Non-BPS fractional branes with Bose-Fermi cancellation in asymmetric orbifolds

We study non-BPS D-branes in the type II string vacua with chiral space-time SUSY constructed based on the asymmetric orbifolds of K3≅T4/ℤ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrm{K}3\cong {T}^4/{\mathrm{\mathbb{Z}}}_2 $$\end{document} as a succeeding work of [21]. We especially focus on the fractional D-branes that contains the contributions from the twisted sector of the ℤ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathrm{\mathbb{Z}}}_2 $$\end{document}-orbifolding. We show that the cylinder partition functions for these fractional branes do not vanish, as in the cases of ordinary non-BPS D-branes. This aspect is in a sharp contrast with the bulk-type branes studied in [21]. We then discuss the extensions of models by including the discrete torsion depending on the spin structures, and investigate whether we obtain the vanishing self-overlaps associated to the fractional branes. The existence/absence of bose-fermi cancellations both in the closed and open string spectra as well as the massless spectra in the twisted sectors crucially depend on the discrete torsion. We find that some choices of the torsion indeed realize the vanishing self-overlaps for the fractional branes, with keeping the vanishing torus partition function intact.


Introduction
Orbifold theories are important and interesting subjects in string theory, yielding various models of string vacua. Besides the symmetric ones, we can work with the asymmetric orbifolds [1], in which the orbifold groups act asymmetrically on the left-and right-moving degrees of freedom, and they provide non-geometric string vacua that are well-controlled. One of interesting studies of asymmetric orbifolds would be the construction of non-SUSY vacua with perturbatively vanishing cosmological constant given e.g. in [2][3][4][5][6][7][8][9][10], and more recently in [11,12] based on simpler cyclic orbifold groups.
Another interesting possibility to achieve non-SUSY vacua with very small cosmological constant would be realized by taking D-branes into account. Even though the space-time SUSY is preserved in the closed string sector, it would be broken by the effect of the non-BPS 'D-brane instantons' (Euclidean D-branes wrapping around internal cycles), if we have sufficiently generic configurations of non-BPS D-branes. If we only have the vanishing O(g 0 s )-contributions to the cosmological constant originating from the open strings, that is, the cylinder partition functions associated to relevant non-BPS D-branes, we would be left with a non-perturbative cosmological constant induced by the instanton effect. This should be exponentially suppressed as long as the string coupling g s is small enough. Such a possibility in a type II theory has indeed been mentioned in [5] in order to explain the JHEP06(2018)125 non-perturbative mismatch of the vacuum energy with that of its heterotic dual. Closely related studies on non-SUSY vacua in heterotic string have also been given e.g. in [13][14][15][16][17][18][19][20].
In the recent paper [21], with this motivation, we studied non-BPS D-branes in some type II string vacua with chiral space-time SUSY easily constructed by the asymmetric orbifolding. What is a simple but crucial fact is that, in these chiral SUSY vacua, any boundary states |· · · , · · · | cannot satisfy the BPS-equation written as Q α + M α βQ β |B = 0, (1.1) just due to the lack of e.g. left-moving unbroken supercharges Q α . This means that any D-branes are non-BPS in these vacua. Nevertheless, as we demonstrated in [21], the 'selfoverlaps' 1 for these branes could vanish generically due to the bose-fermi cancellation. For instance, in the simplest example of the chiral orbifold of T 4 by the involution σ ≡ (−1) F L ⊗ ( − 1 R ) ⊗4 , any non-BPS branes expressed as possess the expected property, where P σ denotes the projection for the asymmetric orbifolding by σ and |B T 4 is an arbitrary boundary states describing a BPS brane in T 4 . We note that these D-brane configurations are fairly generic in the theory T 4 /σ. Preceding studies on non-BPS configurations of D-branes that however realize the bose-fermi cancellation of open string excitations include e.g. [7,22]. Now, in the present paper, we shall proceed to analyze the non-BPS D-branes in the cases of asymmetric orbifolds of K3 ∼ = T 4 /Z 2 , where the Z 2 -orbifolding is defined by the symmetric reflection I 4 ≡ (−1 L ) ⊗4 ⊗ (−1 R ) ⊗4 . For the bulk-type branes |B = √ 2P σ P I 4 |B T 4 such as (1.2), we just arrive at the same conclusion, since the argument given in [21] is generic enough. However, we can still accommodate the fractional branes in these vacua, that is, the boundary states including the contributions from the I 4 -twisted sectors. The main aim of this paper is to examine whether we can also achieve the vanishing self-overlaps for these fractional branes.
The organization of this paper is given as follows: in section 2, we first describe the boundary states defined on the asymmetric orbifold K3/σ ≡ T 4 /I 4 /σ, and analyze the cylinder amplitudes. We especially show that the self-overlaps associated to the fractional D-branes do not vanish generically contrary to the bulk-type branes. Therefore, in section 3, we shall discuss the extensions of models so as to include the discrete torsion [25][26][27] depending on the spin structure. We can no longer regard the relevant conformal model as a simple orbifold; K3/σ ≡ T 4 /I 4 /σ, and physical aspects non-trivially depend on the choice of discrete torsion. We classify the possible string vacua and investigate whether we achieve the vanishing partition functions for the world-sheet of torus as well as cylinder. We find out some choices of discrete torsion indeed realize the expected properties, namely, both of the torus partition function and the cylinder amplitudes for the fractional branes vanish exactly. We also investigate extra massless JHEP06(2018)125 excitations emerging in the twisted sectors both in the closed and open string Hilbert spaces of each model. Furthermore, we consider the 'stable non-BPS branes' in the K3-background given in [22,24], of which NSNS/RR components of the boundary states lie in the untwisted/twisted sectors. So, we discuss what happens for the fractional branes of this type, after performing the σ-orbifolding. We find that relevant cylinder partition functions could vanish if choosing suitably the zero-mode parts of boundary states, in the similar but different way in comparison with [22]. This feature curiously depends on the discrete torsion. We argue that, in some cases, the requirement for vanishing cylinder amplitudes is not compatible with the unitary open string spectra.
In section 4, we summarize the results in this paper and yield a brief discussion.

