The Singularity of Kontsevich's Solution for $QH^{*}(CP^2)$

12 Pages·0.16 MB·English

by  Davide Guzzetti

The Singularity of
2
Kontsevich’s Solution for QH (CP )
∗
Davide Guzzetti
Research Institute for Mathematical Sciences (RIMS)
Kyoto University
Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan
Phone [phone], Fax [phone]
E-mail: [email protected]
3
0
0
2

n
Abstract.
a
J In this paper we study the nature of the singularity of the Kontsevich’s solution of the WDVV
0 equationsofassociativity. Weprovethatitcorrespondstoasingularityinthechangeoftwocoordinates
1 systems of the Frobenius manifold given by the quantum cohomology of CP2.

1
v 1 Introduction
1
1
InthispaperwestudythenatureofthesingularityofthesolutionoftheWDVVequationsofassociativity
0
for the quantum cohomology of the complex projective space CP2. As we will explain in detail below,
1
0 the quantum cohomology of a projective space CPd (d integer) is a Frobenius Manifold which has a
3 structure specified by a solutionto a WDVV equation. In the case ofCP2 sucha solutionwasfound by
0 Kontsevich [20] in the form of a convergentseries in the flat coordinates (t1,t2,t3) of the corresponding
/
h Frobenius manifold:
p
h- F(t):= 12[(t1)2t3+t1(t2)2)]+ t13 ∞ Ak (t3)3exp(t2) k, Ak ∈R (1)
t k=1
a X (cid:2) (cid:3)
m Theseriesconvergesinaneighborhoodof(t3)3exp(t2)=0withacertainradiusofconvergenceextimated
: by Di Francesco and Itzykson [5]. The coefficients A are real and are the Gromov-Witten invariants of
v k
genus zero. We will explain this point later. As for the Gromov-Witten invariants of genus one of CP2,
i
X we refer to [13], where B. Dubrovin and Y. Zhang proved that their G-function has the same radius of
r convergence of (1).
a
As we will explain below, the nature of the boundary points of the ball of convergence of (1) is
important to study of the global structure of the manifold.
In the following, we first state rigorously the problem of the global structure of a Frobenius mani-
fold, then we introduce the quantum cohomology of CPd as a Frobenius manifold and we explain its
importanceinenumerativegeometry. Finally,westudytheboundarypointsoftheballofconvergenceof
Kontsevich’s solution. We prove that they correspond to a singularity in the change of two coordinates
systems.
Ourpaperispartofaprojecttostudyofthe globalstructureofFrobeniusmanifoldsthatwestarted
in [15].
1.1 Frobenius manifolds and their global structure
The subject of this subsection can be found in [9], [10] or, in a more synthetic way, in [15].
The WDVV equations of associativity were introduced by Witten [28], Dijkgraaf, Verlinde E., Ver-
linde H.[6]. They aredifferentialequationssatisfiedbythe primary free energy F(t)intwo-dimensional
topological field theory. F(t) is a function of the coupling constants t := (t1,t2,...,tn) ti C. Let
∈
∂ := ∂ . Given a non-degenerate symmetric matrix ηαβ, α,β = 1,...,n, and numbers q ,q ,...,q ,
α ∂tα 1 2 n
r ,r ,...,r , d, (r =0 if q =1, α=1,...,n), the WDVV equations are
1 2 n α α
6
∂ ∂ ∂ F ηλµ ∂ ∂ ∂ F = the same with α, δ exchanged, (2)
α β λ µ γ δ
1
∂ ∂ ∂ F =η , (3)
1 α β αβ
E(F)=(3 d)F + (at most) quadratic terms, (4)
−
where the matrix (η ) is the inverse of the matrix (ηαβ) and the differential operator E is E :=
αβ
n Eα∂ , Eα :=(1 q )tα+r , α=1,...,n, and will be called Euler vector field.
α=1 α − α α
FrobeniusstructuresfirstappearedintheworksofK.Saito[25][26]withthenameofflat scructures.
P
ThecompletetheoryofFrobeniusmanifoldswasthendevelopedbyB.Dubrovinasageometricalsetting
for topological field theory and WDVV equations in [7]. Such a theory has links to many branches of
mathematics like singularity theory and reflection groups [25] [26] [12] [9], algebraic and enumerative
geometry[20][22],isomonodromicdeformationstheory,boundaryvalueproblemsandPainlev´eequations
[10].
