THE MODULAR GROUP FOR THE TOTAL ANCESTOR POTENTIAL
OF A SIMPLE ELLIPTIC SINGULARITY.
TODOR MILANOV AND YEFENG SHEN
Abstract. In a series of papers Krawitz, Milanov, Ruan, and Shen have verified
the so called Landau-Ginzburg/Calabi-Yau (LG/CY) correspondence for simple
elliptic singularities E N(1,1) (N = 6, 7, 8). As a byproduct it was also proved thatthe orbifold Gromov–Witten invariants of the orbifold projective lines P13,3,3 , P12,4,4 ,and P12,3,6 are quasi-modular forms on an appropriate modular group. While themodular group for the first orbifold is easy to find, namely it is Γ(3), the modulargroups for the other two were left unknown. The goal of this paper is to prove thatthe modular groups in the remaining two cases are respectively Γ(4) and Γ(6).

Contents
1. Introduction
2. The total ancestor potential in singularity theory
3. Modularity and monodromy
References

1
4
9
15

1. Introduction
Let W(x) = x1a1 + x2a2 + x3a3 be a Fermat polynomial whose exponents (a1 , a2 , a3 ) are
given by one of the following triples (3, 3, 3), (2, 4, 4), or (2, 3, 6). Since such a W
defines a hypersurface in C3 that has a simple-elliptic singularity at x = 0, we will
sometimes refer to it as elliptic Fermat polynomial. We assign weights qi = 1/ai toeach variable xi , so that W becomes a quasi-homogeneous polynomial of degree 1.
1.1. Formulation of the main results. Let H = C[x1 , x2 , x3 ]/(W x1 , W x2 , W x3 ) be
the Jacobi algebra of W. Given a triple e = (e1 , e2 , e3 ) of non-negative integers weput φe (x) = x1e1 x2e2 x3e3 . We choose a set E of exponents e, s.t., the monomials φe (x)project to a basis of H. More presicely, put
E = {(e1 , e2 , e3 ) | 0 ≤ ei ≤ ai − 2}.

(1)

It will be convenient also to decompose E = {(0, 0, 0), m} t Etw , where Etw corresponds to monomials of non-integral degree and φm (x) is the monomial of degree
12

TODOR MILANOV AND YEFENG SHEN

1. Let us denote by Σ ⊂ C the set of all marginal deformations
f (σ, x) = W(x) + σ φm (x),σ ∈ Σ,

s.t., f (σ, x) has only one critical point. The hypersurfaces
Xσ,λ = {x ∈ C3 | f (σ, x) = λ}form a smooth fibration on Σ × (C \ {0}), while the homology (resp. cohomology)groups H2 (Xσ,λ ; C) (resp. H 2 (Xσ,λ ; C)) form a vector bundle equipped with a flat
Gauss–Manin connection. We fix a reference point, say (0, 1), and let
h = H2 (X0,1 ; C),h∨ = H 2 (X0,1 ; C)

be the reference fibers. The parallel transport around λ = 0 induces a monodromytransformation J ∈ GL(h), which commutes with the monodromy action of π1 (Σ)
on h. In other words, we have a monodromy representationρ : π1 (Σ) → GL(h0 ) ⊕ GL(h,0 ),where h0 is the J-invariant subspace and h,0 is the direct sum of all eigenspaces of
J with eigenvalue , 0. Put ρ = (ρ0 , ρ,0 ) and let
ρ,0 : π1 (Σ) −→ GL(h,0 )/hJi,
the map induced from ρ,0 . Using an explicit computation we will check that
Ker (ρ0 ) ⊂ Ker (ρ,0 ).

(2)

It will be nice if one can find a conceptual explanation and determine if (2) issatisfied for other normal forms W of the simple elliptic singularities. Using (2) we
get an induced homomorphism
ρW : eΓ(W) −→ GL(h,0 )/hJi,
where eΓ(W) = Im(ρ0 ). Put Γ(W) := Ker(ρW ).
Theorem 1.1. The total ancestor potential AWσ (~; q) of the simple elliptic singularity W transforms as a quasi-modular form on Γ(W).
The definition of the total ancestor potential in singularity theory as well as theprecise meaning of the quasi-modularity will be recalled later on. Our second result
can be stated this way.
Theorem 1.2. If W is an elliptic Fermat polynomial of type E6(1,1) , E7(1,1) , or E8(1,1) ,then Γ(W) is respectively Γ(3), Γ(4), or Γ(6).

