Cyclic Hodge integrals and loop Schur functions

Abstract

We conjecture an evaluation of three-partition cyclic Hodge integrals in terms of loop Schur functions. Our formula implies the orbifold Gromov-Witten/Donaldson-Thomas correspondence for toric Calabi-Yau threefolds with transverse A n singularities. We prove the formula in the case where one of the partitions is empty, and thus establish the orbifold Gromov-Witten/Donaldson-Thomas correspondence for local toric surfaces with transverse A n singularities.

Donaldson–Thomas theory and resolutions of toric A-singularities

“…Similarly, if d j−1+i (μ l ) increases then we must be passing from (A.I.a) to (A.I.b) (with the same value a). This results in a factor of −q l+1 · · · q k , agreeing again with (16).…”

“…By the combinatorial description of this factor which we derived above, we see that if d j−i (μ l ) increases then we must be passing from (A.I.b) to (A.I.a) (with an increase by one of the value a). But the discrepancy in these factors is −q k+1 · · · q n−1 q 0 · · · q l , agreeing with (16). Similarly, if d j−1+i (μ l ) increases then we must be passing from (A.I.a) to (A.I.b) (with the same value a).…”

“…In the case of toric targets with transverse A-singularities, the top equivalence is a theorem of Maulik-OblomkovOkounkov-Pandharipande [11] (more generally they proved it for all toric 3-folds) and the equality on the right is the main theorem of this paper. The bottom conjectural equivalence was made explicit in [15,16] and proved in the case where the toric orbifold is a local surface.…”

“…In [14], we prove that the main result of [1] along with the results of this paper and the correspondence of gluing algorithms of [16] are sufficient to deduce the bottom equality and, hence, allow us to "complete the square" for Z a toric Calabi-Yau 3-orbifold with transverse A-singularities. In particular, the bottom equivalence provides strong structural results about GW(Z) that are not obvious from a purely Gromov-Witten perspective.…”

“…In our related works [13][14][15][16], we heavily rely on the algebrocombinatorial structure of this formula. For completeness, we reproduce the formula here.…”