Abstract

We propose a new way to compute the genus zero Gopakumar-Vafa invariants for two families of non-toric non-compact Calabi-Yau threefolds that admit simple flops: Reid’s Pagodas, and Laufer’s examples. We exploit the duality between M-theory on these threefolds, and IIA string theory with D6-branes and O6-planes. From this perspective, the GV invariants are detected as five-dimensional open string zero modes. We propose a definition for genus zero GV invariants for threefolds that do not admit small crepant resolutions. We find that in most cases, non-geometric T-brane data is required in order to fully specify the invariants.

Highlights

  • For integers known as GV invariants, or BPS invariants ngβ, where g is the genus of a curve, and β ∈ H2(X3) its homology class

  • We propose a new way to compute the genus zero Gopakumar-Vafa invariants for two families of non-toric non-compact Calabi-Yau threefolds that admit simple flops: Reid’s Pagodas, and Laufer’s examples

  • We have introduced a new way to calculate genus zero Gopakumar-Vafa invariants for non-compact CY threefolds that are C∗-fibrations, or Z2-orbifolds of C∗fibrations

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Summary

GV invariants

The topological A-model [12] is a simplified model of closed strings that counts holomorphic maps from the worldsheet into a Calabi-Yau threefold target space. The non-perturbative part of the free energy is recast into an object that counts curves in the target space. The conjecture stems from the fact that, in M-theory, these integers are counting BPS states that are realized as M2-branes wrapping holomorphic curves. Notice that β runs over all classes, and can run over multiples of a generator of H2 Those are interpreted as bound states of coincident M2-branes wrapping a curve. The curves will be rigid, so we will only have hypermultiplet content in our 5d theories

Warmup case: the conifold
Reid’s pagodas
Non-simple flops: the generalized conifold
Generalities
Orientifolds
Laufer’s examples
Morrison-Pinkham example
Non-resolvable singularities: an intriguing family of threefolds
T-brane data
Conclusions
Full Text
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