JHEP05(2020)146 4) , = 1 ] the N 41 Springer = 12 The- April 2, 2020 C May 27, 2020 April 29, 2020 : : : Received Published 4)

JHEP05(2020)146 4) , = 1 ] the N 41 Springer = 12 the- April 2, 2020 c May 27, 2020 April 29, 2020 : : : Received Published 4). Accepted , known as the sextet group. = 12 CFT explored in detail c 24 M = 4 superconformal algebra. The N Published for SISSA by https://doi.org/10.1007/JHEP05(2020)146 b [email protected] has led to the study of K3 sigma models with large , 24 M 1) is larger than that preserving (4 , . 3 and Gregory W. Moore ]. The present paper shows that in both the GTVW and a 2003.13700 41 The Authors. 1) superconformal symmetry suﬃces to deﬁne and decompose the elliptic c Conformal Field Models in String Theory, Conformal Field Theory, Discrete

, Special conformal field theories can have symmetry groups which are interest- , [email protected] = 24 two-dimensional conformal field theory (CFT) constructed by Frenkel, Lep- c Enrico Fermi Institute and Department5620 of Ellis Physics, Ave., University Chicago of ILNHETC Chicago, 60637 and Department of126 Physics Frelinghuysen and Rd., Astronomy, Piscataway Rutgers NJ University, 08855, U.S.A. E-mail: b a Open Access Article funded by SCOAP symmetry group preserving (4 Keywords: Symmetries ArXiv ePrint: (The sextet group isRamond-Ramond also known sector as of the thewhich holomorph GTVW in of model the turn is hexacode.) leadsMoreover, related to Building to (4 a on the [ role Miracle forgenus Octad the of Generator Golay a code K3 as sigma a model group into of characters symmetries of of the RR states. ories the construction ofquantum superconformal error correcting generators codes. can Thegroups be automorphism in groups understood of the via these CFT codes thesupercurrent preserving lift of theory to the the of symmetry GTVW superconformal modelimplies generators. our the result, symmetry In combined group the with is case a the result of maximal of the subgroup T. Johnson-Freyd of by Duncan and Mack-Crane.genus The Mathieu and moonshine the connection Mathieusymmetry between groups. group the A K3 particular elliptic K3superconformal CFT symmetry with was a studied maximal inand symmetry beautiful group Wendland work preserving by [ (4 Gaberdiel, Taormina, Volpato, Abstract: ing sporadic finite simpleof groups. a Famous examples includeowsky the and Monster Meurman, symmetry and group the Conway symmetry group of a error correction Jeffrey A. Harvey Moonshine, superconformal symmetry, and quantum JHEP05(2020)146 24 27 28 34 44 28 diag diag 40 43 25 /Z /Z 6 6 38 S S : : 6 6 16 18 14 20 6 = 1 supercurrent 4 N 9 – i – 21 34 13 21 6 ) within SU(2) Q 24 47 P 3 22 23 47 4) supersymmetry within Spin(4) 1) supersymmetry within Spin(4) = 4 supercurrents to codes , , defines a superconformal current 37 22 = 1 supercurrent within a group that includes parity- N 13 P N 11 5 1 4) superconformal algebra in the GTVW model , reversing operations 5.4.1 Stabilizer5.4.2 of (4 Stabilizer of (4 B.1 Superconformal algebras 7.1 Relation between conway and GTVW superconformal currents 5.4 Extending the automorphism groups to include left- and right-movers 5.5 The “new”5.6 twined elliptic genera Stabilizer of an 4.4 Evading a no-go theorem 5.1 The stabilizer5.2 group within Further symmetries5.3 of Ψ: lifting The the stabilizer hexacode group of automorphisms Im( 4.1 Statement of4.2 Mathieu moonshine The Mukai4.3 theorem Quantum Mukai theorem 3.1 The GTVW3.2 model (4 3.3 The hexacode3.4 representation of the Why the3.5 image of The relation of the 2.1 Codes2.2 related to the Hamming The code hexacode2.3 The hexacode2.4 and the Golay Quantum code codes A The automorphism group of theB hexacode and Supersymmetry the conventions even Golay code 6 RR states and the7 MOG construction of Superconformal the symmetry Golay and code QEC in Conway moonshine 5 Symmetries of supercurrents 4 Relation to Mathieu moonshine 3 Supercurrents, states, and codes Contents 1 Introduction 2 Codes and error correction JHEP05(2020)146 3 48 K = 24 Monster CFT c in the elliptic genus of 24 ) M 5.18 ( ] and the discovery of Calabi-Yau com- 1) superconformal symmetry might shed 55 , – 53 – 1 – = 12 Conway theory can be used to describe a ] and the multitudinous possibilities of (alleged) = 24 CFT underlying Monstrous Moonshine can c 90 ]. These sigma models provide a compactification c 35 ] have led to the idea of a huge landscape of string vacua and 26 50 3 sigma models [ K ] of string theory certainly involved beautiful and exceptional structures 10 SU(2) × ], the discovery of the heterotic string [ = 12 SCFT with Conway symmetry can be used to describe compactification of type 51 Our motivations for the investigations in this paper were multifold. We wanted to c SU(2) understanding of symmetries of conformal field theories and stringunderstand models. the origin of moonshineand for hoped the that study sporadic of group symmetriessome preserving light (4 on this puzzle. We were also intrigued by our realization that error correcting less well understood butelliptic closer genus to of physical realityof have superstring been theory observed to in sixdualities the spacetime in study dimensions four, of and five the playby and a a six central spacetime small role dimensions. in butUmbral) various Despite devoted moonshine string ten community phenomena years of remain of physicists persistent mysterious, and effort hinting mathematicians at the an Mathieu important (and gap in our the II superstring theory toand two dimensions. an Moreover, anti-holomorphic the version holomorphic compactification of of the the heteroticnot string a to good description two of dimensions. the Of real world. course two More recently, dimensions moonshine is phenomenon that are of view of thein second the class landscape of of physicists string andmatical structures. compactifications are that thus Certainly involve interested among beautiful in theseare and special studying amongst exceptional points special the mathe- the points most onesbe associated beautiful. used to moonshine to The describe a compactification of the bosonic string to two dimensions. Similarly, Subsequent developments involving newstanding constructions of D-branes of in Calabi-Yauflux string spaces, compactifications theory the [ [ under- no clear principle thatuniverse. would The select authors one of special the point present in paper this are landscape philosophically inclined as towards describing the our point damental laws that governstructure. it are String based theory ontion some [ provides beautiful evidence and forpatifications exceptional both [ mathematical points ofand view. connections between Anomaly mathematical objects cancella- and the demands of physical consistency. 1 Introduction Physicists who wonder abouttwo the classes. ultimate structure Theensemble of first of our class possible universe believes universes. can that be The our divided second universe into class is believes chosen that at our random universe from and a the huge fun- D Quaternions give a distinguished basis for the (2; 2) representation of C Proof that there are no nonzero solutions of JHEP05(2020)146 ] 41 ]. As ]. We but we 41 ] and of 98 3 elliptic quantum , 24 18 K 97 ]. We point M 4) K3 sigma , , 35 85 1) superconformal , 1) superconformal sym- = 12 Conway Moonshine , c 4) preserving transformations , ]. There are also code VOAs 49 , the hexacode and the relation of , 4 F 23 = 4 superconformal algebra [ N = 1 superconformal structures on a large class – 2 – N ] 67 1) sigma models smoothly connected to (4 , ]. In [ 65 – 62 in understanding the structure of superconformal field theories is new. ]. However as far as we are aware the role we have found for and the elliptic genus of K3 sigma models is based on a decomposition 24 83 , M ] but there are many subtleties in the analysis which have kept us from 77 , 21 69 , 25 The outline of this paper is as follows. Following this introduction, the second section Throughout the text we indicate several directions where further research might be Connections between classical error correcting codes and CFT have a long history. 1) sigma models see [ , sporadic group of the elliptic genusout into that characters this of decomposition the metry depends and only that on the thesymmetry group is existence of in of general symmetry (4 larger transformations than preserving the possible (4 groups of (4 these to the Golay codequantum error via correcting the codes Miracle anding Octad highlight a code Generator particular that (MOG). one will We Qbit alsomain play quantum results. briefly a error We correct- discuss role recall the in GTVWare our model, related later to and quantum explain analysis. codes. that In theexplains Section supercurrents section four the in reviews three this some relevance model we aspects to present of this Mathieu one moonshine paper. of and our The Mathieu moonshine observation regarding the of VOA’s were classified.connections to If quantum error-correcting the codessee main in if message these that of examples. turns out this It to will paper be be is the interesting case. correct,collects to a there number should of be useful facts regarding the field informative and useful. In additionof to these, what our is work the clearly raises modulimodels. the space In interesting of question particular, (4 oneTheorem would like of to Gaberdiel, know Hohenegger, if(4 there and is Volpato an in analog this of context. the Quantum For Mukai some literature on CFT are linked tosuch special a quantum connection error correcting forstudied codes. the in superconformal We [ suspect generatorpresenting that such in there an the is analysis version here. also of the Monster CFT error correcting codes Our study of codes inquantum the error GTVW correcting model codes providesplays play a a important nice central roles. example role where In in bothof particular understanding superconformal classical the the generators and in classical symmetries both of hexacode the that GTVW model model while and the the construction Many special lattices cancourse be from constructed a from classical latticespecial error one codes correcting can codes and constructwhich [ special a utilize VOAs, lattice classical see VOACFTs codes leading for [ to to example embed a [ nontrivial connection CFTs between into tensor products of simpler will henceforth refer to this modelbut as elements the of GTVW the model in modelwill recognition were become of actually the clear analysis constructed in in andwere [ discussed able later to earlier sections, enlarge [ the wegenus possible still beyond symmetry do those transformations that previously not can considered. understand act on the the origin of codes appear naturally in the analysis of the special K3 sigma model discussed in [ JHEP05(2020)146 ]. ]. A 91 (2.2) , n, k 81 , is a linear 50 matrix such , n q F n 18 , × 7 k C ⊂ is a set of vectors in : they are words with n . One can always find a q F ) as ] for mathematically inclined which is a k 46 G , codewords of length 32 of a linear code is the dimension of C k ] and find a role for the Golay code in (of length ] (2.1) 41 m are called ,A k identity matrix. We use the convention that mG C I with coefficients in k = 1 supercharge. Four appendices summarize = – 3 – G = [ × c ] as well as the Magma Computer Algebra system , regarded as an alphabet. If k N G q 47 F the . The main examples we encounter in this paper are k classical q-ary code I = 4. The dimension under grant PHY 1520748 and from the Simons Foundation q 1 ) encode the message n linear code such that the generator matrix takes the form = 2 and . The vectors in the set ) matrix and ] for physicists. C n q k q ] can be specified by a generator matrix F − is spanned by the rows of 70 , ]. Section five uses the relation of the supercurrents to codes to study the n C n, k (of length ( 15 40 c × be a prime power. A k q a A Let For some reviews of moonshine we can recommend [ Any opinions, findings, and conclusions or recommendations expressed in this material are those of the 1 letters drawn from the finite field author(s) and do not necessarily reflect the views of the National Science Foundation. with code words linear codes with the linear subspace spanned bycode the of codewords. type Such [ codesthat are the code said to becode of equivalent to type [ ples that will play a role in our laterthe analysis. vector Useful space references includen [ subspace it is called a 2 Codes and errorCodes, correction both classical andgive quantum, the play basic definitions a of central classical role and quantum in error our correcting analysis. codes and In provide exam- this section we at the Stanford Instituteand for Shamit Theoretical Kachru Physics, in andfor particular, GM finite for would group hospitality. like theory We in towhich have our thank made analysis was the use [ made SITP, of available theedges support GAP through from package a the grant NSF (#399639). from GM is the supported Simons by Foundation. the DOE under JH grant acknowl- DOE-SC0010008 to Rutgers. C. Hull, T. Mainiero, D.conversations and Morrison, correspondence. S.-H. Shao, We especially A.A. Taormina thank and Taormina, M. and K. K. Wendland Gaberdiel,We for T. Wendland gratefully helpful for Johnson-Freyd, acknowledge numerous useful theGrant and hospitality No. important of remarks PHY-1066293) on the where the Aspen this draft. work Center was for initiated. Physics Part (under of NSF this work was also done readers and [ Acknowledgments We thank D. Allcock, A. Banerjee, D. Freed, I. Frenkel, D. Harlow, T. Johnson-Freyd, symmetry groups of theWe GTVW focus model on the thatthis particular preserve model, K3 various sigma namely superconformal model therebilizer structures. of is group [ an of isomorphism abackground between material, left-right supersymmetry the symmetric conventions Golay and code some technical and details. a certain sta- classified in [ JHEP05(2020)146 ] for (2.6) (2.7) (2.3) (2.4) p x n, k, d = 4] binary is we obtain , ¯ x ⊥ } C 1 → , x 4 is the number of x . A useful fact is 2 , x ⊥ c 3 C , 1 , x = c 4 . , x C + with generator matrix. [56] 3 [145] , 2 , x F + are simply the elements of the [34] ∈ C} ) (2.5) 1 , c 2 [136] u . , , x , c 2 . [12] 1 .      all , c C ] , x ( k , 4 [235] d − , is defined to be x 2 n c = 0 C ) then a generator matrix for 3] code over + 6= [1256] ,I letters of , 1 ¯ 2 , u 4 tr [246] ,c · 2.1 x k , ¯ , A x ∈C – 4 – + . 2 − : d 1 ,c 1 0 0 00 1 1 1 0 0 00 1 0 0 1 1 00 0 0 1 0 1 1 1 1 1 [3456] 1 n q x [146] , c = [ { F ,      ) between any two code words ∈ 2 H = ) = min . The code is called self-dual if x , c C [236] G [1234] 1 { C ( , = 1 superconformal symmetries in a K3 sigma model we , c d ( = d N ⊥ dim [245] 4] Hamming code which can also detect (but not correct) errors C , , − differ. The Hamming distance of a code is the minimum distance n [123456] 4] whose sixteen codewords can be listed as: 2 , , c [135] is one has the conjugation (Frobenius) automorphism ⊥ and C q ). 1 F . in canonical form the first c m 3.9 G of type [6 =1 ] code with Hamming distance has generator matrix given by eq. ( N S C . The dual or orthogonal code to a code n, k α If we simply drop the seventh bit of the Hamming code we obtain instead a [6 The Hamming code is a binary linear [7 We now give a number of examples of codes that play a role in our later analysis. The (Hamming) distance In the field p is also called the parity check matrix of the code = Here we are usinghave a adopted notation this where specific we choicee.g. list in equation only order ( to the facilitate entries comparison that to are later 1 expressions. for See each word. We in two bits. linear code. Relabelling thea entries code as The Hamming code can famouslybit to detect the and code correct gives any the single [8 bit error. Adding a parity The codes which governstudy the below are closely related to the renowned Hamming code. The Hamming distance isfor often a included [ in the specification of a code by writing2.1 [ Codes related to the Hamming code places at which between any two codewords, The dimension of that if H message word q so that with JHEP05(2020)146 + 4 to F (2.8) (2.9) (2.11) (2.12) (2.13) (2.10) for all are the i σ = 0 x to be the additive where the . This automorphism 3 2 σ is defined by taking 1 i x . To define this we first , ∗ 4 2 + 4 F σ F → i , and takes takes 1 x σ } 1 (2.14) 3 σ i → and 1 . + 4 , } 2 F ¯ ω . σ , i 3] spanned by ω ω π . . → , ω ω 1) := [3456] 1) := [246] 1) := [56] x ¯ , ω ω , , , , = ¯ = = 1 = 0 − 1 Q 1 1 0 1 = ¯ = 1 = , x , , , ω ω ω σ ↔ 0 i ¯ ¯ 1 1 0 ω ω ω is a field of characteristic two, so 2 – 5 – { , , , + + + ¯ + ¯ ¯ ω ω ω ω and 4 1 0 0 } → : , x 1 1 , , , ω = F 1 } 1 x 0 1 0 1 4 , , , SU(2) generated by i F { { ⊂ = → { . Since some properties of this field might not be familiar = (0 = (0 = (0 4 1 2 3 1 Q Q F (Frobenius) automorphism w w w 2 is a code of type [6 Z . Note that 2 =4 Z N S ⊕ of this truncated Hamming code will also play a role in describing 2 has a considered as an Abelian group with the + law. As an Abelian group Z =4 4 4 N F F S for + 4 F In our analysis of the GTVW K3 sigma model an important role is played by a group Recall that The multiplicative Abelian group law for nonzero vectors A subcode = 4 supercurrents. , and there is no distinction between with group composition givenwith by kernel matrix given multiplication. by There the is subgroup a homomorphism to homomorphism from theconsider quaternion the subgroup quaternion of group standard SU(2) Pauli to matrices. This is an 8 element group. Explicitly: preserves the additive and multiplicative identities 0 x be the multiplicative identity and We write it is isomorphic to To define theidentity Abelian and group law for addition of vectors we take 0 2.2 The hexacode The hexacode utilizes the field to physicists we briefly review them here. We think of it concretely as the set N JHEP05(2020)146 (2.21) (2.19) (2.18) (2.15) (2.16) (2.17) in our π of h ! 4 0 π/ i e 2 . 4 1 )Ω π/ − i 0 x ( − )Ω e h with the quaternion group: 2 x 4 ( − . ) (2.20) F h . ! y 3 ¯ : ω / 2 + ∗ 4 1 1 iσ . This mirrors the action of multipli- = Ω 1 1 F 1 x iσ x − 2 − ( 0) = 1 )Ω = Ω iσ

h x = ^ = ) )Ω )Ω → x, ω 2 ( = x x 1 ( is an order three rotation around the axis 1 z ! ( ( ! √ h x, y h h 3 ! ! 1 ( 1 2 → 1 0 i 0 -valued cocycle R − = 0 1 − − ) = 2 0 i – 6 – i 0 y 1 0 0 1 − 0 i − Z Ð ! , x

