Geometry and Analytic Theory

of Frobenius Manifolds

Boris Dubrovin

Abstract. Main mathematical applications of Frobenius manifolds are in the theory of Gromov - Witten invariants, in singularity theory, in differential geometry of the orbit spaces of reflection groups and of their extensions, in the hamiltonian theory of integrable hierarchies. The theory of Frobenius manifolds establishes remarkable relationships between these, sometimes rather distant, mathematical theories.

1991 Mathematics Subject Classification: 32G34, 35Q15, 35Q53, 20F55, 53B50

WDVV equations of associativity is the problem of finding of a quasihomogeneous, up to at most quadratic polynomial, function F (t) of the variables t = (t^{1}, …, t^{n}) and of a constant nondegenerate symmetric matrix (η^{αβ}) such that the following combinations of the third derivatives c_{αβ}^{γ}(t) := η^{γϵ}∂_{ϵ}∂_{α}∂_{β}F (t) for any t are structure constants of an asociative algebra A_{t} = span (e_{1}, …, e_{n}), e_{α} · e_{β} = c_{αβ}^{γ}(t)e_{γ}, α, β = 1, …, n with the unity e = e_{1} (summation w.r.t. repeated indices will be assumed). These equations were discovered by physicists E.Witten, R.Dijkgraaf, E.Verlinde and H.Verlinde in the beginning of ’90s. I invented Frobenius manifolds as the coordinate-free form of WDVV.

1. Definition of Frobenius manifold (FM). 1.1. Frobenius algebra (over a field k; we mainly consider the case k = C) is a pair (A, < , >), where A is a commutative associative k-algebra with a unity e, < , > is a symmetric nondegenerate invariant bilinear form A × A → k, i.e. < a · b, c >=< a, b · c > for any a, b ∈ A. A gradation of the charge d on A is a k-derivation Q : A → A such that < Q(a), b > + < a, Q(b) >= d < a, b >, d ∈ k. More generally, graded of the charge d ∈ k Frobenius algebra (A, < , >) over a graded commutative associative k-algebra R by definition is endowed with two k-derivations Q_{R} : R → R and Q_{A} : A → A satisfying the properties Q_{A}(αa) = Q_{R}(α)a + αQ_{A}(a), α ∈ R, a ∈ A < Q_{A}(a), b > + < a, Q_{A}(b) > −Q_{R} < a, b >= d < a, b >, a, b ∈ A. 1.2. Frobenius structure of the charge d on the manifold M is a structure of a Frobenius algebra on the tangent spaces T_{t}M = (A_{t}, < , >_{t}) depending (smoothly, analytically etc.) on the point t ∈ M. It must satisfy the following axioms.

FM1. The metric < , >_{t} on M is flat (but not necessarily positive definite). Denote ∇ the Levi-Civita connection for the metric. The unity vector field e must be covariantly constant, ∇e = 0.

FM2. Let c be the 3-tensor c(u, v, w) :=< u · v, w >, u, v, w ∈ T_{t}M. The 4-tensor (∇_{z}c)(u, v, w) must be symmetric in u, v, w, z ∈ T_{t}M.

FM3. A linear vector field E ∈ V ect(M) must be fixed on M, i.e. ∇∇E = 0, such that the derivations Q_{F unc(M)} := E, Q_{V ect(M)} := id + ad_{E} introduce in V ect(M) the structure of graded Frobenius algebra of the given charge d over the graded ring F unc(M) of (smooth, analytic etc.) functions on M. We call E Euler vector field.

Locally, in the flat coordinates t^{1}, …, t^{n} for the metric < , >_{t}, a FM with diagonalizable (1,1)-tensor ∇E is described by a solution F (t) of WDVV associativity equations, where ∂_{α}∂_{β}∂_{γ}F (t) =< ∂_{α} · ∂_{β}, ∂_{γ} >, and vice versa. We will call F (t) the potential of the FM (physicists call it primary free energy; in the setting of quantum cohomology it is called Gromov - Witten potential [KM]). 1.3. Deformed flat connection ˜∇ on M is defined by the formula ˜∇_{u}v := ∇_{u}v + z u · v. Here u, v are two vector fields on M, z is the parameter of the deformation. (In [Gi1] another normalization is used ˜∇ ↦ h̄ ˜∇, h̄ = z^{−1}.) We extend this to a meromorphic connection on the direct product M × C, z ∈ C, by the formula ˜∇_{d/dz}v = ∂_{z}v + E · v − z^{−1}μ v with μ := 1/2(2 − d) · 1 − ∇E, other covariant derivatives are trivial. Here u, v are tangent vector fields on M × C having zero components along C ∋ z. The curvature of ˜∇ is equal to zero. This can be used as a definition of FM [Du3]. So, there locally exist n independent functions t̃_{1}(t; z), …, t̃_{n}(t; z), z ≠ 0, such that ˜∇ dt̃_{α}(t; z) = 0, α = 1, …, n. We call these functions deformed flat coordinates.

