The Orbifolds of Permutation-Type as
Physical String Systems
at Multiples of c = 26
III. The Spectra of ĉ = 52 Strings
M.B.Halpern∗
Department of Physics, University of California
and Theoretical Physics Group,
Lawrence Berkeley National Laboratory
University of California, Berkeley CA 94720 USA
November 15, 2018
Abstract
In the second paper of this series, I obtained the twisted BRST systems
and extended physical-state conditions of all twisted open and closed
ĉ = 52 strings. In this paper, I supplement the extended physical-state
conditions with the explicit form of the extended (twisted) Virasoro
generators of all ĉ = 52 strings, which allows us to discuss the physical
spectra of these systems. Surprisingly, all the ĉ = 52 spectra admit an
equivalent description in terms of generically-unconventional Virasoro
generators at c = 26. This description strongly supports our prior
conjecture that the ĉ = 52 strings are free of negative-norm states,
and moreover shows that the spectra of some of the simpler cases
are equivalent to those of ordinary untwisted open and closed c = 26
strings.
∗halpern@physics.berkeley.edu
http://arxiv.org/abs/0704.1540v1
Table of Contents
1. Introduction
2. The Extended Virasoro Generators of ĉ = 52 Strings
3. First Discussion of the ĉ = 52 String Spectra
4. Equivalent c = 26 Description
of the ĉ = 52 Spectra
5. Conclusions
1 Introduction
Opening another chapter in the orbifold program [1-11,12-15], this is the third
in a series of papers which considers the critical orbifolds of permutation-
type as candidates for new physical string systems at higher central charge.
In the first paper [16] of this series, we found that the twisted sectors of
these orbifolds are governed by new, extended (permutation-twisted) world-
sheet gravities – which indicate that the free-bosonic orbifold-string systems
of permutation-type can be free of negative-norm states at critical central
charge ĉ = 26K. Correspondingly-extended world-sheet permutation super-
gravities are expected in the twisted sectors of the superstring orbifolds of
permutation-type, where superconformal matter lives at higher multiples of
critical superstring central charges.
In the second paper [17] of the series, we found the corresponding twisted
BRST systems for all sectors of the free-bosonic orbifolds which couple to the
simple case of Z2-twisted permutation gravity, i.e. for all the twisted strings
with ĉ = 52 matter. The new BRST systems also implied the following
extended physical-state conditions for the physical states {|χ〉} of each of the
ĉ = 52 strings:
|χ〉 = 0, m ∈ Z, u = 0, 1 (1.1a)
, L̂v
m− n+ u−v
L̂u+v
m+ n + u+v
(1.1b)
δm+n+u+v
The algebra in Eq. (1.1b) is called an order-two orbifold Virasoro algebra (or
extended, twisted Virasoro algebra) and general orbifold Virasoro algebras
[1,18,9,12,16,17] are known to govern all the twisted sectors of the orbifolds
of permutation-type at higher multiples of c = 26.
The set of all ĉ = 52 orbifold-strings is a very large class of fractional-
moded free-bosonic string systems, including e.g. the twisted open-string
sectors of the orientation orbifolds, the twisted closed-string sectors of the
generalized Z2-permutation orbifolds and many others (see Refs. [16,17] and
Sec. 2). Starting from the extended physical-state conditions (1.1) (and
a right-mover copy of (1.1) on the same {|χ〉}for the twisted closed-string
sectors) this paper begins the concrete study of the physical spectrum of
each ĉ = 52 string.
As the prerequisite for this analysis, I first provide in Sec. 2 the explicit
form – in terms of twisted matter fields – of the extended Virasoro generators
, u = 0, 1
of all ĉ = 52 strings. This construction allows us
to begin the study of the general ĉ = 52 string spectra in Sec. 3. The
same subject is further considered in Sec. 4, where I point out that all
the ĉ = 52 spectra admit an equivalent description in terms of generically-
unconventional Virasoro generators at c = 26. This description allows us to
see clearly a number of spectral regularities which are only glimpsed in Sec. 3,
including strong further evidence that the critical orbifolds of permutation-
type can be free of negative-norm states. Moreover, although the generic
ĉ = 52 spectrum is apparently new, we are able to show that some of the
simpler spectra are equivalent to those of ordinary untwisted open and closed
critical strings at c = 26.
Based on these results, the discussion in Sec. 5 raises some interesting
questions about these theories at the interacting level, and speculates on
the form of the extended physical-state conditions for more general orbifold-
strings of permutation-type. I will return to both of these subjects in suc-
ceeding papers of the series.
2 The Twisted Virasoro Generators
of ĉ = 52 Strings
As emphasized in Ref. [17], the universal form of the twisted BRST systems
and the extended physical-state conditions (1.1) are consequences of their
origin in Z2-twisted permutation gravity, which governs all twisted ĉ = 52
matter.
There are however many distinct ĉ = 52 strings, including the twisted
open-string sectors of the orientation orbifolds [12,13,15-17]
U(1)26
, H− = Z2(w.s.)×H (2.1)
and the twisted closed-string sectors of the generalized Z2-permutation orb-
ifolds [15-17]
U(1)26 × U(1)26
, H+ = Z2(perm)×H
′ (2.2)
as well as the generalized open-string Z2-permutation orbifolds and their T -
duals [15-17]. For the orientation orbifolds in Eq. (2.1), I remind that H− is
any automorphism group of the untwisted closed string U(1)26 which includes
world-sheet orientation-reversing automorphisms. Indeed the twisted open-
string orientation-orbifold sectors correspond to the orientation-reversing au-
tomorphisms, which have the form τ− × ω, ω ∈ H where the basic automor-
phism τ− exchanges the left- and right-movers of the closed string and ω is
an extra automorphism which acts uniformly on the left- and right-movers
of the closed string. Similarly, the automorphism group H+ of the general-
ized Z2-permutation orbifolds in (2.2) is generated by elements of the form
τ+×ω, ω ∈ H
′, where the basic automorphism τ+ exchanges the two copies
of the closed string and the extra automorphism ω again acts uniformly on
the left- and right-movers of each closed string. In both cases, the extra
automorphisms ω in τ × ω may or may not form a group (see the examples
at the end of this section).
The spectra of different ĉ = 52 strings are characterized by their extended
(twisted) Virasoro generators, all of which can in fact be written in the
following unified form:
Gn(r)µ;−n(r)ν(σ)
× (2.3a)
×:Ĵn(r)µv
Ĵ−n(r),ν,u−v
m− p−
+ u−v
+δm+u
,0 ∆̂0(σ)
Ĵn(r)µu
, Ĵn(s)νv
= (2.3b)
δn(r)+n(s),0modρ(σ)δm+n+n(r)+n(s)
Gn(r),µ;−n(r),ν(σ)
, Ĵn(r)µv
(2.3c)
Ĵn(r)µ,u+v
m+ n+
+ u+v
∆̂0(σ) =
dim[n(r)]
(2.3d)
dim[n(r)] = 26. (2.3e)
Each set of extended Virasoro generators in Eq. (2.3a) satisfies the order-two
orbifold Virasoro algebra (1.1b) at ĉ = 52, and the current algebras in Eq.
(2.3b) are of the type called doubly-twisted in the orbifold program.
For those unfamiliar with the program, I first give a short summary of the
standard notation in the result (2.3) – followed by the derivation of the result.
As in the extended Virasoro generators themselves, the indices u, v with
fundamental range ū, v̄ ∈ {0, 1} describe the twist of the basic permutations
τ∓ in each H∓. For each extra automorphism ω(σ) in each τ∓ × ω(σ), the
spectral indices {n(r)} and the degeneracy indices {µ ≡ µ(n(r))} of each
twisted sector σ are determined by the so-called H-eigenvalue problem [3,5,6]
of ω(σ)
ω(σ)a
U †(σ)b
n(r)µ
= U †(σ)a
n(r)µ
−2πin(r)
ρ(σ) , ω(σ) ∈ H or H ′ (2.4a)
ω(σ)a
ω(σ)b
Gcd = Gab, Gab = G
(2.4b)
a, b = 0, 1, . . . , 25, n(r) ∈ (0, 1, . . . , ρ(σ)− 1) (2.4c)
where G is the untwisted target-space metric of U(1)26. The quantity ρ(σ)
is the order of ω(σ) and all indices {n(r)µ} are periodic modulo ρ(σ), with
{n(r)} the pullback to the fundamental region and dim[n(r)] the size of the
subspace n(r). The index r is summed once over the fundamental region in
Eqs. (2.3a), (2.3d) and (2.3e). The twisted metric G.(σ) and its inverse G
are defined in terms of the unitary eigenvectors U(σ) of the H-eigenvalue
problem
Gn(r)µ;n(s)ν(σ) = χn(r)µχn(s)νU(σ)n(r)µ
U(σ)n(s)ν
Gab (2.5a)
= δn(r)+n(s),0modρ(σ)Gn(r)µ;−n(r),ν(σ) (2.5b)
Gn(r)µ;n(s)ν(σ) = χ−1
n(r)µ
n(s)ν
GabU †(σ)a
n(r)µ
U †(σ)b
n(s)ν
(2.5c)
= δn(r)+n(s),0modρ(σ)G
n(r),µ;−n(r),ν (2.5d)
where G is again the untwisted metric and the χ’s are essentially-arbitrary
normalization constants. Finally, the standard mode normal-ordering in Eq.
(2.3a) is:
:Ĵn(r)µu
Ĵn(s)νv
:M (2.6)
Ĵn(s)νv
Ĵn(r)µu
Ĵn(r)µu
Ĵn(s)νv
It follows that the quantity ∆0(σ) in Eqs. (2.3a) and (2.3d)
Ĵn(r)µu
|0〉σ = 0 (2.7a)
→ L̂u
|0〉σ = ∆̂0(σ) δm+u
,0|0〉σ (2.7b)
is the conformal weight of the scalar twist-field state |0〉σ of sector σ.
