Double Field Theory and Pseudo-Supersymmetry

Supersymmetric bosonic backgrounds governed by first-order BPS equations, can be realised in a much broader setting by relaxing the requirement of closure of the superalgebra beyond the level of quadratic fermion terms. The resulting pseudo-supersymmetric theories can be defined in arbitrary spacetime dimensions. We focus here on the ${\cal N}=1$ pseudo-supersymmetric extensions of the arbitrary-dimensional bosonic string action, which were constructed a few years ago. In this paper, we recast these in the language of generalised geometry. More precisely, we construct the action and the corresponding supersymmetry transformation rules in terms of O($D$)$\times$O($D$) covariant derivatives, and we discuss consistent truncations on manifolds with generalised $G$-structure. As explicit examples, we discuss Minkowski$\times G$ vacuum solutions and their corresponding pseudo-supersymmetry. We also briefly discuss squashed group manifold solutions, including an example with a Lorentzian signature metric on the group manifold $G$.


Introduction
Supersymmetry provides a powerful tool for probing aspects of physics that would other wise be beyond the limits of computability. One important example is that the second-order non-linear field equations of Einstein gravity or supergravity can be reduced to first-order equations in certain circumstances, namely when there exist supersymmetric bosonic back grounds that admit one or more Killing spinors. Beyond the classical level, supersymmetry severely restricts quantum corrections and allows some non-perturbative results to be ob tained.
A feature of supersymmetry is that it implies restrictions on the dimension of the space time. In particular, beyond 11 dimensions it is not possible to find any supersymmetric ex tension of gravity without adding higher spin fields, and this rules out having a Lagrangian description. One might therefore conclude that supersymmetry would in general be of no help in the study of theories in arbitrary higher dimensions. However, these theories can still possess a pseudo-supersymmetry [1][2][3][4], which is a weaker notion of supersymmetry that only involves fermionic terms up to second order, in the action and the transformation rules. This is in fact sufficient in order to be able to derive many of the useful features of conventional supersymmetric theories, including the existence of pseudo-Killing spinors in certain backgrounds. Thus pseudo-supersymmetry still allows the second-order field equa tions for the bosonic fields to be reduced into first-order Bogomol'nyi-Prasad-Sommerfield (BPS) conditions in such backgrounds. Hence, it can provide a powerful tool in the study of solutions of theories of gravity coupled to matter in arbitrary dimensions.
A well-established framework for exploring the landscape of supergravity vacua is pro vided by (exceptional) generalised geometry [5][6][7][8][9][10], and the closely related double or ex ceptional field theory (DFT/ExFT) [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] 1 . Particularly interesting for our work are the supersymmetric extensions of bosonic DFT [33][34][35][36][37][38][39][40]. All these approaches share one defining property, namely the unification of local diffeomorphisms with form-field gauge transforma tions into one unified symmetry group. In the most basic setup, the metric and the two-form B-field potential of a bosonic string action are combined into the generalised metric, giving rise to the O(n,n) symmetry of DFT. Equivalently, this structure is captured by the gen eralised tangent bundle T M ⊕ T * M of generalised geometry. Note that in general, DFT is capable to capture backgrounds which go beyond supergravity and generalised geome try [41,42]. Here, however, we shall be concerned