Symplectic Manifolds and Gromov-Witten Invariants.

SYMPLECTIC MANIFOLDS AND GROMOV-WITTEN INVARIANTS

1 NUMERICAL PROBLEMS ON SYMPLECTIC MANIFOLDS AND GROMOV-WITTEN INVARI-

ANTS

Problem 1. Consider a compact symplectic manifold (M, ω)with a symplectic form ωsuch that

the first Chern class c1(T M)is equal to 3[ω]. Let Jbe a compatible almost complex structure on

a) If A, B are homology classes in H2(M;Z)such that A= 5[ω]and B= 2[ω], calculate the

Gromov-Witten invariant NM

0,3(A, B), the number of degree 3 holomorphic maps from P1to M

representing the homology class Awith 3 marked points mapping to B.

b) If Mis a K3 surface and the space of stable maps M0,3(M, A)parametrizing degree 3 stable

maps from genus 0 curves with 3 marked points in class Ais compact, prove that the Gromov-

Witten invariant NM

0,3(A, A)≥1.

Solution 1.

a) Given A= 5[ω]and B= 2[ω], since A= 5[ω]and B= 2[ω], we have A·B= (5[ω]) ·(2[ω]) =

10 RMω∧ω= 10ω(M). Therefore, NM

0,3(A, B) = 103= 1000.

b) Since Mis a K3 surface, it has trivial canonical bundle, implying that the space of stable maps

M0,3(M, A)is compact. By the degeneration formula, NM

0,3(A, A) = PA=A1+A2NM

0,3(A1, A2)≥1,

where A1, A2represent decompositions of Aas a sum of two classes. Therefore, NM

0,3(A, A)≥1.

2 STABLE MAPS AND VIRTUAL FUNDAMENTAL CLASSES IN GROMOV-WITTEN THEORY

Problem 1. Consider a symplectic manifold (M, ω)where M=CP 2is the complex projective

plane with the Fubini-Study symplectic form ω. Let [pt]denote the fundamental class of a point in

a) Compute the Gromov-Witten invariant n0,3([CP 1]×[pt]×[CP 1]) where [CP 1]denotes the

fundamental class of a 1-dimensional projective space.

b) Verify that the virtual dimension of the moduli space of stable maps in CP 2with 2 marked

points and degree 1 is zero.

c) Calculate the Gromov-Witten invariant n0,2([CP 1]×[CP 1]).

Solution 1. a) By the definition of the Gromov-Witten invariant, we have:

n0,3([CP 1]×[pt]×[CP 1]) = Z[M0,3(CP 2,1)]

where [M0,3(CP 2,1)] is the moduli space of stable maps from a genus 0 curve with 3 marked

points to CP 2of degree 1. By the dimension formula, this moduli space has virtual dimension

equal to dim(CP 2)+1−3 = 3 −2=1.

Thus, n0,3([CP 1]×[pt]×[CP 1]) = R[M0,3(CP 2,1)] 1counts the number of degree 1 stable maps

from a degree 1 curve with 3 marked points to CP 2. In this case, with 3 marked points, we need

to have all 3 maps constant maps to pt, and hence the answer is 1.

b) The virtual dimension of the moduli space of stable maps in CP 2with 2 marked points and

degree 1 is given by dim(CP 2) + 1 −2=3. However, for a genus 0 stable map this would indicate

a negative virtual dimension. Since the virtual dimension cannot be negative, it implies that such

curve configurations do not exist in this context.

c) Similarly, using the same reasoning, the virtual dimension of the moduli space [M0,2(CP 2,0)]

of stable maps in CP 2with 2 marked points and degree 0 would be -1. Therefore, there are no

such stable maps, and n0,2([CP 1]×[CP 1]) = 0.

3 THE SUBTOPIC: "SYMPLECTIC TOPOLOGY IN GROMOV-WITTEN THEORY"

Problem 3. Let Mbe a closed symplectic manifold of dimension 2nwith a symplectic form ω.

Consider the Gromov-Witten invariant GWd(M), which counts the number of rational curves in the

homology class d∈H2(M;Z).

Suppose we have a symplectic embedding ι:B2n(r),→M, where B2n(r)denotes the closed

ball of radius r∈R>0centered at the origin.

a) Show that there exists a neighborhood Uof ι(B2n(r)) in Msuch that for any d∈H2(M;Z),

the Gromov-Witten invariant GWd(U)is well-defined.

b) Calculate the Gromov-Witten invariant GWd(U)in the case when M=CP 2(complex pro-

jective plane) and d= 3[pt](the homology class of 3 points).

Solution 3. a) To show that the Gromov-Witten invariant GWd(U)is well-defined in a neigh-

borhood Uof ι(B2n(r)), we need to consider the compactness of moduli spaces of pseudoholo-

morphic curves representing the homology class d. By choosing Usmall enough, we can ensure

that there are no "bubbling" or "breaking" of curves in the moduli space, which guarantees that the

Gromov-Witten invariant is well-defined.

b) In the case of M=CP 2and d= 3[pt], the Gromov-Witten invariant GWd(U)can be calcu-

lated as follows:

Since d= 3[pt], we are looking for rational curves in CP 2that pass through 3 general points.

These curves are genus 0 curves, i.e., holomorphic spheres. The Gromov-Witten invariant in this

case counts the number of such curves that pass through the chosen 3 points.

The Gromov-Witten invariant GW3[pt](U)turns out to be 1, as there is only one rational curve of

degree 3 passing through 3 general points in CP 2(up to reparametrization).

Therefore, GW3[pt](U)=1.

4 "THE ROLE OF VIRTUAL FUNDAMENTAL CYCLES IN GROMOV-WITTEN THEORY"

Problem 4. Consider a closed symplectic manifold Mwith a symplectic form ωand let [µ]be a

virtual fundamental class of a moduli space of genus gand n-pointed stable maps in M. Suppose

we are interested in computing the Gromov-Witten invariant ⟨τk1(γ1), . . . , τkn(γn)⟩g,d where each γi

is a homology class in H∗(M),dis the degree, and kiare the degrees for the evaluation maps.

a) Write down the definition of the Gromov-Witten invariant ⟨τk1(γ1), . . . , τkn(γn)⟩g,d.

b) Using the virtual fundamental class [µ], explain how the computation of the Gromov-Witten

invariant is related to intersection theory on M.

Solution 4. a) The Gromov-Witten invariant ⟨τk1(γ1), . . . , τkn(γn)⟩g,d is defined as the integral

over the moduli space of stable maps:

⟨τk1(γ1), . . . , τkn(γn)⟩g,d =Z[Mg,n (M,d)]

ev∗

1(γ1)∪. . . ∪ev∗

n(γn),

where evi:Mg,n(M, d)→Mare the evaluation maps and Mg,n(M, d)is the moduli space of

genus g,n-pointed stable maps in Mwith degree d.

b) The computation of the Gromov-Witten invariant is related to intersection theory on Mthrough

the use of the virtual fundamental class [µ]. The virtual fundamental class of the moduli space

provides a way to define oriented intersection numbers of cycles in the ambient space M. By

integrating the pullbacks of the homology classes γiover the moduli space with respect to the

virtual fundamental class, we effectively compute the intersection products of these classes in M

weighted by the moduli space’s virtual dimension and other relevant data. Thus, the Gromov-Witten

invariants capture intersection-theoretic information and play a crucial role in studying the geometry

of symplectic manifolds.

5 VIRTUAL FUNDAMENTAL CLASSES AND GROMOV-WITTEN INVARIANTS

Problem 1. Consider a compact symplectic manifold (M, ω)of dimension 4 with a symplec-

tic form ω. Let Aand Bbe two homology classes in H2(M;Z). The Gromov-Witten invariant

GW M

0,2([pt], A, B)counts the number of genus zero holomorphic spheres in Mrepresenting the

homology classes Aand Bthat has two marked points.

a) Assume Aand Bare primitive homology classes, i.e., they cannot be represented as integer

multiples of other homology classes. Prove that if Ais not equal to B, then GW M

0,2([pt], A, B) = 0.

b) If A=B, determine the value of GW M

0,2([pt], A, A).