2. Examples of FMs appeared first in 2D topological field theories [W1, W2, DVV]. 2.0. Trivial FM: M = A_{0} for a graded Frobenius algebra A_{0}. The potential is a cubic, F_{0}(t) = \frac{1}{6} < 1, (t)^{3} >, t ∈ A_{0}. Nontrivial examples of FM are 2.1. FM with good analytic properties. They are analytic perturbations of the cubic. That means that, in an appropriate system of flat coordinates t = (t′, t′′), where all the components of t′ have Lie_{E}t′ = const, all the components of t′′ have Lie_{E}t′′ ≠ const, we have F (t) = F_{0}(t) + ∑_{k, l≥0} A_{k,l}(t′′)^{l}e^{k t′} and the series converges in some neiborghood of t′′ = 0, t′ = −∞. 2.2. K.Saito theory of primitive forms and Frobenius structures on universal unfoldings of quasihomogeneous singularities. Let f_{s}(x), s = (s_{1}, …, s_{n}) be the universal unfolding of a quasihomogeneous isolated singularity f(x), x ∈ C^{N}, f(0) = f′(0) = 0. Here n is the Milnor number of the singularity. The Frobenius structure on the base M ∋ s of the universal unfolding can be easily constructed [BV] using the theory of primitive forms [Sai2]. For the example [DVV] of the A_{n} singularity f(x) = x^{n+1} the universal unfolding reads f_{s}(x) = x^{n+1} + s_{1}x^{n−1} + … + s_{n}, M = C^{n} ∋ (s_{1}, …, s_{n}). On the FM e = ∂/∂s_{n}, E = ∑(k + 1)s_{k}∂/∂s_{k}, the metric has the form

< ∂_{s_{i}}, ∂_{s_{j}} >= −(n + 1) res_{x=∞} \frac{∂f_{s}(x)/∂s_{i} ∂f_{s}(x)/∂s_{j}}{f′_{s}(x)}  (2.1)

the multiplication is defined by

< ∂_{s_{i}} · ∂_{s_{j}}, ∂_{s_{k}} >= −(n + 1) res_{x=∞} \frac{∂f_{s}(x)/∂s_{i} ∂f_{s}(x)/∂s_{j} ∂f_{s}(x)/∂s_{k}}{f′_{s}(x)} . (2.2)

Documenta Mathematica · Extra Volume ICM 1998 · II · 315–326

This is a polynomial FM. The deformed flat coordinates are given by oscillatory integrals

t̃_{c} = \frac{1}{\sqrt{z}} ∫_{c} e^{z f_{s}(x)}dx  (2.3)

Here c is any 1-cycle in C that goes to infinity along the direction Re z f_{s}(x) → −∞. 2.3. Quantum cohomology of a 2d-dimensional smooth projective variety X is a Frobenius structure of the charge d on a domain M ⊂ H^{∗}(X, C)/2πiH^{2}(X, Z) (we assume that H^{odd}(X) = 0 to avoid working with supermanifolds, see [KM]). It is an analytic perturbation in the sense of n.2.1 of the cubic for A_{0} = H^{∗}(X) defined by a generating function of the genus zero Gromov - Witten (GW) invariants of X [W1, W2, MS, RT, KM, Beh]. They are defined as intersection numbers of certain cycles on the moduli spaces of stable maps [KM]

X_{[β],l} := {β : (S^{2}, p_{1}, …, p_{l}) → X, given homotopy class [β] ∈ H_{2}(X; Z)} .

The holomorphic maps β of the Riemann sphere S^{2} with l ≥ 1 distinct marked points are considered up to a holomorphic change of parameter. The markings define evaluation maps p_{i} : X_{[β],l} → X, (β, p_{1}, …, p_{l}) ↦ β(p_{i}).

F (t) = F_{0}(t) + ∑_{[β]≠0} ∑_{l} ⟨e^{t′′}⟩_{[β],l} exp ∫_{S^{2}} β^{∗}(t′)

⟨e^{t}⟩_{[β],l} : = \frac{1}{l!} ∫_{X_{[β],l}} p_{1}^{∗}(t) ∧ … ∧ p_{l}^{∗}(t)  (2.4)

for t = (t′, t′′) ∈ H^{∗}(X), t′ ∈ H^{2}(X)/2πiH^{2}(X, Z), t′′ ∈ H^{∗≠2}(X). This potential together with the Poincaré pairing on T M = H^{∗}(X), the unity vector field e = 1 ∈ H^{0}(X), the Euler vector field E(t) = ∑(1 − q_{α})t^{α}e_{α} + c_{1}(X), t = t^{α}e_{α}, e_{α} ∈ H^{2q_{α}}(X) gives the needed Frobenius structure. The deformed flat coordinates are generating functions of certain “gravitational descendents” [Du5], see also [DW,

