Abstract
We study aspects of two-dimensional nonlinear sigma models with Wess-Zumino term corresponding to a nonclosed 3-form, which may arise upon dimensional reduction in the target space. Our goal in this paper is twofold. In a first part, we investigate the conditions for consistent gauging of sigma models in the presence of a nonclosed 3-form. In the Abelian case, we find that the target of the gauged theory has the structure of a contact Courant algebroid, twisted by a 3-form and two 2-forms. Gauge invariance constrains the theory to (small) Dirac structures of the contact Courant algebroid. In the non-Abelian case, we draw a similar parallel between the gauged sigma model and certain transitive Courant algebroids and their corresponding Dirac structures. In the second part of the paper, we study two-dimensional sigma models related to Jacobi structures. The latter generalise Poisson and contact geometry in the presence of an additional vector field. We demonstrate that one can construct a sigma model whose gauge symmetry is controlled by a Jacobi structure, and moreover we twist the model by a 3-form. This construction is then the analogue of WZW-Poisson structures for Jacobi manifolds.
Highlights
On the other hand, Dirac sigma models bear a strong relation to gauging models of strings in backgrounds of a metric G and a closed 3-form H
We study aspects of two-dimensional nonlinear sigma models with WessZumino term corresponding to a nonclosed 3-form, which may arise upon dimensional reduction in the target space
We demonstrate that one can construct a sigma model whose gauge symmetry is controlled by a Jacobi structure, and we twist the model by a 3-form
Summary
The propagation of strings in target spacetimes M is described by nonlinear sigma models. In adapted coordinates xμ = (xi, Φ) where the Killing vector is ∂/∂Φ, this assumption translates to the metric and 3-form being independent of Φ This corresponds to a Kaluza-Klein reduction, where the Ansatz for the metric G takes the characteristic form (in terms of the line element) ds2 = Gij(x)dxidxj + GΦΦ(x)(dΦ + a)2 ,. The action (2.2) can have target space symmetries generated by a set of vector fields ρa = ρμa (X)∂μ that satisfy a non-Abelian Lie algebra [ρa, ρb] = Cabcρc. Note that the gauge indices are raised and lowered with the G-metric (essentially the Killing form), which in a basis of Lie algebra generators {T α} we denote as kαβ. These conditions and their interpretation will be revisited in more detail in the gauged version of the theory
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