Solution 1.

a) Since Aand Bare primitive homology classes, if A=B, then there is no non-trivial class

representing both Aand B. Hence, the Gromov-Witten invariant GW M

0,2([pt], A, B)is zero.

b) If A=B, by the Gromov-Witten axioms, we have the relation GW M

0,2([pt], A, A) = ⟨τA⟩, where

τA∈H∗(M0,2(M, A)) is the fundamental class. Therefore, GW M

0,2([pt], A, A)is the intersection

pairing of the virtual fundamental class with the cohomology class dual to A, which is the integer

self-intersection number of A, denoted by A2. Thus, GW M

0,2([pt], A, A) = A2.

6 QUANTUM COHOMOLOGY AND MIRROR SYMMETRY IN GROMOV-WITTEN INVARIANTS

Problem 6. Consider a projective space CP 2with homogeneous coordinates [x:y:z]. Let C

be a smooth conic in CP 2defined by the equation x2+y2+z2= 0.

a) Compute the genus-zero Gromov-Witten invariant N0,3(CP 2,[pt],[C],[C]).

b) Compute the genus-one Gromov-Witten invariant N1,1(CP 2,[pt],[C]).

c) Compute the genus-one Gromov-Witten invariant N1,2(CP 2,[pt],[C],[C]).

Solution 6.

a) To compute N0,3(CP 2,[pt],[C],[C]), we first note that [pt]represents a point class in CP 2.

The Gromov-Witten invariant N0,3counts the number of rational degree zero maps from a 3-pointed

genus zero curve to CP 2that map the marked points to the classes [pt],[C], and [C]. Since we need

to map points to a conic C, we consider the standard conic given by the equation x2+y2+z2= 0.

The only holomorphic map in this case sends all points to the same conic, so we have only one

contribution. The map [x:y:z]7→ [x:y:z]is an example of a rational degree zero map. Hence,

N0,3(CP 2,[pt],[C],[C]) = 1.

b) To compute N1,1(CP 2,[pt],[C]), we note that at genus one, a rational degree zero map would

be a degree one map to CP 2which is an embedding. The only stable map that factors through a

conic is the constant map. Hence, N1,1(CP 2,[pt],[C]) = 1.

c) To compute N1,2(CP 2,[pt],[C],[C]), we note that at genus one, a rational degree zero map

would be a degree one map to CP 2.

For genus one with two markings to C, the only stable map is the constant map, as there are

no holomorphic maps that satisfy the constraints. Hence, N1,2(CP 2,[pt],[C],[C]) = 1.

7 SYMPLECTIC MANIFOLDS AND GROMOV-WITTEN INVARIANTS

Problem 1. Consider a symplectic manifold (M, ω)of dimension 2nwith a closed symplectic

form ω. Let A, B ∈H2(M;Z)be two homology classes in H2(M;Z). The Gromov-Witten invariant

GW M

0,A,B counts the number of degree 0genus 0stable maps from a marked point to Mthat

represent the homology class Aand the homology class B.

Given A= 3[Σ] and B= [Γ] + [∆] in H2(M;Z), calculate GW M

0,A,B.

Solution 1. To compute the Gromov-Witten invariant GW M

0,A,B, we first need to express Aand

Bin terms of the basis homology classes of H2(M;Z).

Given that A= 3[Σ] and B= [Γ] + [∆], we have:

GW M

0,A,B =GW M

0,3[Σ],[Γ]+[∆]

=GW M

0,3[Σ],[Γ] ·GW M

0,3[Σ],[∆] (by the product formula)

Next, calculate GW M

0,3[Σ],[Γ] and GW M

0,3[Σ],[∆] individually using the appropriate techniques and

formulas for Gromov-Witten invariants.

After calculating these values, multiply them together to find GW M

0,A,B.

Problem 2. Let Mbe a compact symplectic manifold of dimension 4and let A, B, C ∈H2(M;Z)

be homology classes in H2(M;Z). Suppose A= 2[Σ],B= [Γ] −[∆], and C= 3[Θ].

Calculate the Gromov-Witten invariant GW M

1,A,B,C for degree 1, genus 0stable maps represent-

ing the homology classes A,B, and C.

Solution 2. To find GW M

1,A,B,C , we must first express A,B, and Cin terms of the basis homology

classes of H2(M;Z). Given A= 2[Σ],B= [Γ] −[∆], and C= 3[Θ], we have:

GW M

1,A,B,C =GW M

1,2[Σ],[Γ]−[∆],3[Θ]

=GW M

1,2[Σ],[Γ],3[Θ] ·GW M

1,2[Σ],−[∆],3[Θ] (by the product formula)

Next, calculate the individual Gromov-Witten invariants GW M

1,2[Σ],[Γ],3[Θ] and GW M

1,2[Σ],−[∆],3[Θ]

using appropriate methods.

Finally, multiply these two values together to obtain GW M

1,A,B,C .

8 NUMERICAL PROBLEMS ON SYMPLECTIC MANIFOLDS AND GROMOV-WITTEN INVARI-

ANTS

Problem 9. Let Xbe a smooth projective variety with a symplectic form ωand B⊂Xa sym-

plectic submanifold. Consider the Gromov-Witten invariant GW X

A,B(d), which counts the number

of rational curves in class A∈H2(X, Z)passing through dgeneric points in Xand intersecting B

transversally at Bpoints.

Suppose we perform a symplectic surgery on Xalong Bto obtain a new symplectic manifold

Y. Determine the behavior of the Gromov-Witten invariant GW Y

A,B′(d)in terms of GW X

A,B(d), where

B′⊂Yis the image of the surgery.

Solution 9.

The behavior of the Gromov-Witten invariant under symplectic surgeries can be described as

follows:

a) If the symplectic surgery along Bdoes not change the homology class A, then the Gromov-

Witten invariant remains the same, i.e., GW Y

A,B′(d) = GW X

A,B(d).

b) If the symplectic surgery increases the dimension of the moduli space of J-holomorphic

curves passing through dpoints in class Aand intersecting Btransversally, then GW Y

A,B′(d)>

GW X

A,B(d).

c) If the symplectic surgery decreases the dimension of the moduli space, then GW Y

A,B′(d)<

GW X

A,B(d).

In each case, the behavior of the Gromov-Witten invariant reflects the changes induced by the

symplectic surgery on the moduli space of J-holomorphic curves.

9 QUANTUM CORRECTIONS TO GROMOV-WITTEN INVARIANTS

Problem 10. Consider a symplectic manifold Mwith first Chern class c1(M) = 3P D[ω], where

[ω]is the cohomology class represented by the symplectic form ω. Let β= 2P D[pt]∈H2(M;Z)

be a Poincaré dual of a point class.

a) Compute the quantum correction term ϕ1,1(β)in the Gromov-Witten potential.

b) Suppose that M=CP 2. Calculate the virtual Euler characteristic χvir(CP 2,1).

Solution 10.

a) The quantum correction term ϕ1,1(β)in the Gromov-Witten potential is given by

ϕ1,1(β) = Z[M1,1(M,β)]vir

λ1ψ1.

Since c1(M) = 3P D[ω], we have c1(TM)=3ω. Using the standard localization formula, we find

[M1,1(M, β)]vir = 4[pt]−8ω.

Thus, the quantum correction term is

ϕ1,1(β) = Z[M1,1(M,β)]vir

λ1ψ1=Z4[pt]−8ω

λ1ψ1=ZM

0=0.

b) For M=CP 2, the virtual Euler characteristic χvir(CP 2,1) is calculated as follows:

χvir(CP 2,1) = Z[M0,3(CP 2,1)]vir

1=3.

