Unit:
Department of PhysicsLibrary of the School of Science
Author:
Panopoulos Pantelis
Dissertation committee:
Kωνσταντίνος Σφέτσος, Καθηγητής, Τμήμα Φυσικής, Εκπα
Αλέξανδρος Κεχαγιάς, Καθηγητής, ΣΕΜΦΕ, ΕΜΠ
Δημήτριος Τσίμπης, Καθηγητής, Τμήμα Φυσικής, Παν/μιο Λυών
Αλέξανδρος Καρανίκας, Καθηγητής, Τμήμα Φυσικής, Εκπα
Γεώργιος Διαμάντης, Αν. Καθηγητής, Τμήμα Φυσικής, Εκπα
Βασίλειος Σπανός, Αν. Καθηγητής, Τμήμα Φυσικής, Εκπα
Άρης Μουστάκα, Αν. Καθηγητής, Τμήμα Φυσικής, Εκπα
Original Title:
Μη διαταρακτικές μέθοδοι και συναρτήσεις μονοτονίας σε διδιάστατες θεωρίες πεδίου
Translated title:
Non perturbative methods and C-functions in two dimensional Quantum Field Theories
Summary:
In the following thesis we present a set of new results concerning integrable deformations of two dimensional Quantum Field Theories. The approach contains novel features and techniques of non-perturbative calculations, giving a more in insightful view and understanding of this broad area. The class of deformations we are dealing with, are called λ-deformations. These deformations enjoy a set of symmetries appearing to be significant, since when exploiting them, numerous of quantum non-perturbative results arise.
To serve our purpose, we start with an overview of Conformal Field Theory (CFT). Our aim is to describe the necessary tools for the reader, to become familiar with the basic notions of CFT that take place in our calculations. This contains a gentle introduction in two-dimensional CFT, engaging the reader with calculations of correlation functions in a general set up. Next we describe in some detail the famous WZW-models as an intermediate step to understand λ-deformations. The presentation contains single WZW models as well as coset constructions usually called Goddard-Kent-Olive.
To proceed, we give an outlook of monotonicity theorems in QFT. These theorems are related with the definition of functions that capture the effective degrees of freedom as the theory flows from high energies (UV) to low energies (IR). These functions decrease monotonically in accordance with our physical intuitions, since in this motion of the QFT in coupling space the degrees of freedom are lowered. These functions are not globally defined in all dimensions even though they all come under the name of C-theorems. This is because they are related with the form of the energy momentum two-point function, which differs and becomes more complicated as the number of dimensions increase. We first describe in considerable detail the Zamolodchikov C-theorem in two dimensions as a useful background for the evaluation of the λ-deformed C-functions. During the proof, the necessary set of notions concerning geometry in coupling space is developed, as a useful tool for anomalous dimension calculations for several operators. We also discuss the F-theorem in three dimensions and the a-theorem in four dimensions for completeness but without entering deep within.
Then, we introduce a detailed presentation of λ-deformations that will appear in our treatment. The special features of these integrable theories are high-lighted, among which, the invariance under duality symmetry transformations. These discrete transformations are crucial since when exploiting them, non-perturbative calculations are feasible giving a full control of the flow.
The core of our presentation follows. On chapter 6 we utilize the coupling space geometry and the exact β-function to calculate the anomalous dimensions of a wide class of operators in λ-deformed theories. This procedure comes with a set of steps taking place in the following manner: a) Starting with a λ-deformed theory we introduce a new term with a coupling (say λ΄) which can be seen as a perturbative source term. Then, the action contains two coupling constants and the coupling space becomes two dimensional. b) For this new action we calculate the β-functions for both couplings using the heat-kernel method. These β-functions are exact in λ and first order in λ΄. c) We then use a relation between the β-functions and the anomalous dimensions in coupling space and construct the anomalous dimension matrix of operators. At this point, the Zamolodchikov metric (the coupling space metric) is important d) Assuming that there is no mixing of operators as λ΄ approaches zero, we arrive at the anomalous dimension of the operator under study. The λ-deformed models we are focused on contain single λ-deformations, λ-deformations of different algebra levels k and of mutual type. For these theories, we first calculate the anomalous dimensions of single chiral and anti-chiral current operator and then we study chains of chiral, anti-chiral and mixed operators constructed out of these single currents. Our results are checked using perturbation theory appearing to be in full agreement with the Taylor expansions of the exact expressions. Note, that in our derivations, Feynman diagrams and standard perturbation theory are not used, making this method comprehensive and attractive by giving a useful perspective of coupling space geometry in calculating effective results of QFT.
Finally on chapter 7 we present a detailed calculation of the C-function for several λ-deformed theories. The useful expression for our derivation is the one relating the C-function with the β-function. Having in our grasp the β-functions for the models under study and using the Callan-Symansik equation which implies a general form for the C-function, we compute the C-function exact in λ and for large k. Our computation contains single λ-deformation, λ deformations for different level of k and coset spaces. Also it respects the monotonicity property and on fixed points it takes the values of the central charge of the CFT.
Main subject category:
Science
Keywords:
Conformal Field Theory, Correlation Functions, λ-deformations, anomalous dimensions, β-functions, C-function
