Lecture 5. Toda Lattices and the Mirror Conjecture
The example of the mirror partner for CP^1 can b e generalized to the manifold
x of complete flags in cr+l as follows.
Consider the following "2-dimensional Toda lattice" with ( r+ 1) ( r+ 2) / 2 verticesand r(r + 1) edges:
U1 l V1
- ~ •
l U 2 l V2
- ~ • ~ •
l l U 3 l
l l Ur l Vr
- ~ • ~ • ~ •
For each edge E of the lattice we introduce a complex variable y,. For each 1 x 1-
square
Ya. ~.
y'Y l l Yf3
. ~.
Y6
we impose the "commutativity" relation YaY/3 = Y'YY6· These relations determine
the variety Y of complex dimension r + r(r + 1)/2 in the space with coordinates
y,. Using the notation u i , Vi (as shown on the diagram) for the variables y, cor-
responding to the edges next to the d iagonal, we fiber Y over the space B with
coordiriates q1, ... , qr:
For q 1 ... qr i- 0 all the relations together mean that all y, are non-zero and that
their logarithms satisfy the Kirchhoff law: t he voltage drops ln y, accumulate to
0 over a closed contour, and ln qi determine the voltage drop between neighbor
diagonal vertices. Thus one can express y , via the vertex "potentials" - variables
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