Current algebra and conformal discrete series

Volume 200, number 4 PHYSICS LETTERS B 21 January 1988 CURRENT ALGEBRA AND CONFORMAL DISCRETE SERIES David KASTOR J, Emil MARTINEC 2 Enrico Fermi Institute and Department o f Physics, University of Chicago, Chicago, IL 60637, USA and Zongan QIU 3 Institute for Advanced Study, Princeton, NJ 08540, USA Received 27 August 1987 We describe a large class of two-dimensional conformal field theories based on a current algebra construction of Virasoro representations due to Goddard, Kent, and Olive. The basic tool is a generalization of the Feigin-Fuchs representation. All the theories are organized by chiral algebras, the simplest examples being the Virasoro and super-Virasoro algebras. Conformally invariant two-dimensional GFT's describe universality classes of two-dimensional critical phenomena, and are the building blocks of classical string ground states. The classification of conformal field theories appears to be a formidable task. Known examples include gaussian models, affine algebras [ 1 ], parafermion algebras [ 2 ]. various discrete series [3-6], and orbifolds [7] of these whenever the model admits a finite group action. Belavin, Polyakov, and Zamolodchikov [ 3 ] proposed an approach to the problem via the conformal bootstrap: starting from a Hilbert space representation of the Virasoro algebra, one looks for an algebra of field operators in Hilbert space consistent with crossing symmetry. Little progress has been achieved along these lines, due to the difficulty of formulating and solving the crossing symmetry constraints. The Landau Institute school has also been pursuing the study of theories organized by chiral algebras of various sorts - affine algebras [1], parafermionic algebras Supported in part by DOE grant DE-FG02-84ER-45144. Submitted in partial fulfillment of the requirements for a PhD in Physics at the University of Chicago. Supported in part by DOE grant DE-AC02-80ER-10587, an NSF Presidential Young Investigator Award, and the Alfred P. Sloan Foundation. 3 Supported in part by DOE grant DE-AC02-76ER-02220. 434 [ 2 ], conformal and superconformal algebras [ 4,5 ], etc. The idea is that the Virasoro algebra by itself is not sufficiently restrictive; one needs additional assumptions in order to make progress. An important feature of these theories is that null vectors in the algebra impose linear differential equations on the correlation functions [ 3,1 ]. The correlation functions are given by contour integral representations of the solutions to these differential equations [ 8-11 ]. Often the contour representation is generated by an auxiliary, non-unitary QFT, the Feigin-Fuchs theory [ 8,9]. The distinguishing feature of the FF theory is the existence of conformally invariant screening charges q+_=~j+_ such that the null states are of the form [ 12,10] (~null(Z) =IkIk f j ± (Zk) ~ ( Z ) . (1) For instance, the basic discrete series [3,4] of models with c = 1 - 6 / ( L + 2 ) ( L + 3) is represented in terms of a free scalar field ~0 with T=-½(a~)2+ie~oO2~o, c = l - 1 2 a o 2, j+=exp(io~+~), a+=O~o+ ax/a~o+2, (~pq = exp (iO~pq{0), O~pq= ½(1-p)o~+ +½(1-q)o~_ , (2) 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) Volume 200, number 4 PHYSICS LETTERSB and 1 <~p<~L+ 1, 1 <~q<~L+2. Solutions to the differential equations have the form %a 21 January 1988 like construction for the integrable systems, away from criticality. They find lattice models for SU2(K) × SU2(L)/SU2(K+L) for arbitrary levels K, L. This work motivated our search for the conformal field theory whose Hibert space G K O had built. The values of c and critical exponents h take the values = l ~ ( z , - zj)a~,~,ap,~, i<j c -K+L+-------~ ×I~q~dwk ( z i - w k ) ....... +-(wk-wl) '~+-~'+- , k.I d%a hpq - obeying the anomalous charge conservation law Zpiqi-Jr-R0~+ - l - S 0 ~ = 2 0 / 0 , and labelled by choice of contours (g~; the crossing symmetric correlation functions are given by monodromy invariant combinations hab ~a ( g) oj, (z). The screening charges are in general necessary in order to have non-vanishing correlations within the restricted set of charges 0/pq. A representation theoretic construction of the FQS basic discrete series (at the level of characters ~ ) has been given by Goddard, Kent, and Olive [ 13 ]. They note that the decomposition of the product of SU2 current albebras, one et level K and one at level 1, under the diagonal SU2 at level K + 1, contains the discrete series