JHEP10(2019)204 2 G Springer October 2, 2019 August 23, 2019 October 21, 2019 : : : , case. For all of these 2 Received Accepted G Published b,c Published for SISSA by https://doi.org/10.1007/JHEP10(2019)204 and Alexander Otto a [email protected] , . 3 Suvajit Majumder and Spin(7). Our central result is a construction of mirrors of Spin(7) 2 a 1905.01474 G The Authors. c String Duality, Conformal Field Models in String Theory, Superstring Vacua
We study mirror symmetry of type II strings on manifolds with the exceptional , [email protected] 31 Caroline St N, Waterloo,Department ON of N2L Physics, 2Y5, University Canada Waterloo, of ON Waterloo, N2L 3G1, Canada E-mail: [email protected] Mathematical Institute, University ofOxford, Oxford OX2 Woodstock Road, 6GG, U.K. Perimeter Institute for Theoretical Physics, b c a Open Access Article funded by SCOAP feature appearing in the examples we analyse is theKeywords: possibility of frozen singularities. ArXiv ePrint: to the pair offor non-compact such manifolds geometric they mirror are constructions,orbifolds glued we in give from. several a new To CFTmodels provide instances analysis non-trivial we for of checks mirror both find maps themirror possible for symmetry, Spin(7) assignments Joyce and and confirm of the the discrete consistency with torsion the phases, geometrical work construction. out A novel the action of Abstract: holonomy groups manifolds realized as generalized connectedmanifolds, sums. mirrors In of parallel such to twisted Spin(7) connected manifolds sum can be found by applying mirror symmetry Andreas P. Braun, On mirror maps for manifolds of exceptional holonomy JHEP10(2019)204 ]) 2 and 2 G ] (see also [ 1 21 31 ]. In particular, [ 1 27 20 22 26 orbifold 2 – 1 – 15 G 5 to be mirror is that the dimensions of the spaces of (or Spin(7)) mirror symmetry. A necessary condition ∨ 7 2 14 4 M G 11 and 25 16 18 24 19 9 M 12 14 (or Spin(7) manifolds) manifolds may result in equivalent physical 2 4 G 24 1 example 4.3.1 Example4.3.2 1 Example 2 3.2.2 Example 2 3.2.1 Example 1 2 G 4.2 A mirror4.3 map for GCS Examples Spin(7) manifolds 3.3 Mirror symmetry 4.1 Constructing Spin(7) manifolds as generalized connected sums 3.1 Smoothing3.2 of the orbifold Discrete torsion and cohomology 2.1 Smoothing2.2 of the orbifold Constraints on2.3 discrete torsion Cohomology of2.4 smoothings Mirror symmetry 2.5 Realization as2.6 a TCS TCS mirror map determined the extension of theon worldsheet manifolds superconformal algebra of for exceptional stringsgating holonomy propagating and on pointed different outtheories that in type a II phenomenon string called for theories any propa- pair of manifolds 1 Introduction The study of stringSpin(7) theory from on the worldsheet manifolds perspective with goes the back exceptional to holonomy [ groups B Discrete torsion analysis for the C Discrete torsion analysis for the Spin(7) orbifolds A Discrete torsion and modular invariance 4 Spin(7) mirror maps for connected sums 3 Spin(7) examples Contents 1 Introduction 2 A JHEP10(2019)204 . 2 ]. 1 G 7 ], a S , ), in ]. In (1.1) (1.2) 6 19 16 , 1.1 context, ] in [ 15 2 5 G – 3 manifolds were 2 G , ]. As shown in [ i.e. this duality must map . 1 18 ) + 1 ) ∨ ∨ manifolds to have such fibrations, so manifolds as twisted connected M M 2 ( fibre, 2 ( is exchanged for its mirror [ G 3 4 − 3 G and Spin(7) manifolds were shown b b T 2 1) worldsheet CFTs agree. As shown X G , ) + ) + fibrations for some of Joyce’s examples ∨ ∨ fibration, so that it maps type IIA (IIB) 4 4 T M M = (1 ( ( T 2 2 or b b N must satisfy 3 – 2 – T ∨ ) = ]. This is analogous to mirror symmetry in the M M 16 , ). This map can be described as being the result of ( fibre in question is understood as the product of the ) + 1 = 3 b 4 M 1.1 M manifolds as admitting a limit in their moduli space in which an T on each side). Mirror maps for TCS ( 2 4 − ) + 1 G b S M ( 2 ) + ], where the presence of the B-field prevents the occurrence of manifolds allow a second class of mirror maps satisfying ( ]. b ] have given a construction of 2 M 20 11 ( 14 G 2 – b 12 ], it is not expected on general grounds for or Spin(7) manifolds, but also feature in the duality between M-Theory 17 2 ], where it was shown that applying a mirror map to both of the acyl G (times a circle ], mirror maps for type II strings on 8 ], see also [ 16 , collapses. ) denotes the dimension of the space of anti self-dual four-forms. 3 X 15 10 M T , ( manifold which satisfies ( 9 4 − 2 , b G 7 ], the number of exactly marginal operators simply equals the number of geometric , Calibrated torus fibrations are not just interesting in the study of mirror symmetry Interestingly, TCS More recently, [ This was made explicit for the first time in the context of Joyce orbifolds [ As discussed in [ 1 fibration [ 6 1 3 of Calabi-Yau, it might be betterassociative to speak of such completely analogous construction existsof for dual acyl tops, Calabi-Yau threefolds whichmirrors. in makes An terms it intriguing of feature possible pairs smooth of to both geometries give classes to large ofcontext singular numbers mirror of maps of ones K3 is concrete [ that surfaces examplesextra they [ massless of sometimes states. map case one of the acyllikewise Calabi-Yau be threefolds understood carries an from elliptic threetype fibration, T-dualities IIA this along strings mirror a map tohypersurfaces can type or IIB complete or intersections vicecombinatorial in construction versa. toric using varieties, For pairs compact mirror of Calabi-Yau families reflexive manifolds have polytopes which an [ are elegant performing four T-dualities along astrings calibrated to IIA (IIB)Strominger-Yau-Zaslow strings. (SYZ) fibres The of the acyl Calabi-Yau threefolds times the circles which only one of the two acyl Calabi-Yau threefolds sums (TCS) by gluingthreefolds appropriate pairs offound asymptotically in cylindrical [ (acyl)Calabi-Yau Calabi-Yau threefolds, together withTCS T-dualities on the product circles, leads to another manifolds, where the mirrorT map can beto understood arise from from T-dualities T-dualities alongin along calibrated [ a calibrated Likewise, a pair of Spin(7) mirrors where In parallel to the well-studied case of mirror symmetry for type II strings on Calabi-Yau in [ moduli together with the degreesthis of freedom implies associated the with equality the of B-field. the In sum the of the second and third Betti numbers for mirrors: exactly marginal operators of the extended JHEP10(2019)204 2 ]. G 29 and 3 T ], the two 16 mirror maps 2 G manifolds yields ], there is a large manifolds and we 2 2 G 26 , G ]. Furthermore, these 25 9 , 6 ] that the 28 with a flat connection can be ] for TCS 3 T 16 . Different assignments of discrete , and Spin(7) manifolds, it becomes an A 15 , so that it is tempting to exploit this 2 2 G G ] for explorations of M-Theory on TCS manifolds. 24 and Spin(7) manifolds can be compared with 2 manifolds which appear as sections of precisely – – 3 – 2 2 G manifolds. As argued in [ 22 G G 2 G ]. GCS Spin(7) manifolds are glued from two non-compact 27 fibrations also play a crucial role for the determination of the ] (see also [ 4 21 T fibration relevant for mirror maps. 4 ). ] signals the correspondence to topologically different smoothings of the T 30 1.2 manifolds in [ 2 ] are associated with non-trivial automorphisms of the extended superconformal G 16 , Treating new examples of orbifolds, and linking the associated mirror maps with a It is a central motivation of the present work to enlarge the class of models where a Given geometric constructions for mirrors of Applying the M-Theory duality map to heterotic strings on TCS fibrations. Furthermore, it has subsequently been shown [ 15 4 associates CFTs on different smoothby geometries. providing a The action free-fieldautomorphisms of realization of mirror of this maps algebra the can whichestablish extended be are a superconformal found induced link algebra, by a to andtwisted sequence the connected finding of sums. TCS T-dualities. mirror Finally, maps to the orbifolds in question must be described as Possible assignments of discretemodular torsion invariance for phases the can partition be function.arise constrained A and short by introduction can the to be howtorsion requirement such analysed will of constraints in is turn presentedto lead in cohomology to [ appendix a differentorbifold spectrum geometry. of Mirror symmetry RR can ground change states, the which discrete via torsion phases, the so map that it also geometrical construction of mirrorsresults for obtained from the worldsheet, and we complete thisgeometry task requires for several several new steps examples. toon be the completed. orbifolds we Crucially, are the considering definition involves of an string assignment theory of discrete torsion phases [ complimentary approaches result inT the same mirror mapsof and [ identify the same algebra of the worldsheet theory for TCS interesting question if theseaccomplished can for be a recovered fewthat using of they worldsheet can the methods. be examplesgeometries treated of This can from also Joyce. has first be been decomposed principles These as in are twisted string resolutions connected theory of sums. [ orbifolds, As shown so in [ manifolds with the holonomystructure groups SU(4) to and construct candidatesblocks. of This is mirror an manifoldswill analogue follow by of a using similar the path mirrors strategyindeed to used for define satisfy in Spin(7) ( the [ mirrors building in the present work, and show that they superpotential of M-Theory onclass TCS of associative submanifolds of TCS the coassociative M-Theory duals on Spin(7) manifoldsized with connected a sum decomposition (GCS) similar in to [ TCS, called general- surfaces and heteroticused string fibrewise theory to on findSYZ examples a of fibration three-torus lower-dimensional of dualities. Calabi-Yauon This threefolds TCS was to exploited find formanifolds). examples the Furthermore, of heterotic duals of M-Theory and heterotic string theory. The duality in seven dimensions between M-Theory on K3 JHEP10(2019)204 . ]. 1 2 4 are + (2.1) 3 X → − X ] are realized by 4 ], as well the action 5 ]. . 