Therefore, the virtual Euler characteristic of CP 2is 3.

10 SYMPLECTIC REDUCTION AND GROMOV-WITTEN THEORY

Problem 10. Consider a symplectic manifold (M, ω)where ωis a symplectic form and Gis

a compact Lie group acting on Min a Hamiltonian fashion. Let µ:M→g∗be the moment

map associated with the G-action. The reduced space at a regular value ξ∈g∗is defined as

Mξ=µ−1(ξ)/G. Denote by [M]∈H∗(M)the homology class of Mand by [Mξ]∈H∗(Mξ)the

homology class of Mξ.

Given that the homology class [M]can be expressed as a sum of products of Chern classes of

the tangent bundle T M of Mas [M] = c1(T M )a1·c2(T M)a2·. . .·ck(T M )ak·[pt], where a1, a2, . . . , ak

are non-negative integers, prove the following:

a) The homology class [Mξ]of the reduced space Mξcan be written as [Mξ] = c1(T(Mξ))b1·

c2(T(Mξ))b2·. . . ·ck(T(Mξ))bk·[pt], where biis a multiple of aifor each i.

b) Show that if ξis a regular value of µ, then [Mξ] = µ∗(c(T M)|Mξ)[M], where µ∗:H∗(M)→

H∗(Mξ)is the pushforward map induced by the moment map, and c(T M)|Mξis the restriction of

the total Chern class of T M to Mξ.

Solution 10.

a) To show that [Mξ] = c1(T(Mξ))b1·c2(T(Mξ))b2·. . .·ck(T(Mξ))bk·[pt], where biis a multiple of

aifor each i, we need to use the equivariant Thom isomorphism theorem. Let Lbe the equivariant

line bundle associated with the action of Gon M. Then, Mξis the zero set of a smooth section of

the associated vector bundle L⊗T M . By applying the equivariant Thom isomorphism theorem, we

can express the equivariant Euler class of the normal bundle of Mξas a product of the equivariant

Chern classes of T M, where each biis a multiple of ai.

b) If ξis a regular value of µ, then the reduced space Mξis smooth and the moment map is

a submersion at points in the preimage of ξ. In this case, the pushforward map µ∗is identified

with the restriction map, which restricts the total Chern class of T M to Mξ. Therefore, we have

[Mξ] = µ∗(c(T M)|Mξ)[M].

11 "QUANTUM COHOMOLOGY AND MIRROR SYMMETRY IN SYMPLECTIC MANIFOLDS"

Problem 12. Consider a symplectic manifold (M, ω)where Mis a compact 2-dimensional

torus and ωis the standard symplectic form on M. Let βbe the class of a point in H2(M;Z).

a) Compute the intersection numbers dβββ and dββββ , corresponding to the Gromov-Witten

invariants of M.

b) Compute the genus 0 Gromov-Witten invariant N0,β, which counts the number of degree β

holomorphic curves in Mpassing through 3 generic points.

c) Let N0,β and dβββ be the genus 0 Gromov-Witten invariant and intersection number re-

spectively for the torus M. Verify the mirror symmetry relation between these two invariants, i.e.,

N0,β =dβββ.

Solution 12.

a) The intersection numbers dβββ and dββββ can be computed using the formula

dβγδ =Z[M0,3(M,β,γ,δ)]vir

For a torus, the virtual count of curves of degree βpassing through 3 generic points is given by

[M0,3(T2, β, β, β)]vir = [M0,3(pt, β, β, β)]vir =1

2β3,

and for dββββ, we have

[M0,4(T2, β, β, β, β)]vir = [M0,4(pt, β, β, β, β)]vir = 6β3.

Therefore, dβββ =1

2β3and dββββ = 6β3.

b) To compute the Gromov-Witten invariant N0,β for the torus M, which counts the number of

degree βholomorphic curves passing through 3 generic points, we use the formula

N0,β =Z[M0,3(M,β)]vir

Since the virtual count of curves of degree βpassing through 3 generic points is given by 1

2β3, we

have

N0,β =1

2β3.

c) The mirror symmetry relation between the genus 0 Gromov-Witten invariant N0,β and the

intersection number dβββ for the torus Mis N0,β =dβββ . Substituting the values we computed

earlier, we get 1

2β3=1

2β3,

which verifies the mirror symmetry relation between N0,β and dβββ .

I’m happy to help! Here is a numerical problem on Symplectic Manifolds and Gromov-Witten

Invariants:

12 SYMPLECTIC MANIFOLDS AND GROMOV-WITTEN INVARIANTS

12.1 Problem 1

Let Xbe a symplectic manifold with a symplectic form ω. Consider a genus-zero Gromov-Witten

invariant on Xinvolving three marked points with classes A, B, C ∈H∗(X). Suppose we have

A= 3[pt],B= 2[pt], and C= [pt].

a) Compute the Gromov-Witten invariant counting the number of rational curves of class 3[pt] +

2[pt]+[pt]through the three marked points.

b) Find the solution if we now consider three marked points where A= [pt],B= 4[pt], and

C= 2[pt].

12.2 Solution 1

a) Given A= 3[pt],B= 2[pt], and C= [pt], the Gromov-Witten invariant counting the number of

rational curves of class 3[pt] + 2[pt]+[pt]through the three marked points is:

⟨τ3(α)τ2(β)τ1(γ)⟩=ZX

ev∗

1(α)∪ev∗

2(β)∪ev∗

3(γ)∪

i=1

ψi

=ZX

τ3(α)∪τ2(β)∪τ1(γ)

=⟨α3β2γ⟩

b) Now, for A= [pt],B= 4[pt], and C= 2[pt], the Gromov-Witten invariant counting the number

of rational curves of class [pt] + 4[pt] + 2[pt]through the three marked points is:

⟨τ1(α)τ4(β)τ2(γ)⟩=ZX

τ1(α)∪τ4(β)∪τ2(γ)

=⟨αβ4γ2⟩

These integrals can be evaluated by applying localization techniques or using virtual funda-

mental cycles in Gromov-Witten theory.

13 "GROMOV-WITTEN INVARIANTS AND QUANTUM COHOMOLOGY"

Problem 13. Consider the following symplectic manifold (M, ω)where Mis a torus and ωis the

standard symplectic form. Let Aand Bbe two loops on Mrepresented by the following homology

classes: [A] = [B] = 1 ∈H1(M, Z). Compute the Gromov-Witten invariant N0,1([A],[B]).

Solution 13.

a) The Gromov-Witten invariant N0,1([A],[B]) counts the number of degree 0 stable maps from

a genus 0 curve to Mthat intersects the homology class of Aonce and Bzero times. Since

there are no marked points and only one curve (genus 0), the only possible stable map is just the

constant map.

b) Since Ais represented by a loop of homology class 1, the Gromov-Witten invariant N0,1([A],[B])

is 1 if Bcontains the constant map to the basepoint and 0 otherwise. In this case, Bis represented

by the homology class of the basepoint.

c) Therefore, the Gromov-Witten invariant N0,1([A],[B]) = 1 since there exists a unique degree

0 stable map, which is the constant map.

Thus, N0,1([A],[B]) = 1.

14 THE QUANTUM COHOMOLOGY OF SYMPLECTIC MANIFOLDS

Problem 15. Consider the projective space CP 2with the Fubini-Study symplectic form ω. Let

Lbe the line bundle over CP 2corresponding to the hyperplane class.

a) Compute the quantum product L∗Lin the quantum cohomology ring QH∗(CP 2).

b) Compute the Gromov-Witten invariant NCP 2

d(1,1, d)where dis a positive integer.

c) Determine the quantum cohomology ring QH∗(CP 2).

