An Alternative Topological Field Theory of Generalized Complex Geometry

We propose a new topological field theory on generalized complex geometry in two dimension using AKSZ formulation. Zucchini's model is $A$ model in the case that the generalized complex structuredepends on only a symplectic structure. Our new model is $B$ model in the case that the generalized complex structure depends on only a complex structure.Comment: 29 pages, typos and references correcte

A heterotic sigma model with novel target geometry

We construct a (1,2) heterotic sigma model whose target space geometry consists of a transitive Lie algebroid with complex structure on a Kaehler manifold. We show that, under certain geometrical and topological conditions, there are two distinguished topological half--twists of the heterotic sigma model leading to A and B type half--topological models. Each of these models is characterized by the usual topological BRST operator, stemming from the heterotic (0,2) supersymmetry, and a second BRST operator anticommuting with the former, originating from the (1,0) supersymmetry. These BRST operators combined in a certain way provide each half--topological model with two inequivalent BRST structures and, correspondingly, two distinct perturbative chiral algebras and chiral rings. The latter are studied in detail and characterized geometrically in terms of Lie algebroid cohomology in the quasiclassical limit.Comment: 83 pages, no figures, 2 references adde

Toda Fields on Riemann Surfaces: remarks on the Miura transformation

We point out that the Miura transformation is related to a holomorphic foliation in a relative flag manifold over a Riemann Surface. Certain differential operators corresponding to a free field description of $W$--algebras are thus interpreted as partial connections associated to the foliation.Comment: AmsLatex 1.1, 10 page

Deformation Theory of Holomorphic Vector Bundles, Extended Conformal Symmetry and Extensions of 2D Gravity

Developing on the ideas of R. Stora and coworkers, a formulation of two dimensional field theory endowed with extended conformal symmetry is given, which is based on deformation theory of holomorphic and Hermitian spaces. The geometric background consists of a vector bundle $E$ over a closed surface $\Sigma$ endowed with a holomorphic structure and a Hermitian structure subordinated to it. The symmetry group is the semidirect product of the automorphism group ${\rm Aut}(E)$ of $E$ and the extended Weyl group ${\rm Weyl}(E)$ of $E$ and acts on the holomorphic and Hermitian structures. The extended Weyl anomaly can be shifted into an automorphism chirally split anomaly by adding to the action a local counterterm, as in ordinary conformal field theory. The dependence on the scale of the metric on the fiber of $E$ is encoded in the Donaldson action, a vector bundle generalization of the Liouville action. The Weyl and automorphism anomaly split into two contributions corresponding respectively to the determinant and projectivization of $E$. The determinant part induces an effective ordinary Weyl or diffeomorphism anomaly and the induced central charge can be computed.Comment: 49 pages, plain TeX. A number of misprints have been correcte

Fracture Mechanics of Thin, Cracked Plates Under Tension, Bending and Out-of-Plane Shear Loading

Cracks in the skin of aircraft fuselages or other shell structures can be subjected to very complex stress states, resulting in mixed-mode fracture conditions. For example, a crack running along a stringer in a pressurized fuselage will be subject to the usual in-plane tension stresses (Mode-I) along with out-of-plane tearing stresses (Mode-III like). Crack growth and initiation in this case is correlated not only with the tensile or Mode-I stress intensity factor, K(sub I), but depends on a combination of parameters and on the history of crack growth. The stresses at the tip of a crack in a plate or shell are typically described in terms of either the small deflection Kirchhoff plate theory. However, real applications involve large deflections. We show, using the von-Karman theory, that the crack tip stress field derived on the basis of the small deflection theory is still valid for large deflections. We then give examples demonstrating the exact calculation of energy release rates and stress intensity factors for cracked plates loaded to large deflections. The crack tip fields calculated using the plate theories are an approximation to the actual three dimensional fields. Using three dimensional finite element analyses we have explored the relationship between the three dimensional elasticity theory and two dimensional plate theory results. The results show that for out-of-plane shear loading the three dimensional and Kirchhoff theory results coincide at distance greater than h/2 from the crack tip, where h/2 is the plate thickness. Inside this region, the distribution of stresses through the thickness can be very different from the plate theory predictions. We have also explored how the energy release rate varies as a function of crack length to plate thickness using the different theories. This is important in the implementation of fracture prediction methods using finite element analysis. Our experiments show that under certain conditions, during fatigue crack growth, the presence of out-of-plane shear loads induces a great deal of contact and friction on the crack surfaces, dramatically reducing crack growth rate. A series of experiments and a proposed computational approach for accounting for the friction is discussed

Generalized structures of N=1 vacua

We characterize N=1 vacua of type II theories in terms of generalized complex structure on the internal manifold M. The structure group of T(M) + T*(M) being SU(3) x SU(3) implies the existence of two pure spinors Phi_1 and Phi_2. The conditions for preserving N=1 supersymmetry turn out to be simple generalizations of equations that have appeared in the context of N=2 and topological strings. They are (d + H wedge) Phi_1=0 and (d + H wedge) Phi_2 = F_RR. The equation for the first pure spinor implies that the internal space is a twisted generalized Calabi-Yau manifold of a hybrid complex-symplectic type, while the RR-fields serve as an integrability defect for the second.Comment: 21 pages. v2, v3: minor changes and correction

M-theory on eight-manifolds revisited: N=1 supersymmetry and generalized Spin(7) structures

The requirement of ${\cal N}=1$ supersymmetry for M-theory backgrounds of the form of a warped product ${\cal M}\times_{w}X$, where $X$ is an eight-manifold and ${\cal M}$ is three-dimensional Minkowski or AdS space, implies the existence of a nowhere-vanishing Majorana spinor $\xi$ on $X$. $\xi$ lifts to a nowhere-vanishing spinor on the auxiliary nine-manifold $Y:=X\times S^1$, where $S^1$ is a circle of constant radius, implying the reduction of the structure group of $Y$ to $Spin(7)$. In general, however, there is no reduction of the structure group of $X$ itself. This situation can be described in the language of generalized $Spin(7)$ structures, defined in terms of certain spinors of $Spin(TY\oplus T^*Y)$. We express the condition for ${\cal N}=1$ supersymmetry in terms of differential equations for these spinors. In an equivalent formulation, working locally in the vicinity of any point in $X$ in terms of a `preferred' $Spin(7)$ structure, we show that the requirement of ${\cal N}=1$ supersymmetry amounts to solving for the intrinsic torsion and all irreducible flux components, except for the one lying in the $\bf{27}$ of $Spin(7)$, in terms of the warp factor and a one-form $L$ on $X$ (not necessarily nowhere-vanishing) constructed as a $\xi$ bilinear; in addition, $L$ is constrained to satisfy a pair of differential equations. The formalism based on the group $Spin(7)$ is the most suitable language in which to describe supersymmetric compactifications on eight-manifolds of $Spin(7)$ structure, and/or small-flux perturbations around supersymmetric compactifications on manifolds of $Spin(7)$ holonomy.Comment: 24 pages. V2: introduction slightly extended, typos corrected in the text, references added. V3: the role of Spin(7) clarified, erroneous statements thereof corrected. New material on generalized Spin(7) structures in nine dimensions. To appear in JHE