JHEP04(2009)032 to ht oots of 4), and , rm of the (3 April 7, 2009 , : March 15, 2009 August 25, 2008 6) : : , February 22, 2009 rings, D-branes, zed Kac-Moody -orbifolds of the : N Z Published 10), (2 , Accepted Received tric Revised tsenko and Nikulin. The 10.1088/1126-6708/2009/04/032 rstanding the ‘algebra of BPS ) = (1 algebra and are closely related doi: as, N, k ), for ( Published by IOP Publishing for SISSA b Z ( 2 k/ [email protected] , leaving the real simple roots untouched. We 1 , G , ∆ N . The new generalized Kac-Moody superalgebras all 6 T ), arise as additive lifts of Jacobi forms of (integral) weig Z , whose Weyl-Kac-Borcherds denominator formula gives rise ( N 2 G and K. Gopala Krishna k/ a 2. We note that the orbifolding acts on the imaginary simple r 0807.4451 / Black Holes in String Theory, Superstrings and Heterotic St We provide evidence for the existence of a family of generali 2). The square of the automorphic form is the modular transfo , [email protected] 2 and index 1 SISSA 2009 Chennai 600036, India The Institute of MathematicalTaramani, Sciences, Chennai CIT 600113 Campus, India E-mail: Department of Physics, Indian Institute of Technology Madr b a c k/ the unorbifolded GKM superalgebra, possibly (5 generating function of the degeneracy of CHL dyons in asymme heterotic string compactified on to a generalizedautomorphic Kac-Moody forms, superalgebra ∆ constructed by Gri arise as different ‘automorphic corrections’ of the same Lie a genus-two modular form at level states’ in CHL compactifications. Keywords: String Duality ArXiv ePrint: anticipate that these superalgebras will play a role in unde

Abstract: Suresh Govindarajan Generalized Kac-Moody algebras from CHL dyons (GKM) superalgebras, JHEP04(2009)032 4 5 5 7 8 1 3 8 30 14 16 17 19 20 20 23 24 25 26 27 29 10 11 11 13 13 13 15 21 23 25 28 ) 2 N , z ) 1 Z z ( ( 2 2 / k/ 1 , ) 2 (Z) k/ ,II 2 1 ψ k/ A N ( 1 2 G G G W 2-BPS states 4-BPS states / / 3.4.1 Finite3.4.2 Lie algebras Affine3.4.3 Kac-Moody algebras Generalized Kac-Moody algebras 3.3.1 The Weyl group C.1 Additive liftC.2 of Jacobi forms Hecke with operators half-integer and index the additive lift B.1 Basic groupB.2 theory Weak JacobiB.3 forms Additive lift of Jacobi forms with integer index A.1 Genus-one thetaA.2 functions Genus-two theta constants 4.1 Constructing the4.2 modular form ∆ The GKM4.3 superalgebra The GKM4.4 superalgebra The GKM4.5 superalgebras Interpreting the Jacobi form 3.1 From Cartan3.2 matrices to Lie Cartan algebras subalgebra 3.3 Weyl groups 3.4 The Weyl denominator formula 2.1 BPS multiplets 2.2 Counting 1 2.3 Counting 1 2.4 Product formulae for the modular forms C The additive lift with character at level B Jacobi cusp forms 5 Conclusion and outlook A Theta functions 4 Denominator formulae for ∆ 3 Generalized Kac-Moody algebras Contents 1 Introduction 2 Background on CHL strings JHEP04(2009)032 ) 1 2 ), 12 33 31 ]. Z 2.7 ( 2 ( 5 + 1). = 4 su- N leads to ) are T- ], Jatkar ( 2 ] that the that play 6 9 / N 6 m T , q N 5 ) is given by 2 1 , is generated by 6 m ]. Further, 64 = 2 q 4 T · n,ℓ,m + 2) = 24 ( e d k ondence has not been q , ), at level pactified on 2 ] has provided enormous Z e 1 cter) of weight 5, ∆ ( at the GKM superalgebra, q k eracy croscopic description of the 2 1 a [ , whose Weyl-Kac-Borcherds tence of non-renormalization m 1 arges. A key feature of the cuses on four-dimensional su- . Among these, the 7 and ( s ra of BPS states to this GKM lar form is invariant under an ℓ G , ound for testing and developing al to the Weyl denominator formula. r 5 n , ype IIA/IIB string theories [ ) = ( q 3 = 4 supersymmetry. Such theories , ) N = 2 n,ℓ,m ], Gritsenko and Nikulin have shown the 4 N n,ℓ,m ( d = 4 supersymmetry but the gauge symmetry 0 > ) – 1 – N ] however for some recent progress). One of our ], Dijkgraaf, Verlinde and Verlinde proposed that X 7 3 ). In ref. [ n,ℓ,m ( Z , = (Z) ) 2 -orbifolds with Z ( N k/ 64 Z 4 BPS multiplet of states. 10 / ). It has been anticipated by Harvey and Moore [ Φ Z ( 64 of the same form defined by Gritsenko and Nikulin [ 2-BPS states is independent of moduli — there is no ‘jumping’ / , for 10 / 10 ]. These are called the CHL compactifications. In ref. [ 8 4-BPS states in the heterotic string compactified on ) is 1 / Z ( 5 , the Siegel upper-half space and ( 2 H ∈ ) suitably embedded in Sp(2 Z Z , A family of asymmetric orbifolds of the heterotic string com The spectrum of 1 We choose a normalization for the modular forms that is natur D.1 Analyzing the modular forms 1 , must play such a role in this example. However, this corresp 1 heterotic compactifications that preserve typically have a triality of descriptions as heterotic and t is of reduced rank [ which squares to give Φ duality invariant combinations offormula is electric that and it magneticSL(2 is ch also S-duality invariant i.e., the modu a Siegel modular form of weight 10. More precisely, the degen For instance, our ∆ persymmetric theories have provided anew fairly ideas. robust The playgr high degreetheorems of as supersymmetry ensures well the as exis perstring the compactifications, existence the of CHL dualities. orbifolds, This with paper fo denominator formula gives rise to a modular form (with chara entropy of supersymmetric black holes by Strominger and Vaf 1 Introduction The advent of D-branes, and in particular, the successful mi D Explicit formulae for ∆ fully realised for this example(see ref. [ where motivations has been to understandsuperalgebra. the relation of the algeb also happens to be the size of a 1 the degeneracy of 1 existence of a generalized Kac-Moody (GKM) superalgebra, algebra of BPS states mustG form an algebra and one suspects th and Sen constructed a family of genus-two modular forms, Φ impetus to the counting of BPS states in a variety of settings phenomenon. In a remarkable paper [ a role analogous to Φ JHEP04(2009)032 , 2 N G , by N 4-BPS G / ) and (ii) ). Of course, Z Z ( are identical by Gritsenko ( k k 1 N Φ G / G ). In section 4.5, ) that generates the Z Z ce of dyonic charges as ( ( k 2 2 +2 ). e k Φ k/ / ) Z light-like, whose multi- . In other words, one ( ) from the modular form 2 ions to the Lie algebra Nn 1 uctory section, section 2 bras. In section 4, which ) for the ,II or the existence of Siegel atical background leading q k/ 1 G KM superalgebras, ]. The main result of this ty given for e show that these modular ,II nd the counting of 1 ever, different. For instance, Jacobi forms of half-integral A − for GKM superalgebras, generated by 1 1 10 A bra (1 by the ∆ 5. These constructions parallel ) [ not 4 , 2 Z − 3 , k , -orbifolds of the fake Monster Lie ) as the additive lift of weak Jacobi N n q Z = 2 N ) are indeed given by the denominator ), which: (i) square to Φ − Z is a primitive light-like simple root, have Z N ( ( 2 (1 0 2 N η k/ ), of SL(2 k/ ∈ for Y n ) reflecting the fact that the orbifolding breaks N – 2 – ( Z N 1 = , G ), at level n q Z , where ( , that are closely related to the GKM superalgebra ) 0 2 0 N tη k/ G tη ( ). This implies that the Weyl group is identical as well. m ,II ], it is another closely related modular form 1 1 N 9 ]. ∈ A t X ( 12 g − . ) given by the formula: 1 0 N tη ]. ( 4 1, this Weyl group is no longer the symmetry group of the latti ⊂ G m ) N > ,II 1 ) associated with a rank three Cartan matrix A ( ] by Niemann [ = 1. The reason is that the lattice of dyonic charges is ,II g 1 11 N A ( has to the real roots of g and Nikulin [ plicities are determined implicitly by the modular form ∆ Note that this formula correctly reproduces the multiplici imaginary roots of the form multiplicity ). In particular, we observe We will show that the modular forms ∆ Z However, for ( 2. The real simple roots (and hence the Cartan matrix 1. All the algebras arise as (different) automorphic correct 4. There are also other imaginary simple roots, which are not 3. The multiplicities of the imaginary simple roots are, how 2 5 ∆ as was shown by Jatkar and Sen [ it was for constructed by Gritsenko and Nikulin (we call the superalge dyons in the theory.to Section the Weyl-Kac-Borcherds 3 denominator providescontains formula the for the necessary GKM main mathem alge resultsmodular of forms the with paper, character, we ∆ provide evidence f the construction of a family of GKM algebras as forms of half-integral index.forms arise as In the sections Weyl-Kac-Borcherds denominator 4.2, formula 4.3 and 4.4, w formula for GKM superalgebras, lattice of dyonic charges and their degeneracies. algebra [ appear as the denominator formula for The modular group is a subgroup of Sp(2 we attempt to provide a physical interpretation for the weak the S-duality group to the sub-group, Γ paper is to provide for evidence the existence of a family of G with imaginary simple roots whose multiplicities are given The organization ofprovides the the paper necessary physical is background as on CHL follows: strings a After the introd showing the existence of modular forms, ∆ JHEP04(2009)032 , 1 S (2.2) (2.1) × 1 e and six , S ) matrix  × is dual to H ) that we m 4 S Z µν 1 unit of shift T ( e S F 2 ) dimensional 20) associated as ar, , k/ m (6 + /N L × 6 ultiplets of these 1 µν × T . This generalizes e S F )  N ) × ) m H 4 on 4. m forms ∆ S ion in section 5. Several is a (6 + T ) orbifolds provide us with with the charge quantiza- ds: ] SO( ppendix B. In appendix C, 20) lattice gets mapped to µν N Re( ,m ) is the S-duality symmetry , , we review the use of weak 16 F Z 1 4 ansion for the modular forms × Z ption, the bosonic part of the , , 2 at level is a (6 + + / is a Nikulin involution combined ) 15 SO(6 and , µν M N SL(2 L SO(6) Z 13 ML ), µ . The (4 = ⊂ × 1 ) ,m S N LML F × LM ( ML ∂ 1 T 1 U(1) µν µ SL(2) . This leads to the CHL compactifications of e S F ∂ 4 M  = 4 supersymmetry. Writing – 3 – ) T × / and its asymmetric Tr( H
N 3 8 1 ) S 6 and K Z T + ; M Im( H 2 ) gauge fields. 1 4 ,m = 4 vector multiplets each containing a gauge field and ) = ¯ S m − H µ T N ∂ S involution of the Narain lattice of signature (4 M SO(6 H 2) N S × Z µ − ) 2 Im( . ]. The heterotic string compactified on ∂ 1 9 N e S ( − 1 + 1)] Γ R  N orbifold given by the transformation corresponding to a 1 g ( / − N shift of = 4 supergravity multiplet with the graviton, a complex scal √ Z x ) is the T-duality symmetry and Γ 4 /N N ) in the type IIA theory and the orbifolding d is a Lorentzian metric with signature (6 Z = ([48 Z ]. The fields that appear at low-energy can be organized into m ) that were used in arriving at the results presented in secti , L Z 3 graviphotons; and m six scalars. and a simultaneous Z 10 , m, ( The low-energy theory consists of the following bosonic fiel = The moduli space of the scalars is given by 1 K 2 ( e S (i) the S ∗ k/ (ii) an existing result at level 1 and proves the modularity of the with the 1 In terms of the variables that appear in the heterotic descri low-energy effective action (up to two derivatives) is [ that is manifest intion the [ equations of motion and is compatible index. We conclude with a summary of our results and a discuss mathematical appendices have beenJacobi included. forms in In thewe particular construction discuss of the Siegel additive modular lift forms of in Jacobi a forms with index 1 SO(6 have constructed. Appendix D∆ gives the explicit Fourier exp various symmetries. where H valued scalar field satisfying the type IIA string compactified on vector field strength of the (6 + with the heterotic string compactified on in interest in this paper [ consider the 2 Background on CHLThe strings heterotic string compactified on four-dimensional compactifications with JHEP04(2009)032 3 = + 2 e H q (2.3) (2.4) (2.5) aS . ( t is ).  tates in a N . The BPS 2 → 4-BPS mass ) ) is the sub- . m / m H q i ). The charges N q ) ( S L states in a mul- ) from the charge 1 N m ( L ∝ q 1 m Γ + H e q =0 mod ¯ · q S ∈ c e M +
( q ) y dualizing the antisym- -duality, T e e m 2-BPS states appear when S c d and q q a b ), where Γ ( / q = 1 and more appear when )( and . Equivalently, we can write N ) vector. Thus, the electric H N − L ( ¯ = S Z 1 N ). 1 ) 2 m allel) i.e., ; + 2 − q A m ! + q , e ,m m ) . There exist many more invariants M 2 e , M q ( ] for . Under q L Z q 2 + T · )( 2 m with H ) 14 ∈ + e L q S M =1 mod m T q m q d + A M − q 2-BPS formula ( H · / = S T m M m e ) associated with these vector fields are also = 4 supersymmetry admit two kinds of BPS ( ). Further, the electric and magnetic charges ·Q· a q q q m + T Z · 2 e in subsequent formulae that appear in the paper. – 4 – )( A ) q e -duality group, Γ N q e e m L q S q q ( and q ) with , m, , the terms inside the square root appearing in the h − L H
Z m Q→ H ]: S 2