MODULAR GROUPS AND SIMPLE ELLIPTIC SINGULARITIES

3

1.2. Applications to Gromov–Witten theory. The LG/CY correspondence wasproposed by Chiodo and Ruan [9]. In our settings it can be stated this way. A tripletof non-zero complex numbers (g1 , g2 , g3 ) ∈ (C∗ )3 is called a diagonal symmetry of
W if
W(g1 x1 , g2 x2 , g3 x3 ) = W(x1 , x2 , x3 ).
The diagonal symmetries form a group GW . It contains the element√√√

JW = diag(e2π −1q1 , e2π −1q2 , e2π −1q3 ),qi = 1/ai .

eW in the weighted projective plane
The equation W = 0 defines an elliptic curve X
2
CP (c1 , c2 , c3 ), where qi = ci /d for a common denominator d. The action of theeW := GW /hJW i on X
eW is faithful and the quotient XW := X
eW /GeW is an orbgroup G
1ifold projective line Pa1 ,a2 ,a3 . The LG/CY correspondence predicts that the GW invariants of XW can be obtained from the so called Fan–Jarvis–Ruan–Witten (FJRW)
invariants of the pair (W, GW ) via analytic continuation and a certain quantizatied
symplectic transformation (c.f. [9]).
X
The orbifold GW invariants of X = P1a1 ,a2 ,a3 are defined as follows. Let Mg,n,dbe the moduli space of degree-d stable maps from a genus-g orbi-curve, equippedwith n marked points, to X. Let us denote by π the forgetful map, and by evi theevaluation at the i-th marked point
π

X

evi

Mg,n ←− Mg,n,d −→ IX ,where IX is the inertia orbifold of X. Let Eff(X) ⊂ H2 (X; Z) be the cone of effective
curve classes. By definition the Novikov ring is the completed group algebra ofEff(X). In our case, since H2 (X; Z) = Z · [X], where [X] is the fundamental class ofX, we may identify the Novikov ring with the space of formal power series C[[q]].
X
The moduli space is equipped with a virtual fundamental cycle [Mg,n,d ], s.t., themaps∗ΛXg,n : HCR(X; C[[q]])⊗n −→ H ∗ (Mg,n ; C)

defined by
ΛXg,n =∞
X

qd ΛXg,n,d ,

n
Y
 X

ΛXg,n,d (α1 , . . . , αn ) := π∗ [Mg,n,d ] ∩ev∗i (αi )
i=1

d=0

∗
form a CohFT with state space the Chen-Ruan cohomology HCR(X; C[[q]]). The
ancestor GW invariants of X are by definition the following formal series:
Z
∞
X
D
E
d
τk1 (α1 ), . . . , τkn (αn )=qΛXg,n,d (α1 , . . . , αn ) ψk11 · · · ψknn ,
(3)
g,n

d=0

Mg,n

4

TODOR MILANOV AND YEFENG SHEN

where ψi is the i-th psi class on Mg,n , αi ∈ H, and ki ∈ Z≥0 . For more details on
orbifold Gromov–Witten theory we refer to [7]. The total ancestor potential of X isby definition the following generating series
∞
D
E 
X
X
~g−1 q(ψ1 ) + ψ1 , . . . , q(ψn ) + ψn,
Aq (~; q) = exp
g,n

g,n=0

P
k
∞
where q(z) = ∞k=0 qk z and {qk }k=0 is a sequence of formal vector variables with∗values in HCR (X; C). The generating function is a formal series in q0 , q1 + 1, q2 , . . . .
Following the ideas of Krawitz–Shen [25] and Milanov–Ruan [27], one can obtain a very precise correspondence between the total ancestor potentials AW
σ andX
Aq (see [29]). Let us briefly explain this correspondence. Recall that the curve
Eσ = { f (σ, x) = 0} ⊂ P2 (c1 , c2 , c3 )is called elliptic curve at infinity. Let us think of Σ as a punctured P1 and let us
select a puncture p, s.t., the j-invariant j(Eσ ) → ∞ as σ → p. For example, if W
is the Fermat polynomial of type E7(1,1) , then p = −2, 2, or ∞. If W is the Fermat
polynomial of type E8(1,1) , then p is a solution to 4p3 + 27 = 0. The main statement
is that there exists a function f (σ) on P1 holomorphic near σ = p, s.t., under thesubstitution q = f (σ), the total ancestor potential AXq coincides with AW
σ . LetW
us point out that the definition of Aσ requires a choice of a primitive form in thesense of K. Saito [33]. Part of the statement is that there exists a primitive form,s.t., the identification holds. Combining the mirror symmetry theorem of [29] with
Theorem 1.1 and Theorem 1.2 we get
Corollary 1.3. If X is one of the orbifolds P13,3,3 , P12,4,4 , or P12,3,6 , then the Gromov–
Witten invariants (3) are quasi-modular forms respectively on Γ(3), Γ(4), or Γ(6).
2. The total ancestor potential in singularity theory
Let S = Σ × Cµ−1 . We fix a coordinate system on S, such that the coordinates ofs ∈ S are indexed by the exponents E (cf. (1)) in such a way that sm = σ ∈ Σ andse ∈ C for e , m. The miniversal deformation of W can be given by the following
functionX
F(s, x) = W(x) +se φe (x),(4)
e∈Ewhere the domain of F is X = S × C . The marginal deformations f (σ, x) are
obtained from F(s, x) by restricting sm = σ and se = 0 for e , m.
It is well known (see [20, 35]) that Saito’s theory of primitive forms (cf. [33])gives rise to a Frobenius structure (cf. [12]) on S. In this section the goal is torecall the key points in the construction of this Frobenius structure and then use
the higher-genus reconstruction formalism of Givental to define the total ancestor
potential of W.
3

