SIMPLE SINGULARITIES AND INTEGRABLE HIERARCHIES

SIMPLE SINGULARITIES AND INTEGRABLE HIERARCHIES ALEXANDER B. GIVENTAL AND TODOR E. MILANOV To Alan Weinstein on his 60-th birthday Abstract. The paper [11] gives a construction of the total descendent potential corresponding to a semisimple Frobenius manifold. In [12], it is proved that the total descendent potential corresponding to K. Saito s Frobenius structure on the parameter space of the miniversal deformation of the A n 1 -singularity satisfies the modulo-n reduction of the KP-hierarchy. In this paper, we identify the hierarchy satisfied by the total descendent potential of a simple singularity of the A, D, E-type. Our description of the hierarchy is parallel to the vertex operator construction of Kac Wakimoto [17] except that we give both some general integral formulas and explicit numerical values for certain coefficients which in the Kac Wakimoto theory are studied on a case-by-case basis and remain, generally speaking, unknown. 1. The ADE-hierarchies. According to Date Jimbo Kashiwara Miwa [6] and I. Frenkel [10], the KdV-hierarchy of integrable systems can be placed under the name into the list of more general integrable hierarchies corresponding to the ADE Dynkin diagrams. These hierarchies are usually constructed (see [16]) using representation theory of the corresponding loop groups. V. Kac and M. Wakimoto [17] describe the hierarchies even more explicitly in the form of the so called Hirota quadratic equations expressed in terms of suitable vertex operators. One of the goals of the present paper is to show how the vertex operator description of the Hirota quadratic equations (certainly the same ones, even though we don t quite prove this) emerges from the theory of vanishing cycles associated with the ADE singularities. Let f be a weighted-homogeneous polynomial in C 3 with a simple critical point at the origin. According to V. Arnold [1] simple singularities of holomorphic functions are classified by simply-laced Dynkin diagrams: A N, N 1 : f = xn+1 1 N + 1 + x2 2 2 + x2 3 2, D N, N 4 : f = x 2 1 x 2 x N 1 2 + x 2 3, E 6 : f = x 3 1 + x 4 2 + x 2 3, E 7 : f = x 3 1 + x 1 x 3 2 + x 2 3, E 8 : f = x 3 1 + x 5 2 + x 2 3. Let H = C[x 1, x 2, x 3 ]/(f x1, f x2, f x3 ) denote the local algebra of the critical point. We equip H with a non-degenerate symmetric bilinear form (, ) by picking a weighted - homogeneous holomorphic volume ω = dx 1 dx 2 dx 3 and using the residue pairing: (ϕ, ψ) := Res 0 ϕ(x) ψ(x) ω f x1 f x2 f x3. This material is based upon work supported by National Science Foundation under Grants No. DMS-0072658 and DMS-0306316. 1

2 ALEXANDER B. GIVENTAL AND TODOR E. MILANOV Let H = H((z 1 )) be the space of Laurent series f(z) = k Z f kz k in one indeterminate z 1 (i.e. finite in the direction of positive k) with vector coefficients f k H. We endow H with the symplectic form Ω(f,g) := 1 2πi (f( z),g(z)) dz. The polarization H = H + H where H + = H[z] and H = z 1 H[[z 1 ]] is Lagrangian and identifies H with the cotangent bundle space T H +. The Hirota quadratic equations are imposed on asymptotical functions of q = q 0 +q 1 z+q 2 z 2 +... H +. By an asymptotical function we mean an expression of the form Φ = exp g 1 F (g) (q) g=0 where usually F (g) will be formal functions on H +. By definition, vertex operators are elements of the Heisenberg group acting on such functions. Given a sum f = fk z k (possibly infinite in both directions) one defines the corresponding vertex operator of the form e Ω(f,q)/ e f+ = exp{ k 0( 1) k+1 a f a 1 kq a k/ } exp{ k 0 fk a / qk}. a a Here f a k, qa k are components of the vectors f k,q k in an orthonormal basis. We will make use of the vertex operators Γ φ (λ) corresponding to 2-dimensional homology classes φ H 2 (f 1 (1)) Z N and defined as follows. Take f := k Z I (k) φ (λ)( z)k where di (k) /dλ = I(k+1) φ φ, and I ( 1) φ (λ) H is the following period vector: (I ( 1) φ (λ), [ψ a ]) := 1 ψ a (x) ω 2π φ f 1 (λ) df. The cycle φ is transported from the level surface f 1 (1) to f 1 (λ), and ψ a are weighted-homogeneous functions representing a basis in the local algebra H. 1 The functions (I (k) φ, [ψ a]) are proportional to the fractional powers λ ma/h k 1 where h is the Coxeter number and m a = 1 + h deg ψ a are the exponents of the appropriate reflection group A N, D N or E N. The lattice H 2 (f 1 (1)) carries the action of the monodromy group (defined via morsification of the function f) which is the reflection group with respect to the intersection form of cycles. The form is negative definite, and we will denote, the positive definite form opposite to it. Let A denote the set of vanishing cycles, i.e. the set of classes α H 2 (V 1 ) with α, α = 2 such that the reflections φ φ α, φ α belong to the monodromy group. The Hirota quadratic equation of the 1 As it follows, for instance, from [13], the integral on the R.H.S. depends only on the class [ψ a] H.

SIMPLE SINGULARITIES AND INTEGRABLE HIERARCHIES 3 ADE-type takes on the form [ ] dλ (1) Res λ= a α Γ α (λ) Γ α (λ) (Φ Φ) = λ (2) k 0 a α A ( m a h + k)(qa k 1 1 qa k )( qk a N(h + 1) 12h 1 1 q a k (Φ Φ)+ ) (Φ Φ). The tensor product sign means that the functions depend on two copies q and q of the variable q, and the objects on the left of refer to q = q while those on the right to q = q. The equation can be interpreted as follows. Set q = x + y,q = x y. and expand (1,2) as a power series in y. Namely, rewrite the vertex operators: Γ α (λ) Γ α (λ) = exp{ 2( 1) k+1 f a 1 k 1/2 y a k } exp{ f a k 1/2 y a k } where the coefficients fk a (respectively fa 1 k ) are proportional to negative (respectively positive) fractional powers of λ. The residue sum (which should be understood here as the coefficient at λ 0 ) can therefore be written as a power series y m P m ( y ) in y with coefficients P m which are differential polynomials. Also, Φ Φ = Φ(x + y)φ(x y) can be expanded into the Taylor power series in y with coefficients which are quadratic expressions in partial derivatives of Φ(x). Finally the operator in (2) assumes the form 2 a,k (m a/h + k)yk a yk a. Equating coefficients in (1,2) at the same monomials y m we obtain a hierarchy of quadratic relations between partial derivatives of Φ(x). In particular, the equation corresponding to y 0 shows that (3) a α = α A N(h + 1) 12h is a necessary condition for consistency of the hierarchy (i.e. for existence of a non-zero solution Φ). According to C. Hertling (see the last chapter in [15]) for any weighted - homogeneous singularity the expressions N(h + 1)/(12h) and h 2 a m a(h m a )/2 coincide. Therefore the operator on the R.H.S. of the Hirota equation is twice the Virasoro operator 2 a,k ( m a h + k) ya k y a k + a m a (h m a ) 4h 2. The coefficients a α actually depend only on the orbit of the vanishing cycle α under the action of the classical monodromy operator defined by transporting the cycles in f 1 (λ) around λ = 0 and acting as one of the Coxeter elements in the reflection group. In fact the root system A consists of N such orbits with h elements each. Summing the vertex operators within the same orbit acts as taking the average over all h branches of the function λ 1/h. Thus the total sum does not contain fractional powers of λ when expanded near λ =. 2 In a sense it corresponds to the vector field λ λ in the Lie algebra of vector fields on the line see Section 7 for further information about this.

