Jhep12(2020)193

JHEP12(2020)193 Springer October 8, 2020 : December 30, 2020 November 16, 2020 November 17, 2020 : : : Received Revised Accepted Published Published for SISSA by https://doi.org/10.1007/JHEP12(2020)193 [email protected] , . 3 2009.11769 . -symmetry have been classiﬁed by Gaberdiel, Hohenegger, and Volpato. The Authors. 23 R . More recently, automorphisms of K3 sigma models commuting with Discrete Symmetries, Extended Supersymmetry, Gauge Symmetry, String c M 23

We consider symmetries of K3 manifolds. Holomorphic symplectic automor- , M [email protected] possible groups, all of which are realized in the moduli space. The groups are all SU(2) × 40 NHETC and Department of126 Physics Frelinghuysen and Rd., Astronomy, Piscataway Rutgers NJ University 08855, U.S.A. E-mail: Open Access Article funded by SCOAP ArXiv ePrint: list of subgroups of Keywords: Duality phisms of K3 surfacesieu have group been classified,SU(2) and observed toThese be groups are subgroups all of subgroups ofand the the classify Conway Math- the group. possible We hyperkähler fill isometry in groups a of small K3 gap in manifolds. the There literature is an explicit Abstract: Anindya Banerjee and Gregory W. Moore Hyperkähler isometries of K3 surfaces JHEP12(2020)193 21 5 16 24 8 2 19 . 14 9 B 15 compact oriented simply connected four- 4 13 – 1 – 16 6 8 20 smooth 1 13 will be a 17 X 1 16 In the literature one will find slightly different definitions of what is meant by “K3 With the exception of an example discussed in appendix A.3 The LeechA.4 lattice and its The automorphism genus group of A quadratic form group A.1 Lattices andA.2 discriminant groups Lattice embeddings 4.1 Classical vs.4.2 quantum symmetries Components of4.3 the low energy Application gauge to group heterotic/M-theory duality 3.1 Condition 13.2 implies condition 2 Condition 23.3 implies condition 1 Alternative formulations3.4 of condition 2 Finding of the lemma explicit list of isometry groups 1 manifold,” and the distinctionsThroughout between the this definitions paper willmanifold. be very important to us here. 1 Introduction and conclusion The study of K3 manifoldsand has physics. led to many Among fascinatinging their developments discrete in many both symmetry exceptional mathematics groups. propertiessymmetries of these This K3 manifolds paper manifolds: exhibit fills we interest- classify a the surprising hyperkähler gap isometries of in such the manifolds. literature on the C The possible hyperkähler isometries of K3 surfaces B Some K3 surfaces with infinite holomorphic symplectic automorphism 5 Future directions A Some relevant definitions and results from the theory of lattices 4 Implications for physics 3 Relating hyperkähler isometry groups to subgroups of the Conway group Contents 1 Introduction and conclusion 2 Some easy relations between the lists JHEP12(2020)193 . ) K to ) (1.1) X,J ( for an is not a X,J . When ( ) B G which is an is made we is a smooth X, g ( J X g → X we always mean the such that such that the canon- where : ) ) J ) have interesting groups f is always a subgroup of ) ) that acts trivially on the X, g X,J satisfying the quaternionic ( X, g ( ) ( 3 X,J ]. , X, g ( ( 2 26 , X,J ( acts as a group of symplectic auto- . = 1 k → G i J ) , i ijk J X,J + we mean a pair . A choice of holomorphic trivialization of ij . We will refer to this as the Xiao-Hashimoto :( δ ) such that – 2 – is a Riemannian manifold such that there is a f − such that there is an ) G . Some K3 surfaces ], chapter 15 for a nice exposition of known results. ) = that appear as the hyperkähler isometry group of X,J ( C j 18 ” we mean a diffeomorphism ; X, g ) of groups. J G ( i . The isometry groups of K3 manifolds are certainly of X ) . The aim of this paper is to identify what groups can J ( K3 surface i HolSymp 0 J X, g , L ( 2 K3 manifold . ]. See [ X, g surface ) = ( H below. as a i 3 admits an integrable complex structure 30 , 5 ) J K ∗ X, g , that is, X ( f 26 , X X,J ( 21 some , . We will refer to this as Mukai’s list [ .” The hyperkähler isometry group of ) ) isomorphism classes 19 hyperkähler isometry group for some specific , ]. is the subset of is the set of finite groups is holomorphically trivializable. When a choice of such a is the list of groups X, g 2 X,J such that ( 16 ( 30 , the , K g K3 manifold 14 14 one can assume that such a trivialization has been chosen. It is unique up to ) HKIsom max HolSymp HolSymp surface L some morphisms of list [ L proper subgroup of any other finite group which also has a symplectic action on that L While hyperkähler isometry groups are always finite groups it is possible for On the other hand, by a We will assume that X,J These are lists of 3. 2. 1. ( 2 have an infinite groupexample). of holomorphic In symplectic this automorphismsautomorphisms paper (see of we appendix K3 will surfaces. alwayscould Thus, consider compile: there finite are groups several of lists holomorphic of symplectic “K3 symmetry groups” we isometries of the group of all isometriesinterest, of but unfortunately wefrom will one not remark have in much section to say about them in this paper aside By a “hyperkähler isometryisometry of of and do appear aswe groups speak of of hyperkählermaximal isometries group for of some such K3 isometries. manifold Otherwise we use the term “a group of hyperkähler hyperkähler metric on triplet of parallel integrablerelations complex structures try of a K3That surface” is, one a couldone-dimensional mean holomorphic cohomology a diffeomorphism holomorphic symplecticof automorphism holomorphic of symplectic automorphismsliterature and [ these have been extensively studied in the ical bundle will refer to thedefines pair the structure ofof a holomorphic symplectic manifold.overall In scale, this and paper the when we choice speak of scale will not affect our considerations. By a “symme- JHEP12(2020)193 . C (1.2) (1.3) (1.4) gerbe which U(1) ConwayFix ]. L 16 ]. More precisely, 15 -field” or “ B ) is proper. This is the main as a proper sublist — it is the 1.2 is a flat “ , but as we just noted, we do not B HolSymp ConwayFix HKIsom L IsomHK superconformal algebra. The GHV list ⊂ L ∩ L 4) (considered up to conjugacy) whose action ConwayFix ⊂ L , 0 (4 ⊂ L for our notation for lattices.) Co HKIsom – 3 – HolSymp A L HKIsom is not a proper subgroup of some other subgroup ⊂ L L and contains = ]. The known results on symplectic automorphisms of HKIsom G L ]. That project is closely related to the work of Gaberdiel, 30 , 13 and max HolSymp HKIsom – 19 L , } L ConwayFix 11 as a target manifold. (Here 0 ]. Indeed, the GHV theorem gives yet another list of groups as- { L ) 17 9 , . (See appendix 9 6= has a nontrivial sublattice of fixed vectors. This list was compiled G G Λ Λ . The techniques we use are lattice-theoretic, and are closely related to X, g, B = Λ . It is obvious that ( ) 4 if H Λ is the list of groups which are maximal groups of isometries of some hyper- X, g ( with 0 is also a sublist of IsomHK ConwayFix kähler know much about this last list of groups. L Co This paper was motivated by some questions raised in a project with J. Harvey investi- Our work here was mostly motivated by physical questions about heterotic/M-theory We will show that More nontrivially, we can relate these lists to another list of groups, As we will discuss in the next section, simple considerations show that ∈ L 4. ⊂ SigAut 1, the remainder offor discussions the and paper collaboration isthe on unaltered. draft. related matters, We We as alsoand also well discussion. thank thank as G. This J. useful Höhn, workRutgers. Harvey remarks D. is related and supported to D. by Huybrechts, the Morrison and US G. DOE Mason under for grant useful DOE-SC0010008 comments to list of “classical” or “geometrical” automorphisms of K3 sigmaAcknowledgments models. We thank the refereepaper. for pointing Fortunately, as out we an explain error in in the our comments following statement our of revised lemma version 1 of in lemma v1 of this sociated to K3 manifolds, namely,sigma the model list with of automorphismconnection.”) groups By of automorphism the group two-dimensional weconformal mean field an theory automorphism commuting ofL