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Vertex Operators for Affine Algebras and Superalfeberas 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.
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