4 ALEXANDER B. GIVENTAL AND TODOR E. MILANOV The exact values of the coefficients a α can be described as follows. To a vector β H 2 (f 1 (1), C) C N, associate the meromorphic 1-form on C N (4) W β := 1 2 γ A 2 d γ, x β, γ γ, x. Let w be an element of the reflection group and α and β = wα be two roots. Then (5) a β /a α = exp w 1 κ κ W α = γ A κ, γ α,γ 2 /2 β,γ 2 /2, where κ C N denotes an eigenvector of the classical monodromy operator M with the eigenvalue exp(2πi/h). The R.H.S. does not depend on the path connecting κ with w 1 κ since W α is closed with logarithmic poles on some mirrors and with periods which are integer multiples of 2πi. It does not depend on the normalization of κ since W α is homogeneous of degree 0. Also, the identity (see i.g. [5], Section V.6.2) γ, x 2 = 2h x, x γ A implies that i P x a / x a W α = 2h and shows that M 1 κ W κ α = h 1 M h κ W κ α = 4πi so that a Mα = a α as expected. While the ratios of a α are determined by (5), the normalization of a α is found from (3) which says that the average value of a α is (h + 1)/12h 2. Later we give two other description of the coefficients a α as certain limits and as explicit case-by-case values. Conjecture. The Hirota quadratic equation (1 5) coincides (up to certain rescaling of the variables qk a ) with the corresponding ADE-hierarchy of Kac Wakimoto [17]. In Section 8 we confirm this conjecture in the cases A N, D 4 and E 6. 3 2. The total descendent potential. The second goal of this paper is to generalize to the ADE-singularities the result of [12] that the total descendent potential associated to the A n 1 -singularity in the axiomatic theory of topological gravity is a tau-function of the nkdv (or Gelfand-Dickey) hierarchy. According to E. Witten s conjecture [22] proved by M. Kontsevich [18], the following generating function for intersection indices on the Deligne Mumford spaces satisfies the equation of the KdV-hierarchy: 4 (6) D A1 = exp g,m g 1 m! M g,m i=1 m (ψ i + q k ψi k ). In the axiomatic theory, the total descendent potential is, by definition, an asymptotical function of the form D = exp g 0 g 1 F (g) (q) k=0 3 There has been a new development in the subject which leads, in particular, to the proof of the conjecture: motivated partly by this paper, E. Frenkel found a simple formula for the analogues in the Kac Wakimoto theory of the coefficients a α. 4 Here ψi is the 1-st Chern class of the line bundle over M g,m formed by the cotangent lines to the curves at the i-th marked points.

SIMPLE SINGULARITIES AND INTEGRABLE HIERARCHIES 5 where F (g) are formal functions on H + near the point q = 1z. (Here 1 is the unit element in the local algebra H.) This convention called the dilaton shift is already explicitly present in (6). The formal functions F (g) called the genus g descendent potentials are supposed to satisfy certain axioms dictated by Gromov Witten theory. The axioms (while not entirely known) are to include the so called string equation (SE), dilaton equation (DE), topological recursion relations (TRR or 3g 2-jet property) and Virasoro constraints. According to [14], the genus-0 axioms SE+DE+TRR for F (0) are equivalent to the following geometrical property ( ) of the Lagrangian submanifold L H = T H + defined as the graph of df (0) : ( ) L is a Lagrangian cone with the vertex at the origin and such that tangent spaces L to L are tangent to L exactly along zl. In other words, the cone L is swept by the family τ H zl τ of isotropic subspaces which form a variation of semi-infinite Hodge structures in the sense of S. Barannikov [3]. According to his results, this defines a Frobenius structure on the space of parameters τ. In the case of ADE-singularities (and, more generally, finite reflection groups) the Frobenius structures have been constructed by K. Saito [20]. Consider the miniversal deformation f τ (x) = f(x) + τ 1 ψ 1 (x) +... + τ N ψ N (x), where {ψ a } form a weighted-homogeneous basis in the local algebra H, and ψ N = 1. The tangent spaces T τ T to the parameter space T C N are canonically identified with the algebras of functions on the critical schemes crit(f τ ): τ a f τ / τ a mod ( f τ / x). The multiplication on the tangent spaces is Frobenius with respect to the following residue metric: ( τa, τb ) τ := ( 1 ψa (x)ψ b (x) ω 2πi )3 f τ x 1 f τ x 2 f τ x 3. The residue metric is known to be flat and together with the Frobenius multiplication, the unit vectors τ N and the Euler vector field N E := (deg τ a ) τ a τ a, deg τ a = 1 (m a 1)/h, a=1 forms a conformal Frobenius structure on T (see [7]). On the other hand, the condition ( ) involves only the symplectic structure Ω on H and the operator of multiplication by z and thus admits the following twisted loop group of symmetries: L (2) GL(H) = {M End(H) ((1/z)) M( z) M(z) = 1}. According to a result from [14], when the Frobenius structure associated to the cone L is semisimple, one can identify L with the Cartesian product L A1... L A1 of N = dimh copies of the cone L A1 defined by the genus 0 descendent potential F (0) = lim 0 ln D A1. The identification is provided by a certain transformation M τ from (a completed version of) L (2) GL(H) whose construction depends on the choice of a semisimple point τ. A number of results in Gromov Witten theory suggests that the higher genus theory inherits the symmetry group L (2) GL(H) (see [11, 14]). This motivates the

6 ALEXANDER B. GIVENTAL AND TODOR E. MILANOV following construction of the total descendent potential of a semisimple Frobenius manifold. Adopt the following rules of quantization ˆ of quadratic hamiltonians. Let {..., p a,..., q b,...} be a Darboux coordinate system on the symplectic space (H, Ω) compatible with the polarization H = H + H. Then (q a q b )ˆ = q a q b /, (q a p b )ˆ = q a / q b, (p a p b )ˆ = 2 / q a q b. This gives a projective representation of the Lie algebra L (2) gl(h) in the Fock space. The central extension is due to [ ˆF, Ĝ] = {F, G}ˆ+ C(F, G) where C is the cocycle satisfying C(p α p β, q α q β ) = { 1 if α β, 2 if α = β and equal 0 for any other pair of quadratic Darboux monomials. Introduce the total descendent potential as an asymptotical function: D := C(τ) ˆM τ [D A1... D A1 ], where ˆM := exp(lnm)ˆ, and C(τ) is a normalizing constant possibly needed to keep the R.H.S. independent of the choice of a semisimple point τ. This definition has been tested in [11, 12] and is known to agree with the TRR, SE, DE and the Virasoro constraints. Here is a more explicit description of M τ and C(τ) in the form applicable to Frobenius manifolds of simple singularities. Consider the complex oscillating integral J B (τ) = ( 2πz) 3/2 e fτ(x)/z ω. Here B is a non-compact cycle from the relative homology group lim H m(c m, {x : Re(f τ /z) C}) Z N. C We will use the notation 1,..., N for partial derivative with respect to a flat (and weighted - homogeneous) coordinate system (t 1,..., t N ) of the residue metric. We treat the derivatives z a J B as components of a covector field z a J B dt a T T which can be identified with a vector field via the residue metric and via its Levi-Civita connection with an H-valued function J B (z, τ). According to K. Saito s theory these functions satisfy in flat coordinates the differential equations (7) z a J = ( a )J together with the homogeneity condition: (8) (z z µ + z 1 E )J = 0 where µ = µ is the diagonal operator with the eigenvalues 1/2 m a /h. The latter equation yields an isomonodromic family of connection operators τ = z µ/z + (E )/z 2 regular at z = and turning into z µ/z at τ = 0. According to [8], there exists a (unique in the ADE-case) gauge transformation of the form S τ (z) = 1 + S 1 (τ)z 1 + S 2 (τ)z 2 +... (i.e. near z = ) conjugating τ to 0 and such that S τ ( z)s τ(z) = 1. It satisfies the homogeneity condition (z z + L E )S τ = [µ, S τ ]. On the other hand, let τ be semisimple. Then the functions f τ have N nondegenerate critical points x (a) (τ) with the critical values u a (τ) and the Hessians B