q ¯ + S c d a b L 16 ∈ + 1 − 4-BPS states are determined in terms of their charges by , e M ∝ / 2 m ( Q≡ q H 4-BPS state is max( 15 q ( e T e / S  form the doublet. Also note that the Lorentzian inner produc , q q , 2 ( 2 e H m = q ¯ S q q L N 1 − 2 1 − ) matrices BPS H Z − ± 2-BPS states can be obtained as a specialization of the 1 S , 2 and / / 4-BPS multiplets that preserve four supercharges(with 64 s ) vector fields transform as a SO(6 1 ). e / (resp. magnetic charges =
d q m 2 e + q M 2-BPS multiplets that preserve eight supercharges (with 16 BPS H / − 4 cS / ( 1 /
) -duality transformation now is 2 group of SL(2 . We will not indicate the appearance of b transform as a doublet under the vectors (resp. co-vectors) of SO(6 metric tensor) to form the complex scalar charges ± 1. e S One can form three T-duality invariant scalars, q M To be precise, 2. The (6 + 1. The heterotic dilaton combines with the axion (obtained b 3 L 4-BPS mass formula vanish leading to the 1 T e / mass formula for 1 1 formula given above. When are quantized such that The vectors. These transformthe as triplet a as triplet a of symmetric the matrix: S-duality group q the electric and magnetic charges are parallel (or anti-par The square of the mass of a 1 N > 2.1 BPS multiplets Four-dimensional compactifications with states: (i) 1 tiplet) and (ii) 1 due to the discrete nature of the T-duality group [ multiplet). The masses ofmeans the of 1 the BPS formula [ JHEP04(2009)032 6 0. T > (2.8) (2.6) (2.7) ) 2 ℓ ) be the ectric and − . Then, one ave no horizon ℓ is given by a nm n,ℓ,m = ( m ) (see appendix d 3 ] m q 19 = 2 modular forms with πiz · – 2-BPS states. Such and (4 e 4 2 m / , 17 N q ges corresponds to the q ]. Let ar and Sen generalized Z ) denote the degeneracy ) 3 n ( ∈ , d ) is [ and markable leap of intuition, 24 ℓ ) = exp(2 rged 1 τ/N n ) el modular form of weight 10 harge ( ( Z tein-Hawking entropy of large 3 s ) m e with the states of the CHL ( 1, the pure electric and magnetic d z k . c charges. ( 64 ( ), 10 η ≥ f 2 ntropy of large blackholes with torsion ]. Let = 2 Φ +2 2 k 16 24 10 2 m [ ) respectively. From this we see that − πiz ) — the fractional charges arise from . = k q 1 τ 2 2-BPS multiplet/heterotic string state ( n,m z 24 − n , 4-BPS states [ m 24 / ( T ) η ) = 0, where it factorizes as ) -orbifolds of the heterotic string on / s n η τ ). These are level- 3 2 × ( ℓ N N z τ z 2 4 = 2 +2 r η ( ( Z ) √ ) k = 2 T η 2 i = exp(2 2 e ) k n ( ( 2 e q q r πz f q )= ) – 5 – Nτ N τ 0 implies ), and ( ( )= 1 η > 24 (10) ) ≡ ) ) = (2 πiz N 1 πinτ f n,ℓ,m ) Z 2 z ( ( τ ( d ( η ) 10 0 . Another check of this formula is that this modular form k 10), ( > Φ 3 n,ℓ,m , ) z f ) replaced by 0 = exp(2 ) exp( X n → q ↔ ( 2 lim n,ℓ,m z ( 1 d τ/N ) has a double zero at ) = (1 . Then the generating function of z ( 2 ) Z / : =0 and 2-BPS blackholes is more complicated to understand as they h k ∞ ( 8 ( 2 / X n 2 4-BPS states are necessarily dyonic in character with the el 10 N, k -BPS states -BPS states f H / 4 2 H / / 1 1 ∈ 4-BPS states with charges ) / 3 2 z z 2 1 z z + 2). For ( k = ( transformation which exchanges electric and magnetic char 2 Z In a development that has spurred activity in this area, Jatk Z A more precise statement is that the formula reproduces the e 4 In this limit, one seesstates the — appearance of these the are degeneracies generated of by magnetic charge vectors beingDijkgraaf, Verlinde linearly and Verlinde independent. (DVV) proposedis that In the a generating a Sieg function re of the degeneracies of 1 The degeneracy of purely magnetically charged states with c similar formula with The modular form Φ weight ( the geometric action on (as well as the ones that we discuss later) matches the Bekens dyonic blackholes in the limit of large electric and magneti As we saw earlier, 1 2.3 Counting where one. The degeneracy of 1 We will now consider the counting of purely electrically cha 2.2 Counting of heterotic string states carrying charge (B.1) for further details) and ( the DVV proposal to the case of asymmetric with degeneracy of 1 the twisted sectors in the CHL orbifolding. Every 1 has degeneracy 16 = 2 has and hence have vanishing Bekenstein-Hawking entropy. electrically charged states areorbifold of in the one heterotic string to compactified on one correspondenc JHEP04(2009)032 . ) 2 ∞ = 0 i , z ) as 1 ), for 1 while ]. Z (2.11) (2.10) z ) that z , Z ( 21 ( 1 ) = N ∞ k , i 1 ( k, 0 z φ 20 G )= n — we have 1 z , there are two ) (2.9) ]. One has 3 9 N z (higher derivative) ( ) ). Such corrections 2 k . ( ). They also provided R )=0 and ( f 1 2.1 1 dular form [ Z z ) ( N , /z ) = 0 to ( e BH entropy formula for k 2 1 2 4-BPS dyons is generated e z z Φ / /N 1 , − z ive lift of a weak Jacobi form. ( N 3 ) z k ( ) are embedded into Sp(2 n are invariant under the larger group ). Let us call this sub-group of f = Z , ) suitably embedded in the group N , =0 mod 2 3 ]. ( ). This is subgroup of ) ) ˜ z 1 9 N 2 e Z ( N ). Then, for prime ( 1 ( k SL(2 πz , 0 , c N e ( Φ G (2 2 and level for Γ ∈ corresponding to ( J      1 k 2 ∩ − ) 1 b − 0 1 − N /z z J k H N 2 c d 0 a b – 6 – +1 ( − Γ z 24 ≡ ) 0 c c N ) = ≡ N − ) also defined by Jatkar and Sen [ Z 2 = b a ( ) √ ˜ Z z i 0 0 0 0 d k ( k − (see appendix B). The weak Jacobi form N k Φ ( , = 1 mod . We will be dealing with the modular form, Φ      J ∞ 10), this matches the DVV formula. ) = ( a ∞ i , 1 i 7→ Z ) behaves differently at the two cusps. Its behavior at ( /z k Z ! ( 1 ) as well as Γ e Φ ) = k ) = − 3 3 N ) = (1 ˜ Φ 0 c d a b z ( z 0, it has the right factorization property: 1 = → ). exchange on their embedding to match our notation.)

2 lim 1 Z z ) is a modular form of Γ → N, k ˜ 3 z , ]. They proposed that the degeneracy of 1 z Z 2 ( 9 z k 5. ↔ , ˜ Φ 7 [ ). Further, let Γ Sp(2 3 1 , , z 5 N ⊂ ( , 2-BPS blackholes, the formula leads to a prediction for 0 ) 3 = 2 / , (Note that this is not quite the embedding given by Jatkar-Se G N N ( 1 5 ), = 2 corrections to the low-energy effective action given in eq. ( lead to a non-zero entropyEinstein using gravity Wald’s generalization that of agrees th with the prediction from the mo Note that for ( G Z ) for , N It turns out that the modular forms constructed by Jatkar-Se N 5 (i) It is invariant under the S-duality group Γ ( (ii) In the limit 0 (iv) For 1 (iii) It reproduces the entropy for large blackholes [ Γ preserving the cusp at ( inequivalent cusps in the upper-half plane for an explicit construction of theThe modular constructed form modular using form the has addit the following properties: carried out a The S-duality group Γ by a Siegel modular form of weight with Thus, the modular form Sp(2 It is not hard to see that the above change of variables maps ( preserves the cusp at ( follows: that generates with the additional condition is captured by the modular form Φ JHEP04(2009)032 ] 3 z 25 and , (2.13) (2.14) ] and N 23 and ote that ) be the Z 16 1 / z ] 3 . n,ℓ ) 35 ( K 2 , ) (2.12) c 5, there is only 3 , 22 nm,ℓ ( 3 , z 2 ) is a weight two , ]. For the cases of 2 , defined by [ c τ ) n by the Eisenstein ]), any weak Jacobi  ( 3 , z 35 ) and ) N [ 1 = 2 N 35 z N Z ) 5 as the multiplicative 2 τ, z ( ( πipz , / ( ) ) were provided by two n,ℓ m N 2 α 1 3 3 ( , s D5 system in Taub-NUT e N Z , 1 ℓ 0 ( Z ⋉Z c K ) r / φ k n, one has [ f the CHL orbifold. ) 2 .1 in [ 3 n (i) the low-energy dynamics ) Φ = 2 q K , z N ( ( N ( 1 ) and ( ]. They showed that it can be ∗ 0 N z − S ( 6 B ˆ 24 . ) independently admits a product ) E 1 B.9 N N  =Γ × = 0. Let Z )+ Z Z ) for / ) J 2.12 / 3 τ ∈ ) 2 Z 3 ( K τ, z ( ) using the 4D-5D correspondence with -th symmetric product of , z N k Y ( K p ( 1 p 1 Z , 0 S n,ℓ,m z and ( 2 is the spacetime elliptic genus [ ( E k is of the form 2 − × are constants. When T ∞ 2 φ ) J =0 i ∞ ] ) ) × p X ) that differs by on an overall sign from the one ], David and Sen derived the product represen- – 7 – 3 ) ) of Γ N st ) τ Z = N ( 1 23 ( ( ( nm,ℓ z ) ( k τ B 0 in the ‘electric’ and ‘magnetic’ variables )= τ, z ( 1 ×E N 3 ( c ( ) )  /η 1 , z α , ]. ) N and 2 m ( 0 2 ) /N s φ ) 23 ℓ , z 1 N , z ( ) as well as Φ r N 1 z 1 = 1 (c.f. the DVV formula) as both the cusps get identified ( ) are as defined in eq. ( ( z n z ]. To make use of this formula, we need to determine the Z A ( k q ( ( A ) N f 1 k 17 symmetric τ, z N − ϑ ˜ 2 ( Φ Z )= 1 / 1 − , 5 system in Taub-NUT space [ and 3 k 0  D − K φ τ, z Z ) which is the generating function of dyonic degeneracies. N ( N ) ( ∈ ∗ ) Z − 1 S ( N ) is the ‘second-quantized elliptic genus’ of , ) = [ N ˆ is the elliptic genus of the 1 k E 3 ( 0 Y 2 √ φ e Φ i ) n,ℓ,m D , z , z ) and 2 1 N z Z ]. In a subsequent paper [ 1. Note that each of the terms in eq. ( ). , z ( / qrs τ, z 1 2 3 ) = ( Z ( 22 z T ≡ , K 1 is the Nikulin involution associated with the type IIA dual o Z ( , , ( × ) ) ( 2 p )= k N st N N S − 17 Z Z E Z Z e Φ E φ ( / / k 3 3 such weight two modular form with constant coefficient = 1 give We will obtain product representations for Φ Φ K K The ( ( 6 ∗ 0 S S ˆ used by David-Jatkar-Sen [ one modular form at level this point is not relevant for most of this paper as it is E the well-studied representation — for details see [ interest, the groups have two cusps at relevant Jacobi form. According to Aoki-Ibukiyama (Prop. 6 written as the productin of Taub-NUT space, three (ii) contributions thespace arising and center from: of (iii) the mass dynamics dynamics of of D1-branes the along D1- form of weight zero and index one of Γ tations for modular form E even though it is where Note that we use a normalization for Φ coefficients of the Fourier expansion about the two cusps. The where where (Borcherds) lift of a Jacobi form groups [ 2.4 Product formulaeProduct for the representations modular for forms a closely related modular form under SL(2 JHEP04(2009)032 ] 1) ] : 17 the 12) − Fur- , ( ≤ 1 (2.17) (2.15) (2.18) (2.16) 7 ) c +1) as , n 2 N (0). This ) ( 1 2.13 1 / c 2) = − / r = ) n , ) N q ( er of consistency , N ebra can also be ( ) − B B + 2) (1 = 0. Comparing the +2) = 24 +( τ, z 2 k ) ( k ) + 5 1 r m N , ), provides all the terms ( ) 0 5 to the orders ( n , Z q φ N A ur product formula. ( 3 w that the two expansions ( , k s. − , A (0) = ( 2 2 + 1 1 ) with (1 c = 2 − + 1 1 N +2 implies k N 2.13 N ) ) other than its normalization, i.e., ]) given below with eq. ( ). Note that ( τ ar, for informing us about this relationship. ∞ 2 )+ ( i = 22 nN ℓ ) q ) N − ( τ, z N − ( ( α . Thus, our problem is reduced to fixing 1 ,  = 2. These two conditions did not need too 2 nm (1 1) = 0 and ) with an anti-symmetric bilinear map [ 2 − ) as the Fourier coefficients of another weak − − , N g φ , B ( k ) Nτ 2 – 8 – ) ) ( which we do now following the procedure given n,ℓ n τ ( ]. The product representation of the Jacobi form, η ) ( q 2 9 τ, z ] for ) [ c ( N at the cusp at ln , c ( ) that has been given by David-Jatkar-Sen in [ + 1 N 2 1 − N s 22 ( r 2 ϑ Z B − 2) α ( N (1 ) k +2 − τ k =1 ∞ = ( ) Y k 2 n and η + 1 ) 2 ) are all even integers for − ) ) N ln ( Nτ 1 N N ( (  − A n,ℓ η τ r A ( 2 is given in terms of (4 ∂ (0) = ( 2 )= − 1 ), that generates the additive lift for Φ c − s 1) k c . It is also important, for our later discussion, to note that i ) − (1 τ τ, z 12 τ, z N ∞ ( ( , c ( ( i ) qr η 1 π 1 , N ) and k, ( 0 = φ b φ ) = n,ℓ ) = ( 1) = 2 τ 1 satisfying the Jacobi identity. A finite dimensional Lie alg ( c − τ, z ) ( ( g 1 1 ]. N ( c ) = 1. The expansion at the other cusp provides an infinite numb k, ), thus enables us to fix the coefficients in eq. ( ) that appear with coefficient α 22 φ 7→ ∞ Z i ( τ, z ( g Recall that The product expansion for Φ k ( ) We thank Justin David for useful discussions and in particul 1 7 N × ( k, product form (referred to as the Hodge anomaly in [ in Φ much detail about the weight two modular form series: in ref. [ naively appears to be ofare a indeed different the form. same providing However, one us can with sho an additional check on o which match the results given in ref. [ ther, it gives an alternate formula for φ two constant coefficients, Jacobi form α we see that checks of the existence of the multiplicative lift. about the cusp at where the argument of g implies that and similarly, the constant term at the same cusp implies that we have checked. We believe that this is true3 to all order Generalized Kac-Moody algebras 3.1 From Cartan matricesA to Lie Lie algebra algebras is defined as a vector space coefficients usual. For instance, the coefficient of JHEP04(2009)032 8 is a (3.2) B -th row is said to i A ) (and on , satisfying where A I , the second ( 1 g ∈ A ] and we refer the DB i ; 4 j =    f gebra the condition 2 2 , for ) symmetric matrix A ij i r 1). − − rices a h we call ng the axioms on the × − algebra 2 2 2 − r − r − efiniteness of the determi- ]= 2 2 2 2 ich are infinite dimensional. , and j i − − such that ble, ( f ,f j . ) is a sub-algebra of the GKM , i    al class of Kac-Moody algebras, i D e positive definiteness of the de- h e zable matrices. A matrix 6= ,II ≡ 1 i ; [ A j ( ,II is at least ( e 1 g matrix ij A A a 0 (3.1) ons have been discussed in [ → ֒ > ]= } =0 for ! j i { 2 j , e f satisfying the following conditions: A i − – 9 – ij h r a 2 2 2 − ; [ det grading — the algebra has bosonic and fermionic elements − 1 ) j 2