MODULAR GROUPS AND SIMPLE ELLIPTIC SINGULARITIES

52.1. Saito’s theory. Let C ⊂ X be the critical variety of F, i.e., the support of thesheaf
OC := OX /h∂ x1 F, ∂ x2 F, ∂ x3 Fi.
Let q : X → S be the projection on the first factor. The Kodair–Spencer map (TS is
the sheaf of holomorphic vector fields on S)
TS −→ q∗ OC ,∂/∂si 7→ ∂F/∂si mod (F x1 , F x2 , F x3 )is an isomorphism, which implies that for any s ∈ S, the tangent space T s S isequipped with an associative commutative multiplication • s depending holomorphically on s ∈ S. If in addition we have a volume form ω = g(s, x)d3 x, where
d3 x = dx1 ∧ dx2 ∧ dx3 is the standard volume form; then q∗ OC (hence TS as well)is equipped with the residue pairing:Z1
ψ1 (s, y)ψ2 (s, y)ω,(5)(ψ1 , ψ2 ) =3(2πi)
F y1 F y2 F y3Γwhere y = (y1 , y2 , y3 ) is an unimodular coordinate system for the volume form, i.e.,ω = d3 y, and Γ is a real 3-dimensional cycle supported on |F xi | =  for 1 ≤ i ≤ 3.
Given a semi-infinite cycle
A ∈ lim H3 (C3 , (C3 )−m ; C)  Cµ ,(6)(C3 )m = {x ∈ C3 | Re(F(s, x)/z) ≤ m}.

(7)

←−

where
Put
JA (s, z) = (−2πz)

−3/2

Z
zdS

eF(s,x)/z ω,(8)A

where dS is the de Rham differential on S. The oscillatory integrals JA are bydefinition sections of the cotangent sheaf TS∗ .
According to Saito’s theory of primitive forms [33, 36], there exists a volume
form ω such that the residue pairing is flat and the oscillatory integrals satisfy asystem of differential equations, which in flat-homogeneous coordinates t = (te )e∈E
have the formz∂e JA (t, z) = ∂e •t JA (t, z),(9)where ∂e := ∂/∂te and the multiplication is defined by identifying vectors and covectors via the residue pairing. Using the residue pairing, the flat structure, and the
Kodaira–Spencer isomorphism we have the following isomorphisms:
T ∗ S  T S  S × T 0 S  S × H.
Due to homogeneity the integrals satisfy a differential equation with respect to theparameter z ∈ C∗(z∂z + E)JA (t, z) = Θ JA (t, z),
(10)

6

TODOR MILANOV AND YEFENG SHEN

where
E=

µ−2
X

de te ∂e ,(de := deg te = deg se ),i=−1

is the Euler vector field and Θ is the so-called Hodge grading operator!
1∗∗
Θ : TS → TS , Θ(dte ) =− de dte .
2
The compatibility of the system (9)–(10) implies that the residue pairing, the multiplication, and the Euler vector field give rise to a conformal Frobenius structure ofconformal dimension 1. We refer to B. Dubrovin [12] for the definition and moredetails on Frobenius structures.
2.2. Primitive forms for simple elliptic singularities. The classification of primitive forms in general is a very difficult problem. In the case of simple elliptic
singularities however, all primitive forms are known (see [33]). They are givenby ω = d3 x/πA (σ), where πA (σ) is a period of the elliptic curve at infinity. Let
us recall also that πA (σ) can be expressed in terms of a period of the GelfandLerey form d3 x/dF as follows. We embed C3 in the weighted projective space
P3 (1, c1 , c2 , c3 ) via xi = Xi /X0ci , i = 1, 2, 3. The Zariski closure of the Milnor fiber
Xσ,1 is X σ,1 = Xσ,1 ∪ Eσ , therefore we have a tube map
L : H1 (Eσ ; Z) → H2 (Xσ,1 ; Z)
which allows us to write√d3 x= 2π −1
πA (σ) :=L(A) d f
Z