Ifwedefinec (t):=∂ ∂ ∂ F(t),cγ (t):=ηγµc (t)(sumoverrepeatedindicesisalwaysomitted
αβγ α β γ αβ αβµ
inthepaper),andweconsideravectorspaceA=span(e ,...,e ),thenweobtainafamilyofcommutative
1 n
algebras A with the multiplication e e := cγ (t)e . Equation (2) is equivalent to associativity and
t α· β αβ γ
(3) implies that e is the unity.
1
Definition: AFrobenius manifoldisasmooth/analyticmanifoldM overCwhosetangentspaceT M at
t
any t M is an associative, commutative algebra with unity e. Moreover,there exists a non-degenerate
∈
bilinearform< , >definingaflatmetric(flatmeansthatthecurvatureassociatedtotheLevi-Civita
connection is zero).
We denote by the product and by the covariant derivative of < , >. We require that the
· ∇ · ·
tensors c(u,v,w) :=< u v,w >, and c(u,v,w), u,v,w,y T M, be symmetric. Let t1,..,tn be
y t
· ∇ ∈
(local) flat coordinates for t M. Let e := ∂ be the canonical basis in T M, η :=< ∂ ,∂ >,
α α t αβ α β
∈
c (t) :=< ∂ ∂ ,∂ >. The symmetry of c corresponds to the complete symmetry of ∂ c (t) in
αβγ α β γ δ αβγ
·
the indices. This implies the existence of a function F(t) such that ∂ ∂ ∂ F(t)=c (t) satisfying the
α β γ αβγ
WDVV (2). The equation (3) follows from the axiom e=0 which yields e=∂ . . Some more axioms
1
∇
are needed to formulate the quasi-homogeneity condition (4) and we refer the reader to [9] [10] [11]. In
this way the WDVV equations are reformulated in a geometrical terms.
We first consider the problemof the localstructure ofFrobenius manifolds. A Frobenius manifold is
characterized by a family of flat connections ˜(z) parameterized by a complex number z, such that for
z = 0 the connection is associated to < , ∇>. For this reason ˜(z) are called deformed connections.
∇
Let u,v T M, d T C; the family is defined on M C as follows:
∈ t dz ∈ z ×
˜ v := v+zu v,
u u
∇ ∇ ·
∂ 1
˜ v := v+E v µˆv,
d
∇dz ∂z · − z
d d
˜ =0, ˜ =0
d u
∇dz dz ∇ dz
where E is the Euler vector field and
d
µˆ:=I E
− 2 −∇
is an operator acting on v. In flat coordinates t=(t1,...,tn), µˆ becomes:
d
µˆ=diag(µ ,...,µ ), µ =q ,
1 n α α
− 2
provided that E is diagonalizable. This will be assumed in the paper. A flat coordinate t˜(t,z) is a
solution of ˜dt˜∇=0, which is a linear system
∇
∂ ξ =zC (t)ξ, (5)
α α
µˆ
∂ ξ = (t)+ ξ, (6)
z
U z
(cid:20) (cid:21)
where ξ is a column vector of components ξα = ηαµ ∂t˜, α = 1,...,n and C (t) := cβ (t) , :=
∂tµ α αγ U
Eµcβ (t) are n n matrices.
µγ × (cid:0) (cid:1)
(cid:0) (cid:1)
2
The quantum cohomology of projective spaces, to be introduce below, belongs to the class of semi-
simple Frobenius manifolds, namely analytic Frobenius manifolds such that the matrix can be diag-
U
onalized with distinct eigenvalues on an open dense subset of M. Then, there exists an invertible
matrix φ0 = φ0(t) such that φ0Uφ−01 = diag(u1,...,un)=:U,Mui 6= uj for i 6=j on M. The systems (5)
and (6) become:
∂y
=[zE +V ] y (7)
i i
∂u
i
∂y V
= U + y, (8)
∂z z
(cid:20) (cid:21)
where the row-vector y is y := φ ξ, E is a diagonal matrix such that (E ) = 1 and (E ) = 0
0 i i ii i jk
otherwise, and
∂φ
Vi := ∂u0 φ−01 V :=φ0 µˆ φ−01,
i
As it is proved in [9] [10], u ,...,u are local coordinates on . The two bases ∂ , ν = 1,...,n and
1 n M ∂tν
∂ ,i=1,...,narerelatedbyφ accordingtothe linearcombination ∂ = n (φ0)iν ∂ . Locallywe
∂ui 0 ∂tν i=1 (φ0)i1 ∂ui
obtain a change of coordinates, tα = tα(u), then φ = φ (u), V = V(u). The local Frobenius structure
0 0 P
of is given by parametric formulae:
M
tα =tα(u), F =F(u) (9)
wheretα(u),F(u)arecertainmeromorphicfunctionsof(u ,...,u ),u =u ,whichcanbeobtainedfrom
1 n i j
6
φ (u) and V(u). Their explicit construction was the object of [15]. We stress here that the condition
0
u =u is crucial. We will further comment on this when we face the problem of the global structure.
