260 A. GIVENTAL, A TUTORIAL ON QUANTUM COHOMOLOGYTv corresponding to the vertices v of the lattice: Y< = exp(Tv+ -Tv_) where v+ and
v_ are respectively the head and the tail of the edge E. In particular, the variables
Tv corresponding to all under-diagonal vertices form a coordinate system on the
covering of the complex torus Yq. We put
Wq = AvdTvand define f q as the restriction to Yq of the functionf = L Y< = L eTv+(<)-Tv_(<).
all edges ETheorem 5. 1.^1 Complex oscillating integrals
I = r efq/nWq
lrcYqsatisfy the differential equations D 1 I =. .. = Dr+il = 0 where Di(!iq8/8q,q) are
the quantum conservation laws of the quantum Toda lattice associated with the
group SLn+l·
Corollary 5 .2. The family (Yq, fq) generates in T* B the invariant Lagrangian
variety D 1 (p,q) = ... = Dr+ 1 (p,q) = 0 of the classical Toda lattice.Exercise. Check that in the case r = 1 the theorem agrees with t he example of
the mirror partner for CP^1.
Even the corollary is not quite obvious. We will prove it by induction on
the number of diagonals in our 2-dimensional lattice. Let us recall that that the
operator D := ,>..r+l+D 1 >..r + ... +Dr+l is the characteristic polynomial of the matrix
introduced in the section 3. Denote t 0 , ti, ... tr the vertex variables Tv corresponding
to the diagonal vertices. Since of /ati =Vi - Ui+l and qi= UiVi, we need to prove
that the characteristic polynomial of the following matrix equals ,>..n+l at critical
points of fq:
-U1 U1V1 0J
-1 V1 - U2 U2V2 0
Ar+1 =^0 -1 V2 - U3 U3V3
0 - 1
The matrix Ar+l factors into the product UV of the following square matrices:
U1^0 - 1 V1 0(^1) U2 0 0 - 1 V2 0
U=
(^0 1) U3 0
,V=
0 0 -1 V3 0
(^0 1) Ur 0 0 - 1 Vr
0 1 0 0 - 1
(^1) See A. Givental, Stationary phase integrals, quantum Toda lattices, fiag manifolds and the mirror
conjecture.