32 , 7 1 2 ) for the mirror geometries 31 − X − 1.2 ]). In order to do that, we need 6 1 2 is a shorthand for ] (‘example 5’). It is based on a + 4 17 X 1 2 − of the present work we study two − 5 3 + + + − X , and 1 2 4 1 2 + + + − − − − X ] (see also [ + ), which precisely matches with the discrete ] allowed for a straightforward determination 4 ] (‘example 4’). Furthermore, the description Z X – 4 – 4 , 16 3 k + − X → , with an action on the coordinates M ( 4 2 3 X ]. As only submanifolds of real dimension four can , the resulting check of ( indicates that the corresponding coordinate is sent to Z 2 ], the possible assignments of discrete torsion precisely ], they have been first presented as ‘example 5’ in [ 4 2 1 9 H + − − X T − 10 , 16 fibration. In section , 6 1 9 4 + − + + + + X T , and manifolds in [ 7 2 T 2 γ β α G σ presents a general exposition of how the GCS construction can be indicates a shift 4 1 2 , we consider Joyce orbifolds which can be smoothed to manifolds with , we will complete these tasks for a set of models which are free quotients under the group Γ = + 1 giving examples studied in [ 3 2 i 7 2 T example X G 2 ∼ i G ]. Curiously, not all resolutions which have been constructed in [ X The appendices contain a brief introduction to discrete torsion phases and modular Finally, section In section In section In the 33 Here minus itself, while 2.1 Smoothing ofLet the us orbifold first analyseorbifold the by resolving topological the properties singularitiesto of as first in the evaluate [ the smooth fixed limit(s) point obtained set under from the this action of the orbifold group. In this section wequotient consider of the following example from [ precisely agrees with the worldsheet results. invariance for strings on orbifolds, as well technical details of2 the examples we are treating. A used to define mirror mapsthe identification for of Spin(7) a manifolds. fibrationholds by While independently. our We construction then proceed is to motivateddecomposed describe by as how the a examples GCS studied in and section verify that our geometric construction of GCS Spin(7) mirrors the worldsheet perspective inbe [ calibrated for Spin(7)dualities manifolds, along a such calibrated mirrorsuch examples, maps which first must appeared be asof ‘example associated mirror 1’ symmetry with and on ‘example four 2’ such in T- geometries. [ of the orbifolds consideredAs in we [ will see, thesein orbifolds are [ an example thethe ‘extra-twisted’ connected set sums of described consistent assignments of discrete torsion phases. holonomy Spin(7). Some Joyce orbifolds and their mirror maps have been previously from match the different resolutionsof found such in models [ asof TCS torsion in thetorsion homology phases in group the orbifold string theory, as expected from [ JHEP10(2019)204 . 2 Z / (2.2) (2.3) each. 3 2 . With 2 we find Z / Z . For such k 3 / 4 2 ˆ 3 M Z / 7 T . The contribution i sector, the action of (two of the extended 2 , and 8 T times by 7). Each of the singu- 4 Z 3 3 3 α αβ , / k h T T X 3 / sector, we have the same T ) 2 2 } β 8 1 case. Here we have a further and 2 γ Γ) = (0 2) { / , / X 7 2 s, which are then further identified singular set in quotient T C 3 )( 3 × ]. We then exploit the isomorphism k 3 , b k 3) 1) or (0 2 29 − , , b each. For the 3 or (T 2 } ) = (1 ) = (1 action has a singular set consisting of 8 T ) = 4 + ) = 35 1 3 2 Z – 5 – 3 3 k k / γ Z T T T 3 ˆ ˆ / 8 8 { M M . This leads to 8 orbits consisting of 2 T )( T 3 ( ( / 3 2 2 3 2 T b b αβ , b C )( 2 3 b × ( , b δ 3 2 b elements in orbit under Γ ( δ s). Hence, the 3 3 3 3 T T T manifold obtained by making the first choice 16 16 16 2 fixed T G singular set just reduces the fundamental domain for γ 2 sector as these two sectors are isomorphic, up to the permutation of indices . Each of these three elements fixes 16 T σ γ γ β α α 8. is free, resulting in two orbits of 8 T ... i , and 2 β = 0 , The only group elements of Γ which act non-freely and hence give rise to singularities We shall begin by studying the twisted sectors corresponding to various elements of the The net contribution to the singular set is obtained by adding up the separate contri- α k identification under the action of 2 β, γ, σ Z The action of directions in the are under the action ofh the rest ofresult the as orbifold the group.and In fixed case point of labels. the The analysis is a bit different for the the orbifold elements in a basis of highest weight states in a particular sector. However, orbifold group. If the orbifoldthe element corresponding has sector fixed can points then in befixed its further points. action decomposed on into In the sub-sectors general, parent localisedhave manifold, one at discrete those would torsion signs. start by For this, figuring we out need which to of write the down the sectors representation can matrices actually for Let us redo the analysis bystring studying theories, the we string have theory the living additional on degreeconsistent the of orbifold freedom with to switch modular on discretebetween invariance torsion the phases constraints Ramond-Ramond [ sectorfind ground the Betti states numbers and for the the target resulting resolution(s). space cohomology to for 2.2 Constraints on discrete torsion The two choices for thecompact latter smooth case come from different possible smoothings. Denoting the smooth limits obtained bythe resolving Betti the numbers singularities for present thelarities in orbifold is our are locally orbifold. given modelled byto First on ( the of T Betti all, numbers from resolving these singularities is: butions from the differentthe orbifold singular actions, set giving at us hand, a we are total now of ready 4 to T compute the Betti numbers for the different JHEP10(2019)204 . , ) γ 2 σ αβ αβ ( 2 (2.7) (2.4) (2.5) γ σ f , and β ), , α , and αβ 2 ( σ γ , f sectors, as all the αβ γ , and β , ) α ! , implying that the oscillator 4 1 2 × αβ 4 ( ) I γ ) f . 0 γ ) (2.6) αβ ) ( 2 ( B 2 σ γ αβ σ ( f 2 f ( γ γ f f 4 =1 8 γ × X =1 f 4 8 8 =1 γ I twisted sector as follows: X X f f ) 0 γ γ – 6 – ) = ( γ ) = ) = ( αβ γ γ 1 f ( ( , discrete torsion arises in the representation of the αβ 2 sector, discrete torsion arises in the representation of i f
, as is expected from modular constraint requirements. σ αβ B 2 8 2 ,i σ 8 σ × = f 8 labels the different irreducible representations/orbits. 8 αβ I ) from the X f ) 2 αβ generator: f αβ . We can now use the S transform relations to connect this 4 γ =1 σ 4 ( γ ( X f H ,..., | . The corresponding phases are denoted by γ × 2 2 2 f αβ 4 γ 2 , I σ σ ) f γ ( = 1 2 sector, we have the first instance of discrete torsion. We find that and = there are no discrete torsion choices in these sectors. σ γ 4 labels the irreducible representations, and i=1,2 labels the two signs f γ , αβ 3 = αβ 2 αβ analogous to the , 2 σ 2 f σ 2 σ , in this sector, we have discrete torsion signs appearing in the representation H sectors in order to figure out the modular trace constraints in the as shown in appendix | H | in the γ γ = 1 αβ -sector: , 2 β γ σ αβ f -sector: As reviewed in appendix A, one can constrain the discrete torsion signs by imple- In our case, the ground state contributions come from the -sector: αβ -sector: , and - and 2 2 -sector: 2 σ the element Using modular invariance of partition function under the S-transformation gives element discrete torsion sign with where in a given irreducible representation. σ Invariance of the partition function under the S transformation implies menting modular invariance ofelements the with partition non-zero functions. tracescorresponding in to For those the that, elements. above we These representations utilize are and given theαβ by probe the orbifold the group elements twisted sector matrix corresponding to the γ there are two orbifoldrepresentations: generators with possiblerespectively, discrete where torsion signs in their irreducible modes are half-integer in thatσ direction. Aside from thesesectors. sectors, Let we need us to then consider The find details which of of the this above analysis mentioned can sectors beα have found discrete in torsion appendix signs. get away by workingground with state a spectrum few (and hence ofthe the the discrete cohomology), twisted torsion and signs sectors ones for that only: are the needed ones ground to state that constrain contributingremaining contribute sectors. group to elements the act on at least one direction as + as our main aim is to obtain the cohomology of the resulting orbifold smoothings, we can JHEP10(2019)204 , i } X 2 as (2.8) (2.9) 7) , and / (2.13) (2.12) (2.10) (2.11) , 1 β 5 , , , with the α 2 i , = 0 0 | (1 5 , X 6) , . Here the label 4 3 } , i 2 X 6) 0 α , , | f α 4 i (1 7 + , 0 , | 2. The orbifold invariant (3 3 6) / , and , k + ) , 2 . . . ψ i 0 ψ 4 7) ˜ 2 , X n ψ 1 , + k i + i 3 5 , ψ , ψ , 1 1 + ) to get the Betti number contri- dX (1 k + (3 X i 0 ; , , ψ ∧ i ψ 2.8 twisted sector is built on the highest 7) 0 7) | , , : ... α 5 = ( 5 d + , , α ∧ ψ 3 i + f α , c i (2 1 + ψ i , = 1 = 7 0 ψ | (1 , 7 u 4 u b + 6) sector. Firstly, following in the same line of 2 i + b b dX , ψ 5) 4 ψ α – 7 – , a + corresponding to the 7 bosonic coordinates , = = 1 i + 4 ψ i i ' , 0 u 3 u (2 ψ 0 b b 3 , ψ | , ; n 7) ; i + i (2 , 0 α , 6 | f α , i 7) k + 0 , . . . ψ (1 | ψ 6 , 1 i + j , i + + 5) 3 ψ ψ ψ , , i + 4 (2 , ψ ; , sector and use a similar isomorphism statement as in the untwisted α (1 f α 7) , α ; i , i 0 with vanishing momentum and winding modes, by acting with the 3) 6 )) to get the Betti numbers contribution. | 0 , , | 2 5 α ], we should identify the vacuum state in this sector with the exceptional f α , , 9 i 2.11 0 (1 (4 | ∈ { ∈ { ) ) 7. We can generate the RR ground states by acting on the vacuum manifolds, the only unfixed, non-trivial Betti numbers are the second and third 2 enumerates the two irreducible representations corresponding to , 2 i, j, k ,..., G The list of Γ invariant RR ground states in the Aside from the untwisted sector, the smoothings for the orbifold will alsoLet get us contri- look at the contribution from ( = 1 . We have Majorana-Weyl spinors a, b, c, d e ( = 1 . Now, we shall evaluate such twisted sector contributions. α creation operators built from theonly Majorana difference Weyl fermions is as thatf in we the only untwisted sector. havein The zero appendix modes B. along They are given by: divisor Σ resolving theinvariant states particular in the singularity (asector 2-form case (in in ( this case). Then,weight we states list the bution to the Betti numbersγ from the twisted sectors with RR ground states, i.e. argument as in [ For ones. Now on, we will only mention contributions to those. As such, we can usebution the from the list untwisted of sector: invariant states from ( us to use the following identification: Now the isomorphism between RR ground states and the target space cohomology allows set of RR ground states is given by: with the following the triples and the 4-tuples of indices: We are ready to derive theing cohomology its of the isomorphism possible with resolutions theH of RR the ground orbifold by states. exploit- Leti us first consider thecreation untwisted operators sector, built out of the zero modes, 2.3 Cohomology of smoothings JHEP10(2019)204 . α γ f (2.18) (2.14) (2.21) (2.20) (2.15) (2.16) ) as: sectors, we 2.19 β ) is: ) = 1 for all αβ 2 ( = 1, they are: and σ γ ), and ( γ . Now, we can read ( f α f γ α ; (2.19) f f γ γ f γ 2.18 f γ i 1, the orbifold invariant i 0 ) = α 0 | . The isomorphism state- − | f } 6 + αβ Σ 6 + 1 4 (2.17) = ( l . ψ , ψ ), that constrain the discrete − γ sector is isomorphic to the ∧ 2 γ ) 3 4 + f 4 f + , n ψ i β − 2 ψ 2.7 αβ 2 (1,2,3) , + − ) = 1 ) = l ( ψ γ dX ; f ∈{ ), ( αβ αβ ) = 1 positive signs of γ ∧ ( ( ) = f γ 3 ; l γ γ γ i 2.6 0 = 2 2 = 16 =1 f f γ 8 ,k 0 f γ · · γ αβ | ... 