Asymmetric orbifold without discrete torsion
In this section we shall study the asymmetric orbifold without discrete torsion. We start with the type II string vacuum compactified on the 4-dim. torus along the X 6,...,9 -directions at the symmetry enhancement point with SO(8) 1 . The total orbifold group is Z 2 × Z 2 generated by I 4 : X i → −X i (i = 6, . . . , 9) and the 'chiral reflection'; 2 9). The total Z 2 × Z 2 -oribifold is simply identified as

The torus partition function
Let us first consider the partition function of torus T 4 ≡ T 4 [SO (8)]: and we introduce the symbol as the supersymmetric chiral blocks for free fermion. Therefore, the modular invariant partition function on T 4 compactification is written as where Z 6d bosonic (τ,τ ) denotes the partition function of the bosonic sector of uncompactified space-time R 5,1 .

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where we explicitly exhibit the contribution Tr I 4 -twsited σq L 0 − c 24 qL 0 − c 24 , and the omitted terms · · · are uniquely determined by modular transformations. To derive the second line, we made use of the familiar identity θ 2 θ 3 θ 4 = 2η 3 . We will later discuss various modifications of this orbit by including the discrete torsion. It is worthwhile to note the massless spectrum in the I 4 -twisted sector, which is directly extracted from the combination of orbits (2.11) Here we adopted a schematic (probably, standard) expression to exhibit the traces with various twisted sectors. For instance, The numbers of NS-NS and R-R massless states in the I 4 -twisted sector are read from the q-expansions of (2.11), which we denote as 'N

The cylinder partition function
In this subsection, we analyze the cylinder partition functions of which both ends are attached to the same D-brane by using the method of boundary states. First, we introduce the 'bulk-type branes' defined in K3 ∼ = T 4 /I 4 as the (BPS) GSO-projected boundary state |B bulk composed only of the I 4 -untwisted sector; (2.14)