I comment briefly on the derivation of the unified form (2.3) of the
ĉ = 52 extended Virasoro generators. Essentially this result was given for
the twisted open-string sectors of the non-abelian orientation orbifolds in
Subsecs. 3.4, 3.5 of Ref. [12], and that result is easily reduced for our
abelian case U(1)26/H− in Eq. (2.1). With a right-mover copy of the ex-
tended Virasoro generators (and u → ̂ = 0, 1), the result also hold for
the twisted closed-string sectors of the generalized Z2-permutation orbifolds
(U(1)26 × U(1)26)/H+ in Eq. (2.2). This follows by the substitution
G → G, u
(2.8)
into the known results for the ordinary Z2-permutation orbifolds with trivial
H ′ (see Ref. [perm] and Subsec. 4.2 of Ref. [16]). Finally, a single copy of
the unified form (2.3) holds as well for each twisted sector of the generalized
open-string Z2-permutation orbifolds (U(1)
26 × U(1)26)open/H+ and all pos-
sible T -dualizations of each of these sectors. This conclusion follows because
the left-mover extended Virasoro generators of the closed-string orbifolds for
each H+ are the input data for the construction of the correponding open-
string orbifolds [14], and the twisted-current form of each set of extended
Virasoro generators is independent of T-dualization [15]. The branes, quasi-
canonical algebra and non-commutative geometry of the twisted open strings
[13-15,16,17] depend of course on the particular T-dualization, but these will
not be needed here.
In what follows I will consider each twisted ĉ = 52 string separately, but
the reader may find it helpful to bear in mind the complete sector structure of
these orbifold-string systems as labelled by the elements of the automorphism
groups H∓. Given a particular extra automorphism ωn ∈ H or H
′ of order
n, one may list the following low-order examples:
(1; τ∓) (2.9a)
(1; τ∓ × ω2) (2.9b)
(1, ω3, ω
3; τ∓, τ∓ × ω3, τ∓ × ω
3) (2.9c)
(1, ω24; τ∓ × ω4, τ∓ × ω
4) (2.9d)
(1, ω26, ω
6; τ∓ × ω6, τ∓ × ω
6, τ∓ × ω
6). (2.9e)
For the generalized Z2-permutation orbifolds (τ+) all of these sectors are
twisted closed strings at ĉ = 52, while all the sectors of the generalized open-
string Z2-permutation orbifolds (τ+) and their T-dualizations are twisted
open strings at ĉ = 52. For the orientation orbifolds (τ−) the sectors before
the semicolon are twisted closed strings at c = 26 (which form an ordinary
space-time orbifold) while the sectors after the semicolon are twisted open
strings at ĉ = 52. More generally, orientation orbifolds always contain an
equal number of twisted open and closed strings. In all cases, the twisting is
of course trivial for sectors corresponding to the unit element.
3 First Discussion of
the ĉ = 52 String Spectra
To frame this discussion, I remind [1] the reader that the Virasoro primary
states of our orbifold CFT’s are defined by the integral Virasoro subalgebra
(generated by {L̂0(m)}) of the extended Virasoro algebra. Then the extended
physical-state conditions (1.1a) tell us that all the physical states {|χ〉} of
each ĉ = 52 orbifold-string are Virasoro primary
L̂0(m > 0)|χ〉 = 0 (3.1)
but only a small subset of these primary states are selected by the rest of the
physical-state conditions:
L̂0(0)−
|χ〉 = L̂1
|χ〉 = 0. (3.2)
In what follows, I will refer to the L̂0(0) condition in Eq. (3.2) as the spectral
condition, since it will determine the allowed values of momentum-squared
for each ĉ = 52 string.
The space of physical states of each orbifold-string is then much smaller
than the space of states of the underlying orbifold conformal field theory. For
the experts, I remark in particular that the extended physical-state condi-
tions generically disallow the characteristic sequence [19] of Virasoro primary
states known as the principle-primary states [1,9]. This follows first by the
spectral condition (which fixes the conformal weight), and second because
the physical-state condition {L̂u
≃ 0} is stronger than the
principle-primary state condition [1,9]
|p.p.s.〉 = 0, u = 0, 1, m > 0 (3.3)
which does not extend to m = 0.
I turn now to concretize the spectral condition of each twisted ĉ = 52
string, using the explicit form (2.3) of its extended Virasoro generators. For
this, recall [12,15] first that these generators contain in general two kinds
of commuting zero modes (dimensionless momenta), namely {Ĵ0µ0(0)} and
{Ĵρ(σ)/2,µ,1(0)}, where the latter is relevant only when the order ρ(σ) of ω(σ)
is even. In what follows, I often refer to these zero modes collectively as
{Ĵ(0)}. It is then natural to define the “momentum-squared” operator P̂ 2
as follows:
L̂0(0) =
−P̂ 2 + R̂(σ)
+ ∆̂0(σ) (3.4a)
P̂ 2 ≡ −
G0µ;0ν(σ)Ĵ0µ0(0)Ĵ0ν0(0) + (3.4b)
,µ;− ρ(σ)
,ν(σ)Ĵρ(σ)/2,µ,1(0)Ĵ−ρ(σ)/2,ν,−1(0)
R̂(σ) ≡
r,µ,ν
Gn(r)µ;−n(r),ν(σ)× (3.4c)
×:Ĵn(r)µu
Ĵ−n(r),ν,−u
Here the primed sum in the “level-number” operator R̂(σ) indicates omission
of the zero modes.
With this decomposition, the spectral condition in Eq. (3.2) takes the
simple form:
P̂ 2|χ〉 =
P̂ 2(0) + R̂(σ)
|χ〉 (3.5a)
P̂ 2(0) ≡ 2
δ̂0(σ)− 1
(3.5b)
δ̂0(σ) =
dim[n(r)]
≥ 0. (3.5c)
Although I will continue the discussion primarily in this form, in fact Eqs.
(3.4a) and (3.5a) hold only for the twisted open-string sectors of the orbifolds.
For the twisted closed-string sectors, we also have right-mover copies of the
extended Virasoro generators (2.3), and a corresponding right-mover copy of
the extended physical-state conditions (1.1) on the same {|χ〉}. For simplicity
I will limit the discussion of these sectors here to the case of decompactified
zero modes, for which it is appropriate to equate the left and right movers
ĴR(0) = ĴL(0) = 1√
Ĵ(0) → R̂R(σ) = R̂L(σ) (3.6)
where the last equality is level-matching in each twisted sector. Keeping the
same definition of the operator P̂ 2 in Eq. (3.4b), the correct closed-string
ĉ = 52 spectral condition is then obtained by the substitution
P̂ 2 → 1
P̂ 2 (3.7)
in both Eqs. (3.4a) and (3.5a). These identifications, and hence P̂ 2
→ 2P̂ 2
can be used at any point in the discussion below to obtain the corresponding
closed-string results.
Returning to the open-string case, one simple solution of the extended
physical-state conditions is the ground state |0, Ĵ(0)〉σ of twisted sector σ:
R̂(σ)|0, Ĵ(0)〉σ = L̂u
|0, Ĵ(σ)〉σ = 0 (3.8a)
P̂ 2|0, Ĵ(0)〉σ = P̂
(0)|0, Ĵ(0)〉σ, P̂
(0) = −2 + 2δ̂0(σ) (3.8b)
This is the “momentum-boosted” twist-field state (see Eq. (2.7)) of that
sector, with ground-state mass-squared P̂ 2
. Moreover Eq. (3.4c) and the
commutator (2.3c) give the increments
∆(P̂ 2) = ∆(R̂(σ)) = 4
(3.9)
obtained by adding the negatively-moded current
Ĵn(r)µu
to any previous state. The precise content of these excited levels must of
course be determined from the remainder of the extended physical-state con-
ditions.
I continue this discussion with some specific examples of ĉ = 52 strings,
beginning with the simplest twisted open-string orientation-orbifold sectors
[12,13,15,16,17]:
ω = 1l : ρ = 1, n = 0, U = 1l, G = G, Ĵ0au
(3.10a)
∆(P̂ 2) = 4
∣m+ u
∣ (u = 0 is DD, u = 0 is ND) (3.10b)
ω = −1l : ρ = 2, n = 1, U = 1l, G = G, Ĵ1au
m+ u+1
(3.11a)
∆(P̂ 2) = 4
∣m+ u+1
∣ (u = 0 is DN, u = 0 is NN). (3.11b)
In these cases, the extra automorphisms ω act uniformly on the labels a =
0, . . . 25 and G is the untwisted target space metric in Eq. (2.4b). Although
both twisted strings have (26+26) = 52 matter degrees of freedom, note that
each example has only one of the two types of zero modes {Ĵ(0)}: 26DD zero
modes {Ĵ0a0(0)} for ω = 1l and 26NN zero modes {Ĵρ/2,a,1(0)} for ω = −1l.
In both cases, the momentum-squared (3.4b) has the schematic form
P̂ 2 = ηabĴa(0)Ĵb(0), η =
0 −1l
(3.12)
where η = −G is the standard (west-coast) 26-dimensional target-space met-
ric. Then we compute from Eqs. (3.5b) and (3.5c) that both strings share
the same tachyonic ground-state mass-squared
δ̂0(σ) = 0, ∆̂0(σ) =
, P̂ 2(0) = −2 (3.13)
and the first excited state of each is massless:
Ĵ0a1
Ĵ1a0
|0, Ĵ(0)〉σ = 0 for ω =
(3.14)
For this level, I have checked that the L̂1
≃ 0 gauge eliminates the
longitudinal parts of the 26-dimensional “photons”, and moreover the L̂1
and L̂0 (1) gauges together eliminate the negative-norm states at the next
level:
+ βĴ(−1)
|0, Ĵ(0)〉σ, P̂
2 = 2. (3.15)
Since the increments ∆(P 2) in Eqs. (3.10b) and (3.11b) are even integers,
we are led to suspect that the spectra of these two twisted ĉ = 52 strings are
nothing but the spectrum of an ordinary open c = 26 string in disguise 1. I
will return to this question in the following section.