SIMPLE SINGULARITIES AND INTEGRABLE HIERARCHIES 7 a (τ). The local coordinate system {u a } (called canonical) diagonalizes the product and the residue metric: / u a / u b = δ ab / u b, ( / u a, / u b ) τ = δ ab 1 a / ub. Define an orthonormal coordinate system Ψ(τ) : C N T τ T = H, Ψ(q 1,..., q N ) = q a a / u a, and put U τ = diag[u 1 (τ),..., u N (τ)]. Stationary phase asymptotics of the oscillating integrals J Ba, a = 1,..., N, near the corresponding critical points x (a) yield a fundamental solutions to the system (7),(8) in the form Ψ(τ)R τ (z)e Uτ /z, R τ = 1 + R 1 (τ)z + R 2 (τ)z 2 +..., R t τ( z)r τ (z) = 1. The matrix series R τ satisfies the homogeneity condition (z z + L E )R τ = 0 and, according to [11], an asymptotical solution with this property is unique up to reordering or reversing the basis vectors in C N. Define c(τ) := 1 2 τ a R aa 1 (τ)du a as the local potential of the 1-form R aa 1 du a /2 (which is known to be closed [7]). In the above notations, the total descendent potential of the ADE-singularity assumes the form (9) D = e c(τ) Ŝ 1 τ Ψ(τ) ˆR τ e Uτ ˆ z D N. The R.H.S. is known to be independent of τ (see [11]) and defines D (up to a constant factor) as an asymptotical function of q = q 0 + q 1 z + q 2 z 2 +... in the formal neighborhood of q = τ z with semisimple τ. Our main result is the following theorem. Theorem 1. The total descendent potential (9) of a simple singularity satisfies the corresponding Hirota quadratic equation (1 5). In Section 4, we discuss Hirota quadratic equations of the KdV -hierarchy. The plan for the proof of Theorem 1 is to reduce the Hirota quadratic equations for D to those for D A1 by conjugating the vertex operators in (1,2) past the quantized symplectic transformations from (9). In Section 5, we describe the results of such conjugations by quoting corresponding theorems from [12]. The residue in (1) is computed in Section 6 and is compared with (2) in Section 7. The case-bycase tables for the coefficients a α are presented in Section 8. A key to all our computations is the phase form and its properties discussed in next section. 3. The phase forms and the root systems. Consider a flat family of cycles φ H 2 (fτ 1 (λ)) in the non-singular Milnor fibers and define the period vector I (0) φ (λ, τ) H by (10) (I (0) φ (λ, τ), ( 1) ω a) := a. 2π df τ φ fτ 1 (λ) It is a multiple-valued vector function on the complement to the discriminant which turns into I (0) φ (λ) from Section 1 at τ = 0.

8 ALEXANDER B. GIVENTAL AND TODOR E. MILANOV The phase form W α,β (defined in [12], Section 7) is given by the formula W α,β (λ, τ) := N (I α (0) (λ, τ) I (0) β (λ, τ), τ a) dτ a. i=1 It is a multiple-valued 1-form on the complement to the discriminant and depends bilinearly on the cycles α, β (to be chosen in (f0 1 (1) and transported to fτ 1 (λ)). According to [12], the phase forms have the following properties. (1) d W α,β = 0. (2) L λ+ N W α,β = 0, i.e. W is determined by the restriction W α,β (τ) := W α,β (0, τ), W α,β (λ, τ) = W α,β (τ λ1). (3) L E W α,β = 0. (4) Near a generic point of the discriminant T the form W α,β becomes single-valued on the double cover and has a pole of order 1 on D (since I α (0) have a pole of order 1/2). (5) δ γ W α,β = 2πi α, γ β, γ, where γ is the cycle vanishing over a generic point of the discriminant, and δ γ is a small loop going twice (in the positive direction defined by complex orientations) around the discriminant near this point. Proposition 1. W αβ = 1 2 γ A d γ, x α, γ β, γ γ, x. Proof. The phase form W α,β becomes single-valued on the Chevalley cover representing T as the quotient of C N = H 2 (f0 1 (1), C) by the monodromy group. The properties (1) and (4) show that it has at most logarithmic pole on the mirrors γ, x = 0. The property (5) controls the residues on the mirrors. The difference of the L.H.S. and the R.H.S. has to be a holomorphic 1-form, homogeneous of degree 0 by to the property (3), and therefore vanishes identically. Corollary 1. i E W α,β = α, β. Indeed, the Euler vector field becomes h 1 x a xa on the Chevalley cover, so that the equality follows from γ A α, γ β, γ = 2h α, β. This is one more general property of phase forms established in [12]. Corollary 2. The phase form W β of Section 1 coincides with W β,β. Remark. The inverse to the Chevalley quotient map is given by the period map τ [ω/df τ ] H 2 (fτ 1 (0), C) H 2 (f0 1 (1), C) C N. The periods I α (0) are defined via the differential of the inverse Chevalley map and therefore represent parallel translations of the cycles α considered as covectors in C N. The value of phase form W α,β, which is also a covector, is constructed as the Frobenius product α β of covectors (defined by the isomorphisms T τ T Tτ T based on the residue metric). Thus the formula α β := 1 2 γ A α, γ β, γ γ, x γ

