The Orbifold-String Theories of Permutation-Type: I. One Twisted BRST per Cycle per Sector

We resume our discussion of the new orbifold-string theories of permutation-type, focusing in the present series on the algebraic formulation of the general bosonic prototype and especially the target space-times of the theories. In this first paper of the series, we construct one twisted BRST system for each cycle $j$ in each twisted sector $\sigma$ of the general case, verifying in particular the previously-conjectured algebra $[Q_{i}(\sigma),Q_{j}(\sigma)]_{+} =0$ of the BRST charges. The BRST systems then imply a set of extended physical-state conditions for the matter of each cycle at cycle central charge $\hat{c}_{j}(\sigma)=26f_{j}(\sigma)$ where $f_{j}(\sigma)$ is the length of cycle $j$.

Opening a new, more phenomenological chapter of the program, I have recently proposed [16][17][18][19][20] that a very simple subset of these orbifolds, called the orbifold-string theories of permutation-type, can provide new physical string theories at multiples of conventional critical central charges.
This class of theories begins by choosing the conformal field theory A(H) to be a set of copies of any critical string (hence only Abelian currents), and there are many of these, including the bosonic prototypes [16] U (1) 26K H + and orbifold-superstring generalizations of these. In this notation, U(1) 26 is the critical bosonic closed string at central charge 26 and U(1) 26 L,R are the left-and right-movers of U(1) 26 . The automorphism group H(perm) K is any permutation group on K elements (the copies), the non-trivial element of Z 2 (w.s.) is the exchange L ↔ R of the chiral components, and the automorphism group H ′ 26 operates uniformly on each copy of U(1) 26 , including U(1) 26 L and U(1) 26 R . The divisors H ± can be any subgroup of the direct products shown. The sets shown in Eq. (1.1a) are called the generalized permutation orbifolds [9,15,16,18,20], with every sector σ being a twisted closed string at central chargeĉ(σ) = 26K. The second sets in Eq. (1.1b) are the orientation orbifolds [12,13,[15][16][17][18][19], with an equal number of twisted open strings atĉ(σ) = 52 and twisted closed strings atĉ(σ) = 26. The closed-string sectors of the orientation orbifolds form the ordinary space-time orbifolds U(1) 26 /H ′ 26 , which we will discuss separately elsewhere. The last sets in Eq. (1.1c) are the so-called "open-string generalized permutation orbifolds" with all sectors atĉ(σ) = 26K. These sets are constructed along with their branes in Refs. [15,16] from the left-movers of the generalized permutation orbifolds. Theĉ = 52 open-string sectors of the orientation orbifolds are included among the T -dual families of [U(1) 52 /H + ] open , but this is the only case in the third set whose closure to closed strings has been fully studied.
In fact, there exist other bosonic prototypes (see App. B of this paper), but our discussion in the present series will be limited to the sets shown in Eq. (1).
A central issue in all these theories is the presence of extra negative-norm states in the conformal field theories of the twisted sectors. But the orbifold program constructs the twisted sectors of any orbifold from the untwisted sector and in string theory we do not expect that orbifoldization would create negative-norm states where there are none in the untwisted sectors [21]. This observation implies that the twisted sectors of the new string theories in fact possess new extended world-sheet geometries-including new twisted worldsheet permutation gravities [16], new twisted BRST systems [17], new extended physicalstate conditions [17][18][19][20] and new gauges at the interacting level [19,20]. All these phenomena are associated with the existence of the so-called orbifold Virasoro algebras [1,2,9,[16][17][18][19][20] (see Eqs. (4.3) and (8.1b)), which appear universally in every sector of the orbifolds of permutation-type. It is possible , but so far demonstrated only in subexamples [19,20], that the new world-sheet geometries can eliminate all negative-norm states in the orbifold-string theories of permutation-type.
A closely-related issue is the target space-time interpretation of the new string theories, especially since detailed analysis of the physical states of theĉ(σ) = 52 twisted sectors has found an equivalent, reduced formulation [17] of all theĉ(σ) = 52 spectral problems at reduced central charge c(σ) = 26! In the reduced formulation, one sees only (a unitary transformation of) the conventional number of negative-norm states subject in fact to the conventional physical-state condition.
