

Proof. Putting E = ∑_{α} E^{α}∂_{α} + Eα̇eα̇, one obtains

[E, ∂_{α}] = −e_{α} ⟺ ∑_{β} [(∂_{α}E^{β})∂_{β} + (∂_{α}Eβ̇)eβ̇] = ∂_{α},

[E, eα̇] = −\frac{1}{2}eα̇ ⟺ ∑_{β} [(eα̇E^{β})∂_{β} + (eα̇Eβ̇)eβ̇] − 2Eα̇∂_{α} = \frac{1}{2}eα̇,

implying ∂_{α}E^{β} = ∂_{α}Eβ̇ = eα̇E^{β} = eα̇Eβ̇ = 0 for all α ≠ β as well as ∂_{α}Eα̇ = 0, eα̇E^{α} = 2Eα̇ and eα̇Eα̇ = 1/2. Hence Eα̇ = \frac{1}{2}(θα̇ + cα̇) and E^{α} = u^{α} + c^{α} + θα̇cα̇ for some even constants c^{α} and odd constants cα̇.

Given a scaling field E, we can and will normalize the canonical coordinates so that E = ∑_{α} [u^{α}∂_{α} + \frac{1}{2}θα̇eα̇].

4.7.2. Definition. A scaling vector field E on a semisimple pre-Frobenius supermanifold M is called an Euler field if Lie_{E}(g) = Dg for some constant D, that is,

E(g(X, Y)) − g([E, X], Y) − g(X, [E, Y]) = Dg(X, Y),  (4.11)

for all vector fields X, Y.

4.7.3. Proposition. If E is an Euler field on a semisimple pre-Frobenius supermanifold M, then the potential Ψ satisfies

EΨ = (D − 1)Ψ + const,

or, equivalently

Eηα̇ = (D − \frac{3}{2})ηα̇.  (4.12)

Proof. Write (4.11) for (i) X = ∂_{α}, Y = ∂_{β}, (ii) X = eα̇, Y = ∂_{β} and (iii) X = eα̇, Y = eβ̇ to find that

(i) ⇔ Eg(∂_{α}, ∂_{β}) + 2g(∂_{α}, ∂_{β}) = Dg(∂_{α}, ∂_{β}),
(ii) ⇔ Eg(eα̇, ∂_{β}) + \frac{3}{2}g(eα̇, ∂_{β}) = Dg(eα̇, ∂_{β}),
(iii) ⇔ 0 = 0,

which imply equation (4.12) and

Eη_{α} = (D − 2)η_{α}.  (4.13)

However, the latter equation is not independent — applying eβ̇ to both sides of (4.12), one easily obtains E(eβ̇ηα̇) = (D − 2)eβ̇ηα̇ implying (4.13).

4.7.4. Corollary. If E is an Euler field on a semisimple pre-Frobenius supermanifold M, then

Eγ_{μ̇ν̇} = −\frac{1}{2}γ_{μ̇ν̇}.  (4.14)

Proof.

Eγ_{μ̇ν̇} = E (\frac{eμ̇ην̇}{2\sqrt{ημ̇ην̇}} )

= \frac{(Eeμ̇ην̇)}{2\sqrt{ημ̇ην̇}} − \frac{(Eημ̇)(eμ̇ην̇)}{4ημ̇\sqrt{ημ̇ην̇}} − \frac{(Eην̇)(eμ̇ην̇)}{4ην̇ \sqrt{ημ̇ην̇}}

= (D − 2)γ_{μ̇ν̇} − \frac{1}{2}(D − \frac{3}{2})γ_{μ̇ν̇} − \frac{1}{2}(D − \frac{3}{2})γ_{μ̇ν̇}

= −\frac{1}{2}γ_{μ̇ν̇}.

4.7.5. Nullness of the ∘-identity. In the presence of an Euler field, the flatness of e implies that either D = 2 or g(ē, ē) is identically zero, i.e. the identity ē ∈ (T M, ∘) is everywhere a null vector. Indeed,

g(ē, ē) = g(∑_{α} ∂_{α}, ∑_{β} ∂_{β}) = ∑_{α} η_{α},

while equations (4.8) and (4.13) imply (D − 2) ∑_{α} η_{α} = 0.

4.8. Geometry on T_{1}. Let M be a semisimple pre-Frobenius supermanifold. Define a new splitting of the SUSY-extension (4.1) as follows

\begin{aligned}s̃: & T_{0} & ⟶ & T M, \\ & ∂_{α} mod T_{1} & ⟶ & ẽ_{α} := ∂_{α} − ∑β̇ \frac{eβ̇ηα̇}{2ηβ̇} eβ̇. \end{aligned}

4.8.1. Lemma. The splitting s̃ decomposes T M into a direct sum T_{1} ⊕ s̃(T_{0}) of isotropic submodules.

Proof. The restriction of the Egoroff metric to s̃(T_{0}) is

g(ẽ_{α}, ẽ_{β}) = g(∂_{α} − ∑_{μ̇} \frac{eμ̇ηα̇}{2ημ̇} eμ̇, ∂_{β} − ∑_{ν̇} \frac{eν̇ηα̇}{2ην̇} eν̇)

= −δ_{αβ}η_{α} + \frac{eβ̇ηα̇}{2} + \frac{eα̇ηβ̇}{2}

= 0.

Note for future reference that g(s̃(e_{α}), eβ̇) = g(s(e_{α}), eβ̇) = δ_{αβ}ηα̇.

