BKM Lie superalgebra for the Z_5 orbifolded CHL string
aa r X i v : . [ h e p - t h ] N ov HRI/ST/1012
BKM Lie superalgebra for the Z -orbifolded CHL string K. Gopala Krishna ∗ Max-Planck-Institut f ¨ u r Mathematik,Vivatsgasse 753111 Bonn, Germany Abstract
We study the Z -orbifolding of the CHL string theory by explicitlyconstructing the modular form e Φ generating the degeneracies of the -BPS states in the theory. Since the additive seed for the sum form is aweak Jacobi form in this case, a mismatch is found between the modularforms generated from the additive lift and the product form derived fromthreshold corrections. We also construct the BKM Lie superalgebra, e G ,corresponding to the modular form e ∆ ( Z ) = e Φ ( Z ) / which happens to bea hyperbolic algebra. This is the first occurrence of a hyperbolic BKM Liesuperalgebra. We also study the walls of marginal stability of this theory indetail, and extend the arithmetic structure found by Cheng and Dabholkarfor the N = 1 , , N = 4 , N > N = 5 and 6 models. ∗ [email protected] Introduction
There has recently been renewed interest, and considerable activity, in under-standing the area phrased by Harvey and Moore as ‘the algebra of BPS states’ [1, 2]. Our interest in the present paper is in constructing the ‘algebra’ in thecontext of the CHL models. Harvey and Moore considered BPS states in stringtheory with N = 2 spacetime supersymmetry and showed that the thresholdcorrections in the N = 2 heterotic string compactifications are determined interms of the spectrum of the BPS states [2]. They found a connection betweenthe threshold correction integrals and infinite product representations of holo-morphic automorphic forms studied by Borcherds earlier [3, 4]. Later Borcherds,using the regularization of the integral given by Harvey and Moore, constructeda generalization of the Rankin-Selberg method to obtain automorphic forms onGrassmannians which have singularities along sub Grassmannians [5]. In par-ticular, for the case of unimodular lattices R ,s , the results of [3, 4] for familiesof holomorphic automorphic forms could be re-derived using the general methodof [5] in much simpler fashion.In the context of counting -BPS states in string theory, these very automor-pic forms appear as generating functions of dyonic degeneracies. The degeneraciesof BPS states preserving one fourth supersymmetry in a class of N = 4 super-symmetric string theories in four dimensions are found to be generated by themodular forms obtained from the generalized theta correspondence, while the de-generacies of those preserving half the supersymmetry are given by cusp forms ofΓ ( N ). The generalized theta correspondence of Borcherds, in this context, givesthe threshold integral starting from the elliptic genera of K
3. In the prototypicalexample of the type II string theory compactified on K × T or equivalently,the heterotic string compactified on T , the degeneracies of the -BPS statesare generated by the weight 12 cusp form for Γ, η ( τ ) , while the degeneraciesof the -BPS states are generated by the Igusa cusp form, Φ ( Z ), of weight10 for Sp (2 , Z ). More generally, when this theory is considered with the com-pactification space orbifolded by a Z N -group whose action preserves the N = 4supersymmetry, the degeneracies of the -BPS states are found to be generatedby genus-one cusp forms of Γ ( N ), while the degeneracies of the -BPS statesare generated by genus-two Siegel modular forms of suitable level N subgroupsof Sp (2 , Z ).The fact that the degeneracies of the BPS states are generated by modularforms obtained from the generalized theta correspondence gives rise to the pos-sibility that they may have a BKM Lie superalgebra associated with them sincesuch structures were found in the construction of Borcherds. Indeed, the squareroot of the weight 10 Igusa cusp form of Sp (2 , Z ) also occurs as the denominatoridentity of a rank 3 Borcherds-Kac-Moody (BKM) Lie superalgebra, denoted G ,constructed by Gritsenko and Nikulin [6]. Along similar lines, BKM Lie super-algebras (all, like G , of rank 3) have been found to exist corresponding to the1enus-two Siegel modular forms occurring in a family of four-dimensional N = 4supersymmetric string theories, known as the CHL strings [7–10]. The fact thatfor the orbifolded theories there exists more than one cusp, and hence more thanone infinite product expansion of the modular form at each of the cusps, leadsto the existence of more than one familiy of BKM Lie superalgebras associatedwith the orbifolded models. This is similar to the case of the fake monster super-algebra where the BKM Lie superalgebras associated to the two cusps at level1 and 2 are distinct and very different from each other although the denomina-tor identities of the two algebras are the expansions of the same function, albeitabout different cusps. For the CHL family, this leads to two different families ofBKM Lie superalgebras with very distinct structures and properties. A list of allthe modular forms arising in the CHL models that have a corresponding algebrastructure can be found in the “periodic table” of BKM Lie superalgebras listedby Govindarajan [10].The main challenge is to make the algebras constructed relevant to the phys-ical theory from which they arise for which one needs to find relations betweenthe two that can help study one from the other. In what is hopefully the firststep towards setting up a dictionary between the algebraic and physical sides ofthe BPS state counting, it was observed by Cheng and Verlinde that the walls ofmarginal stability of the -BPS states are in one-to-one correspondence with thewalls of the Weyl chambers of the corresponding BKM Lie superalgebras [11].This correspondence is an indication that the BKM Lie superalgebras are notmere academic constructions from the modular forms, but should actually berelated to physical aspects of the CHL theory.That the counting of BPS states should have an algebraic structure associatedto them is both intriguing and promising for the theory. Intriguing, for the reasonsof their origin are not clearly known and are not such as to be foreseen at the levelof the action of the theory, and promising for the possibilities it presents to knowmore about the microscopic side and also for unearthing other deeper structuresof the theory. Already newer and unexpected relations are being uncovered basedon these ideas, like the moonshine conjecture for M [12, 13] etc..In this work we study the Z -orbifolded CHL theory and the BKM Lie su-peralgebra structure arising in it. We also extend Sen’s study of the walls ofmarginal stability [14] of the -BPS states in the CHL models. Organization of the paper
The organization of the paper is as follows. In section 2, we discuss the detailsof the counting of BPS states by providing a brief introduction to the settingof the problem. In section 3, we provide the details of the construction of therelevant modular forms that are the generating functions of the half and quarterBPS states in the CHL models in general, and in the Z -orbifold in particular.In section 4, we discuss the BKM Lie superalgebras arising from these modular2orms. We show that the ‘square root’ of the modular forms constructed in section3 appear as the denominator formulae of BKM Lie superalgebras. We show thatthe BKM Lie superalgebra for the modular form e ∆ ( Z ) = e Φ ( Z ) / exists andis a hyperbolic one with an infinite number of real simple roots. The BKM Liesuperalgebra has two sets of roots which are copies of one another. In section 5,we study the walls of marginal stability of the -BPS states in the CHL models,and study the N = 5 case in section 5 .
