Weighted Hurwitz Numbers, τ -Functions, and Matrix Integrals
79
G(z) =
∞
i=1
(1 + zc i ) = 1 +
∞
j =1
g j z
j
G(z) =
∞
i=1
(1 − zc i )
−1
= 1 +
∞
j =1
g j z
j .
(4)
The weight for a branched covering with ramification profiles (μ (1) , . . . , μ (k) ) is
defined to be:
W G (μ
(1) , . . . , μ
(k) ) :=
1
k!
σ ∈S k
1i 1 <··· c
∗ (μ (1) )
i σ (1)
· · · c
∗ (μ (k) )
i σ (k)
W
G (μ
(1) , . . . , μ
(k) ) :=
(−1)
k
i=1 ∗ (μ (i) )+k
k!
σ ∈S k
1i 1 ···i k
c
∗ (μ (1) )
i σ (1)
· · · c
∗ (μ (k) )
i σ (k)
.
(5)
Weighted double Hurwitz numbers H d
G (μ, ν), H d
G
(μ, ν) for n-sheeted branched
coverings of the Riemann sphere having a pair of unweighted branch points
(Q 0 , Q ∞ ), with ramification profiles of type (μ, ν), and k additional weighted
branch points (Q 1 , . . . , Q k ) with ramification profiles (μ (1) , . . . , μ (k) ) are
defined as:
H
d
G (μ, ν) :=
d
k=1
μ (1) ,...μ (k)
k
i=1 ∗ (μ (i) )=d
W G (μ
(1) , . . . , μ
(k) )H (μ
(1) , . . . , μ
(k) , μ, ν),
H
d
G
(μ, ν) :=
d
k=1
μ (1) ,...μ (k)
k
i=1 ∗ (μ (i) )=d
W
G (μ
(1) , . . . , μ
(k) )H (μ
(1) , . . . , μ
(k) , μ, ν),
where
denotes the sum over all partitions other than the cycle type of the identity
element (1) n . If Q ∞ is not a branch point; i.e. ν = (1) n , we have a weighted single
Hurwitz number
H
d
G (μ) := H
d
G (μ, (1)
n ).
(6)
Two cases of particular interest are: rational weight generating functions:
G c,d (z) :=
L
l=1 (1 + c l z)
M
m=1 (1 − d m z)
(7)
and quantum weight generating function (quantum exponential):