Ho, Gi1] t̃_{α}(t; z) = ∑_{p=0}^{∞} ∑_{[β],l} ⟨z^{μ+p}z^{c_{1}(X)}τ_{p}(e_{α}) ⊗ 1 ⊗ e^{t′′}⟩_{[β],l}e^{∫_{S^{2}} β^{∗}(t′)}, α = 1, …, n = dim H^{∗}(X), μ(e_{α}) = (q_{α} − d/2)e_{α}, The definition of the descendents ⟨τ_{p_{1}}(a_{1}) ⊗ τ_{p_{2}}(a_{2}) ⊗ … ⊗ τ_{p_{l}}(a_{l})⟩_{[β],l} see in [W2], [KM]. The definition of GW invariants can be extended on a certain class of compact symplectic varieties X using Gromov’s theory [Gr] of pseudoholomorphic curves, see [W2, MS, RT].

3. Classification of semisimple FMs. 3.1. Definition. A point t ∈ M is called semisimple if the algebra on T_{t}M is semisimple. A connected FM M is called semisimple if it has at least one semisimple point. Classification of semisimple FMs can be reduced, by a nonlinear change of coordinates, to a system of ordinary differential equations. First we will describe these new coordinates. 3.2. Canonical coordinates on a semisimple FM. Denote u_{1}(t), …, u_{n}(t) the roots of the characteristic polynomial of the operator of multiplication by the Euler vector field E(t) (n = dim M). Denote M^{0} ⊂ M the open subset where all the roots are pairwise distinct. It turns out [Du2] that the functions u_{1}(t), …, u_{n}(t) are independent local coordinates on M^{0} ≠ ∅. In these coordinates ∂_{i} · ∂_{j} = δ_{ij}∂_{i}, where ∂_{i} := ∂/∂u_{i}, and E = ∑_{i} u_{i}∂_{i}. The local coordinates u_{1}, …, u_{n} on M^{0} are called canonical. 3.3. Deformed flat connection in the canonical coordinates and isomonodromy deformations. Staying in a small ball on M^{0}, let us order the canonical coordinates and choose the signs of the square roots ψ_{i1} := \sqrt{< ∂_{i}, ∂_{i} >}, i = 1, …, n. The orthonormal frame of the normalized idempotents ∂_{i} establishes a local trivialization of the tangent bundle T M^{0}. The deformed flat connection ˜∇ in T M^{0} is recasted into the following flat connection in the trivial bundle M^{0} × C × C^{n}

˜∇_{i} = ∂_{i} − z E_{i} − V_{i}, ˜∇_{d/dz} = ∂_{z} − U − z^{−1}V,  (3.1)

other components are obvious. Here the n × n matrices E_{i}, U, V = (V_{ij}) read (E_{i})_{kl} = δ_{ik}δ_{il}, U = diag (u_{1}, …, u_{n}), V = Ψ μ Ψ^{−1} = −V^{T} where the matrix Ψ = (ψ_{iα}) satisfying Ψ^{T} Ψ = η is defined by ψ_{iα} := ψ_{i1}^{−1}∂t_{α}/∂u_{i}, i, α = 1, …, n. The skew-symmetric matrices V_{i} are determined by the equations [U, V_{i}] = [E_{i}, V].

Flatness of the connection (3.1) reads as the system of commuting timedependent Hamiltonian flows on the Lie algebra so(n) ∋ V equipped with the standard linear Poisson bracket

∂_{i}V = {V, H_{i}(V; u)} , i = 1, …, n  (3.2)

with the quadratic Hamiltonians H_{i}(V; u) = \frac{1}{2} ∑_{j≠i} \frac{V_{ij}^{2}}{u_{i}−u_{j}} , i = 1, …, n. For the first nontrivial case n = 3 (3.2) can be reduced to a particular case of the classical Painlevé-VI equation. The monodromy of the operator ˜∇_{d/dz} (i.e., the monodromy at the origin, the Stokes matrix, and the central connection matrix, see definitions in [Du3, Du5]) does not change with small variations of a point u = (u_{1}, …, u_{n}) ∈ M. 3.4. Parametrization of semisimple FMs by monodromy data of the deformed flat connection. We now reduce the above system of nonlinear differential equations to a linear boundary value problem of the theory of analytic functions. First we will describe the set of parameters of the boundary value problem. 3.4.1. Monodromy at the origin (defined also for nonsemisimple FMs) consists of:

- a linear n-dimensional space V with a symmetric nondegenerate bilinear form < , >, a skew-symmetric linear operator μ : V → V, < μ(a), b > + < a, μ(b) >= 0, and a marked eigenvector e_{1} of μ, μ(e_{1}) = −d/2 e_{1}. In main examples the operator μ will be diagonalizable.