with the most conservative case, where the section condition is satisfied globally. In this case DFT is just a rewriting of supergrav ity, and it is completely equivalent to generalised geometry. Still, there are two important advantages of this rewriting: First, supersymmetry variations have a much simpler form and second, abelian T-duality becomes a manifest symmetry of the string's low-energy effective target space action. A natural question in this context is whether pseudo-supersymmetry permits a similar treatment. We answer this question in the affirmative, and demonstrate that it is possible to extend the existing results of supersymmetric DFT from ten dimensions to arbitrary dimensions.
An important application is the construction of pseudo-supergravity vacua. In partic ular, we combine the technique of consistent truncations with pseudo-supersymmetry to show how the field equations in arbitrary dimensions can be simplified significantly. More precisely, in the examples we consider, the consistent truncation renders the field equations algebraic but still quadratic. Clearly this is already a major simplification, but still quadratic equations with multiple variables can be hard to solve. A similar problem arises in the clas sification of Lie algebras, whose Jacobi identity is a quadratic constraint. In low dimensions, it is possible to solve it, and this gives rise to a complete classifications of real Lie algebra up to six dimensions [43]. Beyond that, solving the quadratic constraint becomes forbiddingly complicated. A similar situation is encountered in the prototypical example of consistent truncations in DFT, namely in generalised Scherk-Schwarz reductions [44][45][46][47][48][49]. Compared to 1 There has been a considerable amount of original work in this field and therefore we only reference a few key contributions here, which is of course highly subjective. We refer to the reviews [29][30][31][32] for a complete list of references. a standard geometric reduction on a group manifold with isometry group G L × G R , which retains the singlets under either G L or G R , the consistent reductions in DFT allow one to retain all the gauge bosons of the complete isometry group. This is a much more compli cated reduction, because of the potentially dangerous trilinear coupling of massive spin-2 modes to bilinears constructed from the G L × G R Yang-Mills bosons [50]. The existence of a consistent reduction of the (n + D) bosonic string to a D-dimensional group manifold keeping all of the G L × G R gauge bosons was conjectured in [51], with further supporting evidence found in [52]. A complete proof of the consistency was obtained in [53], utilizing the O(D, D) formulation of (n + D)-dimensional bosonic string [54]. Combining a gener alised Scherk-Schwarz reduction with pseudo-supersymmetry, we show how the quadratic field equations for the remaining fields can be reduced, in appropriate backgrounds, to linear equations. Because of the less restrictive nature of pseudo-supersymmetry, in comparison to ordinary supersymmetry, this can be done in arbitrary spacetime dimensions.
The paper is organized as follows. In section 2, we give a short review of the N = 1 pseudo-supersymmetric theory. In section 3, we reformulate it in terms of generalised ge ometry and then spell out the conditions for the existence of a consistent truncation. In sections 4 we explicitly construct solutions of the form (Minkowski) D−dim G × G, including a description in the framework of generalised geometry. In section 5 we discuss their pseudo supersymmetry, both in standard field theory and in generalised geometry. In an appendix, we construct an example of a (Minkowski) D−dim G × G background, for G = SO(5), where the metric on the group is squashed. It turns out to have Lorentzian signature.

