Supercoset CFT’s for String Theories on Non-compact Special Holonomy Manifolds

Extended Abstract

It is known that an arbitrary (m − 1)-dimensional Einstein spaceX _m _ ₁possesses a Ricci flat metric on its m-dimensional coneC(X _{m_1}) of the form

$$d{{s}^{2}} = d{{r}^{2}} + {{r}^{2}}ds_{{{{X}_{{m - 1}}}}}^{2}, $$

(Lin)

where r is the radial coordinate, and the special holonomies onC(X _m _ ₁₎originate from the “weak special holonomies” onX _m _₁. To be more precise, theSU(n) Sp(n)G2 andSpin(7) holonomies on the coneC(X _{m_1})are in one to one correspondence with theSasaki-Einstein (m =2n)tri-Sasakian (m =4n)nearly Kähler (m =7) andweakG2(m= 8) structures onX _{m_1}respectively. This fact is very useful to systematically construct special holonomy manifolds with conical singularities, because the Einstein homogeneous spacesK _{m_1}=G/Hendowed with these geometrical structures are well known from the old days of Kaluza-Klein supergravity (SUGRA).

On the other hand, there also exists extensive literature on the worldsheet approaches to these conical backgrounds in string theory. Early literature is for the conifold and K3-singularity and more recent studies are in the case ofSU(n)holonomiesemphasizing the role ofN= 2 Liouville theory and the holographically dual descriptions based on the (wrapped) NS5-brane geometry. All of these cases possess the worldsheetN= 2 superconformal symmetry and have been discussed from the viewpoints of the “non-compact extensions” of Gepner models. There are also several related results from the spacetime view points, but with the RR-flux at infinity.

While these constructions in the case of theN= 2 supersymmetry have been rather successful, it is difficult to construct string vacua on the conical backgrounds withSpin(7) and G2 holonomies, which possess at most theN= 1 worldsheet SUSY. Partial attempts to construct the string vacua ofSpin(7) and G2 holonomies with conical singularities have been given in [1, 2]. General structure of string theory on manifolds withSpin(7) and G2 was discussed in [3] in particular from the point of view of the existence of extended chiral algebras. There are also several results [4] for the CFT constructions of compact G2 andSpin(7) manifolds based on the geometrical method of Joyce.

The main purpose of this paper is to give a systematic way of constructing special holonomy manifolds with conical singularities based on the solvableN =1 SCFT’s, which may be regarded as a natural generalization of the construction in theN =2 category mentioned above. Our strategy is quite simple:We formally replace an Einstein homogeneous space X = G/H by an N=1supercoset CFTM = (GxSO(n)) /H(n = dimG —dimH) based on the affine Lie algebra of G and H. SO(n) stands for n free fermions. We then add the N = 1 Liouville sector in place of the radial degrees of freedom.We may call our construction as the “CFT cone” as opposed to the original geometrical cone construction. Of course, one should keep in mind that the coset CFT (WZW model) ofG/His not identical to the non-linear cr-model with the target manifoldG/Hbecause of the presence of NSBfield in WZW models. Nevertheless, as we will see in the following, taking the supercoset CFT associated with the Einstein homogeneous space provides a very good anzats for the superstring vacua of special holonomy. We completely classify these coset constructions (at least for the cosetsGIwith compact simple groupsG)which include the models found in [1, 2] as well as many of the vacua in theN =2 category presented in [5, 6, 7, 8, 9, 10]. Among other things, we will find that our CFT cone approach leads to the right amount of worldsheet and spacetime SUSY’s as expected from geometrical grounds in many examples.