i j
6
The dependence of the system on u is isomonodromic. This means that the monodromy data of
the system (8), to be introduced below, do not change for a small deformation of u. Therefore, the
coefficients of the system in every local chart of are naturally labeled by the monodromy data. To
M
calculatethefunctions(9)ineverylocalchartonehastoreconstructthesystem(8)fromitsmonodromy
data. This is the inverse problem.
We briefly explain what are the monodromy data of the system (8) and why they do not depend on
u (locally). For details the reader is referred to [10]. At z =0 the system (8) has a fundamental matrix
solution (i.e. an invertible n n matrix solution) of the form
×
∞
Y (z,u)= φ (u) zp zµˆzR, (10)
0 p
" #
p=0
X
whereR =0ifµ µ =k >0,k N. Atz = thereisaformaln nmatrixsolutionof(8)given
αβ α β
− 6 ∈ ∞ ×
by Y = I + F1(u) + F2(u) +... ez U where F (u)’s are n n matrices. It is a well known result that
F z z2 j ×
thereexishtfundamentalmatrixsiolutionswithasymptoticexpansionYF asz [2]. Letl beageneric
→∞
oriented line passing through the origin. Let l be the positive half-line and l the negative one. Let
+
−
Π andΠ be twosectorsinthe complex planeto the left andto the rightofl respectively. There exist
L R
unique fundamental matrix solutions Y and Y having the asymptotic expansion Y for x in Π
L R F L
→∞
and Π respectively [2]. They are related by an invertible connection matrix S, called Stokes matrix,
R
such that Y (z)=Y (z)S for z l . As it is proved in [10] we also have Y (z)=Y (z)ST on l .
L R + L R
∈ −
Finally, there exists a n n invertible connection matrix C such that Y =Y C on Π .
0 R R
×
Definition: The matrices R, C, µˆ and the Stokes matrix S of the system (8) are the monodromy data
of the Frobenius manifold in a neighborhood of the point u=(u ,...,u ). It is also necessary to specify
1 n
which is the first eigenvalue of µˆ, because the dimension of the manifold is d = 2µ (a more precise
1
−
definition of monodromy data is in [10]).
The definitionmakessense because the datado notchangeif uundergoesa smalldeformation. This
problem is discussed in [10]. We also refer the reader to [17] for a general discussion of isomonodromic
deformations. Here wejust observethat since a fundamentalmatrix solutionY(z,u)of (8)alsosatisfies
(7),thenthemonodromydatacannotdependonu(locally). Infact, ∂Y Y 1 =zE +V issingle-valued
∂ui − i i
in z.
3
The inverse problem can be formulated as a boundary value problem (b.v.p.). Let’s fix u = u(0) =
(u(0),...,u(0)) such that u(0) = u(0) for i = j. Suppose we give µ, µ , R, an admissible line l, S and
1 n i 6 j 6 1
C. Some more technical conditions must be added, but we refer to [10]. Let D be a disk specified by
z <ρ for some small ρ. Let P and P be the intersectionof the complement of the disk with Π and
L R L
| |
Π respectively. We denote by ∂D and ∂D the lines on the boundary of D on the side of P andP
R R L R L
respectively; we denote by ˜l and ˜l the portion of l and l on the common boundary of P and P .
