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47
Generalized complex structures and Lie brackets
, 2004
"... We remark that the equations underlying the notion of generalized complex structure have simple geometric meaning when passing to Lie algebroids/groupoids. Contents ..."
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We remark that the equations underlying the notion of generalized complex structure have simple geometric meaning when passing to Lie algebroids/groupoids. Contents
Topological strings in generalized complex space
, 2006
"... A twodimensional topological sigmamodel on a generalized CalabiYau target space X is defined. The model is constructed in BatalinVilkovisky formalism using only a generalized complex structure J and a pure spinor ρ on X. In the present construction the algebra of Qtransformations automatically ..."
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Cited by 37 (1 self)
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A twodimensional topological sigmamodel on a generalized CalabiYau target space X is defined. The model is constructed in BatalinVilkovisky formalism using only a generalized complex structure J and a pure spinor ρ on X. In the present construction the algebra of Qtransformations automatically closes offshell, the model transparently depends only on J, the algebra of observables and correlation functions for topologically trivial maps in genus zero are easily defined. The extended moduli space appears naturally. The familiar action of the twisted N = 2 CFT can be recovered after a gauge fixing. In the open case, we consider an example of generalized deformation of complex structure by a holomorphic Poisson bivector β and recover holomorphic noncommutative Kontsevich ∗product.
Generalized complex geometry, generalized branes and the Hitchin sigma model
 JHEP 0503 (2005) 022 [arXiv:hepth/0501062
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Dirac sigma models
, 2004
"... We introduce a new topological sigma model, whose fields are bundle maps from the tangent bundle of a 2dimensional worldsheet to a Dirac subbundle of an exact Courant algebroid over a target manifold. It generalizes simultaneously the (twisted) Poisson sigma model as well as the G/GWZW model. The ..."
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Cited by 19 (3 self)
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We introduce a new topological sigma model, whose fields are bundle maps from the tangent bundle of a 2dimensional worldsheet to a Dirac subbundle of an exact Courant algebroid over a target manifold. It generalizes simultaneously the (twisted) Poisson sigma model as well as the G/GWZW model. The equations of motion are satisfied, iff the corresponding classical field is a Lie algebroid morphism. The Dirac Sigma Model has an inherently topological part as well as a kinetic term which uses a metric on worldsheet and target. The latter contribution serves as a kind of regulator for the theory, while at least classically the gauge invariant content turns out to be independent of any additional structure. In the (twisted) Poisson case one may drop the kinetic term altogether, obtaining the WZPoisson sigma model; in general, however, it is compulsory for establishing the morphism property.
CourantNijenhuis tensors and generalized geometries
, 2006
"... Nijenhuis tensors N on Courant algebroids compatible with the pairing are studied. This compatibility condition turns out to be of the form N + N ∗ = λI for irreducible Courant algebroids, in particular for the extended tangent bundles T M = TM ⊕ T ∗ M. It is proved that compatible Nijenhuis tensor ..."
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Cited by 15 (1 self)
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Nijenhuis tensors N on Courant algebroids compatible with the pairing are studied. This compatibility condition turns out to be of the form N + N ∗ = λI for irreducible Courant algebroids, in particular for the extended tangent bundles T M = TM ⊕ T ∗ M. It is proved that compatible Nijenhuis tensors on irreducible Courant algebroids must satisfy quadratic relations N 2 − λN + γI = 0, so that the corresponding hierarchy is very poor. The particular case N 2 = −I is associated with Hitchin’s generalized geometries and the cases N 2 = I and N 2 = 0 – to other ”generalized geometries”. These concepts find a natural description in terms of supersymplectic Poisson brackets on graded supermanifolds.
Tduality with Hflux: Noncommutativity, Tfolds and G x G structure
, 2006
"... Various approaches to Tduality with NSNS threeform flux are reconciled. Noncommutative torus fibrations are shown to be the openstring version of Tfolds. The nongeometric Tdual of a threetorus with uniform flux is embedded into a generalized complex sixtorus, and the nongeometry is probed ..."
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Various approaches to Tduality with NSNS threeform flux are reconciled. Noncommutative torus fibrations are shown to be the openstring version of Tfolds. The nongeometric Tdual of a threetorus with uniform flux is embedded into a generalized complex sixtorus, and the nongeometry is probed by D0branes regarded as generalized complex submanifolds. The noncommutativity scale, which is present in these compactifications, is given by a holomorphic Poisson bivector that also encodes the variation of the dimension of the worldvolume of Dbranes under monodromy. This bivector is shown to exist in SU(3) × SU(3) structure compactifications, which have been proposed as mirrors to NSNSflux backgrounds. The two SU(3)invariant spinors are generically not parallel, thereby giving rise to a nontrivial Poisson bivector. Furthermore we show that for nongeometric Tduals, the Poisson bivector may not be decomposable into the tensor product of vectors.
Mtheory on eightmanifolds revisited: N = 1 supersymmetry and generalized Spin(7) structures
, 2005
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The Hitchin Model, PoissonquasiNijenhuis Geometry and Symmetry Reduction
, 2007
"... We revisit our earlier work on the AKSZlike formulation of topological sigma model on generalized complex manifolds, or Hitchin model, [20]. We show that the target space geometry geometry implied by the BV master equations is Poisson– quasi–Nijenhuis geometry recently introduced and studied by Sti ..."
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We revisit our earlier work on the AKSZlike formulation of topological sigma model on generalized complex manifolds, or Hitchin model, [20]. We show that the target space geometry geometry implied by the BV master equations is Poisson– quasi–Nijenhuis geometry recently introduced and studied by Stiénon and Xu (in the untwisted case) in [44]. Poisson–quasi–Nijenhuis geometry is more general than generalized complex geometry and comprises it as a particular case. Next, we show how gauging and reduction can be implemented in the Hitchin model. We find that the geometry resulting form the BV master equation is closely related to but more general than that recently described by Lin and Tolman in [40,41], suggesting a natural framework for the study of reduction of Poisson– quasi–Nijenhuis manifolds.
Poisson sigma model on the sphere
, 2007
"... We evaluate the path integral of the Poisson sigma model on sphere and study the correlators of quantum observables. We argue that for the path integral to be welldefined the corresponding Poisson structure should be unimodular. The construction of the finite dimensional BV theory is presented and ..."
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We evaluate the path integral of the Poisson sigma model on sphere and study the correlators of quantum observables. We argue that for the path integral to be welldefined the corresponding Poisson structure should be unimodular. The construction of the finite dimensional BV theory is presented and we argue that it is responsible for the leading semiclassical contribution. For a (twisted) generalized Kähler manifold we discuss the gauge fixed action for the Poisson sigma model. Using the localization we prove that for the holomorphic Poisson structure the semiclassical result for the correlators is indeed the