The Eynard-Orantin recursion and equivariant mirror symmetry for the projective line
aa r X i v : . [ m a t h . AG ] J u l THE EYNARD-ORANTIN RECURSION AND EQUIVARIANTMIRROR SYMMETRY FOR THE PROJECTIVE LINE
BOHAN FANG, CHIU-CHU MELISSA LIU, AND ZHENGYU ZONG
Abstract.
We study the equivariantly perturbed mirror Landau-Ginzburgmodel of P . We show that the Eynard-Orantin recursion on this model en-codes all genus all descendants equivariant Gromov-Witten invariants of P .The non-equivariant limit of this result is the Norbury-Scott conjecture [30, 6],while by taking large radius limit we recover the Bouchard-Mari˜no conjectureon simple Hurwitz numbers [2]. Contents
1. Introduction 21.1. Main Results 21.2. Non-equivariant limit and the Norbury-Scott conjecture 31.3. Large radius limit and the Bouchard-Mari˜no conjecture 3Acknowledgment 32. A-model 32.1. Equivariant cohomology of P P P S -operator 72.6. The A-model R -matrix 92.7. Gromov-Witten potentials 92.8. Givental’s formula for equivariant Gromov-Witten potential and theA-model graph sum 103. B-model 123.1. The equivariant superpotential and the Frobenius structure of theJacobian ring 123.2. The B-model canonical coordinates 133.3. The Liouville form and Bergman kernel 143.4. Differentials of the second kind 143.5. Oscillating integrals and the B-model R -matrix 153.6. The Eynard-Orantin topological recursion and the B-model graphsum 203.7. All genus mirror symmetry 214. The non-equivariant limit and the Norbury-Scott conjecture 244.1. The non-equivariant R -matrix 244.2. The Norbury-Scott Conjecture 255. The large radius limit and the Bouchard-Mari˜no conjecture 28Appendix A. Bessel functions 30 Appendix B. The Equivariant Quantum Differential Equation for P Introduction
The equivariant Gromov-Witten theory of P has been studied extensively. In[31, 32], Okounkov-Pandharipande completely solved the equivariant Gromov-Wittentheory of the projective line and established a GW/H correspondence between thestationary sector of Gromov-Witten theory of P and Hurwitz theory. In [23],Givental derived a quantization formula for all genus descendant potential of theequivariant Gromov-Witten theory of P (and more generally, P n ). In the non-equivariant limit, these results imply the Virasoro conjecture of P .The Norbury-Scott conjecture [30] relates (non-equivariant) Gromov-Witten in-variants of P to Eynard-Orantin invariants [13] of the affine plane curve { x = Y + Y ∶ ( x, Y ) ∈ C × C ∗ } . In [6], P. Dunin-Barkowski, N. Orantin, S. Shadrin, and L.Spitz relate the Eynard-Orantin topological recursion to the Givental formula forthe ancestor formal Gromov-Witten potential, and prove the Norbury-Scott con-jecture using their main result and Givental’s quantization formula for all genusdescendant potential of the (non-equivariant) Gromov-Witten theory of P . It isnatural to ask if the Norbury-Scott conjecture can be extended to the equivariantsetting, such that the original conjecture can be recovered in the non-equivariantlimit.1.1. Main Results.
Our first main result (Theorem 1 in Section 3.7) relates equi-variant Gromov-Witten invariants of P to the Eynard-Orantin invariants [13] ofthe affine curve { x = t + Y + Qe t Y + w log Y + w log Qe t Y ) ∶ ( x, Y ) ∈ C × C ∗ } where t , t are complex parameters, w , w are equivariant parameters of the torus T = ( C ∗ ) acting on P , and Q is the Novikov variable encoding the degree ofthe stable maps to P (see Section 2.2). The superpotential of the T -equivariantLandau-Ginzburg mirror of the projective line is given by W w t ∶ C ∗ → C , W w t ( Y ) = t + Y + Qe t Y + w log Y + w log Qe t Y , so Theorem 1 can be viewed as a version of all genus equivariant mirror symmetryfor P . We prove Theorem 1 using the main result in [6] and a suitable version ofGivental’s formula for all genus equivariant descendant Gromov-Witten potentialof P n [23] (see also [29]).Our second main result (Theorem 2 in Section 3.7) gives a precise correspondencebetween (A) genus- g , n -point descendant equivariant Gromov-Witten invariants of P , and (B) Laplace transforms of the Eynard-Orantin invariant ω g,n along Lef-schetz thimbles. This result generalizes the known relation between the A-modelgenus-0 1-point descendant Gromov-Witten invariants and the B-model oscillatoryintegrals. YNARD-ORANTIN RECURSION AND EQUIVARIANT MIRROR SYMMETRY FOR P Non-equivariant limit and the Norbury-Scott conjecture.
Taking thenon-equivariant limit w = w =
0, we obtain W t ( Y ) = t + Y + Qe t Y . which is the superpotential of the (non-equivariant) Landau-Ginzburg mirror forthe projective line. We obtain all genus (non-equivariant) mirror symmetry for theprojective line.In the stationary phase t = t = , Q =
1, the curve becomes { x = Y + Y − ∶ ( x, Y ) ∈ C × C ∗ } , and Theorem 1 specializes to the Norbury-Scott conjecture [30]. (See Section 4.2for details.)1.3. Large radius limit and the Bouchard-Mari˜no conjecture.
Let w = t = q = Qe t , we obtain x = Y + qY + w log Y which reduces to x = Y + w 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 + w 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 w = − X = e − x , we obtain the Lambert curve X = Y e − Y . In this limit, Theorem 1 specializes to the Bouchard-Mari˜no conjecture [2] relatingsimple Hurwitz numbers (related to linear Hodge integrals by the ELSV formula[8, 20]) to Eynard-Orantin invariants of the Lambert curve. (See Section 5 fordetails.)In [1], Borot-Eynard-Mulase-Safnuk introduced a new matrix model represen-tation for the generating function of simple Hurwitz numbers, and proved theBouchard-Mari˜no conjecture. In [12], Eynard-Mulase-Safnuk provided anotherproof of the Bouchard-Mari˜no conjecture using the cut-and-joint equation of simpleHurwitz numbers. Recently, a new proof of the ELSV formula and a new proof ofthe Bouchard-Mari˜no conjecture are given in [4].
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 ∗ ) act on P by ( t , t ) ⋅ [ z , z ] = [ t − z , t − z ] . Let C [ w ] ∶ = C [ w , w ] be the T -equivariant cohomology of a point: H ∗ T ( point; C ) = C [ w ] . BOHAN FANG, CHIU-CHU MELISSA LIU, AND ZHENGYU ZONG
Equivariant cohomology of P . The T -equivariant cohomology of P isgiven by H ∗ T ( P ; C ) = C [ H, w ]/⟨( H − w )( H − w )⟩ where deg H = deg w i =
2. Let p = [ , ] and p = [ , ] be the T fixed points. Then H ∣ p i = w i . The T -equivariant Poincar´e dual of p and p are H − w and H − w ,respectively. Let φ ∶ = H − w w − w , φ ∶ = H − w w − w ∈ H ∗ T ( P ; C ) ⊗ C [ w ] C [ w , w − w ] Then deg φ α = φ α ∪ φ β = δ αβ φ α , So { φ , φ } is a canonical basis of the semisimple algebra H ∗ T ( P ; C ) ⊗ C [ w ] C [ w , w − w ] . We have φ + φ = , ( φ α , φ β ) ∶ = ∫ P φ α ∪ φ β = δ αβ ∫ P φ α = δ αβ ∆ α , α, β ∈ { , } , where ∆ = w − w , ∆ = w − w . Cup product with the hyperplane class is given by H ∪ φ α = w α φ α , α = , . Equivariant Gromov-Witten invariants of P . Suppose that d > g − + n >
0, so that M g,n ( P , d ) is non-empty. Given γ , . . . , γ n ∈ H ∗ T ( P , C ) and a , . . . , a n ∈ Z ≥ , we define genus g , degree d , T -equivariant descendant Gromov-Witten invariants of P : ⟨ τ a ( γ ) . . . τ a n ( γ n )⟩ P ,Tg,n,d ∶ = ∫ [M g,n ( P ,d )] vir n ∏ j = ψ a j j ev ∗ j ( γ j ) ∈ C [ w ] where ev j ∶ M g,n ( P , d ) → P is the evaluation at the j -th marked point, which is a T -equivariant map. We define genus g , degree d primary Gromov-Witten invariants: ⟨ γ , . . . , γ n ⟩ P ,Tg,n,d ∶ = ⟨ τ ( γ ) ⋯ τ ( γ n )⟩ P ,Tg,n,d . Let t = t + t H . If 2 g − + n >