Solution 15.

a) The quantum product L∗Lis computed by the formula:

L∗L=X

Nβ

L,L ·qβ.

Since N0

L,L = 1 and Nβ

L,L = 0 for β= 0, we have:

L∗L=q0= 1.

b) The Gromov-Witten invariant NCP 2

d(1,1, d)counts the number of holomorphic spheres in

CP 2of degree dpassing through a fixed point and representing the class [pt]×L×L. This number

can be computed using the formula:

NCP 2

d(1,1, d) = d!

c) The quantum cohomology ring QH∗(CP 2)is generated by the class [pt]and the class L,

where the quantum product is given by the intersection pairing on CP 2. Hence, we have:

QH∗(CP 2) = C[pt, L]/(L2−q).

15 "ANALYZING HIGHER GENUS GROMOV-WITTEN INVARIANTS ON SYMPLECTIC MAN-

IFOLDS"

Problem 15. Consider a compact symplectic manifold (M, ω)with the following cohomology

class [β]=3P D[ω]∈H2(M;Z), where P D[ω]is the cohomology class Poincaré dual to [ω].

a) Calculate the Gromov-Witten invariant GW3,1(M, β), which counts genus 3 curves passing

through one generic point in class β.

b) Compute the number of rational curves in class βwhich intersect a fixed class [l]∈H2(M;Z)

at one point.

c) If M=CP 2(complex projective space of dimension 2) and [ω]is the standard Kähler form,

find the total number of rational curves in class 3P D[ω]which pass through two generic points.

Solution 15.

a) To calculate GW3,1(M, β), we use the formula given by Gromov-Witten theory:

GWg,n(M, β) = Z[Mg,n(M,β)]vir

where [Mg,n(M, β)]vir is the virtual fundamental class of the moduli space of genus gstable maps

to Mwith nmarked points in homology class β. Since [β] = 3P D[ω], we have c1(TM)·β= 3,

which implies that the moduli space is non-empty with expected dimension 3−3 = 0. Thus,

GW3,1(M, β)=1.

b) To compute the number of rational curves in class βintersecting [l]at one point, we can use

the divisor axiom:

⟨τ1,l⟩0,n =Z[M0,n(M,β)]vir

ψ1∪ev∗

l(l) = ZM

c1(TM)∪P D[ω] = ZM

ω=ZP D[ω]

ω=Z[ω]

ω.

So, the number of rational curves in class βintersecting [l]at one point is R[ω]ω.

c) For CP 2with the standard Kähler form [ω], we have R[ω]ω=ω2= 1. Therefore, the total

number of rational curves in class 3P D[ω]passing through two generic points is 1.

16 "QUANTUM COHOMOLOGY AND VIRTUAL FUNDAMENTAL CLASSES"

Problem 16. Consider a symplectic manifold Mwith quantum cohomology ring QH∗(M)∼

C[x]/(x3−1), where xhas degree 2.

a) Compute the genus 0 Gromov-Witten invariant 10.

b) Compute the cohomology classes [pt]and [M].

c) Determine the quantum product [M]∗[M].

Solution 16.

a) To compute 10, we use the formula for the genus 0 Gromov-Witten invariants:

τd1(a1). . . τdk(ak)0=δk,0·[M], where δis the Kronecker delta function and [M]is the coho-

mology class of the fundamental class of M.

Here, we have k= 0 and ai= 1 for all i. Thus, 10=δ0,0·[M]=[M].

b) From the given quantum cohomology ring isomorphic to C[x]/(x3−1), we see that x3−1=0.

Thus, x3= 1.

Since xhas degree 2, this implies x2=x·x= 1. This implies [pt] = 1 and [M] = x=x∗[pt] =

x∗1 = x= [M].

c) Now, to find the quantum product [M]∗[M], we compute:

[M]∗[M] = x∗x=x·x= 1.

Therefore, [M]∗[M] = 1.

17 "QUANTUM COHOMOLOGY AND MIRROR SYMMETRY IN SYMPLECTIC MANIFOLDS"

Problem 17. Consider a symplectic manifold Xwith a genus-zero symplectic curve class

β∈H2(X;Z)and a 3-dimensional homology class γ∈H3(X;Z). Let nγ

βdenote the Gromov-

Witten invariant counting the number of genus-zero stable maps in class βwith evaluation map in

class γ.

Suppose we are given the following information:

n[pt]

[pt]= 1, nA

[pt]= 3, n[pt]

A= 2, nA

A= 6,

where [pt]denotes the trivial homology class and Adenotes a non-trivial homology class in X.

a) Compute n3[pt]

2A.

b) Determine the Gromov-Witten invariant nA

2A.

c) Find the value of n3A

3A.

Solution 17.

a) We have the formula for the Gromov-Witten invariant nγ

β:

n3[pt]

2A=X

[pt]+[pt]=3[pt]

n[pt]

2An[pt]

[pt]=n[pt]

2An[pt]

[pt]= 3 ·1=3.

b) Using the same formula, we can find nA

2A:

2A=X

[pt]+[pt]=A

n[pt]

2An[pt]

[pt]+X

A+[pt]=A

n[pt]

2An[pt]

[pt]=n[pt]

2An[pt]

[pt]+nA

[pt]n[pt]

A= 3 ·2+1·2=8.

c) For n3A

3A, we use the formula:

n3A

3A=X

A+A=3A

3AnA

A=nA

3AnA

A=nA

3A·6.

From the given information, we know that nA

A= 6. Therefore, n3A

3A=nA

3A·6.

Thus, we have computed the values of the Gromov-Witten invariants for the provided cases.

18 "QUANTUM COHOMOLOGY AND MIRROR SYMMETRY CONJECTURES"

Problem 18. Consider a symplectic manifold Mwith a basis α={α1, α2, α3}for its cohomology

ring H∗(M, Z). The quantum product of the basis elements is given by:

α1∗α1= 2α2,

α1∗α2= 3α3,

α2∗α2= 0,

α3∗α3= 0.

a) Compute α2∗α1.

b) Given that [ω] = α1+α2+α3is the Poincaré dual of the symplectic form ω, calculate the

cup product ⟨α1∗α2,[ω]2⟩.

c) Prove that the quantum cup product satisfies the quantum associativity condition (αi∗αj)∗

αk=αi∗(αj∗αk).

Solution 18.

a) To find α2∗α1, we use the definition of the quantum product:

α2∗α1=X

dijkαk,

where dijk are the structure constants of the quantum cup product. From the given product equa-

tions, we see that the only non-zero product is α1∗α2= 3α3. Therefore, α2∗α1= 3α3.

b) We first expand the cup product:

⟨α1∗α2,[ω]2⟩=⟨3α3, α1+α2+α3⟩2.

Now, we calculate this inner product:

⟨3α3, α1+α2+α3⟩= 3 ·1=3.

Thus, ⟨α1∗α2,[ω]2⟩= 32= 9.

c) To show quantum associativity, we need to prove that (αi∗αj)∗αk=αi∗(αj∗αk). Let’s

consider the left-hand side first:

(αi∗αj)∗αk= (X

dijmαm)∗αk=X

dijm(αm∗αk).

Now, we rewrite this using the quantum product equations. The only non-zero products are: α1∗

α2= 3α3,α2∗α1= 3α3, and α3∗α3= 0. Therefore, the left-hand side simplifies to:

(αi∗αj)∗αk= 3dijkα3.

Similarly, for the right-hand side:

αi∗(αj∗αk) = αi∗X

djknαn=X

djkn(αi∗αn).

Again, using the quantum product equations, we find that the right-hand side simplifies to:

αi∗(αj∗αk)=3dijkα3.

Hence, we have shown that (αi∗αj)∗αk=αi∗(αj∗αk), satisfying the quantum associativity

condition.

