the eynard-orantin recursion and equivariant mirror symmetry for the projective line

8/21/2019 The Eynard-Orantin Recursion and Equivariant Mirror Symmetry for the Projective Line

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arXiv:1411

.3557v1

[math.AG

]13Nov2014

THE EYNARD-ORANTIN RECURSION AND EQUIVARIANT

MIRROR SYMMETRY FOR THE PROJECTIVE LINE

BOHAN FANG, CHIU-CHU MELISSA LIU, AND ZHENGYU ZONG

Abstract. We study the equivariantly perturbed mirror Landau-Ginzburgmodel ofP1. We show that the Eynard-Orantin recursion on this model en-

codes all genus all descendants equivariant Gromov-Witten invariants of P1.The non-equivariant limit of this result is the Norbury-Scott conjecture [25,5],while by taking large radius limit we recover the Bouchard-Marino conjecture

on simple Hurwitz numbers [2].

Contents

1. Introduction 21.1. Main Results 21.2. Non-equivariant limit and the Norbury-Scott conjecture 21.3. Large radius limit and the Bouchard-Marino conjecture 3Acknowledgment 32. A-model 32.1. Equivariant cohomology ofP1 32.2. Equivariant quantum cohomology ofP1 42.3. The A-model canonical coordinates and the -matrix 5

2.4. The S-operator 62.5. The A-modelR-matrix 72.6. Gromov-Witten potentials 72.7. Giventals formula for equivariant Gromov-Witten potential and the

A-model graph sum 83. B-model 113.1. The equivariant superpotential and the Frobenius structure of the

Jacobian ring 113.2. The B-model canonical coordinates 123.3. The Liouville form and Bergman kernel 123.4. Differentials of the second kind 133.5. Oscillating integrals and the B-modelR-matrix 143.6. The Eynard-Orantin topological recursion and the B-model graph

sum 193.7. All genus mirror symmetry 194. The non-equivariant limit and the Norbury-Scott conjecture 224.1. The non-equivariant R-matrix 224.2. The Norbury-Scott Conjecture 245. The large radius limit and the Bouchard-Marino conjecture 26Appendix A. Bessel functions 28Appendix B. The Equivariant Quantum Differential Equation for P1 30

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2 BOHA N FA NG, CHIU-CHU MELISS A L IU, A ND ZHE NGYU ZON G

References 31

1. Introduction

Okounkov-Pandharipande completely solved the equivariant Gromov-Witten the-ory of the projective line and established a GW/H correspondence between thestationary sector of Gromov-Witten theory ofP1 and Hurwitz theory [26, 27].

The Norbury-Scott conjecture [25] relates Gromov-Witten invariants of P1 toEynard-Orantin invariants [10] of the affine plane curve {x = Y + 1

Y (x, Y)

C C}. In [5], P. Dunin-Barkowski, N. Orantin, S. Shadrin, and L. Spitz relatethe Eynard-Orantin topological recursion to the Givental formula for the ancestorformal Gromov-Witten potential, and prove the Norbury-Scott conjecture usingtheir main result and Giventals formula for all genus descendant Gromov-Wittenpotential ofPn [19].

1.1. Main Results. Our first main result (Theorem1in Section3.7) relates equi-variant Gromov-Witten invariants ofP1 to the Eynard-Orantin invariants [10] ofthe affine curve

{x=t0 + Y+et

1

Y+ w1 log Y + w2 log

et1

Y) (x, Y) C C}

where w1, w2 are equivariant parameters of the torus T= (C)2 acting on P1. The

superpotential of the T-equivariant Landau-Ginzburg mirror of the projective lineis given by

Wwt C C, Wwt (Y) = t0+ Y + e

t1

Y + w1 log Y + w2 log

et1

Y ,

so Theorem1 can be viewed as a version of all genus equivariant mirror symmetry

for the projective line. We prove Theorem 1 using the main result in [5] and asuitable version of Giventals formula for all genus equivariantdescendant Gromov-Witten potential ofPn [19] (see also[24]).

Our second main result (Theorem2in Section3.7) gives a precise correspondencebetween (A) genus g , n-point descendant equivariant Gromov-Witten invariants ofP1, and (B) Laplace tranforms of the Eynard-Orantin invariant g,n along Lefshetzthimbles. This generalizes the known relation between the A-model genus 0 1-pointdescendant Gromov-Witten invariants and the B-model oscillatory integrals.

1.2. Non-equivariant limit and the Norbury-Scott conjecture. Taking thenon-equivariant limit w1= w2= 0, we obtain

Wt(Y) = t0 + Y +

et1

Y .

which is the superpotential of the (non-equivariant) Landau-Ginzburg mirror forthe projective line. We obtain all genus (non-equivaraint) mirror symmetry for theprojective line.

In the stationary phaset0 =t1 =0, the curve becomes

{x=Y + Y1 (x, Y) C C},and Theorem1 specializes to the Norbury-Scott conjecture [25]. (See Section4.2for details.)


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EYNARD-ORANTIN RECURSION AND EQUIVARIANT MIRROR SYMMETRY FOR P1 3

1.3. Large radius limit and the Bouchard-Marino conjecture. Let w2= 0,

t0= 0 and q= et1 , we obtain

x=Y + qY +w1 log Y

which reduces to

x=Y + w1 log Y

in the large radius limit q 0. The C-equivariant mirror of the affine line C isgiven by

W C C, W(Y) = Y + w1 log Y.In the large radius limit, we obtain a version of all genus C-equivariant mirrorsymmetry of the affine line C.

In particular, let w1= 1 and X= ex, we obtain the Lambert curve

X= Y eY .

In this limit, Theorem1specializes to the Bouchard-Marino conjecture [2]relatingsimple Hurwtize numbers to invariants of the Lambert curve defined by the Eynard-Orantin topological recursion. (See Section5 for details.)

In [1], Borat-Eynard-Mulase-Safnuk introduced a new matrix model represen-taion for the generating function of simple Hurwtiz numbers, and proved theBouchard-Marino conjecture. In[9], Eynard-Mulase-Safnuk provided another proofof the Bouchard-Marino conjecture using the cut-and-joint equation of simple Hur-witz numbers.

Acknowledgment. We thank P. Dunin-Barkowski, B. Eynard, M. Mulase, P.Norbury, and N. Orantin for helpful conversations. The research of the authors ispartially supported by NSF DMS-1206667 and NSF DMS-1159416.