i h Z f : = . The Lie algebra ij I δ of rank (1) 1 ∈ (1) 1 are the matrices obtained from A by deleting the } ˆ A A ]= } = (ad i j ) generated by the generators , i, j { j → ,f ֒ e A j A i ( ij ,...,r e  g a 6= 2 2 , − i  1 1 ) { i = = 0 , e , that we construct in this paper. = ). , where 1 for ji N ]=0; [ A I A a G j (ad 0 , 9 ∈ 0 , h ⇔ i ≤ ,...,r h Z [ i, j in the place of positive definiteness. Thus, for an affine Lie al A> = 0 ∈ = 2 , ), = 1 A ii ij ij ij i a a a a Affine Lie algebras are obtained by requiring positive semi-d More general classes of Lie algebras are obtained by weakeni Our main focus will be on the following sequence of Cartan mat A superalgebra is an algebra with a The symmetric condition can be extended to include symmetri 9 8 = ( (i) (ii) (iv) det (iii) satisfying a graded Lie bracket. The required generalizati be symmetrizable if there exists a non-degenerate diagonal symmetric matrix. A defined through its Cartan matrix. Given a real, indecomposa reader to it. one defines a Lie algebra the following conditions for Cartan matrix. In particular, relaxingterminant of condition the (iv) Cartan on matrix th Relaxing yields conditions Kac-Moody (i) algebras and (iv), wh onethe obtains generalized the Kac-Moody most gener (GKM) algebras. nant of occasion simply on the determinant is, for all The Cartan matrix thus uniquely defines the Lie algebra which The first matrix in the sequence leads to the finite rank one Lie and column. For an affine Lie algebra, the rank of leads to the affine Lie algebra superalgebras, JHEP04(2009)032 . . 1 − h L (3.4) (3.5) (3.8) (3.3) (3.6) (3.7) + called 0 h , that is h and h = + ]: ∈ k L , k ntegral linear 26  ). The Lie algebra ,... 3 A , ( , one has . 2 g , (1) 1 w generator to of ˆ e general root structure. A = 1 from a generalized Cartan to a finite Lie algebra. As ization from affine to the . ] = 0 is an element, . } n the characterization of their 0 0 lasses of GKM algebras that h

erent from the affine case and α α 6= 0, is called the root system. ie algebra and the set of roots simultaneous adjoint action of k, d g or all negative. These sets are ∈ [ 0 α and are denoted by implies the root space also gets c> + ra. For , g h I. L nα ∀ h ∈ M ∈ α + g, , in this case are defined to be the h ] = 0 + 1 ) i, j h α L , d ( i M 1) α h [ Cartan subalgebra such that α − for g , α 0 ]= n − satisfying ( α L ) with the following Lie brackets [ i , ij – 10 – ∈ 0 M α h, g a α [ , from the above definition of the set of positive (1) 1 α with the constant | A = (1) 1 1) ( ]=+ , of all g )= b g α A L j − , d g is called the ∈ h 0 n c α , ( L } g i α i [ ∈ { α M α h + ( , ]= { = 1 = , consequently, modifying α ∗ g g h α, d [ ] = 0 , nα ) , d 0 i is called a (real) root if the eigenspace α α [ ∗ . This decomposition gives a grading on the Lie algebra + h − 1 ∈ are called the simple roots and the set of roots generated as i α L , to the Lie algebra ( . d i α ∪ n + α , of the elements  + , is not empty. The set L h L α = − = + = L L -span − Z L The affine Lie algebras are obtained by ‘adding’ a root The class of infinite dimensional algebras that are obtained . An element The root space is generated by elements defined by decomposes into eigenspaces, calledh root spaces, under the The elements 3.2 Cartan subalgebra The Thus, the Cartan matrix of an affine Lie algebra is degenerate, there combinations of these withcalled coefficients the either set all of positive, positive and negative roots respectively, central, i.e., it commutes with all elements of the Lie algeb The roots are defined on The non-degeneracy of the Cartan matrix is fixed by adding a ne the derivation, modified accordingly. The set of positive roots, For the affine Kac-Moody algebra, roots, we have and matrix, known as generalizedIn Kac-Moody general algebras, there havegeneralized is a class no of mor Kac-Moody simplecan algebras, way be as to there constructed are characterize due manyCartan to the c matrices. the general high The root degree systems of are arbitrariness also in considerably diff union of the set offor positive which roots of the finite-dimensional L JHEP04(2009)032 . , r A ,II ) in 1 (3.9) . For A (3.11) (3.10) 3.2 2 further Z yperbolic oots. The given in ( ) be the three real ,II ree hyperbolic Kac- 3 1 A , δ 2 j . , is defined as the group ix , δ 6= A 1 ank two. At rank greater an ten. The embedding of given by the matrix i ]) δ s that have been classified. ots, and the multiplicity of = 1, this group is 28 simple roots r perbolic Kac-Moody algebras. I ections and thus these groups as a sub-algebra and falls into , , 7 ∈ ]. 3 1 and . Further, the Cartan matrix, (1) 1 real S , 28 = 2, the affine Weyl group is the I ˆ A ⋊ r , i ∈ ), the Weyl group is generated by i ) = 0 i α | ∀ ) ,II 3.2 ) ij i 1 i a | A ] (see also [ , α , one has ( i 4 = 1, β, α A α ( W i ( 2 α 2 ), of a Cartan matrix, – 11 – w A − )= is the normal sub-group of translations generated ( Z β , ) has a nicer presentation as a normal subgroup of Z = 1 when W ) as a sub-algebra [ ) 2 (2 | ,II )= 1 ,II ij ,II 1 a β 1 A | ( ( A i A ) is one which contains ( P GL 2+ α ( g W
w ), where ,II j W 1 , with no further relations. For α i A w α ( i ⋉Z g w α 2 2, there are no relations. w Z ), is generated by these three elementary reflections with no ,II ]. More recently, Feingold and Nicolai have shown that this h |≥ 1 ij 27 A a ( ). ) denote the three reflections generated by the three simple r | 2 3 α W w , w 1 2 is the group of permutations of the three real simple roots. ) which we describe later. One has [ α Z 3 w , w , ( 3, these groups are infinite dimensional. For 1 is any root. It is easy to see that S , w (2 has been used by Feingold and Frenkel to study another rank th ≡ β The GKM algebra For the three Cartan matrices given in eq. ( t = 2 (1) 1 ˆ where generated by elementary reflections associated with the the roots can be hard to determine. can have imaginary simple roots in addition to real simple ro Kac-Moody algebra contains elementary reflections, some detail as it plays an important role in this paper. Let ( 3.3 Weyl groups The Weyl group of a Cartan matrix, simple roots whose Gram matrix (matrix of inner products) is a class of GKMThere algebras called are hyperbolic infinitely Kac-Moody manythan algebra hyperbolic two but less Kac-Moody than eleven, algebras thereThere at exist are a r finite no number of hyperbolic hy Kac-Moody algebras at rank higher th A and Moody algebra [ r semi-direct product ( We now discuss the Weyl group associated with the Cartan matr P GL Let ( Weyl group, by relations. It turns out that 3.3.1 The Weyl group determines further relations, ifare any, Coxeter amongst groups. the For symmetric basic matrices refl Further, when JHEP04(2009)032 l ]. 4 (3.15) (3.14) (3.13) (3.18) (3.17) (3.16) (3.12) and a funda- . such that all 3 ic Kac-Moody 3 δ ). Consider the δ Q + Z Z is a sub-lattice of , Z ⊗ ⊕ (2 II II ⊕ 2 2 for all real simple roots δ 2 . / M M δ ) } . Z i 3 . + ∈ P GL , , δ ⊕ Z i ! 3 2 ρ 1 δ f , δ ⊕ 0 0 0 1 Z − 1 δ . . Consider the time-like region = 1 )=( 2

, i = , 1 , the Siegel upper-half space [ + are: , − 2  2 , , i } f . Z i II ↔ ρ, δ 0 0 2 ritsenko and Nikulin (which differs by a + f H R II 1 2 − M = ) = 2 ≤ < ∈ = 2 f M − 3 t of the paper. 1 + ) ) , f II 3 i z ∩ Z = ,f in the following way: ) 3 M ! ) + 3 f i x,x i x, δ ). The root lattice ( ( 3 , δ δ ( 2 , δ 3 f 0 1 1 0 , δ 2 | 2 | − i f 2 δ

, f 1 z , , δ ( ) and 1 R ) explicit, we choose to write the roots in terms 2 = Z which is an element 1 – 12 – + δ − Z ↔ − R 2 ,II ⊗ ( , + 2 1 3 f ∈ II 3 A (2 W 3 f ( )= )= z x  , f i , δ M 2 through the condition ( Z { 3 W − ρ = f ∈ = ! + ρ, δ = P GL ,f ( x Z − re + 2 2 { 1 0 0 0 f V 2 f L ( f Z

2 i.e., it is one-half of the sum over real simple roots. The = . This is equivalent to / 10 ) II + ↔ 3 = 2 δ 2 , which is given by the region bounded by the spaces orthogona 1 iV M δ − II + lattice Weyl vector + f + 2 R ) which are related to the 1 M δ , 2 2 − + R ) provide a basis for Minkowski space denote the future light-cone in the space and has a ,f 2 1 ∈ 3 δ − ]. The Weyl group of this algebra is isomorphic to + II be the lattice ( ,f Z ,f V happens to be the root lattice of another rank three hyperbol 28 2 0 M , 3 = ( f , 1 0 , refers to the Weyl group ,f ρ 1 27 M 2 f M . Let W 1 , . 2 In order to make the map to Let The three real simple roots define the root lattice 0 The standard convention is to define , R 1 10 . However, in this subsection, we reproduce the notation of G i algebra [ δ sign) as we wish to compare with their results in the later par The non-vanishing inner products among the elements M positive real roots are then given by of a basis ( where One has following identification: in be such that to the real simple roots. mental polyhedron, Thus, ( the real simple roots satisfy The lattice JHEP04(2009)032 3 S (3.19) (3.22) (3.20) (3.21) city not 1. The ± = ] behaves better ρ bc − − . ) . ρ ( 0 ad , ! under the action of the positive roots, needs to 1 w 1 ρ An alternate definition is , character formula for the M − oup acts as a permutation one needs to include imag- ] ρ um side of the denominator with 1 2 0 ∈ . The Weyl group has the f positive roots. Thus, one . imensional Lie algebra. 12 . −
)

e algebras, the multiplicity of N ρ N ! ( = c d 1 a b w [ ]) is that [ 3 − e is defined to be half the sum of all δ 2 det ) 30 multiplying the form given here. We write 0 ρ 1 1 11 ) and − w ρ , w Z e

