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Deposited in DRO: 17 September 2018 Version of attached le: Published Version Peer-review status of attached le: Peer-reviewed Citation for published item: Taormina, Anne and Wendland, Katrin (2018) 'Not doomed to fail.', Journal of high energy physics., 2018 (09). 062. Further information on publisher's website: https://doi.org/10.1007/jhep09(2018)062

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Durham University Library, Stockton Road, Durham DH1 3LY, United Kingdom Tel : +44 (0)191 334 3042 | Fax : +44 (0)191 334 2971 https://dro.dur.ac.uk JHEP09(2018)062 may g ], J. Har- Springer 26 April 30, 2018 August 7, 2018 ”[ : : September 3, 2018 September 12, 2018 : : Revised Received Accepted must have double the order Published g Published for SISSA by https://doi.org/10.1007/JHEP09(2018)062 , An uplifting discussion of T-duality b . 3 and Katrin Wendland 1708.01563 a The Authors. c Conformal Field Theory, Discrete Symmetries, Sigma Models

In their recent manuscript “ , [email protected] . It is an interesting question, whether or not “geometric” symmetries are affected g Mathematics Institute, University ofErnst-Zermelo-Straße Freiburg, 1, D-79104 Freiburg, Germany E-mail: [email protected] Centre for Particle Theory,Lower Department Mountjoy, of Stockton Mathematical Road, Sciences, Durham University DH1 of 3LE, Durham, U.K. b a Open Access Article funded by SCOAP symmetry surfing the moduli space of K3 theories doKeywords: not differ from their lifts. ArXiv ePrint: extra phases. If doomed toof fail, then in some cases,by the these lift findings. of Inof the elementary present linear note, algebra: weare answer “geometric” not this symmetries doomed question of to in fail. toroidal the conformal Consequently, negative, and field by in theories means particular, the symmetry groups involved in vey and G. Moore haveconstructions reevaluated a as mod an obstruction two condition tomal appearing the field in description theories asymmetric of by means orbifold certain of symmetries“doomed automorphisms of of to toroidal the fail” underlying confor- condition charge lattice. determineslift to whether The a relevant or symmetry not in such the a corresponding lattice toroidal conformal automorphism field theory without introducing Abstract: Anne Taormina Not doomed to fail JHEP09(2018)062 ], 4) , 25 , 24 = (4 N ], this raises 27 , 23 ], it is shown that vice 26 ]. The Torelli theorems for 11 – 1 = 6, where the charge lattice may be c = c 4 – 1 – ] by J. Harvey and G. Moore, which show that ] allow a description of the symmetries of these 26 17 – 13 ]. 19 , 18 6 ], on the one hand, and of toroidal conformal field theories [ 23 – Symmetries are a driving force in many areas of mathematics and theoret- ] and K3 surfaces [ 20 12 More precisely, this discussion concerns symmetries of a toroidal conformal field theory At the basis of this approach lies, roughly speaking, the description of the moduli spaces lattice which preserve the parametersymmetries point of in at the least moduliother double space, words, the but the charge lattice which order of can suchthe for a only conformal theory the lift field does to corresponding not theory. fully There conformal capturehowever the are acts field symmetry non-trivially also group theory. on cases of some where winding-momentum In a fields lift associated of to invariant the charge same order exists, which directly related to thesome lattice interesting of questions odd about the integral traditional cohomology discussions of of symmetries the for target these models. [ which induce an actionpoint on in the the Grassmannian underlyingversa, description charge for of certain lattice the toroidal fixing conformal moduli the field space. theories, respective there In parameter exist [ automorphisms of the charge toroidal and of K3 theoriesmore in important terms are of the suchthe recent lattice symmetries