) = ω → (0 y ) = Ω ( x ) = Ω ) = Ω x ) = ) = ) = ) = h ¯ ω ) ω ω ω (1 + i) (1 + i) xω x (1 (0 x ( (¯ ( ( ω 1 2 1 2 ( h h ( h h h h h h i) i) . − 2 ) = ) = − ) = x x Ω x (1 ω ω (1 − 1 2 ( (¯ ω 1 2 (¯ h h − ), we can deﬁne a = h is the additive identity,

1 1) so it permutes − y 2.15 , which permutes the 3 elements of 1 or , Ω = ω x 1 so Ω . This cocycle has some nice properties: − 4 F = ∈ 3 through (1 cation by It is 1 if Ω is an SU(2) matrix. Its action on Ω Returning to ( We collect here a few useful properties of Ω x, y 1. 2. 1. for Note the second equation immediately follows from the first since In the above There is also a relation of the multiplicative structure of computations: The sequence does not split. We will make a specific choice of section JHEP05(2020)146 6 4 ). F ) we 2.16 1 (2.29) (2.28) (2.22) (2.23) (2.27) (2.24) (2.25) (2.26) , 0 , (0 , ). Then one ) 0 , y , 2 1 x , ( (0 ) , ) ) by Ω and use ( , y 0 1 , x 0 ( 2.20 , , . ) ) ) = 2 3 , y . . 2 x , y ) = (1 2 . x 3 x, y ( + ( x )    = 0 6= 0 1 1 , x y ) ) ) ) so this makes manifest that we have + 2 , a three-dimensional subspace of 2 ) 1 − − x x ) ) ¯ 6 ω ) x, y 3 1 1 , y x , , ω ( , x ω ω , 1 1 1 1 x H (1 ( (¯ 1 ¯ 1 ω ( , x x x, y + 1 3 3 3 − − x , ω , , ( 2 ( 2 ,x ,x ,x 1 1 1 +1 − 1 1 1 2 2 2 y , , , ) = , x 1 1 ( 0 1 0 1 ,x ,x ,x y − − x 1 1 1 , , , x ) = ) = x x x 1 0 0    2 – 7 – , , , ) we simply conjugate ( y ) = , y x, ω = 2 ) := ω = Φ = Φ = Φ + x ( where we also analyze the automorphism group of y x, x ), 1 ( = (0 = (0 = (1 2.23 4 5 6 x,y ( + 3 x x x A 1 2 3 1 ,x b b b ) such that, if 2 x, y x 2.15 6 . ( ( ,x 1 x Φ , . . . , x 1 x depends linearly on 3 ,x 2 we have, from ( ,x 1 x 6= 0 x, y is bimultiplicative: is “permutation invariant” The diagonal values are For Note that Φ We now turn to a discussion of the hexacode To prove permutation invariance ( 4. 3. 2. 5. automatically get a basis A full list ofcomputations) the is 64 shown hexacode in words appendixthe (which hexacode. can occasionally be quite helpful whena doing three-dimensional subspace. In fact, taking ( then uses the symmetry properties of consisting of codewords ( To prove the bimultiplicative propertyzero, we so first we note canthan that assume an all it explicit arguments is check are obvious of nonzero. if the any 9 We argument are cases is not to estabish aware of any better proof JHEP05(2020)146 (2.38) (2.37) (2.30) (2.31) (2.32) (2.33) (2.34) (2.35) = 0 for all i y, x h coordinate and 0 th 4] code is isomorphic i , -valued inner product: 4 so that so that we have a decom- F x 4] binary linear code defined ⊥ 6 in the , H 3 (2.36) . ≤ i, j ,A ) = 0 ) = 1 i . 1 ≤ j y ω ω 6 2 i b (¯ ( 2 1 F ¯ comp x 6 ij π π F = A = 0 6 → =1    i → X is related to the quantum code interpretation i ⊕ H 6 3 to the span of + ¯ 1 j ω ω , i 6 + 4 π 2 := e + 4 – 8 – ¯ 1 , u F ω ω , H i F i : u = 1 1 1 h ∼ = ij is defined by the set of π i : = 1 δ 0 0 1 6 4 x, y u ∗ 6 i π be the vector with a 1 h F 0 0 1 = , H i i ). ) = 0 ) = 1 i 0 1 0 e e j (0 (1    2.7 , b π π i ) that do not induce a bit-flip map to 0 and those that induce e x h ( h to the 64 code words of the hexacode clearly produces the 16 words of a binary do not span an isotropic subspace. Nevertheless, since the hexacode is isotropic π i 4] code and a short computation shows that the resulting [6 e , . It is not hard to check that the hexacode is self-dual: it is maximal isotropic. 6 It is useful to note another maximal isotropic subspace The dual, or orthogonal, code There is nice relationship between the hexacode and the [6 We can make an inner product space with a nondegenerate ∈ H we can try to modify the span of and then the desired We can do thiselsewhere. as Clearly we follows. have Let But the y position into maximal isotropic subspaces: Applying linear [6 to the one defined in equation ( The conceptual reason behind this choicediscussed of below: operators a bit-flip map to 1. We can extend this in an obvious way to such that earlier. To see this we first define a homomorphism of Abelian groups corresponding to the generator matrix JHEP05(2020)146 (2.43) (2.44) (2.45) (2.39) (2.40) (2.46) (2.42) . Each 2 F . 2           F 1 1 0 0 1 0 0 1 ∈ comp 6           α H . So if ij 1) = 1) = δ 4 , , ω F . That is, ω (1 2 (¯ ∈ = + F + g g α ij × x . A 4 4 2 F F 6 array of elements of           → 0 0 1 1 0 1 1 0 . There are four “interpretations” = 0 (2.41) 4 2 × 2 )           24 2 ) ) i F F j 4 2 1 ¯ F ω ] refers to as “even interpretation of , , ω F , )) × to = 0 , u ¯ 6 1 ω i 18 → 4 + 4 , , , ω 0) = 0) = → u ji as a 4 , , , 6 F F h 6 ) 2 A , ω , ω , ω ω x 2 (1 24 2 (¯ F 0 0 ¯ ( ω F + + , , F + , + 0 g × ¯ ω 0 g ij – 9 – × , , , 4 ij ¯ ω A δ F + 4 ]. This is a particularly efficient way of thinking : F + = = (¯ = (0 = (0 , . . . , g 18 :( i           ) ij 1 2 3 g j 1 1 1 1 1 1 0 1 0 δ + u u u f ,           , u i 1 b x h ( ] the MOG was connected to the space of R-R ground states + 1) = 1) = g , , 41 applied to a “decorated digit” in ω (0 ( + + + g ) = ( g g 6 , 6 corresponds to what SPLAG [ will serve as a maximal isotropic complementary space i           + 0 0 0 0 0 1 0 1 u g is in fact a group homomorphism           , . . . , x 1 + ] which can be found in [ g , 20 1 0) = 0) = x , , ( ω + (0 ( f Now we define a group homomorphism The map The MOG is based on a map from digits in + + g g It is very usefulcolumn to is represent the the result vectors of in Note that hexacode digits” and is defined by: of a particular K3connection. sigma model and our later analysis willof elaborate the on, digits and and clarify, these this can be summarized by a pair of maps 2.3 The hexacodeIn and the this Golay section code (MOG) we [ rephrase slightlyabout the the Golay description code. of In the [ “Miracle Octad Generator” then so the span of There are many solutions to this, but a simple one is to take implies that JHEP05(2020)146 (2.52) (2.51) (2.47) (2.49) (2.50) ) as the 6 ) with           x, 0 1 0 1 0 0 0 0           , . . . , even Golay code 1 1) = 1) = , , ω (1 (¯ − − g g 6 )           2 1 0 1 0 1 1 1 1 F p           as the word and ( × 1 0 0 1 ¯ ω ). . We will call it the 4 on the decorated words ( 6 4 ) + = 0 mod 2 (2.48) 2 24 F F ( + 1 0 1 0 x, F ω α 0) = 0) = f x, , , ∈ ∈      ( ω 1 0 0 0 ) (1 + 6 1 1 0 1 0 =1 (¯ ) 6 g X 6 G ⊂ α −      − g , g 0 0 0 0 0 6 = – 10 – vectors and forms a subgroup of the Golay code p ) := 11 1 1 0 0 1 , . . . , x 1 x,           ( x , . . . , x 0 0 1 0 1 1 1 0 0 0 0 0 0 1 − 6           g , = ( 1 x ∈ H ( w 1) = 1) = x which is displayed along the bottom of the table. , , ω ¯ ω (0 ( ω − − 1 g g 0 1 . This is a set of 2 are homomorphisms. , viewing + G + f           1 1 0 1 0 0 0 1           and for an illustration of this construction. We will refer to elements + 1 g . Illustration of the MOG construction of a Golay code word from an even interpretation 0) = 0) = , , ω (0 ( The complement of the set of even Golay code words inside the set of all Golay code The first nontrivial fact is that the image of − decorated words − g g so and we then define words is the set ofwords odd is Golay to code introduce words. the A column good vector way to parametrize the odd Golay code is an index twoand subspace denote of it the by Golaybecause code as decorations. We also denote decorated words as ( of the hexacode word 0 See table Table 1 JHEP05(2020)146 i = as ,Z n is a 6 i H (2.53) (2.54) (2.58) (2.57) E ,Y i X , we have known as the error correcting ∈ E 24 . The holomorph i M A E is an ! 1 H -Qbit Hilbert space − n ) (2.56) ⊂ 0 1 0 ·

C while preserving the quantum 1 ) ] (see also the very informative if, for every 6 i ). The automorphism group of = − ) (2.55) 18 E , )) 6 ϕ . x ( w, A.10 ( . In some cases we will consider codes p P ( ,Z − + C , ij ) x ! α i ( x ( + and the holomorph of the hexacode. (See − = 1 f , . . . , g i 0 − 0 + ) P ϕ 1 j

G = 1 mod 2 ( ) = – 11 – E is a nonzero Hermitian matrix. , + α = † f i 1 as the Pauli matrices x, x ij of error operations ( ( α 6 n =1 PE + − E ) = X α f g H · ,Y x, and w ( = ! + C f w is deﬁned in equation ( · 0 1 1 0 ϕ