1. In section 5 .
2, we study the BKM Liesuperalgebra e G in relation to the correspondence between the walls of marginalstability of the -BPS states and the walls of the Weyl chamber of e G . We use thiscorrespondence to label the roots of the algebra from the corresponding labelingof the walls, and from this derive the Cartan matrix of the algebra e G . We alsoshow the invariance of the modular form under the Weyl group of the algebra. Insection 6, we study the arithmetic structure in the walls of marginal stability ofthe -BPS states in the CHL models. This is similar to that found by Cheng andDabholkar in [8] for the N = 1 , , N = 4 , , N = 1 , , Although initial progress on microscopic couting goes back some time beforeit, the starting point for us, in the direction we are interested in, will be thework of Dijkgraaf, Verlinde and Verlinde. More than a decade ago, Dijkgraaf,Verlinde and Verlinde (DVV) proposed a microscopic index formula for countingthe degeneracy of -BPS dyons in heterotic string theory compactified on a six-torus [15]. It is known that the degeneracy of electric -BPS states, which can beunderstood as the states of the heterotic string with the supersymmetric sectorin the ground state, is generated by 1 /η ( τ ) . There is an SL (2 , Z ) electric-magnetic duality symmetry, which implies that the magnetic -BPS states, whichnecessarily arise non-perturbatively as solitonic states, are also generated by thesame modular form. To generalize this to the dyonic states, DVV’s basic ideawas to think of a dyonic state carrying electric and magnetic charges as a boundstate of an electric heterotic state with a dual magnetic heterotic state and usingthis picture to construct a modular form which counts the degeneracy of the -BPS states as a generalization of the one that counts the degeneracies of the -BPS states. The appropriate generalization turned out to be a genus-two Siegelmodular form of weight 10, which is the unique cusp form for the modular group Sp (2 , Z ). Intuitively, the 2 × -BPS generating functions. Thedegeneracy, D ( n, ℓ, m ), of a dyonic state carrying charges ( n, ℓ, m ) = ( q e , q e · m , q m ) are generated by the Siegel modular form as64Φ ( Z ) = X ( n,ℓ,m ) D ( n, ℓ, m ) q n r ℓ s m , (2.1)where Z ∈ H , the Siegel upper-half space and ( q e , q e · q m , q m ) are theT-duality invariant combinations of electric and magnetic charges. The aboveconstruction has since then been extended to other settings in four-dimensionswith N = 4 supersymmetry, notably the family of CHL strings and type IIcompactifications [16, 17]. We study a particular case of the CHL orbifoldings,the Z -orbifolding, in this work.The CHL orbifolds arise as a family of asymmetric Z N -orbifolds of the het-erotic string preserving the N = 4 supersymmetry of the unorbifolded theory.Four-dimensional compactification of string theory with N = 4 supersymmetryhas three perturbative formulations in terms of toroidally compactified heteroticstring and type IIA/B string theory compactified on K × T . Consider theheterotic string compactified on a six-torus, T = T × S × e S . The generatorof the Z N -orbifolding acts by a 1 /N shift along the circle S and a simultaneous Z N -involution of the Narain lattice, Γ , , associated with the T . On the dualtype II side, the orbifolding action corresponds to an order N automorphism of K /N translationalong one of the S of the T .Upon orbifolding, the Z N action gives rise to twisted states in the theory andthe vector multiplet moduli space for the theory gets modified to (cid:0) Γ ( N ) × SO (6 , m ; Z ) (cid:1)(cid:31)(cid:18) SL (2) U (1) × SO (6 , m ) SO (6) × SO ( m ) (cid:19) . (2.2)The group SO (6 , m ; Z ) is the T-duality symmetry group where m = [48 / ( N +1)] − ( N ) ⊂ P SL (2 , Z ) is the S-duality symmetry group that is manifestin the equations of motion and is compatible with the charge quantization. Noticethat the orbifolding breaks the S-duality group from SL (2 , Z ) to the subgroupΓ ( N ).Extending the counting of states from the toroidally compactified heteroticstrings as given by DVV, to the CHL orbifolds, Jatkar and Sen constructed twofamilies of genus-two Siegel modular forms, e Φ k ( Z ) and Φ k ( Z ), for the CHL modelswith Z N -orbifoldings, when N is prime [16]. The weight, k , of the modular formis related to the orbifolding group Z N by ( k + 2) = 24 / ( N + 1) (where N isprime and ( N + 1) | Sp (2 , Z ) is broken downto a smaller subgroup and hence there is more than one cusp. The expansionabout each cusp gives a different family of algebras. The family of modularforms, e Φ k ( Z ), are the generating functions for the degeneracies of -BPS states4n the CHL models, while the family Φ k ( Z ) generates the degeneracies for thetwisted dyonic states [10, 18]. From these modular forms, the dyon degeneracyis given by a relation of the type (2.1). Govindarajan and Krishna [9] extendedthis work by constructing the modular forms generating dyon degeneracies forcomposite N . In particular, the case of Z -orbifolding was explicitly worked outand the corresponding modular forms, e Φ ( Z ) and Φ ( Z ), were constructed in [9].Merging the two families and extending the above constructions, Govindarajanrecently constructed the family of modular forms, Φ ( N,M ) k ( Z ), which generatethe degeneracies of the Z M -twisted dyonic states in the CHL Z N -orbifolds [10].They also incorporate the two families, e Φ k ( Z ) and Φ k ( Z ), as particular cases andcorrespond to Φ (1 ,M ) k ( Z ) and Φ ( N, k ( Z ), respectively, in the list. This completesthe general construction of the modular forms generating the dyonic degeneraciesin the CHL theories.Simultaneously, as one constructs the modular forms, one is also interestedin exploring if a BKM Lie superalgebra structure, such as was seen for Φ ( Z ),exists for the other modular forms constructed in the context of the CHL strings.BKM Lie superalgebras for the N = 1 , , N > Z N -orbifolded CHL theories,the algebraic side of the theory has not been studied in detail. In this workwe propose to study the N = 5 orbifolding of the CHL theory and construct thealgebraic structure corresponding to it. We also provide support for the existenceof similar algebras for N = 6. We begin by first discussing the construction of themodular forms that generate the degeneracies of the -BPS states in the theory. Z -orbifolded CHLtheory -BPS states and the additive lift In DVV’s construction the counting of -BPS states in the toroidally compactifiedheterotic string formed the starting point for the construction of the genus-twoSiegel modular form generating the degeneracies of the -BPS states. The count-ing of the degeneracy of -BPS states of a given electric charge is mapped to thecounting of states of the heterotic string with the supersymmetric right-moversin the ground state [20–22]. Let d ( n ) represent the number of configurations ofthe heterotic string with electric charge such that q e = n . The level matchingcondition, n = q e = N L −