2. A-model

Let T= (C)2 act on P1 by

(t1, t2) [z1, z2]=[t11 z1, t12 z2].Let C[w] = C[w1, w2]be the T-equivariant cohomology of a point: HT(point;C)=C[w].2.1. Equivariant cohomology of P1. The T-equivariant cohomology of P1 isgiven by

HT(P1;C)= C[H, w](Hw1)(H w2)where deg H= deg wi= 2. Let p1=[1, 0]and p2=[0, 1]be the Tfixed points. ThenH

pi = wi. The T-equivariant Poincare dual ofp1 and p2 are H w2 and H w1,

respectively. Let

1 = H w2

w1 w2

, 2 = H w1

w2 w1

HT(P1;C)C[w] C[w, 1w1 w2

]Then deg = 0,

= ,

So{1, 2} is a canonical basis of the semisimple algebraHT(P1;C)C[w] C[w, 1

w1 w2

].


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We have1 + 2= 1,

(, ) = P1 = P1 = , , {1, 2},where

1 = w1 w2, 2 = w2 w1.

Cup product with the hyperplane class is given by

H = w, =1, 2.

2.2. Equivariant quantum cohomology ofP1. TheT-equivariant quantum co-homology ofP1 is

QHT(P1;C)= C[H, w, q](Hw1)(H w2) qwhere deg H= deg wi= 2, deg q=4.

The (non-equivariant) quantum cohomology ofP1 is

C[H, q]H2 qLet

1(q) = 12+

H w1+w22(w1 w2)1 + 4q(w1w2)2 ,

2(q) = 12+

H w1+w22(w2 w1)1 + 4q(w1w2)2 .

Then(q) (q)=(q),

where is the quantum product. So

{1

(q

), 2

(q

)} is a canonical basis of the

semi-simple algebra

QHT(P1;C)C[w, 11(q) ]where 1(q) is defined by (1). We also have(q), (q) =1 (q), (q)=1, (q) (q)

= 1, (q)= P1

(q)= (q) ,

where

1(q) =(w1 w2)1 + 4q(w1 w2)2 ,2

(q

) =

(w2 w1)

1 +4q

(w1 w2)2 = 1

(q

).

Quantum multiplication by the hyperplane class is given by

H = w1 +w2 +

(q)2

, =1, 2.

Finally, we take the non-equivariant limit w2=0, w1 0+. We obtain:1(q)= 1

2+

H

2

q, 2(q)= 1

2

H

2

q,
http://-/?-http://-/?-


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1(q)=2q, 2(q)= 2q,H 1

(q

)=

q1

(q

), H 2

(q

)=

q2

(q

).

These non-equivariant limits coincide with the results in [29,Section 2].

2.3. The A-model canonical coordinates and the -matrix. Let{t0, t1} bethe flat coordinates with respect to the basis{1, H}, and let{u1, u2}be the canon-ical coordinates with respect to the basis{1(q), 2(q)}. Then we have q = et1and

u1 =

1

2(1 w1 +w2

1(q)) t0 + 11(q) t1 ,

u2 =

1

2(1 w1 +w2

2(q)) t0 + 12(q) t1 ,du1 = dt0 +

1

2

(1

(q

)+w1 + w2

)dt1,

du2 = dt0 + 12(2(q) +w1 + w2)dt1.

The above equations determine the canonical coordinates u1 and u2 up to a con-stant in C[w1, w2, 1

w1w2 ]. Giventals A-model canonical coordinates(u1, u2) arecharacterized by their large radius limits:

(1) limq0

(u1 t0 w1t1)=0, limq0

(u2 t0 w2t1)=0.For {1, 2}and i {0, 1}, define i by

du

(q

)=

1

i=0

dti i ,

and define the -matrix to be

= 10 20 11 21 .Then du1

1(q)du22(q)= dt0 dt1 ,

0 = 1(q) , 1 = (q) +w1 + w22(q) .

Let

1 =(1) 01 (1) 11(1) 02 (1) 12 be the inverse matrix of , so that

1

i=0(1) i i = .Then (1) 0 = (q) w1 w2

2

(q) ,(1) 1 = 1(q) .Let q= et

1

. We take the non-equivariant limit w2= 0, w1 0+:u1 =t0 + 2

q, u2 =t0 2

q,


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= 1

2

et

14

1et

14

et14

1et

14

,

1 = 12 et1

4 et1

41et

141et

14 .These non-equivariant limits agree with the results in [29,Section 2].

2.4. The S-operator. TheS-operator is defined as follows. For any cohomologyclasses a, bHT(P1;C), (a, S(b))=a, b

z P1,T0 .

Here the double bracketis defined in the following way. For any a1,, an Z0and any 1,, n H

T(P1;C), let

1

a1 ,, nan

P1,T

0 =m=0

d=0

1

m![Mg,n+m(P1,d)]vir

n

j=1

evj

(uaj

)ajj

m

i=1

evi+n

(t

)where t =t01 + t1H. The T-equivariant J-function is characterized by(J, a)=(1, S(a))for anya HT(P1).

Let1 = w1 w2,

2 = w2 w1.

We consider several different (flat) bases for HT(P1;C):(1) The canonical basis: 1=

H w2

w1 w2

, 2= H w1

w2 w1

.

(2) The basis dual to the canonical basis with respect to the T-equivariantPoincare pairing: 1 =11,

2 =22.

(3) The normalized canonical basis1=

11, 2=

22, which is self-dual:

1

= 1, 2

= 2.(4) The natual basis: T0= 1, T1= H.(5) The basis dual to the natual basis: T0 =H, T1 =1.

For , {1, 2}, defineS(z) =(, S()).

ThenS(z)=(S(z))is the matrix1 of the S-operator with respect to the orderedbasis(1, 2):(2) S()= 2

=1

S(z).

For i

{0, 1

}and

{1, 2

}, define

S i(z) =(Ti, S(

)).Then(S i) is the matrix of the S-operator with respect to the ordered bases(1,2) and(T0, T1):(3) S()= 1

i=0

TiS i(z).1We use the convention that the leftsuperscript/subscript is the row number and the right

superscript/subscript is the columnnumber.


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We have

zJ

t

i=

2

=1

S i

(z

).

By [17, 23], the equivariant J-function is

J= e(t0+t1H)z(1 +

d=1

qd

dm=1(H w1 + mz)dm=1(H w2 + mz)).For =1, 2, define

J =Jp = e(t0+t1w)z d=0

qd

d!zd1

dm=1( +mz) .Then

zJ

t0 =J=

2

=1

J, z J

t1=z

2

=1

J

t1.