p . The hexacode acts by ). In fact the sextet group is exactly the same as the holomorph 6 = 24 .S X M component of : 3 ) and 6 6 th i H can be shown to be exactly the maximal subgroup of The sextet group is nicely described in [ Let H be a Hilbert space. A Hilbert subspace 2 24 Aut( for the definition of the holomorph of a group.) The hexacode itself is 2 and S 6 ∈ A is the projector onto and the set of error operations will consist of tensor products of elements ϕ P ∈ H n ⊗ When the above condition is satisfied it is possible to construct a quantum operation The QECs we consider are constructed as subspaces of the The MOG construction of the Golay code gives a nice connection between the auto- Now, the odd Golay code words are within ) w We thank D. Allcock for explaining this to us. that detects and then corrects the quantum errors 2 2 + C which act on the R information carried by states inthat the can quantum code only detect but not correct quantum errors. ( quantum code with respect to a set where Quantum error correcting (QEC)transmission codes and are processing designed ofsubset to of quantum the detect information. space of and completely Thematrices. correct positive, set non-trace We errors increasing define of maps in a on error the the QEC operations space as ofDefinition. follows: density G sextet group. Wikipedia article on of the hexacode. 2.4 Quantum codes for where morphism group of theappendix even Golay code an Abelian group and itshas automorphism the group structure is 2 described in appendix with JHEP05(2020)146 . . . i } 1 = S ]]. 0 C has G { Q I (2.62) = (2.61) (2.59) (2.60) acting 00000 | S ⊗ I n, k, d I V ⊗ ⊗ X i. Explicitly X ⊗ we will refer to Z ⊗ generated by the ⊗ Z S 1 and i i i . (We have Z k < n by cyclic permutation. . This code is of type S is the intersection ⊗ } } . This leads to the basis 1 i S Z X Q for particular choices of M i 01111 00101 11000 | with S ). Thus we say it is of type [[ I + n V 10100 Z, X,Y,Z i − | i i − | d | H { } + ]. The code distance of a quantum = i Y, Z i stabilized by i 91 11110 10111 00110 E n that is not the identity operator on H Y, E Y, 01001 i i − | i − | i − | | is shorthand for + are obtained from X, i are n-fold tensor products of elements of i X, i 4 XZZXI IXZZX XIXZZ ZXIXZ , 00011 01100 11011 n is embedded in 3 , – 12 – ]], where the double square brackets are used to = = = = G 2 k X, I, 1 2 3 4 as Pauli operators and we define the weight of a 10010 i − | i − | i − | M | XZZXI n, k M M M M a ]) { + I, E i i 88 the subspace of = ]] and has code distance 1, so we avoid this case.) Many important quantum error 01010 11101 10001 S | Q − I, V n, k + − | − | 00000 | { . Note that and 1 4 = 5 n ⊗ 1 ) = ) rather than the identity operator ( G G 2 i C L 0 | X,Y,Z Qbits. The Pauli group acting on a single Qbit has a matrix representation n and summing over all elements of the stabilizer group i the two by two identity matrix. (The quaternion group be a subgroup of 3]] and can be viewed as a stabilizer code with stabilizer group SU(2): I , An important example is the smallest QEC capable of detecting and correcting an Elements of the general Pauli group Stabilizer QEC codes are an important class of quantum codes that have much in com- If the code subspace of dimension 2 contains the group element S acting on 1 11111 ∩ | , S 1 n Here as is usual inon the the QEC 5-Qbit literature space ( An orthonormal basis foror the code subspace canvectors (see be e.g. constructed section by 10.5.6. starting of with [ correcting codes can be constructed as subspaces of the form arbitrary single one Qbit[[5 error, that isfour a elements code with and will be of some use later.) Let if consisting of the Pauli matrices along with multiplicative factors of with G distinguish quantum codes from classicalcode codes is as the in minimum [ weight ofIf a a Pauli quantum operator code in is of type [[ mon with classical linearG codes. In the stabilizer formalism one considers the Pauli group Pauli operator to bePauli the operator number ( of entriesweight two. in the tensor product that have athe non-trivial code as a quantum code of type [[ We refer to these tensor products JHEP05(2020)146 ] 2 q 40 (3.1) (3.2) ]. It is a 8. Strictly 41 and can be I/ α = symmetry with value are in the R 0 (3) target [ L ρ 2 Z / SU(2) 4 × T L turns out to be simply 51 ) (2.63) allowed by the results of [ i q ] shows that there are several ¯ 1 20 is independent of 6 41 ˜ s ⊗ 2 M ˜ V i ⊗ | : F 1 1 8 | ⊗ ˜ V ∈ ] T. Mainiero defines several notions s + V ⊗ x 79 i 1 ¯ 0 ∈S ) ˜ s ⊕ V s, ( i ⊗ | 4]] code which is one-dimensional and is con- 0 0 , ⊕ ˜ | V – 13 – 0 ( 6 where , = ⊗ 2 ≤ 1 0 √ V α 3]] code and an ancillary Qbit as = , ≤ a five-term polynomial in 1 GTVW = , 4]] , = 1.) H is the subgroup defined by demanding that the components 0 , H 6 2 R according to whether the states of lowest [[6 F 4) sigma model with a very specific 1 Ψ × , ] notes that this code state is maximally entangled in that the V 6 2 for all 1 a priori F we will describe the structure of the group in excruciating detail.) 91 x and + . There is also a [[6 0 S ⊂ i α 5.4.1 V L s 0 | 5 = ˜ and is the Spin(8) maximal torus. Reference [ , which is i ⊗ 6 2 α 4 L F s J. Preskill [ X 0 T | ∈ = ˜ s i obey s, L 1 ˜ s | Recall that the SU(2) level 1 WZW model has affine SU(2) s, where a tilde denotes right-movers.to In be the GTVW model we take instead the Hilbert space where of singlet or doublet of thehas global a SU(2) Hilbert symmetry. space The of product states of on six such the models circle therefore given by incarnations of this model andof the one six we SU(2) find most level convenientthe 1 is T-duality an WZW invariant extension point models. of a product (Equivalently, the product ofboth six factors Gaussian at models level at will 1. be The labeled unitary by highest weight representations of affine level 1 SU(2) The GTVW model isdistinguished a K3 (4 because it(GHV). has (In the section maximal groupThe 2 target In this section weMoonshine show that phenomena the can spectracorrecting and be codes. supercurrents rephrased of in several SCFTs terms that of exhibit 3.1 both classical and The quantum GTVW model error for Mainiero’s “GNS cohomology,”mologies while are the similarly quite Poincarépolynomialshas simple. for unusual other In correlation other related properties. words: ho- the state, while maximally3 entangled Supercurrents, states, and codes speaking this istect, an but error not detecting correct,of rather “entanglement any than homology,” single a error Qbit homological correctingmeasures generalization error. correlations code of between entanglement in In many entropy bodynomial that [ which for operators it in can a de- given state. The Poincarépoly- structed from the basis of the [[5 Remark. density matrix obtained by tracing over any 3 Qbits is totally random, and JHEP05(2020)146 ]. R F = 1 48 + L k F 1) − 7], although , ]. A modern 21 ]. See the 2019 TASI 78 , . However, the fields are 74 , 2 )-graded. One grading is 2 F ) (3.3) 73 F z , (¯ 43 ⊕ ˜ X · 2 ˜ in terms of six free bosonic fields 2 F i √ 6 can be identified with ( e we define (anti-) holomorphic vertex = πx ) is ( 6 i } ˜ e ˜ 1 V 3.2 ∈ { ] has also been used to formulate theories of self-dual . . In terms of the original supersymmetric K3 ˜ 3 2 2 WZW model. α , 2. To give an explicit construction we use the – 14 – are simply addition in 1 , s / 6 = 1 ) ) α 1 4 V z R with the sum of NSNS and RR sectors of the sigma ( , P ] shows that 1 distinguishes the NS-NS sector from the R-R sector, 0 × X , · V 41 = 2 i y √ SU(2) = 0 GTVW e y × 4) supersymmetry. The left-moving N=4 supercurrents can H global symmetry of the bosonic CFT, couple to an invertible = , L extensions” of a conformal field theory is that one is defining a 2 1. As we will see, the theory of codes pervades this model, and it V 2 ], and play a crucial role in formulating tachyon-free models [ Z Z 92 , symmetry of the product theory. The resulting theory has a super- 87 2 Z 6. We then note that for In our case, the original bosonic CFT is the product of six SU(2) level and the other is in the (SU(2) -extensions of vertex operator algebras go back to the very beginning of 2 3 ··· x 6 L Z , are left- and right-moving fermion numbers. Thus, the GSO-projected version 2 ]. It has recently been explored and extended further in [ , 4) superconformal algebra in the GTVW model , L,R 100 SU(2) F , = 1 8 = 1 SU(2) WZW models. ∈ I It is useful to note that the space of states ( k This procedure, which in fact goes back to [ , 6 3 I 1 (For simplicity we will dropanalysis cocycle below.) factors Any as linear they combination will of play these nofields crucial operators [ direct gives role anlectures in (anti-) by our Y. holomorphic Tachikawa forthis a recent pedagogical literature. introduction. We thank Shu-Heng Shao for guiding us through Frenkel-Kac-Segal construction of level oneX affine SU(2) operators of conformal dimension 6 3.2 (4 The K3 sigma modeltherefore has be (4 expressed inprimary the fields WZW of language. conformal The dimension currents must 3 be (anti-) holomorphic diagonal global VOA as its “chiralbosonic algebra.” CFT. The GSOWZW models, projection and of the the GTVW− theory model is reproduces the the spin original extension based on the diagonal element superstring theory [ They have also playedinterpretation a of these role “ intheory previous that investigations depends into on Moonshineidentity both a [ conformal nonanomalous structure andtopological spin field structure. theory The sensitive procedure to is spin to structure known as the Arf theory, and gauge the where of this model, thatsix is, the GSO projected K3 sigmaRemark. model is equivalent to the product of we will not make use of that fact. provided by sigma model we canmodel. identify The quantum number respectively. Moreover reference [ expansion since the fusion rules for only mutually local up towith sign: a super-chiral the algebra GTVW (a modelmutual super-vertex-operator is locality algebra) not at in a worst which standard theis CFT modules worth but all noting have an that extension the spectrum itself can be described by a code of type [12 either 0 or 1. One easily checks that the space of fields is closed under operator product JHEP05(2020)146 , i σ i 2 (3.7) (3.8) (3.9) (3.4) (3.5) (3.6) − = i with J Ψ V corresponds to [1236] V 2 is a linear space / − 1 2 . ¯ ¯ Ψ Ψ − 0 V V [1245] G − 1 1 = 2 2 2 − − Q i i , i i i[134] and then we define is an ordered basis for a Qbit in + 0 + i − G = = | symmetry transformation the general ) i 1 2 = −i} 6 V R ¯ ¯ i([246] + [235] + [136] + [145]) ( − − 2 ) Q Q i[145] + [135] + [146] + i[12] c ¯ (2) has matrix representation Q = 4 ( i[156] − 6 2 − − − −i − , c Z su − 6 2 − | ∈ + + i[45] + [35] + [46] + i[2] Z , G i ∈ + + + + + + + + X − P = – 15 – | | | − , which in turn is proportional to i[136] 1 1 = i ⊗ · · · ⊗ | = ¯ 1 2 ¯ Q − 1 Q = + s provide a natural basis. We will find it useful to , s ] = Ψ Ψ i[36] | V ∅ + V V + [ − V G 1 {| i[235] + i[56] + i[34] + [245] + [236] [145] = Q = 1 1 [3456] = − 1 2 2 − − Q i i and the i[246] = = we can write 6 − 1 2 6 Q Q ⊗ ] are easily used to show that the N=4 supercurrents of the K3 sigma ) 2 [13456] + i[1246] + i[1235] 41 ] we identify, C ] + i[123456] + ([1234] + [3456] + [1256]) + i([12] + [56] + [34]) − ( 41 ∅ + ([135] + [245] + [236] + [146]) ∈ ] + [3456] [1] we can express these currents as: s ∅ − B Ψ := [ = i[23456] + [234] + [256] = [ = = i[123456] + [1234] + [1256] 2 1 2 1 3. If , ¯ ¯ Ψ Ψ Ψ Ψ 2 , For reasons that will be explained later, we are particularly interested in automor- The notation here is the following: the integers denote the position of a down-spin in The results of [ To compare to [ = 1 current is proportional to 4 = 1 We will show in thein next a section simple that way this by seemingly a complicated quantum state error is detecting in code fact on governed six Qbits. phisms of the K3 sigma modela preserving unitarity one constraint holomorphic then, N=1 supercurrent. upN to If we a impose general SU(2) and so forth. the tensor product of six up/down spins, all other spins being up. Thus, for example, where model can be writtenof as appendix very special states in the six Qbit system. Using the conventions which a basis of anti-Hermitiani generators of to be the corresponding primary field. of complex dimension 2 identify this vector space withthe the natural Hilbert spin space basis of six Qbits. The basis where, for a single tensor factor, primary field so the space of holomorphic primary fields of dimension 3 JHEP05(2020)146 . .) 6 P ◦ ) into (3.16) (3.14) (3.15) (3.11) 2.60 ) (3.10) , 6 ) . 3 ⊗ ) ) x 2 3 below. y + C . We will denote the ¯ ω + . 5 2 2 in fact defines a group 6 ¯ ω Z x 5.4.1 2 h End(( y + ∈ H ∈ + ω ) can be extended to vectors ) (3.13) . (Again, some readers will 2 1 ) 2 ω Z 6 w 1 x , x , w 1 (3.12) 2.15 ( 3 1 . , y + to construct a rank one projector h x ) w 3 → 1 α y ⊗ + 6 w ) 2.2 ( , y ) , + ω 5 + 4 α h ) = 1 supercurrent 2 ) ω x F x 6 x 2 ( 2 . There is a non-split sequence ( ( is isomorphic to 6 y h N + , w Q Z π + 1 ⊗ ) 6 =1 ¯ → ω 2 Y α ) 1 w ¯ ω 4 ( w (SU(2) 1 6 x , x – 16 – χ ). Z ( Q 3 + , y ) = h 3 x 2 for the quotient by 1 ⊂ y → ) = w ⊗ 2.20 + 2 , w ( Z 2 + ) 1 2 /Z h w 3 Z 2 6 x ( w x y ( h ( Q . (Some readers will prefer to use the notation SU(2) + ) → χ h ) = + 1 then we define is exactly equal to one. That is, 1 2 SU(2) defined in equation ( 1 1 /Z w := ⊗ 6 4 6 6 w ( , x with cocycle ⊂ ( ) F , y 6 3 h Q 2 3 h π ) to lie in the group ) ∈ x Q Q , x ∈ 1 ( , y ) w 2 an important point throughout this paper is that while the group 2 . In the latter case, the cocycle is the 6-tuple of cocycles defined h ( 6 w ) is the center, that acts ineffectively on six Qbits. It is the subgroup → 6 ( h , x , y 6 2 w of the cocycles ( ⊗ 4 h Q 1 1 ( Z := SU(2) ) F x y h .) 1 ∈ : 6 6 ∼ = x ◦ , . . . , x ( h = ( = ( ) 1 )) h Q 6 and write 6 x 2 1 x product 6 w w ( = ( Q ) := w w (SU(2) ( by acts on the space of six Qbits, it does not act effectively. There is a subgroup ). This distinction will be of crucial importance in section as: h Z defines a section of , . . . , h 6 6 ) 2 . If h 1 We now come to a central claim for this paper: when restricted to the hexacode the We may consider The section 6 4 2.20 x F , w on the Hilbert space of six Qbits. Then our state Ψ will be a vector in the image of ( 1 h This fact isw both surprising and significant. To prove it we write two hexacode words in ( cocycle defined by homomorphism: Note that this is( to be distinguished from the cocycles generated by the group elements and given by the prefer to write where of six signs whosequotient product by is SU(2) +1.Similarly, The we group will denoteSU(2) the embedding of six copies of the quaternion group ( Important remark: SU(2) We are going to use theP hexacode discussed in section in 3.3 The hexacode representation of the JHEP05(2020)146 j j 6= = i i ) and (3.24) (3.22) (3.18) (3.19) (3.23) 2.32 that for a 3.4 ) 2 y , ω 1 x ) ( ) (3.17) 2 j ) y 2 ) ) ) , y y 6 4 1 2 ). For the terms with ω i s )) = 0 and therefore 0 ¯ ω y y F x ) ω , + ¯ w defined in equation ( ¯ ( ω i, j 3 1 ( ∈ #¯ , ω , y ) ω x π h 1 2 i) x ( 2 ¯ ω x x =1 3 y , − ) Y ω ω · 3 ( i,j 2 ( (¯ i s x 0 (1 + 0 ) (3.20) 6 ) ) , y , , ω ω 0 w 1 2 | 1 1 1 (¯ ( # ) = ) y y x x x 3 h ) ( ( ( ¯ ω w y 3 (i) 6 ( , , ω s i 0 1 2 h ) , y + = = = 1 x x ∈H 3 ) = 1 (3.21) x X ) is diagonal or off-diagonal, respectively. So 2 #1 ) = ( w ω ω x y x ∈H ) for each pair ( P 2 (¯ ( ( π ( j X 1) w | y – 17 – ) induces a bit-flip. One now checks that: 1 6= 0 we have Tr( + h ) ) ) ) 64 ¯ x ω − 1 4 is a superconformal current. 1 2 3 Tr( x 1 ( ( , , by w ( s i s 1 = h 0 , y , y , y , y χ x V 1 2 3 3 ax P , #0 x x x ω x ( Ψ = = ( ( ( ( P i ) ) ) + ) 6 1 2 3 2 2 0 Im y | ) = 1 x ) , y , y , y x ∈ 1 2 3 x ( , ω + ( x x x ) is to use the homomorphism s 1 χ ( ( ( h 1 x x ) that ω ( (¯ 3.22 . Recall that this homomorphism is distinguished because it is zero ) 6 2 3.15 ) , y + 4 1 F x ( is a rank one projection operator. We will show in section P One way to check ( As a check we note that indeed Ψ can in fact be written as: It follows from ( it is zero or one according to whether where its extension to ( or one according to whether the section Therefore suitably normalized spinor is a projection operator. Note that for and it is similar for the other pairs. and using the permutation propertyconsider we see these all multiply to 1. As an example of we have in all and similarly for the othergroup two nontrivial together factors. terms Now, of gathering the all the form terms together we Now we use the bimultiplicative property to expand out JHEP05(2020)146 ). ). α, i A.6 (3.27) (3.28) (3.25) (3.31) (3.30) = ( is a Pauli ··· the vertex A A P ++ 4] described in , T 2 6. (Later we will s 12 1 } z ¯ ≤ s 0 2 0 α κ ¯ ω + ! ≤ ¯ ω 1 B ¯ − J ω is order four: in the image of ¯ A ω i i 1 1) Ψ (3.26) 6 6 , J = 1 superconformal generator. ⊗

¯ ω 2 1 (3.29) ⊗ ) is the energy-momentum tensor s ¯ 2 2 ω N 2 1 (6) C U T ) √ 0 12 ( AB 2 0 , z ⊗ · · · ⊗ σ Σ 6 = ∈ ¯ 1 ) ω 2 (i 2 ¯ s s α ¯ . We use a composite index ω ( U U 2 , ) i z ¯ γ ¯ ω ⊗ = 6. is exactly the code of type [6 σ X α<β tr ). Mainiero computed its entanglement ho- ¯ ω ( 1 6 2 c s 2 1 F ¯ ω κ ⊗ ⊗ · · · ⊗ ¯ ω := 2.62 1 – 18 – + ⊂ we have the operator product expansion ¯ (1) s U 0 ) A 6 ) 6 0 J 2 ⊗ ! ⊗ , ⊗ · · · ⊗ 2 ) H σ 0 2 s ( induce corresponding symmetries of Ψ. Using these 0 i i π A C 2 12 1 1 = 1 0 ( − A z Σ = (i = (1 0 1

