JHEP02(2021)039 f Springer October 6, 2020 , and we prove February 3, 2021 : superconformal : 24 December 18, 2020 : F = 12 c lattice. holomorphic SCFT, the Received and Roberto Volpato 8 Published e E Accepted = 12 , c Published for SISSA by , with any such choice of supercurrent, https://doi.org/10.1007/JHEP02(2021)039 , the holomorphic 24 Daniel Persson 24 F F c,d [email protected] , , with the admissible choices labeled by the semisimple Lie 24 F [email protected] , Natalie M. Paquette, . 3 a,b 24 2009.14710 F The Authors. Conformal and W Symmetry, Conformal Field Models in String Theory, c

We study some special features of , [email protected] supercurrent on supersymmetric version of the chiral CFT based on the = 1 = 1 Dipartimento di Fisica esezione Astronomia ‘Galileo di Galilei’, Padova, Università Via di Marzolo PadovaE-mail: 8, & Padova INFN, 35131, Italy [email protected] 1200 E California Blvd, Pasadena,School of CA Natural 91125, Sciences, U.S.A. Institute1 for Einstein Advanced Drive, Study, Princeton, NJDepartment 08540, of U.S.A. Mathematical Sciences, University of Gothenburg andChalmers Chalmers Tvargata University 3, of Gothenburg Technology, SE-412 96, Sweden Department of Mathematics and805 Statistics, Sherbrooke McGill St University, W, Burnside Montréal,Department Hall, Quebec of H3A Physics, 2K6, McGill Canada 3600 University, Rue Ernest University, Rutherford Montréal, Physics QCWalter Building, H3A Burke 2T8, Institute Canada for Theoretical Physics, California Institute of Technology, b c e d a f Open Access Article funded by SCOAP Keywords: Conformal Field Theory, BRST Quantization ArXiv ePrint: N algebras of dimension 24.can We be also obtained discuss via how orbifoldingN from another distinguished Abstract: field theory (SCFT)superalgebras given of by “physical” states 24 of chiralthat a chiral they free superstring all compactified fermions. have on thefamily We structure of of construct new Borcherds-Kac-Moody eight examples superalgebras. different of This Lie such produces superalgebras. a The models depend on the choice of an Fun with Sarah M. Harrison, JHEP02(2021)039 34 10 27 16 36 4 32 28 27 23 – 1 – SVOAs: a general construction 37 30 = 1 39 structures N 42 34 8 as a Borcherds-Kac-Moody superalgebra 17 fE 8 1 = 1 g V 7 20 A N = g is a BKM superalgebra 34 4 11 35 g 14 1 from orbifolds of 24 6.4 Root multiplicities and denominator formulas 6.1 Construction 6.2 Description of6.3 real roots Weyl group 5.1 Generalities on5.2 BKM superalgebras Proof5.3 that Simple roots5.4 and Weyl vector Denominator and superdenominator 4.1 Super4.2 vertex algebras BRST cohomology 4.3 Lie superalgebra of physical states 2.1 Construction 2.2 Partition functions F 3.1 Generalities 3.2 Examples 1 Introduction Quantum field theories of twoand dimensional fermions nevertheless are among have the remarkablyconformal simplest field to rich write theory physics. down, (CFT),in the For terms critical example, of point perhaps of a the the free 2d simplest Majorana Ising 2d fermion. model, can More be described elaborate fermionic CFTs appear as edge B Details about cohomology 7 Conclusions & future directions A Multivariable Jacobi forms 6 The example of 5 The Lie superalgebra 4 BKM superalgebras from 3 Contents 1 Introduction 2 The SVOA and its JHEP02(2021)039 ] , 3 ], ]. 3 31 54 , ] and 52 , is the , super- 4 8 , 25 15 , 2 51 E ˆ g ] and [ Λ 24 = 1 ⊕ 48 , 3 2 N ] and section (5) ] in his proof of (2) 7 19 , where c , so 8 c su E 18 , , Λ ⊕ 5 / 4 ]. 8 15 , R (5) 39 2 ], and Thompson [ (6) – c su c 14 sp 37 and no weight-1/2 fields. First , , 3 2 ⊕ ) = 12 As has been emphasized in several 2 (2) c 1 ], which studied quantum mechanical fermions c su (2) 55 ⊕ These theories were classified in [ c su ( 5 2 superconformal structure which is stabilized ⊕ . It is notable because it is one of three so- (7) 4 24 c so – 2 – = 1 F (4) have been used to furnish constructions of a special , c N su ]. 2 f\ ⊕ . 21 ) ], as well as in the classification of symmetry-protected , , V 2 1 2 3 53 ⊕ ) (2) and 3 c su 8 ]. Two-dimensional Bose-Fermi duality also relates fermionic ( (3) fE 49 ⊕ c su V 3 ( (3) , ], it has a unique 8 c su 18 ⊕ ) ⊕ 2 3 (2) (5) c su : this is the theory of 8 chiral bosons compactified on c ( so 8 : this is the unique holomorphic SCFT with root lattice, and their 8 fermionic superpartners. : this is a theory of 24 free chiral fermions. One can build an fE f\ 8 24 which we describe in section by Conway’s largest sporadic group. F conformal structure by takingallowed a choices linear are combination classifiedthese of by generates cubic semisimple an Lie affine Fermi Kac-Moody terms, algebras algebra, and of of the dimension which 24. there are Each eight possibilities: of V E V discussed in [ On the face of it, these three theories are quite different — they have notably different In this paper we study a system of 24 free chiral fermions in 2d. This is a holomorphic In this paper, we will rather emphasize intricate Lie algebraic structures hidden in Interesting work in a similar spirit appeared recently in [ A self-dual SVOA W isFor one reviews that of is moonshine, rational see, and e.g., the [ unique irreducible W-module (up to isomorphism). 3. 1. 2. 1 2 3 superalgebra. BKM algebrasthe were monstrous originally moonshine conjectures introduced of by Conway and Borcherds Norton [ [ valued in (gauged) Lie algebras. constructions and symmetry groups. However, asby is gauging described symmetries, in [ one canrespectively, move the from theories one to thetype other. of Furthermore, in infinite-dimensional [ Lie superalgebra known as a Borcherds-Kac-Moody (BKM) superconformal field theory, or super-vertex12 operator and algebra (SVOA), which with we centralcalled refer charge self-dual to SVOAs throughout with centralare as charge given by 12. (up to isomorphism): the deceptively simple physicscorners of of the 2d mathematical free physicsand fermions. landscape references (see, therein), to systems give withspecial an certain properties incomplete distinguished and list, numbers intertwine [ of withbe fermions several in can species enjoy various of modular symmetryassociated objects; automorphic structures forms. our present interest in will a system of 24 chiral fermions and some topological (SPT) phases [ CFTs to lattice CFTs oftheory. bosons, Recent which works appear on in, dualities haveto e.g., also understand toroidal shed the compactifications light on of rich subtle physics string discrete of invariants required two dimensional fermions [ modes in quantum Hall systems [ JHEP02(2021)039 24 ,F BPS 8 fE ,V f\ V spacetime superconformal with Kac-Moody supercurrent, and g Kac-Moody algebra = 1 4 = 1 N N ] we expect that we may ], and this paper, occur as , we review the construction 45 , 2 48 , can be obtained from orbifolds 44 , 31 24 42 F ]. They suggested that BPS states in 33 , ], where it was found that 32 supercurrent. The main results of this work 43 , satisfies the conditions of a BKM superalgebra – 3 – 42 = 1 , the allowed choices of g structure on ]. tw 24 17 N ), , similar to the constructions of BKM superalgebras F = 1 3 24 F ), taken to be one of these theories. N 4.3 V,W ]. Furthermore, given the close relation between ). , and the construction has been used to shed light on the physical mentioned above. For each choice of 30 \ (see section 5.4 V 8 with f\ V fE ¯ in section W V we construct a corresponding BKM superalgebra 2 ⊗ The worldsheet string theory for this construction employed the Monster and 24 V 5 all naturally occur as (chiral halves) of worldsheet CFTs at special points 8 F ] for a proposal for the appearance of a BKM algebra in string theory in a quite different (see section 24 fE 13 g V ,F f\ , its canonically-twisted module We prove that the Lie(see superalgebra theorem We provide an infinite producteach formula for the Borcherds-Weyl-Kac denominator for We show that all choicesof of the SVOA ,V The outline of the rest of the paper is as follows. In section An interesting example of this proposal, similar in spirit to the present study, was Besides the fact that we construct a new family of examples of (super)-BKM algebras, One of the goals of the present paper is to describe the construction of a family of BKM 8 24 That is, a Borcherds-Kac-MoodySee algebra. also [ • • • 4 5 F fE of context, building on the pioneering results of [ We expect that thealgebras BKM acting superalgebras on constructed spacetime inprecise BPS in [ states upcoming at work [ such specialvia points. orbifolding, by This analogy isuncover a with new notion the 2d we constructions spacetime make intensor string [ product dualities by considering worldsheet theories which are a Monster BKM. vertex operator algebra interpretation of the genus zeroV property of monstrous moonshine.in the Similarly, moduli the theories space of maximally supersymmetric type II string compactifications to 2d. — at least in(or some contain contexts one — as this a subalgebra). algebra would bestudied a by generalized the laststates three named in authors a [ second quantized heterotic string theory furnished a natural module over the of which there areto only elucidate very the connection few betweenwas explicit BKM originally constructions, algebras envisaged and by one BPS Harvey of andstring states Moore our in and [ string long-term field theory, interests theories which is with extended supersymmetry should form an algebra, and that structure for symmetry determined by the choiceare of threefold: elucidates connections between modular functions,the the monster physics sporadic of simple 2d group, CFT. and superalgebras based on the theory based on The monster BKM arises from BRST quantization of a chiral bosonic string theory and JHEP02(2021)039 ) ≤ z ( we 1 (2.5) (2.1) (2.2) (2.3) (2.4) G 5 , j λ i algebra. λ sublattice. if and only 1 supercurrent 12 12 D (24) structure can be = 1 c , with OPE so ) currents z N . We conclude with ( = 1 6 24 as a single SVOA, even N . , so it must be of the form ]. An 24 = 276 2 ij . We give more details on F ,
/ 26 δ ) and the supercurrent g , . . . , λ 3 2 24 ) z 2.2 chiral free fermions. In mathe- ( = 2 , 1 : ) = 0 ) λ 24 z jkl k we explain the general construction , structures. c λ kim :( 4 . The j w c i ) and the relative BRST cohomology ikl 2 λ ij c i / − δ jlk = 1 ∂λ A λ 1 with a choice of i c z superconformal structure. In section . Finally, two appendices provide further : k,l λ X N : + 24 7 ∼ algebra corresponding to the ijk F , given by ) = 1 ). c 1 =1 24 i X form a vector module for the w kjm – 4 – ]: ( B . In section c is a super BKM-algebra. In this section we also N 1 2 j 8 X i,j,k 24 26 (24) = 12 λ g lik -cofinite super vertex operator algebra (SVOA) of − . The stress tensor ( F i ) 6 c fE 2 c c so z C structures C ( V − + i SVOAs. We then go on to construct a family of BKM chiral free fermions, ) = superconformal algebra at central charge ∈ λ z = ( 24 klm ijk Kac-Moody algebra, which is the bosonic (even) subVOA G T = 1 c = 1 c = 1 . In our definition of SCFT (or SVOA) we do not include 1 ijk 12 N N N c ( , with the (24) , we describe how have conformal weight with a choice of 12 k i 3 c X so Z subalgebra. For this reason, we refer to λ 24 Kac-Moody symmetry in the following section F of central charge 8 1 = 1 ˆ A 24 N from F ) generate an g 2.3 structures in free fermion theories were classified in [ , generate an 24 = 1 . By bosonization, the same SVOA can be described as a lattice model based on the ≤ N This theory is generated by 24 F if the following conditions are satisfied [ must be a linear combination of conformal primaries of weight for some totally antisymmetric defined by ( i < j of odd unimodular lattice The space of odd (fermionic) states in and stress-energy tensor, with respect to which CFT type of centralthe charge choice of an though, as discussed below, it admits different 2.1 Construction The starting point of ourtheory construction (SCFT) is a simplematical holomorphic language, chiral this superconformal is field a self-dual the example with a brief discussion ofdetails open about questions multivariable in Jacobi section for forms physical (section states in our theories (section 2 The SVOA and its obtained from orbifolds of the SVOA of BKM superalgebras from superalgebras prove our main theorem,discuss showing that the denominator and super denominator formulas of its character. In section JHEP02(2021)039 g 24 free (2.6) (2.7) (2.8) (2.9) (2.10) (2.11) (2.12) . It is 24 , . . . , t g 1 coincides t , given by a 0 ) J ·|· . ( i ∨ g , /h is a direct sum of . 2 ab based on the finite , ) δ is orthonormal with g z algebra generated by , then the restriction ˆ g ( OPE does not contain i 24 . In the following, we = 24 ∨ g T } h a 0 (0) ,..., bkj J (24) c G { descendants of the ,..., =1 ) ..., c so a z ajk } ( c a ... = 1 G . J = 1 and =1 { n (0) + )) currents X k j,k N u (0) + J 1 2 k 24 . , a λ l − abk ..., in the basis t ) c )Ad( z t abk + on the algebra of zero modes jkl c =1 24 :( ajk X k g ), the ic )) = ) b k ic n 0 =1 i (0) z z ; the second is equivalent to the requirement X k λ J a ·|· − 2.5 j on the finite dimensional Lie algebra ( – 5 – ] = i z J + Tr(Ad( λ k g = : . The OPEs 1 2 is the complex Lie algebra generated by 2 ) has dual Coxeter number , t ab j )Ad( λλλλ z j δ g i λ a jk ·|· t 0 = currents are superconformal primaries. Furthermore, i [ ) ( g ajk are the structure constants of a semisimple Lie algebra J ) and ( a λ c g 0 (0) = (0) = ), the set of currents ) b J a u ijk 2.4 λ 252 =1 λ | c on the Killing form. Notice that if 2.8 ) 24 ) t (0) = X ( b z z j,k are singled out as the Ad( ( g -dimensional subalgebra of the ( J i Tr(Ad( a 2 ) generate an affine Kac-Moody algebra ) , where G i z 24 J z 1 2 ) − ( g ( i z a a ( = is such that the long roots have square length J ⊕ J a is a i g ) = J ) g satisfying ( z = b g 0 ( transform in the adjoint representation with respect to the algebra a J g | ) and that a ijk J a (and any rank), i.e. 0 c λ J 2.5 ( 24 term in the OPE reproduces the stress-energy tensor . Thus, g , while the remaining 1 a − λ z The SVOA admits a canonical non-degenerate invariant bilinear form These conditions imply that of dimension where we used ( will often drop thesimple subscript components of the Killing form to With this choice, the Cartan-Killingwith form the bilinear form induced by the Zamolodchikov metric, since the Zamolodchikov metric. Byrespect ( to this bilinearization form. for In the theconvenient Cartan-Killing following to form sections, choose we will need to choose a normal- show that the currents fermions the free fermions of zero modes. the zero modes of fermion bilinears satisfy the OPE which shows that the Lie algebra Given a choice of any singular term withthat four the fermions g with commutation relations The first condition is equivalent to the requirement that the JHEP02(2021)039 , , 0 k ∨ g 2 h , . We 2 , = (2.13) (2.17) (2.18) (2.14) (2.15) (2.16) 1 12 2 | ˆ A has four θ | 4 , /D at level 3 12 , 0 , the affine ˆ ∗ 12 , we have that C i g D 4 1 ∨ g , . D V 2 h 1 ˆ g , A 2 , 7 ⊕ 3 1 . Using the string . 