So

S i(z)= z Jti .Following Givental, we define

S i(z) =S i(z) exp n=1

B2n

2n(2n 1)( z )2n1.Then

S 0(z) = 1exp t0 + t1wz n=1 B2n2n(2n 1)( z )2n1 d=0 qd

d!zd1

dm=1( +mz)

S 1

(z

) =

1

exp

t0 + t1w

z

n=1

B2n

2n

(2n 1

)( z

)2n1

,

w d=0 qd

d!zd1dm=1( +mz) + d=1 q

d(d 1)!zd 1dm=1( + mz).2.5. The A-model R-matrix. By Givental [19], the matrix(S i)(z) is of theform

S i(z)= 2

=1

i R (z)euz =(R(z)) i euz,

where R(z)=(R (z))=I+ k=1 Rkzk and is unitary, andlimq0

R (z)=exp n=1

B2n

2n(2n 1)( z )2n1.2.6. Gromov-Witten potentials. Introducing formal variables

u=a0 uaza,

t =1

i=0

tiTi= t01 + t1H,

where

ua=2

=1

ua(q).


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Define

FP1,T

g,n(u, t

)=

a1,...,anZ0

m=0

d=0

1

n!m![Mg,n+m(P

1

,d)]vir

n

j=1

evj(uaj)

ajj

m

i=1

evi+n(

t

).

We define the total descendent potential ofP1 to be

DP1,T(u)=exp(

n,g

hg1FP1,T

g,n(u, 0)).Consider the map Mg,n+m(P1, d) Mg,n which forgets the map to the targetand the last m marked points. Let i =

(i) be the pull-backs of the classesi, i=1,n, from Mg,n. Then we can define

FP1,T

g,n(u, t) = a1,...,anZ0

m=0

d=0

1

n!m![Mg,n+m(P1,d)]vir

n

j=1

evj(uaj)ajj mi=1

evi+n(t).Let the ancestor potential ofP1 to be

AP1,T(u, t)=exp(n,g hg1FP

1,Tg,n(u, t)).

2.7. Giventals formula for equivariant Gromov-Witten potential and theA-model graph sum. The quantization of the S-operator relates the ancestorpotential and the descendent potential ofP1 via Giventals formula. Concretely, wehave (see[20])

DP1,T(u)=exp(FP1,T1 )S1AP1,T(u, t)

where FP1,T

1 denotesn FP1,T

1,n(u, 0) at u0 = u, u1= u2 = = 0 and S is the quan-tization [20] ofS. For our purpose, we need to describe a formula for a slightly

different potential: FP1,T

g,n(u, t)the descendent potential with arbitrary primaryinsertions.

Now we first describe a graph sum formula for the ancestor potential AP1,T

(u, t

).

Given a connected graph , we introduce the following notation.(1) V() is the set of vertices in .(2) E() is the set of edges in .(3) H() is the set of half edges in .(4) Lo() is the set of ordinary leaves in .(5) L1() is the set of dilaton leaves in .

With the above notation, we introduce the following labels:

(1) (genus)g V() Z0.(2) (marking) V(){1, 2}. This induces L() = Lo() L1(){1, 2}, as follows: ifl L() is a leaf attached to a vertex v V(), define

(l

)=

(v

).

(3) (height)k H

(

) Z0.

Given an edge e, leth1(e), h2(e)be the two half edges associated to e. The orderof the two half edges does not affect the graph sum formula in this paper. Givena vertex v V(), let H(v) denote the set of half edges emanating from v. Thevalency of the vertex v is equal to the cardinality of the set H(v): val(v)=H(v).A labeled graph=(, g , , k) is stable if

2g(v) 2 + val(v)>0for all v V().


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Let (P1) denote the set of all stable labeled graphs=(, g , , k). The genusof a stable labeled graph

is defined to be

g() = vV()g(v) + E() V() + 1= vV()(g(v) 1) + ( eE()1) + 1.Define

g,n(P1)={=(, g , , k)(P1) g ()=g, Lo()=n}.Given {1, 2}, define

u(z)=a0

uaza.

We assign weights to leaves, edges, and vertices of a labeled graph(P1) asfollows.

(1) Ordinary leaves. To each ordinary leafl Lo() with (l)={1, 2} andk

(l

)=k Z0, we assign:

(Lu)k(l)=[zk](=1,2 u

(z)(q)R (z)).(2) Dilaton leaves. To each dilaton leaf l L1() with (l) = {1, 2} and

2k(l)=k Z0, we assign(L1)k(l)=[zk1](

=1,2

1(q)R (z)).

(3) Edges. To an edge connected a vertex marked by {1, 2} to a vertexmarked by {1, 2} and with heights k and l at the corresponding half-edges, we assign

E,k,l

(e

)=

[zkwl

] 1

z + w

(,

=1,2

R

(z

)R

(w

).

(4) Vertices. To a vertex v with genus g(v) = g Z0 and with marking(v)=, withn ordinary leaves and half-edges attached to it with heightsk1,...,kn Z0and m more dilaton leaves with heightskn+1, . . . , kn+m Z0,we assign

Mg,n+m

k11 kn+mn+m .

We define the weight of a labeled graph (P1) to bew() =

vV()((v)(q))2g(v)2+val(v)

hH(v)k(h)g(v)

eE()E(v1(e)),(v2(e))k(h1(e)),k(h2(e))(e)

lLo()

(Lu

)(l)k(l)

(l

) lL1()

(L1

)(l)k(l)

(l

).

Then

log(AP1,T(u, t))= (P1)

hg()1w()Aut() =g0 hg1 n0 g,n(P1) w()Aut() .Now we describe a graph sum formula for FP

1,Tg,n(u, t)the descendant potential

with arbitrary primary insertions. For =1, 2, let

(q) = (q)(q).


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10 BOHAN FANG, CHIU-CHU MELISSA LIU, AND ZHENGYU ZONG

Then1(q), 2(q)is the normalized canonical basis ofQHT(P1;C), theT-equivariantquantum cohomology ofP1. Define

S(z) =((q), S((q))).Then(S

(z)) is the matrix of the S-operator with respect to the ordered basis(1(q),2(q)):

(4) S((q))= 2=1

(q)S(z).