A ∈ ¯ ¯ γ s 0 4]] defines a superconformal current Σ 2 2 , 0 1 0 , { A √ ) by noting that the group of automorphisms of the hexacode , s P X 1 [[6 1 s = Ψ ) = κ : 3.22 1 6 α + U in appendix (0 2 1 s 3 12 3 1 − z ¯ s H π ∼ × ) 3 label a basis of three generators of SU(2) and 1 0 4]] discussed near equation ( 2 , is a superconformal current with the one-dimensional code defined by Ψ is closely related to the standard code z H as the column number in the Miracle Octad Generator.) Then Σ ≤ 0 ( correspond to the 18 generators of SU(2) s above: as promised that code determines the , 2 i α s V A V ≤ ) 2.1 J 1 z ( Given any two states Finally, we note that the image We can now prove ( 1 s V The conjugate spinor is defined by where and all operators onHere the 1 r.h.s. are evaluatedinterpret at matrix for the column where 3.4 Why the imageWe now of show that, for aoperator suitably normalized state where Remark. of type [[6 mologies and found themone to to be the identical. other. Indeed, Mainiero there found is that a local transformation relating So the sixtyfour hexacodeconcerned words it map appears to one sixteen just different has terms. to As check these farsection by as hand. the phases are denoted by symmetries it suffices to checkThen that note we that, get suitable restricted vectors to from the the hexacode seed the codewords fiber ( above 0 JHEP05(2020)146 α, β then (3.32) (3.33) (3.35) (3.34) so that P to be a 6 s Im ⊗ ) V 2 ∈ C s [246]) (3.37) ··· − + + ) imply similar symme- to two cases: first, if T A.6 12 ¯ ss z AB 1 2 6 = 1. + ≤ B to achieve the desired OPE of the ss J s A J [235] + [145] + [136] X s α < β 2 , − [245]) and using the symmetry properties of the † √ ≤ 12 i ∂X AB s i − z [123456] + [56]) Σ ) shows that the conditions for e − ¯ s = − 2 ∼ = – 19 – B.3 s depends on our normalization of SU(2) currents. [236] ¯ Ψ = iΨ 3 X α<β 1 J = − 2 1 J [1234]) + ( κ = 0 1 1 κ ) are, as far as we are aware, all independent so there s − s = 6 corresponds to ¯ + c 3.34 AB 3 12 ¯ ss z Σ ¯ s [1256] = 0 the above equation simplifies to ∼ − s ) 2 ] [146] + [135] ] + [1256] + [1234]) + ([235] + [145] + [136] + [246]) (3.36) A Ψ = 0. We will now proceed to check that these equations indeed z ∅ ∅ − ( Σ s s i([146] + [135] + [236] + [245]) [12] + i[3456] + ([34] AB V ) Σ + i([ + i( + i([ − − † 1 z 2. Specializing to ( represent bit-flip and/or phase-flip errors on two Qbits, this is indeed the s ] are real vectors. Then √ V Ψ = [12] + i[3456] + ([34] + [123456] + [56]) Ψ = / 1) 2) AB , , (2 (2 = 1 , , abc . . . 9 = 135 independent equations on 64 variables. Nevertheless, if 1 code, and not the classical code. For example. 1) 1) Ψ = 0 iff Ψ , , κ × (1 (1 AB Σ Σ 6 2 The symmetry properties of the hexacode mentioned near ( The quadratic equations in ( ¯ ΨΣ All the classical code wordsorthogonal are to the same Ψ. as in Ψ, but, remarkably, these vectorstry are properties in fact of Ψ thatare allow in us the to map same the couple general then case by of permuting Σ within couples and using the cyclic symmetry quantum error-detecting property ofquantum the code! Note that this is really a property of the so hold. Since Σ once these equations aresupercurrent. satisfied In one the can present normalize case are indeed the equations are satisfied.the To basis see [ this, we define a real structure on ( Comparison with linesuperconformal one current of are that equation ( Pauli matrices so that ¯ With the convention (again dropping cocycle factors) We find is a symmetric matrix. The constant JHEP05(2020)146 0 P i, j span ) and (3.40) (3.42) (3.41) (3.38) i ω ) ( x , and let h ( x π | 3] subcode of ¯ ΨΨ = 16i, we , = 6. are in different c ). This defines an 0 α, β P Ψ is orthogonal to Ψ in the [6 . 3 maps every term of Ψ AB ) 3 , 1 , . , w 2 , ω 2). But in this case both 0 . Finally, since , 2.1 ) anticommutes with , w , β 1 ¯ ω (1 w i , h = 1 superconformal symmetry we 6= 6 . The four supercurrents will span i ¯ ω i ) (3.39) 6 2 , 6 + α i ) = (1 | N w 6 + ) + | ⊕ ( | = 4 superconformal algebra. ω ) y + ) i, j h | 2 ¯ ω = (0 P + ω ) ) x 6 N x P with ¯ 2 ω ,j P + P u P ∈H (3 X , 1 + ) w , and that = 1 + P ,i 1 1 is closed under vector addition and therefore AB y 1 − P − (1 – 20 – Ψ 16 1) and ( . In fact, it has 16 elements and is therefore a (1 + ( ( ( x 6 P ) , 6 Σ 0 0 0 0 x 1 (1) H -symmetry group. Note that it commutes with P P P P , H ) P := 1 R + x (1 , = = = = representation. Focusing on the first Qbit, 0 P h 0 6 1 2 1 2 and therefore each has sixteen elements. It follows that , ) = (1 ¯ ¯ R 0 6 H Ψ Ψ Ψ Ψ = Ψ defines a superconformal current with , = 4 supercurrents to codes i, j 4 , H 0 1 4 , is the span of the vectors π/ 4 i Ψ = 0 for all F N in ]Ψ 6 2 − denote the set of hexacode words whose first digit is 2 e 0 6 F ∈ AB = 4 supercurrents in terms of the hexacode. = 2 subalgebra in the x 6 = (0 H = πT ⊂ 1 H i x N N ¯ ΨΣ s u ) − 0 6 let ) is an SU(2) H 0 4 ( P F with π ) is a projector to a two-dimensional subspace of Im( s . In fact, we have ∈ = exp[ is a linear subspace of x 0 V 2 x P P is a projection operator and its image is 4-dimensional. The first SU(2) factor 0 6 0 + H will be interpreted as the P 0 6 are the cosets of P ( x 6 1 2 the truncated Hamming code, as described in section embedding of an H The image For any nonzero Now, For each ).) 1. 2. ω (¯ (Note that Ψ h Remarks. the image of in SU(2) and therefore Im( a two-dimensional space so the representation is Note that two-dimensional subspace. One choice of basis is The can easily describe the for these cases.conclude Thus that 3.5 The relationNow of that the we have understood the code underlying couples then we need to check orthogonality to and then we needinvolve only bit-flips check and ( one easilyto checks a that vector a orthogonal pair to of every bit-flips term on in 1 Ψ. So, rather trivially, Σ of the 3 Pauli matrices we reduce to the two cases above. Second, if JHEP05(2020)146 2 / that (4.3) (4.4) (4.5) (4.1) (4.2) = 1 C ` is not the 4, degeneracy / 24 1 5 M ≥ h 24 2 2 c/ ) with / / − 0 ˜ L h, ` 24 runs over unitary irreps of ¯ = 0 = 1 = 1 q 3 2 0 c/ j / , ` , ` , ` − ˜ zJ R 0 i 4 4 4 =1 4 are nonzero but there is a general π L / / / ˜ ` , 2 j / q 1 1 (4.6) ] is that the virtual ] is that there exist an infinite set 4 1 e 3 ˜ 0 / R 35 = 1 = 1 24 ≥ zJ > 45 i =1 ⊗ , c/ π h h h > n ˜ h i i ; 2 − representations have the property that i h e 0 R 39 3 0 , D L 2 ) arises from consideration of the elliptic 24 J q 2 ) ⊗ i ) 1 − 0 π j n with 3 0 M 36 ; 2 − ˜ 4.2 , i J        e τ, z 2 ) + / ( ,H D 11 i 3 0 =0 = 2 ˜ ` – 21 – h,` =1 J / , χ 4. i,j ˜ ( ` 1 , 4 ) i( / corresponds to a pair ( , 24 4 R / ⊕ π g 0 / 2 ( j c/ i =1 = =1 − = 1 ˜ h D ge ,H ˜ h 0 ; ; or : 0 i i L , h i Tr 0 RR q RR 24 D D ) := Tr 3 0 H H H J i M ) are characters of irreducible representations of the (left- i := X π z, τ 2 i when ( e τ, z = . Moreover, these ) D ( } i h,` 2 24 ) := Tr h,` χ χ ( / ) is any finite-order automorphism of a K3 sigma model M 1 R C 4) superconformal algebra then the twisted character , z, τ , 0 ( Tr g E Aut( ∈ { . Here 4 is, in some natural (but thus far unexplained) way, a representation of ` g / ∈ = 4. In particular 1 g N = 4 algebra: h > 4 and runs over unitary irreps of leftmoving N=4 and / N i 1 R with The outcome of further investigations [ The remarkable Mathieu Moonshine conjecture [ i h > On the Kummer locus the spaces 4) superconformal algebra and admits an isotypical decomposition 5 , of representations of the group expectation that these spaces vanish off the Kummer locus. by the order of moving) must be a weight zero index one Jacobi form for a congruence subgroup of index determined Therefore, if commutes with the (4 remarkable because, thanks to the quantumquotient Mukai of theorem described a below, group ofsignificance (4,4)-preserving of automorphisms the of any virtualgenus. single degeneracy The K3 space Witten sigma index ( model. ofalgebra R-sector The irreducible is: representations of the N=4 superconformal space for the finite simple group the character-valued extension of the elliptic genus exhibits modular properties. This is where rightmoving for 4.1 Statement ofThe Mathieu moonshine RR subspace of(4 the space of states of a K3 sigma model is a representation of the 4 Relation to Mathieu moonshine JHEP05(2020)146 , j ; i 24 4). as a D / with (4.9) (4.7) M 1 has a 24 , 2 23 4 M 24 / / M M =1 ,` 4 / ) = (5 +1 ¯ h representations. n as a subgroup, could χ h, . We note however 24 i )) 24 M D g are true, not virtual, ( M 2 , n n 2 H H H (Tr ⊕ 2 =1 , ∞ on any given degeneracy space X n 0 G H + on the spaces 2 / ⊕ 24 1 ) (4.8) , =1 1 M ,` 4 z, τ H / ( g ⊕ =1 ]. E h 0 , but all the other the function , 4)-preserving automorphism. Moreover χ 1 2 on a set of 24 elements. (The group , ) 75 · , 24 ) = g H – 22 – 23 ( 2 2 42 M ⊕ / − M z, τ 1 , 0 ∈ ( , 0 g = 0 as a (4 g H ˆ 4) supersymmetry. The answer is, remarkably, that φ 4)-preserving automorphism of the CFT. However, we 2 , H C , / dimensional subspace of the states of ( 4, thus explaining why these are not true 4)-preserving automorphism group that has a quotient 1 / ]. Once a K3 surface is endowed with a complex structure , , ∗ ∼ = 0 + Tr below is a powerful no-go theorem that implies that such ) 80 H every = 1 C =0 i 4.3 h ,` 4)-preserving automorphism group 3; 4 , / and acted on K ( =1 1 ∗ 24 h is truly a (4 · χ H M g 3 However, the quantum Mukai theorem of Gaberdiel, Hohenegger, and ) g ∈ 6 ( − . 0 , g 0 24 23 H M = as if 0 , ] A. Taormina and K. Wendland discuss how a certain maximal subgroup of 0 95 ) := Tr H K3 sigma model with a (4 is isomorphic to any subgroup preserving one element.) z, τ The most natural way to explain the Mathieu Moonshine phenomenon would be to find ( We stress that all that is needed is that some quotient of the automorphism group contains 6 g 23 ˆ φ K3 sigma modelsthe that groups preserve are (4 precisely the subgroups of thesubgroup. Conway The group action that ofmight, the preserve in (4 general, sublattices have of act a on kernel. the Note massless that states where a different quotient, not containing M 4.3 Quantum Mukai theorem Motivated by the discovery(GHV) of gave Mathieu a Moonshine, characterization Gaberdiel, of Hohenegger, the and potential Volpato automorphism groups of supersymmetric Any holomorphic automorphism must preserve thesesays five that components. all The groups Mukai ofat theorem holomorphic least 5 symplectic orbits automorphisms in arenatural the subgroups action natural of action as of a permutation subgroup acting on a set with 24 elements. The subgroup When a K3 surface istheorem characterizes given the a possible complex groups structuresurfaces. of it holomorphic For a is symplectic nice automorphisms holomorphic review of symplectic.the see K3 [ 24-dimensional The cohomology Mukai space has a Hodge decomposition Volpato, reviewed in section an explanation of Mathieuhypotheses of Moonshine the cannot quantum work. Mukai theorem. Thus,4.2 one must relax The some Mukai of theorem the the octad group, acts on aSome 45+45 extensions of this work appear in [ some that contains in those cases where stress again that there is nothat known natural in action [ of representations) such that for transforms (where JHEP05(2020)146 acts as a must g 24 24 M M 20) to ques- , 1)-preserving , as a subgroup of 24 which can be used to M 24 (20). A clever argument = 4 primaries with left- M O ∈ × N g ]. In order to prove the the- (4) ]. Clearly such a special point 40 41 /O 20) , ]. 2) and right-moving quantum numbers (4 / O 94 , = 1 93 ] for related remarks.) ]. Our approach makes it clear that a maximal – 23 – 93 = 1 supersymmetry. We are thus led to the idea 41 n, ` N 4 + / = 1 ]. See also [ h 66 ]. None of the relevant groups contains 61 ] to characterize a K3 sigma model as a choice of positive definite 4 , 2). To make sense of this idea one would need some ope-like algebra 3 1) supersymmetry, and that these symmetry groups contain 4) supersymmetry. In order for the Witten index to make sense, / .) The Quantum Mukai Theorem is thus a powerful no-go statement in , , ]. There has been some success with this approach in the context of 1 , 24 ], but not yet in the context of Mathieu moonshine. A second approach is 57 M ) are in fact true automorphisms of at least some K3 sigma model but do not , 89 = 0 56 below, using the twined elliptic genera, that, at least for the GTVW model, the τ, z , ` ( 4) supersymmetry. Thus, we are imagining that the 4 g . This is closely related to the GTVW model [ , / 5.5 ˆ φ 20 The project described in this paper began with the observation that one could relax Two ways of relaxing the hypotheses have been explored in the past, but, at least Now the subgroups of the Conway group that preserve sublattices of the Leech lattice Among the rank four Höhn-Masongroups there is a distinguished maximal subgroup, M = 1 : 8 ˜ h still commute with some right-moving that there might becommute very with symmetric (4 K3quotient sigma group. models While this with idea largesection was an symmetry important groups motivation for that our work, we will argue in pursued by A. Taormina and K. Wendland [ the assumption that the relevant groupwith of (4 automorphisms of the K3 sigmadefine model commutes commute with (4 structure on these BPSfollowing states. [ We couldmoonshine call [ this the “algebrato of attempt BPS to “combine” states”in the approach the symmetries of moduli different space. K3 sigma This models is at the different “symmetry-surfing” points approach that has been vigorously thus far, have onlynot as met an with automorphismof partial of the the success. full subspace conformal Onemoving field of quantum theory approach BPS but numbers is states.” only ( to( as posit an The “automorphism that BPS states are the 4.4 Evading aGiven no-go the theorem powerful no-goproceed theorem by of relaxing GHV, one any of explanation the of hypotheses Mathieu in Moonshine the must theorem. 2 deserves special attention.different derivation Using of the the main relationsubgroup result of of of the [ Mathieu supercurrents group, namely toautomorphisms the sextet of codes group, the acts we model. as a will group of give (1 a have been tabulated ina [ quotient. (Also,subgroups of although this isthe search less for relevant, an many explanation of groups Mathieu cannot Moonshine. be embedded as orem one follows [ four-dimensional subspace of the Grassmannian transfers the problem from ations question about about actions spaces onis of the elegantly indefinite Leech summarized signature lattice, in (4 a [ positive definite lattice of rank 24. (The proof the Leech lattice of rank greater than or equal to four [ JHEP05(2020)146 6 24 M (5.2) . Let (4.10) SU(2) ) where =1 6 α ⊂ h, )) 6 α 6 Q 1) case. The x . Thus a copy , ( 6 h 24 for unitarity. α ∈ H ) c/ ∈ H 6 ≥ w h , . . . , x . Once one admits infinite 1 24 x − M , 4 representations. Unfortunately, / = 1 representations the only new 24 1 (5.1) 1) =1 , h N ; )Ψ = Ψ for all M (4 i can be written as ( → = 1 and ( w 1) supersymmetry. This would indeed D ( 6 6 , α 4 h Q Q − 6 =1 → )Ψ = Ψ (5.3) + Y (Ψ). We claim this is the non-abelian group α , 6 6 4 w /

( Q Q 1) preserving symmetries will not explain ) were true h – 24 – on the groundstate and , 1) =1 α , ) h 6 → ; 6 4.6 (4 i F for Ψ that x Q ) we would like to lift this to the group Z 6 =1 D ( 4 have nonvanishing Witten index. Working out the 1) Y α / h = 1 superconformal group are labeled with ( − 6 → := 3.11 3.22 N 1 = 1 1) , h (4 i , . . . , ) D 1 x ( h 1 4) preserving to (4 = 4 representations in terms of , . Then we need to solve 6 4 N F ∈ ) 6 ) and the expression is a group of symmetries of the supercurrent. (Ψ) = in the spirit of looking for larger automorphism groups by reducing the amount 6 6 Q 3.15 Q , . . . , x 1 ⊂ x Stab 6 To prove this, we note that the elements of Now, recalling the definition ( The statement of Mathieu Moonshine is only slightly altered in the (4 H = ( is a sign given by the action of ( w We now determine the stabilizer group Stab As a first stepfrom to eq. determining ( the symmetriesof of the supercurrent we note that itof follows bit-flip and phase-flip errors. Recall that into the subgroup of theinterested symmetry in group the that group stabilizes preserving thebut certain supercurrents. will superconformal also We are structures comment mainly in on the the GTVW SCFTs5.1 model with moonshine for the The Conway stabilizer group. group within numbers of virtual representations Moonshine becomes unsurprising. 5 Symmetries of supercurrents The construction of supercurrents from quantum error correcting codes provides new insight phisms of K3 sigmabe models possible that if commute all with thebecause (2 representations in the ( massless onesreps contains are infinitely virtual, many massive and N=2many massive reps, the representations the branching into virtual virtual of representations representations make of massless infinitely N=4 reps to N=2 branching rules for point is that the virtual degeneracy spaces relevant to the elliptic genus are now Remark: of preserved supersymmetry it is natural to ask if one could consider instead the automor- symmetry. So the mystery of Mathieu Moonshine remains. irreducible representations of the Only the representations with enhancement from (4 JHEP05(2020)146 (5.9) 5.4.1 denote 4 that acts ˆ g such that 6 A ). Therefore we note that 2 0 ,..., C ). This follows g 1 6 g ) et. seq. above) can be written as ⊗ ) 6 4 2 F Ψ then the sign must 2.35 C . For ∈ P w 1) (5.7) )Ψ = ⊗ w ! 0 (5.4) ( 1 1 h − → ⊗ we have 6 )) (5.6) (5.8) 1 1 1 ) (5.5) . This follows because the vectors 1 6 )Ψ, so these must in fact be linearly 6 w w w

