Topological strings, quiver varieties, and Rogers-Ramanujan identities. (English) Zbl 1440.14261
Summary: Motivated by some recent works on BPS invariants of open strings/knot invariants, we guess there may be a general correspondence between the Ooguri-Vafa invariants of toric Calabi-Yau 3-folds and cohomologies of Nakajima quiver varieties. In this short note, we provide a toy model to explain this correspondence. More precisely, we study the topological open string model of \({\mathbb {C}}^3\) with one Aganagic-Vafa brane \({\mathcal {D}}_\tau \), and we show that, when \(\tau \le 0\), its Ooguri-Vafa invariants are given by the Betti numbers of certain quiver variety. Moreover, the existence of Ooguri-Vafa invariants implies an infinite product formula. In particular, we find that the \(\tau =1\) case of such infinite product formula is closely related to the celebrated Rogers-Ramanujan identities.
MSC:
| 14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |
| 14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |
| 11P84 | Partition identities; identities of Rogers-Ramanujan type |
| 05E05 | Symmetric functions and generalizations |
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