Todor E. Milanov and
Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA. e-mail: milanov@math.stanford.edu
Hsian-Hua TsengDepartment of Mathematics, University of Wisconsin-Madison, Van Vleck Hall, 480 Lincoln Drive, Madison, WI 53706-1388, USA. e-mail: tseng@math.wisc.edu
Abstract
Let Mk,m be the space of Laurent polynomials in one variable 
, where k,m 
1 are fixed integers and 
. According to B. Dubrovin [B. Dubrovin, Geometry of 2d topological field theories, Integrable systems and quantum groups (Montecatini Terme 1993), Lect. Notes Math. 1620, Springer, Berlin (1996), 120–348.], Mk,m can be equipped with a semi-simple Frobenius structure. In this paper we prove that the corresponding descendent and ancestor potentials of Mk,m (defined as in [A. Givental, Semisimple Frobenius structures at higher genus, Internat. Math. Res. Notices 23 (2001), 1265–1286.]) satisfy Hirota Quadratic Equations (HQE for short).
Let 
k,m be the orbifold obtained from 
1 by cutting small discs D1 
{|z| 
ε} and D2 {|z–1| ε} around z = 0 and z = ∞ and gluing back the orbifolds D1/
k and D2//m in the obvious way. We show that the orbifold quantum cohomology of k,m coincides with Mk,m as Frobenius manifolds. Modulo some yet-to-be-clarified details, this implies that the descendent (respectively the ancestor) potential of Mk,m is a generating function for the descendent (respectively ancestor) orbifold Gromov-Witten invariants of k,m.
There is a certain similarity between our HQE and the Lax operators of the extended bi-graded Toda hierarchy, introduced by G. Carlet in [G. Carlet, The extended bigraded Toda hierarchy, J. Phys. A 39 (2006), no. 30, 9411–9435.]. Therefore, it is plausible that our HQE characterize the tau-functions of this hierarchy and we expect that the extended bi-graded Toda hierarchy governs the Gromov-Witten theory of k,m.
