Doubling, T-Duality and Generalized Geometry: a simple model

Published for SISSA by Springer Received: April 25, 2018 Revised: July 31, 2018 Accepted: August 20, 2018 Published: August 29, 2018 Vincenzo E. Marotta,a,c Franco Pezzellab and Patrizia Vitalea,b a Dipartimento di Fisica “E. Pancini”, Università di Napoli Federico II, Complesso Universitario di Monte S. Angelo Edificio 6, via Cintia, 80126 Napoli, Italy b INFN-Sezione di Napoli, Complesso Universitario di Monte S. Angelo Edificio 6, via Cintia, 80126 Napoli, Italy c Department of Mathematics, Heriot-Watt University, Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, U.K. E-mail: , , Abstract: A simple mechanical system, the three-dimensional isotropic rigid rotator, is here investigated as a 0+1 field theory, aiming at further investigating the relation between Generalized/Double Geometry on the one hand and Doubled World-Sheet Formalism/Double Field Theory, on the other hand. The model is defined over the group manifold of SU(2) and a dual model is introduced having the Poisson-Lie dual of SU(2) as configuration space. A generalized action with configuration space SL(2, C), i.e. the Drinfel’d double of the group SU(2), is then defined: it reduces to the original action of the rotator or to its dual, once constraints are implemented. The new action contains twice as many variables as the original. Moreover its geometric structures can be understood in terms of Generalized Geometry. Keywords: Differential and Algebraic Geometry, String Duality ArXiv ePrint: 1804.00744 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP08(2018)185 JHEP08(2018)185 Doubling, T-Duality and Generalized Geometry: a simple model Contents 1 2 The isotropic rigid rotator 2.1 The Lagrangian and Hamiltonian formalisms 5 5 3 Poisson-Lie groups and the double Lie algebra sl(2, C) 3.1 Para-Hermitian geometry of SL(2, C) 9 13 4 The dual model 4.1 The Lagrangian and Hamiltonian formalisms 16 17 5 A new formalism for the isotropic rotator: the doubled formulation 5.1 The Lagrangian formalism 5.2 Recovering the standard description 5.3 Recovering the dual model 5.4 The Hamiltonian formalism 5.5 The Poisson algebra 5.6 Poisson-Lie simmetries 20 21 22 24 27 28 32 6 Conclusions and outlook 33 1 Introduction Generalized Geometry (GG) was first introduced by N. J. Hitchin in ref. [1]. As the author himself states in his pedagogical lectures [2], it is based on two premises: the first consists in replacing the tangent bundle T of a manifold M with T ⊕ T ∗ , a bundle with the same base space M but fibers given by the direct sum of tangent and cotangent spaces. The second consists in replacing the Lie bracket on the sections of T , which are vector fields, with the Courant bracket which involves vector fields and one-forms. The construction is then extended to general vector bundles E over M so to have E ⊕E ∗ and a suitable bracket for the sections of the new bundle. The formal setting of GG has recently attracted the interest of theoretical physicists in relation to Double Field Theory (DFT) [3]. We shall propose in this paper a model whose analysis can help to establish more rigorously a possible bridge between the two through the doubled world-sheet formalism that generates DFT. DFT has emerged as a proposal to incorporate T-duality [4, 5], a peculiar symmetry of a compactified string on a d-torus T d in a (G, B)-background, as a manifest symmetry of the string effective field theory. In order to achieve this goal, the action of this field theory has to be generalized in such a way that the emerging carrier space of the dynamics –1– JHEP08(2018)185 1 Introduction be doubled with respect to the original. What makes T-duality a distinctive symmetry of strings is that these latter, as extended objects and differently from particles, can wrap non-contractible cycles. Such a wrapping implies the presence of winding modes that have to be added to the ordinary momentum modes which take integer values along compact dimensions. T-duality is an O(d, d; Z) symmetry of the dynamics of a closed string under, roughly speaking, the exchange of winding and momentum modes and establishes, in this way, a connection between the physics of strings defined on different target spaces. Despite the preamble, which gives credit to the strings related literature for focusing on the geometrical content of the doubled world-sheet and DFT, the interest for the subject is relevant in the broad area of field theory when one deals with duality symmetries of the dynamics which are not manifest at the level of the action. A few remarks that clarify the philosophy of the paper are here in order. First of all, it is worth stressing again that, in the framework of string theory, the doubling takes place in the D-dimensional target space M of the non-linear sigma model underlying the string action, by introducing new fields x̃i (σ, τ ), which are dual to xi (σ, τ ), with i = 1, . . . , D. From this point of view, a first analogy with Generalized Geometry is straightforward, by identifying xi , x̃i with sections of a generalized bundle E ⊕ E ∗ over the world sheet of the string. Secondly, it is only when the target space is considered as the configuration space of the effective field theory we are going to deal with, that the doubling is reinterpreted as a doubling of the configuration space. Actually, the original non-linear sigma model has no doubled coordinates, but what is doubled are the field coordinates. When the effective field theory derived from the Polyakov string action is considered, then the dual fields xi , x̃i are seen as coordinates of the carrier space of the effective dynamics, which corresponds to the string target space. DFT is thus formulated in terms of the background 1 Let us observe here that we retain the name doubled world-sheet since this has become of common use, but actually it is the string target-space which is doubled and not the world-sheet. –2– JHEP08(2018)185 DFT is supposed to be an O(d, d; Z) manifest space-time effective field theory description coming from a manifestly T-dual invariant formulation of a string world-sheet, i.e. from a doubled world-sheet.1 In fact, a formulation of the world-sheet action of the bosonic string, in which T-duality is manifest, was already initially proposed in ref.s [6–8] and, later, in [9–16] (see also more recent works in [17–22]). This string action must contain information about windings and therefore it is based on two sets of coordinates: the usual ones xa (σ, τ ) and the “dual” coordinates x̃a (σ, τ ), (a = 1, . . . , d) conjugate to the winding modes. In this way the O(d, d; Z) duality becomes a manifest symmetry of the world-sheet action. A corresponding doubling of all the D space-time degrees of freedom (vielbeins in this case, not only relatively to the compact dimensions) in the low-energy effective action first occurred in refs. [23–26] where, a manifestly O(D, D; R) form of the target-spa (...truncated)