mf-
LAPP-TH-222/88 MAY 1988
Vertex operators for affine algebras and superalfeberas
Luc FRAPPAT, Paul SORBA L.A.P.P., BP 909, 74019 Annecy-le-Vieux, France
Antonino SCIARRINO Dipartimento di Scienze Fisiche, Université di Napoli and I.N.F.N., Sezione di Napoli 80125 Napoli, Italy
Abstract
We emphasize the role of the boson-fermion correspondence in two dimensionnal conformai field theory for the realization of level one representations of affine untwisted and twisted Kac- Moody algebras via vertex operators. Using also the boson-boson correspondence, vertex operators for contragredient affine superalgebras can be constructed.
Contribution to the 3rd Annecy Meeting in Theoretical Physics on "Conformai Field Theory and Related Topics", March 14-16, 1988
L. A.P.P. Ill' W) I -41)1') \S\I( \-l I-NII I \ < I I)I \ • III I IMIOM >li:U2.l- «III I\.W5IWM • Il I I (OIMI 50 2" 1. INTRODUCTION
Kac-Moody algebras }) constitute today a corner stone in the study of two dimensional conformai field theories 2), and in this context the role of vertex operators has become more and more important 3X From a pure mathematical point of view the construction of vertex operators associated to a simple affine Lie algebra provides a realization of the basic or level one representation of this algebra. Moreover such objects directly appear in string theory where they are linked to the emission and absorption of a string state during its propagation 4). More precisely, in order to evaluate integrals on the world sheet associated to a string diagram with incoming and outcoming closed strings, one is led, using the invariance of the action under a conformai scale transformation of the metric on the world sheet, to transform the string diagram into a compact world sheet. Then an incoming or outcoming string is projected onto a point of this surface. It is natural to associate to such a point a local operator describing the string propagation in the 1+1 dimensional field theory, this operator having the quantum number of the string, which have been conserved in the above conformai transformation. Such an operator obeying covariance properties under Poincaré, conformai and reparametrizations of the world sheet coordinates transformations, is called the vertex operator of the string. Its general form is 5) V(k,a,T) = f$XM,...) eikX (1.1) where X^ = XM(a,x) (|i=l,...,D) are the string coordinates in the D-dimensional space, O" characterizes a point on the closed string and X is the proper time of the string. The function f(3X^,...) depends only on the derivatives of XM-, while the factor e^X reflects the property of the wave function of an external state of momentum k to be multiplied by eika under the translation XM- -» X^ + aM. Actually it is this last part which occurs in the algebraic study of affine algebras. Let us note also that a vertex operator has conformai dimension 1 under the Virasoro algebra, which in particular ensures the transformation of a physical state into another physical state. Vertex operators show up explicitly in the gauge treatment of the heterotic string model 6X Here the gauge groups G = Eg x Eg or Spin(32)/Z2, allowing the cancellation of Lorentz as well as Yang-Mills anomalies, are introduced in a natural way in the toroidal compactification processus fer degrees of freedom of the left-handed bosonic sector XLV = Xv(a-x) (v = !,...,26) - let us recall that XRM- = XM(C+T) (|X = 1 10). A judicious choice of the compactification radius allows to enlarge the initial abelian gauge group (U(l)L.)d=16 to one of the two above mentionned gauge groups G. For such a purpose, one considers the torus T'6 = Rl6/pl6 where F16 is the 16 dimensional lattice of Q, the Lie algebra of the group G under consideration. Then an explicit construction of the gauge group generators acting on the Fock space of the states in the L sector of the heterotic string can be obtained owing to the Frenkel-Kac-Segal construction 7) of the level one representation for the affine simply laced algebras tjW in terms of vertex operators. Actually the above considered vertex operators involve only the internal symmetry degrees v = 11,...,26. In order to obtain a 4 dimensional string model, it is necessary to compactify six more dimensions in the 10 dimensional 5Vfio space-time 8). This can be done for example by compactifying the dimensions |i,v = 5,..., 10, that is transforming M\Q into 9^4 x !% where % is a Calabi-Yau manifold or an orbifold. In this last approach, one has 1K^ = T6/© where CD is a discrete group acting on the torus T6. One can naturally induce an action of (D on the gauge degrees of freedom. This leads to an explicit breaking of the gauge group, and to the apparition of twisted sectors, i.e. sectors in which the string obeys boundary conditions which are periodic modulo a 1D element : X(ze2iK) = Cu X(z) with coe 2>andz = a+iT (1.2)
