Double Field Theory description of Heterotic gauge symmetry enhancing-breaking
Gerardo Aldazabal, Eduardo Andres, Martin Mayo, Victor Penas
aa r X i v : . [ h e p - t h ] A ug Double Field Theory description of Heterotic gaugesymmetry enhancing-breaking
G. Aldazabal a,b,c , E. Andr´es a,c , M.Mayo a,b , V. Penas a,b a G. F´ısica CAB-CNEA,Centro At´omico Bariloche, Av. Bustillo 9500, Bariloche, Argentina. b CONICET c Instituto BalseiroAv. Bustillo 9500, Bariloche, Argentina.
Abstract : A Double Field Theory (DFT) description of gauge symmetry enhancing-breaking in the heterotic string is presented. The construction, based on previous resultsfor the bosonic string, relies on the extension of the tangent frame of DFT. The fluxesof a Scherk-Schwarz like generalized toroidal compactification are moduli dependent andbecome identified with the structure constants of the enhanced group at fixed “self-dual”points in moduli space. Slight displacements from such points provide the breaking ofthe symmetry, gauge bosons acquiring masses proportional to fluxes. The inclusion offermions is also discussed.September 23, 2018 ontents SU (2) × SO (32) × U (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 SO (34) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Introduction
The possibility of understanding gauge symmetry enhancement from a Double Field The-ory (DFT) perspective was addressed in various recent articles [1, 2, 3]. The discussionwas done in the context of the bosonic string since, even if ill defined, it is the simplestexample in several aspects and allows to identify the relevant ingredients. In the presentnote we follow similar steps as in [3] in order to describe the gauge symmetry enhancement(and breaking) in the heterotic string from a DFT-like formulation.Gauge symmetry enhancement is a very stringy phenomena associated to the factthat the string is an extended object and, therefore, it can wind around non-contractiblecycles. String states are thus characterized by a stringy quantum number, the so-calledwinding number, counting the number of times that the cycle is wrapped by the string.The exchange of winding and momentum states (accompanied by a transformation ofmoduli fields) leads to T-duality invariance, a genuine stringy feature.At certain moduli points (fixed points of T-duality transformations) vector bosonstates in some combinations of windings and momenta become massless and give rise toenhanced gauge symmetries (see for instance [4, 5]). Of course, the effective low energytheory, where massive states are neglected, can be described by an usual gauge field theoryLagrangian, containing gravity, with no reference to any windings. An intriguing aspectis that this field theory somehow encodes information about stringy effects. Moreover,even if gauge symmetry breaking is achieved as usual, with some scalar fields acquiringvevs, this higgsing process must encode information about moduli away from the fixedpoint.Interestingly enough, this effective theory close to self-dual points originated in thebosonic string, can be embedded [3] into a DFT-like formulation. In DFT (we will bemore precise below) the internal configuration space includes, besides the usual spacecoordinates dual to KK momenta, new coordinates dual to winding states and therefore,coordinates are doubled. This DFT rewriting allows to highlight the stringy aspects ofthese gauge theories. Actually, in a generalized Scherk-Schwarz [6, 7] compactification ofthis DFT the fluxes, computed from an internal vielbein depending on doubled coordi-2ates, appear to depend on moduli and become the structure constants of the enhancedgroup at fixed points. We show below that this rewriting also works for the bosonicsector of a toroidally compactified heterotic string. Moreover, we show that by invokingsupersymmetry, a corresponding fermionic sector can also be introduced.In Section 2 we present a brief discussion of symmetry enhancement and show theDFT rewriting of heterotic string theory effective action close or at the enhancing points.It is also shown how breaking of gauge symmetry is encoded into the moduli dependenceof fluxes. A simple illustration for the case of circle compactification is provided. Ideaspresented in [3] are recurrently used throughout the article.The introduction of fermions is discussed in Section 3. In particular we show thatif the gaugings in shift matrices of gauged supergravities, associated to fermionic massterms, are replaced by Scherk-Schwarz (moduli dependent) fluxes, the masses of fermionsare in correspondence with their bosonic partners, as expected from supersymmetry.Several details are presented in the Appendices. In Appendix A a quick introductionto DFT and generalized Scherk-Schwarz like compactification is provided with emphasisin the heterotic case where the ingredients needed in our construction are highlighted.For a more complete introduction to DFT we provide some original references in [8] andrefer the reader to some reviews [9, 10, 11] (where a more extensive list of references canbe found). In Appendix B a brief account of heterotic string features needed for ourdiscussion is presented.Concluding remarks and a brief outlook are presented in Section 4.