Thus, for example, in spite of the half-integral moding of itsĉ = 52 twisted open-string sector, the simplest orientation-orbifold string system with has been shown to be equivalent [18], even at the interacting level [19], to the ordinary untwisted 26-dimensional open-closed string system. In this sense, "orientation orbifolds include orientifolds" [19] in a rising sequence of ever-more twisted open-closed string systems, with the familiar untwisted critical prototype at the bottom of the hierarchy. Similarly, the pure cyclic permutation-orbifold string systems H + = Z K , K = prime have been shown [20] to be equivalent to special modular-invariant collections of ordinary closed 26-dimensional strings, while more general H ′ 26 ⊂ H + corresponds to generically new permutation-orbifold string theories. In succeeding papers of this series, we will demonstrate that more general choices of H ′ 26 ⊂ H ± can in fact describe a large variety of target spaces in the new theories, including in particular Lorentzian target space-times with sector-dependent dimensionality D(σ) ≤ 26! Our task in this first paper of the new series is to begin the quantization of the full list of theories in Eq (1.1). More precisely, we present here the BRST quantization [22,23] of the generalized permutation orbifold-string theories (1.1a) and their open-string counterparts (1.1b), all of whose sectors live at matter central chargeĉ(σ) = 26K. (As noted above the twisted open-string sectors of the orientation-orbifold string theories are included in the case K = 2, and in fact the BRST quantization of each closed-string in U(1) 26 /H ′ 26 is quite ordinary.) This paper in scope and in detail is therefore a generalization of the BRST quantization ofĉ = 52 matter in Ref. [17]. In particular, we follow Ref. [17] in obtaining the general BRST quantization directly from the principle of local isomorphisms [3][4][5][6][7][8][9][10][11][12][13][14][15], leaving an equivalent derivation from the extended actions of the theories (see Ref. 16 and App. B) for another time and place.
The mathematics needed for the generalization is minimal. In orbifold theory, each twisted sector σ corresponds to an equivalence class σ of the divisor, and an appropriate language here is the familiar [7,9,16,18] cycle-basis for each sector σ of the general permutation group H(perm) K : (1.3) In this notation, f j (σ) and N(σ) are respectively the length of cycle j and the number of cycles in sector σ, whilej indexes within each cycle. For the convenience of the reader, the cycle-data for the elements of Z K and S K are included in the text. At any stage of the development, the results of Ref. [17] can be obtained from the cycle data for the single non-trivial element (σ = 1) of H(perm) 2 = Z 2 or Z 2 (w.s.).
The results presented here include the twisted ghost and BRST systems of each cycle j of each twisted sector σ of all these orbifold-string systems, including in particular the extended BRST algebra of sector σ [Q j (σ),Q l (σ)] + = 0, ∀ j, l in sector σ (1.5) which was conjectured in Ref. [17]. In a convention we follow throughout this paper, these are the BRST charges of the open-string analogues (1.1c) only, and right-mover copies of each result must be added to describe the generalized permutation orbifolds (1.1a). The BRST systems then imply the extended physical-state conditions on the matter of cycle j in sector σ where {Lĵ j } are the orbifold Virasoro generators (see Eq.  [16]. Because they descend universally from the twisted permutation gravities of Ref. [16], the BRST systems presented here depend only on the permutation groups H(perm) K or Z 2 (w.s.). The automorphism groups H ′ 26 in Eq. (1.1) are however encoded in the explicit forms of the orbifold Virasoro generators {Lĵ j } of the matter, which will be constructed in the next paper of this series. There we will also discuss the equivalent, reduced formulation of the physical states of each cycle at reduced cycle central charge c j (σ) = 26, and begin our study of the target space-time structure of the new string theories.
