İki boyutlu kuantum alan teorisinde sonsuz konform simetrisi

Abstract

Bu tezde, kütlesiz, iki boyutlu etkileşen alan teorileri incelenmiştir. Bunların en önemli özelliği konform dönüşümleri n sonsuz parametreli gurubu altında değişmeziliği (invariance)dır. Operatör cebrini oluşturan yerel alanların, Virasoro cebrinin indirgenemez temsillerine göre sınıflandırılabildiği ; ve korrelasyon fonksiyonlarının konform değişmezlikle belirlenen konform bloklardan elde edilebildiği gösterilmiştir. Dejenere temsillerle ilgili, tamamen çözülebilir konform teoriler incelenmiştir. Bu teorilerde, anomali boyutları ve lineer diferansiyel denklem sistemlerini sağlayan korrelasyon fonksiyonları tam olarak bulunabilmektedir. Bulunan anomali boyutlarına karşılık gelen, ve kapalı bir cebir oluşturan yerel alanlardan minimal konform alan teorisi kurulabilir. Bu teoriler, faz geçişi noktalarında Cphase transition points} iki boyutlu termodinamik sistemlerin kritik davranışlarını belirler. Minimal teorilerin en basiti, iki boyutlu isi ng modelidir. Dotsenko, S. tarafından incelenen diğer bir minimal teori, yani Z Potts modeli de teze eklenmiştir.

We present an investigation of the massless, two-dimensional, interacting field theories. Their basic property is their invariance under an in-finite-dimensional group of conformal trans-formations. It is shown that the local -fields forming the operator algebra can be classified according to the irreducable representations of Virasoro algebra, and that the correlation functions are built up of the "conformal blocks" which are completely determined by the conformal invariance. Exactly solvable conformal theories associated with the dejenerate representations are analyzed. In these theories the anomalous dimensions are known exactly and the correlation functions satisfy the system of linear diferantial equat i ons. It is also shown that there is infinite number of special quantum conformal invariant theories, which contain only a finite number of basic operators. Such minimal theories of whose anomalous dimensions are found from simple algebraic equations, govern the critical behaviour of 2--di mensi onal thermodynamical systems near the phase transition points. The simpliest example of minimal theories is 2-D Ising model, and another example is Z Potts model too. a In more detail, we shall show the following. (i) The components the stress-energy tensor T (Ç) a© represent the generators of the con-formal group *§. Stress energy tensor is trace! ess and symmetric in two dimensional conform quantum field theory. The algebra of these generators is the central extension of the algebra £ and coincides with the Virasoroalgebra £ The value of the central charge c is the parameter of the theory. (ii) Among the fields A.(Ç) forming the operator- algebra, there are some primery fields » z ? £. of coordinate transformation. Here A and A are real n n non-negative parameters. In fact, the combinations d ~A -+A and s ~A -A are the anomalous scale dimension n n n n n n and the spin of the field d> respectively. The spin s of a local field can take an integer or half-integer value only. We sahall often refer to the quantities. A and A as» to the dimensions of the field. The simpliest example of the primary field is the i dendi ty operator I. A nontrivial theory involves more than one primary field and the index n is introduced to distinguish between them. >JI (iii) A copmlete set of the fields A (Ç) consist of J conf ormalf ami 1 iesCjû"1 each corresponding to a certain primary field 1 and, in some sense, serves as the n ancestor of the family. each conformal family also contains infinitely many other secondary fiel ds (descendants). Dimensions of these secondary fields form integer spaced series. < k>., - < k> -r r A = A + k, A = A + k n n n n where k, k- 0, 1, 2,.... Variations of any secondary field A e Ld> 1 under the infinitesimal conformal transformations -> 2 + £ ( Z ) are expressed linearly in terms of representations of the same conformal family Zd> 1. So, each conformal family n corresponds to some representation of the conformal group $. In accordance with.§ ~ r ® r definition, this representation is a direct product Zip 1 = V « V n n n where V and V are representations of the Virasoro n n algebra £. The representation V is known as the Verma c ri modulus over the Virasoro algebra. In general, these representations are irreducable. (iv) Correlation functions of any secondary fields vn can be expressed in terms of the correlators o-f the corresponding primary -fields by means o-f special linear dif f erential operators. Therefore all information about the conformal quantum field theory is accumulated in the correlators of the primary field . Therefore, the boostrap n xn equations (i. e. the associativity condition for the operator algebra) can be reduced to equations imposing constraints upon these coefficients and the dimensions A n of the primary field. (vi ) At a given value of the charge c there are infinitely many special values of the dimension A such that the representation C0.3 proves to be degenerate. The most important property of the corresponding "degenerate" primary field