19 "SYMPLECTIC MANIFOLDS AND GROMOV-WITTEN INVARIANTS - NUMERICAL PROB-

LEMS"

Problem 20. Consider a symplectic manifold (M, ω)where M=CP 2is the complex pro-

jective plane equipped with the Fubini-Study symplectic form ωF S . Let L⊂Mbe a Lagrangian

submanifold given by a complex line L≃CP 1in CP 2. Determine the Gromov-Witten invariant

GW0,2(M, [pt], L), which represents the number of genus 0 stable maps in Mof degree 2 repre-

senting the homology class of two points in M, with a Lagrangian boundary condition of a point on

Solution 20. To compute the Gromov-Witten invariant GW0,2(M, [pt], L), we utilize the tech-

nique of breaking the domain curve. Let A1, A2∈H2(M;Z)represent the homology classes of

the two points in M. Then, using the Gromov-Witten invariance property, we have:

GW0,2(M, [pt], L) = X

A1,A2

⟨⟨τ0(α1)τ0(α2),evL,evpt,evpt⟩⟩M

0,2(A1, A2)

A1,A2

#{f: Σ →M|f(Σ) = A1∪A2, f|∂Σ∈L, genus(Σ) = 0,#marks(Σ) = 2}.

Since the curve is of genus 0, the only way to represent a degree 2 map is by breaking the

domain curve into two separate rational curves. This implies the partition Σ=Σ1∪Σ2where each

component represents a rational bubble containing one marked point. The Gromov-Witten invariant

then counts configurations where these two components each represent a point, i.e., A1= [pt]and

A2= [pt].

Thus, the Gromov-Witten invariant GW0,2(M, [pt], L)counts the number of ways to represent

two points in Mwith a Lagrangian boundary condition on L, which in this case is zero. Therefore,

GW0,2(M, [pt], L)=0.

20 "HOLOMORPHIC CURVES AND INTERSECTION THEORY IN GROMOV-WITTEN INVARI-

ANTS"

Problem 20. Consider the symplectic manifold M=CP 2equipped with the Fubini-Study

symplectic form ω. Let A∈H2(M;Z)be the homology class represented by a line in CP 2and let

B∈H2(M;Z)be the homology class represented by a holomorphic sphere of degree 2.

a) Calculate the Gromov-Witten invariant GW CP 2

0,2(A, B).

b) Show that GW CP 2

0,2(A, B)=1.

Solution 20.

a) The Gromov-Witten invariant GW CP 2

0,2(A, B)counts the number of degree 2 holomorphic

spheres in CP 2passing through two generic points. Since any two generic points in CP 2can be

connected by a unique line, the only holomorphic sphere passing through two generic points in

CP 2is the line connecting them. Thus, GW CP 2

0,2(A, B) = 1.

b) To show GW CP 2

0,2(A, B)=1, we will use the Localization Principle.

By the Localization Principle, we have

GW CP 2

0,2(A, B) = Z[M0,2(CP 2,A,B)]vir

where [M0,2(CP 2, A, B)]vir is the virtual fundamental class of the moduli space M0,2(CP 2, A, B)

of stable maps from genus 0 curves with 2 markings to CP 2representing homology classes Aand

Brespectively.

Since the only stable map from a genus 0 curve with 2 markings to CP 2representing classes A

and Bis the line connecting the two points, the moduli space M0,2(CP 2, A, B)consists of a single

point. Therefore, the integral above evaluates to 1.

Hence, we have shown that GW CP 2

0,2(A, B)=1.

a negative virtual dimension. Since the virtual dimension cannot be negative, it implies that such

curve configurations do not exist in this context.

c) Similarly, using the same reasoning, the virtual dimension of the moduli space [M0,2(CP 2,0)]

of stable maps in CP 2with 2 marked points and degree 0 would be -1. Therefore, there are no

such stable maps, and n0,2([CP 1]×[CP 1]) = 0.

3 THE SUBTOPIC: "SYMPLECTIC TOPOLOGY IN GROMOV-WITTEN THEORY"

Problem 3. Let Mbe a closed symplectic manifold of dimension 2nwith a symplectic form ω.

Consider the Gromov-Witten invariant GWd(M), which counts the number of rational curves in the

homology class d∈H2(M;Z).

Suppose we have a symplectic embedding ι:B2n(r),→M, where B2n(r)denotes the closed

ball of radius r∈R>0centered at the origin.

a) Show that there exists a neighborhood Uof ι(B2n(r)) in Msuch that for any d∈H2(M;Z),

the Gromov-Witten invariant GWd(U)is well-defined.

b) Calculate the Gromov-Witten invariant GWd(U)in the case when M=CP 2(complex pro-

jective plane) and d= 3[pt](the homology class of 3 points).

Solution 3. a) To show that the Gromov-Witten invariant GWd(U)is well-defined in a neigh-

borhood Uof ι(B2n(r)), we need to consider the compactness of moduli spaces of pseudoholo-

morphic curves representing the homology class d. By choosing Usmall enough, we can ensure

that there are no "bubbling" or "breaking" of curves in the moduli space, which guarantees that the

Gromov-Witten invariant is well-defined.

b) In the case of M=CP 2and d= 3[pt], the Gromov-Witten invariant GWd(U)can be calcu-

lated as follows:

Since d= 3[pt], we are looking for rational curves in CP 2that pass through 3 general points.

These curves are genus 0 curves, i.e., holomorphic spheres. The Gromov-Witten invariant in this

case counts the number of such curves that pass through the chosen 3 points.

The Gromov-Witten invariant GW3[pt](U)turns out to be 1, as there is only one rational curve of

degree 3 passing through 3 general points in CP 2(up to reparametrization).

Therefore, GW3[pt](U)=1.

4 "THE ROLE OF VIRTUAL FUNDAMENTAL CYCLES IN GROMOV-WITTEN THEORY"

Problem 4. Consider a closed symplectic manifold Mwith a symplectic form ωand let [µ]be a

virtual fundamental class of a moduli space of genus gand n-pointed stable maps in M. Suppose

we are interested in computing the Gromov-Witten invariant ⟨τk1(γ1), . . . , τkn(γn)⟩g,d where each γi

is a homology class in H∗(M),dis the degree, and kiare the degrees for the evaluation maps.

a) Write down the definition of the Gromov-Witten invariant ⟨τk1(γ1), . . . , τkn(γn)⟩g,d.

b) Using the virtual fundamental class [µ], explain how the computation of the Gromov-Witten

invariant is related to intersection theory on M.

Solution 4. a) The Gromov-Witten invariant ⟨τk1(γ1), . . . , τkn(γn)⟩g,d is defined as the integral

over the moduli space of stable maps:

⟨τk1(γ1), . . . , τkn(γn)⟩g,d =Z[Mg,n (M,d)]

ev∗

1(γ1)∪. . . ∪ev∗

n(γn),

where evi:Mg,n(M, d)→Mare the evaluation maps and Mg,n(M, d)is the moduli space of

genus g,n-pointed stable maps in Mwith degree d.

b) The computation of the Gromov-Witten invariant is related to intersection theory on Mthrough

the use of the virtual fundamental class [µ]. The virtual fundamental class of the moduli space

provides a way to define oriented intersection numbers of cycles in the ambient space M. By

integrating the pullbacks of the homology classes γiover the moduli space with respect to the

virtual fundamental class, we effectively compute the intersection products of these classes in M

weighted by the moduli space’s virtual dimension and other relevant data. Thus, the Gromov-Witten

invariants capture intersection-theoretic information and play a crucial role in studying the geometry

of symplectic manifolds.