A larger subset of twisted ĉ = 52 strings is the following. For a particular
twisted sector σ, suppose that ω = ±1l acts uniformly on a set of d labels
a = 0, 1, . . . d − 1, d ≥ 4 while a non-trivial element ω(perm) of some per-
mutation group acts non-trivially on the other 26 − d spatial labels. Then
Eqs. (2.4),(2.5) and standard results [3,5-7,9] in the orbifold program give
the following explicit form of the extended Virasoro generators (2.3) in this
sector:
∆0(σ) + (3.16a)
Gab(d)
:Ĵǫav
p+ v+ǫ
Ĵ−ǫ,b,u−v
m− p + u−v−ǫ
fj(σ)
fj(σ)−1
×:Ĵ̂jv
p+ ̂
fj(σ)
Ĵ−̂,j,u−v
m− p− ̂
fj(σ)
+ u−v
δ̂0(σ) =
fj(σ)−1
fj(σ)
fj(σ)
fj(σ)
≥ 0 (3.16b)
fj(σ) = 26− d, 4 ≤ d ≤ 26. (3.16c)
Here ǫ = 0 or 1 for ω = 1l or −1l, G(d) is the restriction of the flat target-space
metric (2.4b) to the first d labels, fj(σ) is the size of the jth cycle in ω(perm),
1The spectra of these two ĉ = 52 strings look even more familiar in terms of the
dimensionful momenta k ≡ Ĵ(0)/
, where α′
is the conventional open-string Regge
slope.
and the previous cases with δ̂0(σ) = 0 are included when d = 26. The half-
integer moded currents in the second term of (3.16) satisfy the twisted current
algebra (2.3b) with G → G(d). For the permutation-twisted currents in the
last term of (3.16), I have used the standard relation (n(r)/ρ(σ)) = (̂/fj(σ))
and (the inverse of) the twisted metric [3,5-7,9]
̂j;l̂l
(σ) = δjlfj(σ)δ̂+l̂,0modfj(σ) (3.17)
which also determines the twisted current algebra (2.3b) for these currents.
Using Eq. (3.16b), we see that the non-trivial element of Z2 on two labels
also gives δ̂0(σ) = 0 and a P̂
= −2 ground state, but a non-trivial element
of Z3 on three labels gives a slightly-raised ground state
δ̂0(σ) =
, ∆̂0(σ) =
, P̂ 2(0) = −
(3.18)
and no photons.
Given the cycle-structure {fj(σ)} of any extra automorphism w(perm)
(see e.g. Eq. (3.4) of Ref. [16]), it is straightforward to evaluate the sum in
Eq. (3.16b). As an illustration, one finds the simple tachyonic ground-state
mass-squares
5 ≤ (d = prime) ≤ 23 : P̂ 2(0) = −
(d− 2 +
26− d
) (3.19)
in twisted sectors which correspond to the action of any non-trivial element
of the cyclic group Zλ of prime order on 3 ≤ (λ = 26−d) ≤ 21 spatial labels.
The result (3.19) includes Eq. (3.18) when d = 23, but does not extend to
the cases d = 26, 24 with P̂ 2(0) = −2 discussed above. I remind that this
result applies only to the open orbifold-strings, while twice these values of
P̂ 2(0) are obtained for the closed-string versions.
Further analysis of the ĉ = 52 strings, including the “larger subset” of
examples (3.16), is found in the following section.
4 Equivalent c = 26 Description
of the ĉ = 52 Spectra
In fact, there exists an entirely equivalent description of all the ĉ = 52 string
spectra in terms of generically-unconventional Virasoro generators at c = 26.
To obtain the c = 26 description, I first define the relabelled (unhatted)
operators
Jn(r)µ
2m+ u+
2n(r)
≡ Ĵn(r)µu
, u = 0, 1 (4.1a)
L(2m+ u) ≡ 2L̂u
(4.1b)
in terms of the hatted operators above. This 1−1 map is recognized as
a modest generalization of (the inverse of) the order-two orbifold-induction
procedure of Borisov, Halpern and Schweigert [1]. SinceM ≡ 2m+u, u = 0, 1
covers the integers once, we then find from (2.3) the explicit form of the c = 26
generators:
L(M) = δ̂0(σ)δM,0 + (4.2a)
r,µ,ν
Gn(r)µ;−n(r),ν(σ)
:Jn(r)µ
2n(r)
J−n(r),ν
M −Q−
2n(r)
δ̂0(σ) =
dim[n(r)]
(4.2b)
[L(M), L(N)] = (M −N)L(M +N) + 26
M(M2 − 1)δM+N,0 (4.2c)
L(M), Jn(r)µ
2n(r)
2n(r)
Jn(r)µ
M +N +
2n(r)
(4.2d)
Jn(r)µ
2n(r)
, Jn(s)ν
2n(s)
(4.2e)
= δn(r)+n(s),0modρ(σ)δM+N+2(n(r)+n(s)ρ(σ) ),0
Gn(r)µ;−n(r),ν(σ).
The expression (4.2b) for δ̂0(σ) is the same as above, and the mode-normal
ordering in Eq. (4.2a)
:Jn(r)µ
2n(r)
Jn(s)ν
2n(s)
:M (4.3)
2n(r)
Jn(s)ν
2n(s)
Jn(r)µ
2n(r)
2n(r)
Jn(r)µ
2n(r)
Jn(s)ν
2n(s)
follows from the ĉ = 52 ordering (2.6) because the map (4.1) preserves the
sign of all arguments.
I emphasize that the c = 26 Virasoro generators in Eq. (4.2) are generically-
unconventional because the twisted matter is now summed over the fractions
{2n/ρ} instead of the conventional orbifold-fractions {n/ρ}. This distortion
of the “extra twist” is the price we must pay in order to unwind the “basic
twist” associated to the basic permutations τ∓ of H∓.
The map (4.1) also tells us that the ĉ = 52 momenta {Ĵ(0)} and the
c = 26 momenta {J(0)} are identical, and we may record
J(0) = Ĵ(0) : J0µ(0) = Ĵ0µ0(0), Jρ(σ)/2,µ(0) = Ĵρ(σ)/2,µ,1(0) (4.4a)
P 2 = P̂ 2 (4.4b)
G0µ:0ν(σ)J0µ(0)J0ν(0) +
,µ;− ρ(σ)
,ν(σ)Jρ(σ)/2,µ(0)J−ρ(σ)/2,ν(0)
where the ĉ = 52 form of P̂ 2 was given in Eq. (3.4b). Similarly, the “level-
number” operator R(σ) in the decomposition of L(0) is the same
L(0) = −1
P 2 +R(σ)
+ δ̂0(σ) (4.5a)
R(σ) = R̂(σ) (4.5b)
r,µ,ν
Gn(r)µ;−n(r),ν(σ)×
×:Jn(r)µ
2n(r)
J−n(r),ν
2n(r)
where the ĉ = 52 form of R̂(σ) was given in Eq. (3.4c).
By itself, the inverse orbifold-induction procedure (4.1) is only a rela-
belling of the operators of the permutation-orbifold CFT’s. The central point
of this discussion however is that for the orbifold-string theories – restricted
by the extended physical state conditions (1.1) – the map also gives us a
completely equivalent c = 26 description of the physical spectrum of each
ĉ = 52 orbifold-string. Indeed, it is easily checked that both components
ū = 0, 1 of the ĉ = 52 extended physical-state condition (1.1a) map directly
onto the simpler and in fact conventional physical-state condition
L(M ≥ 0)|χ〉 = δM,0|χ〉 (4.6)
in the 26-dimensional description! A right- mover copy of Eq. (4.6) on the
same physical states {|χ〉} is similarly obtained in the equivalent c = 26
description of the closed orbifold-strings.
I emphasize that the physical states {|χ〉} of the 26-dimensional descrip-
tion (4.6) are exactly the original physical states (1.1a) of the ĉ = 52 string.
Indeed, each physical state |χ〉 can be regarded as invariant under the map,
or each can now be rewritten in 26-dimensional form. In further detail, Eqs.
(4.5) and (4.6) give the same spectral condition P 2 ≃ P 20 + R(σ), the same
physical ground state 2
|0, J(0)〉σ ≡ |0, Ĵ(0)〉σ, P
0 = P̂
0 = −2 + 2δ̂0(σ) (4.7)
and each negatively-moded hatted current in any physical state can be re-
placed according to Eq. (4.1a) by the corresponding unhatted current mode.
Note finally that the commutator (4.2d) and the decomposition (4.5a) give
the 26-dimensional increment
∆(P̂ 2) = ∆(R(σ)) = 2
2n(r)
(4.8)
which results from the addition of Jn(r)µ
2n(r)
to any previous
state. With M = 2m+ n, these are recognized as the same increments (3.9)
obtained in the ĉ = 52 description.
As simple examples, consider the “larger subset” (3.16) of ĉ = 52 strings
– whose equivalent c = 26 physical state condition (4.6) now involves the
following subset of the c = 26 Virasoro generators (4.2):
L(M) = δM,0δ̂0(σ) +
Gab(d)
:Jǫa(Q + ǫ)J−ǫ,b(M −Q− ǫ):M +
fj(σ)
fj(σ)−1
:J̂j
Q + 2̂
fj(σ)
J−̂,j
M −Q− 2̂
fj(σ)
:M (4.9a)
δ̂0(σ) =
fj(σ)−1
fj(σ)
fj(σ)
− 2̂
fj(σ)
(4.9b)
2Although it is not directly relevant in either description of the ĉ = 52 strings, one
notes that the conformal weight of the scalar twist-field state |0〉
of sector σ has now
shifted from ∆̂0(σ) to δ̂0(σ) in the c = 26 description.
a, b = 0, . . . , d− 1,
fj(σ) = 26− d, 4 ≤ d ≤ 26. (4.9c)
Recall for the larger subset that ǫ = 0, 1 corresponds in the symmetric theory
to the action of the extra automorphism ω = ±1l on the first d ≥ 4 labels
{a}, while fj(σ) is the length of the j-th cycle of the extra permutation
ω(perm) which acts on the remaining 26 − d spatial labels. Shifting the
dummy integer Q by the integer ǫ, we note that the second term in Eq.