Toroidal compactification of the SO (32) (or E × E ) heterotic string to d space-timedimensions leads to a generic gauge group G L × U (1) − dR (2.1)3here the left group G L is generically a product of non-abelian and abelian gauge groups.The rank of G L is r L = 16 + 10 − d = 26 − d originated from the 16 Cartan generatorsof the ten dimensional gauge group plus the r = 10 − d vector bosons coming from leftcombinations of the KK reductions of the metric and the antisymmetric tensor. Differentgauge groups do appear when moving along moduli space. At generic points in modulispace G L = U (1) − dL while a point of maximum enhancement leads to G L = SO (52 − d )for the SO (32) string case.We present some basic details in Appendix B. Let n = n c + r L = dim G L be thedimension of G L at some moduli point with n c denoting the number of charged generators.The effective low energy theory will thus be a G L × U (1) − dR gauge theory coupled togravity and the Kalb-Ramond antisymmetric tensor field in d dimensions. There arealso ( n c + 26 − d )(10 − d ) scalars. Thus, the counting of degrees of freedom leads to: d corresponding to graviton plus B field, n c + 36 − d vectors from G L × U (1) − dR and( n c + 26 − d )(10 − d ) scalars. Recall that the number of scalar fields corresponds to(26 − d )(10 − d ) moduli plus n c (10 − d ) extra scalars that should become massive atgeneric points where the broken gauge group is U (1) − dL × U (1) − dR .It is interesting to notice that the total number of degrees of freedom coincides with dim O ( d + n, d + r ) O ( d + n ) × O ( d + r ) = d + d ( n c + 36 − d ) + ( n c + 26 − d )(10 − d ) . (2.2)Indeed, this coset-like writing provides a clue of how to express the effective theory in aDFT-like form as discussed in Appendix A.Following similar steps as presented in [1, 3] for the bosonic string case, we proposean expression for such an action and then discuss its specific features. Namely, S eff = 12 κ d Z d d x √ ge − ϕ (cid:20) R + 4 ∂ µ ϕ∂ µ ϕ − H µνρ H µνρ − H AB F Aµν F Bµν + 18 ( D µ H ) AB ( D µ H ) AB − V (cid:21) . (2.3)Here V = − f ABK f LC D (cid:0) H AL H BC H KD − H AL η BC η KD + 2 η AL η BC η KD ) − Λ (2.4)is a scalar potential where the last two terms are just constants. The scalars parametrizethe coset O ( n,r ) O ( n ) × O ( r ) of dimension ( n c +26 − d )(10 − d ). The indices can be conveniently split4n a L-R basis (named a C base) as A = ( a, ˆ I ) where a = 1 , . . . r L , r L + 1 , . . . r L + n c = n =dim G L index runs over the left group G L . In addition the ˆ I = 1 , . . . r index corresponds tothe Right U (1) r group. The index contractions are performed with η AB , the O ( r L + n c , r )invariant metric η AB = r L + n c − r . (2.5) H AB is the (so-called) internal generalized metric encoding information about scalarfields. R is the d -dimensional Ricci scalar and F Aµν and H µνρ F B = dA B − √ f CDB A C ∧ A D H = dB + F C ∧ A C − √ f ABC A A ∧ A B ∧ A C , (2.6)are the gauge field and B field strengths.The covariant derivative of the scalars is( D µ H ) AB = ( ∂ µ H ) AB + 1 √ f K LA A Lµ H KB + 1 √ f K LB A Lµ H AK . (2.7)Finally, the f ABC = η AK f K BC are completely antisymmetric constants. Interestinglyenough this action can be interpreted as a generalized Scherk-Schwarz reduction of a DFT-like action, as we briefly sketch in Appendix A, the constants f K BC being the generalizedfluxes of the compactification . There are ( r + n )( r − n )( r − n )3! such fluxes which must satisfythe quadratic constraints f [ ABK f K ] CR = 0 . (2.8)If indices are allowed to transform then the action is globally invariant under O ( n c + 26 − d, − d ) and it can be identified with the bosonic (electric) sector of a half-maximalgauged supergravity action [12, 13, 14, 15].In spite of the fact that this huge number of gaugings was explored in several situations,its physical interpretation deserves further investigation. For instance, if we restricted to a = 1 , . . . , r = 10 − d , and in r = 6 dimensions, the above counting of fluxes wouldcorrespond to the 220 gaugings of electric sector of O (6 ,
6) gauged supergravity. These Recall that other kinds of fluxes like f A could be present [7, 9, 22] as shown in Appendix A. Here weset them to zero since they are not relevant for our discussion. M a, ˆ I with dimG L × r = ( n c + 26 − d )(10 − d ) independent degrees of freedom. Namely we write H AB C = δ AB + H (1) AB + 12 H (2) AB + . . . (2.9)such that matrix elements vanish unless H (1) a ˆ I = M a, ˆ I , H (1)ˆ Ia = M Ta, ˆ I (2.10) H (2) ab = ( M M T ) ab , H (2)ˆ I ˆ J = ( M T M ) ˆ I ˆ J . Moreover, we make a specific choice for flux values (therefore breaking the globalsymmetry), by identifying them with the gauge group structure constants. Namely, f ABC = f abc G L structure constants0 otherwise , where f abc is the subset of all possible fluxes (with Left indices) reproducing the structureconstants of the G L group algebra. When couplings are adequately adjusted the aboveaction reduces to the G L × U (1) − dR gauge theory action S = 12 κ d Z d d x √ ge − ϕ (cid:18) R + 4 ∂ µ ϕ∂ µ ϕ − H µνρ H µνρ (cid:19) − (cid:18) δ ab F aµν F bµν + δ ˆ I ˆ J ¯ F ˆ Iµν ¯ F ˆ Iµν − g d √ α ′ M a ˆ I F aµν ¯ F ˆ Iµν (cid:19) − D µ M a ˆ I D ν M a ˆ I g µν + O ( M ) , (2.11)reproducing the bosonic sector of heterotic string low energy theory at a fixed point. Here a labels the Left gauge group (generically non- Abelian) generators with vector bosons A aLµ and ˆ I = 1 , . . . r the Abelian group U (1) ˆ I associated to vector bosons A ˆ IRµ . The scalarfields live in the ( dimG L ) ˆq = adjoint representation of G L and carry zero vector charge ˆq = (ˆ q , . . . , ˆ q r ) = 0 with respect to U (1) rR right group. Thus, the covariant derivative in62.7) becomes D µ M a ˆ I = ∂ µ M a ˆ I + g d f kla A lLµ M k ˆ I , (2.12)where g d = κ d q α ′ . Notice that no scalar potential is generated for this choice of structureconstants.In the next section we show, in the context of DFT, how gauge symmetry breakingcan be achieved by allowing structure constants to depend on moduli, as expected fromstring theory. In string theory the structure constants can be read out from 3-point vertex vector bosonoperators. For the Cartan generators, which we label with the index ˇ I L = ( i = 1 , . . . r ; I =1 , . . . V ( ˇ I L ) ∝ ∂ z y ˇ I ˜ ψ µ e iK.X , whereas for charged operators we have V ( l L ) ∝ e il L .y ( z ) e iK.X where l ˇ IL are theLeft internal momenta defined in (B.3), and X µ ( z ) and K µ are the space-time coordinateand momentum, respectively. Recall that the internal momenta depend on specific valuesof KK momenta p m and winding numbers ˜ p m as well as on the Λ weights P I . Weencode these values into a “generalized momentum vector”ˇ P = ( P ; P I ) , with P = ( p m , ˜ p m ) . (2.13)Let us encode denote by Φ = ( g, B, A ) a generic moduli point. Since momenta dependon moduli fields we actually have l L = l ˇ P L (Φ) and similarly, l R = k ˇ P R (Φ).At specific points Φ in moduli space and for certain values of ˇ P such that k ˇ P R (Φ ) = 0 , ( l ˇ P L (Φ )) = 2 (2.14)gauge symmetry enhancement occurs (B.5). At these points l (ˇ P ) L (Φ ) ≡ α (ˇ P ) (2.15)become the roots α (ˇ P ) of the G L gauge group. Notice that there is an associated root toeach of the n c possible values of ˇ P , satisfying the massless vector condition (2.14). For the sake of clarity we concentrate in the SO (32) string but same conclusions are valid for the E × E heterotic case with lattice Λ × Λ . h V L ( l (ˇ P ) L ) V L ( l (ˇ P ) L ) V L ( l (ˇ P ) L ) i ∝ f α (ˇ P α (ˇ P α (ˇ P (Φ) E ( ǫ , K ; ǫ , K ; ǫ , K ) , (2.16)where E ( ǫ i , K i ; i = 1 , ,
3) is a Lorentz invariant antisymmetric function of polarizationvectors ǫ µi ( K i ) and space-time momenta K µi . The constants f α (ˇ P α (ˇ P α (ˇ P (Φ) are antisym-metric and vanish unless internal momentum is conserved, namely ˇ P = − ˇ P − ˇ P . At aself-dual point Φ = Φ this indicates that structure constants f α α α vanish unless α + α is a root . In this case, we can normalize by setting f α (ˇ P α (ˇ P α (ˇ P (Φ) = 1. Momentumconservation also implies that, at the self-dual point, amplitudes mixing Left and Rightindices vanish. However, away from the fixed point, the vertices develop a dependence on l R , V L ( l = ( l L , l R )) ∝ e il ˇ P L (Φ) .y ( z )+ l ˇ P R (Φ) . ¯ y (¯ z ) e iK.X and therefore mixing now occurs. In fact,it is found that the only non vanishing amplitudes are h V L ( l (ˇ P ) ) V L ( l ( − ˇ P ) ) V L ( ˇ I ) i ∝ l ( ˇ P ) L (Φ) ˇ I ; h V L ( l (ˇ P ) ) V L ( l ( − ˇ P ) ) V R ( ˆ I ) i ∝ l ( ˇ P ) R (Φ) ˆ I . Following [3], we propose to identify the amplitude coefficients with some algebra structureconstants, even (slightly) away from the fixed point Φ . Namely we set f α (ˇ P ) α ( − ˇ P ) ˇ I L (Φ) = l ( P ) L (Φ) ˇ I , f α ( P ) α ( ˇ − P ) ˆ I (Φ) = l ( ˇ P ) R (Φ) ˆ I (2.17)with the other constants being obtained as permutations, and we propose the algebra (cid:2) E α , E − α (cid:3) = l ( α )ˇ IL H ˇ I + l ( α )ˆ IR ˆ H ˆ I (cid:2) H I , E α (cid:3) = l ( α ) IL E α (cid:2) E α , E α (cid:3) = f α α α E α (cid:2) ˆ H I , E α (cid:3) = l ( α ) IR E α . (2.18)We have used α = α ( P ) to alleviate the notation and, as we found above, f α α α = 1 if α = α + α is a root and vanishes otherwise. All other commutators vanish. It is easyto show that (2.18) satisfies Jacobi identities and, therefore, defines a Lie algebra.Recall that, at the self-dual point where k αR (Φ ) = 0 and f α − α ˆ I = l ( α )ˆ IL = α ˆ I , thealgebra reduces to to the gauge algebra of G L group in the Cartan-Weyl basis. Forinstance [ E α , E − α ] = α ˆ I H ˆ I for charged generators E α and Cartan generators H ˆ I , asexpected. Interestingly enough, by performing a linear combination of generators it canbe shown that there is still an underlying G L algebra. To visualize the linear combination8et us define a double Cartan operator H A = (cid:0) H ˇ I , ˆ H ˆ I (cid:1) and the double (moduli dependent)momentum L ( α ) A = (cid:0) l ( α )ˇ IL , l ( α ) IR (cid:1) . The algebra (2.18) can now be written as (cid:2) E α , E − α (cid:3) = L ( α ) A H A (cid:2) H A , E α (cid:3) = L ( α ) A E α (cid:2) E α , E α (cid:3) = f α α α E α . (2.19)It is worth observing that an O ( r L , r R ) transformation can be performed over thedouble Cartan generator, namely the one given by the inverse