1, implies that we need to count the number of stateswith total oscillator number N L = ( n + 1). The generating function for suchstates is 16 η ( τ ) = ∞ X n = − d ( n ) q n , (3.1)5here the factor of 16 accounts for the degeneracy of a -BPS multiplet – this isthe degeneracy of the Ramond ground state in the right-moving sector. For theorbifolded models, this partition function gets modified because of the presence oftwisted sectors and one needs to add the contribution from the different sectors toget the correct partition function. Sen has studied the degeneracy of the -BPSstates in the orbifolded models and showed that, up to exponentially suppressedterms (for large charges), the leading contribution arises from the twisted sectors.It turns out, for the Z N -orbifolded theories with N being prime and subject tothe constraint ( k + 2) = 24 / ( N + 1), the generating functions for the -BPS statesare just the unique cusp forms for Γ ( N ). In [9] extending and generalizing Sen’sresult, a more general ansatz for the generating function for the -BPS stateswas given based on the relation of the symplectic automorphisms of K M . The generating functions for the degeneracy of the -BPS states in the orbifolded models, for both prime and composite values of N ,were found to be given by multiplicative η -products, of weight k + 2, associatedwith specific (balanced) cycle shapes corresponding to the conjugacy classes ofthe 24-dimensional permutation representation of M . A detailed discussion ofthe cycle shapes and η -products leading to the generating functions of half-BPSstates can be found in [9,10]. Taking into account the fact that the electric chargeis quantized such that N q e ∈ Z in the Z N -orbifolded theories, one finds thatthe degeneracies of the -BPS states are generated by the η -products as16 g ρ ( τ /N ) ≡ ∞ X n = − d ( n ) q n/N , (3.2)for the Z N CHL orbifold. The subscript ρ corresponds to the M cycle shapefrom which the multiplicative η -product g ρ ( τ ) is constructed. The generatingfunctions g ρ ( τ ) are cusp forms of Γ ( N ) for the Z N -orbifolded theory just like η ( τ ) is a cusp form of the modular group Γ. This is not unexpected because,for the orbifolded theories, the S-duality group is no longer SL (2 , Z ) but is brokendown to a smaller subgroup, Γ ( N ). Correspondingly, the cusp forms generatingthe -BPS degeneracies are broken down from the cusp form of Γ to cusp formsof its subgroups.The degeneracy of the -BPS states in the N = 5 case are generated by theweight 4 cusp form 1 /η ( τ ) η ( τ / . The degeneracy is given by16 η ( τ ) η ( τ / = ∞ X n = − d ( n ) q n/ . (3.3)Analogous to the DVV case, this η -product is an input into the genus-two Siegelmodular that generates the -BPS degeneracies. It forms a part of the seed forthe additive lift generating the genus-two Siegel modular form as an infinite sumvia its Fourier-Jacobi expansion. 6he product of g ρ ( τ ) with ϑ ( z ,z ) η ( z ) gives a weak Jacobi form of weight k ,index 1 and level Nφ k, ( z , z ) = ϑ ( z , z ) η ( z ) g ρ ( z ) = X n,ℓ a ( n, ℓ ) q n r ℓ , (3.4)which is the additive seed for generating the modular form Φ ( Z ). The modularform generating the -BPS degeneracies, e Φ k ( Z ), is given by expanding the mod-ular form Φ k ( Z ) about another inequivalent cusp. The modular form e Φ k ( Z ) isrelated to the modular form Φ ( Z ) as e Φ k ( Z ) = z − k Φ k ( e Z ) , (3.5)with ˜ z = − /z , ˜ z = z /z , ˜ z = z − z /z . The additive seed for e Φ k ( Z ) is thus given by the weak Jacobi form φ k, ( − z , z z )and the genus-two Siegel modular form is generated from it after summing overthe index m for all values of m ≥ e Φ k ( z , z , z ) = X m ≥ e πimz z − k e − πimz /z φ k,m ( − z , z z ) . (3.6)For the Z -orbifold, the additive seed is given by φ , ( − z , z z ) = ϑ ( z , z ) η ( z ) × g ρ ( z /N ) η ( τ ) η ( τ / . (3.7)Following the procedure of Jatkar and Sen [16], the genus-two Siegel modularform e Φ ( Z ) would then be obtained as e Φ ( z , z , z ) = X m ≥ e πimz ϑ ( z , z ) η ( z ) η ( τ / . (3.8)However, the seed for the N = 5 case is a weak Jacobi form, and the validityof the above procedure is not guaranteed. One needs to verify the expansion ofthe modular form independently from a different procedure.The generalized thetacorrespondence gives another method to obtain the same modular form, this timeas an infinite product. This is useful not just as a check for the modular formconstructed via the additive lift, but also in interpreting the modular form as theWeyl-Kac-Borcherds denominator formula of the BKM Lie superalgebra. Nowwe discuss the product representation of the modular form e Φ k ( Z ).7 .2 Product formulae Product representations for the genus-two Siegel modular forms, Φ k ( Z ) and e Φ k ( Z ),can be obtained from string threshold correction computations [1, 2, 23–25] andfor the modular forms occurring in the CHL theories was computed by David,Jatkar and Sen [26] using essentially the same method as [25]. Upon evaluatingthe integral and requiring its invariance under the duality transformations oneobtains the modular forms Φ k ( Z ) and e Φ k ( Z ) as an infinite product given in termsof the coefficients of the Fourier expansion of the twisted elliptic genera of K R ,s giving holomorphic automorphic forms as infiniteproducts on the hermitian symmetric space G (2 , s ) (Theorem 13 .