3 ˆ ) at level A 2 algebra 5 q , ·|· (5) ) 3 0 ,C ( (2) g ˆ , the dual Coxeter number is 1 , c so B 0 g c A su m , ) yields the commutation 3 . + , , simple (dim m ⊕ n n 5 2 + : ˆ 2.8 ) , 4 so + n 2 1 g ) c .u m ˆ ,B (5) A ) (6) .u J 3 t 2 1 ) , lattice. This VOA t as the Neveu-Schwarz (NS) sector c su 2 c sp A abc ˆ 2 A c 24 ) for the normalization, one has 12 3 , A , , F 2 2 D c 3 based on a , 2 X ˆ + (Ad( Comparing these equations, we see that 2.11 + (Ad( i B ⊕ 0 k i ∨ g ) n n ˆ g 6 h , the OPE ( 2 (2) − ) 1 − + ). , ,B i − m, c su g n 4 2 (2) m, δ , n (ˆ δ − 2 i G − ab ⊕ c su 2.11 ˆ 2 – 6 – δ ab z ⊕ G m, ( 5 ) a 3 δ δ n , u 0 | 2 = J ⊕ is given by the direct sum of the adjoint and vector g t ab (7) 0 ˆ ( ) 4 Z B ˆ g based on the ∨ g ,B 2 u 24 | ∈ c nδ so | h 2 4 θ n fermions generate an algebra t F | (4) 12 ( , 0 k g D P , 5 k ] = c , su . The embedding index is the ratio of the levels for the corresponding V 2 4 ) b m 0 ,A ] = ˆ ⊕ A g dim ) 3 1 ] = , m ,J ) = 2 3 A a m n z , u 3 , of simple components of dual Coxeter number ⊕ ( J n ) (dim [ (2) t a 2 , u [ i , 3 n 3 1 J g so c t su i ˆ [ ( A ,A ⊂ (3) ⊕ 4 , 0 3 2 ⊕ 3 g c su , there are eight distinct possibilities for ˆ 3 ( = . A 0 ∨ g g (3) h ,A , , = 24 8 1 8 3 c su , ⊕ g A 3 2 representations. While ) ˆ ⊕ 2 A is given by 3 is the length of the long roots. With the choice ( dim ˆ g (2) 2 (24) , | (5) θ 2 c | , su so c 8 1 ( so is the sum is the bosonic lattice VOA Finally, this SVOA admits a unique (up to isomorphism) canonically twisted module For In terms of modes , which also admits an invariant non-degenerate bilinear form ˆ The same conclusion can be reached by noticing that, for a simple algebra More generally, the relations are A 7 6 g must have level tw 24 24 0 ˆ where hence the formula. the embedding index of embedding of affine algebras.g Since can label the four modulestheir as ‘adjoint’, ‘vector’, ‘spinor’ and ‘conjugate spinor’ in terms of F theory terminology, we will oftenand refer to to the its SVOA twistedF module as the Ramondirreducible (R) modules sector. which are Recall in that one-to-one the correspondence even with subalgebra the of cosets that is, The corresponding affine algebras are, i.e. the levels of the simple components equal the dual Coxeter numbers. when the Killing form isif normalized as in ( algebra relations We recall that, for anthe affine commutation Kac-Moody relations algebra read JHEP02(2021)039 0 0 h ∨ g → L h ) so h , the : ) (2.19) (2.20) (2.21) (2.22) (2.23) and z i 0 ( 2.11 , L ) G be the set w , where | 0 r , the root has ξ ∨ g ( C , namely: πi (2) as in ( r /h 2 2 ∼ = C e su g of the Lie algebra ) n ∗ q ∼ = , and in particular h h (NS sector) and its the root lattice, by ·|· ) be the simple roots, 2 ( h . ∗ / ⊂ ∗ 24 ) 3 h ∈ , h supercurrent w F n, w | + ) ( ξ ξ ⊂ ∈ ( w ∆ | } i ξ . f r , i πi NS ( = 1 2 Z ) α c ∀ . With this Killing form, the e πi , w ∗ Z | 2 n +1 i ξ h N e Z q ( n and on ) defines an isomorphism ) n has zero kernel in the Ramond ) = 2 2 P πi q ∈ , . . . , α ∨ g ). For example, for 2 ) -tuple of eigenvalues (the charges) 1 0 1) H ) 2.5 Q e ∨ g and r = ( α ∨ i n, w G n − i have length-squared ∈ ( ( /h , q h g α g ˜ . To keep track of both the 2 n, w ) ( g g R τ ( g P Q c and w ∈ P g X of | w P NS Z . The n, w 0 we are studying, we will omit the subscript on the c ∈ ∗ 1 2 ∈ ( Z , but g g X g 9 1 2 2 g h w R . ) = P r c ∈ g Z X of ∈ ∈ n g g ∈ X P – 7 – ) w X P P n w ∨ r = , ∈ Z ⊂ X { 1 2 Z = w  ) ∈ Z X , . . . , w ∨ g =  = F is the rank of n The Killing form ( ∈ 1 F g ∗ X Q n ) 1) ) w ( 8 = Q depends on the normalization of the Killing form, which is not g , . . . , α i ( . For any choice of the 1) ∨ . We denote by − g = 0 ∨ 1 2 i (R sector).  ( − g Q / α ≡ (  ( ( 1 πiξ πiξ tw 24 2 2 w πiξ πiξ ≡ F e e be a triangular decomposition. Let 2 2 g e e c c = rank( 24 24 + P c c 24 24 − − r g implies that the zero mode 0 0 − − L L 0 0 ⊕ 1 2 q q L L be the coroots. We normalize the Killing form h   q q − the coroot lattice and by h   is even, so that ⊕ 0 is a weight h R NS NS R -modules, with opposite fermion number. This description immediately ∈ ∨ g L − ) 12 g ⊂ Q T r T r T r T r ∨ r ∨ r D = ∨ i , so that one can identify = V α 2 0 ) = ) = ) = ) = g Z G i 2 = 1 , . . . , α , . . . , α / τ, ξ τ, ξ τ, ξ τ, ξ P ∨ 1 ∨ 1 ( ( ( ( 2 ˜ α -modules, the canonically twisted module can be identified with the direct sum of the α R R We can choose the basis vectors of the SVOA to be simultaneous eigenstates of First we describe our notation. Let us choose a Cartan subalgebra = ( Occasionally, when it is clear from context which We notice that the isomorphism f φ φ NS NS 12 8 9 , which we often keep implicit, simply identifying and let ∨ g φ φ D ∗ root, coroot and weight lattices. completely standard (long rootslength do not have length eigenvalues, we introduce characters depending on h coroot lattice and of the Cartanfor generators is the dual CoxeterQ number of its dual lattice (the weight lattice). g of positive roots, where and that the long roots in each simple component and negative fermion number. 2.2 Partition functions In this section, wecanonically compute twisted the module partition functions of the SVOA two spinor shows that the lowestthere conformal are weight no in statesrelation the of Ramond weight sectorsector, is and therefore establishes an isomorphism between the components with positive V JHEP02(2021)039 ρ 2 # / L ): )  g ) r α | (2.29) (2.28) (2.24) (2.25) (2.26) (2.27) − ξ (     πi   2 ) ) . − α α | | = (24 ρ e ξ ξ k )) ( ( q N , and that the πi πi , 2 2 ρ τ, ξ − − 2 ( , )) 1 + e e ) 1 2 1 2  α f so that the Ramond | NS , with − − ξ φ n n =1 ( τ, ξ ∞ N Y ( k q q ± τ, = 0 R  2 ( ) − ) φ 0 = 2 2 r/ α 1 1 + | θ 2 1 G ρ ξ τ, ξ ( + L ( 12+ =     -dimensional representation of πi copies of the representation ∆ ) ) ) ker denotes the Weyl vector of 2 ∈ τ α α 2 12 ! NS e | | ( α g 2 ξ ξ n dim φ r/ ( ( F η P ( q Q 2 : the space of highest weight vectors πi πi 1 2 1) r 2 2 2 ∈ ) e e − r/ 2 τ 1 + = ( 1 2 1 2 α . fermionic zero modes corresponding to 2  (2 − − + ! ± r n n η ∆ =0 q q F 1 ∞ 2 r Y ∈ n α ) = 0 1) " − πiξ 2 − + 1 1 + P = 2 2 -representation must be self-conjugate, because – 8 – ( e τ, ξ   ∆ 1 2 g ( c . In particular, Y ∈ 24 + + )) )) )) ± ˜ R ρ α = ∆ ∆ α α α − 1 φ | | | r 0 Y Y is unique, so it must be isomorphic to its dual. Thus, ∈ ∈ ) ρ ξ ξ ξ L α α ( ( ( m πiξ q r r 24 2 q τ, τ, τ,  

F e ( ( ( 1 2 1 2 2 4 3 c R 24 − − θ θ θ − n n (1 + T r + + + 0 q q 12 12 12 L ) ) ) ∆ ∆ ∆ =1 q ∞ τ τ τ − . Finally, this ∈ ∈ ∈ Y ( ( ( m α α α g ) =

1 + 1 η η η q   Q Q Q 2 r     τ, ξ 2 2 2 NS 2 ( ) r/ r/ r/ ρ ± =1 =1 | ∞ ∞ T r Y Y ξ n n R 0) 0) 0) ( 2 2 φ / / πi τ, τ, τ, for the definition of the multivariable theta functions. 1 1 2 ) = ( ( ( 2 4 3 − − − θ θ θ A q e q , which is isomorphic to the direct sum of τ, ξ g ( = = = ± ) = ) = ) = NS τ, ξ τ, ξ τ, ξ φ ( ( ( In the following, we also need the linear combinations The last equality follows because, as discussed above, R f φ NS NS φ φ and giving the partition functions on the eigenspaces of the fermion number. the canonically twisted module of the highest weight must be the opposite of the lowest, and therefore equal to states forms an irreducible modulethis for description, the it Clifford is algebra easyweights of to in fermionic check zero this that modes. the representationmultiplicity of Using difference is either between the the the lowest or highestis sum highest and weight itself is over the a the lowest module positivethe for Cartan roots, the generators i.e. Clifford of subalgebra of spaces with positive andwe negative use fermion the number factthe are that algebra isomorphic. the Ramond In groundwhose this states highest weight computation, form is the a Weylbeing vector the number of positive roots. To show this, we first notice that the space of ground and See appendix A direct calculation then gives (here JHEP02(2021)039 to can only ]) . 2 (2.33) (2.34) (2.35) (2.36) (2.37) (2.30) (2.31) (2.32) w w g [ / P 1 D, ∈ ( X w c . ≡ 4 and ) n n, w , ( are non-zero only when ]) X ) . Since there are only a c w ) } , M, ([ ) ) ∨ g can be restricted to ∨ g w n, w g m | Q ( ξ ( Q is over integers in the Ramond w ( depend on i ≥ − ( X i c πi ) n g ≥ − 2 + . ) + e g . w ]) n ) w | in the NS sector, or over the coset ) w n, w q w g w ( ) g w | g ) ( ([ ∈ Q X 0 P w 0 w c m ( is a root of = 0 − | + w 0 n, w ) ≥ ⊂ w ( n | ∨ g − ρ w w ( n 2 ≥ ( X Q ) g n 0 ( if if otherwise c    − 2 n ∪ Q /i g w | g g n – 9 – P    0 P 1 r 0 ∈ , the coefficients Q X w ∈ ] w (          = 2 w { , and the sum over ; we will sometimes write ±} [ := n ) ⇒ X D ∨ g g = max ) = , we can also give a bound on the discriminant that is ±} ,R will correspond to the (respectively, even and odd) root ⇒ ˜ ) Q Q , given the absence of Ramond states of weight − ( ∨ g g , w ± ,R M ) = P R Q /i (0 ± ( c g ]) = min NS ∈ − τ, ξ , P , these functions admit a Fourier expansion /i ( w 6= 0 g w w ([ X ∈ { NS ) A , and over half-integers in all other cases. The sum over in P and 6= 0 c φ ˜ m − R,NS ) − X w n, w ( NS of NS for all n, w c X ] ( we show that the coefficients c = w X g , one has [ NS,R, c A X ) = 0 = 0 in the R sector. More generally, the sum over NS, , w n g (0 P ∈ { ± ⊂ R X c g The coefficients In particular, when In appendix As shown in appendix Q + while multiplicities for the BKM superalgebra that we will construct in section Finally, for where where is the minimal squarefinite number length of of cosets vectors in independent in of the the class coset of through the discriminant and on the class emphasize this dependence. be reduced to aρ sum over the root lattice where sector or when JHEP02(2021)039 . 8 f\ = 1 V fE V (2.38) (2.39) (2.40) (2.41) (2.42) (2.43) N or can be 8 24 ... fE F + V ... q + q structure, namely 2048 . As a consequence, − by any such automor- = 1 f\ 2 K3 sigma models with / 8 + 2048 V 1 structure on ... N q 2 fE 4) / by an orbifold procedure, + or ... , 1 ... V q 2 with a given choice of 8 f\ + = 1 q + 2 24 V fE 2 = (4 q structure, via orbifolds of q N F V SVOA, we will next explain how 24 + 276 or superconformal structure inherited N 24 − 8 = 1 F 2 + 49152 / fE + 24 + 276 q = 1 1 N + 98304 q V + 49152 that are closely related to symmetries of 2 / q is exactly the one induced by the parent q N 8 1 = q 24 fE 24 = F V 24 2) ) can be obtained from the SVOAs – 10 – . = 4096 ] it was shown that there is a certain set of su- 24 0) = 2048 τ supercurrent of 2 supersymmetric nonlinear sigma models proposed ( . Similar relationships between the chiral SVOA  τ/ 24 24 8 ( τ, η 24 ) ( 24 F η ) 2 . +1 4) τ ) fE − τ τ , ( , and that the orbifold of = 1 τ 4 R 8 η = (  V 0) + 24 (2 φ η = 0 T η ) and the non-chiral  η N 0) = 24 + 2048 fE τ, = (4 or  c f\ ( − 24 12 τ, structure on V c ( 24 ± − V N f\ = 0 0) = − R − L = 2 0 V φ  τ, q L = 1 NS (  c 24 q F SVOA. φ + c 24 F − 1) R N 8 0 0) = − 1) φ L , we get the same formulas for all choices of 0 − fE ( q L τ, − ]. In particular, in [   ( q ( V = 0   − (essentially self-dual SVOAs. 50 ξ , NS NS R R NS φ 20 , and the , = 12 = 12 f\ is constructed in this way, there is no c c from orbifolds of 15 V 0) = Tr 0) = Tr 0) = Tr 0) = Tr , 24 ]), the orbifolds discussed in the next section will be a precursor to various spacetime τ, τ, τ, τ, or 24 F ] it was shown that the SVOA ( ( ( ( 12 , ˜ with This result is interesting in view of the correspondence between chiral vertex operator Having established some basic properties of the If we set all F 30 8 R R 15 2 f φ φ NS NS s\ fE φ φ in [ persymmetry preserving automorphisms of supersymmetric sigma models on phism is either theV SVOA where the group ofV symmetries we quotienttheory. by In preserves this theobtained section, superconformal from we the current will of show that all choicessuperalgebras of and non-chiral by an orbifold bysymmetries a did not cyclic preserve groupwhen the of symmetries.from However, the original in SVOA. both This raisessuperconformal cases, the structure question this whether can group be of obtained from upon the 3 In [ one can obtain this theory,Though including we the choice do of see not [ undertake adualities full relating string string theories theoretic with construction different in perturbative worldsheet this descriptions work based (though and showing that In particular, JHEP02(2021)039 . . . - 8 λ ∈ 8 1 8 δ 8 G V ) for fE A 8 0) λ /E (3.3) (3.2) (3.1) fields V ) = V fE admits ,..., λ R 7→ − · . Since g V of fields of 24 8 8 ( λ ,..., ⊗ δ πiα 0 = 1 V 8 fE 2 24 , /D i e ∗ E 8 1 supercurrent V , . It would be ( , i D 0 e be the fields of 7→ 24 ∈ − F = 1 λ 8 , V : V 0) E , these i i N ], it was shown that . ,..., is a supersymmetric := ∈ δ ) i ) . To check that this is 50 16+ but acts by ∂X λ , ,..., Z i 8 λ = (0 0 2 8 , i ψ . 16+ , i 15 , ∼ λ fE 8 1 : λ D 0 ∈ e group of symmetries which 8+ V V A , , ) i i λ i ( , where ∈ 8 e i Z e x ) − 2 λ 8 V λ ]; in [ -twisted sector is the V 8 : =1 = (1 δ D i ∈ i X , with can be obtained from an orbifold , and e U(1) SVOA, with the supercurrent . Consider a symmetry := i 20 | } α fermions 8 8 V structure corresponding to , + 8 =1 i i x i . The standard X 8 24 e : 8 8 χ 12 {  F 8+ ( 8 of the SVOA acting trivially on the /D =1 := i X λ fE ∗ Z = 1 8 = 1 , for all ,..., ∪ i 2 | , λ V e D i 8 + structure, by a symmetry that commutes V 8 N − ψ N free fermions, notice that there are -invariant D 1 2 Z = 1 V δ i , where  i e ∈ := = 24 = 1 8 , , another of the four cosets of V i ) ∈ 8 i i 8 , and 8 λ – 11 – N ) ψ 8 E D 8 corresponds to : -invariant subalgebra structure on ∂X with the ,..., . The δ δ + , 8 is a subgroup of a 8 i =1 24 i X α ,..., D , . . . , x 2 the structure constants of ψ = 1 F = 1 is again an 1 Z i , . . . , x ∼ , together with the x : 1 , i 8 ( = 1 , so that ijk N i i x δ ∼ = . Furthermore, since 8 lattice as c e Z i ( h ( 8 i ± R 8 , ( δ Z i V ∂X = h E i ∈ by 8 = can be written (up to normalization) as ψ ∂X 8 8 2) : D ) with and , respectively, generating , / G D i i 2 fE 1 . In particular, 8 , with its standard =1 is one of the four cosets in / 8 ψ 2.3 ψ i X 8 V + preserving automorphisms for which the orbifold is 2 8 λ (up to normalization). χ ∼ fE /E , it turns out that the orbifold can be described as a lattice SVOA based D : ) (and therefore preserves the supercurrent) and acts by 8 ,..., V supercurrent. Let i . This is a symmetry of order i G 2 = 1 R Z + 8 / . The group ψ 1 ⊗ χ , and ∼ = D , N G ∂X , i , namely 1 i 8 = 1 ) 2 2 , 8 + generate the whole SVOA. Set ψ / E / 2 ( i SVOA have also been explored in previous works [ ∂X D 2 1 N / χ / 8 1 ∈ P + 1 E = (1 : ∈ α α = 6 ( ∼ χ The orbifold of Let us first consider the case of c λ ∪ , is the standard basis of G 8 = ¯ 8 Then, the supercurrent which is of the form ( on the odd unimodularactually lattice the same as the SVOAof generated weight by Z weight inherited from the parent theory.lattice The SVOA based onmodule the corresponding lattice to theD coset We note that that acts trivially on for supercurrent preserves some and We use a description of the Let us show that eachof choice the of with the weights is they can be relatedsome via further a certainvery reflection interesting procedure. to understand Here, whatof we the the show meaning correspondence. that of this result is on3.1 the sigma model Generalities