2 X i f ( ) ) . As the 6 ), ( 0 + γ l l ,k ∧ , f | 1 = 2 3 = 6 f α iff iff 1 2 2 ) ψ 1 · · f i 4 + 2.5 ) = 2 + , , − − ψ γ αβ ψ ; (k ( ( – 8 – 2 + dX } = 2 = 2 γ ). This is because the exceptional divisor in both = 1 = 2 ψ f αβ ; 2 α 3 α ' 2 3 γ 3 γ − σ γ 8 αβ f α f γ δb δb 1 + (8 1 + (8 ; ( δb δb f α i · · γ γ i 0 f γ f l l | 0 i =1 | ) = 4 0 + n | 4 αβ i + ψ = 2 = 2 X 2 6 + = 1; = 0; αβ σ irreducible representation) labelled by 2 γ 3 γ ( f ψ 2 γ 2 γ ; γ 2 δb δb f ∼ . . . ψ γ δb δb (1,2),(1,3),(2,3) from the above list as: f γ ; 1 i + i γ ) implies that the only possible solution is i α 0 ψ f γ | ∈ { i δb ) 2 0 + 2.6 | 2 . On the other hand, for ψ ,i γ 1 f ; (i = 1, ( } ) we get: 2 -sector, things are different as there are discrete torsion signs present. Firstly, σ ). Thus, we can re-label the fixed point indices such that: is the highest weight state representing the irreducible basis for the different γ 2.7 represents the exceptional divisor arising due to the resolution of the singularity γ 1,2,3 αβ f γ α ( i f γ 0 ∈{ | f . For 2 For the σ Thus, the resulting Betti number contribution for 2 as before. Incase this of case, the discrete the torsioncases highest sign corresponds of weight to state a is 2-form.can read again Using off the mapped the same to Betti identification a numbers as from in 2-form the the for list either of invariant states in ( where orbits indexed by states are: For the Betti number contributions, we need the state-cohomology identification statement Now let us list the orbifold invariant states as before. For Note that we are forcedsigns by this relation to have an even number of positive discrete torsion we need to solvetorsion for phases arising the in trace thisof sector. relations There in are ( two distinctAlso cases from corresponding ( to the sign off the Betti numbers where the factor ofsector, 2 we find comes the from same the contributions there. index ment can be again used to get the following identification: where Σ corresponding to the orbit ( where i JHEP10(2019)204 l γ 2 1. ˆ − M along (2.24) (2.22) (2.23) ) = 2 6 σ ( ψ γ α, β, γ 5 f l l 2 ψ ), we get: 3 ) must be even. − ψ ): − 2.21 2.17 7 ψ 2.22 4 ) = 35 ψ l 3 1. Here, we don’t get any ) and ( 2 ψ ], but it is, to the authors − − 7 ,X) of conformal dimensions − , is then obtained by omitting 34 = 4 + 2 = 2.15 ψ φ ˆ 7 l 2 N 4 ψ σ . The net Betti number for ψ ), ( 5 l 2 2 ψ ψ ˆ 2 orbifolds. 1 M 2.12 ψ 2 ψ G − + 6 = 0 + 2 + 2 == 4 7 + 6 + 6 = 19 6 : 5 ψ 2 β 3 β ψ j 4 ψ 5 4 δb δb ψ ψ 2 ψ = 0 + 2 + 2 + 2 = 7 + 6 + 6 + (16 ∂ψ 2 3 j – 9 – + + ψ 2 γ 3 γ ) be called ψ ψ ψ 2 α 3 α 1 2 + : δb δb ψ αβ δb δb ψ 7 ( + + 7 γ =1 ψ + + + − j f X 2 β 3 β 6 6 7 2 u 3 u 2 ψ δb δb b b ψ 1 ψ / 4 6 1 ψ + + ψ ψ − 3 ) = ) = 2 α 3 α 3 + : sector in our previous computation in ( ψ ˆ ˆ , not all of the possible geometries are realized. This ultimately ψ 5 N N δb δb j 1 2 ( ( γ ψ 2 3 ψ ψ + + 4 b b 2.1 ∂X ψ 2 u 3 u + − j 1 b b 7 7 ψ ψ j ψ ∂X 5 6 + ) = ) = : l l positive signs of ψ ψ 3 2 2 ∂X 3 -twisted sector as no invariant states can be constructed with 5 } l ] j ˆ ˆ ψ 7 =1 γ ψ ψ M M 2 Φ j X ψ 1 ( ( 4 ψ = 1 superconformal algebra generated by the bosonic stress tensor T along 2 3 2 ψ ψ 4. These are all the contributions from the twisted sectors as the rest of the 1 b b 4. Although every single one of these models corresponds to one of the resolu- 7 G, / =1 G, X j X − ψ { − N ...... = 1 = = = = [ = )=(3/2,2) and their superpartners K,M. Their free field representation is 2 2 2 2 2 2 X = 0 G G G G G G = 0 The extended chiral algebra for a string moving on a compact G2 manifold con- Now let us move onto the other case where we have T l l Φ , h G X K M φ h T-duality transformations on multiple suitably chosen coordinates. sists of a with its fermionic( counterpart G, extended by currents ( fixed points must be leftfor intact. strings The on freezing orbifoldsknowledge, of the in singularities first is the time a presence it well-known of has phenomenon been discrete2.4 observed torsion for [ Mirror symmetry Let us now have a look at the realisation of mirror symmetry in our G2 example through What we have heresmoothed, is but a some scenario where are the frozen. orbifold singularities In could particular, only all partially of be the singularities located at the the action of mirror maps on these models westates from leave the such an investigationThe to Betti future numbers for work. the partiallythe smoothed contribution solution, from say the for tions discussed in section stems from the constraint thatIt the would number be of interesting discrete to torsion understand sign this in mismatch ( better, but as our main interest is in sectors all have at leastsmoothing one for extended 2 direction withcan half-integer be modes. obtained by Let addingwith the up the that resultant contributions coming from from the the three untwisted twisted sectors sector. Using ( for JHEP10(2019)204 , and (2.29) (2.25) (2.26) (2.27) (2.28) + 4 ) should I αβ ( γ f -twisted sector, , flips the signs of γ + 3 I inverting all of them αβ . 2 } − ), should be considered 4 G I + 6) , , M 4 } 2.23 , 3 7) 2 , , ], it should come as no surprise G 4 − 9 (1 , K } , . (3 ) 5) 6) } , 2 , , . ) intact and 4 5 6) 7) G l αβ + , , , , 2 ( X 3 2 5 5 γ αβ l − , , , , f ( 2 8 γ ˆ ˆ (2 (1 (3 (2 4 2 0 f M M , , , , ) do not change. In the ˜ G ψ 2 − 2 0 7) 7) 6) 5) Φ ) σ → → ˜ , , , , ψ ( 2 l l 5 6 4 4 4 0 γ σ 2 2 – 10 – , , , , f ( 2 ψ ˆ ˆ 3 5 γ 2 0 G M M (2 (1 , , f : : , , ψ G ) should flip signs irrespective of the fixed point label (1 (4 + − 3 3 7) 3) , , are precisely those T-duality triples that generate the = = I I , , αβ 2 2 7) 7) 6 2 3 ( , , , , G ) on the right-movers (left chiral algebra stays invariant). σ ) has to remain invariant as well. This implies that + + γ I αβ does not flip any of the three labels. As a result, we can 4 6 T 2 f (1 (1 , , 2 σ { { 2 3 representation matrix remaining invariant under the triples ( , σ -twisted sector (knowing that it has discrete torsion phases , , 2.25 γ 2 2 − 3 γ f = = G (1 (2 σ I { { + − generator, i.e. 3 3 I I 2 = = σ + − 4 4 mirror I I keeping the discrete torsion signs + generator in the 4 I αβ ) arising in its representation), we observe a need for flipping all or none of the We can combine any two of the above transformations to get a new transformation The resultant action of these composite T-dualities on the discrete torsion phases can αβ , with ( γ − 4 f A mirror automorphism of this algebra is given by: which T-dualizes along 4I of the 7 coordinates. These transformations split into Furthermore, the case with frozen singularities,self-mirror. represented by Note ( that allversa of because these of maps the take odd type number IIA of strings fermionic to modes type being IIB T-dualized. strings and vice remain the same; while for f. On the otherof hand, T-dualities the implies that write the elements in terms of the RR zero modes: The above expressions imply that under the set of T-transformations in discrete torsion signs uponcorresponding action of to these the transformations.the On RR the zero other modescoordinate hand, surviving the labels are signs (2,4) labelled while the indices (2,4,6). Now that these are the samecoordinates). triples which were found there (after an appropriatebe relabelling of understood through theof following the line of argument. If we focus on the representation one can check that themirror elements automorphism of eq. ( As the present model is a quotient of the one considered in [ Defining two sets of triples of coordinate indices as follows: JHEP10(2019)204 3 is S as 3 (2.34) (2.33) M along (2.30) (2.31) S can be 6 − ) using the as defined ,X X 4 The further 2.32 are given as a ,X manifold. 2 . In particular, 2 k 2 can be described ]. The upshot of X X G k M 16 ˆ M ) and the coordinate 6 which enjoy fibration . The gluing to . Likewise, I 2 ,X 2 manifolds are constructed X X X 2 , is G manifold. However, it can be 1 + 1 − ). The acyl Calabi-Yau threefold . S S 7 2 G factors in the decomposition of by the free involution generated by × S ,X . 1 k 5 l are approximated (metrically) by the − 2 S 1 2 X extra-twisted connected sum M ) for an interval l − ) z z ,X 2 8 η η which is naturally found by representing parametrizing the non-compact direction = 1 (2.32) 3 . Hence the base of the K3 fibration on X I ˆ ˆ 3 5 2 M M 6 | S sin( → − → − # cos( 2 T × ,X X has coordinates ( z 1 1 2 | → → iφ iψ 1 ∓ z z e e l l − + X S 2 2 parametrizing the non-compact direction. Now, 1 + – 11 – 2 = = manifold X ˆ ˆ S | 6 : M M × 1 1 2 : : 2 z z z 2 × , with | X σ i G of both acyl Calabi-Yau threefolds + − 4 4 hence only acts on the coordinates + S I I 2 X S γ, α σ h / = R ) and the coordinate along 6 M × as a TCS has been discussed in detail in [ , also with 5 i k ) parametrizing the ,X T 4 4 M discussed above is not a TCS γ, β X h ,X with the coordinates ( k / 7 ˆ M . In their asymptotic regions R /γ 4 ,X × turns this into an example of an 5 and a cylinder, i.e. S T , as appropriate for gluing these two acyl CYs to a TCS 5 4 2 T ,X σ of the K3 fibration apparent in the TCS decomposition of X S 3 3 is ] S ,X 1 1 − 14 S , X has coordinates ( The freely acting involution The example can be described as This is in fact the same freely-acting involution of 12 2 . The realization of + + 2 In this presentation,parametrization the decomposition into two solid tori can be seen by solving ( quotient by and acting with however the base of thealong K3 fibration on the base it acts by shiftingappearing the in coordinates the on TCS both construction, of where the it is glued from two solid tori. (smoothing) of X and ( X described as appropriate hyper-K¨ahlerrotation on the K3 surfaces constructed as a quotientσ of another this analysis is that the K3 fibres for a pair ofby K3 asymptotically surfaces cylindrical Calabi-Yauproduct threefolds of then done by identifying those asymptotic regions by a diffeomorphism which induces am 2.5 Realization asLet a us TCS now discuss howas the orbifold a treated extra above twisted andas its connected [ smoothings sum. Twisted connected sum Now as there are eventransformations number do of not fermionic alter modes, the theto sign mirror IIA of maps GSO and corresponding projection, IIB to mapping these respectively. type IIA and IIB strings In particular, these act as JHEP10(2019)204 . 3 2 π to S G ≤ (2.35) η is the S (2.38) (2.39) (2.36) (2.37) , which ≤ . In the k 2 Z . M π/ X and X 2 and ) . π/ , the quotient must )) S ≤ ( − ] in two regards: first ]. The result is that manifolds , so that we can see 1 η , S Z 1 2 12 16 ] for more details). The ( ≤ 1 G H , . . . π 13 2 h s are simply swapped, as in → s 2 1 ) S Z ) + / + X ( Z 1 − 1 ( , S 1 1 , , of the TCS 2 , × for K3 surfaces H h | − π k π 1 + I k V ˆ − is still a two-torus. The second difference + + S M was described in [ s 2 N × for acyl Calabi-Yau threefolds = 4 = 10 = φ ) again. ). The present example of a extra-twisted ϑ ψ Z | | | k ) = 4 1 ∓ ∩ s 2 ) + 2( × # / = Im → | 2.1 → − S Z with which these tori are identified, together M I + invariant. In the example discussed here, the − 2.31 Z N N + φ K ψ × 1 − | | ( in ( N ϑ – 12 – K × V | N 1 S by | ∩ parametrizes the interval 0 , 2 2 , : σ = S + + η × 1 + h S 2 | ) | S N σ ), the TCS decomposition and the action of the TCS 1 + | − + M = S × S and K K 2.1 ( | | manifolds forms Φ 1 π , κ 2 1 + 2 2 X \ in ( | factors, so that G H G ... + manifold is then determined from 2 1 σ V S 2 K → is glued from two pieces | ]. There, the central idea was to construct a mirror of a TCS G asymptote to ) ) is simply the map = = 23 + 2( found by gluing in a single K3 fibre (see [ M . Note that 16 3 2 2 appears, which means that the two , X 2.33 b b X ( κ 1 X 15 π/ , + 1 2 = b H ϑ have the range 0 manifold 2 ψ G are free quotients of are such that = ker and V K ]. This construction differs from the usual TCS construction [ φ X Omitting the action of 33 Here, decomposed as a being glued fromIn the two this solid parametrization, tori described ( by the above for 0 which is nothing but the half-shift induced by compactification of topology of the resulting where can also be described as acting separately on themirror acyl map Calabi-Yau on threefold theboth resulting We can now describe a mirrorof map the in approach this of context, which [ manifold is found by by applying a mirror slightYau generalization symmetry threefolds to in either the both, TCSconnected decomposition sum or is ( to constructed one as of a the free two quotient acyl Calabi- with an appropriate altering ofkeep the the hyper-K¨ahlerrotation acting canonically on defined the‘trivial’ K3 choice surfaces the standard TCS construction. 