characters K) I K+2 K+ 1) V ZI,,,-,)/z(q)zI ) ( q ) = Z Zl,,-~)/2(q)Zm,,(q). ,'7=1 (3) Here le{0, ½} denotes the isospin j = 0 , ½representation of level 1 SU2, and m, n denote the isospins of the level K, K + 1 representations; Zm, v is the Virasoro character of the highest weight state (9.... The G K O construction guarantees unitarity, since the affine algebra modules are unitary for half-integral j~< 1K. Similarly [ 13], SU2(K) ×SU2(2)/SU2(K+2) generates the N = 1 supersymmetric descrete series [4,5]; and generally, given an embedding l)c g we obtain representations of the Virasoro algebra generated by moments of T g - Th [ 13 ]. Another remarkable aspect of these systems is a connection to integrable systems. Andrews, Baxter, and Forrester [ 14] have constructed lattice spin systems with commuting transfer matrices which reduce to the basic discrete series at criticality. More recently, the Kyoto group [ 15 ] discovered a GKO~IA character of a highest weight irrep is the trace z(q) = ~ is>(~l qt-"[~) over the irrep of the cbiral algebra. + ' [(K+L+2)p-(L+2)q]2-K 2 l(K-2l) + - 4K(L+2)(K+L+2) K(K+2) ' l = ½ l ( p - q ) mod2KI, l<~p<~L+l, O<~I<~½K, (4) I<~q<~K+L+I. K = 1 is the basic discrete series; K = 2 is the supersymmetric discrete series [the second term in hpq is the ground state energy of the Neveu-Schwarz ( l = 0, 1 ) and Ramond ( l = ½) sectors]. The G K O method works for any coset pair of algebras g/b, because Tg - Th is a stress tensor with c = c g - Ch. Therefore we expect conformal field theories with the corresponding T, c, and characters Z- Our purpose is to outline their construction at the level of correlation functions; details will be presented in a longer paper in preparation [ 16 ]. The basic idea is to find a generalized Feigin-Fuchs construction which automatically builds in the null vector structure. This construction relies on the vertex operator construction of SU2(K) [2]. Briefly, SUz(K) is realized on a free scalar field ~o (on a circle of radius x f ~ ) and a "parafermion" theory; the diagonal current is bosonized, and the others are realized non-linearly in terms of the scalar and the parafermions ~u_+ (0q/+ =0, h± = 1 - 1/K) J3 =ix/K0~0, J_+ =x/-J~q/± exp( + i ~ / x ~ ) For K = I the parafermion theory is trivial; exp( + i x ~ ~) are currents by themselves. The highest weight states (9lmof the current algebra, with isospin j = l and j3=m factor into free field and parafermion contributions (glm = exp(im~0) Olm • Each highest weight representation contains an infinite number of Virasoro highest weights. From the current algebra structure we deduce the operator product algebra of the parafermion theory. Corre435 Volume 200, number 4 PHYSICS LETTERS B lation functions of the parafermion system are just those of the current algebra fields (9Zmup to a rational function [the contribution of the exponentials exp (imp0) ]. The level K parafermion correlations respect a ZK symmetry (9~,~--.exp(2nil/K) (gb,. The Z2 parafermion is the Ising model fermion; the Z3 parafermion theory is the three-state Ports model. We next observe that for the list (4) the accumulation points of c and certain hvq are the conformal weights of current algebra highest weights. Moreover, for the K = 1 discrete series we have the FF construction (2), and for K = 2 there is a supersymmetric FF theory with screening charge q+ = ~dz d0 exp[ic~+ ~p(z, 0)] =~dz gt exp[ic~ +~0(z)]. This suggest that the CFT's (4) are realized by taking the vertex construction of level K SU2 and turning on a background charge ao on the scalar ~0, such that the off-diagonal currents j+ become the screening charges of the discrete series models. This is indeed the case. In fact, there are two FF constructions, since the GKO construction is symmetric in K and L. Let us first dicuss the Z~ parafermion construction ( K < L ) . Then the scalar ~0 has T = -- ½( O~O)2 + ioto02 ~O , K . 