Z

d3 x
.
ResEσ
df
A

Let us point out that when σ = 0 the image of the tube map L is precisely h0 andthe monodromy representation ρ0 coincides with the monodromy representation of
the elliptic pencil Eσ , σ ∈ Σ.
2.3. Givental’s higher-genus reconstruction formalism. Following Givental we
introduce the vector space H = H((z)) of formal Laurent series in z−1 with coefficients in H, equipped with the symplectic structureΩ( f (z), g(z)) = resz=0 ( f (−z), g(z))dz.
Using the polarization H = H+ ⊕ H− , where H+ = H[z] and H− = H[[z−1 ]]z−1 weidentify H with the cotangent bundle T ∗ H+ . The goal in this subsection is to define
the total ancestor potential of W.

MODULAR GROUPS AND SIMPLE ELLIPTIC SINGULARITIES

7

2.3.1. The stationary phase asymptotics. We fix a primitive form ω = d3 x/πA (σ)
and let t = (te ) be flat coordinates, defined near s = 0, s.t., under the Kodaira–Spencer isomorphism T 0 S  H we have ∂e = φe (x). Since, πA is a multi-valued
analytic function on S, the flat coordinates te are also multi-valued analytic functions on S.
Let s ∈ S be a semi-simple point, i.e., the critical values ui of F (1 ≤ i ≤ µ) form
locally near s a coordinate system. Let us also fix a path from 0 ∈ S to s, so that wehave a fixed branch of the flat coordinates. Then we have an isomorphismpp X ∂uiΨ s : Cµ → H, ei 7→ ∆i ∂ui = ∆i
φe ,
∂te
e∈E
where ∆i is determined by (∂/∂ui , ∂/∂u j ) = δi j /∆i . It is well known that Ψ s diagonalizes the Frobenius multiplication and the residue pairing, i.e.,pei • e j = ∆i ei δi, j , (ei , e j ) = δi j .
Let Sss be the set of all semi-simple points. The complement K = S \ Sss is ananalytic hypersurface also known as the caustic. It corresponds to deformations,s.t., F has at least one non-Morse critical point. By definition
Sss → HomC (Cµ , H),s 7→ Ψ s

is a multi-valued analytic map.
The system of differential equations (9) and (10) admits a unique formal asymptotical solution of the type
Ψ s R s (z)eU s /z ,R s (z) = 1 + R s,1 z + R s,2 z2 + · · ·where U s is a diagonal matrix with entries u1 (s), . . . , uµ (s) on the diagonal andR s,k ∈ HomC (Cµ , Cµ ). Alternatively this formal solution coincides with the stationary phase asymptotics of the following integrals. Let Bi be the semi-infinite cycleof the type (6) consisting of all points x ∈ C3 such that the gradient trajectories of−Re(F/z) flow into the critical value ui . ThenZ
−3/2(−2πz)zdS
eF(s,x)/z ω ∼ eui (s)/z Ψ s R s (z)ei as z → 0.
Bi

We refer to [18] for more details and proofs.
2.3.2. The quantization formalism. Let us fix a Darboux coordinate system on H
given by the linear functions qek , pk,e defined as follows:∞ XX
f(z) =(qek φe zk + pk,e φe (−z)−k−1 ) ∈ H,
k=0 e∈Ee

where {φ }e∈E is a basis of H dual to {φe }e∈E with respect to the residue pairing.
If R = eA(z) , where A(z) is an infinitesimal symplectic transformation, then web as follows. Since A(z) is infinitesimal symplectic, the map f ∈ H 7→
define R

8

TODOR MILANOV AND YEFENG SHEN

Af ∈ H defines a Hamiltonian vector field with Hamiltonian given by the quadratic
function hA (f) = 12 Ω(Af, f). By definition, the quantization of eA is given by thebdifferential operator ehA , where the quadratic Hamiltonians are quantized according
to the following rules:(pk0 ,e0p

k00 ,e00∂2)b= ~ e0 e00 ,∂qk0 ∂qk00

00

00

00

(pk0 ,e0 qek00 )b= (qek00 pk0 ,e0 )b= qek00

∂
0,
∂qek0

0

00

0

00

(qek0 qek00 )b= qek0 qek00 /~.

