Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cohomology ring of unitary $N=(2,2)$ full vertex algebra and mirror symmetry

Yuto Moriwaki

TL;DR

The paper provides a rigorous VOAs-based framework for 2d $N=(2,2)$ SCFTs, defining holomorphic/topological twists and introducing cohomology rings $H(F,d_B)$ and $H(F,d_A)$ that encode 2d TFTs and Frobenius-structure. It establishes unitarity-driven refinements to the cohomology (real $\mathbb{R}^2$-grading), proves Hodge duality and T-duality relations, and develops a spectral-flow construction that links the twisted sectors to CY-type invariants such as the Witten genus. A central contribution is the proposal and partial realization of Hodge-theoretic mirror symmetry at the SCFT/VOA level via mirror algebras $\hat{F}$, supported by tests on abelian varieties and a special K3. The framework aims to connect Calabi–Yau geometry with rigorous algebraic structures in full VOAs, offering a route to proving Hodge-theoretic mirror symmetry through sigma-model-inspired CY-type VOAs and their induced cohomological dualities. Overall, the work lays a mathematically precise foundation for extracting geometric invariants from unitary $N=(2,2)$ full VOAs and for exploring mirror phenomena through cohomology, dualities, and spectral flow in a fully algebraic setting.

Abstract

The mirror symmetry among Calabi-Yau manifolds is mysterious, however, the mirror operation in 2d N=(2,2) supersymmetric conformal field theory (SCFT) is an elementary operation. In this paper, we mathematically formulate SCFTs using unitary full vertex operator superalgebras (full VOAs) and develop a cohomology theory of unitary SCFTs (aka holomorphic / topological twists). In particular, we introduce cohomology rings, Hodge numbers, and the Witten index of a unitary $N=(2,2)$ full VOA, and prove that the cohomology rings determine 2d topological field theories and give relations between them (Hodge duality and T-duality). Based on this, we propose a possible approach to prove the existence of mirror Calabi-Yau manifolds for the Hodge numbers using SCFTs. For the proof, one need a construction of sigma models connecting Calabi-Yau manifolds and SCFTs which is still not rigorous, but expected properties are tested for the case of Abelian varieties and a special K3 surface based on some unitary $N=(2,2)$ full VOAs.

Cohomology ring of unitary $N=(2,2)$ full vertex algebra and mirror symmetry

TL;DR

The paper provides a rigorous VOAs-based framework for 2d SCFTs, defining holomorphic/topological twists and introducing cohomology rings and that encode 2d TFTs and Frobenius-structure. It establishes unitarity-driven refinements to the cohomology (real -grading), proves Hodge duality and T-duality relations, and develops a spectral-flow construction that links the twisted sectors to CY-type invariants such as the Witten genus. A central contribution is the proposal and partial realization of Hodge-theoretic mirror symmetry at the SCFT/VOA level via mirror algebras , supported by tests on abelian varieties and a special K3. The framework aims to connect Calabi–Yau geometry with rigorous algebraic structures in full VOAs, offering a route to proving Hodge-theoretic mirror symmetry through sigma-model-inspired CY-type VOAs and their induced cohomological dualities. Overall, the work lays a mathematically precise foundation for extracting geometric invariants from unitary full VOAs and for exploring mirror phenomena through cohomology, dualities, and spectral flow in a fully algebraic setting.

Abstract

The mirror symmetry among Calabi-Yau manifolds is mysterious, however, the mirror operation in 2d N=(2,2) supersymmetric conformal field theory (SCFT) is an elementary operation. In this paper, we mathematically formulate SCFTs using unitary full vertex operator superalgebras (full VOAs) and develop a cohomology theory of unitary SCFTs (aka holomorphic / topological twists). In particular, we introduce cohomology rings, Hodge numbers, and the Witten index of a unitary full VOA, and prove that the cohomology rings determine 2d topological field theories and give relations between them (Hodge duality and T-duality). Based on this, we propose a possible approach to prove the existence of mirror Calabi-Yau manifolds for the Hodge numbers using SCFTs. For the proof, one need a construction of sigma models connecting Calabi-Yau manifolds and SCFTs which is still not rigorous, but expected properties are tested for the case of Abelian varieties and a special K3 surface based on some unitary full VOAs.

Paper Structure

This paper contains 26 sections, 83 theorems, 321 equations, 3 figures.

Table of Contents

  1. Preliminary
  2. $N=2$ superconformal algebra
  3. $N=2$ vertex operator superalgebra
  4. $N=(2,2)$ full vertex operator superalgebra and twisted modules
  5. Unitarity of $N=(2,2)$ full VOA
  6. Classification of twists
  7. 2d Topological field theory and Frobenius superalgebra
  8. Cohomology ring of unitary $N=(2,2)$ full vertex operator superalgebra
  9. Cohomology ring of $N=(2,2)$ full VOA
  10. Inequalities from unitarity and supersymmetry
  11. Volume form, Hodge duality and Frobenius algebra
  12. A,B-twists, mirror symmetry and T-duality
  13. Spectral flow twist of full vertex operator superalgebra
  14. Structure of full $\mathcal{H}$-vertex operator algebra
  15. Constant subalgebra of generalized full vertex algebra
  16. ...and 11 more sections

Key Result

Proposition 1.10

The automorphism group $\mathrm{Aut}\,(V_c^{N=2})$ is generated by $\alpha$ and $\{\exp (i \lambda J(0))\}_{\lambda \in \mathbb{C}^\times}$ and isomorphic to $\mathbb{C}^\times \rtimes \mathbb{Z}_2$.

Figures (3)

  • Figure 1:
  • Figure 2:
  • Figure 3:

Theorems & Definitions (179)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Remark 1.8
  • Definition 1.9
  • Proposition 1.10
  • ...and 169 more