SIMPLE SINGULARITIES AND INTEGRABLE HIERARCHIES 9 defines on (C N ) a family of commutative associative multiplications depending on the parameter x. 5 It would be interesting to find a representation-theoretic interpretation of this structure defined entirely in terms of the root system A. We prove several further properties of phase forms needed in our computations. Proposition 2. In the case of ADE-singularities, suppose that β has integer intersection indices with all α A and is invariant under the monodromy around a discriminant-avoiding loop γ. Then γ W β,β 2πi Z. Proof. This is Proposition 1 from Section 7 of [12]. It would be interesting to find out if the property remains valid for non-simple singularities. We will see in Section 7 that the coefficients a α introduced in Section 1 can be equivalently defined via the following limits b α. Start with choosing (τ 1,..., τ N ) = 1 = (0,..., 0, 1) in the role of the base point in T and identify A with the set of vanishing cycles in H 2 (f 1 1 (0)) = H 2(f0 1 (1)). Let us also fix τ T such that f τ is a Morse function, and let u will be one of the critical values of f τ so that τ u1. We may assume that τ (u + 1)1 / and that the straight segment connecting τ (u + 1)1 with τ u1 does not intersect. For each α A, pick a discriminant-avoiding path γ α connecting 1 with τ (u + 1)1 and further with τ u1 along the straight segment and such that α becomes the vanishing cycle when transported along γ α from 1 to τ u1. Assuming that integration of the phase form is performed along this path we put (11) b α := lim ε 0 exp { τ (u+ε)1 1 W α,α ε Proposition 3. The limit exists and does not depend on the choice of the path of integration provided that the path terminates at a generic point of the discriminant and that the cycle α transported along the path vanishes over this point. Proof. We may assume that u = u 1 is the first of the canonical coordinates U = (u 1,..., u N ), and therefore u 1 = 0 is the local equation of the discriminant branch. Since α is vanishing at the end of the path, the period vector I α (0) has the following expansion (here 1 i stand for the standard basis vectors in C N ): Ψτ 1 I α (0) ±2 ( (λ, τ) = 1 1 + (λ u 1 ) ) a i (U)1 i + o(λ u 1 ). 2(λ u1 ) 1 2 dt t Since Ψ(1 i ) = i / u i, we have (1 i 1 j, / u k ) = δ ij δ ik, and therefore W α,α = (I (0) α I (0) α, / u k )du k λ=0 = 2du1 u 1 + 4a1 (U)du 1 + O( u 1 ). We see that the integral u 1=0 u W 1= 1 α,α diverges the same way as 0 1 2dt/t so that the difference converges. This proves the existence of the limit. Removing this singular term we find that the integral [4a 1 (U)du 1 + O( u 1 )] vanishes along any path inside the discriminant branch u 1 = 0. This shows that the limit b α is locally constant as a function of the path s endpoint on the discriminant, and therefore globally constant due to the irreducibility of the discriminant. Finally, 5 We are thankful to V. A. Ginzburg who explained to us that this is a special case of a family of Frobenius structures constructed by A. P. Veselov [21]. }

10 ALEXANDER B. GIVENTAL AND TODOR E. MILANOV precomposing a path with a discriminant-avoiding loop γ with trivial monodromy of the cycle α does not change b α thanks to Proposition 2. Corollary. a α /a β = b α /b β for all α, β A. Proposition 4. Let δ ε be a small loop of radius ε around the discriminant near a generic point τ u1, and let α, β = ±1, where β is the cycle vanishing at this point. Then lim ε 0 δ ε W α,α = πi. Proof. We have α = ±β/2 + α where α is invariant under the monodromy around δ ε. Expanding I α (0) = I (0) α near λ = u as in the proof of Proposition 3 we find W α,α = δ ε δ ε ± I(0) β/2 du 2u + O( u) du = πi + O( ε) πi. δ ε In fact this property has been already used in [12]. 4. Two forms of the KdV-hierarchy. Consider the miniversal deformation of the -singularity in the form f u (x) := (x 2 1 + x2 2 + x2 3 )/2 + u. The vanishing cycle α can be identified with the real sphere (x 2 1 + x2 2 + x2 3 ) = 2(λ u). The period is dx 1 dx 2 dx 3 = d 4 df u dλ 3 π(2(λ u))3/2 = 4π 2(λ u). α Since (1, 1) = Resdx 1 dx 2 dx 3 /x 1 x 2 x 3 = 1, we have I α ( 1) (λ, u) = 2 2(λ u), and more generally, I (k) ±α (λ, u) = ±2(d/dλ)k (2(λ u)) 1/2, k Z. The Coxeter transformation swaps α and α and so a α = a α = (h + 1)/12h 2 = 1/16. The equation (1,2) in this example assumes the form (12) Res λ= dλ λ where [ ± (13) l = k 0 Γ ±α (λ) A1 Γ α (λ) ] (Φ Φ) = 16 (l + 1 ) (Φ Φ), 8 2k + 1 (q k 1 1 q k ) ( qk 1 1 qk ). 2 Here we use the notation A1 Γ φ (λ) to single out the vertex operators Γ φ (λ) of the -singularity. In order to identify the condition (12) for Φ with the KdV hierarchy in [16, 17] corresponding to the root system, we denote 2λ by ζ, rescale the variables by q k = (2k + 1)!!t 2k+1 and put x m = (t m + t m)/2, y m = (t m t m)/2 where m = 1, 3, 5,... In this notation l = my m ym, and (12,13) becomes [ Res dζ P ζ m ym ζ e4 e 2 P ζ m m ym 1 8 mym ym ] Φ(x + y) Φ(x y) = 0. This coincides with the equation (14.13.1) in [16] characterizing tau-functions Φ of the KdV hierarchy. Another form of the Hirota quadratic equation for Φ is based on the representation of the KdV-hierarchy as the mod 2-reduction of the KP-hierarchy (see [16], Section 14.11). It can be rephrased (see [12]) as the condition [ ] A (14) 1 Γ ±α/2 (λ) A1 Γ α/2 (λ) dλ ± (Φ Φ) has no pole in λ. λ ±

SIMPLE SINGULARITIES AND INTEGRABLE HIERARCHIES 11 Indeed, in the previous notations this can be rewritten as the property ym Φ(x + y)φ(x y) contains no ζ m for odd m > 0. e 2 P ζ m ym e P ζ m m This coincides with the mod 2-reduction of the KP-hierarchy of the Hirota equation (14.11.5) in [16]. According to a result from [16], Section 14.13, this condition is actually equivalent to (12). In Section 6 we will use the fact that (according to Kontsevich s theorem) the function Φ = D A1 satisfies both forms (12) and (14) of the KdV-hierarchy. 5. Symplectic transformations of vertex operators. Generalizing the construction of Section 1, introduce the vertex operator Γ φ τ(λ) corresponding to the vector f H[[z, z 1 ]] of the form Here I (0) φ f φ τ (λ) := k Z I (k) φ (λ, τ)( z)k. is the period vector introduced in Section 3, and I (k) φ := d k I (0) φ /dλk as before. For k < 0 the integration constants are taken equal 0 so that I (k) φ the homogeneity conditions: (λ λ + L E ) I (k) φ (λ, τ) = (µ 1 2 k) I(k) (λ, τ). φ satisfy In particular Γ φ 0 coincides with the vertex operator Γφ from Section 1. We state below several results about behavior of the vertex operators under conjugation by some symplectic transformations and refer to Sections 5, 6, 7 of [12] for the proofs. Theorem A (see Proposition 2 in [12]). { Ŝ τ Γ φ 1 0 (λ) Ŝ 1 τ = exp 2 τ λ1 λ1 W φ,φ } Γ φ τ(λ) We have to stress here that in order to compare the vertex operators Γ φ 0 (λ) and Γ φ τ(λ) one needs to transport the cycle φ from f0 1 1 (λ) to fτ (λ) along a path in T connecting λ1 = (0,..., λ) with τ λ1 = (τ 1,..., τ N λ) and avoiding the discriminant corresponding to singular levels fτ 1 (0). It is assumed in the formulation of the theorem that the integral of the phase form is taken along this very path. Similar conventions apply to other formulas of this and following sections involving integration of phase forms. Now let the cycle φ H 2 (fτ 1 (λ) be written as the sum φ = φ, β β/2 + φ where φ, β = 0. Here β is the cycle vanishing at a non-degenerate critical point of the function f τ with the critical value u and transported to fτ 1 (λ) along a discriminant-avoiding path connecting τ λ1 and τ u1. Theorem B (see Proposition 4 in [12]). { φ, β τ u1 Γ φ τ(λ) = exp 2 τ λ1 W β,φ } Γ φ 2 β τ (λ) Γ φ,β τ The integral here is taken along the path terminating on the discriminant where the phase form is singular. However the singularity is proportional to (λ u) 1/2 and is therefore integrable. (λ)