The algebra Q of the initial gauge group G breaks then to a subalgebra Q0 left invariant by the automorphism induced on Q by co. Let us note that by imposing twisted boundary conditions on the vertex operators, one obtains a representation of the Kac-Moody algebra CjO) equivalent to the starting one if co is an inner automorphism of Q. If CO corresponds to an outer automorphism of Q, one will be led to a representation of a twisted Gf-^ subalgebra Cf
The above considerations motivate a general study of vertex operators for affine simple algebras, starting from the Frenkel-Kac-Segal construction for simply laced K.M. algebras. One can distinguish two main approaches. In the first one, twisted boundary conditions aie imposed on the vertex operators of a simply laced algebra (^1X Such a study 9) allows to show explicitly the breaking of the symmetry due to an automorphism co of the root lattice of Q, to compute the basic Kac-Moody representation and in particular consider in detail the zero-mode space 10). As already mentioned, when co belongs to the Weyl group of Q, the obtained representation is equivalent to the "untwisted" one. This is no more the case when co is not an inner automorphism and one has then a direct construction of the vertex operator representation for the twisted K.M. algebras. We will denote this approach "bosonic" in contrast with the second one, also called "fermionic" approach, which will be the main subject of this lecture. Indeed a fundamental property of two dimensional conformai field theories which directly occurs in the construction of affine algebras representations via vertex operators is the boson- fermion correspondence 11X It allows in particular to get classes of four dimensional string 1 1 models, either from purely bosonic fields ^) or from purely fermionic ones ^)- it is this basic property we would like to keep as a guiding line in the following. In such a framework vertex representation of K.M. algebras can be written in terms of bilinear in fermions. Starting from simply laced algebras, the folding of Dynkin diagrams helps for the construction of the vertex operators for non simply laced algebras 14\ Moreover the folding of the extended Dynkin diagrams - or Dynkin diagrams associated to ^W - will allow us to build the vertex operators for twisted affine algebras 15X Let us add that results related to these two approaches can be found in ref. 16. Finally the boson-boson correspondence joined to the fermion-boson one gives an explicit expression of the vertex operators for the affine orthosymplectic superalgebras 17). Note that a current algebra treatment of symplectic bosons which arise in constructing superconformai ghosts of fermionic string theories leads to the use of affine superalgebras 18). General properties on superalgebras and of their affine extensions 19) can then be used to generalize this construction for the other basic untwisted and twisted affine superalgebras 20). The plan of the seminar is as follows. We start by recalling in sec. 2 some basic definitions and notations about the Kac-Moody algebras. Then sec. 3 is devoted to the fermionic construction of vertex operators for affine algebras. After a summary of the boson- fermion correspondence, we apply it to the vertex operator construction of the different types of affine algebras using illustrative examples : 50(2Z)O) for the simply laced algebras, 50(2/+I)W for the non simply laced ones and SO(2/)(2) for the twisted ones. The interested reader is invited to consult ref. 15 for the general case. Vertex operator construction for affine superalgebras is summarized in sec. 4, where after presenting the boson-boson correspondence, the case of the level one representation of the orthosymplectic superalgebras OSp(MIN)W is worked out. In the same spirit as in section 3, we restrict ourselves to a specific case and just indicate the main steps for the generalization of this construction to any other affine contragredient superalgebras. Finally we conclude with a few words about possible further developments.