with the the two-dimensional the techniques used by [ K3 surfaces were reproduced using the kind of techniquesgating used Mathieu Moonshine here [ in [ Hohenegger and Volpato [ and as a consequence we learnresult that of each this of paper: the inclusions the in list ( of possible hyperkähler isometryduality groups is as given well in appendix asphysics questions in section related to Mathieu Moonshine. We discuss the relations to H and in fact is the list subgroups ofon the the Conway Leech group lattice by Höhn and MasonG and we refer to it as Höhn and Mason’s list [ JHEP12(2020)193 . ) 3 C R 19) ; and that , ⊗ (2.1) (2.2) X (3 ( we can . By a )) 2 ) = 0 Z TX P R H 2 ; ∈ , and there f X, i ⊂ ( X J preserves the 2 ( = . Such vectors 2 P ) 2 H v, w 2 H R in the quadratic e − ; for P = X Aut( we may associate a ( ) that contains no root 2 2 . with ) α ⊂ R H R ; ⊗ )) Jv, w e, f , associated to on (19) ( Z X 19 3 with , ] ( g 3 i ∗ O ⊕ 2 α X, Γ f ω U [ ( × H 2 ) = corresponds to a choice of two- ⊂ ⊕ H ⊂ 2 ( P J to simplify the notation. ⊕ v, w + Z ( P SO(3) 1) O ) are smooth. Since we only consider 19 ω / , . Indeed, given a compatible complex − 3 . Thus, the set of such diffeomorphisms ) )) ( 2.1 8 P 19 II R ; E − , X for ∼ = 3 ( 2 – 4 – Γ 19 (+ , and hence any isometry group of a hyperkähler H 3 ( + R II (3) O such that the action of ∼ = \ O is to be identified with the subspace spanned by the ) f )) Z S Z and inducing that Kähler class. It follows that to does not contain any vectors is an even unimodular integral lattice of signature ; (with unit volume) may be identified with the double coset by choosing, say, unit volume. Therefore, the moduli space ) . ⊥ J ) X ) X, Z ( of signature ( P P ; , unique up to diffeomorphism and an overall choice of volume. 2 2 ) ) is a one dimensional space. One can choose any nonzero vector in X, g X H X, g H ⊂ R ] this is the full group of (isotopy classes of) orientation-preserving ( ( ( ( ; 2 P + Z . Not all the metrics in ( X, g S 18 X a compatible complex structure ( H , O ( X 7 2 we can define a Kähler form by acting trivially on a positive 3-plane P ) H is the standard hyperbolic plane, spanned by inside 1 (Γ) S , we can associate an oriented, positive-definite 3-plane, 1 X, g + . It is interesting to contrast that with the formulation of an isometry. The compatible with ( ) ). ⊥ O ) II P 1.1 ⊂ = X, g is the span of the classes of the three Kähler forms . In the text we will simply write X, g ( as a Kähler class. Thanks to Yau’s theorem there then exists a unique hyperkähler G ( U but might act as a nontrivial isometry of P P = 1 Thanks to the above facts, the hyperkähler isometry groups we seek are precisely the Next, the lattice Conversely, given an oriented positive 3-plane To P Actually, coarse moduli stack. ∩ f 3 · ⊥ latter would be aset diffeomorphism must form a discretemetric subgroup on a of K3 manifold must be a finite group. where e groups vectors in are called root vectors. and consequently there is an isomorphism of lattices: On the left wehas quotient an by orientation-preserving an action indexresult on two of the Donaldson subgroup [ oriented positivediffeomeorphisms 3-planes of in smooth manifolds in this paper weso will that always the restrict orthogonal attention to space subspaces metric associate a HK manifold of possible K3 manifolds To see this, given dimensional subspace real and complex componentscomplement to of the holomorphicS symplectic form, while the orthogonal Then exists a natural orientationpreserve for ( this basis, where the Kähler classeshyperkähler are manifold oriented so as to As we have mentioned, ouressentially approach lattice-theoretic. to We understanding explain the this symmetries point of first. K3 manifoldsreal is vector space structure for 2 Some easy relations between the lists JHEP12(2020)193 . . It will into (3.1) (3.2) (3.3) (2.3) J . G L 1 , HKIsom 1 However, is trivial. II action will ) 4 ⊂ L . . Since the . L ⊥ . It is conse- G ( into ) M M P G D Z acts trivially on acts trivially on N ⊂ M G G √ max HolSymp . Our notation for S = L = ( Λ L G and ) M such that R ; ∨ ) = X N , ( L 2 ( ) ι . ⊕ H acts as a group of automorphisms N ∨ A acts trivially on the corresponding . Generically, such a ) Q , q ⊂ G ∨ ) L ) ( ⊗ using “glue vectors.” To be precise, we is a positive 3-plane. In general S ι N N . ( β so that ] ∈ ∗ R N M ⊕ X, g n D ( f β on the discriminant group ( n ∨ ⊕ M ⊂ ]. Now, if we posit that ) G ) ⊕ ⊕ ∈ G ∨ → L X L will not be an automorphism of f ( ` S is a group of hyperkähler isometries of the K3 ]) = [ ( ) 28 ι – 5 – N , ι ` . L G = := ([ G R ⊕ 22 , q ˜ [ β ψ ˜ ) β P ∨ ) L ⊕ ( M L ( S D ι ]. Using the discriminant group one can construct an preserves the Kähler cone, and the Kähler cone is convex . A more substantive claim is that is a primitive embedding of an even integral lattice = :( -action on ∗ so that 28 ˜ setwise, so f ψ . Suppose that a group , P G M , . The span of the real and imaginary parts of a compatible β of automorphisms of the Leech lattice acts as a maximal finite group of holomorphic symplectic au- ) 22 M M G [ and the action of → 0 HolSymp G ∈ L } where the orthogonal complement is taken within L, q X,J Co 0 f : from the sublattice ( { − ⊂ L ⊥ ι ) M = L ( ι G ) = associated to L N = ) HKIsom q ( L Suppose ∗ N and hence acts trivially on any two-dimensional subspace ψ X, g is self-dual there is an isometric isomorphism of discriminant groups ( , however, for acts as a group of automorphisms on P β Let M G such that As a preliminary, we will repeatedly make use of the following elementary Now, any hyperkähler isometry group for group As an example, provided by the referee, consider a primitive embedding of , we can always induce a L 4 ∨ . Choose any Kähler class The fiber product isN isomorphic to not preserve the sublattice isomorphic copy of consider the lattice and take the fiber product, i.e. the vectors Proof. lattice such that Lemma 1. an even unimodular lattice of Then We now come to ato useful pair the of Conway lemmas, where group lattices we and relate related groups concepts of can hyperkähler be isometries found in appendix is an invariant Kähler class.manifold Therefore, 3 Relating hyperkähler isometry groups to subgroups of the Conway holomorphic symplectic form isS a positive 2-plane not fix so that 3-plane quently a finite group offollows holomorphic that symplectic automorphisms forTo any see compatible this supposetomorphisms that of some JHEP12(2020)193 )) G 1) L and − are ( then . So (3.4) ( is an is the x 1 G N ∨ of rank G with no Γ Γ L − = . We can such that G R x (mod ∈ L primitively G Λ · ∨ is the Leech x ⊗ x , provided g L M N Γ Λ acting trivially = M , where . Thus, ∈ G ⊂ x ) = y M · y . Consider an even P ) . But then for every ∈ g ⊕ L .) with ) ( Aut(Γ) , where 5 x x . As an example of this ∨ . We choose a sublattice ( D ∈ ≥ − M N − G (Γ) x are equivalent: R ⊕ ) it follows that · to y ∨ } Aut(Λ) G ) g ) 0 Aut ⊕ L , there exists , we have, L { ∈ . Next we embed isomorphism ( ( ∨ ∨ ) as the orthogonal complement of ⊂ ι x ι ). The conditions of lemma ) ) ( = G L ) = ( L L ( · L G M ( ( y G ι ι g D is an Z L such that there exists an even integral acts as an automorphism group. Let ⊕ → with an invariant sublattice n ∈ ∈ = ψ Z / x G ( x Λ / L, x . The main step will be to embed ⊥ v to act trivially on : Q Γ n -action on − . (In particular, − acts trivially. This implies that the induced ι is trivial. ) G G } . Since ( ∼ = v y -action from ) → ) – 6 – L G · q ) G ) by L 1) ⊕ M ( g − , namely G , because − , x D 2.2 Γ ( 5 25 , (Λ · Λ( 1 ∈ − and a finite group D g acts trivially on -action on the sublattice isomorphic to II y : Γ acting trivially on a positive 3-plane = G ,R will act trivially on the discriminant group of on which G , then for any q ⊕ ) but also v (3 x Γ G L { ( Γ (Γ) action on − = 5 + D . Now, for any such . In the next section, we prove that condition 1 implies condition v ∈ ) G v · O 2 L v g . Our strategy for doing this will be to use the formulation of the ( acts on the Leech lattice ι ⊥ such that a finite group Λ − P ∈ G v . Therefore, the natural Γ is trivial. We can then construct · ) for the definition of the invariants. x so that , so the ) M . g ) ∨ − G N and A.3 L . Then, ( N x 0 M D M · The following two conditions on a finite group ∈ g We now amplify on the need for Co ∈ y ) = be the sublattice of (mod y is a subgroup of and discriminant form ⊂ , and as we have seen, L can be used to induce a x ( G G ι , we have ⊕ root vectors in G R lattice with invariants G Γ ∨ Γ also acts trivially on = G x We now prove lemma Over the next sections, we will often invoke the following chain of arguments: we begin See equation ( N ∼ = x 2. 