3, [5]). The cor-respondence relates a holomorphic (vector valued) modular form to automorphicforms on O ,s given as an infinite product with the coefficients coming from thevector valued modular form. In this case the log of the automorphic form isobtained from the generalized theta correspondence and the weight of the auto-morphic form is given by the zeroth coefficient in the Fourier expansion of thevector valued modular form. In the present case the vector valued modular formis the twisted elliptic genus for K Z N group.The twisted elliptic genus for a Z N -orbifold of K F a,b ( τ, z ) = 1 N Tr RR,g a (cid:16) ( − ) F L + F R g b q L ¯ q ¯ L e πızF L (cid:17) , ≤ a ≤ ( N − , (3.9)where g generates the Z N transformation and q = exp(2 πiτ ). These are weakJacobi forms of weight zero, index one and level N [26]. A weak Jacobi form, F ,s ( τ, z ), of Γ J ( N ) , can be written as [27] F , ( τ, z ) = N A ( τ, z ) , (3.10) F ,s ( τ, z ) = a s A ( τ, z ) + α N,s ( τ ) B ( τ, z ) , s = 0 , (3.11)where α N,s ( τ ) is a weight-two modular form of Γ ( N ) and A ( z , z ) = X i =2 (cid:18) ϑ i ( z , z ) ϑ i ( z , (cid:19) , B ( z , z ) = (cid:18) ϑ ( z , z ) η ( z ) (cid:19) . (3.12)For prime N , the modular forms α N,s ( τ ) at weight two is generated by the weighttwo holomorphic Eisenstein series of Γ ( N ), E ( τ ), defined from the weight twonon-holomorphic modular form of SL (2 , Z ) as E N ( τ ) = 1 N − (cid:16) N E ∗ ( N τ ) − E ∗ ( τ ) (cid:17) = iπ ( N − ∂ τ (cid:2) ln η ( τ ) − ln η ( N τ ) (cid:3) , (3.13) The Jacobi group, Γ J = SL (2 , Z ) ⋉ H ( Z ), is the sub-group of Sp(2 , Z ) that preserves thecusp at z = i ∞ . H ( Z ) is the Heisenberg group. The group Γ J ( N ) corresponds to the congru-ence subgroup obtained by considering the congruence subgroup Γ ( N ) in place of SL (2 , Z ). N indicatesthe level and not the weight of the Eisenstein series, which is two. Also, theaction of an SL (2 , Z ) element on the weight 0 modular forms, F a,b ( τ, z ), is givenby F a,b (cid:16) aτ + bcτ + d , zcτ + d (cid:17) = exp (cid:16) πi cz cτ + d (cid:17) F cs + ar,ds + br ( τ, z ) , (3.14)for a, b, c, d ∈ Z and ad − bc = 1. Thus, for prime N , using (3.10) and (3.14) onecan get the Fourier coefficients of the weight zero weak Jacobi forms, F ( r, s ) forall r, s , knowing the weight two Eisenstein series, E N ( τ ), at level N . The Fourierexpansion of the Jacobi forms are F a,b ( τ, z ) = X m =0 X ℓ ∈ Z + m,n ∈ Z /N c a,bm (4 n − ℓ ) q n r ℓ , (3.15)where r = exp(2 πiz ). Using the Fourier coefficients, the product formula forthe modular form, e Φ ( Z ), generating the degeneracies of -BPS states in the Z -orbifolded model is given by e Φ ( Z ) = q / rs Y a =0 Y ℓ,m ∈ Z ,n ∈ Z ± a (cid:16) − q n r ℓ s m (cid:17) P b =0 ω ∓ bm c ( a,b ) (4 nm − ℓ ) (3.16)where ω = exp( πı ) is a fifth-root of unity, and c ( a,b ) (4 nm − ℓ ) are the Fouriercoefficients of the twisted elliptic genera, F ( a,b ) ( z , z ) given in (3.15). Constructing the modular form via two different methods to generate it as aninfinite sum and an infinite product affords a non-trivial check for the validity ofthe methods. It is also necessary to construct the BKM Lie superalgebra fromthe modular form via the denominator formula. Comparing the modular formsgenerated from the additive and multiplicative lifts, we find a minor mismatchbetween the two expansions. The additive and multiplicative expansions do notmatch with each other. In such a case, one needs another way of verifying whichof the two expansions is the one generating the dyon degeneracies.The correspondence between the roots of the BKM Lie superalgebra, con-structed from the square roots of the modular forms, and the walls of marginalstability of the -BPS states provides another minor check for the modular formsconstructed from the additive and multiplicative lifts. Under the correspondence,the BKM Lie superalgebra constructed from the modular form should have a one-to-one correspondence between its real simple roots and the walls of marginalstability of the -BPS states for the model. Comparing with the analysis coming9rom the correspondence with the walls of marginal stability suggests that theproduct formula gives the correct modular form and one needs to add terms tothe modular form generated from the additive lift, to make it match the onegenerated from the multiplicative lift and give the correct correspondence withthe walls of marginal stability. The case of the N = 5 CHL theory is differentfrom the N = 1 , , , N < Z -orbifolded theory are infinite innumber, and divided into two chambers separated by two limit points. Theappearance of a second chamber is not seen in the N ≤ N = 5 theory. Taking the correspondence on the BKM Liesuperalgebra side, one finds that the two chambers generate two sets of equivalentroots which are in one-to-one correspondence with each other. While the modularform constructed from the product formula contains all the roots coming fromboth the chambers, the modular form constructed from the additive lift does notseem to contain the roots arising from the new chamber, while containing all theroots from the old chamber. We now discuss the above ideas in detail, startingwith the BKM Lie superalgebras. BKM (Borcherds-Kac-Moody) Lie superalgebras are the most general class of (in-finite dimensional)-Lie algebras [29,30]. They were first constructed by Borcherdsin the context of Conway and Norton’s moonshine conjecture for the Monstergroup, extending the theory of Kac-Moody algebras. In the present situation,BKM Lie superalgebras appear in relation to the modular forms (more pre-cisely, their square roots) constructed in the previous section via their Weyl-Kac-Borcherds denominator identities.It was already known that the genus-two Siegel modular form, Φ ( Z ), con-structed in the context of generating dyonic degeneracies [15] or from the thresh-old corrections [25] was related to a BKM Lie superalgebra constructed by Grit-senko and Nikulin [6]. So, when the modular forms generating the degeneracyof dyonic states in the CHL models were constructed, it was natural to look fora possible BKM Lie superalgebra structure related to them. Progress in thatdirection was carried out in [8–10, 19] where