Quantum geometry, stability and modularity

Abstract

By exploiting new mathematical relations between Pandharipande-Thomas (PT) invariants, closely related to Gopakumar-Vafa (GV) invariants, and rank 0 Donaldson-Thomas (DT) invariants counting D4-D2-D0 BPS bound states, we rigorously compute the first few terms in the generating series of Abelian D4-D2-D0 indices for compact one-parameter Calabi-Yau threefolds of hypergeometric type. In all cases where GV invariants can be computed to sufficiently high genus, we find striking confirmation that the generating series is modular, and predict infinite series of Abelian D4-D2-D0 indices. Conversely, we use these results to provide new constraints for the direct integration method, which allows to compute GV invariants (and therefore the topological string partition function) to higher genus than hitherto possible. The triangle of relations between GV/PT/DT invariants is powered by a new explicit formula relating PT and rank 0 DT invariants, which is proven in an Appendix by the second named author. As a corollary, we obtain rigorous Castelnuovo-type bounds for PT and GV invariants for CY threefolds with Picard rank one.

Non-perturbative topological string theory on compact Calabi-Yau 3-folds

“…This was observed in [24] and recently proved rigorously in [25] using vanishing results in Donaldson-Thomas theory, see also [70] for a proof in the case of the quintic. As the curves of degree β of maximal genus g max (β) are smooth, the associated invariants can be obtained via the formula…”

“…In particular, a gap condition occurs at some conifold loci, which can be used as boundary condition to fix the holomorphic ambiguity and solve these theories to high genus [24]. Boundary conditions arising at the z = 0 MUM point have recently been better understood [25], as have the critical points at 1/z = 0 in [35], see also [25]. This advance has made it possible to fix the holomorphic ambiguity to higher genus than achieved in [24].…”

“…Using the boundary conditions ( 131), (133), and (135), together with boundary conditions at the conifold at z = µ and at the orbifold at z = ∞, both of which we shall discuss below, one can solve the quintic to genus g = 53 [24]. 15 The maximal genus to which one can obtain the topological string amplitudes solely by imposing the bounds described above are given in [24,25] for all hypergeometric models.…”

“…To fix the holomorphic ambiguity to even higher genus in the class of hypergeometric models, one can invoke modularity of the D4D2D0 rank 1 BPS indices and wall crossing transitions to the rank 1 D6D2D0 Donaldson-Thomas invariants that capture the BPS invariants of the topological string [25]. This permits predicting the value of invariants n g max (β)−∆,β near the Castelnuovo bound (so far only for small ∆ ∈ ) and imposing these values as constraints on the topological string amplitudes.…”

“…Armed with the explicit trans-series solutions for the compact case, we verify this property in detail in the case of compact, one-parameter Calabi-Yau manifolds. For this we rely on the important progress in the calculation of perturbative topological string series started by [24], and recently developed in [25]. We find that our trans-series describe with very high precision the asymptotic behavior of the topological string free energies at large genus.…”