W-cebirleri

Abstract

Bu tezde, amacımız literatürdeki çalışmaları tarayarak konformal boyutları (spinieri) k 3.a olan primary alanları bulmaktır. Bunun için Toda Alan Teorilerine geniş olarak yer verdik, I İkinci bölümde, Virasoro Cebrinin ilk ' olarak Sicim Teorilerinde ortaya çıktığını, W-Cebirlerinin ise ilk olarak 2 boyutlu Konformal Alan Teorilerinde ortaya çıktığını ve standart örneğinin' de Zamolodhickov'un W-Cebirleri olduğunu gösterdik. K d> üçüncü bölümde ***-+f e 1J -1 denklemini sağlayan Toda Alanlarının çözümlerinde elde edeceğimiz ~ İ serbest alanlarını Mansfield 'in çözümlerinden doğrudan aldık. (X ) ve v (X ). fonksiyonlarının Wronskian koşulunda yazdığımızda ( N+l ). dereceden bir dif. denklemi sağladığını gösterdik. Bu denklemde türevlerin katsayıları bize ü(jj) Kuantum potansiyellerini verdi, ve bunu SU(3) ve SU(4) grupları için Toda Alan Modellerini yazarak onların çözümlerinden Zamolodhickov' un spin 3,4- primary alanlarını elde ettik. Ek kısmında Virasoro Cebrinin tanıtılması ve W-Cebrinin komütasyon bağıntılarının çıkarılışını sunmaya ayırdık. IV

SUMMARY (. W-ALGJEHRAS ) In this thesis, our aim is to review some very important work. This work tries to find primary fields of conformal dimensions(spins)k=3,...n built out of free fields and forming a closed algebra. By closed we^mea^ closed in the usual non-linear sense, the OPE of any two W `,W should be re^xpressible in terms of differential polynomials of the other- W s with no dependence on the <t>'s. Clearly,, a good condidate for W is some combination of (.d<P) ', (&4>) & 4>, anything containing a total of k derivatives. One might write down the most general such expression and then determine the coefficente so as to obtain a primary field of dimension k and compute the OPE s among the W ' s. This is clearly reasonable only for the W{3) -algebra, However, having figured out what W 3' is for the W -algebra. One observes an underlying structure related to the weights of SU(3). This motivates the following guess: the W 'e are obtained by considering the following differantial equation of order, n (*~ eN+lA)(d_ eNA)---(d` eıA)Va= ° where e^.. »eN+1 are Unit Vectors. We define the following Vector in the Weight space N A(X) = E >^P,(X) for AN i=l x x wIn section-3 we study in detail the classical solutions of the field theories. Our general form of the Classical solution is an explicit sum of products of functions of the left-and right-moving coordinates. Each of these functions, may be expressed, generically, as a V&onskian of a set of basic functions. These functions are shown to obey differantial equations whose coefficents-called generalised potentials-turn out to generate the extended Virasoro symetries. The bosonic Toda field theories obey extended Virasoro symmetries which involve generators of spins higher than two, and their quantisation gives a systematic treatment of generalised Conformal bosonic models. The Toda field is a multicomponent field in two space-time dimensions satisfying a generalisation of the Liouville equation. The general solution was written by Lesnov and Saveliev [1]. Ki1*i + 1 V*-^i = e with X- = ± {a±r),â±da±âT K is the Cartan matrix of some simple Lie algebra g. For a simply laced algebra (all roots hove the same length, as for AN,DN,EN)E is a Symmetric matrix and an action is 1 KU*J S ~/ d</.dT( ± a 4, k a <p.+ E. e 1J J ) The Canonical energy-momentum tensor has a trace equal to two times the potential (above the action) The tensor can be improved to yield the traceless energy-momentum tensor, T±± = v v= 2To* - Vi*ij Vj-2f *l *i The Toda field theories are thus conformally invariant, the conformal transformations being generated either by the ++ or the - component of T £he stress-energy tensor... For AN : K... = 26^-6^^ ifj = 1,...,N VIHere we concentrated on the algebras A`(SU(N+1)). For the algebi-as A`, the N fundamental weights X, correspond to the representations by antisymmetric rank- i tensors which have dimensions (N+l) i/(i! (N+l)-i) ! ). We use the solution that is found by A.N. Lesnov and M.V. Saveliev [1]..'`i = «x (g d`»'', ( `*(**> x.> Here X.are the N fundamental weight vectors and (X.>denotes the highest weight state in the corresponding x ± ± representation space, and the f. (X )are 2N arbitrary functions of the light -cone variable. The left-moving part of the exp(-#. ) contains the factor. N -<f>. ^(X*) = n ft (X+) 1J 1 j = l J If we call VX (X+) V1(X+)M((X+)X1> = J v2 (X+) WX+> Vll-4>. e 1 = *(X )V(X+) After writing the free fields (#. ) in terms of x and y, putting these fields in equat!.on a ao, 2*N `^+1 we find v2 e = xx'w' for N=2 e = XX'X`W'V` in the same way we define e l = a*'...*'1`1* W'...^i-D For A.,, the series simply stops since the w 's and the x 's N a a each have unit Wronskian [6]. XX'. -.X ¥V' -. -V = 1 One sees that exp(-#N.) equals 1 and the field #h.-j Vanishes. This Wronskian condition clearly shows that the y satisfy a linear differantial equation of order (N+l) by considering the ratios of solutions v a a i- a... a i- a -L a -L- y, =0 a=l,...,N+l IN £2 *1 vl a Ylllf.=f. (X ) are arbitrary functions of the light-cone + variable X, and. p.. denotes the inverse of the Cartan matrix K. The functions 1/f.and 1/y.are moved to the left of the differential operators, *4: = T7<*+Pi>- a-k'--kl8-f,uri) and P, = - i 1 fi The resulting differential equation of order (N+l) reads (d+P1+...+PN-^>ljPi)x...at(*+P1-*>liP1)(*-*>liPi)¥'a=0 ty/hen we def ine the following vector in the weight space N A(X) = E / P,(X), for AN i=1 i i IN The differential equation then simply becomes (d_eN+lA) (d_eNA) *.. <d-eiA>Va=0 The standard form is (-dN+1+ ü2dN 1+ ü3«N 2 +---+0N+1)Va = 0 ££.When we solve this differential equation for N=2, A9=SU(3) U.?.=2e- A'+e2A'~e0Ae1 A-e^Ae^-e-Ae. A U. 3. = ( e -A` +e§A ' e, A+e` Ae`Ae 1 A-e~Ae 1 A ' -e` Ae 1 A ' ) Similarly, for H-3, M»W one can ob*ain U(4> These quantum potentials are not primary fields. The definition of primary field is If we have U` 9V #V T(Z)U,_ = - *- ? + s- + - *- +. UJ (z-w)* (z-w)^ (3-w) *U2 The second term ( 9) shouldn't appear in this OPE u-wr (Operator Product Expansion). So, we subtract the derivative of U,2) from Ug l t W(3)`ü(3)~`^~ü2 this new expression is called Zomolodchikov's spin-3 operator, X