1. 5 ∼ = G ∈ , and prove that the group = -action on · hence and hence v automorphism of on acts trivially on the discriminant group of if Recall that we denote theinto the K3 Leech lattice lattice ( Leech lattices as a subquotient of 2. In the subsequentgives section, us we a show that bijection. condition 2 implies3.1 condition 1. Together, this Condition 1 implies condition 2 procedure, our next lemma tellsK3 us lattice, that can suitable be subgroupslattice. “transferred” of The to transfer isomorphic works groups both ways. Lemma 2. with any even unimodular lattice L into a different evensatisfied unimodular and lattice we can use it to extend the N demonstrate this using thethere argument is in a theg proof above, but ing reverse: if Remark. unimodular lattice N G JHEP12(2020)193 . , ) 0 ) Π ⊥ 25 G , n Co 1 (3.7) (3.5) (3.6) (Γ o ` II Λ ( + ∼ = , and that ) = has no root Z G to be in ∞ ) is the fixed ) Aut n . To do this . (The Leech (Γ G 25 25 , 1 , ` 25 is generated by Co acts trivially on , 1 1 ) is equivalent to (Γ 1 − 2 ι G II II Z . We claim that II 3.5 and = where } ⊂ ) 2 r . On the other hand, ) G · ) Z = 0 G into in one of the chambers. we need only check the 25 n × . Thus, our first step will , r 1 (Γ G Π ) · ι 19 Γ 24) II 25 v ∈ . However, , with no fixed-point sublattice | as a group of automorphisms ) 1 Rank(Γ ≤ u 25 G G , 25 . Therefore ( II − 1 Γ , . ) . ( Aut( 1 (Γ ,..., ) ) ∞ ι G + 2 ) II , whose walls are the hyperplanes G 25 , II , 0 G ∈ (Γ 1 1 Co ∈ C , r Aut (Γ D ) = 22 on v o II ` ( { G ∼ = ∼ = are orthogonal primitively embedded lat- G + W ) ) = had no root vectors. The complement of the Rank(Γ ) + is a disjoint union of connected components G G 25 r ∼ = G Γ , G − Aut R = (70; 0 1 ) (Γ H Γ ) – 7 – ⊗ n ⊂ 25 Rank(Γ D G II , and Γ 1 25 , G as a subgroup of D 1 G II ( acts nontrivially on Rank(Γ ( Γ Aut( ] on the conditions for the existence of a primitive II G using + and let G > > ` 28 = r G , and then argue that there exists and embedding which 4 Γ 26 Aut 25 V , 1 . These are the roots that satisfy defines . Since II . This latter inversion symmetry has no nonzero fixed vectors, G is negative definite and of rank x 25 to define an action of , Γ 1 ). Hence the required embedding exists. G ). Written out in our case this becomes: 1 into Γ ], chapter 27) II → − 6 G A.2 A.5 , (where the orthogonal complement is taken within Γ . Otherwise it would follow that x r ⊥ ] chapter 27) that Leech roots ) 6 is the famous null vector H G . It follows that there must be some vector . Since 25 , (Γ 1 ι A action on II is the Weyl group, generated by reflections in the root vectors, and G ∈ Π := is a nontrivial lattice, and hence n chambers W Now, (again see [ Now, choose any root vector The Now we wish to show that we can take the embedded copy Using the techniques outlined in remark 1, we can demonstrate that Let us first check for the existence of an embedding of Π for the set of is a group of automorphisms of the subquotient. r where is the affine automorphismdistinguished group one, of the the LeechH fundamental lattice. Weyl Amongst chamber theroots chambers, are there in is 1-1 a correspondence with vectors in the Leech lattice.) Moreover, it is known hyperplanes in the vector space called the transformation but cannot sit inside vectors, since, by hypotheses of our theorem we can then use lemma so that point sublattice. it is known (seeis [ the subgroup of transformations preserving the future lightcone and that the embedding remainsG an embedding after passing to the quotient by the discriminant group of The second equation follows because tices in an even unimodular lattice so that Now we simply use ( in appendix sufficient condition ( We now express this in terms of be to try to embed descends to an embedding into the subquotient. we use Nikulin’sembedding theorem of 1.12.2 an even [ lattice into an even unimodular lattice. The full theorem is described where JHEP12(2020)193 . , u Z 1 to 1) n , it − also / − (3.9) (3.8) . By ( W ⊥ (3.10) (3.11) 1)) } ∞ G so that n ) − ∈ Λ ( Λ , q Co v G → 3 would have , ⊂ ⊥ (Λ -action, as a 5 ι n Γ G G . Furthermore, : − ) π G R ( contains a vector (Γ { . We use Nikulin’s ι q , Π 5 = to the rest of 5) − ⊥ ≥ G } R -action from q Π ) ( 2 G ⊕ − G leaves an invariant sublattice for any non-trivial , U 5 Λ , it follows that φ such that ⊕ n − . But , together with its ⊥ 25 1) , ) = G n ,R 1 0 − acts simply transitively on the set of Γ . ( C (3 ⊂ ( 8 II Γ { . v W E ⊥ in -plane. By modifying the group using → = ∩ , Π (Γ) 3 ∼ = 0 } ) = 2 + Rank(Γ G + 1) C q G Γ and discriminant form +3 O − − whose action on ( – 8 – , 5 ,R . Since ) 5 0 G 0 − R C ≥ , so it follows that the projection R . Co :Λ ) − G ) Rank(Γ ι fixes the null vector II G Γ ⊂ G − / (Γ (24 ∈ ∞ G → ) then its orthogonal complement within , ι n − and remark 1 to extend the Co 1) 3.9 is the coinvariant lattice. Therefore, the sublattice of vectors 1 . But this implies 19 − above. Finally, we note that ) = 24 ( , Π G G 1 (3 1)) ∈ { := Rank(Λ does not appear to be a very practical condition. In this section (Λ − :Λ n ( acts trivially on . Since 2 ι G R rk ∞ ∞ contains a positive definite (Λ ι Co G Co ⊂ G and 0 with rank using our lemma has definite signature such that Λ ∈ C G )) , and therefore, Γ Γ G n n ⊂ Now, we can use lemma Thus, we have obtained an embedded copy of First, we can again use Nikulin’s embedding theorem 1.12.2b to state that the existence (Γ ι G ( But condition 2b posits the existence of such aall lattice, of therefore there isfixed an under embedding. all of if necessary, we can arrange the group is in embedding is the existencediscriminant of form an even to integral beIf lattice there the with the were orthogonal appropriate aninvariants complement signature embedding of and ( the embedded copy of Λ embedding theorem 1.12.2b to establish the existence of the embedding, Recall that Nikulin’s theorem states that the only obstruction to the existence of the which completes the proof. 3.2 Condition 2Now implies suppose condition we have 1 a subgroup has no kernel when restricted to sublattice of the Leech lattice.π We can now extend the action of chambers, we can choosein our the embedding fundamental of Weylfollows chamber. that Because fixes since that of a suitable even integral lattice is equivalent to the existence of a primitive embedding From Nikulin’s 1.12.2b weexistence of know an that even such integral a lattice primitive with embedding complementary is invariants equivalent to the 3.3 Alternative formulations ofCondition condition 2 of 2 lemma ofwe lemma give some alternative formulations of condition 2. JHEP12(2020)193 . is 2 G G we , in α g Γ ] and (3.12) (3.13) α > 6= 30 appears of which 0 2 . The list such that 0 g to appear 0 0 ) − G 0 lattice 0 Co R Co G 3 with = ⊂ K X, g ) ( R 0 H − (up to conjugacy) is 2) X, g ( G − and the Λ R that fix a positive 3-plane Λ . Then of course any proper ) (Γ) 1)) = (2 + is problematic because one must X, g − O condition for the existence of the ( ( turns out to be a proper subgroup G 0 = 2 (Λ G ) ) − α ` G G . One of the reasons the quantity that fixes it. Therefore, for any group − 2 (Λ (Λ ` ` G necessary 1)) ≥ − ≥ − ( 6 α – 9 – R 2 G -Sylow subgroups of the discriminant group and p − (Λ := + ` R α − ) as follows: finite group +3 ,R 5 constitute the list of finite groups compiled by Xiao [ − ]. They are all subgroups of the hyperkähler isometry group 2 K3 manifold, to subgroups of the Conway group R 14 largest II ) as the condition ( − ` maximality 3.11 smooth ) + condition for the desired primitive embedding is the condition acts as a group of hyperkähler isometries. It might be that +3 ,R G . 