Non-supersymmetric D-branes with vanishing cylinder amplitudes in asymmetric orbifolds

We study the type II string vacua with chiral space-time SUSY constructed as asymmetric orbifolds of torus and K3 compactifications. Despite the fact that all the D-branes are non-BPS in any chiral SUSY vacua, we show that the relevant non-geometric vacua of asymmetric orbifolds allow rather generally configurations of D-branes which lead to vanishing cylinder amplitudes, implying the bose-fermi cancellation at each mass level of the open string spectrum. After working on simple models of toroidal asymmetric orbifolds, we focus on the asymmetric orbifolds of T2 × ℳ, where ℳ is described by a general N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 4 SCFT with c = 6 defined by the Gepner construction for K3. Even when the modular invariant partition functions in the bulk remain unchanged, the spectra of such non-BPS D-branes with the bose-fermi cancellation can vary significantly according to the choice of orbifolding.


Introduction
String theories on the non-geometric backgrounds may induce interesting features which are not realized in the standard geometric compactifications. One of the salient aspects of such non-geometric vacua would be the vanishing cosmological constant without unbroken SUSY. This is in contrast to our experiences in ordinary geometric string vacua that the SUSY-violation generically gives rise to cosmological constant at the breaking mass scale (string scale, typically). The attempts of the construction of non-SUSY vacua with vanishing cosmological constant have been initiated by [1][2][3] based on some non-abelian orbifolds, followed by closely related studies e.g. in [4][5][6][7][8][9]. More recently, several non-SUSY vacua with this property have been constructed as asymmetric orbifolds [10] by simpler cyclic groups in [11,12]. Studies of non-SUSY vacua in heterotic string theory have also been presented e.g. in [13][14][15][16][17][18][19][20].
In this paper, we would like to focus on similar interesting aspects of non-BPS Dbranes in simple models of non-geometric type II string vacua. Let us first recall that the BPS D-branes are described by the boundary states satisfying the BPS-equation, where Q α ( Q β ) denotes the left(right)-moving space-time supercharges and M α β are some c-number coefficients. Through this paper, we express boundary states by |· · · , · · · | . JHEP08(2017)082 long as it inherits the structure of the bose-fermi cancellation in the bulk torus amplitude. As shown in the following sections, it is indeed possible to find simple models of such asymmetric orbifolds, and thus plenty of boundary states with the vanishing self-overlap. In section 2, we study toroidal models and consider several asymmetric orbifoldings preserving 8 supercharges coming only from the right-mover.
In section 3, which is the main part of this paper, we shall discuss less supersymmetric models constructed as the asymmetric orbifolds of the backgrounds, where M is described by a general N = 4 superconformal field theory (SCFT) withĉ(≡ c 3 ) = 2, which geometrically describes compactifications on K3 with particular moduli. The relevant asymmetric orbifolds are defined by the twisting, where (−1 R ) ⊗2 is the chiral reflection on the T 2 -sector (X 4,5 -directions), 5), (1.5) and σ M denotes an involution on the M-sector, which is allowed to act asymmetrically on the N = 4 superconformal algebra (SCA). As we will clarify later, one obtains in this way the chiral SUSY vacua with the 4-dim. N = 1 SUSY (4 supercharges). We then classify the possible gluing conditions for the boundary states, which are decomposed into the Ishibashi states [40] for each N = 4 unitary irreducible representations (irrep.'s), and examine whether or not their self-overlaps vanish. The spectra of the non-BPS boundary states with this property non-trivially depend on the choice of the twist operator σ M , even in the cases when the modular invariant partition functions remain unchanged; different σ M 's may lead to the same partition functions in the bulk.

Toroidal asymmetric orbifolds
In this section we shall focus on the simpler cases, namely, the asymmetric orbifolds of tori realizing the chiral SUSY vacua of type II string, in order to show how the strategy outlined above is implemented. The discussion is straightforwardly extended to the case of K3 in the next section, though it is technically a little more involved.

Asymmetric orbifold
Let us first consider the asymmetric orbifold of the 4-dim. tours T 4 , which would be the simplest model that has the desired properties. We assume the torus is along the X 6,...,9directions and at the symmetry enhancement point with SO(8) 1 . We thus denote it as

