Non-geometric BPS branes on T-folds

We give a detailed (microscopic) description of the geometric and non-geometric fundamental branes and their bound states in Type II superstring compactifications preserving N=6 supersymmetry. We consider general boundary states that couple to the twisted sector and compute the relevant annulus amplitudes. We check consistency of the construction by relating the transverse channel, corresponding to closed-string tree-level exchange, with the direct open-string loop channel. Focussing on the Type IIA frame, we show that D0-D4 have the expected tension for a geometric brane, while the non-geometric D2-D6 boundary states have a tension equal to $1/\sqrt{K}$ the one of a geometric brane for the $\mathbb{Z}_K$ orbifold. This is consistent with Fricke T-duality of the N=6 model.


Introduction
A large, probably the dominant, part of string and brane configurations, ranging from vacuum configurations to black hole micro-states, may not admit a geometric description [1,2].The simplest possibility is represented by configurations dubbed "T-folds" that require T-duality transformations in order to relate different locally geometric patches.Such vacua can be defined in perturbative string theory as asymmetric orbifolds [3][4][5][6][7].Genuinely non-perturbative constructions, involving S-duality or U-duality transformations, are also conceivable [8][9][10][11] and have attracted some attention recently [12,13].
Geometric branes can be defined on T-folds from the orbifold projection of branes of the parent theory that are invariant under the asymmetric orbifold action.Such branes never couple to the twisted sector of the closed string theory.A first microscopic description of D-branes in a trivial asymmetric orbifold was proposed in [17] for the Z 2 orbifold reflection on T 4 preserving N = 8 supersymmetry.The fundamental 1/2 BPS branes were constructed and couple to the twisted sector.This analysed was extended to Z 2 orbifolds of the bosonic string in [18,19].The construction of these boundary states strongly relies on the enhanced affine Kac-Moody symmetry one has at specific symmetric points of the Narain moduli space.It is desirable to identify which branes exist at generic points in moduli space to understand the non-perturbative dynamics of string theory.
In order to get more control on the expected set of branes, it is useful to consider BPS branes in a supersymmetric theory.The largest is the amount of supersymmetry the more stringent are the consistency constraints.For this reason one can start by focussing on the largest nonmaximal supersymmetry, i.e. 24 supercharges, that results from asymmetric orbifolds of Type II superstrings [4].These backgrounds can be interpreted as a T-folds, with a T 4 fibered over T 2 with transition functions that are T-dualities.
Aim of the present investigation is to give a detailed (microscopic) description of the geometric and non-geometric fundamental BPS branes and their bound states in this class of backgrounds.This line of investigation was initiated in [14,15] and recently revived in [16] in connection with higher derivative terms in the low-energy effective action for Type II superstrings with N = 6 supersymmetry.In particular, in [16], it has been shown that only Z 2 and Z 3 orbifolds give rise to consistent vacuum configurations, thus ruling out Z 4 or higher order abelian groups.
We will derive the relevant annulus amplitudes both in the 'transverse' channel, corresponding to closed-string 'tree-level' exchange between (different) branes, and in the 'direct' channel, corresponding one-loop amplitudes, coding the spectrum of open-string excitations.
To this end we will first briefly review the construction of the consistent N = 6 Type II theories and write the relevant torus partition function, expressed in terms of (super-)characters, in Section 2. For definiteness we work in the Type IIA frame, whereby the relevant branes are bound states of D0, D2, D4 and D6.
Focussing first on the Z 2 case, we identify and describe geometric and non-geometric fundamental branes in Section 3. We consider general boundary states that couple to the twisted sector and compute their annulus partition function.In Section 4, we then show that geometric D2-D4-D6 branes can be viewed as bound-states of fundamental (non-geometric) branes.In Section 5 we pass to consider the Z 3 asymmetric orbifold and describe the geometric D0-D4 brane as well as the non-geometric D2-D6 brane.Our conclusions and an outlook are contained in Section 6.In an Appendix we carefully determine the allowed R-R charges and discuss the issue of rank reduction due to the presence of a discrete (non-dynamical) NS-NS antisymmetric tensor [20,21].

The torus partition function
We consider type II string theory on a vacuum preserving N = 6 supersymmetry.The theory is obtained as an asymmetric orbifold that combines a Z K rotation acting on the left-moving fields along T 4 as [6,7] for i = 6, 7, as well as a shift on the circle S 1 of radius R 5 acting on the bosonic coordinate as This is the simplest configuration of a T-fold [1], for which one has locally a T 4 fibered over S 1 , with the transition functions that are T-dualities of the T 4 world-sheet fields.