0, define ⟪ τ a ( γ ) , . . . , τ a n ( γ n )⟫ P ,Tg,n ∶ = ∑ d ≥ Q d ∞ ∑ ℓ = ℓ ! ⟨ τ a ( γ ) ⋯ τ a n ( γ n ) τ ( t ) ⋯ τ ( t )´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ℓ times ⟩ P ,Tg,n + ℓ,d Suppose that 2 g − + n + m >
0. Given γ , . . . , γ n + m ∈ H ∗ T ( P ) , we define ⟨ γ z − ψ , . . . , γ n z n − ψ n , γ n + , . . . , γ n + m ⟩ P ,Tg,n + m,d ∶ = ∑ a ,...,a n ≥ ⟨ τ a ( γ ) ⋯ τ a n ( γ n ) τ ( γ n + ) ⋯ τ ( γ n + m )⟩ P ,Tg,n + m,d n ∏ j = z − a j − j . YNARD-ORANTIN RECURSION AND EQUIVARIANT MIRROR SYMMETRY FOR P In particular, if n + m ≥ ⟨ γ z − ψ , . . . , γ n z n − ψ n , γ n + , . . . , γ n + m ⟩ P ,T ,n + m, = z ⋯ z n ( z + ⋯ + z n ) n + m − ∫ P γ ∪ ⋯ ∪ γ n + m where we use the fact M ,n + m ( P , ) = M ,m + n × P , and the identity ∫ M ,k ψ a ⋯ ψ a k k = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ ( k − ) ! ∏ kj = a j ! if a + ⋯ + a k = k − , , otherwise . We use the second line of (1) to extend the definition of the correlator in the firstline of (1) to the unstable cases ( n, m ) = ( , ) , ( , ) , ( , ) : ⟨ γ z − ψ ⟩ P ,T , , ∶ = z ∫ P γ ⟨ γ z − ψ , γ ⟩ P ,T , , ∶ = ∫ P γ ∪ γ ⟨ γ z − ψ , γ z − ψ ⟩ P ,T , , ∶ = z + z ∫ P γ ∪ γ Suppose that 2 g − + n + m > n >
0. Given γ , . . . , γ n + m ∈ H ∗ T ( P ) , we define ⟪ γ z − ψ , . . . , γ n z n − ψ n , γ n + , . . . , γ n + m ⟫ P ,Tg,n + m ∶ = ∑ d ≥ ∑ ℓ ≥ Q d ℓ ! ⟨ γ z − ψ , . . . , γ n z n − ψ n , γ n + , . . . , γ n + m , t , . . . , t ´¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¶ ℓ times ⟩ P ,Tg,n + m + ℓ,d . Let q = Qe t . Then for m ≥ ⟪ γ , . . . , γ m ⟫ P ,T ,m = ∑ d ≥ q d ⟨ γ , . . . , γ m ⟩ P ,T ,m,d = δ m, ∫ P γ ∪ ⋯ ∪ γ m + q m ∏ i = ( ∫ P γ i ) . Equivariant quantum cohomology of P . As a C [ w ] -module, QH ∗ T ( P ; C ) = H ∗ T ( P ; C ) . The ring structure is given by the quantum product ∗ defined by ( γ ⋆ γ , γ ) = ⟪ γ , γ , γ ⟫ P ,T , , or equivalently, γ ⋆ γ = γ ∪ γ + q ( ∫ P γ )( ∫ P γ ) . where ∪ is the product in H ∗ T ( P ) , and q = Qe t . In particular, H ⋆ H = ( w + w ) H − w w + q. The T -equivariant quantum cohomology of P is QH ∗ T ( P ; C ) = C [ H, w , q ]/⟨( H − w ) ⋆ ( H − w ) − q ⟩ where deg H = deg w i =
2, deg q = P is C [ H, q ]/⟨ H ⋆ H − q ⟩ BOHAN FANG, CHIU-CHU MELISSA LIU, AND ZHENGYU ZONG
Let φ ( q ) = + H − w + w ( w − w )√ + q ( w − w ) ,φ ( q ) = + H − w + w ( w − w )√ + q ( w − w ) . Then φ α ( q ) ⋆ φ β ( q ) = δ αβ φ α ( q ) , So { φ ( q ) , φ ( q )} is a canonical basis of the semi-simple algebra QH ∗ T ( P ; C ) ⊗ C [ w , ( q ) ] where ∆ ( q ) is defined by (2). We also have ( φ α ( q ) , φ β ( q )) = ( ⋆ φ α ( q ) , φ β ( q )) = ( , φ α ( q ) ⋆ φ β ( q )) = δ αβ ( , φ α ( q )) = δ αβ ∫ P φ α ( q ) = δ αβ ∆ α ( q ) , where ∆ ( q ) = ( w − w )√ + q ( w − w ) , ∆ ( q ) = ( w − w )√ + q ( w − w ) = − ∆ ( q ) . Quantum multiplication by the hyperplane class is given by H ⋆ φ α = w + w + ∆ α ( q ) φ α , α = , . Finally, we take the non-equivariant limit w = w → + . We obtain: φ ( q ) = + H √ q , φ ( q ) = − H √ q , ∆ ( q ) = √ q, ∆ ( q ) = − √ q,H ⋆ φ ( q ) = √ qφ ( q ) , H ⋆ φ ( q ) = − √ qφ ( q ) . These non-equivariant limits coincide with the results in [35, Section 2].2.4.
The A-model canonical coordinates and the Ψ -matrix. Let { t , t } bethe flat coordinates with respect to the basis { , H } , and let { u , u } be the canon-ical coordinates with respect to the basis { φ ( q ) , φ ( q )} . Then ∂∂u = ( − w + w ∆ ( q ) ) ∂∂t + ( q ) ∂∂t ,∂∂u = ( − w + w ∆ ( q ) ) ∂∂t + ( q ) ∂∂t ,du = dt + ( ∆ ( q ) + w + w ) dt ,du = dt + ( ∆ ( q ) + w + w ) dt . YNARD-ORANTIN RECURSION AND EQUIVARIANT MIRROR SYMMETRY FOR P The above equations determine the canonical coordinates u and u up to a con-stant in C [ w , w , w − w ] . Givental’s A-model canonical coordinates ( u , u ) arecharacterized by their large radius limits:(2) lim q → ( u − t − w t ) = , lim q → ( u − t − w t ) = . For α ∈ { , } and i ∈ { , } , define Ψ αi by du α √ ∆ α ( q ) = ∑ i = dt i Ψ αi , and define the Ψ-matrix to be Ψ ∶ = [ Ψ Ψ Ψ Ψ ] . Then [ du √ ∆ ( q ) du √ ∆ ( q ) ] = [ dt dt ] Ψ , Ψ α = √ ∆ α ( q ) , Ψ α = ∆ α ( q ) + w + w √ ∆ α ( q ) . Let Ψ − = [ ( Ψ − ) ( Ψ − ) ( Ψ − ) ( Ψ − ) ] be the inverse matrix of Ψ, so that ∑ i = ( Ψ − ) iα Ψ βi = δ βα . Then ( Ψ − ) α = ∆ α ( q ) − w − w √ ∆ α ( q ) , ( Ψ − ) α = √ ∆ α ( q ) . Let Q = q = e t . We take the non-equivariant limit w = w → + : u = t + √ q, u = t − √ q, Ψ = √ ( e − t / − √ − e − t / e t / √ − e t / ) , Ψ − = √ ( e t / e − t / √ − e t / − √ − e − t / ) . These non-equivariant limits agree with the results in [35, Section 2].2.5.
The S -operator. The S -operator is defined as follows. For any cohomologyclasses a, b ∈ H ∗ T ( P ; C ) , ( a, S ( b )) = ⟪ a, bz − ψ ⟫ P ,T , . The T -equivariant J -function is characterized by ( J, a ) = ( , S ( a )) for any a ∈ H ∗ T ( P ) .Let χ = w − w , χ = w − w . We consider several different (flat) bases for H ∗ T ( P ; C ) : BOHAN FANG, CHIU-CHU MELISSA LIU, AND ZHENGYU ZONG (1) The canonical basis: φ = H − w w − w , φ = H − w w − w .(2) The basis dual to the canonical basis with respect to the T -equivariantPoincare pairing: φ = χ φ , φ = χ φ .(3) The normalized canonical basis ˆ φ = √ χ φ , ˆ φ = √ χ φ , which is self-dual:ˆ φ = ˆ φ , ˆ φ = ˆ φ .(4) The natural basis: T = T = H .(5) The basis dual to the natural basis: T = H − w − w , T = α, β ∈ { , } , define S αβ ( z ) ∶ = ( φ α , S ( φ β )) . Then S ( z ) = ( S αβ ( z )) is the matrix of the S -operator with respect to the orderedbasis ( φ , φ ) :(3) S ( φ β ) = ∑ α = φ α S αβ ( z ) . For i ∈ { , } and α ∈ { , } , define S ˆ αi ( z ) ∶ = ( T i , S ( ˆ φ α )) . Then ( S ˆ αi ) is the matrix of the S -operator with respect to the ordered bases ( ˆ φ , ˆ φ ) and ( T , T ) :(4) S ( ˆ φ α ) = ∑ i = T i S ˆ αi ( z ) . We have z ∂J∂t i = ∑ α = S ˆ αi ( z ) ˆ φ α . By [21, 28], the equivariant J -function is J = e ( t + t H )/ z ( + ∞ ∑ d = q d ∏ dm = ( H − w + mz ) ∏ dm = ( H − w + mz ) ) . For α = ,
2, define J α ∶ = J ∣ p α = e ( t + t w α )/ z ∞ ∑ d = q d d ! z d ∏ dm = ( χ α + mz ) . Then z ∂J∂t = J = ∑ α = J α φ α , z ∂J∂t = z ∑ α = ∂J α ∂t φ α . So S ˆ αi ( z ) = z √ χ α ⋅ ∂J α ∂t i . Following Givental, we define ̃ S ˆ αi ( z ) ∶ = S ˆ αi ( z ) exp ( − ∞ ∑ n = B n n ( n − ) ( zχ α ) n − ) . We use the convention that the left superscript/subscript is the row number and the right superscript/subscript is the column number.
YNARD-ORANTIN RECURSION AND EQUIVARIANT MIRROR SYMMETRY FOR P Then ̃ S ˆ α ( z ) = √ χ α exp ( t + t w α z − ∞ ∑ n = B n n ( n − ) ( zχ α ) n − ) ⋅ ( ∞ ∑ d = q d d ! z d ∏ dm = ( χ α + mz ) )̃ S ˆ α ( z ) = √ χ α exp ( t + t w α z − ∞ ∑ n = B n n ( n − ) ( zχ α ) n − ) , ⋅ ( w α ∞ ∑ d = q d d ! z d ∏ dm = ( χ α + mz ) + ∞ ∑ d = q d ( d − ) ! z d ∏ dm = ( χ α + mz ) ) . The A-model R -matrix. By Givental [23], the matrix ( ̃ S ˆ βi )( z ) is of theform ̃ S ˆ βi ( z ) = ∑ α = Ψ αi R βα ( z ) e u β / z = ( Ψ R ( z )) βi e u β / z , where R ( z ) = ( R βα ( z )) = I + ∑ ∞ k = R k z k and is unitary, andlim q → R βα ( z ) = δ αβ exp ( − ∞ ∑ n = B n n ( n − ) ( zχ β ) n − ) . Gromov-Witten potentials.