results automorphisms of is [ toroidal notin immediate. conformal terms field The of theories theirsuperconformal cannot, action field in on theories general, at the be central underlying charges fully charge described lattice. Applied to toroidal extension of the description of thecomplex moduli surfaces spaces in of terms for hyperkählerstructures of the Grassmannians, respective It which are is modelled thus on the natural secondintegral to cohomology. expect cohomology the to properties allowThough of for well automorphisms rooted a of in the generalization mathematics, underlying to lattices the the of very conformal description field of theory general setting. symmetries both of targets are complex surfaces [ of K3 theories [ on the other, ascohomology Grassmannians of that the are complex modelled two-dimensional target on manifold. the even This or may odd be viewed part as of an the total of Mathieu Moonshine andcomplex its tori relations [ tocomplex K3 surfaces theories in terms [ ofogy. automorphisms Inspired of the by respective thesethe lattices theorems, investigation of the integral and cohomol- use classification of of lattice symmetries automorphisms of has superconformal been field extended theories to whose Introduction. ical physics. They are, in particular, central in the investigation of the recent phenomenon Contents 1 Geometric symmetries of K32 theories Not doomed to fail JHEP09(2018)062 , . ], ]. 6 23 26 The 2 ]. 23 = 6 gives ], and in a c -orbifolding, 2 29 , ]. The former = Z 28 c 33 , symmetry surfing 19 , is the group which, by 18 24 M ] and that allows to combine of 34 8 A ]. This map is described in terms : ], and it had been taken into account 4 2 23 Z 19 ]. Hai Siong Tan used a similar approach 31 , – 2 – 30 -orbifold conformal field theories. By constructing 2 Z ] provides the first piece of evidence that the relevant 8 -orbifolding procedure thereby induces a map between the from distinct points in the moduli space of K3 theories [ 2 Z 4) superconformal field theory at central charges , ]. However, in this note we come back to a remark made by J. Harvey 26 1). The latter phenomenon of “non-trivial phases” is compatible with the = (4 − N Both phenomena could potentially cast some doubt on traditional descriptions 1 ] to determine consistent asymmetric orbifold group actions. The condition can be ], we have shown that the maximal subgroup We are not able to predict the full scope of consequences for Mathieu Moonshine that These findings could well be of direct relevance for the discussion of Mathieu Moon- 9 We thank G. MooreNote for that emphasizing at this this point, point one to needs us. to work on a 2: 1 cover of the moduli space of toroidal superconformal 32 1 2 this maximal subgroup, our work [ field theories in ordereq. to (1.17)]. keep the target space orientation, as is required for the resulting K3 theories [ The latter is ageometric technique symmetry that groups we haveIn proposed [ first inmeans [ of symmetry surfing, combinesof all complex geometric symmetries tori induced in by thethe the corresponding symmetries leading order massive representation which contributes to Mathieu Moonshine, for the resulting K3 theories. Aof failure a of toroidal a theory lattice may automorphism descend to to fully the capture corresponding amay K3 symmetry follow theory. from [ and G. Moore in the first version of their recent paper, addressing rise to a K3respective moduli theory. spaces The whichof was the determined underlying lattices inodd of [ and integral cohomology, the and even it integralsymmetries cohomology requires of of a the complex transition underlying two-tori between toroidal by the conformal means of field triality theories [ thus induce symmetries of lattice automorphism is doomed. shine, independently of theniques general in doubts describing that symmetries theyany of may toroidal cast conformal on field the theories. use Indeed, of by lattice