The Stokes Phenomenon and Some Applications

Multisummation provides a transparent description of Stokes matrices which is reviewed here together with some applications. Examples of moduli spaces for Stokes matrices are computed and discussed. A moduli space for a third Painlev\'e equation is made explicit. It is shown that the monodromy identity, relating the topological monodromy and Stokes matrices, is useful for some quantum differential equations and for confluent generalized hypergeometric equations.


Introduction
Consider a linear differential equation in matrix form y ′ + Ay = 0 where the entries of the matrix A are meromorphic functions defined in a neighbourhood of, say, z = ∞ in the complex plane. A formal or symbolic solution can be lifted to an actual solution in a sector at z = ∞, having the formal solution as asymptotic behavior.
In 1857, G.G. Stokes observed, while working in the middle of the night and not long before getting married, the phenomenon that this lifting depends on the direction of the sector at z = ∞ (see [Ra, St] for more details). This is the starting point of the long history of the asymptotic theory of singularities of differential equations. The theory of multisummation, the work of many mathematicians such as W. The first section is written for the convenience of the reader. One considers a singular matrix differential equation y ′ + Ay = 0 at z = ∞ and recalls its formal classification, the definition of the Stokes matrices and the analytic classification. The theory is illustrated by the confluent generalized hypergeometric equation p D q . We rediscover results from [DM, Mi] as application of the monodromy identity. In the second section moduli spaces for Stokes maps are discussed. It is shown that totally ramified equations have very interesting Stokes matrices. Quantum differential equations coming from Fano varieties are studied in §3. Again the computability of Stokes matrices is the theme. Certain moduli spaces, namely Okamoto-Painlevé spaces, corresponding to Painlevé equations are discussed in §4, including an explicit calculation of a monodromy space for P III (D 7 ).
1 Formal and analytic classification, Stokes maps 1.1 Terminology and notation Let k be a differential field, i.e., a field with a map f → f ′ (called a derivation) satisfying (f + g) ′ = f ′ + g ′ and (f g) ′ = f ′ g + f g ′ . The field of constants of k is {f ∈ k| f ′ = 0}. In this paper we suppose that the field of constants of k is C and that k = C.
The most important differential fields that we will meet are C(z), K := C((z −1 )) and K := C({z −1 }), i.e., the field of the rational functions in z, the formal Laurent series in z −1 and the field of the convergent Laurent series in z −1 . The latter is the field of germs of meromorphic functions at z = ∞. In all cases the differentiation is f → f ′ = df dz (sometimes replaced by f → δ(f ) := z df dz in order to make the formulas nicer).
A matrix differential equation y ′ + Ay = 0 with A a d × d-matrix with coordinates in k gives rise to the operator ∂ := d dz + A of k d , where d dz acts coordinatewise on k d and A is the matrix of a k-linear map k d → k d . Write now M = (M, ∂) for k d and the operator ∂. Then this object is a differential module.
Indeed, a differential module over k is a finite dimensional k-vector space M equipped with an additive map ∂ : M → M satisfying ∂(f m) = f ′ m + f ∂(m) for any f ∈ k and m ∈ M. If one fixes a basis of M over k, then M is identified with k d (with d = dim M) and the operator ∂ is identified with d dz + A. Here A is the matrix of ∂ with respect to the given basis of M.
