A Worldsheet Perspective on Heterotic T-Duality Orbifolds

Asymmetric heterotic orbifolds are discussed from the worldsheet perspective. Starting from Buscher's gauging of a theory of D compact bosons the duality covariant description of Tseytlin is obtained after a non-Lorentz invariant gauge fixing. A left-over of the gauge symmetry can be used to removed the doubled constant zero modes so that D physical target space coordinate remain. This can be thought of as the worldsheet realization of the strong constraint of double field theory. The extension of this description to the heterotic theory is straightforward as all results are written in terms of the invariant and the generalized metrics. An explicit method is outline how to obtain a generalized metric which is invariant under T-duality orbifold actions. It is explicitly shown how shift orbifolds lead to redefinitions of the Narain moduli. Finally, a number of higher dimensional T-folds are constructed including a novel asymmetric Z6 orbifold.


Introduction and conclusions Motivation
The heterotic string [1][2][3] provides an unified framework to describe all interactions among elementary particles and gravity. Compactifications of this theory to the four dimensions we experience today may be performed on six-dimensional symmetric toroidal orbifolds [4,5] without giving up full string computability. Unfortunately, such orbifolds and their related Calabi-Yau compactifications leave many (geometric) moduli unfixed. This impairs predictability of the resulting models as many physical parameters will be ultimately related to undetermined values of these moduli.
Untwisted moduli of such compactifications may be removed by considering asymmetric orbifold actions treating left-and right-moving string coordinate fields differently by modding out an intrinsically stringy duality symmetry [6]. In the most elementary realization it occurs in a compactification on a circle where the theory at a certain radius R is identified with another compactification on a circle with radius 1/R (in string units). In a single construction this is only possible if the radius itself is fixed at the string scale R = 1. More in general T -duality orbifolds result in quotient spaces often referred to as asymmetric orbifolds [7]. As such they can be considered as fully computable non-geometric string backgrounds [8][9][10] or so-called T -folds [11,12]. Asymmetric orbifolds have been considered by various research groups [13][14][15][16][17][18][19][20][21][22]; more recent works are e.g. [23][24][25][26]. Asymmetric twists in free fermionic models to stabilize untwisted moduli were exploited in [27]. The present work focusses on the heterotic setting only, possible dualities to type-II non-geometric compactifications were considered in [28].
In a recent paper [29] a comprehensive framework to study asymmetric toroidal compactifications of the heterotic string was provided. That work mainly focussed on their construction as generalizations of Narain toroidal compactification modded out by T-duality group elements. The current paper is complementary to that one in the sense, that it aims to give an explicit duality covariant bosonic worldsheet description of Narain orbifolds. Asymmetric compactifications of the heterotic string are normally discussed on the level of the torus partition function only, without specifying the underlying bosonic worldsheet theory. An explicit worldsheet description is provided by the free-fermionic formulation of the heterotic string [30][31][32]. This description predominantly accommodates 2 duality symmetries. In order to obtain an explicit bosonic worldsheet description a Buscher's gauging is used and subsequently gauged fixed in such a way that a duality covariant description by Tseytlin is obtained at the expense of manifest worldsheet Lorentz invariance. The basic structure is first exposed for a worldsheet theory of D bosonic fields and after that extended to the heterotic theory with D right-moving and D + 16 left-moving degrees of freedom.
In addition, this paper gives an extensive description of shift heterotic orbifolds and show that they all can be viewed as Narain torus compactifications. A computational method is provided how the new moduli can be determined from the original ones and the applied shift actions. It is demonstrated in various concrete examples that the orbifold shifts may be geometric or non-geometric and may be accompanied by actions on the gauge part of the Narain lattice.
Finally, since the number of explicitly known truly stringy higher dimensional T-folds is very limited, for examples see e.g. [33,34], this paper provides a number of non-trivial higher dimensional T-fold examples illustrating the developed methodology.

Outline of the paper's main results
The main results presented in this paper have been structured as follows: Section 2 starts from the textbook worldsheet action of D compact bosons. After a Buscher's gauging is applied, a gauge fixing is chosen that breaks manifest Lorentz invariance in favor of achieving a duality covariant doubled worldsheet theory first considered by Tseytlin. A residual gauge symmetry is uncovered that removes the doubling of the constant zero modes. The remaining constant zero modes then have the interpretation of the physical target space coordinates. (This may be thought of as a worldsheet realization of the strong constraint of double field theory [35][36][37].) The worldsheet supersymmetry transformations for this doubled worldsheet theory are derived. In addition, the oneloop partition function is obtained for the doubled theory. Here the boundary terms, which provided the equivalence between the original theory of D compact bosons and Tseytlin's formulation, lead to an additional phase factor that cancels out any dependence on the "doubled winding numbers". This ensures that the partition function of the doubled theory is identical to that of the original theory of D compact bosons.
Section 3 discusses the generalization of these results to the heterotic theory. This generalization is straightforward since all results in Section 2 have been written in terms of the generalized metric and the duality invariant metric which have natural extensions in the heterotic context. On the level of the partition function this reproduces the known results for Narain compactifications of the heterotic string.
Section 4 develops the description of orbifold twisting of the Narain theory from the worldsheet point of view. This section recalls the Narain space group description introduced in [29] and provides a novel way to explicitly construct invariant 2 -gradings and associated generalized metrics of Narain orbifolds. If the action is asymmetric, i.e. not isomorphic between the right-and left-movers, so called T-folds are obtained. From the boundary conditions of the worldsheet coordinate fields the full orbifold partition function can be computed. One complication is that it is not a priori clear that an invariant generalized metric exists for a given finite orbifold action on the Narain lattice. If not, no asymmetric orbifold can be associated to this action. Here an explicit procedure is outlined how such an invariant generalized metric can be obtained.
Section 5 considers special orbifold actions that have trivial twist parts. It is shown that orbifold shift actions do not lead to new geometries but rather modify the moduli of the Narain compactification. In particular, an explicit description is provided how the new moduli can be computed from the old ones. This is illustrated for a number of simple yet interesting cases such as geometrical and non-geometrical shifts combined with non-trivial Wilson lines.
Finally, Section 6 provides a number of examples of higher dimensional T-folds. The first two examples are Narain orbifolds obtained by applying the basic T-duality twist to all D compact dimensions. One acts only on the right-movers hence it fixes all moduli but is only supersymmetric in D = 4 or 8 dimensions, while the other only acts on D left-movers preserving all supersymmetries in any dimension and the Wilson lines are left free. In a further example a 6 action is realized in an asymmetric way. This provides one of the first explicitly known examples of higher order asymmetric orbifolds in four dimensions. This paper is concluded with three Appendices that provide some technical background for the results obtained in this work. Appendix A introduces the notation used to evaluate partition functions. Appendix B describes the underlying modular transformations. Finally, Appendix C gives further details of the description of Narain moduli and their transformations as uncovered in [29].