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where α µ n ,α µ n are the left-and right-moving modes of X µ , and ψ µ r andψ µ r are the modes of ψ µ andψ µ , respectively. The sign factor (µ) = ±1 depends on whether the µ-direction is Dirichlet or Neumann, which won't play any roles of importance in the following arguments. |B 0 denotes the zero-mode component, which should be I 4 -invariant, and N is the normalization factor to be determined by the open-closed correspondence. P GSO denotes the standard GSO projection operator which chirally acts on the, say, left-mover only. In other words, (2.14) is nothing but the I 4 -invariant combinations of the BPS boundary states under the T 4 -compactification.
The relevant cylinder partition function, which we call the 'self-overlap' of boundary states is just evaluated as bulk B| e −πsH (c) |B bulk = 0, (2.15) where s and H (c) ≡ L 0 +L 0 − c 12 denotes the closed string modulus and Hamiltonian for the cylinder. Then we obtain the vanishing self-overlap because |B bulk is BPS.
In the previous paper [21], we studied the D-branes in the asymmetric orbifold by the involution σ described by the boundary states of the form; where |B denotes an arbitrary BPS brane in the unorbifolded theory and P σ ≡ 1 + σ 2 is the σ-projection. The numerical factor √ 2 is necessary due to the Cardy condition. As already mentioned, such boundary states |B never satisfy the BPS equation, since we at most possess the chiral SUSY after making the σ-orbifolding. Nonetheless, as addressed in [21], the self-overlaps vanishes rather generically To be more precise, it is true for any boundary states written as (2.16) with the 'bulk-type' branes (2.14) in T 4 /I 4 . However, this is not the whole story. One can still accommodate the fractional branes in the orbifold T 4 /I 4 , that is, the boundary states including the contributions from the I 4 -twisted sector. 4 To do so, we have to consider the boundary states of the type (2.16) with |B ≡ |B frac given by (for the transverse part); where U and T denotes the untwisted sector and the twisted sector, and the untwisted sectors have the momentum along the ith compact direction. Now, the self-overlap of boundary state |B ≡ √ 2P σ |B frac is evaluated as follows;

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and In this way, we have shown that the self-overlap (2.20) does not vanish for generic fractional branes. We will discuss the possibility to change this feature by extending the σ-orbifold so as to include the discrete torsion in the next section. It is also valuable to examine the massless open string spectrum associated to such non-BPS branes. Indeed, by combining (2.21) and (2.22) and making the modular transformation s = 1/t, we obtain 3 Asymmetric orbifolds with discrete torsions

Inclusion of discrete torsion
In the previous section, we have studied the asymmetric orbifold written schematically as We would like to discuss various extensions of this model so as to include the relative phase factors among the relevant orbifold twists, I 4 and σ, which are broadly known as the 'discrete torsion' [25][26][27]. We especially investigate whether we gain the vanishing selfoverlaps associated to the fractional branes in such modified vacua. It is a novel point here that we shall adopt the discrete torsion that depend on the spin structures. In other words we allow relative phases among the orbifold twists by I 4 and σ as well as the GSO twisting, which can be also regarded as a Z 2 -orbifolding. Thus, the most general form of torus partition function is written schematically as In this expression, (α, β) and (a, b) again characterize the I 4 and σ twists, while s L , s R ≡ NS, NS, R express the chiral spin structures. Both of the total building block Z (s L ,s R ) (α,β),(a,b) (τ,τ ) and the phase factor (discrete torsion) (s L ,s R ) (α,β),(a,b) should keep the modular covariance. Especially we require (s L ,s R ) (α,β),(a,b) to be invariant on each modular orbit. δ(s L ), δ(s R ) are the standard sign factors associated to the chiral GSO projection; δ(NS) = 1, δ( NS) = δ(R) = −1.
(We omit the factor δ( R) = ±1 due to the existence of R 5,1 -sector.) To avoid unessential complexities, we further assume that the modular orbits Z[1], Z[I 4 ], and Z[σ] remain unchanged. This assumption obviously implies ). However, we still have the possibilities to replace This 'degeneracy' is partly removed if considering the sectors (a, b) = (α, β), in other words, if focusing on the modular orbit Z[σI 4 ]. We obtain two inequivalent vacua by choosing the phase factor as (1) (a,b) are uniquely determined by the modular invariance.) The first case corresponds to the I 4 (or (−1) F L I 4 )-orbifolding, whereas the second one does to Table 1. Summary of the possible phase choices.
In fact, the original orbit Z[σI 4 ] respects the right-moving chiral SUSY, whereas the modified one Z[σ I 4 ] breaks the right-moving SUSY although the bose-fermi cancellation happens in the left-mover. See appendix B for the explicit forms of these orbits. For each of I 4 and I 4 -orbifolding, we can still include the non-trivial phases (s L , (1,0),(0,1) . This uniquely determines the phase factors for the sectors with (α, β) = (0, 0), (a, b) = (0, 0), and (α, β) = (a, b), which lie on the same modular orbit. We first count how many choices are allowed for this phase. The conformal invariance implies that the it should be factorized as ( holds, the relevant phases are determined by choosing L (NS), L (R), R ( NS) and R (R). We also note a manifest symmetry for (s L , s R ) by Thus, we end up with the 2 4 × 1 2 = 8 independent choices of (s L , s R ), explicitly summarized in the table 1.
In the following, we shall separately investigate the vacua of each case; (i) (s L , s R ) = 1, We then briefly discuss about the remaining four cases, that is, what happens if including the overall minus sign into (s L , s R ).

