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The Ramanujan Journal

, Volume 48, Issue 2, pp 399–421 | Cite as

Topological strings, quiver varieties, and Rogers–Ramanujan identities

  • Shengmao ZhuEmail author
Article
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Abstract

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.

Keywords

Topological strings Ooguri–Vafa invariants Quiver varieties Rogers–Ramanujan identities 

Mathematics Subject Classification

14N35 14N10 11P84 05E05 

Notes

Acknowledgements

The author would like to thank Professor Ole Warnaar for useful discussions [71], and showing him some insights about Formula (57).

Funding Funding was provided by NSFC (Grant No. 11201417).

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Authors and Affiliations

  1. 1.Center of Mathematical SciencesZhejiang UniversityHangzhouChina

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