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.
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K.A. Intriligator, D.R. Morrison and N. Seiberg, Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nucl. Phys. B 497 (1997) 56 [hep-th/9702198] [INSPIRE].
N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [INSPIRE].
C. Closset, S. Giacomelli, S. Schäfer-Nameki and Y.-N. Wang, 5d and 4d SCFTs: Canonical Singularities, Trinions and S-Dualities, JHEP 05 (2021) 274 [arXiv:2012.12827] [INSPIRE].
M. Van Beest, A. Bourget, J. Eckhard and S. Schäfer-Nameki, (5d RG-flow) Trees in the Tropical Rain Forest, JHEP 03 (2021) 241 [arXiv:2011.07033] [INSPIRE].
F. Apruzzi, C. Lawrie, L. Lin, S. Schäfer-Nameki and Y.-N. Wang, 5d Superconformal Field Theories and Graphs, Phys. Lett. B 800 (2020) 135077 [arXiv:1906.11820] [INSPIRE].
L. Bhardwaj and P. Jefferson, Classifying 5d SCFTs via 6d SCFTs: Arbitrary rank, JHEP 10 (2019) 282 [arXiv:1811.10616] [INSPIRE].
C. Closset, M. Del Zotto and V. Saxena, Five-dimensional SCFTs and gauge theory phases: an M-theory/type IIA perspective, SciPost Phys. 6 (2019) 052 [arXiv:1812.10451] [INSPIRE].
A. Collinucci, F. Denef and M. Esole, D-brane Deconstructions in IIB Orientifolds, JHEP 02 (2009) 005 [arXiv:0805.1573] [INSPIRE].
F. Apruzzi, C. Lawrie, L. Lin, S. Schäfer-Nameki and Y.-N. Wang, Fibers add Flavor, Part I: Classification of 5d SCFTs, Flavor Symmetries and BPS States, JHEP 11 (2019) 068 [arXiv:1907.05404] [INSPIRE].
F. Apruzzi, C. Lawrie, L. Lin, S. Schäfer-Nameki and Y.-N. Wang, Fibers add Flavor, Part II: 5d SCFTs, Gauge Theories, and Dualities, JHEP 03 (2020) 052 [arXiv:1909.09128] [INSPIRE].
L. Bhardwaj, On the classification of 5d SCFTs, JHEP 09 (2020) 007 [arXiv:1909.09635] [INSPIRE].
E. Witten, Topological Sigma Models, Commun. Math. Phys. 118 (1988) 411 [INSPIRE].
R. Gopakumar and C. Vafa, M theory and topological strings. 1, hep-th/9809187 [INSPIRE].
R. Gopakumar and C. Vafa, On the gauge theory/geometry correspondence, Adv. Theor. Math. Phys. 3 (1999) 1415 [hep-th/9811131] [INSPIRE].
R. Gopakumar and C. Vafa, M theory and topological strings. 2, hep-th/9812127 [INSPIRE].
M. Aganagic, A. Klemm, M. Mariño and C. Vafa, The topological vertex, Commun. Math. Phys. 254 (2005) 425 [hep-th/0305132] [INSPIRE].
S. Katz and D.R. Morrison, Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups, J. Alg. Geom. 1 (1992) 449 [alg-geom/9202002].
S.H. Katz, Genus zero Gopakumar-Vafa invariants of contractible curves, J. Diff. Geom. 79 (2008) 185 [math/0601193] [INSPIRE].
W. Donovan and M. Wemyss, Noncommutative deformations and flops, Duke Math. J. 165 (2016) 1397 [arXiv:1309.0698] [INSPIRE].
W. Donovan, Contractions of 3-folds: deformations and invariants, arXiv:1511.01656.
Y. Toda, Non-commutative width and gopakumar-vafa invariants, arXiv:1411.1505.
H.B. Laufer, On cp1 as an exceptional set, Recent Developments in Several Complex Variables. (AM-100) (1981) 261.
S. Cecotti, C. Cordova, J.J. Heckman and C. Vafa, T-Branes and Monodromy, JHEP 07 (2011) 030 [arXiv:1010.5780] [INSPIRE].
E. Witten, Phase transitions in M-theory and F-theory, Nucl. Phys. B 471 (1996) 195 [hep-th/9603150] [INSPIRE].
A. Collinucci and R. Valandro, The role of U(1)’s in 5d theories, Higgs branches, and geometry, JHEP 10 (2020) 178 [arXiv:2006.15464] [INSPIRE].
A. Collinucci and R. Savelli, T-branes as branes within branes, JHEP 09 (2015) 161 [arXiv:1410.4178] [INSPIRE].
M. Reid, Minimal models of canonical 3-folds, Adv. Stud. Pure Math. 1 (1983) 131.
P.S. Aspinwall and D.R. Morrison, Quivers from Matrix Factorizations, Commun. Math. Phys. 313 (2012) 607 [arXiv:1005.1042] [INSPIRE].
C. Curto and D.R. Morrison, Threefold flops via matrix factorization, math/0611014.
M.R. Douglas and G.W. Moore, D-branes, quivers, and ALE instantons, hep-th/9603167 [INSPIRE].
J. Karmazyn, The length classification of threefold flops via noncommutative algebras, arXiv:1709.02720.
A. Collinucci, M. Fazzi and R. Valandro, Geometric engineering on flops of length two, JHEP 04 (2018) 090 [arXiv:1802.00813] [INSPIRE].
A. Collinucci, M. Fazzi, D.R. Morrison and R. Valandro, High electric charges in M-theory from quiver varieties, JHEP 11 (2019) 111 [arXiv:1906.02202] [INSPIRE].
H.C. Pinkham, Factorization of birational maps in dimension 3, Amer. Math. Soc., Providence, RI, U.S.A. (1983).
G. Brown and M. Wemyss, Gopakumar-Vafa invariants do not determine flops, Commun. Math. Phys. 361 (2018) 143 [arXiv:1707.01150] [INSPIRE].
C. Closset, S. Schäfer-Nameki and Y.-N. Wang, Coulomb and Higgs Branches from Canonical Singularities: Part 0, JHEP 02 (2021) 003 [arXiv:2007.15600] [INSPIRE].
A. Collinucci, S. Giacomelli, R. Savelli and R. Valandro, T-branes through 3d mirror symmetry, JHEP 07 (2016) 093 [arXiv:1603.00062] [INSPIRE].
A. Collinucci and R. Savelli, F-theory on singular spaces, JHEP 09 (2015) 100 [arXiv:1410.4867] [INSPIRE].
A. Collinucci, M. De Marco, A. Sangiovanni and R. Valandro, Higgs branches of 5d rank-zero theories from geometry, arXiv:2105.12177 [INSPIRE].
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Collinucci, A., Sangiovanni, A. & Valandro, R. Genus zero Gopakumar-Vafa invariants from open strings. J. High Energ. Phys. 2021, 59 (2021). https://doi.org/10.1007/JHEP09(2021)059
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DOI: https://doi.org/10.1007/JHEP09(2021)059