The parameter space of the Feigin-Fuks representations of the N=4 SU(2) superconformal algebras is studied from the viewpoint of the specral flow. The 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.

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) 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) 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) superconformal algebras are defined by the form of the operator product expansions (OPE) among operators given by the energy-momentum tensor , the SU(2) local nonabelian generators , and the iso-doublet and -antidoublet supercharges and :

(1) |

where denote SU(2) triplets in the diagonal basis, while the superscripts (subscripts) label SU(2) doublet (antidoublet) representations. The symmetric tensors are defined in the diagonal basis as , while the antisymmetric tensors are similarly defined as , etc., and otherwise zero. The Pauli matrices are given by . The corresponding notations in terms of the isospin raising and lowering by a half unit for the fermionic operators are given by . The symmetric delta function has the standard meaning with .

Here is a remark in order. The algebras Eq.(1) have a global SU(2) symmetry. As a result, one can consider the infinite number of independent twisted algebras labeled by corresponding to the conjugate classes of the global automorphism[10]. The FF representations of the -extended algebras are discussed in a separate paper[16]. The present paper will be restricted to the case of .

As noted by Schwimmer and Seiberg [10], the SU(2) gauge symmetry gives the inner automorphism

(2) |

for the N=4 SU(2) superconformal algebras, while the boundary conditions are imposed by use of one parameter as

(3) |

This phase can be gauged away through the use of the local automorphism Eq.(2) by the choice [10].

For the convenience of our later use we shall give here the N=4 SU(2) superconformal algebras in terms of Fourier components:

(4) |

where the following Fourier modings are to be given to complete the definition of the algebras:

The equivalence of the algebras which differ by the value of the parameter can be shown by expressing the generators of the algebras twisted by in terms of the generators of the (Ramond) algebra through the relations given by Eq.(2) with the choice of :

(5) |

or by expressing the generators of the algebras twisted by in terms of the generators of the (Neveu-Schwarz) algebra through the relations given by Eq.(2) with the choice of :

(6) |

Since the algebra for each choice of is equivalent to each other through the inner automorphism mentioned above, we basically consider in the following the typical choices of and , and compare their resulting consequences. The value in Eq.(3) corresponds to the Ramond (R) sector with integral and , while that of to the Neveu-Schwarz (NS) one with half-integral values of them. The spectral flow between the NS and R sectors is obtained either by putting in Eq.(5) or by putting in Eq.(6).

For a general value of we consider the R-type raising operators that are given as follows:

(7) |

where , and we have used the following notation: for and for . The above choice corresponds to taking the raising operators among the generators of the Ramond ( sector to be the normal ones as are usually chosen, which can be obtained just by putting in Eq.(7).

But we should note that for the NS sector with , the above choice of Eq.(7) amounts to taking the raising operators to be the tilted ones that are obtained by putting in Eq.(7) rather than the ordinary raising operators that are commonly used for the NS sector and are given by putting in the following NS-type conditions:

(8) |

where . This implies that the hws’s usually constructed for the NS sector by the condition of being killed by all the raising operators obtained by putting in Eq.(8) do not correspond to the hws’s of the R sector, but rather the latter should be reconstructed properly from the former if one started with the hws’s of the NS sector. This procedure will be exemplified for some relevant cases later.

We take the Cartan subalgebra to be , and the lowering operators to be the remaining generators. Then we define a highest weight representation (hwrep) of the algebras Eq.(4) to be one containing a highest weight state (hws) vector such that

(9) |

and

(10) |

for all raising operators .

Our FF representations of the SU(2) superconformal algebra are constructed in terms of four bosons and four real fermions forming a pair of complex fermion doublet and antidoublet ( or ) under SU(2):

(11) |

whose mode expansions are given by

(12) |

and

(13) |

(14) |

In the following we mostly consider the R case () only instead of the R-type case () for clarity and simplicity.

The commutators for the Fourier modes are

(15) |

for the bosons, while for the fermions they are for the NS-type case ()

(16) |

or for the R-type case ()

(17) |

The corresponding propagators for the boson fields are given by

(18) |

while those for the fermions are given by

(19) | |||||

where we have defined the NS-type vacuum as

(20) |

while for the zero modes of the R sector () we have used the following definition of normal-ordering [17, 18, 19, 20]:

(21) |

where

(22) |

As to the non-zero modes is defined to reduce to the usual definition of normal-ordering where creation operators stand to the left of annihilation operators with appropriate sign factors.

As will be shown later, our formalism with the NS-type vacuum allows one to obtain the generic expression for the conformal weights which is valid top from the NS () sector further down to the R () sector, reproducing the right value . Note, however, that the expressions Eq(19) for the fermion propagators are not continuously connected at . This is simply because the two definitions of the normal ordering for the NS-type sector and that for the R sector are discontinuous due to the presence of the zero modes in the latter. We believe that what we have presented is the best one can do for the generic treatment for the conformal weights. of the well-known additional constant for the Ramond conformal weight at the point

Now the FF representations are given with the parameters

(23) |

as follows:

SU(2) Kac-Moody currents

(24) |

where

(25) |

with [21]

(26) |

The boundary conditions Eq.(3) and the spectral flow Eq.(6) for are obviously valid in Eq.(24) when we have

(27) |

which is valid[16] for any conformal state in our FF representations, as will be clear from our later discussion on vertex operators.

Total energy-momentum tensor

(30) |

where the boundary conditions Eq.(3) and the spectral flow Eq.(6) for and are explicitly seen to be valid.

Now we consider the following basic vertex operators[7, 8]

(31) |

Note that the following OPE relations hold[7, 8]:

(32) |

A primary state with conformal dimension and SU(2) spin in the NS sector is obtained by operating the vertex operator on the NS-type vacuum as

(33) |

where the conformal weight in the NS sector is given by

(34) | |||||

The last identity plays a crucial role in the following discussions.

Now the primary state Eq.(33), being a hws vector in the NS sector when , also stands for a NS-type hws vector for the generators given by Eqs.(24),(28),(29) and (30). To be more explicit, we have

(35) |

where the state is written as

(36) |

Later on we find that and , and that can also be rewritten as it should be like

(37) |

Next we consider the R sector where the ground state vacua are quadruply degenerate and are defined by

(38) |

and

(39) |

Let us also note that

(40) |

and

(41) |

Therefore we have an iso-doublet and two iso-singlets .

We find that is the highest weight Ramond (hw R) vacuum which satisfies the hws conditions of being annihilated by all the raising operators given by Eq.(7) with . Also we note that

(42) |

A hws vector in the Ramond sector can be constructed as follows. We first apply the vertex operator on the hw R vacuum to obtain a primary state in the R sector as

(43) |

where generically denotes the eigenvalues of the ground state vacua and . Here we define for later use the primary states with other values of as

(44) | |||||

Then we get a hws vector by putting in Eq.(43):