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The orbifold group is generated by a single element which acts as the chiral reflection on the right-mover, X i R → −X i R , ψ i R → −ψ i R (i = 6, . . . , 9), accompanied by the twisting of the space-time fermion number (−1) F L on the left-moving fermions, that is, the sign-flip of arbitrary states in the left-moving R(amond)sector. Closely related asymmetric orbifolds adopting slightly different setting have been analyzed in the bulk [11,12] for non-supersymmetric string vacua with vanishing cosmological constant. The analysis below follows these references.
We simply assume that σ 2 acts on the untwisted Hilbert space as an involution for the free bosons X i R , whereas we naturally have two possibilities on the fermionic sector; (i) σ 2 = 1, (ii) σ 2 = (−1) F R , depending on the definition of the Ramond vacua or the way of bosonization to introduce the spin fields (see also section 3.1). Here, the operator (−1) F R just acts as the sign flip on any states in the right-moving R-sector. We separately examine theses two cases: (i) σ 2 = 1 (on the untwisted Hilbert space). In this case, the modular invariant is written as ǫ [4] (a,b) χ D 4 (a,b) (τ ) χ A 1 (a,b) (τ ) 4 (a ∈ 2Z + 1, or b ∈ 2Z + 1). (2.3) where Z 6d bosonic (τ,τ ) denotes the partition function of the bosonic sector of uncompactified space-time R 5,1 . The building blocks χ D 4 (a,b) , χ A 1 (a,b) (τ ) 4 , h (a,b) and f (a,b) are evaluated for the sectors of X 6,...,9 L , X 6,...,9

R
, the left-moving fermions, and the right-moving fermions, respectively, where the subscript (a, b) labels the sectors with the spatial and temporal twists by σ given in (2.2). They are obtained first for the (0,1) sector with one temporal twist, and then for other sectors by the modular transformation. Their explicit forms are summarized in appendix A. (See (A.7), (A.12), (A.14), (A.17).) The phase factor ǫ [r] (a,b) is defined in [22], and explicitly written as It is quite useful to note that the combination ǫ is organized so as to be modular covariant with respect to (a, b), where X r denotes the suitable Lie algebra lattice of rank r presented in [22]. Namely, any modular transformation defined by A ∈ SL(2; Z) acts simply on the subscript (a, b) as JHEP08(2017)082 (a, b) −→ (a, b)A. We note that this is an order 4 orbifold due to the existence of the phase factor (2.4) despite σ 2 = 1| untwisted , which would be a typical feature in asymmetric orbifolds.
The right-mover preserves 1/2 space-time SUSY, whereas the left-moving space-time SUSY is completely broken. In fact, it is obvious that σ ≡ (−1) F L ⊗ (−1 R ) ⊗4 cannot preserve any left-moving supercharges in the even a sector, which are essentially those in the unorbifolded theory. Furthermore, if we had a left-moving supercharges belonging to the sector a = 1, we should obtain the equality of the partition functions However, it is easy to see that this is not the case, when observing the explicit forms of relevant partition functions. We can similarly show the absence of supercharges in the a = −1 sector. On the other hand, half of untwisted supercharges in the right-mover are σ-invariant, as in the familiar supersymmetric orbifold Now, let us move on to the discussion on the non-BPS D-branes. As already pointed out, no D-brane can preserve space-time SUSY. Nevertheless, rather general 'bulk-type branes' lead to the vanishing self-overlap. 1 In fact, consider the bulk-type brane written as an orbifold projection, where |B 0 stands for the GSO-projected boundary state describing any BPS D-brane in the unorbifolded theory on R 5,1 × T 4 [D 4 ], and P = 1 2 (1 + σ) is the projection operator onto the invariant sector under the twist. As described in the introduction, P commutes with the Virasoro operators and maintains the conformal invariance. The overall normalization factor √ 2 has been determined by the Cardy condition. By definition, we have 0 B| e −πsH (c) |B 0 = 0, since |B 0 is BPS. Moreover, explicit computation gives Again f (0,1) (is) is defined in (A.14). We thus obtain Although the left-and right-movers are correlated in the boundary states due to the conformal invariance, the twist thereon still leads to the same function f (0,1) as in the bulk, which is regarded as a remnant of the bulk computation. In this way, we have successfully shown that the present string vacuum possesses the desired property to have the non-BPS D-branes with vanishing self-overlaps. Because of the overall factor in |B , its coupling to the gravitons (tension) is √ 2 times that in the unorbifolded theory. The coupling to the RR-particles (RR charge) is also multiplied by √ 2. By the modular transformation, the standard open string excitations in the original theory are found to remain in the self-overlap of the unorbifolded part |B 0 . These are common features for all the non-BPS branes with the vanishing self-overlaps treated in this paper.

