[Paper Review] Closed String Tachyons, Non-Supersymmetric Orbifolds and Generalised McKay Correspondence

Yang‐Hui He|ArXiv.org|Jan 22, 2003

Black Holes and Theoretical Physics38 references9 citations

TL;DR

This paper extends the study of closed string tachyon condensation to non-supersymmetric orbifolds of $\mathbb{C}^2$ by classifying finite subgroups of $GL(2;\mathbb{C})$ and generalizing the Hirzebruch-Jung continued fraction resolution for these groups. It establishes a generalized McKay correspondence linking the tachyon condensation process to partial resolutions of singularities, revealing that the decay of non-supersymmetric orbifolds geometrically corresponds to resolution via generalized continued fractions and identifying conditions under which discrete torsion arises in product groups.

ABSTRACT

We study closed string tachyon condensation on general non-supersymmetric orbifolds of C^2. Extending previous analyses on Abelian cases, we present the classification of quotients by discrete finite subgroups of GL(2; C) as well as the generalised Hirzebruch-Jung continued fractions associated with the resolution data. Furthermore, we discuss the intimate connexions with certain generalised versions of the McKay Correspondence.

Motivation & Objective

• To extend the analysis of closed string tachyon condensation beyond Abelian orbifolds to general non-supersymmetric $\mathbb{C}^2/\Gamma$ orbifolds with $\Gamma \subset GL(2;\mathbb{C})$.
• To classify finite subgroups of $GL(2;\mathbb{C})$ using a construction from $SL(2;\mathbb{C})$ ADE groups and central extensions.
• To generalize the Hirzebruch-Jung continued fraction resolution method to non-Abelian and non-Gorenstein orbifolds.
• To establish a generalized McKay correspondence that links tachyon condensation to geometric resolution of singularities in non-supersymmetric settings.
• To identify conditions under which discrete torsion arises in product groups, particularly in $AD$ and $AE_{6,7}$ types, and to explore their implications for D-brane probes and moduli spaces.

Proposed method

• Constructing all finite subgroups of $GL(2;\mathbb{C})$ via a surjective homomorphism $\psi: Z \times SL(2;\mathbb{C}) \to GL(2;\mathbb{C})$, where $Z$ is the center of $GL(2;\mathbb{C})$, and forming semidirect products using isomorphisms between quotient groups.
• Defining each group $\Gamma$ as a quadruple $(G_1, N_1; G_2, N_2)$, where $G_1/N_1 \cong G_2/N_2$, to systematically generate all non-cyclic finite subgroups of $GL(2;\mathbb{C})$.
• Applying the inverse toric algorithm to resolve the orbifold singularities via generalized Hirzebruch-Jung continued fractions, extending methods previously used for Abelian groups.
• Using worldsheet renormalization group (RG) flow in two-dimensional conformal field theory to model tachyon condensation as a flow toward an IR fixed point corresponding to a resolved geometry.
• Analyzing the chiral ring structure in the CFT to geometrically interpret the tachyon condensation process as a partial resolution of the orbifold singularity.
• Computing the Schur multiplier $M(G)$ for product groups to determine the presence of discrete torsion, using the formula $M(G_1 \times G_2) \simeq M(G_1) \times M(G_2) \times \mathrm{Hom}_{\mathbb{Z}}(G_1/G_1', G_2/G_2')$.

Experimental results

Research questions

• RQ1 How can the classification of finite subgroups of $GL(2;\mathbb{C})$ be systematically extended beyond $SL(2;\mathbb{C})$ ADE groups to include non-supersymmetric orbifolds of $\mathbb{C}^2$?
• RQ2 What is the generalized Hirzebruch-Jung continued fraction resolution for non-Abelian, non-Gorenstein orbifolds, and how does it relate to tachyon condensation?
• RQ3 How does the generalized McKay correspondence emerge in the context of non-supersymmetric orbifolds, and how does it relate to the geometric resolution of singularities?
• RQ4 Under what conditions does discrete torsion arise in product groups such as $A_n E_7$ or $A_9 E_6^{(III)}$, and what are its physical implications for D-brane probes and gauge theories?
• RQ5 Can the tachyon condensation process in the closed string sector be fully described via worldsheet RG flows, and how does it lead to a transition to a supersymmetric Calabi-Yau geometry?

Key findings

• All finite subgroups of $GL(2;\mathbb{C})$ are classified via a quadruple construction $(G_1, N_1; G_2, N_2)$, where $G_1/N_1 \cong G_2/N_2$, and the group is realized as $\Gamma = \psi(G_1 \times_\phi G_2)$, with $\psi$ the natural homomorphism from $Z \times SL(2;\mathbb{C})$ to $GL(2;\mathbb{C})$.
• The generalized Hirzebruch-Jung continued fraction resolution is extended to non-Abelian groups, providing a geometric description of the tachyon condensation process as a partial resolution of the orbifold singularity.
• The tachyon condensation process is shown to correspond to an RG flow in the worldsheet CFT, with the IR fixed point representing a resolved, supersymmetric geometry.
• Discrete torsion arises in product groups such as $A_n E_7$ (for $n$ odd) and $A_9 E_6^{(III)}$, with the latter admitting a $\mathbb{Z}_3$ discrete torsion, leading to three disconnected components in the quiver diagram when the NS-NS B-field is turned on.
• The Schur multiplier computation confirms that only $AD$ and $AE_{6,7}$ types support non-trivial discrete torsion, with $M(G)$ determined by $\mathrm{Hom}_{\mathbb{Z}}(G_1/G_1', G_2/G_2')$ when the individual $M(G_i)$ are trivial.
• The generalized McKay correspondence is established as a link between the representation theory of $\Gamma \subset GL(2;\mathbb{C})$ and the resolution of singularities via continued fractions, extending the classical correspondence to non-supersymmetric settings.