Doubled Target Space Torus
The Minkowskian worldsheet is parameterized by the worldsheet time σ 0 and space σ 1 coordinates. The starting point of the description of D bosons X T = X 1 , . . . , X D ), the internal coordinates fields, on the worldsheet is the action where G is a D-dimensional metric of and B an anti-symmetric tensor on a target space torus T D . Throughout this work these background quantities are taken to be constant. The target space torus periodicities are encoded in integral lattice identifications the geometrical aspects of the torus T D have already been taken into account by the metric G in the worldsheet action (2.1). A duality covariant description of this theory can be obtained following Buscher's gauging procedure [38] and subsequently choosing an appropriate gauge [39]: The coordinate fields X are promoted to possess the following gauge transformations where λ T = λ 1 , . . . , λ D ) are general functions on the worldsheet. To ensure invariance of the action (2.1) the derivatives are promoted to gauge covariant ones (where µ = 0, 1) which are gauge invariant, provided that the gauge fields A µ themselves transform as This gauging would remove all physical bosons from the worldsheet. To avoid this, the gauged action is complemented by a Lagrange multiplier field X , which enforces that the gauge field is pure gauge: The Lagrange multiplier fields X satisfy similar periodicities as the coordinates X themselves: so that the charges are integral in the Euclidean theory, hence the periodicities (2.7) of X result in a trivial phase exp{−2πim T q} = 1 in the path integral. A gauge, that makes the duality manifest, is [39]: 1 Clearly, this gauge breaks manifest Lorentz invariance on the worldsheet. In this gauge Inserting this in the action (2.6) and performing a partial integration to remove the derivative ∂ 1 on the remaining gauge field component A 0 , shows that the equation of motion of A 0 is algebraic: Eliminating all A 0 dependence using this expression, shows that the action can now be cast in the Tseytlin's form [40,41] by two further partial integrations: where Y T = X T X T combines the coordinates X and the dual coordinates X in a single 2Ddimensional vector. In addition, the generalized metric H and the O(D, Given the periodicities, (2.2) and (2.7) of the coordinates fields X and their duals X, the doubled coordinates Y are subject to the periodicities 1 Another gauge choice would be A0 = 0; but for the current purposes this choice would be less convenient.
Just as the periodicities of the dual coordinates X were enforced by charge quantization (2.8), the periodicities of X can be understood in the same fashion, as by a duality transformation the roles of the coordinates X and their duals X can be interchanged. Consequently, the Tseytlin action (2.11) is invariant under the M ∈ O η (D, D; ) duality transformations since by definition M T η M = η and the lattice 2D is mapped to itself: In addition, the generalized metric and the O(D, D; Ê)-invariant metric (2.12) satisfy the following properties This allows to define a 2 -grading (2.16) The one but last relation implies that Z itself is an element of the duality group with real coefficients: In the derivation of (2.11) three partial integrations were performed. Since the coordinate fields X and duals X are quasi-periodic (but not periodic) in general, the resulting boundary terms do not automatically vanish. In particular, because of (2.7) the first term gives a boundary contribution which can be set to zero by a further gauge fixing. Indeed, the gauge fixing (2.9) does not fix the gauge completely; there are residual gauge transformations with gauge parameters λ = λ(σ 0 ) , which are functions of the worldsheet time σ 0 only. Using this residual gauge transformation, a further gauge fixing can be enforced. In the combined gauge (2.9) and (2.19) the boundary action reduces to Since the gauge transformation (2.5) of the gauge fields involves derivatives, even this does not fix the gauge completely: Constant shifts λ = λ 0 in (2.3) are still allowed. This means that the doubled coordinates Y used in (2.11) are uniquely defined up to constant shifts in D directions for some M ∈ O η (D, D; ) defining the used duality frame and λ 0 ∈ Ê D . In other words of the 2D constant zero-modes of the doubled coordinate fields Y only D are physical, assuming that (2.1) should be taken as the starting point of the worldsheet description.

Worldsheet Supersymmetry
The worldsheet action (2.1) can be extended to with D right-moving real fermions ψ . Here left-and right-moving coordinates (σ,σ) and their associated derivatives (∂,∂), were introduced, so that the two-dimensional measure can be written as d 2 σ = dσ 1 dσ 0 = dσdσ . This action is invariant under the supersymmetry transformations δX = ǫ e −1 ψ and δψ = −ǫ e∂X , (2.24) where e is a vielbein associated the metric G = e T e . Applying the Buscher's gauging to this action leads to where DX = ∂X + A , DX =∂X + A and F = F 10 =∂A − ∂A. This action is still supersymmetric, provided that the transformations (2.24) are extended to δX = ǫ e −1 ψ , δψ = −ǫ e DX and δ X = −ǫ e T + Be −1 ψ . (2.26) Since the Tseytlin's form of the action was obtained by using the gauge A 1 = 0, the supersymmetry variation of ψ then becomes Hence, using the doubled coordinate Y , the supersymmetry transformations can be cast in the form: These transformations can also be obtained directly from the duality covariant action (2.11) extended with the right-moving fermions ψ: where the left-and right-moving derivatives ∂ and∂ of the doubled coordinates Y have been separated. Since the action of the right-moving fermions ψ involve the left-moving derivative ∂ only, any supersymmetry that involves these fermions can only be related to the term involving the operator 1 2 H − η) since that term also contains this derivative. By introducing the generalized vielbein where η is the Minkowskian metric with signature (D, D), the matrices can be related to the left-and right-projections defined by the 2 -grading Z leading to the projection operators Inserting (2.32) in the action (2.29) leads to This suggests the following supersymmetry transformations in terms of two 2D × D-matrices U and V. These transformations leave the action (2.34) invariant, provided, that U = −V. Moreover, given the form of the right-moving projector P R , given in (2.33), one may set Making this choice, the supersymmetry transformations (2.28) are recovered using the expression of the generalized vielbein (2.30). The closure of the supersymmetry transformations on the fields ψ and Y reads