5 should also be on our list, or it might be for every other } − ) ⊂ 0 ) is just the statement that R q 0 G II − . ( G , 3.12 gives us a way to relate the finite groups that can be subgroups of the hyperkähler R + sufficient 5 ` 2 ⊗ − that appears on the Höhn-Mason list,there exists no larger group associated with the Γ In the Höhn-Mason list each of the fixed point sublattices In the list of groups associated with both the Leech lattice In order to produce a useful list we must address the following subtlety. Suppose Next, Nikulin’s criterion 1.12.2d gives a acts as a group of hyperkähler isometries in fact The computation is simply ,R 1. 6 ⊂ 0 (3 P introduce a notion of which case G of the full hyperkähleron isometry our group. list. Once Inabove again this question lattice latter techniques in case come terms we to of the do maximality rescue not of because want subgroups we of can rephrase the of some K3 manifold. is the hyperkähler isometry groupsubgroup of some K3 manifold as the full hyperkähler isometry group of some other K3 manifold 3.4 Finding theLemma explicit list of isometryisometry groups group of a of groups that satisfy lemma separately by Hashimoto [ so useful is that1.12.3 it a is tabulated inThe the application tables of of Nikulin’sthen Höhn theorem check and to subsidiary the Mason. conditions cases on Bythese the can Nikulin’s be corollary difficult or tedious to verify. It is very useful to define the quantity so that ( { required embedding ( switching the sign of the quadratic form we get an even integral lattice with invariants JHEP12(2020)193 , ? G G G . fix R Γ is a Γ G that ⊗ Λ ∈ H H , then H 19 Λ , w H 3 . Further, and Γ ⊂ should not . Now, let H . In fact, as G also contains H G G G 0 Γ P is a proper sub- . , so G sublattice of . This implies that G Γ . Also, let H G but not . Put differently, any Γ H G G by a small angle about as proper ) for each such that P and G n are related as 0 is a proper sublattice of of which q Λ G ) P H 19 Γ , (Γ) sublattice of 3 + that they fix, we must also settle (Γ O . Put differently, any group is also fixed by + R G will also be a positive 3-plane (Notice ⊂ O that appears on Hashimoto’s maximal G will be fixed by 0 ⊗ proper 0 as P H G 19 is not isomorphic to P , G 3 Γ H that is a proper subgroup of another allowed Γ . If however, Γ R G – 10 – ⊗ can be realized as maximal hyperkähler isometry 19 that is fixed by both , H 3 still fix the same set of positive 3- planes in Γ R to obtain a new 3-plane G ⊗ and w as a proper subgroup must fix a 19 that is not fixed by the action of the group G , ] (and proved again by us below) the genus determines the . We can then rotate the plane 3 and G that are subgroups of G and Γ P 14 has a fixed point sublattice that is not isomorphic to the fixed Γ H v H ∈ H , both G v and , can Λ G G , and also lists the genus of the quadratic form ( as a proper subgroup must fix a H ) that contains G . H H X,J ( . The two lattices may be isomorphic to each other, in which case is a proper subgroup, which fixes the same sublattice G Γ group, which fixes thecontains same sublattice imal sub-list” consisting ofon groups Hashimoto’s such list. that athe Moreover, fixed same genus if point then lattice two theremaximal appears or is group. just more a once maximal Therefore, groups group forsub-list,there have (under any exists fixed inclusion) group no and point larger we lattices just finite in list group the group face noted on page 33isomorphism of class of [ the fixed point lattice. We can thus compile a “Hashimoto max- G Hashimoto’s list of groups that act holomorphically-symplectically on some K3 sur- We now prove two lemmas that will help us pin down the list of possible hyperkähler We are considering the situation where If two groups A second subtlety we must dispense with concerns the distinction between fixed sub- ⊂ 2. be a positive 3-plane in H with are actually boosts,proper but sublattice this does of notgroups affect for our two argument). different Thus, positive whenever 3-planes. isometry groups. the 2-plane generated by non-zero lattice vectors. Thewe resultant can 3-plane perform a rotationany by null an directions. arbitrarily This smallthat ensures angle the that so the above that plane the 2-plane 3-plane need does not not be cross of definite signature, so that the rotations we work but not by we can find aP vector be a vector in the 3-plane that this cannot happen. Γ the same set of positivethen 3-planes we in want to show that in fact, we can always find a positive 3-plane that is fixed by the following question: ifisometry we have group a group point sublattice of If this is so,appear then on every the list positive of 3-plane maximal fixed isometry by groups. In the following paragraphs, we will show lattices and fixed linear subspacesisometry of groups the to ambient vector the space. positive In 3-planes associating hyperkähler in JHEP12(2020)193 , . . ⊥ G G 19 , L , so Λ Λ 3 such G Γ (3.14) (3.15) ) ∼ = Λ 19 , H 3 , and must also Λ Γ (Γ G . Therefore, + → Γ , O with also appears on . Also, since the 0 1) 2 ⊂ G − . ≥ ( Co H 2 G α ⊂ ≥ :Λ )) H as a proper subgroup and ι and G primitively embeds in 5 can be extended (by identity . Since the Höhn-Mason list H (Λ exists, and by the maximality ) G G ≥ is in the same genus as D Λ G ⊂ Γ ( Λ ) to the whole of ` G Γ L (Λ Γ → ∼ = G Λ , ι − → , G H in R Γ 1) ) :Γ = G − ι ( α G (Λ ι corresponds to a triple with an isomorphic such that such that acts trivially on the discriminant group of -action on . Therefore, :Λ 0 ) G ι G G – 11 – G (Γ) Co , this would imply that . Now, we can consider the possibility that there (Γ + that appears on the Höhn-Mason list as part of the ) ι 2 G O is a primitive embedding, G ι (Λ ⊂ ι 5 and H ∼ = appears on Hashimoto’s maximal sublist, then ) must be a corresponding triple appearing in Hashimoto’s ≥ Γ ) , then we resort to direct inspection of the Höhn-Mason and must be on the Höhn-Mason list, and it can only correspond is an element of Hashimoto’s maximal sublist. G G , the induced G ) Γ (where , then the primitive embedding, . G 1 (and not a larger group H) on the Höhn-Mason list. Only one G 2 = 2 (Λ ) 3 (Γ ⊥ as a proper subgroup and G α rk is a subgroup of ι and appears on the Höhn-Mason list, the maximality condition of which , the primitive embedding α > ,L = H G 2 ) := G . Then, by lemma and (group, fixed point sublattice, orthogonal sublattice) with , G (Γ) R ⊂ G L ) 5 + 2 and Γ (Λ G ι O G 5 ≥ Λ must be isomorphic to ∼ = , ). We need to verify that exists. Thus, ⊂ ≥ R Consider a finite group If a finite group := G G G H Λ the lattice. Now,Λ by inspection, we seeproving that lemma Hashimoto’s maximal sublist (weΓ call the correspondingcoinvariant coinvariant sublattice sublattice on the Hashimoto maximalBy sublist. lemma condition of Hashimoto’s maximalthat sublist, there exists no consists of all thelattice, isomorphism classes ofto fixed the group point sublatticeslattice of has the the same Leech genus as on Hashimoto’s list, so in this case the genus fixes Then, by lemma on the orthogonal complementG of exists a finite group Γ However, tells us that this is notH possible. Therefore, we conclude that no such finite group the Hashimoto lists. Doing so, we can verify that the group R necessarily exists. By remark 1, If G, We divide the proof into two parts: By lemma G, L ( ( Case (a): Case (b): maximal sublist. Proof. triple, Then is its orthogonal sublattice in Lemma 3. Lemma 4. appear on the Höhn-Mason list. Proof. Höhn-Mason list is a complete list of isomorphism classes of fixed point sublattices of the JHEP12(2020)193 . 5 8 2 Z Z , of 5 ≥ (3.16) (3.17) A α : may be 4 ) 2 G and . Therefore, ∼ = 5 (Γ . Notice that G ι as a subgroup ) 20 Λ ≥ 5 ) M Z ∼ = G X, g ( H (Λ Λ isometry. Similarly, 10 D ) . The corresponding coinvariant ⊥ 2 ) λXY W Z G ≥ α (Λ + 4 . 4 ∼ = 4 and , so any group that has as a proper subgroup. G W 5 Λ 10 + G , ) D ≥ 4 G Z R – 12 – (Γ + ι 4 as a proper subgroup, can have and lemma ∼ = as a subgroup of its hyperkähler isometry group, will Y 3 G 5 G + Z Λ 4 that contains G, X ( appear on Xiao’s list of groups and hence have symplectic H , 8 3 Z P are isomorphic. must appear on the Höhn-Mason list as the triple, that contains and ) G . 