Introducing formal variables u = ∑ a ≥ u a z a , where u a = ∑ α = u αa φ α ( q ) . Define F P ,Tg,n ( u , t ) ∶ = ∑ a ,...,a n ∈ Z ≥ ⟪ τ a ( u a ) ⋯ τ a n ( u a n )⟫ P ,Tg,n = ∑ a ,...,a n ∈ Z ≥ ∞ ∑ m = ∞ ∑ d = Q d n ! m ! ∫ [M g,n + m ( P ,d )] vir n ∏ j = ev ∗ j ( u a j ) ψ a j j m ∏ i = ev ∗ i + n ( t ) . We define the total descendent potential of P to be D P ,T ( u ) = exp ( ∑ n,g ̵ h g − F P ,Tg,n ( u , )) . Consider the map π ∶ M g,n + m ( P , d ) → M g,n which forgets the map to the targetand the last m marked points. Let ¯ ψ i ∶ = π ∗ ( ψ i ) be the pull-backs of the classes ψ i , i = , ⋯ n , from M g,n . Then we can define¯ F P ,Tg,n ( u , t ) ∶ = ∑ a ,...,a n ∈ Z ≥ ∞ ∑ m = ∞ ∑ d = Q d n ! m ! ∫ [M g,n + m ( P ,d )] vir n ∏ j = ev ∗ j ( u a j ) ¯ ψ a j j m ∏ i = ev ∗ i + n ( t ) . Let the ancestor potential of P be A P ,T ( u , t ) = exp ( ∑ n,g ̵ h g − ¯ F P ,Tg,n ( u , t )) . Givental’s formula for equivariant Gromov-Witten potential and theA-model graph sum.
The quantization of the S -operator relates the ancestorpotential and the descendent potential of P via Givental’s formula. Concretely, wehave (see [24]) D P ,T ( u ) = exp ( F P ,T ) ˆ S − A P ,T ( u , t ) where F P ,T denotes ∑ n F P ,T ,n ( u , ) at u = u, u = u = ⋯ = S is the quan-tization [24] of S . For our purpose, we need to describe a formula for a slightlydifferent potential: F P ,Tg,n ( u , t ) —the descendent potential with arbitrary primaryinsertions.Now we first describe a graph sum formula for the ancestor potential A P ,T ( u , t ) .Given a connected graph Γ, we introduce the following notation.(1) V ( Γ ) is the set of vertices in Γ.(2) E ( Γ ) is the set of edges in Γ.(3) H ( Γ ) is the set of half edges in Γ.(4) L o ( Γ ) is the set of ordinary leaves in Γ.(5) L ( Γ ) is the set of dilaton leaves in Γ.With the above notation, we introduce the following labels:(1) (genus) g ∶ V ( Γ ) → Z ≥ .(2) (marking) β ∶ V ( Γ ) → { , } . This induces β ∶ L ( Γ ) = L o ( Γ ) ∪ L ( Γ ) → { , } , as follows: if l ∈ L ( Γ ) is a leaf attached to a vertex v ∈ V ( Γ ) , define β ( l ) = β ( v ) .(3) (height) k ∶ H ( Γ ) → Z ≥ .Given an edge e , let h ( e ) , h ( 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 ( Γ ) , 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 ⃗ Γ = ( Γ , g, β, k ) is stable if2 g ( v ) − + val ( v ) > v ∈ V ( Γ ) .Let Γ ( P ) denote the set of all stable labeled graphs ⃗ Γ = ( Γ , g, β, k ) . The genusof a stable labeled graph ⃗ Γ is defined to be g (⃗ Γ ) ∶ = ∑ v ∈ V ( Γ ) g ( v ) + ∣ E ( Γ )∣ − ∣ V ( Γ )∣ + = ∑ v ∈ V ( Γ ) ( g ( v ) − ) + ( ∑ e ∈ E ( Γ ) ) + . Define Γ g,n ( P ) = {⃗ Γ = ( Γ , g, β, k ) ∈ Γ ( P ) ∶ g (⃗ Γ ) = g, ∣ L o ( Γ )∣ = n } . Given α ∈ { , } , define u α ( z ) = ∑ a ≥ u αa z a . We assign weights to leaves, edges, and vertices of a labeled graph ⃗ Γ ∈ Γ ( P ) asfollows.(1) Ordinary leaves.
To each ordinary leaf l ∈ L o ( Γ ) with β ( l ) = β ∈ { , } and k ( l ) = k ∈ Z ≥ , we assign: ( L u ) βk ( l ) = [ z k ]( ∑ α = , u α ( z )√ ∆ α ( q ) R βα ( − z )) . YNARD-ORANTIN RECURSION AND EQUIVARIANT MIRROR SYMMETRY FOR P (2) Dilaton leaves.
To each dilaton leaf l ∈ L ( Γ ) with β ( l ) = β ∈ { , } and2 ≤ k ( l ) = k ∈ Z ≥ , we assign ( L ) βk ( l ) = [ z k − ]( − ∑ α = , √ ∆ α ( q ) R βα ( − z )) . (3) Edges.
To an edge connecting a vertex marked by α ∈ { , } to a vertexmarked by β ∈ { , } and with heights k and l at the corresponding half-edges, we assign E α,βk,l ( e ) = [ z k w l ]( z + w ( δ α,β − ∑ γ = , R αγ ( − z ) R βγ ( − w )) . (4) Vertices.
To a vertex v with genus g ( v ) = g ∈ Z ≥ and with marking β ( v ) = β , with n ordinary leaves and half-edges attached to it with heights k , ..., k n ∈ Z ≥ and m more dilaton leaves with heights k n + , . . . , k n + m ∈ Z ≥ ,we assign (√ ∆ β ( q )) g − + n + m ∫ M g,n + m ψ k ⋯ ψ k n + m n + m . We define the weight of a labeled graph ⃗ Γ ∈ Γ ( P ) to be w (⃗ Γ ) = ∏ v ∈ V ( Γ ) (√ ∆ β ( v ) ( q )) g ( v )− + val ( v ) ⟨ ∏ h ∈ H ( v ) τ k ( h ) ⟩ g ( v ) ∏ e ∈ E ( Γ ) E β ( v ( e )) ,β ( v ( e )) k ( h ( e )) ,k ( h ( e )) ( e ) ⋅ ∏ l ∈ L o ( Γ ) ( L u ) β ( l ) k ( l ) ( l ) ∏ l ∈ L ( Γ ) ( L ) β ( l ) k ( l ) ( l ) . Then log ( A P ,T ( u , t )) = ∑ ⃗ Γ ∈ Γ ( P ) ̵ h g (⃗ Γ )− w (⃗ Γ )∣ Aut (⃗ Γ )∣ = ∑ g ≥ ̵ h g − ∑ n ≥ ∑ ⃗ Γ ∈ Γ g,n ( P ) w (⃗ Γ )∣ Aut (⃗ Γ )∣ . Now we describe a graph sum formula for F P ,Tg,n ( u , t ) —the descendant potentialwith arbitrary primary insertions. For α = ,
2, letˆ φ α ( q ) ∶ = √ ∆ α ( q ) φ α ( q ) . Then ˆ φ ( q ) , ˆ φ ( q ) is the normalized canonical basis of QH ∗ T ( P ; C ) , the T -equivariantquantum cohomology of P . Define S ˆ α ˆ β ( z ) ∶ = ( ˆ φ α ( q ) , S ( ˆ φ β ( q ))) . Then ( S ˆ α ˆ β ( z )) is the matrix of the S -operator with respect to the ordered basis ( ˆ φ ( q ) , ˆ φ ( q )) :(5) S ( ˆ φ β ( q )) = ∑ α = ˆ φ α ( q ) S ˆ α ˆ β ( z ) . We define a new weight of the ordinary leaves:(1)’
Ordinary leaves.
To each ordinary leaf l ∈ L o ( Γ ) with β ( l ) = β ∈ { , } and k ( l ) = k ∈ Z ≥ , we assign: ( ˚ L u ) βk ( l ) = [ z k ]( ∑ α,γ = , u α ( z )√ ∆ α ( q ) S ˆ γ ˆ α ( z ) R ( − z ) βγ ) . We define a new weight of a labeled graph ⃗ Γ ∈ Γ ( P ) to be˚ w (⃗ Γ ) = ∏ v ∈ V ( Γ ) (√ ∆ β ( v ) ( q )) g ( v )− + val ( v ) ⟨ ∏ h ∈ H ( v ) τ k ( h ) ⟩ g ( v ) ∏ e ∈ E ( Γ ) E β ( v ( e )) ,β ( v ( e )) k ( h ( e )) ,k ( h ( e )) ( e ) ⋅ ∏ l ∈ L o ( Γ ) ( ˚ L u ) β ( l ) k ( l ) ( l ) ∏ l ∈ L ( Γ ) ( L ) β ( l ) k ( l ) ( l ) . Then ∑ g ≥ ̵ h g − ∑ n ≥ F P ,Tg,n ( u , t ) = ∑ ⃗ Γ ∈ Γ ( P ) ̵ h g (⃗ Γ )− ˚ w (⃗ Γ )∣ Aut (⃗ Γ )∣ = ∑ g ≥ ̵ h g − ∑ n ≥ ∑ ⃗ Γ ∈ Γ g,n ( P ) ˚ w (⃗ Γ )∣ Aut (⃗ Γ )∣ . We can slightly generalize this graph sum formula to the case where we have n ordered variables u , ⋯ , u n and n ordered ordinary leaves. Let u j = ∑ a ≥ ( u j ) a z a and let F P ,Tg,n ( u , ⋯ , u n , t ) ∶ = ∑ a ,...,a n ∈ Z ≥ ∞ ∑ m = ∞ ∑ d = m ! ∫ [M g,n + m ( P ,d )] vir n ∏ j = ev ∗ j (( u j ) a j ) ψ a j j m ∏ i = ev ∗ i + n ( t ) . Define the set of graphs ˜ Γ g,n ( P ) as the definition of Γ g,n ( P ) except that the n ordinary leaves are ordered . Let { l , ⋯ , l n } be the ordinary leaves in Γ ∈ ˜ Γ g,n ( P ) and for j = , ⋯ , n let ( ˚ L u j ) βk ( l j ) = [ z k ]( ∑ α,γ = , u αj ( z )√ ∆ α ( q ) S ˆ γ ˆ α ( z ) R ( − z ) βγ ) . Define the weight˚ w (⃗ Γ ) = ∏ v ∈ V ( Γ ) (√ ∆ β ( v ) ( q )) g ( v )− + val ( v ) ⟨ ∏ h ∈ H ( v ) τ k ( h ) ⟩ g ( v ) ∏ e ∈ E ( Γ ) E β ( v ( e )) ,β ( v ( e )) k ( h ( e )) ,k ( h ( e )) ( e ) ⋅ n ∏ j = ( ˚ L u j ) β ( l j ) k ( l j ) ( l j ) ∏ l ∈ L ( Γ ) ( L ) β ( l ) k ( l ) ( l ) . Then ∑ g ≥ ̵ h g − ∑ n ≥ F P ,Tg,n ( u , ⋯ , u n , t ) = ∑ ⃗ Γ ∈ ˜ Γ ( P ) ̵ h g (⃗ Γ )− ˚ w (⃗ Γ )∣ Aut (⃗ Γ )∣ = ∑ g ≥ ̵ h g − ∑ n ≥ ∑ ⃗ Γ ∈ ˜ Γ g,n ( P ) ˚ w (⃗ Γ )∣ Aut (⃗ Γ )∣ . B-model
The equivariant superpotential and the Frobenius structure of theJacobian ring.
