Soutenance mémoire
The target space point of view
Supergravity appears as a low energy description of string theory, obtained when asking for the Weyl invariance of the σ-model Lagrangian. Being a one-loop calculation, the corresponding results should always be checked against higher order corrections in α 0 . On the other hand, as we have already stressed many times above, WZW models (just like the asymmetric deformations we study in this work) only receive corrections in terms of the level of the algebra (or, in this language, on the overall volume of the manifold). This implies that the target space description at one loop in α0 is automatically correct at all orders. From this point of view, Wess–Zumino–Witten models describe the motion of a string on a group manifold geometry. The background fields are completed by a NS-NS three form H = dB (Kalb-Ramond field) and a constant dilaton Φ = Φ0. The target space analysis is greatly simplified by the fact that the geometric quantities are all naturally expressed in terms of group theoretical objects. Let us consider for concreteness the case of a compact group G, whose Lie algebra is generated by ht α i and has structure constants f α βγ. The metric for the group manifold can be chosen as the Killing metric (the choice is unique up to a constant in this case) and it is then natural to use the Maurer–Cartan one-forms as vielbeins.
Geometry of squashed groups
In order to describe the squashed group manifolds that we obtain via asymmetric deformation we need to generalize the discussion on group manifold geometry presented in Sec. 2.2. Let { ˆθα } be a set of one-forms on a manifold M satisfying the commutation relations [ˆθβ,ˆθγ]=fαβγ ˆθα (3.37) as it is the case when ˆθα are the Maurer–Cartan one-forms of Eq. (2.57) and fαβγ the structure constants for the algebra. We wish to study the geometry of the Riemann manifold M endowed with the metric g = gαβ ˆθα ⊗ ˆθβ . (3.38) In general such a metric will have a symmetry G × G 0 where G is the group corresponding to the structure constants f αβγ and G 0 ⊂ G. The maximally symmetric case, in which G 0 = G is obtained when g is G-invariant, i.e. when it satisfies fαβγgαδ + fα δγgαβ = 0. (3.39) for compact groups this condition is fulfilled by the Killing metric in Eq. (2.56). The connection one-forms ωαβ are uniquely determined by the compatibility condition and the vanishing of the torsion. Respectively: dgαβ − ωγαgγβ − ωγβgγα = 0 (3.40)d ˆθα + ωαβ ∧ ˆθβ = T α = 0 (3.41) As it is shown in [MHS88], if gαβ is constant, the solution to the system can be put in the form ωαβ = −Dαβγ ˆθγ (3.42) where Dαβγ=1/2fαβγ−Kαβγ and K α βγ is a tensor (symmetric in the lower indices) given by: Kαβγ =12gακ fδκβgγδ +12gακ fδ κγgβδ. (3.43) Just as in Sec. 2.2 we define the curvature two-form Rij and the Riemann tensor Curvature tensors on squashed groups which now reads: Rαβγδ =Dαβκ fκγδ + DακγDκβδ − DακδDκβγ (3.44) and the corresponding Ricci tensor: Ricβδ = Dαβκ fκαδ − DακδDκβα (3.45) In particular for gij ∝ δij, K = 0 so that we recover the usual Maurer–Cartan structure equation Eq. (2.63) and the expressions in Eqs. (2.65). Let us now specialize these general relations to the case of the conformal model with metric given in Eq. (3.26). T
The deformation as a gauging
In this section we want to give an alternative construction for our deformed models, this time explicitly based on an asymmetric WZW gauging. The existence of such a construction is not surprising at all since our deformations can be seen as a generalization of the ones considered in [GK94]. In these terms, just like J¯J (symmetric) deformations lead to gauged WZW models, our asymmetric construction leads to asymmetrically gauged WZW models, which were studied in [QS03]. A point must be stressed here. The asymmetric deformations admit as limit solutions the usual geometric cosets that one would have expected from field theory, as results of a gauging procedure. So, why do we need to go through this somewhat convoluted procedure? The reason lays in the fact that string theory is not the usual point particle field theory. A left and a right sector are present at the same time and they cannot be considered separately if we don’t want to introduce anomalies. Now, gauging the left action of a subgroup, i.e. the symmetry G ∼ GH, which would directly give the geometric coset we are studying, would precisely introduce this kind of problems. Hence we are automatically forced to condider the adjoint action G ∼ H−1GH [Wit91]. The key idea then, as it will appear in this section, is that when G is semisimple and written as the product of a group and a copy of its Cartan torus, the left and right action can be chosen such as to act on the two separate sectors and then be equivalent to two left actions. Instead of a general realization, for sake of clearness, here we will give the explicit construction for the most simple case, the SU (2) model, then introduce a more covariant formalism which will be simpler to generalize to higher groups, in particular for the SU (3) case which we will describe in great detail in the following. To simplify the formalism we will discuss gauging of bosonic CFTs, and the currents of the gauge sector of the heterotic string are replaced by compact U(1) free bosons. All the results are easily translated into heterotic string constructions.
