hep-th/0507242 DAMTP-2005-68 Operator Algebra in Logarithmic Conformal Field Theory Jasbir Nagi [email protected] DAMTP, University of Cambridge, Wilberforce Road, Cambridge, UK, CB3 0WA Abstract For some time now, conformal field theories in two dimensions have been stud- ied as integrable systems. Much of the success of these studies is related to the existence of an operator algebra of the theory. In this note, some of the extensions of this machinery to the logarithmic case are studied, and used. More precisely, from M¨obius symmetry constraints, the generic three and four point functions of logarithmic quasiprimary fields are calculated in closed form for arbitrary Jordan rank. As an example, c = 0 disordered systems with non-degenerate vacua are studied. With the aid of two, three and four point functions, the operator algebra is obtained and associativity of the algebra studied. arXiv:hep-th/0507242v2 27 Sep 2005 1 1 Introduction Since the studies of Belavin, Polyakov and Zamolodchikov [1] and Zamolodchikov and Zamolodchikov [15], Conformal Field Theory in two dimensions (2dCFT) has been heav- ily studied, in particular as an integrable quantum field theory. This means that, in principle, the model is exactly solvable, i.e. all of the correlation functions can be de- termined, and hence the S-matrix. Central to this idea is the notion of an ‘operator algebra’, where the operators of the theory are endowed with a product, so that a prod- uct of two operators may be realized as a linear combination of other operators in the theory. Thus, the calculation of an N-point function can be reduced to one of a linear combination of N − 1 point functions. Given a quantum field theory, one might ask - what is the operator algebra? In general, this is a difficult question to answer, but in 2dCFT, [1] demonstrated how to obtain it. The method depends crucially on the form of two, three and four point functions of ‘primary’ fields of the theory. Primary fields of the theory are fundamental fields of the theory, in the sense that all of the other fields can be obtained by action of the symmetry algebra on the primary fields. Logarithmic conformal field theories (LCFTs) are a class of 2dCFT that have been studied heavily over the last 10 years. They differ from more traditional 2dCFTs by having logarithms in correlation functions, and having elements of the Cartan subal- gebra which are non-diagonalizable - most notably the dilation operator L0 is non- diagonalizable, and hence must be represented in terms of Jordan blocks. Due to these complexities, obtaining an understanding of the 2, 3, 4 point functions and the operator algebra has been tricky, although much work has been done towards this, including, [2][3][4][5][6][10][11][12][13]. In this note, using the constructions developed in [13] (which are reviewed in section 2), the three and four point functions of logarithmic quasiprimary fields are calculated (sections 3 and 4 respectively). It should be stressed that in the calculation of these correlators, no assumptions on the operator algebra are made, and no ansatz is used, all of which are often used in the literature. The calculations are valid for arbitrary Jordan block size, even when the different Jordan blocks inside the correlator are of different rank. The only input is the conformal Ward identities corresponding to the M¨obius group, which are subsequently solved for, yielding general solutions. In order to try and find an operator algebra using these 2, 3, 4 point functions, the case of c = 0 systems with non-degenerate vacua (that is, the vacuum does not have a logarithmic partner - for the degenerate vacua case, see [7][11]) is analyzed, which pertains to two dimensional systems with quenched disorder [8][9]. This is an unusual 2 LCFT, in that the vacuum does not belong to a Jordan block, whereas the stress energy tensor does. Indeed, there are questions as to whether or not such an unusual LCFT can be consistent. In section 5, it is found that from just assuming that the vacuum is non- degenerate, that c = 0, and that the stress energy tensor (a.k.a. the Virasoro generator) has a logarithmic partner (34), (35), with help from the three point function of section 3, one is, remarkably, able to find the entire operator algebra, which corresponds to the one found in [11]. In section 6, a partial study of associativity is then conducted, which, as is the norm in 2dCFT, comes down to studying four point functions. No inconsistencies are found. 