0 must be on this list. The only caveat is that the lattice G Λ 5 The same finite groups appear in Hashimoto’s maximal sublist and the 4 ) Γ Z Co , G G ⊂ and (Γ Γ ι for some group G H G, Γ H Λ Follows directly from lemma order 960. This isof the K3 symplectic surfaces automorphism known group asfollowing of the quartic a Dwork in K3 pencil. surface The in general the member family of this family is the has the same fixed point sublaticeIn as fact, the semidihedral the group listintersection of of of order hyperkähler 16. the isometryappearing groups groups on appearing Höhn of and on K3 Mason’s the surfaces list with can Hashimoto-Xiao the be list, condition that seen and rk as the the groups actions on a K3So, surface. any K3 However, surface these thathave groups has it are as not a on properhas subgroup. Höhn the and From same a Mason’s fixed comparison list. of point of its sublattice quadratic hyperkähler forms, as isometry we see group that will also have at least manfiolds. The list inclusdes thein eleven the maximal appendix. Mukai groups. The full list is given but are not fullgroups hyperkähler isometry like group of any K3 manifold of) finite groups that can and do act as full hyperkähler isometry groups of K3 The largest group of hyperkähler isometries on our list is the group Here are some examples of finite groups that have a symplectic action on a K3 surface, By inspection of the Höhn-Mason list we find there are forty (isomorphism classes However, the maximality condition of Hashimoto’s maximal sublist implies that no This list of groups is the list of hyperkähler isometry groups of K3 manifolds. The 3. 2. 1. This proves lemma listed as larger group the triple, ( Leech lattice, Hashimoto list. Remarks. Proof. full list is given in the appendix, and includes 40 out of the 81 entries in the full Xiao- Main theorem. entries in the Höhn-Mason list thatsublattices satisfy JHEP12(2020)193 ) is , v 1 ξ , we (4.1) (4.2) 3 so should − X, g, B f ( = σ = 1 λ . The sigma the map i 20 Σ , v M . Note that, for 1 : of order 192 (see h . For Π (extended linearly 4 as follows. Define 8 4 ) 2 S ) A , we have the Fermat Z : -field can be identified R and use it to construct : be the positive 4-plane 2 3; ) B 3; 4 ) 4 . = 0 v K 2 R i K ( 20 λ ∼ = ( ∗ 3; ∗ ∼ = M group. ,V,B H . We have . The K H 384 B, ω ) ( + (Σ 20 ∈ F R ∗ 192 R ]. For ) . ] − h T M 4 H 3; 4 4 ∈ Σ ]. The theory can be described as a v ω is positive. Finally, we need to show , v K 0 ⊂ ∈ ( 2 V ]. In this paper we are using the same v preserves the space 0 + 13 9 2 Π , ω , v v ∗ R ) := H 0 − 10 f v ω ! ∈ ], the geometric symmetries of the orbifold . This is the four-plane of the Aspinwall- ( 2 2 ∈ v V Σ [ 13 V = ( 1 3 − ∈ v , ξ K ) . We let 2 – 13 – Z = 2 for every 2 ω R U B i , 4 := while 3

supersymmetry is isomorphic to for i , ξ ) = 0 K 4 . Therefore + 4) ) ( Z i ξ , ) ∗ h B ω v, v B ( 3; -field. The hyperkähler metric is specified by choosing a h ω (4 and a volume factor H ξ ( = B ] but the question we address is slightly different. K ) ( 9 , ξ . Now, we consider the triple → R 4 B 4 ) ∗ ξ ].) The data needed to specify the sigma model consists of a R H 3; h := 1 + f torus SCFT. Geometrically, the Kummer surface corresponding :Σ 4 K 3; 27 4 ( ξ ξ , group. As verified in [ torus SCFT do not generate the ]). It would be interesting to see if this SCFT can alternately be 2 K D ( 2 4 orbifold has the symmetry group , 20 H 2 10 1 will be a classical symmetry if it preserves the Lagrangian. So, D 2 H M ⊂ Z f / Σ 4 and the vectors 4 D ) is: 4 T ) ξ R is a generator of orbifold of a 3; is positive definite. But this follows easily since when restricted to 2 v K ( generate the theory of the Z to the section 4.1 of [ described as a sigma modelThis on would the require quartic K3 the surface geometric from symmetries the Dwork of pencil some family. description of the model to quartic, which has symplecticget automorphism a group quartic K3 with symplecticIt automorphism is group interesting to notemodel that that commutes the with largest possiblemodel automorphism with group this of symmetry a was K3 studied sigma in [ and its solution is necessarily a Kummer surface [ Π ∗ We can now compare classical and quantum symmetries. For a sigma model Let us recall a few basic facts about supersymmetric sigma models on K3 manifolds. H that an isometry and moreover a diffeomorphism be an isometry such that spanned by Morrison theorem. The quadraticto form on and we can use this to check that where span the a copy of the hyperbolic plane positive 3-plane with an element of a positive definite 4-plane inside the full cohomology space It is interesting to comparetechniques our that result were with used that in of [ [ (See, for example, [ hyperkähler metric, and a flat 4 Implications for physics 4.1 Classical vs. quantum symmetries JHEP12(2020)193 . 2 6 G 2 / Λ ≥ ) (4.5) (4.6) (4.3) (4.4) s . and α 5 + 7 r ≥ ( and necessarily R ≤ 5 ) ) q ≥ . By averaging p, q . The necessary + ( f } R p q -theory to ( − if ) IIA R, − r, s ( ) 1 have an important physical is simply q 24 , to be smooth the connected ) − → G and there is an exact sequence 0 , Z ) ) { X 22 3; R -theory and K − gauge fields for which those isometries are ( ∗ U(1) 1)) = M H 4 B ( ). By contrast the same necessary con- H (24 − − 0 ( π ) ,R − G → A.2 , G +4 → (Λ 20 R 1) (Λ , I ` containing more information than − II ( ≥ – 14 – G gauge G in this embedding is an even lattice with the in- → Λ , Λ H R ) = (4 is necessary and sufficient. But in the classical case G G ⊥ . Then is isomorphic to → 4 Λ Λ 4 1)) to exist (which is what we work with) is the requirement 22 ≥ ≥ to compactify both − might not be induced by a hyperkahler isometry. Such 19 ( gauge R ) , R 3 G field must simply be in the space fixed by H Π U(1) Γ (Λ B X, g ι and we consider a fixed point sublattice in the Leech lattice ( → ( 0 I 1 , where , whose existence in turn proves the existence of the embedding, Co ) . , is the existence of an even integral lattice with invariants } ) ) q , q ⊂ R Z 0 − , G 3; 3; 4 R, ( K K ( ( − ∗ ∗ gauge potential arises from the harmonic modes of the 3-form potential in the H H ,R 4 22 { → → , , we know this subspace has positive dimension.) By contrast automorphisms of is necessary, but not sufficient. Rather it is the pair of conditions 1) 1) , and is a nontrivial restriction on G 5 U(1) 2 Thus, in the quantum case The orthogonal complement of According to Nikulin the only obstruction to the existence of the embedding, Suppose that − − ( ( ≥ ≥ G G component of the gauge group The respective supergravity theories. We are concerned here with the group of components. 4.2 Components ofWe the can low use energy a K3 gaugedimensions, manifold group respectively. In these casesinterpretation the symmetry related groups to the componentsnoncompact of directions. the gauge In group either of case, the effective since theory we in assume the Λ R that is necessary and sufficient. as a result of another ofembeds Nikulin’s primitively theorems: in an an even even unimodular lattice lattice with with signature signature variants Λ Now, we can prove that there always exists the following primitive embedding, which is always automatically true, bydition equation for ( the embedding into α theory. Let us compare the conditions for classical andwith quantum invariants symmetries. condition of Nikulin for the embedding classical symmetries. (The over the K3 lattice thatsymmetries preserve are quantum symmetriesmodel of action the invariant sigma and model. consequently are They not do obviously not symmetries leave the of sigma the quantum all hyperkahler isometries there will exist a space of JHEP12(2020)193 ) 8 5 ⊥ R = E F . It > (4.7) -field X L gauge ) . Note F C G ) H ( and then 0 π 0 X, g ( Co Rank(Λ ⊂ on the left and with G is the rank of G G Λ d . It is interesting to ask 23 of this lattice into the one requires an embedding M R F ]. Here, we encounter examples of hyperkähler isometries. This ) = 5 to be hyperkähler. The G ) 20 g . In this paper we have only been 7 ) X, g )) X, g ( Rank(Λ G (Λ . The discussion of this paper makes clear that D HKIsom( ( ` – 15 – ≥ ) = 5 3 G . Thus, CSS compactifications are dual to all the M- 2 ≥ ` Rank(Λ − dimensional Minkowski space, where = 5 + 2 d α -theory we can compactify on the 11-dimensional manifold M . When there is an isometric embedding G := Λ lattice. The obstruction to this is just The finite groups studied here are all subgroups of L 8 . We should choose the Riemannian metric F E X ] a distinguished set of Narain compactifications of the heterotic string called CSS × For the CSS compactification procedure with In the case of 12 6 The hyperkähler isometries will preserve the covariantly constant spinors and hence commute with the , 7 1 gauge groups. A possible futurecan direction is be therefore generalized to to understand ifa include the K3 all CSS background. possible construction discrete gauge symmetry groups of M-theory on spacetime supersymmetries. which is equivalent to theory compactifications with many M-theory compactifications, namelywill those in corresponding to generalprocedure not does have not capture heterotic all duals the Narain given compactifications by with the interesting disconnected CSS procedure. Thus, the CSS can be obtained using this simple construction. into the consider lattice we can formon a the Narain right. lattice The usingtimes result the the is orthogonal E8 a lattices decompactification torusis of to of the interest heterotic to string know on how the many Leech of torus the discrete nonabelian gauge group compactifications In [ (Conway Subgroup Symmetric) compactificationsnice were compactifications examined. exhibiting These largeconnected are nonabelian particularly component. gauge symmetry The groups essential with idea Abelian is to choose a subgroup hierarchy of masses in standardwhere model the discrete phenomenology group [ iskind an of interesting exotic finite symmetry group.explain groups It mass discussed would hierarchies. be here interesting could to be see used if in the string4.3 phenomenology to Application to heterotic/M-theory duality subgroup does have agroup nice that physical commute meaning: with the it spacetime isRemark. supercharges. the group ofif components this of continues to the be gauge thethat case discrete when generalizing symmetries, to both the full global isometry groups and of gauged, have been explored in explaining the M must be flat andcompactification space hence become must gauge be groupsshould of trivial. the be lower-dimensional identified theory In so withable Kaluza-Klein the to theory isometry say isometry group something groups of about of the the subgroup JHEP12(2020)193 , is 2 ) of a (5.1) (A.2) (A.1) ) X, g ( together . X, g ) ( Z L can then be / -linear maps ( Q Z D (Γ) + O will always be a finite 1 ) as the set of . For an integer lattice, we → ∨ Q X, g ( Q L denote the minimal number of → ) → ∨ L ) ( L ` × ) X, g ( ∨ L have a well-known ADE classification. contains no root vectors. In lemma L /L . G : ∨ Γ (3) L Isom . We let Q O Rank( Z K3 isometry groups. It would be even more → / – 16 – ≤ ) ) := Q ) L ( L . In particular, Isom will descend to a set of generators of ( → X, g D ` ( ∨ ) (3) L L ( O potential D comes equipped with a nondegenerate, integral, symmetric : L . Note that q ) . We can define its dual lattice HKIsom L Z ( → D → 1 L × to classify all L ) : Q . The discriminant group is defined to be the finite Abelian group . This inherits the bilinear form X, g ∨ ( Z L is a finite subgroup of is even the discriminant group inherits a bilinear form valued in to ⊂ Q L L L Finally (we owe this question to D. Huybrechts) it would be interesting to see whether Second, as we mentioned in the introduction the full isometry group Isom First, in this paper we have limited attention to smooth K3 manifolds. The notion with a quadratic refinement generators of the group This follows because any basis for from have When In this section, we collect some resultsA.1 and theorems that we Lattices used and inThe discriminant this data groups work. of an integerbilinear lattice form by some K3 manifold. these results can be derived using the twistor formulationA of hyperkähler geometry. Some relevant definitions and results from the theory of lattices where group. Moreover,Combining the this finite and investiging subgroups theof of extension HKIsom problem oneinteresting could to extend determine our the full classification list of hyperkähler isometry groups that are in fact realized K3 manifold is certainly ofa interest. normal Now subgroup the of hyperkähler the isometry full group isometry HKIsom group, and sits inside the group extension, instead of embedding intoone of the the Leech 23 latticeidentified Niemeier with we lattices subgroups with should of root findthis the vectors. a out automorphism Finite we groups primitive subgroups would of embedding of needour these an into knowledge lattices. analog it of In is the order not Höhn-Mason available. to list carry for all the Niemeier lattices. To We mention two avenues for further investigation. of a hyperkähler metricand makes it sense as would well bemanifolds, for nice we K3 to surfaces must extend that relax the have the classification ADE condition singularities, to that include these. For such singular K3 5 Future directions JHEP12(2020)193 , : ⊥ Z G / L = 1 L Q (A.6) (A.4) (A.5) (A.3) s = r D → : obstruction q . For only r,s II coinvariant sublattice .) A surprising and nontrivial s to be the triple ] theorem 1.10.2) states that an ) ) L ≥ } can be primitively embedded into 28 , will sometimes be denoted as L L ( ( − } , q L (acting as a lattice automorphism), ` ` is a quadratic function ) then the orthogonal complement L q ) . It therefore follows that a necessary . ≥ > ` G ( ] that we used in our arguments. They } − ) ) − and q ⊥ r, s, q s s acts trivially. The 28 ) , ` { and , ` r − within G + ) + + G ` 0 s, L ≥ L G ( r r (where ( ( ( L invariants of ≥ − + + } ` – 17 – ` − − s := ( − { ) ) − G is the existence of an even integral lattice with such − − r, ` ` ` L − r, s, q Γ ` { ) := − + + , . L 0 + ( + + U ` ` ` into with invariants I ( ( { ≥ = L L ) with signature r 1 , D ) exists if Γ − 1 ( . We define the denote the dimensions of the maximal positive and negative D R + ) II ` L ⊗ ( and the discriminant group is the trivial group there is a unique such that is acted on by a group (mod 8) and, s > ` − L 0 q ` L + , r s > 0 sign and ≥ ) and L ) = to denote the sublattice on which ( , s 0 s 0 + G would have invariants ` − L ≥ r Γ r > ( r Nikulin’s result 1.12.3 gives a simple sufficient condition for the existence of an em- If an even integral lattice For any lattice Let 1. 2. It also turns out that there is a necessary condition result is Nikulin’s resultto theorem the 1.12.2(b) existence of which a statesuse primitive that this embedding. result this So several is the times condition the in is the necessary and paper. sufficient.bedding. We We must have an even unimodular lattice inside condition for an embedding of invariants. (In particular, rather trivially, A.2 Lattice embeddings We now describe some ofare the conditions results for of the Nikulinprimitive existence [ embeddings and of uniqueness even of lattices lattices into with even unimodular particular lattices. invariants, and we use defined to be the orthogonal complement of Here, the notation signature (modWhen 8) refers to theeven Arf unimodular invariant lattice, of a up finite towe quadratic also isomorphism, form. use and the we notation will denote it as on a finite Abelian group definite subspaces of An important result in latticeeven theory integral lattice (see, with for invariants example [ JHEP12(2020)193 p − ) is a s (A.9) (A.7) (A.8) S (A.10) (A.11) + r ( , i.e., the if the fol- 0 2 ) q ⊕ r, s ( (2) (2) θ q 6= 2 )) ) q 2 2 ) ) , we can define its p-Sylow q ∗ p ∗ 2 ) p Z Z D q ( (( D ), and ( ` . mod S A.8 ≥ mod -Sylow subgroups of the discriminant ) is saturated, i.e. such that ) p ) whose discriminant form is isomorphic p p factor, then, ))( q q 0 ))( 2 p ) s this lattice is unique. Also, discr p q A.8 D p ( q ⊕ ( q (2) ( p + D K = 0 su ( ( K r > ` ( ` p ( q 0 – 18 – s − ) discr + s discr means that we can decompose the discriminant form 0 ± ± r is not unique, but we will not need to go into this + p = q r ) = | ( 2 | q q q ( There exists a primitive embedding of an even lattice with ) saturates the bound ( |D q K |D D ( ` for which the bound ( = 2 p ≥ p , we have the condition, -adic lattice of rank that divide the order of the group ) ) p 0 p q p s into another even, unimodular lattice with invariants . The notation D p ) + ( the lattice q is a ` 0 , q D ) r 0 is the restriction of the discriminant form to the p-Sylow subgroup. Note ( p q . ) = , s p ( = 2 − 0 0 q p into the direct sum, ) r s K p . It turns out that for odd primes q s p + D q + 0 r r (For subtlety for our application.) determinant of the Gram matrix of the Lattice discriminant form does not split off an ( Here to for all subgroups on where that the bound of condition (1) implies that ( For all primes If the even prime For all odd primes Condition (2), as described, is difficult to check explicitly, but if, for all odd primes 3. 