4.8.2. Distinguished connections on T_{1} and T_{0}. The decomposition T M = T_{1} ⊕ s̃(T_{0}) induces a projection

p_{1} : T M → T_{1}.

Then, if ∇ : T M → T_{1}^{∗}⊗T M is the Levi-Civita covariant differential of the Egoroff metric, one may define the operators

˜∇^{1} : T_{1} → T_{1}^{∗} ⊗ T_{1},  ˜∇^{0} : T_{0} → T_{1}^{∗} ⊗ T_{0}

as the compositions

˜∇^{1} : T_{1} ⟶^{i} i(T_{1}) ⟶^{∇} T_{1}^{∗} ⊗ T M ⟶^{Id⊗p_{1}} T_{1}^{∗} ⊗ T_{1},
˜∇^{0} : T_{0} ⟶^{s̃} s̃(T_{0}) ⟶^{∇} T_{1}^{∗} ⊗ T M ⟶^{Id⊗p} T_{1}^{∗} ⊗ T_{0},

where i and p are defined in (4.1).

Remarkably, the connections ˜∇^{1} and ˜∇^{0} are essentially one and the same thing:

4.8.3. Lemma. ˜∇^{1}(XΠ) = ( ˜∇^{0}X)Π, ˜∇^{0}(Y Π) = ( ˜∇^{1}Y)Π, for any X ∈ T_{0}, Y ∈ T_{1}.

Proof. Since

p_{1}(e_{α}) = ∑_{β̇} \frac{eβ̇ηα̇}{2ηβ̇} eβ̇,

one obtains from (4.5)

˜∇_{eμ̇}eα̇ = δ_{μ̇α̇} ∑_{β̇} \frac{eβ̇ηα̇}{2ηβ̇} eβ̇ + \frac{eμ̇ηα̇}{2ηα̇} eα̇ − \frac{eα̇ημ̇}{2ημ̇} eμ̇.  (4.15)

Analogously,

˜∇_{eμ̇}e_{α} = p (∇_{eμ̇}(∂_{α} − ∑_{β̇} \frac{eβ̇ηα̇}{2ηβ̇} eβ̇))

= p(∇_{eμ̇}∂_{α}) + ∑_{β̇} \frac{eβ̇ηα̇}{2ηβ̇} p(∇_{eμ̇}eβ̇)

= \frac{eμ̇ηα̇}{2ηα̇} e_{α} + δ_{μ̇α̇} ∑_{β̇} \frac{eβ̇ηα̇}{2ηβ̇} e_{β} − δ_{μ̇α̇} \frac{η_{μ}}{ημ̇} e_{μ} + \frac{eμ̇ηα̇}{2ημ̇} e_{μ}

= \frac{eμ̇ηα̇}{2ηα̇} e_{α} + δ_{μ̇α̇} ∑_{β̇} \frac{eβ̇ηα̇}{2ηβ̇} e_{β} − \frac{eα̇ημ̇}{2ημ̇} e_{μ}.

Then the required statement follows.

From now on we use one symbol ˜∇ to denote covariant differentials ˜∇^{1} on T_{1}, ˜∇^{0} on T_{0} and ˜∇^{1} ⊕ ˜∇^{0} on T = T_{1} ⊕ T_{0}.

4.8.4 Metrics on T . The Egoroff metric g gives rise to

(i) an even metric h on T_{0}, h(X, Y) := g(ΠX, s̃(Y)), for any X, Y ∈ T_{0} (note that g(ΠX, s̃(Y)) = g(ΠX, s(Y)));

(ii) an odd metric g̃ on T as the pullback of g relative to the isomorphism s̃: T → T M. Note that that T_{0} and T_{1} are isotropic and g̃(X, Y) = g(s̃(X), Y) = g(s(X), Y) for any X ∈ T_{0}, Y ∈ T_{1}.

Due to the isomorphism ⊙^{2}(T_{0}^{∗}) = ⊙^{2}(ΠT_{1}^{∗}) = Λ^{2}(T_{1}^{∗}), the metric h on T_{0} can also be viewed as an even non-degenerate skew-symmetric form on T_{1}. Explicitly, h is given by

h(e_{α}, e_{β}) = δ_{αβ}ηα̇, h(e_{α}, eβ̇) = 0, h(eα̇, eβ̇) = δ_{α̇β̇}ηα̇,

while g̃ satisfies

g̃(e_{α}, e_{β}) = 0, g̃(e_{α}, eβ̇) = δ_{αβ}ηα̇, g̃(eα̇, eβ̇) = 0.

4.8.5. Frobenius property. The triple (T , ∘, g̃) obviously satisfies

g̃(X, Y) = θ(X ∘ Y)

for any X, Y ∈ T , where the odd 1-form θ is defined by

θ = PδΨ,

δ being the SUSY-differential and P the parity change functor [Ma3]. In particular, one has

g̃(X ∘ Y, Z) = g̃(X, Y ∘ Z)

for any X, Y, Z ∈ T (which is a defining property of the so-called Frobenius algebras [D], [H]).