In this case we are interested in obtaining the BRST systems of the generalized permutation orbifolds and their open-string analogues so we begin with K copies of the operator-product form of the untwisted BRST system given for K = 2 in Ref. [17]. This includes the composite operators J I (z) = :ψ I (z)ψ I (z): , I = 0, . . . , K − 1 (2.2a) as well as their operator products, e.g., (2.3b) (2.3c) Here {ψ,ψ} are K copies of the ghosts, taken as half-integer moded Bardakci We remind the reader that here, and throughout the paper the steps are shown as above only for a single set of local operators, which is adequate to describe the open-string analogues in Eq. (1.1c). A second set of right-mover operators is required to describe the closed strings of the generalized permutation orbifolds (1.1a). To obtain the twisted sectors of the orbifolds we will apply the principle of local automorphisms [3][4][5][6][7][8][9][10][11][12][13][14][15] (including monodromies), but for the open-string analogues this is only a simple shortcut known [15][16][17][18] to give the correctlytwisted open-string mode algebras 1 .
The automorphic response of all these fields {A} to an arbitrary element of the general permutation group of order K is The next step in the orbifold program is the construction for each operator A the corresponding eigenfield A, whose automorphic response A ′ to each ω(h σ ) is diagonal: Here U(σ) is the unitary eigenmatrix of the eigenvalue problem of each element ω(h σ ) ∈ H(perm) K , which is known [7,9,16,18] in the cycle-basis of each element: N(σ) is the total number of cycles j in each ω(h σ ), whileĵ indexes within each cycle j of length f j (σ) ≥ 1. I have also included the standard periodicity convention (2.5c) within each cycle (which the eigenfields inherit from the eigenvalue problems) andj is the pullback ofĵ to its fundamental region.
In orbifold theory, each sector σ corresponds to an equivalence class σ of the automorphism group, so we need to choose one representative ω(h σ ) of each equivalence class of defines an equivalence class of the symmetric group H(perm) where f j (σ) is any divisor of K. In the cyclic groups f j (σ) is also the order of each ω(h σ ).
The results of Ref. [16] can therefore be obtained from the results of this paper by choosing the non-trivial element σ = 1 of H(perm) 2 = Z 2 with the single cycle-length f 0 (1) = 2 (and j →ū = 0, 1). More generally, the untwisted sector σ = 0 is described by the unit element for all H(perm) K . The quantities {χ} in the eigenfields (2.5a) are normalizations, which I choose here as for BRST currents f j (σ) otherwise.
The last (square-root) convention here is the standard choice for permutation-orbifold matter [7,9,[16][17][18] in the orbifold program, and the other conventions for the fermionic operators are consistent with the choice in Ref. [17] for the non-trivial element of H(perm) 2 = Z 2 with f j (σ) = 2. It is then straightforward to compute the operator-products of the eigenfields in terms of the eigenfields themselves (see the remark after Eq. (3.5)).
In this last step, and only here, do we change from the untwisted Hilbert space to the twisted Hilbert space of sector σ. The final result for the operator-product form of the twisted BRST system is presented in the following section.
For perspective, we also mention an equivalent, alternate route (via the principle of local isomorphisms and a monodromy decomposition) to the same twisted fieldsÂ. This route and the one described above via eigenfields are both summarized in the so-called commuting diagrams [3,5,11] of the orbifold program.

Operator-Product Form of the Twisted BRST's
The twisted fields of cycle j in sector σ are as followŝ where :. . .: is still operator-product normal ordering and I have included the monodromies and periodicities of each twisted field in Eqs. (3.1e) and (3.1f) respectively The fields in this list include the twisted reparametrization ghosts {ψ,ψ}, the twisted ghost currents {Ĵ}, the extended ghost stress-tensors {θ G } and the twisted BRST currents {Ĵ B }. Finally {θ} and {θ t } are respectively the extended matter stress-tensors and the extended total stress-tensors of cycle j in sector σ.
