Toda equations for Hurwitz numbers

We consider ramified coverings of P^1 with arbitrary ramification type over 0 and infinity and simple ramifications elsewhere and prove that the generating function for the numbers of such coverings is a tau-function for the Toda lattice hierarchy of Ueno and Takasaki.

Note:

10 pages

References(16)

Figures(0)

[1]

The character of the infinite wedge representation

S. Bloch
,
A. Okounkov

- Adv.Math. 149 (2000) 1-60
•
e-Print:
- alg-geom/9712009

[2]

Mirror symmetry and elliptic curves, The Moduli Space of Curves, R. Dijkgraaf, C. Faber, G van der Geer (editors), Progress in Mathematics, 129, Birkhäuser

R. Dijkgraaf

[3]

On Hurwitz numbers and Hodge integrals

T. Ekedahl
,
S. Lando
,
M. Shapiro
,
A. Vainshtein

e-Print:
- math/9902104

[4]

Hodge integrals and Hurwitz numbers via virtual localization

T. Graber
,
R. Vakil

e-Print:
- math/0003028

[5]

The Virasoro conjecture for Gromov-Witten invariants, Algebraic geometry: Hirzebruch 70 (Warsaw,), 147-176, Contemp. Math., 241, Amer. Math. Soc., 1999

E. Getzler

[6]

Combinatorial enumeration, John & Sons

I.P. Goulden
,
D.M. Jackson

[7]

The Gromov-Witten potential of a point, Hurwitz numbers, and Hodge integrals

e-Print:
- math/9910004

[8]

Two-dimensional QCD and strings

David J. Gross
(
- Princeton U.
)
,
Washington Taylor
(
- MIT, LNS
)

- Strings 1993:0214-225,
- In *Berkeley 1993, Proceedings, Strings '93* 214-225, and MIT Cambridge - CTP-2250 (93/10,rec.Nov.) 12 p. Princeton U. - PUPT-1431 (93/10,rec.Nov.) 12 p
•
e-Print:
- hep-th/9311072