|
SIGMA 16 (2020), 059, 31 pages arXiv:1812.06200
https://doi.org/10.3842/SIGMA.2020.059
Mirror Symmetry for Nonabelian Landau-Ginzburg Models
Nathan Priddis a, Joseph Ward b and Matthew M. Williams c
a) Brigham Young University, USA
b) University of Utah, USA
c) Colorado State University, USA
Received September 24, 2019, in final form June 12, 2020; Published online June 27, 2020
Abstract
We consider Landau-Ginzburg models stemming from groups comprised of non-diagonal symmetries, and we describe a rule for the mirror LG model. In particular, we present the non-abelian dual group $G^\star$, which serves as the appropriate choice of group for the mirror LG model. We also describe an explicit mirror map between the A-model and the B-model state spaces for two examples. Further, we prove that this mirror map is an isomorphism between the untwisted broad sectors and the narrow diagonal sectors for Fermat type polynomials.
Key words: mirror symmetry; Landau-Ginzburg models; Calabi-Yau; nonabelian.
pdf (504 kb)
tex (37 kb)
References
- Artebani M., Boissière S., Sarti A., The Berglund-Hübsch-Chiodo-Ruan mirror symmetry for K3 surfaces, J. Math. Pures Appl. 102 (2014), 758-781, arXiv:1108.2780.
- Basalaev A., Takahashi A., Werner E., Orbifold Jacobian algebras for exceptional unimodal singularities, Arnold Math. J. 3 (2017), 483-498, arXiv:1702.02739.
- Berglund P., Henningson M., Landau-Ginzburg orbifolds, mirror symmetry and the elliptic genus, Nuclear Phys. B 433 (1995), 311-332, arXiv:hep-th/9401029.
- Berglund P., Hübsch T., A generalized construction of mirror manifolds, Nuclear Phys. B 393 (1993), 377-391, arXiv:hep-th/9201014.
- Bott C.J., Comparin P., Priddis N., Mirror symmetry for K3 surfaces, arXiv:1901.09373.
- Chiodo A., Iritani H., Ruan Y., Landau-Ginzburg/Calabi-Yau correspondence, global mirror symmetry and Orlov equivalence, Publ. Math. Inst. Hautes Études Sci. 119 (2014), 127-216, arXiv:1201.0813.
- Chiodo A., Ruan Y., Landau-Ginzburg/Calabi-Yau correspondence for quintic three-folds via symplectic transformations, Invent. Math. 182 (2010), 117-165, arXiv:0812.4660.
- Comparin P., Lyons C., Priddis N., Suggs R., The mirror symmetry of K3 surfaces with non-symplectic automorphisms of prime order, Adv. Theor. Math. Phys. 18 (2014), 1335-1368, arXiv:1211.2172.
- Comparin P., Priddis N., Equivalence of mirror constructions for K3 surfaces with non-symplectic automorphism, arXiv:1704.00354.
- Ebeling W., Gusein-Zade S.M., On the orbifold euler characteristics of dual invertible polynomials with non-abelian symmetry groups, arXiv:1811.05781.
- Ebeling W., Gusein-Zade S.M., A version of the Berglund-Hübsch-Henningson duality with non-abelian groups, arXiv:1807.04097.
- Ebeling W., Gusein-Zade S.M., Dual invertible polynomials with permutation symmetries and the orbifold Euler characteristic, SIGMA 16 (2020), 051, 15 pages, arXiv:1907.11421.
- Ebeling W., Takahashi A., Mirror symmetry between orbifold curves and cusp singularities with group action, Int. Math. Res. Not. 2013 (2013), 2240-2270, arXiv:1103.5367.
- Fan H., Jarvis T., Ruan Y., The Witten equation, mirror symmetry, and quantum singularity theory, Ann. of Math. 178 (2013), 1-106, arXiv:0712.4021.
- Fan H., Jarvis T.J., Ruan Y., A mathematical theory of the gauged linear sigma model, Geom. Topol. 22 (2018), 235-303, arXiv:1506.02109.
- Francis A., Jarvis T., Johnson D., Suggs R., Landau-Ginzburg mirror symmetry for orbifolded Frobenius algebras, in String-Math 2011, Proc. Sympos. Pure Math., Vol. 85, Amer. Math. Soc., Providence, RI, 2012, 333-353, arXiv:1111.2508.
- Francis A., Jarvis T., Priddis N., A brief survey of FJRW theory, in Primitive Forms and Related Subjects (Kavli IPMU, 2014), Advanced Studies in Pure Mathematics, Vol. 83, Mathematical Society of Japan, Tokyo, 2019, 19-53, arXiv:1503.01223.
- Francis A., Priddis N., Schaug A., Borcea-Voisin mirror symmetry for Landau-Ginzburg models, Illinois J. Math. 63 (2019), 425-461, arXiv:1708.05775.
- Guéré J., A Landau-Ginzburg mirror theorem without concavity, Duke Math. J. 165 (2016), 2461-2527, arXiv:1307.5070.
- He W., Li S., Li Y., $G$-twisted braces and orbifold Landau-Ginzburg models, Comm. Math. Phys. 373 (2020), 175-217, arXiv:1801.04560.
- He W., Li S., Shen Y., Webb R., Landau-Ginzburg mirror symmetry conjecture, J. Eur. Math. Soc., to appear, arXiv:1503.01757.
- Intriligator K., Vafa C., Landau-Ginzburg orbifolds, Nuclear Phys. B 339 (1990), 95-120.
- Krawitz M., FJRW rings and Landau-Ginzburg mirror symmetry, Ph.D. Thesis, University of Michigan, 2010.
- Kreuzer M., Skarke H., On the classification of quasihomogeneous functions, Comm. Math. Phys. 150 (1992), 137-147, arXiv:hep-th/9202039.
- Lee Y.-P., Priddis N., Shoemaker M., A proof of the Landau-Ginzburg/Calabi-Yau correspondence via the crepant transformation conjecture, Ann. Sci. Éc. Norm. Supér. (4) 49 (2016), 1403-1443, arXiv:1410.5503.
- Mukai D., Nonabelian Landau-Ginzburg orbifolds and Calabi-Yau/Landau-Ginzburg correspondence, arXiv:1704.04889.
- Priddis N., Shoemaker M., A Landau-Ginzburg/Calabi-Yau correspondence for the mirror quintic, Ann. Inst. Fourier (Grenoble) 66 (2016), 1045-1091, arXiv:1309.6262.
|
|