(4.9a) is a set of ordinary Virasoro generators for d untwisted bosons with
the ordinary current algebra
[Ja(Q), Jb(P )] = G
ab QδQ+P,0 (4.10)
for both values of ǫ. The currents in the third term satisfy the twisted current
algebra (4.2e) with the permutation-twisted metric (3.17), and the value of
δ̂0(σ) in Eq. (4.9b) is only a slightly-rewritten form of that given in Eq.
(3.16b).
We are now in a position to confirm our suspicions in the previous sec-
tion about the simplest orbifold-strings, described earlier at ĉ = 52 by the
extended Virasoro generators:
:Ĵǫav
p+ v+ǫ
Ĵ−ǫ,b,u−v
m− p+ u−v−ǫ
,0, u = 0, 1, ǫ = 0, 1. (4.11)
These are now equivalently described by the choice d = 26 in Eq. (4.9), in
which case only the second (ordinary) term of Eq. (4.9a) is non-zero – and
then the equivalent physical-state condition (4.6) verifies that the physical
spectrum of each of these particular twisted ĉ = 52 strings is indeed equiva-
lent to that of an ordinary untwisted c = 26 string! These cases include the
open-string orientation-orbifold sectors corresponding to τ− × (ω = ±1l) in
Eq. (3.10) and their T-duals, as well as the twisted closed-string sectors of
the generalized Z2-permutation orbifolds corresponding to τ+ × (ω = ±1l).
Additionally, consider the following special cases of the extended Virasoro
generators (3.16) at ĉ = 52
= δm+u
+ (4.12)
Gab(24)
:Ĵǫav
p+ v+ǫ
Ĵ−ǫ,b,u−v
m− p+ u−v−ǫ
:Ĵ̂v
p+ ̂+v
Ĵ−̂,u−v
m− p+ u−v−̂
which result when the extra automorphism in the symmetric theory acts as
ω = ±1l on the first d = 24 labels and the non-trivial element of a Z2 on the
remaining 2 spatial labels. I have noted in Sec. 3 that δ̂0(σ) = 0 for these
cases as well, and indeed the equivalent c = 26 description (4.6) and (4.9)
at d = 24 now shows that the open and closed orbifold- strings of this type
also have the spectrum of ordinary untwisted c = 26 strings. The common
thread for the orbifold-strings in Eqs. (4.10) and (4.12) is that they are at
most half-integer moded, so that the shift {n/ρ} → {2n/ρ} gives integer
moding in the c = 26 description.
Beyond these simple cases, the ĉ = 52 strings are apparently new – with
δ̂0(σ) 6= 0, unfamiliar ground-state mass-squares, and fractional modeing
(and increments) in either description.
5 Conclusions
We have discussed the physical spectrum of the general ĉ = 52 orbifold-string,
as well as an equivalent but unconventionally-twisted c = 26 description
of the twisted ĉ = 52 matter. The equivalent c = 26 description holds
only for the orbifold-string theories – restricted by the extended physical-
state conditions (1.1) – and not in the larger Hilbert space of the underlying
orbifold conformal field theories.
In general we have found that the spectra of these orbifold-string systems
are unfamiliar. One simple and unexpected conclusion however is that, as
string theories restricted by the extended physical-state conditions, the single
twisted ĉ = 52 sector of each of the simplest orbifolds of permutation-type
(see Eq. (2.9))
(1; τ∓) (5.1a)
(1; τ∓ × ω2), ω
2 = 1 (5.1b)
have the same physical spectra as ordinary untwisted c = 26 strings. No such
equivalence is found of course in the half-integer moded Hilbert space of the
full orbifold CFT’s. The list in Eq. (5.1) includes the simplest orientation
orbifolds (with τ−) and their T-duals, as well as the simplest generalized
Z2-permutation orbifolds (with τ+).
For the simplest orientation orbifolds in particular, the string theories in
Eq. (5.1) consist of an ordinary unoriented closed string (the unit element)
at c = 26 and a ĉ = 52 twisted open string whose physical spectrum is
equivalent to that of an ordinary untwisted c = 26 critical open string. Since
both the closed- and open-string spectra of these simple orientation orbifolds
are equivalent to those of the archtypal orientifold (without Chan-Paton fac-
tors), we are led to suspect that orientation orbifolds include orientifolds. I
will return in the next paper of this series to consider this question at the
interacting level, where we will also be able to ask about the decoupling of
null physical states. Following that, I will consider in a succeeding paper the
corresponding situation and modular invariance for the simplest permutation
orbifold-string systems.
More generally, we have seen that there are many other orientation orb-
ifolds, open-string Z2-permutation orbifolds and generalized Z2-permutation
orbifolds whose ĉ = 52 spectra show fractional modeing in both the ĉ = 52
and the c = 26 descriptions. These include in particular the orbifolds in Eq.
(2.9) when the order n of the extra automorphism is greater than two.
There is more to say about no-ghost theorems for the general twisted
ĉ = 52 string. The original intuition [16] was that the doubled gauges ū = 0, 1
of the extended physical state condition (1.1) could remove the doubled set
of negative-norm states (time-like modes) of the ĉ = 52 strings – which
are also associated with ū = 0, 1. For the simplest ĉ = 52 strings in Eq.
(5.1), this intuition is certainly born out [20]. More generally, the equivalent
c = 26 description of each spectrum shows that both aspects of the doubling
are indeed eliminated at the same time, leaving us with the conventional
physical state condition (4.6) and only a single set of time-like modes. This
is clearly visible in the set of examples (4.9), where the only time-like modes
(a = 0) are included in the second term. For the general ĉ = 52 string,
the reader should bear in mind that the twisted metric G in Eq. (4.2) is
only a unitary transformation (2.5) of the untwisted metric G with a single
time-like direction. Although not yet a proof, and illustrated here only for
ĉ = 52, I consider this a stronger form of the original arguments [16] that
all the critical orbifolds of permutation-type should be free of negative-norm
states.
The next question I wish to address is the following: I have empha-
sized that the equivalent c = 26 Virasoro generators (4.2) are generically-
unconventional, being summed over the matter-field fractions {2n/ρ} instead
of the conventional orbifold fractions {n/ρ}, but are they actually new Vira-
soro generators? I do not know the answer to this question in general, but
at least some of them can in fact be re-expressed by further mode-relabeling
in terms of more familiar Virasoro generators. As examples, consider the
special case of the “larger subset” (4.9) when ω(perm) is one of the elements
of order λ of each cyclic group Zλ. (These are the particular, single-cycle
elements of Zλ with f0(σ) = λ.) When λ is odd, one finds that the first
and third terms of (4.9) can in fact be re-expressed in terms of the conven-
tional Virasoro generators associated to a twisted sector of an ordinary cyclic
permutation orbifold U(1)λ/Zλ [christ]
Lλ(M) =
Q+ ̂
M −Q− ̂
:M (5.2)
+δM,0
, c = λ = 2l + 1
where I have relabeled the currents J̂ ≡ J̂0. To obtain this result from (4.9),
one needs the fact that {2̂/λ} ≃ {̂/λ} modulo the integers when λ is odd.
This observation is consistent with the ground-state mass-squares for prime
λ in Eq. (3.19). When λ is even, I have also checked that the first and third
terms of (4.9) can be re-expressed as the sum of two identical commuting
Virasoro generators of this type
Lλ(M) = Lλ
(M) + L̃λ
(M), c = λ = 2l (5.3)
each of which is associated to a twisted sector of U(1)λ/2/Zλ/2. This result
is also obtained by relabeling the modes modulo the integers, and provides
us with another way to understand that the ground-state mass-squared is
unshifted when ω(perm) is the non-trivial element of a Z2.
My final remark is a conjecture, that the extended physical-state condi-
tions for the twisted strings at ĉ = 26λ, λ prime will in fact read
m+ ̂
|χ〉 = 0, ̂ = 0, 1, . . . , λ− 1 (5.4a)
âλ ≡
13λ2 − 1
(5.4b)
m+ ̂
n+ l̂
m− n+ ̂−l̂
̂+l̂
m+ n + ̂+l̂
+ (5.4c)
+ 26λ
m+ ̂
m+ ̂
̂+l̂
where Eq. (5.4c) is an orbifold Virasoro algebra [1,18,9] of order λ. This form
includes the correct generators {L̂̂} corresponding to the classical extended
Polyakov constraints of Ref. [16], and includes the correct value â2 = 17/8
studied here for the ĉ = 52 strings. I obtained the system (5.4) by requiring
(as we now know for λ = 2) that it map by the inverse of the order-λ
orbifold-induction procedure [1] to the conventional physical-state condition
(4.6) with â1 = 1 at c = 26. One way to test this conjecture would be the
construction of the corresponding twisted BRST systems [17] for these higher
values of ĉ.
Extensions to include winding number and twisted B fields at ĉ = 52 are
also deferred to another time and place.
Acknowledgements
For helpful information, discussions and encouragement, I thank L. Alvarez-
Gaumé, K. Bardakci, I. Brunner, J. de Boer, D. Fairlie, O. Ganor, E. Gi-
mon, C. Helfgott, E. Kiritsis, R. Littlejohn, S. Mandelstam, J. McGreevy, N.
Obers, A. Petkou, E. Rabinovici, V. Schomerus, K. Schoutens, C. Schweigert
and E. Witten. This work was supported in part by the Director, Office of
Energy Research, Office of High Energy and Nuclear Physics, Division of
High Energy Physics of the U.S. Department of Energy under Contract DE-
AC02-O5CH11231 and in part by the National Science Foundation under
grant PHY00-98840.