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Absence of tachyonic instability. Let us briefly check that no open string tachyons emerge in the cylinder amplitude, In fact, the piece 0 B| e −πsH (c) |B 0 is just the same as the familiar cylinder amplitude associated to the BPS brane, whereas In the last line, the first and second terms are identified as the NS and R-sector amplitudes in the open string channel, of which leading terms are obviously massless. We then obtain 16 pairs of massless bosonic and fermionic states from the orbifolded part, even though no supercharges in the closed string sector preserve the boundary state |B . We can similarly show the absence of open string tachyons in the cylinder amplitudes with the bose-fermi cancellation also for other models discussed below.
(ii) σ 2 = (−1) F R (on the untwisted Hilbert space). In this case, σ acts as a Z 4action already on the untwisted sector, and the modular invariant is slightly modified as The fermionic chiral block h (a,b) is again given in (A.17), while f (a,b) , given in (A.18), is slightly modified from f (a,b) due to the relation σ 2 = (−1) F R . The left-mover has no space-time SUSY as in the first model. At first glance, it seems that the right-moving SUSY is also broken, because all of the supercharges in the untwisted sector are projected out by (−1) F R . However, it is found that (NS,R)-massless states appear in the a = 2 twisted sector, suggesting the existence of new 8 supercharges. These states possess the opposite chirality to the case (i), because the orbifolding by (−1) F R acts like the T-duality transformation (see e.g. [21,23]). In the end, we indeed obtain a chiral SUSY vacuum. One can check that the partition function vanishes after summing up a, b ∈ 2Z, although each f (a,b) (τ ) is not necessarily vanishing. The non-BPS D-branes with vanishing self-overlaps are given by the formula similar to (2.7), but including the contribution from the a = 2 twisted sector ; where is a suitably defined boundary state lying in the a = 2 sector, which contains the right-moving Ramond ground states with the opposite chirality as addressed above. (Obviously, we have no solutions of the boundary states in the a = ±1 sectors.) We return to this point shortly, but here just note σ 2 acts as (−1) F R −1 on the a = 2 sector, rather than (−1) F R , which is read off from the expression of f (a,b) (τ ) in (A.18). The a = 2 NSNS-sector is thus projected out by the P 4 -action, while the a = 0 RR-sector drops off. We are then left with the P 2 -projection of the 'opposite BPS' boundary state, which accounts for the second line of (2.12). In other words, if we consider the type IIA (IIB) vacuum of this asymmetric orbifold, |B (2.14) In fact, in the second case (ii), we can resolve the orbifold group as 2 and by the relation suggested in [21,23], we obtain the above equivalence (2.14). Given this equivalence, a way to construct |B (a=2) 0 is tracing back the relation in (2.12), as mentioned above. The observation here is used to reduce the number of the cases to be analyzed in the following sections.

Asymmetric orbifold
The point of the construction in the previous subsection is rather general as described in the introduction, and various generalizations would be possible. As an example where the open-string boundary condition is more relevant, we next focus on a case of the 5-dim. torus T 5 along the X 5,...,9 -directions. To be more specific, we begin with the following compactification: We consider where S 1 [A 1 ] denotes the circle with the self-dual radius.
We just consider S 1 R , that is, the circle compactification with an arbitrary radius R.
The total modular invariant is given in the form, (2.20) Here Z 5d bosonic denotes the contribution from the bosonic part of R 4,1 . In the second line, we have combined ) and the free fermion chiral blocks f (a,b) , g (a,b) are summarized in appendix A. As already mentioned, the modular covariance of Z T 4 ×S 1

(a,b)
is assured due to the phase factor ǫ  (2.19) describes the first case (i). The modular invariants for the remaining cases are easy to construct. Namely, we only have to replace the chiral blocks f (a,b) , g (a,b) in (2.20) with the ones given in (A.18), (A. 19) suitably. However, as mentioned at the last part in the previous subsection, the cases (ii), (iii) reduces to the first case (i) as in (2.14), and the case (iv) corresponds to a non-SUSY vacuum, which is beyond the scope of this work. Therefore, it is enough to focus on the simplest case (i). We can pick up any BPS boundary states |B 0 in the unorbifolded theory on T 4 × S 1 , and define the non-BPS brane |B by the orbifold projection in the same way as (2.7). It is not difficult to show that |B has the vanishing self-overlap as long as |B 0 satisfies the general gluing condition (for the T 4 × S 1 -directions) given by

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where α i L,n , α i R,n and ψ i L,r , ψ i R,r denote the bosonic and fermionic oscillators (including the bosonic zero modes), and M i j is an arbitrary SO(3)-matrix. 3 To show this fact, it is useful to note the relation, for any |B 0 satisfying (2.21). We thus obtain similarly to (2.8), leading to the vanishing cylinder amplitude, B| e −πsH (c) |B = 0. We add a comment: in the model of (2.1) the vanishing self-overlaps have been achieved for arbitrary BPS boundary states |B 0 in the unorbifolded theory. On the other hand, in the current case, |B 0 defined by (2.21) is restricted to (M i j ) ∈ SO(3) rather than (M i j ) ∈ SO(4). If adopting a different orbifold action instead of (2.18), say, , we still obtain the same modular invariant, yielding the equivalent spectrum of closed string states. Moreover, it is obvious to have an essentially equivalent spectrum of non-BPS branes with the vanishing self-overlaps, in which (M i j ) ∈ SO(3) appearing in (2.21) should act on the X 6,8,9 R -directions this time. This fact is not surprising, of course. However, in the next section, we will see the examples in which the spectra of the non-BPS branes with vanishing self-overlaps would notably depend on the choice of orbifolding, while the modular invariant partition functions remain unchanged.