2.6 TCS mirror map purely act by shifts on the for compact subsets is that the gluing now involves an angle of all, the where neck region in which in [ JHEP10(2019)204 . ]: − 3 are I 16 . As − (2.45) (2.46) (2.43) (2.44) (2.40) (2.41) (2.42) X N ∩ + N are given by with the holon- . Note that the . Repeating the s 2 s 2 . Z Z V V / )) : k − k on Z ˆ M ( M 3 1 , with the image under T 2 e ∧ h ˆ M and . We hence find that are unchanged. In the present ) + 2 ∧ + T ˆ and Z M N ( + 1 4 , . I 2 e , . for ∨ ∨ k − − k k h | k k V V k k − k − − . As in the analysis of the Joyce orbifold − produces a 7-manifold − N = 2 = 10 = # # s 2 | | | , so that V ) = 2 ∩ ) + 2( Z | e ∨ − + + = 8 + e − V ) = 4 + ) = 35 V Z N N K X | ) = 27 + ) = 12 | | ( N ) = 4 + ) = 35 – 13 – ∨ ∨ K k k | ∧ ∧ − 1 | k k ∩ = , = ˆ ˆ ˆ ˆ 2 e ˆ ˆ M M N M M + + ∧ M M + ∨ ( ( h ( ( | ( ( | 2 3 2 3 ∩ ˆ N ˆ 3 2 e b b | − b b M e M + b b ’). In the present case, both the elliptic fibrations on + become fibrations by K 2 K | | N | G + , with the image under X | ∨ ∨ e + ˆ ˆ M K M | , the Betti numbers of the free quotients by the freely acting for (‘barely = = 23 + 2( 1 X 3 2 k s 2 S b b × = + | o Z 2 − b X N indicates taking the even subspace under the involution ] then motivates to consider two types of mirrors of ∩ e 16 + does not act on the K3 fibres of N | ], we can associate and s 2 e Z 16 ], the only non-trivial ingredient in the mirror construction is given by We are now ready to discuss the action of the TCS mirror maps. The fact that Given the data of Orbifolding and the SYZ fibrations on 16 does not act on the K3 fibres at all, we can just quote the result of the analysis of [ 2 whereas As in [ σ whereas self-mirror indicates that thein same [ is true for so that we recover where group case, we have that associated with applying four T-dualities, as well as associated with applying three T-dualities. omy group SU(3) X analysis of [ JHEP10(2019)204 4 δ 4 α 1) S , 2 is = 1 ... 1 dX / (3.1) , 8 and 1 ∧ 1 b 3 , − γ = 1 ]). = , i which are 5 have fixed β dX 0 acts on the 8 b = (0 , ∧ is also 2 sets for T i β α 2 αβ δ d } , 2 S α / dX . 1 and . ∧ , 1 δ αβ 0 ’s for the directions , which has no fixed 8 4 i , S 1) and + , γ T also consists of 4 copies X dX ∈ { 1 , i , β β and 1 S 7 X 7 , , 2 d δ + + α acts freely, and so groups S X − , i = (1 γ i 2 for some S 6 6 2 c c γ, δ / , + h X β − S + 1 , 1) (‘Example 2’). Once again, i 5 , 5 α 5 2 2 c d 1 S X orbifolds above into a family of compact − − − − + + + + , X − − 1 has fixed points 4 2 , acts freely and so we find that the singular → Z α fixed points, grouping the 256 points into 64 is not invariant, while / i 4 i 8 3 + orbifolds in which the generators X X – 14 – = (0 T 8. However, we must also consider the action of ]: αβ 4 2 5 i dX Z d 3 ... α, β, δ 3 2 / d ∧ h 8 ’s. The subgroup + + 2 X } − T ]. In total, the Betti numbers will receive a contribution 1 = 1 , the only composite element with fixed points, they are 5 dX i 2 2 { 0) and 2 αβ c ∧ , / + 1 X 4 1 for . In either example, a simple calculation shows − i , 4 2 T } 0 X Z dX , 2 1 1 1 2 2 / c d coordinates as [ − ’s, and we apply the same reasoning to find that ’s. In a similar way, we see that the set 1 + + + + − − − − 8). For X 4 -fixed points, , } − 2 8 − acts freely on the 0 1 γ = (1 / T T ... i 1 i . The two cases we will study have c δ γ β α { } ∈ { / 1 → γ, δ i = 5 4 , i h 0 i 1, giving 16 T X X that Spin(7) is defined to leave invariant (see e.g. equation (1) of [ − ∈ { 8 . For the i } R act on the 1 4 2 , d ’s by free for i 4 Z = 14. For example, c { i T / ’s. Lastly, 4 4 X 4 b For each single generator, the fixed points correspond to 16 Now consider example 1, we would like to understand its resolutions and so must first T T In this section weof will Γ analyse = two 3 Spin(7) examples fixed into 4 orbits of of set gets reduced toof 2 unchanged by the group action (For example, and the 256 points the other group generators on a given set of fixed points. For the case of 4-form on understand its singular set.points, In as this case, all it otherpoints. is combinations So, clear involve the that singular only set consists of 5 sectors, from the orbifold itself, asthe well fixed as points. contributions Before forminvariant resolution, the under the resolutions the of cohomology group the consistsand singularities of at the classes of is. In particular, the 14 4-forms are precisely the 14 elementary ones appearing in the 3.1 Smoothing of the orbifold First, let us reviewSpin(7) the manifolds as resolution described of in [ the where (‘Example 1’), and shorthand for JHEP10(2019)204 , 3 γ b )’s, } (3.2) , the 1 β by 1, S { 2 ) for a 4- by 2 and / b } 4 ζ 3 1 b and B -fixed points ( γ { α × / S ) 4 ζ } B 1 ’s ’s ’s ’s ( action converts this } } } } . 1 1 1 1 × { , the neighbourhoods ’s ’s ’s ’s γ δ } / ) 4 4 4 4 4 { { { { 4 ζ S } T T T T / / / / 1 B 4 4 4 4 2 2 4 2 ... T T T T 64 points 64 points , the { 0 and 4 ζ 4 4 4 4 / 4 B γ ∈ { S T . ] we will relate the distribution × 9 singular set in quotient singular set in quotient 4 may be resolved in two inequivalent T , j 150) γ ) , consist of 64 ( j turns this into the neighbourhood of a 2 ], these come from type (ii) singularities, 16 5 αβ by 6. For , − αδ S ’s ’s ’s ’s by 2, and the other increasing 4 } } } } b 1 1 1 1 4 154 ’s ’s ’s ’s b , 4 4 4 4 – 15 – { { { { T T T T ) = (12 / / / / 16 4 4 4 4 4 8 8 4 8 j, 4 points 4 points T T T T by 1. In total, we find that upon resolution the non- , b 4 4 4 4 3 4 by 1 and b , b 1 by 2 and 2 b b 3 ( elements in orbit under Γ elements in orbit under Γ b ) = (10 + 4 )’s, which arise from type (i) singularities, and increase , b ]. The action of the rest of the group then orders these 16 type } ’s ’s ’s ’s ’s ’s ’s ’s 3 by 1, 5 4 4 4 4 4 4 4 4 1 , b 2 T T T T T T T T 2 b { b ( / 16 16 16 16 16 16 16 16 ), but in fact the action of 4 ζ 256 points 256 points } singular set singular set B 1 ( × { δ δ γ γ β β α α 4 be the number of type (iv)’s we resolve in the first way, we find a family of 5 αβ αβ / acts freely). In summary, the fixed set for example 1 looks like: by 6. Lastly, the neighbourhoods of T 4 ζ } i 4 4 B b ( αβ . In the language of Proposition 3.1.1 in [ ... h 4 × ζ 0 4 B Let us now consider example 2. In this case, all sectors except the Lastly, we consider the neighbourhood of each singular point. For now acts trivially. Starting from a neighbourhood T by 4. As a result, including the contribution from the unresolved orbifold and letting ∈ { 4 (here, singularity of type (iv)(iv) [ singularities into 4 orbits. In summary, the singular set of example 2 is given by contribute the same singularities,αδ and thus Betti numbers,to as example 1. However, for consist of 4 by 4 and which are of type (iii)trivial and Betti increase numbers of the manifold are: neighbourhoods of the fixed lociball are a total ofwhich 8 upon copies resolution of increases ( 3.2 Discrete torsionIn and this cohomology section weCFT. will Once recover again, theof following resolution the a allowed of similar discrete the analysis torsioncohomology orbifold signs used of (which by the in will orbifold studying [ mostly resolutions. the be derived associated in the appendix) to the b j Spin(7) manifolds with Betti numbers given by: In this case, the singularities inducedways, by one the action increasing of JHEP10(2019)204 } α 15 8 , 7 (3.4) (3.3) , 2 and , 4 . Using ,..., , , 3 on the g { = 1 αβ i = 0 β g g = f f k γ 0; I , for 0 ∈ | , for i g k j + i g ψ , as well as f i + + 4 δ 1). The cohomology of , 0; , α , γ 1 f (where sectors: 0 , , | sectors, for which there are 1 = β i + β , δ , ψ ) for the action of α αβ . β f . ( and α = (0 g and f α ) i i f α g α d γ ( f =1 β invariant states. In doing so we find 4 f α 0; X f sectors to be the same: 4 2 , , ψ 0 Z | β =1 4 g β 1) and l i + X f , g ψ 1 f ], we need to find linear combinations of these ) = 16 and , k + 9 β 1 0; ) = ψ – 16 – ( α , , β j + 0 ( αβ | ψ f α f i i + + = (1 ) we obtain i =1 δ c =1 64 4 X α αβ X f f for } on these states, which in fact must be equal and we call it 8 , , we find the following results for the discrete torsion phases: β . In other words, the 64 possible signs get grouped into 4 sets of , ψ 6 α , C g f 4 ) in the other direction. In this case modular invariance requires i , both running from 1 to 4 and thus contributing 4 ground states , g and α β 2 f in these sectors, the 16 fixed points get organized into 2 sets of 8, ( in each case the 16 twisted sectors get ordered into 4 sets of 4 labelled f { β α ) = 0; f , running from 1 to 64. We have a discrete torsion degree of freedom for = , ψ 0 | and δ g α, β I αβ k i + ( in this sector, we have 256 fixed points organized into 64 sets of 4 labelled α f g k ψ f f ∈ j + +4 i sectors: 0; ψ α ) such that when acted on by the raising operators sectors: , f δ ). We can then label the fixed points such that i + 0 δ β | ψ or and α, β sector: With this in mind, we can now determine the contribution to the cohomology from γ ( γ and and = αβ f for states of the following form: the different twisted sectors.no We discrete begin torsion first phases. with Insets the either of case, 8 the states. 168 Following fixed the states point/twist methods fields in in get each [ grouped ofg into the 2 two sectors (we call such a combination by a number both the action of and get 16 signs which must allmust be correspond equal, to and how we how distribute we the distribute signs such signs in across the these sets of 16 γ however there is no discretework torsion with. sign available and so we onlyαβ have 2 ground states to the distribution of these signs within the the analysis in appendix α by numbers each. In each sector,states we and have a a sign discrete torsion sign We begin with Examplethe 1, smooth where limit of thein orbifold each must twisted be sector. relatedwhich to In only the arise particular, ground in this states twistedwe means sectors of only we associated RR need are ground with to studying group states consider zero contributions elements from acting momentum the non-freely. states, generators Thus 3.2.1 Example 1 JHEP10(2019)204 ) or (3.6) (3.7) (3.5) β ( α f invariant 4 2 . In partic- Z } singularities, 1 , β 0 act diagonally on , . Thus, identifying ∈ { and β g i 1, we get states with i l . So, we have 1 state α g γ, δ − β f , and take i and . However, there are 2 = 0; or , g α αβ 0 dX i | α l + αβ = f ψ g k + with 0; , ψ i + 0 j + | ψ and ψ i + } k 4 4 k , 3 − , is the number of signs we take to be +1, and , ψ we find the contribution (recalling that the k 2 g i k = 16 = 2 = 8 = 12 , k i 1 2 g 3 g 4 g g { – 17 – = = 16 = 6 f 4 αβ dX δb δb δb 2 g 3 g 4 g = δb 0; ' , sector is identical), we recall that our 16 twist fields δb δb δb β 0 | I i + β . In total, there is one state of the first form, 4 of the ψ j + δ or ψ discrete torsion phases to be -1 meant choosing 16 of the i + } (the or 8 β 4. , is the untwisted ground state and each α γ ] and 7 i 9 or , , ψ = 0 6 | α g , ,..., g i 5 g { sectors are more interesting, as they involve the choice of a discrete f = 0 with the constant 0-form, each = and 0; taking values in the same possible index set. So, we would find 1 state k where i , αβ i α g 0 0 i | I I | 0 | ∈ and ∈ i l and . ) β δ β i + a 2 form as in [ , the 256 fixed points were organized into 64 sets of 4. There are no oscillators ’s with is again the number of positive signs we find that this sector contributes: , ψ or or i + ( ' α )’s to be -1 for consistency. When we do so, because i k αβ ψ α γ i, j, k, l i, j, k, l g Q i = = ) to be +1 we get states: f α, β For The ( α g g ( 0; αβ β , f f 0 We must also includestates the contribution from theular, untwisted we sectors, identify which arisesstates. from As the expected, we find that they are in one to one correspondence with the invariant that choosing one of the the states within the 64Thus, orbits, if we find we cannot create any invariant linear combination. to use asinteresting raising things operators, to so notebut here. we rather only Firstly, a this have 4and state states form so should (loosely, not the we be can associated interpreted think form as of as a these a 2 as product form, products of of 2 forms, giving a 4 form). Secondly, recall for Where with no oscillators, 61 with and 2, 3 and 1with with no 4. oscillators, 4 When with we 1,we take and make it 4 the to with same 3. be identifications If of states with forms as before, we find the contribution: were grouped into 4 setsthe of appropriate 4 signs and between follow a states similar in procedure one to orbit, above. we Provided find we that choose when we choose for torsion sign. Beginning with second, 6 of the 3rd,| 4 of the 4thdistribution and of one signs of within the each 5th sector for must each be of the same): Where JHEP10(2019)204 , ]! α 5 , , = (3.9) (3.8) γ 4, we (3.10) i γ g 8. The i γ , -twisted γ f 7 f k < = +1, we for , 0; αβ . The only 4 p γ , g 0; f , δ , 0 | 0 C | 7 + = 3 7 + ψ i ψ 3 + . are the same as in 4 + } for 4 . αβ 1). This model is very i ) + , , ψ , but consistency of the ... γ ψ 1 , ψ ( γ 0 , i γ 1 and αδ i γ , f γ f αδγ δ ∈ { 4, some of the 64 f , 0; =1 β , 0; . When we choose 8 = (0 , 0 γ , and αδ X | i f 0 , k k < f | α d singularities. Thus, when ) 8 + = 4. At first, we may naively believe 4 + k αδ ψ = ≡ 8 ψ k 3 , + αβ ) j g 3 + δ ) = 2 − 2 δ ( 0) and − sectors all receive a discrete torsion sign — 48 αδγ for , , ψ ( , 1 αδγ p g γ γ j f , f i . 0 δb αδγ γ + 4 , = 8 = 16 = – 18 – f γ =1 k γ 3 γ 4 γ 2 i , ψ 4 2 0; γ ) = , X , and δb δb δb γ αδγ f = (1 f 0 i | αδ i γ sector will have zero momentum states to contribute 0; ( c αδ f 8 , + γ , f 0 γ | sector are organized into 4 sets of 4 labelled by a number 0; which are equal to 1, we find that the contribution from γ , ) = 4 ) = (10 8 + γ δ 0 γ k | , ψ ( ψ f γ ) = 7 7 M + + γ f δ i ( )( γ γ 4 f f =1 4 . We then fill out the ground states with , b γ X 0; 1 we find the states: 3 γ f , i , ψ resolving 16 of the associated 4 − 0 , b γ | ), and so the total Betti numbers are, after adding up each 2 γ f 4 2 b i = 3 + ( γ Z 0; not γ f , / , ψ f 8 0 : , and so we build states from linear combinations of states within the orbits, | γ 0; } T , ψ i , ( 4 γ αβ 0 i γ | f i be the number of H 8 γ + 0; f j ψ , and ,..., 0 4 + 1 0; | δ , ψ The 16 fixed points of the Rather interestingly, this does not actually completely agree with the result in [ , 4 + 0 | ∈ { γ ψ , γ Letting this sector to the cohomology of the associated resolution is: whereas when which we call discrete torsion phases comerepresentation from requires the actionsfind of that the only invariant states are: Once again, of theseto 3 the cohomology, only and the example the 1. contributions f however, modular invariance constrains themsigns so must that be we the may same set within the each distribution sector: of these 3.2.2 Example 2 Now we analyze example 2,similar where to examplecrucial 1, difference the being that full the analysis for this sector is done in appendix find that this issectors not could not quite provide the usas case. with saying states, we as Recall are they that weredo when not not invariant. actually have We a then complete interpretorbifold resolution this — and rather, the we have remaining only singularities a partial are resolution frozen of due the to the presence of discrete torsion. β Rather, it only reduces towe the have expected found result a when set of other new resolutions, but on returning to the previous analysis we classes in JHEP10(2019)204 - T (3.11) (3.12) . algebra } 2 8) G , 4 means we map: 7 ) (3.13) . , 2 2 6 8 G , G k < , (5 ∂ψ ,M 8 ), of the , ) 2 j G 8) 2 , ∂X K 2.24 7 1 2 − , − . 4 k , algebra invariant, without ), ( − , ) 2 2 j 8 2 G (3 G 2 ψ G , 8 − ,X ,M 8) 2 X 2 42 + 28 , 2 G ]. Written this way, it is easy to 5 G 1 ∂ 154 Φ , k, , 1 2 4 8 , − ,K , , : 2 16 − + 2 (1 G 8 2 j, G , 48 G ,X ∂ψ 8) ,G 2 j, 8 , M 2 G 6 ψ G + , , − Φ : T ]. Once again, a choice of , we can then explicitly realize this automor- 3 , k 2 ( 8 – 19 – 5 8 , 2 1 2 algebra invariant and so map the Spin(7) algebra G G sector and some of the singularities are frozen by X (1 ) = (10 + 2 − → ∂ψ . This can be done using the combinations of 4 K 4 , 8 : 8 G ) j, ,G . Doing so, we see that this combination maps the 2 − ψ ψ 8 8) 2 αβ 8 , G , M 2 1 2 = 4 we get: G 7 : G X or T )( mirror automorphism by combining the , k ∂X + ) = (2 + 2 4 8 Φ ,M 8 2 8 2 8 2 8 , , b j,k G G X ψ 3 ∂X (1 2 ∂X M 8 ∂X , , b : G ,K 2 ψ )( 2 b 8) 1 2 4 . When ( G , ] = k 5 + Φ , b + + : , 3 2 2 ,X 2 3 2 G G , b , ], which leave the G 2 G G, X and 9 T G X b (2 Φ ( j , , = [ = = = 2 8) G T G , X M 6 represents the space we obtain after the different resolutions of the orbifold, , ,G 2 4 , j,k G T (2 M ( { ] combined with T-duality along The starting point is the Spin(7) superconformal algebra, the generators of which can 9 Another option is toaffecting use the an terms automorphism involving thatdualities found leaves in the [ together with a T-dualityalgebra directly along back on toin itself. [ Using combinationsphism of as the 3-direction such T-dualities aautomorphism: found duality. We find that the following 7 combinations generate our Spin(7) The algebra these operatorswork out satisfy the is analogue worked of out the in [ be obtained from the generatorsas ( follows: 3.3 Mirror symmetry In this section, weobtained will in briefly example discuss 2. mirrorthe symmetry Example (partially for 1 or the can fully smooth be resolved) Spin(7) models treated manifolds obtained analogously, there with are the self-mirror. result that all of which are thecannot Betti create numbers certain found statesdiscrete in in torsion. the [ where parametrized by and thus the total cohomology is given by: JHEP10(2019)204 in αβ β = 4, (3.16) (3.14) (3.15) or k . or . β } } α , 4) leave them sector, and . α 4) , ’s. Our first , 3 + } , α i 0 3 , I ˜ , 2 ψ 6) 6) 2 , , , )). Applying the , 5 5 (1 , , (1 , 2 4 and in the , α, β , , ( 6) i 0 β 7) , (1 (3 ψ , αβ 5 , , f 6 , . . , 2 7) 7) ) ) 3 , , , = , β α 5 5 ( (1 3 these are dualities between ( , , α sector, (2 , α β , 2 4 f α ), while those in f f 2 . , , δ H β , 7) | ( ) · · 1 , (2 (1 γ δ β , 6 7 4 0 0 f , , ( , ˜ ˜ . γ ψ ψ = 3 7) 8) f 7 3 0 0 , = 0 , , ˜ ˜ ψ ψ j,k in the 7 0 6 7 αβ 6 2 (2 k sector: 0 0 ˜ , , f αβ − ψ ˜ ˜ 4 j,k , α 4 6 ψ ψ 4 0 H α parity that we are interested in. A quick , , 5 1 0 0 | ˜ ψ 7) M M ˜ ˜ α ψ ψ 7 0 , (1 (5 8 4 0 0 5 αδ ψ , , → → , ψ ψ 4 0 7 3 0 0 4 8) 8) ψ – 20 – , in the , , ψ ψ j,k j,k 1 4 7 5 6 2 0 0 (2 β , , M M ψ ψ , : : 4 4 = 5 1 0 0 , , 6) γ ψ ψ − + f γ , (1 (3 I I ]. However, for 1 1 5 H , , 16 16 | , 10 4 8) 8) = = , , , αδ . 7 6 β α (3 , , k f β f α , 2 3 H H , , | | sector: 7) α β , (1 (1 β ] and show that these agree with our results obtained above. 6 , , , 4 27 8) 8) , , , sector. We find, for 6 5 sector, for which it is the (1 , , , 4 3 γ αβ , , 7) ’s (i.e. it can be represented by , (2 (2 ψ effectively swap the discrete torsion signs 5 { { , sector we have no zero modes, and so there is no representation of − acting in the 2 = = I , in the + − α (1 αβ I I β { We would now like to see how these maps act on any discrete torsion signs. To do and 4 Spin(7) mirror mapsIn for this connected section sums we considernected mirror sums maps (GCS) [ for Spin(7) manifolds realized as generalized con- Both of these mapsthese take are type the IIA dualities string foundsingular theory in manifolds, to [ which type were IIAof not and T-dualities found IIB which in to change their IIB. analysis. When Note there is no combination Those in alone. Interestingly, nonesectors. of Thus these we 14 have the combinations set change of the dualities: signs in the T-dualities, we find that the 14 possible combinations group into two sets: In the terms of the α and for focus is on the calculation gives: Next, we want to consider the representation of algebra in this way: so, we construct representations of the operators in terms of the on to itself once again. The following 7 index sets generate an automorphism of the Spin(7) JHEP10(2019)204 t ]. 4 + β ]. 27 ]. By (4.2) (4.3) (4.4) (4.1) 35 → − 27 t and , it fol- , we may Z 2 + I β × 1 and as Γ for Γ a finite S / 3 with neck region 8 × X T − 3 Z given as a resolution , X -th cohomology group ) holds in the present case. − i ) Z ] that there exists a Ricci Z 4.2 4 e for a Calabi-Yau threefold ( b 4 27 I b + ) ] × 3 e manifold Z b 1 , ) + 2 27 1 . S + S Z ) G manifold ( . × 3 o Z 3 2 × b 3 , b 3 3 1 G X + as the generalized connected sum , recovering the construction of [ S X X 2 o ) + ( ( Y b i i Z . Second, compactifications of heterotic × Z ( H H + − 3 2 ] argued that one finds a decomposition such b 4.3 Z 3 + 4 + can be realized as (a resolution of) a quotient 2 e → → b n n 27 − ) ) − – 21 – # 2 + + + Z Z Z are found to be [ holds. and an acyl , , + 2 − 3 − 4 − 3 + − 3 1 + Z Z n n n + β Z Z Z ( ( = + + + i i 8 + 2 + 2 − 3 − Z H H − ] allow precisely such a decomposition. We reviewed a n n n 5 manifolds : : acts as an anti-holomorphic quotient on ) = ker 2 + i i 2 + − ) = 0 ) = ) = ) = G β β Z ) below in section Z Z Z Z Z ( . Furthermore, we have assumed that the images of ( ( ( ( manifolds should have a lift to M-Theory on a Spin(7) manifold 3 2 1 2 3 4 ) + 1 = and identifying the isomorphic neck regions 4.1 b b b b 2 with asymptotic neck region Z can be globally described as a resolution of an anti-holomorphic H Z 1 ( G + S Z 4 − , and an asymptotically cylindrical Z b I × manifold, the authors of [ − 2 Z , the Betti numbers of 2 G are the dimensions of the odd/even subspaces of the in which the Z e i 2 / b ) Z / R are the kernels of the restriction maps ) and × i . Taking R ) on the M-Theory side and checked the equivalence of the spectra of light fields in a under the action I i o n 3 . In this case, 3 b × By using the fact that there exists a single covariantly constant spinor on For an acyl Calabi-Yau fourfold and an interval 4.1 X This last assumptions is slightly weaker than the assumptions made for technical simplicity in [ R × 3 X 3 3 3 X lows that following the same analysis presented there, it is straightforward to see that ( and of are surjective and that Here of ( map to a TCS as ( few examples. Finally, acyl ( on quotient of a suitably chosen Calabi-Yau fourfold First of all, the examplessubgroup of of Spin(7) manifolds Spin(7) realizeddecomposition as given resolutions such in of as [ ( string theory on TCS which can also be decomposed into two pieces. By applying an appropriate fibrewise duality then form a compact eight-dimensional manifold Based on a number offlat observations, it metric has of been conjectured holonomy in Spin(7) [ on such manifolds. The evidence for this is as follows. As a preparation, letThe us building briefly blocks review the fromCalabi-Yau construction which fourfold of such GCS Spin(7) Spin(7) areX manifolds formed of are [ a asymptoticallyX cylindrical 4.1 Constructing Spin(7) manifolds as generalized connected sums JHEP10(2019)204 , . ) ∨ ∨ − 2 N · Z Z Z . 