Note that the quantization defines a projective representation of the Poisson Liealgebra of quadratic Hamiltonians:b G]
b = {F, G}b+ C(F, G),[F,
where F and G are quadratic Hamiltonians and the values of the cocycle C on a pairof Darboux monomials is non-zero only in the following cases:



0
00
1 if (k0 , e0 ) , (k00 , e00 ),C(pk0 ,e0 pk00 ,e00 , qek0 qek00 ) = (11)
2 if (k0 , e0 ) = (k00 , e00 ).
2.3.3. The total ancestor potential. By definition, the Kontsevich-Witten tau-functionis the following generating series:
Z
n
Y

X 1
g−1
~
(q(ψi ) + ψi ) ,(12)
Dpt (~; q(z)) = expn!
Mg,n i=1g,nP
where q(z) = k qk zk , (q0 , q1 , . . .) are formal variables, ψi (1 ≤ i ≤ n) are the first
Chern classes of the cotangent line bundles on Mg,n . The function is interpreted asa formal series in q0 , q1 + 1, q2 , . . . whose coefficients are Laurent series in ~..
Let s ∈ Sss be a semi-simple point. Motivated by Gromov–Witten theory Givental introduced the notion of the total ancestor potential of a semi-simple Frobenius
structure (see [18]). In our settings the definition takes the formcs Rbs eU s /z
A s (~; q) := Ψ
d

µ
Y

p
Dpt (~ ∆i (s); i q(z) ∆i (s))

(13)

i=1

where
q(z) =∞ X
X

qek zk φe ,iq(z) =

k=0 e∈E∞
Xi

qk zk .

k=0

cs is interpreted as the change of variables
The quantization Ψ
µ
X
i=1

i

q(z)ei = Ψ−1s q(z)

i.e.

i

X
p
qk ∆i =(∂e ui ) qek .
e∈E

(14)

MODULAR GROUPS AND SIMPLE ELLIPTIC SINGULARITIES

9

3. Modularity and monodromy
The flat coordinates are multi-valued analytic functions on S. In this section
we will compute their monodromy under analytic continuation. Once this task iscompleted the proof of Theorem 1.1 will be easy.
3.1. Picard–Fuchs equations. We make use of the so called geometric sections(see [3])Zd3 xΦe (σ, λ) :=xe∈ H 2 (Xσ,λ ; C),(15)
df
where e = (e1 , e2 , e3 ), ei ∈ Z≥0 , and xe := x1e1 x2e2 x3e3 . The geometric sections (15) withe ∈ E give rise to a trivialization of the vanishing cohomology bundle. The Gauss–
Manin connection corresponds to a system of Fuchsian differential equations known
as Picard–Fuchs equations. It is enough to solve this system at λ = 1, because thehomogeneity of f (σ, x) yields the following simple relationΦe (σ, λ) = λdeg(e) Φe (σ, 1),P
where deg(e) := deg(xe ) = i ei qi . The Picard–Fuchs equations have the formX∂σ Φe (σ, 1) =Ge,e0 (σ) Φe0 (σ, 1),
(16)
e0 ∈E

where G = (Ge,e0 ) is a square matrix of size µ = |E|, whose entries are holomorphic
functions on Σ. Let us denote by F = (Fe0 ,e00 (σ)) a fundamental solution to theabove system, i.e., F is a non-degenerate matrix satisfying ∂σ F = G · F. We haveXΦe (σ, λ) = λdeg(e)Fe,r (σ) Ar ,
(17)
r∈E

where Ar , r ∈ E are multi-valued flat sections.
The system (16) is block-diagonal in the following sense
Ge0 ,e00 , 0⇒deg(e0 ) − deg(e00 ) ∈ Z.

Therefore the matrix F is also block-diagonal. Since analytic continuation around
λ = 0 corresponds to the classical monodromy transformationJ, we get that the√∨−2π −1 deg(r)vectors {Ar }r∈E give an eigenbasis of h , i.e., J(Ar ) = e
Ar .
3.2. Flat coordinates. We follow the idea of [27], except that we will avoid theuse of explicit formulas. It is convenient to introduce the following multi-index
notation. We will be interested in sequences κ = (κe )e∈E\{m} , where κe are nonnegative integers. Put
Y
Y sκe
X
sκ
e
e1 e2 e3 κeκx :=(x1 x2 x3 ) ,:=, dκ := deg(sκ ) =κe de ,κ!
κ!
ee∈E\{m}e∈E\{m} ewhere recall that de = degse = 1 − deg(xe ).