One-Loop Partition Function
To determine the one-loop partition function by computing the path integral on the worldsheet torus, a quantum version of the Tseytlin's action (2.11) is required. As this is a gauge fixed action, following the standard BRST-procedure the full quantum Euclidean action reads where h are D Lagrange multipliers enforcing the gauge A 1 = 0 and b, c form D associated ghost systems. The fields h and A 1 can be trivially integrated out without leaving a trace. The one-loop periodic boundary conditions for the ghosts b, c and the quasi-periodicities for the doubled coordinate fields Y with N, N ′ ∈ 2D , are solved by the off-shell mode expansions using the definitions (A.12) and (A.14). The prime on the sum denotes the sum over all integers r, r ′ ∈ excluding the zero-mode r = r ′ = 0 contribution. Inserting these mode expansions in the remaining path integral and evaluating the infinite dimensional integrals over the mode coefficients b r r ′ , c r r ′ and Y r r ′ leads to an expression involving infinite products: (2.42) Notice that the infinite product factors ′ r due to the ghosts and the ∂ 1 -derivative on Y in the worldsheet action (2.39) cancel. In fact, since this is a pure constant, i.e. not τ -dependent, infinite factor, it may be dropped from the path integral altogether. 2 Using the properties (2.15) and (2.16) it follows, that is independent of the moduli G and B for any two complex constants a, b ∈ . (A derivation of this result in a more general Narain context can be found in appendix C.3.) Consequently, the remaining infinite product factor reduces to which can be expressed in terms of the Dedekind eta-function η(τ ) using (B.5).
The phase (−1) N T η N ′ in (2.42) is modular invariant by itself. The appearance of this phase may seem somewhat surprising, since computing the partition function for X directly would not lead to this phase factor. The cause of this can be traced back to the observation that the worldsheet actions (2.1) and (2.11) are equivalent to each other up to the boundary contribution (2.20), which can be evaluated on the Euclidean worldsheet torus to using the worldsheet torus coordinates x, y defined in (A.7). This thus leads to precisely the same phase in the path integral and hence they cancel out. (In any event, since partition functions are defined from the path integral up to modular invariant phases, we are always free to include this factor once more, so that it cancels out.) Including this phase means that the sum over N ′ is trivial giving rise to an infinite constant factor, which may subsequently be dropped. Observe that this is very different to what happens to the quantum numbers n ′ ∈ D that label the worldsheet boundary conditions in the τ -direction in the original theory of a D-dimensional target space torus. In that case n ′ are physical, as they can be interpreted as the Poisson resummed Kaluza-Klein numbers. In the doubled formalism both winding and Kaluza-Klein quantum numbers are contained in N simultaneously and hence there is no need for N ′ .
The partition function can then finally be written as It is possible to express this partition function as in terms of left-and right-moving momenta, P L = P L P , P R = P R P and P = E N , (2.48) are defined using the left-and right-moving projectors (2.33) and the generalized vielbein (2.30).

Relation to Double Field Theory
Double field theory [35][36][37] is an attempt to obtain a duality covariant target space description of string theory. There the dimension D of a target space torus is doubled to 2D and the generalized metric is assumed to be the metric on the doubled torus. However, since only D coordinates are physical, a so-called strong constraint is being implemented by hand to remove D of the 2D doubled coordinates.
The doubled worldsheet description of strings on a D-dimensional torus presented here should not be confused with a worldsheet theory where the target space is a torus of the dimension 2D. There are several important differences that appeared by performing the Buscher's gauging procedure and the subsequent gauge fixing: Hence, in particular, the removal of D of the 2D doubled torus coordinates is not enforced by hand, but rather is a left-over consequence of the gauge fixing procedure, which did not fix the gauge completely. The other consequences mentioned here have no interpretation in the target space theory: A (none Lorentz invariant) worldsheet action on an one-loop torus with two different cycles is simply not part of the target space description.

Narain Lattice Worldsheet
The extension to the heterotic string theory can be obtained by replacing the O η (D, D; Ê)-invariant metric η (as given in (2.12)) and the generalized metric H by their heterotic counter parts: where the 16-dimensional Cartan metric g 16 is given by (C.2), and with C = B + 1 2 A T A , in the Tseytlin's action. This leads to where the generalized coordinate vector Y T = X T X T χ T is extended to include 16 bosonic gauge degrees of freedom χ , satisfying the torus periodicities ) except for the last one, which reflects that there is a mismatch in left-and right-moving bosonic degrees of freedom in the heterotic theory.

Worldsheet Supersymmetry
The action (3.3) can be written as Also in the Narain case a generalized vielbein can be defined by (C.7) satisfying (C.10). Hence, the matrices are related to left-and right-projection operators P L = 1 2 (½ + Z) and P R = 1 2 (½ − Z) and and can therefore be used to rewrite the action (3.6) as Following the second method of identifying the supersymmetry transformations discussed in Subsection 2.2 one obtains since the inverse generalized vielbein is given by (C.9). Again, the closure of the supersymmetry transformations on the fields ψ and Y can be expressed as

One-Loop Partition Function
The properties (3.5) imply, that the determinant relation (2.43) changes to which is again independent of all moduli; in this case G, B, A . This is derived in appendix C.3, The full one-loop partition function is then given by In the Narain partition function (3.14c) there is only the sum over N but not over N ′ , just like in the doubled partition function (2.46): A similar phase factor, as discussed there, has been included to ensure that the sum over N ′ just results in an irrelevant constant infinite factor. The partition function of the non-compact bosons representing Minkowski space in light cone gauge is modular invariant by itself: Finally, the modular properties of the Narain partition function read Hence, the full partition function is modular invariant. The modular transformations of Z Mink and Z Ferm are rather standard and follow directly from the properties of the Dedekind and the theta-functions recalled in Appendix B. The modular transformations of the Narain partition function as given here are less standard and therefore it is instructive to explain them in more detail: The first relation (3.17) follows upon using that under τ 1 → τ 1 +1 the Narain lattice part in (3.14c) is invariant up to a factor exp{2πi 1 2 N T ηN }. Given the form (3.1) of η, it follows that N T ηN ∈ 2 so that this factor is simply equal to unity. Hence, only each of the 16 factors η(τ ) give rise to the non-trivial phase in the first relation (3.17); see the modular transformation (B.4) of the Dedekind function.
The second equation in (3.17) results from a Poisson resummation, which can be cast in the form since both η and η −1 are integral as |det η| = 1. Applying this to the Narain lattice sum gives upon shifting the integration variables Y . The integral over Y can be evaluated to using (3.12) and assuming that D is even. The sum over M is identical to the original sum over N except for the kernel