4.8.6 Proposition. ˜∇h = 0, ˜∇g̃ = 0.

Proof. Let us view h as, for example, a skew-form on T_{1}. Then

( ˜∇_{eμ̇}h)(eα̇, eγ̇) = eμ̇h(eα̇, eγ̇) − h( ˜∇_{eμ̇}eα̇, eγ̇) + h(eα̇, ˜∇_{eμ̇}eγ̇)

= δ_{α̇γ̇}eμ̇ηα̇ − \frac{1}{2}δ_{α̇γ̇}eμ̇ηα̇ + \frac{1}{2}δ_{μ̇γ̇}eα̇ημ̇ − \frac{1}{2}δ_{α̇μ̇}eγ̇ηα̇

− \frac{1}{2}δ_{α̇γ̇}eμ̇ηα̇ + \frac{1}{2}δ_{α̇μ̇}eγ̇ηα̇ − \frac{1}{2}δ_{μ̇γ̇}eα̇ημ̇

= 0.

Analogously one checks other statements.

4.9. Odd identity. The •-identity ε is said to be flat if ˜∇ε = 0.

4.9.1. Proposition. ˜∇ε = 0 ⇔ ∑α̇ ηα̇ = const.

Proof.

˜∇_{eμ̇} (∑_{α} eα̇) = ∑_{β̇} \frac{eμ̇ηβ̇}{2ηβ̇} eβ̇ − \frac{∑_{α} eα̇ημ̇}{2ημ̇} eμ̇ + ∑_{β̇} \frac{eμ̇ηβ̇}{2ηβ̇} eβ̇

= \frac{η_{μ}}{2ημ̇} − \frac{∑_{α} eα̇ημ̇}{2ημ̇} eμ̇

= \frac{eμ̇(∑_{α} ηα̇)}{2ημ̇}eμ̇

= 0.

4.9.2. Proposition. Flatness of ε implies flatness of e, or, equivalently,

∑_{α̇} ηα̇ = const ⇒ ∑_{α} η_{α} = const.

Proof.

∑_{α̇} ηα̇ = const ⇔ eβ̇ (∑_{α̇} ηα̇) = 0

⇔ η_{β} = ∑_{α̇:α̇≠β̇} eα̇ηβ̇

⇒ ∑_{β} η_{β} = ∑_{α̇,β̇:α̇≠β̇} eα̇ηβ̇ = 0.

4.9.3. Orthogonality of flat identities. In the presence of an Euler field, the flatness of the odd identity ε implies that either D = 3/2 or g(ē, ε) is identically zero, i.e. the even and odd identities ē, ε ∈ (T M, ∘) are everywhere g-orthogonal. Indeed,

g(e, ε) = g(∑_{α} ∂_{α}, ∑_{β̇} eβ̇) = ∑_{α̇} ηα̇,

while Proposition 4.9.1 and equation (4.12) imply (D − 3/2) ∑α̇ ηα̇ = 0.

Since g̃(e, ε) = g(s(e), ε) = g(ē, ε), one may reformulate the above observation as the g̃-orthogonality of the identities (e, ε) ∈ (T , ∘).

4.10. Uniqueness and flatness of ˜∇. The main justification for introducing the connection ˜∇ comes from the following result.

4.10.1. Proposition. Let M be a semisimple pre-Frobenius supermanifold. Then the associated connection ˜∇ is flat if and only if Ψ satisfies the equation (4.3) and

∑_{β̇:β̇≠μ̇,ν̇} eβ̇γ_{μ̇ν̇} = eμ̇γ_{μ̇ν̇} + eν̇γ_{μ̇ν̇}  (4.16)

for all μ̇ ≠ ν̇.

Proof is a straightforward but lengthy calculation.

4.10.2. Corollary. Let M be a semisimple pre-Frobenius supermanifold. If the associated connection ˜∇ is flat, then M is semisimple Frobenius.

Proof. By Proposition 4.5.3, it will suffice to show that equation (4.16) implies equation (4.4). This is established by the following calculation:

∑_{α} ∂_{α}γ_{μ̇ν̇} = ∑_{α̇,β̇} eα̇eβ̇γ_{μ̇ν̇}

= 2 ∑_{α̇} eα̇eμ̇γ_{μ̇ν̇} + 2 ∑_{α̇} eα̇eν̇γ_{μ̇ν̇}

= 4e_{μ}γ_{μ̇ν̇} − 2 ∑_{α̇} eμ̇(eα̇γ_{μ̇ν̇}) + 4e_{ν}γ_{μ̇ν̇} − 2 ∑_{α̇} eν̇(eα̇γ_{μ̇ν̇})

= 4e_{μ}γ_{μ̇ν̇} − 2eμ̇(2eμ̇γ_{μ̇ν̇} + 2eν̇γ_{μ̇ν̇}) + 4e_{ν}γ_{μ̇ν̇} − 2eν̇(2eμ̇γ_{μ̇ν̇} + 2eν̇γ_{μ̇ν̇}) = − 4(eμ̇eν̇ + eν̇eμ̇)γ_{μ̇ν̇} = 0.

4.10.3. Corollary. Let M be a semisimple pre-Frobenius supermanifold. If ε is flat and Ψ satisfies the equation (4.3), then ˜∇ is flat as well (implying that M is Frobenius).

Proof. Assume eμ̇(∑β̇ ηβ̇) = 0, or, equivalently,

η_{μ} = ∑_{β̇:β̇≠μ} eβ̇ημ̇.