10

TODOR MILANOV AND YEFENG SHEN

Let us define a block-diagonal matrix C = (Ce,r (s))e,r∈E , whose entries are holomorphic functions on S
Xsκ
Ce,r (s) =ce,κ (σ) ,(18)κ!
κ:d =dκrwhere the functions ce,κ (s) are defined from the identity
!Z
Z −∞
3
X
λ deg(xκ )+1
κ d x
− 23
eλ
dλ
x
=
ce,κ (σ) Φe (σ, 1),
(−2π)
df
0
e∈E

(19)

where the integration path in the first integral is the negative real axis and the 2-nd
one is interpreted as a cohomology class in H 2 (Xσ,1 ; C). The identity is obtained by
performing successively integration by parts until the degree of the monomial xκ isreduced to some number in the interval [0, 1]. In particular, since each integration
by parts decreases the degree by an integer number, the sum on the RHS is over alle ∈ E, s.t., de − dκ ∈ Z. It follows that the matrix C is block-diagonal. Given cycles
αr ∈ h∨ , r ∈ E, we define the following multi-valued analytic functions on S:

D
E
D
E
tm (s) = C0,m (s) Φ0 (σ, 1), αm + Cm,m (s) Φm (σ, 1), αm π−1
A ,

D
E
D
E
t0 (s) = Cm,0 (s) Φm (σ, 1), α0 + C0,0 (s) Φ0 (σ, 1), α0 π−1
A ,
D
 X
E
Ce,r (s) Φe (σ, 1), αr π−1
tr (s) =r ∈ Etw .
A ,e∈E:de =dr

Note that by definition Cm,m (s) = 0 and C0,m (s) is a constant independent of s.
Proposition 3.1. There are cycles {αr }r∈E that form an eigenbasis for the classicalmonodromy J, s.t.,(i) The functions te (s), e ∈ E form a flat coordinate system on S.
(ii) We have ∂0 = 1 and (∂0 , ∂m ) = 1, where ∂e := ∂/∂te .
(iii) The following identity holds (compare with the definition of t0 (s)):

D
E
D
E
1 X
(∂e0 , ∂e00 )te0 te00 = Cm,0 (s) Φm (σ, 1), αm + C0,0 (s) Φ0 (σ, 1), αm π−1A .
2 e0 ,e00 ∈E
Proof. Let σ ∈ Σ be an arbitrary point. We fix a path in Σ from 0 to σ. Our goal is
to construct flat coordinates in a neighborhood of σ. Given a basis of cycles {αr }r∈E
we denote by ασ,1r ∈ H2 (Xσ,1 ; C) the parallel transport of αr . The polynomial f (σ, x)
is weighted homogeneous, so there is a natural C∗ -action on C3 , s.t., f (σ, c · x) =c f (σ, x) for every c ∈ C∗ . Using this action we define
Ar = {(λz) · y | λ ∈ (−∞, 0],y ∈ ασ,1r }.

MODULAR GROUPS AND SIMPLE ELLIPTIC SINGULARITIES

11

Note that Aσ,zr is a semi-infinite cycle of the type (6), so the corresponding oscillatory integral is convergent. Using the Fubini’s theorem we get
Z
Z −∞
Z
P
e (1−de ) −de ω−3/2
F(s,x)/z−3/2λdλ(−2πz)eω = (−2πz)e (λz)
e e∈E\{m} se x λ z
σ,z
σ,1
df
Ar
0
αr
The exponential in the 2-nd integral on the RHS isX sκκxκ λdeg(x ) z−dκ ,κ!
κwhere the sum is over all sequences κ = (κe )e∈E\{m} of non-negative integers. Substituting the above expansion we get! κZ
X Z −∞s −dκω
λ 1+deg(xκ )−3/2 −1/2e λdλ(−2π) z
xκ
z
.
σ,1
κ!
df
0
αr
κ
Comparing with formula (19) we get the following formula for the oscillatory integral


D
E 
X −d sκ X1−1κce,κ (σ) Φe (σ, 1), αr πA  .
JAr (s, z) = z 2 d 
z
(20)
κ!
κ
e∈E
The oscillatory integrals JAr (s, z) are solutions to the differential equations (9) and(10). On the other hand near z = ∞ these equations have a fundamental solution of
the type S t (z)zΘ , where S t (z) = 1 + S t,1 z−1 + · · · and S t,k ∈ HomC (H, H). Therefore,
we can choose the cycles {αr } in such a way that