12 ALEXANDER B. GIVENTAL AND TODOR E. MILANOV Let us recall that the columns of the matrix R τ in the asymptotical expansion Ψ(τ)R τ (z)exp(u/z) correspond to non-degenerate critical points of the Morse function f τ with the critical values u i (τ). Let β i be the cycle vanishing over u i. Theorem C (see Proposition 3 in [12]). where (Ψ(τ) ˆR τ ) 1 Γ cβi τ (λ) (Ψ(τ) ˆR τ ) = e c2 W i/2 [ 1 ( A1 Γ cβ (λ)) (i) 1 ], u i W i := τ u i 1 τ λ1 ( 2 dt N ) W βi,β i τ N u i t N, and A1 Γ cβ u (λ) is the vertex operator of the -singularity with the miniversal deformation x2 1 2 + x2 2 2 + x2 3 2 + u corresponding to the c-multiple of the vanishing cycle. The behavior of I (0) β i near λ = u i is described by the asymptotics Ψ 1 (τ) I (0) β i (λ, τ) = 2 2(λ ui ) (1 i +...) where 1 i is the i-th basis vector in C N, and the dots mean higher order powers of λ u i. Respectively, the vertex operator of the -singularity is more explicitly defined by the series f C[[z, z 1 ]] of the form f = k Z d k 2c dλ k ( z) k, 2(λ u) where the branch of the square root should be the same as in the above asymptotics. The subscript (i) indicates the position of the vertex operator in the tensor product operator acting on the Fock space of functions of (q (1),...,q (N) ) = Ψ 1 (τ)q. The integrand in the formula for W i considered as a 1-form in the space with coordinates (t 1,..., t N ) identical to parameters of the miniversal deformation, while the notation τ = (τ 1,..., τ N ) is reserved for expressing the limits of integration. The phase form W has a non-integrable singularity at t = τ u i 1 which happens to cancel out with that of the subtracted term so that the difference is integrable. Finally, the following result is the special case of Theorem A corresponding to the -singularity. Theorem D (see Proposition 3 in [12]). { e (u/z)ˆ A1 Γ cβ u (λ) e (u/z)ˆ = exp c2 2 λ 2 dt λ u t i } Γ cβ 0 (λ) In fact this result can be obtained more directly using Taylor s formula. Indeed, for any analytic function I (0) we have [ ] e u/z I (k) (λ)( z) k e u/z = I (k) (λ + u)( z) k k Z k Z provided that u does not exceed the convergence radius of I (0) at λ. Thus the transformation in the theorem effectively consists in the translation λ u λ along an origin-avoiding path. The integral in the exponent should be taken along this path.

SIMPLE SINGULARITIES AND INTEGRABLE HIERARCHIES 13 6. The residue sum. In this section, we compute the residue sum [ ] dλ (15) Res λ= b α Γ α λ 0(λ) Γ α 0 (λ) D 2 α A assuming that the coefficients b α are defined as in Proposition 3. Introduce the total ancestor potential A τ := Ŝ τ D = e c(τ) Ψ(τ) ˆR τ e (Uτ /z)ˆ D N. Applying Theorem A of the previous section we find that (15) can be rewritten as [ ] (16) Res λ= λdλ c α Γ α τ (λ) Γ α τ (λ) A 2 τ, where c α = lim ε 0 exp { α A τ (u+ε)1 τ λ1 W α,α ε assuming that α H 2 (fτ 1 (λ)) vanishes at λ = u when transported along the path of integration of the phase form. Note that the factor λ 1 dλ in (15) is replaced by λdλ in (16) due to Corollary 1 from Section 3 which shows that exp { λ1 1 W α,α } = exp { α, α λ 1 1 dt t 2 dt t } } = λ 2. The ancestor potential A τ = exp (g 1) F τ (g) is a tame asymptotical function in the following sense: F τ (g) considered as a formal function of t a k = qa k + δ k1δ an satisfy r F τ (g) t a1 k 1... t ar t=0 = 0 whenever k 1 +... + k r > 3g 3 + r. k r This follows from the analogous property of D N, from the invariance of D A1 under the string flow exp(u/z)ˆ and from the upper-triangular property of ˆR τ. We refer to Proposition 5 in [12] for the proof. It is also shown in Section 8 of [12] that for tame asymptotical functions Φ the vertex operator expressions Γ φ τ (λ) Γ φ τ (λ) Φ 2 can be considered not only as series expansions in fractional powers of λ near λ =, but also as multiple-valued analytical functions defined over the entire range of λ and ramified only on the discriminant. Moreover, the sum in (16) is manifestly invariant under the entire monodromy group (= the ADEreflection group). Therefore the sum is actually a single-valued differential 1-form on the complement to D. Thus the residue (16) at λ = coincides with the sum of residues at the critical values λ = u i of the function f τ. Our next goal is to take u = u i and compute the residue. In a neighborhood of λ = u, the monodromy group reduces to Z 2 generated by the reflection σ in the hyperplane orthogonal to two vanishing cycles which we denote ±β. First, consider the summand in (16) corresponding to a σ-invariant cycle α A. The period vectors I α (k) (λ, τ) are therefore single-valued analytic functions near λ = u. In particular, lnc α, which differs from a constant by τ λ1 τ (u+1)1 W α,α, is analytic too. We conclude that λ c α Γ α τ (λ) Γ α τ (λ) A 2 has no pole at λ = u..

14 ALEXANDER B. GIVENTAL AND TODOR E. MILANOV Next, consider a pair of cycles α ± A transposed by σ and having intersection indices ±1 with β. We have α ± = α ± β/2 where σα = α. We use Theorem B to replace Γ α± τ with Γ α τ Γ ±β/2 τ and then commute Γ ±β/2 τ across Ψ ˆR exp(u/z)ˆ using Theorems C and D. The terms from (16) corresponding to α = α ± turn into ] λ dλ [Γ α τ (λ) Γ α τ (λ) (Ψ(τ) ˆR τ e (Uτ/z)ˆ Ψ(τ) ˆR (Uτ (17) τ e /z)ˆ) ( (i) A...1 d 1 ± Γ ±β/2 0 (λ) A1 Γ β/2 0 (λ)) 1... ( D N D N ). The coefficients d ± here are { (18) ± d ± = lim ε 0 exp τ (u+ε)1 τ λ1 τ (u+ε)1 τ λ1 ε 2dt W α±,α ± 1 t ± ε dt W β/2,β/2 + u λ 2t τ u1 W α,β+ τ λ1 λ u λ } dt. 2t We have to emphasize that all integrals here except the first one are taken along a short path near λ = u making β vanish while in the first integral this path is precomposed with a loop transforming α ± to β. Let us take λ = u + 1 for the base point for such a loop γ ± and rearrange the first integral as W α±,α ± γ ± τ (u+ε)1 τ (u+1)1 τ λ1 W β,β + W α±,α ±. τ (u+1)1 Combining this with W α±,α ± = W α,α ± W α,β + W β/2,β/2 we can rewrite the exponent in (18) as (19) (20) (21) W α±,α ± γ ± τ (u+ε)1 τ (u+1)1 τ λ1 τ (u+1)1 ε 2dt W β,β ± 1 t τ (u+ε)1 W α,β τ (u+1)1 ε τ (u+ε)1 W α,α + W β/2,β/2 + τ (u+1)1 ε u λ 1 dt 2t ε dt 2t 1 λ The integrals in (21) add up to λ 1 dt/2t and contribute λ 1/2 to the coefficients d ±. The sum in (20) is a function of λ analytic near λ = u (since α is σ-invariant) and is the same for both cycles α ±. The values of (19) may depend on the cycle α ± but are independent of λ. We claim that in the limit ε 0 the difference is an odd multiple of πi. Indeed, transporting α along the composition γ γ+ 1 yields α +. On the other hand 2W α,β = W α+,α + W α,α. Thus the difference of the two values of (19) can be interpreted as W α,α along a loop γ ε starting and terminating at τ (u + ε)1 and transporting α to α +. Let us compose it with a small loop δ ε of radius ε around λ = u. Since α + transports along this loop back to α, the composite integral γ εδ ε W α,α 2πiZ due to Proposition 2 and does not depend on ε. Our claim follows therefore from Proposition 4. u λ dt 2t dt 2t.

