Hodge Diamonds of the Landau–Ginzburg Orbifolds

Alexey Basalaev ; Andrei Ionov

Symmetry, integrability and geometry: methods and applications, Tome 20 (2024)

Résumé

Consider the pairs $(f,G)$ with $f = f(x_1,\dots,x_N)$ being a polynomial defining a quasihomogeneous singularity and $G$ being a subgroup of ${\rm SL}(N,\mathbb{C})$, preserving $f$. In particular, $G$ is not necessary abelian. Assume further that $G$ contains the grading operator $j_f$ and $f$ satisfies the Calabi–Yau condition. We prove that the nonvanishing bigraded pieces of the B-model state space of $(f,G)$ form a diamond. We identify its topmost, bottommost, leftmost and rightmost entries as one-dimensional and show that this diamond enjoys the essential horizontal and vertical isomorphisms.

Détail

Keywords: singularity theory, Landau–Ginzburg orbifolds.

Bibliographie

[1] Arnold V.I., Gusein-Zade S.M., Varchenko A.N., Singularities of differentiable maps, v. I, Monogr. Math., 82, The classification of critical points, caustics and wave fronts, Birkhäuser, Boston, MA, 1985 | DOI | MR | Zbl

[2] Basalaev A., Ionov A., “Mirror map for Fermat polynomials with a nonabelian group of symmetries”, Theoret. and Math. Phys., 209 (2021), 1491–1506, arXiv: 2103.16884 | DOI | MR | Zbl

[3] Basalaev A., Ionov A., “Hochschild cohomology of Fermat type polynomials with non-abelian symmetries”, J. Geom. Phys., 174 (2022), 104450, 28 pp., arXiv: 2011.05937 | DOI | MR | Zbl

[4] Basalaev A., Takahashi A., “Hochschild cohomology and orbifold Jacobian algebras associated to invertible polynomials”, J. Noncommut. Geom., 14 (2020), 861–877, arXiv: 1802.03912 | DOI | MR | Zbl

[5] Basalaev A., Takahashi A., Werner E., Orbifold {J}acobian algebras for exceptional unimodal singularities, Arnold Math. J., 3 (2017), 483–498, arXiv: 1702.02739 | DOI | MR | Zbl

[6] Basalaev A., Takahashi A., Werner E., “Orbifold Jacobian algebras for invertible polynomials”, J. Singul., 26 (2023), 92–127, arXiv: 1608.08962 | DOI | MR

[7] Berglund P., Henningson M., “Landau–Ginzburg orbifolds, mirror symmetry and the elliptic genus”, Nuclear Phys. B, 433 (1995), 311–332, arXiv: hep-th/9401029 | DOI | MR | Zbl

[8] Berglund P., Hübsch T., “A generalized construction of mirror manifolds”, Nuclear Phys. B, 393 (1993), 377–391, arXiv: hep-th/9201014 | DOI | MR | Zbl

[9] Chiodo A., Ruan Y., “LG/CY correspondence: the state space isomorphism”, Adv. Math., 227 (2011), 2157–2188, arXiv: 0908.0908 | DOI | MR | Zbl

[10] Clawson A., Johnson D., Morais D., Priddis N., White C.B., “Mirror map for Landau–Ginzburg models with nonabelian groups”, J. Geom. Phys. (to appear) , arXiv: 2302.02782 | DOI | MR

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

[12] Ebeling W., Gusein-Zade S.M., “A version of the Berglund–Hübsch–Henningson duality with non-abelian groups”, Int. Math. Res. Not., 2021 (2021), 12305–12329, arXiv: 1807.04097 | DOI | MR | Zbl

[13] Ebeling W., Takahashi A., “Variance of the exponents of orbifold Landau–Ginzburg models”, Math. Res. Lett., 20 (2013), 51–65, arXiv: 1203.3947 | DOI | MR | Zbl

[14] Francis A., Jarvis T., Johnson D., Suggs R., “Landau–Ginzburg mirror symmetry for orbifolded Frobenius algebras”, String-Math 2011, Proc. Sympos. Pure Math., 85, American Mathematical Society, Providence, RI, 2012, 333–353, arXiv: 1111.2508 | DOI | MR

[15] Griffiths P., Harris J., Principles of algebraic geometry, Wiley Classics Lib., John Wiley Sons, New York, 1994 | DOI | MR | Zbl

[16] Hertling C., Kurbel R., On the classification of quasihomogeneous singularities, J. Singul., 4 (2012), 131–153, arXiv: 1009.0763 | DOI | MR | Zbl

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

[18] Ionov A., “McKay correspondence and orbifold equivalence”, J. Pure Appl. Algebra, 227 (2023), 107297, 11 pp., arXiv: 2202.12135 | DOI | MR | Zbl

[19] Kaufmann R.M., “Orbifolding Frobenius algebras”, Internat. J. Math., 14 (2003), 573–617, arXiv: math.AG/0107163 | DOI | MR | Zbl

[20] Kaufmann R.M., “Singularities with symmetries, orbifold Frobenius algebras and mirror symmetry”, Gromov–Witten theory of Spin Curves and Orbifolds, Contemp. Math., 403, American Mathematical Society, Providence, RI, 2006, 67–116, arXiv: math.AG/0312417 | DOI | MR | Zbl

[21] Krawitz M., FJRW rings and Landau–Ginzburg mirror symmetry, Ph.D. Thesis, The University of Michigan, 2010 | MR

[22] Kreuzer M., “The mirror map for invertible LG models”, Phys. Lett. B, 328 (1994), 312–318, arXiv: hep-th/9402114 | DOI | MR

[23] Kreuzer M., Skarke H., “On the classification of quasihomogeneous functions”, Comm. Math. Phys., 150 (1992), 137–147, arXiv: hep-th/9202039 | DOI | MR | Zbl

[24] Milnor J., Orlik P., “Isolated singularities defined by weighted homogeneous polynomials”, Topology, 9 (1970), 385–393 | DOI | MR | Zbl

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

[26] Orlik P., Solomon L., Singularities. {II} {A}utomorphisms of forms,, Math. Ann., 231 (1978), 229–240 | DOI | MR | Zbl

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

[28] Saito K., “Quasihomogene isolierte Singularitäten von Hyperflächen”, Invent. Math., 14 (1971), 123–142 | DOI | MR | Zbl

[29] Shklyarov D., “On Hochschild invariants of Landau–Ginzburg orbifolds”, Adv. Theor. Math. Phys., 24 (2020), 189–258, arXiv: 1708.06030 | DOI | MR | Zbl

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

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

[32] Yu X., “McKay correspondence and new Calabi–Yau threefolds”, Int. Math. Res. Not., 2017 (2017), 6444–6468, arXiv: 1507.00577 | DOI | MR | Zbl