1/2 O~o=(2(L+Z)(L+K+2) ) such that c= 1 - , (5) 12C~o 2. We find screening charges q+ = ~ ~_+ exp(ioz+ ~0) , __(2(L+2) o~_a+=\ ,,/2 \K(K+L+2)J ' ( 2 ( K + L + 2 ) ~ '/2 ~ ] (6) (gvq= exp(iapq~O)q)lm , (7) and primary fields O~pq= ½ ( 1 - P ) a + +½(1-q)~x_ , l~<p~<L+l, l=m, I~q<~L+K+I, O~m,l~½K. subject to charge conservation iap,q,+Ra+ +Sc~_ = 2c%. Examples will be worked out in ref. [ 16 ] (see also refs. [ 9,17 ]). There is also an alternate construction with ZL parafermions. We simply repeat ( 5 ) - ( 8 ) with K*-*L; the relation between the assignments of quantum numbers is m= IP-ql, l = p ' - 1, q=q', p - 1 = IP' - q ' mod 2Lp, where primes denote the Z~ quantum numbers, and IP-ql <g. Fields with l ¢ m are descendants in an algebra that generalizes the Virasoro and super-Virasoro algebras. For p - q > K the dimensions hvq and hp,q, only agree up to integers as a result of the algebra structure. The null states for SU2(K)× SU2(L)/SU2(K+L) arise from known algebras for L = 1 (Virasoro) [ 3,41, 2 (super-Virasoro) [4,5], and 4 (spin 4 currents×Virasoro) [2], and similarly for the SU3 ZF theories (spin 3 currents×Virasoro) [10]. The screening charges guarantee the existence of null states of an algebra. For K = 1 (2) the screening charges commute with the (super)Virasoro algebra; therefore, if ~ is highest weight, then so is (1). However, this field is not on the list (7), (8), so it must be a descendant as well. The property of being simultaneously highest weight and descendant characterizes a null state. Thus we search for new algebras for K > 2 by looking for fields which commute with the screening charge. The algebra which generalizes (super) Virasoro is generated by ~T.]=0~0~1,0 "Jl-i I ( K + 2)C~000~.0 + i½(O~+ -- a _ )~1,o , OI,O.-~A~I/KOI,1, (8) The exponential of the scalar ~o contributes the first 436 term in hvq, eq. (4); the parafermion field contributes the second. Non-vanishing correlation functions are built from up to normalization. Here ¢~.o is the energy operator of the lowest dimension in the parafermion system, and ~Lo is a related field: with m=mvq=½1p-qmod2KI, 21 January 1988 ~l,O--~At-l/K_l~)l,l, where A, A* are modes of the parafermion fields gt+, gt_. For K = 2 the last term is absent, since ~Lo is the same as 0q~Lo. The fields in the hvq table for K < L Volume 200, number 4 PHYSICS LETTERSB which are absent for K > L are descendants in the level K ~ algebra. The discrete series are representations of both the level K and level L ~ algebras, and it is sufficient to restrict attention to fields that are highest weight under both. This structure is already apparent in the tricitical Ising model, which has both a Virasoro and super-Virasoro FF construction; the h = 3 supercurrent (the descendant of the identity) appears as (_0~.3of the Virasoro construction. This feature is generic - the closed subalgebra generated by (9~,3 in the basic ( K = 1 ) discrete series are the generalized currents of the level L construction. Finally, we note that as L--+~ for fixed K we approach the SU2(K) current algebra. In this limit, the screening charge become the zero modes of the currents, and the current g-' reduces to :j"ea:, where (9a is the adjoint highest weight field of the current algebra, treated as a holomorphic field (g-' commutes by construction with the zero modes fja of the currents, and :fO": is the only invariant of the right dimension). The characters for the chiral algebras are given by a generalization of the G K O construction [13 ] for the super-Virasoro characters. The raw materials for building the characters of the chiral algebras are the branching functions [ 18 ] defined by 21 January 1988 where OLpq ( t ) = t i p . _ q ( t ) [2(K+L+2)(L+2)t+ (K+L+2)p- (L+2)q] 2-K 2 4K(K+L+2)(L+2) mp.q( t) = I ½ ( P - q ) + ( L + 2 ) t (mod K) I , and the functions Cl,~)(z) are the SUz(K) string functions [18]. For c = 0 ( K = 0 ) , Zl.l v =1, and (9) gives identities on level L string functions. Each algebra has two sectors: p - q even (integer l), and p - q odd (half-integer 1), analogous to the NS and R sectors of super-Virasoro. Within each sector we group collections of fields according to their various l values, the simplest example being supermultiplets in the case K = 2 (but note that generally there is no global subalgebra!). From the character formulas we conjecture a determinant formula [ 19] for the K th chiral algebra. Fields with different l values are orthogonal, so the matrix of inner products is block diagonal in l. The determinant in each block is det M Kj = const. [I ( h - hpq)P(. . . . -- rpq) , p,q; p--q=21 mod 2 rpq <~tlp,q )~tK)(z)xIL)I)/2(Z)=E)~lpq( q V Z ) X { qK- -+l )L/ 2) ( Z )" (9) In the case K = 2 , /~{0, ½, 1} the NS supercharacter is the sum of the branching functions f o r / = 0 and l= 1 and the R character is the l= ½ branching function. In general we will have characters of the chiral algebras given by K/2 where _~+m 2 _.. 