Non-geometric Backgrounds Based on Topological Interfaces

We study simple models of the world-sheet CFTs describing non-geometric backgrounds based on the topological interfaces, the `gluing condition' of which imposes T-duality- or analogous twists. To be more specific, we start with the torus partition function on a target space S^1 [base] x (S^1 x S^1) [fiber] with rather general values of radii. The fiber CFT is defined by inserting the twist operators consisting of the topological interfaces which lie along the cycles of the world-sheet torus according to the winding numbers of the base circle. We construct the partition functions involving such duality twists. The modular invariance is achieved straightforwardly, whereas `unitarization' is generically necessary to maintain the unitarity. We demonstrate it in the case of the equal fiber radii. The resultant models are closely related to the CFTs with the discrete torsion. The unitarization is also physically interpreted as multiple insertions of the twist/interface operators along various directions.


Introduction
theory is yet to be uncovered. As an attempt in this direction, we shall investigate the modular invariants involving the conformal interfaces. Our discussion below is based on an observation that the topological interfaces induce the twists associated with the symmetries and dualities, which may yield an exact CFT description of T-folds and analogous non-geometric backgrounds. We shall see that this is indeed the case by explicitly constructing a novel type of the modular invariants describing such non-geometric CFT models. For the discussions on the world-sheet conformal interfaces in the context of string theory, see for instance [16,24,25,26,27].
More specifically, we shall focus on the CFT models defined on the background, (1.1) how our unitarization is physically interpreted as the multiple insertions of the twist operators consisting of the topological interfaces.
In section 5, we present a summary and discussion.

Partition Functions of Compact Bosons
The partition function of a free boson compactified on the circle with radius R should be where Z R (τ | ν) represents a contribution from the winding sector specified by ν. Its modular property is expressed as When the radius R can be written as R = √ k, (k ∈ Z >0 ), the partition function Z R (τ ) is rewritten in terms of theta functions, 1 (2.4) More generally, if R = L K , (K, L ∈ Z >0 ) holds (the rational CFT cases), we obtain Ls](τ ), (2.5) with the notation, Z (k) [u, v](τ ) := 1 |η(τ )| 2 Θ u+v,k (τ )Θ −u+v,k (τ ).
(2.6) 1 Our conventions of theta functions are summarized in Appendix A.

Orbifolding
We introduce the operators corresponding to the following two types of orbifolding: (i) Z N -action of translation : First, we define τ (N ),R γ as the operator linearly acting on the function (2.2) as with arbitrary c i , ν i . Acting on Z R (τ ), the operator 1 We set above γ = L(aτ + b) ∈ Λ/NΛ with a, b ∈ Z N by assuming N, L are coprime. This form of the action turns out to be useful for our later analysis.
(ii) Z N -action of 'dual translation' : We define τ (N ),R γ as the operator linearly acting on the function (2.2) as where we introduced the symbol, ν, γ := 1 τ 2 Im(νγ). (2.11) This time, the operator 1 N j∈Z N τ (N ),R j acts on Z R (τ ) as the projection onto the states with the winding w ∈ NZ. We then have an identity, . We also note a schematic equivalence, (2.14) where T denotes the T-duality transformation, R → 1 R .
We shall later use the abbreviations τ , as long as there is no fear of confusion.