Chiral SUSY vacua as asymmetric orbifolds of T 2 × K3
In this section, we shall study less supersymmetric cases with M = K3 in the background (1.3), The strategy to construct the non-BPS D-branes with vanishing self-overlaps is the same as in the previous section. The discussion is however a little more involved. We thus first summarize the relevant asymmetric orbifolds in the bulk in subsection 3.1. We then concentrate on the examples of the Gepner construction in subsection 3.2. For these subsections, we follow [24] where the modular invariant partition functions of related asymmetric orbifolds ('mirrorfolds') are constructed. (See also [25].) Based on these set-ups, we construct the non-BPS D-branes in subsection 3.3, which is the main part of this section 3.

Asymmetric orbifolds of T 2 × K3 with chiral SUSY
To begin with, we assume that the M-sector is described by a general N = (4, 4) SCFT withĉ ≡ c 3 = 2, not reducing to the toroidal models. We denote the relevant N = 4 SCA [26] by L n (Virasoro), J α n ( SU(2) 1 ) with α = 1, 2, 3, G a r with a = 0, 1, 2, 3. Recall that the total R-symmetry is given by SO(4) ∼ = SU(2) c × SU(2) f , in which the inner symmetry JHEP08(2017)082 ('color SU(2)') is generated by the affine currents J α , whereas the global SU(2)-symmetry ('flavor SU(2)') is an outer one. G 0 is a singlet of SU (2) We also assume that N = 2 SCA is embedded into the N = 4 one in the standard fashion by identifying the N = 2 U(1) R -current as and The generators of integral spectral flows U ±1 are identified with the remaining SU(2)currents and the half-spectral flows U ±1/2 define the Ramond sector.
Let us now consider the asymmetric orbifolding of (3 We first note the action of (−1 R ) ⊗2 on the world-sheet fermions, which we assign to ψ 4 R , ψ 5 R . We bosonize them as 4 It fixes the action of (−1 R ) ⊗2 on the R-sector and we find We next consider the M-sector. We would like to suitably choose the orbifold twisting σ M so as to obtain a 4-dim. vacuum with N = (0, 1)-chiral SUSY unbroken. Obviously σ M should be an automorphism of both left and right-moving N = 4 SCAs. Furthermore, since working on superstring vacua in the NSR-formalism, σ M should satisfy the following conditions: (i) σ M preserves T (energy-momentum tensor) and G 0 , which is necessary for the BRSTinvariance.
(ii) σ M keeps the Ramond sector intact so as to be compatible with U ±1/2 . This means that the automorphism σ M has to satisfy The same conditions are required for the right-mover. 4 If we had adopted the bosonization for the combinations, e.g. ψ 2 R +iψ 4 R and ψ 3 R +iψ 5 R , the L.H.S of (3. 6) would have been the identity. However, we shall not consider this possibility here in order to respect the super-Poincare symmetry in R 3,1 , and (3.5) is the unique choice. We also simply assume that (−1 R ) ⊗2 is involutive on the bosonic coordinates X 4 R , X 5 R (on the untwisted Hilbert space) in this paper, even though we have more general possibilities if utilizing the fermionization of them as discussed in [12].

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Let us now introduce the automorphisms σ and we assume that they are involutive on the whole Hilbert space; σ We also set σ (α) L obviously acts on the N = 4 SCA in the same way as (3.7), but it is no longer involutive; σ To complete the definition of σ

Recalling the simple relation
L . It obviously acts on each N = 4 chiral current in the same way as σ L are anti-commutative on the R-sector due to (3.8), we find (σ and σ L acts on the half-spectral flows U ±1/2 as R (α = 1, 2, 3) for the right-mover are defined in the same way. Now, let us specify the possible orbifold actions σ ≡ σ M ⊗ (−1 R ) ⊗2 . We again have four possibilities (i) as in the previous section. However, all the space-time SUSY are broken in the fourth case, and the second and third cases reduce to the first case by the chirality flip; IIA ←→ IIB as mentioned in subsection 2.1. It is thus enough to consider the first case such that σ is involutive, that is, the cases with σ ≡ σ We shall especially focus on the following three cases; R . Of course, we have to examine whether they are actually compatible with the modular invariance. In the next subsection, we explicitly confirm in the case of the Gepner construction that the asymmetric orbifolding by σ ≡ σ M ⊗ (−1 R ) ⊗2 constructed this way yields superstring vacua with modular invariance. In all the three cases, the space-time SUSY from the left mover is broken to achieve the 4-dim. N = (0, 1) chiral SUSY.

