Abstract
We compute genus zero correlators of hybrid phases of Calabi-Yau gauged linear sigma models (GLSMs), i.e. of phases that are Landau-Ginzburg orbifolds fibered over some base. These correlators are generalisations of Gromov-Witten and FJRW invariants. Using previous results on the structure of the of the sphere- and hemisphere partition functions of GLSMs when evaluated in different phases, we extract the I-function and the J-function from a GLSM calculation. The J-function is the generating function of the correlators. We use the field theoretic description of hybrid models to identify the states that are inserted in these correlators. We compute the invariants for examples of one- and two-parameter hybrid models. Our results match with results from mirror symmetry and FJRW theory.
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References
M. Bertolini, I.V. Melnikov and M.R. Plesser, Hybrid conformal field theories, JHEP 05 (2014) 043 [arXiv:1307.7063] [INSPIRE].
M. Bertolini and M. Romo, Aspects of (2, 2) and (0, 2) hybrid models, Commun. Num. Theor. Phys. 14 (2020) 325 [arXiv:1801.04100] [INSPIRE].
H. Fan, T.J. Jarvis and Y. Ruan, The Witten equation, mirror symmetry and quantum singularity theory, arXiv:0712.4021 [https://doi.org/10.48550/arXiv.0712.4021].
E. Clader, Landau-Ginzburg/Calabi-Yau correspondence for the complete intersections X3,3 and X2,2,2,2, Adv. Math. 307 (2017) 1 [arXiv:1301.5530] [INSPIRE].
A. Chiodo and J. Nagel, The hybrid Landau-Ginzburg models of Calabi-Yau complete intersections, in Topological recursion and its influence in analysis, geometry, and topology, volum 100 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, U.S.A. (2018), p. 103 [arXiv:1506.02989].
E. Clader and D. Ross, Sigma models and phase transitions for complete intersections, Int. Math. Res. Not. 2018 (2018) 4799 [arXiv:1511.02027] [INSPIRE].
E. Clader and D. Ross, Wall-crossing in genus-zero hybrid theory, Adv. Geom. 21 (2021) 365.
Y. Zhao, Landau-Ginzburg/Calabi-Yau correspondence for a complete intersection via matrix factorizations, Int. Math. Res. Not. 2022 (2022) 11796 [arXiv:1903.07544] [INSPIRE].
A. Givental, A mirror theorem for toric complete intersections, in Topological field theory, primitive forms and related topics (Kyoto, 1996), volume 160 of Progr. Math., Birkhäuser, Boston, MA, U.S.A. (1998), p. 141 [alg-geom/9701016].
J. Knapp, M. Romo and E. Scheidegger, D-brane central charge and Landau-Ginzburg orbifolds, Commun. Math. Phys. 384 (2021) 609 [arXiv:2003.00182] [INSPIRE].
D. Erkinger and J. Knapp, Sphere partition function of Calabi-Yau GLSMs, Commun. Math. Phys. 394 (2022) 257 [arXiv:2008.03089] [INSPIRE].
H. Fan, T. Jarvis and Y. Ruan, A mathematical theory of the gauged linear sigma model, Geom. Topol. 22 (2018) 235 [arXiv:1506.02109] [INSPIRE].
E. Clader, F. Janda and Y. Ruan, Higher-genus wall-crossing in the gauged linear sigma model, Duke Math. J. 170 (2021) 697 [arXiv:1706.05038] [INSPIRE].
S. Guo, F. Janda and Y. Ruan, A mirror theorem for genus two Gromov-Witten invariants of quintic threefolds, arXiv:1709.07392 [INSPIRE].
S. Guo, F. Janda and Y. Ruan, Structure of higher genus Gromov-Witten invariants of quintic 3-folds, arXiv:1812.11908 [INSPIRE].
M.-X. Huang, A. Klemm and S. Quackenbush, Topological string theory on compact Calabi-Yau: modularity and boundary conditions, Lect. Notes Phys. 757 (2009) 45 [hep-th/0612125] [INSPIRE].
E. Sharpe, Predictions for Gromov-Witten invariants of noncommutative resolutions, J. Geom. Phys. 74 (2013) 256 [arXiv:1212.5322] [INSPIRE].
E. Witten, On the structure of the topological phase of two-dimensional gravity, Nucl. Phys. B 340 (1990) 281.
R. Dijkgraaf, H.L. Verlinde and E.P. Verlinde, Notes on topological string theory and 2D quantum gravity, in the proceedings of the Cargese study institute: random surfaces, quantum gravity and strings, (1990) [INSPIRE].
R. Dijkgraaf and E. Witten, Mean field theory, topological field theory, and multi-matrix models, Nucl. Phys. B 342 (1990) 486.
E. Witten, Topological sigma models, Commun. Math. Phys. 118 (1988) 411 [INSPIRE].
E.P. Verlinde and H.L. Verlinde, A solution of two-dimensional topological quantum gravity, Nucl. Phys. B 348 (1991) 457 [INSPIRE].
S. Cecotti and C. Vafa, Topological antitopological fusion, Nucl. Phys. B 367 (1991) 359 [INSPIRE].
D. Cox and S. Katz, Mirror symmetry and algebraic geometry, American Mathematical Society, Providence, RI, U.S.A. (1999) [https://doi.org/10.1090/surv/068].