Let Y be coordinates on C ∗ . The T -equivariant superpotential W w t ∶ C ∗ → C is given by W w t ( Y ) = Y + t + qY + w log Y + w log qY , where q = Qe t and Y = e y . In this section, we assume w − w is a positive realnumber. The Jacobian ring of W w t isJac ( W w t ) ≅ C [ Y, Y − , q, w ]/⟨ ∂W w t ∂y ⟩ = C [ Y, Y − , q, w ]/⟨ Y − qY + w − w ⟩ YNARD-ORANTIN RECURSION AND EQUIVARIANT MIRROR SYMMETRY FOR P Let B ∶ = q ∂W w t ∂q = qY + w . The Jacobian ring is isomorphic to QH ∗ T ( P ; C ) if one identifies B with H Jac ( W w t ) ≅ C [ B, q, w ]/⟨( B − w )( B − w ) − q ⟩ . The critical points of W w t are P , P , where P α = w − w + ∆ α ( q ) , α = , . Endow a metric on Jac ( W w q ) by the residue pairing ( f, g ) = ∑ α = Res Y = P α f ( Y ) g ( Y ) ∂W w t ∂y dYY . By direct calculation, we have ( B, B ) = w + w , ( B, ) = ( , B ) = , ( , ) = . We denote b = , b = B and b i by ( b i , b j ) = δ ij . These calculations show thefollowing well-known fact. Proposition 3.1.
There is an isomorphism of Frobenius manifold QH ∗ T ( P ; C ) ⊗ C [ w ] C [ w , w − w ] ≅ Jac ( W w t ) ⊗ C [ w ] C [ w , w − w ] . We denote Jac ( W w t ) ⊗ C [ w ] C [ w , w − w ] by H B . The Dubrovin-Givental connec-tion is denoted by ∇ Bv = z∂ v + v ● on H B ∶ = H B (( z )) .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 coordinates in this subsectionby the critical values of the superpotential W w t , and find the constant difference tothe A-model coordinates that we set up in earlier sections.Let C w t = {( x, y ) ∈ C ∶ x = W w t ( e y )} be the graph of the equivariant superpoten-tial. It is a covering of C ∗ given by y ↦ e y . Let ¯Σ ≅ P be the compactification of C ∗ with Y ∈ C ∗ ⊂ P as its coordinate. At each branch point Y = P α , x and y havethe following expansion x = ˇ u α − ζ α ,y = ˇ v α − ∞ ∑ k = h αk ( q ) ζ kα , where h α ( q ) = √ α ( q ) . Note that we define ζ α by ζ α = ˇ u α − x , which differs fromthe definition of ζ in [9, 14] by a factor of √ − u α = t + w α t + ∆ α ( q ) − χ α log χ α + ∆ α ( q ) . Since ∂ ˇ u α ∂t = ,∂ ˇ u α ∂t = qP α + w = w + w + ∆ α ( q ) , we have(6) d ˇ u α = du α , α = , . Recall that lim q → ∆ ( q ) = w − w , so in the large radius limit q →
0, we havelim q → ( ˇ u α − t − w α t ) = χ α − χ α log χ α . (7)From (6),(7), and (2), we conclude thatˇ u α = u α + a α , α = , , where a α = χ α − χ α log χ α . The Liouville form and Bergman kernel. On C w t , let λ = xdy be the Liouville form on C = T ∗ C . Then dλ = dx ∧ dy . LetΦ ∶ = λ ∣ C w t = W w t ( e y ) dy = ( e y + t + qe − y + ( w − w ) y + w log q ) dy. Then Φ is a holomorphic 1-form on C . Recall that q = Qe t and Y = e y . DefineΦ ∶ = ∂ Φ ∂t = dYY , Φ ∶ = ∂ Φ ∂t = ( qY + w ) dYY . Then Φ , Φ descends to holomorphic 1-forms on C ∗ which extends to meromorphic1-forms on P . We have ● Φ has simple poles at Y = Y = ∞ , andRes Y → Φ = , Res Y → ∞ Φ = − . ● Φ − w Φ = − qd ( Y − ) is an exact 1-form.Let B ( p , p ) be the fundamental normalized differential of the second kind on¯Σ (see e.g. [19]). It is also called the Bergman kernel in [13, 14]. In this simplecase ¯Σ ≅ P , we have B ( Y , Y ) = dY ⊗ dY ( Y − Y ) . Differentials of the second kind.
Following [9, 14], given α = , d ∈ Z ≥ , define dξ α,d ( p ) ∶ = ( d − ) !!2 − d Res p ′ → P α B ( p, p ′ )(√ − ζ α ) − d − . Then dξ α,d satisfies the following properties.(1) dξ α,d is a meromorphic 1-form on P with a single pole of order 2 d + P α .(2) In local coordinate ζ α near P α , dξ α,d = ( − ( d + ) !!2 d √ − d + ζ d + α + f ( ζ α )) dζ α , where f ( ζ α ) is analytic around P α . The residue of dξ α,d at P α is zero, so dξ α,d is a differential of the second kind. YNARD-ORANTIN RECURSION AND EQUIVARIANT MIRROR SYMMETRY FOR P The meromorphic 1-form dξ α,d is characterized by the above properties; dξ α,d canbe viewed as a section in H ( P , ω P (( d + ) P α )) . In particular, dξ α, is dξ α, = √ − √ α ( q ) d ( P α Y − P α ) . Then we have d ( Φ dW ) = d ( Y ( Y − P )( Y − P ) ) = P − P d ( P Y − P − P Y − P ) = √ − √ ( q ) dξ , + √ − √ ( q ) dξ , = √ − ∑ α = Ψ α dξ α, ,d ( Φ dW ) = d ( q + w Y ( Y − P )( Y − P ) ) = P − P d ( q + P w Y − P − q + P w Y − P ) = √ − ( q ) (√ ∆ ( q ) ( qP + w ) dξ , − √ ∆ ( q ) ( qP + w ) dξ , ) = √ − ((√ ∆ ( q ) + w + w √ ∆ ( q ) ) dξ , + √ (√ ∆ ( q ) + w + w √ ∆ ( q ) ) dξ , ) = √ − ∑ α = Ψ α dξ α, . We have(8) ( d ( Φ dW ) d ( Φ dW ) ) = √ − ( dξ , dξ , ) , √ − − ( d ( Φ dW ) d ( Φ dW ) ) = ( dξ , dξ , ) . Oscillating integrals and the B-model R -matrix. For α, β ∈ { , } , i ∈ { , } and z >
0, defineˇ S α i ( z ) ∶ = ∫ y ∈ γ α e W w q ( Y ) z Φ i = − z ∫ y ∈ γ α e W w q ( Y ) z d ( Φ i dW ) , where γ α is the Lefschetz thimble going through P α , such that W w q ( Y ) → −∞ nearits ends. It is straightforward to check that ∑ i = b i ˇ S α i is a solution to the quantumdifferential equation ∇ B f = α = ,
2. We quote the following theorem
Theorem 3.2 ([7, 22, 23]) . Near a semi-simple point on a Frobenius manifold ofdimension n , there is a fundamental solution S to the quantum differential equationsatisfying the following properties (1) S has the following form S = Ψ R ( z ) e U / z , where R ( z ) is matrix of formal power series in z , and U = diag ( u , . . . , u n ) 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 of e ∑ ∞ i = A i − z i − where A k are constantdiagonal matrices. Remark 3.3.
For equivariant Gromov-Witten theory of P , the fundamental solu-tion S in Theorem 3.2 is viewed as a matrix with entries in C [ w , w − w ](( z ))[[ q, t , t ]] .We choose the canonical coordinates { u α ( t )} such that there is no constant term byEquation (2) . Then if we fix the powers of q, t and t , only finitely many terms inthe expansion of e U / z contribute. So the multiplication Ψ R ( z ) e U / z is well definedand the result matrix indeed has entries in C [ w , w − w ](( z ))[[ q, t , t ]] . Remark 3.4.
For a general abstract semi-simple Frobenius manifold defined overa ring A , the expression S = Ψ R ( z ) e U / z in Theorem 3.2 can be understood in thefollowing way. We consider the free module M = ⟨ e u / z ⟩ ⊕ ⋯ ⊕ ⟨ e u n / z ⟩ over the ring A (( z ))[[ t , ⋯ , t n ]] where t , ⋯ , t n are the flat coordinates of the Frobenius manifold.We formally define the differential de u i / z = e u i / z du i z and we extend the differentialto M by the product rule. Then we have a map d ∶ M → M dt ⊕ ⋯ ⊕ M dt n . Weconsider the fundamental solution S = Ψ R ( z ) e U / z as a matrix with entries in M .The meaning that S satisfies the quantum differential equation is understood by theabove formal differential.In our case, the multiplication in the A-model fundamental solution S = Ψ R ( z ) e U / z is formal in z as in Remark 3.3. On B-model side, we use the stationary phase ex-pansion to obtain a product of the form Ψ R ( z ) e U / z . The multiplications Ψ R ( z ) e U / z on 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 [24] and state it as the following fact.
Proposition 3.5.
The fundamental solution matrix { ˇ S α i √− πz } has the following as-ymptotic expansion where ˇ R ( z ) is a formal power series in z ˇ S α i ( z )√ − πz ∼ ∑ γ = Ψ γi ˇ R αγ ( z ) e ˇ uαz . Proof.