of δ IJ A Jn δ nm − A Jm δ JI δ nm G nm − B nm − R A In A Im − A Jm δ JI δ nm − G nm − B nm − R A In A Im , (2.20)such that L ( α ) is mapped to ˇ P (see (2.13)) and H to new double Cartan’s H leading to (cid:2) E α , E − α (cid:3) = ˇ P ( α ) A H A (cid:2) H A , E α (cid:3) = ˇ P ( α ) A E α (cid:2) E α , E α (cid:3) = f α α α E α . (2.21)This final algebra has the same form independently of moduli values. Furthermore sincethe algebra (2.18) and (2.21) are isomorphic, due to (2.20), we conclude that the algebraat the self-dual point is the same at all other (neighborhood) points.In generalized Scherk-Schwarz like compactifications of DFT, the generalized fluxes f ABC are defined from the generalized algebra satisfied by the internal frame (A.13). Letus assume for the moment that a specific choice of frame exists such that these fluxesare the structure constants found in (2.18). Once these fluxes are identified we mustreplace them into the action (2.3). The output is that the resulting action is the gaugebroken symmetry action where vector bosons and scalars acquire masses proportional tostructure constants mixing left and right indices, namely f α ( P ) α ( − P ) ˆ I (Φ). We start by inspecting the couplings between vectors and scalars arising from kineticterms in (2.3). By keeping the first term in the internal metric expansion (2.9), H AB C =9 AB + H (1) AB + . . . we find that D µ H AB D µ H AB ≈ ∂ µ M AB f K LA δ KB A µL = 4 ∂ µ M a ˆ I f aL ˆ I A µL (2.22)= 4 ∂ µ M a ˆ I f ab ˆ I A µb . (2.23)Here we have used the exapnsion into Left and Right indices A = ( a, ˆ I ), we have used themetric (2.5), the antisymmetry of f ABC and the fact that the only non vanishing fluxesare of the form f abc , f ab ˆ I .The conclusion is that, for a given vector boson A bµ , there is a combination of ˆ I =1 , . . . , r = 10 − d , would-be Goldstone bosons scalar fields f ab ˆ I M a ˆ I ≡ f α − α ˆ I M α ˆ I = l ( P ) R (Φ) ˆ I M α ˆ I ( whenever f ab ˆ I = 0). We have recast the expression in a Cartan-Weyl basisby recalling that the only non vanishing fluxes (away from the point of enhancement)containing a Cartan index are of the form f α − α ˆ I .Interestingly enough, this combination arises as a conformal anomaly contribution inthe OPE of energy momentum tensor with scalars whenever these scalars become massive,away from the fixed point (see [1] for a bosonic string example). This indicates that thecombination l ( P ) R (Φ) ˆ I M α ˆ I ( K ) of internal R-momentum and scalar polarizations, must beset to zero. Let us see, as an example, how vector bosons and scalar masses arise.
In order to read the vector boson masses we must just look at quadratic terms in the scalarkinetic term. Thus, following similar steps as above but now keeping just the constantterm in the internal metric expansion (2.9) H AB = δ AB + . . . we find18 ( D µ H ) AB ( D µ H ) AB ≈
18 ( f RLB δ KA + f RLA δ KB ) η RK ( f P SA δ P ′ B + f P SB δ P ′ A ) η P P ′ A Lµ A µS = 2 18 ( f RLB f P SA δ P ′ B + f RLB f P SB δ P ′ A ) δ KA η P P ′ η RK A Lµ A µS = − f ˆ IaL f ˆ IaS A Lµ A µS = − ( f α − α ˆ I (Φ)) | A α | , Alternatively, such a combination of scalar vertex operators must be included into a new massive,anomaly free, vector field. m A α given by m A α = X ˆ I ( f α − α ˆ I (Φ)) = l L (Φ) . (2.24) From a DFT point of view, the scalar masses arise from quadratic terms in scalar fluc-tuations in the scalar potential. Thus, by inserting the expansion (2.9) into the scalarpotential (2.4) we find: − f ABC f DEF H (1) AD H (1) BE δ CF − f ABC f DEF H (2) AD (cid:0) δ BE δ CF − η BE η CF (cid:1) . (2.25)We notice that, due to the relative minus sign between left and right indices in η AB (see(2.5)) the second term vanishes unless indices organize as δ be δ ˆ I ˆ J leading to12 X α, ˆ I f α − α ˆ I f α − α ˆ I H (2) αα = 14 X α, ˆ I, ˆ J ( f α − α ˆ I ) | M α ˆ J | = 14 X α, ˆ J m α | M α ˆ J | , (2.26)where m α = X ˆ I ( f α − α ˆ I (Φ)) = m A α (2.27)is the mass (square) of the scalar field M α ˆ J , coinciding with the vector boson mass.On the other hand, the first term contribution in (2.24) leads to12 X α (cid:16) X ˆ I f α − α ˆ I m α ˆ I (cid:17) . (2.28)However, this contribution is irrelevant since (cid:16) P ˆ I f α − α ˆ I m α ˆ I (cid:17) is the Goldstone bosoncombination.Let us stress that the obtained masses coincide with the masses computed from stringmass formula (B.3). 11 .2 Examples Here we discuss a simple illustration of the above construction in the simplest case ofcompactification on a circle of radio R . In this case (B.3) reads K IL = P I + RA I ˜ p (2.29) k L = r α ′ (cid:2) pR + ˜ p ˜ R − P.A − R A.A ˜ p (cid:3) k R = r α ′ (cid:2) pR − ˜ p ˜ R − P.A − R A.A ˜ p (cid:3) . A massless state requires k R = 0 ( ¯ N F = 1) and then k L = √ α ′ ˜ p ˜ R . SU (2) × SO (32) × U (1)A possible set of massless vectors is provided by choosing p = ˜ p = ±
1, by setting the radioto its self-dual value R = √ α ′ = ˜ R and A I = 0. Together with the massless vector stateassociated to KK compactification (with p = ˜ p = 0 = P I ) mode these massless vectorslead to an SU (2) left group.In addition, an SO (32) group associated to the weights P = ( ± , ± , . . .
