Dissertation committee:
Κωνσταντίνος. Σφέτσος, Καθηγητής του Τμήματος Φυσικής του ΕΚΠΑ
Γεώργιος Διαμάντης, Αναπληρωτής Καθηγητής Φυσικής του ΕΚΠΑ
Αναστάσιος Πέτκου, Καθηγητής του Τμήματος Φυσικής του ΑΠΘ
Φώτιος Διάκονος, Καθηγητής του Τμήματος Φυσικής του ΕΚΠΑ
Ιωάννης Παπαδημητρίου, Επίκουρος Καθηγητής του Τμήματος Φυσικής του ΕΚΠΑ
Βασίλειος Σπανός, Αναπληρωτής Καθηγητή του Τμήματος Φυσικής του ΕΚΠΑ
Νικόλαος Τετράδης, Καθηγητής του Τμήματος Φυσικής του ΕΚΠΑ
Summary:
This thesis provides a better understanding of a large class of two dimensional integrable theories that are constructed as generalizations of the well known lambda models. They appear as multiparametric deformations of products of conformal field theories (CFTs), with the key property that every CFT is assigned with a different level $k_i$. The operators driving away the models in consideration from the conformal points couple the Kac-Moody currents of adjacent copies of algebras and induce non trivial RG flows towards well defined IR fixed points. Below we present the structure of the thesis at hand, which is organized into three main parts. In the first and second part, which are mainly bibliographic, we introduce the important concepts and formalism necessary for a better understanding of the third part which contains original research results.
In particular, the part one introduces the bosonic sigma models. We study both closed and open strings and describe how the conformality on the worldsheet restricts
the dynamics of the target space and D-branes respectively. We then continue with the analysis of classical integrable theories focusing on the mathematical tools of integrability and the principal chiral model (PCM). Both the group and symmetric space cases is discussed in detail, focusing on their integrability structures and RG flows. These models are well known building blocks of string theory solutions with appropriate Ramond-Ramond fields and have played a crucial role in applications of the gauge/gravity duality. Besides integrable models, exact conformal sigma models are of main importance in this thesis. Such theories provide us with consistent string backgrounds in their own right. We study two main examples, the WZW and gauged WZW model, focusing entirely on their lagrangian formulation. A brief overview of the asymmetric cosets is additionally given. Lastly we consider open strings in the PCM and WZW models. A detailed analysis of their reformulation towards the definition of consistent boundary conditions and their geometric realization as D-branes is given. In the case of conformal/integrable theories such boundary conditions preserve their conformality/integrability respectively. As examples we present all the conformal brane geometries for a product of WZW models and the integrable ones for the PCM.
On the second part of this thesis we consider the well known $\l$-deformations for both the group and symmetric space cases. As before we study their integrable structure and RG flows, which reveal that they interpolate between exactly conformal WZW models in the UV and the non-abelian T-dual of the PCM towards the IR. A notable number of techniques has been developed, which vary from traditional field theory to geometric ones, for the calculations of various physical quantities. Specific cases of the $\l$-models have been embedded in ten dimensional solutions of type-II supergravity leading to potential generalizations of the gauge/gravity duality. Lastly we give the construction of integrable D-brane configurations. We show that the geometric picture of D-branes in WZW models as twisted conjugacy classes persist in the deformation. We obtain such configurations by applying the integrability techniques of the previous chapter for the PCM.
In the third and last part we construct the generalized $\l$-models, mentioned in the first paragraph. It is divided into two main chapters. The first chapter is devoted in the analysis of our models at the IR points towards a complete realization of the corresponding CFTs. Using the beta functions of the deformation parameters, we see that our model flow towards $2^{N-2}$ different IR points and the expression of the central charges, derived using the C-function of Zamolodchikov, reveal that the conformal algebra of the corresponding CFTs depend on the order of the WZW levels. A careful analysis of the central charge for the cases $N>2$ lead to the conclusion that its expression is not enough for the univocal determination of the conformal algebras. Thus, turning our attention to the lagrangian formulation, we construct gauge invariant
actions which, after fixing the gauge, describe the corresponding IR CFTs. The subgroup gauged
at each case, is a different, anomaly free, subgroup of the global $G_L × G_R$ symmetry of $N$
WZW models. This procedure lead to the conclusion that the left and right sector of each of the
IR CFTs is based on different products of coset and affine type conformal symmetries. Despite
this asymmetry, the left and right central charge are the same and in agreement
with the central charge read from the exact in the deformation parameters C-function. Furthermore, using the symmetries, we see that there are CFTs defined at different fixed
points which are related by a transformation, the generalized parity transformation. A geometric representation of the CFT lagrangians in terms of polygons, reformulate the parity transformation as a reflection in terms of a specific perpendicular bisector and leads to an easy classification of the inequivalent IR CFTs. In the second chapter we embed integrable brane configurations in the generalized $\l$-models. To achieve this we generalize the boundary monodromy method applied in the PCM and ordinary $\l$-deformations in order to find boundary conditions that do not have an analog in the single group valued sigma models. Doing so, we find that the richer structure of our generalized theories reflect on the variety of the integrable conditions. Next, we proceed to the geometrical realization of them as $D$-branes. Due to the complexity of the fields present in the boundary equations we apply the sigma model approach, a method based on the modification of the corresponding lagrangians, in order to incorporate the boundary effects, and the variation principle. As a result we find that all the conformal brane geometries known in the literature for a product of WZW models survive the generalized deformations. They consist of the well known $G$-conjugacy classes,
twisted $G$-conjugacy classes by a permutation automorphism (permutation branes)
and the newest class known as generalized permutation branes. Subsequently, we study the properties of the
aforementioned brane geometries, especially of those embedded in the backgrounds
interpolating between the UV and IR fixed points, studied in the previous chapter. Finally, as an example, we considered the lowest dimensional generalized permutation brane embedded in the deformed $SU(2)_{k_1} × SU(2)_{k_2}$ CFT and we extracted its induced fields.
Keywords:
Conformal field theories, String theory, Integrable sigma models, Integrable deformations, D-branes