1
(21)
JAr (s, z) = S t (z)zΘ dtr = z 2 −dr dtr + z−1 S t,1 dtr + · · · ,where t = (tr ) is a flat coordinate system. Note that dr > 0 for r , m and dm = 0.
Therefore, we have11

JAr (s, z) = z 2 −dr dtr + z 2 −1 δr,m S t,1 dtm + · · · ,where the dots stand for terms involving higher order powers of z−1 . Let us choose
the flat coordinates in such a way that ∂0 = 1 and (∂m , ∂0 ) = 1, then we have1  X
S t,1 dtm = S t,1 1 = t = d(∂e0 , ∂e00 )te0 te00 .
2 e0 ,e00 ∈E
Finally we get


X


1
JAr (s, z) = z d z−dr tr + z−1 δr,m(∂e0 , ∂e00 )te0 te00  + · · · .
212e0 ,e00 ∈E

All statements in the Proposition follow by comparing the above formula with (20).
12

TODOR MILANOV AND YEFENG SHEN

3.3. The monodromy of the flat coordinates. Let us choose the fundamental matrix F of the Picard–Fuchs equations (see (17)) in such a way that {αr }r∈E and {Ar }r∈E
are dual bases. Furthermore, since C0,m αm is a tube cycle we can find B ∈ H1 (E0 ; C)such that L(B) := C0,m αm , so we haveπB (σ)tm =.
πA (σ)
The flat coordinate t0 is such that ∂/∂t0 = 1. Therefore, the coefficient in front of s0
in t0 (s) must be 1. On the other hand, using formulas (18) and (19) we get
C0,m (s) = c0,0 ,C0,0 (s) = c0,0 s0 + · · · ,where the dots stand for at least quadratic terms in s. It follows that L(A) = C0,m α0 .
Let γ be a loop in Σ based at the refernce point σ = 0. Let us denote by [ρ,0 (γ)]e,r
the matrix of the linear operator ρ,0 (γ) in the basis {αr }r∈Etw , i.e.,Xρ,0 (γ)αr =[ρ,0 (γ)]e,r αe .
e∈Etw

Since the monodromy representation ρ,0 commutes with the classical monodromy
J and {αr } is an eigenbasis for J, the matrix [ρ,0 (γ)] is block diagonal[ρ,0 (γ)]e,r , 0⇒

de = dr .

Similarly let us denote by [ρ0 (γ)] the matrix of ρ0 (γ) in the basis {αm , α0 }
ρ0 (γ)αm = [ρ0 (γ)]m,m αm + [ρ0 (γ)]0,m α0 ,
ρ0 (γ)α0 = [ρ0 (γ)]m,0 αm + [ρ0 (γ)]0,0 α0 .
The space h0  H1 (E0 , C) is equipped with a symplectic form that comes from theintersection pairing. By continuity, the linear transformation ρ0 (γ) is a symplectictransformation, i.e., the matrixa b[ρ0 (γ)]m,m [ρ0 (γ)]0,m 
T



 ∈ SL2 (C).
c d := [ρ0 (γ)] =  [ρ (γ)][ρ0 (γ)]0,0 
0m,0An immediate corollary of Proposition 3.1 is the transformation rule for the flat
coordinates under the analytic continuation along γ.
Corollary 3.2. The analytic continuation along a loop γ transforms the flat coordinates as follows
a tm + be,tm =c tm + dXcet0 = t0 +(∂e0 , ∂e00 ) te0 te00 ,2(c tm + d) e0 ,e00 ∈E
tw
X
e
tr =
[ρ,0 (γ)]e,r te , r ∈ Etw .
e∈Etw :de =dr

MODULAR GROUPS AND SIMPLE ELLIPTIC SINGULARITIES

13

3.4. Monodromy of the asymptotical operator. Recall the notation from Section
2.3.1. Let us identify the space of linear operators HomC (Cµ , H) with the space ofsquare matrices of size µ by fixing the bases φe (x) := xe (e ∈ E) and ei (1 ≤ i ≤ µ)respectively in H and in Cµ , i.e.,X
Aei =[A]e,i φe .
e∈EThe asymptotical operator Ψ s R s eU s /z can be viewed as a matrix with entries formal
asymptotical series. Let us fix a loop γ in Σ. We would like to find out how theoperator changes under the analytic continuation along γ. The answer can be statedin the following way. Let M(γ, t) ∈ HomC (H, H) be the operator whose matrix is∂tr[M(γ, t)]r,e = (ctm + d)−1
− z c δe,m δ0,r ,∂etewhere c, d, and ete are determined via γ as it was stated in Corollary 3.2. Analytic
continuation along γ transforms the sequence of critical values (u1 (s), . . . , uµ (s))
via some permutation p. Let us denote by P(γ) ∈ HomC (Cµ , Cµ ) the linear operator
whose matrix is given by[P(γ)]i, j = δi,p( j) .
Proposition 3.3. The analytic continuation along the loop γ transforms the asymptotical operator Ψ s R s eU s /z intoT