SIMPLE SINGULARITIES AND INTEGRABLE HIERARCHIES 15 We conclude that d ± = ±d 0 (λ)λ 1/2 where d 0 is a non-vanishing analytic function near λ = u. Now we use the fact that D A1 is a tau-function of the KdVhierarchy (14) to conclude that the factor in (17) of the form ± ± dλ λ [ A1 Γ ±β/2 0 (λ) A1 Γ β/2 0 (λ) ] (D A1 D A1 ) is everywhere analytic in λ. The same remains true after application of the operator (Ψ ˆRe (U/z)ˆ) 2. The vertex operator Γ α τ Γ α τ is analytic near λ = u since α is σ-invariant. Thus (17) has no pole at λ = u and contributes 0 to the residue sum. Finally, consider the summands in (15) with α = ±β. Applying Theorems C and D, we transform the corresponding summands from (16) to the form ( ) (Ψ(τ) ˆR (U/z)ˆ) 2 ( ) (i) τ e...d 2 eλdλ A1 Γ ±β 0 A1 Γ β 0 D 2 D 2..., where e = exp { ε 1 2dt t ± u λ 2dt λ ε t u λ } { 2dt λ = exp t 1 } 2dt = λ 2. t The contribution of these terms to the residue sum (16) at λ = u i can be calculated using the form (12) of the KdV-hierarchy for D A1 and is equal to 16 (Ψ ˆRe (U/z)ˆ) 2 l (i) (D N ) 2, where l (i) = (...1 l 1...). In order to justify this conclusion, recall from the end of Section 5 that conjugation by exp(u i /z) act as translation λ u i λ. Also, since D A1 is tame, the vertex operator expression in (12) yields a meromorphic 1-form in λ with a singularity only at λ = 0. Thus the residue in (12) at λ = is the same as at λ = 0. Let us summarize our computation. Proposition 5. The residue sum (15) is equal to ( N (22) 16 (e c Ŝ 1 (U/z)ˆ) 2 Ψ ˆRe 8 + N i=1 l (i) ) (D N ) 2. 7. The Virasoro operator. Functions of the form Φ Φ belong to a Fock space which is the quantization of the symplectic space H H, the direct sum of two copies of (H, Ω). Respectively the operator (23) ( m a h + k)(qa k 1 1 qk)( a qk a 1 1 qk a ) k 0 a in (2) is the quantization of a certain quadratic hamiltonian Ω(Df,f)/2 on H H. Let us describe the infinitesimal symplectic transformation D explicitly. Introduce the Virasoro operator l 0 := z z + 1/2 µ. 6 Since µ = µ, the operator l 0 : H H is anti-symmetric with respect to Ω, and the corresponding 6 The name comes from the property of the operators lm := l 0 zl 0 z...zl 0, (z repeated m times, m = 1,0, 1,2,...) to form a Lie algebra isomorphic to the algebra of formal vector fields x m+1 / x on the line and participating in the formulation of the Virasoro constraints (see [11, 14]).

16 ALEXANDER B. GIVENTAL AND TODOR E. MILANOV quadratic hamiltonian reads 1 (l 0 f( z),f(z)) dz = ((k+ 1 4πi 2 µ)f k, ( 1) k f 1 k ) = N ( m a h +k)qa k pa k. k 0 k 0 a=1 Comparing this with (23) we conclude that [ ] l0 l (24) D = 0 End(H H). l 0 l 0 The expression (2) on the R.H.S. of the Hirota equation is proportional to ˆDD 2 = ˆM 2 ( ˆM 2 ) 1 ˆD( ˆM 2 )(D N ) 2, where ˆM = e c(τ)ŝ 1 τ Ψ(τ) ˆR τ e (Uτ /z)ˆ. Note that ˆM 2 is the quantization of a blockdiagonal operator B := [ M 0 0 M ] [, and B 1 M DB = 1 l 0 M M 1 l 0 M M 1 l 0 M M 1 l 0 M Proposition 6. M 1 l 0 M = N i=1 ( A1 l 0 ) (i). Proof. We have S(z z + 1 2 µ)s 1 = z z + 1 2 µ + E z, since (z z + L E )S = µs Sµ and a S = z 1 a S. Next, in the canonical coordinates E = u i / u i, and therefore Ψ 1 (z z + 1 2 µ + E z )Ψ = z z + 1 2 V + U z, where V := Ψ 1 µψ = Ψ 1 L E Ψ. Furthermore, the differential equations a (ΨRe U/z ) = z 1 ( a )(ΨRe U/z ) translate into (d+ψ 1 dψ)r = z 1 (du R R du). This implies (L E +V )R = z 1 (UR RU), which together with the homogeneity condition (z z + L E )R = 0 shows that R 1 (z z + 1 2 V + U z )R = z z + 1 2 + U z. ]. Finally e U/z (z z + 1 2 + U z )eu/z = z z + 1 2. Proposition 7. ˆM 1ˆl 0 ˆM = (M 1 l 0 M)ˆ+ trµµ /4. Proof. The quadratic hamiltonians for z z + 1/2, µ, V, lns, (E )/z, U/z contain no p 2 -terms, and the quadratic hamiltonians for z z +1/2, µ, V, lnr contain no q 2 - terms. Therefore, in the quantized version of the previous computation, the only point where the cocycle C makes a non-trivial contribution is: ˆR 1 ( U z )ˆ ˆR = (R 1 U R)ˆ + C. z Let A = lnr, B = U/z. Then the quadratic hamiltonian of BA AB contains no q 2 -terms (since R z=0 = 1). We have therefore d dt e tâ ˆBe tâ = e tâ ( ˆBÂ Â ˆB)e tâ = e tâ (BA AB)ˆ e tâ + C(B, A) = [ e ta (BA AB)e ta]ˆ + C(B, A) = d [ e ta Be ta]ˆ+ C(B, A). dt

