Special Holonomy and Two-Dimensional Supersymmetric Sigma-Models (original) (raw)

Superstrings and manifolds of exceptional holonomy

Selecta Mathematica-new Series, 1995

The condition of having an N=1N=1N=1 spacetime supersymmetry for heterotic string leads to 4 distinct possibilities for compactifications namely compactifications down to 6,4,3 and 2 dimensions. Compactifications to 6 and 4 dimensions have been studied extensively before (corresponding to K3K3K3 and a Calabi-Yau threefold respectively). Here we complete the study of the other two cases corresponding to compactification down to

Supersymmetry and Lorentzian holonomy in various dimensions

Journal of High Energy Physics, 2004

We present a systematic method for constructing manifolds with Lorentzian holonomy group that are non-static supersymmetric vacua admitting covariantly constant light-like spinors. It is based on the metric of their Riemannian counterparts and the realization that, when certain conditions are satisfied, it is possible to promote constant moduli parameters into arbitrary functions of the light-cone time. Besides the general formalism, we present in detail several examples in various dimensions.

Euclidean supersymmetry, twisting and topological sigma models

Journal of High Energy Physics, 2008

We discuss two dimensional N-extended supersymmetry in Euclidean signature and its R-symmetry. For N = 2, the R-symmetry is SO(2) × SO(1, 1), so that only an A-twist is possible. To formulate a B-twist, or to construct Euclidean N = 2 models with Hflux so that the target geometry is generalised Kahler, it is necessary to work with a complexification of the sigma models. These issues are related to the obstructions to the existence of non-trivial twisted chiral superfields in Euclidean superspace.

The geometry of supersymmetric sigma-models

We review non-linear σ-models with (2,1) and (2,2) supersymmetry. We focus on off-shell closure of the supersymmetry algebra and give a complete list of (2, 2) superfields. We provide evidence to support the conjecture that all N = (2, 2) non-linear σ-models can be described by these fields. This in its turn leads to interesting consequences about the geometry of the target manifolds. One immediate corollary of this conjecture is the existence of a potential for hyper-Kähler manifolds, different from the Kähler potential, which does not only allow for the computation of the metric, but of the three fundamental twoforms as well. Several examples are provided: WZW models on SU (2) × U (1) and SU (2) × SU (2) and four-dimensional special hyper-Kähler manifolds.

Counterterms, Holonomy and Supersymmetry

Deserfest, 2006

The divergence structure of supergravity has long been a topic of concern because of the theory's non-renormalizability. In the context of string theory, where perturbative finiteness should be achieved, the supergravity counterterm structures remain nonetheless of importance because they still occur, albeit with finite coefficients. The leading nonvanishing supergravity counterterms have a particularly rich structure that has a bearing on the preservation of supersymmetry in string vacua in the presence of perturbative string corrections. Although the holonomy of such manifolds is deformed by the corrections, a Killing spinor structure nevertheless can persist. The integrability conditions for the existence of such Killing spinors remarkably remain consistent with the perturbed effective field equations.

Almost special holonomy in type IIA and M-theory

Nuclear Physics B, 2002

We consider spaces M 7 and M 8 of G 2 holonomy and Spin(7) holonomy in seven and eight dimensions, with a U (1) isometry. For metrics where the length of the associated circle is everywhere finite and non-zero, one can perform a Kaluza-Klein reduction of supersymmetric M-theory solutions (Minkowski) 4 × M 7 or (Minkowski) 3 × M 8 , to give supersymmetric solutions (Minkowski) 4 × Y 6 or (Minkowski) 3 × Y 7 in type IIA string theory with a nonsingular dilaton. We study the associated six-dimensional and seven-dimensional spaces Y 6 and Y 7 perturbatively in the regime where the string coupling is weak but still non-zero, for which the metrics remain Ricci-flat but that they no longer have special holonomy, at the linearised level. In fact they have "almost special holonomy," which for the case of Y 6 means almost Kähler, together with a further condition. For Y 7 we are led to introduce the notion of an "almost G 2 manifold," for which the associative 3-form is closed but not co-closed. We obtain explicit classes of non-singular metrics of almost special holonomy, associated with the near Gromov-Hausdorff limits of families of complete non-singular G 2 and Spin(7) metrics.

String and M-theory deformations of manifolds with special holonomy

Journal of High Energy Physics, 2005

The R 4-type corrections to ten and eleven dimensional supergravity required by string and M-theory imply corrections to supersymmetric supergravity compactifications on manifolds of special holonomy, which deform the metric away from special holonomy. Nevertheless, in many such cases, including Calabi-Yau compactifications of string theory and G 2-compactifications of M-theory, it has been shown that because of associated corrections to the supersymmetry transformation rules, the deformation remains supersymmetric. Here, we consider Spin(7) compactifications of string theory, showing that once again supersymmetry survives, despite the loss of Spin(7) holonomy in the deformed background. We also consider the deformations of Spin(7) backgrounds in M-theory, showing that supersymmetry is also maintained in that case. Finally, we consider the intrinsically M-theoretic case of compactifications on 10-manifolds of SU(5) holonomy. We obtain the corrected solutions, and show that these too are supersymmetric.