We present a construction of superconformal field theories for manifolds with Spin(7) holonomy. Geometrically these models correspond to the realization of Spin(7) manifolds as anti-holomorphic quotients of Calabi-Yau fourfolds. Describing the fourfolds as Gepner models and requiring anomaly cancellation we determine the resulting Betti numbers of the Spin(7) superconformal field theory. As in the G2 case, we find that the Gepner model and the geometric result disagree.

We study M-theory on two classes of manifolds of Spin(7) holonomy that are developing an isolated conical singularity. We construct explicitly a new class of Spin(7) manifolds and analyse in detail the topology of the corresponding classical spacetimes. We discover also an intricate interplay between various anomalies in M-theory, string theory, and gauge theory within these models, and in particular find a connection between half-integral G-fluxes in M-theory and

Using D2-brane probes, we study various properties of M-theory on singular, non-compact manifolds of G2 and Spin(7) holonomy. We derive mirror pairs of N = 1 supersymmetric three-dimensional gauge theories, and apply this technique to realize exceptional holonomy manifolds as both Coulomb and Higgs branches of the D2-brane world-volume theory. We derive a “G2 quotient construction” of non-compact manifolds which admit a metric of G2 holonomy. We further discuss the moduli space of such manifolds, including the structure of geometrical transitions in each case. For completeness, we also include familiar examples of manifolds with SU(3) and Sp(2) holonomy, where some of the new

We investigate nonperturbative effects in M-theory compactifications arising from wrapped membranes. In particular, we show that in d = 4, N = 1 compactifications along manifolds of G2 holonomy, membranes wrapped on rigid supersymmetric 3-cycles induce nonzero corrections to the superpotential. Thus, membrane instantons destabilize many M-theory compactifications. Our computation shows that the low energy description of membrane physics is usefully described in terms of three-dimensional topological field theories, and the superpotential is expressed in terms of topological invariants of the 3-cycle. We discuss briefly some applications of these results. For example, using mirror symmetry we derive a counting formula for supersymmetric three-cycles in certain Calabi-Yau manifolds. July

We show various vanishing theorems for the cohomology groups of compact hermitian manifolds for which the Bismut connection has (restricted) holonomy contained in SU(n) and classify all such manifolds of dimension four. In this way we provide necessary conditions for the existence of such structures on hermitian manifolds. Then we apply our results to solutions of the string equations and show that such solutions admit various cohomological restrictions like for example that under certain natural assumptions the plurigenera vanish. We also find that under some assumptions the string equations are equivalent to the condition Riemannian manifolds equipped with a closed form have found many applications in various branches of mathematics and physics. In physics, the classical example is that of manifolds equipped with a closed two-form which describe gravity in the presence of a Maxwell field. More recently, Riemannian or pseudo-Riemannian manifolds M equipped with (closed) forms

Using mirror pairs (M3,W3) in type II superstring compactifications on Calabi-Yau threefolds, we study, geometrically, F-theory duals of M-theory on seven manifolds with G2 holonomy. We first develop a way for getting Landau Ginzburg (LG) Calabi-Yau threefolds W3, embedded in four complex dimensional toric varieties, mirror to sigma model on toric Calabi-Yau threefolds M3. This method gives directly the right dimension without introducing non dynamical variables. Then, using toric geometry tools, we discuss the duality between M-theory on S1 ×M3 Z2 with G2 holonomy and F-theory on elliptically fibered Calabi-Yau fourfolds with SU(4) holonomy, containing W3 mirror manifolds. Illustrating examples are presented.

If X and Y are a mirror pair of Calabi--Yau threefolds, mirror symmetry should extend to an isomorphism between the type IIA string theory compactified on X and the type IIB string theory compactified on Y , with all nonperturbative effects included. We study the implications which this proposal has for the structure of the semiclassical moduli spaces of the compactified type II theories. For the type IIB theory, the form taken by discrete shifts in the Ramond-Ramond scalars exhibits an unexpected dependence on the B-field. (Based on a talk at the Trieste Workshop on S-Duality and Mirror Symmetry.) Mirror Symmetry and the Type II String David R. Morrison a a Department of Mathematics, Box 90320, Duke University, Durham, NC 27708-0320, USA If X and Y are a mirror pair of Calabi--Yau threefolds, mirror symmetry should extend to an isomorphism between the type IIA string theory compactified on X and the type IIB string theory compactified on Y , with all nonperturbative effect...

Compact manifolds of G2 holonomy may be constructed by dividing a seven-torus by some discrete symmetry group and then blowing up the singularities of the resulting orbifold. We classify possible group elements that may be used in this construction and use this classification to find a set of possible orbifold groups. We then derive the moduli Kähler potential for M-theory on the resulting class of G2 manifolds with blown up co-dimension four singularities.

We find a one-parameter family of new G2 holonomy metrics and demonstrate that it can be extended to a two-parameter family. These metrics play an important role as the supergravity dual of the large N limit of four dimensional supersymmetric Yang-Mills. We show that these G2 holonomy metrics describe the M theory lift of the supergravity solution describing a collection of D6-branes wrapping the supersymmetric three-cycle of the deformed conifold geometry for any value of the string coupling constant. June

Using two dimensional (2D) N = 4 sigma model, with U(1) r gauge symmetry, and introducing the ADE Cartan matrices as gauge matrix charges, we build ” toric ” hyper-Kahler eight real dimensional manifolds X8. Dividing by one toric geometry circle action of X8 manifolds, we present examples describing quotients X7 = X8 U(1) of G2 holonomy. In particular, for the Ar Cartan matrix, the quotient space is a cone on a S 2 bundle over r intersecting WCP 2 (1,2,1) weighted projective spaces according to the Ar Dynkin diagram.