Topological Interfaces
Consider two CFTs with central charge c = 1, which are denoted by 'CFT 1 ' and 'CFT 2 ', and described by free bosons X 1 , X 2 compactified on the circle with radius R 1 , R 2 , respectively. We assume that there exist two positive integers k 1 , k 2 such that We denote the oscillators of the free bosons X i as α i n ,α i n , and the Fock vacua as |n, w; (i) , n, w; (i)| whose left and right momenta are given by Then, one has the topological interface operator I (−) 12;(k 1 ,k 2 ) that glues the world-sheet of CFT 2 with that of CFT 1 [16,24] Note that the zero-mode part G whereas those on the oscillator part are (2.20) 2 We simply set to zero the 'moduli' parameters of the topological interface operators through this paper.
We also set to plus a possible sign which could appear in the oscillator part, since it is absorbed by exchanging the left and right movers. It is understood that the oscillators α 1 n ,α 1 n act on the left side of G 12;(k1,k2) , whereas α 2 n ,α 2 n on the right side.
Namely, I (−) glues two theories related by T-duality. We also note that when k 1 > 1 or k 2 > 1, the image of I (−) 12;(k 1 ,k 2 ) is strictly smaller than the whole Hilbert space of CFT 1 . In other words, I (−) 21;(k 2 ,k 1 ) ·I (−) 12;(k 1 ,k 2 ) is not equal to the identity of CFT 2 , but is rather identified with the projection operator whose image is the invariant subspace satisfying (2.19).
There is another type of the topological interface operator I (+) gluing two free boson theories. It is given by taking the T-duality of the CFT 1 in the above construction, where S 1 R denotes the circle with radius R. The condition (2.15) is then replaced with and its explicit form is given by The gluing conditions for I (+) are written as From the above gluing conditions, one readily finds that the left and right energy-momentum tensors are separately preserved across the interface operators. Thus, I 12;(k 1 ,k 2 ) indeed represent topological interfaces [15,17], which can be deformed freely on the world-sheet. These are special cases of more general u(1)-preserving conformal interfaces [16,24]. The general gluing conditions are concisely written by O(1, 1) matrices. The superscripts (±) stand for the connected components of O(1, 1). The topological interfaces which do not preserve the u(1)-symmetries have also been discussed in [31].
Generally, there are important subclasses of the topological interfaces [18,19]. One is named the group-like defect. This class of the interfaces implements the symmetries of the CFT. The other, which includes the former, is the duality defect. This class implements the orderdisorder dualities or orbifold equivalences. In our case, I 12;(k 1 ,k 2 ) with k 1 k 2 = 1 is group-like and generates the T-dual symmetry. All other I (±) 12;(k 1 ,k 2 ) are duality defects [31,26]. Accordingly, our topological interfaces result in two different classes of the 'duality' twists.

Construction of the Models
Now, let us start the construction of the models describing non-geometric backgrounds based on the topological interfaces.

Basic Set Up
We start with a conformal system consisting of free bosons whose target space is given by where it is assumed that Then, the condition (2.15) is satisfied when setting We would like to make a 'twisted' compactification of this system implemented by the operator, . (3.5) The permutation operator P in the above acts as where |α 1 , |β 2 express arbitrary states of CFT 1 and CFT 2 , respectively. We note that I (−) is regarded as a 'self-interface' of the fiber CFT on S 1 R 1 × S 1 R 2 , and thus any product I (−) m is well-defined on its Hilbert space. More explicitly, the zero-mode part of I (−) is given as As mentioned above, the topological interface I (−) induces the duality transformations. Especially, I (−) (0) glues the zero-mode parts so as to interchange the KK momentum (winding) of the S 1 R 1 -theory with winding (KK momentum) of the S 1 R 2 -theory. In this way, we observe non-geometric nature due to the duality twist by (3.5). 3 Now, our ansatz of the partition function twisted by (3.4) is expressed in the form, Here, the base part is expanded by the contribution from each winding sector specified by λ, and the fiber part is correlated according to this base winding λ. Adopting this winding basis, the twist operator (3.4) induces a shift of the winding due to T 2πR b as well as the duality twist caused by I (−) . Every time the string wraps around the base cycle, the fiber part thus receives an additional duality twist by I (−) . Consequently, the contribution from the fiber sector with the winding λ = m ∈ Z (temporal winding) is given by where N λ is a normalization constant.
The torus partition functions with multiple insertions of the general u(1)-preserving conformal interfaces have been evaluated in [32]. The evaluation of (3.9) is much simpler and, once it is obtained explicitly, we can uniquely determine the fiber partition function with general winding Z fiber (w,m) (τ ) ≡ Z fiber λ=wτ +m (τ ) so that the total partition function (3.8) becomes modular invariant. This means that Z fiber λ (τ ) should possess the modular properties, Here, we should note that λ and τ are treated as independent variables. One may rephrase (3.10) in terms of the alternative notation Z fiber (w,m) (τ ) as Based on these relations, one can readily generate the general building blocks Z fiber (w,m) (τ ) from Z fiber (0,m) (τ ). We shall also assume the 'parity invariance', which seems physically natural and in accord with the twist operator (3.4). In particular, the normalization constants should satisfy We later choose these constants suitably.
From now on, let us analyze concrete examples.