K. Hori et al., Mirror symmetry, volume 1 of Clay mathematics monographs, American Mathematical Society, Providence, RI, U.S.A. (2003).
H. Fan, T.J. Jarvis and Y. Ruan, The Witten equation and its virtual fundamental cycle, arXiv:0712.4025 [INSPIRE].
M. Kontsevich and Y. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Commun. Math. Phys. 164 (1994) 525 [hep-th/9402147] [INSPIRE].
E.R. Sharpe, Discrete torsion and gerbes. 1, hep-th/9909108 [INSPIRE].
E.R. Sharpe, Discrete torsion and gerbes. 2, hep-th/9909120 [INSPIRE].
T. Pantev and E. Sharpe, GLSM’s for gerbes (and other toric stacks), Adv. Theor. Math. Phys. 10 (2006) 77 [hep-th/0502053] [INSPIRE].
T. Pantev and E. Sharpe, String compactifications on Calabi-Yau stacks, Nucl. Phys. B 733 (2006) 233 [hep-th/0502044] [INSPIRE].
C. Vafa, Quantum symmetries of string vacua, Mod. Phys. Lett. A 4 (1989) 1615 [INSPIRE].
E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].
H. Jockers et al., Two-sphere partition functions and Gromov-Witten invariants, Commun. Math. Phys. 325 (2014) 1139 [arXiv:1208.6244] [INSPIRE].
P.S. Aspinwall and M.R. Plesser, Decompactifications and massless D-branes in hybrid models, JHEP 07 (2010) 078 [arXiv:0909.0252] [INSPIRE].
C. Vafa, String vacua and orbifoldized L-G models, Mod. Phys. Lett. A 4 (1989) 1169 [INSPIRE].
K. Intriligator and C. Vafa, Landau-Ginzburg orbifolds, Nucl. Phys. B 339 (1990) 95.
W. Lerche, C. Vafa and N.P. Warner, Chiral rings in N = 2 superconformal theories, Nucl. Phys. B 324 (1989) 427 [INSPIRE].
A. Caldararu et al., Non-birational twisted derived equivalences in Abelian GLSMs, Commun. Math. Phys. 294 (2010) 605 [arXiv:0709.3855] [INSPIRE].
P. Candelas et al., Mirror symmetry for two parameter models. 1, Nucl. Phys. B 416 (1994) 481 [hep-th/9308083] [INSPIRE].
S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces, Commun. Math. Phys. 167 (1995) 301 [hep-th/9308122] [INSPIRE].
M. Herbst, K. Hori and D. Page, Phases of N = 2 theories in 1 + 1 dimensions with boundary, arXiv:0803.2045 [INSPIRE].
M. Kreuzer and H. Skarke, On the classification of quasihomogeneous functions, Commun. Math. Phys. 150 (1992) 137 [hep-th/9202039] [INSPIRE].
M. Kreuzer and H. Skarke, All Abelian symmetries of Landau-Ginzburg potentials, Nucl. Phys. B 405 (1993) 305 [hep-th/9211047] [INSPIRE].
N. Addington and P.S. Aspinwall, Categories of massless D-branes and del Pezzo surfaces, JHEP 07 (2013) 176 [arXiv:1305.5767] [INSPIRE].
P.S. Aspinwall, M.R. Plesser and K. Wang, Mirror symmetry and discriminants, arXiv:1702.04661 [INSPIRE].
T. Schimannek, Modular curves, the Tate-Shafarevich group and Gopakumar-Vafa invariants with discrete charges, JHEP 02 (2022) 007 [arXiv:2108.09311] [INSPIRE].
K. Hori and J. Knapp, Linear sigma models with strongly coupled phases — one parameter models, JHEP 11 (2013) 070 [arXiv:1308.6265] [INSPIRE].
D. Favero and B. Kim, General GLSM invariants and their cohomological field theories, arXiv:2006.12182 [INSPIRE].
Z. Chen, J. Guo and M. Romo, A GLSM view on homological projective duality, Commun. Math. Phys. 394 (2022) 355 [arXiv:2012.14109] [INSPIRE].
J. Guo and M. Romo, Hybrid models for homological projective duals and noncommutative resolutions, Lett. Math. Phys. 112 (2022) 117 [arXiv:2111.00025] [INSPIRE].
Acknowledgments
We would like to thank Alessandro Chiodo, Ilarion Melnikov, Robert Pryor, Mauricio Romo, Emanuel Scheidegger, Thorsten Schimannek, and Eric Sharpe for helpful discussions and collaboration on related projects. JK thanks MATRIX Institute and Sorbonne Université for hospitality. We also would like to thank the anonymous referee for helpful comments and for providing an explanation of the selection rule (2.10) that has been added to a revised version of the article. DE was supported by the Austrian Science Fund (FWF): [P30904-N27]. JK is supported by the Australian Research Council Discovery Project DP210101502 and the Australian Research Council Future Fellowship FT210100514.
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ArXiv ePrint: 2210.01226
David Erkinger has now left academia.
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Erkinger, D., Knapp, J. On genus-0 invariants of Calabi-Yau hybrid models. J. High Energ. Phys. 2023, 71 (2023). https://doi.org/10.1007/JHEP05(2023)071
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DOI: https://doi.org/10.1007/JHEP05(2023)071