An interesting mix
A particular kind of asymmetric deformation is what we will call in the following double deformation [KK95, Isr04]. At the Lagrangian level this is obtained by adding the following marginal perturbation to the WZW action: δS = δκ2Zd2z J ¯J + HZd2 z J ¯I; (4.84)J is a holomorphic current in the group, ¯J the corresponding anti-holomorphic current and ¯I an external (to the group) anti-holomorphic current (i.e. in the right-moving heterotic sector for example). A possible way to interpret this operator consists in thinking of the double deformation as the superposition of a symmetric – or gravitational – deformation (the first addend) and of an antisymmetric one – the electromagnetic deformation. This mix is consistent because if we perform the κ deformation first, the theory keeps the U(1) × U(1) symmetry generated by J and ¯J that is needed in order to allow for the H deformation. Following this trail, we can read off the background fields corresponding to the double deformation by using at first one of the methods outlined in Sec. 3.1 and then applying the Kaluza-Klein reduction to the resulting background fields. The final result consists in a metric, a three-form, a dilaton and a gauge field. It is in general valid at any order in the deformation parameters κ and H but only at leading order in α 0 due to the presence of the symmetric part. Double deformations of AdS3 where J is the time-like J 3 operator have been studied in [Isr04]. It was there shown that the extra gravitational deformation allows to get rid of the closed time-like curves, which are otherwise present in the pure J 3 asymmetric deformation (Eq. (4.15)) – the latter includes Gödel space. Here, we will focus instead on the case of double deformation generated by space-like operators, J 2 and ¯J 2
Discrete identifications in asymmetric deformations
Our analysis of the residual isometries in purely asymmetric deformations (Sec. 3.1) shows that the vector ξ (Eq. (4.108a)) survives only in the hyperbolic deformation, whereas ξ in Eq. (4.108b) is present in the parabolic one. Put differently, non-extremal BTZ black holes allow for electric deformation, while in the extremal ones, the deformation can only be induced by an electro magnetic wave. Elliptic deformation is not compatible with BTZ identifications. The question that we would like to address is the following: how much of the original black hole structure survives the deformation? The answer is simple: a new chronological singularity appears in the asymptotic region of the black hole. Evaluating the norm of the Killing vector shows that a naked singularity appears. Thus the deformed black hole is no longer a viable gravitational background. Actually, whatever the Killing vector we consider to perform the identifications, we are always confronted to such pathologies. The fate of the asymmetric parabolic deformation of AdS3 is similar: there is no region at infinity free of closed time-like curves after performing the identifications.
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Table des matières
1 Introduction
2 Wess-Zumino-Witten Models
Wess-Zumino-Witten models constitute a large class of the exact string theory solutions which we will use as starting points for most of the analysis in the following. In this chapter we see how they can be studied from different perspectives and with different motivations both from a target space and world-sheet point of view.
2.1 The two-dimensional point of view
2.2 The target space point of view
3 Deformations
In this rather technical chapter we describe marginal deformations of Wess-Zumino-Witten models. The main purpose for these constructions is to reduce the symmetry of the system while keeping the integrability properties intact, trying to preserve as many nice geometric properties as possible.
3.1 Deformed WZW models: various perspectives
3.2 Background fields for the asymmetric deformation
3.3 Geometry of squashed groups
3.4 A no-renormalization theorem
3.5 Partition functions
3.6 The deformation as a gauging
4 Applications
In this chapter we present some of the applications for the construction outlined above. After an analysis of the most simple (compact and noncompact) examples, we describe the near-horizon geometry for the BertottiRobinson black hole, show some new compactifications and see how Horne and Horowitz’s black string can be described in this framework and generalized via the introduction of an electric field.
4.1 The two-sphere CFT
4.2 SL(2, R)
4.3 Near horizon geometry for the Bertotti-Robinson black hole
4.4 The three-dimensional black string revisited
4.5 New compactifications
5 Squashed groups in type II
In this chapter we start deviating from the preceding ones because we will no longer deal with WZW models but with configurations in which the group manifold geometry is sustained by RR fields. In particular, then, we see how the squashed geometries can be obtained in type II theories by Tdualizing black brane configurations.
5.1 SL(2, R) × SU(2) as a D-brane solution
5.2 T duality with RR fields
5.3 The squashed sphere
6 Out of the conformal point: Renormalization Group Flows
This chapter is devoted to the study of the relaxation of squashed WZW models further deformed by the insertion of non-marginal operators. The calculation is carried from both the target space and world-sheet points of view, once more highlighting the interplay between the two complementary descriptions. In the last part such techniques are used to outline the connection between the time evolution and the RG-flow which is seen as a large friction limit description; we are hence naturally led to a FRW-type cosmological model.
6.1 The target space point of view
6.2 The CFT approach
6.3 RG flow and friction
6.4 Cosmological interpretation
7 Hyperbolic Spaces
In this chapter we investigate type II and M-theory geometries written as direct products of constant-curvature spaces. We find in particular a class of backgrounds with hyperbolic components and we study their stability with respect to small fluctuations.
7.1 M-theory solutions
7.2 Stability
7.3 Type IIB backgrounds
8 Conclusions and further perspectives
A Table of conventions
B Explict parametrizations for some Lie groups
B.1 The three-sphere
B.2 AdS3
B.3 SU (3)
B.4 USp (4)
C Symmetric deformations of SL(2, R)
D Spectrum of the SL(2, R) super-WZW model
Bibliography
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