88
J. Harnad
Then τ (H q ,β)
X
becomes a ratio of Wronskian determinants
τ
(H q ,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 )
.
(44)
Matrix Integral Representation of τ (G,β) ([X]): Quantum Case
It similarly follows [7] that
τ
(H q ,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 q (ln(X)),
(45)
where Z dμ (q,n) (X) =
M∈Nor
n×n
Cn
dμ (q,n) (M)e
trY M ,
and dμ (q,n) (M) := (Δ(ζ )
2 det(A H q ,n (M))
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 )
(46)
with eigenvalues ζ i ∈ C supported on the contour C n .
Acknowledgments This work was partially supported by the Natural Sciences and Engineering
Research Council of Canada (NSERC) and the Fonds de recherche du Québec, Nature et
technologies (FRQNT).
References
1. A. Alexandrov, G. Chapuy, B. Eynard, J. Harnad, Weighted Hurwitz numbers and topological
recursion: an overview. J. Math. Phys. 59(081102), 1–20 (2018)
2. M. Bertola, J. Harnad, Rationally weighted Hurwitz numbers, Meijer G-functions and matrix
integrals. J. Math. Phys. 60(103504), (2019)
3. M. Guay-Paquet, J. Harnad, 2D Toda τ -functions as combinatorial generating functions. Lett.
Math. Phys. 105, 827–852 (2015)
4. M. Guay-Paquet, J. Harnad, Generating functions for weighted Hurwitz numbers. J. Math. Phys.
58, 083503 (2017)
5. J. Harnad, Weighted Hurwitz numbers and hypergeometric τ -functions: an overview, in AMS
Proceedings of Symposia in Pure Mathematics, vol. 93, 289–333 (2016)
6. J. Harnad, A.Yu. Orlov, Hypergeometric τ -functions, Hurwitz numbers and enumeration of
paths. Commun. Math. Phys. 338, 267–284 (2015)
J. Harnad
Then τ (H q ,β)
X
becomes a ratio of Wronskian determinants
τ
(H q ,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 )
.
(44)
Matrix Integral Representation of τ (G,β) ([X]): Quantum Case
It similarly follows [7] that
τ
(H q ,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 q (ln(X)),
(45)
where Z dμ (q,n) (X) =
M∈Nor
n×n
Cn
dμ (q,n) (M)e
trY M ,
and dμ (q,n) (M) := (Δ(ζ )
2 det(A H q ,n (M))
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 )
(46)
with eigenvalues ζ i ∈ C supported on the contour C n .
Acknowledgments This work was partially supported by the Natural Sciences and Engineering
Research Council of Canada (NSERC) and the Fonds de recherche du Québec, Nature et
technologies (FRQNT).
References
1. A. Alexandrov, G. Chapuy, B. Eynard, J. Harnad, Weighted Hurwitz numbers and topological
recursion: an overview. J. Math. Phys. 59(081102), 1–20 (2018)
2. M. Bertola, J. Harnad, Rationally weighted Hurwitz numbers, Meijer G-functions and matrix
integrals. J. Math. Phys. 60(103504), (2019)
3. M. Guay-Paquet, J. Harnad, 2D Toda τ -functions as combinatorial generating functions. Lett.
Math. Phys. 105, 827–852 (2015)
4. M. Guay-Paquet, J. Harnad, Generating functions for weighted Hurwitz numbers. J. Math. Phys.
58, 083503 (2017)
5. J. Harnad, Weighted Hurwitz numbers and hypergeometric τ -functions: an overview, in AMS
Proceedings of Symposia in Pure Mathematics, vol. 93, 289–333 (2016)
6. J. Harnad, A.Yu. Orlov, Hypergeometric τ -functions, Hurwitz numbers and enumeration of
paths. Commun. Math. Phys. 338, 267–284 (2015)