[Paper Review] Quantization and ``theta functions''

Andrei Tyurin|ArXiv.org|Apr 11, 1999

Homotopy and Cohomology in Algebraic Topology18 references3 citations

TL;DR

This paper generalizes classical theta functions to higher-rank vector bundles on Abelian varieties and K3 surfaces using geometric quantization, constructing a special basis in spaces of conformal blocks. It extends Mumford's theta function theory via holomorphic geometry, providing a framework that unifies conformal field theory with algebraic geometry and offers new tools for mirror symmetry.

ABSTRACT

Geometric Quantization links holomorphic geometry with real geometry, a relation that is a prototype for the modern development of mirror symmetry. We show how to use this treatment to construct a special basis in every space of conformal blocks. This is a direct generalization of the basis of theta functions with characteristics in every complete linear system on an Abelian variety (see Mumford's "Tata lectures on theta" cite(Mumford)). The same construction generalizes the classical theory of theta functions to vector bundles of higher rank on Abelian varieties and K3 surfaces. We also discuss the geometry behind these constructions.

Motivation & Objective

• To extend the classical theory of theta functions with characteristics on Abelian varieties to higher-rank vector bundles.
• To construct a canonical basis in spaces of conformal blocks using geometric quantization.
• To unify holomorphic and real geometry in a way that models modern mirror symmetry.
• To generalize Mumford's 'Tata lectures on theta' to higher-rank settings on K3 surfaces and Abelian varieties.
• To provide a geometric framework for understanding conformal blocks through holomorphic line bundles and projective geometry.

Proposed method

• Utilizes geometric quantization to link holomorphic geometry with real geometry, forming the foundation of the construction.
• Applies the theory of conformal blocks in algebraic geometry to define vector spaces equipped with a natural basis.
• Constructs a special basis analogous to theta functions with characteristics, but in higher-rank vector bundles.
• Employs holomorphic line bundles and projective embeddings to generalize the classical theta line bundle construction.
• Uses the geometry of Abelian varieties and K3 surfaces to extend the classical theta function framework beyond line bundles.
• Applies techniques from algebraic geometry, including complete linear systems and moduli of vector bundles, to ensure the basis is canonical and geometrically meaningful.

Experimental results

Research questions

• RQ1 How can the classical theory of theta functions on Abelian varieties be generalized to higher-rank vector bundles?
• RQ2 What is the geometric structure underlying the conformal blocks in higher-rank settings?
• RQ3 How does geometric quantization provide a canonical basis in spaces of conformal blocks?
• RQ4 In what way do K3 surfaces and Abelian varieties support a generalized theta function theory beyond line bundles?
• RQ5 What is the role of holomorphic geometry in constructing such bases, and how does it relate to mirror symmetry?

Key findings

• The paper constructs a canonical basis in every space of conformal blocks, analogous to the basis of theta functions with characteristics on Abelian varieties.
• This basis arises naturally from geometric quantization, linking holomorphic and real structures in algebraic geometry.
• The construction generalizes Mumford's theta function theory to higher-rank vector bundles on Abelian varieties and K3 surfaces.
• The method provides a direct generalization of classical theta functions using holomorphic line bundles and projective geometry.
• The framework offers a new geometric interpretation of conformal blocks, rooted in algebraic geometry and compatible with mirror symmetry principles.
• The results establish a bridge between conformal field theory and algebraic geometry through a unified, geometrically natural construction.