0) appears(underlining meaning permutation over the 16 entries) and the corresponding 16 Cartanoscillators. Therefore, at this moduli point, the enhanced gauge group is SO (32) L × SU (2) L × U (1) R . In the notation of (2.19) this full set of massless states corresponds toˇ P SU (2) × SO (32) × U (1) = ( ± , ±
1; 0 , . . . ; 0) , (0 , ± , ± , . . . SO (32) L × SU (2) L × U (1) R to SO (32) L × U (1) L × U (1) R or to U (1) L × U (1) R depending on the direction of the moduli space onwhich we move.For instance, by sliding away from the self-dual radio, charged SU (2) vectors becomemassive with mass square m − with m − = q α ′ a − = R − R , where a ± are defined in (A.16).The algebra (2.19) becomes (cid:2) E + , E − (cid:3) = 2( a + H + a − H ¯3 ) (cid:2) E P , E − P (cid:3) = P I H I (cid:2) H , E ± (cid:3) = ± a + E ± (cid:2) H I , E P (cid:3) = P I E P (cid:2) H ¯3 , E ± (cid:3) = ± a − E ± (cid:2) E P , E P (cid:3) = f P P P E P . (2.30)12he subindices ± denote the two roots of SU (2), the subindex 3 denotes the correspondingCartan whereas f P P P are the structure constants of SO (32) where P I are the roots and H I the Cartan generators. At the self-dual radio we have a − = 0, a + = 1 and the SU (2)gauge algebra is recovered.By turning on Wilson lines A I the group is broken to U (1) L × U (1) R . The algebrabecomes (cid:2) E + , E − (cid:3) = (cid:0) − A (cid:1) H + (cid:0) A (cid:1) H ¯3 + A I H I (cid:2) E P , E − P (cid:3) = P I H I − ( P · A ) H − ( P · A ) H ¯3 (cid:2) H , E ± (cid:3) = ± (cid:0) − A (cid:1) E ± (cid:2) H I , E P (cid:3) = P I E P (cid:2) H ¯3 , E ± (cid:3) = ± (cid:0) A (cid:1) E ± (cid:2) E P , E P (cid:3) = f P P P E P (cid:2) H I , E ± (cid:3) = ± A I E ± (cid:2) H ¯3 , E P (cid:3) = − ( P · A ) E P . (2.31) As discussed, the vector boson masses are identified with the structure constants mixingLeft and Right indices. Therefore we find that SU (2) charged vectors A ± µ acquire a mass m SU (2) = | f ¯3 ±± | = A whereas SO (32) charged vectors masses are m SO (32) = | f ¯3 P − P | = | P · A | . As discussed in the general case, the above commutators satisfy Jacobi Identitiesand define an SU (2) × SO (32) algebra now involving massive states. Let us recall thatfrom DFT perspective the algebra is obtained through generalized Lie derivatives of thetwists E A ( Y ). The explicit twist for the SU (2) sector is given in (A.15). SO (34)Other enhanced groups can be obtained at different points in moduli space. points inmoduli space can lead to different enhancements. For instance, by choosing [4] ˜ R = √ α ′ and RA = ( − , . . .
0) we notice that for ˜ p = 0 massless states are obtained if P =( ± , ± , . . . SO (32) root, if KK momenta p = − P is selected. Moreover,the SO (32) weights P = ( ± , ± , . . . , (0 , . . . , (2 , . . .
0) combined with ˜ p = ± l L = ( ± ; ± , . . .
0) states that combined with the SO (32) roots lead to massless states withcharged operators associated to l L = ( ± ; ± . . .
0) corresponding to the well known SO (34)enhanced group [4].Recall thet our description holds at the neighborhood of the SU (2) × SO (32) × U (1)point (defined by a specific choice of generalized momentum ˇ P SU (2) × SO (32) × U (1) and modulifields) or SO (34) point with different generalized momenta ˇ P SO (34) and moduli fields but13t is not possible to continuously interpolete between both points. The action (2.3), for d = 4, is nothing but the N = 4 bosonic (electric) sector of ageneric gauged supergravity theory (see for instance [17, 12, 13, 14]). We then see that,Scherk-Schwarz reduction of DFT provides a way of deriving this gauged supergravitysector.Inclusion of the magnetic sector requires considering EFT or an extension of the initialglobal group. The inclusion of fermions from a DFT point of view was considered in severalworks [18] and, in particular, a Scherk- Schwarz like reduction was proposed in [20] in thecontext of the superstring.The aim of the present section is to show that the mechanism of gauge symmetryenhancing- breaking through moduli dependent fluxes, found for the bosonic sector, isreproduced in the fermionic sector.By invoking supersymmetry we conclude that the fermionic sector is just the fermionicsector of gauged supergravities discussed in the literature. We first concentrate in the N = 4 case in four dimensions and discuss its generalization later on. Therefore, wemust deal with the global symmetry group O (6 + n, ψ µi andgaugini λ aj read [17, 12] e − L f.mass = 13 g A ij ¯ ψ µi Γ µν ψ µj + ig A aij ¯ ψ µi Γ µ λ aj + A abij ¯ λ aj λ aj + h.c. , (3.1)where the matrices A ij , A aij , A abij are known as shift matrices. Here indices i spanthe spinorial representation of SO (6) or, equivalently, the 4-dimensional representationof SU (4), the universal cover of SO (6). SO (6) vectors v ˆ m can be recast in terms ofthe antisymmetric combinations of spinorial representations, or, equivalently in terms ofantisymmetric SU (4) six dimensional representation v ij through v ˆ m ( γ ˆ m ) ij = v ij where v ij = v [ ij ] and v ij = ( v ij ) ∗ = 12 ǫ ijkl v kl . (3.2)14he shift matrices are known to depend on scalars through the coset representatives U A ¯ A ( x ) defining the scalar matrix (A.11). For internal indices such matrix reads H AB ( x ) = δ ¯ A ¯ B U A ¯ A U B ¯ B . (3.3)with U A ¯ A ( x ) ≡ ( U Aa ; U A ˆ I ) = ( U Aa ; U Aij ) , (3.4)and where the SO (6) vector index ˆ I was expressed in terms of the spinor indices ij in thelast term.The shift matrices then read (see for instance[17, 