82
J. Harnad
τ
(G,β) (t, s) =
∞
d=0
μ,ν,
|μ|=|ν|
β
|μ|+d H
d
G (μ, ν)p μ (t)p ν (s).
(17)
It is thus a generating function for the numbers H d
G (μ, ν) of weighted n-fold
branched coverings of the sphere, with a pair of specified branch points having
ramification profiles (μ, ν) and genus given by the Riemann–Hurwitz formula
2 − 2g = + (ν) − d, d =
k
i=1
∗ (μ
(i) ).
(18)
Corollary 1 (Hypergeometric KP τ-Functions as Generating Functions for
Weighted Single Hurwitz Numbers) Set: s = β −1 t 0 := (β −1 , 0, 0, . . . ).
Then the series
τ
(G,β) (t, β
−1 t 0 ) := τ
(G,β) (t) =
λ
(h(λ))
−1 r
(G,β)
λ
s λ (t)
=
∞
d=0
μ
β
d H
d
G (μ)p μ (t)
is a KP τ -function which is a generating function for weighted single numbers
H d
G (μ) for |μ|-fold branched coverings of the sphere, with a branch point having
ramification profile (μ) at Q 0 and genus given by the Riemann–Hurwitz formula.
2 − 2g = |μ| + − d.
(19)
3 Wronskian and Matrix Integral Representation
of τ (G,β) ([X])
In [2, 7] new matrix integral representations were derived for the τ -functions
that serve as generating functions for rationally and quantum weighted Hurwitz
numbers. The main result is that, using Laurent series and Mellin-Barnes integral
representations of the adapted bases for the respective elements of the infinite
Grassmannian corresponding to these cases, the τ -functions may be expressed as
Wronskian determinants or as matrix integrals.
J. Harnad
τ
(G,β) (t, s) =
∞
d=0
μ,ν,
|μ|=|ν|
β
|μ|+d H
d
G (μ, ν)p μ (t)p ν (s).
(17)
It is thus a generating function for the numbers H d
G (μ, ν) of weighted n-fold
branched coverings of the sphere, with a pair of specified branch points having
ramification profiles (μ, ν) and genus given by the Riemann–Hurwitz formula
2 − 2g = + (ν) − d, d =
k
i=1
∗ (μ
(i) ).
(18)
Corollary 1 (Hypergeometric KP τ-Functions as Generating Functions for
Weighted Single Hurwitz Numbers) Set: s = β −1 t 0 := (β −1 , 0, 0, . . . ).
Then the series
τ
(G,β) (t, β
−1 t 0 ) := τ
(G,β) (t) =
λ
(h(λ))
−1 r
(G,β)
λ
s λ (t)
=
∞
d=0
μ
β
d H
d
G (μ)p μ (t)
is a KP τ -function which is a generating function for weighted single numbers
H d
G (μ) for |μ|-fold branched coverings of the sphere, with a branch point having
ramification profile (μ) at Q 0 and genus given by the Riemann–Hurwitz formula.
2 − 2g = |μ| + − d.
(19)
3 Wronskian and Matrix Integral Representation
of τ (G,β) ([X])
In [2, 7] new matrix integral representations were derived for the τ -functions
that serve as generating functions for rationally and quantum weighted Hurwitz
numbers. The main result is that, using Laurent series and Mellin-Barnes integral
representations of the adapted bases for the respective elements of the infinite
Grassmannian corresponding to these cases, the τ -functions may be expressed as
Wronskian determinants or as matrix integrals.