The Erwin Schrodinger International Pasteurgasse ESI Institute for Mathematical Physics A Wien Austria The Top ological GG WZW Mo del in the Generalized Momentum Representation Anton Yu Alekseev Peter Schaller Thomas Strobl Vienna Preprint ESI May Supp orted by Federal Ministry of Science and Research Austria Available via httpwwwesiacat ETHTH TUW PITHA ESI hepth The Top ologi cal GG WZW Mo del in the Generalized Momentum Representation Anton Yu Alekseev Institut fur Theoretische Physik ETHHonggerb erg CH Zuric h Switzerland Peter Schaller Institut fur Theoretische Physik TU Wien Wiedner Hauptstr A Vienna Austria y Thomas Strobl Institut fur Theoretische Physik RWTH Aachen Sommerfeldstr D Aachen Germany May Abstract We consider the top ological gauged WZW mo del in the general ized momentum representation The chiral eld g is interpreted as a counterpart of the electric eld E of conventional gauge theories The gauge dep endence of wave functionals g is governed by a new gauge co cycle We evaluate this co cycle explicitly using the GW Z W machinery of Poisson mo dels In this approach the GWZW mo del is reformulated as a Schwarz typ e top ological theory so that the ac tion do es not dep end on the worldsheet metric The equivalence of this new formulation to the original one is proved for genus one and conjectured for an arbitrary genus Riemann surface As a bypro duct we discover a new way to explain the app earence of Quantum Groups in the WZW mo del On leave of absence from Steklov Mathematical Institute Fontanka StPetersburg Russia email alekseevitpphysethzch y email tstroblplutophysikrwthaachende Intro duction In this pap er we investigate the quantization of the gauged GG WZW mo del in the generalized momentum representation The consideration is inspired by the study of twodimensional YangMills and BFtheories in the momentum representation The problem of quantization of gauge theories in the momentum repre 1 sentation has b een attracting attention for a long time While in the connection representation the idea of gauge invariance may b e implemented in a simple way g A A we get a nontrivial b ehaviour of the quantum wave functions under gauge transformations in the momentum representation Indeed one can apply the following simple argument The wave functional in the momentum represen tation may b e thought of as a functional Fourier transformation of the wave functional in the connection representation Z Z A E D A exp i tr E A i i Taking into account the b ehaviour of A and E under gauge transformations g 1 1 g A g g g A i i i g 1 E g E g i i we derive R 1 g i tr E g g i i E E e We conclude that the wave functional in the momentum representation is not invariant with resp ect to gauge transformations Instead it gains a simple phase factor E g which is of the form Z 1 E g tr E g g i i The innitesimal version of the same phase factor Z E tr E i i 1 The authors are grateful to Prof RJackiw for drawing their attention to this pap er and for making them know ab out the scientic content of his letter to Prof D Amati corresp onds to the action of the gauge algebra It is easy to verify that satises the following equation g E g h E g E h This prop erty assures that the comp osition of two gauge transformations with gauge parameters g and h is the same as a gauge transformation with a parameter g h Equation is usually referred to as a co cycle condition It establishes the fact that is a oneco cycle of the innite dimensional gauge group A oneco cycle is said to b e trivial if g E g E E for some In this case the gauge invariance of the wave function may b e restored by the redenition ~ i(E ) E e E An innitesimal co cycle E is trivial if it can b e represented as a function of the commutator E E E It is easy to see that the co cycle is nontrivial Indeed let us cho ose b oth a a E and having only one nonzero comp onent E and in the Lie algebra Then the commutator in is always equal to zero whereas the expression is still nontrivial As a consequence also the gauge group co cycle is nontrivial On the other hand on some restricted space of values of the eld E the co cycle may b ecome trivial generically if we admit nonlo cal expressions for This is imp ortant to mention as one may rewrite the integrated Gauss law as a triviality condition on the co cycle Let us parametrize as E exp iE which is p ossible whenever and insert this ex R 1 2 pression into The result is precisely with tr E g g In fact i i eg in two dimensions the wave functions of the momentum representation 2 More accurately one obtains only mo d But anyway this mo dication of is quite natural in view of the origin of the co cycle within Alternatively