The operator products of the twisted fields are also obtained in detail as follows. We begin with the basic operator products of the twisted ghosts and their currentŝ and continue with the operator products involving the extended stress tensors: Finally, we give the operator products which involve the twisted BRST currents: We should also mention that the extended matter stress tensors are independent of the ghost systemsθĵ j (z, σ)Ĝl l (w, σ) = :θĵ j (z, σ)Ĝl l (w, σ): whereĜ can beψ,ψ or any composite thereof. Eqs. (3.1) thru (3.4) are a complete description at the operator-product level of the twisted BRST systems of twisted sector σ. The composite forms and operator products of the (intermediate) eigenfields {A} can in fact be read from these equations by replacinĝ A → A and the monodromies (3.1e) by the automorphic responses (2.5b). Note also that the dynamics of twisted sector σ is semisimple with respect to the cycles, as seen in the earlier conformal-field-theoretic study of the permutation orbifolds [9].
We close this section with some remarks about the operator-product algebra of the extended stress tensors and their associated ordinary Virasoro subalgebras. From Eqs. (3.3a-c), one reads the central charges of the extended stress-tensors of cycle j in sector σ In fact these "cycle" central charges are the central charges of the ordinary (Virasoro) stresstensorsθ 0j ,θ G 0j andθ t 0j of cycle j, whose operator products have the schematic (Virasoro) formθ for each of the three types. More familiar central charges are obtained for the physical stress tensors of sector σθ all three of which share the schematic (Virasoro) operator product: The cycle-sums in (3.9b,c) were evaluated with the sum rule in Eq. (3.1g), and we recognize 26K in Eq. (3.9b) as the matter central charge of each sector of any permutation orbifold on K copies of U(1) 26 .

The Mode Algebras of Twisted Sector σ
Consulting the monodromies (3.1e) and the conformal-weight terms {∆/(z − w) 2 } in the operator products with the extended stress tensors, we define the modes {Â(m +ĵ f j (σ) )} of each twisted field as follows: The periodicity (4.1g) of the modes is a consequence of the periodicity (3.1f) of the twisted fields.
We turn next to the mode algebras of the extended stress tensors, which include the following three extended Virasoro algebras: [Lĵ j (m +ĵ f j (σ) ),Ll l (n +l f l (σ) )] These algebras are also called general orbifold Virasoro algebras [1,2,9,[16][17][18][19][20], and in particular the algebra of the matter generators {Lĵ j } at fixed j is an orbifold Virasoro algebra of order f j (σ). Each of the three orbifold Virasoro algebras contains a so-called integral Virasoro subalgebra for each cycle j in sector σ which exhibit the three cycle central charges discussed in the previous section. Because the cycle dynamics is semisimple, there are also three physical Virasoro subalgebras for each sector σθ Of course, the extended Virasoro generators of the matter are independent of the ghost systems [Lĵ j (m +ĵ f j (σ) ),Ĝl l (n +l f l (σ) )] = 0 (4.6) whereĜ can beĉ,b or any composite thereof. This statement includes Eq. (4.3d) as a special case.
We conclude with the operator-product normal-ordered forms of the composite mode operators themselveŝ but these forms are not as useful as the mode-ordered forms of these operators given in Sec. 7.

One BRST Charge per Cycle per Sector
In the discussion above, we have constructed the modes {Ĵ B jj (m +ĵ/f j (σ))} of one twisted BRST current per cycle per sector for the general bosonic orbifold-string system of permutationtype. We may then define exactly one BRST charge per cycle per sector Q j (σ) ≡Ĵ B 0j (0), j = 0, 1, . . . , N(σ) − 1 (5.1) as the unique zero mode (m =ĵ = 0) of each twisted BRST current. As a consequence, the results above imply the following algebra of the BRST charges with the other operators in sector σ as well as the previously conjectured algebra [17] of the BRST charges themselves: The extended BRST algebra in Eq. (5.3a) follows directly from Eq. (5.2c), and comprises one of the central results of this paper. In addition to the implied nilpotency (5.3b) of each BRST charge, I note the following two consistency checks among the algebras (5.2), (5.3),(4.3c),(4.6) and (4.7b). Both checks involve using Eq. (5.2b) at j = l as a definition of the total orbifold Virasoro generators {L t lj }. First, adding to this the BRST algebra (5.3a), we find that the algebra (5.2a) is implied. Second, an independent derivation of the total orbifold Virasoro algebra (4.3c) is obtained as follows: Start on the left side of Eq. (4.3c) and use (5.2b) for one of the total Virasoro generators in the commutator. Then sequential application of the algebras (5.2b), (4.6), (4.7b) and a final application of Eq. (4.2b) gives the right side of Eq. (4.3c). These are of course generalizations of the consistency checks of the ordinary, untwisted BRST system (see e.g. Ref. [23]) and the twisted BRST system ofĉ = 52 matter [17].