Concrete examples: Gepner construction
Let us consider the generic Gepner construction [27,28] for K3, that is, the superconformal system defined by where M k denotes the N = 2 minimal model of level k (ĉ ≡ c 3 = k k+2 ), and we set We start with the diagonal modular invariant for simplicity. We have to make the Z Norbifolding that renders the total U(1) R -charge (in the NS-sector) integral, where I := {(ℓ 1 , m 1 ), . . . , (ℓ r , m r )} denotes the collective label of the primary state in , and the twisted sectors of orbifolding are identified with the 'spectral flow orbits' by the actions U n (n ∈ Z N ) with See [29] for more detail.
The relevant Hilbert space for the K3 sector (before imposing the GSO projection) is schematically expressed as I, * is uniquely determined by the half-spectral flow in the standard manner. 5 Note that the left-right symmetric primary states lie in the n = 0 sector, but we also have many asymmetric primary states generated by the spectral flows. As already mentioned, the N = 2 superconformal symmetry withĉ = 2 is enhanced to the N = 4 by adding the spectral flow operators, which are identified with the SU(2) 1 currents J ± ≡ J 1 ± iJ 2 in the N = 4 SCA [29]. Accordingly, the chiral parts of H (s,s) Gepner are decomposed into irreducible representations of N = 4 SCA at level 1, that are classified as follows [30,31]: These are non-degenerate representations whose vacua have conformal weights h. The vacuum of C The relevant character formulas are summarized in appendix A. Now, let us construct the asymmetric orbifolds by the involution σ. Since a detailed account of closely related asymmetric orbifolds has been given in [24], based on [25], we here briefly describe the relevant construction.
Since the most non-trivial part σ M has the form σ  16) where the N = 2 involution σ N =2,(i) L acts as in each minimal factor M k i . On the other hand, σ L acts on the N = 4 SCA as the automorphism (3.7). We still have to define how it acts on the N = 4 primary states |v L . A simple choice would be given as where J ± L ≡ J 1 L ± iJ 2 L are the SU(2) currents in the N = 4 SCA, which turns out to be compatible with the modular invariance.
By these definitions and the fact that σ As before, Z 4d bosonic denotes the contribution from the bosonic part of R 3,1 , which is related with neither the σ-twisting nor the GSO-projection. Those for the various σ-twisted sectors Z (a,b) (a, b ∈ Z 4 ), which are crucial in our arguments, are described in the following way: where F  [32][33][34][35]. The chiral blocks for other spin structures are determined by the 1/2-spectral flows and by incorporating the suitable sign factors to impose the GSO condition. (See [29] for more detail.) We also adopted the concise notation θ [s] (τ ) := θ 3 (τ ), θ 4 (τ ), θ 2 (τ ), iθ 1 (τ )(≡ 0) for s = NS, NS, R, R respectively. The modular invariant coefficients N I, I are straightforwardly determined due to the Gepner construction, which are independent of a, b, and the overall factor 1/4 originates from the chiral GSO projection.
where we set in the odd sectors are slightly non-trivial. We can determine them in a way parallel to that presented in [24,25]. We here briefly describe the results, which depend on the spectrum of the level k i in (3.11) as follows:

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(i) At least one of k i 's is odd. In this case, the modular invariant coefficients are very simple, (ii) All k i 's are even. In this case, N in (3.12) is even, and we set Then, the relevant coefficients are given by One can directly confirm that the Z (a,b) (τ,τ ) in the odd sectors (3.21) show the suitable modular covariance by using the modular transformation formulas given in (B.7). Note that σ We make a few comments: • As already mentioned, we are considering the orbifolding by σ = σ (α) for various α, β, and obtain the equivalent spectra of closed string states in all these models. However, this fact does not necessarily imply that they are equivalent string vacua. Indeed, it turns out that they have quite different D-branes, as we elucidate in subsection 3.3.
• One finds that the contributions from the (R, * ) or ( * , R)-sectors do not appear in the building block (3.21) with a ∈ 2Z and b ∈ 2Z + 1. This fact is actually expected so as to achieve the modular invariance. It is not difficult to confirm that this is indeed the case in almost all the Gepner models for K3 due to the basic properties of the twisted characters. (See appendix B.) The exception is only the (4) 3 type, in which there would exist a non-vanishing (R, NS) contribution 6 that could spoil the modular invariance. However, by suitably fixing the sign ambiguity of σ (α) L on the Ramond vacua with Q = 0, one can avoid this possibility still in the (4) 3 -model.