one might regard also a multiplicative co cycle exp i right from the outset cf App endix A 1 are supp orted on some conjugacy classes E x g x E g x with sp ecic 0 values of E But away from these sp ecic conjugacy classes and in partic 0 ular in the original unrestricted space of values for E the general argument of the co cycle b eing nontrivial applies More details on this issue may b e found in App endix A It is worth mentioning that in the ChernSimons theory a co cycle app ears in the connection representation as well g i(Ag ) A e A The co cycle A g is ususally called WessZumino action It is intimately related to the theory of anomalies Recently a co cycle of typ e has b een observed in twodimensional BF and YMtheories In this pap er we consider the somewhat more complicated example of the gauged WZW GWZW mo del for a semisimple Lie group Like the BF theory it is a twodimensional top ological eld theory for a detailed account see It has a connection oneform gauge eld as one of its dynamical variables and p ossesses the usual gauge symmetry However there is a complication which makes the analysis dierent from the pattern In the GWZW mo del the variable which is conjugate to the gauge eld and which shall b e denoted by g in the following takes values in a Lie group G instead of a linear space So we get a sort of curved momentum space We calculate the co cyle which governs the gauge dep endence of wave GW Z W functions in a g representation and nd that it diers from the standard expression We argue that while the co cycle corresp onds to a Lie group G our co cycle is related to its quantum deformation G In the course q of the analysis we nd that the GWZW mo del b elongs to the class of Poisson mo dels recently discovered in This theory provides a technical to ol for the evaluation of the co cycle GW Z W Let us briey characterize the content of each section In Section we develop the Hamiltonian formulation of the GWZW mo del nd canonically conjugate variables and write down the gauge invariance equation for the wave functional in the g representation Section is devoted to the description of Poisson mo dels A two dimensional top ological mo del of this class is dened by xing a Poisson bracket on the target space Using the Hamiltonian formulation the top ology of the spacetime b eing a torus or cylinder we prove that the GWZW mo del is equivalent to a certain Poisson mo del coupled to a top ological term S that has supp ort of measure zero on the target space of the eld theory The target space of the coupled Poisson mo del is the Lie group G We start from the GWZW action evaluate the Poisson structure on G and discover its relation to the theory of Quantum Groups The origin of the term S is considered in details in App endix B In Section we solve the gauge invariance equation and nd the gauge dep endence of the GWZW wave functional in the g representation This provides a new co cycle Calculations are p erformed for the Poisson GW Z W mo del without the singular term In App endix C we reconsider the problem in the presence of the top ological term It is shown that in the case of G SU at most one quantum state is aected We compare the results with other approaches In some nal remarks we conjecture that the Poisson mo del coupled to S gives an alternative formulation of the GWZW mo del valid for a Riemann surface of arbitrary genus We comment on the new relation b etween the WZW mo del and Quantum Groups which emerges as a bypro duct of our consideration Hamiltonian Formulation of GWZW Mo del The WZW theory is dened by the action Z Z k k 1 1 2 1 1 3 tr g g g g d x tr d dg g WZW g where the elds g take values in some semisimple Lie group G and indices are raised with the standard Minkowski metric The case of a Euclidean metric may b e treated in the same fashion Some remarks concerning the second term in may b e found in App enidx B The simplest way to gauge the global symmetry transformations g 1 l g l is to intro duce a gauge eld h taking its values in the gauge group the action 1 GW Z W h g WZW hg h 1 1 is then invariant under the lo cal transformations g l g l h hl With 1 the celebrated PolyakovWigmann formula and a h h where GWZW can b e brought into the standard form o 1 GW Z W g a a WZW g + R k 1 1 1 2 tr a g g a g g a g a g a a d x + + + + 4 + In the course of our construction of GWZW a a dx a dx has b een + 2 sub ject to the zero curvature condition da a This condition results also from the equations of motion arising from So further on we treat a as unconstrained elds taking their values in the Lie algebra of the chosen gauge group In order to nd a Hamiltonian formulation of the GWZW