Genus zero Gopakumar-Vafa invariants from open strings

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.

[1]  R. Savelli,et al.  F-theory on singular spaces , 2014, 1410.4867.

[2]  M. Douglas,et al.  D-branes, quivers, and ALE instantons , 1996, hep-th/9603167.

[3]  C. Curto,et al.  Threefold flops via matrix factorization , 2006, math/0611014.

[4]  W. Donovan.  Contractions of 3-folds: deformations and invariants , 2015, 1511.01656.

[5]  Lakshya Bhardwaj.  On the classification of 5d SCFTs , 2019, Journal of High Energy Physics.

[6]  M. Wemyss,et al.  Gopakumar–Vafa Invariants Do Not Determine Flops , 2017, 1707.01150.

[7]  M theory and topological strings. 2. , 1998, hep-th/9809187.

[8]  K. Intriligator,et al.  Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces , 1997, hep-th/9702198.

[9]  Lakshya Bhardwaj,et al.  Classifying 5d SCFTs via 6d SCFTs: arbitrary rank , 2018, Journal of High Energy Physics.

[10]  F. Denef,et al.  D-brane Deconstructions in IIB Orientifolds , 2008, 0805.1573.

[11]  Ling Lin,et al.  Fibers add flavor. Part I. Classification of 5d SCFTs, flavor symmetries and BPS states , 2019, Journal of High Energy Physics.

[12]  Yi-Nan Wang,et al.  Coulomb and Higgs branches from canonical singularities. Part 0 , 2020, Journal of High Energy Physics.

[13]  Ling Lin,et al.  5d superconformal field theories and graphs , 2019, Physics Letters B.

[14]  D. Morrison,et al.  High electric charges in M-theory from quiver varieties , 2019, Journal of High Energy Physics.

[15]  Henry B. Laufer.  ON CP1 AS AN EXCEPTIONAL SET , 1981 .

[16]  R. Savelli,et al.  T-branes through 3d mirror symmetry , 2016, 1603.00062.

[17]  V. Saxena,et al.  Five-dimensional SCFTs and gauge theory phases: an M-theory/type IIA perspective , 2018, SciPost Physics.

[18]  C. Vafa,et al.  The Topological Vertex , 2003, hep-th/0305132.

[19]  R. Gopakumar,et al.  On the Gauge Theory/Geometry Correspondence , 1998, hep-th/9811131.

[20]  Gorenstein Threefold Singularities with Small Resolutions via Invariant Theory for Weyl Groups , 1992, alg-geom/9202002.

[21]  M. Reid.  Minimal Models of Canonical 3-Folds , 1983 .

[22]  P. Aspinwall,et al.  Quivers from Matrix Factorizations , 2010, 1005.1042.

[23]  S. Schäfer-Nameki,et al.  (5d RG-flow) trees in the tropical rain forest , 2020, Journal of High Energy Physics.

[24]  C. Vafa,et al.  T-branes and monodromy , 2010, 1010.5780.

[25]  Edward Witten,et al.  Topological sigma models , 1988 .

[26]  M. Wemyss,et al.  Noncommutative deformations and flops , 2013, 1309.0698.

[27]  J. Karmazyn.  The length classification of threefold flops via noncommutative algebras , 2017, Advances in Mathematics.

[28]  M. Fazzi,et al.  Geometric engineering on flops of length two , 2018, 1802.00813.

[29]  R. Savelli,et al.  T-branes as branes within branes , 2014, 1410.4178.

[30]  Yi-Nan Wang,et al.  5d and 4d SCFTs: canonical singularities, trinions and S-dualities , 2020, Journal of High Energy Physics.

[31]  Yukinobu Toda.  Non-commutative width and Gopakumar–Vafa invariants , 2014, 1411.1505.