Thus a differential module is "a matrix differential equation where the basis is forgotten" and a matrix differential equation is the same as a differential module with a given basis. We will use differential operators, i.e., elements of the skew polynomial ring k[∂] (where ∂ stands for d dz ) defined by the rule ∂f = f ∂ + f ′ . Instead of ∂ we sometimes use δ := z∂. Then δf = f δ +δ(f ) (in particular δz = zδ +z).
Let M be a differential module. The ring k[∂] acts from the left on M. For any element e ∈ M, there is a monic operator L ∈ k[∂] of smallest degree such that Le = 0. The element e is called a cyclic if L has degree d = dim M. Cyclic elements e exist and the corresponding operator L can be seen as a scalar differential equation. Further L determines the module M.
In practice one switches between differential modules, matrix differential equations and differential operators.
1.2 Classification of differential modules over K := C((z −1 )) The classification of a matrix differential equation z d dz + A over K (note that we prefer here z d dz ) goes back to G. Birkhoff and H.L. Turritin. This classification is somewhat similar to the Jordan normal of a matrix. However it is more subtle since z d dz + A is linear over C and is not linear over K.
We prefer to work "basis free" with a differential module M and classify M by its solution space V with additional data forming a tuple (V, {V q }, γ). If dim M = d, then we want the solution space, i.e., the elements w with δw = 0 to be a C-vector space of dimension d. Now we write δ : M → M instead of ∂ because d dz is replaced by z d dz . In general {m ∈ M|δ(m) = 0} is a vector space over C with dimension < d. Thus we enlarge K to a suitable differential ring U and consider {w ∈ U ⊗ K M | δw = 0}.
This differential ring U (the universal Picard-Vessiot ring for K) is build as follows. We need a linear space of "eigenvalues" Q := ∪ m≥1 z 1/m C[z 1/m ] and symbols z λ with λ ∈ C, log z, e(q) with q ∈ Q. The relations are z a+b = z a · z b , z 1 = z ∈ K, e(q 1 + q 2 ) = e(q 1 ) · e(q 2 ), e(0) = 1. And we define their derivatives by the formulas (z a ) ′ = az a , log(z) ′ = 1, e(q) ′ = qe(q) (note that ′ stands for z d dz and that the interpretation of e(q) is e q dz z ). This universal Picard-Vessiot ring is U : The Galois group of the algebraic closure ∪ m≥1 C((z −1/m )) of K is ∼ = Z and is topologically generated by the element γ given by γz λ = e 2πiλ z λ for all λ ∈ Q. The algebraic closure of K lies in U. One extends γ to a differential automorphism of U by the following formulas (corresponding to the interpretation of the symbols) γz a = e 2πia z a for all a ∈ C, γ log(z) = 2πi + log(z), γe(q) = e(γq).
For every differential module M over K, its solution space, defined as V := ker(δ, U ⊗ M) has "all solutions" in the sense that dim C V = dim K M and the canonical map U ⊗ C V → U ⊗ K M is an isomorphism.
Put V q := ker(δ, U q ⊗ M), then V = ⊕ q V q is a decomposition of the solution space. Further the action of γ on U induces a γ ∈ GL(V ) such that γV q = V γq for all q.
Theorem 1.1 (formal classification) The functor M → (V, {V q }, γ) is an equivalence of the Tannakian categories of the differential modules over K and the category of the tuples (V, {V q }, γ).
This "Tannakian" property means that the functor of the theorem commutes with all constructions of linear algebra applied to modules, including tensor products. Suppose M induces the tuple (V, {V q }, γ). Then q is called an eigenvalue of M if V q = 0. Further the map γ ∈ GL(V ) is called the formal monodromy of M. §1.3 illustrates the computation of the tuple (V, {V q }, γ).