the two families of rank 3 BKM Liesuperalgebras, that correspond to the two modular forms e Φ k ( Z ) and Φ k ( Z ) forthe N= 2 , , g be a BKM Lie superalgebra and W its Weyl group. Let L + denote theset of positive roots of the BKM Lie superalgebra and ρ the Weyl vector. Then,the WKB denominator identity for the BKM Lie superalgebra g is Y α ∈ L + (1 − e − α ) mult( α ) = e − ρ X w ∈W (det w ) w ( e ρ X α ∈ L + ǫ ( α ) e α ) , (4.1)where mult( α ) is the multiplicity of a root α ∈ L + . In the above equation, det( w )is defined to be ± w is the product of an even or oddnumber of reflections and ǫ ( α ) is defined to be ( − n if α is the sum of n pairwiseindependent, orthogonal imaginary simple roots, and 0 otherwise. In the caseof BKM Lie superalgebras the roots appear with graded multiplicity – fermionicroots appear with negative multiplicity while bosonic roots appear with positivemultiplicity.The BKM Lie superalgebras for the N = 1 , , A ,II = − − − − − − . (4.2) A ,II = − − − − − − − − − − − − . (4.3) A ,II = − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − . (4.4)The BKM Lie superalgebra for the N = 4 model is of parabolic nature withan infinite number of real simple roots in the algebra. This was in agreementwith the fact that the CHL theory with Z -obrifolding was known to have aninfinite number of walls of marginal stability [14] and there exists a one-to-onecorrespondence between the walls of marginal stability and the real simple roots11f the corresponding BKM algebra. The Cartan matrix of the BKM Lie superal-gebra e G is given by A (4) = ( a nm ) where a nm = 2 − n − m ) , (4.5)with m, n ∈ Z .For the Z -obrifold, one finds that there exists a BKM Lie superalgebra, againwith an infinite number of real simple roots, but of hyperbolic nature. This is thefirst instance of a hyperbolic BKM Lie superalgebra. It is given by the Cartanmatrix (see section 5.2, eq. (5.18)) A (5) = ( a nm ) where a nm = − (cid:16) Λ − n Λ − m + Λ − m Λ − n (cid:17) + ( − d ( x m )+ d ( x n ) , where m, n ∈ Z , Λ and Λ are the roots of the equation r − · r + 1 = 0 and d ( x m ) = 0 if x m ∈ B L ∪ B R and d ( x m ) = 1 if x m ∈ B C is the grading on thetwo sets of roots of the algebra, and B L ∪ B R and B C are the two chambers inthe fundamental domain of the walls of marginal stability. We will come back todiscuss the algebra e G in more detail again after studying the walls of marginalstability for the -BPS states in the model. We will see a detailed constructionof the above given Cartan matrix and also point out certain peculiar aspects ofthe algebra in relation to the ones constructed before it. Now we come to an important idea which establishes a relation between the alge-bras discussed in the previous section and the string theories they are constructedfrom. Such a bridge is important because our final aim is to understand the originof these algebras and the role played by them in the theory. Hence one needsto construct a dictionary between the algebraic and physical sides. One suchrelation was found by Cheng and Verlinde [11] between the walls of marginalstability for the -states and the walls of the Weyl chambers of the algebras. Inthis section we study this in the case of the Z -orbifold before studying the moregeneral case.The walls of marginal stability for the -BPS states in N = 4 supersymmetrictheories was originally studied by Sen [14](see also [31, 32]). The CHL theory forthe Z -orbifold has an infinite number of walls of marginal stability just like the N = 4 case. We will eventually see that this is part of a pattern that exists acrossthe CHL theories with orbifolding groups Z N for N = 4 , , N , however, we will firststart with a detailed study of the N = 5 model.For dyons with a given set of charges, the moduli space of N = 4 supersym-metric theories contains subspaces – of co-dimension one, known as the walls of12arginal stability – on which the dyon becomes marginally unstable against de-cay into two -BPS states respecting the conservation of charge and mass acrossthe subspace. The spectrum of the -BPS states therefore changes discontinu-ously as one moves through a wall of marginal stability in the moduli space. Thedegeneracy formula is thus valid only in a limited domain in the moduli space.As one crosses a wall of marginal stability, one needs to take into account thefact that a -BPS state may split into two -BPS states under suitable circum-stances. However, the change in the expression for the degeneracy across the wallis not very drastic and is such that the partition function for the -BPS degenera-cies formally remains the same, but the point in the Siegel moduli space aroundwhich one should series expand the partition function to extract the degeneracieschanges as one moves across a wall. From the point of view of the BKM Liesuperalgebras, it is a change in the Weyl chamber of the algebra, and the set ofreal simple roots.The walls are, in general, given by complicated dependence on all the moduliin the moduli space. However, for the sake of simplicity, one can study them inthe axion-dilaton moduli space fixing all the other moduli. The walls of marginalstability would then be curves in the axion-dilaton plane (which is modelled bythe complex upper half-plane). The curves for the walls of marginal stabilityare found to be circles and straight lines in the upper half-plane which intersectonly on the real λ -axis or at i ∞ , but no where in the interior of the upper half-plane. The points of intersection have a universal nature, in that, although thequalitative features of the curves depend on the charges and other moduli, thepoints of intersection on the real λ -axis depend only on N .One can study the walls of marginal stability in a fundamental domain inthe upper half plane by restricting the value of Re( λ ) to the interval [0 , λ ) = 0 , λ )-axis between [0 , N = 1 , , ,
1] interval with theintersection points given by the following set of intercepts on the real line:( , ) , ( , , ) , ( , , , , ) . (5.1)A fundamental domain is then given by restricting to the region bounded by thesesemi-circles and the two walls connecting λ = 0 , − and . The fundamental domainsfor the N = 1 , , N >