- A linear operator R : V → V satisfying the following properties: (1) R = R_{1} + R_{2} + … where R_{k}(V_{λ}) ⊂ V_{λ+k} for the root decomposition of V = ⊕_{λ}V_{λ}, μ(v_{λ}) = λv_{λ} for v_{λ} ∈ V_{λ}. (2) {Rx, y} + {x, Ry} = 0 for any x, y ∈ V where {x, y} := ⟨e^{πiμ}x, y⟩ .

3.4.2. Stokes matrix is an arbitrary n × n upper triangular matrix S = (s_{ij}) with s_{ii} = 1, i = 1, …, n. We treat it as a bilinear form < a, b >_{S}:= a^{T} Sb, a, b ∈ C^{n}. 3.4.3. Central connection matrix is an isomorphism C : C^{n} → V satisfying < a, b >_{S}=< Ca, e^{πiμ}e^{πiR}Cb > for any a, b ∈ C^{n}. The matrices S and C are defined up to a transformation S ↦ DSD, C ↦ CD, D = diag (±1, …, ±1). 3.4.4. Riemann - Hilbert boundary value problem (RH b.v.p.). Let us fix a radius R > 0 and an argument 0 ≤ φ < 2π. Denote ℓ = ℓ_{+} ∪ ℓ_{−} the oriented line ℓ_{+} = {z | arg z = φ}, ℓ_{−} = {z | arg z = φ + π}. It divides the complex z-plane into two halfplanes Π_{right} and Π_{left}. For a given u = (u_{1}, …, u_{n}) with u_{i} ≠ u_{j} for i ≠ j and for given monodromy data we are looking for: (1) n × n matrix-valued functions Φ_{right}(z), Φ_{left}(z) analytic for |z| > R and z ∈ Π_{right} and z ∈ Π_{left} resp., continuous up to the boundaries |z| = R or z ∈ ℓ and satisfying Φ_{right/left}(z) = 1 + O (1/z) for |z| → ∞ within the correspondent half-plane Π_{right/left}; (2) n × n matrix-valued function Φ_{0}(z) (with values in Hom(V, C^{n})) analytic for |z| < R and continuous up to the boundary |z| = R, such that det Φ_{0}(0) ≠ 0. The boundary values of the functions must satisfy

Φ_{right}(z)e^{zU} = Φ_{left}(z)e^{zU}S for z ∈ ℓ_{+}, |z| > R;
Φ_{right}(z)e^{zU} = Φ_{left}(z)e^{zU}S^{T} for z ∈ ℓ_{−}, |z| > R;
Φ_{right}(z)e^{zU} = Φ_{0}(z)z^{μ}z^{R}C for |z| = R, z ∈ Π_{right};
Φ_{left}(z)e^{zU}S = Φ_{0}(z)z^{μ}z^{R}C for |z| = R, z ∈ Π_{left}.

The branchcut in the definition of the multivalued functions z^{μ} and z^{R} is chosen along ℓ_{−}. For solvability of the above RH b.v.p. we have also to require the complex numbers u_{1}, …, u_{n} to be ordered in such a way, depending on φ, that

R_{jk} := {z = −ir(ū_{j} − ū_{k})|r ≥ 0} ⊂ Π_{left} for j < k.  (3.3)

Denote U(φ) ⊂ C^{n} the set of all points u = (u_{1}, …, u_{n}) with u_{i} ≠ u_{j} for i ≠ j satisfying (3.3). Let U_{0}(φ) be the subset of points u ∈ U(φ) such that: (1) the RH b.v.p. is solvable and (2) all the coordinates of the vector Φ_{0}(0)e_{1} are distinct from zero. It can be shown (cf. [Mi], [Mal]) that the solution Φ_{right/left} = Φ_{right/left}(z; u), Φ_{0} = Φ_{0}(z; u) of the RH b.v.p depends analytically on u ∈ U_{0}(φ). Let Φ_{0}(z; u) = ∑_{p=0}^{∞} ϕ_{p}(u)z^{p}. Denote (only here) ( , ) the standard sum of squares quadratic form on C^{n}. Choose a basis e_{1}, e_{2}, …, e_{n} of eigenvectors of μ, μ(e_{α}) = μ_{α}e_{α}, μ_{1} = −d/2, and put η_{αβ} :=< e_{α}, e_{β} >, (η^{αβ}) := (η_{αβ})^{−1}. Theorem 1 [Du2, Du3, Du5]. The formulae

t_{α}(u) = (ϕ_{0}(u)e_{α}, ϕ_{1}(u)e_{1}), t^{α} = η^{αβ}t_{β}, α = 1, …, n,
F = 1/2 [(ϕ_{0}t, ϕ_{1}t) − 2(ϕ_{0}t, ϕ_{1}e_{1}) + (ϕ_{1}e_{1}, ϕ_{2}e_{1}) − (ϕ_{3}e_{1}, ϕ_{0}e_{1})]