We define a new weight of the ordinary leaves:

(1) Ordinary leaves. To each ordinary leafl Lo() with (l)={1, 2} andk(l)=k Z0, we assign:

(Lu

)k(l)=[z

k

]( ,=1,2u

(z

)(q)S

(z)R(z) ).

We define a new weight of a labeled graph(P1) to bew() =

vV()((v)(q))2g(v)2+val(v)

hH(v)k(h)g(v)

eE()E(v1(e)),(v2(e))k(h1(e)),k(h2(e))(e)

lLo()

(Lu)(l)k(l)(l)

lL1()(L1)(l)

k(l)(l).Then

g0

hg1 n0

FP1,T

g,n(u, t)= (P1)

hg()1w()

Aut

(

) =

g0

hg1 n0

g,n(P1)

w()

Aut

(

).

We can slightly generalize this graph sum formula to the case where we have norderedvariables u1,, un and n orderedordinary leaves. Let

uj =a0

(uj)azaand let

FP1,T

g,n(u1,, un, t) = a1,...,anZ0

m=0

d=0

1

m![Mg,n+m(P1,d)]vir

n

j=1

evj ((uj)aj)ajj mi=1

evi+n(t).Define the set of graphs g,n(P1) as the definition of g,n(P1) except that the nordinary leaves areordered. Let{l1,, ln} be the ordinary leaves in g,n(P1)and for j = 1,, n let

(Luj)

k(lj)=[zk

]( ,=1,2uj

(z

)(q)S

(z)R(z)

).Define the weight

w() = vV()

((v)(q))2g(v)2+val(v) hH(v)

k(h)g(v) eE()

E(v1(e)),(v2(e))k(h1(e)),k(h2(e))(e)

n

j=1

(Luj)(lj)k(lj)(lj)

lL1()(L1)(l)

k(l)(l).


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Then

g0hg1 n0 FP1,T

g,n(u1,, un, t)= (P1)hg()1w

(

)Aut() =g0hg1 n0 g,n(P1)w

(

)Aut() .3. B-model

3.1. The equivariant superpotential and the Frobenius structure of theJacobian ring. Let Y be coordinates on C. The T-equivariant superpotentialWwt C

C is given byWwt(Y) = Y + t0 + qY +w1 log Y +w2 log qY ,

where q= et1 and Y =ey. In this section, we assume Re

(w1 w2

)>0 and Im

(w1

w2)0. The Jacobian ring ofWw

t is

Jac(Wwt)C[Y, Y1, q, w]Wwty= C[Y, Y1, q, w]Y qY +w1 w2Let

B =qWwt

q=

q

Y+w2.

The Jacobian ring is isomorphic to QHT(P1;C) if one identifies B with HJac(Wwt) C[B,q,w](B w1)(B w2) q.

The critical points ofWwt are P1, P2, where

P= w2 w1 + (q)2 , =1, 2.

Endow a metric on Jac(Wwq) by the residue pairing(f, g)= 2

=1

ResY=Pf(Y)g(Y)

Wwty

dY

Y .

By direct calculation, we have

(B, B)= w1 + w2,(B, 1)=(1, B)=1,(1, 1)=0.We denote b0 = 1, b1 = B and b

i by

(bi, bj

) = ij . These calculations show the

following well-known fact.Proposition 3.1. There is an isomorphism of Frobenius manifold

QHT(P1;C)C[w] C[w, 1w1 w2

]Jac(Wwt)C[w] C[w, 1w1 w2

].We denote Jac(Wwt)C[w] C[w, 1w1w2 ] by HB. The Dubrovin-Givental connec-

tion is denoted by Bv =zv + v on HB =HB((z)).


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3.2. The B-model canonical coordinates. The isomorphism of Frobenius struc-tures automatically ensures their canonical coordinates are the same up to a permu-tation and constants. We fix the B-model canonical coorindates in this subsectionby the critical values of the superpotential Wwt , and find the constant difference tothe A-model coordinates that we set up in earlier sections.

Let Cwt ={(x, y) C2 x=Wwt(ey)}be the graph of the equivariant superpoten-tial. It is a covering ofC given by y ey. Let P1 be the compactification ofC with Y C P1 as its coordinate. At each branch point Y =P, x and y havethe following expansion

x= u 2,

y= v k=1

hk(q)k,where h1

(q

)=

2(q) .

The critical values are

u =t0 +wt1+(q) log + (q)

2 .

Sinceu

t0 = 1,

u

t1 =

q

P+ w2=

w1 +w2 + (q)

2 ,

we have

(5) du =du, =1, 2.

Recall that limq0 1

(q

)= w1 w2, so in the large radius limit q 0, we have

limq0(u t0 wt1)= log .(6)From (5),(6), and (1), we conclude that

u =u + a, =1, 2,

where

a= log .

3.3. The Liouville form and Bergman kernel. On Cwt , let

=xdy

be the Liouville form on C2 =TC. Then d =dx dy. Let =

Cwt

=Wwt

(ey

)dy=

(ey + t0 + qe

y+

(w1 w2

)y +w2 log q

)dy.

Then is a holomorphic 1-form on C. Recall that q= et1

and Y =ey. Define

0 =

t0=

dY

Y ,

1 =

t1=(q

Y +w2)dY

Y .

Then 0, 1descends to holomorphic 1-forms on Cwhich extends to meromorphic

1-forms on P1. We have


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14/32


We have

(7) d

(0dW

)d( 1dW)= 1

2 d1,0

d2,0 , 21

d

(0dW

)d( 1dW)= d1,0

d2,0 .3.5. Oscillating integrals and the B-model R-matrix. For , {1, 2}, i {0, 1}and z > 0, define

S i(z) = y

eWwq(Y)z i= z

y

eWwq(Y)z d( i

dW),

where is the Lefschetz symbol going through P, such that Ww

q(Y) nearits ends. It is straightforward to check that 1i=0 biS i is a solution to the quantumdifferential equation Bf= 0 for =1, 2. We quote the following theorem

Theorem 3.2 ([6, 18, 19]). Near a semi-simple point on a Frobenius manifold ofdimensionn, there is a fundamental solutionSto the quantum differential equation

satisfying the following properties(1) Shas the following form

S= R(z)eUz,whereR(z)is matrix of formal power series inz , andU=diag(u1, . . . , un)is a matrix formed by canonical coordinates.

(2) If S is unitary under the pairing of the Frobenius structure, then R(z) isunique up to a right multiplication ofe

i=1A2i1z

2i1whereAk are constant

diagonal matrices.