N = M = 1 case
We first focus on the simplest case of N = M = 1. This case also serves as a guide for the general cases that we discuss later. It turns out that it suffices to set N λ = 1 in this case. We also set R 1 = 1/R 2 = R f . We then evaluate Z fiber (0,m) in (3.9) for m ∈ 2Z + 1 and m ∈ 2Z − {0}, separately.
m ∈ 2Z + 1 sectors : We first consider the m = 1 case. For the zero-mode part, we obtain (3.14) In this calculation, only the 'diagonal part' of CFT 1 ⊗ CFT 2 with p (1) survives in the trace under the insertion of I (−) . This eventually leads to the zero-mode spectrum of a compact boson of radius √ 2R f , where the KK momenta are restricted to even numbers, that is, r ′ = 2r ∈ 2Z.
The oscillator part also only includes the diagonal part of CFT 1 ⊗ CFT 2 , namely, n,ñ=1 (3.16) Combining (3.14) and (3.16), we obtain In the second line we made use of the Poisson resummation. It is easy to confirm that we reach the same result for any m ∈ 2Z + 1: The interpretation of the odd sectors as the diagonal part becomes important in the later discussions.
m ∈ 2Z − {0} sectors : The even sectors are easy to compute, since we have I (−) 2 = 1 for the N = M = 1 case. We then obtain general winding sectors : We finally determine all the winding sectors that are compatible with the modular invariance.
Let λ ≡ wτ + m ∈ Λ be the winding of the base circle. Then, it turns out that the partition functions in question are given by (3.20) • λ ∈ 2Λ + 1 : • λ ∈ 2Λ + τ + 1 : The total partition function is obtained by substituting these results into (3.8).
Note that the present model would be identified with an example of the T-folds, that is, the non-geometric backgrounds based on the T-duality twists. We emphasize that this model includes a continuous modulus R f , while most examples in the literature, of which torus partition functions are precisely calculable, are well-defined only at special points of the moduli space (say, the self-dual radius of the circle). Generally, the T-folds lie at the fixed points of the moduli space under the T-duality twists [1]. In our setting, this is translated into the fact that the twist operator I (−) , which is composed of the topological interface operators, is a 'self-interface' acting within one Hilbert space. Indeed, the Kähler modulus of the T 2 compactification is fixed under the twist induced by I (−) , though the complex structure modulus is not.

General N , M
We next consider more general cases of It seems again the simplest to first consider the sectors with the base winding (0, m). Relevant calculations are similar to those in the N = M = 1 case, but we have a few differences: • For the m ∈ 2Z + 1 sectors, we find the zero-mode spectrum of radius √ 2R 1 with the restriction that (KK momentum) ∈ 2NZ and (winding) ∈ MZ, which means • For the m ∈ 2Z − {0} sectors, we obtain the square of the partition function of a compact boson of radius R 1 with the restriction that (KK momentum) ∈ NZ and (winding) ∈ MZ, which means contrary to the N = M = 1 case. This is because I (−) 2 acts as a projection operator when k 1 > 1 or k 2 > 1.
The restrictions of the zero mode spectra given above are suitably achieved by inserting the orbifolding operators τ , (2.10). Namely, we obtain the fiber partition functions for the temporal winding sectors as: Here, i = 1 or 2. Both expressions are the same due to the relation (3.24). This also implies that the result in (3.29) is rewritten in a symmetric form with respect to R 1 and R 2 . Moreover, we should note that the normalization constants of the interface operators appearing in (2.18) have been absorbed into the redefinition of N m in these expressions (3.28), (3.29). For the special case of m = 0, we of course obtain The total partition function is now determined by the modular invariance. For notational convenience, we introduce, for ∀ λ ∈ Λ. We also define the following set where the symbol , is defined in (2.11).
With these preparations, we can write down the partition functions of the fiber CFT in general winding sectors as follows: (3.33) Here, i = 1 or 2. For λ = 0, the fiber partition function is given by (3.30). Note that for general λ follows from that for λ = m. We next determine the normalization factors N λ for ∀ λ ∈ Λ. We shall choose these constants to be the smallest positive numbers such that the q-expansion of Z fiber λ (τ ) is written in the form, for every λ. We note, for instance, the following q-expansion of the function always includes fractional coefficients taking the values in 1 N M Z up to some phase factors. We thus simply set It is easy to confirm that, with this choice of N λ , the fiber partition function Z fiber λ (τ ) is actually written in the form (3.36). We later discuss the validity of these normalization constants (3.38), after introducing the 'unitarized model' in section 4. The above choice is also regarded as specifying the weights of the summation over the winding sectors, which is equivalent to that over the interfaces along various cycles, as discussed shortly in the next subsection.
Let us finally comment on a useful rewriting of the relevant partition functions. If recalling the analysis for the simplest case N = M = 1, the partition functions for the odd sectors are expected to be interpreted as the 'diagonal parts' of the even sectors. We here demonstrate this is indeed the case. Set for ∀ λ ∈ Λ. In terms of this, for the even sector ∀ λ ∈ 2Λ − {0}. Moreover, the partition function (3.28) for the sector to preserve the modular covariance, as is illustrated in Appendix B. Thus, by the modular transformation, (3.41) extends to the case with general λ ∈ Λ − 2Λ: In this way we can write down the total partition function in a compact form 4 It is also easy to confirm that (3.43) actually reduces to the total partition function given in section 3.2, when setting N = M = 1.