12])gravitini-gravitini: A ij ∝ ( U Akl ) ∗ U Bik U C jl f ABC gravitini − gaugini : A aij ∝ U Aa ( U Bik ) ∗ U C jk f ABC gaugini − gaugini : A abij ∝ U Aa U Bb U C ij f ABC , (3.5)where we have used f ABC to denote the electric sector gaugings f + ABC , the + subindexindicating the electric sector[12]. In order to read vector masses we need to keep theconstant term in the expansion of U A ¯ A in scalar fluctuations (see (3.3)) reproducing themetric expansion H AB = δ AB + O ( M ). Therefore ( U Ab ; U A ˆ I ) = ( δ Ab ; δ A ˆ I ) with U Aij = δ A ˆ m ( γ ˆ m ) ij . (3.6)By replacing this expansion into shift matrices expressions we find A ij ∝ δ A, ˆ a ( γ ˆ a ) ∗ kl δ B, ˆ I ( γ ˆ I ) ik δ C, ˆ c ( γ ˆ c ) jl f ABC = ( γ ˆ a ) ∗ kl ( γ ˆ I ) ik ( γ ˆ c ) jl f ˆ a ˆ I ˆ c A aij ∝ ( γ ˆ I ) ∗ ik ( γ ˆ c ) jk f a ˆ I ˆ c A abij ∝ ( γ ˆ c ) ij f ab ˆ c . (3.7)By identifying the gaugings f ABC with the fluxes defined above and by using thatfluxes involving more than one right index vanish ( f ˆ a ˆ I ˆ c = f a ˆ I ˆ c = 0) we find that, gravitiniremain massless, as expected (same is valid for dilatini). On the other hand, gauginimasses are proportional to f ab ˆ c , so having the same masses as their vector boson super-partners, vanishing at the self-dual enhancing points. Together with scalars and vectorbosons they fill up the N = 4 vector supermultiplet multiplet.15et us argue that the discussion presented here for d = 4 extends to other dimensions.In fact, for half-maximal theories, the scalars form a coset G/H = SO ( d, d + n ) /SO ( d ) × SO ( d + n ) and are encoded in a coset representative U ¯ AA = ( U Aa ; U A ˆ I ) as in (3.4) where A is now a G = SO ( d, d + n )-vector index, a is a SO ( d )-vector index and ˆ I is a SO ( d + n )-vector index. The index ˆ I is expressed in terms of spinor indices since fermions transformunder H = Spin ( d ) × SO ( d + n ). From the full set of possible gaugings it is still possible tochoose a subset parametrized by an antisymmetric G -tensor, namely, f ABC . For instance,in d = 4 the full set of gaugings is parametrized by ξ αA and f αABC ( α = ± is the electri-magnetic index) and we have restricted to ξ αA = 0 and f + ABC = f ABC , f − ABC = 0. Thesame applies in other dimensions . The fermion shift matrices would couple to scalarsthrough the embedding tensor and therefore they will necessarily have the same form asin (3.5) but where i, j indices span the spinorial representation of Spin ( d ). The readershould be aware that the actual mass terms of bilinear fermions are linear combinationsof the above terms including scalars factors.As before when the gaugings are the ones coming from an enhancement point ofthe string moduli space the structure constant (the fluxes) will take the values of theprevious section. In these points, the gravitini shift matrices are zero and supersymmetryis preserved. Away from the point of enhancement scalars vectors and fermions organizedinto a massive supermultiplet. In the present work we have shown how DFT can provide an interesting descriptionof the gauge symmetry enhancing-breaking process that occurs in the heterotic stringsat specific points of moduli space. The construction relies on previous ideas used todescribe this process in the bosonic string case. The three key ingredients encodingenhancing information are: a global O ( n , n ) invariant gauged (super) gravity action, ascalar fluctuation expansion of a generalized scalar metric and the presence of generalized, If we wanted it to also hold in d = 9 and d = 8 we must necessarily include vector multiplets, thus, n ≥
1. Otherwise, f ABC = 0, see [17, 22]. O ( n, r ) (where n = dim G L is the dimension of the enhanced group at the fixed point and r = 10 − d the number of compact dimensions) and by identifying 3-form fluxes f ABC (Φ) with theinternal momenta of the string. Recall that indices are conveniently written as A = ( a, ˆ I )with a = 1 , . . . n, ˆ I = 1 , . . . r . At a point of enhancement Φ the only non vanishing fluxesare those with only Left indices f abc (Φ ) reproducing the structure constants of the G L group. Away from this point, mixed indices give rise to non-vanishing fluxes f ab ˆ I (Φ).These mixed indices fluxes govern the vector boson masses, the scalar masses and thestructure of the would be Goldstone bosons. It is worth emphasizing that this structureexactly matches the string theory results with the correct full dependence on moduli fields.By invoking supersymmetry, a fermionic sector can also be included. In particular wehave shown that moving away from the enhancing point Φ produces the expected massesfor gaugini partners of massive vector bosons while keeping supersymmetry unbroken.Let us address some open questions. In DFT the generalized fluxes appear as gen-eralized Lie derivatives, involving internal coordinates Y M , of generalized internal framevectors E A ( Y ). In a generic construction, the internal coordinates transform in the vectorrepresentation of the global group O ( n , n ), namely M = 1 , . . . , n + n and the same isvalid for the frame index A . However, it appears that in order to reproduce the abovefluxes, just a dependence on the “true” internal Left and Right 16 + r + r = 36 − d coordinates, associated to string coordinates would be needed. In fact this was shown tobe the case for some specific examples in [1, 3] (see also [2]) for the bosonic string case. Ina similar line of reasoning a dependence on Y = ( Y I , y IL , y ˆ IR ) with I = 1 , . . .
