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 σ of the general case, verifying in particular the previously-conjectured algebra $ {\left[ {{{\hat{Q}}_i}\left( \sigma \right),{{\hat{Q}}_j}\left( \sigma \right)} \right]_{+} } = 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 ĉj(σ) = 26fj(σ), where fj(σ) 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 closed string (hence only Abelian currents), and there are many of JHEP11(2010)031 these, including the bosonic prototypes [16] U ( 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 appendix B of this paper), but our discussion in the present series will be limited to the sets shown in eq. (1.1).
A central issue in all these theories is the presence of extra negative-norm states in the conformal field theories of the twisted sectors. (These "ghost" states are formally associated to the time-like direction of each untwisted closed-string copy in the untwisted sector, but these ghosts are removed in the untwisted sector by the physical state conditions and gauges of each copy, leaving K decoupled copies of the ordinary ghost-free closed string. ) We also know that 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 physical 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 world-sheet permutation gravities [16], new twisted BRST systems [17], new extended physical-state 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 twisted sector of the orbifolds JHEP11(2010)031 of permutation-type. It is therefore 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. The reduced formulation is therefore central in providing a strong suggestion that at least some of these theories can describe conventional (single-time) target space-times, with no ghosts in their physical spectra.
As noted above, some simple subexamples of these phenomena have already been studied. 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 at the interacting level 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 by construction that more general choices of H ′ 26 ⊂ H ± can in fact describe a variety of (single-time) target spacetimes in the new theories, including in particular Lorentzian target space-times with sectordependent dimensionality D(σ) ≤ 26! In the succeeding papers we will also discuss further evidence for the no-ghost conjecture [16] in the Lorentzian systems, and point out that these theories can sometimes contain a unique graviton (e.g. the orientation-orbifold string theories) or -more often -multiple, presumably-decoupled gravitons (as in the untwisted closed-string sectors for higher K).
Following the development of refs. [16][17][18][19][20], 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 orbifoldstring 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 JHEP11(2010)031 derivation from the extended actions of the theories (see ref. [16] and appendix 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 : 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 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. (4.3a)) with cycle central chargê c j (σ) and the quantityâ f j (σ) is called the intercept of cycle j in sector σ. These results generalize those of ref. extended Virasoro constraints {Lĵ j = 0} obtained from the extended actions of these theories in ref. [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-times 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 as well as their operator products, e.g., Here {ψ,ψ} are K copies of the ghosts, taken as half-integer moded Bardakci the c = −26 ghost stress-tensor and the BRST current respectively. The independent quantities {T } are the copies of the c = 26 matter stress tensor while {T t } are the total stress-tensor copies, each with c = 0. Finally, as seen in eq. (2.3), the symbol : . . . : is operator-product normal ordering. 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 openstring 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 correctly-twisted 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 of j 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

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defines an equivalence class of the symmetric group H(perm) K = S K . The cyclic group 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 (andj →ū = 0, 1). More generally, the untwisted sector σ = 0 is described by the unit elementj The quantities {χ} in the eigenfields (2.5a) are normalizations, which I choose here as 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)). The final step in the orbifold program is an application of the principle of local isomorphisms [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], which maps the eigenfields {A} of sector σ to the twisted fields {Â} of sector σ: (2.10) There are two components to this principle. First, the twisted fields are locally isomorphic to the eigenfields, that is, they have the same operator products. Second, the monodromies of each twisted fieldÂÂĵ is the same as the diagonal automorphic response (2.5b) of the corresponding eigenfield A. 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.

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3 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. 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

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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) :  (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)-(3.3c), 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) stress-tensorsθ 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 σ

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all three of which share the schematic (Virasoro) operator product: The cycle-sums in (3.9b), (3.9c) 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:ψĵ The periodicity (4.1g) of the modes is a consequence of the periodicity (3.1f) of the twisted fields.
Then by standard contour methods [5] we find the mode algebra of cycle j in each twisted sector σ, beginning with the twisted ghosts and their currents ĉĵ j m +ĵ f j (σ) ,bl l n +l f l (σ)
We turn next to the mode algebras of the extended stress tensors, which include the following three extended Virasoro algebras: 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