2. BASIC DEFINITIONS AND NOTATIONS
Let us briefly recall the definition of untwisted and twisted affine algebras. A Kac-Moody Gf-V constructed from a simple Lie algebra Q is the loop algebra £0) = C[t,H]® £ © Ck © Cd (2.1) in which we denote by Qt.t"1] the algebra of Laurent polynomials in the complex variable t, k is the central extension term and d is the derivation. Commutation relations among generators of ^1) are : m n m+n [t ® a, t ® b] = t ® [a,b] + m(a,b)ôm+n k (2.2a) [d , tm ® a] = -m tm ® a (2.2b) where a,b are Q generators and (a,b) denotes the usual Killing form on Q. A twisted affine algebra ^(m) (m*l) is defined with the help of an outer automorphism t of Q of order m (i.e. m is the smallest positive integer such that xm = 1) in such a way that its elements tn ® a = ltln eine ® a = f(6) ® a (2.3) satisfy f(9 + 2K) ® a = f(0) ® x(a) (2.4) Setting ^={ae$ I x(a) = e2i^m a} (2.5) where the quantities e2mk/m (k=0,...,m-l) are the m eigenvalues of x, one has for Ç(m) the decomposition m-1 n+k m Çftn) = £ t / ® £k (2.6) k=0, neZ and for Ç = C0 ® Q\ © ... © Çm-\ the Z/mz gradation :
[&, Q\] c &+i (modm) (2.7) Remark that (^m) is actually a subalgebra of (^1) (this is obvious when replacing in eq. (2.2) the t variable by zm in order for the Q^ generators to appear vtth integer powers in the complex variable). In the Cartan Weyl basis, the commutation relations of an affine algebra ^O ) of rank I are
[H'm , HJn] = m Ôm+n 5ij
[H*m , E«n] = ai E«m+n fe(a,P) Ea+P if a+p is a root
[Eam , EPn] = \ a.Hm+n + m5m+n if a = -p ^O otherwise (2.8)
[d, H*m] = -m Him [d , E%] = -m E% [d,k]=O for i,j = \,...JL and m,n e Z. The Cartan subalgebra of C^) is generated by the 1+2 generators H'o (i=l,...,/), k and d. a Therefore to the operator E n will be associated the root (a,0,n) and to H'n the root nS = (0,0,n) where Ô is the isotropic direction. We recall that if ai,...,a^ is a system of simple roots of Q, a simple root system of ^1) is provided by the l+\ vectors (ai,0,0), ..., (0^,0,0) and (-0,0,1) where 0 is the highest root of Q. 3. VERTEX OPERATORS FOR AFFINE ALGEBRAS
3.1. Boson-fermion correspondence 1^
Let us consider a system of two fields a(z) and a*(z) of conformai dimension 1/2 with an action of first order S = J a*(z,z)âa(z,z)dzdz (3.1) where 3 = d/dz and d = d/dz. The equations of motion 3a(z,z) = 3a*(z,z) = 0 have the solution in the complex plane C*
a(z,z") = a(z) = X amz-m-l/2 (3.2a) m
a*(z,z) = a*(z) = £ a*mz-m-i/2 (3.2b) m Assume first that the fields a(z) and a*(z) are fermionic i.e. have anticommutation relations given by am a n + a n am — Om+n (3-3) These relations induce singularities in the product of two fields at the same point, which are suppressed by the intro-duction of a normal ordering :
(a*m an if m < n :a*man:=
^- an a*m if m > n The stress energy tensor (SET) of the system is given by Tbc(z) = \£ :3a*(z) a(z) - a*(z) 3a(z): = ^ Tbcm 2^"2 (3-5) m whose components satisfy the commutation relations of the Virasoro algebra with central charge Cbc = l: [Tbcm f Tbcn] = (m.n) Tbcm+n +1L(m3 . m) Sm+n (3.6)
Using the U(I) global symmetry of the action S a(z) -> e-'g a(z) and a*(z) -> e'8 a*(z) the Noether theorem furnishes an associated conserved current
(X(Z) = -:a*(z) a(z): = ^am z-m-1. (3.7) m Usual techniques of OPE lead to the commutation relations : [am > an] = am+n . tam » a nl = " a m+n (3.8)
which implies that, (X0 being interpreted as a charge operator, a(z) and a*(z) have charge 1 and - 1 respectively. Moreover, the components of oc(z) satisfy the commutation relations of an oscillator algebra :
