Higher genus relative Gromov–Witten theory and double ramification cycles

Abstract

We extend the definition of relative Gromov–Witten invariants with negative contact orders to all genera. Then we show that relative Gromov–Witten theory forms a partial CohFT. Some cycle relations on the moduli space of stable maps are also proved.

Holomorphic anomaly equation for and the Nekrasov-Shatashvili limit of local

“…The holomorphic anomaly equation fixes the -dependence of P 2 and so determines P 2 up to some ambiguity living in the (2 − 1)-dimensional vector space −(2 −2) • Q[ , ] 6 −6 . The coefficient of 1/ 2 −2 con and the (2 − 3) vanishings of the coefficients of 1/ con for 1 ≤ ≤ 2 − 3 predicted by (25), combined with the fact that the -expansion of P 2 has no constant term, uniquely fix this ambiguity and so determine P 2 completely.…”

“…It would be interesting to understand how (25) and ( 26) are related through Theorem 1.2. We leave this question open.…”

“…For simplicity, we assume all vertices on Γ are stable. Denote the forgetful maps by the following: As mentioned above, [25,Theorem 4.1] can be generalized to moduli with a disconnected domain using the same proof. Let be the arithmetic genus of the disconnected curve corresponding to the graph Γ.…”

“…The disconnected version of the theorem implies that the following identity The right-hand side decomposes into a product of such expressions according to the connected component of domain curves, because the disconnected moduli in orbifold theory is a product of connected moduli, and − * * L decomposes into a sum accordingly. Applying the original connected version of [25,Theorem 4.1] to each factor, we conclude that the right-hand side is nothing but the pushforward of a product of 1 − Ψ ∞ ∩ [M ∼ ( )] vir , where Ψ ∞ should be treated as the target psi-class of M ∼ ( ).…”

“…Let us recall the content of the theorem. In [25,Theorem 4.1], we consider 0 , , which is the th root stack of := P ( ⊕ O). After the root stack construction, there are two invariant substacks D 0 , D ∞ under the fiberwise C * action.…”