The Physical States of Cycle j in Sector σ
For each sector σ of each orbifold of permutation-type, we use the BRST charges (5.1) to define the physical states {|χ(σ) j } of cycle j as follows: bĵ j ((m +ĵ f j (σ) ) ≥ 0)|χ(σ) j = 0 (6.1b) cĵ j ((m +ĵ f j (σ) ) > 0)|χ(σ) j = 0 (6.1c) Using then Eqs. (5.2b) and (6.1a,b), we find also that the physical states are also annihilated by the non-negative modes of the total orbifold Virasoro generatorŝ which is consistent because these generators have zero central extension. The role of thê c-condition in Eq. (6.1c) will be noted in the following section.
Summing on the cycles, we find that the physical states are also annihilated by the total BRST charge of sector σQ (σ) ≡ jQ j (σ),Q 2 (σ) = 0 (6.3a) but this condition is less useful than the BRST conditions for each cycle in Eq. (6.1a).

Mode-Ordered Form of the Twisted Ghost Systems
To further analyze the physical states, we need more explicit forms of the operators in the twisted ghost systems. For this purpose, I define the following mode normal-ordered product whereÂ andB can be eitherĉ orb.
Then the mode expansions in Eq. (4.1) straightforwardly give the following relation between the operator-product normal-ordered quadratics and the mode normal-ordered quadratics :ψĵ j (z, σ)ψl l (w, σ): = :ψĵ j (z, σ)ψl l (w, σ): M +δ jl δĵ +l,0 mod f j (σ) (∆j j (z, w) − 1 z−w ) (7.2a) which are sufficient to obtain the mode-ordered forms of the relevant operators of the ghost system. We begin with the mode-ordered form of the twisted ghost currents {Ĵ G }: In what follows we use only these mode-ordered forms {Ĵ G } as the properly-ordered ghost currents. Since they differ from the original ghost currents {Ĵ} only by a c-number shift, we may in fact readĴ →Ĵ G in all the mode algebras of Section 4. Using the definition (7.1) of the mode-ordering, we can for example write out the twisted ghost charge of cycle j in full detail: Note that the terms involving the zero modesĉ 0j (0) andb 0j (0) of the twisted ghosts are isomorphic to the standard ghost-charge operator of ordinary (untwisted) BRST. With Eqs. (3.1b) and (7.2), we also obtain the mode-ordered form of the extended ghost stress-tensors and their orbifold Virasoro generators: In what follows, I shall refer to the quantityâ f j (σ) in Eq. (7.7c) as the intercept of cycle j in each sector σ. Mode-ordered expressions can also be obtained for the twisted BRST currents {Ĵ Ĝ jj } and the BRST charges {Q j }, but these will not be needed in the present development.
As an application of the mode-ordered forms (7.5b) and (7.7b), we may use both theb and theĉ conditions (6.1b,c) on the physical states of cycle j to compute directly that We emphasize that these results require attention to the definition (7.1) of mode-ordering. In particular, the explicit form (7.6) of the twisted ghost charge is needed to see that the physical states of cycle j have ghost number (− 1 2 ), uniformly, as in ordinary untwisted BRST (see e.g. Ref. [25])).
For completeness, we finally give the action on the physical states of the ghost current modes and ghost Virasoro generators of sector σ where {L G σ (m)} satisfies the Virasoro algebra (4.5c) atĉ G (σ) = −26K, N(σ) is the number of cycles in sector σ and the product states {|χ(σ) } are defined in Eq (6.3b).

The Extended Physical-State Conditions
The final step in this paper is the transition from the action of the ghost operators to that of the matter operators on the physical states.