General construction of boundary states with vanishing self-overlaps
Let us present our main studies. Namely, we investigate how we achieve the vanishing self-overlaps in the current models of asymmetric orbifolds, (3.26)

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We begin with specifying the boundary conditions in the (unorbifolded) M-sector that characterize general BPS D-branes. 7 Naively, any boundary conditions preserving the N = 4 superconformal symmetry with an arbitrary twisting by automorphism may be allowed, which are schematically expressed as in [39] by Here, A I r , A I r are the chiral currents and g denotes any (inner or outer) automorphism of the N = 4 SCA. However, since we are working on the physical boundary states in the RNS superstrings, we still have to impose the following conditions: (i) |B preserves G 0 -symmetry without any twisting, which is necessary for the BRSTinvariance.
(ii) |B contains the correct components of the RR-sector compatible with the above definition of U ±1/2 . This means that the automorphism g in (3.27) has to satisfy g · J 3 = J 3 , or g · J 3 = −J 3 .
Thus, at least generically, the allowed twisting g by the N = 4 automorphism is restricted and we eventually obtain the following two types of gluing conditions: A-type: In the above, R(θ) denotes the SO(2)-rotation with the angle parameter θ, and R(θ) ≡ R(θ)σ 3 ∈ O(2). The relevant Ishibashi states [40] are characterized by the N = 4 irrep. classified in the subsection 3.2 as well as the gluing conditions given above, and should satisfy e.g.
This shows why θ is restricted to discrete values θ = 2πr N . The B-type Ishibashi states are similarly constructed.
The Ishibashi states in the RR-sector are obtained by the half-spectral flow from the NSNS ones, (3.32) We note the correspondence of the representations, (3.33) The R-massive rep. C (R) h is generated by doubly degenerated vacua with conformal weight h belonging to an SU(2)-doublet, as opposed to the NS-one C (NS) h . As in the previous analyses on the toroidal models, generic D-branes in our asymmetric orbifold (3.26) are expressed by the boundary states in the form of the orbifold projection with σ 2 = 1, where |B 0 is a (GSO-projected) boundary state describing a D-brane in the unorbifolded theory and P ≡ 1+σ 2 . We assume that |B 0 describes a half-BPS brane with the Dirichlet conditions for all the transverse coordinates along R 3,1 × T 2 [D 2 ], just for convenience. Namely, |B 0 is expanded by the Ishibashi states given above for the M-sector and the JHEP08(2017)082 self-overlap is schematically written as actually vanishes. Therefore, to achieve the vanishing cylinder amplitudes in the asymmetric orbifolds (3.26), it is enough to examine whether or not the amplitude 0 B| σe −πsH (c) |B 0 vanishes. From now on, we examine this problem in each case of (1) σ M ≡ σ R , as addressed before. We set θ r ≡ 2πr N , (r ∈ Z N ) in the following.
(1) σ M = σ It is worthwhile to emphasize that this relation does not depend on the angle parameter θ r at all. Thus, the amplitude from each component of Ishibashi state is eventually evaluated by the σ (2) R -twist irrespective of θ r , yielding the N = 4 twisted character, , (3.37) or trivially vanishing one. We summarize necessary formulas for the N = 4 twisted characters in appendix B. In this way, we obtain for the NSNS-sector, where (−1) f L denotes the twisting for the GSO projection. The fact that (−1) f L σ Minsertion leads to the equal amplitude is obvious from the boundary conditions for the fermionic currents G a (z), (a = 0, 1, 2, 3). We also recall that σ includes (−1 R ) ⊗2 , which just makes the free fermion contribution from the (transverse part of) R 3,1 ×T 2 [D 2 ]-sector proportional to θ 3 η θ 4 η for the NSNS-sector, JHEP08(2017)082 η . On the other hand, the contributions from the RR-sector trivially vanish due to free fermion zero-modes along either of the R 3,1 or Combining all the contributions and taking account of the GSO-projection, we finally obtain In this expression 8 the summation is taken over all the spin 0 irrep.'s, that is, C Thus, the net effect of the twist is just a phase factor for the RR-component of boundary state. Incorporating also the R 3,1 ×T 2 -sector, the RR-component of the overlap again drops off due to the fermionic zero-modes, and we obtain the following amplitude instead of (3.39), At least for generic Gepner models, the R.H.S of (3.42) does not vanish for any value of the moduli parameter θ r . In fact, R.H.S of (3.42) does not depend on θ r , and for a generic rep. r i . Rephrasing more physically, the D-brane tension has been modified by the σ-insertion, while the RR-charge remains the same as in case (1). This causes the mismatch of amplitudes for the graviton and RR-particle exchanges. In this way, we conclude that all of the D-branes in the second case have non-vanishing self-overlaps, as one expects from the general features of non-BPS D-branes.