4.6 / 4 N (4.7) (4.8) (4.9) (4.5) (4.6) ) 2 1 which (4.10) T . R − 4 3 o T × b , the circle . 3 1 realized as 2 holds. S X Z 2 o ) × M b ) + 2 2 e = ( R − b = Z − × . In particular, 4 e + Z b I ∨ 3 2 o b × X 1 . manifold and S = ( 2 3 e + 2( b × ∨ − G 3 − ) + 1 ∨ Z = n 3 sits inside 2 ∨ 3 o X 3 ) suggests how such a structure Z b as Z + 3 / ( X . ∨ 3 has a SYZ fibration by 4 − 4.1 2 − 1 b , and a must produce another manifold n X ) ) ) 1 I + S 3 ∨ ∨ ∨ S Z Z × ) + × Y Y Y + × 1 3 ( ( ( ∨ 1 inside a Spin(7) manifold has dimension 3 in the neck region. On 1 1 1 Z , , , ∨ S − Z + 1 3 2 1 X N ( # Z h h h S 2 × + 3 + b SYZ fibre of 3 = Z n # 3 X – 22 – ) = ) = ) = − + = ∨ T + . For both of these compact geometries, there are Y Y Y Z ( ( ( I Z 2 + Y 1 1 1 # n , , , × ) + 1 = 1 2 3 = − along h h h 3 ) holds. Our main task is to work out how the topology Z Z ∨ + 2 ( + X Z 4 − = 4 + 4.2 times a circle Z b 5 while the n 3 1 M . This can be done as follows. There is a compact Calabi-Yau The GCS decomposition ( X S 1 3 ) + along 4 Z Z ( ]. − 8+ 2 6 Z b − 2 3 ], exactly marginal deformations of Spin(7) sigma models are counted SYZ fibre of 1 3 fibration [ )+1 = T 4 Z ( realized T 4 − ), a mirror is given by b Y ), so a mirror map for a Spin(7) manifold 4.1 is related to that of )+ 4.5 In the following, we will collect some evidence for this proposal by showing that ( This motivates the following construction: for a Spin(7) manifold realized as a GCS ∨ Z The moduli space of a Cayley (calibrated) four-cycle We would like to thank Michele del Zotto for suggesting this construction. ( 4 5 Z 2 b so that we can at best hope to approximate such a fibration in a collapsed limit. by gluing two copies of mirror maps which act in the usual way, i.e. of fourfold realized by gluing two copies of which are glued along ais neck constructed region from with an is antiholomorphic isomorphic involution to of indeed holds for this construction.fying assumptions In under order which to ( prove this, we will stick to the same simpli- We hence expect to find a Spin(7) mirror byas performing in four ( T-dualities along this Furthermore, such a mapcalibrated can be themight result of be an realized. applicationbecomes of the The four acyl T-dualities Calabi-Yausimply along becomes fourfold a the product As shown inby [ ( such that Here, we have used that for anti-holomorphic involutions 4.2 A mirror map for GCS Spin(7) manifolds and we can compute JHEP10(2019)204 , ) 3 e ). b − ∨ ∨ . It Z Y + 3 ( must M 2 o (4.15) (4.16) (4.12) (4.13) (4.14) (4.17) (4.11) 1 ( X , b 2 + 3 2 b and in terms n h + ∨ 2 e and + +2 ) + b Z Z , the Mayer- ∨ − ) = 4 + Z n Y Y M 3 ( , 23 and ( 1 , 2 + 2 b 3 − − Z h n ) 3 + ) = X 2 − . M ( ( n 1 , must also be left invariant 3 . , 2 b ) h 2 + 3 − . ∨ ) + 4 n n 2 3 3 23 ) + Z M ) + + X / ( 3 + 2 − ( M . 3 2 − ( 1 ) b , X 1 2 4 + 3 − . Furthermore, n 2 ( b Y S n 1 n + h , ( , 1 2 ) + Z × ∨ + b h ∨ M 2 − + 2 ∨ Z 2 3 3 n 3 e ) + 4 M 2 3 # b X 3 ) + 1 ( ) is preserved, it follows that − 3 ∨ 2 3 + ) X + + 2 + b ) gives ( Z X X = 2 e 2 o Y 1 . ( ( , ( b b 1 – 23 – 2 + 1 ∨ 1 + = ∨ ) = 0 there is the relation , 4.8 − , 4 ) = 1 n 2 h b manifold Z Z ∨ Y h h 1 6 ( 2 ) = ) = M # Y 1 ( , ) for GCS Spin(7) manifolds is preserved under the + 2 + + 2 G ∨ − 2 3 ) + 1 for anti-holomorphic involutions, this expression )+ M M . and b 3 h 4 + 3 + 3 + 4 + 2 ( ( Z ) = 4.5 − 2 3 n n n n + X X b b Y Z ( = ( ( Z ) + 1 1 1 , 1 3 , ∨ , 2 1 3 M h must leave the above expression invariant. As this mirror map h ) = 2 ) = 2 ) = 2 ( h M 2 now have the decompositions Y Y Y ) = b . Here, the Mayer-Vietoris sequence yields Y ( ( ( ∨ ) + , and 3 4 2 ∨ 3 b b b M I Y ( X M 1 ( × , 1 ) holds. 3 , 1 , so that . Using these relations is the key to find the topology of 1 h I ∨ S 3 and h 4.6 X is the same for × × ∨ = ) + ∨ ∨ 3 3 3 + ∨ Y to 3 e n b X X Y 3 ( + 1 X , 2 o 1 b h is a Calabi-Yau fourfold and + . 2 e Note that we have not provided an explicit construction of mirrors for Altogether, we have shown that the expressions Let us now discuss Let us now work out the resulting relations in detail. Starting with Y b Z are all left invariant underthen an follows application that of the themirror mirror expression map, map ( i.e. acting ( on but only used the topological constraints they have to satisfy to arrive at this conclusion. Under the mirror map actingAs on the is preserved by the mirrorunder map. the mirror It map hence acting follows on that implies that The mirror map acting on also maps also be invariant under the mirror map acting on As so that Vietoris sequence for the decomposition ( glued along of Furthermore, glued along and JHEP10(2019)204 , i − is ∨ + 1), Z 7 is a Z , with and α, γ 1 ) + 1 X h ζ (4.20) , (4.19) 3 . Two 2 3 / 1 i 6 Z X , and act non- ( / T 1 , + γ 2 3) Z = h = (0 , where α, β, γ K 3 ]. This would is the Nikulin h d I ˜ / and X × σ 19 ) + × R 3 3 0) simultaneously α 1 , and a resolution of X × K S 8 ( 3 7 , 1 1) and , × X 1 T , 3 1 h ˜ , = X on 1 ) (4.18) = , + , but the definition of 2 3 e ) = (10 3 ˜ Z Z b is given by L, ζ X ( ] + = (1 ˜ 2 Y , which has already been studied 2 o c − r, a, δ ˜ 36 . i b Y of b ∨ − + Z + Y 2 e . b 2 . Resolving the orbifold , only two generators Z 1 1 8 / we have [ S 2 1 , which has ) = 19 ) = 19 = ) = 8 ) = 24 ) = 24 3 3 Z S Y Y Y 7 0). It can be described as ( / and hence ( ( ( X X , – 24 – does not change under the mirror map. × 1 1 1 ( ( X . Clearly, 8 , , , ∨ 1 1 1 2 3 2 3 1 3.2.1 , , , 2 + 1 2 S with h h h Z X n M h h / 3 × ) . At X 3 i + 2 R 1 8 b X 4 + ) = (10 × n = 3 ) = 7 X M r, a, δ X ( a coordinate on the i b 8 , we find an acyl Calabi-Yau fourfold X 1 8 , so that we can identify the neck region as ], the GCS construction of Spin(7) manifolds is closely related to the 6 ≤ and 27 7 I X ...X 1 . can be glued to form an Calabi-Yau orbifold X ], in which Spin(7) manifolds are found by resolving anti-holomorphic quotients + 4.2 ˜ is the (real three-dimensional or empty) fixed locus of the involution and Z 35 acting as the Nikulin involution with invariants ( L ]. In their terminology, this case is the fourfold ‘model B’, in which 2 Restricting As shown in [ It is of course straightforward to describe mirrors of Z 37 the on both K3 surfaces. The topology of the resolution copies of in [ involution with invariants ( trivially on a coordinate on produces the Calabi-Yau threefold in section 4.3.1 Example 1 Let us first study theand example start of section by describingcutting its the GCS orbifold decomposition. along Such a decomposition can be found by manifolds in general, and the ones considered4.3 here in particular. Examples In this section weshow revisit that the the two mirror examples map of found Spin(7) there manifolds agrees studies with in the section GCS mirror map described above possible twist. Thisresolution potentially can constrains be which chosen antiholomorphic to involutions construct and which work of [ of Calabi-Yau fourfolds. This offers another possible perspective on mirror maps of Spin(7) the orbifold singularities ofdoes ( not depend oncase the there details exists of a the resolution antiholomorphic of involution chosen. Furthermore, in where five-dimensional tops as hasin been turn done allow for to acyl derivewhich combinatorial Calabi-Yau in formulae threefolds for turn in the must [ topological imply invariants that of furthermore involves specifying an antiholomorphic action of It should be possible to give a construction of acyl Calabi-Yau fourfolds from projecting JHEP10(2019)204 ) ) 2 = 1). G are , 3.8 − 1 4.20 1 8 ˜ (4.24) , (4.23) (4.26) (4.25) (4.21) (4.22) Z 1 ≤ , manifold 7 2 . The neck x = 11. This 1 8 G = (0 given in ( 2 o d is again one of = b are self-mirror, manifold 7 Z ˜ 2 M X is M G )). As we have seen + 0) and Z , 3.8 and 1 # = 8 and , + 0 Y 2 e , Z b . Two copies of this orbifold found by setting 1 = S = 4 in ( + = (1 Y Z k and an acyl c 1 S , l . l is such that − ∨ 3 Z X ) = 19 ) = 19 , which has = 2 = 6 = 6 3 3 = 2 = 8 = 76 = ) = 12 ) = 43 which has a unique resolution to a 2 − 3 − 4 − ) = 8 + ) = 47 X X on . 2 + 3 + 4 + – 25 – with ( ( M M n n n n n ˜ n n n 1 1 ( ( Z Z , , M i 4 1 3.2.2 2 3 1 2 and its resolution M M b b ) to recover the Betti numbers of ( ( are different in this example, but = h h 2 3 = + b b α, γ ∨ ˜ ˜ 4.2 h Z 7 M Z / 6 X orbifold T and 2 at ]. Its resolution is not unique and produces nine distinct = 4 G − δ 3 ˜ Z ˜ X being a coordinate on the product , we find the product of an 8 1 8 X ≥ 7 orbifolds X 2 , with ]) i with Betti numbers G 4 l M ). α, γ, δ ) = 288. The relevant data of the decomposition h Y / ( 4.21 R χ The acyl The acyl Calabi-Yau fourfolds As a check, one can now use ( Restricting with (see [ × 6 the elementary examples of [ manifolds the same as inand the ( first example, so that we already know their topological data, ( Let us now study theWe example can of proceed section in theregion same is way again as formed for as the first example and cut along so that and our Spin(7) mirror map gives 4.3.2 Example 2 (note that the complete resolutionfrom corresponds the to CFT setting analysisconclusion this is Spin(7) reached manifold should by be applying considered the self-mirror. GCS mirror The same map: both Furthermore, the action of determines that T can be glued to theM compact and JHEP10(2019)204 g of g ), so (A.3) (A.1) (4.27) ]. String 4.27 as before = 11. We 29 ∨ + 2 o b in ( composed of Z l e = H − + 8 Z = 8 and → 2 e l b = 4 (again, only the case k = 1 (A.2) . We have and Z bc j − , = 2 ]. Crucially, the definition of string ad g l 9 , . H | for every non-trivial group element 2 2 0 g,f g is again such that h l/ l/ ) H . 3 H , we need to study the action of other group 2 h − − f j g ( X ) for − l/ g ⊕ b 4 H ) setting g – 26 – Z = = 8 = 8 = on a = 4.2 h g 1 4 2 − 3 − 4 − = g ( n n n d H ] H ∨ j g = | c Z h 29 7 h X ) from ( at ) = 3.8 h δ ( g by a group Γ is built from the untwisted sector n T . This action in general involves the assignment of phases, g H . This means that the GCS mirror map replaces l must be even and that on − l 8 g 8. The action of M 6= refers to the usual action of h in the g-twisted sector as expected from the orbifold h = ... 