, ! = (2 det( 1 ) is the image of 0 ρ − ( ) matrices: ∈W 1 0 0 w Z , through its inner product with all real simple roots . X w i , P GL ρ , r

∀ – 13 – (2 ∈ = ), = i
2 ! α , δ δ i − δ P GL ( e , A 0 1 1 0 is then given by 1 2 , w = 1. The three elementary reflections that generate . Further, T −

0 , α A ! 1 3 1 )= = ) · + i 0 M L 1 r 1 0 + N ∈ − 2 1 ρ, δ 1 · L α − r ∈ − ∈ Y r

A α P N 2 1 = ) is generated by → = ( ) is given by the integral matrices 1 = δ and so on. Further, the imaginary roots appear with multipli Z N 2 0 , ρ w r 3.11 im + (2 L = ∪ 1 P GL re + 2 − ]. The key observation (due to MacDonald [ of the Weyl group. r L is a formal exponential and the Weyl vector 29 w ) are given by the following = α e ,II + 1 Also, the definition of the Weyl vector as half the sum over all L Conventionally, the denominator formula is written with There is another definition of the Weyl vector A ( 11 12 . It is the vector that satisfies ( i than either of the terms. Recall that an element of the Weyl gr Recall that following action: mentioned in eq. ( δ The norm of a matrix in a form that is more suitable to affine Kac-Moody algebras. given by Lepowsky and Milneformula which [ is tailored to writing the s where element 3.4.2 Affine Kac-MoodyThe algebras first twist occurs forinary the affine roots, Kac-Moody specifically algebras, thosehas where with zero norm, into the set o positive roots, i.e., satisfying necessarily equal to one.imaginary For roots instance, is for equal non-twisted to the affin rank of thebe underlying regulated finite due d to the infinite number of such positive roots W 3.4 The Weyl denominator3.4.1 formula Finite LieThe algebras Weyl denominator formulatrivial is representation. a For specialisation finite Lie of algebras, the one Weyl has JHEP04(2009)032 , ]. 31 ∈W (3.25) (3.26) (3.24) (3.28) (3.27) (3.23) w 1 α , +1) n for all l invariant and 
(2 1 , which explains w n − + − r ,... e ras, after including L n 3 0 q , . α = 2 , , − o − +1) i 1 − n w L , = 1 ) L ts a bit more involved. A Φ − 1 i es the set Φ
algebras can be found in [ n α ∈ −h r w ginary roots do not appear in fication. For the class of GKM +1)(2

1 e 1) ) Φ n ities that are hard to determine. − ( − ) 0 α n ( n − w ( q 1 ∼ h e nα . − − ] obtains contribution, only when a Z e − )] 2 ρ ∈ w 0 + / det( 1 α X n

1 1 ( − nα − α − . Note that w ) − + ) n ∈W e ρ 1 w X n r 1) L w ( − α q − 2 w − ∈ α = [ − 2) . Putting it all together into the denominator +1) / n ) ): α 2 , from the above definition of the set of positive n 1 + ( α )(1 (1 n – 14 – , − L 1 (2 0 n (1) 1 ∈ n ⋉Z τ, z ( X = b α ( α nα − A q 2 =1 1 ∞ mult( − e Y 2 1 n + 1) Z
ϑ 0 n e 2 α 0 L α , the above identity is equivalent to the Jacobi triple / − 1) α 1 − 1 1) ∩ e )= − 1) n − − − ) ρ ( r − n ( − qr 8 − n Z (2 ( / + ( w 1 L 1 ∈ n − 1 ( = X n q − e − w e 1 , is the Weyl-Kac denominator formula + ρ = − α Z L + , nα = ∈ − ) ∈ ) = Y L X n e 0 α w )(1 α 1 Φ = τ, z ( + nα 1 and 1 − ] iϑ e r α 0 ( 29 − n = nα is the sum of elements of the set Φ  − 0 for affine Lie algebras as the imaginary roots turn out to be Wey i e α = w − w − e + Φ h L (1 1 The denominator formula that works for affine Kac-Moody algeb For the affine Kac-Moody algebra, ≥ Y n identity gives [ hence cancel out in the above equation. the imaginary roots in and the Weyl group is isomorphic to roots, we have of all roots (not necessarily positive). Thus, [ New kinds of roots appear hereThe and ‘sum have nontrivial side’ multiplic of the denominator formula also requires modi where 3.4.3 Generalized Kac-MoodyFor algebras the case ofdetailed discussion generalized of Kac-Moody the Weyl denominator algebras, formula for the GKM story ge the half disappearing in thethe set r.h.s. Φ of the above formula. Ima Setting identity involving the theta function positive root gets mapped to a non-positive root. So one defin Using this definition, we can see that JHEP04(2009)032 1 ), ), G Z Z ( , icity (4.1) (3.31) (3.29) (3.30) 10 Sp(2 ary roots ⊂ ) , N . We will verify ( ) , i 0 η ( n ) ]. The Weyl group G i w α e 34 ) P . i − simple roots. pairwise independent, eyl-Kac-Borcherds de- α w ( is ructing the set o representations that are ǫ n Φ i − + δ real linear independence of the −h i L L e t pairwise orthogonal imag- n ∈ ) ter formula for Kac-Moody la of a GKM superalgebra X artan subalgebra is centrally ll consider superalgebras, the ∈ i η α the Weyl group of the GKM, he ). The modular form, Φ e definition of the denominator ) , whose denominator formulae ρ ll defined [ e the denominator identity was Z ht( P e α ( N . We extend this correspondence W ( ( 5 1) 2 G 1 = ) w − − η . Z ) ( w ( ) Ω w 5 i

is the sum of ∈ Z is the product of an even or odd number w X η ), of suitable groups, ( + ) α k Φ Z w ] (det h L ( w if 2 34 ∈ ∈W =Φ k/ n X )= w α (det 2 (Z) ρ 1) ) σ n ( – 15 – 2 ρ − Z W w − the Weyl vector, ( k/ = ∈ e X 2 ) of an element ρ − w ) with character, ∆ + η k/ = Z ρ L = , ) ∆ α ) ∩ α ) ] − mult( L 32 ) mult( ( α ) w α − e − = 1 depending on whether ) continues to hold with roots appearing with graded multipl e − ± w − ) is defined to be ( Φ (1 3.29 α ( (1 + ]. These GKM algebras contain, in addition to real and imagin ǫ L + ∈ L Y 34 α ∈ Y – α 32 , is the set of positive roots, 11 + L ) is defined to be As before we can cast it into a slightly different form by const For the above class of generalized Kac-Moody algebras, the W w give rise to modular forms with character, ∆ to argue for the existence of new GKM superalgebras, with the property constructed by Borcherdsalgebras generalizing [ the Weyl-Kac charac algebras with a bilinear form that is almost positive definit that generates the degeneracy of dyons is equal to ∆ is related to a modular form of Sp(2 continues to be generated only by elementary reflections of t – fermionic roots appear with negative multiplicity. of Kac-Moody algebras, imaginary simple rootsidentity and is a suitabl required toextended correct modifies for the these. root Also, spacesimple the roots and is requires way crucial some the for care C the as denominator the identity to be we where Ω is defined asinary the simple sum roots of and all the the height possible ht( sets of distinc nominator identity becomes [ that the modular formsindeed that compatible we construct with have the product denominator and formula. sum 4 Denominator formulae for ∆ Gritsenko and Nikulin have shown that the denominator formu where det( of reflections and and the denominator identity takes the form [ orthogonal imaginary simple roots, andabove 0 formula, otherwise. eq. As ( we wi such that JHEP04(2009)032 ⊂ ) ) to 10). ] by (4.2) (4.4) (4.6) (4.3) (4.5) , Z N 17 ( ( k 1 , Γ ) = (1 , = 3, we need a N, k 2 2 ted modular form N / m/ s 2 . ℓ/ keeping in mind that r 2 2 ) when ( / z , = 3). We find that for Z 2 n/ ), enables us to define the ) can be obtained as the ( q +2) ). ℓ/ ormula generates Φ 5 N for 15 in the Fourier expansion Z k . r Z ( (
πiz 2 ( ) 2 2 ) ℓ d ≤ 2 )), while for z . n/ , Z k/ , , 2 and index 1 q k/ ( Nτ 2 . ( d ) m k/ mn nm C η θ B.16 and 2 g n,ℓ τ / ( ) = exp(2 2 4) g 2 ) 2 − − . This happens to be the square root of k for Z k Y 2 ( ( d ) can be written as a product of six even πiz ) 1 =1 mod 2 2 H =0 mod3 =1 mod3 =2 mod3 z ) τ Z m k/ ( ( d ) of weight ) d d d 1 mod 2 3 ( X η ∆ Z ≡ N χ 0 1 1 ) ( – 16 – ( 2 1 ) − n,ℓ θ = exp(2 τ, z      ( r 1 1 ). A discussion of the additive lift has been provided 64 )= n,ℓ,m X θ i.e., ( 2 ] and product formulae were provided in subsequent Z | ( )= d 9
assumption of this paper. Recall that the modular forms , z k d 3 )= 1 ). ( d )= ) and z 0 − Z Z ψ ( ( > , 2 χ ) 3 / τ, z πiτ main 1 ( , ) is the trivial one (see eq. ( ) can be written as a product of all ten even genus-two theta ∆ 2 X 2 )= d / Z ( 1 n,ℓ,m d ( k/ , ( ( χ 5 2 ): ψ ψ ≡ k/ Z χ ) used here differs from the one defined by David-Jatkar-Sen [ ( ). This generalises the Maaß formula for ∆ ) ψ 2 3 Z ) = exp(2 Z ( 1 ( k/ k 2 πiz k/ πiz ]. Further, the dyon degeneracies are given by a closely rela ∆ ) by Jatkar and Sen. This differs in the way the S-duality group 22 Z , ( ) and more to confirm the non-trivial character for k = exp(2 17 Z 5, the character ), is embedded in Sp(2 ˜ = exp(2 Φ s ( , Z k ) were first constructed in [ q , Note that in the following we will write It is known that ∆ Z ( = 2 k A proof of the additive lift has been given in appendix We have found experimentally that the modular form ∆ additive lift of the Jacobi cusp form of Γ 4.1 Constructing the modular form SL(2 called (Note that the Φ non-trivial character an overall sign.) This is the papers [ where Φ We however do not expect such a formula for the other ∆ We have experimentally verified that the square of the above f with genus-two theta constants: constants. As we show in the appendix, ∆ these become two of the three coordinates on N for Φ fairly high order (all terms which appear with values of in appendix B. The Fourier expansion additive lift for ∆ the Jacobi form that generates Φ JHEP04(2009)032 . ], ), ]). ) Z ) is 32 ( 12 (4.7) , 3 ) can ,II 1 ˜ nm,ℓ Z ∆ 11 ( ( 2 A 2 c ( 1 2 g k/  ) is one with N ) Z = 3, we obtain ( m 2 (3)(see appendix s 0 ℓ N The resolution lies r b G n q ( ) even though our original vector does not change Z ), which we call ( ), to − with their multiplicities. xist in the mathematical ρ 2 — it is the one given by Z Z Gritsenko and Nikulin for 1 ( s-two theta constants (see + ( 6  r the Lie algebra 5 )), giving us the Weyl vector ots whereas in our situation, L 13 ˜ Z Φ mula of a GKM superalgebra, ) corresponds to the sum side ∈ . ) is given by the product of six (see Theorem (1.6) in ref. [ tions. This combined with the ρ, z . The Z ( 4.4 Y ( ence of ∆ 3 ] have studied the ring of modular N πı n,ℓ,m − D. Ramakrishnan for a useful conversation. 35  × ) ) integers as they provide the multplicity D 3 . The product formula for ∆ formed us that Gritsenko has independently con- − N nm,ℓ ( G 1 det( ). Following the ideas of Borcherds [ even c Z 1 2  Is there a contradiction? (  5 – 17 – m ). ) are )= s Z ℓ = 3, they show that modular forms with character, ( r M ), we are ready to verify whether they can be the 2 ]. ( n Z 4 to illustrate the method before moving on to the new q N ψ 1 nm,ℓ ( ), considered by Aoki-Ibukiyama and the restriction of ( v 2 1 G show that this is indeed true. 2 − G M c k/ ( 1 C ψ  v Z ]). Aoki and Ibukiyama [ ∈ can be identified with exp( ) and 36 ) and it gives us the positive roots 2 Y , / 4. At level 1 n,ℓ,m , s , we show that the Jacobi form that generates ∆ 35 . It appears to be true to the orders that we have checked. The (3), necessarily have odd weight. However, for 3 2.13 2 ) transforms with a different character 2 nm,ℓ 0 , / + / ( Z 1 1 ), that appears in the transformation of ∆ b 1 ( G L A.1 s r c 2 2 2 = 2 M = 2, we have been able to show that ∆ / / ∈ ( 1 1 Γ r q N
N 2 v / 1 q A B C D )= = Z ) and we repeat their method [ ( 2 However, one needs to verify that the Maaß formula ( With the construction of ∆ It is of interest to ask whether these modular forms already e Z M We thank Prof. Aoki for encouraging us to believe in the exist ( k/ 5 13 . We thus see that the Weyl vector is independent of ∆ B.4). In appendix the character, in the fact that ∆ common factor of various roots in ρ even genus-two theta constants. In fact, the square root of during the orbifolding process in the work of Niemann either can also be writtenappendix as A.2 and the ref. product [ of six (different) even genu In the work of Niemann,the the three orbifolding real changes the simple simple roots ro are unaffected by theof orbifolding the denominator formula.∆ This is the procedure adopted by 4.2 The GKM superalgebra literature. For forms at levels the product of the character, for be obtained from eq. ( For this to make sense as theit product is side necessary of that a denominator for denominator formula of a GKM superalgebra non-trivial character which isadditive consistent lift with shown our in appendix expecta a modular form with even weight, ∆ approach initially had no mathematical rigor.structed He a has modular also form in of weight two at level 3. We also thank We first consider the example of GKM superalgebras. As it stands, the denominator formula fo not related to the the automorphic form ∆ JHEP04(2009)032 . 2 a le II H (4.8) (4.9) M ∼ ) as a + R + ,II 1 ) becomes ∩ )! ıV A ) ( II ,z ]. We would gebra. There + g ) 3.29 4 η M 1 ( , + 2 ρ nce of imaginary ( has 14 R w — again fermionic ( 1 πı + G that the automorphic es only the difference − . L ) each with multiplicity a e ). The other terms are 2 ) ormula ). 2 ) f g the multiplicities from τ f η ( nd Nikulin [ 3 ( ,II rs. There are two primitive ) to the space 1 +2 τ do not write out the detailed m II 3 + 2 (10) ( A erators and hence is more like f mionic. For instance, one has ( η f 3 M 2 g senko and Nikulin. In particular, one f + p − R 2 − ∩ = 0) and the other arising from the , by adding imaginary simple roots X − = II 2 1 f η 9 − G M ) f ∈ k η q − − . We have deliberately rewritten the sum as and (2 belongs to the subspace ) + 2 ,z 2 (1 ) − V N ρ − . These multiplicities are determined from the )(an explicit expansion is given in appendix D) ( f ) and (2 ∈ f – 18 – Z 2 Y w k 1 Z ( f , 2 ( z ∈ 2 5 πı = f of these primitive vectors and multiplicities given by ) + − t + 2 e η q ) 3 ( ) determines the positive roots ) 2 tη α f ( 0 − Z 2 ) m ( z f tη w 5 ( ). mult( + m  ) leads to a simple expression for light-like imaginary simp 2 ) det( 3.29 N 0: (2 f Z ∈ 3 ( 6= 0). The first term thus arises as the sum side of the Lie algebr t X α,z z 5 ( < ∈W η X − πı w ) = − 1 in eq. ( )

