By Saemann C.
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Extra info for Aspects of twistor geometry and supersymmetric field theories within superstring theory (hep-th 0603098)
A holomorphic vector bundle E of rank k over a manifold M with dim M = n is a (k + n)-dimensional complex manifold E endowed with a holomorphic projection π : E → M satisfying the conditions (i ) π −1 (p) is a k-dimensional complex vector space for all p ∈ M . (ii ) For each point p ∈ M , there is a neighborhood U and a biholomorphism φU : π −1 (U ) → U × k . The maps φU are called local trivializations. (iii ) The transition functions fU V are holomorphic maps U ∩ V → GL(k, ). Holomorphic vector bundles of dimension k = 1 are called line bundles.
59) P 5: P 5 : c1 (z0 , . . , z5 ) = c2 (z0 , . . 60) where c1 and c2 are homogeneous cubic polynomials. §11 Calabi-Yau manifolds from vector bundles over P 1 . A very prominent class of local Calabi-Yau manifolds can be obtained from the vector bundles O(a) ⊕ O(b) → P 1 , where the Calabi-Yau condition of vanishing first Chern class amounts to a+b = −2. To describe these bundles, we will always choose the standard inhomogeneous coor1 dinates λ± on the patches U± covering the base P 1 , together with the coordinates z± 2 in the fibres over the patches U .
8 Rigid Calabi-Yau manifolds. There is a class of so-called rigid Calabi-Yau manifolds, which do not allow for deformations of the complex structure. 5, as it follows that the mirrors of these rigid Calabi-Yau manifolds have no K¨ ahler moduli, which is inconsistent with them being K¨ ahler manifolds. 2, §13. A ten-dimensional string theory is usually split into a four-dimensional theory and a six-dimensional N = 2 superconformal theory. For a theory to preserve N = 2 supersymmetry, the manifold has to be K¨ ahler, conformal invariance demands Ricciflatness.
Aspects of twistor geometry and supersymmetric field theories within superstring theory (hep-th 0603098) by Saemann C.