M(γ, t) Ψ s R s eU s /z P(γ),

where for a linear operator A : H → H we denote by T A the transpose of A withrespect to the residue pairing (·, ·).
Proof. Let us denote by Ii (s, z)(1 ≤ i ≤ µ) the stationary phase asymptotic of theoscillatory integralZ−3/2(−2πz)
eF(s,x)/z d3 x.
Bi

By definition the asymptotical operator is defined by the following identity:(φe , Ψ s R s eU s /z ei ) = z∂e Ii (s, z)π−1
e ∈ E, 1 ≤ i ≤ µ.
A ,The analytic continuation along γ transforms the above matrix into X ∂tr−1−1ze∂e I p(i) (s, z)π−1
(ct
+
d)
=
z∂r I p(i) (s, z)π−1,mAA (ctm + d)
e
∂
t
e
r∈E
where e∂e := ∂/∂ete . Note that
X ∂tr

∂tm 
z∂r (ctm + d)−1 = zδe,m− c (ctm + d)−2 = −zc δe,m
∂e
te
∂e
tm
r∈E

(22)

14

TODOR MILANOV AND YEFENG SHEN

and


−1
I p(i) (s, z)π−1
A = z∂0 I p(i) (s, z)πA .
Hence the RHS of (22) becomesX∂tr(ctm + d)−1z∂r (I p(i) (s, z)π−1)− zcδe,m z∂0 (I p(i) (s, z)π−1
A
A ).
e
∂
t
e
r∈E
It remains only to check that the above expression coincides withX
(M(γ, t)φe , Ψ s R s eU s /z P ei ) =[M(γ, t)]r,e z∂r (I p(i) (s, z)π−1
A ).



r∈E

Let denote by J(γ, t) ∈ HomC (H, H) the linear operator whose matrix is the Jacobian of the transformation t 7→ et (see Corollary 3.1)
[J(γ, t)]r,e =

∂tr
.
∂e
te

Let us introduce also the linear operatorX(γ, t) = 1 −cz
φm • s=0 ,ctm + dwhere φm • s=0 : H → H is the operator of multiplication by φm in the Jacobi algebra
H. Note that X(γ, t) is a symplectic transformation (of H).
Proposition 3.4. Analytic continuation transformsb t) At )(~(ctm + d)2 ; J(γ, t)q),
At (~; q) 7→ (X(γ,b t) and then we rescale ~ and q.
where we first apply the operator X(γ,We may assume that P(γ, t) = 1 because P is a permutation matrix, so its quanb will leave the product of Kontsevich–Witten tau functions invariant. Put
tization P
M = M0 + M1 z. Then we haveT

e Re
e U/z ,
MΨReU/z = Ψ

where

e = M0−1 Ψ,Ψe = Ψ−1 M0 T MΨR.
R

The quantization is in general only a projective representation. However, the quantization of the operators Ψ−1 M0 T MΨ and R involves quantizing only p2 and p qterms. Since the cocycle (11) on such terms vanishes we get
e
b
(R)b=
(Ψ−1 M0 T MΨ)bR.
The operators M0 and Ψ are independent of z and their quantizations by definition
are just changes of variables. Hencee R)b=ec0 −1 (M0 T M)b(ΨR)b.
(Ψ
M

MODULAR GROUPS AND SIMPLE ELLIPTIC SINGULARITIES

15−2
under analytic
By definition ∆−1
i is (∂ui , ∂ui ), which gains a factor of (ctm + d))continuation. The ancestor potential (13) is transformed into
c0 −1 (M0 T M)b At ((ctm + d)2 ~; (ctm + d)q) .
(23)
MNote thatM0−1 = (ctm + d) J(γ, t)−1 ,M0 (T M) = X(γ, t).

It remains only to notice that the rescaling(~, q) 7→ ((ctm + d)2 ~, (ctm + d)q)
commutes with the action of any quantized operator.



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