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The Srní lectures on nonintegrable geometries with torsion
 Arch. Math. (Brno
, 2006
"... Abstract. This review article intends to introduce the reader to nonintegrable geometric structures on Riemannian manifolds and invariant metric connections with torsion, and to discuss recent aspects of mathematical physics—in particular superstring theory—where these naturally appear. Connections ..."
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Abstract. This review article intends to introduce the reader to nonintegrable geometric structures on Riemannian manifolds and invariant metric connections with torsion, and to discuss recent aspects of mathematical physics—in particular superstring theory—where these naturally appear. Connections with skewsymmetric torsion are exhibited as one of the main tools to understand nonintegrable geometries. To this aim a a series of key examples is presented and successively dealt with using the notions of intrinsic torsion and characteristic connection of a Gstructure as unifying principles. The General Holonomy Principle bridges over to parallel objects, thus motivating the discussion of geometric stabilizers, with emphasis on spinors and differential forms. Several Weitzenböck formulas for Dirac operators associated with torsion connections enable us to discuss spinorial field equations, such as those governing the common sector of type II superstring theory.
Deformations of generalized calibrations and compact nonKähler manifolds with vanishing first Chern class
, 2002
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Killing Spinor Equations In Dimension 7 And Geometry Of Integrable G_2Manifolds
, 2008
"... We compute the scalar curvature of 7dimensional G2manifolds admitting a connection with totally skewsymmetric torsion. We prove the formula for the general solution of the Killing spinor equation and express the Riemannian scalar curvature of the solution in terms of the dilation function and the ..."
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Cited by 41 (0 self)
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We compute the scalar curvature of 7dimensional G2manifolds admitting a connection with totally skewsymmetric torsion. We prove the formula for the general solution of the Killing spinor equation and express the Riemannian scalar curvature of the solution in terms of the dilation function and the NS 3form field. In dimension n = 7 the dilation function involved in the second fermionic string equation has an interpretation as a conformal change of the underlying integrable G2structure into a cocalibrated one of pure type W3.
Special metric structures and closed forms
, 2004
"... In recent work, N. Hitchin described special geometries in terms of a variational problem for closed generic pforms. In particular, he introduced on 8manifolds the notion of an integrable PSU(3)structure which is defined by a closed and coclosed 3form. In this thesis, we first investigate this P ..."
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Cited by 33 (4 self)
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In recent work, N. Hitchin described special geometries in terms of a variational problem for closed generic pforms. In particular, he introduced on 8manifolds the notion of an integrable PSU(3)structure which is defined by a closed and coclosed 3form. In this thesis, we first investigate this PSU(3)geometry further. We give necessary conditions for the existence of a topological PSU(3)structure (that is, a reduction of the structure group to PSU(3) acting through its adjoint representation). We derive various obstructions for the existence of a topological reduction to PSU(3). For compact manifolds, we also find sufficient conditions if the PSU(3)structure lifts to an SU(3)structure. We find nontrivial, (compact) examples of integrable PSU(3)structures. Moreover, we give a Riemannian characterisation of topological PSU(3)structures through an invariant spinor valued 1form and show that the PSU(3)structure is integrable if and only if the spinor valued 1form is harmonic with respect to the twisted Dirac operator. Secondly, we define new generalisations of integrable G2 and Spin(7)manifolds which can be transformed by the action of both diffeomorphisms and 2forms. These are defined by special closed even or odd forms. Contraction on the vector bundle T ⊕ T ∗ defines an inner product of signature (n, n),
On types of nonintegrable geometries
"... Abstract. We study the types of nonintegrable Gstructures on Riemannian manifolds. In particular, geometric types admitting a connection with totally skewsymmetric torsion are characterized. 8dimensional manifolds equipped with a Spin(7)structure play a special role. Any geometry of that type a ..."
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Cited by 30 (2 self)
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Abstract. We study the types of nonintegrable Gstructures on Riemannian manifolds. In particular, geometric types admitting a connection with totally skewsymmetric torsion are characterized. 8dimensional manifolds equipped with a Spin(7)structure play a special role. Any geometry of that type admits a unique connection with totally skewsymmetric torsion. Under weak conditions on the structure group we prove that this geometry is the only one with this property. Finally, we discuss the automorphism group of a Riemannian manifold with a fixed nonintegrable Gstructure. Contents
HETEROTIC SUPERSYMMETRY, ANOMALY CANCELLATION AND EQUATIONS OF MOTION
, 2009
"... We show that the heterotic supersymmetry (Killing spinor equations) and the anomaly cancellation imply the heterotic equations of motion up to two loops in dimensions five, six, seven, eight if and only if the connection on the tangent bundle is an instanton. For heterotic compactifications in dimen ..."
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Cited by 25 (1 self)
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We show that the heterotic supersymmetry (Killing spinor equations) and the anomaly cancellation imply the heterotic equations of motion up to two loops in dimensions five, six, seven, eight if and only if the connection on the tangent bundle is an instanton. For heterotic compactifications in dimension six this fixes an unique choice of the connection on the tangent bundle in the α′ correction of the anomaly cancellation.
Intersecting Brane Solutions in String and MTheory
 HEPTH/9803116 J.M.FIGUEROAO’FARRIL, “INTERSECTING BRANE GEOMETRIES” HEPTH/9806040 P.K.TOWNSEND, “BRANE THEORY SOLITIONS”, HEPTH/0004039
, 2008
"... We review various aspects of configurations of intersecting branes, including the conditions for preservation of supersymmetry. In particular, we discuss the projection conditions on the Killing spinors for given brane configurations and the relation to calibrations. This highlights the close connec ..."
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Cited by 18 (2 self)
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We review various aspects of configurations of intersecting branes, including the conditions for preservation of supersymmetry. In particular, we discuss the projection conditions on the Killing spinors for given brane configurations and the relation to calibrations. This highlights the close connection between intersecting branes and branes wrapping supersymmetric cycles as well as special holonomy manifolds. We also explain how these conditions can be used to find supergravity solutions without directly solving the Einstein equations. The description of intersecting branes is considered both in terms of the brane worldvolume theories and as supergravity solutions. There are wellknown simple procedures (harmonic function rules) for writing down the supergravity solutions for supersymmetric configurations of orthogonally intersecting branes. However, such solutions involve smeared or delocalised branes. We describe several methods of constructing solutions with less smearing, including some fully localised solutions. Some applications of these supergravity solutions are also considered – in particular the study of black holes and gauge theories.
Generalised G2manifolds
 Comm. Math. Phys
, 2006
"... We define new Riemannian structures on 7manifolds by a differential form of mixed degree which is the critical point of a (possibly constrained) variational problem over a fixed cohomology class. The unconstrained critical points generalise the notion of a manifold of holonomy G2, while the constra ..."
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Cited by 18 (2 self)
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We define new Riemannian structures on 7manifolds by a differential form of mixed degree which is the critical point of a (possibly constrained) variational problem over a fixed cohomology class. The unconstrained critical points generalise the notion of a manifold of holonomy G2, while the constrained ones give rise to a new geometry without a classical counterpart. We characterise these structures by the means of spinors and show the integrability conditions to be equivalent to the supersymmetry equations on spinors in type II superstring theory with bosonic background fields. In particular, this geometry can be described by two linear metric connections with skewsymmetric torsion. Finally, we construct explicit examples by using the device of Tduality. 1