side c JHEP02(2021)039 . , k j 24 g by G λ k ) = F . In i . In ij (3.4) (3.5) α ⊕ θ J 24 is also ) have N πir ( j 2 F θ , which fields in = e B P and dual 2 -invariant g ijk 2.16 → / is obtained σ c . With our k 1 ≡ i k ρ has the same g J B ) i P Q λ k has a ‘quantum ( ⊕ σ B = 7→ structures on g supercurrent and such that projects out most i P λ 24 can be obtained from ∈ = 1 F -twisted sector by = 1 8 k r ρ ) — by an automorphism of N structure. Then, one uses the δ of to be an inner automorphism . fE k N corresponding to a root θ act by σ V ⊕ . The symmetry α 2.16 = 1 = 2 A σ , with Weyl vectors fields corresponding to the Cartan = λ , then the CFT k g 2 i N ρ g , / δ . h 1 12 k is be the current superpartner of the free — whose supersymmetric descendants 24 g dim g i i F 12 is again J λ fields corresponding to non-trivial roots of listed in ( i dim ) = with its 2 . In order to preserve the supercurrent k Q g 8 / ) automatically ensures that there are at most 8 h ρ 1 | g ) = k fE – 12 – of them — this is the number of spin fE k ρ by giving all the other V 8 ( ρ V supercurrent, the induced supercurrent in | 8 fermions B k k ρ X fE ( = 1 24 be a Lie algebra automorphism acting as which preserves a given V is also preserved by this symmetry. The condition that -preserving cyclic group, we know that the orbifold of , it must preserve the structure constants θ g ) = k N 24 . Explicitly, let to act on the fermion ρ λ | = 1 F g j ρ σ λ ( i by multiplying the states in the into simple components N λ g B ijk , and let c -invariant fermions must commute (i.e., they must be contained in acts on the free fermions in -invariant 24 σ . Then, the Weyl vector of δ 8 σ k i,j,k projecting out all spin ∨ g , and to act trivially on the spin h via an orbifold by a cyclic group fields, leaving at most α P σ ,..., λ 2 ) has a A ∼ / . This condition can be achieved by taking α | 1 8 such that the orbifold of and acts on = 1 ρ A G ( i δ Q πi fields surviving the orbifold projection, because the algebras listed in ( , 2 i on the currents. Then, we let the symmetry spin as e 2 is an automorphism of λ / j 1 θ 24 J N by the ‘quantum symmetry’ will give back . Furthermore, the currents that are supersymmetric descendants of these ) = ij in a given Cartan torus acting non-trivially on all non-zero roots. We take the symmetry A symmetry To implement this construction, we need a symmetry There are similar orbifolds of 8 8 α θ g λ j fE fE -invariant, because it resides in the untwisted sector. By applying this general procedure spin ( , and such that the orbifold is consistent, can be constructed as follows. Let so that σ be the decompositionCoxeter of numbers normalization of the Killing form, the Freudenthal-de Vries strange formula reads 8 rank at most of g Since implies that the superpartners of some Cartan subalgebra of the Lie algebra descendants of the it is sufficient that (currents) generate one ofthe the corresponding Lie Lie algebra algebras fermion P V of the V fermions must commute with each other — this is what happens with the supersymmetric symmetry’ order particular, if Q to the case we arethrough interested an in, orbifold then by if an we can show that order to describe them,cyclic it group is of easier symmetriesthat of to the orbifold implement theory the is isomorphicfact procedure to that in orbifolds by reverse, cyclic i.e.from groups are to the ‘invertible’. CFT find This means a that if a CFT JHEP02(2021)039 , - n of α σ ]). πi +1 on 2 the n ± ≤ σ 46 e λ i σ (3.8) (3.6) (3.7) i α i m , where i Z m i P N 1 , which are 2 P g , then the . 1 2 = 1 , so this formula α acts trivially only ; since + | ≤ i i σ r ∆ m | i , so we conclude that ∈ +1 . 2 P -twisted NS states have n / α Z σ πi 1 2 N 1 e 2 corresponding to the Cartan in the twisted sector in terms acts non-trivially on all i , the ∈ . This standard formula can be acts by multiplication by λ n for all component, we have Z σ 2 i , one has that L 1 r k 2 N σ k i g / ∨ g 10 1 1 . The latter case can be easily ruled ∈ is satisfied (see, for example, [ h ) = 1 P , fields ; then, for any root free fermions with a symmetry i ρ f\ ) 0 1 2 n | r 2 fields of spin − k n ρ | ≤ L V , / θ ( ) 2 | in the 1 1 2 α ρ 24 or | ( algebra, α ρ leaves invariant a number of superconformal = is a consistent holomorphic SVOA of central ) = 1 ( -twisted NS sector are valued in 24 , . . . , α n , and the level matching condition is satisfied. ] = 2 . . . , n ≤ | k 1 1 σ 2 F σ Z σ A θ ) ) α . In particular, if we take | − , α N 1 k α 8 by multiplication by – 13 – Z | | = 1 2 ρ by ,L ρ ρ i α N 1 fE 1 ( ( 2 g , 24 + L i V [ < F ∆ + ) = ( πir ∈ 0 X k 2 i 2 α θ r | . With our normalization for the Killing form, we have ± i 1 2 k ρ e ( g P = 1 2 algebra, the automorphism 2 i contains no such fields. Finally, we verified in a case-by-case r n fields corresponding to the Cartan subalgebra are preserved by f\ A i theory. 2 X V / 8 for all positive roots. In particular, 1 1 2 for every simple root 1 fE i . In general, for a theory of α V σ g for negative roots) for the < ) n , while α g | ρ ( ≥ − i < is the highest root of , so the only possibilities are acting with eigenvalues m 0 k i 12 θ by requiring that the relation N P 0 For example, for an We conclude that the orbifold of In order to check that the orbifold is consistent, one needs to check the level-matching This is automatic by one of the definitions of the Coxeter number. is the order of L . Below we shall consider some explicit examples. 10 i on the Cartan subalgebra.supersymmetric Besides descendants, the the symmetry currents inprimary the currents. Cartan These subalgebra canλ of be easily determined since we know the eigenvalues of the orbifold is the on the root space automorphism acts on the root(and spaces charge out: the orbifold theory containssubalgebra at of least the spin analysis that the orbifold theory never contains Thus, the conformal weights are valued in We stress that it isdoes not not necessarily imply true that that the conformal weights of the ground states are always obtained, for example, by writingof the normal Virasoro ordered generators products of fermionicin generators, and fixing the normal orderingApplying constant this formula to our case, we obtain condition, i.e. to verify thatN the levels of the order conformal weights valued in twisted ground states have conformal weight exactly so that so that only thethe spin orbifold projection. where subalgebra. Notice that, for each positive root JHEP02(2021)039 ; . 2 8 / 16 1  fE 9 10 : V g  , , e ,..., spin  ,  which re- 2 5  be the set 3 4  7 ,  16 10 e } = 1 g 24 e ,     , λ 4 5 F 2 3  7 8 e supercurrent to a o  ,  2 5   5 6 3 4 , e , e   e  , ,  , − e  o e 3 e , = 1 10  a  , 2 3 1 4  , on the 4 5 , 1 3    7 8  λ , and   1 6 e   e 2 5   8 e N  1 3 4 , , e  3 4 , 2 5 e  e ,  ,  e   e   − fE  , 2 3 e ,   e 1 4 e 1 5 8 ,   10 , 4 5 , -twisted (NS) sector, the 3 4 , as well as the currents V  , which together form a  2 3 e      5 2 2 ,  σ  e be the orbifold symmetry  e 4 1 e e 3 4 , 2 5 e  /   , e k 16 , , , to  g , 2 3  g e  1 . Here we use the abbreviation    e λ } 11 12 ,  . We take the automorphism    e 1 4 k e σ 5 8 , 1 4 5 g , 4 5 2 3 , e g 3 4  ≤   ,  24  −   e   σ 1 5   -preserving e , e , 1 4  e e e 3 4 , k F 2 5 e 1 g  , , , 1 3 , of each choice of semisimple Lie 1 ,    e  −   ⊕   e . In the ,  − ,  e e 7 1 3 e 3 8 , k , 12 4 5 , 3 5 1 , 1 4 ,   1 . Let of spin g = {   for a summary of these eigenvalues     1 5    e − e with 1 4 e e , 1 3 e 3 4 a < b g Eigenvalues 1 5 e 24 ,  , -preserving orbifolds of , , , 1 + ,    1 σ  e  , and to act by F  a  1 3  e e , 24   e − e 8 3 8 , , 4 5  ≤ , 3 5 , λ , 5 1 4 F  1 12  1 e   = 1   1 5 in   , 1 3 −  e e − 1 e 1 4 1 , 1 5 e  ,  , , e , , 1 g  , e 1 − e  ,  1  N , ,  , , − e ,..., e  1 8 3 5 1 1 1 , − 3 5 , −  , , , fields 1 b  1  −  1 5 1 12 1  1 , e whenever − e – 14 – λ e fermions corresponding to a Cartan subalgebra n 1 5   ,  1 , − 28 + 120 = 148 8 , , = 1 . See table , , 1 k e − e    1 k 1 ,  g 2 a e − 1 8 , a g 1 3 5 n 3 5 , , / λ 1 , λ 1  1 λ  , which relate  k 1 1  , e , e k − e 1 ∪ + , 1 g , , , ∪ a 1 1 σ  1 = − λ  3 5  can be decomposed into sets of eigenvalues which act on each , = g spin  1 , and , supercurrent determined by g e λ g 1 8 8 σ , λ 1  of ≤ = 1 g , for each choice of λ 8 N . As explained in the previous section, these eigenvalues can be on the g fE 8 3 14 21 21 10 24 15 σ , such that V algebra of dimension a < b g +1 with Dimension , such that ≤ k k 24 g (16) 1 g k F so , . We have already studied this case, albeit beginning with an orbifold of ⊕ 8 1 . + , which we denote by ⊕ b 8 1 = A λ 2 3 2 4 3 2 1 πix 3 A to act by 2 g + (8) C . Eigenvalues for symmetries A A A A B B G = e a of fermions corresponding to non-zeroThe roots, untwisted sector which contains weλ the denote by so g let us consider the sameσ algebra from an orbifold of . The 24 eigenvalues Below we give more details of these orbifolds for several choices of Lie algebra 8 • ) := x fE Simple Lie algebra for each choice of simple Lie algebra which arises in our construction. produce the SVOA which relates of 24 eigenvalues of computed independently for eachalgebra simple component simple component of e( 3.2 Examples In this section, we summarize the action of Table 1 V JHEP02(2021)039 , . ζ πi 3 2 128 (9) − from = 32 , the - and e , and of the , for a su i 5 6 σ = 256 1) 2 gr 8 (4) | (and vice ≤ 2 2 , while the / × − su Z , and 1 Z 9 -eigenvalue − ⊕ + 3! = 84 (8 corresponding + σ 1 a, algebra. In the 5 6 / -representation 1 2 i , -invariant. This λ . The invariant 7 a < b -twisted sectors, . The eigenvalue (1) λ ) 4) σ 3 · 3 1) . It follows that, i ∈ , u ; there are (9) σ 1 8 ≤ A 4 r · × ⊕ − / su 1 (16 currents and the on the twisted sectors 9 × − -dimensional represen- , fields corresponding to ⊕ σ 1 b (16 - and (10) (4 with 24 λ 2 ) = 3 σ 1 a / (16) (1) , 2 here). We have a total of 2 so ) / 1 λ u , and -dimensional representation i 1 so are moded in 2) 6 ⊕ − ⊕ algebra. The untwisted / 24 r with multiplicity × ¯ ω a − λ (6) -eigenvalue z (6) (4 πi 3 a < b (8) obtained by acting on the ground ,..., 2 σ + so , so e a r . Half of them have a so 9 + 10(1 1 λ 1) , additional currents, commuting with ∼ = from each twisted sector combine to on the = 1 ∼ = 2 on the subset of the 6 r -eigenvalue of the states, half of these − ≤ i a σ σ 8 1 4) corresponding to the Cartan subalgebra P and the (4) , / × − 128 gr (9) A i on the simple roots of i u | Z u current algebra. = 6 λ i eigenvalue) in the / ⊕ gr = 8 + | 1 ⊕ + (10 . We fix the action of 2 a − g 1 6 E × + 4(1 ζ / , , together with the six currents λ ω c, 1 2 on each of the spin (6) – 15 – − (10) λ 1) a < b < c − 6 4) (1) so 1 a, / so πi 3 / × ≤ u 1 λ 2 and transforming in a spinor representation of the − ⊕ e . The ground state level can be easily computed to be 1 ⊕ (6 ω b, + Z , (4(1 b λ i = has an eigenvalue distribution . Together, the untwisted with multiplicity 1 4 (6) λ 6 (6) ∈ fields and one more from each of the two twisted sectors, / − 2 + gr 1 b ω 1 fields. From each twisted sector, we have six additional , so they must be degenerate with multiplicity so r | untwisted sector currents. The ground states of the so a σ 2 λ 6 − -invariant and half have . This means that in each of the − are moded in λ / / 2 -invariant currents, transforming in a i a 0 − ω a, σ 1 1 / σ a ω a λ . There are no zero modes, so that there is a unique ground λ − 1 act as it did for = λ 2 λ with (note that we do not require ω c, / ζ σ λ . The untwisted sector currents are 84 − 9 act trivially on the six 6 9 a r / spin- act by multiplication by . Since λ 1 = 1 σ free fermions with eigenvalues ≤ . The ground states must form a representation of the Clifford algebra -twisted sector). Thus, the level of the twisted sector ground state − 8 σ 2 2 ω b, have a mode expansion (4) 16 = 64 currents, forming an 10 σ algebra. λ 3) u fermions are: = 1 · a, b, + 6 / 8 / a 2 generated by 4 zero modes 2 . We let ⊕ 1 E . The untwisted λ (1 3 1 ≤ / i 2) − 1 4 15 + 45 + 16 = 76 1 untwisted spin- A generated by / . We let 16 1 ω a, (8) 3 gr 3 2 6 λ | , and let × , (1 (10) A 3 1 ground states are have modes A so ¯ ω b 1 4 so A λ 18 ∗ -twisted sector have level − (16) = = 3 -invariant currents of the form -twisted sector, the -twisted currents form a copy of the ω a a modes of the while the other half havein eigenvalue such a way that there are of distribution in the formtation (multiplicity of the freecurrents form fermions the is algebra total of σ degenerate ground states, forming a representation of the Clifford algebra of the zero together with the form the g to for a total of σ form state each of the two twisted sectors, combine into an 15 + 81 = 96 σ versa in the is state in each of thegives twisted sectors, and we can choose them to be and by multiplication by simple roots. Therefore,of the spin eigenvalues of with multiplicity λ after the orbifold projection,the there will be so σ g λ 16 of the Since each of these256 zero modes changes the fermions • • JHEP02(2021)039 , 3 1 16 10 , (8) 10 ( has 9 , fields so 1 σ − 2 algebra, -twisted ,..., / = 3 . On the 2 1 g σ n -eigenvalue (1) component. = 1 dimensional σ u i can be easily . Currents in 2 algebra in the algebra. ) for , i ⊕ B σ 8 -twisted ground Z 2 1 n = 16 (4) E 1 12 , (6) σ 4 × − of them) or from the su 2 , and in all cases we so 10 ∈ 6 g 6 , (4 ⊕ i ). This is sufficient to ( -twisted sectors (in the r , 0 × 3 ) from the i -invariant spin σ = 2 structure. The procedure 10 2 3 +1 (10) σ × n , case. The symmetry so 1 2 = 1 (4 , 2 - and for , ) to form the 1 6 σ B N 2 1 , 1) , and form a 2 1 3 (4) 1 2 , G , where the ) and the i -invariant currents. In total, we have su 11 = × − πr , = σ (4) 2 2 -eigenvalues ⊕ 7 fermionic zero modes. The (6 (and more than ± , σ g 2) su with an e 8 , 5 / -twisted sector, form the ‘fermionic’ , . As a consistency check, we also verified ⊕ 24 2 1) 8 (10) (1 σ SVOAs: a general construction fields, while the 1 4 6 = 72 × fE so = 1 in each of the = 12 2 (10) × × V . / c n (2 – 16 – 1 g 8 64 so = 1 component, and for 2 2+10 N 7 G 12 × 6 states in each twisted sector. Next, one needs to project currents in the . As expected, the level matching condition is satisfied. The currents, as expected. In particular, the fields is less than orbifolds complete, we will now turn to a uniform construction 2 2 / 2 representation of from the fermions, corresponding to the Cartan subalgebra of 1 / 24 = 6 × 1 3 4 F 4 6) 6 n representation of , , with central charge 1 2 ¯ 4) as the orbifold theory. , for V , (10 fermions, the eigenvalues are 8 5 12 1 2 , . We performed a similar analysis for all choices of & the rest. For the other algebras, in particular with non-simply-laced fE , (16 20 1 3 8 -invariant fields define a consistent SVOA. However, determining the right 8 V 2 and fixes , σ -invariant subspace. On each twisted sector, the action of , B σ 1 and 12 2 -twisted sector are obtained by acting on the ground states with one of the 12 = 4 2 , G 4) -invariant spin 1 4 σ n , σ -twisted ground states have level = 2 that there is ais choice exactly of phases forobtained which the number of that the phases explicitly is usually complicated,for