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	Introduction
	The Twisted Virasoro Generators   of =52 Strings
	First Discussion of    the  = 52 String Spectra
	Equivalent c = 26 Description    of the  = 52 Spectra
	Conclusions

  In the second paper of this series, I obtained the twisted BRST systems and
extended physical-state conditions of all twisted open and closed $\hat{c} =
52$ strings. In this paper, I supplement the extended physical-state conditions
with the explicit form of the extended (twisted) Virasoro generators of all
$\hat{c} = 52$ strings, which allows us to discuss the physical spectra of
these systems. Surprisingly, all the $\hat{c}=52$ spectra admit an equivalent
description in terms of generically-unconventional Virasoro generators at
$c=26$. This description strongly supports our prior conjecture that the
$\hat{c}=52$ strings are free of negative-norm states, and moreover shows that
the spectra of some of the simpler cases are equivalent to those of ordinary
untwisted open and closed $c=26$ strings.

Introduction
2. The Extended Virasoro Generators of ĉ = 52 Strings
3. First Discussion of the ĉ = 52 String Spectra
4. Equivalent c = 26 Description
of the ĉ = 52 Spectra
5. Conclusions
1 Introduction
Opening another chapter in the orbifold program [1-11,12-15], this is the third
in a series of papers which considers the critical orbifolds of permutation-
type as candidates for new physical string systems at higher central charge.
In the first paper [16] of this series, we found that the twisted sectors of
these orbifolds are governed by new, extended (permutation-twisted) world-
sheet gravities – which indicate that the free-bosonic orbifold-string systems
of permutation-type can be free of negative-norm states at critical central
charge ĉ = 26K. Correspondingly-extended world-sheet permutation super-
gravities are expected in the twisted sectors of the superstring orbifolds of
permutation-type, where superconformal matter lives at higher multiples of
critical superstring central charges.
In the second paper [17] of the series, we found the corresponding twisted
BRST systems for all sectors of the free-bosonic orbifolds which couple to the
simple case of Z2-twisted permutation gravity, i.e. for all the twisted strings
with ĉ = 52 matter. The new BRST systems also implied the following
extended physical-state conditions for the physical states {|χ〉} of each of the
ĉ = 52 strings:
|χ〉 = 0, m ∈ Z, u = 0, 1 (1.1a)
, L̂v
m− n+ u−v
L̂u+v
m+ n + u+v
(1.1b)
δm+n+u+v
The algebra in Eq. (1.1b) is called an order-two orbifold Virasoro algebra (or
extended, twisted Virasoro algebra) and general orbifold Virasoro algebras
[1,18,9,12,16,17] are known to govern all the twisted sectors of the orbifolds
of permutation-type at higher multiples of c = 26.
The set of all ĉ = 52 orbifold-strings is a very large class of fractional-
moded free-bosonic string systems, including e.g. the twisted open-string
sectors of the orientation orbifolds, the twisted closed-string sectors of the
generalized Z2-permutation orbifolds and many others (see Refs. [16,17] and
Sec. 2). Starting from the extended physical-state conditions (1.1) (and
a right-mover copy of (1.1) on the same {|χ〉}for the twisted closed-string
sectors) this paper begins the concrete study of the physical spectrum of
each ĉ = 52 string.
As the prerequisite for this analysis, I first provide in Sec. 2 the explicit
form – in terms of twisted matter fields – of the extended Virasoro generators
, u = 0, 1
of all ĉ = 52 strings. This construction allows us
to begin the study of the general ĉ = 52 string spectra in Sec. 3. The
same subject is further considered in Sec. 4, where I point out that all
the ĉ = 52 spectra admit an equivalent description in terms of generically-
unconventional Virasoro generators at c = 26. This description allows us to
see clearly a number of spectral regularities which are only glimpsed in Sec. 3,
including strong further evidence that the critical orbifolds of permutation-
type can be free of negative-norm states. Moreover, although the generic
ĉ = 52 spectrum is apparently new, we are able to show that some of the
simpler spectra are equivalent to those of ordinary untwisted open and closed
critical strings at c = 26.
Based on these results, the discussion in Sec. 5 raises some interesting
questions about these theories at the interacting level, and speculates on
the form of the extended physical-state conditions for more general orbifold-
strings of permutation-type. I will return to both of these subjects in suc-
ceeding papers of the series.
2 The Twisted Virasoro Generators
of ĉ = 52 Strings
As emphasized in Ref. [17], the universal form of the twisted BRST systems
and the extended physical-state conditions (1.1) are consequences of their
origin in Z2-twisted permutation gravity, which governs all twisted ĉ = 52
matter.
There are however many distinct ĉ = 52 strings, including the twisted
open-string sectors of the orientation orbifolds [12,13,15-17]
U(1)26
, H− = Z2(w.s.)×H (2.1)
and the twisted closed-string sectors of the generalized Z2-permutation orb-
ifolds [15-17]
U(1)26 × U(1)26
, H+ = Z2(perm)×H
′ (2.2)
as well as the generalized open-string Z2-permutation orbifolds and their T -
duals [15-17]. For the orientation orbifolds in Eq. (2.1), I remind that H− is
any automorphism group of the untwisted closed string U(1)26 which includes
world-sheet orientation-reversing automorphisms. Indeed the twisted open-
string orientation-orbifold sectors correspond to the orientation-reversing au-
tomorphisms, which have the form τ− × ω, ω ∈ H where the basic automor-
phism τ− exchanges the left- and right-movers of the closed string and ω is
an extra automorphism which acts uniformly on the left- and right-movers
of the closed string. Similarly, the automorphism group H+ of the general-
ized Z2-permutation orbifolds in (2.2) is generated by elements of the form
τ+×ω, ω ∈ H
′, where the basic automorphism τ+ exchanges the two copies
of the closed string and the extra automorphism ω again acts uniformly on
the left- and right-movers of each closed string. In both cases, the extra
automorphisms ω in τ × ω may or may not form a group (see the examples
at the end of this section).
The spectra of different ĉ = 52 strings are characterized by their extended
(twisted) Virasoro generators, all of which can in fact be written in the
following unified form:
Gn(r)µ;−n(r)ν(σ)
× (2.3a)
×:Ĵn(r)µv
Ĵ−n(r),ν,u−v
m− p−
+ u−v
+δm+u
,0 ∆̂0(σ)
Ĵn(r)µu
, Ĵn(s)νv
= (2.3b)
δn(r)+n(s),0modρ(σ)δm+n+n(r)+n(s)
Gn(r),µ;−n(r),ν(σ)
, Ĵn(r)µv
(2.3c)
Ĵn(r)µ,u+v
m+ n+
+ u+v
∆̂0(σ) =
dim[n(r)]
(2.3d)
dim[n(r)] = 26. (2.3e)
Each set of extended Virasoro generators in Eq. (2.3a) satisfies the order-two
orbifold Virasoro algebra (1.1b) at ĉ = 52, and the current algebras in Eq.
(2.3b) are of the type called doubly-twisted in the orbifold program.
For those unfamiliar with the program, I first give a short summary of the
standard notation in the result (2.3) – followed by the derivation of the result.
As in the extended Virasoro generators themselves, the indices u, v with
fundamental range ū, v̄ ∈ {0, 1} describe the twist of the basic permutations
τ∓ in each H∓. For each extra automorphism ω(σ) in each τ∓ × ω(σ), the
spectral indices {n(r)} and the degeneracy indices {µ ≡ µ(n(r))} of each
twisted sector σ are determined by the so-called H-eigenvalue problem [3,5,6]
of ω(σ)
ω(σ)a
U †(σ)b
n(r)µ
= U †(σ)a
n(r)µ
−2πin(r)
ρ(σ) , ω(σ) ∈ H or H ′ (2.4a)
ω(σ)a
ω(σ)b
Gcd = Gab, Gab = G
(2.4b)
a, b = 0, 1, . . . , 25, n(r) ∈ (0, 1, . . . , ρ(σ)− 1) (2.4c)
where G is the untwisted target-space metric of U(1)26. The quantity ρ(σ)
is the order of ω(σ) and all indices {n(r)µ} are periodic modulo ρ(σ), with
{n(r)} the pullback to the fundamental region and dim[n(r)] the size of the
subspace n(r). The index r is summed once over the fundamental region in
Eqs. (2.3a), (2.3d) and (2.3e). The twisted metric G.(σ) and its inverse G
are defined in terms of the unitary eigenvectors U(σ) of the H-eigenvalue
problem
Gn(r)µ;n(s)ν(σ) = χn(r)µχn(s)νU(σ)n(r)µ
U(σ)n(s)ν
Gab (2.5a)
= δn(r)+n(s),0modρ(σ)Gn(r)µ;−n(r),ν(σ) (2.5b)
Gn(r)µ;n(s)ν(σ) = χ−1
n(r)µ
n(s)ν
GabU †(σ)a
n(r)µ
U †(σ)b
n(s)ν
(2.5c)
= δn(r)+n(s),0modρ(σ)G
n(r),µ;−n(r),ν (2.5d)
where G is again the untwisted metric and the χ’s are essentially-arbitrary
normalization constants. Finally, the standard mode normal-ordering in Eq.
(2.3a) is:
:Ĵn(r)µu
Ĵn(s)νv
:M (2.6)
Ĵn(s)νv
Ĵn(r)µu
Ĵn(r)µu
Ĵn(s)νv
It follows that the quantity ∆0(σ) in Eqs. (2.3a) and (2.3d)
Ĵn(r)µu
|0〉σ = 0 (2.7a)
→ L̂u
|0〉σ = ∆̂0(σ) δm+u
,0|0〉σ (2.7b)
is the conformal weight of the scalar twist-field state |0〉σ of sector σ.