Pseudo-Supersymmetrised Bosonic String
As described in [1] one can construct a pseudo-supersymmetric fermionic extension of the bosonic string Lagrangian, in a completely arbitrary dimension D. That is to say, there exist supersymmetry-like transformation rules that leave the Lagrangian invariant, modulo terms beyond the quadratic order in fermions. In practice many of the desirable features of super symmetry, such as the existence of Killing spinors in bosonic backgrounds, BPS conditions and first-order equations, do not directly depend upon the full closure of the transformations. This means that all the useful consequences of having fermionic symmetries in bosonic back grounds will equally well arise in the much larger arena of pseudo-supersymmetric theories.

Lagrangian and pseudo-supersymmetry transformation rules
The Lagrangian for the pseudo-supersymmetric extension of the bosonic string in an arbi trary dimension was constructed in [1], where it was presented both in the Einstein frame and in the string frame. Here, we reproduce the result from [1] in the string frame, with the following notational changes. Firstly, we denote the spacetime dimension by D rather than d, since in this paper d will be reserved to denote the generalised dilaton of DFT. Secondly, in order to harmonise our notation with some of the DFT literature, we perform the rescalings on the fermion fields and pseudo-supersymmetry parameter, and finally, we make the replace ment Γ a → −Γ a , which of course preserves the Clifford algebra. With these replacements, the D-dimensional pseudo-supersymmetric Lagrangian of [1], in the string frame, becomes and the pseudo-supersymmetry transformation rules are given by Note that δψ µ may be re-expressed in terms of a torsionful connection as The constant β, which is either +1 or −1 depending on the dimension D and the spinor representation, characterises the symmetry property of the gamma matrices, It is listed for each dimension and representation in table 1 in the appendix A. Many fur ther properties of spinors in diverse dimensions are summarised in our notation in [1]. All coefficients in (2.2) and (2.3) were determined by the requirement that the Lagrangian be invariant under the pseudo-supersymmetry transformations, provided that one neglects fermionic terms that would arise from higher fermionic powers in the Lagrangian or pseudo supersymmetry transformations. It was shown in [4] that, just like in the case of the supersymmetry transformations for ten-dimensional N = 1 supergravity, the integrability conditions obtained by taking commutators of the pseudo-supersymmetry transformations on a bosonic background are satisfied if the full set of field equations for the D-dimensional bosonic string are satisfied.

Adding a conformal anomaly term
As was shown in [4], one can also add a "conformal anomaly" term to the Lagrangian. In the string frame, after performing the rescalings (2.1) and the replacement Γ a → −Γ a detailed above, the additional terms in the Lagrangian take the form There are associated additional terms in the fermion transformation rules, given by Note that the fermionic extension of the conformal anomaly term in (2.6) really requires a doubling of the fermionic degrees of freedom. This is most easily stated in dimensions D = 2 mod 8, where we can choose β = −1 and the basic spinors of the pseudo-supersym metrised bosonic string would be both Majorana and Weyl (with ψ µ and ǫ being chiral, and λ anti-chiral). The fermionic terms in (2.6) would vanish under these conditions, but will be non-vanishing if the chirality constraints on the fermions are removed. In cases where β = +1, the first two fermionic terms in (2.6) will vanish identically, if the spinors are Majorana or symplectic-Majorana. In these cases, one can still pseudo-supersymmetrise the conformal anomaly term if one doubles the number of fermions, by adding an additional doublet index, All the previous fermion bilinears in the Lagrangian will now have α and β indices contracted with δ αβ . The terms in L c , on the other hand, will have the α and β indices contracted with ǫ αβ . An ǫ αβ should also be inserted in the extra terms (2.7) in transformation rules for ψ µ and λ.

Generalised Geometry and Pseudo-Supersymmetry
It is possible to simplify the Lagrangian (2.2) considerably by introducing the generalised dilaton and its superpartner Furthermore, we unify the frame field and the B-field by introducing the generalised frame field with the components and ι ea B = e µ a B µν dx ν . Each of these 2D components is a generalised vector on the gener alised tangent space T M ⊕ T * M . After this identification and the redefinitions above, the pseudo-supersymmetry transformation rules (2.3) and (2.7) can be written in the compact form play a crucial role. They are defined by [8,34] ∇ and as we will see in the next subsection, they also have very nice properties when it comes to consistent truncations. Like the pseudo-supersymmetry transformation rules, also the action (2.2) simplifies considerably once written in string frame and after applying the redefined fields and the adapted covariant derivatives, where β is defined by the use of charge conjugation matrix C in (2.5). In ten dimensions, this Lagrangian matches the one of N =1 Double Field Theory [34] af ter implementing the solution of the section condition which removes the dependence on the coordinates conjugate to string winding modes and choosing the parameter β = −1. The in teresting observation here is that this result even holds in arbitrary dimensions once we drop the additional constraints imposed by supersymmetry in favour of pseudo-supersymmetry.
The conformal anomaly terms (2.6) can also be formulated in the language of generalised geometry, To satisfy the pseudo-supersymmetry property, it follows from (2.7) that the variation rule of ρ needs to be modified by δ extra in (3.4).