0 ∨ l h = 0 M We are now ready to discuss the GCS mirror map for l = 4 corresponds to a complete resolution). to guarantee modular invariance. Astypically the decomposes twisted as sector associated with a group element where group action on the coordinates.of These Γ discrete and torsion furthermore phases must must satisfy form [ a representation theory on orbifolds in generaltheory involves an on assignment a of orbifold discrete of Γ-invariant torsion states, phases as [ well asΓ. a To find twisted the sector stateselements in the twisted sectors through the PGS D scholarship. A Discrete torsion andTo modular set the invariance stage and outlinediscrete our strategy, torsion let for us review strings a few on basic facts orbifolds about following (generalized) [ Gauge Theories and Geometry” (HIGGSBNDL). ThePerimeter Institute research for of Theoretical AO Physics. was supported Researchthe at by Government Perimeter the Institute of is Canada supported through by Development the Canada Department and of by Innovation,Development, Science the and Job Province Economic of Creation Ontario and through Trade. the Ministry AO of also Economic acknowledges the support of NSERC Acknowledgments We would like toand thank Ashoke Michele Sen for delis discussions Zotto, and supported Marc-Antoine inspiration by Fiset, related the Sakura to this Sch¨afer-Nameki ERC project. Consolidator The Grant work 682608 of “Higgs APB bundles: Supersymmetric This data again reproduces ( k and that it reproduces the CFT results now find that for JHEP10(2019)204 ) 2 h ). G ( g g ( f (A.4) (A.5) h f ) in our model, . h g ; ( h g f Z Γ ∈ , this is not the case for X ) (A.6) 0 h,g g 7 1 2 τ | g ( + − − ; , f X − 1 g Γ g ; g | h Z f Z ≡ 6 2 1 + + − X − ) = 24 c/ 5 + 1) = /τ ) for all − + − + − 1 τ X 0 h ( , a different assignment of phases ) by linking them to the phases L − ( e g orbifold ; 0 ( ¯ g q h f g f h ( 4 1 2 ; 2 ] for a more detailed discussion. For the 24 g Z g + + − 9 f X + c/ G Z − ) = 0 → 3 L h – 27 – → ( ) + − − + X g gq τ f ) ( τ ( Z h 2 1 2 H Z + − + X + Tr Γ ) = 1 ∈ . We work out the representation matrices for the orbifold + 1) = + − + + /τ X . For the sake of brevity, we have omitted the details of these X 2 h,g τ 1 8 ( ) | − 1 2 1 Γ Z ( | γ β α S ]. Of course, these still have to form a representation of Γ. σ Z 9 = ( ) = ¯ 8 q T q, ) = 1 follows, and ( g or ( Z g has several fixed points labelled by 7 f ) is possible [ g refers to the partition function component restricted to the h twisted sector, as 1 g S g ; f h Z = ( 7 The orbifold we are interested in is defined by For the examples discussed in this paper, the computation of partition functions is sig- Although modular invariance for bosonic strings at one loop is sufficient to guarantee If we choose to include such ‘generalized’ discrete torsion phases T model introduced in section elements in the highesthave weight discrete states torsion signs of showing the up, whichfrom different are the twisted then S-transformation. constrained sectors. by trace These relations coming matrices will computations. B Discrete torsion analysisIn for this appendix, the we derive the necessary conditions on discrete torsion phases for the again present a caveatexamples we to are this presenting, analysis, thesethem to subtleties see known are [ smooth alleviated geometries by obtained the by fact a that smoothingnificantly we simplified of can by the match the orbifolds fact inon that question. all of the elements of the orbifold group act diagonally more general assignments. However,provide us studying with solutions necessary to conditions,we which the are will above be only constraints enough going at for tobelieved our least study that purposes. partition modular Furthermore, functions invariance of ofsufficient the bosonic for bosonic strings. modular string Although invariance partition it of function is is the generally necessary full and superstring theory, higher genus amplitudes which constrains possible assignments of the modular invariance at higher genus if where in the summation in the middle. Modular invariance then implies from which for each modular invariance must bewritten reconsidered. as The partition function for our models can be e.g. in case JHEP10(2019)204 . } α 2 f α , ; i ; 1 j (B.1) (B.2) (B.3) | { 1 2 1 2 = , , 1 2 α 1 2 , f , 1 2 0 ∼ ∼ , α α 2 f α f α × i i 0 2 E α α α 4 8 α α α α I | | f α f α f α α f α f α f α f α i i i f α i i i i 2 3 7 8 4 5 6 7 8 , where index ×
0 0 ; 2 α ; I f α ↔ | ↔ | ↔ | i ! ↔ ↔ | ↔ | ↔ | ↔ | 2 j 4 0 | 0 α α α α α α α α , × × 0 0 0 f α f α f α 0 , f α f α f α f α f α 2 4 i i i 1 2 i i i i i I 1 2 I 1 5 6 , 2 1 2 3 4 | | | , | | | | | 2 4 1 2 0 × × 0 00 0 0 0 0 2 4 I I = = ∼ ∼ α α α α f f α α f α f α = = sectors are relevant because they are the H H i i ): , we get two 8D irreducible representations | | α α 3 7 i 7 γ 2 | | f f α α β 2 σ H H | | ,X 2 6 β ; – 28 – ; σ , and β, γ, σ ,X β . They can be labelled by the different values of 1 2 h and j=1,2,. . . ,8 enumerates the different choices of 4 1 2 , α α α α α , , 5 f α f α f α f α X 0 0 α 0 i i i i , H . Now each of these twisted sectors localised at the , X 2 4 6 8 1 2 0 is the space spanned by the highest weight states 0 0 }} H 1 2 ↔ | ↔ | ↔ | ↔ | α , f α 0 0 0 ∼ ∼ choices. H α α α α 0 . The f α f α f α f α H 5 α α 8 i i i i 0 0 0 0 0 0 f α f α 8 H 1 3 5 7 × αβ i i X | | | | × 8 ∈ { 2 2 6 8 I | | σ I and i = = α X = = f α ; α α f f α α α α and H f f α α -twisted sector can be decomposed into 16 smaller sectors corresponding H H ): 2 0 | | H H α α | | 7 , f σ 0); γ α 0 γ , α . , , 0 L ,X , 1 2 The 6 0 1 1 0 αβ = (0 , ,X γ α ∼ ∼ 5 , H α α β α f α f ,X i i We can then assign coordinate labels for the basis states As discussed in the section 2, we do not need to analyse all the different twisted sectors , 4 5 1 | | α sector. X ( where H = The representation matrices for the generators after removing spurious phases are where In this basis, the orbifold generators act as follows: corresponding to the two corresponds to the twothe choices other of 3 fixed-point coordinates ( to the fixed points{ of the actionfixed points, of have ato highest find weight the state representation ofweight matrices states. zero for momentum Under the the and free orbifold zero action elements winding. of in the We want basis of these highest in this orbifold. Instead, weof would focus on theonly particular ones sectors twisted that under contribute thethe to action others the are ground needed state only spectrum in of orderα the to orbifold fix string the theory, discrete and torsion phases. JHEP10(2019)204 2 α = σ γ ). (B.6) (B.7) (B.4) (B.5) (B.8) f 7 sector γ ,X 6 ; ! ). Under the ,X ; 2 7 3 0 0 , 1: 1 1 2 ,X 0 2 1 ,X 5 , 2 , γ 0
f γ 2 1 , X ,X H , where index ) = 1 2 = | − 3 γ 2 2 f γ γ σ σ i f γ ). Now the irreducible ( choices as before, and ,X j 7 5 H γ | ∼ ∼ 1 | 7 f + X 2 3 4 X ) coordinate labels. Let us ,X X αβ αβ σ 4 αβ αβ f f 4 = i i = 6 1 2 3 6 2 ; = id. = | | 7 ,X ,X 2 ) and j=1,2 enumerates the two X + 2 3 γ 5 × X f γ = 2 I ; H 5 ,X 1 ,X | ,X ; 6 2 − 3 2 γ = 1 2 X ∼ n 0 × , X ; , γ 2 γ 0 mixes the 2 case in the fixed point coordinate labels. f γ ,X 1 2 I f γ 0 γ , 1 ) i 4 f γ , , H γ 2 1 2 | α 2 − H 1 2 1 2 | yields X , X | σ γ ( 1 2 β 2 − − ; γ f 4 labels the different irreducible representations ; 3 = – 29 – ασ − 1 4 ): H X ∼ ∼ ∼ 5 ) = -sector, = γ ,... = f γ 2 α 2 αβ αβ αβ i α 2 αβ , σ αβ f αβ f αβ f ,X 7 2 2 ( i i i 3 − X σ γ 2 5 8 X | | | f X = 1 sector, there are 16 highest weight states in the each. ↔ | on the coordinates is given by: 1 5 γ = α ∼ αβ + ; f γ X f i ; X γ γ αβ f γ 1 , 1 2 f γ did in the | i , 1 2 H | 0 1 αβ β 0 , | αβ f , . So, no discrete torsion phase arises in this sector either. β , , and coordinate labels ( αβ i 1 2 2 0 = 1 2 , we get eight 2D irreducible representations corresponding to the , and 1 2 j , , i | 3 γ X ; 2 2 1 2 1 − f γ X H H sector only differs from the 1. The relation | , 1 α β ∼ ∼ ∼ = X twisted sector, lowest energy states are labelled by the half-integer mode α, β, σ γ : f γ h The action of αβ αβ αβ ) = 7 f αβ f αβ f αβ corresponds to the 8 choices of ( H Analogous to the The i i i | αβ X 1 4 7 } α | | | αβ 8 ( γ f ,..., Removing spurious phases by exploiting commutation relations of the representation Then the action of the orbifold generators can be obtained from their action on the Let us now assign coordinate labels for the basis states In the sector. taking values in 2 sector. sector. , 6 1 αβ where So there are two choices of discrete torsion signs available in each irreducible representation. matrices, we get: coordinate labels, as follows: free action of two choices for { choices for Here they are givenbehaves the by same two way choices as mixes each the for choices the for set ofγ coordinates ( which can be identified by their fixed point coordinate labels ( β n representations are 8D, andassign the are basis states spanned bycorresponding the to the ( choices for ( JHEP10(2019)204 . ), γ 4 (B.9) ,X (B.10) (B.11) 2 ,X 6 n 1 2 2 − × 0 , 4 2 1 2 I × 4 I 2 − × 0 0 2 = 1, i=1,2; as is expected I 2 ∼ 7 2 σ + 7 2 × X 0 0 0 σ 2 + ; i X I 4 ) = 4 | × 2 6 1 2 γ 4 ( × 6 I ; 00 00 0 0 X + 2 ) + I αβ X γ i f ( 1 2 2 5 , σ 5 − = X 1 2 + X − = αβ . We can then have the following assign- f αβ 4 1 2 1 2 γ 4 H 1 2 matrix: X | − ∼ ; 2 X + γ 2 σ σ 3 i ; 3 − 3 – 30 – X | + ! X ) ; γ 2 1 2 ( 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 on the coordinates is given by: 0 2 4 1 2 H sector, the lowest energy states are labelled by X − 2 1 × X + 0 4 − 0 0 I H αβ , = 1 2 αβ 2 1 αβ 1 + σ 0 0 0 β on the coordinates is given by: 2 f H X + X 2 ; 0 0 0 0 0 0 4 σ H ∼ ! × 2 2 4 , where j=1,2,3,4: αβ 0 I σ H σ 2 2 ) each taking values in ) i σ = 4 σ 0 2 γ i 0 | n H ( j , | αβ 2 ;
), each of which takes two values. The irreducible representation, as f αβ αβ 6 n 1 f H | = The action of 2 1 , n β
5 α , 2 1 -twisted sector, the lowest energy states are labelled by the two half-integer The action of = = 2 ,X sector analysis and the S-transform relations. σ 4 αβ αβ f αβ f αβ γ ∼ H H ,X 2 | | 3 sector. σ γ α i 1 In the | ,X sector. 2 αβ 2 2 X Just as( the casededuced with from the action the of the orbifold generators, corresponds to the labels ( Note that the only non-trivial trace involving aσ discrete torsion sign in this sector is for the following 4D representation matrices: Looking at the action of the different generators on the basis states as listed above, we get winding numbers ( ment of basis states Here, the discrete torsion sign showsfrom up our in the σ at their action on the basis states just as we did before for the previous sectors: In this basis, we can write the representation matrices for the orbifold elements by looking JHEP10(2019)204 . }} and 1 2 γ , (B.12) (B.13) 4: 0 , β , ∈ { ; α ,..., i ; 2 , 1 4 X , 3 4 3 4 2 , = 1 , × ): 1 4 0 2 1 2 4 , I αβ 1 2 2 − 2 σ ,X 8 × 3 f 0 0 + + − + 2 , these get grouped into 0 X H I i ∼ ∼ ,X 2 0 0 2 H 7 × αβ αβ 1 2 0 0 0 γ, δ αβ αβ 2 2 2 h + − + 2 2 I σ σ 0 0 0 X − ,X H f σ f σ 1 i i 2 3 6 0 0 0 0 0 0 × , as outlined in section 3. These 00 00 0 0 | | H X 6 2 ( α I + − − + { X ; = ; = 5 1 2 1 2 as follows: 1 4 3 4 − + αβ X − − αβ 2 , , αβ 2 1 4 σ γ αβ 2 f 1 4 3 4 σ , 2 σ , , f σ H 4 3 4 . Here we have 16 twist fields in one-to-one | 1 2 1 2 H , + + + − α | β X orbifold where the group generators – 31 – 1 2 2 − − , where j=1,2,. . . ,8, and 4 2 ; σ αβ Z 3 1 2 αβ 2 ; / 2 + + − σ ∼ ∼ ∼ 8 8 X − f σ 0 H × i T 8 j αβ αβ αβ I | αβ αβ αβ 2 2 2 0 0 2 ) H 2 2 2 σ σ σ f σ f σ f σ γ + − + − X i i i ( 0 0 0 H 5 8 2 | | | αβ 2 0 0 0 0 0 0 1 1 2 H σ We begin with f − + − X − ; ; 3 4 = = , δ γ β α 3 4 1 4 1 4 , , αβ αβ αβ αβ , 2 2 3 4 1 4 2 2 σ σ 1 2 f f σ σ , , 1 2 1 2 H H sectors. − | | γ α ∼ ∼ ∼ αβ αβ αβ αβ αβ αβ αβ 2 2 2 2 2 2 σ σ σ σ f σ f σ f and i i i 7 1 4 First, let us recall the definition of the orbifold we are interested in — we focus on After absorption of spurious phases via commutation relations and normalization of | | | β act as: , δ In this appendix,constraints we for will the Spin(7) explicitlylogic orbifold determine as in in section the appendix 3 allowed B. — discrete the general torsion structureexample phases follows 2 and the from section same 3. This is a C Discrete torsion analysis for the Spin(7) orbifolds states, we get a discrete torsion sign arising in and can be organised in the basis Under the action of4 the sets rest of of 4, thethese corresponding group, representations to in come 4 particular with irreducibleand by a our representations goal highest of is weight the tohighest state orbifold find weight of the group. states. matrix zero representations Each momentum of of and the group winding, element in the basis of such Let us now determine the allowed discreteα torsion phases. correspondence with the fixedstates points are of labelled the by action the of choice of a coordinate set JHEP10(2019)204 } α α f α 4 H and . In | α (C.4) (C.1) (C.2) (C.3) δ . So, ) and f α α 4 ,..., × H }} 1 4 . 1 2 α permutes I f , · γ ⊕ 0 ∈ { ) 4 0 i β = ( . ∈ { 3 α α is: 0 0 f i H ! δ 2 X 0 0 0 α α f α f α = 4. i i 1 0 0 0 0 0 0 3 4 and ) to: β ): 4 ... corresponding to one of 8 4 dimensional. It is then γ × ↔ | ↔ | 4 . } , and so we have 256 fixed = I i ,X 4 fixed points, and so we can αβ . Explicitly, we set: α α = 1 , 7 f αβ , f α f α α 2) 3 i i = f α β }} / H 1 2 f H 1 2 | | 0) 1 X ,X , where the label | , , αβ , 4 6 ,..., α 0