3, this picture does not terminate – one needs an infinite numberof semi-circles to obtain a closed domain. The walls of marginal stability for the Z -orbifold was studied in [9, 14] where it was found that the intercepts of thewalls on the Re( λ )-axis are given by( , , , , , . . . , − n +1 − n , − n − n − , . . . , , . . . , n +12 n +1 , n +14 n . . . , , , , , ) . (5.2)13 =1 N=2 N=3
Figure 1: Fundamental domains/Weyl chambers for N = 1 , , N = 4 is given in Figure 2. Weyl chamber for β β −1 α −1 β α N = 4 α Figure 2: Fundamental domains/Weyl chambers for N = 4 Z -orbifold For the N = 5 case, following Sen’s method [14] to find the location of thewalls, one again obtains an infinite number of walls in the fundamental domain.However, the fundamental domain is divided into three regions, denoted B R , B L ,and B C , where for the cases N = 2 , , B R and B L which were symmetric about the point , and for N = 1, only onedomain. The appearance of a new region, B C , happens only for N ≥
5. For the N = 2 , , was a limit point for the set of walls starting at 0and 1. For N = 5, one has two limit points given by (1 ± q ) correspondingto the three chambers that the fundamental domain is divided into.14he intercepts of the walls, on the Re( λ )-axis in the three domains are givenby B L = (cid:8) A n B n , B n · A n +1 (cid:9) , n ∈ Z + , (5.3)starting at the intercept , B R = (cid:8) A n +1 B n , B n +1 · A n +1 (cid:9) , n ∈ Z + , (5.4)starting at the intercept , where, A n = ( √ )Λ n − ( √ )Λ n , B n = ( √ )Λ n + ( −√ )Λ n , (5.5)while the intercepts of the walls on the Re( λ )-axis in the domain B C are given by B C = (cid:8) ˜ A n ˜ B n , ˜ B n +1 · ˜ A n (cid:9) , (5.6)where, ˜ A n = ( √ − √ )Λ n + ( √ √ )Λ n , ˜ B n = ( −√ )Λ n + ( √ )Λ n , (5.7)and Λ , Λ in all the equations above are given byΛ = √ , Λ = −√ . (5.8)The limit points of these ratios of sequences are as expected by Sen in his analysisof the walls of marginal stability [14].Since there are an infinite number of walls, having a systematic way of labelinghelps in studying them. The labeling is also very important to write down theCartan matrix of the associated BKM Lie superalgebra. It is clear from thestructure of the walls that we will need two separate sequences to label the walls.Following [9] we label the walls by two sequences, α n and β n , indexed by aninteger, n . The sequences, α n and β n , denote the semi-circles in B L ∪ B R withintercepts given by α n = (cid:18) B n − A n · A n B n (cid:19) and β n = (cid:18) A n +1 B n B n · A n (cid:19) . (5.9)It will turn out that these correspond to the real simple roots of the BKM Liesuperalgebra e G . Note that α and β represent the two straight lines at Re( λ ) =0 , In fact, the two chambers B L and B R can be considered as one chamber by allowing n ∈ Z instead of Z + , and from here on we think of B L ∪ B R as one connected chamber. N = 4 case, one can interpret the fundamental domain as aregular polygon with an infinite number of edges and with an infinite-dimensionaldihedral group, D ∞ , as its symmetry group. In this case, there are two polygonscorresponding to the two sets of chambers B L ∪ B R and B C . Each of the polygonshas two infinite-dimensional dihedral symmetry groups denoted D (1) ∞ and D (2) ∞ .The group D (1) ∞ is generated by two generators: a reflection y and a shift γ givenby: y : α n → α − n , β n → β − n − and γ : α n → α n +1 , β n → β n − , (5.10)satisfying the relations y = 1 and y · γ · y = γ − . The transformation γ ∈ e Γ ( N ) permutes the set of walls in B L ∪ B R , and hence the real simple roots of theBKM Lie superalgebra, just like for the N < B C , is the same as given in eq. (5.10) for the walls inthe chamber B L ∪ B R . The transformation γ is realised as a Γ (5) matrix as γ = (cid:18) − N − N (cid:19) = (cid:18) − − (cid:19) [8].The transformation, δ , that exchanges the walls α n and β n generates a second Z defined as follows: δ = (cid:18) − (cid:19) : α n ←→ β n . (5.11)The transformations ( γ, δ ) together generate the other dihedral group denoted D (2) ∞ .Similarly, choosing a labeling for the roots occurring in the chamber B C , thewalls can be written as two infinite sequences, ˜ α n and ˜ β n , labelled by an integer n as ˜ α n = (cid:18) ˜ B − n +1 ˜ A − n +1 · ˜ A − n ˜ B − n +1 (cid:19) and ˜ β n = (cid:18) ˜ A n +1 ˜ B n +2 ˜ B n +1 · ˜ A n +1 (cid:19) , (5.12)where ˜ A n and ˜ B n are as given in eq. (5.7). The chamber B C also has a dihedralsymmetry group given by D (1) ∞ as defined in (5.10). The second dihedral group, D (2) ∞ , generated by δ and γ is also a symmetry of the chamber B C , with its actionon the roots as given in (5.11) with α n , β n replaced by e α n and e β n . Thus, therewould appear to be two seperate chambers given by B L ∪ B R and B C , each aninfinite polygon with the same set of dihedral symmetry groups.This can be understood as follows. Although the intercepts on the real λ -axis in the two sets B L ∪ B R and B C look completely different, following thecorrespondence on the BKM Lie superalgebra side one can see that the two The extended S-duality group, e Γ ( N ), is defined by including a Z parity operation to theS-duality group Γ ( N ). For N = 1, this is the group P GL (2 , Z ) [11]. The generator y is notrealized as an element of a level 4 subgroup of P GL (2 , Z ) and thus is not an element of theextended S-duality group. This is similar to what happens for N = 2 , e G , the sets ofroots arising from the two polygons are in exact one-to-one correspondence witheach other. With the labeling introduced earlier, for each root α n in the polygon B L ∪ B R , there exists a corresponding root e α n in the polygon B C and similarlywith the roots β n and e β n . Further, computing the element in the Cartan matrixcorresponding to the inner product between any two roots in each of the twochambers, one finds that( x n , x m ) = ( e x n , e x m ) , x i ∈ B L ∪ B R , e x i ∈ B C . (5.13)That is, the inner product between any two roots in the polygon B L ∪ B R is thesame as that for the corresponding two roots in the polygon B C . Thus, one seesthat the two polygons are identical to each other and give rise to two identicalcopies of the roots in the BKM Lie superalgebra with the same inner products.Hence