so this in practice example. we takeof We the verified following that, shortcut forconclude any that choice the of orbifold the theory phases, is the total number in each twisted sector.the Using the number algebra of of spin- on fermionic the oscillators, one candetermined determine up to an overall phase.the The level general matching theory condition of orbifolds is tells satisfied, us that, there when exists a choice for these phases such are From these data, onestates can (NS compute sector), the obtaining conformalfor weights of the degeneracy of the ground states is determined by the number of fermionic zero modes g components, the analysisprocedure is by slightly describing moreorder one complicated. example, the Weremaining illustrate the general 76 + 64 + 72 + 64 =together 276 with the obtained from the OPE ofuntwisted two sector spin combines with the (16 sector (in the representation of the Clifford algebra ofdistribution the on the groundthe states is half-integrally moded fermions, whichof have them). Therefore, we get σ With the analysis of • 4 BKM superalgebras from In this section, we describeholomorphic a SVOA procedure to constructis a heavily BKM inspired superalgebra by starting from the a definition of physical states in superstring theory — it is, in of BKM superalgebras for each choice of JHEP02(2021)039 - ) 8 ]. f\ z ⊕ ( 10 15 = 1 = 1 V − (4.1) (1) N N u . Note ,X of their (the NS ) of states ] for the 12 of central = z is a ] that was ( given by a  48 has central ; the other (1) g , and there 1 2 dimensional + X,ψ 7 int NS int , F  2 tot int X gh 8 V 5 V NS V 1) V V V , int and , and its canoni- NS ≡ − = 24 V ( on the SVAs, such 1) ⊆ f\ F , int NS  lattice, and has been V X,ψ (1 1 2 V 8 NS V  E for int . The canonically twisted NS gh V 2 . Up to a choice of the 2 V = 0 / and the Conway module 12 1 . -dimensional compact manifold, ⊗ 8 g V − 3 8 described in section G of signature fE X,ψ 1 V , 24 -dimensional for 1 V and the ‘ghost’ SVA based on the F Γ 24 8 ⊗ . , where 15 ]. We will briefly describe the main steps g fE is int int R 31 , which are non-abelian and depend on the V 2 V V g / 1 = in [ ] for the proofs of most statements. . The space – 17 – f\ gh f\ of central charge 48 , lattice SVOA V V V 8 ⊗ = 40 , are described in section X,ψ , E -grading given by a fermion parity V is itself a product of an ‘internal’ SVOA m V V 2 31 G V Z m . The basic fields are two chiral free bosons free fermion SVOA -eigenvalue) V = has central charge 0 X,ψ R L 24 tot m etc. V is the lattice SVOA V V gh the possible algebras V V -dimensional for 24 0 ≡ ]. The name ‘internal’ comes from the idea of compactifying the F 18 gh NS and ; for ,V 8 8 int fE . The matter SVA fE (though with a certain ambiguity). We again use the physics parlance: the and a ‘space-time’ SVA V V V 15 F ≡ − 12 1) choice of the supercurrent module (Ramond sector) is denoted by The ‘uncompactified’ directions aresector) represented based by on the an evencally SVA twisted unimodular module lattice of conformal weightfor ( superpartners will be relevantgenerate in a the finite-dimensional following. Liefor algebra The zero modes of these currents studied in [ dimensional spacetime of a type IIwhose superstring on corresponding an non-linear sigmahowever model that is the a SCFT standardsector, of while superstring central our construction charge construction has in this also article an is anti-holomorphic chiral. The space The ‘internal (Neveu-Schwarz) sector’ of the superstringself-dual theory SVOA (holomorphic SCFT) of central charge supercurrent, there areOne only is three given possibletwo by such are the SVOAs, the up supersymmetric to isomorphism [ int − In more detail, the various factors are as follows: Each of these SVAs has a NS ( • • V ity vertex superalgebras are theRamond (R) Neveu-Schwarz sector. (NS) For sectoras this and reason, the we will twisted often module put is a subscript the where the ‘matter’ SVA charge charge is a canonically twisted module on which one can choose an action of the fermion par- 4.1 Super vertex algebras For this construction, weproduct need to consider a super vertex algebra (SVA) of Borcherds’ construction ofinspired the by fake bosonic Monster and stringspecific example Monster theory. where Lie The algebrasgeneralized [ main to the steps example have where in been this developed construction and in refer [ to [ a sense, a ‘chiral version’ of that construction — and is a supersymmetric generalization JHEP02(2021)039 ) = z ( ikX = 1 and (the m e (4.6) (4.7) (4.8) (4.9) (4.2) (4.3) (4.4) (4.5) µ (4.10) V X,ψ N gh R ψ T V dimensional (momentum . 1 , respectively; µ 0 and the 2 α / ) 1+1 , 2 z = k 2 ( / ) µ . The fields matter sector z gh , . . +1 P T r :( and − +1 = 3 µ 2 −} −} n z mn . / c , , r 2 − is as the lattice of vectors 1 γ as the ∂ψ z + + − 1 ν µ n , . The mode expansions are , , c 1 + ψ ) ) 1 = and have weight in the Ramond sector ∈ { ∈ { Z Γ Z z X,ψ : z X − ∈ ( ∈ ( µ − r V X 1 4 n k . The stress energy tensor ψ . The chiral bosons alway appear = = 0 + X,ψ − X,ψ k − ν −} ) = G ) + T ) = and 2 , + z , ψ , µ z z ( η ( − 2 + + γ c / , :( int ) + ) + and ψ 1 = , µ z z in the R sector. The vertex operators µ V ) − ∈ { ( ( 1 r ν gh z = 0 V V − k µ − V ∂X n 2 2 :( µ z , T G = 0 / µ − µ k µ r − −− 3 µ ≡ – 18 – z ν n k , or exponentiated in the form of vertex operators η ψ − µν µ n − r 1 ). The stress-energy tensor ) = ν ∂X ) = are given by η α z gh ∂X = − z z : NS + gh ) n µ ( ≥ ( z Z Z V ≡ with quadratic form b . z r X 1 2 ∈ ψ V ∈ m m ( n X Z β ++ n r µ Z , T G η ∈ k ν represent the light-cone coordinates in a X n ± = µ + X,ψ ⊕ = 15 k Z ) = ) = ) =: − µ X G X superalgebra with central charge c Z ∈ z z z that are eigenstates of the zero modes n r ( ( ( ≡ ) = i ∂ ∈ . A convenient description of ,X µ z is a SVA generated by the anticommuting bosonic fields 2 1 k ( i∂X , ) + ψ | k b X,ψ X,ψ = 1 1 ) = gh with eigenvalues X Γ z T G ( V µ N m, n β ∈ with metric are superconformal descendants of in the NS sector and in the NS sector 1 k , µ i∂X 2 2 1 , with matter stress-energy tensor / / ) = ( supercurrent R − ∂X X,ψ = 1 = 1 , k = 1 supercurrent V ν ν + for each k are superconformal primaries with conformal weight ⊗ N ( We refer to the product of the SVAs = 1 ≡ int ikX ikX where canonically twisted module of The ghost sector and their superpartners, the commuting fermionic fields N and total central charge and generate an e the fields V and where correspond to states operators) of Roughly speaking, space-time either with derivatives e k and their superpartners, the free fermions • JHEP02(2021)039 , Z ∈ φ φ n ∈ and p even ∂ p (4.19) (4.15) (4.16) (4.17) (4.18) (4.11) (4.12) (4.13) (4.14) is un- 2 , / m i i 3 p 0 | | , 1 . The fields -eigenvalues. − 0 15 , have , L φ : 2 − is strictly larger ; therefore, only 2 generated by the i m having the lowest φ p e = | i c β∂γ p η∂ξ and a chiral scalar = 0 | 2 : . and / 1 2 − 3 2 1 gh tot / φ c − generates a lattice vertex V − 2 η, ξ : and ∂ p + 1 φ ⊗ -invariant vacuum γ . It is often useful to have an annihilate 0 gh ) ) ) − 1 m . z T C γ ( ≥ − − φ V , ∂β + bγ ηe :( ≡ 2 have arbitrarily low ∂φ∂φ and , while . 1 2 = c 2 1 X,ψ i gh i − as a subalgebra of a SVA generated by β 0 0 PSL(2 − T γ ) | − V | : z -modules, starting from a state n : have conformal weights -modules are related to each other in the ( ) + pφ ⊗ and 2 e β, γ βγ γ / V βγ b 1 b∂c b∂c φ have charge β∂c = T γ X,µ ( − . More precisely, the fields 3 2 i β 2 : V = 2 : p and Z -module. It is easy to see that only for – 19 – | − − . − ⊗ ∂ξe of conformal weight gh ∈ : β ) 2 , by γ : βγ z and T − c η = int c ( ) n m ) b c + V β − ) , , r , r > p and ∂b z ∂b φ ξ, η, φ m and = n 2 ∂β β :( m :( T ( L ξ = 0 = 0 e (R sector), satisfying tot − − n i i − = , while p p Z V | | P = = T r r -dimensional lattice, and always appears with derivatives are given by superVirasoro algebra with central charge + +1 ≡ β γ ) = eigenvalues are bounded from below with 1 ) 1 2 gh z gh ) = ( z 0 , all positive modes of both tot T T z ( NS ∈ } L ( = 1 gh V obey the same OPE as 2 gh p T G / G N 3 superVirasoro subalgebra with central charge -module built starting from the − η, ξ , βγ 2 and their superpartners have charge = 1 / 1 or in exponentials c γ N , respectively. One can define a ghost quantum number, with respect to which − 0 The , 2 1 / is the picture number of the and 1 and eigenvalue (ground state). The different larger algebra generated by p {− in this case the bounded from below, sinceMore the generally, one states can consider different (NS sector) or tensor becomes in terms of these fields.than the Note one that generated the by SVA generated by The fields algebra based on a n > or odd depending on whether they act on the NS or the R sector. The stress energy b − c alternative description of the superghosts two anticommuting fields and form an supercurrent The full SVA contains a stress-energy tensor JHEP02(2021)039 . . 1 X,µ R in the 11 V (4.23) (4.24) (4.25) (4.26) (4.20) (4.21) (4.22) on the 0 G ⊗ 2 ker int R , V 0 and compute the β Z (R sector) = , f\ ⊕ 0 ker 0 ¯ m V , R has order Z :) ) V ,G ⊗ , and then to further i F 0 ∼ = 0 f\ gh 1) 1 , β +1 V , 0 ,G 1 − γG 0 ( , Γ ... ,L (: . , β 0 ∈ gh . 0 R 1 2 b Q ) . ) gh V −→ k ) − ,L + ( G ) k 0 ⊗ n 0 p ( b k , k + n h ( p :) m and shifts the ghost number by , H + R C k V +1 + gh ker tot n p X,ψ R , we have a complex = p, k ∩ V C = ( G 1 cT k,p,n , 1 + k gh ⊕ ⊕ + (: R tot Γ Q R 1 2 V + −→ V V ∈ ( ) ) = ⊗ G + tot NS k k k ⊕ ( V ( m r = – 20 – ) n p is just the product X,µ i C G R ≡ C 0 1 r gh , V , leaving bosons fixed and multiplying fermions by 1 − tot G Γ ,L γ ⊗ Q 0 V tot ∈ GSO −→ + b k r denote the subalgebras generated by the corresponding h V int ) V R X ⊕ m i k V ( 0 G ker + = 1 (NS sector) or to the kernel of = − ∩ m n ,G = 0 C n p 0 L + and momentum , and the momentum tot G n R ,L , β Z p tot 0 V − 0 NS 2 1 b c ...C V ,L ∈ n 0 X b p = ( h ] we will study a non-chiral superstring theory based on = C 30 and Q and Z i 0 supercurrent ∈ ,L n 0 b = 1 h N Strictly speaking, it is not clear that one needs to impose the extra restrictions to , is the product of the fermion number operators on the single factors. The canonically 11 1 , which is integral in the NS sector and half-integral in the Ramond sector. We now and we define the corresponding cohomology spaces as R sector, though itare is a insensitive choice to that thisbecome is simpler. restriction, sometimes In but made [ for computationsphysical convenience. states of The without the states imposing cohomology at this of nonzero extra momentum R states sector at condition. zero-momentum can which is a nilpotentFor operator each that picture commutes number with Notice that the NSp and the Ramond sectorintroduce can the be BRST distinguished charge by their picture number where elements. Onnumbers this space, we introduce some gradings given by the ghost and picture subspace which is the eigenspace of therestrict total to fermion number the with kernel eigenvalue of Ramond sector. 4.2 BRST cohomology The next step onsuperstring the construction, path is to to obtaining perform the a chiral GSO physical projection, states, i.e. mimicking to the consider usual only the even of the Ramond sectors,For where each we of defined these theextended SVA, the matter to action Ramond an of sector actionthis the on expansion; fermion we the number assume operator Ramond that on sector. a the choice algebra There has been can is made, be a so certain that ambiguity in choosing The fermion number operator− on twisted module (Ramond sector) of and JHEP02(2021)039 ) ]. k ( 23 and (NS 2 with / 2 1 (4.27) (4.28) 1 / ) 1 − 1 − . k − ( H , m n = p G -eigenvalue V 0 p H 6= 0 L 2 with , pairing int because the module is V 6= 0 H ) · , k , = 0 · , so the theorem generalizes ( ! k 1 2 int , the no-ghost theorem ensures and with positive (respectively, V ) + k h 2 ( 2 might have a non-trivial kernel and 6= 0 k , k the matter sector is a free module +1 . k n − p X ! restricts to a non-degenerate form on

H 1 2 , ]; an alternative proof is given in [ − For H 6= 0 int + R ) → 40 ) = 0 · 6= 1 V 2 k ) , k 2 · might be degenerate. We will deal with the ( k k , we get a non-degenerate bilinear form on n ( -eigenvalue n ( ∼ = ) p 0 − n k p H – 21 – ( ! (0) L

2 H 2 1 2 and / / : − 1 . Therefore, at least at nonzero momentum, one is 1 1 + 1 − − int X NS with 2 6= 0 H H 6= 0 V 2 -invariant vacuum state of the matter SVA k (R sector). It is also reasonable to expect that one can , there is a ‘picture raising operator’ homomorphism k ) V k ∼ = , combining this bilinear form with the spectral flow iso- − denote the components of the canonically twisted module 2 C → ) , /

) , the homomorphism n, k k 3 ) For h ( + 6= 0 k ( − int 1 ( R = 0 , − k 1 2 ). Similarly, there is an isomorphism of vector spaces − V int 2 / R k Z 3 / PSL(2 H V ∼ = 1 1 − + ) − H denotes the component of the internal SVOA 1 2 For each k . For ( and ) : ) . For = 2 ) ∈ k h ) / ( k p 1 h 1 2 There is a non-degenerate bilinear form − , which is defined in terms of the bilinear forms on the matter and ( X ( 1 − ( − ) 2 . In the Ramond sector, this bilinear form non-degenerately pairs 2 / ) + k + / H int 1 int k R NS 3 − 2 1 ( − 2 1 V ( V − k case separately in the following. 1 p H of the internal SVOA 1 − annihilate the H − n − 2 − 1 H 6=0 2 and negative fermion number (the latter condition is automatically satisfied, since = − = 0 int k k − R where V where h h the induced bilinear formk on that there is an isomorphism of vector spaces ghost vertex algebras.