Point Group and Orbifold Action
Orbifolds are obtained by enforcing invariance of the theory under the action of a finite point group P. The total number of elements in the point group is denoted by |P|. The action (3.3) is preserved by orbifold transformations of the form: on any vector y (not just the Narain coordinate fields Y ), provided that for all θ ∈ P and N ∈ 2D+16 . The final condition results from the requirement, that the doubled torus periodicities (3.4) need to be respected. The last two conditions imply that The identity element θ = 1 has R 1 = ½ and V 1 = 0. The composition of point group transformations If the order of the point group element θ ∈ P is denoted as |θ|, it follows that θ |θ| [Y ] ∼ Y . Writing this out explicitly, leads to where the projectors P θ , on the directions in which R θ act trivially R θ P θ = P θ R θ = P θ , are defined as Since the action is invariant under a shift Y 0 of the origin of the coordinate system defined by Y , the vectors V θ may be redefined as These transformation may be used to set a certain number of components of the vectors V θ to zero; but these are never in the directions in which R θ act trivially.

Space Group
When the point group P is combined with the lattice identifications the so-called space group S is obtained. The space group is parameterized by element g = (θ, N ) ∈ S where θ ∈ P and N ∈ 2D+16 . The space group is generated by the elements Hence, a general element of the space group acts as: Notice that the order in which these actions are applied is important here, since does not equal (θ, N )[Y ] defined above. Indeed, since the space group acts on any vector y not just the Narain coordinate fields Y , the general composition rule reads where the composition of the orbifold actions (4.4) has been used. The space group S is in general non-commutative (even if the point group P is). Two space group elements g = (θ, N ) and (4.12)

Narain Orbifold Fixed Points
Fixed points Y fix of a space group element g ∈ S are defined by the condition By bringing the first term on the right to the left-hand-side shows that this equation only has solutions provided that P θ (V θ + N ) = 0, hence the fixed point condition becomes which can be solved by the expression: To derive that this determines the fixed points, first observe that while the matrix ½ − R θ is not invertible, it is invertible on the subspace defined by P ⊥ θ : Only one of the eigenvalues, exp(2πi j/|θ|) for j = 0, . . . , |θ| − 1, of R θ is equal to unity, but that one is excluded by this projection operator. On the corresponding subspace one can show that by making a general power expansion in R θ of the left-hand-side and multiplying this by ½ − R θ and requiring that this equals P ⊥ θ . This determines the expansion uniquely up to adding an arbitrary constant to P θ . This constant is fixed by requiring that projecting with P θ should give zero.

Orbifold Compatible Residual Gauge Symmetry
Not all fixed points identified by this equation are physical. Only those fixed points that cannot be removed by the residual gauge transformations (2.21) uncovered in Section 2 are physically distinct. Given the boundary conditions for the doubled coordinate fields, the constant gauge parameters Λ 0 are subject to where L ∈ 2D+16 . In order that this gauge symmetry is compatible with the orbifold action, the constant zero model gauge parameters have to be modified to 3 where Λ fix are particular solutions to (4.17) which are non-vanishing only if L = 0 and have to be of the form (λ 0 , 0, 0). In addition to the particular solution the homogeneous part of (4.17) admits an arbitrary solution using P P , that projects on the invariant subspace of the whole point group P. For symmetric orbifolds the second term in (4.18) is trivial (or points only in none orbifolded directions) and Λ fix define the fixed points on the original torus which can be removed by the gauge symmetry. For asymmetric orbifolds Λ fix does not lie in the subspace (Ê D , 0, 0) and is therefore not admissible.

Construction of an Orbifold Compatible Generalized Metric
The first condition in (4.2) is not so much a condition on the integral representation matrices R θ , but rather an existence condition of an appropriate generalized metric H and therefore the Narain where η is given in (C.10), of the point group P has been constructed. Hence, these matrices may be displayed as (4.20) Next, consider the matrix M inv constructed as from a generic real 2D + 16 × 2D + 16-matrix M 0 , such that M inv is invariant under the full point group P. Indeed, under any element θ of the point group, this generalized vielbein M inv maps to itself: since θ ′ ∈ P and θ ′′ = θ ′ θ ∈ P both label the full point group P . Thus the matrix M inv would be a candidate for an invariant vielbein E inv , from which the Narain moduli G, B, A could be read off, provided that one chooses it such that it satisfies (C.11). However, solving this non-linear constraint can prove difficult and is in fact not necessary to determine these moduli. Indeed, define the following invariant 2 -grading It squares to the identity, because η does, and it is point group invariant: ). An invariant generalized metric can then be obtained from this by the simple relation This form ensures that H inv satisfies the quadratic constraint (3.5) automatically, since Z inv squares to the identity. The other constraint that H inv is symmetric is not implemented and hence needs to be enforced afterwards. From the generalized metric (4.25) or the 2 -grading (4.23) the Narain moduli G, B, A can be read off using (3.2) or (C.6): If one starts with a completely generic M 0 , this procedure determines both the values of the frozen moduli as well as the unconstraint ones in the form of free parameters.
In order to obtain an appropriate representation R θ ∈ O(D; Ê) × O(D + 16; Ê) one may proceed as follows: 1. Block diagonalize all elements R θ of the point group P simultaneously over the real numbers using a similarity transformation U ∈ GL(2D + 16; Ê). This procedure always works, since the point group P has finite order and hence the matrices R θ lie in the compact part of O η (D, D + 16; Ê). However, this does not necessarily lead to a valid generalized metric H: It might happen that M inv is not invertible and the 2 -grading (4.23) cannot be defined. Secondly, it might happen that the metric G read off from (4.25) using the explicit form (3.2) is not positive definite. This means that the distribution of the blocks chosen in the second step is not appropriate and another distribution should be considered. The procedure outlined here may seem to be somewhat awkward in light of the issues that the generalized metric H is not automatically symmetric or that the metric G is not necessarily positive definite. However, these are minor concerns as compare to the situation before: In [29] the moduli were constraint by a coupled set of Riccati matrix equations (see (5.17) in that reference) for which the very existence of solutions is unclear and the explicite determination of such solutions is a highly non-trivial task. Therefore, that here a concrete procedure is unfolded might be considered as a definite step forward in the construction of asymmetric orbifolds. In Appendix 6 examples are given in which this procedure has been executed to obtain the most general generalized metrics for certain asymmetric orbifolds.
In the following it is assumed that the generalized metric H, the vielbein E and the 2 -grading Z are invariant under the orbifold action and hence the subscript inv is dropped. In particular, the generalized vielbein E satifies for all θ ∈ P .