2 Review of Logarithmic Primaries Logarithmic conformal field theories are conformal field theories that are characterized by L0 being non-diagonalizable and logarithms appearing in correlation functions. To this end, one can try and alter the definition of a primary field to accommodate the non-diagonal behaviour, and see if logarithms come out. The author should stress that it is not obvious to him if all logarithmic conformal field theories can be realized in this way. Consider an action of the Virasoro algebra on fields φi(z), i = 0 ...N − 1 given by m ∈ Z, m m+1 [Lm,φi(z)] = z (m + 1)(hφi(z)+ φi+1(z)) + z ∂φi(z) i =0 ...N − 2(1) m m+1 [Lm,φN−1(z)] = z (m + 1)hφN−1(z)+ z ∂φN−1(z). (2) Now, the φi+1 term in (1) prevents this from being a collection of N primary fields of conformal weight h - indeed as it stands there is only one primary field. Acting on the vacuum |0i, and considering z = 0, one finds L0|φii = h|φii + |φi+1i for i =0 ...N − 2, L0|φN−1i = h|φN−1i (3) and thus the primary field corresponds to the eigenvector of the Jordan block. In light of this, one can construct a vector vφ(z) out of the φi(z), and rewrite (1)(2) as m m+1 [Lm, v(z)] = z (m + 1)(h + J)vφ(z)+ z ∂vφ(z) (4) where J is a rank N nilpotent matrix, that is satisfies J N = 0, J N−1 =6 0. Now, one might try to integrate up (4), to obtain a geometric object, and one finds that v(z) can 3 be realized as a section of the formal rank n vector bundle whose transition functions are generated by dzh+J (see [13] for more details). Now, given transition functions for a vector bundle, one is always free to rewrite everything in terms of G-bundles. In the case at hand, this translates to defining N−1 N−i−1 φ(z,J) := φi(z)J (5) i=0 X in which case (4) reads m m+1 [Lm,φ(z,J)] = z (m + 1)(h + J)φ(z,J)+ z ∂φ(z,J) (6) and defines a logarithmic primary field φ of weight h and rank N. If (6) only holds for m ∈ {−1, 0, 1}, then φ is a logarithmic quasiprimary field. It should be emphasized that (6), (4) and the pair (1)(2) are equivalent ways of describing the same thing. Whilst (4) might be a more convenient realization when studying representation theory, (6) is more convenient for studying the operator algebra and correlation functions, and hence will be used here. As is usual in conformal field theory, one can restrict to m ∈ {1, 0, −1} to obtain the action under the Lie algebra of the M¨obius group. Since the L0, L±1 annihilate the vacuum, these can be used to give readily solvable Ward identities for the correlation functions. For example, one has on the two point function hφ(z,J) ⊗ ψ(w,K)i, h[Lm,φ(z,J)] ⊗ ψ(w,K)i + hφ(z,J) ⊗ [Lm, ψ(w,K)]i =0 (7) which can be solved [13] to yield hφ(z,J) ⊗ ψ(w,K)i = C(J, K)(z − w)−2(I⊗Ih1+J⊗I) (8) where, for the two point function to be non-zero, one must have the conformal weights of φ and ψ identical, i.e. h1 − h2 =0, as well as (J − K)C = 0, where C is a ‘function’ of the J, K, i.e. has an expansion N−1,M−1 i j C = Ci,jJ ⊗ K . (9) i=0,j=0 X Note, for the particular case of N = M = 2, that is J 2 =0= K2, one has the two point 4 function (surpressing tensor products) hφ(z,J)ψ(w,K)i =(z − w)−2h1 (J + K)a + JK(b − 2a log(z − w)) (10) where a and b are arbitrary, yielding the logarithms, as promised. For higher rank Jordan blocks, the solution, when expressed in components, can become quite unwieldy, with powers of logarithms all over the place. For the remainder of this note, the tensor products will be surpressed for clarity. 3 Three point function Consider the three point function hφ1(x, J)φ2(y,K)φ3(z, L)i = f(x,y,z,J,K,L) (11) where J M = 0, J M−1 =6 0, KN = 0, KN−1 =6 0, LP = 0 and LP −1 =6 0 for some M,N,P ∈ Z with M,N,P ≥ 2. For the purposes of this calculation, the co-ordinates t = x − y, u = y − z, v = z + x (12) are useful. The L−1 condition then becomes ∂ ∂ ∂ ∂ + + f =2 f =0. (13) ∂x ∂y ∂z ∂v Hence f = f(t, u, J, K, L). The L0 and L1 conditions then read ∂ ∂ h + h + h + J + K + L + t + u f =0 (14) 1 2 3 ∂t ∂u (v + u + t)(h1 + J)+(v + u − t)(h2 + K)+(v − u − t)(h3 + L) (15) ∂ ∂ +(u + v)t +(v − t)u f =0 ∂t ∂u respectively.