⊕ In particular, and morphism ⊕ immediately to the casenot we are free considering. in It thatL fails case: for there are relations corresponding to theH fact that This theorem isThe proved in proof section onlyfor 3 uses the of the superalgebra [ fact generatedalgebra. that by This the for is negative true modes for any of critical the internal matter SVOA superVirasoro sector) and restrict to these canonical picturesinformation, at and zero we momentum do without so losing throughout any this interesting text. which is an isomorphismled if to consider only the cohomology groups in the ‘canonical pictures’ Let us recall some results about this cohomology: Equivalence with light-cone quantization. Bilinear form. Canonical ghost number. Picture changing. JHEP02(2021)039 ) ) X,ψ R that 4.30 4.32 V (4.31) (4.32) (4.33) (4.34) (4.29) (4.30) ). The ⊗ ), ( X,ψ NS int R = 0 V has a rep- V i sector needs ) 4.31 ⊗ k = ( u, k 1 int | , we simply get NS m = 0 R , 1 − V  V 0 , k A particularly use- 5 8 H . = 0 is given by the zero 0 = is given by states of −  ) 24 ) ≥ 1 2 h m F k m 0 NS (  ( , it is convenient to per- n, r > L V , r 1 − k = ∀ + int  0 1 − R R V H int V admit (possibly non-unique) . The outcome is that in the ], respectively. In superstring V ∼ = n > ) = B ∀ k ) 31 (  , h implies 2 . It is easy to see that states of 2 1 ( -closed and therefore define classes / 2  1 + , / Q int = 0 1 i , − R 1 − = 0 i V 2 R H ] and [ i k = 0 V ] that every class in u, k | i 48 ) are χ, k 2 , 46 | u, k for | φ φ/ in the matter SVA χ, k − m 0 − | 4.32 = 0 e e k in the matter Ramond sector G 1 , since m r 1 , i – 22 – c c k ), ( G 24 are the statements that the BRST quantization for F u, k = 0 = | i i 4.31 m r 6= 0 G k χ, k χ, k | | ) = 0 = k and such that m n i  ( supercurrent. For null momentum L 2 1 2 1 cases, we refer to [ / , the condition 1 − u, k 5 8 − f\ | 1 − = 1 m n V − m 0 H L L m 0 N  L It is clear from the previous observations that the and ) and satisfying ( = is a state of momentum 8 is a state of momentum -exact). Similarly, all classes in 2 i ) i of the fE . Vice versa it can be shown [ 4.30 Q ) m 0 V k V 0 ) is known as the ‘Old Covariant Quantization’ in superstring theory. χ, k G ( u, k | | ( G 1 1 − 4.34 H or ( to be considered separately,apply since in most this of case. thecomputation. theorems Fortunately, it we The is mentioned analysis easy above is to do find described not the in cohomology appendix groups by a direct (since definition of the space of physical states in terms of states satisfying ( might be representatives of the form where with fermion number i.e. it is athe form superconformal ( primary ofin weight resentative of this form, but possibly more than one (i.e. some of the states ( satisfies ful set of representativesthe for form the cohomology classes in where theory, the isomorphisms for non-zero momentum is equivalent tois actually the an light-cone isometry, since quantization. itand preserves This the on isomorphism bilinear the forms SVOA on (and the cohomology its groups module). mode form a case-by-case analysis. For For the negative) fermion number. The isomorphism Zero momentum. Cohomology representatives from old covariant quantization. JHEP02(2021)039 ] . . , 2 ). / C 24 48 1 F − , with (4.43) (4.44) (4.41) (4.42) (4.35) (4.36) (4.37) (4.38) (4.39) (4.40) 4.30 1 for , with , one has by 2 ) 24 , + 1 mod 2 0 1 F = 24 k p , 0 + , N = is spanned by + 2 k  of physical states. ( n 1 2 1 int , in the sense that (0) − = 1  V Q -exact states, so that | 1 m − phys q u + Q int + | NS H H n p at picture number V . C . One has . . } int )) →  + 2 k = 0 . When V 1 2 ) ( 0  = dim 2 f\  k k u, Qv 1 2 / in ( . However, the representatives of 1 { V v ,  2 1 int 0 m − q 24 R } ) / +1 | V is defined by F = C H −} 1 u int u NS , 1 | ) × V is ⊕ − k and BRST charge . In particular, ,...,N + int 1) ( ) b ) u, Xv ( n k p k V | − . int { X (0) ( ( u ∈ { = 1 C | 1 1 n p V i = − 1 − + ( 1) ∈ C } and H } : (0) = dim H − u 2 ( – 23 – 8 } / 1 . Notice that these states are of the form ( -closed and there are no ∼ = , , 1 (0) = 1 (0) = dim (0) = 1 1 = ( 1 1 1 { f\ 1 Q fE − Γ 0 1 2 − − − Xu, v Qu, v } ∈ V (0) V H { { k 1 H H H ,, µ a are i − ⊕ = u, v = = i for i C { dim 0 = 0 } } | dim dim dim | (0) 1 int 1 1 c c = 0 are the fields of weight V i φ − u, v u, v φ of an element -graded versions of skew-symmetry and Jacobi identity, and is phys − { { C − 2 N e Z H e Q Z 2 X 2 2 , in particular when / / / -picture, the cohomology is non-zero only at ghost number 1 , so there is no cohomology for 1 int Z µ and − a − 2) ,...,N V ψ v / 8 = 0 1 | ∈ fE = 1 u −  | ( V a 1 2 ,  . a ] for the cases ], we define a Lie bracket for 2 v int / R 31 41 -picture (NS sector) the cohomology is non-zero only at degrees 3 In the V − = 8 1) structure on − The description of Ramondand states [ is adim bit more complicated,or and we refer to [ where N In fact, all statesthere in is an isomorphism of ( = 1 where the parity This Lie bracket satisfies compatible with the picture changing operator Let us now exhibit theThe structure starting of Lie point superalgebraFollowing is on [ the to space define a Lie superalgebra structure on the graded complex N the cohomology classes do dependof physical on states this that choice we and, will most define importantly,4.3 in the the superalgebra next section Lie depends superalgebra on of this physical choice. states To conclude, the space of physical states is given by Notice that the dimensions of the cohomology spaces do not depend on the choice of the JHEP02(2021)039 ] on 48 X,ψ and NS ) 0 V ), i.e. 0 (4.47) (4.48) (4.49) (4.45) (4.46) deter- k g ⊗ ) + k 4.30 , one has k int − ( NS 24 ( 1 V p F − n − m 2 = is a subalgebra . In particular, − q -picture version = ) 2 -closed, and the + − 0 + ) 0 n p k k Q m , H NS ( , ) H int V g + )). Then k 2 V ( of the form ( / k i → 1 6=0 ( 1 ) with k g . 1 ) p 0 4.32 ) k − ) . ( χ, k k k H L , depends on the choice of | 6=0 ( 1 k ( ] , = 2 k ; when one of the elements ( ), ( 2 1 − , defined by / , ⊕ . m ) q 0 / n → ) p q 0 1 [ 1 g L H k is defined by k γ H ) 1 H . When − ( 0 = ] (0) 4.31 ∈ ∈ ) 1 k 0 + + g × 0 H p g ( , k g i 1 ) 1 p . k , ( g if otherwise 1 , k ( [ 1 i q 6=0 u, v u, v ( H 1 Γ k -exact. f\ , where the even and the odd compo- n M ∈ 1 p χ, k H − if if otherwise | 1 ) k × Q V 0 L 2 -closed states is still χ, k g H H ∈ ) | / k : = 1 2 Q k ˜ ⊕ u v ⊕ (see eqs. ( or / ( m − + 1 } 1 → 0 1 0 0 is not an isomorphism in this case. In [ 8 ) , 2 X g g 0 G m − k g ) { / 1 ( 0 H ], proposition 5.17 for a proof); bilinear forms k fE 1 G and = c k : = (0) on ( +1 48 + V ) – 24 – u 2 2 = q } = g k / ) / , k + ( 1 is i 1 ( k 1 { p 1 p 1 ( 1 − 1 − are odd (in the Ramond sector), then the bracket h·|·i 1 Xu 1 H − induces a well-defined bracket (which, by slight abuse (see [ H int H with 1 H 1 − χ, k v -closed state is | H 1 } − V ∈ H φ , b g × Q . → } ∈ 1 { , and therefore the bracket − u i H , ) e 1 ) and k } ∈ X H 1 Γ is just the zero mode of the current corresponding to the ( is a state in the matter vertex algebra that non-degenerately pairs ) u, v (0) M ∈ 2 χ, k u 2 { k / i 0 | u, v / u, v Xc 1 H ) 2 (˜ 3 { X ) / u, v 1 − = · 1 1 − 0 ( − = ,    χ, k 0 H · m − Xu | the bilinear form is in general not defined, because the picture          H ( g 1 : G on classes : − Xu = ] = ] -exact and a } b ( i (0) , X Q 2 { v | / u, v u, v 1 [ [ u are given, respectively, by the Neveu-Schwarz and by the Ramond physical 1 , where h − i 1 H and then use the bracket g ) supercurrent. χ, k | , so this case is simpler. This form is symmetric when restricted to k ( φ matter state and 1 0 − ], it was proven that when 1 0 = 1 e ) is even (in the NS sector), then one needs first to map it to its H (0) = g 1 u 1 31 c The bilinear form This is made more explicit if we take representatives of One then defines a Lie superalgebra g ∈ , and in particular it is the derived subalgebra N coincides with g = (0) = 0 ] 1 , changing operator and [ of g antisymmetric when restricted to On As a consequence, weight mines a non-degenerate bilinear form that is a superconformal primary of weight so that Xu the picture changing operator the u In other words, when[ both (say, states: The Lie bracket of notation, we denote bythe the BRST same cohomology symbol) —bracket between the a bracket between nents The latter property ensures that JHEP02(2021)039 , ) r ∼ = = 24 a F (24) , we int 4.39 i , (4.55) (4.51) (4.52) (4.53) (4.54) (4.50) 0 is also so V a = | 1 J ⊂ c int φ g h·|·i − V e 2 . We see that, / 1 g ± − . As described in currents a ψ . For λ ∨ r 24 is isomorphic to of the finite dimensional ; this properties follows fields g g g , . . . , α 2 . ∨ 1 Q ∈ of / in terms of momentum (i.e., ∗ α 1 is the rank of , h is actually a maximal abelian g ) r (0) ⊂ k h . A maximal abelian subalgebra 0 ( − further states of the form ( g g g , x, y, w ˜ Q , which is a finite dimensional Lie ) on the two states ∈ ,P 24 g , , ⊕ + (0) ) , where h , X ⊕ 0 P Z ˆ Let us specialize to the case where k 1 g 2 ( corresponding to the space-time currents ⊕ − )). Thus for all ⊕ g ⊕ + 2 b i 2 -grading for , and no other generator with nonzero mo- m, n, w g . An obvious consequence is that the only 0 i 1 r ( Z 0 ⊕ , Q | ] k 0 (1) 1 are the ones in the zero momentum component 1 ∈ 2.31 g u ∼ = is ˆ k Γ , with generators ± − (1) − Z g – 25 – g g α ⊕ w, y u u u ∈ [ Q P , there are ˜ | µ Q , whose eigenvalues are the space-time momenta (0) ∈ M = = = x k 24 1 ± 0 w h m,n ⊕ g . In particular, the even and odd components are (0) = i F g h component α . We conclude that has rank and generated by g 0 0 1 , | = h g = ˜ ] = (see eq. ( 1 2 g Q 2 + 0 = ). Acting by / ⊕ i g ∈ , u 1 P g is the union of the root lattice ⊕ y = 0 µ ± | − Q ± µ ] g (1) := Γ ψ Z P P k [ P u + P 2 g / ⊕ ρ 1 w, x Q . Thus, ⊆ (0) = 0 [ m − g 1 h Z ) g g G of the Cartan algebra, we can now introduce a finer grading for the ∈ Q of definite momentum ∨ r ) + g ρ -picture version of these states correspond to the ( states 0 ∈ (again, . Thus, the zero momentum subalgebra ∪ with values in the lattice 1 u g g is a Cartan subalgebra for , . . . , α g 24 m, n, w ∨ 1 Q g α . The and its translate := = ( ⊂ , the zero modes of these currents generate a semi-simple Lie algebra , that are superconformal descendants of the weight is given by g g (0) ˆ k 2 h . These operators obey the commutation relations ˜ 24 1 Q − 1 . The zero modes are − (0) . When the internal SVOA is eigenvalues), by taking into account the eigenvalues with respect to the remaining 0 ± , and consider the even H , k g µ 24 + (0) ,..., where graded as Here, algebra subalgebra for the whole while our construction providedP a natural generators superalgebra of where the zero momentum oddmentum component can commute with section of dimension with the abelian component with elements generators commuting with both g in 1 get the weight ∂X k invariant, meaning that from analogous properties of the bilinear formCartan subalgebra on and the root vertex multiplicities. algebras. F subalgebra of on a superspace satisfying this property are called supersymmetric. The form JHEP02(2021)039 ) ) ). , the 2.28 2.36 (4.65) (4.61) (4.62) (4.63) (4.64) (4.56) (4.57) (4.58) (4.59) (4.60) ∨ r in the m, n, w w ( 1 g of -eigenvalues , . . . , α . ∨ r ] ∨ 1 ) w α [ and ) which extends the , . . . , α nm, w ∨ 1 ( (0) α − M, 0 = 0 , R g superalgebra: see ( m, n, w , i , c ≤ g ( g ) , − 0 g ) and Q , on g g ) P w | g = 1 ) = g n | Q w , | − Q ) w } ⊂ N + w P nm, w . R ∈ h h·|·i ( ρ . + ( 1) − ⊗ 6= 0 + ( nm, w = ) 0 ∈ , ( , general properties of the coefficients ) ∨ g i , w , one has ) implies that NS + ˆ k ≥ mn + Z c Q ( R mn + 1 2 int ( , w 1 2 c g R 2.2 P 2 r | -eigenvalue − | V 2.35 Z ( ⊕ − mn 0 ∈ + ) = m, n, w g ∈ ( = L − ) =    P n 1 Q h i n P g of the root and on the class ∈ g R g Z – 26 – Q . With this choice, one has mn, w ) ˆ k with ∈ ( ⊕ g + { mn, w w ρ ( − | M ⇒ + 24 and section ∈ m, n, w the subsets of even and odd roots, respectively. The ± m,n w | := F int int w are the Fourier coefficients of the Jacobi forms ( P NS R A 1 ) 1 ) R V V , ˆ ) = ∼ = ∆ + ( ˆ ∆ 1 1 := g n, w − ∪ int NS ( . This form satisfies n, w m, n, w R mn 0 6= 0 and V ( g = nm, w 2 h ˆ ) int ( ∆ 0 NS i int − ≡ h R ± V ) = dim ) = dim ˆ − ∆ on ≡ V R 2 P c g ˆ k | ˆ ) ∆ . We denote by are orthogonal to + ·|· g m, n, w ( P ( − and h , with Q m, n, w m, n, w g is real-valued, with signature ( ( g . Furthermore, the condition ( P ) restricted to the real spaces 0 1 ) ∈ g g R ∨ g ) dim Q ⊗ is a constant depending on the choice of the and h·|·i ( g dim dim nm, w 0 + /i ( Q g P − ). As discussed in appendix P m, n, w ( := NS M > c ∗ R ≡ 2.29 h Since all (super)ghosts and superconformal generators commute with ˆ k and quotient where Here, and ( of Jacobi forms implydepend that only the on dimension the of norm the root spaces w the analogous component in the twisted module ing. As a consequence, if we denote by the component of the SVOA bilinear form and equivalence between BRST and light-cone quantization is compatible with this finer grad- for the set of roots of Cartan-Killing form and both The bilinear form onspace, the defines cohomology, an when invariant non-degenerate restricted bilinear to form the zero momentum space NS and JHEP02(2021)039 ) ˆ ˆ < ∆ ∆ α ( .A 0 ∈ ∈ g (5.1) ] = 0 , with is real α α γ ∈ = 0 G − R x y H and for all n on . being finite ) x, G ) 0 [ x to α α such that ( < ], a root G h·|·i g for all ) ) on the norm of α . (ad 47 h·|·i h ∈ ( can be decomposed . ∞ x α = 0 4.65 G + G y n of ) is such that x −→ →±∞ α n root G i ), such that (ad y α | ∈ such that that we constructed in the pre- be a (complex) Lie superalgebra. . α x , i.e. if for all h 1 g G 2 g and and negative if G n ⊂ x 0 ] = 0 ⊕ + roots are allowed to have non-positive β , with each eigenspace 0 contains all roots. is called a i > H and if G β H ∗ ) R | h x, G α = H H [ ( h = 0 α such that the restriction of n ∈ simple G ∼ = i R ∗ α R β – 27 – | ], and we begin by describing this below before + 2 H H Let α , then i | 47 h ) is not a root, so that β | h β ( we have (a regular element) that is not orthogonal to any root h there is only a finite number of roots α nα | are orthogonal to each other. ˆ as a Borcherds-Kac-Moody superalgebra ∆ R 0 = if it is not of finite type. The bound ( + 1 < H i g ∈ | G β of eigenspaces for ) β ∈ , nα (possibly depending on h N > α n ( h n + if it acts locally nilpotent on γ and G | α β 0 . Moreover, if ). Furthermore, | 0 ⊕ ∗ R G admits a real form nα H with infinite type of infinite type or of zero norm that are both positive or both negative, i ≤ + γ H ∼ = ], corollary 2.5.11). β . A root is called positive if β | R h finite type α 47 , then for any α, β h H 0 . < N ]). In our case, it is useful to use an alternative characterization of BKM su- ) > | β ) i 47 ( is a Borcherds-Kac-Moody superalgebra. h g α ( | G α ∈ α one has for all roots (so that and such that for| all as a direct sum dimensional. A non-zero eigenvalue respect to which , there is an integer h For any The algebra There is an element There is a non-degenerate, supersymmetric, invariant bilinear form There is a self-centralizing even subalgebra y g 5. 3. 4. 2. 1. ∈ Then, Theorem 1 ([ Suppose that the following conditions are satisfied: with Thus, for sufficiently large and First, we list some relevantis definitions. said According to to be definition of y 2.3.17 of [ root is said to bethe of roots implies that a root of positive norm is necessarily of finite type. Indeed, if peralgebras, which was givenembarking by on Ray the [ proof. 5.1 Generalities on BKM superalgebras In this section wevious will section prove is that a thethe Borcherds-Kac-Moody Lie usual (BKM) superalgebra Kac-Moody superalgebra. algebrasnorm. BKM because algebras the They differ can from example be [ defined in terms of Chevalley-Serre generators and relations (see for 5 The Lie superalgebra JHEP02(2021)039 . ; , 1 if ˆ g ∆ ˆ ∆ 0 6= 0 ≥ ∈ ). If ∈ (5.3) (5.4) (5.2) ) < N . 