E(t) = ∑_{α=1}^{n}(1 + μ_{1} − μ_{α})t^{α}∂_{α} + ∑_{α}(R_{1})_{1}^{α}∂_{α}

define on U_{0}(φ) a structure of a semisimple FM F r(V, < , >, μ, e_{1}, R, S, C). Any semisimple FM locally has such a form. 3.5. Remark. The columns of the matrices Φ_{0}(z; u)z^{μ}z^{R} and Φ_{right}(z; u)e^{zU} correspond to two different bases in the space of deformed flat coordinates. The first basis is a deformation, z → 0, of the original flat coordinates, t̃ = [t + O(z)]z^{μ}z^{R}. The second one, defined only in the semisimple case, corresponds to a system of deformed flat coordinates given by oscillatory integrals (see (2.3) and Section 6 below). 3.6. Global structure of semisimple FMs and action of the braid group on the monodromy data. Let B_{n} be the group of braids with n strands. We will glue globally the FM from the charts described in n.3.4 with different S and C. So, for brevity, we redenote here the charts F r(V, < , >, μ, e_{1}, R, S, C) =: F r(S, C). The charts will be labelled by braids σ ∈ B_{n}. By definition in the chart F r(S^{σ}, C^{σ}) the functions t^{α}(u), F (u) are obtained as the result of analytic continuation from F r(S, C) along the braid σ. The action S ↦ S^{σ}, C ↦ C^{σ} of the standard generators σ_{1}, …, σ_{n−1} of B_{n} is given by S^{σ_{i}} = KSK, C^{σ_{i}} = CK where the only nonzero entries of the matrix K = K^{(i)}(S) are K_{kk} = 1, k = 1, …, n, k ≠ i, i+1, K_{i,i+1} = K_{i+1,i} = 1, K_{i,i} = −s_{i,i+1}. Let B_{n}(S, C) ⊂ B_{n} be the subgroup of all braids σ such that S^{σ} = DSD, C^{σ} = CD, D = diag(±1, …, ±1). Theorem 2 [Du3, Du5]. Any semisimple FM has the form

M = ∪_{σ∈B_{n}/B_{n}(S,C)}F r(V, < , >, μ, e_{1}, R, S^{σ}, C^{σ})

where the glueing of the charts is given by the above action of B_{n}. 3.7. Tau-function of the isomonodromy deformation and elliptic GW invariants. Like in n.2.2, the genus g GW invariants can be defined in terms of the intersection theory on the moduli space X_{[β],l}(g) of stable maps β : C_{g} → X of curves of genus g with markings [KM, Beh]. It turns out that, assuming semisimplicity of quantum cohomology of X, the elliptic (i.e., of g = 1) GW invariants can still be expressed via isomonodromy deformations. To this end we define, following [JM], the τ-function τ(u_{1}, …, u_{n}) of a solution V (u) of the system (3.2) by the quadrature of a closed 1-form d log τ = ∑_{i=1}^{n} H_{i}(V (u); u)du_{i}. We define G-function of the FM by G = log(τ/J^{1/24}) where J = det (∂t^{α}/∂u_{i}) = ± ∏_{i=1}^{n} ψ_{i1}(u). Theorem 3 [DZ2]. For an arbitrary semisimple FM the G-function is the unique, up to an additive constant, solution to the system of [Ge] for the generating function of elliptic GW invariants satisfying Lie_{e}G = 0, Lie_{E}G = const. 3.8. Problem of selection of semisimple FMs with good analytic properties of n.2.1 is still open. Experiments for small n [Du3] show that such solutions are rare exceptions among all semisimple FMs. Analyticity of the G-function near the point t′ = −∞, t′′ = 0 imposes further restrictions on M [DZ2]. To solve the problem one is to study the behaviour of solutions of the RH b.v.p. in the limits when two or more among the canonical coordinates merge. At the point t′ = −∞, t′′ = 0 all u_{1} = … = u_{n} = 0.

4. Examples of monodromy data. 4.1. Universal unfoldings of isolated singularities. The subspace M_{0} ⊂ M consists of the parameters s for which the versal deformation f_{s}(x) has n = dim M distinct critical values u_{1}(s), …, u_{n}(s). These will be our canonical coordinates. The monodromy at the origin is the classical monodromy operator [AGV] of the singularity, the Stokes matrix coincides with the matrix of the variation operator in the Gabrielov’s distinguished basis of vanishing cycles (see [AGV]; we may assume that dim x ≡ 1 (mod4)). 4.2. Quantum cohomology of Fano varieties. The following two questions are to be answered in order to apply the above technique to the quantum cohomology of a variety X. Problem 1. When does the generating series (2.4) converge? Problem 2. For which X the quantum cohomology of X is semisimple?