By the stationary phase expansion,ˇ S αi ( z ) ∼ √ πze ˇ uαz ( + a αi, z + a αi, z + . . . ) , it follows that { ˇ S αi } can be asymptotically expanded in the desired form (noticethat Ψ is a matrix in z -degree 0). In particular, by (8)ˇ R αβ ( z ) ∼ √ ze − ˇ uαz √ π ∫ γ α e W w tz dξ β, . Following Eynard [9], define Laplace transform of the Bergman kernel(9) ˇ B α,β ( u, v, q ) ∶ = uvu + v δ α,β + √ uv π e u ˇ u α + v ˇ u β ∫ p ∈ γ α ∫ p ∈ γ β B ( p , p ) e − ux ( p )− vx ( p ) , where α, β ∈ { , } . By [9, Equation (B.9)],(10) ˇ B α,β ( u, v, q ) = uvu + v ( δ α,β − ∑ γ = ˇ R αγ ( − u ) ˇ R βγ ( − v )) . Setting u = − v , we conclude that ( ˇ R ∗ ( u ) ˇ R ( − u )) αβ = { ∑ γ = ˇ R αγ ( u ) ˇ R βγ ( − u )} = δ αβ . This shows ˇ R is unitary. (cid:3) Following Iritani [26] (with slight modification), we introduce the following def-inition.
YNARD-ORANTIN RECURSION AND EQUIVARIANT MIRROR SYMMETRY FOR P Definition 3.6 (equivariant K-theoretic framing) . We define ̃ ch z ∶ K T ( P ) → H ∗ T ( P ; Q )[[ w − w z ]] by the following two properties which uniquely characterize it. (a) ̃ ch z is a homomorphism of additive groups: ̃ ch z ( E ⊕ E ) = ̃ ch z ( E ) + ̃ ch z ( E ) . (b) If L is a T -equivariant line bundle on P then ̃ ch z ( L ) = exp ( − π √ − ( c ) T ( L ) z ) . For any
E ∈ K T ( P ) , we define the K-theoretic framing of E by κ ( E ) ∶ = ( − z ) − ( c ) T ( T P ) z Γ ( − ( c ) T ( T P ) z ) ̃ ch z ( E ) where ( c ) T ( T P ) = H − w − w . By localization, property (b) in the above definition is characterized by ι ∗ p α κ ( O P ( l p + l p )) = ( − z ) − χαz Γ ( − χ α z ) e − lαπ √− χαz , α = , , where ι p α ∶ p α → P is the inclusion map.The following definition is motivated by [15, 17]. Definition 3.7 (equivariant SYZ T-dual) . Let
L = O P ( l p + l p ) be an equivari-ant ample line bundle on P , where l , l are integers such that l + l > . We definethe equivariant SYZ T-dual SYZ ( L ) of L to be the oriented graph in Figure 1 below.We extend the definition additively to the equivariant K-theory group K T ( P ) . +∞ + ( l − ) πi −∞ + (− l − ) πi (− l − ) πi ( l − ) πi Figure 1.
SYZ ( O P ( l p + l p )) in C πi − πi +∞ + πi −∞ − πi −∞ exp SYZ (O P ( p )) in C SYZ (O P ( )) in C ∗ Figure 2.
The equivariant SYZ T-dual of O P ( p ) in C and the(non-equivariant) SYZ T-dual of O P ( ) 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 that z, q, w − w ∈ ( , ∞ ) . Then for any L ∈ K T ( P ) , (11) ∫ y ∈ SYZ (L) e W w tz dy = ⟪ , κ ( L ) z − ψ ⟫ P ,T , . (12) ∫ y ∈ SYZ (L) e W w tz ydx = − ⟪ κ ( L ) z − ψ ⟫ P ,T , . Here dx = d ( W w t ( y )) .Proof. The left hand side of (11) is ∫ y ∈ SYZ (L) e W w tz dy = − z ∫ y ∈ SYZ (L) e W w tz yd ( W w t ) . By the string equation, the right hand side of (11) is ⟪ , κ ( L ) z − ψ ⟫ P ,T , = ⟪ κ ( L ) z ( z − ψ ) ⟫ P ,T , . So (11) is equivalent to (12).It remains to prove (11) for
L = O P ( l p + l p ) , where l + l ≥
0. We will expressboth hand sides of (11) in terms of (modified) Bessel functions. A brief reviewof Bessel functions is given in Appendix A. The equivariant quantum differentialequation of P is related to the modified Bessel differential equation by a simpletransform (see Appendix B).Let γ l ,l be defined as in Appendix A. ∫ SYZ (L) e W w tz dy = ∫ SYZ (L) exp ( z ( e y + t + qe − y + w y + w ( t − y ))) dy = e z ( t + w t ) ∫ γ ℓ ,ℓ exp ( z ( e y − iπ + qe iπ − y + ( w − w )( y − πi ))) dy = ( − ) w − w z e t z + w + w z t ∫ γ ℓ ,ℓ exp ( − √ qz cosh ( y − t ) + w − w z ( y − t )) dy = ( − ) w − w z e t z + w + w z t ∫ γ ℓ ,ℓ exp ( − √ qz cosh ( y ) + w − w z y ) dy By Lemma A.1, ∫ γ l ,l exp ( − √ qz cosh ( y ) + w − w z y ) dy = π sin ( w − w z π ) ( e − πil w − w z I w − w z ( √ qz ) − e − πil w − w z I w − w z ( √ qz )) = − ∑ α = e − πil α χαz π sin ( χ α z π ) I χαz ( √ qz ) . Therefore, the left hand side of (11) is ∫ SYZ (L) e W w tz dy = − e t z + w + w z t ∑ α = e −( l α − ) πi χαz π sin ( χ α z π ) I χαz ( √ qz ) . Recall from Section 2.5 that J α = ⟪ , φ α z − ψ ⟫ P ,T , = χ α ⟪ , φ α z − ψ ⟫ P ,T , . YNARD-ORANTIN RECURSION AND EQUIVARIANT MIRROR SYMMETRY FOR P We have J α = e ( t + t w α )/ z ∞ ∑ d = q d d ! z d ∏ dm = ( χ α + mz ) = e ( t + t w α )/ z ∞ ∑ m = ( √ qz ) m Γ ( χ α z + ) m !Γ ( χ α z + m + ) = e t z + w + w z t z χαz Γ ( χ α z + ) I χαz ( √ qz ) κ ( L ) = ∑ α = ( − z ) χα − z + Γ ( − χ α z ) e − lαπ √− χαz φ α . So the right hand side of (11) is ⟪ , κ ( L ) z − ψ ⟫ P ,T , = ∑ α = ( − z ) χα − z + Γ ( − χ α z ) e − πilαχαz J α χ α = − e t z + w + w z t ∑ α = ( − ) χα − z e − πilαχαz π sin ( χ α z π ) I χαz ( √ qz ) = − e t z + w + w z t ∑ α = e −( l α − ) πi χαz π sin ( χ α z π ) I χαz ( √ qz ) (cid:3) Remark 3.9.
Definition 3.6 (equivariant K-theoretic framing) and Definition 3.7(equivariant SYZ T-dual) can be extended to any projective toric manifold. In [16] , the first author uses the mirror theorem [21, 28] and results in [26] to extendProposition 3.8 to any semi-Fano projective toric manifold. The left hand side of (11) is known as the central charge of the Lagrangian brane
SYZ ( L ) . Proposition 3.10.
The A and B-model R -matrices are equal R αβ ( z ) = ˇ R αβ ( z ) . Proof.
By the asymptotic decomposition theorem of the S -matrix (Theorem 3.2),weonly have to compare at the limit q = , t = S and ˇ S are unitary. Noticethat Ψ has an non-degenerate limit at q =
0, then it suffices to show that˜ S ˆ αi e − u α / z ∣ q = ,t = ∼ √ − πz ˇ S α i e − ˇ u α / z ∣ q = ,t = . The Lefschetz thimble γ is { Y ∣ Y ∈ ( −∞ , )} . While the Lefschetz thimble γ couldnot be explicitly depicted, we could alternatively consider the thimble γ ′ = { Y ∣ Y ∈ ( , ∞ )} for z < ∫ e W w t / z dy . The integral yields the sameasymptotic answer once we analytically continue z < z >
0, since the stationaryphase expansion only depends on the local behavior (higher order derivatives) of W w t at the critical points.So letting Y = − T z for α =
2, or Y = − qT z for α = e − ˇ u α / z ˇ S α = e − ∆ α ( q ) z ( χ α + ∆ α ( q ) ) χαz ( − z ) − χαz ∫ ∞ e − T e − qTz T χαz − dT. Taking the limit q → √ − πz e − ˇ u α / z ˇ S α ∣ q = = √ − πz e − χαz ( − χ α z ) χαz Γ ( − χ α z ) ∼ √ χ α exp ( − ∞ ∑ n = B n n ( n − ) ( zχ α ) n − ) ∼ ˜ S ˆ α e − u α / z ∣ q = . Here we use the Stirling formulalog Γ ( z ) ∼
12 log ( π ) + ( z − ) log z − z + ∞ ∑ n = B n n ( n − ) z − n . Notice that ˇ S α = z ∂∂t ˇ S α = z ∫ γ α e W w t / z ( qY + w ) dYY , and similar calculation shows (letting Y = − T z if α = Y = − qT z if α = √ − πz e − ˇ u α / z ˇ S α ∣ q = ∼ w α √ χ α exp ( − ∞ ∑ n = B n n ( n − ) ( zχ α ) n − ) ∼ ˜ S ˆ α e − u α / z ∣ q = . (cid:3) Notice that the matrix ˇ R is given by the asymptotic expansion. This theoremdoes not imply ˜ S ˆ αi e − u α / z = √− πz ˇ S α i e − ˇ u α / z , which are unequal.3.6. The Eynard-Orantin topological recursion and the B-model graphsum.