Models with the I 4 -orbifolding
As we discussed above, the models of our interests are classified by a single phase factor (s L , s R ) ≡ (s L ,s R ) (1,0),(0,1) . We first focus on the I 4 -orbifolding. In all the cases, the cylinder amplitudes (self-overlaps) associated to D-branes inherited from any BPS bulk-type branes on K3( ∼ = T 4 /I 4 ) vanish according to the arguments given in [21].
(1-ii) (s L , s R ) = (−1) F L . In this case the string vacuum has again N = (0, 1) spacetime SUSY. It is a crucial difference from the first case that the orbifold action is no longer factorized, when taking the spin structures into account. This means that the relevant vacua possess non-trivial discrete torsion.
What happens for the cylinder amplitudes of the fractional branes? Due to the extra sign factor = (−1) F L , σ acts on the I 4 -twisted components of the boundary state as as opposed to the first case. We then obtain instead of (2.22) The part [· · · ] expresses the contribution from world-sheet fermions, of which cancellation implies the bose-fermi degeneracy.
Furthermore, one can examine the massless spectra in the I 4 -twisted sectors for both of the closed and open strings in the same way as the previous section. To examine the closed string massless spectra, it is convenient to generalize (2.11) to that with the discrete torsion (s L , s R ) ≡ αβ F L γ F R , (α, β, γ = ±1); .

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On the other hand, we can extract the massless open string spectrum in the twisted sector as in (2.23). As easily seen, the second line of (2.23) is now replaced with and we find n (1-iv) (s L , s R ) = (−1) F L +F R . In this case, we obtain Z cyl frac (is) = 0, as in the case (1-i), since (−1) F L +F R |B = (−1) F L |B holds.
The right-moving space-time SUSY is broken in the I 4 -twisted sector, while the leftmoving SUSY is already broken in the untwisted sector (i.e. with no I 4 -twisting). Nevertheless, we find that the cosmological constant vanishes at the one-loop; Z torus (τ,τ ) = 0. In fact, all the partition functions for the sectors of (α, β) = (0, 0) and (α, β) = (a, b) vanish due to the bose-fermi cancellation in the right-mover, whereas the (α, β) = (0, 0) sectors show the left-moving cancellation; where the plus and minus in the double signs correspond respectively to (1-ii) and (1-ii)' and Z zero-modes (is) expresses the zero-mode contribution for the bosonic T 4 -sector. One can directly read off the open string spectrum by making the modular transformation s = 1/t. Denoting simply Z zero-modes (is) = t 2 Z zero-modes (it), we obtain

(3.12)
Consequently, focusing on the twisted open string sector (the second term), we find that extra massless excitations appear for the (1-ii)'-case, while they do not for the (1-ii)-case as already mentioned. Again, the other cases are similarly investigated, and it turns out that the numbers 16 and 0 are exchanged for n (s) tw when comparing the 'undashed' and the 'dashed' cases.
All these aspects of constructed vacua are summarized in table 2.

Models with I 4 -orbifolding
We next consider the I 4 ≡ (−1) F R I 4 -orbifolding to realize the K3 ∼ = T 4 /Z 2 -sector. In this case, as opposed to the I 4 -orbifolding, the σ I 4 -projection removes all the supercharges in the right-mover. We then obtain string vacua with no supercharges. We recall the fact that the GSO-condition is effectively reversed in the I 4 -twisted sector (see e.g. [22,23]), in comparison with that of I 4 .
To examine the I 4 -massless spectra, it is also convenient to derive the formula corresponding to (3.5) (we again set (s L , s R ) = αβ F L γ F R , α, β, γ = ±1);
On the other hand, the fractional branes yield non-vanishing self-overlaps because (2-iii) (s L , s R ) = (−1) F R . In this case, we again obtain Z torus (τ,τ ) ≡ 0. However, as opposed to the previous case, the bose-fermi cancellation appears in the rightmover in the I 4 -twisted sector. Again, we find Z cyl frac (is) = 0, because of the identity (−1) F R |B = (−1) F L |B for any boundary states.
Similarly to the above cases, We find N (2-iv) (s L , s R ) = (−1) F L +F R . This case would be the most curious.
We have Z torus (τ,τ ) ≡ 0 due to the bose-fermi cancellation in both of the left and right movers despite the absence of unbroken SUSY. We also achieve the vanishing self-overlaps for any fractional branes, as in the case (2-i).
We similarly obtain N Finally, let us again discuss the inclusion of overall minus factor, namely, the 'dashed cases'. The relevant arguments are almost parallel, and the aspects of bose-fermi degeneracies remain unchanged. On the other hand, the GSO-conditions are opposite to the 'undashed cases'. We summarize these results in table 3.