Remark 3.3. For equivariant Gromov-Witten theory ofP1, the fundamental solu-tionSin Theorem3.2is viewed as a matrix with entries inC

[w, 1

w1

w2

]((z

))[[q, t0, t1

]].

We choose the canonical coordinates{u(t)}such that there is no constant term byEquation (1). Then if we fix the powers ofq , t0 andt1, only finitely many terms inthe expansion of eUz contribute. So the multiplicationR(z)eUz is well definedand the result matrix indeed has entries inC[w, 1

w1w2 ]((z))[[q, t0, t1]].Remark 3.4. For a general abstract semi-simple Frobenius manifold defined overa ring A, the expression S=R(z)eUz in Theorem3.2can be understood in the

following way. We consider the free moduleM=eu1z eunz over the ringA((z))[[t1,, tn]]wheret1,, tn are the flat coordinates of the Frobenius manifold.We formally define the differential deu

iz =euizdui

z and we extend the differential

to Mby the product rule. Then we have a map d M M dt1 M dtn. Weconsider the fundamental solution S=R

(z

)eUz as a matrix with entries in M.

The meaning thatSsatisfies the quantum differential equation is understood by theabove formal differential.

In our case, the multiplication in the A-model fundamental solutionS= R(z)eUzis formal inz as in Remark3.3. On B-model side, we use the stationary phase ex-pansion to obtain a product of the formR(z)eUz. The multiplicationsR(z)eUzon both A-model and B-model can be viewed as matrices with entries in M. Andtheir differentials are obvious the same with the formal differential above.

We repeat the argument in Givental[20] and state it as the following fact.


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Proposition 3.5. The fundamental solution matrix{ S i2z } has the following as-ymptotic expansion whereR

(z

)is a formal power series in z

S i(z)2z

2

=1

i R (z)e uz .

Proof. By the stationary phase expansion,

S i(z) 2ze uz (1 + a i,1z + a i,1z2 + . . . ),it follows that{S i} can be asymptoically expanded in the desired form (noticethat is a matrix in z -degree 0). In particular, by (7)

R (z) ze uz2 eWwtz d,0.Following Eynard[8], define Laplace transform of the Bergman kernel

(8) B,(u,v,q) = uvu + v

,+uv2

euu+vu

p1p2

B(p1, p2)eux(p1)vx(p2),where , {1, 2}. By [8,Equation (B.9)],(9) B,(u,v,q)= uv

u + v(, 2

=1

R ( 1u)R (1v )).Setting u = v, we conclude that(R( 1

u)R( 1

u)) ={2=1 R ( 1u)R ( 1u)} =

. This shows R is unitary.

Following Iritani[22] (with slight modification), we introduce the following def-inition.

Definition 3.6 (equivariant K-theoretic framing). We definechz KT(P1)HT(P1;Q)[[w1w2z]] by the following two properties which uniquely characterize it.(a)chz is a homomorphism of additive groups:

chz(E1 E2)=chz(E1) + chz(E2).(b) IfL is aT-equivariant line bundle onP1 then

chz(L)=exp 21(c1)T(L)z

.For anyE KT(P1), we define theK-theoretic framing ofE by

(E

)=

(z

)1 (c1)T(TP1)

z

(1

(c1

)T

(TP1

)z )chz

(E

)where(c1)T(TP1)=2H w1 w2.By localization, property (b) in the above definition is characterized by

p(OP1(l1p1 + l2p2))=(z)1z (1 z)e 2l1z , =1, 2,where p p P1 is the inclusion map.

The following definition is motivated by[12,14].


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Definition 3.7(equivariant SYZ T-dual). LetL = OP1(l1p1 + l2p2) be an equivari-ant ample line bundle onP1, wherel1, l2 are integers such thatl1+l2> 0. We definethe equivariant SYZ T-dualSYZ(L)ofL to be the oriented graph in Figure 1 below.We extend the definition additively to the equivariant K-theory group KT(P1).

++(2l2 1)i

+(2l1 1)i

(2l1 1)i

(2l2 1)i

Figure 1. SYZ(OP1(l1p1 + l2p2)) in Ci

i

++ i

i exp

0 1

SYZ(OP1(p2)) in C SYZ(OP1(1)) in C

Figure 2. The equivariant SYZ T-dual ofOP1(p2) in C and the(non-equivariant) SYZ T-dual ofOP1(1) in C.

The following theorem gives a precise correspondence between the B-model os-cillatory integrals and the A-model 1-point descendant invariants.

Theorem 3.8. Suppose thatz,q, w1 w2

(0,

). Then for anyL KT

(P1

),

(10) ySYZ(L)

eWwtz dy=1,(L)

z P1,T0,2 .

(11) ySYZ(L)

eWwtz ydx= (L)

z P1,T0,1 .

Heredx =d(Wwt(y)).Proof. The left hand side of (10) is

ySYZ(L)

eWwtz dy=

1

zySYZ(L)

eWwtz yd(Wwt).

By the string equation, the right hand side of (10) is

1,(L)z P1,T0,2 = (L)z(z )P1,T0,1 .So (10) is equivalent to (11).

It remains to prove (10) forL = OP1(l1p1+l2p1), wherel1+l2 0. We will expressboth hand sides of (10) in terms of (modified) Bessel functions. A brief reviewof Bessel functions is given in Appendix A. The equivariant quantum differentialequation ofP1 is related to the modified Bessel differential equation by a simpletransform (see AppendixB).


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Remark 3.9. Definition3.6(equivariant K-theoretic framing) and Definition3.7(equivariant SYZ T-dual) can be extended to any projective toric manifold. In[16],we use the mirror theorem[17, 23] and results in [22] to extend Proposition3.8 toany semi-Fano projective toric manifold. The left hand side of (10) is known asthe central charge of the Lagrangian braneSYZ(L).Proposition 3.10. The A and B-model R-matrix are equal

R (z)= R (z).Proof. By the asymptotic decomposition theorem of the S-matrix (Theorem3.2),we

only have to compare at the limit q= 0, t0= 0 since both Sand Sare unitary. Noticethat has an non-degenerate limit at q=0, then it suffices to show that

S i euz

q=0,t0=0 1

2z

S i euz

q=0,t0=0.

The Lefschetz thimble 2 is{YY (, 0)}. While the Lefschetz thimble 1could not be explicitly depicted, we could alternatively consider the thimble 1 ={YY (0,)} for z < 0 of the oscillating integral eWwtzdy. The integral yieldsthe same asymptotic answer once we analytically continuate z 0, sincethe stationary phase expansion only depends on the local behavior (higher orderderivatives) ofWwt at the critical points.