The confluent generalized hypergeometric equation
is this equation in operator form. We assume that 1 ≤ p < q and that the complex parameters µ j , ν j are such that µ 1 , . . . , µ p are distinct modulo Z.
We regard p D q as element of K[δ] and start by factoring The two factors almost commute and there is a factorization in the opposite direction means that the differential modules over K defined by the two operators are isomorphic). This yields solutions f j := z −µ j for j = 1, . . . p. They form a basis of the C-vector space V 0 with eigenvalue q 0 = 0. The action of the formal is equivalent to δ σ − z (up to a sign) with σ := q − p. This operator factors over K(z 1/σ ) and one finds the eigenvalues q 1 = z 1/σ , q 2 = ζz 1/σ , . . . , q σ = ζ σ−1 z 1/σ with ζ = e 2πi/σ .

Now we can describe the solution space
Choose a basis e 1 of the 1-dimensional space V q 1 . Put e 2 := γe 1 , e 3 := γe 2 , . . . , e σ := γe σ−1 . Then V q j = Ce j for j = 1, . . . , σ. Finally γe σ = e 2πiλ e 1 with λ := 1 2 (σ + 1) + p j=1 µ j − q j=1 ν j . This follows from a computation of γ on f 1 ∧· · ·∧f p ∧e 1 ∧· · ·∧e σ and the determinant δ The above coincides with the formula of the formal monodromy in [Mi], p 373. In [Mi] and [DM] an explicit basis of formal or symbolic solutions of p D q is constructed and the computation of the formal monodromy and, later on, of the Stokes matrices is with respect to this basis. Our basis f 1 , . . . , f p , e 1 , . . . , e σ is not unique. More precisely, the above tuple has a non trivial automorphism group G ∼ = (C * ) p+1 . The elements of G are given by f j → α j f j for j = 1, . . . , p and e j → α p+1 e j for j = 1, . . . , σ (and (α 1 , . . . , α p+1 ) ∈ G). We will see that the group G/C * acts non trivially on the entries of the Stokes maps.

Stokes maps and the analytic classification
Let M be a differential module over K. Then K ⊗ M is a differential module over K and induces a tuple (V, {V q }, γ). For two eigenvalues q,q of K ⊗ M one considers special directions e 2πid , d ∈ R, called singular for the difference q −q. Those are the d such that e (q−q) dz z (this is the solution of y ′ = (q −q)y) has maximal descent to zero for z := re 2πid and r > 0, r → 0.
The tuple (V, {V q }, γ) is the formal classification of M, i.e., the classification of K ⊗ K M. Now we consider the classification of M itself. In the sequel, we use multisummation as a black box. It is a rather technical extension of the classical Borel summation of certain divergent power series. It has the property that for any v ∈ V , the sum d (v) has asymptotic expansion v (one has to add some conditions to make sum d unique).
For a singular direction d one takes real numbers d − < d < d + close to d and defines the Stokes map . One can show that this tuple has the additional properties: Here, every ℓ ∈ Hom(V q , Vq) is seen as an element of End(V ) by the sequence One considers the (Tannakian) category of the tuples with the properties (*) and (**).

Theorem 1.2 (the analytic classification)
is an equivalence of the Tannakian categories of the differential modules over K and the above category of the tuples (V, {V q }, γ, {St d }) satisfying (*) and (**).
The above theorem is, in contrast with the formal classification, a deep and final result in the asymptotic theory of linear differential equations. The irregularity of Malgrange, irr(M) of the differential module M is defined by One observes that the dimension of the space of all possibilities for Stokes maps with a fixed formal tuple is equal to irr(M) (see also §2).
A useful result, obtained by the above description of the Stokes maps, is the following.

Proposition 1.3 (The monodromy identity)
The topological monodromy of M is conjugated to γSt ds . . .
One cannot claim that the topological monodromy is equal to this product since one has to identify V with the local solution space at a point near z = ∞ and that can be done in many ways.

Stokes matrices for p D q
The main observation is the following. The monodromy identity yields complete formulas for the Stokes matrices of p D q if we are allowed to choose a suitable basis of V . More precisely, by choosing multiples of the basis f 1 , . . . , f p , we can normalize some of the entries of the Stokes maps to be 1. The others are then determined by the monodromy identity. This seems to work under the assumption that µ 1 , . . . , µ p are distinct modulo Z and that the equation is irreducible.
We note that the differential Galois group of p D q is in fact the differential Galois group of p D q as equation over the field of convergent Laurent series K. This group does not depend on the the choice of multiples of f 1 , . . . , f p . For its (rather involved) computation, see [Mi, DM], the formal classification and the characteristic polynomial of the topological monodromy at z = 0 (or equivalently at z = ∞) suffice. We illustrate this by the easy example The formal solution space is the direct sum of three 1-dimensional spaces V 0 ⊕ V z 1/2 ⊕ V −z 1/2 with basis f 1 , e 1 , e 2 . There is only one singular direction in the interval [0, 1), namely d = 0. The topological monodromy at z = 0 is that of the operator 3 j=1 (δ + ν j − 1) and has eigenvalues e −2πiν j , j = 1, 2, 3. The formal monodromy is γ(f 1 ) = e −2πiµ f 1 , γe 1 = e 2 and γ(e 2 ) = e −2πiλ e 1 . The product of the formal monodromy and the unique Stokes ma- e 1 e 1 x 1,0 0 e 2 x 0,2 e 2 x 1,2 e 2 0 1 0   .
The exceptional case x 1,0 x 0,2 = 0 cannot be handled in this way. Using J.-P. Ramis result that the differential Galois group is generated as algebraic group by the formal monodromy, the exponential torus and the Stokes matrices, one concludes that x 1,0 = 0 or x 0,2 = 0 implies that the equation is reducible. See [DM] for a complete description of all cases and all differential Galois groups. We remark that the monodromy identity for p D q is explicitly present in [DM].