This transition is now effected in a single step, using Eq (7.8b) and the fact (6.2) that the physical states are annihilated by the non-negative modes of the total orbifold Virasoro generators {L t } withĉ t j (σ) = 0. The result is the following extended physical-state conditions of cycle j in sector σ: These conditions on the twisted matter of the new string theories are a central result of the paper. We remind that the extended Virasoro algebra (8.1b) of each cycle j is called an orbifold Virasoro algebra of order f j (σ) at cycle central chargeĉ j (σ) = 26f j (σ), and the interceptâ f j (σ) of cycle j in sector σ descends directly from Eq. (7.8b) of the ghost system. The extended physical state conditions (8.1a) are the operator analogues of the classical extended Virasoro constraintŝ θĵ j (z, σ) =Lĵ j (m +ĵ f j (σ) ) = 0, ∀ĵj in sector σ (8.3) associated to the general permutation gravities of the extended actions given for these theories in Ref. [16]. Further remarks on the action formulations are included in App. B.
In this sense the cycle central charges {ĉ j (σ)} and the cycle-intercepts {â f j (σ) } are the fundamental numbers obtained in our BRST quantization, and both quantities are universal in that they depend only on the length f j (σ) of cycle j in sector σ. underscores that both quantities increase monotonically with cycle-length.
Using the cycle basis of each element ω(h σ ) ∈ H(perm) K , the twisted matter of each sector σ of each orbifold-string of permutation-type is described by cycle-collections of our result in Eq. (8.1). I illustrate this here with some simple examples.
The first case in the table f j (σ) = 1 includes ordinary untwisted critical string theory. Copies of ordinary string theory are encountered here for example in the untwisted sectors σ = 0 (the trivial element of H(perm) K and H ′ 26 ) of all the orbifold-string theories of permutation-type: With a set of right-mover copies and the relabeling j → I, this is recognized as the physicalstate conditions of the original K copies U(1) 26K of the ordinary untwisted closed string U(1) 26 , i.e., the starting point of the orbifold program in this case. There are no cycles of unit length in the non-trivial twisted sectors σ = 1, . . . , K − 1 of H(perm) K = Z K , but cycles of unit length occur frequently in the elements of H(perm) K = S K -where they also describe unit-cycle strings withĉ j (σ) = 26 and unit intercept. We emphasize however that these unit-cycle strings are not ordinary strings for non-trivial elements of the space-time automorphism group H ′ 26 . Indeed, for example, each sector σ of the ordinary space-time orbifolds U(1) 26 /H ′ 26 (the closed-string sectors of each orientation-orbifold string system) is also described by the ordinary physical-state condition (8.4) at K = 1.
The simplest example of the second case f j (σ) = 2 is the single twisted sector σ = 1 of H(perm) 2 = Z 2 or Z 2 (w.s.), where we find With the relabelingj →ū = 0, 1, this collection is recognized as the two extended physicalstate conditions of twistedĉ = 52 matter in Refs. [16][17][18][19][20]. These systems appear for example in the twisted open-string sectors of the orientation-orbifold string theories [15][16][17][18][19] or, adding right-mover copies, in the generalized Z 2 -permutation orbifolds [15][16][17][18]20]. The result (8.5) is included in the extended physical-state conditions for the twisted sectors σ = 1, . . . , K − 1 of Z K , K = prime, where f j (σ) = K for each sector: The extended physical-state conditions for prime cyclic groups were conjectured in Ref. [18] and verified at the interacting level in Ref. [20]. The cycle-bases for the K −1 twisted sectors of the general cyclic group H(perm) were described in Eq. (2.7). Then for example we find the following extended physicalstate conditions for the three twisted sectors σ

Conclusions
We have completed a BRST quantization of the bosonic prototypes of the generalized permutation-orbifold string theories and their open-string analogues where H ′ 26 is any automorphism group of the critical closed string U(1) 26 and H(perm) K is any permutation group on the K copies. The matter of each sector σ of these orbifolds lives at sector central chargeĉ(σ) = 26K. We remind that the orientation-orbifold string systems are also described in our quantization: The twisted open-string sectors of these theories are included in the open-string analogues at K = 2 andĉ(σ) = 52, while the twisted closedstring sectors are the ordinary space-time orbifolds U(1) 26 /H ′ 26 atĉ(σ) = 26. These last cases correspond to the trivial element of Z 2 (w.s.) -and hence ordinary physical-state conditions for each sector. The extended action formulations [16] of the bosonic prototypes are reviewed in App. B, which also points out a generalization of the orientation-orbifold string systems.