Interface Operator I (−) along Various Cycles
In our argument so far, λ = wτ + m ∈ Λ has represented the winding along the base circle. 5 From the point of view of the fiber sector, which is correlated to the base sector according to λ, one may also regard it as specifying the homology cycles of the world-sheet torus along which the interface lies. This would become evident, once we take the standard parameterization of the homology cycles so that they are labeled by γ ∈ Λ(≡ Zτ + Z) and the 'basic spatial (temporal) cycle' is associated with γ = 1 (γ = τ ). To be more specific, let us consider the interface along the cycle γ ∈ Λ created by the operator I (−) in (3.5). We denote the corresponding fiber partition function with this interface inserted by 'Z fiber [I (−) γ ](τ )'. It is quite natural to interpret Z fiber λ (τ ) given in (3.33) and (3.34) to be this partition function for γ = λ ∈ Λ up to the normalization factor N λ ; namely, (3.44) Indeed, for λ = m ∈ Z, the left side reduces to (3.9), which represents the m-fold insertion of the interface operator along the basic spatial cycle. By the modular transformation, such a picture extends to the case of general λ. We take (3.44) to define the homology cycles for the interfaces in the following discussion.

Relation to the 'I (+) -twisted' Models
It would be a natural question what happens if we consider the models twisted by I (+) in place of I (−) . As mentioned in section 2.3, I (+) is obtained by taking the T-duality for the CFT 1 as in (2.21). This means that In the special case withR 1 = 1/R 1 = R 2 and k 1 = k 2 = 1, the interface operator I (+) composed of I (+) becomes the permutation operator P. Therefore, the I (−) -model for the Along the fiber direction, this indeed acts as the permutation (X 1 , X 2 ) → (X 2 , X 1 ).
Let us evaluate the partition function of this permutation orbifold. It is convenient to introduce the new coordinates, Then, the permutation P acts as Namely, P acts as the ordinary reflection orbifold with respect to the X − -direction. The insertion of P into the trace only leaves the Fock vacua with On the other hand, the X + -direction is compactified onto the circle with the radius √ 2R 2 , but the zero-mode spectrum is constrained as KK momentum ∈ 2Z, winding ∈ Z, (3.55) due to the condition (3.54). Thus, we eventually obtain the fiber partition function for the sector λ ≡ m ∈ 2Z + 1, We also obtain Z fiber for λ = m ∈ 2Z, since P 2 = 1.
Based on these results and the modular invariance, we can uniquely determine the fiber partition functions Z fiber λ (τ ) with ∀ λ ∈ Λ, which are equal to those of the I (−) -model with R 1 = 1/R 2 and N = M = 1, as is anticipated. In this way, we have seen that the simplest case N = M = 1 reduces to a geometrically realized model, that is, the orbifold by (3.51). A similar construction of the partition function involving the permutation is given in [9]. However, in generic cases NM > 1, our models would be still non-geometric even in the I (+) -picture, because the gluing condition (2.25) non-trivially restricts both of the KK and the winding spectra.