Hopefully, in the semisimple case the convergence can be proved on the basis of the differential equations of n.3. To our opinion the problem 2 is more deep. A necessary condition to have a semisimple quantum cohomology is that X must be a Fano variety. It was conjectured to be also a sufficient condition [TX], [Man1]. We analyze below one example and suggest some more modest conjecture describing also a part of the monodromy data. 4.2.1. Quantum cohomology of projective spaces. For X = P^{d}: (1) the monodromy at the origin is given by the bilinear form < e_{α}, e_{β} >= δ_{α+β,d+2} in H^{∗}(X) = V = span (e_{1}, …, e_{d+1}), the matrix μ = 1/2diag (−d, 1 − d, …, d − 1, d) and R is the matrix of multiplication by the first Chern class R = R_{1} = c_{1}(X), R e_{α} = (d + 1)e_{α+1} for α ≤ d, R e_{d+1} = 0. With obvious modifications these formulae work also for any variety X with H^{odd}(X) = 0 (see [Du3]).

(2) The Stokes matrix S = (s_{ij}) has the form

s_{ij} = ( \begin{matrix}d + 1 \\ j − i \end{matrix} ) for i ≤ j, s_{ij} = 0 for i > j.  (4.1)

This form of Stokes matrix was conjectured in [CV], [Zas] but, to our knowledge, it was proved only in [Du5] for d = 2 and in [Guz] for any d. (3) The central connection matrix C has the form C = C′C′′, C′ = (C′_{β}^{α}),

C′′ = (C′′_{j}^{β}) where C′′_{j}^{β} = [2πi(j − 1)]^{β−1}/(α − 1)!, j, β = 1, …, d + 1,

C′_{β}^{α} = \frac{(−1)^{d+1}}{(2π)^{\frac{d+1}{2}} id̄} \{ \begin{matrix}A_{α−β}(d), α ≥ β \\ 0, α < β \end{matrix} with d̄ = 1 for d =even and d̄ = 0 for

d =odd where the numbers A_{0}(d) = 1, A_{1}(d), …, A_{d}(d) are defined from the Laurent expansion for x → 0: 1/x^{d+1} + A_{1}(d)/x^{d} + … + A_{d}(d)/x + O(1) = (−1)^{d+1}Γ^{d+1}(−x)e^{−πid̄x}. Observe that (4.1) is the Gram matrix of the bilinear form χ(E, F) := ∑_{k}(−1)^{k}dim Ext^{k}(E, F) in the basis given by a particular full system E_{j} = O(j−1), j = 1, …, d+1 of exceptional objects in the derived category Der^{b}(Coh(P^{d})) of coherent sheaves on P^{d} [Rud]. The columns of the matrix C′′ are the components of the Chern character ch (E_{j}) = e^{2πic_{1}(E_{j})}, j = 1, …, d + 1. The geometrical meaning of the matrix C′ remains unclear. In other charts of the FM S^{σ} and C^{σ} = C′C′′^{σ}, σ ∈ B_{n}, have the same structure for another full system E_{1}^{σ}, …, E_{d+1}^{σ} ∈ Der^{b}(Coh(P^{d})) of exceptional objects, where the action of the braid group (E_{1}, …, E_{d+1}) ↦ (E_{1}^{σ}, …, E_{d+1}^{σ}) is described in [Rud]. Warning: the points of the FM corresponding to the restricted quantum cohomology [MM], where t ∈ H^{2}(P^{d}), do not belong to the chart F r(S, C) with the matrices S and C as above!

4.2.2. Conjecture. We say that a Fano variety X is good if Der^{b}(Coh(X)) admits, in the sense of [BP], a full system of exceptional objects E_{1}, …, E_{n}, n = dim H^{∗}(X). Our conjecture is that (1) the quantum cohomology of X is semisimple iff X is a good Fano variety; (2) the Stokes matrix S = (s_{ij}) is equal to s_{ij} = χ(E_{i}, E_{j}), i, j = 1, …, n; (3) the central connection matrix has the form C = C′C′′ when the columns of C′′ are the components of ch (E_{j}) ∈ H^{∗}(X) and C′ : H^{∗}(X) → H^{∗}(X) is some operator satisfying C′(c_{1}(X)a) = c_{1}(X)C′(a) for any a ∈ H^{∗}(X).

For X = P^{d} the validity of the conjecture follows from n.4.2.1 above. The conjecture probably can be derived from more general conjecture [Kon] about equivalence of Der^{b}(Coh(X)) to the Fukaya category of the mirror pair X^{∗} of X. According to it (see also [EHX, Gi1, Gi2]) the basis of horizontal sections of ˜∇ corresponding to the columns of Φ_{right}(z; u)e^{zU} coincides with the oscillatory integrals of the Fukaya category of X^{∗}. However, we do not know who is the first factor C′ of the connection matrix in this general setting.