[am , an] = m 8m+n (3.9) The action of the SET on the current ot(z) gives bc [T m , Ct1n] = -n ccm+n (3.10) which shows that the current oc(z) has conformai dimension one. If we define T«(z) = \ * Let us call the operators am with m < 0 and a*m with m > 0 creation operators and the operators 1 am with m > 0 and a*m with m < 0 annihilation operators. One generates a Fock space J by action of these operators on a vacuum vector IO>. The Fock space is 1 J = {a*ii ...a*ipaji ... ajq i0> with ii,...,ip < 0 , ji,...,jq > 0} (3.11) By virtue of eq. (3.8) the state a*n ...a*ip aji... ajq IO> has the charge p-q if the vacuum state IO> has charge zero . Therefore two states lm> and an!m> have the same charge and the previous representation does not remain irreducible under the oscillator algebra (3.9) 21X Thus in order to be able to put a correspondence between the fermionic operators a(z) and a*(z) and the bosonic operators oc(z), it is necessary to introduce a "charge split" operator or translation operator q such that el a(z) = :exp Q(z): = U+(z) (3.14a) a*(z) = :exp-Q(z): = U.(z) (3.14b) the dots meaning that the operators ocm with m > 0 have to be moved to right the of the operators am with m < 0 and O0 to the right of q. The U+(z) are called vertex operators. + these dots represent a normal ordering for the bosonic field oc(z) : fccman if m < n anam) ifm = n if m > n This correspondence between the fermion fields a(z) and a*(z) and the bosonic operators ot(z) and q is the boson-fermion correspondence. 3.2. Level k=+l representation of SO(2/)(D Start with an affine Clifford algebra of dimension I 22) : {a'm , a)n} = {ai*m , ai*n} = 0 and {aV , aJ*n} = Sm+n Sy (3.15a) (1 < i,j < I and m,n e Z) and define the generating functions m 'm z" and ai*(z) = £ ai*m z" m (3.15b) m m One introduces the functions (i*j): BJ(z) = :ai(z) aj( m " "il* 7-m m m Ei*J*(z) = :ai*(z) aJ*(z): = £ Ei*J*m z" (3.16) m m Then EU1n, E'J*m, E'*J*m are the SO(2jt)W step operators associated to roots ±£i ±£j and the H'm's are the Cartan operators of 50(2^)^). Now using the boson-fermion correspondence, we will be able to express these fermion fields a'(z) and a'*(z) in terms of vertex operators. Considering the Cartan subalgebra H'm of dimension Jt which satisfy eq. (3.9), and [q*, pi] = iSy where p) = HJ0, we define I Fubini-Veneziano fields Qi(z) = qU pi In z-^ ^-z-n (3.17) The fermion fields a'(z) and a'*(z) are obtained as 1 2 a'(z) = z / : exp ei.Q(z) : cei ai*(z) = zV2 : exp - e'.Q(z) : c.Ei (3.18) where the e''s are the fundamental weights of S0(2^), and c+ei are cocycle operators ensuring the fermionic character of a'(z) and al*(z). Then combining relations (3.16) and (3.18), we obtain in this way the k=+l representation of in terms of vertex operators. 3.3. Level k=+l representation of Let us recall that the root system of SO(2i+l) is constituted by the roots ±£j ±£j and ±e, with 1 < i < I. To construct the generators associated to the short roots ±6;, we extend the m affine Clifford (3.15a) algebra with an auxiliary real fermion field e(z) = ^ em z" such that IHEZ {a'm.en} = {a^m.en} =0 and {em,en}=ôm+n (3.19) Then the step operators associated to the short roots are defined by : E' = :a'(z) e(z): and E'* = :a>*(z) e(z): (3.20) This auxiliary real fermion field e(z) can be written as a vertex operator as before. In fact, using again the boson-fermion correspondence, it is straightforward to realize that a real fermion fiela can be written as zl/2, x e(z) = -T=- (: exp e.Q(z) : ce + : exp - e.Q(z) : c.e) (3.21) V2 where e is some SU(2) root of length squared 1 and ce is the associated cocycle operator. Actually, this formula can be deduced in an elegant way by using the folding of Dynkin diagrams 14X Consider the Dynkin diagram of SO(2^+2) which has a Z2 symmetry associated to an outer automorphism t of the Lie algebra SO(2/+2) (see figure below).Then SO(2/ + 1) can be seen as the subalgebni invariant under this outer automorphism x of SO(2/+2). Indeed, if the simp'e root a of SO(2/+2) is transformed into x(a), then ^(a+x(a)) is x-invariant since x-= \ and appears as a simple root of SO(2^+1) associated to