Geodesic
Mirror map for Fermat polynomials with a nonabelian group of
Teoretičeskaâ i matematičeskaâ fizika, Tome 209 (2021) no. 2, pp. 205-223 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We study Landau–Ginzburg orbifolds $(f,G)$ with $f=x_1^n+\cdots+x_N^n$ and $G=S\ltimes G^d$, where $S\subseteq S_N$ and $G^d$ is either the maximal group of scalar symmetries of $f$ or the intersection of the maximal diagonal symmetries of $f$ with $SL_N(\mathbb{C})$. We construct a mirror map between the corresponding phase spaces and prove that it is an isomorphism restricted to a certain subspace of the phase space when $n=N$ is a prime number. When $S$ satisfies the parity condition of Ebeling–Gusein-Zade, this subspace coincides with the full space. We also show that two phase spaces are isomorphic for $n=N=5$.
Keywords: mirror symmetry, nonabelian symmetry group, singularity theory. 
@article{TMF_2021_209_2_a0,
     author = {A. A. Basalaev and A. A. Ionov},
     title = {Mirror map for {Fermat} polynomials with a~nonabelian group of},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {205--223},
     year = {2021},
     volume = {209},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2021_209_2_a0/}
}
TY  - JOUR
AU  - A. A. Basalaev
AU  - A. A. Ionov
TI  - Mirror map for Fermat polynomials with a nonabelian group of
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2021
SP  - 205
EP  - 223
VL  - 209
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2021_209_2_a0/
LA  - ru
ID  - TMF_2021_209_2_a0
ER  - 
%0 Journal Article
%A A. A. Basalaev
%A A. A. Ionov
%T Mirror map for Fermat polynomials with a nonabelian group of
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2021
%P 205-223
%V 209
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2021_209_2_a0/
%G ru
%F TMF_2021_209_2_a0
A. A. Basalaev; A. A. Ionov. Mirror map for Fermat polynomials with a nonabelian group of. Teoretičeskaâ i matematičeskaâ fizika, Tome 209 (2021) no. 2, pp. 205-223. http://geodesic.mathdoc.fr/item/TMF_2021_209_2_a0/

[1] K. Intriligator, C. Vafa, “Landau–Ginzburg orbifolds”, Nucl. Phys. B, 339:1 (1990), 95–120 | DOI | MR

[2] C. Vafa, “String vacua and orbifoldized $LG$ models”, Modern Phys. Lett. A, 4:12 (1989), 1169–1185 | DOI | MR

[3] E. Witten, “Phases of $N=2$ theories in two dimensions”, Nucl. Phys. B, 403:1–2 (1993), 159–222, arXiv: hep-th/9301042 | DOI | MR

[4] D. Shklyarov, “On Hochschild invariants of Landau–Ginzburg orbifolds”, Adv. Theor. Math. Phys., 24:1 (2020), 189–258 | DOI | MR

[5] H. Fan, T. Jarvis, Y. Ruan, “The Witten equation, mirror symmetry, and quantum singularity theory”, Ann. Math. (2), 178:1 (2013), 1–106 | DOI | MR

[6] N. Priddis, J. Ward, M. M. Williams, “Mirror symmetry for nonabelian Landau–Ginzburg models”, SIGMA, 16 (2020), 059, 31 pp., arXiv: 1812.06200 | DOI | MR

[7] P. Berglund, M. Henningson, “Landau–Ginzburg orbifolds, mirror symmetry and the elliptic genus”, Nucl. Phys. B, 433:2 (1995), 311–332 | DOI | MR

[8] P. Berglund, T. Hübsch, “A generalized construction of mirror manifolds”, Nucl. Phys. B, 393:1–2 (1993), 377–391 | DOI | MR

[9] M. Krawitz, FJRW rings and Landau–Ginzburg mirror symmetry, arXiv: 0906.0796

[10] A. Francis, T. Jarvis, D. Johnson, R. Suggs, “Landau–Ginzburg mirror symmetry for orbifolded Frobenius algebras”, String-Math 2011, Proceedings of Symposia in Pure Mathematics, 85, eds. J. Block, J. Distler, R. Donagi, E. Sharpe, AMS, Providence, RI, 2012, 333–353 | DOI | MR

[11] A. Basalaev, A. Takahashi, “Hochschild cohomology and orbifold Jacobian algebras associated to invertible polynomials”, J. Noncommut. Geom., 14:3 (2020), 861–877 | DOI | MR

[12] A. Basalaev, A. Takahashi, E. Werner, Orbifold Jacobian algebras for invertible polynomials, arXiv: 1608.08962

[13] A. Basalaev, A. Takahashi, E. Werner, “Orbifold Jacobian algebras for exceptional unimodal singularities”, Arnold Math. J., 3:4 (2017), 483–498, arXiv: 1702.02739 | DOI | MR

[14] D. Mukai, Nonabelian Landau–Ginzburg orbifolds and Calabi–Yau/Landau–Ginzburg correspondence, arXiv: 1704.04889

[15] W. Ebeling, S. M. Gusein-Zade, “A version of the Berglund–Hübsch–Henningson duality with non-abelian groups”, Int. Math. Res. Not., 2021:16 (2019), 12305–12329 | DOI | MR

[16] W. Ebeling, S. M. Gusein-Zade, “Dual invertible polynomials with permutation symmetries and the orbifold Euler characteristic”, SIGMA, 16 (2020), 051, 15 pp., arXiv: 1907.11421 | DOI | MR

[17] A. Basalaev, A. Ionov, Hochschild cohomology of Fermat type polynomials with non-abelian symmetries, arXiv: 2011.05937