+ + R L
− −
Let’s consider the following discontinuous b.v.p.: we want to construct a piecewise holomorphic n n
×
matrix function
Φ (z), z P
R R
∈
Φ(z)= Φ (z), z P ,
 L ∈ L
Φ (z), z D
 0
∈
continuous on the boundary of P , P , D respectively, such that
R L 
Φ (ζ)=Φ (ζ) eζUSe ζU, ζ ˜l
L R − +
∈
Φ (ζ)=Φ (ζ) eζUSTe ζU, ζ ˜l
L R −
∈ −
Φ (ζ)=Φ (ζ) eζUCζ Rζ µˆ, ζ ∂D
0 R − − R
∈
Φ (ζ)=Φ (ζ) eζUS 1Cζ Rζ µˆ, ζ ∂D
0 L − − − L
∈
Φ (z) I if z in P .
L/R L/R
→ →∞
The reader may observe that Y˜ (z):=Φ (z)ezU, Y˜(0)(z):=Φ (z,u)zµˆzR have precisely the mon-
L/R L/R 0
odromy properties of the solutions of (8).
Theorem[23][21][10]: If the above boundary value problem has solution for a given u(0) =(u(0),...,u(0))
1 n
such that u(0) =u(0) for i=j, then:
i 6 j 6
i) it is unique.
ii) The solution exists and it is analytic for u in a neighborhood of u(0).
iii) The solution has analytic continuation as a meromorphic function on the universal covering of
Cn diagonals , where “diagonals” stands for the union of all the sets u Cn u =u , i=j .
i j
\{ } { ∈ | 6 }
A solution Y˜ , Y˜(0) of the b.v.p. solves the system (7), (8). This means that we can locally
L/R
reconstructV(u), φ (u) and (9) fromthe localsolution of the b.v.p.. It follows that every localchart of
0
the atlas covering the manifold is labeled by monodromy data. Moreover, V(u), φ (u) and (9) can be
0
continued analytically as meromorphic functions on the universal covering of Cn diagonals.
\
Let be the symmetric group of n elements. Local coordinates (u ,...,u ) are defined up to
n 1 n
S
permutation. Thus,theanalyticcontinuationofthelocalstructureof isdescribedbythe braid group
M
, namely the fundamental group of (Cn diagonals)/ . There exists an action of the braid group
n n
B \ S
itself on the monodromydata, correspondingto the change of coordinate chart. The groupis generated
by n 1 elements β ,...,β such that β is represented as a deformation consisting of a permutation
1 n 1 i
− −
of u , u moving counter-clockwise (clockwise or counter-clockwise is a matter of convention).
i i+1
If u , ..., u are in lexicographical order w.r.t. l, so that S is upper triangular, the braid β acts on
1 n i
S as follows [10]:
S Sβi =A (S) S A (S)
i i
7→
where
(A (S)) =1 k =1,...,n n= i, i+1
i kk 6
(A (S)) = s
i i+1,i+1 − i,i+1
(A (S)) =(A (S)) =1
i i,i+1 i i+1,i
andalltheotherentriesarezero. Foragenericbraidβ theactionS Sβ isdecomposedintoasequence
→
of elementary transformations as above. In this way, we are able to describe the analytic continuation
of the local structure in terms of monodromy data.
Not all the braids are actually to be considered. Suppose we do the following gauge y Jy,
7→
J =diag( 1,..., 1), on the system (8). Therefore JUJ 1 U but S is transformed to JSJ 1, where
− −
± ± ≡
someentrieschangesign. Theformulaewhichdefinealocalchartofthemanifoldintermsofmonodromy
data, which are given in [10], [15], are not affected by this transformation. The analytic continuation of
4
thelocalstructureontheuniversalcoveringof(Cn diagonals)/ isthereforedescribedbytheelements
n
\ S
of the quotient group
/ β Sβ =JSJ (11)
n n
B { ∈B | }
From these considerations it is proved in [10] that:
Theorem [10]: Given monodromy data (µ , µˆ, R, S, C), the local Frobenius structure obtained from
1
the solution of the b.v.p. extends to an open dense subset of the covering of (Cn diagonals)/ w.r.t.
n
\ S
the covering transformations (11).