Then

∑_{β̇:β̇≠μ̇,ν̇} eβ̇γ_{μ̇ν̇} = ∑_{β̇:β̇≠μ̇,ν̇} \frac{eβ̇eμ̇ην̇}{2\sqrt{ημ̇ην̇}} − ∑_{β̇:β̇≠μ̇,ν̇} \frac{(eβ̇ημ̇)(eμ̇ην̇)}{4ημ̇\sqrt{ημ̇ην̇}} − ∑_{β̇:β̇≠ν̇} \frac{(eβ̇ην̇)(eμ̇ην̇)}{4ην̇ \sqrt{ημ̇ην̇}}

= − ∑_{β̇:β̇≠ν̇} \frac{eμ̇(eβ̇ην̇)}{2\sqrt{ημ̇ην̇}} + \frac{e_{μ}ην̇}{2\sqrt{ημ̇ην̇}} − \frac{η_{μ}(eμ̇ην̇)}{4ημ̇\sqrt{ημ̇ην̇}} − \frac{η_{ν}(eμ̇ην̇)}{4ην̇ \sqrt{ημ̇ην̇}}

= − \frac{e_{ν}ημ̇}{2\sqrt{ημ̇ην̇}} + \frac{e_{μ}ην̇}{2\sqrt{ημ̇ην̇}} − \frac{η_{μ}(eμ̇ην̇)}{4ημ̇\sqrt{ημ̇ην̇}} − \frac{η_{ν}(eμ̇ην̇)}{4ην̇ \sqrt{ημ̇ην̇}}

= eμ̇γ_{μ̇ν̇} + eν̇γ_{μ̇ν̇},

so that the equation (4.16) is satisfied.

For later use we give the following characterization of ˜∇:

4.10.4. Proposition. Let M be a semisimple pre-Frobenius supermanifold. A linear connection ∇ : T_{1} → T_{1}^{∗} ⊗ T_{1} satisfies the conditions

(a) ∇h = 0,

(b) ∇_{eμ̇}eα̇ + ∇_{eα̇}eμ̇ = 2δ_{μ̇α̇}∇_{eμ̇}eμ̇,

(c) h(∇_{eμ̇}eα̇, eβ̇) = 0 for any μ̇ ≠ α̇ ≠ β̇ ≠ μ̇ if and only if it is given by (4.15).

Proof. Define Δ_{μ̇α̇β̇} as

∇_{eμ̇}eα̇ = ∑_{β̇} \frac{Δ_{μ̇α̇β̇}}{2ηβ̇} eβ̇.

Then

(a) ⇔ eμ̇ηα̇δ_{α̇β̇} = \frac{1}{2}(Δ_{μ̇α̇β̇} + Δ_{μ̇β̇α̇}),
(b) ⇔ Δ_{μ̇α̇β̇} + Δ_{α̇μ̇β̇} = 2δ_{α̇μ̇}Δ_{μ̇μ̇β̇},
(c) ⇔ Δ_{μ̇α̇β̇} = 0 for all μ̇ ≠ α̇ ≠ β̇ ≠ μ̇,

implying that the only non-vanishing components of Δ_{μ̇α̇β̇} are

Δ_{μ̇α̇α̇} = Δ_{μ̇α̇μ̇} = Δ_{α̇α̇μ̇} = eμ̇ηα̇.

Hence

∇_{eμ̇}eα̇ = \{ \begin{matrix}∑β̇ \frac{eβ̇ηα̇}{2ηβ̇} eβ̇ & for μ̇ = α̇, \\ \frac{eμ̇ηα̇}{2ηα̇} eα̇ + \frac{eμ̇ηα̇}{2ημ̇} eμ̇ = \frac{eμ̇ηα̇}{2ηα̇} eα̇ − \frac{eα̇ημ̇}{2ημ̇} eμ̇ & for μ̇ ≠ α̇, \end{matrix}

implying ∇ = ˜∇.

§5. Supersymmetric Schlesinger equations

5.1. Meromorphic connections with logarithmic singularities. Let M be a complex supermanifold equipped with a SUSY structure (4.1) and let F → M be a locally free holomorphic sheaf on M. Keeping in mind semisimple Frobenius structures, we also assume that M comes equipped with a monomorphism Θ : T_{0} → Λ^{2}T_{1} which, as explained in 4.4, canonically extends any covariant differential ∇ : F → T_{1}^{∗} ⊗ V to a linear connection on F.

Let D be a complex super-submanifold of M of codimension 1|0.

Assume first that D is irreducible and that the associated divisor line bundle [D] is free. Let f be a global basis section of [D]. A holomorphic covariant differential

∇ : F → T_{1}^{∗} ⊗ F

on M \ D is said to be a meromorphic connection with logarithmic singularities along D if there is a holomorphic covariant differential

∇′ : F → T_{1}^{∗} ⊗ F

on M such that

∇ − ∇′ = A\frac{δf}{f}

for some even holomorphic section A ∈ H^{0}(M, F ⊗ F^{∗}).

Note that

a) this definition does not depend on the choice of a particular trivialization f of [D] and hence can be appropriately localized and generalized;

b) the section A restricted to D does not depend on the choices made and hence gives a well-defined element of H^{0}(D, F ⊗ F^{∗}) which is called the residue of ∇ at D;

c) for any local trivialization f of [D], the connection ∇ induces a holomorphic residual connection ∇^{D,f} on F|_{D} associated with the holomorphic covariant differential ∇′|_{D} = (∇ − A\frac{δf}{f})|_{D}; if ∇ is flat, ∇^{D,f} is flat for any f.