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R : the third case is the most subtle one. When translating the σ (1) Linsertion into that of the right-mover similarly to (3.36), (3.41), we have to take account of the R(θ) ( R(θ)) rotation appearing in the gluing conditions (3.29) ((3.28)). For instance, for the B-type gluing condition, we obtain denotes the automorphism acting on the N = 4 SCA rotated by R(θ r ) in the same way as σ (1) R . 9 Obviously the relation (3.43) yields the self-overlap that depends on the parameter θ r , as opposed to the first and second cases. The resultant amplitude does not vanish generically. However, for the special value θ r = ± π 2 , we find σ yielding the cancellation as given in (3.39). The A-type gluing condition is likewise treated.
In this way, we conclude that the D-branes in the third case have the vanishing selfoverlaps only for the gluing conditions with θ r ≡ 2πr N = ± π 2 , which is possible when N ∈ 4Z >0 .
Absence or presence of tachyonic instabilities. Here we would like to further discuss whether the non-BPS branes considered above could include the tachyonic instabilities. Since it is obvious that no closed string tachyons appear in the relevant boundary states, we should examine the open string excitations in the orbifolded sector. Indeed, it is easy to estimate the lightest excitation in the open string channel. By detailed case studies, it would be possible to write down the formulas of the general spectra, which are however beyond the scope of this paper.
Let us first note common features in the orbifolded sector for the above three cases; (i) the RR contribution to the self-overlap vanishes due to the fermionic zero-modes, implying the lack of GSO-projection for the open string Hilbert space, (ii) the twist by (−1 R ) ⊗2 along the T 2 [D 2 ]-direction adds the conformal weight 1 4 to the open string vacua. Now, the estimations for the above three cases are summarized as follows; Case (1)

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and the inequality can be saturated only when all the k i 's are even. Therefore, the lightest open string excitation could be massless when all k i 's are even, and always massive if at least some k i 's are odd. In this way, we conclude that no tachyonic instability emerges in the open string spectrum.
Case (2): σ M again acts on the N = 4 primary states in the same way, whereas it effectively makes the N = 4 SCA invariant, after taking account of the identity (3.41). Thus, the twisted N = 4 character χ [0,1] ( * ; is) ∝ θ 3 θ 4 η 3 (is) for the case (1) has to be replaced with the untwisted one ∝ θ 2 3 η 3 (is) for the NS-sector. Making the modular transformation, the net effect just amounts to the shift by − 1 8 to the R.H.S of (3.45). We thus obtain the inequality and open string tachyons would appear. This result is expected since the open string spectrum is non-supersymmetric in this case.
Case (3): again, σ M acts on the N = 4 primary states as the above two cases. On the other hand, by utilizing (3.43), we find that the net effect on the (right-moving) N = 4 SCA by the σ M -insertion amounts to the SO(2)-rotation with the angle parameter 2θ r on the J 1 R , J 2 R (and G 1 R , G 2 R ) plane, while leaving the other generators intact. Then, the twisted N = 4 character ∝ θ 3 θ 4 η 3 (is) for the case (1) is replaced with ∝ θ 3 (is,0)θ 3 (is,2θr) This implies that open string tachyons would generically emerge except for the special angle θ r = π 2 for N ∈ 4Z >0 , which realizes the bose-fermi cancellation in the open string spectrum as mentioned above.

Points of toroidal orbifolds
Our discussion so far is based mostly only on general properties of the N = 4 SCFT for M. Thus, we would expect that the spectrum of the non-BPS branes with the vanishing overlaps is unchanged over generic points of the moduli space of K3, as long as the asymmetric orbifolding by σ ≡ (−1 R ) ⊗2 ⊗ σ M is well-defined. The points in our argument were: • The global symmetry SU(2) diag preserving G 0 is only identified with an outerautomorphisms of the N = 4 SCA.
• We need to pick up a particular U(1)-subalgebra of the N = 4 SCA to define the Ramond sector by the half-spectral flows, which has been generated by J 3 in the above arguments.