Let’s start from a Frobenius manifold M of dimension d. Let be the open sub-manifold where (t)
M U
has distinct eigenvalues. If we compute its monodromy data (µ = d, µˆ, R, S, C) at a point u(0)
1 −2 ∈M
and we construct the Frobenius structure from the analytic continuation of the corresponding b.v.p. on
the covering of (Cn diagonals)/ w.r.t. the quotient (11), then there is an equivalence of Frobenius
n
\ S
structures between this last manifold and .
M
Tounderstandingtheglobal structureofaFrobeniusmanifoldwehavetostudy(9)whentwoormore
distinct coordinates u , u , etc, merge. φ (u), V(u) and (9) are multi-valued meromorphic functions of
i j 0
u = (u ,...,u ) and the branching occurs when u goes around a loop around the set of diagonals
1 n
u Cn u = u , i= j . φ (u), V(u) and (9) have singular behavior if u u (i =j). We call
ij{ ∈ | i j 6 } 0 i → j 6
such behavior critical behavior.
S
The Kontsevich’s solution introduced at the beginning has a radius of convergence which might be
due to the fact that some coordinates u , u merge at the boundary of the ball of convergence. We will
i j
prove that this is not the case. Rather, there is a singularity in the change of coordinates u t.
7→
1.2 Intersection Form of a Frobenius Manifold
The deformedflat connectionwas introducedas a naturalstructure ona Frobenius manifoldand allows
to transformthe problemofsolvingthe WDVVequationsto aproblemofisomonodromicdeformations.
There is a further natural structure on a Frobenius manifold which makes it possible to do the same. It
is the intersection form. We need it as a tool to calculate the canonical coordinates later.
There is a natural isomorphism ϕ : T M T M induced by < .,. >. Namely, let v T M and
t → t∗ ∈ t
define ϕ(v):=<v,.>. This allow us to define the product in T M as follows: for v,w T M we define
t∗ ∈ t
ϕ(v) ϕ(w):=<v w,.>. In flat coordinates t1,...,tn the product is
· ·
dtα dtβ =cαβ(t) dtγ, cαβ(t)=ηβδcα (t),
· γ γ δγ
(sums over repeated indices are omitted).
Definition: The intersection form at t M is a bilinear form on T M defined by
∈ t∗
(ω ,ω ):=(ω ω )(E(t))
1 2 1 2
·
where E(t) is the Euler vector field. In coordinates
gαβ(t):=(dtα,dtβ)=Eγ(t)cαβ.
γ
Inthe semi-simplecase,letu ,...,u be localcanonicalcoordinates,equaltothe distinct eigenvalues
1 n
of (t). From the definitions we have
U
1 u
du du = δ du , gij(u)=(du ,du )= iδ , η =(φ )2
i· j η ij i i j η ij ii 0 i1
ii ii
Then gij −ληij = uηi−iiλδij and
1
det((gij ληij))= (u λ)(u λ)...(u λ).
1 2 n
− det((η )) − − −
ij
Namely, the roots λ of the above polynomial are the canonical coordinates.
In order to compute gαβ, in the paper we are going to use the following formula. We differentiate
twice the expression
1
Eγ∂ F =(2 d)F + A tαtβ +B tα+C
γ αβ α
− 2
5
which is the quasi-homogeneity of F up to quadratic terms. By recalling that Eγ =(1 q )tγ+r and
γ γ
−
that ∂ ∂ ∂ F =c we obtain
α β γ αβγ
gαβ(t)=(1+d q q )∂α∂βF(t)+Aαβ (12)
α β
− −
where ∂α =ηαβ∂ , Aαβ =ηαγηβδA .
β γδ
2 Quantum Cohomology of Projective spaces
In this section we introduce the Frobenius manifold called quantum cohomology of the projective space
CPd and we describe its connections to enumerative geometry.
It is possible to introduce a structure of Frobenius algebra on the cohomology H (X,C) of a closed
∗
oriented manifold X of dimension d such that
Hi(X,C)=0 for i odd.
Then
H (X,C)= d H2i(X,C).
∗ ⊗i=0
For brevity we omit C in H. H (X) can be realized by classes of closed differential forms. The unit
∗
element is a 0-form e1 H0(X). Let us denote by ωα a form in H2qα(X), where q1 = 0, q2 = 1, ...,
∈
qd+1 = d. The product of two forms ωα, ωβ is defined by the wedge product ωα ωβ H2(qα+qβ)(X)
∧ ∈
and the bilinear form is
<ω ,ω >:= ω ω =0 q +q =d
α β α β α β
∧ 6 ⇐⇒
ZX
It is not degenerate by Poincar´e duality and q +q =d.
