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Conformal Perturbation Theory on K3: The Quartic Gepner Point

Christoph A. Keller

TL;DR

The paper develops a concrete framework for conformal perturbation theory applied to Gepner-model realizations of K3, focusing on the quartic Gepner point (2)^4. By expressing weight shifts as a perturbative series in the modulus coupling λ and regulating higher-order integrals with hard-sphere minimal subtraction, it computes first- and second-order liftings of a set of light states of weight (h,ḣ) = (1/4,1/4). The analysis distinguishes moduli into type A, B, and C and finds that type B moduli lift certain states at first order and, after careful second-order treatment, produce minima in ḣ ≈ 0.16 near λ ≈ ±0.32, while type C moduli yield their own nontrivial second-order effects; overall the results illustrate a toy symmetry-enhancement mechanism near a Gepner point and provide explicit computational steps for correlators and integrals in CY-related rational CFTs.

Abstract

The Gepner model (2)^4 describes the sigma model of the Fermat quartic K3 surface. Moving through the nearby moduli space using conformal perturbation theory, we investigate how the conformal weights of its fields change at first and second order and find approximate minima. This serves as a toy model for a mechanism that could produce new chiral fields and possibly new nearby rational CFTs.

Conformal Perturbation Theory on K3: The Quartic Gepner Point

TL;DR

The paper develops a concrete framework for conformal perturbation theory applied to Gepner-model realizations of K3, focusing on the quartic Gepner point (2)^4. By expressing weight shifts as a perturbative series in the modulus coupling λ and regulating higher-order integrals with hard-sphere minimal subtraction, it computes first- and second-order liftings of a set of light states of weight (h,ḣ) = (1/4,1/4). The analysis distinguishes moduli into type A, B, and C and finds that type B moduli lift certain states at first order and, after careful second-order treatment, produce minima in ḣ ≈ 0.16 near λ ≈ ±0.32, while type C moduli yield their own nontrivial second-order effects; overall the results illustrate a toy symmetry-enhancement mechanism near a Gepner point and provide explicit computational steps for correlators and integrals in CY-related rational CFTs.

Abstract

The Gepner model (2)^4 describes the sigma model of the Fermat quartic K3 surface. Moving through the nearby moduli space using conformal perturbation theory, we investigate how the conformal weights of its fields change at first and second order and find approximate minima. This serves as a toy model for a mechanism that could produce new chiral fields and possibly new nearby rational CFTs.
Paper Structure (30 sections, 120 equations, 2 figures)

This paper contains 30 sections, 120 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Conformal perturbation theory
  3. Shift of conformal dimension
  4. Perturbation theory up to second order
  5. $N=2$ minimal models
  6. Primaries and fusion rules
  7. The bosonic subalgebra and characters
  8. The $k=2$ minimal model and correlation functions
  9. Gepner models
  10. The Quartic Gepner model: $(2)^4$
  11. Expressions for the characters
  12. Chiral primaries
  13. CY extension and correlation functions
  14. Lifting the $h=1/4, \bar{h} =1/4$ fields under type B moduli
  15. First order lifting
  16. ...and 15 more sections

Figures (2)

  • Figure 1: $\bar{h}$ as a function of the coupling constant $\lambda$ for the two lifted fields. Note the minima at $\lambda=\pm0.32$.
  • Figure 2: $\bar{h}$ as a function of the coupling constant $\lambda$ for the two lifted fields.