Calabi-Yau di¤erential equations with a parameter. Gert Almkvist Introduction In this paper we give some examples of Calabi-Yau di¤erential equations whose coe¢ cients depend on a parameter a: We start with De…nition: A Calabi-Yau di¤erential equation is a 4-th order di¤erential equation with rational coe¢ cients y (4) + a3 (x)y 000 + a2 (x)y 00 + a1 (x)y 0 + a0 (x)y = 0 satisfying the following conditions. 1. It is MUM (Maximal Unipotent Monodromy), i.e. the indicial equation at x = 0 has zero as a root of order 4. It means that there is a Frobenius solution of the following form y0 = 1 + A1 x + A2 x2 + ::: y1 = y0 log(x) + B1 x + B2 x2 + :: 1 y0 log2 (x) + (B1 x + B2 x2 + :::) log(x) + C1 x + C2 x2 + ::: 2 1 1 y3 = y0 log3 (x)+ (B1 x+B2 x2 +:::) log2 (x)+(C1 x+C2 x2 +:::) log(x)+D1 x+D2 x2 +::: 6 2 It is very useful that Maple’s "formal_sol" produces the four solutions in exactly this form (though labelled 1 4 ) 2. The coe¢ cents of the equation satisfy the identity y2 =
a1 =
1 a2 a3 2
1 3 a + a02 8 3
3 a3 a03 4
1 00 a 2 3
3. Let t = y1 =y0 : Then q = exp(t) = x + c2 x2 + ::: can be solved x = x(q) = q
c2 q 2 + ::::
which is called the "mirror map". We also construct the "Yukawa coupling" de…ned by d 2 y2 K(q) = 2 ( ) dt y0 This can be expanded in a Lambert series K(q) = 1 +
1 X d=1
1
nd
d3 q d 1 qd