α d α+1
−
LetX =CPd. Lete =1 H0(CPd),e H2(CPd),...,e H2d(CPd)beabasisinH (CPd).
1 2 d+1 ∗
∈ ∈ ∈
For a suitable normalization we have
1
1
 
(η ):=(<e ,e >)= ·
αβ α β ·
·
 1 
 
1 
 
 
The multiplication is
e e =e .
α β α+β 1
∧ −
We observe that it can also be written as
e e =cγ e , sums on γ
α∧ β αβ γ
where
∂3F (t)
η cδ := 0
αδ βγ ∂tα∂tβ∂tγ
n 1
1 1 −
F0(t):= (t1)2tn+ t1 tαtn−α+1
2 2
α=2
X
F is the trivial solution of WDVV equations. We can construct a trivial Frobenius manifold whose
0
points are t := d+1 tαe . It has tangent space H (CPd) at any t. By quantum cohomology of CPd
α=1 α ∗
(denoted by QH (CPd)) we mean a Frobenius manifold whose structure is specified by
∗
P
F(t)=F (t)+ analytic perturbation
0
This manifold has therefore tangent spaces T QH (CPd)=H (CPd), with the same <.,.> as above,
t ∗ ∗
but the multiplication is a deformation, depending on t, of the wedge product (this is the origin of the
adjective “quantum”).
6
3 The case of CP2
To start with, we restrict to CP2. In this case
1
F (t)= (t1)2t3+t1(t2)2
0
2
(cid:2) (cid:3)
which generates the product for the basis e = 1 H0, e H2, e H4. The deformation was
1 2 3
∈ ∈ ∈
introduced by Kontsevich [20].
3.1 Kontsevich’s solution
The WDVV equations for n=3 variables have solutions
F(t ,t ,t )=F (t ,t ,t )+f(t ,t ).
1 2 3 0 1 2 3 2 3
f(t ,t )satisfiesadifferentialequationobtainedbysubstitutingF(t)intotheWDVVequations. Namely:
2 3
f f +f =(f )2 (13)
222 233 333 223
withthenotationf := ∂3f . Asfornotations,thevariablest areflatcoordinatesintheFrobenius
ijk ∂ti∂tj∂tk j
manifoldassociatetoF. Theyshouldbewrittenwithupperindices,butweusethelowerforconvenience
of notation.
Let N be the number of rational curves CP1 CP2 of degree k through 3k 1 generic points.
k
→ −
Kontsevich [20] constructed the solution
1 ∞
f(t ,t )= ϕ(τ), ϕ(τ)= A τk, τ =t3 et2 (14)
2 3 t k 3
3
k=1
X
where
N
k
A =
k
(3k 1)!
−
The A (or N ) are called Gromov-Witten invariants of genus zero. We note that this solution has
k k
precisely the form of the general solution of the WDVV eqs. for n =3, d=2 and r =3 [9]. If we put
2
τ =eX and we define
∞
Φ(X):=ϕ(eX)= A ekX,
k
k=1
X
we rewrite (13) as follows:
6Φ+33Φ 54Φ (Φ )2+Φ (27+2Φ 3Φ )=0 (15)
′ ′′ ′′ ′′′ ′ ′′
− − − −
The prime stands for the derivative w.r.t X. If we fix A , the above (15) determines the A uniquely.
1 k
Since N =1, we fix
1
1
A = .
1
2
Then (15) yields the recurrence relation
k−1 AiAk i i(k i) (3i 2)(3k 3i 2)(k+2)+8k 8
Ak = − − − − − − (16)
6(3k 1)(3k 2)(3k 3)
i=1" (cid:0) − − − (cid:1)#
X
The convergence of (14) was studied by Di Francesco and Itzykson [5]. They proved that
Ak =b ak k−72 1+O 1 , k
k →∞
(cid:18) (cid:18) (cid:19)(cid:19)
and numerically extimated
a=0.138, b=6.1 .
