Volume 222, number 1 PHYSICS LETTERS B 11 May 1989 BOSON CONDENSATION IN SUPERSTRING-INSPIRED MODELS R. MANKA, J. SLADKOWSKI Department of Theoretical Physics, Silesian University, Uniwersytecka 4, PL-40-O07Katowice, Poland and G. VITIELLO Dipartamento di Fisica dell'Universita, 1-84100 Salerno, Italy and Instituto Nationale di Fisica Nucleare, Sezione di Napoli, 1-80125 Naples, Italy Received 27 June 1988; revised manuscript received 15 February 1989 The Hosotani mechanism is discussed in terms of the abelian gauge boson condensation on a non-simply connected manifold. Such a condensation could violate supersymmetry if massive fermions behave as anyons. It is well known that superstring theories predict a gauge symmetry of the form G=E~ X E8 [ 1 ]. The CHSW compactification scenario demands N= 1 su- persymmetry in order to solve the hierarchy problem [ 1,2 ]. This leads to a consistent N= 1 supersymmet- ric E~ X E6 gauge theory. The E 6 gauge factor is sub- sequently broken via the Hosotani mechanism [ 3 ] to obtain a realistic low-energy gauge theory. The aim of this letter is to point out some inconsistencies. The CHSW scenario demands the N--- 1 supersymmetry to survive energies close to the electroweak unifica- tion scale [ 1,2 ]. In general, the Hosotani mechanism does not violate supersymmetry. But this raises the following question: what mechanism chooses the low energy gauge group G' ? If supersymmetry is not bro- ken then for any G' the vacuum energy E vanishes, E=0. to show this let us consider a D= 10 supersym- metric gauge theory with the following field content: the gauge field AM and the gaugino 2. The Lagrange function has the form L= _ 4Jt MN- t l Ta lff'aMN__ ½~aFM(DM,~) a , (la) where FMN -~ OMAN -- ONAM+ g[AM, AN] , (lb) DM2= 0M2 +g[AM, 2] . (lc) It is convenient to use the Cartan-Weyl basis [4] so that AM=A~Hi+Aa~Ea. The vectors Hi span the Cartan subalgebra. The CHSW scenario assumes that M = M~4) X K, where M ~ 4 ) is the ordinary Minkowski space and K is a Calabi-Yau manifold. The gauge field condensation means the vacuum state change [ 5-7 ], that is, AM=//M+AM, (2) where 2M is the background field andAM is the quan- tum fluctuation. Such a gauge field shift is equivalent to the Hosotani mechanism in the case of a non-sim- ply connected internal manifold [ 8]. The "4N field could be decomposed into hlt(xN): ~ Ailz(X)~i(y), A~(x)Eg, (3a) t=l Am(xN)= ~ #(x)A~m(y), #(x)~g, (3b) i=l The field A~(x) and #(x) describe the four-dimen- sional quantum fields. The zero models are given by 1 0m [ ~ 0m~i(y) ] -----0, (4a) 0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division ) 31