5.2. Universal isomonodromic deformation. Consider ℂ^{n|n} with its natural SUSY structure T_{1} ⊂ T ℂ^{n|n} spanned by the vector fields eα̇ = ∂α̇ +θα̇∂_{α}, where (u^{α}, θα̇) are natural coordinates. Let B be the universal covering of ℂ^{n|n} \ (u^{α} − u^{β} − θα̇θβ̇ = 0), where pairwise distinct integers α and β run over 1, …, n.

Consider a supermanifold B × P^{1|1} with the direct product SUSY-structure, and denote by D̃_{α} the inverse image in B × P^{1|1} of the submanifold λ − u^{α} − ξθα̇ = 0 in ℂ^{n|n} × P^{1|1}, where (λ, ξ) are natural coordinates on a big cell of P^{1|1}. Furthermore, define D̃_{∞} = B ×∞ ⊂ B ×P^{1|1}, where ∞ stands for the codimension 1|0 submanifold of P^{1|1} given by λ̂ = 0, where λ̂ = 1/λ.

To any given point (x_{0}^{α}, θ_{0}^{α}) ∈ ℂ^{n|n} one may associate an embedding

\begin{aligned}i_{0} : & P^{1|1} & ⟶ & B × P^{1|1} \\ & (λ, ξ) & ⟶ & (x_{0}^{α}, θ_{0}^{α}, λ, ξ). \end{aligned}

Theorem 5.2.1. Let F^{0} be a locally free sheaf of rank p|q on P^{1|1} and let ∇^{0} be a flat meromorphic connection on it with logarithmic singularities at ∪_{α=1}^{n}D_{α}^{0} ∪ ∞, where D_{α}^{0} ⊂ P^{1|1} is given by λ − u_{0}^{α} − ξθα̇ ; 0 = 0.

Then there exists a locally free sheaf F of rank p|q on B × P^{1|1} and a flat meromorphic connection ∇ on it such that

(a) ∇ has logarithmic singularities at D̃_{α}, α = 1, …, n, and D̃_{∞};

(b) there is a canonical isomorphism i : i_{0}^{∗}(F, ∇) → (F^{0}, ∇^{0});

(c) the data (F, ∇, i) are unique up to unique isomorphism.

Comment on the proof. According to Penkov [P], a pair (E, ∇) consisting of a locally free sheaf E on a supermanifold M and a flat holomorphic connection ∇ on E is uniquely determined by the associated monodromy representation of π_{1}(M_{red}) on E_{red}. This together with the observation made in 2.2.2 about the isomorphism of the first homotopy groups of the underlying classical manifolds, immediately implies that there is a pair (F, ∇) on B × P^{1|1} \ (∪_{α=1}^{n}D̃_{α} ∪ D̃_{∞}) such that the statements (b) and (c) are true outside singularities.

Using a straightforward generalization of the original Malgrange’s arguments, one may extend (F, ∇) to B × P^{1|1} in such a way that (a)-(c) hold.

5.3. Supersymmetric Schlesinger equations. In this subsection we will assume that F^{0} = T ⊗ O_{P^{1|1}} where T is a vector superspace of dimension p|q.

Using the semicontinuity principle as in Section 2.3, one may show that there is an open subset B′ ⊂ B such that F is free on B′ ×P^{1|1}. Moreover, one may identify F on B′ × P^{1|1} with T ⊗ O_{B′×P^{1|1}} compatibly with the respective trivialization of F^{0}. Indeed, one may first trivialize F along D̃_{∞} using the residual connection, and then take the constant extension of each horizontal section along P^{1|1}. Since

δ(u^{ν} − λ − θν̇ξ) = (θν̇ − ξ)d(θν̇ − ξ),

in the chosen trivialization F|_{B′×P^{1|1}} = T ⊗ O_{B′×P^{1|1}} the covariant differential ∇ must be of the form

∇ = δ + ∑_{ν=1}^{n} \frac{A_{ν}(θν̇ − ξ)}{u^{ν} − λ − θν̇ξ} d(θν̇ − ξ)  (5.1)

for some even meromorphic sections A_{ν} ∈ H^{0}(B, F ⊗ F^{∗}).

Theorem 5.3.1. The connection (5.1) is flat if and only if

eμ̇A_{ν} = − \frac{θμ̇ − θν̇}{u^{μ} − u^{ν} − θμ̇θν̇} [A_{μ}, A_{ν}]  (5.2)

eμ̇A_{μ} = ∑_{ν:ν≠μ} \frac{θμ̇ − θν̇}{u^{μ} − u^{ν} − θμ̇θν̇} [A_{μ}, A_{ν}].  (5.3)

or, equivalently,

dA_{μ} = ∑_{ν:ν≠μ} \frac{d(u^{μ} − u^{ν} − θμ̇θν̇)}{u^{μ} − u^{ν} − θμ̇θν̇}[A_{μ}, A_{ν}]  (5.4)

where d is the usual exterior differential on B.

Proof is straightforward.

The equations (5.2) and (5.3) are called supersymmetric Schlesinger’s equations.

5.4. From Frobenius supermanifolds to strict special solutions of Schlesinger’s equations. Let M be a semisimple Frobenius supermanifold with an Euler field E and flat identities ε and ε̄. Define an even linear operator V : T_{1} → T_{1} as follows,

V(X) = p_{1}(∇_{X}E) − \frac{1}{2}(D − \frac{1}{2})X for any X ∈ T_{1},

where ∇ is the Levi-Civita connection and p_{1} : T M → T_{1} the projection defined in 4.8.2.

