We will assume basic knowledge of Gromov-Witten theory. For more information, consult the relevant chapter in this volume. We’ll confine our discussion to the con- crete example of X :=P2. Define M :=SpecC[[y0,y1,y2]]. Let Ti be a positive
generator ofH2i(P2,Z)and let
γ:=y0T0+y1T1+y2T2
With this data, we are able to define theGromov-Witten potential ofP2.
Φ:= ∞
∑
k=0β∈H∑
2(X,Z) 1 k!hγ ki 0,β.This function encodes much of the enumerative information ofP2. Define a constant metricgonM with
g(∂yi,∂yj):= Z
P2 Ti∪Tj
and the connection∇given by the flat sections∂yi. Define a product structure on the
tangent bundle ofM given by
∂yi∗∂yj :=
∑
a,l(∂yi∂yj∂yaΦ)gal∂yl.
This data defines aFrobenius manifold. For much more on these objects, see [35]. Identifying Ti with ∂yi, one can think of ∗ as giving a product structure on
H∗(P2,C[[y0,y1,y2]]). This is known as thebig quantum cohomology ring. The A-
model data encoded in this manifold can be arranged into a function that will arise naturally on the other side of the mirror. To define this function, we’ll need a slight upgrade of the Gromov-Witten invariant, known as thedescendentGromov-Witten invariant.
Definition 7.1 (Descendent Gromov-Witten invariants).Forαi∈H∗(X,C), de- fine hψj1α1, . . .ψjnαni g,β := Z [M¯g,n(X,β)]vir ψ1j1∪. . .∪ψnjn∪ev∗(α1× · · · ×αn).
Here we’ve attached a natural line bundle Li to M¯g,n(X,β) associated to each
marked point xi. The fiber of Li at a point [(C,x1, . . . ,xn)] is the cotangent line
mxi/m2
xi, wheremxi⊆OC,xiis the maximal ideal. Thenψi:=c1(Li)∈H2(M¯g,n(X,β),Q).
Definition 7.2 (Givental’s J-function forP2).JP2 :M×C×→H∗(P2,C)is de- fined as follows: JP2(y0,y1,y2,h¯):=e y0T0+y1T1 ¯ h ∪ T0+ 2
∑
i=0 y2h¯−1δ2,i∑
d≥1ν∑
≥0 hT3d+i−2−ν 2 ,ψ νT 2−ii0,dh¯−(ν+2)edy1 y3d+i−2−ν 2 (3d+i−2−ν)! ! Ti !We can define functionsJi:M×C×→H2i(P2,C)by the decomposition ofJ: JP2 = n
∑
i=0 JiTi7.2 B model
Here we follow the summary of Barannikov’s results [4] as given in [15]. The mirror ofP2is the Landau-Ginzburg model(Xˆ,W), where ˆX :=V(x0x1x2−1)⊆
SpecC[x0,x1,x2]andW=x0+x1+x2.
We consider the universal unfolding of W parametrized by the moduli space SpecfC[[t0,t1,t2]] Wt:= 2
∑
i=0 Witi,and the local systemRonM×C×whose fiber at a point(t,h¯)is the relative ho- mology groupHn(Xˆ,Re(Wt/h¯)0). With this setup, Barannikov uses semi-infinite
variation of Hodge parameters to show the following result. See Chapter 2 of [16] for a discussion of how these structures arise in our particular example. First, there is a unique choice of the following data:
• A (multi-valued) basis of sections ofR,Ξ0,Ξ1,Ξ2, withΞi uniquely defined
moduloΞ0, . . . ,Ξi−1.
• A sectionsofR∨⊗COM×Cdefined by integration of a family of holomorphic forms on ˆX×M×C×of the form
eWt/h¯fdlogx
1∧dlogx2
where ¯his the coordinate onCand f is a regular function on ˆX×M×C×with f|Xˆ×{0}×C× =1 and which extends to a regular function on ˆX×M ×(C×∪
• The monodromy associated with ¯h→he¯ 2πi inR is given, in the constructed
basis, by exp(6πiN), where
N= 0 1 0 0 0 1 0 0 0
• A fiber ofR∨ is identified with the ring C[α]/(α3), with αi dual toΞi. The
selected section sof R∨⊗OM×C× gives us an element of each fiber ofR ∨, which we write as s(t,¯h) = 2
∑
i=0 αi Z Ξi eWt/¯hfdlogx 1∧dlogx2We require that we can write
s(t,h¯) =h¯−(3α) 2
∑
i=0
φi(t,h¯)(αh¯)i
for functionsφisatisfying
φi(t,h¯) =δ0,i+
∞
∑
j=1
φi,j(t)h¯−j
for 0≤i≤2. These conditions place a restriction on the function f. In the above,
¯ h−3α= 2
∑
i=0 (3)i i! (−log ¯h) i αi,which absorbs the multi-valuedness of the integrals.
As a result of these conditions, if we setyi(t) =φi,1(t), the functionsyiform a set of
coordinates onM, limh¯→∞h¯
i
φi(0,h¯) =δ0,i, and we are able to state the following: Proposition 7.3 (Mirror symmetry forP2).Given the above setup, on theCvector spaceC[[y0,y1,y2,h¯−1]],
Ji=φi
See [4] for the part of the statement not involving descendent invariants, and [26] for a more direct proof. The functionsφi,t(t)can be thought of as specifying a new
set of coordinates on the moduli space; it is this change of coordinates that gives the isomorphism of the B-model Frobenius manifold with that arising in the A-model. In Barannikov’s formulation, this change of coordinates is difficult to make explicit and not immediately meaningful. We will see that Gross’s tropical methods make the transition very natural and explicit, providing a tropical interpretation of mirror symmetry.
7.3 Tropical A-model
The story here is the relatively long and extensive history of the tropical computation of Gromov-Witten invariants. See Section 5. It’s important to note that not all of the invariants appearing in theJfunction havea prioritropical interpretations. In particular, tropical versions of descendent invariants of the typehψνT
i,T2, . . . ,T2i0,d
are, fori6=2, a result of the mirror symmetry construction outlined here. The case wherei=2 was previously treated by Markwig and Rau [36].