a tech- description of symmetries of toroidal sigmain models previous [ discussions ofphenomenon, symmetries which in affects conformal thethis field relevant form groups theory before. of [ symmetries,elegant, The has as criterion not it for been makes at it discussed least a in one simple of matter these of two linear phenomena algebra to to occur check is whether very or not a given in [ traced back to thepoint: properties of whenever there involutions exists on aunder charge the vector charge such which has lattice an an which involution, oddexists, fix scalar the or product the lift there with parameter its are is image invariant charge “doomedmultiplied vectors by to whose ( associated fail”, momentum-winding i.e. fields either are only a lift of order four of symmetries in conformal fieldfor theory. a Whether or given not latticesimply either dubbed automorphism of the these is “doomed phenomena encoded towith occurs fail” different by condition. interpretation equation by This K.S. (2.17)slightly condition Narain, different had of M.H. context previously the Sarmadi by been and manuscript J. stated C. Lepowsky [ Vafa [ [ vectors. JHEP09(2018)062 – is 37 The 3 of toroidal ] are doomed 35 = 6 with proper , c geometric symme- geometric symme- 9 (3)-point for two free , ] and that are thus su 8 is devoted to a more , 6 1 34 , geometric symmetries 9 , 6 doomed to fail by the uplifting . We describe the properties of geometric symmetries not – 3 – geometric symmetries ], the cohomology of the chiral de Rham complex of [ 36 , 9 = 4 superconformal algebra at central charge N ]. thus solely means that such symmetries leave invariant a geometric , section 4.1]. We thank G. Moore for this comment. 11 , 26 10 ], yielding additional evidence in favour of the proposal of symmetry surfing. ] have important implications on the discussion of symmetries of K3 theories, 35 ], we need to require compatibility of the symmetries in question with a large 26 geometric 43 , section 4.5]. However, in this note we use plain scientific arguments to prove that needs to be treated with great care. It refers to symplectic automorphisms of finite of K3 theories. In fact, we show more generally that The remainder of this note is divided into two sections. Section We emphasize that symmetry surfing has not been proved to explain Mathieu Moon- Whether or not the symmetries that are relevant for the works [ A lower-dimensional toroidal example is a reflection in a simple root in the 11 3 ] is expected to model this “space of generic states”. In identifying the cohomology of detailed discussion of ourthe notion symmetries of that enter the symmetry surfing proposal of [ bosons, discussed in [ in [ the symmetry surfing programme ismetries not at jeopardized by the misidentification level offindings of geometric of sym- toroidal [ conformal fieldand theories. thus on Nevertheless, the we broader expect programme. that the tations of the “small” multiplicity spaces of everyrepresentations massive [ representation, i.e. without the occurrence of virtual shine, so far, and that this proposal may still ultimately fail, as is extensively discussed theories [ volume limit. We do sothe by model. requiring that Recently, they the leaveof idea invariant a generic of geometric states explaining interpretation has Mathieu of been Moonshinethe further chiral by substantiated de means by of Rham the observation such complex that a for the space K3 cohomology surfaces of indeed decomposes into irreducible represen- inseparably connected with thein idea all that K3 there theories,As is which a already bears space predicted all of in42 the states [ structure that that generically isthe exist chiral relevant de to Rham Mathieu complex Moonshine. with the large volume limit of a topological half-twist of K3 order that also leave invariantthis the B-field, notion viewed excludes as the aclasses, real-valued identification thus two-form. of excluding