I comment briefly on the derivation of the unified form (2.3) of the
ĉ = 52 extended Virasoro generators. Essentially this result was given for
the twisted open-string sectors of the non-abelian orientation orbifolds in
Subsecs. 3.4, 3.5 of Ref. [12], and that result is easily reduced for our
abelian case U(1)26/H− in Eq. (2.1). With a right-mover copy of the ex-
tended Virasoro generators (and u → ̂ = 0, 1), the result also hold for
the twisted closed-string sectors of the generalized Z2-permutation orbifolds
(U(1)26 × U(1)26)/H+ in Eq. (2.2). This follows by the substitution
G → G, u
(2.8)
into the known results for the ordinary Z2-permutation orbifolds with trivial
H ′ (see Ref. [perm] and Subsec. 4.2 of Ref. [16]). Finally, a single copy of
the unified form (2.3) holds as well for each twisted sector of the generalized
open-string Z2-permutation orbifolds (U(1)
26 × U(1)26)open/H+ and all pos-
sible T -dualizations of each of these sectors. This conclusion follows because
the left-mover extended Virasoro generators of the closed-string orbifolds for
each H+ are the input data for the construction of the correponding open-
string orbifolds [14], and the twisted-current form of each set of extended
Virasoro generators is independent of T-dualization [15]. The branes, quasi-
canonical algebra and non-commutative geometry of the twisted open strings
[13-15,16,17] depend of course on the particular T-dualization, but these will
not be needed here.
In what follows I will consider each twisted ĉ = 52 string separately, but
the reader may find it helpful to bear in mind the complete sector structure of
these orbifold-string systems as labelled by the elements of the automorphism
groups H∓. Given a particular extra automorphism ωn ∈ H or H
′ of order
n, one may list the following low-order examples:
(1; τ∓) (2.9a)
(1; τ∓ × ω2) (2.9b)
(1, ω3, ω
3; τ∓, τ∓ × ω3, τ∓ × ω
3) (2.9c)
(1, ω24; τ∓ × ω4, τ∓ × ω
4) (2.9d)
(1, ω26, ω
6; τ∓ × ω6, τ∓ × ω
6, τ∓ × ω
6). (2.9e)
For the generalized Z2-permutation orbifolds (τ+) all of these sectors are
twisted closed strings at ĉ = 52, while all the sectors of the generalized open-
string Z2-permutation orbifolds (τ+) and their T-dualizations are twisted
open strings at ĉ = 52. For the orientation orbifolds (τ−) the sectors before
the semicolon are twisted closed strings at c = 26 (which form an ordinary
space-time orbifold) while the sectors after the semicolon are twisted open
strings at ĉ = 52. More generally, orientation orbifolds always contain an
equal number of twisted open and closed strings. In all cases, the twisting is
of course trivial for sectors corresponding to the unit element.
3 First Discussion of
the ĉ = 52 String Spectra
To frame this discussion, I remind [1] the reader that the Virasoro primary
states of our orbifold CFT’s are defined by the integral Virasoro subalgebra
(generated by {L̂0(m)}) of the extended Virasoro algebra. Then the extended
physical-state conditions (1.1a) tell us that all the physical states {|χ〉} of
each ĉ = 52 orbifold-string are Virasoro primary
L̂0(m > 0)|χ〉 = 0 (3.1)
but only a small subset of these primary states are selected by the rest of the
physical-state conditions:
L̂0(0)−
|χ〉 = L̂1
|χ〉 = 0. (3.2)
In what follows, I will refer to the L̂0(0) condition in Eq. (3.2) as the spectral
condition, since it will determine the allowed values of momentum-squared
for each ĉ = 52 string.
The space of physical states of each orbifold-string is then much smaller
than the space of states of the underlying orbifold conformal field theory. For
the experts, I remark in particular that the extended physical-state condi-
tions generically disallow the characteristic sequence [19] of Virasoro primary
states known as the principle-primary states [1,9]. This follows first by the
spectral condition (which fixes the conformal weight), and second because
the physical-state condition {L̂u
≃ 0} is stronger than the
principle-primary state condition [1,9]
|p.p.s.〉 = 0, u = 0, 1, m > 0 (3.3)
which does not extend to m = 0.
I turn now to concretize the spectral condition of each twisted ĉ = 52
string, using the explicit form (2.3) of its extended Virasoro generators. For
this, recall [12,15] first that these generators contain in general two kinds
of commuting zero modes (dimensionless momenta), namely {Ĵ0µ0(0)} and
{Ĵρ(σ)/2,µ,1(0)}, where the latter is relevant only when the order ρ(σ) of ω(σ)
is even. In what follows, I often refer to these zero modes collectively as
{Ĵ(0)}. It is then natural to define the “momentum-squared” operator P̂ 2
as follows:
L̂0(0) =
−P̂ 2 + R̂(σ)
+ ∆̂0(σ) (3.4a)
P̂ 2 ≡ −
G0µ;0ν(σ)Ĵ0µ0(0)Ĵ0ν0(0) + (3.4b)
,µ;− ρ(σ)
,ν(σ)Ĵρ(σ)/2,µ,1(0)Ĵ−ρ(σ)/2,ν,−1(0)
R̂(σ) ≡
r,µ,ν
Gn(r)µ;−n(r),ν(σ)× (3.4c)
×:Ĵn(r)µu
Ĵ−n(r),ν,−u
Here the primed sum in the “level-number” operator R̂(σ) indicates omission
of the zero modes.
With this decomposition, the spectral condition in Eq. (3.2) takes the
simple form:
P̂ 2|χ〉 =
P̂ 2(0) + R̂(σ)
|χ〉 (3.5a)
P̂ 2(0) ≡ 2
δ̂0(σ)− 1
(3.5b)
δ̂0(σ) =
dim[n(r)]
≥ 0. (3.5c)
Although I will continue the discussion primarily in this form, in fact Eqs.
(3.4a) and (3.5a) hold only for the twisted open-string sectors of the orbifolds.
For the twisted closed-string sectors, we also have right-mover copies of the
extended Virasoro generators (2.3), and a corresponding right-mover copy of
the extended physical-state conditions (1.1) on the same {|χ〉}. For simplicity
I will limit the discussion of these sectors here to the case of decompactified
zero modes, for which it is appropriate to equate the left and right movers
ĴR(0) = ĴL(0) = 1√
Ĵ(0) → R̂R(σ) = R̂L(σ) (3.6)
where the last equality is level-matching in each twisted sector. Keeping the
same definition of the operator P̂ 2 in Eq. (3.4b), the correct closed-string
ĉ = 52 spectral condition is then obtained by the substitution
P̂ 2 → 1
P̂ 2 (3.7)
in both Eqs. (3.4a) and (3.5a). These identifications, and hence P̂ 2
→ 2P̂ 2
can be used at any point in the discussion below to obtain the corresponding
closed-string results.
Returning to the open-string case, one simple solution of the extended
physical-state conditions is the ground state |0, Ĵ(0)〉σ of twisted sector σ:
R̂(σ)|0, Ĵ(0)〉σ = L̂u
|0, Ĵ(σ)〉σ = 0 (3.8a)
P̂ 2|0, Ĵ(0)〉σ = P̂
(0)|0, Ĵ(0)〉σ, P̂
(0) = −2 + 2δ̂0(σ) (3.8b)
This is the “momentum-boosted” twist-field state (see Eq. (2.7)) of that
sector, with ground-state mass-squared P̂ 2
. Moreover Eq. (3.4c) and the
commutator (2.3c) give the increments
∆(P̂ 2) = ∆(R̂(σ)) = 4
(3.9)
obtained by adding the negatively-moded current
Ĵn(r)µu
to any previous state. The precise content of these excited levels must of
course be determined from the remainder of the extended physical-state con-
ditions.
I continue this discussion with some specific examples of ĉ = 52 strings,
beginning with the simplest twisted open-string orientation-orbifold sectors
[12,13,15,16,17]:
ω = 1l : ρ = 1, n = 0, U = 1l, G = G, Ĵ0au
(3.10a)
∆(P̂ 2) = 4
∣m+ u
∣ (u = 0 is DD, u = 0 is ND) (3.10b)
ω = −1l : ρ = 2, n = 1, U = 1l, G = G, Ĵ1au
m+ u+1
(3.11a)
∆(P̂ 2) = 4
∣m+ u+1
∣ (u = 0 is DN, u = 0 is NN). (3.11b)
In these cases, the extra automorphisms ω act uniformly on the labels a =
0, . . . 25 and G is the untwisted target space metric in Eq. (2.4b). Although
both twisted strings have (26+26) = 52 matter degrees of freedom, note that
each example has only one of the two types of zero modes {Ĵ(0)}: 26DD zero
modes {Ĵ0a0(0)} for ω = 1l and 26NN zero modes {Ĵρ/2,a,1(0)} for ω = −1l.
In both cases, the momentum-squared (3.4b) has the schematic form
P̂ 2 = ηabĴa(0)Ĵb(0), η =
0 −1l
(3.12)
where η = −G is the standard (west-coast) 26-dimensional target-space met-
ric. Then we compute from Eqs. (3.5b) and (3.5c) that both strings share
the same tachyonic ground-state mass-squared
δ̂0(σ) = 0, ∆̂0(σ) =
, P̂ 2(0) = −2 (3.13)
and the first excited state of each is massless:
Ĵ0a1
Ĵ1a0
|0, Ĵ(0)〉σ = 0 for ω =
(3.14)
For this level, I have checked that the L̂1
≃ 0 gauge eliminates the
longitudinal parts of the 26-dimensional “photons”, and moreover the L̂1
and L̂0 (1) gauges together eliminate the negative-norm states at the next
level:
+ βĴ(−1)
|0, Ĵ(0)〉σ, P̂
2 = 2. (3.15)
Since the increments ∆(P 2) in Eqs. (3.10b) and (3.11b) are even integers,
we are led to suspect that the spectra of these two twisted ĉ = 52 strings are
nothing but the spectrum of an ordinary open c = 26 string in disguise 1. I
will return to this question in the following section.
