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

TL;DR

The paper develops a non-compact Gepner-like framework by tensoring N=2 Liouville or SL(2,R)/U(1) cosets with N=2 minimal models and applying a U(1) charge projection, yielding modular-invariant vacua that describe non-compact Calabi-Yau manifolds such as ALE fibrations. It shows that the massless closed-string spectrum at these points encodes a conifold-type moduli structure with Liouville sectors controlling deformations and coset sectors controlling resolutions, and it provides detailed open-string diagnostics via Cardy states, Witten indices, and intersection numbers of vanishing cycles. The D-brane analysis reveals a spectrum of compact BPS branes, non-compact branes with both A- and B-types, and a class of non-BPS branes with tachyonic open channels, offering a consistent picture of brane dynamics in these geometries. Across CY3 and CY4, the massless spectrum is restricted to Kahler-type moduli (ac/ca) with cc/aa absent, while K3 exhibits the richer N=4 structure; together these results illuminate conifold physics in non-compact Calabi-Yau compactifications and suggest avenues for further geometric and DBI-based explorations.

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 (Kähler) 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.