16; ˆ I = 1 , . . . r would be expected. Therefore, the tangent space here, spanned by A would account forthe gauge symmetry enhancement, associated to states with non vanishing KK momentaand windings, but the “physical space” would be the string torus (including Γ ). Theexplicit construction for the heterotic string here remains as an open question.Recall that our description is valid close to a given moduli point.When moving fromone point of enhancement to a new point the dimension of the gauge group can drasticallychange and, therefore, the dimension of the tangent space. Even if, as stressed in [3], these17angent directions are not physical dimensions an explanation of how, moving continuouslyfrom one point of enhancement to another could lead to a discrete change in the number ofthese extra tangent dimensions is still lacking. DFT description would presumably requirethe introduction of extra states, mimicking the string theory situation. Following thesuggestions in [3] this could be presumably achieved by considering a sort of generalizedKK expansion on generalized momenta L of the different fields coming into play. Thus,very schematically a vector boson corresponding to a charged generator would read A Lν ( x, Y ) = X L A ( L ) Lν ( x ) e i L M Y M δ ( L ,
1) = X L A ( L ) Lν ( x ) e iK.Y + ik L .y L + ik R .y R δ ( L , , where K I , k mL , k mR are functions of the moduli. Therefore, when moving continuously alongthe moduli space, and for specific values of generalized momenta L in above sum, k R = 0and the associated vector fields A ( L ) Lν ( x ) become massless. The neighborhood of each ofsuch points is what our description would be capturing.In order to address the description of the enhancement process we made a specificchoice of generalized fluxes f ABC with A = ( a, ˆ I ) by keeping just the indices leading to theenhanced gauge group structure constants at the enhancing moduli point and setting theother components to zero. However, it appears interesting to explore the meaning of otherpossible components. In fact, we have already mentioned that, if we look at all indicesrunning from 1 , . . . r + n , namely a = i, ˆ I = 1 , . . . r the corresponding fluxes encode thegeometric and non-geometric (closed string) fluxes discussed in the literature[16]. In thesix dimensional case, these fluxes span the representation of O (6 , Similar expansions were considered in [29] for the bosonic string case and for L = 0. cknowledgments We thankD. Marqu´es, C. Nu˜nez and A. Rosabal for useful discussions. This work wassupported by CONICET grant PIP-11220110100005 and PICT-2012-513. G. A. thanksthe Instituto de F´ısica Te´orica (IFT UAM-CSIC) in Madrid for its support via the Centrode Excelencia Severo Ochoa Program under Grant SEV-2012-0249.
A Heterotic DFT
In this section we briefly present some basic ingredients of DFT in the so called dynamicalfluxes formulation. Details can be found in the references [8, 9, 21, 11, 10].In this formulation, the field degrees of freedom (metric, antisymmetric field, 1-forms)are encoded into generalized frame vectors E ¯ AM that parametrize a coset G/H where G is a duality group. Generalized metric is thus obtained from H = E ¯ A S ¯ A ¯ B E ¯ B , (A.1)where the S ¯ A ¯ B is given by S ¯ A ¯ B = s ¯ a ¯ b s ¯ a ¯ b , (A.2)and s ¯ a ¯ b is the d -dimensional Minkowski metric.A scalar field d incorporates the dilaton. Gauge invariance appears through a gen-eralized Lie derivative L ξ . Transformations under this derivative lead to the dynamicalfluxes.The DFT action reads S = Z dXe − d R , (A.3)with R = F ¯ A ¯ B ¯ C F ¯ D ¯ E ¯ F (cid:20) S ¯ A ¯ D η ¯ B ¯ E η ¯ C ¯ F − S ¯ A ¯ D S ¯ B ¯ E S ¯ C ¯ F − η ¯ A ¯ D η ¯ B ¯ E η ¯ C ¯ F (cid:21) + F ¯ A F ¯ B (cid:20) η ¯ A ¯ B − S ¯ A ¯ B (cid:21) , (A.4)19here the dynamical fluxes F ¯ A ¯ B ¯ C and F ¯ A are defined in terms of the generalized Liederivative L and vielbeins E ¯ A by F ¯ A ¯ B ¯ C = E ¯ CM L E ¯ A E ¯ BM (A.5) F ¯ A = − e d L E ¯ A e − d . (A.6)The field d incorporates the dilaton field and transforms like a measure L V e − d = ∂ P ( V P e − d ) . (A.7)The indices M take values in the fundamental representation of the group G whereasthe flat indices ¯ A run over H . Usually, motivated by the bosonic string constructionthe group G is chosen to be O ( d, d ) with d the space-time dimensions. However theconstruction is more general and we can choose G = O ( d L , d R ) generically containing O ( d, d ) and H = O ( d L ) × O ( d R ). Indices are raised and lowered with a symmetric metricthat can be chosen as η P Q = d L − d R . (A.8)A numerically similar matrix η ¯ A ¯ B is used for flat indices.The dynamical fluxes depend on the generic coordinates spanning a vector repre-sentation of O ( d L , d R ). The above action is generically non invariant under generalizeddiffeomorphisms generated by the generalized Lie derivative unless some constraints areimposed. A consistent solution is given by a generalized Scherk Schwarz reduction wherethe frame is split into a space-time dependent part and an internal one [7, 9, 10]. The Liederivative becoming a gauge transformation plus usual space-time diffeomorphisms.Since we are interested in the description of the heterotic case we perform a specifchoice suitable for its description (see also [9, 23, 24, 25]). Inspired by the coset structurepresented in (2.2) we choose d L = d + n c + 26 − d = d + n and d R = d + 10 − d = d + r .We also choose