β , so that f αβ 2 2 0 clearly act diagonally, 1 f α for all i / / H i : 3 i | i β ,X 0 0 and ) for | 5 ∈ { α ∈ { X α 1 f α 2 αβ ( 0 0 0 3 X = (1 = (1 and H α β ( X and | are the same as they were in e.g. the f f 1 { α α δ 0 0 0 0 0 0 take the form: ,X f α f α , with each α → − 1 i i ) for all αβ i . β 2 4 f αβ αβ β | | X } f αβ ( H X iθ . This groups our states into 64 sets of 4 — . We can choose two of these as labels for our 1 2 | = α 3 7 e H and δ , f and ): 0 – 32 – 0 0 X X αβ 3 =1 , f αβ δ , ≡ H iθ are identified in 4’s, reducing to only 4 degrees of , αβ ∈ { sends and . Once again, the 16 signs get identified in 4’s, and , | 2) 64 f ,X − γ permute the i ! i / β β 1 and β and e ; ⊕ α α 0) 1 δ αβ f α f α X α 5 , , αβ 1 X f αβ 0 00 1 0 0 1 0 00 0 0 = i i ( = Z H 2 4 X { X and | , acts diagonally and the action of αβ γ and f αβ αβ 3 = (0 = (0 α = ↔ | ↔ | β H γ H | α α X . In other words, the 16 dimensional space of these highest α α α f f α α 4 f α α i i f α f α i i H 1 3 X | | | 1 3 | | to be equal,
i . In this case and , δ : 2 permutes α αβ ! f α δ X 0 H H acts trivially, | γ ). 0 α H β (
, and analysis is virtually identical, only differing by the coordinate labelling of states α 5 f , the basis of states then consists of vectors = β X = 1. The requirement that all group elements commute sets the phases in α f α α f α i 2 H H Next, let’s do The Explicitly, in this sector | and γ 1 , we may set them both equal (in order to impose states, and in particulari.e. choose we have a decomposition easy to see that thesector, actions and of so theyβ take the same form (also without discrete torsion phases). For the exchanging of the rolesso of we end up with the 4 sign degreespoints of corresponding to freedom the choices X to 1, and forcesthe all 16 signs wefreedom would expect in (e.g. the fixed points now correspond to the set with Introducing discrete torsion phases, byvectors an the appropriate matrix choice representations of of normalization of the basis In this basis, label each of thethe 4 4 representations choices by of aweight states number decomposes further intoeach a sum of 4represents dimensional one spaces of as the 2-tuples in JHEP10(2019)204 . , ) ) . 5 = } } 7 2 2 X / / γ δ δ δ δ ,X f γ α, β (C.6) 1 (C.7) (C.8) (C.9) 1 (C.5) f δ f δ f δ f δ 5 ( , , i i i i H | 0 0 8 3 4 7 αβ ,X f , 3 ∈ { ∈ { αδγ ↔ | ↔ | ↔ | ↔ | . . , 4) 6 1 δ δ δ δ permutes ,X / f δ f δ f δ f δ = 1 ! 3 4) i i i i X ! 2 sector), and we , β / ,X 6 1 2 5 γ γ X | | | | 4 2 × γ 1 0 f γ f γ , f γ 2 , / i i 3 I equal, and we call X αβ 4 1 H 3 4 | , : / . Choosing a label X i 2 6 2 3 × 0 δ αδ , / 2 ↔ | ↔ | f δ X — thus, the 16 states and I γ γ H . In each of these sectors, | f γ f γ }
5 , , 7 i i 4 γ = (0 = (1 1 2 X / = | | and X δ δ 4) 4) 3 f δ f δ γ / / ,
2 i i permutes f γ 6= 1 3 4 (we use the notation 3 6 : ) are: , , H | | / 5 X and 5 | α 4 4 4 1 γ β × / / X f γ 4 ,X , I H | ∈ { 3 · or . , permutes ) β 5 7 = (3 = (3 δ δ δ δ , ,X 4) 4) and f δ f δ f δ f δ γ γ β X 1 / / f γ f γ i i i i ,X ). 4) 1 3 i i α, β 1 8 2 4 6 X = , , ( / 2 4 γ | | and with basis states labelled by f γ 4 4 3 X 5 αβ and / , / α, β H δ ↔ | ↔ | ↔ | ↔ | ). Only the elements f = ( ( | 4 1 3 X f δ δ , , , 1 , , δ δ δ δ / δ α ( – 33 – f δ f δ f δ f δ αβ f δ H 2 2 1 · γ i i i i f = i X 4) 4) , / / ! f i ) 7 1 3 5 / / | γ γ | | | | δ f γ f γ 1 3 αβ ( have discrete torsion signs in the i i 8 and can easily be constructed, and we find after , , f αβ γ 2 4 4 4 = (1 = (0 = (1 f H β : / / | × δ δ δ β δ f δ f δ f δ ↔ | ↔ | f δ i i i = sector. Here we get 16 fixed points in ( 8 2 5 = H γ γ and | | | γ = (1 = (1 | δ γ f γ f f γ i i . We choose to label each of the two 8 dimensional sets by α γ γ H αβ 1 3 1 f γ f γ | | | f αβ i i X H 1 3 , , β
both permute = 1. The commutation constraints set all | | | . The 8 states : α 2 i , , } 4) , δ δ acts diagonally. We can then turn these into matrix representations, 4 / δ δ δ δ γ f δ f δ f δ f δ / f γ 1 4) 4) ! ): i i i i , 3 / / 5 H αδ 0 5 6 7 8 , | 4 1 3 H and 4 / , , , δ / 3 4 4 ,X 0 permutes α H to represent this choice, the 4 states within each sub sector correspond to , γ ↔ | ↔ | ↔ | ↔ | 1 1 / / f γ ) so that 2 1 3
δ δ δ δ γ } X H / , , f δ f δ f δ f δ | 4 act diagonally, and so we should look at these sectors next. Here we get 16 fixed points with Now let’s do the ∈ { i i i i = 1 when = α 1 2 3 4 5 4 α, β , corresponding to either the case | | | | γ ( } × f γ = (0 = (0 = (1 4 2 αβ , and ,X αβ ,..., H I δ δ δ , | : 7 f 3 δ f δ f δ f 1 · 1 i i i α ) δ X X 1 4 7 f δ | δ | | ∈ { ( ∈ { H sector. sector. | γ γ δ f α So we get 4 sign degrees of freedom with: It is then clear that and after imposing any constraints we find: f the 2-tuples ( The matrix representations areimposing any 8 commutation relations that any discreteγ torsion signs vanish. In this sector, get grouped into 4 sets of 4 labelled by the choices of and the group action is: and f we have an 8 dimensionaland representation them to denote the fact thatend both up with 64 sign degrees ofδ freedom grouped into 2 sets of 8 by the rest of the group, where so that JHEP10(2019)204 , 2 8 β } / 2 has / ... . 1 β + 1 1 , (C.12) (C.13) (C.10) (C.11) retains 3 0 ∈ αγδ . γ X permutes , while f and 3 αδ ∈ { β , , 8 n , → f ) × 6 3 8 αδγ αδγ αδγ αδγ +) − I ! f αδγ f αδγ f αδγ f αδγ X , , · i i i i and ,X 4 4 αδ αδ ) 2 4 6 8 4 f αδ f αδ / / 3 i i 1 3 X 3 4 , , X δ, γ ↔ | ↔ | ↔ | ↔ | 4 4 ( 8 , / / + + 1 ): ↔ | ↔ | X αγδ 3 αδγ αδγ αδγ αδγ f X f αδγ f αδγ f αδγ f αδγ αδ αδ i i i i , . , n f αδ f αδ = (3 = (1 ) with 7 1 3 5 7 . As required, the discrete 1 2 1 2 5 i i | | | | ) 3 4 = 1 2 X − − | | × X +) − get signs in this sector). This γ αδγ αδγ 4 , n , , f αδγ f αδγ
I 7 : , 4 4 δ , and the action i i acts diagonally, and · 6 : 7 / / α 3 6 | | ) · + − permute ,X γ X X γ ) αδγ 6 αδ f αδγ ( δ , and f αδ , , , 3 2. In the lowest energy state we must = (3 = (3 H H αδ | 5 ) ) 1 2 / γ ,X n δ, γ | f and 4 ( β + αδ αδ +) − − β X − f αδ f αδ and 5 , , , i i + 1 4 4 4 αγδ ,X = 3 4 X ) within each set: f α / / / | | 1 Z 4 3 , 3 1 3 γ − − 4 , , , . and X are: X and , n = 4 4 4 } , , X 7 / / / 2 – 34 – ) δ , , / , 3 1 2 1 2 2 +) − 1 αδγ ,X , , 1 X + + X − ! 4 4 = (1 = (1 = (3 , / / both permute X αδ αδ 2 αδγ αδγ αδγ αδγ f αδ f αδ and 2 / δ αδγ f αδγ f αδγ f αδγ f i i αδγ αδγ αδγ i i i i 1 − + f αδγ f αδγ f αδγ 2 4 X 8 5 6 7 i i i = (1 = (1 αδ ). This action groups the 32 states (16 fixed points/twist 2 5 8 takes values in | | | } and ∈ { 2. So we find: αδ αδ ↔ | ↔ | 4 3 1 1 2 ↔ | ↔ | ↔ | ↔ | f αδ f αδ / / i i 3 n and by ( + α 1 , , , ) emphasizes that both X − 3 αδ αδ 1 2 n | | f αδ f αδ ) , } αδγ αδγ αδγ αδγ i i 4 4 αδγ f αδγ f αδγ f αδγ f +) +) − 1 3 δ, γ / i i i i | | = , , , ( . Once again the discrete torsion phases are winding independent, 1 2 3 4 1 4 4 4 and | | | | 3 8
αδ / / / αδγ acts diagonally. This orders the 32 states into 4 sets of 8, labelled by n × αγδ : 1 3 1 } ,..., 8 f ∈ { , , , 4 1 I γ : 4 4 4 / αδ The actions of / / / 3 f αδ = , ∈ { 2, and here we have fixed points in H αδγ 4 / and | . f αδγ / 1 , the twist fields have labels ( (both 7 αδγ = (1 = (3 = (3 H 1 αδ αδγ , δ f αδγ | 7 f X H αδ | αδγ αγδ , δ is short for X f αδ αδγ αδγ αδγ = ∈ { f αδγ f αδγ f αδγ diagonal. After imposing representation constraints, only the generator H i i i 3 | 7 αδγ 1 4 7 γ αδγ n f αδγ | | | α For So, for and and H ,X | 1 5 α αδ After organizing the phases, weno find phases that (again, way, and the 16 signs get organized into 8 sets of 2 across the twist fields labelled by The group action is: organized across the twist fields into 8 sets ofX 2. permutes a number with a phase and hastorsion a sign matrix is independent representation of of the winding number, and the 16 sign degrees of freedom get Here, means the winding number take X fields with two possiblerepresenting winding these numbers sectors. each) Within into each, 8 we sets label of states 4, by with ( a label JHEP10(2019)204 , α (C.16) (C.17) (C.18) (C.14) (C.15) and get: αδ , f α f . = ) =1 4 αδγ α X δ, γ f f ( ) = γ ( αδγ γ f f αδ f ) = 16 =1 . 4 =1 ) X 8 α, β αδγ γ ( f . αδ X ( f ) αβ we find: f +4 . α ( γ β , ) f we may set β ) = 4 f ) α ) = 2 =1 δ g ( ( δ 64 f α ( X γ αβ and f f f γ =1 ) = 4 α, β f β ( α X f k ⇒ =1 δ, γ 4 γ =1 ( ) = 4 +4 X For the rest of the composite group elements, twisted sectors. After taking into account the f γ γ X β f f α αβ 4 ( . For f – 35 – ) = 4 f 4 . α β f αδγ ⇒ ( αβ ⇒ . Taking into account the identification of phases = α =1 ) ) = 4 f α ) ≡ δ τ X α f ( ( τ f αβ and αβδγ α ( γ γ 4 and 4. ). Thus, we may set: f ; f =1 γ 4 ; α β X allow us to absorb all discrete torsion phases. So, no new αδ ), which permits any use, distribution and reproduction in f αδγ αδ , ≡ , I 4 and C.17 Z Z α ! γ γ = ,..., f αγ ) = 16 f ) = ) = αγ = 1 αβδ : H | /τ /τ α β , sectors. We would now like to understand how modular invariance 1 1 α, β f f ( CC-BY 4.0 αγ − − = ( ( αβ f αβγ This article is distributed under the terms of the Creative Commons αδγ α αδ ; , f αδγ In summary, we have found that their are discrete torsion phases in the ; γ γ =1 15 and Z βδ 64 Z and X αβ , f 4 αδ δγ ,..., , , αβ = 0 βγ However, consider the case of Now consider the case of First, let us consider the , k , γ , for Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. where we also used equation ( across the twist fields, we find: So, after setting As a result, after a possible reordering of the labels relates these phases. identification of discrete torsion phases, we find: the analysis follows the above structure andor we equalities find such that imposing as commutation constraints constraints will arise here. Constraints. β αγ JHEP10(2019)204 ]. ]. A B ] (1996) 2 SPIRE SPIRE G B 479 43 IN IN 43 [ B 492 J. Phys. Selecta ][ , ]. , Nucl. Phys. (1996) 507. SPIRE , J. Reine Angew. Nucl. Phys. IN , ]. , ][ Nucl. Phys. 123 arXiv:1207.4470 J. Diff. Geom. , J. Diff. Geom. [ ]. , supergravity on spaces of , ]. hep-th/9404151 (2017) 083 hep-th/9609113 SPIRE . II .I duality , 2 [ 2 IN g g T SPIRE = 11 10 . ][ ]. SPIRE IN D Inv. Math. 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