It was found in [16] that this orbifold is consistent for K = 2 and 3 only.The construction of geometric D-branes was initiated for these models in [14].In this paper we consider more general boundary states that couple to the twisted sector and compute their annulus partition function.
For this purpose we first give some notations and recall the torus partition function.We will start with the Z 2 orbifold and will discuss the Z 3 orbifold in a later section.
The D 4 lattice admits a consistent asymmetric Z 2 action, while a consistent asymmetric Z 3 action requires the lattice A 2 ⊕ A 2 .The D 4 root lattice is isomorphic to the ring H(2) of Hurwitz quaternions while the A 2 ⊕ A 2 root lattice is isomorphic to the ring H(3) of Eisenstein quaternions [16]. 1  The type II torus partition functions can be written in terms of level 1 affine SO(2n) characters (B.1) as [20,21] where we have not included the (divergent) integral over the bosonic (center-of-mass and momentum) zero modes, while denotes the Narain partition function for the Lorentzian lattice II d,d and p L and p R are the dimensionless momenta.It will also be useful to introduce the N = 2 supersymmetric characters with χ j the SU (2) 1 character of spin j (B.2).The notation V, H reminds us that V has the field content of a vector multiplet at the lowest (zero) mass level, while H has the field content of a 'half' hyper-multiplet at the lowest (zero) mass level.The N = 4 supersymmetric character decomposes as By a slight abuse of language, we shall write both the type IIA right-moving fermion sector QA = V8 − C8 and the type IIB one QB = V8 − S8 as (2.7) using that the two are identical up to a permutation of the two SU (2) 1 characters in S 4 and C 4 (B.3).This permutation will be implicit in the definition of the Ishibashi states in type IIA.The Z 2 orbifold is defined at the D 4 symmetric point in Narain moduli space, where the T 4 metric and Kalb-Ramond fields are defined as in (A.2) and the cross components of the metric and Kalb-Ramond fields with one leg on T 2 and one leg on T 4 are equal.For simplicity, we shall assume that the cross components all vanish, such that the Narain partition function factorises as We define (2.9) with T = T 1 + iT 2 the Kähler structure and U = U 1 + iU 2 the complex structure on T 2 .
Then the type II Z 2 orbifold partition function can be written as [20,21] Z (2.10) Although this partition function vanishes, thanks to supersymmetry, it conveniently exhibits the spectrum of the theory.To read the spectrum, we need to distinguish the cases in which the left-momentum on T 4 vanishes or not.If it does not vanish, the two states with opposite leftmomenta are identified, with a sign depending on the left-fermion SU (2) 1 states and the parity of m 2 .When the left-momentum along T 4 vanishes, the right-momentum must be in the D 4 lattice in the untwisted sector.Then the parity of the SU (2) 1 fermion state, the parity of the torus momentum m 2 and the parity of the mode number of the T 4 bosons must be the same.
At the symmetric point one can also make use of the level one affine SO(8) algebra to fermionise the T 4 bosons.The orbifold action only preserves the SO(4) × SO(4) symmetry, so we decompose the Hilbert space into four SU (2) 1 modules.Then the action of the orbifold is defined as (−1) m 2 (−1) 2j ψ (−1) 2jo for j ψ the SU (2) 1 spin in (2.6) and j o the first SU (2) 1 of the T 4 bosons, so that The advantage of this description is that the identification of the non-vanishing momenta along T 4 is made manifest.The interpretations of SU (2) 1 characters as sums over Hilbert spaces of free fermions allows to identify the isomorphism with the Hilbert space of the 'standard' world-sheet fermions.To exhibit more explicitly the Hilbert spaces, let us introduce the following notation for the T 2 partition functions which sum all momenta and winding numbers on T 2 for which the second momentum is either even or odd and the second winding number is either integer or half-integer.Then one has The T-duality group of the theory was identified in [16] as the automorphism group of the lattice to distinguish the five SU (2) 1 characters.We find that in order to act consistently, the Fricke duality must permute χ ψ j ψ and χ o jo , and triality permutes χ v jv , χ s js , χ c jc . 2 We show in Appendix C that Fricke duality is consistent with N = 1 worldsheet supersymmetry.