0 ) 2 > | . This η | ⊥ n 0 ) t ( ) ) γ ≥ h h h α ( ( + ( + γ γ γ if and only 2 | ) = | mn h M > ut m, n, w 0 m such that the ( ) > < γ η | > | 0 L ) = ( η ) h , so that for each ( ( ; then, η γ such that ( g for all α − , that M | g w ) 0 2 satisfies the following t ( L g > i ≤ B ) + ; ≥ . This means that a root such that i , is a BKM superalgebra. γ n | h ) , then ; g | the dual lattice, then there γ M 0 + M h h i ≤ ˆ ) 6= 0 ∆ can be uniquely decomposed ) 4.3 ∗ there are only finitely many , is a non-zero null root of η ⊥ ) mn ) η | > h m | −h ∈ γ h ( P η | ) = 0 Z ) η ( ( ⊂ γ ( h L ⊥ − . + 2 γ ( ⊆ 4.65 γ ) = 0 ∈ g 0 large enough. If is a primitive vector in the lattice > P − h γ ) = ( w P t M 2 η Q | h ) = L | γ < )) must be a root of L )( h ⊕ w η η ) ⊂ ( | ( . This shows that no root is orthogonal η 1 w η | , then , η γ . By ( g | g ) = g 1 η ( + ( − η , then ⊥ ( ˆ ∆ + 0 h 2 − P ≤ ∈ L 2 + ( such that ∩ = Γ ( mn ) L 2 2 , then γ ) one has γ 0 ( n > P P w L – 28 – | > − ∈ = 0 w or mn 4.65 ) is positive large enough t )( ⊥ n i ≡ − = ), that 0 , constructed in section ( η γ g α | i L 0) | B = in the real Cartan subalgebra of h η , . Then, there exists a positive integer ) + 2 | α ( 2 g 4.65 R cannot have opposite sign, since for a root h h m m > n h h and . Any root ≤ n of 6= (0 + ∈ 2 Z is a positive root of by ( is a root with ) ) 2 g , then by ( η be an element of the Cartan subalgebra of η ∈ is a non-zero root of with with g 0 w ( ∆ m , with and ) = 1 0) + h ( ) ˆ g g g g ∆ , h w 2 h ≥ ∈ m, n − ( ˆ ˆ ˆ . If we denote by ∆ ( ∆ ∆ ⊂ m ( L ∈ u R γ α ) ∈ ∨ ∈ ∈ g h 0 LP ≥ is a BKM superalgebra 6= (0 mn ) ) ) , i.e. if = Q ) 2 − ⊂ 0 is not orthogonal to any root, for g , w ) t with , there are only a finite number of roots to be very large, so that, in particular, ∈ + 0 η 0 ∨ h > g ( , L η Q m, n ) w is a positive definite lattice, for each LP ( of is a root of η satisfying this bound, and therefore finitely many roots with m, n, w m, n, w m, n, w such that ( ⊕ − − ) ) N > ⊥ Let = (0 (notice the w 2 ⊥ there is an upper bound h − , where P ) γ = h 0 = ( = ( = ( P n ⊥ Z ∩ not both null, and that for h ∈ ∩ γ γ α γ Z ≥ We take + ∈ u P and P + t ⊕ m, n, w m, n, w m for all if if if Without loss of generality, we can assume that . ( m, n ∈ + tu h 3. 4. 1. 2. 2 m, n = ( = ( P ⊥ L as fixed Since γ proves point 3. γ and only if to Z exists where we used that for γ if so that Let us prove the is a root with Proof. for all non-zero roots element properties: Using the characterization of BKMcan superalgebras prove presented that in the the Lie previousThe superalgebra subsection, following we lemma will prove to beLemma a 1. useful intermediate step. 5.2 Proof that JHEP02(2021)039 , is  be ) for η 0 and L | ) ) 2 g (5.5) (5.6) (5.7) (5.8) η 0 ( h ( − −} , w . , 1 0 ) implies )  , η + < α | η | nL 4.65 ( ) ∈ { h = (0 ( µ , one can choose mn γ a null root with γ are finite dimen- g | , 2 µ ) ∆ ˆ q ∆ . Eq. ( P α ∈ ( 1) − ∈ g 0 . , ) with value g . Let w 0 contains the root lattice ) n ) Q , , , η ∈ | | + 1 ) ) , but not both. Thus, for ∗ R > + 0 η ) α 6= 0 L is a positive null root with h h r 2 h . ( 2( (  ⊕ ( ) m , w α )  √ = 0 ( γ 0 w | η y m, n, w | L , | L ≤ 6= 0 ) = n 2 η w = η y ( ) |  | = ( η or ) ) ) = = (0 − η η = ( η L 2( | | ( α . Let us prove that 1 γ 0 2 η η g q (  w = 0 )( | m, n, w mn L − − w 2 | m 1 ≥ w = ( . The non-degenerate bilinear form satisfies L >  ( . The dual space  g + 1) g ) α q ≥ – 29 – 2 η L | ) y < L − 2 ( η n is real with signature | in the decomposition ( ) is a root of L ) , so that n  + 0 η R ) − ( h 0 α w n + m ( 1 w 6= 0 ( is non-degenerate, supersymmetric, invariant, and g |  , which clearly satisfies the properties in point 4. ≥ ), so that m L ( ) 1 w = , so that either h L nL | ( 6= 0 h·|·i ) ) = α ≥ h ≥ , where h ( ; the latter generate a Cartan subalgebra of the compact real = 0 ( ) g ) constructed in the previous subsection is a self-centralizing even , we can take the real algebra generated by γ mn ∨ r η α | h R ( ( h h w α mn cannot have positive norm. For a lattice of Lorentzian signature, . Now, suppose that g . g as in lemma , we have . We have , . . . , α ) + g , the right-hand side has a minimum at ∨ 1 n L L R . Its restriction to of is a BKM superalgebra. y α h of 1 , and therefore all roots of + 0 g g g 0 ∈ (recall that w ˜ Q w m h ( n m ⊕ L . Suppose first that 1 0 q , The subalgebra are both positive or both negative of non-positive norm, then they belong to the 1 . This implies ) = = > h To complete the proof, we just need to establish point 5. As discussed above, a root We are now ready to prove the main theorem: As for point 4, it is sufficient to prove it for y ( ) = Γ = 0 α h sufficiently large so that α, β ( that the norms ofan the element roots are bounded from above. Asof a infinite regular type element,if in we can take sional. As a realby form the coroots form of the finiteQ dimensional Lie algebra all the required properties: orthogonal to Theorem 2. Proof. subalgebra, and all components sufficiently large for all roots for all roots w As a function of so that Since there are only finitelyL many roots of the form Set α another non-zero root of sufficiently large JHEP02(2021)039 , . , . ) ) 8 1 k } β 0 ) is is is g } g k A for i − . k ∆ θ w j , w ) δ 0) =1 0 ,... m, n ˆ = 0 γ − k n ∆ is the 2 ∈ and | , so it , , i g 0 g ⊕ , w k 0 . Then ∪ { γ , if m, n, w , γ α I , 0 k h ( w 1 , are real ∈ k , = g m, n, g i γ I } { 6= ∈ g ∆ (0 = 0 i ) + (0 i 0 ∈ ) = = (1 } 0 α = ( i n w w . Let us prove ij { are necessarily is not in | + ˆ k α α , . . . , n ⊕ ) be any non-zero A δ . { ) − = and the existence ( h 0 k , ( ˆ ˆ g g ∆ w α m = 1 , w := | for all ∈ 0 − , . . . , r ) , and , k if and only if ) 0 is the sum − < k k requires a case by case g , since of the form , . . . , r , θ | w D g ) ) g = 1 γ − k − h = 0 , θ , = 0 ( i = 1 be such that 0 ) = (1 0 i γ , i − . If , | , and ) m, n, w } w , β g , i | , w ) 1 (1 . The only way to obtain a root (0 } i and the subalgebra generated by α , 0 g , but not both. For such for all g be the highest root of , α h + , = ( k ) 0 ∆ ) , α δ g γ , (1 α 0 γ , then the matrix is a root of . ( , . . . , r , so the subalgebra generated by the = 0 ∈ , ∆ k = (0 and, in the case g − k 1 k − ∪ { satisfies ˆ g , then w ( ˆ g n w + k . When D 0 k , we will show that in the case g ∈ k . This means that no non-zero element − k ˆ θ I D − := (0 ∆ δ g = (0 ⊆ { dimensional and transforms in the adjoint 6 ∈ g ∈ k ≥ or or i − i i , γ θ ∈ k } ˆ ) α ˆ α 24 i ± I + k k ) = 0 ˆ α θ is and D L { = 0 m, n w − ) , w , let – 30 – , + ⊕ 0 , let 0 g m , := , w h , 0 if and only if , be the simple roots of if and only if + k ˆ g (1 ˆ , . . . , n g ∆ = (0 D g , . . . , n cannot commute with := (1 ∆ . But if g γ . Therefore, g ) . Furthermore, one has P ∈ + g = 1 + 0 k α ∈ ∆ , this does not necessarily mean that they form a complete is in = 1 δ ( r w ∆ ∗ is positive if ) g k ∈ h k , can commute with all ⊕ must be a simple root. An analogous result holds for roots of ) , any ∈ , i ≤ 0 ∈ , w ) k generated by ) β w 1 0 | g k , and set w x , g θ k we perform this analysis for the BKM algebra corresponding to α , . . . , α is a root of − h g 1 , . . . , r 0) := (1 − 6 ) , , α w m, n, w as a sum over positive roots is as 0 . For the last statement, it is sufficient to notice that, if , i.e. m, n, w (1 , ) ) ( = 1 , so g g are both isomorphic to the affine Kac-Moody algebra belongs to a non-trivial representation of the finite Lie algebra k g Let g = ( (1 . For each i θ ) and , w ∆ g x γ 0 , − α ∈ m, n, w ( + g , ) , ( . By lemma i w g , so 1 g ∆ (1 , span the space , + g − k , α -dimensional adjoint representation of D ∈ 0 (0 ∆ ⊂ ⊕ , then ∈ , 0 24 γ ) ∈ A root w ± α 0 , . . . , n simple components ( 6= 0 w We stress that, while the simple real roots g L := (0 n , while we leave the other cases to future work. w ∈ = 1 8 1 ⊕ i ˆ k set of real simplethere roots. are infinitely For many example, real in simple section roots. any the form is a set ofthe real Cartan simple matrix roots of the equalcorresponding affine to root Kac-Moody either elements algebra must be isomorphic to a positive root of simple. The space representation of of the form where simple roots of set of simpleThen, roots the of subalgebra of h Proof. A Proposition 1. of α 5.3 Simple roots andIn Weyl this vector section, we discuss someof of a the simple Weyl roots vector. oftreatment. the A BKM algebras In complete description section of all simple roots of cannot commute with allnull generators and of non-zero, oneone has has that that either forms a x automatically satisfies are both null andnull are root proportional of to eachthat other. a Let non-zero element If same connected component of the cone of non-positive norm vectors, so that their product JHEP02(2021)039 , r ∼ = ˆ α ) with (5.9) is an k (5.15) (5.13) (5.14) (5.10) (5.11) (5.12) ( γ cannot ) 2 r / ,..., ρ ) 1 coincides 1 the Weyl ρ − 1 − ˆ α 8 1 , k − , for all the H 1 , ρ , this means A ˆ ρ , 1 , = mn (1 only satisfies the 2 g = (1 ˆ − ρ , with ˆ ρ which is the sum of k ), we obtain = g − .  , one has 2 3 2 3.5 k =1 k 24 ∨ g n k 1  h . F ⊕ i int R − ± k = = V δ | g , ± k ∼ = int δ i is the defining property of a Weyl ) , i h ) = 1 V . k k 1 2 α α ( g . This implies that | θ γ , . below. By ( 2 i the real and imaginary simple roots | ρ i / ) Q k = i α . Given that i 1 . Note that even if β − h ρ ) = 0 γ 2 | 1 α ∈ − g k | 5.4 ρ | 1 2 all / ∨ g γ | 1 i γ ) h 3 H h ρ h α 2 ( , ρ ) = ( ≥ ) = 1 1 2 k i − is spanned by the simple real roots θ ) = ], who showed that the Weyl vector is indeed 1 2 − | α 2 + ( β k | , 8 ∗ – 31 – ρ admits a Weyl vector, then it must be equal to θ 1 ρ – − ( h + ) = | i 5 g γ ρ − , we get 2 ( = 2 ) = α g k | = ( β i ρ Q − i ( ˆ = ( − ρ ( i , | γ for all simple roots ˆ . The Weyl vector obeys the usual property ˆ ˆ ρ ∈ α r ρ k | i k h ∨ g ) 2 ˆ , one has that ρ = 1 g α with lowest weight h h | i , ρ α g 1 ± k h = 1 δ 1 2 − | as the Weyl vector of ) = , ˆ ρ k n ˆ ρ h 1 θ = | i − k . Since the space = be the Weyl vector of the algebra θ α g | ( ˆ k ρ = ( m we prove that the BKM algebra corresponding to h ρ ˆ ρ =1 is nonzero only for 6.4 n k  is a root of non-zero norm, it makes sense to consider the reflection . For P 2 1 1 ˆ ∆ = + ≥ 2 ∈ ρ 2 k γ , we conclude that if the algebra − mn − k  If Let δ . In section g , int R . We conjecture that the Weyl vector exists, and therefore coincides with + k ˆ be obtained as a sum of positive roots, andrespect therefore to it hyperplane is perpendicular simple. to so that theodd Weyl vector simple has root; zero thisV norm. follows from Finally, the wethat fact notice that, that for irreducible representations of the following we referdefining to properties with respectthe to denominator the formula, real as simple we roots, discuss it in may section still be used to construct satisfies the defining propertiesof with respect to with an algebra studiedρ by Borcherds [ other cases as well, but we leave the proof for future work. Based on this conjecture, in vector for the algebra δ To verify that this is actually the Weyl vector of the algebra, one must check that it and The condition that Thus, if we define vector of the simple component Furthermore, with the normalization we have chosen for the Killing form, we obtain JHEP02(2021)039 i α i k . I be an (5.17) (5.18) (5.19) (5.16) 0 ∈ i I P i ≤ γ = | . γ α ) and set h . ) show that the g α ( ˆ ∆ 0 ∈ 2.29 m is even and real (and α satisfy − 12 γ ) γ is the rank of the algebra α ( 4 ) and ( , respectively. Let ≥ + 1 W, mult r ˆ 2.26 ∆ ∈ , of the infinite dimensional BKM + 0 w ) = ˆ W ∆ 1 g , i , where . ∩ k i 1 may then be expanded as be a BKM superalgebra with even and , where the root α k S − γ g \ α 2 1 I r I g ∈ r/ X ∈ i , X ]. Formulas ( i 2 i ⊕ 47 β [ 0 | ) = g – 32 – ) = α 1 α h α ( ) = dim( ( = 0 α = ht . Any root ( g are always even. Indeed, it is well-known that real i i 1 ht ) α g β w( | ] for the proof in the Monster case). ) to be 43 , m α ) α ], we define the Weyl group 0 , respectively. Let us denote the roots by w( g 1 h 47 of g ∩ α g and height 0 g . ∗ h . Following [ ) = dim( ∗ ∈ α h as the group generated by reflections ( 0 ∈ g m α, β β Let us now write down the (super-)denominator formula for our BKM algebra. First With our knowledge of the roots, we can now study some interesting representation Notice that the real roots of It is known that this cannot hold for general BKMs since, for instance, there are known examples of 12 . This implies that there are no odd real roots, i.e. all odd roots We also define the “even height” as BKMs that do not have a Weyl vector. index set, indexing the simple roots and we define the we introduce some generalities.odd components Let We further denote the positive even or odd roots by sides of the denominator identityin is sufficient the to ordinary determine Kac-Moody themodule algebra case, whose itself. the highest Furthermore, Weyl-Kac weightsimple denominator is low-rank is BKMs the itself (see Weyl the [ vector; character again, for this a holds for some particularly character of the trivialthe module, denominator produces identity a contains Weyl-Kac valuablemultiplicities, denominator information and identity. about (real Each the and side imaginary) rootthe simple of spaces, form roots root of of space automorphic the an algebra object. equality in between In question, certain two and nice takes very examples, different like formulations the of Monster BKM, a knowledge given of modular both or theoretic, automorphic functions associated with our BKM5.4 superalgebras Denominator andAs superdenominator in the case of finitevectors or possess Kac-Moody a Lie version (super)algebras, of BKM the (super)algebras Weyl-Kac character with formula Weyl which, when one considers the for all roots in BKM algebrasmultiplicities have of multiplicity all odd roots areg a multiple of algebra therefore automatically of finitecase, the type Weyl and group non-zero preserves the norm). bilinear form As in the finite dimensional where JHEP02(2021)039 . g g (5.26) (5.23) (5.24) (5.25) (5.20) (5.21) (5.22) and the ) was found . . g ) ) ) ) , constructed in ) m, n, w g , ( mn,w mn,w µ mn,w mn,w ( ( ( ( e of distinct pairwise − − ) + − µ mn, w µ R R ( ( NS NS c c 0 . c c , ) ) − ) ) ) ) ht 0 and the right hand side w and the zero momentum w R w w e T e c T 1) e e g n n n n q q − q q ( m m ) = m m p p µ p p X − ˆ ρ − − − det(w)w( det(w)w( (1 + (1 e mn, w (1 (1 . The Weyl vector of ( is parametrized by W W . product side g g = + ∈ ∈ ˜ ˜ g X ) Q Q X 0 R w w ∈ ∈ c Y Y 2.2 , ρ of w w = 1 = 0) 0) ) α 0 0 ) − , , ) = α , α ≥ ≥ ( ) ( α ) 1 Z Z 0 0 ( α 6=(0 6=(0 α ∈ ∈ ( ) ) 1 ( m − – 33 – Y Y m 1 1 ) ) m α coincides with the Weyl denominator formula of m α are the Fourier coefficients of the Jacobi forms m,n m,n m,n m,n ) e ,T ) = ( e ( ( ) α µ α ) ) ) ˆ ρ e e ,w − e − ) ,w ,w (0 µ − ( (1 (0 (0 (1 − that a root + 0 − − ht + 0 (1 + (1 ˆ NS mn, w ˆ ∆ ∆ c + 1 ( 1) + 1 NS NS super-denominator formula ∈ , m 4.3 ∈ ) c c ˆ ) ˆ ∆ ± α ∆ − α ) ) w ∈ ( R ∈ e w w Q constructed in section α c Q α e e µ ˆ − ρ ˆ ) ρ Q X Q − − − ˆ − ρ mn, w (1 e e ( − 0 . Thus the denominator formula provides a product representation τ, ξ and (1 (1 e ( − g + g + `> ) ± ∆ ∆ = R NS Q ∈ ∈ Y Y c φ g T w w ρ ρ ρ − of the denominator formula. The sum side is sometimes referred to as the mn, w − − indexes only the odd roots. Before we can state the denominator formulas we e ) = denominator formula ( and to be α I − ( pqe pqe ) 5 0 . Recall from section ⊆ is the Weyl vector. The sums here are taken over all sums NS m denotes the underlying finite-dimensional subalgebra of c ˆ 5 ρ S τ, ξ ( g Similarly, the product side of the super-denominator formula takes the form Let us now discuss the denominator formulas for the super BKM We now have all the ingredients to state the desired formulas. For any super BKM − NS contribution Here φ in section Combining everything, we deducecomes that the product side of the denominator formula be- section root multiplicities are given by where For obvious