the fields to depend on the coordinates X M = (cid:0) x µ , ˜ x µ , Y M (cid:1) where Y M = ( y m L L , y m R R (cid:1) . M = ( m L , m R ) is an internal index with m L = 1 , ..., n and m R =1 , ..., − d whereas µ = 1 , ..., d . Here r = 10 − d is the number of compact dimensionsand n = n c + 26 − d is the number of extra directions needed to achieve the enhancement.20ote that in section (2) the index M is denoted as A and m L = a , m R = ˆ I . As usual,˜ x µ coordinates are just an artifact and can be dropped away or, in the DFT language,the strong constraint must be used on the space-time part. Explicitly the Scherk-Schwarzreduction ansatz reads E ¯ A ( x, y L , y R ) = U ¯ AA ( x ) E ′A ( y L , y R ) . (A.9)The matrix U encodes the field content in the effective theory, while E ′ is a general-ized twist that depends on the internal coordinates. All the dependence on the internalcoordinates occurs through this twist.By introducing the splitting ansatz (A.9) into the expression (A.1) for the generalizedmetric we can write H = S ¯ A ¯ B U ¯ AA ( x ) E ′A ( Y ) U ¯ BB ( x ) E ′B ( Y ) = H AB ( x ) E ′A ( Y ) E ′B ( Y ) , (A.10)where all the field dependence on space-time coordinates is encoded in H AB ( x ) = S ¯ A ¯ B U ¯ AA ( x ) U ¯ BB ( x ) . (A.11)In particular, when the indices take internal values A, B = 1 , ..., n + r , the matrix of H AB parametrizes the scalar content of the theory.By restricting the expression for the generalized Lie derivative to the specific case ofthe twist it is found that L E ′ A E ′ B = 12 (cid:2) E ′ PA ∂ P E ′ MB − E ′ PB ∂ P E ′ MA + η MN η P Q ∂ N E ′ PA E ′ QB (cid:3) (A.12)[ E ′ A , E ′ B ] = L E ′ A E ′ B = f ABK E ′ K , (A.13)where here all indices are internal. The fluxes f ABK of the generalised Scherk-Schwarzreduction must be constants and must satisfy the constraints f ABC ≡ η AK f ABL = f [ ABC ] , f [ ABK f K ] CR = 0 (A.14)in order for the algebra to close.When replacing above results into the initial DFT action (A.3), the expression for thegauged DFT action (2.3) is obtained. 21et us stress that a specific selection of values for the fluxes f ABC , constructed outfrom the internal frame derivatives (A.12) will be associated to a specific dependence onthe generalized coordinates. For instance, for the extreme case of a coordinate indepen-dent frame leads to an abelian compactification, and corresponds to a KK reduction, forinstance.In particular, it was shown in [1, 3, 2] that, at least for some cases, for the twists E A to reproduce the structure constants of the gauge group the twist only depends on thetrue internal coordinates of the torus. Thus, for a circle compactification of the bosonicstring the twist only depends on the circle coordinate y L and its dual y R . Explicitly[3], E ± = i √ α ′ ( e ∓ iw , ± ie ∓ iw ,
0; 0) E = − i √ α ′ (0 , ,
1; 0)¯ E ˆ3 = − i √ α ′ (0 , ,
0; 1) , (A.15)where w = a + y L + a − y R , ¯ w = a − y L + a + y R and a ∓ = r α ′ R ∓ R ] . (A.16)It is easy to check that, by inserting this twist expression into (A.12) and noticing that theonly contributions to the derivatives come from ∂ A = (0 , , ∂ y L ; ∂ y R ), the SU (2) algebrais reproduced. Here we just assume that there exists a choice of internal coordinates suchthat (A.13) leads to the desired gauge group structure constants and leave the constructionof the explicit twist for future work. B Some Heterotic string basics
We summarize here some string theory ingredients (that can be found in string books)needed in the body of the article. We mainly concentrate in the SO (32) string.For a heterotic string compactified to d space-time dimensions, Left and Right mo-menta are encoded in momentum L = ( l L , l R ) (B.1)defined on a self-dual lattice Γ − d, − d of signature (26 − d, − d ). By writing l ˇ IL =( K IL , k L,m ) with I = 1 , . . . ,
16 and m = 1 , . . . − d = r , the moduli dependent momenta,22ead K IL = P I + RA In ˜ p n (B.2) k L,m = r α ′ (cid:2) p m R + ( g mn − B mn ) ˜ p n ˜ R − P I A Im − R A Im A In ˜ p n (cid:3) k R,m = r α ′ (cid:2) p m R + ( − g mn − B mn ) ˜ p n ˜ R − P I A Im − R A Im A In ˜ p n (cid:3) , where g mn , B mn are internal metric and antisymmetric tensor components, A m are Wilsonlines and p n and ˜ p n are integers corresponding to KK momenta and windings, respectively. P I are Spin (32) weight components.The mass formulas for string states are α ′ m L = 12 l L + ( N B −
1) = 12 K L + 12 k L + ( N B − α ′ m R = 12 k R + ¯ N B + ¯ N F + ˜ E , (B.3)where N, ¯ N are the number of string oscillators, ˜ E = − (0) for NS (R) sector and thelevel matching condition is m L − m R = 0 or, in terms of above notation L = 12 l L − k R = 1 − N B − ¯ N B − ¯ N F − ˜ E . (B.4)In particular, massless charged vectors correspond to L = 1. As is well known, there are10 − d +16 Left gauge bosons corresponding to 16 Cartan generators ∂ z Y I ˜ ψ µ of the originalgauge algebra as well as 10 − d KK Left gauge bosons coming from a Left combinationof the metric and antisymmetric field ∂ z Y i ˜ ψ µ . The 10 − d Right combinations ∂ z X µ ˜ ψ m with m = 1 , . . . − d generate the Right abelian group. These states have k R = 0 and l L = 0, with vanishing winding and KK momenta.Besides these states, a number of different situations arises. At generic points inmoduli space k R = 0 and therefore there are no extra gauge bosons. The gauge groupis then U (1) − dL × U (1) − dR . 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