The gerby Gopakumar–Mariño–Vafa formula

Abstract

We prove a formula for certain cubic Z n -Hodge integrals in terms of loop Schur functions. We use this identity to prove the Gromov-Witten/Donaldson-Thomas correspondence for local Z n -gerbes over P 1 .14N35, 53D45; 05E05

Cyclic Hodge integrals and loop Schur functions

“…When τ = ∅, the correspondence holds by the main theorem in [18]. Therefore, to prove the correspondence for τ = ∅ we need only show that P α ∅,ρ,λ (w a ) satisfies the representation basis analog of (24).…”

“…In [18], this fact is generalized to compute the arbitrary rubber integrals in (43) in terms of wreath Hurwitz numbers. In particular, we have…”

Donaldson–Thomas theory and resolutions of toric A-singularities

“…Multiplying these factors together and combining with the remaining terms in the left side of Proposition 3.5, we obtain exactly the first case of (15). The other two cases are similar.…”

“…where the notation on the left-hand side originates from an interpretation in terms of the representation theory of the generalized symmetric group (see, for example, [15] Section 6). It is easily checked that this quantity is independent of the choice of border strip decomposition.…”

“…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.…”