In some a particular, symmetriesadjective B-field that with are its certainly in shiftsinterpretation. the by realm In integral of the cohomology geometry. context of symmetry surfing, our notion of means of elementary linear algebra:properties symmetry surfing of is lattice automorphisms.closely tied It to is the important facttries to that symmetry appreciate surfing that only thisconformal proposes field statement to theories is combine are nevertries doomed to fail. The very definition of Our arguments have, in the meantime,and been H. Paul vastly [ generalized by M. Gaberdiel, Ch. Keller to fail is an important questionvery for elegant our “doomed programme. to In fail” this condition, note this we question show that may thanks be to answered the in the negative by representations of Mathieu Moonshine might arise intrinsically from conformal field theory. JHEP09(2018)062 - 0, 24 ) a Z M X, -graded ], which , V > ( 2 4 R Z 43 H ∈ ∈ V ] arise precisely υ in the sense that ), 35 , Z -module. The latter 8 24 X, ( M 0 ) of K3 is equipped with H , its volume R geometric X ∈ X, 0 ( ∗ υ = 6, whose diagonally H c . graded representation of the “small” ]. However, in addition one expects a 2 7 . This should yield the corresponding Z 24 × M 2 ]. Indeed, a K3 theory may be specified by Z 23 – 4 – , ). Denoting by 20) that is induced by the intersection form. A R 22 ]. , -orbifold conformal field theories on K3. We suggest 32 X, 2 ] at central charge ( symmetry surfing Z ]. To be compatible with such a large volume limit, the 2 44 H 10 ]. ⊂ ). Here, the cohomology ], we expect such a space of generic states to arise as the large 22 R may then be uniquely specified by an oriented, positive definite 36 , X X, of signature (4 ( with their strongly restrictive modular properties. The existence might then be explained by means of certain symmetry groups of 2 11 ·i , , H 24 is devoted to the proof of our claim that geometric symmetries of 9 h· -module with all this structure. 2 M ∈ ]. In particular, the properties of the underlying symmetries of toroidal 24 ], predicts the existence of a , proposition 3.7 and definition 4.1] indeed carries the structure of a super B M -module was proved by T. Gannon [ 1 35 , 38 24 8 M -field B , section 5], for example). According to A. Kapustin, it thus may be modelled by 9 To characterize such symmetries more specifically, let us recall the description of the As explained in [ Our quest for an explanation of these phenomena has led us to propose that the in [ = 4 superconformal algebra of [ data that determine a hyperkählerstructure onand a its K3 surface the scalar product hyperkählerstructure on three-dimensional subspace Σ symmetries that are relevantthey to are Mathieu induced Moonshine by must geometricthe be symmetries volume. of More the precisely, underlyingof we K3 our require surface, theories these independently according symmetries of to to [ fix a geometric interpretation moduli space of K3 theories, following [ volume limit of aX topological half-twist of thethe space cohomology of of states of theaccording our chiral to K3 de [ theories Rham (denoted complexvertex operator of algebra, the see underlying also [ K3 surface [ as such from the spacesthat of the states of action of K3 theories on thisspace. space The of latter generic is states, our proposal combined of from distinct points of the moduli has not been constructed, soexistence far. of Neither an has a satisfactory explanation been found formodule the in question should ariseis as common a to subspace all of such the theories. spaces of The states representations of constructed K3 in theories [ which character yields the complex ellipticnishes genus a of representation a of K3twisted surface, the twining and Mathieu genera which group simultaneouslyof fur- such an compatible structure of a super vertex operator algebra on this 1 Geometric symmetries ofThe K3 theories MathieuY. Moonshine Tachikawa [ phenomenon,N discovered by T. Eguchi, H. Ooguri and conformal field theoriesdiscussed. and their Section inducedtoroidal actions conformal on field the theories,doomed in respective to particular charge fail. those lattices that Wediscussion are enter remark around symmetry that equation surfing, an (4.54) are alternative, of not just [ as immediate proof follows from the relevant for [ JHEP09(2018)062 ) d = R i (1.1) (1.2) ], are X, , υ ( 0 ∗ υ 35 , h H 9 • , section 1]. , , 9 , eqs. (A.3)– 0 r 8 d,d , footnotes 18, , p 33 9 6 R · ) with by means of the r = 4, with B-field Z . ⊂ ∗ p ) d d is a lattice of rank − X, υ ( 0 l R Λ d . 4 Λ, p ) / R · ⊕ r i H d l ∗ p ⊂ R gp ⊕ on the charge lattice is Λ 2 ; with ( . l must be invariant under our ) , eq. (1.11)], [ B,B g ∈ d h Z o ) = υ gp 23 ) 0 r R Λ ] are thus given by linear maps p − X, ; ∈ ( 0 l 26 µ, λ 0 V p ( λ ( ) := ( matrix, Λ | H ∀ p • ( ) d must also be fixed, see [ + ) g Z λ r × p B B ∈ − ; d l λ + p · 0 e Bλ υ ): µ is equipped with the scalar product – 5 – | e − B . Thus d , d 0 µ ∪ ; R R υ ):( λ ⊕ Γ(Λ ∈ e . The induced action of B Σ + d , µ ∈ ], the symmetries in question in particular induce lattice and e ∈ R Bg ) which is generated by n ) υ e 18 r . It follows that the vector Bλ σ R Γ(Λ = = p =

V ; − υ l ∗ ∈ e X, B i p d,d ( µ Λ ) g ) which leave the above four-dimensional subspace of ∗ ( 0 r R , section 4.1.1]. Z p 2 = ( H 1 ; B, σ 0 l √ 33 p p X, ∀ ( ( − h ∗ , ) σ H r , that is, · ) = p we denote its dual after identification of ). The geometric symmetries of the toroidal superconformal field theory ; e Λ = Λ and B l d g , p R d, d ( ∀ is given by a real, skew-symmetric ⊂ Γ(Λ , eq. (3.3)], and e , we require it to fix the geometric interpretation, i.e. we require that the induced B ∗ 26 ) with d , section 4], and [ ( but solely depends on a choice of complex structure, in all our works we have been 8 All the symmetries that have been used in symmetry surfing, so far [ O V . Here, ∈ e For concrete examples relevant to symmetryWe emphasize surfing, that the the reader respective is symmetrynot referred groups impose to may be [ any non-abelian, additional and difficulties. that this does of signature ( that might potentially beg affected by the findings of [ where we use the(A.5)], standard [ conventions as for example in [ Euclidean metric The corresponding charge lattice then is induced from symmetries ofthe underlying Kummer complex surfaces torus. thatgeometric In in other interpretation words, turn of any descend our suchB symmetry from toroidal is theory symmetries given on of inand some terms by of torus the Λ of even more restrictive on thegeometric symmetries that enter symmetrylattice surfing. automorphism To fixes call both a19], symmetry [ As is explained, for example,automorphisms in [ of invariant, point-wise. To be compatibleindependently with of a the large volume value limit, of lattice this automorphisms. property must Since hold our large volume limit is not only independent of the value 1, the K3 theorydimensional in subspace of question is uniquely specified by the positive definite oriented four- generating pair of vectors for the hyperbolic lattice JHEP09(2018)062 ) 1 1.1 (2.2) (2.1) cyclic Z 2 ) (2.3) 0 , symmetry- ) as in ( is even and ∈ λ e 1 B ` , , and a lattice − gλ 0 , and we find · d,d T µ e λ R g Bλ ) of the corresponding e − = B ⊂ ` 0 g . 