Dimension of the moduli space of a Narain Orbifold
As was argued in [29], the dimension of the untwisted moduli space of a Narain orbifold is given by In particular, when either the left-or right-moving twist representation is trivial while the other is not, the dimension of the moduli space is zero. Hence, this result may be used to confirm if all unconstraint moduli have been identified.

Narain Orbifold Worldsheet Torus Boundary Conditions
On the worldsheet the Narain coordinates Y are periodic up to actions of the space group S: The boundary conditions of the superpartners ψ of the non-compact coordinate fields x are the standard ones: where the spin structures s, s ′ = 0, 1 have been included. The boundary conditions for the rightmoving worldsheet fermions Ψ have to be compatible with the worldsheet supersymmetry transformations (3.10) linking them with the Narain coordinates Y . Using (4.26) one infers that their boundary condtions read Since, the matrices R R θ and R R θ ′ commute, see (4.12), they can be simultaneously diagonalized over the complex numbers to For elements that involve rotations in two dimensions, the eigenvalues come in complex conjugate pairs. Target space spinors are admitted only if, the orbifold twists preserve orientation hence their determinants have to be equal unity.
Only the real twist vector v R θ is relevant to determine how many target space supersymmetries are perserved in the non-compact dimensions. No supersymmetries are preserved if in succession: • an odd number of entries of v R θ are non-zero; • not all non-zero entries can be chosen in opposite signed pairs; • there is only one such pair.
If none of these three conditions are satisfied, a complex twist vector v R θ can be associated to the real twist vector by taking one of the two entries of the opposite signed pairs augmented by a number of zeros such that this complex vector has four entries. By the final condition it follows that at least two entries are non-zero. If there is a choice of these entries such that at least N = 1 supersymmetry is preserved in the d non-compact dimensions.
Because of the inhomogeneous terms, solving the boundary conditions (4.28) of the Narain coordinates Y is more involved. To this end the following ansatz is made Y c (z) = 2π(y 0 + p z +pz) (4.32) where y 0 , p,p are constant vectors and Y q (z) is assumed to satisfy the homogeneous boundary conditions where the twist matrices can be written as complex exponentials in terms of the matrices v θ and v θ ′ which mutually commute. The zero modes contained in Y c (z) can be treated in the following fashion: Inserting the ansatz (4.32) in the boundary conditions (4.28) leads to the requirements (4.35b) The first two conditions imply that both p andp lie in both the invariant subspaces of R θ and R θ ′ , which can be identified by the projectors P θ and P θ ′ given in (4.6): To avoid overcomplicating the notation, the projector on the combined invariant subspaces of R θ and R θ ′ is denoted simply by P without the subscripts θ, θ ′ , i.e. P = P θ,θ ′ = P θ P θ ′ and P ⊥ = ½ − P . Hence, by projecting the two equations in (4.35b) by P the vectors p andp can be determined to be given by: using (A.12).
Using the operator (4.16) two expressions for the constant vector y 0 can be obtained: where y ′ 0 and y ′′ 0 are two arbitrary vectors. Using that the projectors defined in (4.6) are complete, the first terms in these expressions are in independent directions that are undetermined from the other equations. Hence, for both expressions to agree, it follows that and P θ ′ P θ y ′′ 0 = P θ ′ P θ y ′ 0 , which is arbitrary and may be set to zero for convenience. The final two terms in the expressions (4.42) are equal by virtue of the second equation in (4.12): Multiplying it with both projectors P ⊥ θ and P ⊥ θ ′ and using that on the subspace defined by these projectors ½ − R θ and ½ − R θ ′ are invertible, this equality follows.
Putting everything together, the ansatz (4.32) for the Narain coordinates Y on the worldsheet torus can be cast in the form:

Narain Orbifold Partition Functions
The full orbifold partition function can be written as a sum over commuting space group elements g, g ′ ∈ S: Aside from the standard Minkowski space contribution, the various factors arise as follows: Taking into account the boundary conditions (4.29), the fermionic partition function can be expressed as the complex conjugated of the result (A.26) for real left-moving fermions in the Appendix A.4. The result only depends on the point group elements θ, θ ′ ∈ P not on the whole space group elements g, g ′ ∈ S. The resulting expression is given below in (4.52a).
Inserting the expression (4.44) in the worldsheet action leads to a projected lattice sum 46) using the notation introduced in (4.36), in particular (V θ ) = P V θ = P θ,θ ′ V θ is projected on the combined invariant subspace defined by R θ and R θ ′ (and not just R θ ). As before in the Narain lattice case, the opposite phase to the one out front has been included to ensure that the partition function is independent of N ′ . An unwanted consequence of this procedure is that also no orbifold projection by summing over θ ′ is implemented anymore. The phase in front of (4.46) corrects for this and will enforce the orbifold projection when summing over θ ′ . On the subspace defined by P two space group elements g and g ′ commute if their point group projections θ, θ ′ ∈ P commute. Hence, in light of these observations the sum over commuting space group elements in (4.45) can be replaced by a single sum over N and a sum over commuting point group elements [θ, θ ′ ] = 0. Finally, the boundary conditions (4.33) can be solved by the mode expansion Inserting this in the Narain action to evaluate the corresponding path integral gives . (4.48) Using (C.25) with a = r½ + v θ and a ′ = r ′ ½ + v θ ′ this can be written as Since, like R R θ , R R θ ′ , the matrices R θ , R R θ ′ can be diagonalized over the complex numbers, the 2D + 16 × 2D + 16-matrix v θ can be splitted in a right-and left-moving part as and hence for the associated real twist vector v θ = (v R θ , v L θ ). By worldsheet supersymmetry v R θ is the same real twist vector as introduced below (4.30). The vector v L θ = (v L θ 1 , . . . , v L θ D+16 ) are in general independent of those of v R θ ; except for symmetric orbifolds, where they are equal to those of v R θ augmented with 16 zeros. On the subspace defined by P defined in (4.37) both v θ , v θ ′ are integral (which by redefintions of r, r ′ maybe set to zero), hence give infinite product factors that can be written in terms of the Dedekind function via (B.5). On the complementary subspace defined by P ⊥ either v θ or v θ ′ are non-integral, hence are of the form of chiral bosons (A.31). Taking all this into account, the full partition function can be reshuffled to with the right-moving fermionic partition function given by . (4.52a) Combining the projected Narain lattice sum (4.46) with the contributions of (4.49) associated to the subspace defined by P , leads to The remaining contributions of (4.49) give rise to Some additional phase factors have been included in these expressions so as to ensure that these building block all transform covariantly under modular transformations. (Extended discussion on the vacuum phases to obtain modular invariant partition functions can be found in e.g. [30,32,[42][43][44][45].) Applying the projector P to (4.4) shows that consequently, the modular transformation rules become: for all N ∈ 2D+16 . This implies that η should define a invariant metric for a Narain, e.g. even selfdual, lattice of dimension D for all commuting θ, θ ′ ∈ P . For all such lattices D L − D R is dividable by 8. Because of (4.40) also D ⊥ L − D ⊥ R is dividable by 8, hence the phase in the second equation of (4.54c) is trivial. 4 In addition, (4.40) implies that the phases in the τ → τ + 1-transformation combined become trivial.
Finally, to ensure that the full partition function encodes the correct orbifold action, the order of the orbifold elements needs to be checked on the level of the partition function. Since θ ′ = θ ′ θ |θ| , this leads to the requirements for all commuting θ, θ ′ ∈ P . Here it was used that the non-vanishing entries of v R θ and v ⊥ R θ are equal.