e ρ Z η, η − − = 1 3.3.1 ) = 0. They are generated by the formula by  0 ρ + , η L 0 27. Thus, such roots are fermionic and hence we have a superal ∈ Y η α − ) ). The second terms is specific to GKM algebras due to the prese ρ,z ( ) = 0 ,II πı In the above setting, the Weyl-Kac-Borcherds denominator f The product formula for ∆ η 1 Here we write the denominator formula in the notation of Grit − e A (2 14 ( — some are bosonic and others are fermionic. The superalgebr needs to replace sub-algebra and its Weyl group is identical to the one for obtained upon complexification of the cone Gritsenko and Nikulin constructed a superalgebra, where the element (see also section like to commentthe here exponent that in therebetween the is the product a multiplicities of formula subtlea the — issue Witten bosonic the index. in and product fermionic extractin gen formula giv two terms — one arising from the real simple roots ( Negative values of multiplicitym implies that the root is fer are three primitive light-like vectors: 2 9. The action ofvectors the Weyl satisfying group ( generates the remaining vecto — in other words,properties one are adds attained. enough The imaginary imaginary simple simple roots roots such belong g simple roots with ‘multiplicities’ imaginary simple roots ( connection to the automorphic form ∆ roots i.e., ( roots appear with negative multiplicitylist in and the refer exponent. the We diligent reader to the paper by Gritsenko a generated as multiples of the form the above formula. The Maaß formula for ∆ JHEP04(2009)032 = 6. ts — ) and (4.10) k Z ( ] for the 5 ) (given in 12 , generated Z II ( ) we see that ), is not very 5 Z by orbifolding. . Z M ( 1 ( analogous to the 3 G ) such that they 6 1 G ) τ ltiplicity factors such ) and ∆ 3 ( ry, Weyl group, Weyl e given by completely ence we can not track ) Z n,ℓ,m τ . ( (6) of Niemann [ orbifold for which ) and ∆ ( e-to-one for comparison. hat we can construct for 3 η f η ple roots, we observe that ), and Φ result from orbifolding by Z 2 d. The fundamental cone, . The remarkable thing is ( e root of the Siegel modular es ( , these will be the ‘twisted’ Z Z p 5 ction on ( . Below we analyze the Maaß 10 = 4 e roots ) k 2 ) using the additive lift of the weak is the same as for q Z ) are not obviously associated to any ρ 2 as discussed earlier. We also need ( . This implies the lattice generated by − / 3 Z 1 ( G 6 is the same as before. It also implies that )(1 3 k δ q . Next, we can see that all terms that appear Z 1 − ⊕ G = 6 and is the special case of a more general – 19 – ), in the same way as was done for ∆ 2 ) and Φ (1 δ Z k N Z ( ) and the fundamental polyhedron Z are given by the generating function ( ∈ 3 Y k II 2 ⊕ 10 1 G 2 M = δ ( G t ), ∆ Z ]. + q Z ( V ) 12 = 6 = 2 and 0 II tη ( N M ) should appear as the denominator formula of some GKM su- which we give below. remain unchanged from m is the same as for ) albeit with different coefficients. Thus the set of simple roo ) of weight 3 and index 1 Z 2 2 N 2 for k ( Z ( ∈ G 2 G t X , z 3 1 k/ +2 ) in the case of z − k 0 and ( η 1 2 ( N / N 1 , m +2 3 k , containing the the imaginary roots also remains unchanged ψ II ), i.e., ∆ Z M ( k ) appear in ∆ + ). The 0 Z R η ( ( 5 The only difference comes in the generating function of the mu It is remarkable that although the degeneracy of the dyons ar By a straightforward comparison of the Fourier expansions ∆ We have already derived a Maaß formula for ∆ ∩ m II as real and imaginary, of As already observed, the Weyl vector for the lattice the cone defined by the lattice by the spaces boundedM by the real roots, also remain unchange This can be understood from the twisted denominator formula the three real simple roots formula for any different modular forms inthem the are two cases, similar the to underlying such GKM a t degree. The roots — real and imagina 4.3 The GKM superalgebra cycle shape 1 give the space Ωgeneralized of GKM the algebras algebra that we are construct. obtained by Mathematically the orbifold a appendix D) by firstthe focusing Cartan on matrix terms for appearing from real sim Jacobi form ones constructed by Niemann [ to suitably interpret the region of summation of the variabl It might be thought that although Φ in ∆ obtain the multiplicities of the primitive imaginary simpl peralgebra. The first example that we use to verify this is the An important assumption in this paper has been that the squar forms Φ GKM algebras, we coulddirectly nevertheless comparing obtain the the two changesthat as that the given comparison by their at Maaß the expansions level of the modular forms, Φ transparent, in that theThere terms occur appearing terms in indown each the their that fate two as do are we not not orbifold. appear on It in is the only other, when and we compare h ∆ both the expansions areexpansion similar and of suitable square for root comparison of Φ JHEP04(2009)032 5 5 , G 3 , 2 (4.11) (4.12) , orbifold- ear to us = 1 N re are fewer Z N for N G ry simple roots for the . multiplicities. There is ) ) as the orbifolding group mulae for a . bras 0 y of black holes as well as ) η τ, z ge. , its Fourier expansion with egral Fourier expansion and ( 3 ( of the series. This affects the 2 ht-like simple roots for differ- 2 ents of the terms occurring in ies of the imaginary roots. are different from each other. m / H ain unchanged. The orbifolding 1 , Jacobi form. While the formula 2 ∈ πimz k/ Z ). However, these forms (and their 15 ψ ed additive lift using the weak Jacobi form ) Z 2 ( = weak 1 , z with 1 ! ) exp(2 t z 2 ( k q = 7. The weak Jacobi form has half-integral 2 , z ) / 1 1 0 , N z 2 ) and ∆ ( tη ) is one with character and hence we anticipate – 20 – ( k/ Z Z ) as can be seen in appendix D. ( ψ m k,m ( 2 Z . The modular forms leading to these algebras, and 1 φ ) affects the existence of the GKM superalgebra N ( 3 N ∈ N Z t X ( =0 ) of weight G ∞ 1 X m Z − ( 1 k F

