[Paper Review] Consistency of Orbifold Conformal Field Theories on K3

Katrin Wendland|ArXiv.org|Oct 30, 2000

Black Holes and Theoretical Physics20 references3 citations

TL;DR

This paper determines the exact locations of G-orbifold conformal field theories on K3 surfaces within the moduli space of N = (4,4) superconformal field theories, using the known structure of the moduli space and consistency conditions. It explicitly computes the Kummer-type lattices for all such orbifolds (G = ZM, M ∈ {2,3,4,6}, G = bDn, n ∈ {4,5}, G = bT), and uniquely fixes the B-field values along exceptional divisors in the blow-up of orbifold singularities without relying on D-geometry. The results are shown to be consistent with the classical McKay correspondence.

ABSTRACT

We explicitly determine the locations of G orbifold conformal field theories, G=Z_M, M=2,3,4,6, G=\hat D_n, n=4,5, or G the binary tetrahedral group \hat T, within the moduli space M^{K3} of N=(4,4) superconformal field theories associated to K3. This is achieved purely from the known description of the moduli space [AM94] and the requirement of a consistent embedding of orbifold conformal field theories within M^{K3}. We calculate the Kummer type lattices for all these orbifold limits. Our method allows an elementary derivation of the B-field values in direction of the exceptional divisors that arise from the orbifold procedure [Asp95,Dou97,BI97], without recourse to D-geometry. We show that our consistency requirement fixes these values uniquely and determine them explicitly. The relation of our results to the classical McKay correspondence is discussed.

Motivation & Objective

• To determine the precise locations of G-orbifold CFTs on K3 within the moduli space of N = (4,4) SCFTs.
• To derive the Kummer-type lattices for all such orbifold limits (G = ZM, bDn, bT) using lattice-theoretic consistency conditions.
• To uniquely fix the B-field values in the directions of exceptional divisors arising from orbifold resolution, without using D-geometry.
• To establish a direct link between the consistency of orbifold CFTs and the classical McKay correspondence.

Proposed method

• Uses the known description of the moduli space MK3 as O+(Heven(X,Z))\O+(Heven(X,R))/SO(4)×O(20), based on [AM94].
• Applies the isomorphism (B.1) to map four-planes in R4,20 to geometric data (Σ, V, B), where B encodes the B-field.
• Imposes consistency conditions on the orbifold CFT embedding to fix the B-field values along exceptional divisors.
• Performs explicit lattice calculations to determine the Kummer-type sublattices for each group G, using the structure of Heven(X,Z) and its sublattices.
• Uses the discriminant form and embedding theorems (Theorem A.2) to characterize the orthogonal complement of the orbifold-invariant cohomology.
• Relies on the fact that the orbifold limit corresponds to a four-plane in T4,20 that is compatible with the lattice structure and the action of G on cohomology.

Experimental results

Research questions

• RQ1 What are the precise locations of G-orbifold CFTs (G = ZM, bDn, bT) within the moduli space of N = (4,4) SCFTs on K3?
• RQ2 How can the B-field values along the exceptional divisors of the orbifold resolution be uniquely determined from consistency conditions alone?
• RQ3 What is the structure of the Kummer-type lattice for each such orbifold limit?
• RQ4 How does the consistency of the orbifold CFT embedding constrain the geometric interpretation (Σ, V, B) of the corresponding four-plane in the moduli space?
• RQ5 In what way does the resulting B-field structure relate to the classical McKay correspondence?

Key findings

• The B-field values in the directions of the exceptional divisors are uniquely fixed by the consistency requirement of embedding the orbifold CFT into MK3, without recourse to D-geometry.
• For each group G (ZM, bDn, bT), the Kummer-type lattice is explicitly computed as the sublattice of H2(X,Z) generated by the exceptional divisors and the orbifold-invariant cohomology.
• The method provides an elementary derivation of the B-field values, which are found to be rational and explicitly determined by the group action and lattice embedding.
• The results are consistent with the classical McKay correspondence, as the structure of the Kummer lattice matches the representation theory of the corresponding finite group G.
• The four-plane corresponding to the orbifold CFT is shown to lie in a specific position in T4,20, determined by the relative position to the even self-dual lattice H2(X,Z).
• The paper confirms and extends earlier results on Abelian orbifolds (e.g., [Ber88]), showing agreement for cyclic groups G = ZM.