So lettingY = T z for =2, or Y = qTz

for =1,

euzS 0 =e

(q)z ( + (q)

2 )z (z) z

0eTe

q

Tz2 Tz1dT .

Taking the limit q 01

2z eu

zS

0q=0= 1

2z ez (

z)

z (

z)

1

exp(

n=1

B2n

2n(2n 1)( z )2n1) S 0 euz q=0.Here we use the Stirling formula

log(z) 12

log(2) + (z 12) log z z +

n=1

B2n

2n(2n 1)z12n.Notice that

S 1 =z

t1S 0 =z

eW

w

tz( qY +w2)dY

Y ,

and similar calculation shows (letting Y = T z if=2 and Y = qT z

if =1)

12z

euzS 1q=0 w 1

exp(

n=1

B2n

2n(2n 1)( z )2n1) S 1 euz q=0.

Notice that the matrix R is given by the asymptotic expansion. This theoremdoes not imply S i e

uz = 12z S

i euz, which are unequal.


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3.6. The Eynard-Orantin topological recursion and the B-model graphsum. Let g,n be defined recursively by the Eynard-Orantin topological recursion[10]:

0,1= 0, 0,2= B(Y1, Y2)= dY1 dY2(Y1 Y2)2 .When 2g 2+ n>0,

g,n(Y1, . . . , Y n) = 2=1

ResYP Y=Y B(Yn, )

2(log(Y) log(Y))dWg1,n+1(Y, Y , Y1, . . . , Y n1)+ g1+g2=g

IJ={1,...,n1}

IJ=

g1,I+1(Y, YI)g2,J+1(Y , YJ)where Y ,Y P, Wwq(Y)=Wwq(Y) and Y Y.

Following[5], the B-model invariantsg,n are expressed in terms of graph sums.

Given a labeled graph

g,n

(P1

)with Lo

(

)=

{l1, . . . , ln

}, we define its weight

to be

w() =(1)g()1+n vV()

h1222gval(v)

hH(v)k(h)g(v)

eE()B(v1(e)),(v2(e))k(e),l(e)

n

j=1

12

d(lj)k(lj) (Yj)

lL1()( 1

2)h(l)

k(l) .

Here,hk = 112k1 2(2k 1)!!h2k1. In our notation[5, Theorem 3.7] is equivalent

to:

Theorem 3.11 (Dunin-BarkowskiOrantinShadrinSpitz [5]). For2g 2+n>0,

g,n = g,n

(P1

)

w

(

)Aut

(

).

3.7. All genus mirror symmetry. Given a meromorphic function f(Y) on P1which is holomorphic on P1 {P1, P2}, define

(f)= dfdW

= Y2(Y P1)(Y P2) dfdY.

Then (f) is also a meromorphic function which is holomorphic on P1 {P1, P2}.For {1, 2}, let

,0= 11

2

(q) PY P .Then ,0 is a meromorphic function on P

1 with a simple pole at Y = P andholomorphic elsewhere. Moreover, the differential of,0 is d,0. For k >0, define

W

k

=d((1)k

k

(,0)).DefineS (z)= z

y

exz

d,02

, S (L)

(z)= z ySYZ(L)

exz

d,02

.

Then

S (z)= zk+1

yeW(y)z

Wk2

, S (L)

(z)= zk+1ySYZ(L)

eW(y)z

Wk2

(12)


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Therefore,

(13) ySYZ(L) eW(y)z

Wk

2 = zk

1

S

(L

) (z)= zk

1

(q),

(L

)z P1,T

0,2 .

where the last equality follows from Theorem3.8.For =1, 2 and j = 1,, n, let

(14) uj(z)= 2=1

S

(z) uj(z)

(q) .Theorem 1 (All genus equivariant mirror symmetry for P1). For n> 0 and2g 2 + n>0, we have

(15) g,n 12Wk(Yj)=(uj)k =(1)g1+nFP1,Tg,n(u1,, un, t).Proof. We will prove this theorem by comparing the A-model graph sum in the end

of Section 2.7 and the B-model graph sum in the previous section.(1) Vertex. By Section 3.1, we haveh1 (q)= 2(q) . So in the B-model vertex,

h12

=

1(q) . Therefore the B-model vertex matches the A-model vertex.

(2) Edge. By Section 3.6, we know that

B,k,l =[ukvl] uvu + v (, G f(u, q)f(v, q))

=[zkwl] 1z + w (, G f(1z , q)f(1w , q)) .By definition

E,

k,l =[zkwl] 1

z +w (, =1,2 R (z)R (w).But we know that

R (z)=f(1z ).Therefore, we have

B,k,l

= E,k,l

.

(3) Ordinary leaf. We have the following expression fordk (see [15]):

dk =Wk

k1i=0

B,k1i,0W

i .

By item 2 (Edge) above, for k, l Z0,

B,k,l =[zkwl] 1z +w (, =1,2 R (z)R (w)).We also have [z0](R (z))=, .Therefore,

dk =k

i=0

2

=1

[zki]R (z)Wi .


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So under the identification

1

2Wk(

Yj

)=

(uj

)k

The B-model ordinary leaf matches the A-model ordinary leaf.(4) Dilaton leaf. We have the following relation betweenhk and f

(u, q) (see

[15])

hk =[u1k]

1h1 f

(u, q).

By the relation

R (z)=f(1z)and the facth1(q)= 2(q) , it is easy to see that the B-model dilaton leafmatches the A-model dilaton leaf.

Taking Laplace transforms at appropriate cycles to Theorem1 produces a the-orem concerning descendants potential.

Theorem 2 (All genus full descendant equivariant mirror symmetry for P1). Sup-pose thatn> 0 and2g 2 + n> 0. For anyL1, . . . , Ln KT(P1), there is a formalpower series identity

(16)y1SYZ(L1)

ynSYZ(Ln)

eW(y1)z1

++W(yn)zn g,n

=(1)g1 (L1)z1 1

, . . . , (Ln)zn n

g,n .Remark 3.12. By Theorem3.8,

(17) y1SYZ(L)

eW(y1)z1 ydx = (L1)

z1 1P1,T0,1

which is the analogue of (16) in the unstable case(g, n)=(0, 1).Proof of Theorem2. By (14),

uj(z)= 2=1

(q)(q),(q)

z P1,T0,2 uj(z).