)= Z = 5 is generated by a ) ( k N F τ, z -corrections in the low-energy effective action. It is not cl ( 2 1 θ R = 3, one sees that certain multiplicities vanish and thus the are given in terms of a single formula: N can be written as N ). 3 2 z ,z 1 z ( 2 / The generating functions of the multiplicity factors of lig The Cartan matrix, Weyl group, and the set of real and imagina The additive lift for One may wonder what goes wrong when 1 varies. For remain the same for all values of , Interestingly, it was this pattern that lead us to the propos 2 15 N N k/ ψ of the space is reflected only in the4.4 change of the multiplicit The GKM superalgebras vector, the lattice, and the space of the imaginary roots rem obtained from the modular forms that occur in degeneracy for In this section we list the properties of the class of GKM alge hence the denominator identities of the algebras, however, The difference in the denominatorthe identities expansion, is whereas in the the terms coeffici themselves undergo no chan ent values of From the above we see the pattern in the progression of the ing action. G Z terms in the Fourier expansion for ∆ appears to go through, thereholomorphicity are of issues both with the the convergence forms — Φ modular transforms) appear tothe be higher-derivative compatible with the entrop subtleties associated with it. 4.5 Interpreting the Jacobi form whether this meromorphicity of ∆ weight as well ashence half-integral does index. not It seema does to related not lead issue to have — an a the int GKM modular superalgebra form with Φ integer Given a Siegel modular form, but we caution the reader about this possibility. respect to JHEP04(2009)032 ]. ots 40 – ). It 2 ), that 37 , z Z 1 ( sectors in z k re nothing ) are weak ( forms with 2 2 2 at level one. he affine Lie / T 1 , z , 1 2 (1) 1 ve been able to z ˆ ( k/ A ) under which the is a fermionic one ψ Z 0 = 1. The same root k,m , , . However this naive η 2 1 φ 4D-5D lift leads to a 4-BPS dyons [ N N / G lgebra rphic form, Φ — it is these roots that her, we have proposed an dex tor formula. For instance, ) from a weak Jacobi form a of BPS states. for (1) 1 Z — these are reflected in the , of Sp(2 ( ˆ plicity one. The Jacobi forms A J e spacetime and k η lectrically charged states arise )=9 for ∆ 0 s is a small step in understanding η d heterotic string states appear to ( m can also be understood as the removal ). For Siegel modular forms with character →∞ 1, there are two distinct modular forms that ). In the product form for the theta function, B . The Cartan matrix for the two-dimensional ) into three terms. Taking the square-root to ) 2 3 N Z – 21 – z simple — it is the sum of the two real roots. For , z ( G 1 k N > z ( -orbifolds, seems, at first sight, to be in contradiction under the sub-group, Γ 1 not has multiplicity N iθ 2 Z m − f 4-BPS CHL dyons can be interpreted as the Weyl-Kac- / 1 respectively. So we see that the root . As is well known and discussed earlier, the denominator = 2 = 0 vanishes. The Fourier coefficients − 3 sector. N , 0 sectors (in the type II picture) while the other imaginary ro ] observe that the walls of the Weyl chambers for the GKM η m 7 K 2 and index T ⊂ G k ) = 0 0 η (1) 1 ( ˆ A is invariant(see appendix has more light-like imaginary roots than m get mapped to walls of marginal stability for the 1 N ∞ with multiplicity 10. The difference is easy to understand — t 1 ) does not change the separation. We see that two of the terms a G ), the above sum runs over half-integers and we obtain Jacobi G + Z Z ( ( L 2. From the physical point of view, using the 4D-5D lift, we ha ) = 2 5 3 / k/ z ) that we are interested in contain other powers of 5, we find , = 5. N ( In our situation, taking the limit Im( As mentioned earlier, the analysis of David and Sen using the Cheng and Verlinde [ 0 N = 3 extension is not quite correct, since for We thus see that appear in the correctionthe terms imaginary in simple the root sum side of the denomina with our observation that the Weyl group remained unchanged generates the degeneracies ofBorcherds 1 denominator formula foradditive a lift GKM that superalgebra. directly generates Furt the automorphic form it is easy to see the appearance of imaginary roots with multi of Γ For cusp forms, the term the type II picture.arise In from particular, such the light-like simple electrically roots. charge the We connection believe between that the thi GKM superalgebras and the algebr 5 Conclusion and outlook In this paper we have shown that the square-root of the automo half-integral index. fact that the superalgebra This observation, if extended to the such as ∆ subspace of real roots is then the same as the affine Kac-Moody a formula for this algebra is given by appears in of index 1 separation of the product formula for Φ N for show that the real rootsfrom and the spacetime light-like imaginary and roots for e obtain ∆ algebra has one light-like root that is appears that real and light-like simple roots appear from th but the product representation for the weak Jacobi form of in of a real root of the GKM superalgebra necessarily arise from the Jacobi forms of weight cusp Im( JHEP04(2009)032 e = 5 ew. tions N product form ). This might Z ( r product of two = 11 as well and 5 3, the domain has ) is the generating light-like imaginary the algebra of BPS c K3’s whose auto- ). This analysis has τ N ne bounded by walls ( Z olutions of K3 mani- ( boxes). An interesting that the degeneracy of /η k he GKM superalgebras, N > cond one is the one that e de observation should be n Φ 24 ]) and thus they conclude have provided an explicit ) and it is believed that a s for / e same GKM superalgebra 3) alone. It is of interest to 1 Z of 42 q no direct relationship to the ( [ K 0 ar also observe that the ). The first one is related to anding of the role of the roots ( 1 Z , lattice. We anticipate that these vant observation here is that the ( 1 bra. It would be nice to have a 3. is no longer a symplectic Nikulin by Gritsenko and Nikulin in their k , ons (or 3D Young diagrams) which H e ] who find that the Weyl groups of Φ 11 and the walls of the Weyl chambers = 0) Φ 41 Z = 2 N k N ) and Z ( k – 22 – 3) and not on ], Φ = 5. Should one relax the finiteness condition and ] (these are two-dimensional complex surfaces that 9 K N ( 45 ∗ = 5? Is the meromorphicity of the modular form that H N 4-BPS dyons. Walls of marginal stability are precisely wher / involutions of K3 surfaces) gives rise to ∆ (equivalently, Young diagrams with 2 n Z orbifold both from the physical and mathematical point of vi ] and from his results one sees that for N 37 Z = 5 relevant? We do not have any concrete answers to these ques N GKM superalgbra for ]. In particular, they have explicitly constructed ellipti no leads to powers of the Dedekind eta function appearing in the 44 + L , -corrections in the low-energy effective action while the se 43 2 As mentioned in the introduction, our aim has been to address Sen has studied the fundamental domain in the upper half-pla Garbagnati and Sarti have studied symplectic (Nikulin) inv As we have seen, for affine Kac-Moody algebras, the presence of R . BPS states thus seem to be elements of a module that is a tenso N been recently carried out by Cheng and Dabholkar [ study aspects of the states. While we havealgebra uncovered of a BPS nice states algebraicBPS structure, has states appears been related achieved.G to A the related direct problem sum is of two copies of t this degeneracy jumps.applicable The to extension GKM of superalgebras the related Cheng to and the Verlin square-root generates the degeneracy of 1 situation violates a certain finitenessclassification condition of (imposed rank three Lorentzian Kac-Moody algebras copies of (irreducible?)microscopic representations understanding of of these the issues.elliptic superalge genus A of possibly the rele Enriques surface [ the we observed for of marginal stability [ and we leave if for future considerations. folds [ CHL string may exist. In the type IIA picture, the get mapped to the walls of marginal stability for infinite volume (in the Poincar´emetric). Cheng and Dabholk have been constructed by Jatkar-Sen [ arise as fixed-point free that there is look for a GKM superalgebra for morphism groups aredescription the of Nikulin the invariant involutions. lattice andresults its might Further, complementary be they relevant inof improving the our GKM physical underst superalgebras.it leads The to Jatkar-Sen a construction modular hold function (i.e., one of weight the correspoding GKM superalgebras change with generalisation is the generating function of plane partiti involution, it acts non-trivially on of the Weyl-Kac denominator formula. As is well known, provide a hint onfor the K3. appearance of two identical copies ofroots th in function of partitions of JHEP04(2009)032 ), 2 , z (A.5) (A.2) (A.3) (A.4) (A.1) 1 z (  1 0  θ ]. Is there an 47 ≡ , ) 2 46 , z me of this work was 1 , z ) his support. SG would ( . 2 , -Vafa [ ) τ, z ϑ dular group is given by ( . 1 ), ntly answering our numerous ) ϑ iπlb 2 ments that have led to several τ/N e τ ( ( , , z . ) /τ η es to this enriches this story and η 1 ) 2 2 a ) z , 2 2 ( + n ) / , l / πiz (due to MacMahon). This function q (  1 τ, z 1 ). τ ) e ) ( r ) 1 1 ( 2 n τ 1 τ −  τ η ) ( r 2 ( ( ϑ , z ) 2 n η θ 1 2 / a 1 (1 q 4 4 4 12 z 1 + ( 12 ≡ q l =1 − N ∞ ( πi/ iπ/ πi/  Y n ) 2 1 e − 0 1 − πi/ − √ 2 Nπi/ (1 q e e  e e 24 n , z θ Z 1 – 23 – ∈ ) = Q ) = l z X πiτ/ ≡ ( ) = ) = ) = 2 , z ) ∼ 1 e 2 z/τ N /τ ). ϑ /τ )= +1) = D 1 1 − 2 , z + 1 3 + 1 τ ) under modular transformations is given by − η )= − πiz τ ( , z ( z ( τ ( ( 1 /τ, η η ( η ]. 1 z 1 4 Nτ τ, z ) is defined by η ( ϑ ( ( ϑ 48 τ − 1 i η : ( : ( ) is given by ϑ b a 1 η S T h ϑ = exp(2 : : Nτ θ r ( ) and T S η 2 : : , z S T 1 z ) and (  1) mod 2. One has 0 0 , πiτ  (0 θ ∈ ≡ ) 2 = exp(2 a, b , z q 1 The transformations of z ( 3 has a nice product representation with algebraic interpretation for this? Theleads addition to of interesting D2-bran formulae [ Acknowledgments We would like to thank Dileep Jatkarqueries and via Hiroki Aoki email. for patie KGKlike would to like to thank thank thecarried S. out. CERN Kalyana We Theory Rama thank for the Groupimprovements anonymous in for referee our for a critical paper. visit com during which so A Theta functions A.1 Genus-one theta functions The genus-one theta functions are defined by The transformation of the Dedekind eta function under the mo The transformation of appears in the counting of D0-branes in the work of Gopakumar The Dedekind eta function where ϑ JHEP04(2009)032 ). = he Z q ( ], let 1 (A.6) (A.9) (A.8) ) and ) gets (A.10) (A.11)  τ 12 ( 1 1 1 1 9 rm with k Nτ  f ( η ) take values , 2  9 representing ) 0 0 1 1 8 2 , b b 1  2 l , b ,..., + 2 1 , 1 b ,  ) 1 , a N . l ) must transform with 0 1 1 0 τ 7 ( 1 ( ven theta constants are Z a ′ iπ  ( = 0 j t the functions struct the Siegel modular e 2 − 2 . m transformation. ψ )  ) 2 2 ) ). Following Niemann [ S a ). One has 0 0 1 0 6 . In particular, one can show 1 mod Z τ G Z ( + ( (  , 2 . Further, we have defined − l j m on of Jatkar-Sen regarding Φ 2 χ ( θ ) with ψ 1 2 H 2 s Z /  ]: ( 1 4 ) 1 0 0 1 5 ∈ ) ) (A.7) ) transform with character. As these m 2 2 , τ a τ θ τ  ( ) ( ( ,...,N ! Y + ) of SL(2 3 2 1 Z 2 ) under the =1 mod 2 l , z z ). The constants ( +1 12 ( (4) N j 3  )( m ( f ) has to be calculated on a case by case basis m 2 1 ) 1 ψ 0 0 0 1 0 4 πi/ 2 ) can be written out in terms of the even theta θ z z a ) G = 0 τ/N Z p ′ πiz ( Z  ( ( j

+ 12 – 24 – ( =0 9 2 1 η χ + l 3 Y θ ( j , j m = πi/ ( r e e   1 1 = 3, 2 1 1 0 0 64 64 3 Z j ) 1 2  a = exp(2 + N ) = + ) and ∆ j s ) = ) = 1 /τ l + Z N (  Z Z +1) = τ ( 1 , and ( ( 1 2 1 0 0 0 5 2 3 5  τ − ! q ( ( 1 2  η 2 ∆ ∆ j j b b ) and Z as defined in the above table. ψ 2 ψ ∈ ≡

]. This is the basis of our claim that ∆ ) b ) and for 2 ) transforms into )  ) respectively, these two Siegel modular forms will transfo and the character X τ ,l πiz τ 50 = 0 1 0 0 : 1 : ( Z 1 ( l = 0 mod 2. There are ten such theta constants for which we list t ( j ( ] for details). N Nτ  2 T S b and (1) ( ψ b 12 f η , : T a )= a ! b  ]. One finds 2 1 no longer have a diagonal action on the Z 0 0 0 0 0 = exp(2 ( a a 49 T  i r = 7,

, ) and ∆ and 4 a b Z =1 mod ), a h = ( N 1 ′ 1 ! and θ a a b jj m S πiz

The modular functions ∆ The transformations of the eta related functions show us tha 1). Thus, there are sixteen genus-two theta constants. The e , the ten values of constants [ mapped to itself only under the subgroup, Γ Both A.2 Genus-two theta constants We define the genus-two theta constants as follows [ We will refer to the above ten theta constants as One can see that two functions enter theforms weak Jacobi Φ forms that are used to con non-trivial character and is consistent with the observati (0 those for which exp(2 where non-trivial character [ (see chapter 2 of [ their square root canthat transform for with non-trivial character values of where JHEP04(2009)032 ) ∈ Z
, (B.3) (B.4) (B.1) (B.2) c d a b (A.12) 2 matrices × . This group N ) that preserves Z , , . , ) 1  Z Z − d ( ) 9 ∈ = 0 mod 2 2 θ D + ), is generated by Sp(2 ) 1 . The congruence subgroup cz C + Z es expansion. This is also in I ping in mind that the leading Z ( cz ( Z 8 H = λ, µ, κ . θ − C ] in terms of theta constants. T 3 ) )(      35 ). One may wonder if it can also be Z ]. B 0 0 ( , z with Z 7 BC 50 ( b d d + θ 6 ! − 0 0 ) ˜ + 2 Z Φ , as z T Z 1 2 c 0100 0001 a A ) is generated by the embedding of ( ( 6 H      Z cz      A B C D θ ) is the sub-group of Sp(2 , λ AD ≡ , µ

) Z ≡ − b ( d Z – 25 – Z ) = ( · µ κ H ) was constructed experimentally by looking for a + + 3 ) given in ref. [ and θ 1 Z 1 1 c, d Z 4 matrices written in terms of four 2 M ⋉ M ( T ; ( ) 3 000 1 0 0 1 1 0 0 ) cz az λ 6 × Z Z (       a, b , . The SL(2 DC 1 7−→ ( θ 1 ≡ ∞ g = ! ). The Heisenberg group, i ) 7−→ 3 2 as 1 d 16 z z T ) = 2 , we find that the following combination achieves this (to the 3 2 1 + 3 2 = SL(2 H z z / z , z 1 CD )= 1 2 J λ, µ, κ