A larger subset of twisted ĉ = 52 strings is the following. For a particular
twisted sector σ, suppose that ω = ±1l acts uniformly on a set of d labels
a = 0, 1, . . . d − 1, d ≥ 4 while a non-trivial element ω(perm) of some per-
mutation group acts non-trivially on the other 26 − d spatial labels. Then
Eqs. (2.4),(2.5) and standard results [3,5-7,9] in the orbifold program give
the following explicit form of the extended Virasoro generators (2.3) in this
sector:
∆0(σ) + (3.16a)
Gab(d)
:Ĵǫav
p+ v+ǫ
Ĵ−ǫ,b,u−v
m− p + u−v−ǫ
fj(σ)
fj(σ)−1
×:Ĵ̂jv
p+ ̂
fj(σ)
Ĵ−̂,j,u−v
m− p− ̂
fj(σ)
+ u−v
δ̂0(σ) =
fj(σ)−1
fj(σ)
fj(σ)
fj(σ)
≥ 0 (3.16b)
fj(σ) = 26− d, 4 ≤ d ≤ 26. (3.16c)
Here ǫ = 0 or 1 for ω = 1l or −1l, G(d) is the restriction of the flat target-space
metric (2.4b) to the first d labels, fj(σ) is the size of the jth cycle in ω(perm),
1The spectra of these two ĉ = 52 strings look even more familiar in terms of the
dimensionful momenta k ≡ Ĵ(0)/
, where α′
is the conventional open-string Regge
slope.
and the previous cases with δ̂0(σ) = 0 are included when d = 26. The half-
integer moded currents in the second term of (3.16) satisfy the twisted current
algebra (2.3b) with G → G(d). For the permutation-twisted currents in the
last term of (3.16), I have used the standard relation (n(r)/ρ(σ)) = (̂/fj(σ))
and (the inverse of) the twisted metric [3,5-7,9]
̂j;l̂l
(σ) = δjlfj(σ)δ̂+l̂,0modfj(σ) (3.17)
which also determines the twisted current algebra (2.3b) for these currents.
Using Eq. (3.16b), we see that the non-trivial element of Z2 on two labels
also gives δ̂0(σ) = 0 and a P̂
= −2 ground state, but a non-trivial element
of Z3 on three labels gives a slightly-raised ground state
δ̂0(σ) =
, ∆̂0(σ) =
, P̂ 2(0) = −
(3.18)
and no photons.
Given the cycle-structure {fj(σ)} of any extra automorphism w(perm)
(see e.g. Eq. (3.4) of Ref. [16]), it is straightforward to evaluate the sum in
Eq. (3.16b). As an illustration, one finds the simple tachyonic ground-state
mass-squares
5 ≤ (d = prime) ≤ 23 : P̂ 2(0) = −
(d− 2 +
26− d
) (3.19)
in twisted sectors which correspond to the action of any non-trivial element
of the cyclic group Zλ of prime order on 3 ≤ (λ = 26−d) ≤ 21 spatial labels.
The result (3.19) includes Eq. (3.18) when d = 23, but does not extend to
the cases d = 26, 24 with P̂ 2(0) = −2 discussed above. I remind that this
result applies only to the open orbifold-strings, while twice these values of
P̂ 2(0) are obtained for the closed-string versions.
Further analysis of the ĉ = 52 strings, including the “larger subset” of
examples (3.16), is found in the following section.
4 Equivalent c = 26 Description
of the ĉ = 52 Spectra
In fact, there exists an entirely equivalent description of all the ĉ = 52 string
spectra in terms of generically-unconventional Virasoro generators at c = 26.
To obtain the c = 26 description, I first define the relabelled (unhatted)
operators
Jn(r)µ
2m+ u+
2n(r)
≡ Ĵn(r)µu
, u = 0, 1 (4.1a)
L(2m+ u) ≡ 2L̂u
(4.1b)
in terms of the hatted operators above. This 1−1 map is recognized as
a modest generalization of (the inverse of) the order-two orbifold-induction
procedure of Borisov, Halpern and Schweigert [1]. SinceM ≡ 2m+u, u = 0, 1
covers the integers once, we then find from (2.3) the explicit form of the c = 26
generators:
L(M) = δ̂0(σ)δM,0 + (4.2a)
r,µ,ν
Gn(r)µ;−n(r),ν(σ)
:Jn(r)µ
2n(r)
J−n(r),ν
M −Q−
2n(r)
δ̂0(σ) =
dim[n(r)]
(4.2b)
[L(M), L(N)] = (M −N)L(M +N) + 26
M(M2 − 1)δM+N,0 (4.2c)
L(M), Jn(r)µ
2n(r)
2n(r)
Jn(r)µ
M +N +
2n(r)
(4.2d)
Jn(r)µ
2n(r)
, Jn(s)ν
2n(s)
(4.2e)
= δn(r)+n(s),0modρ(σ)δM+N+2(n(r)+n(s)ρ(σ) ),0
Gn(r)µ;−n(r),ν(σ).
The expression (4.2b) for δ̂0(σ) is the same as above, and the mode-normal
ordering in Eq. (4.2a)
:Jn(r)µ
2n(r)
Jn(s)ν
2n(s)
:M (4.3)
2n(r)
Jn(s)ν
2n(s)
Jn(r)µ
2n(r)
2n(r)
Jn(r)µ
2n(r)
Jn(s)ν
2n(s)
follows from the ĉ = 52 ordering (2.6) because the map (4.1) preserves the
sign of all arguments.
I emphasize that the c = 26 Virasoro generators in Eq. (4.2) are generically-
unconventional because the twisted matter is now summed over the fractions
{2n/ρ} instead of the conventional orbifold-fractions {n/ρ}. This distortion
of the “extra twist” is the price we must pay in order to unwind the “basic
twist” associated to the basic permutations τ∓ of H∓.
The map (4.1) also tells us that the ĉ = 52 momenta {Ĵ(0)} and the
c = 26 momenta {J(0)} are identical, and we may record
J(0) = Ĵ(0) : J0µ(0) = Ĵ0µ0(0), Jρ(σ)/2,µ(0) = Ĵρ(σ)/2,µ,1(0) (4.4a)
P 2 = P̂ 2 (4.4b)
G0µ:0ν(σ)J0µ(0)J0ν(0) +
,µ;− ρ(σ)
,ν(σ)Jρ(σ)/2,µ(0)J−ρ(σ)/2,ν(0)
where the ĉ = 52 form of P̂ 2 was given in Eq. (3.4b). Similarly, the “level-
number” operator R(σ) in the decomposition of L(0) is the same
L(0) = −1
P 2 +R(σ)
+ δ̂0(σ) (4.5a)
R(σ) = R̂(σ) (4.5b)
r,µ,ν
Gn(r)µ;−n(r),ν(σ)×
×:Jn(r)µ
2n(r)
J−n(r),ν
2n(r)
where the ĉ = 52 form of R̂(σ) was given in Eq. (3.4c).
By itself, the inverse orbifold-induction procedure (4.1) is only a rela-
belling of the operators of the permutation-orbifold CFT’s. The central point
of this discussion however is that for the orbifold-string theories – restricted
by the extended physical state conditions (1.1) – the map also gives us a
completely equivalent c = 26 description of the physical spectrum of each
ĉ = 52 orbifold-string. Indeed, it is easily checked that both components
ū = 0, 1 of the ĉ = 52 extended physical-state condition (1.1a) map directly
onto the simpler and in fact conventional physical-state condition
L(M ≥ 0)|χ〉 = δM,0|χ〉 (4.6)
in the 26-dimensional description! A right- mover copy of Eq. (4.6) on the
same physical states {|χ〉} is similarly obtained in the equivalent c = 26
description of the closed orbifold-strings.
I emphasize that the physical states {|χ〉} of the 26-dimensional descrip-
tion (4.6) are exactly the original physical states (1.1a) of the ĉ = 52 string.
Indeed, each physical state |χ〉 can be regarded as invariant under the map,
or each can now be rewritten in 26-dimensional form. In further detail, Eqs.
(4.5) and (4.6) give the same spectral condition P 2 ≃ P 20 + R(σ), the same
physical ground state 2
|0, J(0)〉σ ≡ |0, Ĵ(0)〉σ, P
0 = P̂
0 = −2 + 2δ̂0(σ) (4.7)
and each negatively-moded hatted current in any physical state can be re-
placed according to Eq. (4.1a) by the corresponding unhatted current mode.
Note finally that the commutator (4.2d) and the decomposition (4.5a) give
the 26-dimensional increment
∆(P̂ 2) = ∆(R(σ)) = 2
2n(r)
(4.8)
which results from the addition of Jn(r)µ
2n(r)
to any previous
state. With M = 2m+ n, these are recognized as the same increments (3.9)
obtained in the ĉ = 52 description.
As simple examples, consider the “larger subset” (3.16) of ĉ = 52 strings
– whose equivalent c = 26 physical state condition (4.6) now involves the
following subset of the c = 26 Virasoro generators (4.2):
L(M) = δM,0δ̂0(σ) +
Gab(d)
:Jǫa(Q + ǫ)J−ǫ,b(M −Q− ǫ):M +
fj(σ)
fj(σ)−1
:J̂j
Q + 2̂
fj(σ)
J−̂,j
M −Q− 2̂
fj(σ)
:M (4.9a)
δ̂0(σ) =
fj(σ)−1
fj(σ)
fj(σ)
− 2̂
fj(σ)
(4.9b)
2Although it is not directly relevant in either description of the ĉ = 52 strings, one
notes that the conformal weight of the scalar twist-field state |0〉
of sector σ has now
shifted from ∆̂0(σ) to δ̂0(σ) in the c = 26 description.
a, b = 0, . . . , d− 1,
fj(σ) = 26− d, 4 ≤ d ≤ 26. (4.9c)
Recall for the larger subset that ǫ = 0, 1 corresponds in the symmetric theory
to the action of the extra automorphism ω = ±1l on the first d ≥ 4 labels
{a}, while fj(σ) is the length of the j-th cycle of the extra permutation
ω(perm) which acts on the remaining 26 − d spatial labels. Shifting the
dummy integer Q by the integer ǫ, we note that the second term in Eq.