the generator Ea if a = x(a) and -L (E« + ET(«)) if a.T(oc) = 0, Ea and ET(°0 being the generators of SO(2i+2) associated to \2 the roots a and x(oc) of SO(2/+2) respectively. The Dynkin diagram of SO(2/ + I) will therefore be obtained by folding the Z2 symmetric diagram of SO(2/+2), i.e. by transforming each couple (cc,x(a)) into the root ^(a+x(a)) of SO(2/+1). One obtains : o- -0 -OC^J-* 0- -0- -00 SCK22+2) The renerators associated to the short roots ie; are obtained by 1 2 E' = z / :exp ej.Q(z): cei ~(: exp e/+i.Q(z) : ce/+i + : exp - £i+i.Q(z) : V2 (3.22) 1 2 E>* = z / :exp -Ej.Q(z): c.ei ^=-(: exp EM.Q(Z) : Cei+i + : exp - e^+i.Q(z) : c.e/+i) \2 The quantity into brackets being precisely the expression of e(z) in terms of vertex operators. 3.4. Level k=l representation of SO(2i)(2> The twisted affine algebra SO(2/)(2) has as an invariant part the algebra SO(2/+1) and as a twisted part a repre-sentation of this algebra. Therefore, one finds at the Z level the generators E'Jm, E'J*m, Ei*J*m and E'm, E'*m associated to the roots ±£j±ej +mS and ±£i +mS and at the \ Z+^ level the generators E''m, E'»*m associated to the roots ±ei +(m+^)8 (me Z). In order to construct the generators E''(z) associated to the short roots at the Z+^ level, one m introduces another auxiliary fermion field e'(z) = ^ e'm z" such that meZ+1/2 {aim , CnJ = {a'*m , e'n) = 0 and {e'm , e'n} = 6m+n (3.23) Then the step operators E''(z) are defined by the functions E'i = :ai(z) e'(z): and E'1* = :al*(z) e'(z): (3.24) To write explicitely this auxiliary twisted field as a vertex operator 15), one considers the Z2 symmetry of the extended Dynkin diagram of SO(2i+l)(1) and performs a folding on this diagram involving now the affine root. One obtains diagrammatically : -O -OO -** O4O- -O -OO SO(22+1)0) SO(2/)(2) To be able to write down a vertex operator with half-integral moments, one extends the weight lattice A of SO(2/) to a Lorentzian lattice X by adding to A the isotropic direction 5, such that 52 = 0 and 0.Ei = O l If Q'(z) is extended from 1 < i < dim A to 1 < i < dim A such that p.5 = 1 and am.5 = O (m*0) (3.26) one obtains by the folding of the extended Dynkin diagram the expession of the twisted auxiliary fermion field e'(z) : 7I/2 /X Js \ e'(z) = ^=- (:exp (Ei + |).Q(z): cei + :exp (- Ei + |).Q(z): c.ei) (3.27) Generalization: This construction can be easily generalized to all the twisted affine algebras by exploiting the symmetries of the extended Dynkin diagrams of the affine algebras. One obtains the following general structure of the generators : E(OCL,Z) = U(0CL,z) caL E(as,z) = U(as,z) cas Fas(z) (invariant part) E'(ocs,z) = U(as,z) caS raS(z) (twisted part) where U(a,z) denote the vertex operators associated to a. OCL and as represent the long and short roots of the invariant algebra respectively. Fas(z) and r'as(z) are auxiliary fermionic fields whose explicit expressions are obtained by folding the extended Dynkin diagrams of the affine algebras. A delicate problem which is not developed here (see ref. 15) concerns the factorization of the cocycle operators which must be studied carefully in each case. 4. VERTEX OPERATORS FOR SUPERALGEBRAS 4.1. Boson-boson correspondence n) We consider again a system of two fields a(z) and a*(z) as in the previous section 3.1., but we assume now that the fields a(z) and a*(z) are bosonic, i.e. satisfy commutations relations am a n ' a n am = °m+n (4.1) with the normal ordering : |Yman ifm [O1n , Ocn] = -m 5m+n (4.3) - if we define T«(z) = - ^ * a(z) a(z)j, one finds then that Tbc(z) * T«(z). More precisely, one can write Tbc(z) = T«(z) + T"2(z) (4.4) where T"2(z) satisfy the commutation relations of the Virasoro algebra with central charge c=-2. The two SETs Ta and T~2 commute. Therefore a(z) and a*(z) will be expressed as products of two vertex operators, the first one related