Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 024, 25 pages      arXiv:2307.01295      https://doi.org/10.3842/SIGMA.2024.024

Hodge Diamonds of the Landau-Ginzburg Orbifolds

Alexey Basalaev ab and Andrei Ionov c
a) Faculty of Mathematics, National Research University Higher School of Economics, 6 Usacheva Str., 119048 Moscow, Russia
b) Skolkovo Institute of Science and Technology, 3 Nobelya Str., 121205 Moscow, Russia
c) Boston College, Department of Mathematics, Maloney Hall, Fifth Floor, Chestnut Hill, MA 02467-3806, USA

Received July 12, 2023, in final form March 06, 2024; Published online March 25, 2024

Abstract
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.

Key words: singularity theory; Landau-Ginzburg orbifolds.

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