s , ( cz Z ) to be the square-root of 2 ( 2 , z T = / 3 Z 1 g 1 ). ( z ˜ r 3 ∆ Z ( Z 4 ) is the set of 4 BA ) as given below ( ˜ / ) = ( ∆ ) is given by the set of matrices such that 6 1 Z ], the formula for ∆ Z , , q = D Z ˜ Φ , 49 16 T + as Z AB C ) in Sp(2 ) of Sp(2 Z , Let us define The Jacobi group Γ N This section is based on the book by Eichler and Zagier [ ( 16 0 b Following ref. [ satisfying with det( product of six evenagreement theta with constants the that expression had for the Φ correct seri written as the product ofterm six will genus-two go theta constants. as Kee orders that we have verified): squares to give A, B, C, D B Jacobi cusp forms B.1 Basic groupThe theory group Sp(2 G matrices of the form acts naturally on the Siegel upper half space, the one-dimensional cusp The above matrix acts on SL(2 JHEP04(2009)032 . ). J Z = 0 , (B.7) (B.6) (B.8) (B.5) c . ) by any Sp(2 0. J 0 relaxing Z ) , ⊂ ≥ ) i.e., ≥ N J ( 2 N ℓ 0 n ( 0 , − Γ ) is a holomorphic
Z nt ∈ , ]. λµ
45 + c d a b 2 , , 0 and 4 ) , λz ) ℓ Z 2 at ≥ r ( + 2 n , z F n , q 1 1 ) is generated by considering the z k z ) C ( ) 2 N ( λ D k,t 0 → . n,ℓ φ ( , if b + G + preserves the one-dimensional cusp at C f ) Z 3      3 with respect to Sp(2 1 2 Z Z J × + v C t 1 ∈ µ, z X ∈ πitz t ℓ H ) = 0, unless ) are non-vanishing only when + 0100 1000 0001 0010 2 with respect to the Jacobi group Γ n,ℓ – 26 – ) det( z )= n,ℓ (      2 ( k ) : f 2 + M f ≡ , z ( 1 1 , z ). v 3 z ) = exp(2 1 g . The subgroup ( z N and index Z 3 ( ( and character ( , λz z k,t 0 )= k k 1 k,t k φ b ). Jacobi forms of integer index were considered by Eichler ˜ Z z G φ ). In the above definition, one can replace Sp(2 2 φ ↔ · as ℓ Z 1 , or 2 z − M J ( H ) is generated by adding the exchange element to the group Γ 7−→ nt Z F ) Sp(2 , 3 of weight of weight ∈ , z satisfying 2 M C , z ). Further, we will call the corresponding Jacobi group Γ ) = 1. It is easy to see that Γ 1 z exchanging → D ( B.2 2 and 2 ] and extended to half-integral indices by Gritsenko [ + H 2 H is a modular form of weight 50 . : Z H Jacobi form 2 Jacobi forms, the coefficients ∞ C F ∈ H in eq. ( Z on such that the Fourier coefficients, )= N 3 A holomorphic function The full group Sp(2 weak z 1. The function 2. It has a Fourier expansion Siegel modular form of its sub-groups such as Γ B.2 Weak Jacobi forms A function The above matrix acts on is called a for all Im( For This acts on same three sets of matrices with the additional condition th the condition involving (4 mod with det( and Zagier [ JHEP04(2009)032 (B.9)  (B.11) (B.13) (B.12) (B.10) . 5 2 / /  3 3 5 in writing q q / 6  J  q ) r O  ). The Fourier 8 N (
+ O N 0 , 2 − ( q ) q 0 +  . q 2 . τ, z O
r + 16 (  m 2 , 2 1 2 roup Γ s + , q ur modular forms with r 8 r ℓ  2 0 orms. Examples include: q + 8 r ) φ . −  2 n O r 0)  )
 + 4 q 3 , 2 2 2 , z / + 2 1 ) 64 r lift of the Jacobi form. It is q r 1 2
+ z 1 + 1 , z 1 z q q ℓ d ( z 1 5 ( − 26 i + ( i , / z O  N  ϑ 3 2 ( ϑ 2 − r 2 2 r 1 q η 4 d +  r nm  3 18  iϑ − q )+ / r + 112 4 4 r additive f  =2 ).  6 − q + 44 i X 8+ 2 1 r 2 Z τ, z r  64 − ( ( or − r − − 1 26 k )= k O , 2 2 2 2 − d + 36 r r 4+ − ) = 8 − + 2 4 2 , z is the Eisenstein series for Γ ) 2 8 + 12 φ r r 2 1 q − 18 4    r z ] , z ) ) r 6 r  – 27 – (  1 − τ 2 2 + +  2 n,ℓ,m z 51 ( X − (  + r T ( )  | Nτ   +  3 d × 2 ( is N arithmetic + r r / 2 2 ( 5 + η st K 1 + + 6 / 3 α E E 0 q 4 / ∞  3 ln r q i 1 > / ) r 2  and index 1, Maaß constructed a Siegel modular form of q 1 − 42 r  q ) ) = ) = + 1 N k r  +4+ + +2+ X τ 2 2 2 − r 16  ( r r r n,ℓ,m 14 6 r N η ( , z , z 2 2 2 2 − 1 1 − −    ln z z ≡ − ( ( + 72  )= ) 1 1 , , τ 2 0 Z )= )= )= ∂ r + 32 ( 42 φ − + 28 + 12 + 4 τ, z k 1) ( φ r i ) r r 16 6 2 at the cusp at τ, z − τ, z τ, z − Φ r 1 ( ( ( 14 , 12 N ) N 2 1 1 1 − − − ( 0 ( , , 6 , 1 r , − N (2) 0 (5) 0 (3) 0 π φ    ( 0 φ φ φ   φ + + )= τ ( =8+ =6+ =4+ leading to an explicit formula [ ) 1 1 1 k , , , N ( (2) 0 (3) 0 (5) 0 α φ φ φ We will see the appearance of weight zero Jacobi forms of the g The elliptic genus of Calabi-Yau manifolds are weak Jacobi f This procedure is known as the product representations for the modular form Φ known that the ring of Siegel modular forms is generated by fo expansion for and about the cusp at 0 is B.3 Additive lift ofGiven Jacobi a Jacobi forms form with of integer weight index weight with JHEP04(2009)032 = (C.4) (C.2) (C.1) (C.3) M (B.17) (B.14) (B.15) (B.16) ) given ) matrix ivial real N Z ( , 0 ] and b G i.e., 53 ), is generated Γ Z v ) [ ( . , Z 10 ) is a uni-modular b , one has a similar , m 2 3 s J + rance of a non-trivial u u ℓ ) 2 b r 1 0 . ) is trivial, i.e., N u u , + n 5, Jatkar and Sen have ( 1 d rm, Φ q ) induces a character for ) , b 0 ( +2 3 Z k χ
, 1) ( ) = ( , ℓ d B.2 k . 1 − 2 , u ∆ om the additive lift is one with U 2 = 2 0 18 Nz d ) k u nm = ( ( ) ) with character 1 ] η N z )+ Z ! D ( 3 N . When and , f 57 4 , 2 , u η 6= 1 B I − 1 ) + N +  k 2 ) ∗ − 1 2 ) ) 2 0 b Z k u I 1 2 ) is [ Sp(2 d M z 1 C

b b ( denote the corresponding Sp(2 ( b , z ) d, N ⊂ M Γ 1 η Γ )(1+ ∗ 2 respectively as additive lifts of the weight d ( ) given in eq. (  2 v z ) ( , Γ 2 ( u Z 4 ) χ v M 1 N + ≡ = , 2 ] modulo θ ( 0 ) det( 0 if ( 1 otherwise ) – 28 – ) 0 u , z 52 b B ( 1 M G = 6 M ( , let z )= (1+ ( ( Γ 2 n,ℓ,m k X Γ ( 1 v | M 1) )= θ , z d w ) in Sp(2 , v d 1 − ( Z z 0 , ( )= χ )= 1 > = ( = 1 , ) given above. Z 2 ) Siegel modular form i.e, a modular form of · 10 , z ! ! φ X 1 2 1 N ) matrix 0 M I z B.15 0 n,ℓ,m U ( ( ), one defines ( Z − k 1 , T k, ≡ 2 0 ∆ = 7, Jatkar and Sen have constructed a modular form of weight 1 0 B.2 I φ , index 1 Jacobi form of the subgroup Γ U ) Siegel modular form. For levels 2 identity matrix, k

Z N ( Γ Γ N × k v v ]: ]. In this case, one does indeed see the appearance of a non-tr Φ 9 9 ). An explicit expression for N ) is the unique non-trivial real linear character of Sp(2 ( 0 M ) is a real Dirichlet character [ is the 2 b ) be a Siegel modular form of ] ( G d Γ ( Z 2 ). Given an SL(2 56 ( ∈ v I χ – k Z
, When the Jacobi form is one with character, one sees the appea Given a weight 54 Jacobi forms [ A B C D as determined by eq. ( Dirichlet character and the Siegelcharacter. modular form At obtained level fr with character [ by [ formula leading to a level- weights 4, 6, 10 and 12. For instance, the weight 10 modular fo by the Jacobi form of weight 10 and index 1 Dirichlet character in eq. ( C The additive liftLet with ∆ character at level SL(2 constructed modular forms of weight where where where matrix. The embedding of SL(2 k we obtain a level JHEP04(2009)032 ). 2 , z ). It (C.6) (C.7) (C.5) (C.8) (C.9) 1 Z (C.10) (C.11) z , ( 1 (induced has half- . ϑ Γ ∞ (1) 1 w i b ). It is useful A p, Sp(2 be odd else the Z , kulin have shown . 2 must m/ e half-integral follows form s , discussed in the main k 2 . , , y algebra, N , ℓ/ ) ) theta function, 2 . G 2 r 2 2 a GKM superalgebra. t the cusp at ) has the following expan- ined from the lift of Jacobi ) with half-integral powers ) 2 ). ℓ/ ight m/ h half-integral index as they Z , z , z Z he following transformations r s n/ ( 2 and character , 1 1 ( 2 2 with character. This Jacobi 5 q onents. One has 5 z z 4.2 ) 9 / B.15 ( ( ) 2 n/ 2 2 ∆ m/
1 q ℓ d z , z 5 ) via the additive lift. The Fourier ) ( 1 , ) η z Z k,m/ k,m/ 2 ( ( D d ) 5 2 nm ψ ψ n,ℓ 2 ( + h λ g , index , z g Z 1 k k,m/ 1) 1) 1 z ψ C ( − − − 2 0 1 k ) to higher levels. In particular, we construct / 2 and index 1 2 given in eq. ( ) ( ϑ d / / X m> ) =1 mod 2 – 29 – M ) = ( ) = ( ( 2 C.10 h )= Γ n,ℓ z 2 v + X n,ℓ,m X mod 2. + , z ( | 2 1 1 d )= 2 z 0. The modular form ∆ 2 I )= , z =1 mod 2; ( 1 2 0 Z m , z ≥ z , λz / = · 1 > ( 1 1 , ) 2 z 2 z 5 S ( ℓ ( )= M 2 ψ ) character induced by the character of Sp(2 2 X ( / Z − Z 5 1 k,m/ n,ℓ,m ]. ( , , ( 5 k n ψ ∆ k,m/ 57 ψ ∆ ψ )= Z ( ) is a Jacobi form of weight 5 2 ) =1 when S ∆ , z ( 1 ) Γ 2 z ( ) is the SL(2 w appearing where integral powers appeared. Gritsenko and Ni 2 , z 1 s ) = 0 unless 4 M z ) is a modular form with character under the full modular grou ( ( Γ k,m/ Z 2 ] n,ℓ ( ψ w and ( 5 ). The condition that the exponent in the Fourier expansion b 57 g r Γ ∆ Such a modular form admits a Fourier-Jacobi expansion of the The Jacobi form of weight 5 and index 1 k,m/ , v ψ q from the behavior ofof the character. Similarly, one also has t integral exponents. An important point to note is that the we modular form vanishes [ Thus, one sees that Fourier expansion of the Jacobi form abou that this modular form appears as the denominator formula of where by transforms as additive lifts of the Jacobi forms of index 1 C.1 Additive liftSo of far Jacobi we forms have only withforms considered half-integer with examples index integral of index. modular formsappear We will obta in now the consider denominator examples formulae wit for the the GKM algebras, of Notice the similarity with the Maaß formula given in eq. ( to note that We will now generalize the additive lift ( where body of the paper. The simplest example is given by the Jacobi It is a holomorphic Jacobi form of weight 1 form appears as the denominator formula of the affine Kac-Mood expansion of the Jacobi form now involves half-integral exp sion [ generates the Siegel modular form with character, ∆ with JHEP04(2009)032 ] 57 ) = 1. (C.15) (C.16) (C.17) (C.13) (C.18) (C.12) (C.14) α, N and replaced  . δ given above, we 0 + ℓα by the following mz δ γτ , mz m = m , z ′ ′ X and (iv) ( α> δ β ℓ iπℓ \ e + + 1 iπℓ δ N β e γτ ατ X τ iπℓmz and  ) = 1. We will show that ′ δ +2  e , see appendix A] and will ∗ δ 2 9 ατ mz / iπn M , 1 oset 2 for odd e , δ ( n [ 2 k, β Γ iπn nα/δ )=1 and iπnατ ℓ/ e ψ v the procedure due to Maaß [ m/  δ e +2 which implements the condition r ′ = =0 mod ) with integral entries such that (i) δ α 2 ℓ 2: ) 2 ′ ατ + , γ α, N n  n/ N )= γz (  γτ n,ℓ 2 q m m/ , ′ n,ℓ ( 2 α M ( n ) / g γ δ α β ( g 1 1) πim  Γ  k, e g n,ℓ − n,ℓ w nα X ψ ( | k n, ℓ = 1 δ δ X 1 1 g − ( − =0 =0 − ) − ) as follows: k δ δ X X M β β δ α mod 2; (iii) +1 and index n,ℓ – 30 – X ) with Fourier expansion k k k + τ, z ) ,..., ( − − − k 2 α . The coset can be represented by the elements 1 2 δ δ δ I ( , τ, z γτ m )= ( ( χ m = 2 ) for which X m m m ) 0 0 0 X / k,m/ \ ) = 1 ) = 1 ) = 1 = 0 \ 1 τ, z = = = N 1 1 ψ , m 0 ( M ( ′ k, X X X α > α > α > 2 matrices 2 X 0 αδ αδ αδ X ,ℓ α, N α, N α, N ψ / ′ ∈ ( ( ( 1 X × α > X n ” ( 1 1 1 k, | m, β α − − − ψ γ δ k k α β k = 1 ) ) ) ′ “ ,ℓ − ′ m m m k αδ X n ) m = ( = ( = ( ) = ( with ≡ = 1 mod 2); (ii) , ) τ, z ( is a subgroup of Γ m 2 ! ( Consider the coset τ, z be the group of 2 β 1 δ ( 2 2 ) is the trivial Dirichlet character modulo m X m k,m/ . 0 α α ψ X (