Following [16], we denote by Γ D4 0 * (α) the full T-duality group, and by Γ D4 0 (α) its subgroup preserving the twisted and the untwisted sectors.Γ D4 0 (α) is a congruent subgroup of the symplectic group Sp(4, H) over the Hurwitz quaternions.It is the maximal subgroup of O(6, 6, Z) preserved by the Z 2 orbifold.On the contrary, the Fricke duality arises as an accidental symmetry that mixes the twisted and the untwisted sectors.It will play an important role in the following because it maps geometric branes to non-geometric branes.

Geometric and non-geometric fundamental branes
Geometric branes can be defined as the orbifold projection of bound states of a brane and its image under the Z 2 orbifold action in the original type II theory.A simple example in type IIA is to consider a D0-brane at a point (x 4 , x 5 ) in T 2 and its D4-brane image wrapping T 4 at the translated point (x 4 , x 5 +πR 5 ) in T 2 .
For simplicity we assume that the Kalb-Ramond field T 1 = 0 on the torus T 2 , such that the torus partition function factorises as with (3.2) 2 One does not see directly in the partition function that Fricke duality must permute χ ψ j and χ o j in the Z2 invariant sector with m2 even and n2 integer.Nevertheless, it follows by requiring Fricke invariance of three-point functions.Fricke duality relates the interaction between two massive states with m2 odd and a state of the invariant sector m2 even to the interaction between two massive twisted states and a state in the invariant sector with χ ψ j and χ o j quantum numbers exchanged.
Then the transverse channel amplitude (corresponding to closed-string 'tree-level' exchange between two identical D0-D4 branes) only involves momenta along T 2 while the winding number of the closed strings are set to zero.It follows that the D0-D4 brane must be geometric, because it cannot couple to the twisted states that all have a non-vanishing half-integer winding number n 2 .For states with a non-vanishing momentum along T 4 , one identifies the left momentum with the right momentum p L = ±p R .For vanishing momentum along T 4 , the states in Vχ 0 P even 2 + Hχ 1 2 P odd 2 have an even boson mode number in T 4 , while states in Vχ 0 P odd 2 + Hχ 1 2 P even 2 have an odd mode number.This gives the transverse channel annulus amplitude To discuss the generalisation to non-geometric branes, it will be useful to write this amplitude using SU (2) 1 characters as These characters transform under modular inversion τ → −1/τ as and ) , χ 4 such that the direct channel annulus amplitude is The normalisation of the transverse channel amplitude is justified by the interpretation of the direct channel amplitude as a partition function for the open string states.The only contribution to the tension square arises from the NS sector of V and reads [22] so one gets the mass M = e −φ √ α ′ as expected for a geometric brane.The massless content of the theory is just one N = 4 Maxwell multiplet, as expected for a fundamental brane, despite the fact that it only preserves 1/3 of the supersymmetry in N = 6 supergravity.
Although this microscopic description is only valid at a specific locus in moduli, the fundamental D0-D4 brane exists at any point in Narain moduli space Γ D4 0 (α)\O * (8)/U (4).The T-duality group Γ D4 0 * (α) includes the group of unit Hurwitz quaternions H × ⊂ SU (2).One can obtain all the fundamental branes of type D0-D2-D4 by changing the boundary conditions such that the group element u ∈ H × acts on the right sector.In particular u acts on the R-symmetry SU (2) of the N = 2 characters V and H, and on the right-momentum in D * 4 .The open string amplitudes are the all same for identical branes.The generic 1/3 BPS bound states of D0-D2-D4 are more complicated, we will discuss an example in the next section.
Another example of geometric brane is a D2-brane wrapping T 2 and its Z 2 image D6-brane wrapping T 2 × T 4 .Because of the orbifold projection, one would expect nonetheless to have non-geometric branes for which half of T 2 is filled with the D2-brane and the other half by the D6-brane.The transverse channel amplitude for this non-geometric brane then couples to the twisted sector since closed string states with fractional winding can couple a D2-brane wrapping half a T 2 to a D6-brane wrapping the other half.