it is intuitively understandable that they are same as infinite polygonswith the same sets of dihedral symmetry groups.One would also need to find transformations that take the walls from B L ∪B R chamber to the chamber B C . This is given by the Γ (5) matrix, σ , withdeterminant +1 as follows [14] σ = (cid:18) − − (cid:19) : α n ↔ ˜ α − n , β n ↔ ˜ β − n − , where α n , β n ∈ B L ∪B R , ˜ α n , ˜ β n ∈ B C . (5.14)Adding the generator σ one gets a group generated by δ, γ and σ given by therelations σ = 1; σ · γ · σ − = γ − ; δ · σ · δ − · σ − = γ . (5.15)One sees that the relations between the two polygons is not such that onecan consider the union of the two polygons as one single polygon with a singledihedral symmetry group acting on it. This is also reflected in the Cartan matrixof the BKM Lie superalgebra, where the inner product between the two chambersoccurs with a gradation. We will shortly come to discuss the algebra e G , wherewill we see the above mentioned facts. e G and walls of its Weylchambers Here we put together everything about the algebra e G in the context of studyingthe walls of its Weyl chambers. We will use the correspondence with the wallsof marginal stability, and the labeling introduced in the previous section to writedown the Cartan matrix for the algebra and also study its properties.Cheng and Verlinde [11] had observed that the walls of marginal stability forthe -BPS states in the N = 1 model had a correspondence with the walls ofthe Weyl chamber of the BKM Lie superalgebra G . Subsequently, Cheng and17abholkar and Govindarajan and Krishna have shown that for the N = 1 , , -BPS states correspond to the Weyl chambers of a family of rank-three BKM Liesuperalgebras. Each wall (edge) of the fundamental domain is identified with areal simple root of the BKM Lie superalgebra. Recall that each wall correspondsto a pair of rational numbers ( ba , dc ) which are the intercepts of the wall on theRe( λ )-axis. This is related to a real simple root α of the BKM Lie superalgebraas: ( ba , dc ) ↔ (cid:18) a bc d (cid:19) ↔ α = (cid:18) bd ad + bcad + bc ac (cid:19) , (5.16)with ac ∈ N Z and ad, bc, bd ∈ Z . The norm of the root is [11] − α ) = 2( ad − bc ) = 2 . The Cartan matrix, A ( N ) , is generated by the matrix of inner products amongall real simple roots. The Cartan matrices for the N = 1 , , e G , this correspondence goes through even though thenumber of real simple roots are infinite in number and there are two chambers inthe fundamental domain for N = 5. To construct the Cartan matrix of e G , let usorder the real simple roots in the chamber B L ∪ B R into an infinite dimensionalvector X = ( . . . , x − , x − , x , x , x , x , . . . ) = ( . . . , α , β − , α , β , α − , β , . . . )and similarly the real simple roots in the chamber B C into an infinite dimensionalvector as e X = ( . . . , ˜ x − , ˜ x − , ˜ x , ˜ x , ˜ x , ˜ x , . . . ) = ( . . . , ˜ α , ˜ β − , ˜ α , ˜ β , ˜ α − , ˜ β , . . . ) . This is precisely the labeling order we introduced on the walls in the previoussection. Equivalently, let x m = (cid:26) α − m/ or ˜ α − m/ , m ∈ Z β ( m − / or ˜ β ( m − / , m ∈ Z + 1 . (5.17)The Cartan matrix is given by the matrix of inner products a mn ≡ h x n , x m i and is given by the infinite dimensional matrix: A (5) = ( a nm ) where a nm = − (cid:16) Λ − n Λ − m + Λ − m Λ − n (cid:17) + ( − d ( x m )+ d ( x n ) , (5.18)where m, n ∈ Z , Λ and Λ are the roots of the equation r − · r + 1 = 0 and d ( x m ) = 0 if x m ∈ B L ∪ B R and d ( x m ) = 1 if x m ∈ B C is the grading on the twosets of roots of the algebra. 18s mentioned before, one can see that the inner products between any tworoots in X is equal to the inner product between the corresponding two roots in e X . Thus, X and e X are two copies of the same set of roots with the same innerproduct matrices. However, a peculiar thing occurs in taking the inner productof the real simple roots with the Weyl vector ρ . The inner product of the rootswith the Weyl vector ρ satisfies h ρ, x m i = − , ∀ x m ∈ B L ∪ B R and h ρ, x m i = 1 , ∀ x m ∈ B C . (5.19)As seen from the above equation, the real simple roots in the chamber B C seemto have the wrong sign when one takes their inner product with the Weyl vectorwhile the real simple roots in the chamber B L ∪ B R have the correct sign. At thispoint, no explanation for the above fact is known by the author. D (2) ∞ -invariance of e ∆ ( Z )It remains to be proven that e ∆ ( Z ) gives rise to the denominator identity for theBKM Lie superalgebra e G . One needs to show that e ∆ ( Z ) contains all the realsimple roots that one expects from the study of the walls of marginal stability.The D (2) ∞ -generators γ and δ act on the roots x m written as a 2 × γ : x m −→ (cid:18) − − (cid:19) · x m · (cid:18) − − (cid:19) T , (5.20) δ : x m −→ (cid:18) − (cid:19) · x m · (cid:18) − (cid:19) T . (5.21)The matrix γ is denoted by γ (5) in [8]. Under the level 5 subgroup G (5) ∈ Sp (2 , Z ), the modular form e Φ ( Z ) transforms as [16] e Φ ( M · Z ) = { det( CZ + D ) } e Φ ( Z ) , (5.22)where M = (cid:18) A BC D (cid:19) ∈ Sp(2 , Z ) , M · Z = ( A Z + B )( C Z + D ) − , and C = 0 mod 5 . Consider the subgroup of G (5) given by B = C = 0 and A T = D − . Under thissubgroup, eq. (5.22) becomes e Φ ( D T · Z · D ) = (det D ) e Φ ( Z ) . (5.23)Choosing D = γ = (cid:18) − − (cid:19) , one sees that e Φ ( Z ) is invariant. Similarly, when D = δ = (cid:18) − (cid:19) , or D = σ = (cid:18) − − (cid:19) , the modular form e Φ ( Z ) is invariant.19hus, we see that the modular form e Φ ( Z ) is invariant under the action of γ , δ ,and σ , which means that under the action of γ , δ and σ , e ∆ ( Z ) → ± e ∆ ( Z ) . (5.24)One can show that the sign must be +1 by observing that any pair of terms inthe Fourier expansion of e ∆ ( Z ) related by the action of γ (resp. δ, σ ) appearwith the same Fourier coefficient. For instance, the terms associated with thetwo simple roots α and β related by the action of δ appear with coefficient+1. Similarly, the terms associated with the real simple roots β and β − relatedby a γ -translation also appear with coefficient +1. Thus, we see that e ∆ ( Z ) isinvariant under the full dihedral group generated by δ, γ and σ . This providesan all-orders proof that the infinite real simple roots given by the vector X allappear in the Fourier expansion of e ∆ ( Z ).The q → s symmetry of the modular form is equivalent to the symmetrygenerated by the dihedral generator, y , as defined in Eq. (5.10). Here we study the the general structure of the walls of marginal stability forthe CHL models. In [8] Cheng and Dabholkar gave an arithmetic argumentunderlying the walls of marginal stability for the N = 1 , b + da + c , of the successive pair of rationals { ba , dc } starting from ± , . Eachof the successive rows gives the intercepts of the walls of marginal stability forthe N = 1 , N ), doesnot exist for N >