Let ω g,n be defined recursively by the Eynard-Orantin topological recursion[13]: ω , = , ω , = B ( Y , Y ) = dY ⊗ dY ( Y − Y ) . When 2 g − + n > ω g,n ( Y , . . . , Y n ) = ∑ α = Res Y → P α − ∫ ˆ Yξ = Y B ( Y n , ξ ) ( log ( Y ) − log ( ˆ Y )) dW ( ω g − ,n + ( Y, ˆ Y , Y , . . . , Y n − ) + ∑ g + g = g ∑ I ∪ J = { ,...,n − } I ∩ J = ∅ ω g , ∣ I ∣+ ( Y, Y I ) ω g , ∣ J ∣+ ( ˆ Y , Y J ) where Y ≠ P α is in a small neighborhood of P α , and ˆ Y ≠ Y is the other point in theneighborhood such that W w q ( ˆ Y ) = W w q ( Y ) .The B-model invariants ω g,n can be expressed as graph sums [27, 9, 10, 6]. Wewill use the formula stated in [6, Theorem 3.7], which is equivalent to the formulain [9, Theorem 5.1]. Given a labeled graph ⃗ Γ ∈ Γ g,n ( P ) with L o ( Γ ) = { l , . . . , l n } ,we define its weight to be w (⃗ Γ ) = ( − ) g (⃗ Γ )− + n ∏ v ∈ V ( Γ ) ( h α √ ) − g − val ( v ) ⟨ ∏ h ∈ H ( v ) τ k ( h ) ⟩ g ( v ) ∏ e ∈ E ( Γ ) ˇ B α ( v ( e )) ,α ( v ( e )) k ( e ) ,l ( e ) ⋅ n ∏ j = √ − dξ α ( l j ) k ( l j ) ( Y j ) ∏ l ∈ L ( Γ ) ( − √ − ) ˇ h α ( l ) k ( l ) . Here, ˇ h αk = − √− k − ( k − ) !! h α k − . Note that the definitions of ˇ B α,βk,l , ˇ h αk , dξ αk in this paper are slightly different from those in [6]; for example, the definition of YNARD-ORANTIN RECURSION AND EQUIVARIANT MIRROR SYMMETRY FOR P ˇ B α,βk,l in this paper differs from Equation (3.11) of [6] by a factor of 2 − k − l − . In ournotation [6, Theorem 3.7] is equivalent to: Theorem 3.11.
For g − + n > , ω g,n = ∑ Γ ∈ Γ g,n ( P ) w (⃗ Γ )∣ Aut (⃗ Γ )∣ . All genus mirror symmetry.
Given a meromorphic function f ( Y ) on P which is holomorphic on P ∖ { P , P } , define θ ( f ) = dfdW = Y ( Y − P )( Y − P ) dfdY . Then θ ( f ) is also a meromorphic function which is holomorphic on P ∖ { P , P } .For α ∈ { , } , let ξ α, = √ − √ α ( q ) P α Y − P α . Then ξ α, is a meromorphic function on P with a simple pole at Y = P α andholomorphic elsewhere. Moreover, the differential of ξ α, is dξ α, . For k >
0, define W αk ∶ = d (( − ) k θ k ( ξ α, )) . Define ˇ S α ˆ β ( z ) = − z ∫ y ∈ γ α e xz dξ β, √ − , ˇ S κ (L) ˆ β ( z ) = − z ∫ y ∈ SYZ (L) e xz dξ β, √ − . Then ˇ S α ˆ β ( z ) = − z k + ∫ y ∈ γ α e W ( y ) z W βk √ − , ˇ S κ (L) ˆ β ( z ) = − z k + ∫ y ∈ SYZ (L) e W ( y ) z W βk √ − ∫ y ∈ SYZ (L) e W ( y ) z W βk √ − = − z − k − ˇ S κ (L) ˆ β ( z ) = − z − k − ⟪ ˆ φ α ( q ) , κ ( L ) z − ψ ⟫ P ,T , . where the last equality follows from Theorem 3.8.For α = , j = , ⋯ , n , let(15) ˜ u αj ( z ) = ∑ β = S ˆ α ˆ β ( z ) u βj ( z )√ ∆ β ( q ) . Theorem 1 (All genus equivariant mirror symmetry for P ) . For n > and g − + n > , we have (16) ω g,n ∣ √− W αk ( Y j ) = ( ˜ u j ) αk = ( − ) g − + n F P ,Tg,n ( u , ⋯ , u n , t ) . Proof.
We will prove this theorem by comparing the A-model graph sum in the endof Section 2.7 and the B-model graph sum in the previous section.(1) Vertex. By Section 3.1, we have h α ( q ) = √ α ( q ) . So in the B-model vertex, h α √ = √ α ( q ) . Therefore the B-model vertex matches the A-model vertex. (2) Edge. By Section 3.6, we know thatˇ B α,βk,l = [ u − k v − l ] ⎛⎝ uvu + v ( δ α,β − ∑ γ = , f αγ ( u, q ) f βγ ( v, q ))⎞⎠ = [ z k w l ] ⎛⎝ z + w ( δ α,β − ∑ γ = , f αγ ( z , q ) f βγ ( w , q ))⎞⎠ . By definition E α,βk,l = [ z k w l ]( z + w ( δ α,β − ∑ γ = , R αγ ( − z ) R βγ ( − w )) . But we know that R αβ ( z ) = f αβ ( − z ) . Therefore, we have ˇ B α,βk,l = E α,βk,l . (3) Ordinary leaf. We have the following expression for dξ αk (see [18]): dξ αk = W αk − k − ∑ i = ∑ β ˇ B α,βk − − i, W βi . By item 2 (Edge) above, for k, l ∈ Z ≥ ,ˇ B α,βk,l = [ z k w l ]( z + w ( δ α,β − ∑ γ = , R αγ ( − z ) R βγ ( − w ))) . We also have [ z ]( R αβ ( − z )) = δ α,β . Therefore, dξ αk = k ∑ i = ∑ β = ([ z k − i ] R αβ ( − z )) W βi . So under the identification1 √ − W αk ( Y j ) = ( ˜ u j ) αk The B-model ordinary leaf matches the A-model ordinary leaf.(4) Dilaton leaf. We have the following relation between ˇ h αk and f αβ ( u, q ) (see[18]) ˇ h αk = [ u − k ] ∑ β √ − h β f αβ ( u, q ) . By the relation R αβ ( z ) = f αβ ( − z ) and the fact h β ( q ) = √ β ( q ) , it is easy to see that the B-model dilaton leafmatches the A-model dilaton leaf. (cid:3) Taking Laplace transforms at appropriate cycles to Theorem 1 produces a the-orem concerning descendants potential.
YNARD-ORANTIN RECURSION AND EQUIVARIANT MIRROR SYMMETRY FOR P Theorem 2 (All genus full descendant equivariant mirror symmetry for P ) . Sup-pose that n > and g − + n > . For any L , . . . , L n ∈ K T ( P ) , there is a formalpower series identity (17) ∫ y ∈ SYZ (L ) ⋯ ∫ y n ∈ SYZ (L n ) e W ( y ) z +⋯+ W ( yn ) zn ω g,n = ( − ) g − ⟪ κ ( L ) z − ψ , . . . , κ ( L n ) z n − ψ n ⟫ g,n . Remark 3.12.
By Theorem 3.8, (18) ∫ y ∈ SYZ (L) e W ( y ) z ydx = − ⟪ κ ( L ) z − ψ ⟫ P ,T , which is the analogue of (17) in the unstable case ( g, n ) = ( , ) .Proof of Theorem 2. By (15),˜ u αj ( z ) = ∑ β = √ ∆ α ( q )⟪ φ α ( q ) , φ β ( q ) z − ψ ⟫ P ,T , u βj ( z ) . Define the flat coordinates u αj by ∑ α = u αj ( z ) φ α ( q ) = ∑ α = u αj ( z ) φ α ( ) , and a power series in 1 / z S ˆ αβ ( z ) = ⟪ ˆ φ α ( q ) , φ β ( ) z − ψ ⟫ , . Then ˜ u αj ( z ) = ∑ β = (⟪ ˆ φ α ( q ) , φ β ( ) z − ψ ⟫ u βj ( z )) + = ∑ β = ( S ˆ αβ ( z ) u βj ( z )) + . Notice that ( S ˆ αβ ) is unitary, i.e. ∑ γ S ˆ γα ( z ) S ˆ γβ ( − z ) = χ β δ αβ . We have ∑ α = ( S ˆ αγ ( − z ) ˜ u αj ( z )) + = ∑ α = ( ∑ β = S ˆ αβ ( z ) S ˆ αγ ( − z ) u βj ( z )) = u γj ( z ) χ γ . Taking the Laplace transform of ω g,n ∫ y ∈ SYZ (L ) . . . ∫ y n ∈ SYZ (L n ) e W ( y ) z +⋅⋅⋅+ W ( yn ) zn ω g,n = ∫ y ∈ SYZ (L ) . . . ∫ y n ∈ SYZ (L n ) e ∑ ni = W ( yi ) zi ( − ) g − + n ( ∑ β i ,a i ⟪ n ∏ i = τ a i ( φ β i ( ))⟫ g,n ⋅ n ∏ i = ( u i ) β i a i )∣ ( ˜ u j ) βk = √− W βk ( y j ) = ∫ y ∈ SYZ (L ) . . . ∫ y n ∈ SYZ (L n ) e ∑ ni = W ( yi ) zi ( − ) g − + n ( ∑ β i ,a i ⟪ n ∏ i = τ a i ( φ β i ( ))⟫ g,n ⋅ n ∏ i = ( χ β i ∑ α = ∑ k ∈ Z ≥ [ z a i − ki ] S ˆ αβ i ( − z i ) W αk ( y i )√ − ) . Using (14) ∫ y ∈ SYZ (L ) . . . ∫ y n ∈ SYZ (L n ) e W ( y ) z +⋅⋅⋅+ W ( yn ) zn ω g,n = ( − ) g − + n ( ∑ β i ,a i ⟪ n ∏ i = τ a i ( φ β i ( ))⟫ g,n n ∏ i = ( χ β i ∑ α = ∑ k ∈ Z ≥ ([ z a i − ki ] S ˆ αβ i ( − z i )) S κ (L i ) ˆ α ( z i )( − z − k − i ))) = ( − ) g − ∑ β i ,a i ⟪ n ∏ i = τ a i ( φ β i ( ))⟫ g,n n ∏ i = χ β i ( φ β i ( ) , κ ( L i )) z − a i − i = ( − ) g − ⟪ κ ( L ) z − ψ , . . . , κ ( L n ) z n − ψ n ⟫ g,n . (cid:3) The non-equivariant limit and the Norbury-Scott conjecture