5. Intersection form of a FM is a bilinear symmetric pairing on T^{∗}M defined by (ω_{1}, ω_{2})|_{t} := i_{E(t)}(ω_{1} · ω_{2}), ω_{1}, ω_{2} ∈ T_{t}^{∗}M. Discriminant is the locus Σ = {t ∈ M | det( , )_{t} = 0}. On M \Σ the inverse to ( , ) determines a flat metric and, thus, a local isometry π : (M \ Σ, ( , )^{−1}) → C^{n} where C^{n} is equipped with a constant complex Euclidean metric ( , )_{0}. This local isometry is called period mapping (our terminology copies that of the singularity theory where the geometrical structures with the same names live on the bases of universal unfoldings, see [AGV]). The image π(Σ) is a collection of nonisotropic hyperplanes in C^{n}. Multivaluedness of π is described by the monodromy representation π_{1}(M \ Σ) → Iso (C^{n}, ( , )_{0}) (for d ≠ 1 to the orthogonal group O (C^{n}, ( , )_{0})). The image W (M) of the representation is called monodromy group of the FM M. In the semisimple case it is always an extension of a reflection group (see details in [Du5]). Our hope is that, for a semisimple FM M with good analytic properties, the monodromy group acts discretely in some domain Ω ⊂ C^{n}, and M is identified with a branched covering of the quotient Ω/W (M). 5.1. Examples of a FM with W (M)= finite irreducible Coxeter group W acting in R^{n} [Du3]. These are polynomial FMs, M = C^{n}/W, constructed in terms of the theory of invariant polynomials of W. Conjecturally, all polynomial semisimple FMs are equivalent to the above and to their direct sums.

This construction was generalized in [DZ1] to certain extensions of affine Weyl groups and in [Ber] to Jacobi groups of the types A_{n}, B_{n}, G_{2}. For the quantum cohomology of P^{2} the monodromy group is isomorphic to P SL_{2}(Z) × {±1} [Du5].

6. Mirror construction represents certain system of deformed flat coordinates on a semisimple FM by oscillatory integrals I_{j}(u; z) = \frac{1}{\sqrt{z}} ∫_{Z_{j}} e^{zλ(p;u)}dp, ˜∇I_{j}(u; z) = 0, j = 1, …, n having the phase function λ(p; u) depending on the parameters u = (u_{1}, …, u_{n}) defined on a certain family of open Riemann surfaces R_{u} ∋ p realized as a finite-sheeted branched covering λ : R_{u} → D ⊂ C over a domain in the complex plain. The ramification points of R_{u}, i.e., the critical values of the phase function, are u_{1}, …, u_{n}. The 1-cycles Z_{1}, …, Z_{n} on R_{u} go to infinity in a way that guarantees the convergence of the integrals. The function λ(p; u) satisfies an important property: for any two critical points p_{i}^{1, 2} ∈ R_{u} with the same critical value u_{i} the equality d^{2}λ(p_{i}^{1}; u)/dp^{2} = d^{2}λ(p_{i}^{2}; u)/dp^{2} must hold true. The metric < , > and the trilinear form c(a_{1}, a_{2}, a_{3}) :=< a_{1} · a_{2}, a_{3} > are given by the residue formulae similar to (2.1), (2.2). The solutions p = p(u; λ) of the equation λ(p; u) = λ are the flat coordinates of the flat pencil of the metrics ( , ) − λ < , > on T^{∗}M [Du3-Du5].

For the case when generically there is a unique critical point p_{i} over u_{i} for each i and R_{u} can be compactfied to a Riemann surface of a finite genus g, we arrive at the Hurwitz spaces of branched coverings [Du1, Du3].

The construction of the Riemann surfaces R_{u}, of the phase function λ(p; u) and of the cycles Z_{1}, …, Z_{n} is given in [Du5] by universal formulae assuming det(S + S^{T}) ≠ 0. In the quantum cohomology of a d-fold X the last condition is valid for d = even. For d = odd one has det(S − S^{T}) ≠ 0. In this case one can represent the deformed flat coordinates by oscillatory integrals with the phase function λ(p, q; u) = ν(p; u) + q^{2} depending on two variables p, q. The details will be published elsewhere.

7. Gravitational descendents is a physical name for intersection numbers < τ_{m_{1}}(a_{1}) ⊗ … ⊗ τ_{m_{l}}(a_{l}) > of the pull-back cocycles p_{1}^{∗}(a_{1}), …, p_{l}^{∗}(a_{l}) with the Mumford - Morita - Miller cocycles ψ_{1}^{m_{1}}, …, ψ_{l}^{m_{l}} ∈ H^{∗}(X_{[β],l}) [W2], [DW], [KM]. We will describe first their genus g = 0 generating function

