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37
Parallel spinors and connections with skewsymmetric torsion in string theory
, 2008
"... We describe all almost contact metric, almost hermitian and G2structures admitting a connection with totally skewsymmetric torsion tensor, and prove that there exists at most one such connection. We investigate its torsion form, its Ricci tensor, the Dirac operator and the ∇parallel spinors. In p ..."
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Cited by 151 (7 self)
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We describe all almost contact metric, almost hermitian and G2structures admitting a connection with totally skewsymmetric torsion tensor, and prove that there exists at most one such connection. We investigate its torsion form, its Ricci tensor, the Dirac operator and the ∇parallel spinors. In particular, we obtain solutions of the type II string equations in dimension n = 5, 6 and 7.
Geometry of hyperKähler connections with torsion
 Comm. Math. Phys
"... Abstract The internal space of a N=4 supersymmetric model with WessZumino term has a connection with totally skewsymmetric torsion and holonomy in Sp(n). We study the mathematical background of this type of connections. In particular, we relate it to classical Hermitian geometry, construct homogen ..."
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Cited by 66 (9 self)
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Abstract The internal space of a N=4 supersymmetric model with WessZumino term has a connection with totally skewsymmetric torsion and holonomy in Sp(n). We study the mathematical background of this type of connections. In particular, we relate it to classical Hermitian geometry, construct homogeneous as well as inhomogeneous examples, characterize it in terms of holomorphic data, develop its potential theory and reduction theory.
A nogo theorem for string warped compactifications
, 2000
"... We give necessary conditions for the existence of perturbative heterotic and type II string warped compactifications preserving eight and four supersymmetries to four spacetime dimensions, respectively. In particular, we find that the only compactifications of heterotic string with the spin connecti ..."
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Cited by 60 (21 self)
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We give necessary conditions for the existence of perturbative heterotic and type II string warped compactifications preserving eight and four supersymmetries to four spacetime dimensions, respectively. In particular, we find that the only compactifications of heterotic string with the spin connection embedded in the gauge connection and type II strings are those on CalabiYau manifolds with constant dilaton. We obtain similar results for compactifications to six and to two dimensions.
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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Cited by 58 (8 self)
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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.
Vanishing theorems on Hermitian manifolds
, 2003
"... We prove the vanishing of the Dolbeault cohomology groups on Hermitian manifolds with ddcharmonic Kähler form and positive (1, 1)part of the Ricci form of the Bismut connection. This implies the vanishing of the Dolbeault cohomology groups on complex surfaces which admit a conformal class of Herm ..."
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Cited by 25 (7 self)
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We prove the vanishing of the Dolbeault cohomology groups on Hermitian manifolds with ddcharmonic Kähler form and positive (1, 1)part of the Ricci form of the Bismut connection. This implies the vanishing of the Dolbeault cohomology groups on complex surfaces which admit a conformal class of Hermitian metrics, such that the Ricci tensor of the canonical Weyl structure is positive. As a corollary we obtain that any such surface must be rational with c2 1> 0. As an application, the pth Dolbeault cohomology groups of a leftinvariant complex structure compatible with a biinvariant metric on a compact even dimensional Lie group are computed.
Vanishing Theorems and String Backgrounds
, 2000
"... 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 ..."
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Cited by 23 (1 self)
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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 twoform which describe gravity in the presence of a Maxwell field. More recently, Riemannian or pseudoRiemannian manifolds M equipped with (closed) forms
CANONICAL BUNDLES OF COMPLEX NILMANIFOLDS, WITH APPLICATIONS TO HYPERCOMPLEX GEOMETRY
, 2009
"... A nilmanifold is a quotient of a nilpotent group G by a cocompact discrete subgroup. A complex nilmanifold is one which is equipped with a Ginvariant complex structure. We prove that a complex nilmanifold has trivial canonical bundle. This is used to study hypercomplex nilmanifolds (nilmanifolds ..."
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Cited by 20 (7 self)
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A nilmanifold is a quotient of a nilpotent group G by a cocompact discrete subgroup. A complex nilmanifold is one which is equipped with a Ginvariant complex structure. We prove that a complex nilmanifold has trivial canonical bundle. This is used to study hypercomplex nilmanifolds (nilmanifolds with a triple of Ginvariant complex structures which satisfy quaternionic relations). We prove that a hypercomplex nilmanifold admits an HKT (hyperkähler with torsion) metric if and only if the underlying hypercomplex structure is abelian. Moreover, any Ginvariant HKTmetric on a nilmanifold is balanced with respect to all associated complex structures.
Almost contact manifolds, connections with torsion and parallel spinors
 J. reine angew. Math
"... Abstract. We classify locally homogeneous quasiSasakian manifolds in dimension five that admit a parallel spinor ψ of algebraic type F · ψ = 0 with respect to the unique connection ∇ preserving the quasiSasakian structure and with totally skewsymmetric torsion. We introduce a certain conformal tr ..."
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Cited by 18 (0 self)
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Abstract. We classify locally homogeneous quasiSasakian manifolds in dimension five that admit a parallel spinor ψ of algebraic type F · ψ = 0 with respect to the unique connection ∇ preserving the quasiSasakian structure and with totally skewsymmetric torsion. We introduce a certain conformal transformation of almost contact metric manifolds and discuss a link between them and the dilation function in 5dimensional string theory. We find natural conditions implying conformal invariances of parallel spinors. We present topological obstructions to the existence of parallel spinors in the compact case. 1.
Hypercomplex structures on fourdimensional Lie groups
 C. R. Acad. Bulgare Sci
, 1997
"... Abstract. The purpose of this paper is to classify invariant hypercomplex structures on a 4dimensional real Lie group G. It is shown that the 4dimensional simply connected Lie groups which admit invariant hypercomplex structures are the additive group H of the quaternions, the multiplicative group ..."
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Cited by 18 (5 self)
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Abstract. The purpose of this paper is to classify invariant hypercomplex structures on a 4dimensional real Lie group G. It is shown that the 4dimensional simply connected Lie groups which admit invariant hypercomplex structures are the additive group H of the quaternions, the multiplicative group H ∗ of nonzero quaternions, the solvable Lie groups acting simply transitively on the real and complex hyperbolic spaces, RH 4 and CH 2, respectively, and the semidirect product C⋊C. We show that the spaces CH 2 and C⋊C possess an RP 2 of (inequivalent) invariant hypercomplex structures while the remaining groups have only one, up to equivalence. Finally, the corresponding hyperhermitian 4manifolds are determined. 1.