In this section, we consider the non-equivariant limit w = w = The non-equivariant R -matrix. By [23, Section 1.3], R ( z ) = I + ∑ ∞ n = R n z n is uniquely determined by:(1) The recursive relation: ( d + Ψ − d Ψ ) R n = [ dU, R n + ] .(2) The homogeneity of R ( z ) : R n q n / is a constant matrix.The unique solution R ( z ) satisfying the above conditions was computed explicitlyin [35]: Lemma 4.1 ( [35, Lemma 3.1] ) . R n = q − n ( n − ) !! ( n − ) !! n !2 n ( − n √ − ( − ) n + n √ − ( − ) n + ) By Proposition 3.10 , R ( z ) = ˇ R ( z ) . In this subsection, we recover the abovelemma by computing the stationary phase expansion of ˇ S .We assume z, q ∈ ( , ∞ ) , where q = Qe t .ˇ S = ∫ y = +∞ y = −∞ e z ( t + e y − iπ + qe −( y − iπ ) ) dy = e t / z ∫ y = +∞ y = −∞ e − √ qz cosh ( y − t ) dy = e t / z ∫ y = +∞ y = −∞ e − √ qz cosh ( y ) dy = e ( t − √ q )/ z ∫ y = +∞ y = e − √ qz ( cosh ( y ) − ) dy. Let T = √ qz ( cosh ( y ) − ) , then y = cosh − ( + zT √ q ) , dy = q − T − / ¿ÁÁÀ z + zT √ q . YNARD-ORANTIN RECURSION AND EQUIVARIANT MIRROR SYMMETRY FOR P ˇ S = e ( t − √ q )/ z ∞ ∑ n = ( z √ q ) n + ( − / n ) − n ∫ T = +∞ T = e − T T n − / dT = e ( t − √ q )/ z ∞ ∑ n = ( z √ q ) n + ( − ) n ( n − ) !! n !2 n Γ ( n + ) = √ πe ( t − √ q )/ z ∞ ∑ n = ( z √ q ) n + ( − ) n (( n − ) !! ) n !2 n ;ˇ S = z ∂∂t ˇ S = √ πze ( t − √ q )/ z ∞ ∑ n = ( z √ q ) n − / ( + ( + n ) z √ q ) ( − ) n + (( n − ) !! ) n !2 n . Similarly, ˇ S = √ − πe ( t + √ q )/ z ∞ ∑ n = ( z √ q ) n + (( n − ) !! ) n !2 n ;ˇ S = √ − πze ( t + √ q )/ z ∞ ∑ n = ( z √ q ) n − ( − ( + n ) z √ q ) (( n − ) !! ) n !2 n . Therefore, ̃ S ( z ) = √ − πz ˇ S ( z ) , [ z n ] ( ̃ S ( z ) e − U / z ) = ⎛⎜⎜⎝ (( n − ) !! ) √ n !2 n q n + √ − ( − ) n + (( n − ) !! ) √ n !2 n q n + (( n − ) !! ) √ n !2 n q n − − ( n − ) (( n − ) !! ) √ ( n − ) !2 n − q n − √ − ( − ) n ( n − ) !! ) √ n !2 n q n − + ( n − ) √ − ( − ) n + (( n − ) !! ) √ ( n − ) !2 n − q n − ⎞⎟⎟⎠ ,R n = ⎛⎜⎝ − ( n − ) !! ( n − ) !! n !2 n √ − ( − ) n + ( n − ) !! ( n − ) !! ( n − ) !2 n − √ − ( n − ) !! ( n − ) !! ( n − ) !2 n − ( − ) n + ( n − ) !! ( n − ) !! n !2 n ⎞⎟⎠ q − n = q − n ( n − ) !! ( n − ) !! n !2 n ( − n √ − ( − ) n + n √ − ( − ) n + ) The Norbury-Scott Conjecture.
In this subsection, we assume w = w = t =
0. Then ⟪ τ a ( H ) ⋯ τ a n ( H )⟫ P g,n = q ( ∑ ni = a i ) + − g ⟨ τ a ( H ) ⋯ τ a n ( H )⟩ P g,n . Note that when ( ∑ ni = a i ) + − g is not an nonnegative integer, both hand sidesare zero.When 2 g − + n > ω g,n is holomorphic near Y =
0, and one may expand it inthe local holomorphic coordinate ̃ x = x − = ( Y + qY ) − . Theorem 4.2.
Suppose that g − + n > . Then near Y = , ω g,n has the followingexpansion ω g,n = ( − ) g − + n ∑ a ,...,a n ∈ Z ≥ ⟪ τ a ( H ) ⋯ τ a n ( H )⟫ P g,n n ∏ j = ( a j + ) ! x a j + dx j The Norbury-Scott conjecture corresponds to the specialization q =
1, i.e. t = , Q = Proof.
Define ̃ W αk by 1 √ − ̃ W αk = ˜ u αk ∣ t a = ,t a = ( a + ) ! x − a − dx . By Theorem 1, it suffices to show that ̃ W αk agrees with the expansion of W αk near Y = ̃ x = x − .We now compute ̃ W αk explicitly. J = e ( t + t H )/ z ( + ∞ ∑ d = q d ∏ dm = ( H + mz ) ) = e t z ( + t Hz )( + ∞ ∑ d = q d z d ( d ! ) − ( ∞ ∑ d = q d z d ( d ! ) d ∑ m = m ) Hz )) = e t z ( + ∞ ∑ d = q d z d ( d ! ) ) + e t z ( t ( + ∞ ∑ d = q d z d + ( d ! ) ) − ∞ ∑ d = q d z d + ( d ! ) d ∑ m = m ) Hz ∂J∂t = e t z ( ∞ ∑ d = dq d z d − ( d ! ) ) + e t z ( t ( ∞ ∑ d = dq d z d ( d ! ) ) + + ∞ ∑ d = q d z d ( d ! ) ( − d d ∑ m = m )) HS ( z ) = ( H, S ( )) = ( , z ∂J∂t ) = e t z ( t ( ∞ ∑ d = dq d z d ( d ! ) ) + + ∞ ∑ d = q d z d ( d ! ) ( − d d ∑ m = m )) S ( z ) = ( , S ( )) = ( , J ) = e t z ( t ( + ∞ ∑ d = q d z d + ( d ! ) ) − ∞ ∑ d = q d z d + ( d ! ) d ∑ m = m ) S ( z ) = ( H, S ( H )) = ( H, z ∂J∂t ) = e t z ( ∞ ∑ d = q d + z d + d ! ( d + ) ! ) S ( z ) = ( , S ( H )) = ( H, J ) = e t z ( + ∞ ∑ d = q d z d ( d ! ) ) S ˆ αj ( z ) = ∑ i = Ψ αi S i j ( z ) S ˆ11 ( z ) = √ e t z ∞ ∑ n = (√ q ) n + z n ⌊ n ⌋ ! ⌈ n ⌉ ! S ˆ21 ( z ) = √ e t z ∞ ∑ n = ( − √ q ) n + z n ⌊ n ⌋ ! ⌈ n ⌉ !˜ u α ( z ) = ∑ i = S ˆ αi ( z ) t i ( z ) YNARD-ORANTIN RECURSION AND EQUIVARIANT MIRROR SYMMETRY FOR P ˜ u k ∣ t a = = √ ∞ ∑ n = (√ q ) n + ⌊ n ⌋ ! ⌈ n ⌉ ! t k + n , ˜ u k ∣ t a = = √ ∞ ∑ n = ( − √ q ) n + ⌊ n ⌋ ! ⌈ n ⌉ ! t k + n For α = , ̃ W αk = √ − u αk ∣ t a = ,t a = ( a + ) ! x − a − dx = d (( − ddx ) k ̃ ξ α, ) where(20) ̃ ξ , ∶ = − √ − ∞ ∑ n = (√ q ) n + ( n ⌊ n ⌋) x − n − (21) ̃ ξ , ∶ = − √ − ∞ ∑ n = ( − √ q ) n + ( n ⌊ n ⌋) x − n − Recall that(22) W αk = d (( − ddx ) k ξ α, ) By (19) and (22), to complete the proof, it remains to show that, ̃ ξ α, agree withthe expansion of ξ α, near Y = ̃ x = x − = ( Y + qY ) − .Assume that q ∈ ( , ∞ ) . We have P = √ q, ∆ = √ q, ξ , = √ − q / Y − √ q ,P = − √ q, ∆ = − √ q, ξ , = q / Y + √ q , The n -th coefficient in the expansion of ̃ x = ( Y + qY ) − at Y = Y = ̃ x − n − ξ , d ̃ x = − √ − q / Res Y = ( Y + qY ) n − ( − qY ) dYY − √ q = − √ − q / Res Y = ( Y + q ) n − ( Y + √ q ) Y n + dY = − √ − (√ q ) n − ( n − ⌊ n ⌋ ) ξ , = − √ − ∞ ∑ n = (√ q ) n − ( n − ⌊ n ⌋ )̃ x n = − √ − ∞ ∑ n = (√ q ) n + ( n ⌊ n + ⌋)̃ x n + = − √ − ∞ ∑ n = (√ q ) n + ( n ⌊ n ⌋) x − n − which agrees with ̃ ξ , defined in (20).Res Y = ̃ x − n − ξ , d ̃ x = − q / Res Y = ( Y + qY ) n − ( − qY ) dYY + √ q = − q / Res Y = ( Y + q ) n − ( Y − √ q ) Y n + dY = − √ − ( − √ q ) n − ( n − ⌊ n ⌋ ) ξ , = − √ − ∞ ∑ n = ( − √ q ) n + ( n ⌊ n ⌋) x − n − which agrees with ̃ ξ , defined in (21). (cid:3) The large radius limit and the Bouchard-Mari˜no conjecture
In this section, we will specialize Theorem 1 to the large radius limit case. Inthis case, Theorem 1 relates the invariant ω g,n of the limit curve to the equivariantdescendent theory of C . After expanding ξ α, in suitable coordinates, we can relatethe corresponding expansion of ω g,n to the generation function of Hurwitz numbersand therefore reprove the Bouchard-Mari˜no conjecture [2] on Hurwitz numbers.Let w = t = q →
0. Then our mirror curvebecomes x = Y + w log Y. When w = −
1, this is just the Lambert curve. Recall that the two critical points P , P of W w t ( Y ) are P α = w − w + ∆ α ( q ) . Since ∆ ( ) = w − w , P → q →
0. In other words, P goesout of the curve under the limit q → ξ , = √ α ( q ) P Y − P →
0. As a result, W k = d ( θ k ( ξ , )) also turns to zero under the large radius limit.Under the identification √ − W αk ( Y j ) = ( ˜ u j ) αk in Theorem 1, we have ( ˜ u j ) k → q →