1 ) = 2 ( − p · , µ ) 0 g ( ), Γ = Γ(Λ 0 r = 0 λ d γ λ ) p ( = r ] is solely testing the λ • + ) g − = 2. As explained in section O g e 0 p Bλ ; 0 26 . e ` l B ∈ λ Z p g ], J. Harvey and G. Moore find − ) e l ( 2 Bλ g g e · µ B 32 / e 0 ∈ ( − g B. – = g ( 0 λ g ) ): · · r µ p 28 doomed to fail. e − ) = ( 0 ( ( + g B λ p 2 0 , λ 2 ( = 1 λ `/ = − γ √ not ) . + γ 0 0 l e . Then, reevaluating obstructions previously Γ(Λ B g = 4 we learn that the symmetries that have e • ` Bg = g gλ gλ gλ . We thus have 0 ∈ p d ( − · · · . In other words, the “doomed to fail” condi- – 6 – · ) ∗ ) e µ µ 0 Bg = λ λ we thus have Λ Γ: , p + + T = ∗ e − ) Bλ ∈ ) ∈ Λ λ e e ) B B gλ gλ Γ: − µ r e has order g ∈ g · · Bλ − 0 : p ( ∈ 0 0 0 d µ ; γ , eq. (2.17)]: such a lift is doomed to fail if l − µ µ p to lift to an automorphism of order R e p Bλ ∃ ) 26 T γ 2 µ, µ ∈ g . − = = Λ and 0 λ, µ 2 = ( = ( Λ = Λ, g p g + ∈ ) = ( ∀ λ, µ p λ, λ ( ∀ Λ and e gλ Bλ + γ ∈ • − 0 0 e Bλ p µ of Γ which fixes the parameter point of the theory, i.e. which acts on the ). Assume that ( − d 2 γ , and that λ, λ ( 1 ) we furthermore assume that µ e √ B O ( 2 − 1.2 ∈ = 1 of a given symmetry group. However, as noted at the end of section √ r p = ) does not hold. In particular, at ∀ = T , g l e 2.1 p B g We remark that the “doomed to fail condition” of [ -invariant charge vectors [ γ tion ( been relevant for symmetry surfing, so far, are subgroups surfing does involve non-cyclic, in fact even non-abelian symmetry groups. Actually, all In particular, since by assumption, with arbitrary For charge vectors with In particular, we have To show that the geometric symmetriesto of fail, toroidal we conformal may field thus theoriesabove, assume are without by not loss ( doomed of generality that with discussed in different interpretationsthe following or obstruction contexts for [ toroidal conformal field theoryto that leaves invariant winding-momentum fields associated Consider a toroidalautomorphism conformal field theorycharge with lattice charge by means lattice of Γ 2 Not doomed to fail JHEP09(2018)062 . That JHEP G 3, K ] arise in non- has a lift to a 26 G , of our symmetry Commun. Num. g , , eqs. (A.5)–(A.11)]), T 24 26 7→ M . g 0 , . (see [ 0 ]. b G λ · G p, p µ surface and the Mathieu group ∈ ε 3 1) −→ ]. 2 SPIRE ]. − K = IN G } , g ) 1 1 ][ 0 := ( g p SPIRE ( SPIRE Mathieu moonshine in the elliptic genus of Mathieu twining characters for 0 IN IN , γ ) thus implies ][ ) ][ d p, p p ( −→ { ]. for all ( ε O γ 2 Γ – 7 – Notes on the g 1 ∈ ε × g g T SPIRE dyons and the Mathieu group IN :Γ = ] for the classical results). Indeed, with notations as ε Γ: arXiv:1004.0956 ][ 2 [ = 4 g ∈ 48 Γ: doomed to fail and that are discussed in [ T – 0 ), which permits any use, distribution and reproduction in N ◦ ∈ 45 1 arXiv:1005.5415 0 arXiv:1008.3778 as above, g are [ p, p [ T ∀ γ p, p (2011) 91 ∀ surfaces, 20 3 CC-BY 4.0 . K (2010) 623 b arXiv:1006.0221 G (2010) 062 This article is distributed under the terms of the Creative Commons [ 4 ∼ = 10 , appendix A], this means that a lift G 26 Exper. Math. JHEP exists which obeys ] to us very early on, and for very useful discussions. This gave us the chance , (2010) 058 24 3, ), one may use the cocycle 26 G Theor. Phys. 09 K M M.R. Gaberdiel, S. Hohenegger and R. Volpato, M.R. Gaberdiel, S. Hohenegger and R. Volpato, T. Eguchi, H. Ooguri and Y. Tachikawa, M.C.N. Cheng, 2.3 [3] [4] [1] [2] References right context. Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. We thank J.work Harvey [ and G.to Moore set for the record communicatingWe also straight a thank quickly, an regarding preliminary anonymous symmetrynumber referee version of surfing for of carefully questions not reading whose their being the answers manuscript doomed will and to certainly for help fail. raising the a readers to put our work in the In terms of [ group thus yielding Acknowledgments For a geometric symmetry on the charge latticeoperators Γ (see, which for governs example,in ( the [ operator product expansions between vertex abelian global symmetry groups of specialThis conformal field raises theories with the enhanced symmetry. symmetry question group of of whether thethis every respective is geometric conformal indeed field symmetry the theory case group which follows is from isomorphic the to existence of an invariant 2-cocycle examples of symmetries that JHEP09(2018)062 B , 7 B , Proc. ] 894 , (1977) B 694 s 3 3 Commun. (1981) 145. K , 32 K . 42 Nucl. Phys. ]. Phys. 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