Volume 257, number 1,2 PHYSICS LETTERS B 21 March 1991 Spiky projective planes D. Johnston Department of Mathematics, Heriot-Watt University,Riccarton,EdinburghEH I 4 4AS, Scotland, UK Received 12 December 1990 It has been suggested that the so-called c= 1 barrier in subcritical string theory is the result of a Kosterlitz-Thouless transition in the associated Liouville theory. We show that arguments analogous to those used to imply a transition on orientable worldsheets may also be applied to non-orientable worldsheets such as the real projective plane. This means that the barrier is still present for such worldsheets. The recent work on the quantization of the contin- uum Liouville theory in both the light cone [ 1 ] and the conformal gauge [2 ] runs into a barrier at c= 1 beyond which nonsensical results such as complex scaling dimensions appear, perhaps signalling some sort of phase transition. This barrier is also apparent in the work of non-perturbatively quantizing the the- ory by making use of random matrix techniques [ 3 ] and the earlier investigations by Gervais and Neveu using the operator formalism [ 4 ]. It has been sug- gested that the trouble is caused by a Kosterlitz- Thouless (KT) [ 5 ] transition in the Liouville theory which occurs at c= 1 leading to a proliferation of "spikes" [ 6-8 ] or possibly macroscopic tears (if we allow the genus to change) [ 9 ] in the surface. This would then destroy the picture of a functional inte- gral over smooth Riemann surfaces. The initial work on the Liouville theory and the matrix model ap- proach has considered orientable surfaces. It is also possible to consider models in which the worldsheets are non-orientable and the matrix models ensembles which produce these in the non-perturbative ap- proach have recently been explored [ 10,11 ]. In such models a weak coupling l/N (topological) expan- sion of the partition function gives Z=N2Zs2 +NZ~p2 +ZT2 + ... , (1) where we see that odd powers of 1/N, which is effec- tively the string coupling, accompany the non-orient- able contributions to the genus sum such as that from the real projective plane ~tl RP 2. In view of the trou- ble that the even terms in the expansion run into at c= l it would be interesting to enquire, whether this also constitutes a barrier for the odd (non-orienta- ble) terms. It is not inconceivable that the orientable and non-orientable world sheets should undergo a transition at different values of c, though this would leave us in the peculiar situation of being left with half of a perturbation expansion. To investigate the nature of the barrier on a non- orientable surface we shall resort to the same heuris- tic arguments that were first employed to suggest the presence of the KT transition in the XY model. We shall thus employ the continuum approach rather than looking at a matrix model and work in the conformal gauge. The starting point for the calculation is just the fixed area partition function for conformal mat- ter coupled to gravity Z[A] = f D~,f~D~XJ × e x p ( 4--~--~n-n .125--C~d2ax/g(OAO+RO)) ×exp(-Sm[X,~],O(~d2trx/~exp(O)-A), (2) #t It is not clear [ 10,11 ] whether only non-orientable surfaces with odd Euler characteristic appear, at least for the branches of the weak couplingexpansion discovered so far. Thus a Klein bottle, for instance, may not contribute. 0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland ) 51