Conifold Type Singularities, N=2 Liouville and SL(2;R)/U(1) Theories

TL;DR

This paper explores non-compact conformal field theories related to conifold singularities, revealing that such models exhibit only complex structure deformations and analyzing their D-brane spectra and topological features.

Contribution

It demonstrates that coupled N=2 Liouville and SL(2;R)/U(1) theories model conifold-like singularities with specific deformation properties and provides detailed analysis of D-brane spectra and intersection numbers.

Findings

Models exhibit only (c,c), (a,a) massless states.

Deformations correspond to N=2 Liouville and SL(2;R)/U(1) theories.

Computed D-brane intersection numbers and Witten index.

Abstract

In this paper we discuss various aspects of non-compact models of CFT of the type: $ \prod_{j=1}^{N_L} {N=2 Liouville theory}_j \otimes \prod_{i=1}^{N_M} {N=2 minimal model}_i $ and $ \prod_{j=1}^{N_L}{SL(2;R)/U(1) supercoset}_j \otimes \prod_{i=1}^{N_M} {N=2 minimal model}_i $. These models are related to each other by T-duality. Such string vacua are expected to describe non-compact Calabi-Yau compactifications, typically ALE fibrations over (weighted) projective spaces. We find that when the Liouville ($SL(2;R)/U(1)$) theory is coupled to minimal models, there exist only (c,c), (a,a) ((c,a), (a,c))-type of massless states in CY 3 and 4-folds and the theory possesses only complex (K\"{a}hler) structure deformations. Thus the space-time has the characteristic feature of a conifold type singularity whose deformation (resolution) is given by the N=2 Liouville (SL(2;R)/U(1)) theory. Spectra of compact BPS D-branes determined from the open string sector are compared with those of massless moduli. We compute the open string Witten index and determine intersection numbers of vanishing cycles. We also study non-BPS branes of the theory that are natural extensions of the ``unstable B-branes'' of the SU(2) WZW model in hep-th/0105038.