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Hodge diamonds of the Landau--Ginzburg orbifolds

Alexey Basalaev, Andrei Ionov

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.

Hodge diamonds of the Landau--Ginzburg orbifolds

Abstract

Consider the pairs with being a polynomial defining a quasihomogeneous singularity and being a subgroup of , preserving . In particular, is not necessary abelian. Assume further that contains the grading operator and satisfies the Calabi-Yau condition. We prove that the nonvanishing bigraded pieces of the B-model state space of 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.
Paper Structure (24 sections, 19 theorems, 69 equations, 2 figures)

This paper contains 24 sections, 19 theorems, 69 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Hodge diamonds of Calabi--Yau manifolds
  3. Landau--Ginzburg orbifolds
  4. Calabi--Yau/Landau--Ginzburg correspondence
  5. Preliminaries and notations
  6. Quasihomogeneous singularities
  7. Graph of a quasihomogeneous singularity
  8. Graph decomposition of a polynomial
  9. Graph exponents matrix
  10. Invertible polynomials
  11. Symmetries
  12. Fixed loci of the $\boldsymbol{{\mathrm{GL}}_f}$ elements
  13. Age of a $\boldsymbol{{\mathrm{GL}}_f}$ element
  14. Diagonal symmetries and a graph $\boldsymbol{\Gamma_f}$
  15. Diagonal symmetries of an invertible singularity
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $f \in {\mathbb{C}}[x_1,\dots,x_N]$ be a quasihomogeneous polynomial satisfying Calabi--Yau condition and defining an isolated singularity. Then for any $G \subseteq {\mathrm{SL}}_f$, such that $J \subseteq G$ the state space ${\mathcal{B}}(f,G)$ forms a diamond of size $N-2$ in a sense that it

Figures (2)

  • Figure 1: Graphs of $N=3$ quasihomogenous singularities (see Example \ref{['ex: N=3 quasihomogeneous']}).
  • Figure 2: Connected component of $\Gamma_f$ (see Proposition \ref{['prop: graph of f']}).

Theorems & Definitions (38)

  • Theorem 1.1
  • Example 2.1
  • Example 2.2: AGV85
  • Example 2.3: AGV85
  • Example 2.4
  • Proposition 2.5: cf. HK
  • Example 2.6
  • Remark 2.7
  • Example 2.8
  • Proposition 2.9
  • ...and 28 more