Unitarized Models
Let us discuss the unitarity of the modular invariant models we constructed in the previous section. The N = M = 1 model obviously leads to a unitary spectrum as in ordinary Z 2orbifolds. However, in the cases of NM > 1, the spectrum gets non-unitary. For example, one can observe a non-unitary q-expansion in the sector with base winding w = N, 6 that is,

Definition of the Unitarized Model
Therefore, we shall propose the 'unitarized' model for the NM > 1 cases. It turns out that one can construct a simple unitary extension of the present model at least in the cases, which satisfy the condition (2.15). We would like to discuss more general cases with generic R 1 and R 2 satisfying (2.15) in the future work.
Let us proceed with the assumption (4.2). Our construction of the unitarized model is simple. All we have to do is to replace the set S Consequently, the total partition function of the unitarized model is defined by In other words, new twisted sectors are created when gcd{|w|, |m|, K} > 1, which are shown to recover the unitarity of the spectrum. We also note that S (K) [λ] has a periodicity, We discuss later how the new partition function Z fiber λ (τ ) is interpreted to be generated by multiple insertions of I (−) along various cycles of the world-sheet torus.

Proof of Unitarity
We here demonstrate that the proposed model given by (4.8) is unitary. Namely, we show that the total partition function (4.8) is indeed q-expanded only with positive integral coefficients, after making the Poisson resummation with respect to m ∈ Z in λ ≡ wτ + m.
We proceed by three steps: First, since the total partition function looks quite intricate, we analyze a simpler part of the partition function in (4.4). Second, using the result in the first step, we show that the sectors with even w in λ = wτ + m have a q-expansion with positive integral coefficients. Finally, we show a similar statement for the sectors with odd w.
We first examine the simpler partition function Z (0) λ (τ ) defined in (4.4). With this aim it is convenient to introduce the phase factor defined by (4.11) where , τ is defined in (2.11) (see also Appendix A). By the subscript, we have made the τ -dependence explicit to avoid confusion. Since Nα · Z R f (τ ). (4.14) In the second line, we made use of the assumption gcd{N, M} = 1. Note here that τ The phase factor ǫ (N ) j (λ, γ; τ )ǫ (M ) j (λ, γ; τ ) is interpreted as the discrete torsion [30] in the theory of orbifolds, and its insertion does not break the unitarity. Actually, it turns out that the sector of the partition function (4.8) with w even or odd separately leads to a unitary q-expansion. To see this below, it is important to observe that (4.14) is explicitly evaluated as Here, we set α = aτ + b,α =ãτ +b, summed over b ∈ Z N ,b ∈ Z M and used the notation (2.6).
This expression is suited for making the Poisson resummation, since the temporal winding m appears only in the phase factor e 2πim jM a N +j Nã M .
(ii) unitarity of w ∈ 2Z sectors : Now, we are ready to show the unitarity of the model or the partition function in question, that is, Z(τ ) in (4.8). We first focus on the case of λ = wτ + m with a fixed value w ∈ 2Z, and consider the Poisson resummation with respect to m.
For the cases of m ∈ 2Z, the fiber partition function is equal to the square of (4.4), that is, Then, it is straightforward to make the Poisson resummation over m ∈ 2Z with the help of (4.15). Note that the phase factor appearing in (4.15) just shifts the KK momentum along the base circle as where i = 1, 2 labels the contributions from the two factors of Z (0) λ (τ ). We thus obtain (4.18) It is not difficult to see that the sectors with 'non-diagonal' quantum numbers (j 1 , a 1 , . . .) = (j 2 , a 2 , . . .) are expanded in a q-series with positive integral coefficients. Note that the factor 1 2 appearing in (4.18) indeed cancels out due to the degeneracy of relevant contributions. Furthermore, the partition functions for the 'diagonal' sectors with j 1 = j 2 ≡ j, a 1 = a 2 ≡ a, and so on, are rewritten by using the product formula of theta function (A.8) as The 'residual terms' in the above denote some q-series with positive integral coefficients whose explicit form is not important here.
(iii) unitarity of w ∈ 2Z + 1 sectors : Let us examine the remaining cases with w ∈ 2Z + 1. By means of (B.5), we obtain the following: • For m ∈ 2Z, • For m ∈ 2Z + 1, In these evaluations, the phase factors were slightly non-trivial again. In (4.26), for example, we made the following evaluation, This means that the summation over m ∈ 2Z + 1 is written in the same form of q-series as that for m ∈ 2Z, but with an extra sign ±1 in each term. Therefore, the partition function again produces a unitary q-series. Similarly to the twisted sectors of the familiar Z 2 -orbifolds, the negative terms due to the oscillator part cancel with the corresponding positive terms.
In this way, we have succeeded in showing the unitarity of the proposed model (4.8).
We add a few comments: • It is worthwhile to point out that there exists a unique vacuum with minimal conformal weights h =h = 0 (the 'identity state') in the spectrum read off from the partition function (4.25) for the sector w = j =j = a =ã = r = 0. This fact would suggest that our choice of the normalization constants (3.38) is reasonable.
• As clarified in the above discussion, the even sectors with λ ∈ 2Λ are reinterpreted as the orbifold with the discrete torsion. Nevertheless, the total partition function (4.8) is not likely to be described by any model of orbifold in the precise sense, except for the simplest case N = M = 1. This is because the operator D [λ] is hard to be explicitly realized in terms of an automorphism acting on the Hilbert space of CFT.