Supersymmetric Symmetric Orbifolds
The dominant part of the (heterotic) string literature concerns itself with a very special class of orbifold: Orbifolds with actions that are purely geometrical that treat the left-and right-moving coordinate fields identically. Moreover, the actions are such that a certain amount of supersymmetry is preserved.
In the language employed here this means that v L θ has no non-vanishing components on the gauge directions and that the non-vanishing entries of v L θ and v R θ are equal and is simply denoted by v θ . Moreover, supersymmetry implies the existence of a complex structure so that the internal coordinate fields and fermions can be combined to complex entities. In that literature v θ has thus four instead of eight components, so that certain factors of 1/2 in front of the inner products with these quantities here should be removed when translated to the literature on supersymmetric symmetric orbifolds. (Alternatively in the formalism developed here this means that each entry appears twice with opposite signs.) Finally, the gauge shifts V θ lie purely in the left-moving gauge directions of the lattice. In the supersymmetric orbifold literature there is a similar relation like the second relation in (4.56) for V θ . In the formalism here this condition is obsolete as it is already incorporated by the metric η which contains the Cartan metric of E 8 ×E 8 .
The treatment of the Wilson lines is very different in the standard orbifold literature and in the formalism of [29] and this paper. In the standard orbifold literature the (discrete) Wilson lines arise when torus lattice translations are combined with simultaneous shifts on the gauge lattice. In the formalism here the Wilson lines A are part of the data encoded in the generalized metric H that also include the target space torus metric G and the anti-symmetric tensor field background B.

Shift Orbifolds and Lattice Refinements
Shift orbifolds are special types of orbifolds that have trivial twist actions. This means that the orbifold action (4.1) reduces to see (4.5) and (4.56), since the projection P θ = ½ and P ⊥ θ = 0, because the absence of any twist action. Consequently, the fermionic partition function (4.52a) reduces to (3.14b) and (4.52c) is equal to unity. Shift orbifolds are therefore entirely characterized by (4.52b).

Single Shift Orbifolds
where W 2 = W T ηW , characterized by a vector W ∈ 2D+16 whose entries of W are relatively prime w.r.t. K. This leads to a refinement of the standard 2D+16 lattice to the lattice N + k V N ∈ 2D+16 , k = 0, . . . , K − 1 (5.4) subject to the orbifold projection condition resulting from the phases in (4.52b) that involve θ ′ . This means that two things are happening to the lattice at the same time: the lattice is refined by the inclusion of a new lattice vector and make coarser by the orbifold projection condition which is kicking out certain lattice vectors. Hence, in addition to the integral shift vector W there is a second integral associated vector W ′ necessary in order to solve the orbifold projection condition (5.5).

Lattice Decomposition
To solve the projection condition (5.5) it is convenient if the original lattice 2D+16 could be decomposed in the directions of the integral vectors W and W ′ and the rest: where n, n ′ are integers and N ⊥ is a 2D + 16-dimensional vector spanned by 2D + 14 integer vectors that are perpendicular to W and W ′ in the sense that By taking inner products of this ansatz with W and W ′ a linear system for n, n ′ is obtained, which is readily solved where δ denotes the determinant of the system In general the solution for n, n ′ won't be integral; only if δ −1 = δ, as enforced above, this is guaranteed. Hence, from now on it is assumed that the second vector W ′ is chosen such that (5.9) holds. By performing the matrix multiplication in (5.8) expressions for n and n ′ can be found which can be written as n = W W N , Using these vectors and their conjugates two projections, Π and Π ′ , and two nilpotent operators, Ξ and Ξ ′ , can be introduced on W and W ′ , satisfying the algebra 14) The traces of these operators show that Π and Π ′ project on one dimensional subspaces. With this the subspaces parallel and perpendicular to W and W ′ can be easily identified by the projectors Π = Π + Π ′ , and Π ⊥ = ½ − Π , Since the projection operators, Π, Π and Π ⊥ , are all integral and have unit determinant on the subspaces on which they project, this decomposition is invertible over the integers and n, n ′ ∈ and N ⊥ lies in a 2D + 14-dimensional sublattice of 2D+16 .

Construction of the Shift Orbifold Lattice
To avoid arriving at very complicated formulae, the vector W ′ is required to satisfy The first condition, in fact, only restricts W Tη W ′ = ±1, which already implies that δ = −1 by (5.9). By multiplying the whole vector with −1 the sign of this inner product can always be assumed to be positive.
The refined lattice vector can be written as where m = K n + k is an arbitrary unconstraint integer since n ∈ and k = 0, . . . , K − 1. In addition, the constraint (5.5) can be solved as follows by introducing an arbitrary m ′ ∈ , since ≡ means equal up to integers and 1 2K W 2 ∈ by (5.3). The final term is added to ensure that the projection condition can be written entirely in terms of m and m ′ only. Indeed, by multiplying by K and solving for n ′ gives: This shows that the refined lattice vector can be cast in the form (5.24) in terms of unconstraint integers m, m ′ ∈ . By introducing the integral vector N ′ = m W + m ′ W ′ + N ⊥ , this can be written as using the operators (5.12).

