Spectral Flow and Feigin-Fuks Parameter Space of N=4 Superconformal Algebras

The parameter space of the Feigin-Fuks representations of the N=4 SU(2)k superconformal algebras is studied from the viewpoint of the specral flow. The η phase of the spectral flow is nicely incorporated through twisted fermions and the spectral flow resulting from the inner automorphism of the N=4 superconformal algebras is explicitly shown to be operating as identiy relations among the generators. Conditions for the unitary representations are also investigated in our Feigin-Fuks parameter space. Work supported in part by the Monbusho Grant-in-Aid for Scientific Research on Priority Areas 231 “Infinite Analysis”, No. 08211227. e-mail address: [email protected] It is well recognized nowadays that the so-called Feigin-Fuks (FF) representations (or the Coulomb-gas representations) [1, 2] are very important and almost inevitably required tools for investigating the representation theories of the conformal and superconformal algebras. By now we have established the FF representations of the superconformal algebras with higher number of supercharges [3, 4, 5, 6], up to N=4 [7, 8, 9]. On the other hand, the spectral flows resulting from the inner automprphisms of the conformal and superconformal algebras with N=2,3 and 4 were first recognized by Schwimmer and Seiberg [10], and their remarkable implications on the representation theories of the algebras have been discussed by many people [11, 12, 13]. In the present paper we shall study the parameter space of the FF representations of the N=4 SU(2)k superconformal algebras with particular focus on the properties of their spectral flow which is nicely embedded in our FF parameterization [7]. We shall explicitly show how remarkably the spectral flow emerges by use of the identities holding among the FF parameters. Our study not only shows how the spectral flow for the unitary representations [13, 14, 15] of the N=4 SU(2)k superconformal algebras is operating, but also establishes it to hold explicitly in the nonunitary representations by use of the continuous parameters of our FF representations [7]. The N=4 SU(2)k superconformal algebras are defined by the form of the operator product expansions (OPE) among operators given by the energy-momentum tensor L(z), the SU(2)k local nonabelian generators T (z), and the iso-doublet and antidoublet supercharges G(z) and Ḡa(z): L(z)L(w) ∼ 3k (z − w)4 + 2L(w) (z − w)2 + ∂wL(w) z − w , T (z)T (w) ∼ 1 2 kη (z − w)2 + iǫηklT (w) z − w , L(z)T (w) ∼ T (w) (z − w)2 + ∂wT (w) z − w , T (z)G(w) ∼ − 1 2 (σi)abG (w) z − w , T (z)Ḡa(w) ∼ 1 2 Ḡb(w)(σ i)ba z − w ,