[Paper Review] Resolution of Orbifold Singularities in String Theory

Paul S. Aspinwall|ArXiv.org|Mar 21, 1994

Black Holes and Theoretical Physics9 references55 citations

TL;DR

This paper establishes a correspondence between the resolution of orbifold singularities in string theory and the moduli space of N=(2,2) superconformal field theories, using mirror symmetry and toric geometry to analyze the blow-up process for abelian quotients 𝕔²/ℤₙ. It derives exact expressions for the Kähler moduli in terms of twist fields and shows that the size of the exceptional divisor grows with n, with the Calabi-Yau phase becoming singular at a critical radius determined by n.

ABSTRACT

In this paper the relationship between the classical description of the resolution of quotient singularities and the string picture is reviewed in the context of N=(2,2) superconformal field theories. A method for the analysis of quotients locally of the form C^d/G where G is abelian is presented. Methods derived from mirror symmetry are used to study the moduli space of the blowing-up process. The case C^2/Z_n is analyzed explicitly. This is largely a review paper to appear in "Essays on Mirror Manifolds, II".

Motivation & Objective

• To understand how string theory encodes the resolution of orbifold singularities even before classical blow-up is performed.
• To connect the classical geometry of orbifold resolution with the quantum (stringy) geometry of N=(2,2) superconformal field theories.
• To analyze the moduli space of the blowing-up process for abelian quotient singularities 𝕔ᵈ/G using mirror symmetry techniques.
• To provide an explicit field-theoretic description of the Kähler moduli in terms of twist fields for the 𝕔²/ℤₙ case.
• To determine the physical size of the exceptional divisor in the resolved geometry as a function of n.

Proposed method

• Uses mirror symmetry to map the resolution of 𝕔²/ℤₙ singularities to a Landau-Ginzburg model with a specific superpotential.
• Applies star-subdivision of the fan in toric geometry to describe successive blow-ups, each resolving one singularity and producing a ℙ¹ exceptional divisor.
• Derives the Kähler form (B+iJ)₁ as an integral transform involving the Γ-function and hypergeometric series, valid in different convergence regions.
• Expresses the Kähler modulus in terms of the string coupling ψ via the relation z = ψ⁻ⁿ, enabling analytic continuation across phases.
• Uses the closed-form expression for n=2 to validate the general framework and compare with the n>2 case.
• Evaluates the Kähler modulus at the phase transition point z=(n−1)ⁿ⁻¹/nⁿ to determine the size of the exceptional divisor.

Experimental results

Research questions

• RQ1 How does string theory encode the resolution of orbifold singularities before the classical blow-up is performed?
• RQ2 What is the precise field-theoretic realization of the Kähler moduli space in the context of N=(2,2) SCFTs for 𝕔²/ℤₙ?
• RQ3 How does the size of the exceptional divisor in the resolved geometry depend on the order n of the ℤₙ group action?
• RQ4 What is the analytic structure of the Kähler modulus in the twisted sector of the orbifold CFT?
• RQ5 How does the transition between the orbifold phase and the Calabi-Yau phase manifest in the moduli space?

Key findings

• The Kähler modulus (B+iJ)₁ for the 𝕔²/ℤₙ orbifold is given by a hypergeometric series expansion valid for |z| < (n−1)ⁿ⁻¹/nⁿ.
• For small ψ, the Kähler modulus behaves as (B+iJ)₁ = −1/2 + (1/2π)e^{(1/2+1/n)πi}ψ + O(ψ²), showing a phase shift dependent on n.
• At the phase transition point z = (n−1)ⁿ⁻¹/nⁿ, the B-field vanishes (B₁=0), and the Kähler form J₁ takes values: 0 for n=2, 0.11 for n=3, 0.18 for n=4, and 0.22 for n=5.
• For n=2, the Calabi-Yau phase extends all the way to ψ=0, meaning the orbifold and resolved theories differ only in the B-field; for n>2, the exceptional divisor has finite size in the singular theory.
• The exceptional divisor size increases with n, indicating that higher-order orbifolds have larger quantum corrections in the singular limit.
• The method successfully reproduces the Hirzebruch-Jung resolution chain of ℙ¹’s for 𝕔²/ℤₙ via a sequence of star-subdivisions and twist-field analysis.