WN cebirlerinin kuruluşunda genel bir yöntem

Abstract

Bu çalışmada son birkaç senedir parçacık fiziğinin, istatiksel fiziğin ve matematiksel fiziğin en önemli araştırma konularından birini oluşturan Virasoro cebrinin genelleştirilmiş formları üzerinde bir genel yöntem elde edilmeye çalışılmıştır. Bu genel yöntem yardımıyla genelleştirilmiş bir Virasoro cebrine ait kommütatörler doğrudan konform spin alanlarının Fourier modları olan katsayı doğurayları için açık olarak elde edilmişlerdir. Bilindiği gibi bu tür cebirlerin dikkat çekici taraflarından biri, kapsadıkları non-lineer terimlerdir. Çalışmamızda bu temel sorun için de belli bir ifade tarzı getirerek W4 örneğinde uygulamaya çalıştık. Bilindiği gibi W4 cebri Virasoro cebrine 3 ve 4 konform spinine sahip iki doğurayın eklenmesiyle elde edilmektedir ve literatürde nispeten daha az bilinen bir yapıya sahiptir. Çalışmamızda bu yapıyı kommütatörleri açık bir şekilde ortaya koyarak belirledik ve ayrıca tüm Jacobi özdeşliklerinin bu kommütatörler aracılığıyla sağlandığını göstererek elimizdeki W4 cebrinin assosiyativ özelliğini de kanıtlamış olduk. Her ne kadar burada W4 örneğinde kaldıysak da önerdiğimiz genel yöntemin tüm genelleştirilmiş W- cebirleri üzerinde benzer şekilde uygulanması mümkündür.

It is well know, since their first appearance in physics, WN - algebras have the importance as underlying symmetries for two - dimensional conformal field theories which play an essential role both in physics and also in mathematics. They are frequently studied within the framework of conformal bootstrap. They can be studied also in terms of the Fourier modes of primary conformal fields. In this thesis work, we studied W - algebras in the Fourier mode formalism as augmented forms of Virasoro algebra with the inclusion of generators having conformal spins higher than 2. In this framework, we know that Virasoro algebra can be defined as: [Ln, LJ = (n-m) Ln+m + -SL (rf-n) ön+ffl,0 There, c is taken as the central term and 1^ as the Virasoro algebra's generators. As it is known, The primary generators W(s) which have s conformal spin are defined by: [L", K3)] = ( (s-lj- n-m) w± WN algebras that we will study are formed from the extensions of n-2 primary generators which have s=3,4...n spins to the Virosoro algebras having only spin 2 generators and a central term c. From the mathematical point of view, the prominent feature of W-Algebras is their non-linear character, i.e. they contain some terms of multiple products of primary generators. The emphasis here is on the fact that these composite operators are the ones which are fully normal ordered. This brings us back to the very definition of normal orderings for double, triple.. etc products of primary generators. It will be seen in the following that the whole structure of a WN algebra could change depending on the manner of definition for normal ordering of composite operators. Therefore (AB)n=Ş2 8<İ,İ2-İ) (AiBa-i+B^J is a double composite of primary generators i^ and Bn which are Fourier modes of primary fields of conformal spins a and b where Q(i,j)=\ f 1 i>J 1 - o ij We can also define a new composite generator which has the form of triple products of three primary generators. Using the function 8(i/j) and XIr(LL)Q the triple product of three spin 2 primary generators is given by: (L.LL)n=£ 6(i,il-i).(X^t+LiK^j) vxx It is known that WN algebras are in general constructed with the aid of such polynomials and also some structure constants. Then the commutator between two primary- generators has the form İ+J-1 £ ic=o U (i) '. B U) -İ = £ aijk Fijk (n,m) X(k) n> where Fljk(n,m) = < 0 (n-m) {n-m) P{i+j-k-2).P(i+j-l-Jc) i=j,ic=i+j-i i=j',Jc=i+j-2 i=j,k>i+j-2 i*j,k=i+j-l i*j,k>i+j-l with; (i) X(a=0)n,a=Sa+m,0, (ii) Pa(n,m) is a polynomial which the higher total degree of integers n and m is a (iii) X{a)n>m is a primary generator or a normal ordered composite generator which has a total spin a. We therefore define W4 comutators as follows Vlll WB,Wj=a (n-m) Tn+a + p (n-m) (LL) n+m+ (a-ri) PL2{n,m) Ln.a + (n-m) PWl^ju) ara+JB + cw5 (i3)8a+JD#0 and *4 (LL)*aH-P£4(n,iD) La+iB + P&r3(.l2,.m) ^+ffl + PT2(n,m) Tn+m + cwt6(n) 6n+JS/0 and [Tn/rj=Yi (n-in) (XL)n4.m+Y2 (û-in) (ri)n+fl! + y3 (xi-m). (RW)a. (Jn-ja) (L£)'2?*.+ (a-jn)' P^ (22,122). (WL)a+a + (a -in) JPL4. -{n-m) PW3(n,m) Wa+Itt + {n-m) PT2(n,m) Tn+m + cttT(n) All the structure constants and polynomials of these commutators are determined by the aid of the following Jacobi identities: LLk, Aa, BJ 1 + lBm, lLk,AQU + Un, {Bn. Lk] ] h0 xx We have also shown that the Jacobi identities without a 1^ generator are fulfilled in order to complete the assosiativity of W4 algebra.