2. 1. that divide the order of the discriminant group, we have the strict inequality then condition (2) holds.Similarly, condition (3) This is easier, difficult sufficient in general condition, but is an what easier we condition is: use in our proof. invariants ( lowing conditions are satisfied: of an embedding. Thisgroup that involves can conditions be on difficult torely the check. on Fortunately, the these arguments more inresult difficult the present here: conditions. paper do Nevertheless, not for completeness, we recallNikulin, Nikulin’s theorem 1.12.2d. When this inequality is an equality there are further necessary conditions for the existence JHEP12(2020)193 = 1 2 ) s , where, R 2 r Z R root lattice. ( × that have the 1 ) D s 25 , ]. 1 6 II is the characteristic ( + , then the discriminant ) O 24) 2 , to be the hyperplane in q ∼ = r D ) ( ··· R ` , 25 , 2 , a group generated by a single 1 , 2 1 ) = Z II , 0 ( s that have the relation, O s + 0 corresponding to these roots form the r ( s and = (70; 0 − R . We define r starts by looking at its root vectors. These ) 2 n s 25 − , then the two nodes are joined. , + 1 root lattice is – 19 – r = ( = 1 II . Reflections of the lattice vectors about these . r i (2) 3 n . It is an infinite group that is so named because ) , where called the fundamental chamber. It is the chamber s ∞ su 1 r, r . 0 R h − r C ∼ = Co , we draw a node in the diagram. called the Weyl group of the Leech lattice. It turns out to , known as Weyl chambers, with the hyperplanes forming satisfies i R 1 = 1 = ( s C D 2 r i W R n = 2 p s, h is the autochronous group. such that ∞ 25 , , there are an infinite number of nodes. The group of automorphisms 1 Co 25 , we have , r II 1 . They form a set of fundamental roots. Now, there is a distinguished set o has the structure of a direct product, II ∈ W W 25 , r perpendicular to the root 1 ∼ = ) II R 25 , 1 ⊗ . To draw the Coxeter diagram, we need the following relations, Next, the fundamental roots are used to define the Coxeter diagram of the lattice The study of the automorphisms of (3b) If the even prime II not intersect, we join themfor with divergent a hyperplanes. bold line for parallel hyperplanes, and a dotted line when acting on a generic lattice element, the two nodes are unjoined. 25 25 ( , , Both of these relations come from hyperplanes that intersect. If two hyperplanes do For each fundamental root If the hyperplanes correspond to two roots If they satisfy the relation, For any root 1 1 + (4) (1) (2) (3) (0) of the Coxeter diagramit is is called abstractly isomorphicing to translations. the It fullgroup acts automorphism of transitively group on ofO the the fundamental Leech roots. lattice, includ- The full automorphism For the lattice that contains the characteristic vector II additional property that vector of the lattice,called a the distinguished Leech null roots.points vector. on They the There form Leech iswalls lattice. a of a set a one-to-one The of distinguished correspondence hyperplanes fundamental chamber between roots these roots and be a normal subgroupthe of vector the space automorphism into chambers groupthe of walls the of lattice. theroots chambers. The corresponding hyperplanes The divide to Weyl group thegenerators acts for walls transitively of on any theof Weyl one fundamental chambers. roots of The that these we chambers can can use be for convenience. taken Consider as the a roots set of This section reviews some essential material from chapter 26are and vectors 27 of [ II hyperplanes generate a group form must split off aThe factor discriminant corresponding group to of the the element. discriminant form on the A.3 The Leech lattice and its automorphism group JHEP12(2020)193 and A (A.16) (A.15) (A.12) (A.13) of the form), but we AX t ⊕ · · · T ) ] for more details. X 6 is simply a positive definite = q ⊕ · · · ! f 2 y x , q 1 f q . Using this notion of equivalence (with associated matrices 2 f 1 − q g ! ± = and the oddity ⊕ p b c a b q e and ) = qf

6= 2 f ( M are integrally equivalent (A.14) AM ( p ⊕ T ! ) x y M for – 20 – 0 1 1 0 = = ⊕ · · ·

dr e B 2 q 2 p ⊕ f cy g 2 are said to be in the same genus if and only if, p + g ⊕ and p bxy ! and pf ( + 2 f 0 1 1 0 2 ⊕ 1

ax ) f q has integer entries and det is the dimension of the Jordan constituent ⊕ f = = ( ) f q f f M f , and we can generalize this from binary to n-ary integral quadratic forms ( ] for a systematic treatment. det Z 1 6 , − ∈ 5 dim p is the determinant of the Jordan constituent is a p-adic unit form (a term we do not define here, merely understanding the = , we will need further invariants (the type ) = q q q q f f a, b, c n ( = 2 With this Jordan decomposition, we can now define a complete set of invariants to For any quadratic form, there exists a Jordan decomposition over the p-adic integers. Two quadratic forms We define a binary quadratic form using the following equation, = p = det r d ) are said to be integrally equivalent when the matrices are related as, For do not define theseis here. what we We will can readthe now off genus express the of the a tables genus quadratic of using form Höhn-Mason a is and p-adic a Hashimoto. symbol, formal The sum and p-adic over this symbol the for Jordan constituents, • • • above as a decomposition intothe a role set of of building standard blocks formsform. in called the We Jordan classification). refer constituents the For that interested play reader to section 7,characterize chapter the 15 genus of of [ a quadratic form, The decomposition is as follows, where If two forms are integrallynot equivalent, true. then they are in the same genus, but the converse is where the matrix of quadratic forms, we can formulate a classification scheme for quadratic forms. where in a straightforward way.B Two quadratic forms Our proof of the mainwe theorem briefly makes recall use here of a the fewforms. relevant notion facts See of from [ the the genus classification of theory a of lattice. integral quadratic Therefore A.4 The genus of A quadratic form JHEP12(2020)193 (A.17) (A.18) (A.19) is divisible by f ] corollary 2.12, 18 det } 2] and dimension at least n/ ] where they have been [ 4 I,II 128 29 ) , ∈ { < 25 with the invariants, | is required to specify the form, e – f t 2 representing the span of the real Z ) 23 , det R | 3 and hence there can in principle be . This is the source of infinite groups ; where S )) X b dr ( e ( 19) 2 q , H (1 ) (2) r,t,d,I 2 L if the lattice is odd Z ( q – 21 – I ( ) = (2) dr t L b (mod 8 ( II, i acting trivially on or 1 is of signature . i ± rank I X (Γ) ) 4) + R = = = = ; O having indefinite signature, r d t e X f (mod ( where t is the oddity or be a positive 2-plane in 2 1 H S dr t q or in S = 0 is the unique even unimodular lattice over k for , we need to specify the type and the oddity invariants where applicable. We use (2) r,t,d,e 2 / L = 2 1) group Indeed, concrete examples can be found using the K3 surfaces with Picard number We will not require these theorems, as we will only use the result that if two lattices For quadratic forms − p n ( n or “Shioda-Inose surfaces.” Theyinterplay have between also string been theory popularcalled with and “attractive physicists number interested K3 theory in [ page surfaces.” the 317) that Shioda K3 andgroup surfaces (as with Inose well Picard as have number an shown 20 infinite always group (see have of [ an symplectic automorphisms, infinite since automorphism this is a finite index and imaginary parts ofcomplement the to nowhere zeroinfinite holomorphic groups 2-form. of boosts Note in of that holomorphic the symplectic orthogonal automorphisms. 