1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

246 A. GIVENTAL, A TUTORIAL ON QUANTUM COHOMOLOGY

Exercise. Define the genus 0 potential of X as the following formal function of q
and t E H(X):
00 1
F(t, q) =LIL qd(t, t, ... , t)d.
k=O k. d
Solutions to the following exercises can be obtained by slight modification of the
arguments used in the proof of the WDVV, string and divisor equations.
(a) Express the CW-invariants (a 1 , ... , ak)d as Taylor coefficients of F.
(b) On the space H (X), define the quantum cup-product ct depending on t
by

(a Ct b, c) := 8aEM3cF

where BvF means the directional derivative of F as a function of t in the direction
of the vector v E H*(X). Prove that ct provides the cohomology space with the
structure of commutative associative Frobenius algebra with unity 1 at least if X is
a homogeneous Kahler space. Find the degrees of the parameters t , q which make
the quantum cup-product graded.

(c) Show that the connection 'Vn :=lid-"L.a(<Po:ct)/\ (it can be understood as

a connection on the tangent bundle of the ma nifold H *(X) ) is flat for any Ii=/=-0.

(d) Introduce the potential

Saf3 := ~ L.., k!^1 L.., """' q d (¢a, t, ... , t, Ii_ ¢!3 c(k+Z) )d

k=O d

for the gravitational descendents. Prove that (Saf3) is a fundamental solution matrix
for the differential system 'V nS = 0 (at least in the case of homogeneous Kahler
spaces).
( e) Prove the following generalization of the divisor equation: for our basis
(P1, ... , Pr) in H^2 (X) and any a, b, c
a

8p;8a8b8cF =qi oqi 8a8b8cF.

Analyze the relation between the q-deformation c of the cup-product introduced at
the beginning of the section 2 and the t-deformation ct.
( f) Find generalizations of WDVV, string and divisor equations to gravitational
descendents. Show that all the genus 0 descendents (T^1 , ... , Tk)d are determined by
the potential F. Introduce the potential

:F(T) -- """' L.., k! 2_ """' L.., q d (T(c (1) ), ... , T(c (k) ))d

k d
for genus 0 gravitational descendents and try to describe the procedure expressing
:F in terms of F (we refer to B. Dubrovin, The geometry of 2D topological field
theory for the answer).
(g) Introduce higher genus analogies :F 9 of the potential :F and find the higher
genus versions of the string and divisor equations.