2 Pq ¢rtP'-q~ z , rpq- Iaq K np.q=~ l(l+ ) + mpq(mpq+l) K+ 2 K+ 2 ' and P(k) is the coefficient of za in V z,~(z) Z x~o(z), = /=0 2 /K Z - - I ( l + l ) / ( K + 2 ) + m "-q CI .K) ..... (z). where the sum is over integer l values for p - q even and half-integer l values for p - q odd. For the general case of S U z ( K ) × S U 2 ( L ) / S U 2 ( K + L ) the branching functions are given by K/2 V = E C h'~ n(Z) z,~(z) m:0 /rlllq( I ) =/n I~Zp, _ q( I ) =/11 The proof should be similar to the usual one based on screening charges [ 8,12]. The vanishings of M provide another route to the unitarity of the models [4]. The work of [ 13,20,15 ] neatly gives us the modular invariant torus partition functions of these theories. We read off the modular properties from those of the SU2 characters. Modular invariant partition functions are products of SU2 invafiant character metrics Naa 437 Volume 200, number 4 ~ PHYSICS LETTERSB 2 Na, Ns/,Ncc --1 Zab,'( V q)zabc( V q) 6tjLC , generalizing results for the cases K = 1 [ 21,22 ], K = 2 [23]. The matrix Naa (or NEt, in the alternate construction) generalizes the GOS projection of the super-Virasoro case. The general g/I) theory partition functions follow analogously. Higher loop correlation and partition functions are an open question. One must understand how to "factor out" I) in the G K O construction [24], or what is likely easier, to extend the FF construction to arbitrary Riemann surfaces. This is a technical question, not a question of principal; the difficulty is that the null states (1) must be removed from the Hilbert spaces propagating in internal loops. Therefore the characters satisfy linear differential equations (with modular forms as coefficients!) which completely determine them (see ref. [25] for an analysis in the basic discrete series). Clearly a FF construction, which parametrizes the solution space to these equations would give greater satisfaction. The FF construction generalizes still further to an arbitrary coset pair g/I). Gepner [26] has described the vertex operator construction of current algebras at arbitrary level, arbitrary group. The currents have the form j, = (i2x/K/a~,2)a~,.0(o, j ~ , = ~ i = 1..... rank g, ~u,, exp(ia.q~/x/K ) . The bosons ~o span the maximal torus of the group. The unitary representations are described in ref. [27]. Again we turn on a background charge a~o and turn the j,, into screening currents. We defer details to ref. [ 16 ]. We note that Zamolodchikov and Fateev [ 10 ] have worked out the case SU3(K)XSU3(1)/ SU3 ( K + 1) which produces Z3 invariant models. The SU2 models are Z2 invariant. Generally the models should be invariant under the part of the center of g that is preserved by I). Generically one expects only these symmetries, but it may be that a model has more than one construction. For instance, the threestate Potts model occurs in both the SU2(K= 1, L = 3 ) and S U 3 ( K = L = 1) series; thus it carries a Z3 as well as a Z2 symmetry. The discrete symmetries might be exhausted in this way. The Kyoto group [ 15] conjectures that any coset pair produces an integrable system, and have found lat438 21 January 1988 tice models for SUn(K) ×SU,(L)/SUn(K+L). To summarize, we have uncovered a vast class of two-dimensional CFT's based on current algebra, whose correlation functions on the plane have FF contour integral representations. The basic ingredients are a "coset pair" g/D, the vertex construction of 9, and FF screening charges. The models appear to be intimately connected with integrable systems, perhaps due to the integrability of chiral models themselves. The FF construction builds in null vectors, thus all the theories are organized by a chiral algebra and its concomitant null vector differential equations. Although the models arise from current algebra, they contain no currents if I) is maximal; depending on the groups involved, they may not even have discrete invariances. This is particularly important in string theory, where currents [ and certain (1,1) operators] are phenomenologically unacceptable. We gratefully acknowledge helpful discussions with A. Kent and S.H. Shenker. Preliminary reports of our work were presented at SLAC and LBL (June 1987) and at the AMS Summer Institute on Theta functions (July 1987). Appendix (by E. Martinec). Let us return to the question posed in the introduction - how do we classify two-dimensional CFT? Friedan and Shenker [28] have proposed an alternate starting point for the conformal bootstrap - instead of imposing constraints on local Virasoro representations in order to