reasons wethe call sum the side leftdenominator hand function side the of the denominator function. and, in addition, we have the we have the introduce the following sums where orthogonal imaginary simple roots. where JHEP02(2021)039 . , 8 1 2 g = + = 8 ˜ , so Q ; in = (2) ∨ (24) γ ) with have 0 (6.1) , this mn is the 1 2 ⊕ O Q w in the su 2 0 g | ; recall ,..., 1 1 , + A ˆ , − θ ∆ w ⊂ 8 = w 1 n for which ( , and the λ ⊕ 6= 0 8 g = = 1 ∈ n Z ) S i ) k = Γ γ is the Fourier n < | with o = in the simplest 2 , while the odd with zero norm ) = 2 γ ) Q 8 n, w g ) h ρ g ( g | Q P , where ρ is − i ( m, n, w 0 ⊂ with g | NS 2 ) SU(2) c Q / m, n, w , and the first non-zero 8 m, n, w 1 ( = ( 1 1 0 θ − D θ γ n, w λ -th position, m ) have norm at most ( = ( ∼ = free fermions corresponding k , and the lowest − Z produces currents generating γ 8 2 ) one has with norm NS Z = 0 24 c 2 n F and on the norm  , we proceed as follows. If . Thus, it is sufficient to choose a i ∈ w 1 2 g 2 2 ∈ at the γ with negative fermion number and . As explained in appendix | Z i Z is an even or odd integer depending ) γ 1 are the x h 24 ) ∼ = i ∼ = has a symmetry F . i ,..., w τ, ξ λ ∗ g ∗ | g 8 ( X 1 2 8 1 ˜ ˜ Q w | − Q in  D A / / 8 . The dual lattice g g w = NS 8 ⊕ + ( structure in Q Q φ ⊕ with the ρ Z . The corresponding vectors of weight  , where . Thus, all roots valued in the root lattice – 34 – in ∈ 8 1 Z mn 0) = 1 θ ) 2 w w 8 + − (equivalently, , a short vector is given by N -weight 0) = 8 2 1 ∗ , corresponding to a state g ..., ,..., 2 , g , and check for which  ˜ 0 Q 2 Z for all simple components, so that the roots of have (0 , / ∪ , the shortest vector is Z . Furthermore, all even roots 1 g = 1 − , . . . , x ∗ g g 8 ∼ = , 8 1 1 ) = 1 i Q ˜ ⊕ 0 Q 1 ∗ x g , = 2 NS of structure of type / A ( Z i . This means that all non-zero even roots of c . The root lattice of the BKM algebra ˜ g Q , θ 0 g g ∨ ( ˆ | / ∆ Q h = P (0 2 g 0) = 8 is trivial or not in ,..., / ) = = , ∈ − = 1 of the Jacobi form 1 Q g g ⊂ ∗ g ) w i − (0 ) , with Q ˜ NS . In particular, we will discuss the root spaces and their multiplicities λ Q N current. The corresponding finite dimensional Lie algebra − c 8 1 . The Weyl vector is = (0 Q + 8 A = 0 k NS ⊕ ρ + θ c ( = 1  n for each class in = mn, w ρ m, n, w ( Z is an even root, we know that the multiplicity ∪ ( is g 1 2 . The root, coroot and weight lattices are, respectively, N w − g in 0  Q ˆ ∆ NS w 6= 0 . The multiplicities of real roots correspond to = = 1 c = ∈ n theory with 0) ∨ g P g 2 ) h = ˜ , Q are of the form 24 n, 8 . In fact, in this case, the norm 1 2 ( ) F ⊕ For the non-trivial class in For the trivial class in 24 − ) w − | in the trivial class of F Z 0 NS w m, n, w w particular, there are no real roots with even norm. Fourier coefficient is free fermion corresponding to the root there are only a finite number of states to checkc in order to findin all real root multiplicities. to the Cartan subalgebra of have multiplicity ( on whether the class of representative that this is theL number of states of 6.2 Description ofIn real order roots to find( the multiplicities ofcoefficient the real rootsmultiplicity depends of only on the class of is an even lattice isomorphic to the root lattice (2 highest roots are The even roots roots have where 6.1 Construction The preserving the has dual Coxeter number length We conclude this note byconcrete illustrating example: the when formal the properties choicethe of of Lie our algebra BKMs and the Weyl group of this BKM. 6 The example of JHEP02(2021)039 . ) 8 ). ) 8 1 E ) = − E only 2 , and ( . For g , w and in r/ W W 2 m, n, w (1 ± o ) = 2 , and their R = ( 8 only depend ρ c 1 ) = 0 E ) ρ γ is the (unique, − , with shortest , corresponding 8 ∗ g − that have inner 1 are the same for θ Z ˜ 1 n, w Q ) = | ) 1 ( θ 8 / = 1 ρ | ⊕ ) ± . We conclude that E ρ 1 , w g in one class of , n is the subgroup that ( R − 1 1 c Q − (1 ) acts by translations by 1 , and the real roots are is Γ are multiple of w 8 with norm θ 1 ± aff ) + ) 1) θ E 8 R g , i ≤ ( c ρ W E ( γ 6= 0 Q | ( (9 , + ( W ) is strictly larger than n, w 8 γ ) = ( ( is the group of automorphisms in aff h ∈ n ρ 1 : E , | ± 2 g 9 ) ρ W w R vectors. This reflection group is , which is generated by reflections n, w − I W ( c ( g of 1 ± of real roots into the disjoint union R . It is also finitely generated, and the ) = W ) have c 2 . ρ | ) m, n, w ∗ ∗ g g real are invariant under the Weyl group of ( ρ is any simple root, form a copy of the of signature w ˜ ˜ + 1 ˆ Q Q | , and both with multiplicity ∆ factor in ) 1 x / / , 3 2 w Z + ( ) ) 8 9 2 g g I E n = n, w Q Q . The set of simple roots can be identified with + ( 2 ( ) for which – 35 – 1 2 − ± + + i ∈ , ρ R n γ ) imply that all ρ ρ mn , where 1 + c | ( ( 2 0 γ − n x h − , 2.29 1 − . = − x i ˜ ), while the , both of square length . The fact that the coefficients ; equivalently, the multiplicity of odd roots W γ 0 1 ) 1 | = ( θ , as expected. x γ w | ˆ h ) and ( ρ lattice, in the sense that, for any choice of an arbitrary fixed + w 8 ( ρ of reflection automorphisms of are exactly the vectors ) = 1 E 2.26 − 1 − ˜ g W , θ n acts on the set of simple roots of of positive and negative ones, depending on the sign of the product in this lattice. The Weyl group 2 (0 ) and 1 8 − real − ρ E ]. As usual, one splits the set ( is isomorphic to the affine Weyl group of ˆ − ∆ NS c , the set of vectors aff t 10 0 (formulas ( , with the vector . Simple roots are characterized as the vectors of norm x 9 W 2 real + ˜ / W W /W ˆ 1 ∆ , and some elements in this Weyl group exchange a vector generated by reflections with respect to norm 1 = 8 8 , and in particular under = 1 ⊕ 9 , , − As for the odd roots, there are again two classes of 9 1 lattice vectors. Since the multiplicities of both the odd and the even roots of I I in the two classes of (2) /r real 8 2 ˆ The group fixes a given simple rootE (say depend on their norm,of they are actually invariant under the full group of automorphisms product the vectors of thesimple affine root lattice. The full group includes reflections with respect toquotient vectors of norm studied in [ ∆ with a regular element. Therethat is cannot an be infinite number writtengenerate of as simple real sum roots of (i.e. other positive roots positive roots), whose corresponding reflections Let us now consider the Weylwith group respect of the to BKM real algebra roots.up As to discussed isomorphisms) above, the odd even unimodularall root vectors lattice lattice of norm of with a vector of the sameon norm the in the discriminant other class.only As depends a on consequence, their norm 6.3 Weyl group is not asu coincidence: the coefficients to Ramond ground states2 of weight Thus, the odd rootsparticular have maximal there norm are noroots odd cannot real have multiplicity roots.w This is consistent with the observation that odd the even real rootsmultiplicity is of representatives both these representatives, the smallest non-trivial class (equivalently, with JHEP02(2021)039 8 1 A [0]) and , (6.8) (6.4) (6.5) (6.6) (6.7) (6.2) (6.3) : i . It is γ q k | (see ap- θ . γ − 8 . − λ −h 2 k D 2) ( θ is an even or NS q free fermions, − λ 8 2 τ/ i φ ]) 8 ) ( k v NS τ η [ i =: c ( 8 ) η P . k D, τ ) , respectively. Then, , by replacing them ( + q 8 τ (2 , − ( η i∂Y ) structure corresponding NS mn odd = c 2 f ,..., + 1 , ) 8 − is given by ) +1 ) = 1 Z 2 8 Z , counting the negative fermion . In this description, it is easy g = / ) 2 2 = 1 τ, z k X 1 4 n ∈ 4 ( i . We perform a similar splitting − τ, z N ) of − ) , ∈ 8 ) q ( k only the half-integral ones. Here, γ D τ n 8 | 8 ) τ ) i ∂Y D ( i NS 8 , ( q − γ τ D x 3 , by setting + θ D k η h φ ( D i θ v 8 θ + ) = i + k (1 v + − Z τ v i X × odd and the second from the ( v (1 + | f Θ ( , λ i ) P k 8 ) + Θ ∪ θ odd Y =1 8 ∞ ) + Θ Z τ Y ) 8 λ n τ, ξ ) + ( and τ ( 2 τ ∈ D ( 8 / by keeping only the integral powers of ( m, n, τ, z q 1 η Z ) ( even 8 = − – 36 – 8 Θ f SVOA with the q 8 NS = ( even ) D f φ Z γ = 24 ) = τ, z F 4 , f ( , . . . , x 12 ) ) = compactified on 8 1 ) 2 ) = Θ D τ τ τ, ξ D x τ ( 8 ( fermions, corresponding to the Cartan subalgebra of q ( ( ( 3 f θ η τ, z 8 ( is the translate ( NS [0]) 8 φ 8 = , i.e. Z pairs of fermions ) = Θ 4 ) = D 8 D, ) Θ ,...,Y 12 8 τ ( ) τ 1 ( D ( τ , and we used τ, ξ + − ( 3 ( Y θ η v 0) + − odd NS v f contain only integral and half-integral powers of ), c , depending on whether the norm NS Z ) = ]) 2 φ 6.1 τ ,..., v ) + odd ∈ ( [ 0 X f τ , D , f ( i . The remaining γ | k and ) = γ even contains only integral powers of = (1 iY τ f 8 ( v ), with ± −h D e ( even A Θ − ) = f by even . We bosonize the = chiral free scalars τ f 8 1 8 ( NS is the lattice ( k 8 A f c θ D 8 ± or odd integer. Altogether, thefunction multiplicities of even roots are the Fourier coefficients of the pendix Thus, the multiplicities of an even root where we have We recognize this form as the theta decomposition of the Jacobi function of of the function where D where the first factor comes fromtheir the superpartners. free scalars We are interestednumber in states, the which function is obtainedconvenient to from split the theta function as are now interpreted as theto superpartners of obtain the the currents NS partition function In order to findadopt a the different root description multiplicitiesto of for the all rootsby of the algebra,λ it is more useful to 6.4 Root multiplicities and denominator formulas JHEP02(2021)039 ) ]. i of 8 θ i ¯ k (6.9) of 24 W i (6.11) (6.10) SVOA ⊗ 24 P 8 F V Z + considered ], this note . g , all of them 31 4 [ 12 ) N ) τ f\ τ ( ∈ ( m, n, ρ 2 V θ η n free fermions. The )) = ( , ˆ 8 ρ γ of the function n being even or odd. The τ, ξ admits an analytic con- ( is a superalgebra already . − ]) 8 n g 8 8 1 v . D ]). Using this construction, n by two. The form above is A 8 + + 1) ) ], and in example 13.7 of [ 16 v R ) 7 τ 48 − ρ , the group of automorphisms + , ( [ φ τ ) ρ ) , ( (2 ) i 31 η η ρ M γ Conway module | w(ˆ γ n ) + Θ SVOA (holomorphic SCFT) = 8 − Aut( −h e 8 8 = 12 ( ) ) τ, ξ ( n n − c − 8 q q = 12 R D c (1 c − + ρ =1 ∞ Y (1 + (1 n – 37 – ], in section 2 of [ ]) = ) 6 ρ ρ [ =1 ∞ Y = (Θ , n w(ˆ i − 4 γ 4 ) e | ) , times the Ramond sector for is even or odd depending on τ = 8 γ τ 8 ( ( ˆ has two orbits of primitive norm zero vectors, which are ρ 2 4 Z 8 η ) −h θ n ) ( τ + τ − M ( − det(w) ( × 2 ρ R η θ ) c W is obtained simply by dividing 1 2 ∈ X w 8 τ, ξ − ) ( R 8 ], the denominator of the BKM algebra τ φ ( Z 8 η + ρ = . The root which is the maximal even sublattice of the odd unimodular lattice of 8 Θ + . The lattice ], and discussed also in [ R 5 M φ ) = 10) ], this automorphic form was also interpreted as a non-vanishing function on , 6 (2 τ, ξ ( R φ ]. In [ As with our analogous study concerning the As discussed in [ 48 we produced a new family ofdenominators, Borcherds-Kac-Moody labeled superalgebras, by and semisimple their corresponding Lie algebras of dimension 24should and be arbitrary viewed rank. asstring a compactifications whose warm-up internal for worldsheet producing SCFTs are complete given (i.e. by products non-chiral) low-dimensional 7 Conclusions & futureIn directions this note we studied somefree properties fermions, of as the well asstring its worldsheet role as theory. the The internal,proving latter “compactification” the system SCFT monstrous in is moonshine a a chiral conjectures super-analogue super- (see of also Borcherds’ [ method for products. One ofin these this infinite section, products whilein is the [ the other denominator is of thethe denominator the moduli of algebra space the of Enriques BKM surfaces. superalgebra constructed tinuation to a holomorphicof automorphic the form lattice for signature associated to two different expansions of the automorphic form into infinite (Borcherds) with multiplicity additive side of the denominator identity, therefore, in this case reads This analysis shows that theconsidered BKM superalgebra in associated [ to Besides the real simple rootscorresponding described to above, the negative algebra integer contains multiples imaginary of simple roots the Weyl vector The function already a theta decomposition, so thatare the multiplicities the of Fourier odd coefficients roots corresponding to theRamond coset partition function, therefore, is The Ramond sector of the theory is given by the product of the module of the JHEP02(2021)039 Ψ Ψ )), ], who 6.11 8 ]. – 5 ]. We also we showed 34 -preserving , 8 1 43 29 , ]. It would be by an explicit A = 1 8 42 . In this context = Ψ N X g from an orbifold of 24 ]. As just mentioned F 43 , (defined below eq. ( ) 42 , M 29 ) played a role in organizing BPS . In the case of in the “geometric reduction” of the g , with a choice of supercurrent, from 24 1 F 24 F F . We leave this question for future work. g Γ = Aut( ). It would be instructive to explicitly deter- 6 BKM (or – 38 – 8 1 , where the denominator of the BKM superalgebra arises A 8 1 A = can also be expanded along its “level 1 cusp” in which SO(10)) Such peculiar critical string vacua have proved relevant for with a fixed choice of superconformal structure. The non- g Ψ , expanded along the level-1 cusp, coincides with the form of × 24 13 Ψ ]. F structures labeled by 30 = 1 . It would also be interesting to understand what ] for analogous appearances of orbifolds of the Monster and Leech ], i.e. in type II string theory on the Enriques CY yield (SO(2) 8 ]). When embedded into a string theory construction, the BKM / 42 N 36 f\ fE 28 we determined the product sides of the denominator and super-denom- 10) V V , we described how one can obtain , 5.4 3 SVOAs, see [ coincides with a BKM superalgebra already studied by Borcherds [ SO(2 g \ by a cyclic group. These orbifolds will be relevant in studying string theoretic Γ can be interpreted as a counting rational curves, i.e. Gromov-Witten invariants, on . The expression for f\ = 12 c arises as the genus oneFHSV topological model amplitude [ Ψ X fascinating to understand ifstates the in a string compactificationframe, on and the if Enriques it CY,string in could duality. some be perturbative related duality to theTo expand BKM on at the the previous other point, we cusp further of note that the same automorphic form from the expansion alongon the “level 2 cusp”a of moduli a space holomorphic closelysame automorphic related automorphic form form to thatcase it of gives the rise Enriques to a Calabi-Yau denominator threefold. formula of The another BKM-algebra [ space; the expansions can(as each in, be (super)denominators e.g., foralgebras [ different are BKM expected algebras tomodel, be and associated related toabove, different to in perturbative one the descriptions another example of of via the dualities [ was able to determine thesimple additive roots side of the of algebra thesemine (see formulas all section and simple explicitly roots, describethe as the remaining well as the additive sidesA of single the automorphic denominator form identities, can for have distinct expansions at different cusps in moduli dualities (see [ VOAs in a string compactification). In section inator formulas associated withthat the super BKM In section orbifolds of orbifolds of trivial question here is toV determine whether one can obtain We conclude by highlighting a few outstanding questions raised by our study: Related examples which are potentially relevant for this investigation are explored in [ • • • • 13 understanding aspects of moonshine,used including are the moonshine genus modules; zero thisbelieve property, was when these illustrated the vacua, for SVOAs viewed theuseful as Monster toy case machines systems in to for [ produce exploring and explicit understanding BKM BPS-algebras. algebras, can serve as these JHEP02(2021)039 , it X ]. In view , there are f\ 35 V SVOA compactifica- in this context, and to 24 = 12 F ], and refer to those articles c 28 , and 27 g ]) contain information about anoma- 4 as there is for its close cousin ], on the one hand, and certain sub-VOAs of 24 11 [ – 39 – F f\ V . 