By Fricke T-duality, one obtains that the corresponding boundary state is obtained by exchanging the states in χ ψ j ψ and χ o jo , in a similar way as for the permutation branes considered in [18].Recall that the action of Fricke T-duality exchanges χ ψ j ψ and χ o jo and momentum and winding n 2 → m 2 2 , m 2 → 2n 2 , which can be seen to be consistent with (2.13).This is consistent with worldsheet supersymmetry as we show in Appendix C.This gives the transverse channel amplitude The corresponding Ishibashi state is similar to the one constructed in [17] in the trivial orbifold theory, so that Hχ 1 2 χ 4 0 is identified on the left-moving sector to Hχ o The modular inversion of τ gives and (3.6) such that the direct channel amplitude is The only contribution to the tension square arises once again from the NS sector of V and reads so one gets the mass M = e −φ T 2 √ 2α ′ , which is 1 √ 2 the expected mass for a geometric D2-D6 brane.The massless content of the theory is just one N = 4 Maxwell multiplet without additional hyper-multiplets, as for the geometric D0-D4 brane.The factor of 1 √ 2 in the tension is the same as the one found in [18] for a similar asymmetric orbifold of the bosonic string.It is consistent with the Euclidean D-brane instantons corrections obtained in [16], based on supersymmetry and U-duality.The factor of 1 √ 2 follows by Fricke T-duality, because the general D0-D4 charge q ∈ H is mapped under Fricke T-duality to the D2-D6 charge 1 α q with |α| 2 = 2.Although this microscopic description is only valid at a specific locus in moduli space, it is obtained by Fricke T-duality from a geometric brane and we therefore expect this fundamental D2-D6 brane to exist at any point in Narain moduli space.

Geometric D2-D6 brane as a bound state
Let us now consider a D0-D2-D4 brane.The direct construction of the boundary state leads to a supersymmery breaking brane with a scalar tachyon.Just as for the D0-D2 boundary state in type II on a torus, one expects the transverse channel amplitude between D0-D4 and D2-D2 to give a phase associated to the SO(2) For short we define 2) The contribution of V( 12 ) to the tension is the same as for V, while its contribution to the Ramond-Ramond (RR) charge vanishes.It results an attractive force that should lead to the recombination of the branes into a supersymmetric D0-D2-D4 bound state.
The argument z = 1 2 of the SU (2) 1 character in V and H is understood as an order four element iσ 3 ∈ SU (2).The RR charges of the D0-D4 and the D2-D2 branes are related by this order four SU (2) element interpreted as a pure imaginary unit in H × .As a T-duality, it acts on the T 4 sector by exchanging two pairs of SU (2) 1 affine modules, that we choose to be χ o j ↔ χ v j and χ s j ↔ χ c j .It follows that the transverse amplitude between D0-D4 and D2-D2 can only involve non-zero momenta in D 4 and D 4 + v (of respective affine characters O 8 and V 8 ).The transverse channel amplitude is ) ) where p∈D 4 e iπτ (p,p)+πi(µ,p) + p∈D 4 +v e iπτ (p,p)+πi(µ,p) η 4 (4.4) In the representation of the SU (2) 1 modules in terms of free fermions, the T-duality permutes the SU (2) 1 characters χ o j ↔ χ v j and χ s j ↔ χ c j and gives a transverse amplitude associated to a permutation brane [23] A D0-D4→D2-D2 = V( 12 )χ 0 ( 1 2 ) ) ) where χ j (2τ ) appears because the two states in the two SU (2) 1 modules must be the same.One checks indeed that 5 so the two interpretations are compatible.
The direct channel amplitude is which is consistent with the interpretation as a Hilbert space trace, and where we use the notation (B.2) for j = ± 1 4 , in which case χ ± 1 4 = ξ 4 ±1 + ξ 4 ∓3 is not an SU (2) 1 character but a short for the sum of two U (1) 1 characters (B.8).One finds one scalar tachyon coming from O 4 χ 2 − 1 4 that is expected to drive the system to a supersymmetric bound state.
We will determine this bound state at the end of this section, but let us first describe the amplitude between two non-geometric branes.
Using Fricke T-duality one can directly obtain the dual amplitude between a non-geometric D2-D6 and a non-geometric D4-D4 brane wrapping half T 2 .The transverse channel amplitude is ) ) 4 The same Wilson line gives zero if inserted in S8 and C8, and so consistently projects out the states that are incompatible with the mixed Dirichlet-Neumann boundary conditions. 5These identities follow from ϑ 4 (2τ ) .