to the Ta system, the second one related to the T"2 system. The vertex operators related to the Ta system are obtained as before by exponentiating the primitive of the U(I) current a(z) : if we set a(z) = -3 10 T"2(z) is the stress energy tensor of a system with central charge -2. Such a system is obtained by taking two fermion fields T|(z) and i;(z) of conformai dimension 1 and 0 with respect to the SET T-2(z). One has : T-2(z) - - :T](Z) 9C(Z): (4.5) One can define also an U(I) symmetry with the associated current Y(z) = Z Ym z-m = - :T|(Z) Ç(Z): = - 9%(z) (4.6) m with [Ym , Yn] = m ôm+n (4.7) which leads to the vertex operator expressions of the fields Ti(Z) and t,(z) : Tl(z) = :exp -x(z): Ç(z) - :exp x(z): (4.8) Finally, one can write the initial bosonic fields a(z) and a*(z) as : a(z) = :exp <|>(z)::exp -%(z): a*(z) = :exp - 4.2. Level k=-l representation of Sp(2/)(D Consider an affine Weyl algebra of dimension / 22) : [aim , aJn] = [ai*m , afn] = 0 and [aim , aJ*n] = 5m+n Sy (4.10a) (1 < i,j < I and m,n e Z) and define the generating functions a'(z) = £ a'm *-m and a>*(z) = £ ai*m ^m ( m m One introduces the functions : m E'J*(z) = :a»(z) ai*(z): = m E'*J*(z) = :a'*(z) aJ*(z): = £ E'*j*m z"m (4. H ) m ri (z) = ;a (z) â (z); = 2^1 H m z m Then E'Jm, E'J*m, E'*J*mare the Sp(2^)(!) step operators associated to roots ±£i±£j and ±2EJ and the HVs are the Cartan operators of Sp(2/)(!). Using the previous boson-boson correspondence, we will be able to express these boson fields a'(z) and a'*(z) in terms of vertex operators : a'(z) = z1/2 :exp 11 We obtain in this way the level k=-1 representation of Sp(ZZ)W in terms of vertex operators. 4.3. Level k=l representation of OSp(M|N)(^ The two previous constructions - namely the level k=l representation OfSO(M)C1) and the level k=-l representation of Sp(2n)^) - can be put together to obtain a level k=l representation of the superalgebra 17) OSp(MI2n)0) (N=2n). More precisely, the superalgebra 0Sp(MI2n) has as bosonic part the Lie algebra SO(M) x Sp(2n) and as fermionic part the (M,2n) representation of this underlying algebra. Denoting by Y1H2) (1 - i - m)tne elementary fermionic fields for S0(2m) and by ^(z) (1 < j < n) the elementary bosonic fields for Sp(2n), the generators of OSp(2ml2n)0) are defined by : :V±i(z) v±J(z): for SO(2m)0), :^±'(z) DZJ(z): for Sp(2n)0) and ^'(z) ^J(z): for the (2m,2n) representation. In order to construct the generators of OSp(2m+ll2n)(1) one adds as in the sec. 3.3 an auxiliary fermionic field F(z) such that the generators of S0(2m+l)(1) are :\f^(z) \i/±J(z): and -.yi^z) F(z): and the generators of the (2m+ll2n) representation are ^(z) ^1J(Z): and ^1Kz) F(z): . The expressions of the elementary fields \y-l(z), ^(z) and F(z) in terms of vertex operators lead therefore to a writing of the generators of OSp(2ml2n)(1) or OSp(2m+ll2n)(1) as vertex operators. One obtains in this way a level k=l representation of the superalgebra OSp(MQn)(1X 4.4. Generalization 2°) One can start from the vertex operator construction of OSp(MIN)O) in order to obtain those relative to all the affine, untwisted and twisted, contragredient superalgebras. General properties on the structure of superalgebras have been developed 19) and can be directly applied for such a generalization. As examples, regular embeddings of superalgebras are used to described the case of Sl(mln)(1) and folding methods on the extended Dynkin diagrams considered to compute the OSp(2ml2n)(2) and Sl(mln)(2) cases. Let us conclude by emphasizing on the property for vertex operators of affine superalgebras to involve as well as the boson-fermion correspondence as the boson-boson correspondence. We think the above results can be considered as a first step in a more general study of conformai field theories based on superalgebras. 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