Using the cycle-bases of general permutation groups, our central result in this paper is the construction of one twisted BRST system per cycle per sector in each of the orbifold-string systems (9.1) and (9.2). In particular, we have found the extended BRST algebra (5.3a), conjectured in Ref. (17), which gives one nilpotent BRST chargeQ j (σ) for each cycle j in each sector σ of these orbifolds. The twisted BRST systems then imply the extended physical state conditions (8.1) for the matter at cycle central chargeĉ j (σ) = 26f j (σ), where f j (σ) is the length of cycle j. The sector central charges of the matter are recovered as the cycle sumŝ c(σ) = jĉ j (σ) = 26K. The extended physical-state conditions also exhibit another set of fundamental numbers, the so-called cycle-interceptsâ f j (σ) in Eq. (8.1c). We remind that a right-mover copy of the results given here are necessary to describe the twisted closed-string sectors of these orbifolds.
Our results here therefore generalize the construction in Ref. (17) of one twisted BRST system for the non-trivial element of H(perm) 2 = Z 2 or Z 2 (w.s.), with single-cycle length f 0 (1) = 2 and matter central chargeĉ 0 (1) = 52. We emphasize that all aspects of our new twisted BRST systems, including the extended physical-state conditions, are separable in the cycles j of each sector. This is in accord with the extended actions [16] of these theories (see App. B), and with earlier work on the WZW permutation orbifolds [9] with trivial H ′ 26 . In this paper, the generators {Lĵ j (m +ĵ f j (σ) )} of the orbifold Virasoro algebras of the matter have been treated abstractly, so our first task in the next paper of this series will be the explicit construction of these generators in terms of the twisted matter fields. It is in this step that each element of the automorphism group H ′ 26 is encoded. With this information, we will also find the equivalent, reduced formulation of the physical-state problem of each cycle j at reduced cycle-central charge c j (σ) = 26, and begin our study of the target space-times of the orbifold-string theories of permutation-type.
On the basis of these results, one expects that the extended actions of theĉ = 26K open-string analogues of the generalized permutation orbifolds will have the same form as that shown in Eq. (B.2), but with the substitution H + ] open :  [15]. Note that the extended actions (B.1-3) are separable in the cycle-label j, consistent with the results of the text and earlier work in Refs. [9,[16][17][18]. In principle, the twisted BRST systems of this paper can also be derived from these actions by the Faddeev-Popov method.
In fact, there exist other sets of bosonic prototypes of the new string theories, for example the generalized orientation-orbifold string theories which reduce to the orientation orbifolds (B.1a) when K = 1. Detailed construction of these theories for higher K is beyond the scope of this paper, so I confine the discussion here to some preliminary remarks on their structure.
I begin with some simple properties of these theories as CFT's. Like orientation orbifolds, the presence of the world-sheet orientation group Z 2 (w.s.) in the divisor tells us that the generalized orientation orbifolds have an equal number of twisted open-and closed-string sectors, now at sector central chargesĉ = 52K and 26K respectively. In fact, the closedstring sectors (corresponding to the trivial element of Z 2 (w.s.)) of these theories clearly form (two copies of) the generalized permutation orbifold (B.2a), while the sectors corresponding to the trivial element of H(perm) K comprise (K copies of) the ordinary orientation orbifold (B.1a).
It is also clear that the matter currents of the open-string sectors of these theories are triply-twisted, with three spectral fractions corresponding to the elements of H Here the quantitiesĥ (ĵ)j(u) mn andĤ (ĵ)j(u) are the extended metric and density of the worldsheet permutation gravity associated with each equivalence class of the product permutation group Z 2 (w.s.) × H(perm) K . I have not worked out these last structures in detail, and beyond this I will not speculate in this paper.