Volume 254, number 1,2 PHYSICS LETTERS B 17 January 1991 Bosonized actions for anomalous gauge theories on coadjoint orbits David Bar-Moshe, M.S. Marinov ~ and Yaron Oz 2 Physics Department, Technion-lsrael lnstitute of Technology, Haifa 32000, Israel Received 7 September 1990 Starling from the extended Lie algebras generated by the Gauss-law constraints, and using the coadjoint-orbit method, we construct bosonized actions for anomalous gauge theories in two and four space-time dimensions. We show that the anomalous gauge algebras determine the anomalous part of the actions, but not the gauge invariant terms. These are generated by exact cocycles. 1. Introduction Coadjoint orbits of Lie groups possess a symplectic structure, and thus can be viewed as classical phase spaces [1]. The construction of field theories on coadjoint orbits of infinite-dimensional Lie groups has attracted considerable attention recently [2-4 ], in an attempt to gain a better understanding of the structure of these theories by investigating their geo- metrical meaning. Indeed, the construction of WZNW theory and two-dimensional induced gravity on coadjoint orbits of Kac-Moody and Virasoro groups, respectively, revealed a similarity in their structures, and a natural interpretation for the SL(2, ) current algebra underlying the two-dimensional induced gravity has been found. The method ofcoad- joint orbits, combined with the theory of generalized coherent states, has been applied to the construction of various theories of particles and strings as well as topological field theories [ 4 ]. In this paper we study the structure of anomalous gauge theories from the viewpoint of coadjoint orbits combined with the theory of generalized coherent states. Such a geometrical formulation may be help- ful for the quantization and the analysis of the Hil- t Worksupported by the Fund for the Promotion of Reasearch at the Technion. 2 Bitnetaddress: Phr82yo@Technion. bert-space structure of these theories. We will be in- terested to know to what extent the anomalous gauge algebra determines the theory. First, let us briefly describe the construction of the action functional on the coadjoint orbit of a Lie group. Denote by G a Lie group, by f9 its Lie algebra and by c~. the dual space to f#. The coadjoint representation of G is defined as (Ad*(g) X, a) = (X, Ad(g-') a), ( 1) whereg~G, ae ~, X~ f#*,Ad(g) a=gag -~ and (X, a) is the value of the linear functional Xon a. Let Ox be the orbit of an element X~ f#* under the action of G. It has a G-invariant symplectic structure defined by the Kirillov-Kostant two-form [ 1 ], t~x=½(x, [~, .~y]>, (2) where ~/ is a fq-valued one-form satisfying the condition 5X=ad*(~) X. (3) Here 5 denotes the exterior derivative on the coad- joint orbit and ad*(~/) is the infinitesimal transfor- mation of X, (ad*(~/) X, a)=(X, [~¢, a]). Since 12x is closed one has, locally, Ox=~a. The Hamilton action on the coadjoint orbit Ox is defined by 5e= for, where the integration is along a one-dimensional sub- manifold of the coadjoint orbit. In order to get a solution of (3), the orbit is para- 0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland ) 1 15