1 ⊗ · · ⊗ ⊗ · ) u 2 ) commutes with 1 i 0 5 i ( 1 → H 2 g g g − i √ h . Similarly, define . ( ( ( C g ⊗ . That is, there is no matrix such that h h h P P 4 (Ψ). Here it is very useful to observe that 6 4 (Ψ) Ω F ) = 6 = = = F H (as we saw in equation ( 3 := (Ω ⊗ Q 1 1 1 σ ∈ 6 0 1 − − − i 0 5 – 25 – are pure permutations. Letting ˆ g g g g − + w )ˆ )ˆ )ˆ 4 Stab 1 form a linear basis for End(( ∈ H (Ω w w w is the subgroup of the center of SU(2) σ ( ( ( · 6 4 ( 2 → h h h F 2 i u Z 5 0 i ˆ g ˆ ˆ g g Z , . . . , g √ . It is easy to prove that there is no linear operator that ∈ 1 4 and therefore ˆ ). F g , commutes with commutes with → and g w 3 := 0 5 , 1 acts on the six Qbit system in a natural way, and this group g g := (465) v 2 3.20 6 , 5 )Ψ. So these must generate the entire vector space. But there S ˆ comp 6 g 1 ) for : u form a linear basis for ( = 1 w ( 6 ( i h form a linear basis for the complex vector space End( ∈ H h linearly independent vectors 1 4 generates the entire vector space. But every 6 comp 6 u F . 6 4 ). Nevertheless, if we define 6 )Ω and therefore letting ˆ defined in ( ∈ F x x = 2 (¯ and all and therefore ˆ ( . The generators x ∈ H := SU(2) P )Ψ = . Accordingly, ˆ ∈ 3 h with h 6 4 6 4 6 4 ∈ H w 1 A 2 w H ( F F F w = − w u h 1 ) for ∈ ∈ ∈ below we discuss the lift to the full GTVW model. As a preliminary, it is therefore − x + ( w w w 1 A h ) ) = Ω u Finally, we define a lift of To demonstrate this we use the description of the automorphism group of the hexacode Thus we have Now, note that x )Ψ for )Ψ for 5.4.2 ( = ωx w w ( ( ( implements the Frobenius automorphismAh on so that for all h for all in appendix the corresponding permutationsclearly acting have on the factors of the sixfor Qbit all Hilbert space we and useful to discuss whatautomorphisms we of know the about hexacodethe lift Stab projector to operators on the Q-bit system that commute with where the readerineffectively will on recall the 6 that Qbit system. 5.2 Further symmetries ofThe group Ψ: lifting thecan hexacode be lifted automorphisms to a symmetry group of the chiral part of the GTVW model. In sections are at most 4 independent. It follows that ifbe there + is and a since h h w and then JHEP05(2020)146 . Of = 1. 6 to be (5.14) (5.17) (5.10) (5.13) 0 ξ .S F 3 . There- ) implies g Z = +1 for P shows that i i = +1. ) isomorphic ξ 5.14 6 5 Ψ | ξ H F ˆ g | 6 . is in fact isomorphic + P i h 5 and then ( ¯ γ 0 ˆ g . 5 it follows that the group (or any other element of the , . . . , g γ F generate Aut( 0 6 ξ = 0 6= 0 g = ¯ h F 7 (5.12) x x ∗ 0 ,..., (6) , g g ) (5.11) := 6 ) 5 ) 2 SU(2) w , and S x commutes with 5 σ 05 · ) : ,..., = 0 (¯ ξ ∈ (i x 6 H h F F , ( i ) . g g 0 5 h − = 1 , . . . , g ( Ψ (5.15) ξ 1 ∗ 0 i h ( , − g 1 = SU(2) = − ⊗ · · · ⊗ 1 † Ψ = Ψ (5.16) generate a group that commutes with ⊂ ) Ψ = − F – 26 – F Ψ x 6 g Ψ = iΨ (1) F F i ˇ g , a simple direct check shows that ( )ˆ ) ¯ ˆ ˆ γ g ξ g 4 .S h 2 , ˆ w g 5 Ψ = Ψ for σ ( i : 3 = ˆ g Then h g is a third root of unity. We claim that, in fact, 6 1 Ψ = F ) which implies that ˆ 7 2 ˆ · g 2 − = (i 0 ,..., ). stabilize Ψ and generate a group isomorphic to ξ the elements i v 1 σ ¯ γ ) ,..., ˆ g Z g F 0 x A = 1. The same style of argument shows that ˇ g g ( , only an extension of has order two. A direct computation of , )Ω(i 0 5 2 ξ F vh 1 and is one-dimensional we can say that ˆ g σ g − (i P = − . Then, since there are always an even number of nonzero digits in a priori 6 ,..., 2 0 ⊗ = v g v with the transformation ( . Now, for ˆ ∗ i · that is not in ξ F 6 g , and this is sufficient to prove that ˆ = 1 it follows that and therefore 6 3 0 0 , which is, = (56) 4. We must work a bit harder to find g i ξ , 5 . The lifts ˆ ∈ H F 3 ˆ g = 6 , g . is not in the stabilizer group. However we can remedy this by defining ˇ ∗ 2 w 0 .S , ξ 05 3 As discussed in appendix Finally, we note that ˆ It is worth noting that since ˆ Since ˆ Thus, the lifted elements ˆ F We thank T. Johnson-Freyd for pointing this out to us. g ,..., H Z 7 0 = 1 ˆ g course, the “translation” actionsemidirect by product hexacode elements themselves stabilize Ψ and sostabilizes the Ψ. center of SU(2) to in fact so ˆ the product of ˆ Now recall that Ω that h to and compute: for some phase i To prove this we use the reality properties of Ψ. Define the symmetric matrix: when fore, since the image of Define ˆ a hexacode word we do in fact have one can check that JHEP05(2020)146 = 1 to be (5.19) N ∗ 4 F ∈ i x below. ) for i x to a symmetry of the ( 5.4.2 h 6 ], and the relation of the S 67 : and 6 5.4.1 6 factor. The computation, which is th i α involves an extension by the subgroup Ψ = 0 (5.18) Ψ | α 6 of the GTVW model. ) Ψ x 0 h ~ there would be an accumulation point. We ( V 6 h = . The main point though, is that it is a again i x,α C – 27 – Ψ c | to SU(2) u ) within SU(2) | 6 =1 6 ]. Using the methods of [ X α P Ψ ∗ 4 h 67 group. If the stabilizer were discrete and infinite, then, F ∈ X x . Second, including left-movers with right-movers, those (2). Note well that in this sub-section we are not thinking for a definition of the term “holomorph.” 6 finite su A of the chiral algebra 6 S algebra : 6 . 6 S is a discrete group. To show this we consider the : 6 ). See appendix means that the matrix only acts on the 6 6 α H ) ] T. Johnson-Freyd has discussed the automorphism groups of holomorphic x of the center of SU(2) ( 67 SU(2) h lifting from a subgroupZ of SU(2) automorphisms that involve aembedded diagonally nontrivial in permutation the productThese of of aspects hexacode left- will and digits be right-moving carefully must hexacode described holomorphs. be in sections Hol( full GTVW Hilbert space. There are two issues involved when doing this. First, What we have described above as the stabilizer of Ψ is the holomorph of the hexacode, We still must lift the above symmetry group in SU(2) ∈ In [ In fact, the stabilizer is a We can show that the stabilizer group of Ψ is discrete by showing that there are no One nice consequence of the error-detecting code description of Ψ is that the stabilizer 1. 2. u superVOA’s in a large classof of models. the class It of turns models outGTVW considered that model in the to [ GTVW a modelΨ theory is a can of special in 12 case fact MW beis fermions, no exactly one larger the can than holomorph show the that of holomorph the the of hexacode. symmetry the group hexacode. of Given our result above, it which constitutesof a (complicated) algebraic equation for the matrix elements true due to the error correcting properties of Ψ. being a subgroup of thecan compact rule group out SU(2) this possibility byIndeed, noting it that can in fact be the characterized stabilizer as group the is an solutions algebraic to group. where slightly technical, is relegated to appendix within SU(2) generators of the Lie of these matrices multiplicatively! nontrivial solutions of 5.3 The stabilizer groupThe of question Im( now arisesphism as group to SU(2) the nature of the full stabilizer group within the automor- Remarks: JHEP05(2020)146 i .) e 0 (4) ˜ into L O (5.20) (5.21) 6 (4). Of as well ∓ and 2) 0 / 3 SU(2) L , (4) that acts (0 GTVW O ⊂ diag Z ⊕ H /Z 6 current algebra sym- 0 , S 2 : / =1 3 GTVW 6 k R acts trivially so only the symmetry group of the set H R 6 S : \ and taking the determinant of SU(2) 6 6 × SU(2) (4) 6 (4) × 1 =1 O k L ˜ O V L is a set of Clifford generators for Pin ⊗ l is that where image of the determinant permutes the 6 factors. In the GTVW e 6 6 k 0 6 e \ SU(2) 2) S j / e (4). 3 , ⊕ V O (0 GTVW ijkl 6 – 28 – 0 (4). In fact, the groups are canonically isomorphic: if 1 ˜ 3! V P (in(4)) symmetry is an automorphism of the model. (An acts both on the space − where Pin(4) is the double cover of ⊕ H 6 ⊗ R 6 Pin S 6 S (4) is not an automorphism of the entire sigma model. denote the diagonal embedding of 1 : 0 ∼ = , : V O 2 6 / 6 (4) then has a projection to 3 SU(2) GTVW (4) diag 6 + H × Z . When the image is not one, some left- and right-conformal 6 2 1. We will denote this group as: L Z , acts effectively. On the other hand, there is a left-right reflec- − 4) supersymmetry within Spin(4) , LR . 4) supersymmetry. One way to construct such symmetries proceeds . Let 6 , ) and hence 6 S S . The group of symmetries acting effectively on the GTVW spectrum : : ˜ h, h ( 6 R 6 diag → ) there are spinor representations so the symmetry group is in fact a quo- , where the permutation group = 1 WZW model has, famously, Note that Pin(4) 6 ) /Z k ˜ h S SU(2) 3.2 8 6 : h, × 6 L 6 LR Just the way there is a further parity symmetry when one considers ground states of When we turn to the product of six WZW models we clearly have a symmetry group Thanks to Bott periodicity Pin Spin(4) 8 SU(2) We are now in athat good commute position with to (4 determine the group of symmetries of theis GTVW a model set ofcourse, Clifford they generators are for not Pin isomorphic as double covers of Put simply: thefactors. spinor Thus lift the of groupas a P on (in(4)) parity the transformation RR is ground states. diagonally embedded5.4.1 in all six Stabilizer of (4 The subgroup that preserves map is either all +1 or all on spinors. each factor gives a mapweights to on some factorspreserve will the be space exchanged. of potential In supercurrents: particular this group does not, in general is the WZW model for aof single RR factor ground SU(2), states there of isthere the an model. is a In group the GTVW action model of we Pin(4) find spinor representations and SO(4) spectrum ( tient of Spin(4) quotient, denoted SO(4) tion action of theaction model. on the It space exchangesweights the of ( left- states. and ElementsNevertheless, right-movers it in and is the generates a an nontrivial global component symmetry permute of the conformal space of ground states and it can be useful. The SU(2) metry and the globalautomorphism SU(2) of the sigmahence model we only should take certainly theActually, preserve subgroup the the of diagonally the conformal embedded affine weights, center Lie and of group that SU(2) commutes with 5.4 Extending the automorphism groups to include left- and right-movers JHEP05(2020)146 . 6 6 S S that : ) is a (5.23) 6 F S ( p restricted ) the first diag and hence w p → and we will ), that fixes x ( /Z 6 = 4 currents 6 h P 6 ) 6 H S . The elements N . i : H 5 5 6 A . It has 16 elements . First, as we have by combining with Aut( As an example of a 4 6 Spin(4) subgroup of order 4. we will denote its lift F S A ⊂ F , . . . , g 2 : Aut( : F 1 F ) (5.22) Z g 6 p h ∈ . The kernel of × in a way explained in detail 5 , x ϕ diag = 2 2 S 20 ) is a nontrivial element of 5 Z /Z , x ϕ ⊂ M 6 H 3 ). On the other hand, ( 5 x p we are led to consider the group in appendix F must commute with ⊂ ¯ A ( ω is isomorphic to 5 6 R 6 , p g 4 F 6 F x S Spin(4) : . Symmetries of this type that are lifts ∈ H are certainly in ): , ω 6 6 5 w ,R A 0 6 , x . 1 note that if and SU(2) x ( 6 × H – 29 – 6 L 3.3 S ,L 7→ : . 0 6 5 ) 6 ) must be exactly H be the subspace consisting of hexacode words whose . Since we can independently lift a translation by a A ( 4 2 F 6 diag ( Z which preserve all four chiral supersymmetries. Now, p ) (acting either on the left- or the right- movers) that /Z 6 ∼ = 6 w , . . . , x ⊂ H S ( 1 0 6 0 6 h x : on the left- and right-movers. If )) that there is a homomorphism H H :( 6 does not generate a central extension in SU(2) 0 5 we can modify the generator that commute with all four chiral supersymmetries must commute A.10 g Spin(4) i = (35)(46) of appendix and hence 0 5 6 F b g to get, for example: 5 5 = 60 must divide the order of . Let 4 mentioned in section S is isomorphic to · g S 3 : 6 elements = 4 supersymmetry. Thanks to the description of the 3 6 H F diagonally · 6 ,..., × N 1 b g 0 h we see that these translations by . As we have seen, this is a 2-dimensional subspace over H must be 0 to SU(2) 3.5 6 w acting diagonally as a group of automorphisms. is trivial so F At this point, the reader should recall theIn important order remark to lift concerning to the relation Now we must lift these group operations to Now, still working chirally, let us consider the “translation symmetries,” that is, the To begin, we work chirally, and consider lifts of holomorphs of the hexacode to the ) = (25)(34). Now (34)(56) and (35)(46) generate a 0 5 = (34)(56) and F g ( 2 demonstrate below that the further lift to SU(2) of SU(2) then it must act hexacode word to elements of both SU(2) with seen, the lift in section must preserve the first digitdigit of of hexacode words. Thereforefirst when digit acting is 0 with and, as an Abelian group, proper subgroup of to action by SU(2) commute with Now recall (equationsimply ( tracks whatp permutation ofBut digits (34)(56)(25)(34) = the (256)It automorphism has follows order that implements. 4 3 while (35)(46)(25)(34) We = have (23645) has order 5. g nontrivial element of elements of elements of SU(2) with the SU(2) R-symmetry.with the In first the SU(2)of GTVW factor holomorphs in model of the the the productthe hexacode chiral SU(2) hexacode. must R-symmetry be Therefore is lifts wethe identified of begin first holomorphs by digit that determining of preserve the the the subgroup, hexacode. first digit It of is easy to see that We will construct a group ofbelow. symmetries that Since is the relatedexplicit discussion to detail. has several subtleties we will besemidirect going into product excruciating SU(2) and by lifting suitable subgroups of the holomorph of the hexacode to JHEP05(2020)146 . 5 A (5.30) (5.24) (5.26) has the imposes 4 , ,R 4 ) (5.29) 0 6 G σ ; R ) and hence the × H ) to a subgroup of 6 isomorphic to 6 ,L ˜ u 0 6 6 4 2.20 H , H 4 . We will write group has order 3, and that 4 G , 0 5 4 b g 6 G 2 , . . . , z 1 (5.25) ; 1) (5.31) ; (25)(34) b g 1 R ˜ u → R 1 )) ) (5.28) z 6 SU(2) . ) generated by ( F σ 1 x , ; , : (˜ ∈ . More abstractly, 1 L R 5 , ) in equation ( ) ) 5.25 )) ), when lifting 6 A 6 ,R Ω 6 diag 0 u 6 : ˜ , u x 5 (5.27) 6 x, y 1 5.4 ( /Z · ( ) is nontrivial. , . . . , h − 4 2 ) ) 3 1 Ω × H Z R · , . The restriction to x ,..., ) = 1. The multiplication of these group Z 5.25 6 (˜ 1 1 ,L × 1 15 , . . . , z , 0 6 h u , . . . , h 4 2 1 x × ( 1 (˜ ) (˜ H , u , Z 1 has order two, that L 1 h , ∈ H = 2 – 30 – x in equation ( L L z 0 5 Z ( | ) L ) and defines a subgroup of b g · )) 4 6 π h (( 6 , ), and not their product! Altogether then, we have F 6 ( 4 5 2 . Again, we stress that the cocycle defined by this 1 −→ ˜ x F x ) = 5 2 using the notation , 4 ∼ Z , ( 1 |G 1 4 Z → 6 ) 2.20 x , ). ∼ = ( over ) S σ 6 Ω = ∼ , . . . , u 6 ; ,..., h : 4 , , 1 π 1 . The multiplication rule is the usual semidirect product → G 1 4 R u 6 x Z . As discussed near ( ) ) , . . . , h G − S 6 ) b Z (( ϕ 6 R Ω 1 ˜ u ; (34)(56) ; (35)(46) we choose the section , ∈ x , . . . , x (SU(2) 6 R 6 R → 1 ( 1 1 1 , and (˜ ,R Z h σ 1 x , , 6 1 ( 6 6 L 6 L (( ,..., SU(2) ⊂ 1 1 1 u × Z × H (˜ ∈ H . In other words it is understood that we identify: 6 L , = = = ∈ ,L ) L 0 2 4 5 6 ) ) 6 b b b g g g might be nontrivial. 