Interpretation of the Unitarized Model : Multiple Insertions of the Interface Operators
At the last of this section we would like to discuss a possible physical interpretation of the unitarized model described by (4.8), or the fiber partition functions Z fiber λ (τ ) given in (4.5) and (4.6). Let us start by extending the formulas (3.49), (3.50) to the cases of multiple insertions of the interface operator I (−) along different cycles, Now, let us focus on the partition functions Z fiber λ (τ ) given in (4.5), (4.6). We attempt to rewrite them in the forms of (4.31) and (4.32). Recall that the partition function in the even sector (4.5) is obtained by replacing the sets S ( * ) [λ] appearing in (3.33) with their 'hatted' counterparts (4.3), and in the odd sector by acting with the diagonal operator D [λ] . We again begin our analysis with the decomposition λ = n λ λ 0 for ∀ λ ∈ Λ−{0} with λ 0 ∈ Λ 0 , n λ ∈ Z. We also introduce the unique element λ 1 ∈ Λ 0 by requiring the condition λ 1 , λ 0 = 1. Then, making use of the assumption gcd{N, M} = 1, we find  To be more precise, it is straightforward to show The factor 2 of the subscript in the latter operator assures that the cycle is even, though the actual action is implemented also by P

(N )
λ+N M λ 1 due to (3.48). Furthermore, the partition function for the odd sector (4.6) is rewritten in the form of (4.32), that is, These are the equalities we have looked for, and show that the fiber partition functions Z fiber λ (τ ) are interpreted to be generated by multiple insertions of the interface operators along different cycles.