Consistent truncations
The crucial observation for constructing consistent truncations in the N = 1 pseudo-su persymmetric theory is that all relevant quantities like the Lagrangian, the pseudo-super symmetry transformation rules and the field equations can be written in terms of covariant derivatives ∇ (±) µ . To construct them, one starts with an O(D,D) structure which is defined by the invariant metric It encodes the metric and the B-field once it is pulled to the generalised tangent space Note that we here have switched from using Greek indices, µ, ν, . . . , for spacetime coor dinates to Latin indices i, j, . . . . We do this because in DFT, there is a need for capital, doubled, indices as well as lower case, standard, indices, and the Greek alphabet does not lend itself to this distinction. The metrics (3.9) and (3.10) are related by the generalised frame, In order to construct consistent truncations, one restricts the form of the covariant where D A is a second covariant derivative which admits some invariant tensors and thus defines a generalised G-structure [55]. G can be any subgroup of O(D)×O(D); we present some examples later but for the moment we keep the discussion general. To obtain ∇ A from D A the tensor ω AB C has to be fixed. This is done by imposing four constraints on ∇ A [56]: 1. It is compatible with the O(D,D) metric: 2. It is compatible with the generalised metric: 3. It is compatible with integration by parts: 4. It has vanishing generalised torsion, implying where T ABC is the generalised torsion of D A .
These constraint do not fix ω AB C completely. However, all physically relevant quantities like the action, the field equations and the pseudo-supersymmetry transformations use ∇ (±) µ in such a way that the undefined contributions drop out. To present the partially fixed ω AB C , it is convenient to introduce the projectors The constraints above restrict the form of ω ABC to (3.20) Note that there are still unconstrained components of ω ABC left [56]. But these are irrelevant in the calculation of all physically relevant quantities and thus we can safely ignore them. A consistent truncation arises if the action of ∇ A on a tensor invariant under D A gives rise to another (or the same) tensor that is invariant under D A . In this case, the set all these invariant tensors forms a consistent truncation. By using the definition of ∇ A in (3.12), this requirement translates to the two constraints

Minkowski×G Group Manifold Compactifications
It was observed in [51] that the d-dimensional bosonic string with the added conformal anomaly term admits a vacuum solution of the form (Minkowski) D−dim G × G, where G is any semi-simple compact dim G-dimensional Lie group. Here, we shall study the pseudo-su persymmetry of these vacuum solutions. In order to do this, we first need to establish some basic notation and results for group manifold compactifications.

Conventions and geometry for group manifolds
The vacuum solution employs the group manifold G equipped with its bi-invariant metric g mn . This has left-acting and right-acting Killing vectors of the group G, which we denote by K m L a and K m R a respectively. They obey the algebra where f ab c are the structure constants, and c is a scale-setting constant. The Killing vectors may be normalised so that with δ ab being proportional to the Cartan-Killing metric, where C A is the quadratic Casimir of the group G. Conversely, one has It follows that one may view either the K m L a or the K m R a Killing vectors as defining a vielbein e a = e a m dy m . We shall consider the left-invariant vielbein e a = K a R = K a R m dy m . (4.5) Using (4.1), the 1-forms K a L and K a R obey The vielbein (4.5) therefore obeys de a = − 1 2 c f bc a e b ∧ e c , and so the torsion-free spin-con nection, defined by de a = −ω a b ∧ e b and ω ab = −ω ba is therefore given by Note that since we are taking G to be compact and semi-simple, f abc is totally antisymmetric. The curvature 2-forms Θ ab = dω ab + ω ac ∧ ω cb and the Riemann tensor (following from Θ ab = 1 2 R abcd e c ∧ e d ) are then given by 2 Finally, we have the Ricci tensor and Ricci scalar, given by Note that f abc is covariantly constant. The Lorentz-covariant exterior derivative D acts on Lorentz vector as DV a = dV a + ω a b V b , so and since the f abc are constants, and the spin connection is given by (4.7), we have 11) and this vanishes by virtue of the Jacobi identity. Thus it follows that in coordinate indices we also have ∇ m f npq = 0.

