On Mathieu moonshine and Gromov-Witten invariants

Abstract

We provide strong evidence that a large number of CY 3 manifolds are involved in an intricate way in Mathieu moonshine viz. their Gromov-Witten invariants are related to the expansion coefficients of the twined/twisted-twined elliptic genera of K3. We use the conjectured string duality between CHL orbifolds of heterotic string theory on K3 × T 2 and type IIA string theory on CY 3 manifolds to explicitly show this connection. We then work out two concrete examples where we exactly match the expansion coefficients on both sides of the duality.

Topological strings on genus one fibered Calabi-Yau 3-folds and string dualities

“…appears to be relevant [50]. Note that the Hodge numbers of the dual Calabi-Yau are related to the number of vector-and hypermultiplets via have been identified by explicitly comparing the predictions from [51] to the genus zero enumerative invariants of Calabi-Yau threefolds with h 1,1 = 3 [116]. We will show below that those geometries are again genus one fibered with 2-sections and, using the modular bootstrap, extend the comparison to all genera.…”

“…However, the list of gauge backgrounds that has been obtained by [50] is not exhaustive and we expect that a different construction should provide the missing heterotic duals. The two models found by [116] are contained in this list.…”

“…Indeed, using the normal form [121] for reflexive polytopes one can easily check that the two geometries found by [116] respectively correspond to (q 1 , q 2 ) = (2, 2) and (q 1 , q 2 ) = (4, 0). For any of the nine geometries we can consider the K3 fiber with base P 1 A or P 1 B which we will sometimes denote by K3 1 and K3 2 .…”

Heterotic strings on (K3 × T2)/ℤ3 and their dual Calabi-Yau threefolds

“…2 are related by a change of basis. In [8,9] the Gopakumar-Vafa invariants of Calabi-Yau threefolds that exhibit a K3 fibration and a genus one fibration with two-sections haven been compared against the heterotic one-loop calculation for compactifications on (K3 × T 2 )/Z 2 from [5,6]. We note that the matching cases are exactly those where the polarization lattice is I…”

“…The indices in turn had been JHEP04(2020)203 calculated for various compactifications with standard embedding of the spin connection into the gauge connection on (K3 × T 2 )/Z N and some non-standard embeddings when N = 2 [6]. Calabi-Yau duals for some of the N = 2 cases have been proposed in [7][8][9]. In this paper we extend the calculation of the new supersymmetric index to compactifications on (K3 × T 2 )/Z 3 with non-standard embeddings and predict the geometric data of the dual Calabi-Yau manifold.…”