and the direct channel amplitude
Perturbative techniques do not allow directly to determine the supersymmetric boundary state that should emerge from tachyon condensation.However, the unique supersymmetric object with the correct tension square and RR charge is a geometric BPS brane.Repeating the construction of the previous section one obtains the transverse channel amplitude of the geometric D2-D4-D6 brane and its direct channel amplitude . (4.12) In this case there is no distinction between odd and even momenta m 2 , and the massless content of the theory includes both an N = 4 Maxwell multiplet and an N = 2 hyper-multiplet. 6y Fricke T-duality one can now deduce the supersymmetric D0-D2-D4 bound state transverse channel amplitude as being and the direct channel amplitude Note that although the transverse amplitude above does not involve the twisted sector, it is nongeometric in the sense that the left-moving fermion and boson modules χ ψ j and χ o j are permuted in the definition of the Ishibashi state.This provides the two simplest examples of composite BPS branes in the asymmetric orbifold.In this case we could argue from Fricke T-duality what the direct amplitude should be.For more general cases such a short argument will not work.It may nonetheless be possible to determine the composite geometric branes by relying on duality symmetry.One may expect for example that more general D0-D2-D4 brane might be described by a D0-D4 brane in a magnetic field. 7 The Z 3 asymmetric orbifold One can do a similar computation for the Z 3 orbifold at the A 2 ⊕ A 2 symmetric point (A.13) [16].For simplicity we consider again the Narain moduli such that where χ su (3)   p for p = 0, 1, 2 are the three SU (3) 1 characters respectively associated to the trivial, the fundamental and the anti-fundamental representations of SU (3).
The torus partition function in the Z 3 orbifold can be written with ω = e 2πi 3 .It will be convenient to write the partition function in terms of U (1) 1 characters in order to exhibit the modules involved and their chiral algebra symmetry.The level 1 SU (3) characters decompose into the products of level 1 SU (2) characters and level 1 U (1) characters according to where each U (1) 1 character ξ 3 p corresponds to the A 3 modules of the primary fields e ϕ .Recall that for a free field ϕ of radius R = √ N , i.e. defined modulo 2π √ N , one defines the dimension that belong to irreducible modules of the chiral algebra A N generated by the current J = i∂ϕ and the vertex operators V ±2N , for each p modulo 2N [24].
To exhibit more explicitly the modules involved, it is convenient to introduce the notation such that with (5.12) One identifies the 72 irreducible A 3 9 modules of characters labeled by k i ∈ {0, 1} and r, s ∈ {0, 1, 2}.

The geometric D0-D4 brane
We can now consider the transverse channel amplitude for a D0-D4 boundary state.Let us first address the geometric description of the boundary state.For non-zero momentum along T 4 , the orbifold projection identifies the momenta in triplets and one simply gets a 1/3 factor in the amplitude.For vanishing momentum along T 4 , the T 4 free boson mode number must have a fixed congruence modulo 3 and we get It will be useful to write this amplitude using U (1) 1 characters as ) where we introduced (5.17) Using a modular inversion one gets the direct channel amplitude The massless sector of the theory consists in a single N = 4 Maxwell multiplet, as expected for a fundamental brane.One obtains the tension square as for the Z 2 orbifold.

The non-geometric D2-D6 brane
To obtain the transverse channel amplitude of a non-geometric D2-D6 brane, we need to find an isomorphism between the twisted sector and the untwisted sector that allows to define Fricke T-duality.This symmetry between the two sectors is not manifest in (5.11), but we are going to see that the isomorphism is an appropriate outer automorphism of A 3 9 .The 72 irreducible A 3  9 characters (5.13) are not independent.There are related by the obvious automorphisms defined as permutations of the three U (1) 1 modules, and the automorphism ϕ i → −ϕ i that gives (r, s) → (−r, −s).
Fricke T-duality involves yet another automorphism, that can be identified as the field redefinition φi = 1 3 ϕ i − 2ϕ i+1 − 2ϕ i+2 . (5.20) One checks that it preserves the algebra of the U (1) 1 currents because ( 2 3 ) 2 + ( 2 3 ) 2 + ( 1 3 ) 2 = 1 and J i = i∂ φi are level 1 currents.It does not stabilise the subalgebra A 9 × A 9 × A 9 ⊂ A 3  9 because e ±6iϕ i = e ∓4i( φ1 + φ2 + φ3 ) e ±6i φi , ( but it does stabilise A 3 9 .It acts on the irreducible A 3 9 modules of characters (5.13) as The character identity associated to this automorphism is given in (B.12).A Fricke Tduality is defined to act on the momenta and winding as m 2 → 3n 2 and n 2 → m 3 3 [16].One has therefore the symmetry of the partition function (5.11) obtained by exchanging (r, s) → (s, r) on both the T 2 Narain partition function and the A 3 9 characters ξ 3 9 [ r,s k i ].One can identify the twisted sector A 3 9 modules ξ 3 9 [ 0,s k i ] with the untwisted sector modules ξ 3 9 [ s,0 k i ].We show in Appendix C that this isomorphism is consistent with worldsheet supersymmetry.