Massless spectra in the other sectors
To complete the analysis on the closed string massless spectra, we next discuss the other sectors, that is, the untwisted, σ-twisted, and σI 4 -(σ I 4 -)twisted sectors. All of them are found to be independent of the choice of the discrete torsion (s L , s R ).   σ-twisted sector. The analysis on the σ-twisted sector is more non-trivial, since the relevant parts of the partition function depend on the discrete torsion, which we again parameterize as (s L , s R ) ≡ αβ F L γ F R (α, β, γ ∈ ±1). Then, we can evaluate Note that the R-R amplitude does not depend on the discrete torsion, since

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The massless spectra are again extracted from the terms including |θ 2 | 8 , which does not depend on the discrete torsion. Taking account of the bose-fermi degeneracy in this sector, we obtain N (NS,NS) which is again independent of the discrete torsion (s L , s R ).

Comments on unitarity and absence of the tachyonic instability
We here briefly discuss about the unitarity and the absence of tachyonic instability of the string vacua constructed above. The torus partition functions in unitary theories should be q-expanded so that all of the coefficients are positive integers. This is not necessarily self-evident in various twisted sectors, that is, those defined by the I 4 (or I 4 ), σ, σI 4 (or σ I 4 ) twistings. Indeed, we would face up to the opposite GSO-projections in the chiral NS-sectors due to the (−1) F L or (−1) F R -twisting for some choices of the discrete torsion (s L , s R ). This would potentially give rise to tachyonic excitations, and even worse, the q-expansions with negative coefficients. depend on the discrete torsion (s L , s R ). Then, we can confirm that the coefficients of q-expansions are always positive integers regardless of (s L , s R ) (α, β, γ, in other words).
One would be aware of the existence of several terms with h L = 0, h R = 1 in (3.17), and be afraid that tachyonic modes might emerge in the left-mover. However, one has to recall that the level matching condition h L − h R = 0 should be imposed in order to extract the physical spectrum in string theory, while the modular invariance just requires the weaker condition h L − h R ∈ Z. In other words, we have to integrate out the modulus of world-sheet torus, and all of the level mismatch terms are removed after performing the integration of τ 1 ≡ Re τ . 5 In the end, the lightest excitations in this sector are at most massless and no tachyons emerge in this sector.
We can likewise analyze the σ I 4 -twisted sector. The coefficients of relevant qexpansions are found to be always positive integers irrespective of the discrete torsion. We generically have the terms with h L = 1, h R = 0, but they do not survive after imposing the level matching condition h L = h R . given in (3.19) includes negative terms in its q-expansion, when choosing αβ = +1, γ = −1 (in other words, = (−1) F R or (−1) F L +F R ), although only positive terms appear otherwise. Especially the leading terms with negative coefficients appear at h L = 1, h R = 0. However, it turns out that again all of these 'pathological' terms reside only in the level mismatch sectors, and thus do not contribute to the physical string spectrum.
In fact, the relevant term is of the form as ∼ 1 |η| 16 θ 8 2 θ 8 2 − 4θ 4 3 θ 4 4 , and the (complex conjugate of) right-mover is q-expanded as where we made use of the identity θ 2 θ 3 θ 4 = 2η 3 in the first line. This expansion ensures that the above statement is correct. 5 Such 'level mismatch' terms already appear in the asymmetric orbifold T 4 /σ, that is, in the q-expansion In the case of [11], on the other hand, σ includes the shift operator T2πR, and such terms do not emerge as explicitly shown in [11].