20. These are known as “singular K3 surfaces” (a term we suggest should be deprecated) B Some K3 surfaces with infinite holomorphic symplectic automorphism In this appendix weautomorphisms. discuss Let K3 surfaces with infinite groups of holomorphic symplectic k have quadratic forms onthe their two discriminant lattices are groups non-isomorphic. that are not in the same genus, then same genus. There areclass different of theorems quadratic that forms decidetheorem in when is there as a is follows, given a genus unique and equivalence when there3, are there is more only than one one. equivalence class One of such forms in a genus, unless The genus ofisomorphic a lattices quadratic may have form quadratic is forms on an their equivalence discriminant groups relation, that which are in means the that two non- where the symbol, Finally, we will use the following notation where the oddity For JHEP12(2020)193 ∼ 2 and z (B.7) (B.3) (B.4) (B.5) (B.6) (B.1) (B.2) 1 τ and Z are pairwise 1 τ so that + that is a product D Z a, b, c A associated with the √ + X 1 1 z . Then the Shioda-Tate 2 ∼ c/τ E − 1 × z 1 = E . D ) so that the transformation = with 2 √ z 2 ] C 1 A ] 2 − ] + 1 2 z ∈ 2 , 2 [ aτ P b 1 Z = 0 = 0 βz z δz + / ∼ = = c − ) b + + 2 ac ( 2 2 and τ + 1 τ 1 Z ] = + ∼ 1 / E 1 1 αz γz 2 1 z ) α, β, γ, δ [ [ c [ bτ . We will also assume that τ × 2 – 22 – E 0 bτ 1 − + , z τ → → associated to an abelian surface − 1 → 2 ] ] 1 E z = 1 2 2 ( 2 ( X z z a > aτ τ 2 X [ [ τ D √ a are integers and we choose the sign of 2 + with complex multiplication. To be concrete we consider the b 2 − E and a, b, c = × ) , 1 1 0 Z τ E τ + that is a product of elliptic curves, ac < Z 4 ( A / − C . The orbifold is simply 2 Z b 2 := τ τ := + E D Z It is not difficult to find conditions on We start with a Kummer surface are in the upper half plane. So + 2 2 z It is also worth noting that On the elliptic curves we use coordinates from which it follows immediately that here τ relatively prime integers. Note that where example. of two elliptic curves Kummer surface formula implies that there always exists a non-torsion section of the map, Translation by such a sectiongroup of is the an surface automorphism is of an infinite infinite order, group. so Here the we automorphism content ourselves with a simple explicit formula. The Shioda-Tate formulasurface relates to the the Picard rank numberthe of of section the an can Mordell-Weil elliptically define group. fibered anattractive When K3 infinite K3 this surface group rank is of a is automorphisms.abelian rational positive Shioda double surface translations cover and of by Inose a show Kummer that surface an subgroup of the automorphism group). The result is an application of the Shioda-Tate JHEP12(2020)193 2 , α 1 (B.8) (B.9) α (B.12) (B.13) (B.14) (B.10) (B.11) = 0 = 1 ) 2 1 Z γ γ 1 1 1 ! aτ β Z Z aβ 1 Z 1 + − 1 τ − 1 aτ τ Z γ 2 R ( + 1 2 + + γ 2 1 1 β 1 τ ⊂ Z ⊂ aτ Z Z τ 2 aτ 2 ) 2 bβ − γ Z ∈ 2 = 1 ∈ α 2 ⊂ Z β δ 2 τ 1 β − + δ δ ) τ + + 2 2 + 1 βγ Z + 1 γ Z 1 Z 1 1 + 2 1 bα 2 τ α β aγ δ Z – 23 – − τ Z a aτ cβ − ( + := := := := + αδ 1 = + Z + × δ δ 0 γ Z β ( α 2 2 ) Z a R δ α ) is a monoid under matrix multiplication. Therefore, Z × 2 2 ∈ ∈ τ + R . So we must find solutions to the Diophantine conditions: γ α B.8 2 acα 2 + is a ring and δ

, 1 we get a pair of equations that we can use to solve for Z − Z α a 1 1 ( = 1 δ aτ 1 i = 0 α + 2 . Therefore the group is infinite. δ i for Z , γ Z i = β and ∈ R i , δ = 1 i 1 , γ δ i , β i α Now write Finally, we want this to be a symplectic automorphism. The symplectic condition is If we put for any choice of with simply the determinant one condition: so that the setthe of set matrices of in matricesaction ( of descends to the the above orbifold. type that is invertible is a group. Note that the group and is an ideal. Moreover Now, note that is well defined. The result is: JHEP12(2020)193 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M Type 8 6 4 4 5 2 4 2 4 4 4 2 3 2 4 2 2 2 2 3 3 2 3 2 4 2 3 2 2 2 2 2 3 3 2 2 2 3 2 2 α #1 #1 #2 #1 #1 #5 #3 #1 #3 #4 #4 #4 #8 #3 #1 #9 #5 #14 #11 #12 #49 #27 #50 #10 #48 #43 #29 #40 #41 #34 #42 #138 #227 #955 #118 Index #1023 #1493 #1026 #18135 #11357 3 8 S D ) 8 o 5 2 4 4 2 5 3 1 6 4 2 4 Q 3 3 4 8 3 3 S 16 S A 9 , , , a Z 8 8 , 4 5 6 3 4 5 2 3 4 10 12 × (7) 21 Z Z A 3 4 4 D × 192 : 3 2 3 4 ∗ : 192 Q G 2 2 2 2 S S S 4 2 2 25 A A A 4 Q D × 8 ∼ = M F D D S 4 A A A T 8 2 4 H 2 3 4 2 SD L Γ 2 2 4 ∼ = 2 D Q Hol( 72 – 24 – 48 N T 2 3 4 4 6 8 8 8 16 10 12 12 16 18 24 32 32 48 16 20 21 36 36 48 60 64 72 96 48 72 72 192 120 168 192 192 288 360 384 960 Order 9 9 8 8 8 8 8 7 7 7 7 7 6 6 6 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 5 5 5 16 12 12 10 10 10 Rank 9 8 7 6 5 4 3 2 1 # 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 C The possible hyperkähler isometries of K3 surfaces JHEP12(2020)193 ] . ) ∗ ) is ≥ 8 ∗ G 15 5 ( α Q 2 ( A (C.1) ) = 5 O : implies ∼ = G 4 Λ 2 23 192 M T #1493(23 , . Here G and ]. 12 is a group of order 14 ) = D 3 , G 4 3 ( #955(23) 2 2 A , ^ O ∼ = . 3 , i 3 ]. The notation 2 192 ) = 5 and A H G 15 4 yx . , 2 23 ) 4 (Λ = G A M ∼ = ( 2 4 4 ⊂ , 4 e, xy Aut G A = o 3 G y . It is a generalized dihedral group of the = : 2 – 25 – 7 generated by all elements of odd order. It is a , but it is not known whether the extension splits. 2 x 4 2 | 3 G Z implies that , acting via fractional linear transformations and with x, y 7 h by ∗ 23 F 5 = M A 21 . Three distinct groups appear on Höhn-Mason’s list with F . Thus, all three of Xiao’s groups must be on our list. 2 , and in general, exponents represent direct products, positive ) 4 2 ∗ Three appear on Xiao’s list- ≥ Z α is the projective special linear group that acts on a vector space of , which is a bit technical to describe so the reader should consult [ ) , the Mathieu group of a set of 20 elements. ) 7 20 of order 288 has the structure F G , ( We see that the eleven groups in the above list for which rk( 4 M #1023(23 2 . The notation , 4 , ^ O 23 A We use the notation of Hashimoto to denote our groups [ M PSL(2 ) = 6 ⊂ -the semicolon denotes a semidirect product, i.e., the extension splits. . Of these, the last two appear on Mukai’s list, so they must definitely appear on G or denotes an extension of is the semi-dihedral group of order 16. 5 3 G 5 A is the unique nonabelian group of order 21, and also the smallest nonabelian group S (Λ is a Mathieu group, a multiply transitive permutation group on a set of nine elements. 16 is used to represent : (7) 9 .A o 2 21 4 4 4 ) for the definition. The last column uses thethat notation of Höhnis and Mason the [ minimal normalsubgroup subgroup of of F of odd order. It has the following presentation, M L dimension 2 over the finitedeterminant field equal to 1. SD The group 18 which is theelementary abelian semidirect group product of order 9. 2 Hol-denotes the holomorph of a group, i.e., 2 integers denote cyclic groups of that order. 2 isomorphic to 8 , and they have the following GAP nomenclature. For rk Groups of order 192. Q Höhn-Mason’s list with 2 For rk Mukai’s list. are precisely the groupssymplectic that action appear on on a Mukai’s K3 list surface. of maximal finite groups that have a Notation. JHEP12(2020)193 . 20 448 ] M (2012) : 8 2 Z 206 (2018) 145 (2014) 173 Commun. Development of J. Algebra , , 05 75 , (1990) 257 , in 29 -model with -models JHEP arXiv:1309.6528 , pp. 421–540 (1996) σ σ , [ ]. 3 AMS/IP Stud. Adv. Math. 3 , K K ]. Nagoya Math. J. Topology ]. A ]. , , SPIRE , Grundlehren der ]. J. Geom. Phys. Surface and the Mathieu group IN , 3 (2016) 387 ][ surfaces SPIRE SPIRE SPIRE K IN 3 69 IN SPIRE IN ][ K ]. IN ][ ][ Theoretical Advanced Study Institute in ][ Symmetries of Black holes and class groups , in SPIRE surfaces and Mathieu groups IN [ 3 , Dover (1978). Notes on the ]. ]. 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