Define the flat coordinatesuj by

2

=1

uj

(z

)

(q

)=

2

=1

uj

(z

)

(0

),

and a power series in 1zS

(z)=(q),(0)z 0,2.Then

uj(z)= 2=1

(q),(0)z

uj(z)+

=2

=1

S(z)uj(z)+.


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Notice that(S) is unitary, i.e.S(z)S(z)= 1 . We have

2

=1S(z)uj(z)+=2

=1 2

=1 S

(z)S(z)uj(z)=uj (

z

) .Taking the Laplace transform ofg,n

y1SYZ(L1)

. . .ynSYZ(Ln)

eW1(y1)z1

++Wn(yn)zn g,n

=y1SYZ(L1)

. . .ynSYZ(Ln)

eni=1Wi(yi)zi(1)g1+n

i,ai

ni=1

ai(i(0))g,n

n

i=1

(ui)iai(uj)k= 12Wk(yj)=

y1SYZ

(L1

)

. . .ynSYZ

(Ln

)

eni=1Wi(yi)zi

(1

)g1+n

i,ai

n

i=1

ai

(i

(0

))g,n

n

i=1

(i 2=1

kZ0

[zaiki ]Si(zi)Wk(yi)2

.Using (13)

y1SYZ(L1)

. . .ynSYZ(Ln)

eW1(y1)z1

++Wn(yn)zn g,n

=(1)g1+ni,ai

ni=1

ai(i(0))g,n ni=1

(i 2=1

kZ0

([zaiki ]Si(zi))S (Li) (zi)(zk1i ))=(1)g1

i,ai

ni=1

ai(i(0))g,n ni=1

i(i(0), (Li))zai1i=(1)g1 (L1)z1 1 , . . . , (Ln)zn n g,n.

4. The non-equivariant limit and the Norbury-Scott conjecture

In this section, we consider the non-equivariant limit w1= w2= 0.

4.1. The non-equivariant R-matrix. By [19,Section 1.3],R(z)=I+n=1 Rnznis uniquely determined by:

(1) The recursive relation:(d + 1d)Rn=[dU,Rn+1].(2) The homogeneity ofR

(z

): Rnq

n2 is a constant matrix.The unique solution R

(z

)satisfying the above conditions was computed explicity

in [29]:

Lemma 4.1 ([29, Lemma 3.1] ).

Rn= qn2

(2n 1)!!(2n 3)!!n!24n

1 2n1(1)n+12n1 (1)n+1

By Proposition 3.10 , R(z) = R(z). In this subsection, we recover the abovelemma by computing the stationary phase expansion ofS.


23/32


24/32


4.2. The Norbury-Scott Conjecture. In this subsection, we assume w1= w2=t0 =0. Then

a1(H)an(H)P1

g,n = q(n

i=1ai)

2+1ga1(H)an(H)P1

g,n.

Note that when(ni=1 ai2)2+1 gis not an nonnegative integer, both hand sidesare zero.

When 2g 2 + n> 0, g,n is holomorphic near Y =0, and one may expand it inthe local holomorphic coordinatex=x1 =(Y + q

Y)1.

Theorem 4.2. Suppose that2g2+n>0. Then nearY =0, g,n has the followingexpansion

g,n =(1)g1+n a1,...,anZ0

a1(H)an(H)P1g,n nj=1

(aj + 1)!xaj+2

dxj

The Norbury-Scott conjecture corresponds to the specialization q= 0, i.e. t1 =0.

Proof. DefineWk by 12

Wk =uk t0a=0,t

1a=(a+1)!xa2dx

.

By Theorem1, it suffices to show thatWk agrees with the expansion ofWk nearY =0 inx=x1.

We now computeWk explicitly.J = e(t

0+t1H)z(1+

d=1

qd

dm=1(H+ mz)2 )= e

t0

z (1 + t1 Hz)1 +

d=1

qd

z2d

(d!

)2 2(

d=1

qd

z2d

(d!

)2

d

m=1

1

m)H

z)

= et0

z (1 +

d=1qd

z2d(d!)2 )+et0

z (t1(1 + d=1

qd

z2d+1(d!)2 ) 2 d=1 qd

z2d+1(d!)2 dm=1 1m)Hz

J

t1 = e

t0

z ( d=1

dqd

z2d1(d!)2 )+et0

z t1( d=1

dqd

z2d(d!)2 ) + 1 + d=1 qd

z2d(d!)2 (1 2d dm=1 1m)HS00

(z

)=

(H, S

(1

))=

(1, z

J

t1 )=et0

z

t1

(

d=1

dqd

z2d(d!)2

)+ 1 +

d=1

qd

z2d(d!)2

(1 2d

d

m=1

1

m)S10(z)=(1, S(1))=(1, J)=e t0z (t1(1 + d=1

qd

z2d+1(d!)2 ) 2 d=1 qd

z2d+1(d!)2 dm=1 1m)S01(z)=(H, S(H))=(H, zJt1 )=e t0z ( d=0 q

d+1

z2d+1d!(d + 1)!)S11(z)=(1, S(H))=(H, J)=e t0z (1+

d=1

qd

z2d(d!)2 )


25/32


26/32


1,0= 1

1

n=1

(

q

)n 1

2

n 1

n2

xn =

1

1

n=0

(

q

)n+ 1

2

n

n+1

2

xn+1 =

1

1

n=0

(

q

)n+ 1

2

n

n2

xn1

ResY=0xn12,0dx= q

14ResY=0(Y + qY)n1(1 q

Y2) dY

Y +

q

= q14ResY=0(Y2 + q)n1(Y q)

Yn+1 dY

= 11

(q)n 12 n 1n2

2,0= 11

n=0

(q)n+ 12 nn2xn1

5. The large radius limit and the Bouchard-Marino conjecture

In this section, we will specialize Theorem 1to the large radius limit case. Inthis case, Theorem1relates the invariantg,n of the limit curve to the equivariantdescendent theory ofC. After expanding ,0in suitable coordinates, we can relatethe corresponding expansion ofg,n to the generation function of Hurwitz numbersand therefore reprove the Bouchard-Marino conjecture[2] on Hurwitz numbers.

Let w2= 0, t0= 0 and take the large radius limit q 0. Then our mirror curvebecomes

x=Y + w1 log Y.

When w1 = 1, this is just the Lambert curve. Recall that the two critical pointsP1, P2 ofW

w

t

(Y

)are

P= w2 w1 +(q)2 .