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In this way, we conclude that the torus partition functions are suitably q-expanded in the way compatible with unitarity and no tachyonic instability appear for all the string vacua we constructed irrespective of the discrete torsion (s L , s R ).
3.6 Non-BPS fractional branes in the type of [22] Even though our main purpose in this paper has been achieved in the previous analyses, there is another interesting issue. Namely, we would like to focus on the non-BPS fractional branes of the types studied in [22] (see also [23,24]). As opposed to the above cases, they are already non-BPS before taking the σ-orbifolding, and identified with the D-branes wrapped around non-supersymmetric cycles in K3. The boundary states describing them, which we denote |B GS , contain the NSNS components from the I 4 -( I 4 -)untwisted sector, whereas the RR components from the I 4 -( I 4 -)twisted sector with the 'ordinary' ('opposite') GSOprojection. It is notable that these type branes possess the non-vanishing RR-charges and thus could be stable despite their non-BPS properties [24].
Then, the relevant computation of self-overlap of |B GS yields where the first and second terms correspond to the NSNS and RR components, and Z zero-modes (is) denotes the zero-mode part of bosonic coordinates along the T 4 -direction.
For the second equality, we made use of the identities θ 4 3 − θ 4 4 = θ 4 2 and 2η 3 = θ 2 θ 3 θ 4 . According to the arguments given in [22], one can choose the moduli of T 4 as well as the zero-mode part of boundary states |B GS so as to achieve the vanishing cylinder amplitude Z cyl GS (is) ≡ 0. Namely, it is enough to choose them so that Z zero-modes (is) = θ 3 (is) 4 .

(3.24)
In our set-up, we assumed T 4 = T 4 [SO (8)] for the consistency with the asymmetric orbifolding by σ, and thus this equality can be consistently satisfied. Now, let us consider the σ-orbifolding. We again define the GS-type fractional brane in the σ-orbifold as follows; Then, taking account of the insertion of the operator 1 + σ, the cylinder partition function (3.23) should be modified as

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Here, the double sign depends on how the involution σ acts on the RR-components of the boundary state, which lies in the I 4 -( I 4 -) twisted sector. The amplitude (3.26) does not vanish generically. However, one can suitably choose the zero-mode components of relevant boundary states as done in [22]. This aspect is described as follows; (n-ii), (n-iii), (n-i)', (n-iv)'-cases (n = 1, 2). The plus of the double signs in (3.26) is realized. Therefore, the vanishing amplitude is achieved if we assume Z zero-modes (is) = θ 3 (is) 4 + θ 4 (is) 4 .

Conclusions and discussions
In this paper, we have studied non-BPS D-branes in the type II string vacua based on asymmetric orbifolds of K3 ∼ = T 4 /Z 2 as a succeeding work of [21], focusing on the fractional D-branes that contains the contributions from the twisted sector of the Z 2 -orbifolding. We have seen that the cylinder partition functions for these fractional branes do not vanish as opposed to the bulk-type branes studied in [21]. Nextly, as a main analysis, we studied the extensions of models by including the discrete torsion depending on the spin structures, which was a crucial point in this paper. We classified the relevant string vacua with this torsion, and found the vacua that make it possible to gain the vanishing self-overlaps for arbitrary fractional non-BPS branes, with keeping the vanishing torus partition function intact. Namely, in the classification of vacua given in section 3, we have shown that the models (1-ii), (1-ii)' (chiral SUSY vacua), and (2-iv), (2-iv)' (no SUSY vacua) possess the expected properties. (See the tables 2, 3.) We have also analyzed the massless spectra in the twisted sectors and their compatibility with unitarity of each model, which non-trivially depend on the discrete torsion. In some cases, the GSO projection acts on the twisted sector with the 'wrong' sign and the relevant q-expansions include negative coefficients that might spoil the unitarity. However, it turned out that the negative terms only appear in the sectors with level mismatching in JHEP06(2018)125 our case. We thus conclude that all the sting vacua constructed above would be consistent with unitarity. It is a curious issue to what extent such a feature of unitarity can be generic in asymmetric orbifolds. Indeed, the discrete torsion of the type adopted in this paper would offer a novel and simple way of breaking the space-time SUSY with keeping the bose-fermi degeneracies both in the closed and open string one-loop amplitudes, even though the toroidal models worked out here still remain toy models. Therefore, it would be interesting to further clarify the universal features of such vacua and to explore possible generalizations beyond the toroidal models.
Since we start with the type II string on the asymmetric orbifold of K3 ∼ = T 4 /I 4 , it is also an interesting question to ask what the heterotic dual should be. Of course, this is an intriguing and challenging issue because of the existence of discrete torsion among the orbifold actions by I 4 , σ and also the GSO-twisting, which non-perturbatively acts on the Hilbert space of the heterotic side. It would be interesting to discuss about the comparisons or possible relations with various heterotic string vacua with exponentially suppressed cosmological constants studied e.g. in [13][14][15][16][17][18][19][20].