6 diag 6 H and consequently Z SU(2) and Z SU(2) which fits in an extension: ∈ 4 , , . . . , x 4 should really be regarded as ( i = 0 , . . . , z , . . . , u 1 1 ˜ G u 1 via x 1 z , Z u i x 6 u has order 5. (( Now, over Now we choose a section of It is possible to give a very concrete description of the group = ˜ 0 5 b g 1 4 b where ( x elements will clearly involve aextension 6-tuple by of cocycles This section splits theIt sequence is over an amusingg exercise to verify that rule. As mentioned above,a to quotient obtain by a group acting effectively on the CFT we must take where ( elements in with Note that We will now argue that the central extension ( where structure of a nontrivial central extension: SU(2) we encounter an extensionsection by is a 6-tuple ofa the group cocycles ( to the full GTVW model by JHEP05(2020)146 )). can ) of ]. The as a v 0 0 ( 84 (5.33) (5.32) where 2.23 . Note v 11 Z ) is the ) are all is of the LR 20 v 6= ( diag H M v ,L Z 0 6 5.34 : LR = 1”. appears as a In fact, every 9 H . Now denote H 0 0 F ]. Although the v 9 ) for Z 0 v 41 groups 6= ( v LR is the projection and define H ] (5.34) F ) for R 0 v . The group → ) do form a subgroup thanks )) ( 6 6 .) On the other hand, every F 4 2 6= 0 obtained in [ H : ; 1 0 LR vx Z 5.34 = 0 ( 6 R form a subgroup, even when we , each of which is isomorphic to 20 H xy 1 4 ,R ) is indeed nontrivial. First, note , 0 6 xy M 4 over the “part with × we will define sub : G not ) is nontrivial is to note that since = 1. In fact, all elements of L π 5.25 8 ∗ 4 × H , . . . , h ) and or and )) F 1 ) do ,L 6 v 1 z 0 6 ) of ( 5.25 ∈ 6 v , y H y y vx ( and the commutator function is 6 Q 0 LR v ( : x 0 6 = 6= h ( 0 LR H ∈ ( x x , (2) ) shows that every element of H – 31 – p L . The square will be a nonidentity element in is isomorphic to )) ) is isomorphic to ∈ H 6 ) acts, via conjugation, on )) v ,R 0 1 ) x ( 6 0 6 v 6 , . . . , c ( A.14 Z ( − 1 x ) is an involution. Consider the square of the lift of ( For each 1 where provides an example. LR ( as an Abelian group. Similarly, the restriction to the LR π × H , y H ,R 6 2 ,R ◦ 1 10 H 0 6 0 6 Z , . . . , y . This implies the extension is nontrivial. x ) = ,L , . . . , h 1 ( (2) 0 6 ) Z , . . . , h c y p 1 ∼ = ( 1 H × H × H x x, y ) x ( . However the group elements ( ( := ( v 6 c h ,L ,L ( h 0 6 0 6 . Then the commutator of the lifts of elements in (2) and ( [( 6 R defined by the condition π . and the permutation invariance of the cocycle (see equation ( H H subgroup. LR 0 6 ∈ H (2) H ) Z π = 1 6 diag ∈ H = 0 defines subgroups ⊂ Z R ) 1 . Each subgroup 6 0 )) x ) is a nontrivial central extension it is in fact isomorphic to 2 6 the subgroup defined by the fiber of Z is order two so it is impossible to produce a nonidentity element which is a is Abelian it suffices to check if the commutator of lifted group elements x , . . . , x 4 diag (˜ , 1 noncentral 4 Z x 5.25 ,R /Z , . . . , x 0 6 ) 1 6 ⊂ G x is a Q , . . . , h 4 , ) × H 9 1 4 1 × We can now describe the relation to the group 2 A second way to see that the extension ( There are two ways to see that the extension ( to be the kernel of For an explanation ofThe this result relevant isomorphism the was reader might discovered wish by to T. consultTo Remark Johnson-Freyd, be 5 and more in we precise: section thank let 14.3 him of for [ extensive dis- G x ,L 6 9 . Now it is easy to check that 4 (˜ 0 6 , 10 11 4 2 1 4 Q h by cussions and clarifications related to this. G Again, using permutation invariance ofinvolutions. the cocycle, Indeed, the elementssubgroup of the with form ( Z nontrivial automorphism, and similarly for where the square brackets denotethat the the equivalence group class elements underrestrict ( to the ( quotient by to the division by extension ( the 2 ( group of elements Again, a simple perusalgroup of commutator. equation ( ( form: where ( perfect square. Thereforethemselves there perfect exist squares involutions in whosenonidentity element lifts in have squares which are not H is trivial or not. The story is the same for left- and right-movers so we might as well take that every element of any nonidentity element in the subgroup be obtained in thiselement way. in (Note that JHEP05(2020)146 . . ) 0 4 2 4) v =1 , and Z ( ] α R ) α (5.41) (5.40) (5.37) (5.38) (5.39) a LR 2 ) 6 ; H y = 2 0 v α ( -symmetry y R that appears ) obtained by + R α 20 , . . . , h x ) 5.36 M 1 y : we can solve 0 v SU(2) 8 b ( h × L , and is a normal subgroup. A L to write a )) 4 , ) (5.36) that 6 F 1 , that is we can write 4 y ( F v ) acts nontrivially on 13 ) (5.35) there exists a solution. v v we need to solve 0 6= ): ( ⊂ G ( v ) of RR states. Comparing with 0 v α ) ( v 2 0 , . . . , h LR 6= LR vs. the group 2 ; ) v 1 v ( H 2 LR H . y 9 2 20 ( : and this element can be interpreted as 5 H F. Z 20 h 20 0 LR M [( A ] only addresses the commutant of (4 ) with :( ) with M 0 = 1. That subgroup is isomorphic to · : M ∼ = : ): ] v that acts trivially on the space ( ∗ 4 v : : 40 v ) we can restore ); 1 9 1 . It is known ( 4 2 ( ) F 0 R × H ( 0 z 9 2 8 4 6 R v 0 Z 6 , 20 )) v ∈ 2 F 1 ( Z 4 LR Z 6 is noncentral. 0 LR ( 0 0 , 6 H G M ∼ = 8 H – 32 – ∼ = 6 L vx ∼ = 0 LR ( 1 ∼ = v, v with 4 4 20 , , × H 0 × H − 4 4 20 ( M Z G G Z × H Z ] M , . . . , h ⊂ R ) Z 1 ∼ = )) 0 6 4 b vx Z , ( ( ∼ = 1 4 h 4 G ( , , 4 . To see this note that for each L G 6 , . . . , h )) ) ) and the theorem of [ 6 1 x b 1 ( ( ∈ H ; h 1 ( is central. is obtained by restriction of the semidirect product ( , , where in the semidirect product 9 2 x, y 4 L . Each of the subgroups 0 ]. Note again that the 2 , 20( , . . . , h Z 0 4 v )) Z ) can be generated by 6 41 G 1 12 ⊕ a ⊂ and x 6= 2 ( is . On the other hand, ) ( . One can easily check that for ∗ 4 5 2 ) for the RR groundstates in the GTVW model we see that to compare results Z h 2 v α with the subgroup 4 Z , F b Z ; [( 1 4 · 2 ∈ 6.1 Z = ∈ G z , . . . , h z ) α v 1 . Thus, = y 0 http://brauer.maths.qmul.ac.uk/Atlas/v3/misc/M20/ a R v ( ]. The space of RR groundstates decomposes, under the SU(2) F h Finally, we comment on the difference of 2 Now, since the fibration trivializes over + See To make this completely explicit, the main point is to note that for all 41 [( 1) α 13 12 − for some vx ( in accord with [ equation ( we should only considerThis subgroup, the subgroup of replacing The “extra” where only in [ group as ( supersymmetry that acts trivially on the subspace ( for any and hence we obtain and so we can identify We can now make contact with the group complementary group can be taken to be for any pair by conjugation. center of JHEP05(2020)146 . 2 6 u ⊂ as R s S Z e / 6 K that 4 (5.43) S diag (5.45) (5.46) (5.44) R D and R ]. The defines : Z ,p T 4 Z 6 L 6 v 41 p S s × : × = 1, by the 34). The L 6 . L α Z e K α 4) supersymmetry Q is the projection to , the fiber product is p . It is included for the H 4 , 4 → G act without central exten- which is still an involution. 2 . i 0 5 where G 5 } b g : . The symmetries in equation ) . Its lift to SU(2) 0 (5.42) R 2 does not act effectively on the 5 2 diag g F ψ ( A 6 : R → 2 /Z S ψ 0 6 Z , . . . , g : F 0 and H + 1 the factor is Z R : g × 6 h ) = e H α 0 6 α K 1 L ⊂ s s g Z . Since they are purely left-moving they ( R ) → +˜ 1 is the largest subgroup of 6 α ,p = ,L 1 ψ and while less obvious is also a half-period s α SU(2) 0 6 | L → H G ) p α 2 : H s e diag × R K p,T 6 1 =1 , g Z ,p Y 1 α ψ , . . . , L α g L – 33 – 1 e ( → p K { ( × Z = L ∼ = 2 . But then the diagonally embedded subgroup → 6 4 G α , s α 1 2 4 S : G ,ψ 62). This lifts to the element α . 1 as 6 ψ Q ˜ s × e V 1 G ⊗ SU(2) s ] and have a geometrical origin as half period shifts in the V where group elements that involve nontrivial permutations of factors 41 . One can check that that fits in an exact sequence 6 Z S e ] correspond to (the lift of) K : this factor is equal to 1. If ˜ via the scalar 6 41 acts on α s s V R Our symmetry group will be a quotient of the fiber product Z = α × 4) preserving symmetries with the specific transformations studied in [ 14 elements (34)(56) and (35)(46) above correspond to the transformations 41) of [ s . . , L 6 6 (4 respectively in [ orbifold description of thepermutation GTVW (25)(34) sigma corresponds model. to shift. See See their their equation equation (4 (4 Thus these elements generate a group isomorphic to (4 S If ˜ definition of acts ineffectively. Thus,that the we group have identified of is symmetries properly preserving described (4 as acts on Z a fiber product of S The reason we must takeGTVW a spectrum. quotient The is reason that is that because, as we have seen, thesion. lift When of elements combining in left-moversof with Spin(4) right-movers we aim tomust produce act a diagonally subgroup on left- and right-movers. We therefore view Spin(4) benefit of fussbudgets. The subgroupthe of first the hexacode holomorph digit of is the hexacode thea semidirect that subgroup product preserves It is also instructive to compare our description of the generators of the group of Here is a slightly more conceptual description of the group Recall that given groups and homomorphisms 2. 1. 14 Remarks: JHEP05(2020)146 ]. 41 and 2 (5.48) (5.49) (5.47) . Our lattice 0 6 Z because 4 / H 4 D F D ) as well as T 6 i − H F diag , g 3 5 /Z ]. The symmetries Aut( Z ) is then 6 41 S ⊂ ): : 2 5.46 F , . . . , g in [ Z 6 1 j g × t h i × i 2 t 0 g Z in ( R F 23 (5.51) diag ∈ h ∼ = : ϕ · i on the left-movers and right-movers ϕ /Z 6 11 5 (5.50) 1 ,R ϕ · · 6 0 R 7 2 h− e · ): K 3 / . The analog of ( and i · 5 × H 0 R R 1 R · , by the lift of the holomorph of the hexacode ,p e 0 17 4 K ,L 5.25 3 b L g 0 6 R G 3 p h 1) supersymmetry. As discussed above, these · e , : ] to diagonally embedded elements of H – 34 – K × = 2 = 1 right-moving supercurrent then we can drop | 10 L 41 1 , N · ,R e 4 K 0 6 Z = 2 |G H corresponding to or half period shifts in the | / = ∼ = 4 24 v ,R 1 1 39) of [ , , + 6 . 4 M 4 . Call the lift 2 | v 6 G H G ) we replace s is 12 times bigger. In particular, , 2 ∼ = 1 1) supersymmetry within Spin(4) ∈ H as a quotient group, by Lagrange’s theorem. , v can readily be generalized to produce a group of symmetries of 5.46 , 4 4 + , R G 1 4 24 v w G s M / -moving N=4 superconformal algebra. 5.4.1 corresponds to the symmetries denoted 1 , is a normal subgroup of 4 4 Z left G , 4 , corresponding to rotations acting on the Kummer surface 2 G . Specifically we now include elements γ 6 S : and 6 cannot have 1 1 . We will not give a systematic study of the full vector space of such “new” ellip- , of symmetries in terms of the holomorph of the hexacode. Our group γ the symmetries are related in equationdescription (4 is evidently less geometric, but has the benefit of unifying the treatment are nongeometric symmetries. See the discussion at the end of section 4.2 of [ 1 4 , Note that If we only aim to preserve a single 4 G G 5.5 The “new”In twined elliptic this genera section wein define a fewtic elliptic genera. genera associated to some of the “new” elements For comparison note that So, symmetry of the In this sense the group Note that we cannot makethese use have permutations of that other change automorphisms (saying the that first the digit. permutation Because imagemust of of the be the fiber the action product of same) structure such automorphisms of the CFT do not commute with the SU(2) R- The group will have structure analogous to ( the restriction that ourcanonical description symmetries of act ( triviallyto SU(2) on the firstarbitrary translations hexacode digit. Using the 5.4.2 Stabilizer ofThe (4 ideas of section the GTVW model thatshould still commute be with relevant (4 to the degeneracies computed by the elliptic genus. JHEP05(2020)146 (5.60) (5.53) (5.54) (5.55) (5.58) (5.52) that contains . In fact, they 1 RR , f 0 H , f ) ) is the action on the z g ( ˜ ( H 1 ¯ q U , f H . We will make the explicit ) )) (5.56) (5.57) q i 0 6 z ) 0 z z 4 1 ( α ( ( H 0 ( f zJ 6 define 0 1 i / 0 f e f f V 6 π f 1 on the second summand in the 2 ≤ 5 ) 1 H e ⊗ − ) (5.59) z f ) ) ) 6 := := ) α 1 ( ) s e 1 6 ¯ ) g ω 1 ) ) z s , F ( , ω , 0 ≤ ( f )+ τ τ 1 , ¯ 0 ω U α , 2 2 0 ) ) ( f , ω , − R ¯ + 5 , ω τ τ 1 z, z, F 4 , 0 0 ( ( 0 , − , ω ⊗ · · · V , + f ¯ η η ω 1 F (2 (2 ⊗ · · · V 5 ⊕ V 0 1 0 1 L , , 3 2 s , , ω ) f f 1 F ¯ ω ϑ ϑ ) ) s 0 1 α V , ω , ( z z , , ) generated by 1) V – 35 – ( ( e ω ω V = = = 1 0 − ) = 2( = ( f f e z, τ ⊗ H H ) q q ) := ( := (¯ := (0 := (1 )+ α RR z, τ 0 0 ( α α we can define a twisted elliptic genus: ( 1 0 ( H ¯ ω ω ( V ) := ) := zJ zJ 1 . The relevant subspace of the RR states is then . Similarly, define i i 1 V w w V w w , E α α π π h 4 (0). The twisted elliptic genera for group elements with 2 2 1 G z, τ z, τ e e ). To this end, for 1 6= 6= f ( ( ) we begin by isolating the subspace of 1 0 =1 1 4 0 1 V V 6 α β β ) := Tr F F h, := ⊕ Tr Tr 5.52 1 z, τ f ( g ) = ( E is +1 on the first summand and ˜ h = 1 for = 0 for we choose coset representatives of R β β h, 4 F s s F + is normalized to have integral eigenvalues and (0) and L ∈ 0 F 3 0 f x J 1) − := = 2 = 0 and = 1 and 0 is an automorphism in f 0 α α J s s g Now, for Now, for a single Gaussian model we have In order to compute ( If = 1 are in the ring of functions of ( L For example, the elliptic genus itself is just choice: g will be in the linear span of the functions: also let Note that ( expression in square brackets above. where where R-moving groundstates, that is,sector the states subspace with of ( the GTVW state spacethat contains RR where GTVW space of states.a The Jacobi supersymmetric form cancellations of will weight continue zero to and hold index so 1 we for get a suitable congruence subgroup. JHEP05(2020)146 , ) 4 , ), 4 36 , 5.64 5.52 − G (5.62) (5.63) (5.64) 1 35 , and the , 4 G 1 . There is 11 2 n − H ) , the function 1 ··· ). If our “new” 1 then we define, 24 = F 1 F + dim 4 ω F 2 , M 1 1 2 4 / 4 − + 5 1 / F = ¯ , . Looking at ( ∈ − 0 4 +5 0 0 4 1 21 x F / 0 g ∈ G F F χ H ( F − 21 g ( ) = − χ g ). The representations of − ( 1484 n 2. This shows we have not 1 determined by [ every ) 4.6 H 1 − − 1 ) (5.61) ≥ F 4 F / + 1456 ω n 1 4 17 − z, τ / + 5 F = ( χ 0 4 ∗ ∗ 0 17 )) x F x F . These give 4 χ ( n 1 ∗ w − 543 ω ( · 231 770 − H h − 3 45 a = ¯ 1 ⊕ ⊕ 4 − 0 · / + 564 g − ⊕ ) ) 1 1 and 2 4 , x 13 (1;ˆ / F F · 2 χ 23 − 45 231 770 g +(001111))) we find / 13 E 1 4)-preserving automorphism. Moreover as noted x – 36 – , = 0 symmetry of K3 sigma models. We will argue χ = = = = = , 0 + 5 + 5 0 = 1 w 210 = 4 characters: 0 2 1 2 3 ( a 0 0 , / x ) by looking at a few examples. We begin by noting 24 0 is zero. On the other hand, h − F F 1 ,H H H H ) := , N a ( ( 0 4 0 H , M 2 ) 4.8 / + 196 in this representation is 0 ). If we apply this formula to / − − 6 and 9 H 1 as a (4 z, τ 4 , ⊗ H g χ ( / 0 into C ω 9 1 have the property that, for z, τ ) ( 49 ) χ 1 1 ch a,x g 1 1 ≥ = F E F − a,x F F g n define: 3 4 , x / + 56 + + 5 = (1; (Ω = 0 4 5 −E − and 4 0 0 F χ g / x 0 F 0 5 F = 0 ) transform as Jacobi forms - precisely analogous to those in ( F ( ∈ 15 F χ element then we would need to have Tr 2( ) = , with they indeed coincide. a − x n 4.7 − 4 , 24 2 1 = 11 “new” elliptic genera for each of the “old” elliptic genera. z, τ 4 / + 12 ,H ( M 1 2 and , − 0 ,x , / 0 = 0 = 1 = 2 0 1 acted on the CFT ∈ G 6 , χ +3 0 χ a a a Z g a g 24 − E = 1: ∈ ,H M = = 2 0 g a , were an 0 1 0 conform to the expectation ( ∈ ) defined in ( ), for F F ] are no such group element. we see the coefficientcharacter of of every element made some error by confusing traces with supertraces. g H g Next consider Now note that for We now explain that the twined genera associated to the “new” elements in The first few representations One might wonder whether the new group elements we have found are related to a A small computation leads to the following table of “new” twisted elliptic genera for For , 4.8 z, τ not 45 2. 1. ( , 24 g ˆ do the decomposition of φ as if in ( 39 subquotient of thenow still-mysterious that they areM not. Recall the discussion near equation ( the case Note that in principle, 12 JHEP05(2020)146 to = 1 R (5.69) (5.70) (5.71) (5.72) . The N 6 P with b Q SU(2) 24 × of the M L 6 ∈ S = 1 supercurrent g : N 6 exists. ) g α !! !! ( R ) ) ) 6 6 2 ( x x ( ( h h =1 6 L 6 R 6 α and we denote it by 2 matrices over the quaternions. ) 0 1256 (5.65) (5.66) ) 0 25 6 6 51217 (5.67) (5.68) · · ⊕ ⊗ that no such × x x − − ) ( ( 0 0 α h h ( L 24 ) 6 L 6 R ) = 4 ) = 4 ) = ) = 2 M ( g g g g . The stabilizer of the R