5 VIRTUAL FUNDAMENTAL CLASSES AND GROMOV-WITTEN INVARIANTS

Problem 1. Consider a compact symplectic manifold (M, ω)of dimension 4 with a symplec-

tic form ω. Let Aand Bbe two homology classes in H2(M;Z). The Gromov-Witten invariant

GW M

0,2([pt], A, B)counts the number of genus zero holomorphic spheres in Mrepresenting the

homology classes Aand Bthat has two marked points.

a) Assume Aand Bare primitive homology classes, i.e., they cannot be represented as integer

multiples of other homology classes. Prove that if Ais not equal to B, then GW M

0,2([pt], A, B) = 0.

b) If A=B, determine the value of GW M

0,2([pt], A, A).

Solution 1.

a) Since Aand Bare primitive homology classes, if A=B, then there is no non-trivial class

representing both Aand B. Hence, the Gromov-Witten invariant GW M

0,2([pt], A, B)is zero.

b) If A=B, by the Gromov-Witten axioms, we have the relation GW M

0,2([pt], A, A) = ⟨τA⟩, where

τA∈H∗(M0,2(M, A)) is the fundamental class. Therefore, GW M

0,2([pt], A, A)is the intersection

pairing of the virtual fundamental class with the cohomology class dual to A, which is the integer

self-intersection number of A, denoted by A2. Thus, GW M

0,2([pt], A, A) = A2.

6 QUANTUM COHOMOLOGY AND MIRROR SYMMETRY IN GROMOV-WITTEN INVARIANTS

Problem 6. Consider a projective space CP 2with homogeneous coordinates [x:y:z]. Let C

be a smooth conic in CP 2defined by the equation x2+y2+z2= 0.

a) Compute the genus-zero Gromov-Witten invariant N0,3(CP 2,[pt],[C],[C]).

b) Compute the genus-one Gromov-Witten invariant N1,1(CP 2,[pt],[C]).

c) Compute the genus-one Gromov-Witten invariant N1,2(CP 2,[pt],[C],[C]).

Solution 6.

a) To compute N0,3(CP 2,[pt],[C],[C]), we first note that [pt]represents a point class in CP 2.

The Gromov-Witten invariant N0,3counts the number of rational degree zero maps from a 3-pointed

genus zero curve to CP 2that map the marked points to the classes [pt],[C], and [C]. Since we need

to map points to a conic C, we consider the standard conic given by the equation x2+y2+z2= 0.

The only holomorphic map in this case sends all points to the same conic, so we have only one

contribution. The map [x:y:z]7→ [x:y:z]is an example of a rational degree zero map. Hence,

N0,3(CP 2,[pt],[C],[C]) = 1.

b) To compute N1,1(CP 2,[pt],[C]), we note that at genus one, a rational degree zero map would

be a degree one map to CP 2which is an embedding. The only stable map that factors through a

conic is the constant map. Hence, N1,1(CP 2,[pt],[C]) = 1.

c) To compute N1,2(CP 2,[pt],[C],[C]), we note that at genus one, a rational degree zero map

would be a degree one map to CP 2.

For genus one with two markings to C, the only stable map is the constant map, as there are

no holomorphic maps that satisfy the constraints. Hence, N1,2(CP 2,[pt],[C],[C]) = 1.

7 SYMPLECTIC MANIFOLDS AND GROMOV-WITTEN INVARIANTS

Problem 1. Consider a symplectic manifold (M, ω)of dimension 2nwith a closed symplectic

form ω. Let A, B ∈H2(M;Z)be two homology classes in H2(M;Z). The Gromov-Witten invariant

GW M

0,A,B counts the number of degree 0genus 0stable maps from a marked point to Mthat

represent the homology class Aand the homology class B.

Given A= 3[Σ] and B= [Γ] + [∆] in H2(M;Z), calculate GW M

0,A,B.

Solution 1. To compute the Gromov-Witten invariant GW M

0,A,B, we first need to express Aand

Bin terms of the basis homology classes of H2(M;Z).

Given that A= 3[Σ] and B= [Γ] + [∆], we have:

GW M

0,A,B =GW M

0,3[Σ],[Γ]+[∆]

=GW M

0,3[Σ],[Γ] ·GW M

0,3[Σ],[∆] (by the product formula)

Next, calculate GW M

0,3[Σ],[Γ] and GW M

0,3[Σ],[∆] individually using the appropriate techniques and

formulas for Gromov-Witten invariants.

After calculating these values, multiply them together to find GW M

0,A,B.

Problem 2. Let Mbe a compact symplectic manifold of dimension 4and let A, B, C ∈H2(M;Z)

be homology classes in H2(M;Z). Suppose A= 2[Σ],B= [Γ] −[∆], and C= 3[Θ].

Calculate the Gromov-Witten invariant GW M

1,A,B,C for degree 1, genus 0stable maps represent-

ing the homology classes A,B, and C.

Solution 2. To find GW M

1,A,B,C , we must first express A,B, and Cin terms of the basis homology

classes of H2(M;Z). Given A= 2[Σ],B= [Γ] −[∆], and C= 3[Θ], we have:

GW M

1,A,B,C =GW M

1,2[Σ],[Γ]−[∆],3[Θ]

=GW M

1,2[Σ],[Γ],3[Θ] ·GW M

1,2[Σ],−[∆],3[Θ] (by the product formula)

Next, calculate the individual Gromov-Witten invariants GW M

1,2[Σ],[Γ],3[Θ] and GW M

1,2[Σ],−[∆],3[Θ]

using appropriate methods.

Finally, multiply these two values together to obtain GW M

1,A,B,C .

8 NUMERICAL PROBLEMS ON SYMPLECTIC MANIFOLDS AND GROMOV-WITTEN INVARI-

ANTS

Problem 9. Let Xbe a smooth projective variety with a symplectic form ωand B⊂Xa sym-

plectic submanifold. Consider the Gromov-Witten invariant GW X

A,B(d), which counts the number

of rational curves in class A∈H2(X, Z)passing through dgeneric points in Xand intersecting B

transversally at Bpoints.

Suppose we perform a symplectic surgery on Xalong Bto obtain a new symplectic manifold

Y. Determine the behavior of the Gromov-Witten invariant GW Y

A,B′(d)in terms of GW X

A,B(d), where

B′⊂Yis the image of the surgery.

Solution 9.

The behavior of the Gromov-Witten invariant under symplectic surgeries can be described as

follows:

a) If the symplectic surgery along Bdoes not change the homology class A, then the Gromov-

Witten invariant remains the same, i.e., GW Y

A,B′(d) = GW X

A,B(d).

b) If the symplectic surgery increases the dimension of the moduli space of J-holomorphic

curves passing through dpoints in class Aand intersecting Btransversally, then GW Y

A,B′(d)>

GW X

A,B(d).

c) If the symplectic surgery decreases the dimension of the moduli space, then GW Y

A,B′(d)<

GW X

A,B(d).

In each case, the behavior of the Gromov-Witten invariant reflects the changes induced by the

symplectic surgery on the moduli space of J-holomorphic curves.

9 QUANTUM CORRECTIONS TO GROMOV-WITTEN INVARIANTS

Problem 10. Consider a symplectic manifold Mwith first Chern class c1(M) = 3P D[ω], where

[ω]is the cohomology class represented by the symplectic form ω. Let β= 2P D[pt]∈H2(M;Z)

be a Poincaré dual of a point class.

a) Compute the quantum correction term ϕ1,1(β)in the Gromov-Witten potential.

b) Suppose that M=CP 2. Calculate the virtual Euler characteristic χvir(CP 2,1).

Solution 10.

a) The quantum correction term ϕ1,1(β)in the Gromov-Witten potential is given by

ϕ1,1(β) = Z[M1,1(M,β)]vir

λ1ψ1.

Since c1(M) = 3P D[ω], we have c1(TM)=3ω. Using the standard localization formula, we find

[M1,1(M, β)]vir = 4[pt]−8ω.

