T-dualities and Doubled Geometry of the Principal Chiral Model

Published for SISSA by Springer Received: March 15, 2019 Revised: August 13, 2019 Accepted: October 20, 2019 Published: November 11, 2019 Vincenzo E. Marotta,a Franco Pezzellab and Patrizia Vitaleb,c a Department of Mathematics, Heriot-Watt University, Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, U.K. b INFN-Sezione di Napoli, Complesso Universitario di Monte S. Angelo Edificio 6, via Cintia, 80126 Napoli, Italy c Dipartimento di Fisica “E. Pancini”, Università di Napoli Federico II, Complesso Universitario di Monte S. Angelo Edificio 6, via Cintia, 80126 Napoli, Italy E-mail: , , Abstract: The Principal Chiral Model (PCM) defined on the group manifold of SU(2) is here investigated with the aim of getting a further deepening of its relation with Generalized Geometry and Doubled Geometry. A one-parameter family of equivalent Hamiltonian descriptions is analysed, and cast into the form of Born geometries. Then O(3, 3) duality transformations of the target phase space are performed and we show that the resulting dual models are defined on the group SB(2, C) which is the Poisson-Lie dual of SU(2) in the Iwasawa decomposition of the Drinfel’d double SL(2, C). A parent action with doubled degrees of freedom and configuration space SL(2, C) is then defined that reduces to either one of the dually related models, once suitable constraints are implemented. Keywords: Sigma Models, Differential and Algebraic Geometry, String Duality ArXiv ePrint: 1903.01243 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP11(2019)060 JHEP11(2019)060 T-dualities and Doubled Geometry of the Principal Chiral Model Contents 1 2 The Isotropic Rigid Rotator 2.1 The dual model 2.2 The generalized action 6 9 11 3 The Principal Chiral Model 3.1 The Hamiltonian formulation 3.1.1 Poisson-Lie structure 3.1.2 A family of Born geometries 14 16 19 20 4 Poisson-Lie dual models 4.1 The Lagrangian approach 4.1.1 The Hamiltonian description 4.1.2 Dual Born geometry 23 26 27 28 5 Double principal chiral model 5.1 The Lagrangian formalism 5.2 The Hamiltonian formalism 5.3 Recovering the Chiral Model on TSU(2) 29 29 31 31 6 Conclusions and outlook 33 A Poisson brackets 34 1 Introduction Duality symmetries play a fundamental role in String Theory since they provide a powerful tool for investigating the structure of the target spacetime from the string point of view by relating, in the usual sigma-model approach, backgrounds which otherwise would be considered different. The Abelian T-duality [1–3] (where T stands for Target-space) is a well-known example of them. It is a distinctive symmetry of strings since, differently from particles, one-dimensional objects can wrap d non-contractible cycles. This implies the presence of winding modes wa (a = 1, . . . , d) that have to be added to the ordinary momentum modes pa which take integer values along compact dimensions. On a d-torus T d , the Abelian T-duality is an O(d, d; Z) string symmetry under, roughly speaking, the mapping of the radii of the compact dimensions into their inverse, together with the exchange of momentum and winding modes: in this way it establishes a connection between two apparently different but dual target spacetimes. From the sigma model point of view, the necessary condition to work out a dual to some background was, initially, that the latter possess an Abelian group of isometries [4–6] excluding in this way many physically relevant classical string vacua from being considered. –1– JHEP11(2019)060 1 Introduction 1 An isotropic subspace of the Lie algebra d is such that the bilinear form evaluated on any couple of vectors lying in that subspace vanishes; maximally isotropic means that the subspace cannot be enlarged while preserving the property of isotropy. –2– JHEP11(2019)060 After the work in ref. [7], it was understood that T-duality symmetries could also be associated with the non-Abelian isometries of the target manifold and, subsequently, the notion of Abelian and non-Abelian T-duality was extended to the one of Poisson-Lie T-duality [8–10]. Briefly, the term Abelian T-duality refers to the presence of global Abelian isometries in the target spaces of both the paired sigma models; non-Abelian T-duality refers to the existence of a global Abelian isometry on the target space of one of the two sigma-models and of a global non-Abelian isometry on the other. Finally, the Poisson Lie T-duality generalizes the previous definitions to all the other cases, including the one of a pair of sigma models both having non-Abelian isometries in their target spaces. Beyond the string world-sheet action, a category of models that reveal themselves to be very helpful in understanding the above mentioned T-dualities is provided by sigma models whose target configuration space is a Lie group G with g its Lie algebra. These are the so-called Principal Chiral Models (PCM). Studying these models has led to abandoning the requirement of the existence of isometries for the target space as the condition for the existence of dual counterparts. Indeed, the relevant structure in this case reveals to be the one of Drinfel’d double for G together with the well-established notion of Poisson-Lie symmetries [11–14]. The Drinfel’d double of a Lie group G is defined as a Lie group D, with dimension twice the one of G, such that its Lie algebra d can be decomposed into a pair of maximally isotropic sub-algebras, g, g̃ with respect to a non-degenerate invariant bilinear form on d, with g, g̃, respectively the Lie algebra of G and its dual algebra.1 The dual algebra is endowed with a Lie bracket which has to be compatible with existing structures, in a precise sense which will be clarified below. Any such triple, (d, g, g̃), is referred to as a Manin triple. By exponentiation of g̃ one gets the dual Lie group G̃ such that locally D ' G × G̃. The simplest example is the cotangent bundle of any ddimensional Lie group G, T ∗ G ' G n Rd , which we shall call the classical double, with trivial Lie bracket for the dual algebra g̃ ' Rd . For every decomposition of the Drinfel’d double D into dually related subgroups G, G̃, it is possible to define a couple of PCM’s having as target configuration space either of the two subgroups. Hence, every PCM has its dual counterpart for which the role of G and its dual G̃ is interchanged. The set of all decompositions of d into maximally isotropic subspaces (not necessarily subalgebras), plays the role of the modular space of sigma models mutually connected by an O(d, d) transformation. In particular, for the manifest Abelian T-duality of the string model on the d-torus, the Drinfel’d double is D = U(1)2d and its modular space, is in one-to-one correspondence with O(d, d; Z) [10]. In this paper, we are going to show that the target phase space of the SU(2) PCM can actually be replaced by the Drinfel’d double of SU(2), namely the group SL(2, C), without modifying the (...truncated)