Abstract
Poisson-Lie T-duality of the Wess-Zumino-Witten (WZW) model having the group manifold of SU(2) as target space is investigated. The whole construction relies on the deformation of the affine current algebra of the model, the semi-direct sum mathfrak{su}(2)left(mathrm{mathbb{R}}right)overset{cdot }{oplus}mathfrak{a} , to the fully semisimple Kac-Moody algebra mathfrak{sl}left(2,mathrm{mathbb{C}}right)left(mathrm{mathbb{R}}right) . A two-parameter family of models with SL(2, ℂ) as target phase space is obtained so that Poisson-Lie T-duality is realised as an O(3, 3) rotation in the phase space. The dual family shares the same phase space but its configuration space is SB(2, ℂ), the Poisson-Lie dual of the group SU(2). A parent action with doubled degrees of freedom on SL(2, ℂ) is defined, together with its Hamiltonian description.
Highlights
Different backgrounds but yielding the same physics, as it can be seen looking at the mass spectrum
The kind of T-duality discussed so far belongs to a particular class, so called Abelian T-duality, which is characterised by the fact that the generators of target space duality transformations are Abelian, while generating symmetries of the action only if they are Killing vectors of the metric [30,31,32]
We show that such a deformation is possible, which does not alter the nature of the current algebra, nor the dynamics described by the new Hamiltonian
Summary
Let G be a semisimple connected Lie group and Σ a 2-dimensional oriented (pseudo) Riemannian manifold (we take it with Minkowski signature (1, −1)) parametrized by the coordinates (t, σ). Configuration space is the space of maps SU (2)(R) = {g : R → SU (2)}, with boundary condition (2.9), whereas the phase space Γ1 is its cotangent bundle As a manifold this is the product of SU (2)(R) with a vector space, its dual Lie algebra, su(2)∗(R), spanned by the currents Ii: Γ1 = SU (2)(R) × su(2)∗(R). Let us stress here that we are not going to deform the dynamics but only its target phase space description, and in particular its current algebra. This is completely different from the usual deformation approach followed for instance for integrable models. We do not repeat the calculation, performed in [83], the only difference being the choice of τ as a real or imaginary parameter
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