Equivariant orbifold structures on the projective line and integrable hierarchies

Kavli Affiliate: Todor E. Milanov

| First 5 Authors: Todor E. Milanov, Hsian-Hua Tseng, , ,

| Summary:

Let $CP^1_{k,m}$ be the orbifold structure on $CP^1$ obtained via
uniformizing the neighborhoods of 0 and $infty$ respectively by $zmapsto z^k$
and $wmapsto w^m.$ The diagonal action of the torus on the projective line
induces naturally an orbifold action on $CP^1_{k,m}.$ In this paper we prove
that if k and m are co-prime then Givental’s prediction of the equivariant
total descendent Gromov-Witten potential of $CP^1_{k,m}$ satisfies certain
Hirota Quadratic Equations (HQE for short). We also show that after an
appropriate change of the variables, similar to Getzler’s change in the
equivariant Gromov-Witten theory of $CP^1$, the HQE turn into the HQE of the
2-Toda hierarchy, i.e., the Gromov-Witten potential of $CP^1_{k,m}$ is a
tau-function of the 2-Toda hierarchy. More precisely, we obtain a sequence of
tau-functions of the 2-Toda hierarchy from the descendent potential via some
translations. The later condition, that all tau-functions in the sequence are
obtained from a single one via translations, imposes a serious constraint on
the solution of the 2-Toda hierarchy. Our theorem leads to the discovery of a
new integrable hierarchy (we suggest to be called the Equivariant Bi-graded
Toda Hierarchy). We conjecture that this new hierarchy governs, i.e., uniquely
determines, the equivariant Gromov-Witten invariants of $CP^1_{k,m}.$

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