Since 1(0) = w1 w2, P1 0 under the limit q 0. In other words, P1 goesout of the curve under the limit q 0 and 1,0 =

2

(q)P1

YP1 0. As a result,

W1k =d(k(1,0)) also turns to zero under the large radius limit.Under the identification 1

2Wk(Yj)=(uj)k in Theorem1,we have(uj)1k 0

when q 0. On the A-model side, since q= 0, the Smatrix(S(z)) is diagonal.Therefore, we also have(uj)1k 0 whenq 0 under the identification in Theorem1. This means that in the localization graph of the equivariant GW invariants ofP1, we can only have a constant map to p 2 P

1. Since H

p2= w2= 0 and t

0 =0, wecan not have any primary insertions. Therefore, in the large radius limit, we get

FP1,C

g,n(u1,, un; t) = a1,...,anZ0

[Mg,n(P1,0)]virnj=1

evj ((uj)2aj2(0))ajj=

a1,...,anZ0

1

w1Mg,n

n

j=1

(uj)2ajajj g(w1),where

g(u)=ug 1ug1 ++ (1)gg.


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28/32


So

FC,C

= ()=n

Aut

(

)(2g 2 + + n)!(w1)2g2++nHg,

Sn

n

j=1

Zj

(j)

.

When w1= 1, this is just the generating function of Hurwitz numbers.LetWg,n(Z1,, Zn)be the expansion ofg,n(Y1,, Yn)in the coordinateZnear

Y =0. Then we have

Corollary 5.1 (Bouchard-Marino conjecture). For n > 0 and 2g 2 + n > 0, theinvariant Wg,n(Z1,, Zn) for the curvex=Y +w1 log Y satisfies

Z1

0

Zn

0Wg,n(Z1,, Zn)

=(1)g1+n a1,...,anZ0

1

w1Mg,n

n

j=1

ajj

g(w1) nj=1

( j=0

(jw1

)j+ajj !

Zjj)

=(1)g1+n ()=n Aut()Hg,(2g 2 + + n)!(w1)2g2++n Snn

j=1 Zj

(j).

In particular, when w1= 1, the right hand side is the generating function of Hur-witz numbers and the Bouchard-Marino conjecture is recovered.

Appendix A. Bessel functions

In this section, we give a brief review of Bessel functions.The Bessels differential equation is

(18) x2d2y

dx2 + x

dy

dx+ (x2 2)y= 0.

TheBessel function of the first kindis defined by

J(x)= m=0 (1)mm!(m + + 1)(x2 )2m+.TheBessel function of the second kindis defined by

Y(x)= J(x) cos() J(x)sin() .

Whenn is an integer, Yn(x) =limn Y(x).J(x) and Y(x) form a basis of the 2-dimenisonal space of solutions to the

Bessels differential equation (18).Replacingx by ix in (18), one obtains the the modified Bessel differential equa-

tion

(19) x2d2y

dx2 + x

dy

dx

(x2 +2

)y= 0.

Themodified Bessel function of the first kindis defined by

I(x)=iJ(ix)= m=0

1

m!(m + + 1)(x2 )2m+.Themodified Bessel function of the second kindis defined by

K(x)= 2

I(x) I(x)sin() .


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The following integral formulas are valid when R(x)>0:I(x) =

1

0

ex cos cos()d sin

(

)

0

ex coshttdt

K(x) = 0

ex cosht cosh(t)dt= 12t0,0

ex coshttdt

where 0,0 is the real line with the standard orientation:

+

Figure 3. The contour 0,0

eiK

(x

)+ iI

(x

)=

2

eiI

(x

) eiI

(x

)sin()=

ei

2

0

ex cosh ttdt +ei

2

2

0ex cos(i)(i)d(i) + ei

2

0ex coshttdt

= ei

2

0,1

ex coshttdt

where 0,1 is the following contour:

++ 2i

0

2i

Figure 4. The contour 0,1

Therefore,

(20) 0,0

ex coshttdt=

sin()(I(x) I(x))(21)

0,1

ex cosh ttdt=

sin()(I(x) e2iI(x))For any integers l 1, l2 with l1 + l2 0, let l1,l2 be the following contour:

++ 2l2i

2l1i

2l1i

2l2i

Figure 5. The contour l1,l2


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Lemma A.1. For any l1, l2 Z such that l1 + l2 0, we have

(22)

l1,l2

ex coshttdt=

sin() e2l1iI(

x

) e2l2iI(

x

)Proof. We observe that

(23) l1k,l2+k

ex coshttdt=e2ki l1,l2

ex cosh ttdt.

In particular,

l1,l1

ex coshttdt=e21i 0,0

ex coshttdt=

sin()(e2l1iI(x)e2l1iI(x))This proves (22)in case l1 + l2= 0. Ifl1 + l2>0 then

(24) l1,l2=l21

k=

l1

1k,k l21

k=

1

l1

k,k .

Equations (23) and(24) imply

l1,l2

ex coshttdt

= l21k=l1

e2ki 0,1

ex cosh ttdt l21k=1l1

e2ki 0,0

ex cosh ttdt

Equation (22) follows from the above equation and (20), (21).

Appendix B. The Equivariant Quantum Differential Equation for P1

The equivariant quantum differential equation ofP1 is the vector equation

zq d

dqI= 0 qw1w2

1 w1 +w2I

which is equivalent to the following scalar equation:

(25) (zq ddq w1)(zq d

dq w2)I= qI .

Let

I= ew1+w22z

logqy, x=2

q

z .

Then (25) is equivalent to

x2 d

2y

dx2 + xdy

dx (x2 + (w1 w22z )2)y= 0which is the modified bessel differential equation (19) with = w1w2

2z . When

w1 w2 0, any solution to (25)is of the form

I= ew1+w22z

logqc1I1z

(2qz) + c2I2

z

(2qz).

where 1 = w1 w2= 2, and c1, c2 are functions of w1, w2, z.


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3, 525588.

Bohan Fang, Beijing International Center for Mathematical Research, Peking Uni-

versity, 5 Yiheyuan Road, Beijing 100871, China

E-mail address: [email protected]

Chiu-Chu Melissa Liu, Department of Mathematics, Columbia University, 2990 Broad-

way, New York, NY 10027


Zhengyu Zong, Department of Mathematics, Columbia University, 2990 Broadway,

New York, NY 10027


the eynard-orantin recursion and equivariant mirror symmetry for the projective line

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