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Register a new userSigma models with non-commuting complex structures and extended supersymmetry
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We discuss additional supersymmetries for N = (2, 2) supersymmetric non-linear sigma models described by left and right semichiral superfields.
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We describe the conditions for extra supersymmetry in N=(2,2) supersymmetric nonlinear sigma models written in terms of semichiral superfields. We find that some of these models have additional off-shell supersymmetry. The (4,4) supersymmetry introduces geometrical structures on the target-space which are conveniently described in terms of Yano f-structures and Magri-Morosi concomitants. On-shell, we relate the new structures to the known bi-hypercomplex structures.
We discuss two dimensional N-extended supersymmetry in Euclidean signature and its R-symmetry. For N=2, the R-symmetry is SO(2)times SO(1,1), so that only an A-twist is possible. To formulate a B-twist, or to construct Euclidean N=2 models with H-flux so that the target geometry is generalised Kahler, it is necessary to work with a complexification of the sigma models. These issues are related to the obstructions to the existence of non-trivial twisted chiral superfields in Euclidean superspace.
Briefly: Using a novel $(1,1)$ superspace formulation of semichiral sigma models with $4D$ target space, we investigate if an extended supersymmetry in terms of semichirals is compatible with having a $4D$ target space with torsion. In more detail: Semichiral sigma models have $(2,2)$ supersymmetry and Generalized Kahler target space geometry by construction. They can also support $(4,4)$ supersymmetry and Generalized Hyperkahler geometry, but when the target space is four dimensional indications are that the geometry is restricted to Hyperkahler. To investigate this further, we reduce the model to $(1,1)$ superspace and construct the extra (on-shell) supersymmetries there. We then find the conditions for a lift to $(2,2)$ super space and semichiral fields to exist. Those conditions are shown to hold for Hyperkahler geometries. The $SU(2)otimes U(1)$ WZW model, which has $(4,4)$ supersymmetry and a semichiral description, is also investigated. The additional supersymmetries are found in $(1,1)$ superspace but shown {em not} to be liftable to a $(2,2)$ semichiral formulation.
We derive and discuss, at both the classical and the quantum levels, generalized N = 2 supersymmetric quantum mechanical sigma models describing the motion over an arbitrary real or an arbitrary complex manifold with extra torsions. We analyze the relevant vacuum states to make explicit the fact that their number is not affected by adding the torsion terms.
We study non-local non-linear sigma models in arbitrary dimension, focusing on the scale invariant limit in which the scalar fields naturally have scaling dimension zero, so that the free propagator is logarithmic. The classical action is a bi-local integral of the square of the arc length between points on the target manifold. One-loop divergences can be canceled by introducing an additional bi-local term in the action, proportional to the target space laplacian of the square of the arc length. The metric renormalization that one encounters in the two-derivative non-linear sigma model is absent in the non-local case. In our analysis, the target space manifold is assumed to be smooth and Archimedean; however, the base space may be either Archimedean or ultrametric. We comment on the relation to higher derivative non-linear sigma models and speculate on a possible application to the dynamics of M2-branes.
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