Modification of the Generalized Metric
The expression (5.25) seems to suggest, the shift orbifold defines another Narain theory with a modified generalized metric H ′ obtained from an T ∈ O η (D, D +16; É) transformation. Before this transformation can be identified, first it has to be investigated whether the lattice transformation (5.25) modifies η and if so how to correct for that. To this end, define η ′ by By inserting (5.25) here and using that 27) it follows that η ′ can be written as In other words, in terms of the lattice defined by N ′ the metric η has changed to η ′ . Thus, by replacing N ′ by the new integral vector N ′′ = M −1 N ′ , the metric η remains the same. This motivates to define T = T 0 M, which reads explicitly: It may be verified that T ∈ O η (D, D + 16; É). The new generalized metric H ′ is given by This result shows explicitly that a shift orbifolds of a Narain theory is again a Narain theory but with redefined moduli.

Dependence on the Associated Vector
The construction developed here determines the new moduli of the Narain lattice that results from a shift orbifold. However, the procedure depends on a somewhat arbitrary integral associated vector W ′ only constraint to satisfy (5.20

A Simple Shift Orbifold
As a first illustration it is instructive to consider the case in which the target space is a circle of radius R. Hence, the dual circle has radius 1/R in string units, i.e. the corresponding generalized vielbein reads Performing an order-K shift orbifold on this circle leads to the Narain mapping Hence, the generalized vielbein becomes (5. 33) This indicates that the target space circle is shrunk by a factor 1/K while the dual circle is stretched by K at the same time.

Geometrical Shift with a Wilson Line
A Wilson line is often introduced in the orbifold literature as geometrical shift orbifold with an associated shift action on the gauge degrees of freedom. Concretely, consider an order-K i Wilson line on the i-th torus direction where a i is a 16-component integral vector such that The i-th D-dimensional Euclidean basis vector ǫ i satisfies Since W i and W ′ i both involve ǫ i but in different directions in the Narain lattice, it is ensured that we have an unique decomposition of the Narain lattice by (5.6). The vector W ′ i is chosen such that the conditions in (5.20) are fulfilled, hence the operators (5.12) are given by where π i = ǫ i ǫ T i and π ⊥ i = ½ D − π i . This results in the following Narain mapping using (5.29). By using the Oη(D, D + 16; Ê) elements given in (C.14) this can be written as This reduces to the case discussed above when switching off the Wilson line and taking D = 1.
Since shift actions on the Narain space commute, various geometrical shifts with Wilson lines can be combined. This simply leads to a product of the corresponding Narain mappings. If all D dimensions are shift orbifolded, this gives Hence, in particular, if the anti-symmetric tensor B and the Wilson lines A initially were switched off, only a Wilson line background is introduce by this shift orbifold. When only the B-field was zero, but the initial Wilson lines A were not, then the anti-symmetric tensor becomes switched on after this shift orbifold.

Non-Geometric Shift with a Wilson Line
Next, consider an order-K i Wilson line on the i-th dual torus direction where a i satisfies the same properties as in the previous subsection. Hence, the same procedure can be followed as in that subsection leading to where M α is defined in (C.17). Hence, in this case non-geometric gauge moduli are switched on. The reason how this comes about can be easily understood by the following observation. The starting data of this and the previous subsection are related to each other via the T -duality operator I given in (C.16): This is just the statement that T -duality takes geometric shifts to non-geometric ones. Hence, one expects that also the change of the moduli due to the (non-)geometric shift with a Wilson line are related via the same operation. This is indeed the case, since This exemplifies that the interpretation of a certain transformation to be geometric or non-geometric depends on the duality frame chosen.
In this case it is only under certain circumstances possible to determine the new moduli using products of Narain vielbeins: If in the initial case B = A = 0, then However, as soon as either an anti-symmetric tensor or a Wilson line background was switched on already, this does not work any more. To determine the moduli for more general initial Narain backgrounds, one should use the generalized metric instead. The new generalized Narain metric H ′ obtained after performing this non-geometric shift with a Wilson line is given by

Non-Geometric Shift in Two Dimensions
Consider the order-K non-geometric shift for i = j and no action on the gauge degrees of freedom. Here two choices for the vector W ′ are given distinguished by the label i and j, which is used below as well for this purpose, to exemplify possible consequences of different choices of the associated vector W ′ to solve the orbifold constraint.
In the case W ′ i is employed, one finds Hence the Narain mapping becomes This can be written as Hence, performing a non-geometrical shift orbifold in two directions has the effect of switching on the anti-symmetric tensor field B in those directions. If W ′ j is used instead, the results are which can be written as Hence, by making the choice W ′ j for the vector to solve the orbifold constraint, instead the interpretation of performing the same simultaneous shift in a torus direction and a shift in a different dual torus direction leads to a non-geometric β-background.
The two seemingly different situations can again be mapped to each other. Indeed, it is easy to see that where X ij exchanges the directions i and j on the torus and the dual torus simultaneously. Applying these transformations to the moduli mapping T i gives Hence, the interpretation of the moduli depends on the choice of the associated vector W ′ .

Higher dimensional T-fold examples 6.1 Right-Twisted Full T-Duality Narain Orbifolds
Full T-duality orbifolds refer to Narain orbifolds in which the orbifold action is a T-duality transformation in all D compact torus directions. Below two realizations of them are discussed: one in this subsection, one in the next. Even though their starting data are very similar, their properties differ in a number of interesting ways. Consider the Narain orbifold with the 2 T-duality twist given by The associated block-diagonal twist is given by assuming that α, δ and τ are invertible matrices. These expressions imply that using (4.23) and (4.25). Notice that these results are, in fact, independent of the precise form of α, δ and τ (which thus could have been chosen to be α = δ = ½ D and τ = ½ 16 ). This is just a reflection of the fact that this orbifold fixes all moduli. They read: G = ½ D and B = A = 0.
The definition of a Narain orbifold is not complete without specifying the Narain shift vector. Given that the parallel and perpendicular projectors (4.6) take the form is parameterized by integral vectors w R ∈ D and w 16 ∈ 16 . The consistency condition (4.56) takes the form since v L θ = 0 . Since g 16 is even, this is gives a mod-two condition on the D + 16 integers, which is easily solved by requiring that the number of odd integers is even.
The fixed points of this T-fold are determined via the equation (4.14)  On the subspace defined by P ⊥ this can be solved as with m 1 , . . . , m D = 0, 1, leading to 2 D physically distinct fixed points.