SIMPLE SINGULARITIES AND INTEGRABLE HIERARCHIES 17 Integrating in t from 0 to 1 we find C = C(B, A). Since A = R 1 z+o(z), we compute explicitly C = tr(ba)/2 = i Rii 1 u i /2. This expression, which seems to be a function of τ, has to be a constant, and the value of this constant is well-known to be trµµ /4 (see for instance the last chapter in [15]). For the sake of completeness we include the computation. Namely, comparing the z 0 - and z 1 -terms in the equation (L E + V )R = z 1 (UR RU) we find V ij = (u i u j )R ij 1 and respectively Thus we have 1 u i R ii 1 2 = ij i R ii 1 = L E R ii 1 = j u i V ij V ji 2(u j u i ) = ij since V = Ψ 1 µψ and V t = Ψ 1 µ Ψ. (u i u j )R ij 1 Rji 1 = j u j V ij V ji 2(u i u j ) = 1 4 V ij V ji u j u i. V ij V ji = 1 4 trµµ, Remark. Slightly generalizing Propositions 6 and 7 one obtains the following transformation formula (see Theorem 8.1 in [11]) ˆM 1ˆl m ˆM = ˆl(i) i m for the Virasoro operators with m 0. Since (ˆl m δ m,0 /16)D A1 = 0, this implies that N D = ˆMD satisfies the Virasoro constraints [ˆl m δ m,0 tr(µµ /4 + 1/16)]D = 0. In fact this is Corollary 8.2 in [11] specialized to Frobenius structures of weightedhomogeneous singularities. Note that the conjugation l 0 M 1 l 0 M of the off-diagonal blocks in the matrix D yields after quantization ˆM 1ˆl 0 ˆM = (M 1 l 0 M)ˆ (since the cocycle C vanishes on pairs of quadratic hamiltonians corresponding to block-diagonal and block-offdiagonal operators.) Thus ˆB 1 ˆD ˆB = i l(i) trµµ /2. Taking into account that N(h + 1) = m a (h m a ) = 1 ( 1 12h 2h 2 2 2 + µ a)( 1 2 µ a) = 1 2 tr(1 4 + µµ ) a we conclude that the R.H.S. of the Hirota equation (1,2) can be written as a ij ˆM 2 ( N 8 + i l (i) ) (D N ) 2. Comparing this with Proposition 5 we arrive at the following result. Proposition 8. The function D satisfies the Hirota quadratic equation (1,2) with a α = b α /16. Since D = 0, the Hirota equation is thus rendered consistent, and the following corollary completes the proof of Theorem 1. Corollary. The average value 1 Nh b α = α A 4(h + 1) 3h 2. Note that in the proof of Theorem 1 we use neither the Virasoro constraints for D A1 nor the fact that the N factors in D N are the same. The only relevant conditions for D A1 were both forms of the KdV -hierarchy and the tame property of exp(u/z)ˆd A1. Thus we have actually proved the following generalization of Theorem 1.

18 ALEXANDER B. GIVENTAL AND TODOR E. MILANOV Theorem 2. Suppose that tame asymptotical functions Φ 1,..., Φ N are taufunctions of the KdV-hierarchy and remain tame under the string flow Φ i exp(u/z)ˆφ i for all u. Then Φ := e c(τ) Ŝ 1 τ Ψ(τ) ˆR τ e (Uτ/z)ˆ (Φ 1... Φ N ) satisfies the corresponding Hirota quadratic equation (1 5). Remark. Although the condition for Φ i to remain tame under the string flow is quite restrictive, D A1 is not the only tau-function satisfying it. A large class of examples consists of the shifts D A1 (q + a) where a(z) = a 0 + a 1 z + a 2 z 2 +... is a series with coefficients a k which are arbitrary series in such that a 0 and a 1 are smaller than 1 in the -adic norm and a k 0 in this norm as k. 8. The Kac Wakimoto hierarchies. Let us compare the ADE- hierarchies (1 5) with the principal hierarchies of the types A (1) N, D(1) N, E(1) N described in Theorem 1.1 in [17]. The corresponding Hirota equation (1.14) in [17] has the form Res dζ ζ (25) N i=1 P m E 2β g i e + i,m 1/2 y mζ m e P m E β + i, m 1/2 ym ζ m /m Φ(x + y)φ(x y) = 2h m y m + ρ, ρ Φ(x + y)φ(x y). m E + Here ρ is the sum of the fundamental weights of the root system A, and the value ρ, ρ = Nh(h+1)/12 can be found for instance from the tables in [5]. The index set E + = {m a + kh a = 1,..., N, k = 0, 1, 2,...}, and m denote the remainder modulo h. The vertex operators in the sum correspond to a set of roots α i, i = 1,..., N, chosen one from each orbit of some Coxeter element M on the root system A. The coefficients β i,m are coordinates of α i with respect to a basis of eigenvectors H m of the Coxeter transformation M with the eigenvalues exp(2π 1m/h) satisfying the additional normalization condition H m, H m = 1. 7 The coefficients g i are defined via representation theory of affine Lie algebras. The numerical values of g i are computed in [17] in the cases A N, D 4 and E 6. In order to identify the vertex operators in (25) with those in (1,2) let us start with taking ζ = (hλ) 1/h. Then the components of the period vector I α ( 1) i with respect to a suitable basis [ψ a ] H will have the form (26) (I α ( 1) i (λ), [ψ a ]) = β i,ma m 1 a (hλ) ma/h. Indeed, the weighted - homogeneous forms ψ a ω/df represent a basis of eigenvectors for the classical monodromy operator in H 2 (f 1 (1), C). Then it is straightforward to check that the relation k (27) qk a = (m a + rh) t ma+kh r=0 7 As usual, there is a caveat in the case Dl with odd l when the eigenvalue 1 of the monodromy operator has multiplicity 2. The involution of the Dynkin diagram induces an automorphism of the root system which allows one in this case to single out one invariant and one anti-invariant eigenvector, H (i) l 1 and H(a) l 1, orthogonal to each other and normalized by H l 1, H l 1 = 1 each.

SIMPLE SINGULARITIES AND INTEGRABLE HIERARCHIES 19 (together with the standard change x+y = t, x y = t as in Section 4) identifies the vertex operators in (25) with Γ αi Γ αi. Note that replacing α i with any of the h roots from the same M-orbit does not change the corresponding residue in (25) since the new vertex operator would differ from the old one only by the choice of the branch of ζ = (hλ) 1/h. Thus we arrive at the following conclusion. Proposition 9. The choice of the basis {[ψ a ] H} such that (26) holds true and the change of variables (27) identify the Hirota equation (1 5) with the corresponding hierarchy of the form (25) provided that g i = h 3 a αi = h 3 b αi /16. Let us now compute the coefficients b α. First, rewrite the definition (11) as b α = lim e R τ (u+ε)1 ε1 W α,α = lim ε 0 ε 0 γ A y(ε), γ α,γ 2 2 ε 1/h κ, γ α,γ 2 2 = v, α 4 κ, α 4 γ,α =1 x, γ κ, γ, where ε 1/h κ, y(ε) and x are inverse images under the Chevalley map of ε1, τ (u + ε)1 and τ u1 respectively, x is a generic point on the mirror α, x = 0 and v is determined from the expansion y(ε) = x + ε 1/2 v + o(ε 1/2 ). We will use this formula in the case of A and D series. Case A N. The root system consists of the vectors γ ij := e i e j in the space C N+1 with the standard orthonormal basis e 0,..., e N and coordinates z 0,..., z N. Take F(z, τ) = zn+1 N + 1 + t 1z N 1 +... + t N = 1 N (z z i ). N + 1 Let α = e a e b and let t = τ u1 be a generic point on the discriminant. Then the components y i (ε) = x i +ε 1/2 v i +εw i +o(ε) (where x a = x b ) satisfy F(y i, t ε1) = 0 and therefore ε = F(x i, t) + F (x i, t)ε 1/2 y i + F (x i, t)εw i + F (x i, t)ε v2 i 2 + o(ε). We have F(x i, t) = 0 for all i and F (x i, t) = 0 for i = a, b. This implies that v i = ± 2/F (x a, t) for i = a, b and hence α, v = ±2 2/F (x a, t). Thus α, v 4 = 64/F (x a, t) 2. On the other hand, x, γ = ( 1) ( (N + 1)F N 1 (x i x a ) 2 = ( 1) N 1 ) 2 (x a, t). 2 γ,α =1 i a,b The eigenvector κ = (N+1) 1/(N+1) (1, η, η 2,..., η N 1 ) of the Coxeter transformation (z 0,..., z N ) (z 1,..., z N, z 0 ) with the eigenvalue η = exp 2πi/(N +1) is a preimage of t = 1 under the Chevalley map. We find 8 κ, α 4 γ = (N + 1) α,γ =1 κ, 2 (η a η b ) 2 j a(η a η j ) (η i η b ) = i b ( 1) N (N + 1) 4 (η a η b ) 2 η N(a+b) = ( 1) N 1 (N + 1) 4 (2 η a b η b a ). Collecting the results we find b α = 16 1 (N + 1) 2 (2 η a b η b a ). 8 We use here the facts that the product Qk a (ζ e2πik/n ) over all n-th roots of unity except ζ = e 2πia/n is equal to the derivative of z n 1 at z = ζ, i.e. to n/ζ. i=0