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Then, only the restricted SO(2)(⊂ SU(2) diag ) twisting is allowed in the gluing conditions (3.28), (3.29), so as to preserve the Ramond sector Hilbert space. On the other hand, there are special points with the 'symmetry enhancement' in the moduli space, at which more general gluing conditions could be solved. For instance, it has been known [29] that the Gepner model (2) 4 (Kummer surface) is equivalent with the Z 2 -orbifold of T 4 [D 4 , B ij ≡ 0], which is defined as the 4-dim. torus associated to the root lattice of D 4 with the vanishing Kalb-Ramond field. 10 We can reinterpret this system in terms of free bosons and fermions, and thus, the SU(2) diag is explicitly realized by these free fields. In this special case all the choices of orbifold twisting σ M = σ (α) (α, β = 1, 2, 3) lead to equivalent superstring vacua, as in the toroidal models studied in section 2. Especially we find the equivalent spectra of the non-BPS D-branes with the vanishing self-overlaps. Indeed, with the help of free field interpretation, one can straightforwardly solve the following equations for the boundary states, where R(θ, ϕ) denotes an arbitrary SO(3)-rotations.
There also exist the Z 3 , Z 4 , Z 6 -orbifold points within the Gepner models for K3 as discussed in [29]. However, such an enhancement of symmetry does not happen for these points, and SU(2) diag is still identified as outer-automorphisms.

Discussion
We have studied the type II string vacua with chiral space-time SUSY constructed as asymmetric orbifolds, focusing on the D-branes on these backgrounds. The simple but crucial idea in this paper is that all the D-branes are non-BPS in any chiral SUSY vacua. As clarified in sections 2 and 3, one can straightforwardly construct the chiral SUSY vacua based on asymmetric orbifolds which accommodate rather generally the non-BPS D-branes with vanishing cylinder amplitudes. This would be hardly realized in the geometrical compactifications of superstring theory.
We have especially investigated the asymmetric orbifolds of T 2 × M, as well as simpler toroidal models, where M = K3 is described by a general N = 4 SCFT with c = 6 defined by the Gepner construction. We have demonstrated in subsection 3.3 that the spectra of such non-BPS D-branes with the bose-fermi cancellation depend notably on the choice of orbifolding, even when the closed string spectra remain unchanged. This feature is in contrast to those of the toroidal asymmetric orbifolds presented in section 2.
In this respect we note that the most of the analyses on the boundary states given in subsection 3.3 are based only on general properties of the N = 4 SCFT for M, as mentioned in the previous section. Thus, the spectrum of the non-BPS D-branes with vanishing cylinder amplitudes would be unchanged over generic points of the moduli space of K3, as JHEP08(2017)082 long as the asymmetric twist is well-defined. The point in our discussion is summarized in subsection 3.4. The exception would be the orbifold point with symmetry enhancement.
Based on the results in this paper, one may now discuss a possible application to the problem of cosmological constant. As mentioned in the introduction, the cosmological constant induced solely by the non-BPS D-branes would be exponentially suppressed for small string coupling. Furthermore, in a given non-BPS D-brane background, the contributions to the closed-string vacuum amplitude would come only from the diagrams with the external legs sourced by that non-BPS D-brane. The analysis of the loops thus would be much simpler than the case of the bulk SUSY-breaking [1,3,9,20], to control the almost vanishing cosmological constant. It would also be challenging to substantiate the scenario [4], which is based on the analysis on the heterotic dual side and mentioned in the introduction, that the non-BPS D-branes condensate to produce the non-perturbative mismatch of the spectrum. This would also be an interesting problem involving a non-supersymmetric duality. We hope to return to these issues elsewhere.

Acknowledgments
This work is supported in part by JSPS Grant-in-Aid for Scientific Research 24540248 and 17K05406, Japan-Hungary Research Cooperative Program and Japan-Russia Research Cooperative Program from Japan Society for the Promotion of Science (JSPS).

A Summary of conventions
Theta functions. Here, we have set q := e 2πiτ , y := e 2πiz ( ∀ τ ∈ H + , ∀ z ∈ C), and used abbreviations, Bosonic building blocks. Here we summarize the notation of the building blocks used in the main text according to [22]. Associated to the basic representation of (D r ) 1
Fermionic building blocks. To describe the supersymmetric chiral blocks for the free fermions, we introduce the notation In the second line, each term corresponds to the NS, NS, R sectors with keeping this order. These trivially vanish, as is consistent with the space-time SUSY. They satisfy the modular covariance of the form, We next define the non-supersymmetric chiral block twisted by the two component and also for the twisting by (−1) F L , , (a, b ∈ 2Z + 1), Again they satisfy the modular covariance in the same sense as (A.15). We also introduce slightly modified chiral blocks,
Characters for the N = 4 SCA with c = 6. The character formulas of the unitary irrep.'s of the N = 4 SCA with c = 6 (level 1) are given in [30,31], and we exhibit them here. We focus on the NS-sector: The R-sector characters are obtained by the 1/2-spectral flow. Namely, In this appendix we summarize the definitions of the twisted characters of N = 2 and N = 4 superconformal algebras, according to [24,25].
The character formulas for other boundary conditions are just determined by the modular transformations. We denote the spin structures as well as the boundary conditions of σ Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.