obtain a two-dimensional quantum field theory, they ask what global modular forms arise from two-dimensional QFT. We call the study of families of twodimensional QFT partition functions over the moduli space of Riemann surfaces modular geometry. The characters Z and their higher genus analogues form representations of the modular (mapping class) group F. Friedan and Shenker [28] propose that a first step in the classification program is to exhaust the set of modular geometries that consist of finite dimensional representations o f F ; they call these rational conformal field theories. We might then hope to complete these in some appropriate way to some class of infinite dimensional representations. All of our discrete series models are rational conformal field theories, with rational c and critical exponents. Ra- Volume 200, number 4 PHYSICS LETTERS B tionality is a severe constraint. Cardy [21] has proven that theories with c > 1 c o n t a i n an infinite n u m b e r of irreps of Virasoro. Generically these have arbitrarily distributed values of conformal highest weights h (consistent with m o d u l a r invariance, of course). The Virasoro characters would all have different eigenvalues e x p ( 2 g i h ) u n d e r z ~ z + 1. Thus a necessary condition for rationality is that the conformal weights h consist of a finite set {ha}, plus integers. The infinite collection of Virasoro characters must then assemble into a finite set z . ( q ) = E i.h,~,,+zZ, v (q). (Note: this is at the level of chiral characters, before assembling the partition function Z = Z xx.) All of this is extremely unlikely unless there is some larger chiral algebra (typically not current algebra!) organizing the representation content. In think there is a sizeable body of evidence for the following Conjucture. The rational conformal field theories are equivalent to representations of chiral algebras. There is no k n o w n counterexample a m o n g the theories listed in the introduction, a n d our new discrete series certainly fit in; the above argument is rather persuasive. A strong form of the conjectured would be that all rational theories are related in some way (via orbifolds, etc.) to our current algebra discrete series. This is basically just wishful thinking; the evidence consists only of an absence of counterexamples. For examples, the Z~ parafermions of ref. [2] are S U 2 ( K ) / U ( 1 ), or SUK(1 ) × SUx(1 )/SUK(2); the N = 2 discrete series is SU2(K) × U ( 1 ) / U ( 1 ) [6]. What is needed is a procedure for d e t e r m i n i n g the chiral algebra from the m o d u l a r geometry. There is a beautiful local-global interplay between operator algebras a n d m o d u l a r geometry. In one direction, the null vector equations of a local operator algebra determine the global m o d u l a r geometry in terms of sheaves of modules over the ring of differential operators [29]. The differential constraints supplem e n t the algebraic Ward identies of ref. [30] so as to completely determine the correlation functions. Conversely, recent work with Shenker shows that the global m o n o d r o m y of a m o d u l a r geometry defines a set of differential equations [ 31 ] which should have a local interpretation in terms of an operator algebra ( a n d does in the k n o w n examples). The added al- 21 January 1988 gebraic structure of an operator interpretation should prove helpful in sorting out the rational theories. Finally, the connection to non-critical integrable models needs further elucidation. How does the beautiful algebraic structure of the critical theories embed into the Yang-Baxter algebra (or some appropriate generalization) ? References [ 1] V. Knizhnik and A.B. Zamolodchikov, Nucl. Phys. B 247 (1984) 83. [2] A.B. Zamolodchikovand V.A. Fateev, JETP 62 (1985) 215; 63 (1986) 913; Spin 4/3 "Parafermionic" currents in 2d conformal field theory. Minimal models and tricritical Z 3 Potts model, Landau Institute preprint (November 1985), Theor. Math. Phys. to be published; D. Gepner and Z. Qiu, Nucl. Phys. B 285 (1987) 423. [3] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nuel. Phys. B 241 (1984) 333. [4] D. Friedan, Z. Qiu and S.H. Shenker, Phys. Rev. Lett. 52 (1984) 1575; in: Vertex operators in mathematics and physics, eds. J. Lepowsky el al. (Springer Berlin, (1984); Phys. Lett. B 151 (1985) 31. [ 5 ] M. Bershadsky, V. Knizhnik and M. Teitelman, Phys. Lett. B151 (1985) 21. [6] W. 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