Theorem 5.4.1. a). Let fα̇ = eα̇/\sqrt{ηα̇}, then

V(fα̇) = ∑_{β̇:β̇≠α̇} [θα̇γ_{β̇α̇} + uβ̇∂β̇γ_{β̇α̇} − uα̇∂α̇γ_{α̇β̇} + ∑_{γ:γ≠α,β} u^{γ}eγ̇γ_{α̇β̇}] fβ̇.  (5.5)

b). V is symmetric relative to h, i.e.

h(V(X), Y) + h(X, V(Y)) = 0 for any X, Y ∈ T_{1},

c). ˜∇V = 0.

d). V(ε) = \frac{3−2D}{4} ε.

Proof. Items (a) and (c) follow from a straightforward calculation which we omit, while (b) immediately follows from (4.9) and (5.5). To check (c), note that

V(ε) = p_{1}(∇_{ε}E) − \frac{1}{2}(D − \frac{1}{2})ε

= p_{1}(∇_{E}ε + [ε, E]) − \frac{1}{2}(D − \frac{1}{2})ε

= \frac{1}{2}ε − \frac{1}{2}(D − \frac{1}{2})ε

= \frac{3 − 2D}{4} ε.

5.4.2. The structure connection. Let M be a semisimple pre-Frobenius supermanifold with canonical coordinates (u^{α}, θα̇), and let M̌ be a (n + 1|n + 1)- supermanifold M × P^{1|1} \ ∪_{ν}(u^{ν} − λ − θν̇ξ = 0) equipped with a product SUSY structure

Ť_{1} := span(eα̇, e_{ξ}) ⊂ T M̌,

where e_{ξ} = ∂/∂ξ + ξ∂/∂λ. Let pr : M̌ → M be a natural projection and denote E_{0} := _{∑α}u^{α}eα̇ and E_{1} := _{∑α}θα̇eα̇.

The following theorem introduces a so-called structure connection on pr^{∗}(T_{1}) ⊂ Ť_{1} which associates with any semisimple Frobenius structure a 1-parameter solution of Schlesinger’s equations.

5.4.3. Theorem. For any X, Y ∈ pr^{−1}(T_{1}) ⊂ Ť_{1} put

ˇ∇_{X}Y = ˜∇_{X}Y − (V + κId)(E_{0} − λ + ξE_{1})^{−1} • (E_{1} − ξ) • X • Y  (5.6)
ˇ∇_{e_{ξ}}Y = (V + κId)(E_{0} − λ + ξE_{1})^{−1} • (E_{1} − ξ) • Y.  (5.7)

where κ ∈ ℂ.

If M is a semisimple Frobenius supermanifold with an Euler field and flat identities ε and ε̃, then ˇ∇ is a flat connection on pr^{∗}(T_{1}) ⊂ Ť_{1} for any κ.

Proof. If X = eμ̇ and Y = eα̇, the equations (5.6) and (5.7) take the form

ˇ∇_{eμ̇}eα̇ = ˜∇_{eμ̇}eα̇ + δ_{α̇μ̇} \frac{[V(eα̇) + κeα̇](θα̇ − ξ)}{u^{α} − λ − θα̇ξ}

ˇ∇_{e_{ξ}}eα̇ = − \frac{[V(eα̇) + κeα̇](θα̇ − ξ)}{u^{α} − λ − θα̇ξ}.

Then, by Proposition 5.3.1, it will suffice to show that under the conditions stated in 5.4.2 the matrix valued fields

A β̇ ; ν α̇ = −δ_{ν̇α}[V_{α̇β̇} + κδ_{α̇β̇}]

satisfy, for any κ ∈ ℂ, the Schlesinger’s equations

eμ̇A β̇ ; ν α̇ + Γβ̇ ; μ̇δ̇A δ̇ ; ν α̇ − Γδ̇ ; μ̇α̇A β̇ ; ν δ̇ = − \frac{θμ̇ − θν̇}{u^{μ} − u^{ν} − θμ̇θν̇} [A_{μ}, A_{ν}]α̇ ; β̇, μ ≠ ν  (5.8)

eμ̇A β̇ ; μ̇ α̇ + Γβ̇ ; μ̇δ̇A δ̇ ; μ̇ α̇ − Γδ̇ ; μ̇α̇A β̇ ; ν δ̇ = ∑_{ν:ν≠μ} \frac{θμ̇ − θν̇}{u^{μ} − u^{ν} − θμ̇θν̇} [A_{μ}, A_{μ}]β̇ ; α̇,  (5.9)

where

Γβ̇ ; μ̇α̇ = δ_{α̇β̇} \frac{eμ̇ηα̇}{2ηα̇} − δ_{β̇μ̇} \frac{eα̇ημ̇}{2ημ̇} + δ_{μ̇α̇} \frac{eβ̇ηα̇}{2ηβ̇}

are the coefficients of the connection ˜∇, and V_{α̇β̇} are the coefficients of the operator V in the basis eα̇.

It is not hard to show that equations (4.9) and (5.5) imply, for μ ≠ ν,

\frac{θμ̇ − θν̇}{u^{μ} − u^{ν} − θμ̇θν̇} V_{μ̇ν̇} = \frac{eμ̇ην̇}{2ην̇}

which in turn imply that the terms of order κ^{0} in (5.8) and (5.9) are all equivalent to the equation ˜∇V = 0 which follows from 5.4.1(c).