7
We remark that the problem of the exact computation of a and b is open. The result implies that ϕ(τ)
converges in a neighborhood of τ =0 with radius of convergence 1.
a
We remark that as far as the Gromov-Witten invariants of genus one are concerned, B. Dubrovin
and Y. Zhang provedin [13] that their G-function has the same radius of convergenceof (1). Moreover,
they provedthe asymptoticformulaforsuchinvariantsas conjectuderby DiFrancesco–Itzykson. As far
as I know, such a result was explained in lectures, but not published.
The proof of [5] is divided in two steps. The first is based on the relation (16), to prove that
1 1 2
Ak a for k , <a<
k → →∞ 108 3
a is real positive because the A ’s are such. It follows that we can rewrite
k
1
A =bak kω 1+O , ω R
k
k ∈
(cid:18) (cid:18) (cid:19)(cid:19)
The above estimate implies that ϕ(τ) has the radius of convergence 1. The second step is the determi-
a
nation of ω making use of the differential equation (15). Let’s write
A :=C ak
k k
∞ ∞ 1
Φ(X)= Ak ekX = Ck ek(X−X0), X0 :=ln
a
k=1 k=1
X X
The inequality 1 < a < 2 implies that X > 0. The series converges at least for X < X . To
108 3 0 ℜ 0
determine ω we divide Φ(X) into a regular part at X and a singular one. Namely
0
∞ ∞
Φ(X)= d (X X )k+(X X )γ e (X X )k, γ >0, γ N,
k 0 0 k 0
− − − 6∈
k=0 k=0
X X
d and e are coefficients. By substituting into (15) we see that the equation is satisfied only if γ = 5.
k k 2
Namely:
Φ(X)=d0+d1(X X0)+d2(X X0)2+e0(X X0)25 +...
− − −
This implies that Φ(X), Φ(X) and Φ (X) exist at X but Φ (X) diverges like
′ ′′ 0 ′′′
1
Φ′′′(X) , X X0 (17)
≍ √X X →
0
−
On the other hand Φ (X) behaves like the series
′′′
∞
b kω+3 ek(X X0), (X X )<0
− 0
ℜ −
k=1
X
Suppose X R, X <X . Let us put ∆:=X X <0. The above series is
0 0
∈ −
b ∞ (∆k)3+ωe−|∆|k b ∞dx x3+ωe−x
∆3+ω | | ∼ ∆3+ω
| | k=1 | | Z0
X
It follows from (17) that this must diverge like ∆−12, and thus ω =−27 (the integral remains finite).
As a consequence of (15) and of the divergence of Φ (X)
′′′
27+2Φ(X ) 3Φ (X )=0
′ 0 ′′ 0
−
4 The case of CPd
The case d=1 is trivial, the deformation being:
1
F(t)= t2t +et2
2 1 2
8
For any d 2, the deformation is given by the following solution of the WDVV equations [20] [22]:
≥
F(t)=F (t)+ ∞ ∞ ˜ Nk(α1,...,αn) t ...t ekt2
0 n! α1 αn
" #
kX=1 nX=2α1X,...,αn
where
˜
:=
α1X,...,αn α1+...+αn=2Xn+d(k+1)+k−3
Here N (α ,...,α ) is the number of rational curves CP1 CPd of degree k through n projective
k 1 n
→
subspaces of codimensions α 1,...,α 1 2 in general position. In particular, there is one line
1 n
− − ≥
through two points, then
N (d+1,d+1)=1
1
Note that in Kontsevich solution N =N (d+1,d+1).
k k
In flat coordinates the Euler vector field is
∂ ∂
E = (1 q )tα + k
− α ∂tα ∂t2
α=2
X6
q =0, q =1, q =2, ..., q =k 1
1 2 3 k
−
and
d d 2 d 2 d d
µˆ=diag(µ ,...,µ )=diag( , − ,..., − , ), µ =q
1 k α α
−2 − 2 2 2 − 2
5 Nature of the singular point X
0
We are now ready to formulate the problem of the paper. We need to investigate the nature of the
singularity X , namely whether it corresponds to the fact that two canonical coordinates u , u , u
0 1 2 3
merge. Actually, we pointed out that the structure of the semi-simple manifold may become singular
in such points because the solutions of the boundary value problem are meromorphic on the universal
covering of Cn diagonals and are multi valued if u u (i = j) goes around a loop around zero. We
i j
\ − 6
will verify that actually u , u do not merge, but the change of coordinates u t is singular at X . In
i j 0
7→
this section we restore the upper indices for the flat coordinates tα.