20 ALEXANDER B. GIVENTAL AND TODOR E. MILANOV This agrees with Theorem 1.2 in [17] where g i = (N +1)/(2 η i η i ) corresponds to α i = e 0 e i. In particular gk = N + 1 N sin 2 πk N(N + 1)(N + 2) ( ) =. 4 N + 1 12 k=1 The middle expression is a special case of Dedekind sums, and the second equality, which follows from our results, is well known in number theory (see i.g. [4]). Case D N. The root system consists of the vectors ±e i ±e j, i j, where e 1,...e N is the standard orthonormal basis and (z 1,..., z N ) are the corresponding coordinates in C N. The parameters (t 1,..., t N ) in the following family of polynomials N F(z, t) = z 2N + t 2 z 2N 2 + t 3 z 2N 4 +... + t N z 2 + t 2 1 = (z 2 zi 2 ) are identified with coordinates on the Chevalley quotient C N /W. Note that the invariant t N of degree h = 2N 2 is the coefficient at z 2. Let us assume that x is a generic point on the mirror z a ± z b orthogonal to the root α = e a e b, and that t is the corresponding point on the discriminant, so that x a = ±x b and F(±x a, t) = F (±x a, t) = 0. Taking y i (ε) = x i +ε 1/2 v i +εw i +o(ε) and expanding F(y(ε), t 1,..., t N 1, t N ε) = 0 in ε we find εx 2 i = F(x i, t) + F (x i, t)(ε 1/2 v i + εw i ) + F (x i, t)ε v2 i 2 + o(ε). Thus v a = 2x 2 a/f (x a, t), v b = 2x 2 a/f (x a, t) and α, v 4 = 64x 4 a/f (x a, t) 2. Furthermore, F (z, t) z=xa = [2 a i a(z 2 x 2 i) + 4z 2 (z 2 x 2 i)] z=xa = 8x 2 a (x 2 a x 2 i ). a b i a,b i a Using this we find x, α = (x 2 a x2 j ) ( 1)(x 2 i x2 b ) = F (x a, t) 2 (±1)N 2. 64x 4 a α,γ =1 j a,b i a,b Next, the eigenvector κ = (1, η,..., η N 2, 0) of the Coxeter transformation (z 1,..., z N ) (z 2,..., z N 1, z 1, z N ) with the eigenvalue η = exp πi/(n 1) is mapped to (t 1,..., t N ) = (0,..., 0, 1) under the Chevalley map. Assuming first that α = e a e b with a, b < N we find κ, α 4 κ, γ = (η a η b ) 4 (η a ) 2 ( η b ) 2 (η 2a η 2i )(±1)(η 2b η 2i ) = α,γ =1 i a,b,n (±1) N 2 (N 1) 2(ηa η b ) 2 (η a ± η b ) 2 = (±1)N 2 (N 1) 2 (2 ηa b η b a ) (2 ± η a b ± η b a ). Combining with the previous formulas we compute b α = Now let α = e a e N. Then κ, α 4 κ, γ = η 4a α,γ =1 1 (2 ± η a b ± η b a ) (N 1) 2 (2 η a b η b a ) i=1 for α = e a e b. (η 2a η 2i )( η 2i ) = ( 1) N 2 (N 1) i a,n

SIMPLE SINGULARITIES AND INTEGRABLE HIERARCHIES 21 and therefore b α = 1/(N 1). Taking the representatives α 1 = e N 1 e 1,..., α N2 = e N 1 e N 2, α N 1 = e N 1 e N, α N = e N 1 + e N in the orbits of the Coxeter transformation on A, we find g i = (N 1) (2 η i η i ) 2 (2 + η i + η i ) for i = 1,..., N 2, and g (N 1)2 i = for i = N 1, N. 2 The identity g k = (N 1)N(2N 1)/6, which follows from our general theory, agrees with the value of the Dedekind sum 9 N 2 k=1 tan 2 πk ( 2N 2 (N 2)(2N 3) ) =. 3 In the case N = 4 the values g i = 1/2, 9/2, 9/2,9/2 agree with the values of g i found in [17], Proposition 1.3(a). Cases E N. We find g i using the packages LiE and MAPLE to compute the ratios via (5) and then apply the normalizing relation (3). In each case E N, let α 1,..., α N be the simple roots and M be the Coxeter transformation described by the following diagrams: α 1 α 3 α 4 α 5 α 6 M = σ 1 σ 4 σ 6 σ 2 σ 3 σ 5, α 2 α 1 α 3 α 4 α 5 α 6 α 7 M = σ 1 σ 4 σ 6 σ 2 σ 3 σ 5 σ 7, α 2 α 1 α 3 α 4 α 5 α 6 α 7 α 8 M = σ 1 σ 4 σ 6 σ 8 σ 2 σ 3 σ 5 σ 7. α 2 One can check (i.g. using LiE) that all simple roots α 1,..., α N belong to different M-orbits. The following tables represent the values of the corresponding coefficients g i while the values of b αi can be obtained from them as in Proposition 9. Case E 6. We have b αi = g i /108, where g 1 = g 6 = 16 + 8 3, g 3 = g 5 = 16 8 3, g 2 = 7 + 4 3, g 4 = 7 4 3. This agrees with the values of g i found in [17], Proposition 1.3(b). Case E 7. We have b αi = 2g i /729. Put u = cos(π/9). Then g 1 = 27 2 + 36u + 24u2, g 2 = 225 2 + 36u 144u2, g 3 = 3 2, g 4 = 147 2 + 12u 96u2, g 5 = 9 2 72u + 72u2, g 6 = 21 2 48u + 72u2, g 7 = 9 2 + 36u + 72u2. 9 It is essentially the same one as in the A-case since sin 2 x 1 = cot 2 x = tan 2 (π/2 x).