Since there are no terms quadratic in κ in (5.8) and (5.9), it remains to show that the terms of order κ^{1} cancel. Indeed, the terms linear in κ in the l.h.s. of (5.8) are

−δ_{να}Γβ̇ ; μ̇α̇ + δ_{νβ}Γβ̇ ; μ̇α̇ = (δ_{να}δ_{μβ} − δ_{μα}δ_{νβ})\frac{eα̇ηβ̇}{2ηβ̇} ,

while the terms linear in κ in the r.h.s. of (5.8) are

−\frac{θμ̇ − θν̇}{u^{μ} − u^{ν} − θμ̇θν̇} (−δ_{να}δ_{μβ}V_{ν̇μ̇} + δ_{μα}δ_{νβ}V_{μ̇ν̇}) = (δ_{να}δ_{μβ} − δ_{μα}δ_{νβ})\frac{eα̇ηβ̇}{2ηβ̇} .

Analogously one checks that the terms linear in κ in (5.9) also vanish identically.

5.5. Strict special solutions of Schlesinger’s equations. Consider the supermanifold M = ℂ^{n|n} with canonical coordinates (u^{α}, θα̇), an (p|q)-dimensional vector space T and a set of even holomorphic matrix functions A_{ν} : M → End(T), ν = 1, …, n such that the Schlesinger equations (5.4) are satisfied. In particular, summing (5.4) over ν and denoting W = ∑_{ν} A_{ν}, we find dW = 0, i.e. W ∈ End(T).

Theorem 5.4.3 motivates the following definition (cf. Section 2.4):

5.5.1. Definition. A solution to Schlesinger’s equation as above is called strict special if

a). rank T = 0|n;

b). T is endowed with a complex non-degenerate skew-symmetric form h ∈ Λ^{2}(T);

c). W = −V − κId, where κ ∈ ℂ is an arbitrary parameter and V ∈ End(T) is an even operator symmetric with respect to h;

d). for any ν,

A_{ν} = −(V + κId)P_{ν},  (5.10)

where P_{ν} : M → End(T) is a set of even holomorphic matrix functions whose values at any point of M constitute a complete system of orthogonal projectors of rank 1 with respect to h:

P_{ν}P_{μ} = δ_{μν}P_{ν}, ∑_{ν} P_{ν} = Id_{T} , h(ImP_{ν}, ImP_{μ}) = 0  (5.11)

if μ ≠ ν.

e). There is a vector ε ∈ T such that

V(ε) = \frac{3 − 2D}{4} ε  (5.12)

for some D ∈ ℂ and eν̇ := P_{ν}(ε) are nowhere vanishing on M.

Theorem 5.4.3 says that to any semisimple Frobenius supermanifold there is canonically associated a strict special solution of Schlesinger’s equations.

5.6. From strict special solutions to Frobenius supermanifolds. Let (M, T, h, A_{ν}, ε) be a strict special solution of Schlesinger’s equations.

5.6.1. Theorem. These data come from the unique structure of semisimple split Frobenius supermanifold on M, with Euler field and flat identities ε and ε̄.

Proof. Put eν̇ = P_{ν}(ε) ∈ T ⊗O_{M} and identify O_{M}⊗T with T_{1} ⊂ T M by setting eν̇ = ∂/∂θν̇ + θν̇∂/∂uν̇. This transfers h from T to T_{1}. Denote ηα̇ = h(eα̇, eα̇). Define the multiplication in T M by the formulae 4.3.2.

To prove Theorem 5.6.1 it will suffice to show that 1) ηα̇ = eα̇Ψ for an odd function Ψ (potentiality); 2) Ψ satisfies the Darboux-Egoroff equations (4.3) and (4.4) (flatness of the Egoroff metric); 3) the equation (4.12) is satisfied (Euler property); 4) the equations (4.9) and (4.10) are satisfied (flatness of ε̄). That the condition ∑α̇ ηα̇=const (flatness of ε) is satisfied follows immediately from the definition of ηα̇ and the fact that ∑α̇ ηα̇ = h(ε, ε).

Step 1 (potentiality). Since

∑_{α̇} eα̇ = ∑_{α} P_{α}(ε) = Id(ε) = ε

and h(ImP_{α}, ImP_{β}) = 0 if α ≠ β, we have hα̇ = h(ε, eα̇) and hence, in view of 5.5.1(c,d),

h(ε, A_{ν}(ε)) = −h(ε, (V + κId)eν̇) = h(V(ε), eν̇) − κh(ε, eν̇) = \frac{3 − 2D − 4κ}{4}ην̇.

(5.13)

Let ∇ be the unique flat connection in T_{1} ≃ T × M which makes constants sections of T × M horizontal. Obviously, ˜∇ preserves h and satisfies ∇ε = 0. Then derivating (5.13) we find, for every μ ≠ ν,

\frac{3 − 2D − 4κ}{4}eμ̇ην̇ = h(∇_{eμ̇}ε, A_{ν}ε) − h(ε, ∇_{eμ̇}(A_{ν}ε)) = h(ε, (eμ̇A_{ν})ε). (5.14)

Since, in view of (5.2),

eμ̇A_{ν} = − \frac{θμ̇ − θν̇}{u^{μ} − u^{ν} − θμ̇θν̇} [A_{μ}, A_{ν}] = − eν̇Aμ̇ ∀ μ ≠ ν,

we find

eμ̇ην̇ + eν̇ημ̇ = 0 ∀ μ ≠ ν,

or, equivalently,

eμ̇ην̇ + eν̇ημ̇ = 2δ_{μ̇ν̇}η_{μ} ∀ μ, ν,

where η_{μ} := eμ̇ημ̇. Analogous calculations, involving Schlesinger’s equations (5.2) and (5.3), show that

eμ̇η_{ν} − ∂_{ν}ημ̇ = 0, ∂_{μ}η_{ν} − ∂_{ν}η_{μ} = 0.

