Weighted Hurwitz Numbers, τ -Functions, and Matrix Integrals
87
Then τ (G c,d ,β)
X
becomes a ratio of Wronskian determinants
τ
(G c,d ,β)
X
= γ n
n
i=1
x
n−1
i
det
(φ
(c,d,β)
n
) (i−1) (e y j )
1≤i,j,≤n
Δ(e y )
.
(39)
Matrix Integral Representation of τ (G,β) ([X]): Rational Case
It follows [2] that
τ
(G c,d ,β) (
X
) =
β
1
2 n(n−1) (
n
i=1 x
n−1
i
)Δ(ln(x))
(
n
i=1 i!)Δ(x)
Z dμ (c,d,β,n) (X),
(40)
where
Z dμ (c,d,β,n) (X) =
M∈Nor
n×n
Cn
dμ (c,d,β,n) (M)e
trY M
(41)
and
dμ (c,d,β,n) (M) := (Δ(ζ )
2 det(A
(c,d,β)
n
(M))dμ 0 (U )
n
j =1
dζ i
is a conjugation invariant measure on the space of normal matrices
M = UZU
†
∈ Nor
n×n
C n
, U ∈ U(n), Z = diag(ζ 1 , . . . , ζ n )
(42)
with eigenvalues ζ i ∈ C supported on the contour C n .
Wronskian Representation: Quantum Case
For quantum weight generating functions G = H q , and any n ∈ N + , let
φ
(H q ,β)
n
(e
y ) =
C n
A
(c,d,β)
n
(s)e
ys ds,
A H q ,n (z) := (−β)
1−n Γ (1 − n − z)
∞
m=0
(−βq
m )
−z Γ (−β −1 q −m )
Γ (z − β −1 q −m )
.
Define the diagonal matrix Y = diag(y 1 , . . . y n )
X = e
Y , Y = ln(X), x i = e
y i , i = 1, . . . , n,
(43)