24 F , particularly in contrast with the other SVOA. We follow the treatment in [ f\ V 24 F ] for an exploration of boundary states in a bosonic Monster string theory. ] on the other. It may be interesting to explore generalizations of both of these 16 3 [ \ constructions for tions. The BKMs constructed in thisFinally, note it should would play be aalgebras. interesting key to role Various study sporadic in the that symmetrysuperconformal discrete study. groups algebras symmetry have within groups been of shownV our to BKM stabilize extended See [ Though there is not moonshinenumerous modular for coincidences amongfascinating their to McKay-Thompson see series. if/howproperty It the for would full be string theory construction detects the genus zero ogy; certain variants of thislies cohomology (e.g., and [ D-brane states.in It the corresponding would be non-chiral string very constructions. interestingMore to generally, it explore would these be cohomologies in very the interesting to non-chiral better string understand models the D-brane and states their representations under moonshine groups. of these observations, andwould their be connection interesting to to BPSconnect further states our explore in BKMs the to string role curve-counts. theory of on We constructed our BKM algebraschiral construction. from the In a cohomology true of string “physical theory, states” one in must our take the semirelative cohomol- the gravitational threshold correction of the FHSV-model obtained in [ The work of S.M.H. is supported by the National Science and Engineering Council of • • • • A Multivariable Jacobi forms In this sectionthen we use first them recall to obtainfunctions some for some known the useful facts results aboutfor about the proofs multivariable Fourier and coefficients Jacobi details. of forms, the partition and Physics, under Award Number de-sc0011632,PHY-1911298, and and is the currently supported Sivian by Fund.(Vetenskapsrådet), D.P. the grant was grant nr. supported NSF 2018-04760. by the Swedish Research Council correspondences. Canada, an FRQNT newChairs university researchers program. start-up NMP grant, was andCaltech supported the and by Canada by a Research the Sherman U.S. Fairchild Postdoctoral Department Fellowship of at Energy, Office of Science, Office of High Energy Acknowledgments It is a pleasure to thank R. Borcherds, S. Carnahan and V. Gritsenko for very helpful email JHEP02(2021)039 ) ) − C ) 1 2 and `, ` ( ⊗ + (A.8) (A.4) (A.5) (A.6) (A.7) (A.1) (A.2) (A.3) mn z k for the 2 − L )( . ( 1 2 , , defined ) 0 + Z Z × n ( mn , or ±} ( 2 z , 2 0 ∈ H ) πi ) 2 1 2 ,R e Z ≥ SL + L. 2 ( ± on n ) ) 2 ( 1 2 in × ) πi . A Jacobi form of + 2 ) `, ` 3 n SL L e · ( ( and weight 2 , τ, ξ ) 1 2 · ) ˜ ∈ ∈ ( 1 2 R,NS q ( 1 2 − ϕ ) + , Z + ! z ∈ n ( = 1 n X ( b d mn πiτ 2 1 λ, µ c a , n, m πin 2 − 2 q m ( 2 ˜ only depend on e ) πinz

Z e NS,R, 2 2 ∈ ) e ) = is a weak Jacobi form for some = n n X 2 1 1 2 τ, z , or n ) − q 0 1 2 n, ` ]( y NS, Z q ( ) = n b a 1 ∈ c Z ≥ τ, ξ , q [ n X − ( ∈ ) ∈ { − ) y n θ X n ϕ , q = ) n . Furthermore, one can consider Jacobi ) q = N )(1 X  ) τ, ξ ) τ, ξ y mz 1 ( , with bilinear form ). Let us first notice that all such functions ξ,`  ( Z n τ ( ) − 1 g ( ϕ q ϕ +2 y for a subgroup of index )(1+ − L 2 πi ) )) τ − 1 2 y y ∆( 2 2 d 2 τ, ξ n 1 2 e − ξ,ξ + ( SL / q λ,ξ m ( of theta functions times modular function that n )(1 n − ( 1 – 40 – q n cτ X q n mc q ) πi q − )) φ +2( . The Jacobi theta functions is a holomorphic function 1 − − )(1+ τ πi α ] are Jacobi forms of the same index and weight for ) Z | e 1+ n e (1 n, ` 2 q ξ L k   22 ( ( ) λ,λ bm =1 − y c ∞ √   Y d n 1 2 (( y ∗ τ, (1 Z +2 for 1 2  ( − L = . + ∈ i n 1 2 =1 ∈ − ∞ X an and index n θ πim q Y ` n N n L − 2 − q cτ − y + 2  e 1 1) . These definitions can be generalized in the obvious way to 1 2 ∈ / ∆ − /mL  1+ 1 ) = 1 2 − ∈ ∗ − 24 = ( )  y α y ). ) m n L ) ) =  τ + q  n τ, ξ Q ) implies that the coefficients ( µ 1 2 1 8 q ( − d is restricted to, respectively, in η A.2 y ) = ( − ϕ iq + (1 `  ` + ξ A.2 (1 − 1 8 =1 ∞ mτ Y q n λτ ) = =1 cτ = ∞ Y n τ , + and = = i + b ) and ( (the coroot lattice of the algebra d 2 1 2 1 i i and index n only. n ∆( ] = are weakly holomorphic Jacobi forms of index 1 2 1 2 0 0 h ∨ g + + 0 0 + τ θ Z τ, ξ h [ h A.1 Q ( θ θ θ − cτ aτ 2.2 ϕ ∈ . It is called weakly holomorphic if = as defined, for example, in [ τ, z  0 k ) = ) = ) = ) = , with L ]( ϕ m > N b -dimensional even lattice a τ, z τ, z τ, z τ, z Given the elliptic properties of the theta functions Let us now show that the functions The condition ( According to this definition, ‘ordinary’ single-variable Jacobi forms of weight Consider an even positive definite lattice [ 1 ) ( ( ( ( ∈ 4 2 3 1 θ θ θ θ θ `, ` lattice are given by a productdepends on are Jacobi forms of weight in section the properties ( and on the image of index the sum over ( N Jacobi forms with respectforms to of subgroups of half-integral index, at the cost of introducing some sign in the transformation The Jacobi form is called weak, holomorphic, or cusp, if in its Fourier expansion weight satisfying JHEP02(2021)039 , . . 0 0 for ) of ˜ ≥ NS ≥ g (A.9) 2.5 ) P n (A.12) (A.13) (A.14) (A.15) (A.10) (A.11) 0 = and on w ∈ )) | X 0 D α | w w ξ ( + ( ) 0 τ, ( w i | D ξ θ ( , we get the bound , ≡ + ∨ πi g ∆ 2 )) n Q e Y ∈ )) α 2 , ) | . α ), where the coefficients 0 α + ξ | w ]) )) | ( R . 0 µ ξ w w | 2 w τ, ]) ⊗ α 2.30 ([ ( +( i )( g + ( w D m θ α | ([ Q τ q λ , + ) + , m ]) ) ∈ ∆ α ) | n Y ∈ w +2( w [ λ | 2 α τ 2 ≥ − w τ, ξ we must have ) )) λ, µ ≤ ( ( D, ξ α ∀ | | ( ∨ ∨ ) g g D are non negative, except when λ ) + ( λ − Q X w Q ) , possibly up to a sign. We will sometimes (( α | c | + n ) +2( ξ w ∨ ∨ g w + g τ ∆+ ( ) ( ) (coroot lattice) Q ∈ only depend on the discriminant Q n, w in the Fourier expansion is bounded by λ µ w α | )Θ τ, ( + ( | ∨ g λ ( ⇒ ) = 2 τ n w ]) λ i X X ∈ P /i – 41 – ( (( ∈ Q θ ] c 14 g w 0 0 [ πi πi w + w P [ ∈ w n, w − − ⇒ ∆ . ( ) = ( e e has the elliptic properties of a Jacobi form of index X, D, Y ∈ D Z ( µ X h there are only a finite number of vectors D ± ± α | 2 1 c ) ) λ, µ )) α 6= 0 ∨ ∨ g g n ≡ = = α ∈ | Q Q )( ]) ( ( ξ n α )) = D ( w | /i /i [ 6= 0 α g g λ | X X τ, P P , the sum over ( ) ( µ then for all ∈ ∈ i D, ] ] ± θ ( to stress this dependence. The sign is easily recovered by noticing ∆+ w w + c . The general theory of Jacobi forms for lattices tells us that the [ [ R in the quotient , with + ∈ X n, w ]) ∨ g n ( α 6= 0 ∆ = 2 w admits a Fourier expansion of the form ( c λτ w or Q ∈ [ ) = ) ]) α 1) + of − , we get, for all w ] − D, Q ξ [ . ( o τ, ξ ( ( τ, ξ w ( c 1 2 ( [ NS implies that the Jacobi functions admit a theta decomposition , τ, D, X X ( ) ( 6= 0 0 φ i = φ c ∨ g , where we used the identity n ) θ 4 Q + X , ∈ ( is the minimal squared length of a vector in the coset 3 ∆ only depend on , n, w /i Y ∈ ) ( ]) g α a, b c w P = 2 The fact that the coefficients When ([ ∈ n, w i Once again, the relative normalization of the two sides of this identity depends on our choice ( ( ] m for the even lattice 14 X w Killing form. which [ This bound can be also written as which shows that for each given Therefore, if If and on the class use the notation that, by definition, all Fourier coefficients where the sign is This implies that 1 Jacobi forms c for where JHEP02(2021)039 , , 2 ∪ 8 ∨ g Z 0 ⊕ 8 ˜ , so and Q ) ⊕ 8 2 ac ∼ = (B.2) (B.1) Z g Z + Z (A.19) (A.16) (A.17) (A.18) + P . Thus, ∨ g  : ∗ g w ˜ = Q ⊆ ˜ r = (2 Q / ∨ γ g g g r ∨ g Q Q Q := − . This means Q / m ∨ g ∨ g ∈ ∈ γ ˜ Q Q Q ] is given by w m and w − [ ) ⊇ b ) becomes 8 g is a proper sublattice : ∨ g ⊕ Q g ˜  Q − Q . + Z A.15 , : ) 2 1 ) ρ 0 r ) (  γ w and | Z and on the coset of τ, ξ ∪ ξ = m ( 2 , ( ρ ) c g ∨ g g 2 r πi ∈ ˜ w Q Q 2 P | − , so that i + e D/ ∗ , m w x ) q ) w = 8 0 ( − g w ]) ⊕ i and on the class are Jacobi forms with respect to any | g 2 β ˜ 0 )Θ . − Q X Z w ˜ ) ) w ( Q [ are nonzero only for 2 τ ( | n ( ): w ] ) q = 2 8 | τ, z D, w m ∨ g ⊕ [ g w ( ( generated by ( Q lattice), so that we can set Z − R X, Q X n, w r + g c 8 φ ( h − ∈ w ˜ X Q (2 R ∨ g D ∈ ) n c D 0 – 42 – ˜ are Jacobi forms with respect to a lattice that is 2 1 8 Q X 2 w /  ) and g ˜ ) , and Q X and ) = ∈ ∨ m,r g X τ, z ) = ] τ ( ) , one has (R sector). By comparison, one has w Q ( , . . . , x τ, z [ 8 ] X ( 1 : + τ, ξ ∨ g ⊕ w φ + x [ ( ) ˜ ( n, w Q 1 NS ∨ g m ) = ( X, w / ( c φ Q A ) h n g + is c NS = r τ, ξ w c Q ( g m m − = ( ∗ g Θ . In this case, the lattice ˜ X − ˜ Q G depend only on + Q g g n φ r , then P ) ρ − g γ . ( b P ∈ X r . Correspondingly, the theta decomposition ( c ∈ X ):  ∨ n, w different classes rather than just ] ( 1 2 + Q m 8 R , respectively. If the lattice w m [ c 2 g − , and this leads to more stringent conditions on their Fourier coefficients. In only depends on the discriminant + m − P n ∨ ) ( L w ,..., Q 1 2 ⊆ that is even and contained in the dual 1 2 m and . The dual of c 8 g  ∨ g ) n, w ⊕ Q ( ˜ X m,n m Q =  X X + + Z ρ c n, w ρ For example, when In some cases, the functions = ( + ∈ Q 1 2 NS B Details about cohomology In this note, wephysical follow string a states. chiral The version BRST of charge is the given construction by of the relative cohomology of which is an even latticec (isomorphic to the (NS sector) or that, just for thecoefficients NS for sector, using the most naive constraints one needs to compute the  with w of the weight lattice lattice that rather than is a weakly holomorphicFourier coefficients modular form containing all non-trivial information‘finer’ than about the particular, the coefficients is the theta series of the coset where JHEP02(2021)039 , we (B.3) (B.4) in the is pro- 2 i 6= 0 / k eigenvalue 1 | k with repre- 1 0 c 1 L φ and the linear − is proportional − e = i i 2 p k k / ) | | 1 k 1 1 and one linear com- : ( − c c -closed (the latter is 1 β φ φ r i , of weight : , modulo states of the γ Q k − − H | 0 r e e r 1 b 2 2 − γ c / / are obtained by acting by n r φ 1 1 -closed, and they therefore −} γ − − − µ − , 1 : Q e ,...,N γ ψ γ + 2 = 0 / : r 1 ). Therefore, when 0 X = 1 − 3 ∈ { ,...,N, classes in r L γ − are always X a . States with integral : i , i i , i N 0 − = 1 -exact ( 0 0 a | k | | m | 1 v 1 1 Q c m 1 c c c c c m φ φ φ φ m − − − − − ,...,N n e − e e e 1 c in the Ramond sector. Relative cohomology , a 2 2 2 c states / / , µ i / ) 5 8 1 1 = 1 1 i 0 ): , the BRST variation of | -picture with − − 0 m – 43 – − − N a 1 | 1 γ β m 2 γ c 1 , 2 − , all these states are c = φ i = 0 − φ − k − has a | 2 e − 1 (1 and 2 e n k = 0 c states 0 / ( 2 V -closed states in the kernel of φ 1 , i / . 1 2 k 1 0 a − − . k k X Q N m> | b µ e − v 1 1 2 m ), so we are left with ψ c / X = i , while the BRST variation of 1 φ ker i k } a − − | = k 1 on the ground state . . . | 1 e v ), but two of them are in } 1 c 0 1 2 2 i 2 / c . When n φ i k / 1 i | φ Q, c − 1 λ 1 a − k | { − e | c v e 2 1 Q, c φ / 2 c { ; 1 / − φ i 1 e µ − k − with 2 µ − | ψ e / 1 i ψ 1 2 in the NS sectors and c / µ λ µ − φ | 1 1 2 k ψ closed states ( − a − Q − e v 2 / = 1 + 2 i ∼ − a χ with ghost number with ghost number One state again with ghost number One state NS sector. Then, we have with ghost number There are two states Suppose the internal SVOA γ | N µ The zero momentum states in the As discussed in the main text, the cohomology classes for zero momentum have to • • • • k obvious, since therehave are no statesbination with of ghost number combination of sentatives With non-zero null momentum portional to to Notice that and in particular computation using standard techniques and explicit representatives. any operator of weight are automatically included by the GSO projection. There are the following possibilities: (which is equivalent totheory) the is physically given relevant byform semirelative considering cohomology for a non-chiral be treated separately from the nonzero momentum states, but are amenable to a direct where JHEP02(2021)039 , ) + 0 − β K with (B.6) (B.7) (B.8) (B.9) (B.5) (B.10) ker and in distinct has ±i ,...,K − k, V | ). The Ra- K 1 . = +1 = 1 − . + i 0 tot + = 0 , F ≤ i − K i 1) 1 n u − − n ( + − . , there cannot be exact 1) 0 − ±i ( , ± (respectively, ± i , ,...,K ,...,K : ± r , , u i +1 γ 0 = 1 = 1 | r , u : 1 (respectively, − 0 + 2 c r | n 2 , . 1 γ c + c r N φ/ 2 − − ): , i c = 1 = = 1 φ/ e i are isomorphic to each other, with the -closed and represent n equal to : 1 1 1 1 − m 0 Q r − − − − 1 − , i V G ,...,K n i 3 2 = = = F Qe r 0 p p p n 1 1) ≥ γ 1) r (3 − 1) − = 1 X n – 44 – 0 1 2 − = n ( − ( γ i , = 0) = 0) = 0) ( r , + , ] = k k k ±i i = X ( ( ( − 0 + , i i i 0 1 2 + − , u ± 1 u i 0 , u ] = H H H − | states with total fermion number Q, γ 0 n n 1 , u [ | . If we drop the requirement that the states are in c 1 0 1) 1 2 | c dim dim dim 1 2 − Q, γ ,...,K ,...,K φ/ = 0 c [ . There are no states with ghost number ( 2 φ/ − 1 k ± − φ/ , = 1 = 1 ne e ≥ ). ‘space-time’ vertex algebra contains two ground states − . 1 1 i i ± , one has 1 i 0 -fermion number − − n , , 1 γ i n n Qe 0 0 V , u X,ψ γ γ + −i 0 , so that all the states are | , , V 1 m 0 + − c ) Ramond states i i and G 2 − 2 , u , u φ/ are: where the sign denotes space-time spin (and the fermion number). Then, K -picture, the / 0 0 − ker | | 0 1 e k 1 1 2) β 1 c c / 2 2 − 1 n 0 φ/ φ/ ker − ( − − ∩ e e Qγ 0 At higher ghost number, we use For ghost number Let us now consider the Ramond sector. Let us assume that the SVOA b • • to conclude that Thus, all cohomologyisomorphism groups given of by degree and in particular Actually, for all thecontained matter in SVOA we arecohomology considering, classes the (since Ramond there groundstates are states at no ghost are states number with all ghost number for each ghost number all of them with ghostthen number we have states in the ker (respectively, with weight mond sector of the momentum spaces are therefore correspond to distinct cohomology classes. The dimensions of the non-zero cohomology JHEP02(2021)039 2 , φ/ , 132 − . 109 e and in ] (1979) (2018) 010 and 2 Mock 11 = +1 Invariants φ/ 3 (1987) 133 3 tot − F K e Invent. Math. ]. , 1) Invent. Math. . . 111 , − n TASI2017 ( − arXiv:1406.5502 SPIRE . 1 [ IN PoS arXiv:2009.00186 , ][ Vertex operator , Equivariant (1990) 219 J. Algebra , ]. Conformal Field Theories with ]. (1990) 30 130 (2015) 13 ]. 2 83 Bull. London Math. Soc. SPIRE , SPIRE IN IN SPIRE ][ ][ IN [ J. 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