Left-Twisted Full T-Duality Narain Orbifolds
Next, consider another 2 T-duality Narain orbifold with a very similar twist as the one discussed above so that the associated block-diagonal twist now reads In this case the action on the right-movers is trivial. Consequently, there is also no action on the rightmoving fermions nor on the target space spinors. Hence, this T-duality Narain Orbifold necessarily preserves all target space supersymmetries.
In this case the parallel and perpendicular projector (4.6) is given by is parameterized by integral vectors w L ∈ D and w 16 ∈ 16 . The consistency condition (4.56) takes the form Inserting the form (6.14) of v L and using that w 2 L is an integer, this condition may be stated as 1 4 2 w 2 L + w T 16 g 16 w 16 ≡ using (4.21). After determining it's inverse, the invariant 2 -grading can be expressed as (6.20) in terms of the matrices ∆ = α −1 γ τ −1 ρ and ∆ ′ = τ −1 ρ α −1 γ . (The matrix δ dropped out during the computation.) Comparing with the standard form of the 2 -grading (C.6), the following identifications can be made The reason, that there are two expressions for the Wilson lines A obtained here, reflects the fact that the procedure outlined in Subsection 4.2 does not automatically enforce that H is symmetric. Hence this constraint has to be implemented in addition. Doing so shows that all other identifications are consistent and, in particular, From this it may be concluded, that the Wilson lines A represent the only unfixed Narain moduli, while This shows that there needs to be a bound on the Wilson lines in order to ensure that the metric G remains positive definite.
The fixed points of this T-fold are determined via the equation (4.14)  On the subspace defined by P ⊥ this can be solved as where m 1 , . . . m D = 0, 1, 2, 3 labels 4 D distinct points on the doubled torus, which are not all the same as in the previous case, see (6.10). However, in light of the residual gauge symmetry not all are physical. The orbifold compatible residual gauge transformations contain the following constant gauge parameters with m 1 , . . . , m D = 0, 1, leading once more to 2 D physically distinct fixed points.

Supersymmetric 6 T-fold in four dimensions
Most T-folds discussed in the literature are variants of the T-duality orbifolds, like the ones considered above. T-folds based on other duality symmetries than 2 are far less known. In the following a Tfold is discussed based on the 6 group that acts asymmetrically on the left-and right-movers. This example is a higher dimensional extension of the T-fold 6 -V listed in Table 3 of [29]. Consider the 6 -orbifold action given by where δ is a positive integer for now, so that D = 2δ. (Later below the restriction δ = 2 will be considered.) This leads to an asymmetric action: A 2 -symmetry acts on the right-moving sector while a 6 -symmetry on the left-movers. Another way to view this action is by considering the underlying 2 and 3 generators of the subgroups P 2 and P 3 of the point group P: This shows that the 2 action acts symmetrically and only the 3 is asymmetric. In fact, it only affects the left-movers: The metric and anti-symmetric tensor field for the geometry 6 -V were given in Table 3 of [29]. Extending these results to the case here gives: Inserting this in (C.6) shows that the 2 -grading reads By a straightforward computation it may be verified that it is a 6 invariant 2 -grading: Setting δ = 2 leads to a four dimensional T-fold that preserves N = 1 target space supersymmetry in the six non-compact dimensions. To see this in this formalism notice that the real and complex right-moving twist vectors can be chosen such that this Wick-rotation reads Consequently, the Minkowskian derivatives can be mapped to derivatives of these real Euclidean coordinates The Euclidean action S E = −i S is obtained from the Minkowskian action S after applying these substitutions.

A.2 Euclidean Worldsheet Torus
At the one-loop level the worldsheet is a torus T 2 WS defined by the periodicities z ∼ z + 1 and z ∼ z + τ (A. 5) and, consequently, the area of the worldsheet torus equals By expressing this complex coordinate z and its conjugatez as in terms of the real coordinates (x, y), the periodicities can be stated as x ∼ x + 1 and y ∼ y + 1 .
Hence, the functions φ r r ′ (z) = r φ 1 (z) + r ′ φ τ (z) and φ r r ′ (x, y) = r x − r ′ y , (A.12) where r, r ′ ∈ É, possess the following properties on the wordsheet torus are quasi-periodic These mode functions are orthogonal in the sense that

C.1 Properties of the T-duality Group
Narain tori and orbifolds concern themselves with lattice vectors N ∈ 2D+16 . A general lattice transformation is given by In particular, it's inverse reads: where E ′ = E e , B ′ = e T Be + b + 1 2 e T A T a − 1 2 a T Ae and A ′ = Ae + a , (C.15b) using the multiplication rules obtained in [29].
The T-duality operator in all D directions can be introduced as where K is introduced in (C.10). Clearly, it squares to the identity. It can be used to define the where β = −β T ∈ M D×D (Ê) and α ∈ M 16×D (Ê) . where C = BA −1 commutes with Z = η −1 H. By taking the log of the final determinant factor and using the Taylor expansion of the ln-function this expression takes the form ln det ½ + C Z = tr ln ½ + C Z = tr m≥1 −1 2m C 2m + m≥0 1 2m + 1 C 2m+1 Z (C. 20) since Z squares to unity. Using that the left-and right-moving projectors (3.7) satisfy ½ = P L + P R and Z = P L − P R , (C. 21) this can be split in left-and right-moving contributions ln det ½ + C Z = tr L ln ½ + C) + tr R ln ½ + C) ,

C.3 Generalized Metric Independence
where the projected traces tr L/R [X ] = tr X P L/R were introduced. Hence, the initial expression (C. 19) can be written as in terms of determinants det L/R over the D + 16-and D-dimensional subspaces defined by the projections P L/R . This result shows that this determinant is independent of the (fixed or free) moduli contained in the generalized metric H. These determinants may be evaluated in the lattice basis but also in the basis where the orbifold twists are (block) diagonal, e.g. (4.20). Taking A = τ 1 a + a ′ and B = iτ 2 a , (C. 24) for two commuting matrix a and a ′ , this leads to the identity det η (a τ 1 + a ′ ) + H iτ 2 a = det η det L [a τ + a ′ ] det R [aτ + a ′ ] .