Summary and Discussion
In this paper, we studied simple models of non-geometric backgrounds based on the worldsheet CFT equipped with the topological interfaces [15,17]. The topological interfaces there composed the twist operators similar to those in orbifold CFTs, and we performed the twisted compactification of the Scherk-Schwarz type [28,29]. When crossing the defect lines on the world-sheet created by the interface operators, the KK momenta and the winding modes are mixed, which embodies a 'non-geometry' (or the 'stringy geometry', possibly). Requiring the modular invariance, we needed to sum up all the winding sectors with respect to the Scherk-Schwarz circle, or the base space. The fiber CFT, on the other hand, was made up of the worldsheet with the interfaces aligned suitably, so as to correlate in a modular covariant manner to the winding numbers of the base circle.
In our concrete studies, two types of the topological interfaces appeared. One is the grouplike defect and the other, which includes the former, is the duality defect [18,19]. The group-like defect corresponds to the case of N = M = 1 in section 3.2, and resulted in an example of the T-fold, that is, the Scherk-Schwarz compactification twisted by the T-duality. As we clarified in section 3.5, this model is also T-dualized to the permutation orbifold, which manifestly yields a unitary theory.
On the other hand, the cases of NM > 1 given in section 3.3 are described in terms of the duality defects that do not implement symmetries in the strict sense. The obtained partition function (3.43) looks rather intricate and the unitarity is not realized as it stands. As mentioned in section 2.3, the topological interfaces in these cases implement a projection that restricts the zero-mode spectrum, instead of an automorphism of the Hilbert space. In order to maintain the unitarity, or conserve the probability, one may thus need to complement this projection by supplying 'twisted sectors' which may differ from those in ordinary orbifold theories. Quite interestingly, a 'unitarization' was indeed possible at least for the cases of R 1 = R 2 = N M , as demonstrated in section 4. Furthermore, the unitarized model is found to be described by the world-sheet with multiple insertions of the interfaces along different cycles. While the even sectors of this model might be identified with some orbifold with the discrete torsion [30], the total partition function itself does not seem to correspond to any orbifold conformal theory. 8 For future work, if intending the applications to string compactifications, we would have several tasks to be done: • the construction of the unitarized models for general radii R 1 , R 2 satisfying the condition (3.2); • the extension to the case of the fiber CFT on higher dimensional tori; • the supersymmetric extension by means of superconformal interfaces; • the target-space interpretation of the insertions of the interfaces and the resultant models.
Especially, the topological interfaces in N = 1 SCFTs on higher dimensional tori have been elaborated in [26]. It has been clarified there that the duality (or the 'orbifold equivalence') defects which do not preserve the charge lattice of string theory generically correspond to the O(d, d; Q)-group for d-dimensional torus. It is an interesting question whether one can construct the unitary and modular invariant models based on such O(d, d; Q)-interfaces as extensions of the unitarized model proposed in this paper. It would also be interesting to see, as mentioned in [26], that an object representing the 'quasi-symmetry' survives even under the α ′ corrections and plays a definite role in string theory.
Moreover, a natural direction of the future studies would be the extensions to non-trivial cases with N = 2 SCFTs, say, the Gepner models as the fiber CFT, in which one would handle the duality defects inducing the mirror transformations. Of course, one may imagine the models of Scherk-Schwarz compactifications with the self-dual mirror twisting acting on some N = 4 fiber SCFTs, similarly to [36]. These cases may be described by the group-like defects and reduce to asymmetric orbifolds. However, more general duality defects made up of the N = 2 topological interfaces would be capable of producing a much broader class of superstring vacua, which would be fairly non-trivial and curious.
Another interesting direction to be pursued would be the possibility of more general unitary models by multiple insertions of the interface operators. In section 4.3, we chose particular cycles along which the interfaces are aligned, in order to obtain the unitarized model (4.8). However, it is not yet clear what the general principle is, in order for the configuration of the interfaces to realize sensible unitary models. In any case, we hope that this work would serve as a step to construct a novel type of string vacua based on the world-sheets equipped with the topological interfaces or the interface operators composed of them. Here, we have set q := e 2πiτ , y := e 2πiz . We use abbreviations, θ i (τ ) ≡ θ i (τ, 0) (θ 1 (τ ) ≡ 0), Θ m,k (τ ) ≡ Θ m,k (τ, 0). We also set Z (k) [u, v](τ ) := 1 |η(τ )| 2 Θ u+v,k (τ )Θ −u+v,k (τ ), (A.7) which is often used in the main text.
As a preliminary, we start with a general function F (w,m) (τ ) (w, m ∈ Z) of the form such as F (w,m) (τ ) = ℓ,l n,ñ∈Z c(ℓ,l, n,ñ; w, m) q ∆(ℓ)+n q ∆(l)+ñ , (B.1) that possesses the following properties, We identify the 'partition function' whose diagonal part is of our interest as and regard the two factors F (w,m) (τ ) as the contributions from CFT 1 and CFT 2 . 9 9 Of course, a natural extension of the argument given here would be the diagonal part operator acting on a more general partition function such as Z (w,m) (τ ) ≡ F (w,m) (τ ) F (w,m) (τ ), F (w,m) (τ ) = F (w,m) (τ ). In this paper, however, it is sufficient to restrict to the simple case (B.3).