Moduli spaces for the Stokes data
For given formal data F := (V, {V q }, γ) at z = ∞, there exists a unique differential module N over K = C({z −1 }) with these formal data and with trivial Stokes matrices. One considers differential modules M which have formal classification F . The set of isomorphism classes of these modules does not have a good algebraic structure since F has, in general, automorphisms. D.G. Babbitt and V.S. Varadarajan consider instead pairs (M, φ) of a differential module M over K and an isomorphism φ : K ⊗ M → K ⊗ N. Two pairs (M j , φ j ), j = 1, 2 are equivalent if there exists an isomorphism α : M 1 → M 2 such that φ 2 •α = φ 1 . The set Stokesmoduli(F ) of equivalence classes of pairs (M, φ) has been given a natural structure of complex algebraic variety. Babbitt and Varadarajan prove that However, there is, in general, no universal family of differential modules parametrized by Stokesmoduli(F ) ∼ = Spec(C[x 1 , . . . , x m ]). In other words, Stokesmoduli(F ) is, in general, not a fine moduli space for the above family of differential modules.
Indeed, suppose that such a family {M ξ | ξ ∈ Stokesmoduli(F )} of differential modules over K = C({z −1 }) exists. This family is represented by a matrix differential operator z d dz + A in the variable z and with entries in, say, K(x 1 , . . . , x m ). The monodromy identity shows that the eigenvalues of the topological monodromy are algebraic over this field. A logarithm of the topological monodromy is computable from z d dz + A and has again entries in K(x 1 , . . . , x m ). This is, in general, not possible. See §2.1 for a concrete case.
In order to produce a fine moduli space one replaces the differential module M over K = C({z −1 }) by a tuple (M, ∇, φ). Here (M, ∇) is a connection on the projective line P 1 over C which has two singular points 0 and ∞. The point z = 0 is supposed to be regular singular. Further φ is an isomorphism of the formal completion of the connection at z = ∞ (i.e., ]. This prescribed object is a standard lattice (i.e., a C[[z −1 ]]-submodule generated by a basis) of the formal differential module N over C((z −1 )) corresponding to the given According to a theorem of Birkhoff, this "spreading out of M" exists. We will make the assumption that M is a free vector bundle on P 1 . The above description leads to a fine moduli space Mod(F ).
Let Stm : Mod(F ) → Stokesmoduli(F ), denote the map which associates to a tuple (M, ∇, φ), belonging to Mod(F ), its set of Stokes matrices. Known results are (see [vdP-Si]): . Stm is analytic and has an open dense image. (c). The generic fibre of Stm is a discrete infinite set and can be interpreted as a set of logarithms of the topological monodromy.
Comments. The proof of (a) is complicated and the result itself is somewhat amazing. (b) follows from the observations: If the topological monodromy of M is semi-simple, then a tuple (∇, M, φ) with free M exists. Moreover semi-simplicity is an open property. (c) follows from the construction of "spreading out". One needs a logarithm of the topological monodromy in order to construct the connection (M, ∇) on P 1 from the differential module over K = C({z −1 }).
A precise description of the generic fibre seems rather difficult. Moreover, a better moduli space, replacing Mod(F ), which does not require the vector bundle M to be free, should be constructed.