Consistent truncations
Following the discussion in section 3.1, we now construct a consistent truncation on the group manifolds G. To do so, our first objective is to fix an appropriate covariant derivative D A . We impose now, that D A annihilates the generalised frame field E B I . This renders the corresponding generalised geometry parallelisable or, equally, the generalised structure group trivial. More specifically, we have thus determining the corresponding connection Its generalised torsion is given by and therefore we obtain from (3.16) A consistent truncation requires that (3.21) hold. Thus, we have to find a generalised frame field E A I on the group manifold G such that holds. Equivalently stated, the generalised torsion in flat indices must be constant. This problem does not have a unique solution, because there are an infinite number of admissible generalised frame fields that satisfy (4.16) on the Lie group G. For definiteness, we choose here the solution discussed in [53]. It is given by where K R and K L denote the left-and right-invariant vectors fields and their respective duals from subsection 4.1, and ι X B = X m B mn dx n for any vector X . Additionally, we also have to incorporate a B-field whose corresponding H-flux yields For this generalised frame field, we now compute the generalised torsion (4.15) with the non-vanishing components Note that f abc = f ab d δ dc = fābc coincides with the structure coefficients that govern the generators of the Lie group G. They appear here because of the Killing vectors algebra (4.1).
We also need to compute the flux F A , which captures the dilaton, and check that it is constant, as required by the second equation in (3.21). By combining (3.1) with (3.15), we obtain This equation splits into two contributions, for F a and Fā respectively. Let us take a closer look at where we take into account that e a m can be identified with K a Lm . The right-hand side of this relation can be further simplified by using (4.1), yielding The last term vanishes because we take G to be semi-simple. An analogous argument applies to Fā. Hence, we conclude that F A =const. requires a linear dilaton. Finally, the Bianchi identity for D I implies that F AB C F C = 0 (4.23) must hold. Since the generalised torsion F ABC matches the structure coefficients of the isometry group G L × G R , F C is in one-to-one correspondence with an element in the center of this group. But because G is semi-simple, so is G L × G R . Semisimple Lie groups have a trivial center, and therefore only F A = 0 is consistent with the Bianchi identity (4.23). Thus we conclude that the dilaton must be constant in order to give rise to a consistent truncation.

The Minkowski×G vacuum
At this point, it is convenient to change the index labelling conventions and notation a little, and rewrite the Lagrangian in section 2 usingμ,ν . . . world indices in the full D dimensions, and furthermore to place hats on all D-dimensional fields (and gamma matrices). When needed, D-dimensional tangent-space indices will be written asâ,b, . . .. We then use world indices µ, ν, . . . and tangent-space indices α, β, . . . in the (D − dim G)-dimensional spacetime and world indices m, n, . . . and tangent-space indices a, b, . . . in the group manifold G. Thuŝ µ = (µ, m) andâ = (α, a), etc. The D-dimensional bosonic field equations for the bosonic string, including conformal anomaly term, are given in the string frame bŷ We seek a ground-state solution whose metric is a direct sum of a (D − dim G)-dimensional spacetime of maximal symmetry (Minkowski, AdS or dS) times the bi-invariant metric on the group manifold G: dŝ 2 = g µν dx µ dx ν + g mn dy m dy n . (4.27) The dilaton will be assumed to be constant, and taken, without material loss of generality, to vanish. The components of the 3-formMμνρ will also be assumed to vanish except those lying entirely in the group-manifold, and for these we can takê The choice of sign is arbitrary, as far as the bosonic equations of motion are concerned. Our choice of the negative sign is for consistency with the pseudo-supersymmetry; see later. Here f mnp is constructed from the structure constants f abc using the vielbein K a R in the obvious way: It follows that we shall havê Plugging the ansatz into the dilaton field equation (4.24) implies that we can takeφ = 0 if m is given by Thus we have proved that we indeed have a Minkowski×G vacuum solution withφ = 0 andĤ mnp given by (4.28), provided that the coefficient m 2 of the anomaly term is given by (4.31) and that the metric g mn on the group manifold G is chosen as described in section 4.1. An identical conclusion arises from the consistent truncation outlined in the last sub section. It is straightforward to see that a Minkowski space with a constant dilaton is captures by F ABC = 0 and F A = 0. Hence, solving the field equations for the product space Minkowski×G boils down to solving the field equations in the internal space [15]. Here R AB denotes the generalised Ricci tensor and R is the gener alised Ricci scalar . Both admit a very simple expressions for the generalised Scherk-Schwarz truncation we are concerned with According to (4.19), only constructions of F ABC with exclusively P AB or P AB give non-van ishing contributions. For this observation, we immediately see that R AB vanishes (remember F A = 0) as expected. In the same vein, we obtain and therefore recover (4.31) from the second equation in (4.34).

