[Paper Review] String Theory on Calabi-Yau Manifolds

Brian Greene|ArXiv.org|Feb 23, 1997

Black Holes and Theoretical PhysicsPhysics and Astronomy17 references213 citations

TL;DR

This paper provides a comprehensive introduction to quantum geometry in string theory compactified on Calabi-Yau manifolds, emphasizing how duality and non-perturbative effects reveal that classically distinct Calabi-Yau manifolds can be physically equivalent via mirror symmetry. The key contribution is the demonstration that topology change—previously thought impossible in classical geometry—can occur smoothly in string theory through conifold and flop transitions, with the full moduli space of string theory unified across different geometric phases.

ABSTRACT

These lectures are devoted to introducing some of the basic features of quantum geometry that have been emerging from compactified string theory over the last couple of years. The developments discussed include new geometric features of string theory which occur even at the classical level as well as those which require non-perturbative effects. These lecture notes are based on an evolving set of lectures presented at a number of schools but most closely follow a series of seven lectures given at the TASI-96 summer school on Strings, Fields and Duality.

Motivation & Objective

• To elucidate the emergence of quantum geometry in string theory compactified on Calabi-Yau manifolds, particularly through non-perturbative effects.
• To explain how mirror symmetry relates topologically distinct Calabi-Yau manifolds via duality, making them physically equivalent despite differing classical geometries.
• To investigate how topology change—previously forbidden in classical geometry—can occur smoothly in string theory via conifold and flop transitions.
• To unify the moduli spaces of Kähler and complex structure deformations across different geometric phases using toric geometry and conformal field theory.
• To establish the role of duality in connecting strongly coupled string theories to weakly coupled duals, including M-theory and heterotic compactifications.

Proposed method

• Uses N=2 superconformal field theory to analyze the structure of string compactifications on Calabi-Yau manifolds, focusing on chiral primary fields and spectral flow.
• Applies the N=2 superconformal algebra to classify BPS states and understand the role of U(1) charge in mirror symmetry and moduli space structure.
• Employs toric geometry to describe Kähler and complex structure moduli spaces, enabling explicit construction of mirror pairs and resolution of singularities.
• Analyzes moduli spaces by extending the complexified Kähler moduli space to include non-perturbative effects, revealing a unified structure across geometric phases.
• Utilizes asymptotic mirror symmetry and the monomial-divisor mirror map to relate geometric invariants to conformal field theory data.
• Applies duality to map non-perturbative conifold transitions in type II string theory to perturbative transitions in heterotic theories on K3×T², demonstrating the power of duality in resolving singularities.

Experimental results

Research questions

• RQ1 How can two topologically distinct Calabi-Yau manifolds give rise to the same physical string theory?
• RQ2 What is the role of non-perturbative effects in enabling smooth topology change in string compactifications?
• RQ3 How do the moduli spaces of Kähler and complex structure deformations unify in the full non-perturbative string theory landscape?
• RQ4 In what way does mirror symmetry extend beyond perturbative geometry to include non-perturbative corrections?
• RQ5 How does duality resolve the apparent discontinuity of topology change in classical geometry within the framework of quantum string theory?

Key findings

• Mirror symmetry establishes a physical equivalence between topologically distinct Calabi-Yau manifolds, with one being the mirror of the other, even though they are not isomorphic as complex manifolds.
• Topology change via flop transitions is smooth and continuous in string theory, occurring even at the perturbative level, challenging classical geometric intuition.
• Conifold transitions require non-perturbative string effects and are resolved via dual descriptions in heterotic string theory, where they appear as perturbative transitions.
• The full moduli space of type II string theory on Calabi-Yau threefolds is connected and unified, with different geometric phases (e.g., large radius, conifold, small radius) forming a single component through duality.
• The complexified Kähler moduli space must be enlarged to include non-perturbative corrections to achieve isomorphism between mirror pairs, revealing a richer quantum geometry than classical geometry suggests.
• Toric geometry provides a powerful framework to construct and analyze mirror manifolds, with the monomial-divisor mirror map relating geometric invariants to conformal field theory data.