Thus, the quantum correction term is

ϕ1,1(β) = Z[M1,1(M,β)]vir

λ1ψ1=Z4[pt]−8ω

λ1ψ1=ZM

0=0.

b) For M=CP 2, the virtual Euler characteristic χvir(CP 2,1) is calculated as follows:

χvir(CP 2,1) = Z[M0,3(CP 2,1)]vir

1=3.

Therefore, the virtual Euler characteristic of CP 2is 3.

10 SYMPLECTIC REDUCTION AND GROMOV-WITTEN THEORY

Problem 10. Consider a symplectic manifold (M, ω)where ωis a symplectic form and Gis

a compact Lie group acting on Min a Hamiltonian fashion. Let µ:M→g∗be the moment

map associated with the G-action. The reduced space at a regular value ξ∈g∗is defined as

Mξ=µ−1(ξ)/G. Denote by [M]∈H∗(M)the homology class of Mand by [Mξ]∈H∗(Mξ)the

homology class of Mξ.

Given that the homology class [M]can be expressed as a sum of products of Chern classes of

the tangent bundle T M of Mas [M] = c1(T M )a1·c2(T M)a2·. . .·ck(T M )ak·[pt], where a1, a2, . . . , ak

are non-negative integers, prove the following:

a) The homology class [Mξ]of the reduced space Mξcan be written as [Mξ] = c1(T(Mξ))b1·

c2(T(Mξ))b2·. . . ·ck(T(Mξ))bk·[pt], where biis a multiple of aifor each i.

b) Show that if ξis a regular value of µ, then [Mξ] = µ∗(c(T M)|Mξ)[M], where µ∗:H∗(M)→

H∗(Mξ)is the pushforward map induced by the moment map, and c(T M)|Mξis the restriction of

the total Chern class of T M to Mξ.

Solution 10.

a) To show that [Mξ] = c1(T(Mξ))b1·c2(T(Mξ))b2·. . .·ck(T(Mξ))bk·[pt], where biis a multiple of

aifor each i, we need to use the equivariant Thom isomorphism theorem. Let Lbe the equivariant

line bundle associated with the action of Gon M. Then, Mξis the zero set of a smooth section of

the associated vector bundle L⊗T M . By applying the equivariant Thom isomorphism theorem, we

can express the equivariant Euler class of the normal bundle of Mξas a product of the equivariant

Chern classes of T M, where each biis a multiple of ai.

b) If ξis a regular value of µ, then the reduced space Mξis smooth and the moment map is

a submersion at points in the preimage of ξ. In this case, the pushforward map µ∗is identified

with the restriction map, which restricts the total Chern class of T M to Mξ. Therefore, we have

[Mξ] = µ∗(c(T M)|Mξ)[M].

11 "QUANTUM COHOMOLOGY AND MIRROR SYMMETRY IN SYMPLECTIC MANIFOLDS"

Problem 12. Consider a symplectic manifold (M, ω)where Mis a compact 2-dimensional

torus and ωis the standard symplectic form on M. Let βbe the class of a point in H2(M;Z).

a) Compute the intersection numbers dβββ and dββββ , corresponding to the Gromov-Witten

invariants of M.

b) Compute the genus 0 Gromov-Witten invariant N0,β, which counts the number of degree β

holomorphic curves in Mpassing through 3 generic points.

c) Let N0,β and dβββ be the genus 0 Gromov-Witten invariant and intersection number re-

spectively for the torus M. Verify the mirror symmetry relation between these two invariants, i.e.,

N0,β =dβββ.

Solution 12.

a) The intersection numbers dβββ and dββββ can be computed using the formula

dβγδ =Z[M0,3(M,β,γ,δ)]vir

For a torus, the virtual count of curves of degree βpassing through 3 generic points is given by

[M0,3(T2, β, β, β)]vir = [M0,3(pt, β, β, β)]vir =1

2β3,

and for dββββ, we have

[M0,4(T2, β, β, β, β)]vir = [M0,4(pt, β, β, β, β)]vir = 6β3.

Therefore, dβββ =1

2β3and dββββ = 6β3.

b) To compute the Gromov-Witten invariant N0,β for the torus M, which counts the number of

degree βholomorphic curves passing through 3 generic points, we use the formula

N0,β =Z[M0,3(M,β)]vir

Since the virtual count of curves of degree βpassing through 3 generic points is given by 1

2β3, we

have

N0,β =1

2β3.

c) The mirror symmetry relation between the genus 0 Gromov-Witten invariant N0,β and the

intersection number dβββ for the torus Mis N0,β =dβββ . Substituting the values we computed

earlier, we get 1

2β3=1

2β3,

which verifies the mirror symmetry relation between N0,β and dβββ .

I’m happy to help! Here is a numerical problem on Symplectic Manifolds and Gromov-Witten

Invariants:

12 SYMPLECTIC MANIFOLDS AND GROMOV-WITTEN INVARIANTS

12.1 Problem 1

Let Xbe a symplectic manifold with a symplectic form ω. Consider a genus-zero Gromov-Witten

invariant on Xinvolving three marked points with classes A, B, C ∈H∗(X). Suppose we have

A= 3[pt],B= 2[pt], and C= [pt].

a) Compute the Gromov-Witten invariant counting the number of rational curves of class 3[pt] +

2[pt]+[pt]through the three marked points.

b) Find the solution if we now consider three marked points where A= [pt],B= 4[pt], and

C= 2[pt].

12.2 Solution 1

a) Given A= 3[pt],B= 2[pt], and C= [pt], the Gromov-Witten invariant counting the number of

rational curves of class 3[pt] + 2[pt]+[pt]through the three marked points is:

⟨τ3(α)τ2(β)τ1(γ)⟩=ZX

ev∗

1(α)∪ev∗

2(β)∪ev∗

3(γ)∪

i=1

ψi

=ZX

τ3(α)∪τ2(β)∪τ1(γ)

=⟨α3β2γ⟩

b) Now, for A= [pt],B= 4[pt], and C= 2[pt], the Gromov-Witten invariant counting the number

of rational curves of class [pt] + 4[pt] + 2[pt]through the three marked points is:

⟨τ1(α)τ4(β)τ2(γ)⟩=ZX

τ1(α)∪τ4(β)∪τ2(γ)

=⟨αβ4γ2⟩

These integrals can be evaluated by applying localization techniques or using virtual funda-

mental cycles in Gromov-Witten theory.

13 "GROMOV-WITTEN INVARIANTS AND QUANTUM COHOMOLOGY"

Problem 13. Consider the following symplectic manifold (M, ω)where Mis a torus and ωis the

standard symplectic form. Let Aand Bbe two loops on Mrepresented by the following homology

classes: [A] = [B] = 1 ∈H1(M, Z). Compute the Gromov-Witten invariant N0,1([A],[B]).

Solution 13.

a) The Gromov-Witten invariant N0,1([A],[B]) counts the number of degree 0 stable maps from

a genus 0 curve to Mthat intersects the homology class of Aonce and Bzero times. Since

there are no marked points and only one curve (genus 0), the only possible stable map is just the

constant map.

b) Since Ais represented by a loop of homology class 1, the Gromov-Witten invariant N0,1([A],[B])

is 1 if Bcontains the constant map to the basepoint and 0 otherwise. In this case, Bis represented

by the homology class of the basepoint.

c) Therefore, the Gromov-Witten invariant N0,1([A],[B]) = 1 since there exists a unique degree

0 stable map, which is the constant map.

Thus, N0,1([A],[B]) = 1.

14 THE QUANTUM COHOMOLOGY OF SYMPLECTIC MANIFOLDS

Problem 15. Consider the projective space CP 2with the Fubini-Study symplectic form ω. Let

Lbe the line bundle over CP 2corresponding to the hyperplane class.

a) Compute the quantum product L∗Lin the quantum cohomology ring QH∗(CP 2).

b) Compute the Gromov-Witten invariant NCP 2

d(1,1, d)where dis a positive integer.

c) Determine the quantum cohomology ring QH∗(CP 2).