F_{0}(T) = ∑_{[β]} ∑_{l} ⟨e^{∑_{α=1}^{n} ∑_{p=0}^{∞} T^{α,p}τ_{p}(e_{α})}⟩_{[β],l,g=0}. Here T = (T^{α,p}) are indeterminates (the coordinates on the “big phase space”, according to the physical terminology). This function has the form F_{0}(T) = 1/2 ∑ Ω_{α,p; β,q} (t(T)) T̃^{α,p}T̃^{β,q} where T̃^{α,p} = T^{α,p} for (α, p) ≠ (1, 1), T̃^{1,1} = T^{1,1} − 1, the functions Ω_{α,p; β,q}(t) on M are the coefficients of the expansion of the matrix valued function Ω_{αβ}(z, w; t) := (z+w)^{−1} [(Φ_{0}^{T} (w; t)Φ_{0}(z; t))_{αβ} − η_{αβ}] = ∑_{p, q≥0} Ω_{α,p; β,q}(t)z^{p}w^{q}, the vector func-

tion t(T) = (t^{1}(T), …, t^{n}(T))

t^{α}(T) = T^{α,0} + ∑_{q>0} T^{β,q}∇^{α}Ω_{β,q; 1,0}(t)|_{t^{α}=T^{α,0}} + …  (7.1)

is defined as the unique solution of the following fixed point equation t = ∇ ∑_{α,p} T^{α,p}Ω_{α,p; 1,0}(t).

The generating function F_{1}(T) of the genus g = 1 descendents has the form [DZ2], [DW], [Ge] F_{1}(T) = [G(t) + \frac{1}{24} log det M_{αβ}(t, ṫ)]_{t=t(T), ṫ=∂_{T^{1,0}} t(T)} where

G(t) is the G-function of the FM, the matrix M_{αβ}(t, ṫ) has the form M_{αβ}(t, ṫ) = ∂_{α}∂_{β}∂_{γ}F (t)ṫ^{γ}, the vector function t(T) is the same as above. The structure of the genus g = 2 corrections is still unclear, although there are some interesting conjectures [EX] related, in the case of quantum cohomology, to the Virasoro constraints for the full partition function

Z(T; ε) = exp ∑_{g=0}^{∞} ε^{2g−2}F_{g}(T)  (7.2)

ε is a formal small parameter called string coupling constant.

8. Integrable hierarchies of PDEs of the KdV type and FMs. The idea that FMs may serve as moduli of integrable hierarchies of evolutionary equations (see [W2], [Du2], [Du3]) is based on

(1) the theorem of Kontsevich - Witten identifying the partition function (7.2) in the case X = point as the tau-function of a particular solution of the KdV hierarchy.

(2) The construction [Du2, Du3] of bihamiltonian integrable hierarchy of the Whitham type ∂_{T^{α,p}}t = {t(X), H_{α,p}}_{1} = K_{α,p}^{(0)}(t, t_{X}) (the vector function in the r.h.s. depends linearly on the derivatives t_{X}) such that the full genus zero partition function is the tau-function of a particular solution (7.1) to the hierarchy. The solution is specified by the symmetry constraint t_{X} −∑ T^{α,p}∂_{T^{α,p−1}}t = 1. The phase space of the hierarchy is the loopspace L(M) = {(t^{1}(X), …, t^{n}(X)) | X ∈ S^{1}}, the first Hamiltonian structure is {t^{α}(X), t^{β}(Y)}_{1} = η^{αβ}δ′(X − Y), the second one { , }_{2} is determined [Du3] by the flat metric ( , ) according to the general scheme of [DN]. The Hamiltonians are H_{α,p} = ∫ Ω_{α,p; 1,0}(t) dX. Actually, any linear combination { , }_{2} − λ{ , }_{1} with an arbitrary λ is again a Poisson bracket on the loop space since ( , ) and < , > form a flat pencil of metrics on T^{∗}M [Du3, Du4] (this bihamiltonian property is a manifestation of integrability of the hierarchy, see [Mag], [Du4]).

What we want to construct is a deformation of the hierarchy of the form ∂_{T^{α,p}}t = K_{α,p}^{(0)}(t, t_{X}) + ∑_{g≥1} ε^{2g}K_{α,p}^{(g)}(t, t_{X}, …, t^{(2g+1)}) where K_{α,p}^{(g)} are some vector valued polynomials in t_{X}, …, t^{(2g+1)} with the coefficients depending on t ∈ M. All the equations of the hierachy must commute pairwise. The full partition function must be the tau-function of a particular solution to the hierachy. The first g = 1 correction for an arbitrary semisimple FM was constructed in [DZ]. Its bihamiltonian structure is described, for d ≠ 1, by a nonlinear deformation of the Virasoro algebra with the central charge c = 6ε^{2}(1 − d)^{−2}[n − 4tr μ^{2}]. For the FMs corresponding to the ADE Coxeter groups this formula gives the known result [FL] for the central charge of the classical W -algebra of the ADE-type c = 12ε^{2}ρ^{2}, where ρ is half of the sum of positive roots of the corresponding root system.

More recently it has been proved [DZ3] for a semisimple FM that the partition function (7.2) is annihilated, within the genus one approximation, by half of a Virasoro algebra described in terms of the monodromy data of the FM. Acknowledgments. I am grateful to D.Orlov for helpful discussion of derived categories, and to S.Barannikov and M.Kontsevich for fruitful conversations.

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B. Dubrovin, SISSA, Via Beirut, 2-4, I-34013 Trieste, Italy