k,m/ χ )= ψ ) = 1. In obtaining the last line, we have defined m/α Claim: Let M by α, N Note that the following is a Jacobi form of weight det( C.2 Hecke operators andGiven the a index-half additive Jacobi lift form The proof is along lines similar to the one given by Jatkar-Se where not be repeated here. Using the above representation of the c for Jacobi forms of index half to higher levels. we construct a Jacobi formaveraging of procedure. the This same generalizes weight and and index closely follows can rewrite the equation defining δ ( JHEP04(2009)032 ) 2 = 1 ˜ 1 M m/ = 2. ( M Γ k (C.21) (C.22) (C.20) (C.19) w ] for the as 9 is another ˜ S ) = and not as in 2 3 , follow in an 1 . s J . ) as defined in and index ψ ) 2 2 3 M α . ) . ( w q k ( N m/ γ Γ 2 ) ( ψ ) and s  0 τ, z w w χ 2 m/ 2 5 ( N τ, z 2 s ℓ/ ( r and is anti-symmetric ( / 2 0 r 1 2 9 . When the seed Jacobi 2 Γ ℓ/ 3 sformation under = 5. k, Γ z r − n/ v ψ 2 ∈ q k,m/ 2 3 N rucial argument. Given any group, Γ k ] (and Jatkar-Sen [ n/ ↔  r ψ )
q ˜ ˜ b d ˜ ℓ d α 1 57 k ˜ c ) but with ) ˜ a z
, + d ℓ α 2  + 93 can be replaced by averaging over , α 3 nm + r cτ 2 = C.21 S α nm √ g 1 cτ ) (˜ iπmz  1 ˜ 1 e 3 2 90 M g ) = 1, we see that ˜ ) ( and character − M ) ˜ 1 k S 1 ( − 2 qs ( k ) constructed by the averaging procedure α convergent and one obtains a meromorphic − Γ M Γ is a Jacobi form of weight r k , z √ ( ) w 1 w 90 α 2 Γ √ being odd. Taking into account the additional α τ, z z where + ( ( ( ) = not w k 2 2 – 31 – s + ), transforms with character χ ˜ α ˜ S d k,m/ ( = 2 1 √ ) = 2 3 + ) ψ ψ ˜ c ˜ cz 93 2 3 S r τ, z k,m/ k,m/ M ˜ 2 cτ χ 0 ( ( + (Z) q ψ ψ 2 Γ cz 2 − iπ  / ) cτ = 0 n, ℓ, m X 1 w α > − ( . In particular, one has , . 5 2 k/ | 0 1 e  9 2 1 k α r m> 3 2 iπm ψ  n, ℓ, m M X r ˜  α > − d ( | SM 0 e + α z − + > ) and  X ) ˜ cτ ) with (odd) weight r ˜ s d S , 0 1 + ˜ b z √ N ˜ √ d X > ˜ ( =1 mod 2; q ) + + 9 cτ M n,ℓ,m 0 ): ( m ( , √ ) is symmetric under the exchange ˜ b ˜ cτ aτ G α − Γ b d X ( Z  = , one has + + w  ( r n,ℓ,m χ 2 ( )= 2 9 r m / cτ aτ . In other words, averaging over all ]. This is indeed the case in our example for √ 1 Z for all values of k/ √ X  m ( we now prove that k, 56 \ k 2 + 2 ) = )= + X 1 ψ z = 3, the weight of the modular form we have constructed is even , one obtains a formula similar to eq. ( 1 ). The proof follows along the lines of Maaß [ 2 3 ∆ Z \ ψ r X ( 1 − N taking the place of 1 2 N k,m/ w ( r √ X . SM ∈ 1 0 ψ ∆ →− ( ), thus evading the restriction on  ˜ − Γ m Γ 2 S M ) replacing  + z w ∈ 4.5 C.19
When One thus sees that = with 5 c d a b ˜ ∆ eq. ( However, the seed Jacobi form, under form is a weak one,modular then form the [ above sum is is a modular form of We note that ∆ Modularity: D Explicit formulae for ∆ modifications to account for thematrix level.). We only repeat the c is a Jacobi form with character. Let us first consider the tran eq. ( character, elementary fashion. Thus, we see that The transformation under the other generators of the Jacobi Since S matrix in leading to the required result. for odd JHEP04(2009)032   q q √ √  2 2 / / 2 13 3 9 s s  qs + + 2 2 13 √ 2 / / 9 3  + qs q 3 2 sq s sq  √ s √ √ 3 2 2 √ √ 2 11 +  / q    2 3 13 s q q  qs  s q   7 2  √ 2 √ √ 2 2 5  √ / 9 2  + / 2 2 3 r 3 2 3 / / 7 2  13 2 qs  r +  9 7 r / q 2 qs 9 2 − 11 s s 7 2 3 s √ qs  − − r 2 3 √ 2 + + √ sq qs + √ qs  r / 2  − 2 2 3 + q / 2 √ 2 s 2 7 √ 1 11 √ / / 3 r +  / 2 5 s 9 7  r √ r q 3 3 √ q  s r + 5 2 + + s  √ q /   sq sq 2 7  − 2 √ r s s √ 9 2 2 /  q 2 12 √ 13  √ √ + 2 5  / 3 7 2 / 2 5 + q √ √ 2 √ 1  s 5 2 3 2 5  q  r  / q −  2 r 2 9 19 5 r q 2 / 11   + s  qs    s q 2 3 qs 3 2 3 + 6 2  5 q √ 2 2 2 2 + s  r + 4 √ 2 2 / / / + / 5  √ r  + 2 1 1 1 r / 5 5 5 r 2 5 / 3 2 r 2 qs 6 7 2 13  + r + 27 r r 1 3 / + + r √ r /  s 9 5 √ 2 135 q √ s 5 2 3 2 5 sq s 2 19 3 r − − − + + + r − − r  √ − − r + – 32 – sq 2 √ √ 2 2 2 sq 2 3 2 r − 5 / 2 3 2 5 2 3 r r / / /  s / 5 2 7 √ 5 r 2 2 3 3 3 3 3 27 r 2 3 3 √ q r √ q −  √ √ r r r √  r r   sq + 90  − 2 2 3 3 2 − 4  + / r + 5 √ + + +  − r r q 1  + 6 2 7 5 2  2 2  + 12 / r 2 3 − 2 r  6 √ r r r / 2 3 1 / + 54 √ 3 r − / 3 5 3 5 3 r 7 r r r 6 √  r 3 r √ √ √ r − r r  + √ r 2 12 √ √  2 3 / + r 54 3 − + √ 2 1 − − − 5 − r + 9 − / + 4 − r   + r 3 √ 90 5 + 135 r r r r r ) r − 2 r − + 6 r r s √ 3 2 5 − + + √ √ √ √ − − r 4 √ √ ( r 12 5 r 3 2 √ √ √ r s s + r r √ 6 − 135 q 3 r − √ 1 √ 135 − 2 − r √ √ √ + 3 + 5 + 5 90 − + √ √ / r √ q q r − 5 − 2 2 2 r + 27 + − − + 3 3 2 2 / / / 5 √ r 5 + + √ √ 3 r 2  √ 3 3 3 r 27 √ r r 3 r 2 2 2 3 5 5 1 r r r r 90 r 4 4 2 2 3 3   √ 12 r r r √ 27 6 2 2 3 √ − + 2 + − + 12 r r √ 3 r r + 5 − 1 1 2 2 5 5 3 2 2 2 − − − + − + + − 2 − √ √ / / + + − / − 7 − − 3 2 2 2 2 2 2 2 2 3 7 7 5 2 5 7 9 27 / / / / 2 3 r r r r r r 2 2 5 5 − − r 5 5 5 5 − − − − r 9 5 1 r r r − − r 3 r − − r r r r − r − r − √ r r                   √ √ −    + + + + + + + + + + + + + + + + + + = = = 1 2 3 ∆ ∆ ∆ JHEP04(2009)032 ] ) s  2 1 s q , 2 1 √ ↔ , 2 / 1 2 q [ 13 α . s ) ). The real = + ρ 2 , sr nm / 1 8 13 − ant subgroup of −  sq 2 ).) Thus, one has q ). In other words, sr . he terms involving ℓ 3 √ √ qr, r ,II ( = 2  1 in the sum side of the / ctions associated with 2 rated by the three real ame multiplicity. πiz ) /  A 2 Z 11 1 ( / e expansions. , 2 s s 1 3 g / 2 δ 1 s 7 ), and their inner product ( / ). All the modular forms, + 2 3 r 1 / (or equivalently the πi Z 2 r 1 , δ ( / − 2 + r , 3 2 5 1 continue do so in the other / = exp(2 e has norm (2 2 ) δ 11 1 2 / , δ ± 3 / Z q s 1 , 3 1 5 sq ] q δ r ↔ and √ mδ 1 )= symmetry which permutes the three  − + 2 δ 1 n,ℓ,m [ / 2 3 2  − ) and 3 α δ / 1]) and the Weyl vector is S 2 r ( 2 5 ) s 3 , / 2 2 r n 5 πi 1 / = r , 3 − πiz + ) + r e [0 Z 2 − , m r / 2 = 9 1 2 , α δ – 33 – + √ / ( 5 l 3 m 0] ,q πi r , s 2 − − ) with coefficient ℓ / − 1 = exp(2 1 r r Z + − s n ( +( r √ , 2 . Further, one has the identification relating the root 5 q r 1 / 2 5 [0 1 / , see Prop. 2.1]. Their proof makes use of the non-trivial ), √ qr, e nδ 1 are − 4 1 + 9 ), all terms (in the expansion given above) that arise with [ s , α r 1 − 2 2 = ). A formal proof can be given by following Gritsenko and 2 Z 1 / / 0] G / ( ) r πiz ]= 3 1 Z G , 1 ). This is consistent with the observation that the real root 5 ( Z r r 1 , √ 2 Z 2 5 1 , ,q through the relation: ( / δ 2 k/ ( 1 2 [1 / − q + 5 1 m πi α k/ 2 n,ℓ,m s s 2 [ − / ℓ 2 / 5 e = exp(2 / r α 3 r 3 n r appearing in the modular transform ∆ r q )) = q 1 come from the sum side of the Lie algebra 5 2 Γ Z / + 3 ± 3 − v 2 q ρ, 2 / ] to ( ( / 7 ), share the same character suitably restricted to the relev ). Hence, the same proof goes through. 5 , one sees that r πi r Z Z Z − 2 ( , − 2   n,ℓ,m k/ [ + + Nikulin’s argument for simple roots are ( where the root symmetry in the ∆ are unaffected by theimaginary roots orbifolding. related by We Weyl have reflections appear also with verified the that s t α real simple roots. It is easy to see only the Sp(2 modular forms ∆ exp( ∆ coefficient with they arise by the actionsimple of all roots. elements of For the instance,the Weyl simple the group real gene terms roots arising of from Weyl refle denominator formula to extract the roots. character in this notation. Note that we need to pull out an overall factor of (Recall that 4. All the GKM superalgebras have an outer 3. The terms that appear in ∆ 2. In the expansion for ∆ 1. Using the expressions for the real simple roots, ( D.1 Analyzing the modularWe list forms some of the observations that can be made from the abov JHEP04(2009)032 ) 1]. Z ( , 9 in 2 2 , − , ell. [1 cessarily α Mem. , , invariance , 2 ], with norm δ ), 0 in ∆ ]. (2005) 063 Z ( ↔ 3 licities 1 07 string theory δ n,ℓ,m [ α . =4 ]. 1in ∆ ] (2006) 018 JHEP ]. N − , 04 , m,m , SPIRES ]. ]. SPIRES ] [ ]. ]. ). These functions depend only alg-geom/9504006 + 2 — both have multiplicity ] [ JHEP ]. ℓ (1990) 30. , 2 − / ]. 3 nm,ℓ 83 n, s ( SPIRES SPIRES 2 2 SPIRES SPIRES / + c ] [ 5 ] [ SPIRES ] [ ] [ ). The multiplicities of positive roots is r Counting dyons in maps a positive root m ] [ 2 (1997) 181 [ / Maximally supersymmetric string theories in 3 4.7 3 + S q ℓ ) and 1] is mapped to another light-like root hep-th/9505054 119 [ , − – 34 – Adv. Math. hep-th/9609017 [ to 0 [ , , α Four-dimensional string-string-string triality symmetry needs us to verify the 2 / nm,ℓ [0 3 math.QA/0001029 3 ( symmetry can be obtained by considering the prod- ↔ ) as they arise from the Fourier expansions of weak α s S 1 Wall crossing, discrete attractor flow and Borcherds 2 ] 2 c 3 ℓ / hep-th/9607026 S [ hep-th/9510182 hep-th/9508094 1 Algebras, BPS states and strings On the algebras of BPS states [ [ Siegel automorphic form corrections of some lorentzian arXiv:0806.2337 r − hep-th/9601029 2 [ (1995) 2264 / Microscopic origin of the Bekenstein-Hawking entropy (1998) 489 ]. ]. 1 n,ℓ,m q [ nm 75 Amer. J. Math. Dyon spectrum in CHL models α , 197 (2002) 746 [ (1997) 543 (1996) 315 (1996) 125 SPIRES SPIRES (1996) 99 (2008) 068 [ 157 ] [ ] [ ). A similar analysis has been carried out for other terms as w 4 Z ), to another root with the same norm. Thus both these roots ne The Monster Lie algebra ( 2 Some generalized Kac-Moody algebras with known root multip B 484 1 ℓ B 463 B 459 B 379 − ). One can show that under this exchange Phys. Rev. Lett. SIGMA Z Black holes, elementary strings and holomorphic anomaly ( ). These two terms appear with identical multiplicity , nm k Z 10, ( 5 2(4 share the same multiplicity. − Jacobi forms. The action of the outer on the combination (4 determined by two functions For instance, the light-like root uct form of the modular form in eq. ( of ∆ and 1 in ∆ This relates the term ∆ Nucl. Phys. hep-th/0502126 hep-th/0510147 Amer. Math. Soc. algebra [ [ Kac-Moody Lie algebras Nucl. Phys. Commun. Math. Phys. Nucl. Phys. Phys. Lett. D< 5. A practical check for the outer 6. Another proof of the outer [1] A. Strominger and C. Vafa, [7] M.C.N. Cheng and E.P. Verlinde, [2] M.J. Duff, J.T. Liu and J. Rahmfeld, [3] R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, [4] V.A. Gritsenko, V.V. Nikulin, [5] J.A. 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