Example: Unramified cases
F is defined by V = V λ 1 z ⊕ · · · ⊕ V λnz where each V λ j z has dimension one and the λ 1 , . . . , λ n ∈ C are distinct. Further γ is the identity. Then N 0 can be given by the differential operator δ + z · diag(λ 1 , . . . , λ n ). Clearly The universal family is δ + z · diag(λ 1 , . . . , λ n ) + (T i,j ), where for notational convenience T i,i = 0 and the {T i,j } with i = j are n 2 − n independent variables. Thus Mod(F ) is indeed isomorphic to A n(n−1) C . Further one observes that, in general, the matrix L := (T i,j ) has the property that e 2πiL is the topological monodromy at z = 0 (or equivalently at z = ∞). Now the entries of e 2πL are rational in the n(n − 1)-variables of Stokesmoduli(F ). This shows that there is no universal family above Stokesmoduli(F ).
In the case n = 2, the map Mod(F ) → Stokesmoduli(F ) can be made explicit and is shown to be surjective. For n > 2, the above map is "highly transcendental" and we do not know whether it is surjective. The problem is the choice of a free vector bundle M in the definition of Mod(F ).
The problem of explicit computation of the Stokes matrices, i.e., making Stm explicit in this special case, has been studied over a long period and by Now we come to a surprising new case.

Example: Totally ramified cases
The formal case F is essentially V = V z 1/n ⊕V ζz 1/n ⊕· · ·⊕V ζ n−1 z 1/n , where ζ := e 2πi/n , each V ζ j z 1/n has dimension 1 and γ satisfies γ n = 1. The irregularity i =j 1 · 1 · deg(ζ i z 1/n − ζ j z 1/n ) is equal to n − 1 and this is small compared to the unramified case with irregularity n(n − 1). This is responsible for special features of these important examples.
For notational convenience we will consider the case n = 3. The lattice N 0 with formal data F and trivial Stokes matrices can be represented by the differential operator δ + A computation shows that Mod(F ) is represented by the universal family δ +   a 1 0 z 1 a 2 0 0 1 a 3   with a 1 , a 2 , a 3 ∈ C with a 1 + a 2 + a 3 = 0 ( General case of the family δ + The case all a j = 0 corresponds to δ n − z. The case a i = i n − n−1 2n for i = 0, . . . , n − 1 corresponds to all Stokes matrices are trivial.).
Conclusions. For the totally ramified cases F that we consider, the Stokes matrices have explicit formulas in exponentials of algebraic expressions in the entries of the matrix differential operator. This shows in particular, that there is no universal family parametrized by Stokesmoduli(F ). Further Stm : Mod(F ) → Stokesmoduli(F ) is surjective and the fibers correspond to choices of the logarithm of the topological monodromy.
3 Fano varieties and quantum differential equations This part of the paper represents work by John Alexander Cruz Morales and the author [CM-vdP]. A (complex) Fano variety F is a non singular, con-nected projective variety of dimension d over C, whose anticanonical bundle (Λ d Ω) * is ample. There are rather few Fano varieties.
Examples: For dimension 1 only F = P 1 ; for dimension 2: the Fano's are del Pezzo surfaces and ∼ = P 1 × P 1 or ∼ = to P 2 blown up in at most 8 points in general position; for dimension 3 and 4, there are classifications (long lists).

Quantum cohomology and quantum differential equations
We borrow from the informal introduction to the subject from M. A. Guest' book [Gue]. Let F be a Fano variety. On the vector space H * (F ) := ⊕ d i=0 H 2i (F, C) there is the usual cup product • (say obtained by the wedge product of differential forms). Quantum cohomology introduces a deformation • t of the cup product • on H * (F ) for t ∈ H 2 (F, C).
With respect to a basis b 0 , . . . , b s of H * (F, Z) := ⊕ d i=0 H 2i (F, Z), the quantum products b i • t have matrices which are computable in terms of the geometry of F . Let b 1 , . . . , b r be a basis of H 2 (F, Z).
The quantum differential equation of a Fano variety F is a system of (partial) linear differential operators ∂ i − b i • t , i = 1, . . . r acting on the space of the holomorphic maps H 2 (F, C) = Cb 1 + · · ·+ Cb r → H * (F, C) = Cb 0 + · · ·+ Cb s .
For the case r = 1 that interest us, the quantum differential equation reads z d dz ψ = Cψ, where ψ is a vector of lenght s + 1 and C is the matrix of quantum multiplication b 1 • t . The entries of the (s + 1) × (s + 1) matrix C are polynomials in z with integer coefficients. Clearly z = 0 is a regular singular point and z = ∞ is irregular singular. By taking a cyclic vector one obtains a scalar differential equation of order s + 1.