(4.9a) is a set of ordinary Virasoro generators for d untwisted bosons with
the ordinary current algebra
[Ja(Q), Jb(P )] = G
ab QδQ+P,0 (4.10)
for both values of ǫ. The currents in the third term satisfy the twisted current
algebra (4.2e) with the permutation-twisted metric (3.17), and the value of
δ̂0(σ) in Eq. (4.9b) is only a slightly-rewritten form of that given in Eq.
(3.16b).
We are now in a position to confirm our suspicions in the previous sec-
tion about the simplest orbifold-strings, described earlier at ĉ = 52 by the
extended Virasoro generators:
:Ĵǫav
p+ v+ǫ
Ĵ−ǫ,b,u−v
m− p+ u−v−ǫ
,0, u = 0, 1, ǫ = 0, 1. (4.11)
These are now equivalently described by the choice d = 26 in Eq. (4.9), in
which case only the second (ordinary) term of Eq. (4.9a) is non-zero – and
then the equivalent physical-state condition (4.6) verifies that the physical
spectrum of each of these particular twisted ĉ = 52 strings is indeed equiva-
lent to that of an ordinary untwisted c = 26 string! These cases include the
open-string orientation-orbifold sectors corresponding to τ− × (ω = ±1l) in
Eq. (3.10) and their T-duals, as well as the twisted closed-string sectors of
the generalized Z2-permutation orbifolds corresponding to τ+ × (ω = ±1l).
Additionally, consider the following special cases of the extended Virasoro
generators (3.16) at ĉ = 52
= δm+u
+ (4.12)
Gab(24)
:Ĵǫav
p+ v+ǫ
Ĵ−ǫ,b,u−v
m− p+ u−v−ǫ
:Ĵ̂v
p+ ̂+v
Ĵ−̂,u−v
m− p+ u−v−̂
which result when the extra automorphism in the symmetric theory acts as
ω = ±1l on the first d = 24 labels and the non-trivial element of a Z2 on the
remaining 2 spatial labels. I have noted in Sec. 3 that δ̂0(σ) = 0 for these
cases as well, and indeed the equivalent c = 26 description (4.6) and (4.9)
at d = 24 now shows that the open and closed orbifold- strings of this type
also have the spectrum of ordinary untwisted c = 26 strings. The common
thread for the orbifold-strings in Eqs. (4.10) and (4.12) is that they are at
most half-integer moded, so that the shift {n/ρ} → {2n/ρ} gives integer
moding in the c = 26 description.
Beyond these simple cases, the ĉ = 52 strings are apparently new – with
δ̂0(σ) 6= 0, unfamiliar ground-state mass-squares, and fractional modeing
(and increments) in either description.
5 Conclusions
We have discussed the physical spectrum of the general ĉ = 52 orbifold-string,
as well as an equivalent but unconventionally-twisted c = 26 description
of the twisted ĉ = 52 matter. The equivalent c = 26 description holds
only for the orbifold-string theories – restricted by the extended physical-
state conditions (1.1) – and not in the larger Hilbert space of the underlying
orbifold conformal field theories.
In general we have found that the spectra of these orbifold-string systems
are unfamiliar. One simple and unexpected conclusion however is that, as
string theories restricted by the extended physical-state conditions, the single
twisted ĉ = 52 sector of each of the simplest orbifolds of permutation-type
(see Eq. (2.9))
(1; τ∓) (5.1a)
(1; τ∓ × ω2), ω
2 = 1 (5.1b)
have the same physical spectra as ordinary untwisted c = 26 strings. No such
equivalence is found of course in the half-integer moded Hilbert space of the
full orbifold CFT’s. The list in Eq. (5.1) includes the simplest orientation
orbifolds (with τ−) and their T-duals, as well as the simplest generalized
Z2-permutation orbifolds (with τ+).
For the simplest orientation orbifolds in particular, the string theories in
Eq. (5.1) consist of an ordinary unoriented closed string (the unit element)
at c = 26 and a ĉ = 52 twisted open string whose physical spectrum is
equivalent to that of an ordinary untwisted c = 26 critical open string. Since
both the closed- and open-string spectra of these simple orientation orbifolds
are equivalent to those of the archtypal orientifold (without Chan-Paton fac-
tors), we are led to suspect that orientation orbifolds include orientifolds. I
will return in the next paper of this series to consider this question at the
interacting level, where we will also be able to ask about the decoupling of
null physical states. Following that, I will consider in a succeeding paper the
corresponding situation and modular invariance for the simplest permutation
orbifold-string systems.
More generally, we have seen that there are many other orientation orb-
ifolds, open-string Z2-permutation orbifolds and generalized Z2-permutation
orbifolds whose ĉ = 52 spectra show fractional modeing in both the ĉ = 52
and the c = 26 descriptions. These include in particular the orbifolds in Eq.
(2.9) when the order n of the extra automorphism is greater than two.
There is more to say about no-ghost theorems for the general twisted
ĉ = 52 string. The original intuition [16] was that the doubled gauges ū = 0, 1
of the extended physical state condition (1.1) could remove the doubled set
of negative-norm states (time-like modes) of the ĉ = 52 strings – which
are also associated with ū = 0, 1. For the simplest ĉ = 52 strings in Eq.
(5.1), this intuition is certainly born out [20]. More generally, the equivalent
c = 26 description of each spectrum shows that both aspects of the doubling
are indeed eliminated at the same time, leaving us with the conventional
physical state condition (4.6) and only a single set of time-like modes. This
is clearly visible in the set of examples (4.9), where the only time-like modes
(a = 0) are included in the second term. For the general ĉ = 52 string,
the reader should bear in mind that the twisted metric G in Eq. (4.2) is
only a unitary transformation (2.5) of the untwisted metric G with a single
time-like direction. Although not yet a proof, and illustrated here only for
ĉ = 52, I consider this a stronger form of the original arguments [16] that
all the critical orbifolds of permutation-type should be free of negative-norm
states.
The next question I wish to address is the following: I have empha-
sized that the equivalent c = 26 Virasoro generators (4.2) are generically-
unconventional, being summed over the matter-field fractions {2n/ρ} instead
of the conventional orbifold fractions {n/ρ}, but are they actually new Vira-
soro generators? I do not know the answer to this question in general, but
at least some of them can in fact be re-expressed by further mode-relabeling
in terms of more familiar Virasoro generators. As examples, consider the
special case of the “larger subset” (4.9) when ω(perm) is one of the elements
of order λ of each cyclic group Zλ. (These are the particular, single-cycle
elements of Zλ with f0(σ) = λ.) When λ is odd, one finds that the first
and third terms of (4.9) can in fact be re-expressed in terms of the conven-
tional Virasoro generators associated to a twisted sector of an ordinary cyclic
permutation orbifold U(1)λ/Zλ [christ]
Lλ(M) =
Q+ ̂
M −Q− ̂
:M (5.2)
+δM,0
, c = λ = 2l + 1
where I have relabeled the currents J̂ ≡ J̂0. To obtain this result from (4.9),
one needs the fact that {2̂/λ} ≃ {̂/λ} modulo the integers when λ is odd.
This observation is consistent with the ground-state mass-squares for prime
λ in Eq. (3.19). When λ is even, I have also checked that the first and third
terms of (4.9) can be re-expressed as the sum of two identical commuting
Virasoro generators of this type
Lλ(M) = Lλ
(M) + L̃λ
(M), c = λ = 2l (5.3)
each of which is associated to a twisted sector of U(1)λ/2/Zλ/2. This result
is also obtained by relabeling the modes modulo the integers, and provides
us with another way to understand that the ground-state mass-squared is
unshifted when ω(perm) is the non-trivial element of a Z2.
My final remark is a conjecture, that the extended physical-state condi-
tions for the twisted strings at ĉ = 26λ, λ prime will in fact read
m+ ̂
|χ〉 = 0, ̂ = 0, 1, . . . , λ− 1 (5.4a)
âλ ≡
13λ2 − 1
(5.4b)
m+ ̂
n+ l̂
m− n+ ̂−l̂
̂+l̂
m+ n + ̂+l̂
+ (5.4c)
+ 26λ
m+ ̂
m+ ̂
̂+l̂
where Eq. (5.4c) is an orbifold Virasoro algebra [1,18,9] of order λ. This form
includes the correct generators {L̂̂} corresponding to the classical extended
Polyakov constraints of Ref. [16], and includes the correct value â2 = 17/8
studied here for the ĉ = 52 strings. I obtained the system (5.4) by requiring
(as we now know for λ = 2) that it map by the inverse of the order-λ
orbifold-induction procedure [1] to the conventional physical-state condition
(4.6) with â1 = 1 at c = 26. One way to test this conjecture would be the
construction of the corresponding twisted BRST systems [17] for these higher
values of ĉ.
Extensions to include winding number and twisted B fields at ĉ = 52 are
also deferred to another time and place.
Acknowledgements
For helpful information, discussions and encouragement, I thank L. Alvarez-
Gaumé, K. Bardakci, I. Brunner, J. de Boer, D. Fairlie, O. Ganor, E. Gi-
mon, C. Helfgott, E. Kiritsis, R. Littlejohn, S. Mandelstam, J. McGreevy, N.
Obers, A. Petkou, E. Rabinovici, V. Schomerus, K. Schoutens, C. Schweigert
and E. Witten. This work was supported in part by the Director, Office of
Energy Research, Office of High Energy and Nuclear Physics, Division of
High Energy Physics of the U.S. Department of Energy under Contract DE-
AC02-O5CH11231 and in part by the National Science Foundation under
grant PHY00-98840.
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	Introduction
	The Twisted Virasoro Generators   of =52 Strings
	First Discussion of    the  = 52 String Spectra
	Equivalent c = 26 Description    of the  = 52 Spectra
	Conclusions