( ( ( ( ∗ ∗ ∗ ∗ Ψ =1 2 . We would need to ﬁnd a 6 α 45 45 / 1 − 3 231 231 ⊕ ⊕ , ⊗ F ,..., ,..., 0 ⊕ GTVW ⊕ L – 37 – 45 45 ! ! ∼ = Ψ ) ) 231 231 Tr Tr 1 1 ⊕H 2) x x / Tr Tr ( ( 3 , 0 h h , (0 GTVW 1 L 2 1 R / (2), the algebra of 2 3 GTVW ) 0 ) 0 H 1 H 1 ⊕ H we would require, at the massive level x x ⊂ ( ( 0 = 1 supercurrent within a group that includes parity- 0 1 h h 0 , F 1 R 1 L 2 3 N / exists. 3 GTVW to include parity. We do this by extending SU(2) − g H 0

F 5.1 . This is a non-abelian group of order 2 4 indicates that we have included a diagonally-acting parity operation. The F P ∈ i x reversing operations Again, no such it is easy to see from the character table of Let us consider again the character 4 Similarly, for In some discussions of Moonshine authors will distinguish massless and massive states. used in section 4. 3. 6 where subscript above group acts naturally on together with In order to see theQ full symmetry of the RRPin(4) states and we need viewing to Pin(4) Viewed extend this the quaternion way we group are lead to consider a group consisting of elements The potential supercharges live in the subspace of the GTVW space of states: As we will see inRR the next groundstates section, of the the full GTVWright-moving structure supersymmetries model of the only and Golay becomes study codesupercurrent apparent as the based when a stabilizer, symmetry on we within of combine Pin(4) the left- and 5.6 Stabilizer of an JHEP05(2020)146 (6.1) (6.2) ) (5.73) 6 ) 6 SU(2) can ∈ H ) × 6 ∈ H . !! ) ). The resulting ) 6 6 D = 1 supercurrent H !! x , ) ( , . . . , x 6 N 1 h (1 x x 6 L ( U , . . . , x h representation, has the 1 ) 0 × 6 R x 6 6 ) ) x ) 0 ( H R 0 6 , h x ) 1 & ( 6 R ( (1 0 α ( − h ) U ) ) ) ) 6 L

i i SU(2) 2 = ; + +

= 1 & ( 2 × i R ( admits a canonical real structure and i R 2. The main signiﬁcance of this group L ]. It is determined by the quaternionic ,..., | − −i | − 6 =1 =1 R × ! 41 i Y ) that preserves the 6 α ,..., + + 6 ) =1 3 i Y 1 i i − | − −i ⊕ 4 ! x ) −i − | − −i ( 5.73 1 ∼ = × h SU(2) x + + + + + + 4 5 1 L ( / | | | | – 38 – 2 1 & 1 × h ( ( ( ( , ) 0 4 1 R 2 2 2 2 − × L i i / 1 1 1 1 GTVW 5 √ √ √ √ x = 1 & = ) 0 ( 0 H 1 can be determined to be: h i L i L = = = = x 1 R ( 6 P = i i i i 0 h b 3 4 1 2 Q 6 6 | | | | =1 =1 1 L i i Y Y RR

) of SU(2) V 2 ( ; 2 q within ( R Ψ ) = R − Ψ L − L (Ψ also acts on the space of RR ground states according to a pattern governed by 6 P . There will also be a larger group making use of the “rotational” automorphisms b Q R R Ψ Ψ The space of RR ground states, as an (SU(2) We should note that above we have only discussed the “translation symmetries” of − Stab − L L This slightly peculiar basis ofstructure states and appeared in in [ particularunderstanding the is above compatible remarks with more the thoroughly can real consult structure. appendix Readers interested in Now the representation ( the resulting four-dimensional real vectorbe space, identiﬁed as with a the quaternions, representationcanonical of as basis a SU(2) is: representation of the Golay code. Thebegin pattern by emerges explaining when this we distinguished use basis. a special basisstructure: of RR states, so we be of the hexacode. We have not explored this larger6 symmetry in detail. RR states andIn the this MOG section construction weΨ will of show the that Golay the code group ( This is a non-abelianwill group be of order apparent when 2 we consider its action onΨ the RR ground states. determined by Ψ JHEP05(2020)146 ). (6.5) (6.6) (6.7) (6.8) (6.9) (6.3) (6.4) )) = 2.43 0 and x ( h ( ↔ ) we arrive LR ρ 2.53 ) to ( on the distinguished 6 4 , on the canonical basis 4 F !! !! 2.48 ) ) F . This array can, in turn, 6 6 ∈ 2 ∈ x x F ( ( 1. We convert + w 0) deﬁned in equation ( x , h h 1 along the diagonal. We will 6 L 6 R x, ( ) 0 ) 0 i,α i,α + 6 6 g )), for V V )), for x x i i ( x ( ) ) 0 0 w ( 2 h h α R ( α R h Z 6 L 6 R (      α L α L , h , , × ) −

α LR α 6 w x x ρ ( ( (      − 2 h + 0 0 1 1 − g g F + ,..., ,...,      – 39 – 1) 1) × ! ! + − − ) ) 6 array of elements of + 4 ( ( 1 1      F x x )) acts in this basis as the diagonal matrix × ( ( → → h h (1 factors through to an action of an Abelian group 1 L 1 R i,α i,α . , h 4 ) / ) 0 V V 24 2 ) 0 1 , 1 1 (1 F 4 x x ) when acting under the diagonal embedding h / ( ( 0 1 GTVW 0 x h h ( H 1 R 1 L h are neatly encoded in the map on R

6 P 2)) in the distinguished basis we find an action of the Golay code — in the b Q / 4 / (1 1 , 4 ⊗ / 1 GTVW 2) H / )) will act diagonally on this basis, with entries x ( ((1 1. Thus, for example, ( , h ∼ = ) Group elements of the form Next, when we act with the operators ( The action of The Pauli matrices x H ( h acts as Comparing with the descriptionat of one the of Golay our code main in statements: equations ( while group elements acts on the distinguished basis of RR states as: Acting on following sense. of basis of the RRbe sector we identified obtain with a a vector 4 in In this way, the signs appearing in the action of and we summarize that action by the column vector ( represent such a matrix− ↔ by a column vector with entries 0 JHEP05(2020)146 ] s V 29 ) in , 6 . (6.10) 28 H + G = 1 supercurrent “symmetry” within N = 1 supersymmetry 24 N M defined above. This stabilizer . is exactly the largest sporadic 6 P s indeed defines a superconformal b Q in the distinguished basis defined ) s R V Ψ RR V − L (Ψ ]. 6 P such that b Q 29 dimensional space of potential supercurrents s , – 40 – 12 28 Stab within a left-right symmetric version of the quater- R dimensional represention of Spin(24) with a natural Ψ to define a supercurrent. RR V 12 s − ρ V L ∼ = Ψ G . Using the representation theory of the Conway group [ s . This then, gives a new interpretation of an 24. This theory has a Spin(24) automorphism group. In the Ramond 24 M ,..., , and therefore there is a 2 3 2 WZW model and the quaternions, defines the Golay code. In equations, there = 1 6 = i , ] since very similar techniques can be used to construct the 1 i 16 ψ × SU(2) 29 , Consider the stabilizer of The Conway Moonshine module can be thought of as a theory of 24 Majorana-Weyl There are two natural directions in which the above could be extended: first, we have This gives a physical interpretation of the MOG presentationWhat is of the significance of the this result? Golay It code. is well-known that the automorphism group of 28 = 24 current, and moreover, the stabilizerConway of group, Spin(24) acting thus on identifyingpreserving the automorphisms. Conway group What as wethe a will Golay group do code here of andimply is the show, construct required in the identities spinor an for explicitly elementary using way, how the properties of the Golay code spinors sector the ground statesreal form structure. a The 2 vertexh operators associated tolabeled these by states Spin(24) have spinors conformalshowed dimension that there is a distinguished spinor The methods ofin this [ paper shedthat some plays light a on starringand role the explicit in Conway some the Moonshine points analysis module left of implicit those studied in papers. [ We are merely making concrete elements acting eitherpermutation on matrices. the right or on the left.7 The action Superconformal will symmetry then and be QEC in by Conway signed moonshine walls will shed further light on this question. once again left off anythe investigation holomorph of of the the role hexacode.action of on the Second, “rotational” we the symmetries limited canonical Aut( discussion to RR the basis. left-right symmetric One could of course, also consider the same group the Golay code is a K3 sigma model. Wegroup put of “symmetry” a in quotation symmetry marksgroup group. because of It it symmetries” is has is for an not twinedis automorphism clear elliptic possible to genera that and us the the what emerging space implication ideas of such massive in BPS a the states. context “symmetry of It of “generalized a symmetries” and domain Note that, if wesymmetric do action not of consider the the group extension preserving by Ψ parity, then and we only obtain consider the the even left-right- Golay code nion group (or group of errorgroup operators), namely is the a group non-Abelian groupby which, the when acting on is a natural isomorphism JHEP05(2020)146 ) ). x, (7.8) (7.9) (7.1) (7.2) (7.3) (7.4) (7.5) (7.6) ( 2.53 (7.10) (7.11) (7.12) (7.14) (7.13) − f . The key 2 we use the ) to ( ) or F ) 2 let ··· , x, 2.48 by 2 ( 24 2 x + + F l 24 2 f +( F ψ (2) g ∈ k γ ψ w j with itself will define a ψ for i s the cocycle is trivializable 4 ψ V s 2 24 ) and we define F ) ijkl 12 6 γ z , . . . , γ , j 1 2 x, 1 6 tr w w x γ 4 going from top to bottom in the s G ⊂ i 1234 2 α , + γ +( w 1 = 1. Now, for ) 3 + (6) g i < j , w α 24 2 i γ = x 2 γ w T i