Thecanonicalcoordinatescanbecomputedfromtheintersectionform. Werecallthattheflatmetric
is
0 0 1
η =(ηαβ):= 0 1 0
 
1 0 0
 
The intersection form is given by the formula (12):
gαβ =(d+1 q q ) ηαµηβν∂ ∂ F +Aαβ, α,β =1,2,3,
α β µ ν
− −
where d = 2 and the charges are q = 0, q = 1, q = 2. The matrix Aαβ appears in the action of the
1 2 3
Euler vector field
E :=t1∂ +3∂ t3∂
1 2 3
−
on F(t1,t2,t3):
E(F)(t1,t2,t3)=(3 d)F(t1,t2,t3)+A tµtν F(t1,t2,t3)+3t1t2
µν
− ≡
Thus
0 0 0
(Aαβ)=(ηαµηβνA )= 0 0 3
µν
 
0 3 0
After the above preliminaries, we are able to compute theintersection form:
3 [2Φ 9Φ +9Φ ] 2 [3Φ Φ] t1
[t3]3 − ′ ′′ [t3]2 ′′− ′
(gαβ)= 2 [3Φ Φ] t1+ 1Φ 3 
[t3]2 ′′− ′ t3 ′′
 
 t1 3 t3
 
 − 
9
The canonical coordinates are roots of
det((gαβ uη)=0
−
This is the polynomial
1 t1 1
u3 3t1+ Φ′′ u2 3[t1]2 2 Φ′′+ (9Φ′′+15Φ′ 6Φ) u+P(t,Φ)
− t3 − − − t3 [t3]2 −
(cid:18) (cid:19) (cid:18) (cid:19)
where
1
P(t,Φ)= 9t1t3Φ +243Φ 243Φ +6ΦΦ+
′′ ′′ ′ ′
[t3]3 − −
(cid:0)
9(Φ )2+6t1t3Φ+[t1]2[t3]2Φ 3ΦΦ +[t1]3[t3]3 4(Φ)2+54Φ 15t1t3Φ
′′ ′′ ′ ′′ ′ ′
− − − −
It follows that (cid:1)
1
u (t1,t3,X)=t1+ (X)
i t3Vi
(X) depends on X through Φ(X) and derivatives. We also observe that
i
V
1
u +u +u =3t1+ Φ (X)
1 2 3 t3 ′′
As a first step, we verify numerically that u =u for i=j at X =X . In order to do this we need
i j 0
6 6
to compute Φ(X ), Φ(X ) , Φ (X ) in the following approximation
0 ′ 0 ′′ 0
N N N
1 1 1
Φ(X0)∼= Ak ak, Φ′(X0)∼= k Ak ak, Φ′′(X0)∼= k2 Ak ak,
k=1 k=1 k=1
X X X
We fixed N =1000and we computed the A , k=1,2,...,1000exactly using the relation(16). Then we
k
computed a and b by the least squares method. For large k, say for k N , we assumed that
0
≥
Ak ∼=bakk−27 (18)
which implies
7
ln(Ak k2)∼=(lna) k+lnb
The corrections to this law are O k1 . This is the line to fit the data k72Ak. Let
(cid:0) (cid:1)
N N
y¯:= 1 ln(Ak k27), k¯:= 1 k.
N N +1 N N +1
0 0
− kX=N0 − XN0
By the least squares method
lna= Nk=N0 (k−k¯)(ln(Ak k72)−y¯), with error 1
P Nk=N0 (k−k¯)2 (cid:18)k¯2(cid:19)
P 1
lnb=y¯ (lna) k¯, with error
− k¯
(cid:18) (cid:19)
ForN =1000,A isoftheorder10 840. Inourcomputationwesettheaccuracyto890digits. Hereis
1000 −
the results, for three choices of N . The result should improve as N increases,since the approximation
0 0
(18) becomes better.
N =500, a=0.138009444..., b=6.02651...
0
N =700, a=0.138009418..., b=6.03047...
0
N =900, a=0.138009415..., b=6.03062...
0
It follows that (for N =900)
0
Φ(X0)=4.268908..., Φ′(X0)=5.408... , Φ′′(X0)=12.25...
10