Examples of quantum differential equations
for a non singular hypersurface of degree m in P n+m−1 .
for del Pezzo surfaces.
B. Dubrovin is one of the founders of quantum cohomology. One of his conjectures (1996)(1997)(1998)(1999), [Du98,Du99] states that the Gram matrix (G i,j ) of a (good) Fano variety coincides with the " Stokes matrix" of the quantum differential equation of F (up to a certain equivalence of matrices). Here where {E i } is an exceptional collection of coherent sheaves on F generating the derived category D b coh(F ). Further "Stokes matrix " is in fact a connection matrix and, in our terminology, equal to the product d∈[0,1/2) St d (in counter clock order).
For F = P n , this has been verified by D. Guzzetti (1999). There are recent papers [Iri,MT,Tan,Ue1,Ue2] which handle more cases and use the "Laplace type transformation" to a regular singular differential equation.
The contribution of [CM-vdP] is proving Dubrovin's conjecture by computing all St d , using only the formal classification and the monodromy identity, for the cases P n , non singular hypersurfaces of degree m ≤ n in P n , and for weighted projective spaces P(w 0 , . . . , w n ). The equations and the method are closely related to §1.5 and §2.2.

Riemann-Hilbert approach to Painlevé equations
This classical method, related to isomonodromy, was revived and refined by M. Jimbo, T. Miwa and K. Ueno around 1980. The literature on the subject is nowadays impressive.
Certain details of the Riemann-Hilbert approach are worked out (2009) in collaboration with Masa-Hiko Saito, [vdP-Sa]. In collaboration with Jaap Top, refined calculations of Okamoto-Painlevé spaces and Bäcklund transformations were presented for P I − P IV (2009)(2010)(2011)(2012)(2013)(2014), see [vdP-T]. We give here a rough sketch of the ideas and especially of the part where Stokes matrices enter the picture. The starting point is a family S of differential modules M over C(z) with prescribed singularities at fixed points of P 1 . The type of singularities gives rise to a monodromy space R build out of monodromy, Stokes matrices and 'links'.
An example for R: The set S which gives rise to P III (D7) consists of the differential modules M over C(z) which have only 0 and ∞ as singular points. The point 0 has Katz invariant 1/2 and the point ∞ has Katz invariant 1. The monodromy space R consists of the analytic classification (V (0), . . . ) of M at z = 0 and (V (∞), . . . ) at z = ∞ and a connection matrix between these data, the link L : V (0) → V (∞), which describes the relation between the solutions around z = 0 and the the solutions around z = ∞.
Consider the map S → R which associates to each module M ∈ S its monodromy data in R. The fibers of this map are parametrized by some T ∼ = C * and there results a bijection S → R × T . The set S has a priori no structure of algebraic variety. A moduli space M over C, whose set of closed points consists of certain connections of rank two on the projective line, is constructed such that S coincides with M(C). This defines the analytic Riemann-Hilbert morphism RH : M → R. The fibers of RH are the isomonodromic families. There results an extended Riemann-Hilbert isomorphism RH + : M → R × T . From the isomorphism RH + the Painlevé property for the corresponding Painlevé equation follows and the moduli space M is identified with an Okamoto-Painlevé space. Special properties of solutions of the Painlevé equations, such as special solutions, Bäcklund transformations etc., are derived from the extended Riemann-Hilbert isomorphism.
The above sketch needs subtle refinements. One has, depending on the Painlevé equation and its parameters, to add level structure, to forget points, to desingularize R and M, to replace spaces by their universal covering etc., in order to obtain a correct extended Riemann-Hilbert isomorphism. For the remarkable fact that for each Painlevé equation the moduli space for the monodromy R is an affine cubic surface with three lines at infinity, there is not yet an explanation.