Cohomology ring of unitary $N=(2,2)$ full vertex algebra and mirror symmetry (2504.09919v1)

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

This paper by Yuto Moriwaki presents a formalism to define and explore the structure of unitary $N=(2,2)$ full vertex operator algebras (VOAs) and their cohomology rings with the aim to uncover insights into mirror symmetry among Calabi-Yau manifolds through supersymmetric conformal field theories (SCFTs). The text explores the intricate mathematical structures underlying the $N=(2,2)$ SCFTs and proposes methods to potentially prove the existence of mirror symmetry within the framework of theoretical physics.

Mathematical Formulation of SCFT

The paper begins with establishing a rigorous mathematical foundation for supersymmetric conformal field theories utilizing unique structures known as full vertex operator superalgebras (VOAs). It sets out to define cohomology theory associated with $N=(2,2)$ SCFTs, focusing on holomorphic/topological twists which lead to the construction of the cohomology rings, Hodge numbers, and Witten indices. The narrative explains how these entities collectively determine two-dimensional topological field theories and exhibit dualities such as Hodge duality and T-duality, which are central to understanding string theory and mirror symmetry.

Cohomology Ring and Topological Twists

The development of the cohomology ring centers around mathematical constructs such as the twist operations referred to as the A-twist and B-twist in physics. These are represented mathematically as superderivatives acting on elements within the SCFT. The author lays out conditions under which these superdrivatives lead to well-defined cohomology rings. The cohomology rings further underpin the definition of Hodge numbers, furnishing a detailed account of the algebraic structures that mirror different aspects of the more intuitive, geometric mirror symmetry often discussed in string theory.

Spectral Flow Formulation

A significant contribution of this paper is the formulation of the spectral flow within the framework of generalized full vertex operator algebras, which has implications for periodicity properties of SCFTs linked to both the chiral and anti-chiral sectors. This spectral flow, characterized in technical detail by an $R^2$-grading parameter, serves as a tool for exploring dualities and constructing mirrors in theoretical physics. By rigorously defining spectral flow twist as an algebraic operation rather than relying purely on path integral methods, the paper bridges a gap between abstract mathematical formalism and its physical implications.

Implications for Mirror Symmetry

The discussion on mirror symmetry aims to provide a novel approach to recognizing and proving mirror pairs of Calabi-Yau manifolds through the SCFT framework. The proposal of a 'mirror algebra' operation, which offers a way to prove the existence of mirror symmetry by establishing correspondences between specific algebraic structures, is among the paper's most intriguing elements. Through a sequence of steps involving characterizations of sigma models and topological twists, Moriwaki posits a methodology that could extend mathematical understanding of mirror symmetry beyond the traditional geometric interpretation.

Examples and Testing Conjectures

Finally, the paper provides concrete examples, particularly of SCFTs related to abelian varieties and K3 surfaces, to test the proposed conjectures regarding sigma models. Throughout these examples, the author highlights numerical results and algebraic structures which demonstrate the validity of the theoretical constructs, reinforcing the notion that these models can reproduce expected quantum invariants and genus structures, thus affirming the conjectures set forth.

Conclusion and Future Directions

Throughout, Moriwaki's work builds upon the mathematical and theoretical physics literature to offer compelling advancements in understanding SCFTs associated with Calabi-Yau manifolds. By proposing generalized full vertex algebra structures and exploring their implications for mirror symmetry, the paper sets the stage for further exploration into quantum field theories and their interconnectedness with algebraic geometry. The prospect of formal proofs of mirror symmetry through these innovative constructs holds significant promise for future developments in both mathematics and theoretical physics.