We look for possible nonsupersymmetric black hole attractor solutions for type II compactification on (the mirror of) CY 3 (2, 128) expressed as a degree-12 hypersurface in WCP 4 [1, 1, 2, 2, 6]. In the process, (a) for points away from the conifold locus, we show that the existence of a non-supersymmetric attractor along with a consistent choice of fluxes and extremum values of the complex structure moduli, could be connected to the existence of an elliptic curve fibered over C 8 which may also be "arithmetic" (in some cases, it is possible to interpret the extremization conditions for the black-hole superpotential as an endomorphism involving complex multiplication of an arithmetic elliptic curve), and (b) for points near the conifold locus, we show that existence of non-supersymmetric black-hole attractors corresponds to a version of A 1 -singularity in the space Image(Z 6 → R 2 Z2 (֒→ R 3 )) fibered over the complex structure moduli space. The (derivatives of the) effective black hole potential can be thought of as a real (integer) projection in a suitable coordinate patch of the Veronese map: CP 5 → CP 20 , fibered over the complex structure moduli space. We also discuss application of Kallosh's attractor equations (which are equivalent to the extremization of the effective black-hole potential) for nonsupersymmetric attractors and show that (a) for points away from the conifold locus, the attractor equations demand that the attractor solutions be independent of one of the two complex structure moduli, and (b) for points near the conifold locus, the attractor equations imply switching off of one of the six components of the fluxes. Both these features are more obvious using the atractor equations than the extremization of the black hole potential.

The Iwasawa manifold is uplifted to seven-folds of either G 2 holonomy or SU (3) structure, explicit new metrics for the same having been constructed in this work. We uplift the Iwasawa manifold to a G 2 manifold through "size" deformation (of the Iwasawa metric), via Hitchin's Flow equations, showing also the impossibility of the uplift for "shape" and "size" deformations (of the Iwasawa metric). Using results of [1], we also uplift the Iwasawa manifold to a 7-fold with SU (3) structure through "size" and "shape" deformations via generalisation of Hitchin's Flow equations. For seven-folds with SU (3)-structure, the result could be interpreted as M 5-branes wrapping two-cycles embedded in the seven-fold (as in [1]) -a warped product of either a special hermitian six-fold or a balanced six-fold with the unit interval. There can be no uplift to seven-folds of SU (3) structure involving non-trivial "size" and "shape" deformations (of the Iwasawa metric) retaining the "standard complex structure" -the uplift generically makes one move in the space of almost complex structures such that one is neither at the standard complex structure point nor at the "edge". Using the results of [2], we show that given two "shape deformation" functions, and the dilaton, one can construct a Riemann surface obtained via Weierstraß representation for the conformal immersion of a surface in R l , for a suitable l, with the condition of having conformal immersion being a quadric in CP l−1 .1

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We perform a heat kernel asymptotics analysis of the nonperturbative superpotential obtained from wrapping of an M 2-brane around a supersymmetric noncompact three-fold embedded in a (noncompact) G 2 -manifold as obtained in [1], the three-fold being the one relevant to domain walls in Witten's MQCD [2], in the limit of small "ζ", a complex constant that appears in the Riemann surfaces relevant to defining the boundary conditions for the domain wall in MQCD. The MQCD-like configuration is interpretable, for small but non-zero ζ as a noncompact/"large" open membrane instanton, and for vanishing ζ, as the type IIA D0-brane (for vanishing M -theory circle radius). We find that the eta-function Seeley de-Witt coefficients vanish, and we get a perfect match between the zeta-function Seeley de-Witt coefficients (up to terms quadratic in ζ) between the Dirac-type operator and one of the two Laplace-type operators figuring in the superpotential. Given the dissimilar forms of the bosonic and the square of the fermionic operators, this is an extremely nontrivial check, from a spectral analysis point of view, of the expected residual supersymmetry for the nonperturbative configurations in M -theory considered in this work. 1

[3,4]) Meijer basis of solutions and monodromies (at 0,1 and ∞) in the large and small complex structure limits, as well as obtain the mirror hypersurface, which in the large Kähler limit, turns out to be either a bidegree-(6,6) hypersurface in