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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 whereĜ can beĉ,b or any composite thereof. This statement includes eq. (4.3d) as a special case. Other algebras in the twisted ghost systems include the following: . (4.7c) Finally, we have the algebra which involves the modes of the twisted BRST currents: Appendix A includes a technical remark used to simplify the right side of the anticommutator (4.8c) of the BRST current modes. Also, evaluation of this anticommutator at f j (σ) = 2 shows a (non-propagating) typo in eq. (4.4i) of ref. [17], where the last factor before the sums should read (m + n + u+v 2 − 1) instead of (m + n + u+v 2 + 1). 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 section 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 permutation-type. We may then define exactly one BRST charge per cycle per sector 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 Q j (σ),L t ll n +l f l (σ) = 0, ∀ j, l, σ (5.2a) 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.

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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]. 6 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: Using then eqs. (5.2b) and (6.1a), (6.1b), 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 σ: Owing to the relations (5.3a), this condition for the full sector is equivalent to the BRST conditions for each of its cycles in eq. (6.1a).

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7 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 . . , f j (σ)−1 is again the pullback ofĵ to the fundamental region. Comparing to the exact expression∆ we see that the terms given explicitly in the Laurent expansion above are exact forj = 0. Eq. (7.2) then implies the local relations :ψĵ j (z, σ)ψl l (z, σ) : =:ψĵ j (z, σ)ψl l (z, σ) : M +δ jl δĵ +l,0 mod f j (σ)â 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 thê b and theĉ conditions (6.1b), (6.1c) 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 JHEP11(2010)031 Virasoro generators {L t } withĉ t j (σ) = 0. The result is the following extended physicalstate 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 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 appendix 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. The preceding table of numerical values underscores that both quantities increase monotonically with cycle-length.

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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 (8.3) at K = 1.

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where {L σ } are the physical Virasoro generators of sector σ for any H(perm) K and the product-states {|χ(σ) } are defined in eq. (6.3b). With the sum rule in eq. (8.1d), this condition follows simply from our central result (8.1) by summing over the cycles of each sector.

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 Ref. [16] gave the extended actions for theĉ = 52 twisted open-string sectors of the orientation-orbifold string systems mn , which is the twisted metric of Z 2 -permutation gravity associated to the world-sheet orientation-reversing element of Z 2 (w.s.). The explicit forms of the extended diffeomorphisms, the densitiesĤ (u) and the twisted metric G(σ) of H ′ 26 are also given there, as well as the branes of these strings. The generators {L u (m + u 2 )} of the orbifold Virasoro algebras are singly-twisted and the matter currents {Ĵ(m + n(r) ρ(σ) + u 2 )} are doubly-twisted, where {n(r)/ρ(σ)} are the spectral fractions of the elements of H ′ 26 . The twistedĉ = 26 closed-string sectors of the orientation orbifold form the ordinary spacetime orbifold U(1) 26 /H ′ 26 , with ordinary Polyakov gravity, ordinary Virasoro algebras and singly-twisted currents {Ĵ (m + n(r) ρ(σ) )}. Ref. [16] also gave the extended actions of each twistedĉ = 26K closed-string sector of the generalized permutation orbifolds where h (ĵ)j mn is the extended metric of the permutation gravity associated to each equivalence class of H(perm) K . The monodromies, extended diffeomorphisms and the explicit form of the densitiesĤ (ĵ)j in these sectors are also given in ref. [16]. Again, the generators {Lĵ j (m+ĵ f j (σ) )} of the orbifold Virasoro algebra are singly-twisted and the matter currents {Ĵ (m + n(r) ρ(σ) +ĵ f j (σ) )} are doubly-twisted. In these cases of course one obtains a left-and right-mover copy of each algebra.
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.

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By construction these extended actions have the same bulk-invariances and modeing as those of the generalized permutation orbifolds, and these actions reduce for K = f j (σ) = 2 to the actions (B.1) of the open-string sectors of the orientation orbifolds. See also the construction of general twisted open strings in ref. [15]. Note that the extended actions (B.1)-(B.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 closedstring sectors, now at sector central chargesĉ = 52K and 26K respectively. In fact, the closed-string 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 ′ 26 , H(perm) K and Z 2 (w.s.):

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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.