(5.22)The direct channel amplitude then reads The massless content of the theory is just one N = 4 Maxwell multiplet, as for the Z 2 orbifold.The tension square of the non-geometric brane is so that the tension has the expected additional factor of 1 √ 3 predicted by Fricke T-duality [16].As for the Z 2 orbifold, one may argue that a geometric brane can be obtained as a D2-D4-D6 bound state of a D2-D6 brane and two D4-D4 branes.Here we do not discuss in details the relation between the RR charges and the geometric configuration of branes, but we want to stress that the geometric brane and the non-geometric brane cannot have the same RR charge up to a factor of √ 3, because this is inconsistent with the lattice of charges H(3) = A 2 ⊕ A 2 [16]. 9 The transverse channel amplitude is simply obtained by assuming that it does not couple to the twisted sector and one gets
so one finds a factor of √ 3 between the masses of the geometric branes and the non-geometric brane, as expected from Fricke T-duality [16].

Conclusions and outlook
We have given a detailed (microscopic) description of the geometric and non-geometric fundamental branes and their bound states in two classes of T-fold backgrounds with N = 6 supersymmetry.As shown in [16] these correspond to Z 2 and Z 3 asymmetric orbifolds that combine a non-geometric 'T-duality' action on T 4 with a shift along T 2 .
Focussing on the Type IIA framework we have first analyzed the Z 2 case and identified both geometric and non-geometric fundamental branes.After discussing D0-D4 whose tension is the one expected for a geometric brane, we passed to consider a D0-D2-D4 boundary state, whose direct construction leads to a supersymmery breaking brane with a scalar tachyon.We argue by Fricke duality that tachyon condensation leads to a supersymmetric bound state with tension equal to √ 2 times the one of the geometric D0-D4 brane.This bound state is not geometric in the sense that the Ishibashi state combine world-sheet fermions and bosons SU (2) 1 modules.The same construction holds for the non-geometric D2-D6 brane, which has tension equal to 1/ √ 2 the one of the geometric D2-D4-D6 brane.
In the Z 3 case we found similar results for geometric D0-D4 brane and for non-geometric D2-D6 brane.The non-geometric brane involves again a mixing between world-sheet fermions and bosons.One finds a factor of √ 3 between the tension of geometric and non-geometric branes, as expected from Fricke T-duality.
It would interesting to extend our analysis to other T-fold backgrounds with lower or no supersymmetry [6,7] or to S-folds [11][12][13] or U-folds [8][9][10].The price one has to pay is the lack of a 'perturbative' world-sheet description that in the present case proved crucial in checking consistency between open-and closed-string descriptions.Knowledge of the relevant U-duality may be sufficient to determine the lattice of charges as in the N = 6 case [16] Constructions of more general bound states using magnetised D0-D4 branes or T-branes [25,26], that could lead to supersymmetry enhancement, may well provide new useful insights in this endeavour.
where ϑ α are Jacobi (elliptic) theta functions.We also use the SU (2) 1 characters with j = 0 or 1 2 .In particular we have We write the U (1) 1 characters The A 3 9 characters (5.13) transform under modular inversion as ω rs ′ +sr ′ +ss ′ (−1) In this appendix we show that the supersymmetry current is invariant under Fricke duality. 11or this we use a representation of all fields associated to T 4 in terms of free bosons [27], such that the T 4 supersymmetry current is c λ e i(λ,φ(z)) , (C.1) where φ(z) is a vector of six free bosons and ∆ is a set of 16 sin( π K ) 2 vectors of norm square (λ, λ) = 6, that is invariant under the −1 reflection and a Fricke reflection.This gives 16 vectors for Z 2 and 12 for Z 3 .The cocycle c λ is determined such that G(z) is a fermion field and its operator product expansion gives the singular terms For the Z 3 orbifold we use the convention φ = (σ, ϕ 1 , ϕ 2 , σ 2 , ϕ 3 , σ 3 ) , (C.12) where we introduced ϕ 1 = ϕ ψ for convenience and the twelve vectors λ are the rows of the matrix 0 χ s 0 χ c 0 where the two vacua of the bosonic module χ o 1 2 χ v 0 χ s 0 χ c 0 are generated by the SU (2) doublet of twisted boson fields.
N , which gives in particular for any integer p χ ℓ+p) , (B.8) which is not an SU (2) 1 character but the sum of N U (1) 1 characters.The T 4 mode number partition function with the insertion of the Z 3 operator can be written in terms of U (1) 1 characters as follows