Finally, defining the 1-form ω = ∑_{α}[dθα̇(ηα̇ − θα̇η_{α}) + du^{α}η_{α}], it is straightforward to check that the latter three equations are equivalent to dω = 0. Hence ω = dΨ for some odd function Ψ, i.e. ηα̇ = eα̇Ψ.

Step 2 (flatness of the Egoroff metric). Let Ψ be as above and g the associated Egoroff metric. Let us prove that g is flat.

By Corollary 4.10.2, it will suffice to show that the flat connection ∇ coincides with the connection (4.15), i.e. that ∇ satisfies all three conditions of Proposition 4.10.4.

The condition 4.10.4(a) is obviously satisfied.

Since κ is arbitrary, we may assume without loss of generality that W is invertible and rewrite Schlesinger’s equation (5.2) in the form

eμ̇P_{ν} = − \frac{θμ̇ − θν̇}{u^{μ} − u^{ν} − θμ̇θν̇} W^{−1}[A_{μ}, A_{ν}].

Then for every μ̇ ≠ ν̇ we have

∇_{eμ̇}eν̇ + ∇_{eν̇} eμ̇ = ∇_{eμ̇}(P_{ν}ε) + ∇_{eν̇} (P_{μ}ε)

= (eμ̇P_{ν})ε + (eν̇P_{μ})ε

= − \frac{θμ̇ − θν̇}{u^{μ} − u^{ν} − θμ̇θν̇} W^{−1}[A_{μ}, A_{ν}]ε − \frac{θν̇ − θμ̇}{u^{ν} − u^{μ} − θν̇θμ̇} W^{−1}[A_{ν}, A_{μ}]ε

= 0,

or, equivalently,

∇_{eμ̇}eν̇ + ∇_{eν̇} eμ̇ = 2δ_{μ̇ν̇}∇_{eμ̇}eμ̇ ∀ μ, ν.

Thus, it remains to check the condition 4.10.4(c) for all μ̇ ≠ ν̇ ≠ α̇ ≠ μ̇ :

h(∇_{eμ̇}eν̇, eα̇) = h((eμ̇P_{ν})ε, P_{α}ε)

= − \frac{θμ̇ − θν̇}{u^{μ} − u^{ν} − θμ̇θν̇} h(W^{−1}[A_{μ}, A_{ν}]ε, P_{α}ε)

= − \frac{θμ̇ − θν̇}{u^{μ} − u^{ν} − θμ̇θν̇} h(P_{μ}(…) + P_{ν}(…), P_{α}ε)

= 0.

This establishes the flatness of the Egoroff metric.

Step 3 (Euler property). Schlesinger’s equations (5.2)-(5.3) imply

EA_{ν} = ∑_{μ:μ≠ν} (u^{μ}∂_{μ}A_{ν} + \frac{1}{2}θμ̇eμ̇A_{ν}) + u^{ν}∂_{ν}A_{ν} + \frac{1}{2}θν̇eν̇Aν̇

= ∑_{μ:μ≠ν} (−\frac{u^{μ}[A_{μ}, A_{ν}]}{u^{μ} − u^{ν} − θμ̇θν̇} − \frac{θμ̇(θμ̇ − θν̇)[A_{μ}, A_{ν}]}{2(u^{μ} − u^{ν} − θμ̇θν̇)}

+\frac{u^{ν}[A_{ν}, A_{ν}]}{u^{ν} − u^{μ} − θν̇θμ̇} + \frac{θν̇(θν̇ − θμ̇)[A_{ν}, A_{μ}]}{2(u^{ν} − u^{μ} − θν̇θμ̇)})

= − ∑_{μ:μ≠ν}[A_{μ}, A_{ν}].

Then, using (5.13), we find

\frac{3 − 2D − 4κ}{4}Eην̇ = h(ε, (EA_{ν})ε)

= −h(ε, ∑_{μ:μ≠ν}[A_{μ}, A_{ν}]ε)

= −h(ε, [V + κId, (V + κId)P_{ν}]ε)

= − (\frac{3 − 2D}{2} ) (\frac{3 − 2D − 4κ}{4}) ην̇.

Hence Eην̇ = (D − \frac{3}{2})ην̇.

Step 4 (flatness of ε̄). One finds from (5.14):

\frac{3 − 2D − 4κ}{4}(θμ̇ − θν̇)eμ̇ην̇ = (θμ̇ − θν̇)h(ε, (eμ̇A_{ν})ε)

= −(θμ̇ − θν̇)h(ε, \frac{θμ̇ − θν̇}{u^{μ} − u^{ν} − θμ̇θν̇} [A_{μ}, A_{ν}]ε)

= 0.

Hence the equation (4.9) is satisfied. Analogously one checks that (4.10) is valid as well.

This completes the proof of Theorem 5.6.1.

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