Pseudo-supersymmetry of the Minkowski×G vacuum
To check if this background at least partially preserves pseudo-supersymmetry, we need to plug it into the fermionic pseudo-supersymmetry transformation rules, to see whether δψ M and δλ vanish for some subset of the parametersǫ. The calculations can be set up along the same lines as those described in [51] for compactifications of d = 11 supergravity. In particular, it will involve decomposing the spinors of the D-dimensional spacetime into tensor products of spinors in the (D − dim G)-dimensional spacetime and spinors on the group manifold G. See appendix B for a summary of how the Dirac matrices may be decomposed in the various cases of even or odd-dimensional spacetime and internal space.
As a preliminary check, consider the dilatino transformation rule in (2.3), together with the conformal anomaly contribution in (2.7). In the background we are considering, with Φ = 0 and the 3-form given by (4.28), we shall have We shall consider the case β = −1, where, as we discussed before, the conformal anomaly extension is simpler. Usinĝ it follows that if we defineQ ≡ 1 6 f abcΓ abc , (5.3) times the identity matrix. By the Cayley-Hamilton theorem, and noting that trQ = 0, this means thatQ has the eigenvalues with equal numbers of each. Thus with m given, from (4.31), by we see that ifǫ is any of the eigenvectors with eigenvalue −i C A dim G

6
, we shall get δλ = 0. The dilatino transformations suggest therefore that the Minkowski×G background preserves one half of the pseudo-supersymmetry.
To confirm this, we now turn to the gravitino transformation rule. Assuming again that β = −1 we have, from (2.3), In the internal group manifold directions we have Thus, we see that the pseudo-supersymmetry variations of both the dilatino and the gravitino vanish provided thatǫ is an eigenstate ofQ with eigenvalue −i C A dim G

6
, and thatǫ is independent of all the coordinates.
Again, we rederive this result using the relation between generalised geometry and pseudo-supersymmetry established in section 3 where we consider the transformation of the gravitino first. Combining (3.4), (3.12), (3.19) and (4.19) yields where we take into account that partial derivatives onǫ have to vanish. Second, we include the conformal anomaly term, which alters the transformation of the generalised dilatino according to for β = −1. Together, (5.11) and (5.12) yield which leads to the same result as already discussed above.
One may also consider more general vacuum solutions of the form (Minkowski) D−dim G × G s , where G s is a group manifold endowed with a "squashed" metric that, while still being invariant under the left action of the group G L , is no longer invariant under the right action of the full group G R . We have looked at examples where G is taken to be SU (3) or SO(5), and although these can indeed give rise to squashed solutions, we find that there is non surviving pseudo-supersymmetry in these backgrounds. The details of the SO(5) example are described in appendix C.