0. On the A-model side, since q =
0, the S − matrix ( ˚ S αβ ( z )) is diagonal.Therefore, we also have ( u j ) k → q → P , we can only have a constant map to p ∈ P . Since H ∣ p = w = t =
0, wecan not have any primary insertions. Therefore, in the large radius limit, we get F P , C ∗ g,n ( u , ⋯ , u n ; t ) = ∑ a ,...,a n ∈ Z ≥ ∫ [M g,n ( P , )] vir n ∏ j = ev ∗ j (( u j ) a j φ ( )) ψ a j j = ∑ a ,...,a n ∈ Z ≥ − w ∫ M g,n n ∏ j = ( u j ) a j ψ a j j Λ ∨ g ( − w ) , where Λ ∨ g ( u ) = u g − λ u g − + ⋯ + ( − ) g λ g . and λ j = c j ( E ) is the j -th Chern class of the Hodge bundle. At the same time,we also have ˚ S = ( ˆ φ ( ) , ˆ φ ( )) =
1. So ( u j ) k √ − w = ( ˜ u j ) k . Therefore Theorem 1specializes to ω g,n ∣ √− W k ( Y j ) = ( uj ) k √− w = ( − ) g − + n ∑ a ,...,a n ∈ Z ≥ − w ∫ M g,n n ∏ j = ( u j ) a j ψ a j j Λ ∨ g ( − w ) . Now we study the expansion of ξ , near the point Y = Z = e x w . We have ξ , = √ − √ − w − w Y + w . YNARD-ORANTIN RECURSION AND EQUIVARIANT MIRROR SYMMETRY FOR P Since Z = Y e Y w , by taking the differential we have dZZ = Y + w Y w dY. Therefore, ξ , = − √ − √ − w dY dZZ Y . Let ξ , = ∞ ∑ µ = C µ Z µ . near the point Y =
0. Then we have C µ = Res Y → ξ , Z − µ dZZ = − √ − √ − w Res Y → e − µY w dYY µ + = − √ − √ − w ( − µ w ) µ µ ! . Therefore W k = − √ − √ − w w ∞ ∑ µ = ( − µ w ) µ µ ! ( − µ w ) k + Z µ − dZ. On A-model side, let ( u j ) a j = ∞ ∑ µ j = ( − µ j w ) µ j µ j ! ( µ j w ) a j Z µ j j . Then F C , C ∗ g,n ( u , ⋯ , u n ) = ∑ a ,...,a n ∈ Z ≥ − w ∫ M g,n n ∏ j = ψ a j j Λ ∨ g ( − w ) n ∏ j = ( ∞ ∑ µ j = ( − µ j w ) µ j µ j ! ( − µ j w ) a j Z µ j j ) = ∑ a ,...,a n ∈ Z ≥ − w ∫ M g,n n ∏ j = ( − µ j ψ j w ) a j Λ ∨ g ( − w ) n ∏ j = ( ∞ ∑ µ j = ( − µ j w ) µ j µ j ! Z µ j j ) . By the ELSV formula [8, 20], H g,µ = ( g − + ∣ µ ∣ + n ) ! ∣ Aut ( µ )∣ n ∏ j = µ µ j j µ j ! ∫ M g,n Λ ∨ g ( ) ∏ nj = ( − µ j ) = ( g − + ∣ µ ∣ + n ) ! ∣ Aut ( µ )∣ n ∏ j = µ µ j j µ j ! ∫ M g,n Λ ∨ g ( − w )( − w ) g − + n ∏ nj = ( − w − µ j ) . So F C , C ∗ = ∑ ℓ ( µ ) = n ∣ Aut ( µ )∣( g − + ∣ µ ∣ + n ) ! ( − w ) g − + ∣ µ ∣ + n H g,µ ∑ σ ∈ S n n ∏ j = Z µ j σ ( j ) . When w = −
1, this is just the generating function of Hurwitz numbers.Let W g,n ( Z , ⋯ , Z n ) be the expansion of ω g,n ( Y , ⋯ , Y n ) in the coordinate Z near Y =
0. Then we have
Corollary 5.1 (Bouchard-Mari˜no conjecture) . For n > and g − + n > , theinvariant W g,n ( Z , ⋯ , Z n ) for the curve x = Y + w log Y satisfies ∫ Z ⋯ ∫ Z n W g,n ( Z , ⋯ , Z n ) = ( − ) g − + n ∑ a ,...,a n ∈ Z ≥ − w ∫ M g,n n ∏ j = ψ a j j Λ ∨ g ( − w ) n ∏ j = ( ∞ ∑ µ j = ( − µ j w ) µ j + a j µ j ! Z µ j j ) = ( − ) g − + n ∑ ℓ ( µ ) = n ∣ Aut ( µ )∣ H g,µ ( g − + ∣ µ ∣ + n ) ! ( − w ) g − + ∣ µ ∣ + n ∑ σ ∈ S n n ∏ j = Z µ j σ ( j ) . In particular, when w = − , the right hand side is the generating function of Hur-witz numbers and the Bouchard-Mari˜no conjecture is recovered. Appendix A. Bessel functions
In this section, we give a brief review of Bessel functions.The Bessel’s differential equation is(23) x d ydx + x dydx + ( x − α ) y = . The
Bessel function of the first kind is defined by J α ( x ) = ∞ ∑ m = ( − ) m m !Γ ( m + α + ) ( x ) m + α . The
Bessel function of the second kind is defined by Y α ( x ) = J α ( x ) cos ( απ ) − J − α ( x ) sin ( απ ) . When n is an integer, Y n ( x ) ∶ = lim α → n Y α ( x ) . J α ( x ) and Y α ( x ) form a basis of the 2-dimensional space of solutions to theBessel’s differential equation (23).Replacing x by ix in (23), one obtains the the modified Bessel differential equa-tion(24) x d ydx + x dydx − ( x + α ) y = . The modified Bessel function of the first kind is defined by I α ( x ) = i − α J α ( ix ) = ∞ ∑ m = m !Γ ( m + α + ) ( x ) m + α . The modified Bessel function of the second kind is defined by K α ( x ) = π I − α ( x ) − I α ( x ) sin ( απ ) . The following integral formulas are valid when R ( x ) > I α ( x ) = π ∫ π e x cos θ cos ( αθ ) dθ − sin ( απ ) π ∫ ∞ e − x cosh t − αt dtK α ( x ) = ∫ ∞ e − x cosh t cosh ( αt ) dt = ∫ t ∈ γ , e − x cosh t − αt dt where γ , is the real line with the standard orientation: YNARD-ORANTIN RECURSION AND EQUIVARIANT MIRROR SYMMETRY FOR P +∞−∞ Figure 3.
The contour γ , e απi K α ( x ) + iπI α ( x ) = π e απi I − α ( x ) − e − απi I α ( x ) sin ( απ ) = e απi ∫ −∞ e − x cosh t − αt dt + e απi ∫ π e − x cos ( iθ ) − α ( iθ ) d ( iθ ) + e − απi ∫ ∞ e − x cosh t − αt dt = e απi ∫ γ , e − x cosh t − αt dt where γ , is the following contour: +∞ + πi −∞ πi Figure 4.
The contour γ , Therefore,(25) ∫ γ , e − x cosh t − αt dt = π sin ( απ ) ( I − α ( x ) − I α ( x )) (26) ∫ γ , e − x cosh t − αt dt = π sin ( απ ) ( I − α ( x ) − e − απi I α ( x )) For any integers l , l with l + l ≥
0, let γ l ,l be the following contour: +∞ + l πi −∞ − l πi − l πi l πi Figure 5.
The contour γ l ,l Lemma A.1.
For any l , l ∈ Z such that l + l ≥ , we have (27) ∫ γ l ,l e − x cosh t − αt dt = π sin ( απ ) ( e l απi I − α ( x ) − e − l απi I α ( x )) Proof.
We observe that(28) ∫ γ l − k,l + k e − x cosh t − αt dt = e − kαπi ∫ γ l ,l e − x cosh t − αt dt. In particular, ∫ γ l , − l e − x cosh t − αt dt = e ℓ απi ∫ γ , e − x cosh t − αt dt = π sin ( απ ) ( e − l απi I − α ( x ) − e l απi I α ( x )) This proves (27) in case l + l =
0. If l + l > γ l ,l = l − ∑ k = − l γ − k,k − l − ∑ k = − l γ − k,k . Equations (28) and (29) imply ∫ γ l ,l e − x cosh t − αt dt = ( l − ∑ k = − l e − kαπi ) ⋅ ∫ γ , e − x cosh t − αt dt − ( l − ∑ k = − l e − kαπi ) ⋅ ∫ γ , e − x cosh t − αt dt Equation (27) follows from the above equation and (25), (26). (cid:3)
Appendix B. The Equivariant Quantum Differential Equation for P The equivariant quantum differential equation of P is the vector equation zq ddq ⃗ I = ( q − w w w + w ) ⃗ I which is equivalent to the following scalar equation:(30) ( zq ddq − w )( zq ddq − w ) I = qI. Let I = e w + w z log q y, x = √ qz . Then (30) is equivalent to x d ydx + x dydx − ( x + ( w − w z ) ) y = α = w − w z . When w − w ≠
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E-mail address : [email protected] Chiu-Chu Melissa Liu, Department of Mathematics, Columbia University, 2990 Broad-way, New York, NY 10027
E-mail address : [email protected] Zhengyu Zong, Yau Mathematical Sciences Center, Tsinghua University, Jin ChunYuan West Building, Tsinghua University, Haidian District, Beijing 100084, China
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