g g /p 12 Z holomorph of , . . . , x − 1234 1 itself where the first factor acts by (say) left-translation. γ 2 ab cd ef x G = 2 − 1. This shows that :( . To describe these it is useful to arrange a 6-digit word in 1 6 0 D − g ) = ( P H 6 = − we define the P , . . . , x 1 G x indicates the group ( of couples. We can take as generators: that preserve ) acts naturally on m indicates a group with normal subgroup 6 4 p G F and one generator would be pairs A.B 3 denotes a group which is a semi-direct product of , Z B Next, there are some simple permutation symmetries, i.e. subgroups of the natural The hexacode has some very useful symmetries,First and of in all, this being appendix a we linear review subspace the of For any group as 3 couples: ( B ∼ = : 6 0 H flipping H action on structure of it automorphism group Aut( multiplication by any scalar,denote and the if group the of scalar scalar is multiplication nonzero by this nonzero is elements an of automorphism. We where the semi-direct product is definedgroup using Hol( the natural action of Aut( A good example isis the isomorphic group to of the automorphisms holomorph of of real the group In this paperparticular, we adopt notationand for finite groupsA and their extensionsa used full description in requires [ specifying a homomorphism The first term preserves allralities. column The chiralities first and term the can second be term restricted changes to all column chi- A The automorphism group of the hexacode and the even Golay code We can write: projection operator with all column chiralities = onto the space of spinors with all column chiralities = +1 and JHEP05(2020)146 (A.8) (A.9) . One (A.11) (A.10) . One 4 3 16 S S ). ∼ = is a central ∼ = i 2 whose group 5 2 A.14 H 6 H : A ) (A.6) 5 1 ¯ ω x ), ω H , . . . , g 4 6 6 = 0 x 1 x g = 3 0 h x = 3 = ) and checking that the 1 ) (A.7) 3 5 3 , . . . y H + := x 1 x 2 y , x 3 3 2.29 05 x + 4 + x x iff ω H 2 2 , x 6 x 3 6 + + x ) (0 6 + ) = ( x ¯ 2 2 ω ω A 6 1 + , x x x · ω x + 1 3 ∈ H + ¯ ω ω 3 x ω 2 ) ¯ 6 1 ω Z , x x 6 . Thus, + + ¯ x S ω 2 = = ¯ 6 1 1 , . . . , x ω ∼ = x + S 3 3 ¯ 1 ω x x ¯ → i ω 1 y y x = ω 5 , ω ω ) ⊂ x ( 1 6 3 + + · , . . . , x 6 x y 1 2 2 5 H A ω ) = ¯ ) = y y x g ) = ( + – 45 – ω ω ω ω , . . . , g )( = (13)(24)= (35)(46) (A.4) (A.5) 2 (1 ( (¯ 0 ]. 3 3 3 1 y → 3 4 + + ¯ g 1 . Another subgroup of permutation symmetries is ,x ,x ,x g g h ) 68 : Aut( 1 1 2 2 2 2 + 6 1 y y Z p ,x ,x ,x 1 1 ω ω 1 1 1 := y x x x 0 × . Moreover, one can check by direct computation that the 05 2 0 ) = ¯ ) = ) = , . . . , x H Z = Φ = Φ = Φ ω ω 1 H of the automorphism group. One can check directly that the . By computing the lift of two elements in (1 ( (¯ x 6 4 5 3 3 3 3 ∼ = 4 x x x and together these generate a group would be: ,y ,y ,y S Z :( ) (0 1 2 2 2 to three basis vectors in equation ( 1 2 ¯ ω 5 ,y ,y ,y × H is a group of symmetries of the hexacode proceeds by applying ∼ = 4 g 1 1 1 ¯ ω H is just H just tracks how the automorphism permutes the hexacode digits. 3 y y y 0 3 . There is a projection Z p is the entire subgroup ω i p H H 5 ω ) above.) Then, letting p ∼ = = Φ = Φ = Φ 1 , . . . , g is a symmetry note that ( 3 4 5 6 by 1 is the entire set of nonzero words in the hexacode. See eq. ( y y y 5 (1 g H 2.28 6 g 3 , . . . , g × A under 1 H 0 normalizes g 5 h 2 × H H 0 H := H 5 There are further, “nonobvious” automorphisms of the hexacode. An example of such H For a nice discussion see Lecture one of [ 16 extension of commutator vanishes one easily checks that it is a nontrivial central extension, so where the image of Clearly, the kernel of image of Let (See equation ( an automorphism is To prove that orbit of the four “seed codewords” under Clearly way to prove that the generators resulting vectors remain in thesubgroup hexacode. Together with scalar multiplication we obtain a obtained by arbitrary permutationchoice of of generators couples. of This defines a subgroup these generate a group JHEP05(2020)146 is i F (A.14) (A.13) acts as , g 5 F g is identical 2 generated by , . . . , g x 2 1 → → → → so we have now → → g Z → → → → → → → → → → → → → → → h → 1 1 6 ω ω ) (A.12) × 6 0 0 1 1 0 0 S 1 ω ¯ ¯ ω ω ω ω ω ω ω ω x ωω of ¯ 00 2 · x ¯ ¯ ω ω ω ω 0¯ 1 ω 0¯ ¯ . ω ω is an automorphism 1 1 1 ω ω ω ¯ Z 3 p 6 ω ωω ¯ 0 ¯ 0 1 0 1 0¯ 1 ¯ 0 0¯ ω 1 ω 11 ω ω ω ω ω ω ω ¯ ω S Z ω F ¯ ¯ ¯ ω 00 ¯ ω 0¯ ω ω 1 ω 0¯ ω ∼ = · 1 1 1 g ω ω ω ω ¯ ω ,..., ∼ = 1 1 0¯ 0 ¯ 0 0¯ 0 0 1 1 ¯ 3 1 ω ω 1 ¯ ¯ ω ω ω ω ω ω ω ω ω ¯ 1 ω 11 ¯ ¯ ¯ ¯ ¯ ) 1 ω 0¯ ω 0¯ ω 0¯ ω 0¯ ω 1 ω Z ¯ ¯ H x 00 ¯ ω ω 6 (¯ → → → → → → is · ∼ = H → → → → → → → → → → → → and conjugation by 1 1 → → → ω ω 6 i ¯ 0 0 0 1 1 0 0 and ω ω 1¯ 11 11 ω ω ω ω ω ω F H g ¯ ω 00 ω ω 1¯ 0 ω 0 3 1 ¯ 1 1¯ ωω ω ω ¯ ω , g S ¯ 0 0 ¯ 0 1¯ 1 0 1 0 0 ω 1¯ ω ω ω ω ω ω ω ω ω ¯ 5 ω ¯ 0 ω 0 ω ω 1¯ ω 00 1 ¯ 1 1¯ ∼ = ω ω ωω ωω ¯ ) = (56) ω ωω ¯ ¯ 1 0 ¯ 1 0 0 0 0 0 1¯ ω 1¯ 1¯ ω 2 6 ω ω ω ω ω ω ω ω 2 ¯ ωω ω ¯ ¯ ω 1¯ 0 ω 0 ω 0 ω 0 ω ω 1¯ 00 ωω ωω ωω H → → → → , . . . , g . To prove that → → 1 → → → → → → → → → → → → 4 ¯ ¯ ω ω → → → g ¯ ¯ ) and the result follows immediately. Since ω ω F ¯ ¯ ¯ ¯ ω ω ωω ωω , . . . , x · h 01 01 10 10 ¯ ω ω 2 1 ω ω ωω ωω ¯ ¯ 0 ω ω ω x of ¯ ¯ – 46 – ω ω ( 2.28 ¯ ¯ 10 10 01 10 01 01 10 01 ωω ω ωω 11 ¯ H · ω ωω ωω ω ωω ¯ x ¯ ¯ ¯ ¯ ω ω ω ω ω ∼ = ¯ ¯ ¯ ¯ ω ωω 10 ω ωω ω ωω ωω ω ω 00 11 11 11 00 11 11 11 00 01 01 10 10 01 10 ¯ 10 01 ¯ 01 10 ¯ 01 ¯ 11 ωω i → F . Thus, → → → → → → → → → → → → → → → → → → → → → 0 x , g does not commute with H 5 ) = (56) 6 F                                                                (               g 12 , , − , . . . , g 9 , 1 , , . . . , x , 8 1 orbit orbit , g − invariant invariant x orbits 5 0 3 3 , 2 2 orbit g 4 orbit S S 2 :( h ,g ,g 3 − ,g are each 1 1 3 S 1 g g F S exchanges rows g g 2 ) which contains all hexacode words with the exception of the trivial word ,g rows 1 1 g A.14 . On the other hand, 6 ] it is asserted that the full automorphism group Aut( S . ) = (56) is an odd permutation it is clear that the image under In computations it can be useful to have a full list of hexacode words. We provide this Another example of a non-obvious automorphism of 2 99 F , g g ( 1 described in detail the full structure of the automorphismlist group in of eq. the ( hexacode. 00 00 00. Theg words are organized into orbits of all of the nontrivial automorphism of In [ In the second equalityto we have the used Frobenius theof automorphism fact the that hexacode the we nonlinearp again map use equation ( JHEP05(2020)146 (B.1) (B.2) (B.3) 0 ··· n, + + m ) δ w with commutation relations w ) ( − m m 0 z ∂T s, − ,L r + 3 ··· ··· r + r G δ m n G 2 + + ( 2 ) ) 1) L ) / ) 2 c w 3 w w w 12 w w ( − − − ( ( n r − 2 T − − r + T − − r 2 z + z z 1 2 n z z ( ∂G (4 m + r + + + n c G m 12 – 47 – X X 2 4 3 L ). The relation between these two group actions ) ) ) ) ) r + n w w w w s c ) = ) = ( c − 2 3 2 z z + − − − − A.14 G r ( ( 2 for the Ramond or Neveu-Schwarz algebra respectively. 3 2 m z z z L T m G ( ( ( 1 2 ∼ ∼ ∼ + = 2 ) ) ) ] = ] = ( Z } r n s w w w ∈ ( ( ( ,L ,G r T ,G G G ) ) ) r m m z z z or L L G ( ( ( [ [ { T T Z G . 6 ∈ S r and it is quite interesting to note that the automorphisms of the hexacode act on Z ∈ = 1 2d superconformal algebra has generators on sets with six elements is related by an exceptional outer automorphism of the m N 6 S we have Here In terms of OPE’s of the currents: B.1 Superconformal algebras The symmetric group B Supersymmetry conventions two distinct sets of six objects.also The permutes first is the the set no-zeroes ofzero). six hexacode digits. Up words However, (i.e. to the automorphisms those overallcorrespond scale words to there where the are none firstof exactly of six six the lines types digits of of is ( no-zeroes hexacode words. They Remark: JHEP05(2020)146 = 4 (B.6) (B.8) (B.4) (B.5) (C.1) N r + doublet. ab with relations: a m δ R ¯ Q 0 i r m = 1 superconformal s, − + 0 r + ,T b m r G a r N ¯ δ ) − Q ∗ ∝ ,Q 1) ) 2 for unitary theories. Here m n 2 i ab − 5.18 2 r L Q n + ( σ 2 ¯ i ( Q − . In fact, this is a vector in the Z r 0 2 3 1 2 T 0 ] = β n, (4 T ∈ a r + = 2 k α ¯ + + Q ) ] = 0 k m † , 1 r a r ) δ x ) is another SU(2) + G ) (B.7) i ) ( ¯ n ¯ m Q s − Q 1 h ] we note that 1 T , L m + ¯ ∝ ( [ and i Q β r ij i m ,G -symmetry. 41 2 δ x,α − T T k + + R ¯ 0 c + [ , Q 3 i 0 ab 1 are antihermitian. 2 r n, σ 6 =1 m G + i Q 0 ) Q ( = 6 X α ( s 2 J m 2 ∗ 4 k – 48 – 1 2 c α F − − an eigenstate of ∈ r r mδ + X + = ]. There is a natural real structure on this algebra r x G . It is not difficult to show that, up to an overall a − + 2 k n 1 r r ¯ 2( a m r G Q + ∝ not − 34 := Q + + , = 0 Q m 1 1 − G = n b m ¯ L O α s } 33 Q n † + ) s b r Q + ) = + k n r m ¯ = doublet and ( a r Q i i ab m − † T L , r ) Q σ − R r a ( r ab G 2 1 2 nT ijk ¯ m + Q δ m G { i − − G = = 2 ∝ are real they will be represented by Hermitian operators in a unitary ] = ] = ] = ] = ] = ( } } i 1 i j 0 a r a r n n = 4 algebra. We thus require that a linear combination of n s b s b T Q ¯ ) has no nontrivial solutions we write Q ,T ,T ,L ,Q ,Q = 4 superconformal algebra has generators ,Q , N i m i m r a r a m m m = 1 superconformal algebra. Moreover, we require that under the real ) is an SU(2) T N L 5.18 T L L Q Q [ [ ) we have ( [ [ − [ , { { 0 N B.5 ,G + rotation the most general solution is G that completely breaks continuous SU(2) R To make contact with the notation used in [ As explained above we are interested in embeddings of the The small 2 ⊕ C Proof that thereTo are show that no ( nonzero solutions of Since ( Note that this linear combination2 is supercurrents satisfies the structure ( SU(2) Note that since the representation whereas in our conventions algebra into the Here it is traditionalwe are to using parametrize the by defined conventions of by [ JHEP05(2020)146 (C.7) (C.8) (C.2) (C.3) ] with a ] we get ∅ = 0. = 0 ) = 0 (C.6) ) (C.4) ) = 0 ) = 0 ) = 0 ) = 0 ) = 0 ) = 0 6 6 ,a , , α 6 6 4 6 6 5 ¯ , , , , , , ω ¯ ω ) ¯ ω ] is mapped to states [ ¯ ¯ ¯ ¯ ¯ ¯ c ω ω ω ω ω ω c c ω c c c c c c ∅ ( + (56), and this symmetry must be + h + + + + + + , ) = 0 (C.5) 5 6 ,α , 5 5 3 4 3 3 , , , , , , , ··· ¯ and ω ω ¯ ω ¯ ¯ ¯ ¯ ¯ ¯ ω ω ω ω ω ω c c c = 0 = 0 = 0 c c c c c c (34) 1 + , , 5 5 6 ¯ + + ω 2 + , , , + + + + + + , c ¯ ¯ ¯ ω ω ω 4 α α ¯ ω , 4 2 2 2 1 2 c c c ) ) , , , , , , − c ¯ note that [ ω ··· ¯ ¯ ¯ ¯ ¯ ¯ ω ω ω ω ω ω ω c flip (1 + + + c c c c c c (¯ + = ( ( ( + h h 3 4 4 O + flip 1 , , , 2 + + + , 2 , ,α , ¯ ¯ ¯ ω ω ω ,α 3 − − − – 49 – ¯ ω O ¯ , + ω 3 1 1 1 ¯ c c c ω ¯ , , , ω c ) ) ) ¯ c c ω c c ¯ ¯ ¯ ω ω ω 5 5 6 ( c , , , + + + c c c ¯ ¯ ¯ ω ω ω ( ( ( i( + diag 6 6 1 2 1 =1 =1 c c c , , , 1 X X α α and , ¯ ¯ ¯ O ω ω ω − − − − ¯ + + + c c c ω 3 ) ) ) ) , c = 6 3 4 4 2 2 4 := := ¯ , , , , , , , ω i( ¯ ¯ ¯ ¯ ¯ ¯ ¯ c ω ω ω ω ω ω ω O c c c c c c c − − flip diag acts on the spin states in Ψ just by multiplying by a phase, + + + + + + + 3 matrix is nonzero and hence O = O 5 1 2 1 1 1 3 , , , , , , , × 4 , ¯ ¯ ¯ ¯ ¯ ¯ ¯ ω ω ω ω ω ω ω diag ¯ c c c c c c c ω ( ( ( ( ( ( c i( O , Ψ = 0 we get 16 equations. For example, acting on [ 5 , ¯ ω c diag − O = 6 , flips one digit. So these terms cannot interfere. ¯ ω c flip O Similarly, with the bit-flip operator Some of the tedium of writing these equations can be reduced by recalling that Ψ is So, setting The determinant of the 3 just one minus sign.three minus signs. While The states equations like split nicely. [12] For are example the mapped coefficients to of the states following with just one or just reflected in the equations.find that In any case, by adding and subtracting equations we quickly symmetric under permutations of the couples (12) Altogether we 16equations equations. They are not all independent but we find independent while acting on [12] we get: The idea here iswhile that As a sum JHEP05(2020)146 (C.9) (D.2) (D.4) = 0. It x,a c as a real vec- H ) in this paper to x ( h . 6 = 0 = 0 = 0 = 0 2 1 2 1 , , , , ω ω ω ω c c c c ) (D.1) H − − + i + i 1 2 1 , , = 0 (C.10) 2 1 , , − 2 1 1 ! ! 1 1 2 GL( c c q , / c c ! 0 ) (D.3) ! ω q i 0 c 1 0 C + i + i + + 1 0 i 0 1 ( i 0 q 0 i − 1 0 0 1 − 1 2 1 2 2 = , , , , 4 matrix we find it is nonzero and hence SU(2) (D.5)

× / ω ω ω ω

2 2 , → c c c c ) × 1 0 ) this defines a group isomorphism: c H ψ − − q ( + i + i ) := ) := H ) := ) := 1 2 Mat = – 50 – , , ( 1 2 U ω ω , , 1 1 ): ) (0 (1 ( (¯ 1 1 1 U c c , 2 c c i i × → h h h h H ω ( ) , q c − − − − H 1 U → → → → H q = ( i j ( k 1 1 U T , 1 : c T [1] [2] [156] [256] be unit quaternions, and use the choice of matrices -linearly. Restricted to k SU(2) , R j , × i ) is the group of invertible linear transformations regarding H ) denote the group of unit quaternions. There is a standard representation: H ( SU(2) Now let U and extending by taking Here GL( tor space. define a map Let defined by follows that the stabilizer group is a discrete subgroupD of SU(2) Quaternions give a distinguished basis for the (2; 2) representation of Computing the determinant of the relevant 4 Taking into account theidentical symmetry equations of hold the for permutations of the couples, couple we (34) conclude and that for the (56) and hence all the output spin states give us the following equations: JHEP05(2020)146 (D.7) (D.8) (D.9) (D.11) (D.12) (D.13) (D.14) (D.10) SU(2) × can be written as V i ) above. Note the signs in 0 ˙ β 6.2 ; 0 ) (D.6) ) ) is the group of real invertible α ϕ V | V ˙ ) ) β 0 , 0 ˙ H β V GL( ˙ , ∗ β ˙ 0 GL( β ) / α α 2 ) ! u GL( X GL( 1 ¯ ( / / 0 i i w ˙ ¯ − 2 z i β α 3 4 ˙ 0 V ˙ i i − β β α 1 2 canonical, but the difference does not affect | | −| −| 0 ) Xu z w → 1 α, T R 1 | SU(2) representation is: αα u ˙

u β ) H ( ) = ) = ) = that × – 51 – i j : α, k SU(2) ) H = ( ( = ( = ( (1) = ( ˙ X → ϕ in that order vectors in β ϕ ϕ × i ϕ ∗ ϕ U SU(2) ψ,ψ ˙ ˙ α ) β ( β ˙ β ; α X × × α −} α ) X , | X ) H ( + 2 ( { : SU(2) , u U SU(2) 1 R u . This is just the (2; 2) representation with a reality condition. ( is that given above in equation ( V R i} ) is the group isomorphism induced by the isomorphism of real 4 run over | ϕ , ˙ this is i β 3 ˙ β | α, α , i X 2 | . In these terms the SU(2) , i C 1 ∈ {| is a real four-dimensional vector space and GL( V z, w Using this distinguished isomorphism we define Now we claim there is a unique isomorphism of real vector spaces On the other hand, we can define a real representation of SU(2) The basis the last two equations. So,the the way basis the is not Golay code appears. vector spaces. commutes. 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