Doubled Formalism, Complexification and Topological Sigma-Models (original) (raw)
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A topological sigma model of biKähler geometry
Journal of High Energy Physics, 2006
BiKaehler geometry is characterized by a Riemannian metric g ab and two covariantly constant generally non commuting complex structures K ± a b , with respect to which g ab is Hermitian. It is a particular case of the biHermitian geometry of Gates, Hull and Roceck, the most general sigma model target space geometry allowing for (2, 2) world sheet supersymmetry. We present a sigma model for biKaehler geometry that is topological in the following sense: i) the action is invariant under a fermionic symmetry δ; ii) δ is nilpotent on shell; iii) the action is δ-exact on shell up to a topological term; iv) the resulting field theory depends only on a subset of the target space geometrical data. The biKaehler sigma model is obtainable by gauge fixing the Hitchin model with generalized Kaehler target space. It further contains the customary A topological sigma model as a particular case. However, it is not seemingly related to the (2, 2) supersymmetric biKaehler sigma model by twisting in general.
The biHermitian topological sigma model
Journal of High Energy Physics, 2006
BiHermitian geometry, discovered long ago by Gates, Hull and Roček, is the most general sigma model target space geometry allowing for (2, 2) world sheet supersymmetry. By using the twisting procedure proposed by Kapustin and Li, we work out the type A and B topological sigma models for a general biHermtian target space, we write down the explicit expression of the sigma model's action and BRST transformations and present a computation of the topological gauge fermion and the topological action.
Physics Letters B, 1995
It is shown that two dimensional (2d) topological gravity in the conformal gauge has a larger symmetry than has been hitherto recognized; in the formulation of Labastida, Pernici and Witten it contains a twisted "small" N = 4 superconformal symmetry. There are in fact two distinct twisted N = 2 structures within this N = 4, one of which is shown to be isomorphic to the algebra discussed by the Verlindes and the other corresponds, through bosonization, to c M ≤ 1 string theory discussed by Bershadsky et.al. As a byproduct, we find a twisted N = 4 structure in c M ≤ 1 string theory. We also study the "mirror" of this twisted N = 4 algebra and find that it corresponds, through another bosonization, to a constrained topological sigma model in complex dimension one.
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Journal of High Energy Physics, 2007
We gauge the (2, 2) supersymmetric non-linear sigma model whose target space has bihermitian structure (g, B, J ±) with noncommuting complex structures. The bihermitian geometry is realized by a sigma model which is written in terms of (2, 2) semichiral superfields. We discuss the moment map, from the perspective of the gauged sigma model action and from the integrability condition for a Hamiltonian vector field. We show that for a concrete example, the SU(2) × U(1) WZNW model, as well as for the sigma models with almost product structure, the moment map can be used together with the corresponding Killing vector to form an element of T ⊕ T * which lies in the eigenbundle of the generalized almost complex structure. Lastly, we discuss T-duality at the level of a (2, 2) sigma model involving semi-chiral superfields and present an explicit example.
Generalized Kahler geometry and manifest N=(2,2) supersymmetric nonlinear sigma-models
Journal of High Energy Physics
Generalized complex geometry is a new mathematical framework that is useful for describing the target space of N = (2, 2) nonlinear sigma-models. The most direct relation is obtained at the N = (1, 1) level when the sigma model is formulated with an additional auxiliary spinorial field. We revive a formulation in terms of N = (2, 2) semi-(anti)chiral multiplets where such auxiliary fields are naturally present. The underlying generalized complex structures are shown to commute (unlike the corresponding ordinary complex structures) and describe a Generalized Kähler geometry. The metric, B-field and generalized complex structures are all determined in terms of a potential K.
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Journal of High Energy Physics, 2008
We investigate the topological theory obtained by twisting the N = (2, 2) supersymmetric nonlinear sigma model with target a bihermitian space with torsion. For the special case in which the two complex structures commute, we show that the action is a Q-exact term plus a quasi-topological term. The quasi-topological term is locally given by a closed two-form which corresponds to a flat gerbe-connection and generalises the usual topological term of the A-model. Exponentiating it gives a Wilson surface, which can be regarded as a generalization of a Wilson line. This makes the quantum theory globally well-defined.
Potentials for topological sigma models
Physics Letters B, 1991
Topological sigma models with potential terms are obtained by twisting N=2 supersymmetric sigma models. These models exist for manifolds with isometries and potential terms in their actions are built out of the corresponding Killing vector fields. It turns out that, as ordinary topological sigma models, the models resulting after twisting N=2 supersymmetry can be generalized to the case of almost hermitian manifolds.