B Decomposition of Dirac Matrices
In a Kaluza-Klein reduction we need to write the higher-dimensional Dirac matricesΓ A in terms of tensor products of lower-dimensional spacetime Dirac matrices γ α and internal space Dirac matrices Γ a . The way this works depends upon whether the various space(time)s are even-dimensional or odd-dimensional. A table of how the decompositions may be made is given in appendix A of [57]: (even,odd) :Γ α = γ α ⊗ 1l ,Γ a = γ * ⊗ Γ a , (odd,even) :Γ α = γ α ⊗ Γ * ,Γ a = 1l ⊗ Γ a , (even,even) : where the first entry in the pair enclosed in parentheses indicates whether the lower-dimen sional spacetime is even or odd-dimensional, and the second entry indicates whether the internal space is even or odd-dimensional. γ * denotes the chirality operator in even-dimen sional lower-dimensional spacetimes, and Γ * denotes the chirality operator in even-dimen sional internal spaces (with γ 2 * = +1 and Γ 2 * = +1). In the (odd,odd) case the extra factor involving the Pauli matrices σ 1 and σ 2 ensures that theΓ A matrices obey the Clifford alge bra. They are needed because the Dirac matricesΓ A in this case are twice the size of the tensor products of the lower-dimensional and the internal Dirac matrices.

C Squashed Group Manifold Solutions
It is well known that any compact semi-simple group manifold other than SU (2) or SO(3) admits at least one additional, inequivalent, Einstein metric, over and above the standard bi-invariant metric. This raises the possibility that there might exist Minkowski×G vacua in which the metric on the group manifold G is not the bi-invariant one. Such solutions would not necessarily involve a squashed Einstein metric on G, since the form of the 3-form field strength H mnp in the squashed vacuum may also change. One approach to looking for such squashed solutions is to consider families of squashed metrics on G, with an associated deformation of the 3-form field. The families of metrics in question here will be homogeneous, invariant still under the left-acting copy of G, but no longer invariant under the full right action of G. Such metrics can be obtained by rescaling the left-invariant vielbeins by constant factors. A detailed discussion of the construction of squashed Einstein metrics using this procedure can be found, for example, in [58].
One can look for squashed vacuum solutions on a case by case basis. We have checked two examples, one being a family of squashed metrics on the SU (3) group manifold and the other a family of squashed metrics on the SO(5) group manifold. In neither case do we find any squashed vacua in the bosonic string for which the squashed metric on the group manifold is of positive-definite signature. For the case of SO(5) we do find one squashed example for which the metric has Lorentzian signature (that is, (1,9) signature). Since this may be of some interest, we shall present some details below.
In order to obtain a solution of the bosonic string of the form (Minkowski)×G squashed , we need also to construct a 3-form G (3) that is closed and also co-closed (i.e. the 3-form must be harmonic). In the case of the bi-invariant vacuum, we just used a constant multiple of the structure constants f abc . In fact we could write This is manifestly closed, and one can easily verify that it is also co-closed in the bi-invariant metric.
There must always exist an harmonic 3-form regardless of whether the metric is bi-in variant or squashed, since the topological number b 3 (the third Betti number) is equal to 1 regardless of the metric. One way to construct the required harmonic 3-form is by a brute-force Mathematica calculation, starting with a general 3-form and solving for the (constant) components G abc such that dG (3) = 0 = d * G (3) . In fact we find that the harmonic 3-form is exactly the same as the one constructed in eqn (C.6) (i.e. still written using the bi-invariant vielbeinē a ). Of course when one calculates the Hodge dual and d * G (3) (or equivalently, the divergence ∇ a G abc ), the fact that the metric is squashed enters in the calculation. SubstitutingĤ mnp = ±cG mnp and the direct sum of the Minkowski metric and the squashed SO(5) metric (C.3) into the equations of motion for the bosonic string (with dilaton set to zero), we find two inequivalent solutions. Up to scaling, they are We see from (C.3) that the squashed metric on SO (5) in this solution has Lorentzian (1,9) signature, because x 4 is negative.
It was noted in [58] that although various examples of squashed group manifolds were checked and many squashed Einstein metrics were found, all them had either Euclidean signature or else more than one timelike direction. Our squashed SO(5) bosonic string vacuum (C.9) thus provides a first example of a Lorentzian signature group manifold metric arising as a solution in a theory of physical interest.We can, of course, take the flat directions in the vacuum solution to be Euclidean space rather than Minkowski spacetime in this case, so that the signature of the entire higher-dimensional bosonic string spacetime will be (1, D − 1).