Volume 232, number 2 PHYSICS LETTERS B 30 November 1989 SUPERCONFORMAL GHOST CORRELATIONS ON RIEMANN SURFACES OlafLECHTENFELD 1 Physics Department, City Collegeof,Vew York, New York, NY 10031, USA Received 11 September 1989 We give a short summary of chiral bosonization on Riemann surfaces, with emphasis on commuting fermions, so-called (fl, y) systems. Employing Fay's trisec~nt identity a new, fermionic representation for superconformal ghost correlations is found. Es- pecially useful for multi-loop superstring calculations, it completes an earlier proof thai the cosmological constant vanishes up to eight loops. The concept of chiral bosonization [ 1-8 ] has cru- cially improved our understanding of two-dimen- sional conformal field theories. Not only did it pro- duce fundamental insight in their algebraic and geometric structures and interrelations, but boson- fermion equivalence has also provided us with an as- sortment of powerful computational tools [9-11 ]. Its effects can be noticed in statistical physics and string theory, alike. A comprehensive treatment of chiral bosonization can be found in refs. [ 1,4]. The basic idea may be described as follows. Consider a level one U ( 1 ) KaY- Moody system on a Riemann surface Z of genus g, realized by a pair of free conformal fields of dimen- sions 2 and 1-2. Geometrically these are generalized differential forms, i.e. sections of rank ). and 1-2 tensor or spinor bundles over the surface, depending on 2 being integral or odd half-integral [ 12 ]. We de- note them by b(z) and c(z) in the anticommuting case and write [3(z) and 7(z) for commuting fields, fixing).>~ ~ as the spin ofb or//. In both cases we shall call the fieldsfermionic since the system is described by thefirst-order action ~1 Sf=l-Tr d2zbOc(z) °r zr dZz[3~7(z)" (1) Z E Bosonization stales that the fermionic theory ( 1 ) Bitnet address: OLAF@SCI.CCNY.CUNY.EDU ~ For simplicity we choose locally orthonormal coordinates on I;. is equivalent to the holomorphic square root ofa bo- sonic one, given by the second-order action Su=~ d2z(0,O~0+~Q~/gR0)(z), (2) with Q-2).- 1 and e= + 1 or - 1 for anticommuting or commuting fermions. It is important to note that the target space of¢(z) is the unit circle ~q/2~zZ; this compactification insures the necessary holomorphic factorization [3,4]. In the following we shall only display the holomorphic part of bosonic correlators. The equivalence of ( 1 ) and (2) is established by equating their U ( 1 ) currents bc or [37=j= 0¢ which for e = + 1 yields the field relations ~2 b=e ~, c=e -~. (3) The bosonic system has a charge asymmetry of Q=22- 1 units, such that operators e q° have a con- formal dimension of ½eq(q+Q). Hence, Q may be interpreted as a background charge which is to be cancelled by the external charges, Xi q, = Q(g- 1 ), and translates into bosonic language the fermionic zero mode asymmetry #b or [3 zero modes - ~c or y zero modes =Q(g-l), (4) dictated by the Riemann-Roch theorem [ 13 ]. The ~2 Note that we use the conventions ofref. [4] which differ from ref. [ I ] in the signs of 0 and Q. 0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland ) 193