T-Duality and the Moment Map

Klimčı́k, Ctirad; Ševera, Pavol

doi:10.1007/978-1-4899-1801-7_13

Cited by 15 publications

(18 citation statements)

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“…Conventional NATD can be reproduced as a special case where one of the two groups is an Abelian group. Aspects of the PL T -duality and generalizations have been studied in [39][40][41][42][43][44][45][46][47], and concrete applications are given, for example, in [38,[48][49][50][51].Low-dimensional Drinfel'd doubles were classified in [52][53][54], and it was stressed that some Drinfel'd double d can be decomposed into several different pairs of subalgebras g andg , (d, g,g) ∼ = (d, g ′ ,g ′ ) ∼ = · · · . The decomposition is called the Manin triple, and each Manin triple corresponds to a sigma model.…”

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confidence: 99%

Type II DFT solutions from Poisson–Lie $T$-duality/plurality

Sakatani

2019

Progress of Theoretical and Experimental Physics

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String theory has the T -duality symmetry when the target space has Abelian isometries.A generalization of the T -duality, where the isometry group is non-Abelian, is known as the non-Abelian T -duality, which works well as a solution-generating technique in supergravity.In this paper, we describe the non-Abelian T -duality as a kind of O(D, D) transformation when the isometry group acts without isotropy. We then provide a duality transformation rule for the Ramond-Ramond fields by using the technique of double field theory (DFT).We also study a more general class of solution-generating technique, the Poisson-Lie (PL)T -duality or T -plurality. We describe the PL T -plurality as an O(n, n) transformation and clearly show the covariance of the DFT equations of motion by using the gauged DFT.We further discuss the PL T -plurality with spectator fields, and study an application to the AdS 5 × S 5 solution. The dilaton puzzle known in the context of the PL T -plurality is resolved with the help of DFT. IntroductionThe T -duality was discovered in [1] as a symmetry of string theory compactified on a torus.The mass spectrum or the partition function of string theory on a D-dimensional torus was studied for example in [2][3][4][5][6] and the T -duality was identified as an O(D, D; Z) symmetry.It was further studied from a different approach [7,8], and the transformation rules for the background fields (i.e. metric, the Kalb-Ramond B-field, and the dilaton) under the T -duality were determined. In [9,10], the T -duality was understood as an O(D, D) symmetry of the classical equations of motion of string theory. The classical symmetry was clarified in [11] by using the gauged sigma model, and this approach has proved quite useful, for example when we discuss the global structure of the T -dualized background [12]. The transformation rules for the Ramond-Ramond (R-R) fields and spacetime fermions were determined in [13][14][15][16].This well-established symmetry of string theory is called the Abelian T -duality since it relies on the existence of Killing vectors which commute with each other (see [17,18] for reviews).An extension of the T -duality to the case of non-commuting Killing vectors was explored in [19] (see [20,21] for earlier works), and this is known as the non-Abelian T -duality (NATD).Various aspects have been studied in [12,[22][23][24][25][26][27][28][29][30][31][32][33][34][35], but unlike the Abelian T -duality, there are still many things to be clarified. For example, the partition function in the dual model is not the same as that of the original model (see [36] for a recent study), and NATD may rather be regarded as a map between two string theories. The global structure of the dual geometry is also not clearly understood [12]. However, NATD at least generates many new solutions of supergravity, and it can be utilized as a useful solution-generating technique.Under NATD, the isometries are generally broken, and naively we cannot recover the original model from the dual model. However, this issue was re...

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Type II DFT solutions from Poisson–Lie $T$-duality/plurality

Sakatani

2019

Progress of Theoretical and Experimental Physics

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“…The notion of Poisson-Lie symmetry can also be formulated in the Hamiltonian formalism [73,75,[87][88][89]. We may state the following JHEP09(2020)060…”

Section: Jhep09(2020)060mentioning

confidence: 99%

Poisson-Lie T-duality of WZW model via current algebra deformation

Bascone¹,

Pezzella²,

Vitale³

2020

J. High Energ. Phys.

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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 $$ \mathfrak{su}(2)\left(\mathrm{\mathbb{R}}\right)\overset{\cdot }{\oplus}\mathfrak{a} $$ su 2 ℝ ⊕ ⋅ a , to the fully semisimple Kac-Moody algebra $$ \mathfrak{sl}\left(2,\mathrm{\mathbb{C}}\right)\left(\mathrm{\mathbb{R}}\right) $$ sl 2 ℂ ℝ . 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.

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“…35) The bracket of the type { β L , x , β L , y } T * G for x, y ∈ G is more involved. In order to compute it, we need to prove the following formulae where x, y ∈ G and x * , y * ∈ G * .…”

Section: A2 Cotangent Bundle Of a Group Manifoldmentioning

confidence: 99%

Quasitriangular WZW Model

Klimčı́k

2004

Rev. Math. Phys.

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A dynamical system is canonically associated to every Drinfeld double of any affine Kac-Moody group. In particular, the choice of the affine Lu-Weinstein double gives a smooth one-parameter deformation of the standard WZW model. The deformed WZW model is exactly solvable and it admits the chiral decomposition. Its classical action is not invariant with respect to the left and right action of the loop group, however, it satisfies the weaker condition of the Poisson-Lie symmetry. The structure of the deformed WZW theory is characterized by several ordinary and dynamical r-matrices with spectral parameter. They describe the q-deformed current algebras, appear in the definition of q-primary fields and characterize the quasitriangular exchange (braiding) relations. The symplectic structure of the deformed chiral WZW theory is cocharacterized by the same elliptic dynamical r-matrix that appears in the Bernard generalization of the Knizhnik-Zamolodchikov equation, with q entering the modular parameter of the Jacobi theta functions. This reveals a remarkable connection between the classical q-deformed WZW model and the quantum standard WZW theory on elliptic curves. ContentsQuasitriangular WZW Model 681 a The classical action (1.3) makes sense even if this solution θ q exists only locally onM L . Rev. Math. Phys. 2004.16:679-808. Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 04/01/15. For personal use only. Quasitriangular WZW Model 683 model is Kac-Moody Poisson-Lie symmetric. Due to this property, the quantized quasitriangular chiral WZW model will enjoy the q-Kac-Moody symmetry.Remarks. (i) The deformation of the master model (1) exists for every Drinfeld double ofG. However, the two-step symplectic reduction can be performed only ifD is the WZW double (see above). We found with satisfaction that a nontrivial WZW doubleD ofG can indeed be constructed; it is in fact nothing but the complexifi-cationD =G C of the affine Kac-Moody groupG equipped with certain invariant maximally noncompact inner product onD = Lie(D). We shall callD the "affine Lu-Weinstein double", since it turns out to be the natural affine generalization of the standard Lu-Weinstein double D 0 of the group G 0 .(ii) The Weyl alcove A + is usualy viewed as the fundamental domain of the action of the affine Weyl group on the Cartan subalgebra T of G 0 ≡ Lie(G 0 ). The affine Weyl group acts also on the Cartan subalgebraT ofG ≡ Lie(G). The alcovẽ A + is again the fundamental domain of this action. It is also important to note that the second floor chiral Hamiltoniañ

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T-Duality and the Moment Map

Cited by 15 publications

References 7 publications

Type II DFT solutions from Poisson–Lie $T$-duality/plurality

Type II DFT solutions from Poisson–Lie $T$-duality/plurality

Poisson-Lie T-duality of WZW model via current algebra deformation

Quasitriangular WZW Model

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