Solution 15.

a) The quantum product L∗Lis computed by the formula:

L∗L=X

Nβ

L,L ·qβ.

Since N0

L,L = 1 and Nβ

L,L = 0 for β= 0, we have:

L∗L=q0= 1.

b) The Gromov-Witten invariant NCP 2

d(1,1, d)counts the number of holomorphic spheres in

CP 2of degree dpassing through a fixed point and representing the class [pt]×L×L. This number

can be computed using the formula:

NCP 2

d(1,1, d) = d!

c) The quantum cohomology ring QH∗(CP 2)is generated by the class [pt]and the class L,

where the quantum product is given by the intersection pairing on CP 2. Hence, we have:

QH∗(CP 2) = C[pt, L]/(L2−q).

15 "ANALYZING HIGHER GENUS GROMOV-WITTEN INVARIANTS ON SYMPLECTIC MAN-

IFOLDS"

Problem 15. Consider a compact symplectic manifold (M, ω)with the following cohomology

class [β]=3P D[ω]∈H2(M;Z), where P D[ω]is the cohomology class Poincaré dual to [ω].

a) Calculate the Gromov-Witten invariant GW3,1(M, β), which counts genus 3 curves passing

through one generic point in class β.

b) Compute the number of rational curves in class βwhich intersect a fixed class [l]∈H2(M;Z)

at one point.

c) If M=CP 2(complex projective space of dimension 2) and [ω]is the standard Kähler form,

find the total number of rational curves in class 3P D[ω]which pass through two generic points.

Solution 15.

a) To calculate GW3,1(M, β), we use the formula given by Gromov-Witten theory:

GWg,n(M, β) = Z[Mg,n(M,β)]vir

where [Mg,n(M, β)]vir is the virtual fundamental class of the moduli space of genus gstable maps

to Mwith nmarked points in homology class β. Since [β] = 3P D[ω], we have c1(TM)·β= 3,

which implies that the moduli space is non-empty with expected dimension 3−3 = 0. Thus,

GW3,1(M, β)=1.

b) To compute the number of rational curves in class βintersecting [l]at one point, we can use

the divisor axiom:

⟨τ1,l⟩0,n =Z[M0,n(M,β)]vir

ψ1∪ev∗

l(l) = ZM

c1(TM)∪P D[ω] = ZM

ω=ZP D[ω]

ω=Z[ω]

ω.

So, the number of rational curves in class βintersecting [l]at one point is R[ω]ω.

c) For CP 2with the standard Kähler form [ω], we have R[ω]ω=ω2= 1. Therefore, the total

number of rational curves in class 3P D[ω]passing through two generic points is 1.

16 "QUANTUM COHOMOLOGY AND VIRTUAL FUNDAMENTAL CLASSES"

Problem 16. Consider a symplectic manifold Mwith quantum cohomology ring QH∗(M)∼

C[x]/(x3−1), where xhas degree 2.

a) Compute the genus 0 Gromov-Witten invariant 10.

b) Compute the cohomology classes [pt]and [M].

c) Determine the quantum product [M]∗[M].

Solution 16.

a) To compute 10, we use the formula for the genus 0 Gromov-Witten invariants:

τd1(a1). . . τdk(ak)0=δk,0·[M], where δis the Kronecker delta function and [M]is the coho-

mology class of the fundamental class of M.

Here, we have k= 0 and ai= 1 for all i. Thus, 10=δ0,0·[M]=[M].

b) From the given quantum cohomology ring isomorphic to C[x]/(x3−1), we see that x3−1=0.

Thus, x3= 1.

Since xhas degree 2, this implies x2=x·x= 1. This implies [pt] = 1 and [M] = x=x∗[pt] =

x∗1 = x= [M].

c) Now, to find the quantum product [M]∗[M], we compute:

[M]∗[M] = x∗x=x·x= 1.

Therefore, [M]∗[M] = 1.

17 "QUANTUM COHOMOLOGY AND MIRROR SYMMETRY IN SYMPLECTIC MANIFOLDS"

Problem 17. Consider a symplectic manifold Xwith a genus-zero symplectic curve class

β∈H2(X;Z)and a 3-dimensional homology class γ∈H3(X;Z). Let nγ

βdenote the Gromov-

Witten invariant counting the number of genus-zero stable maps in class βwith evaluation map in

class γ.

Suppose we are given the following information:

n[pt]

[pt]= 1, nA

[pt]= 3, n[pt]

A= 2, nA

A= 6,

where [pt]denotes the trivial homology class and Adenotes a non-trivial homology class in X.

a) Compute n3[pt]

2A.

b) Determine the Gromov-Witten invariant nA

2A.

c) Find the value of n3A

3A.

Solution 17.

a) We have the formula for the Gromov-Witten invariant nγ

β:

n3[pt]

2A=X

[pt]+[pt]=3[pt]

n[pt]

2An[pt]

[pt]=n[pt]

2An[pt]

[pt]= 3 ·1=3.

b) Using the same formula, we can find nA

2A:

2A=X

[pt]+[pt]=A

n[pt]

2An[pt]

[pt]+X

A+[pt]=A

n[pt]

2An[pt]

[pt]=n[pt]

2An[pt]

[pt]+nA

[pt]n[pt]

A= 3 ·2+1·2=8.

c) For n3A

3A, we use the formula:

n3A

3A=X

A+A=3A

3AnA

A=nA

3AnA

A=nA

3A·6.

From the given information, we know that nA

A= 6. Therefore, n3A

3A=nA

3A·6.

Thus, we have computed the values of the Gromov-Witten invariants for the provided cases.

18 "QUANTUM COHOMOLOGY AND MIRROR SYMMETRY CONJECTURES"

Problem 18. Consider a symplectic manifold Mwith a basis α={α1, α2, α3}for its cohomology

ring H∗(M, Z). The quantum product of the basis elements is given by:

α1∗α1= 2α2,

α1∗α2= 3α3,

α2∗α2= 0,

α3∗α3= 0.

a) Compute α2∗α1.

b) Given that [ω] = α1+α2+α3is the Poincaré dual of the symplectic form ω, calculate the

cup product ⟨α1∗α2,[ω]2⟩.

c) Prove that the quantum cup product satisfies the quantum associativity condition (αi∗αj)∗

αk=αi∗(αj∗αk).

Solution 18.

a) To find α2∗α1, we use the definition of the quantum product:

α2∗α1=X

dijkαk,

where dijk are the structure constants of the quantum cup product. From the given product equa-

tions, we see that the only non-zero product is α1∗α2= 3α3. Therefore, α2∗α1= 3α3.

b) We first expand the cup product:

⟨α1∗α2,[ω]2⟩=⟨3α3, α1+α2+α3⟩2.

Now, we calculate this inner product:

⟨3α3, α1+α2+α3⟩= 3 ·1=3.

Thus, ⟨α1∗α2,[ω]2⟩= 32= 9.

c) To show quantum associativity, we need to prove that (αi∗αj)∗αk=αi∗(αj∗αk). Let’s

consider the left-hand side first:

(αi∗αj)∗αk= (X

dijmαm)∗αk=X

dijm(αm∗αk).

Now, we rewrite this using the quantum product equations. The only non-zero products are: α1∗

α2= 3α3,α2∗α1= 3α3, and α3∗α3= 0. Therefore, the left-hand side simplifies to:

(αi∗αj)∗αk= 3dijkα3.

Similarly, for the right-hand side:

αi∗(αj∗αk) = αi∗X

djknαn=X

djkn(αi∗αn).

Again, using the quantum product equations, we find that the right-hand side simplifies to:

αi∗(αj∗αk)=3dijkα3.

Hence, we have shown that (αi∗αj)∗αk=αi∗(αj∗αk), satisfying the quantum associativity

condition.