3. Also, from Sen’s analysis, an infinite number of walls areexpected for
N >
3, but the Stern-Brocot tree gives only a finite number ofrationals at any level. Thus, it would appear that either an arithmetic structuredoes not exist for
N >
3, or if one exists it is given by a different kind of series(in place of the Stern-Brocot series) for N ≥ N > N = 4 , models. All these modelshave an infinte number of walls in the fundamental domain. The walls of marginalstability for these models are generated by a pair of linear recurrance relations.We will see that there are many universal properties which are given only asa function of N but the form of these functions across different N remains the The following analysis also holds for the N = 7 case, but only for the chamber B L ∪ B R .The intercepts in the chamber B C is a bit more complicated and the analysis does not give theintercepts in this chamber. λ )-axis are given by two sequences of the form A n = ( N − A n − − A n − , (6.1)and B n = ( N − B n − − B n − , (6.2)where N = 4 , N . One can compute the above sequences by finding solutions to therecurrence equations and using two initial values to fix the sequence. They canalso be written down from a generating function as is explained later . The characteristic equation (equation satisfied by the solution to the ansatz A n = ℓ n )is p ( ℓ ) ≡ ℓ − ( N − · ℓ = − , (6.3)which can also be seen to be of the same form for all N . The solutions to thelinear recurrance depend on the nature of the roots of the characteristic equation.For identical roots, as one has in the case of N = 4, one has the general solution A n = C Λ n + Dn Λ n , (6.4)while for distinct roots, as is the case for N = 5 and 6, one has A n = C Λ n + D Λ n , (6.5)where C and D are fixed from two initial conditions.The walls in the chambers B L ∪ B R and B C are given by intercepts whichcome from linear recurrance sequences like the above. Let us study each caseseparately, before putting together a general picture. The walls of marginal stability for N = 4 The characteristic equation and the (identical) roots for the walls of marginalstability for the N = 4 model are p ( ℓ ) ≡ ℓ − · ℓ + 1 = 0 , with roots Λ = 1 . Using initial conditions to determine the sequence, we have for the series A n and B n A n = n ; B n = 2 n + 1 . (6.6) Sen generates all the walls by the action of the matrix g ≡ (cid:18) − N − N (cid:19) , which translatesthe walls by an even number of steps. Repeated action of g generates all the infinite numberof walls. λ )-axis in thetwo chambers B L ∪ B R are given by B L ∪ B R = (cid:8) A n B n , B n · A n +1 (cid:9) , n ∈ Z , (6.7)One can check that the limit points for both the sequences in (6.7) arelim n →∞ A n B n = lim n →∞ B n · A n +1 = (6.8)in keeping with the fact that the point is the limit point for walls, starting at0 and 1, in the fundamental chamber.As before, it is again convenient to divide the intercepts into two sequences α n and β n and use a labeling which will help in studying them and also writingthe Cartan matrix of the BKM Lie superalgebra. In the notation of [9] the rootsof the BKM Lie superalgebra are given from the intercepts of the correspondingwalls by α n ≡ n B n − · A n , A n B n o ↔ (cid:18) · A n B n − · A n + B n B n − · A n + B n B n − · A n B n (cid:19) β n ≡ n A n +1 B n , B n · A n o ↔ (cid:18) · A n +1 B n · A n A n +1 + B n · A n A n +1 + B n · A n B n (cid:19) (6.9)Using the correspondence between the walls of marginal stability and theroots, one can construct the Cartan matrix for the BKM Lie superalgebra, e G ,by taking the inner product between the roots using their form given in (6.9)and using (6.6) . The Cartan matrix is given by (4.5). This is indeed the sameCartan matrix as obtained by constructing the BKM Lie superalgebra directlyfrom the modular forms via the Weyl-Kac-Borcherds denominator formula. Onecould use this to write down the Cartan matrices of the BKM Lie superalgebrasof the models where direct computation of the modular forms is difficult.The Γ (4) element γ (4) = (cid:18) − − (cid:19) acts as a translation on the interceptstaking α n α n +1 and β n β n − , while the element δ = (cid:18) − (cid:19) acts as areflection on the intercepts exchanging the walls δ : α n ↔ β n . The two elements( γ (4) , δ ) form a dihedral group, which is the symmetry group of the polygonformed by the walls in the fundamental chamber. Walls of marginal stability for N = 6 The characteristic equation for the N = 6 case is given by p ( ℓ ) ≡ ℓ − · ℓ + 1 = 0 . B L ∪ B R chambers are given by B L ∪ B R = (cid:8) A n B n , B n · A n +1 (cid:9) , n ∈ Z , (6.10)where the two sequences A n and B n are given by A n = 12 √ (cid:16) Λ n − Λ n (cid:17) , B n = 1 + √
32 Λ n + 1 − √
32 Λ n . (6.11)The intercepts of the walls in the domain B C are given by B C = (cid:8) e A n · e B n , e B n +1 · e A n (cid:9) , (6.12)where the two sequences e A n , e B n are given as e A n = √ − √ n + √ √ n , e B n = 2 − √
32 Λ n + 2 + √
32 Λ n , (6.13)where in all the equations above Λ = 2 + √ = 2 − √
3, which arenothing but the roots of the characteristic equation (6). One can check that thelimit points for the ratios of the sequences in (6.10) and (6.12) are (1 ± q )which is consistent with Sen’s analysis [14].As before, choosing a labeling in the chambers B L ∪ B R we form two sequencesgiven as α n = (cid:18) B n − A n · A n B n (cid:19) and β n = (cid:18) A n +1 B n B n · A n (cid:19) (6.14)and in the chamber B C as e α n = e B − n +1 e A − n e A − n +1 e B − n +1 ! and e β n = e A n +1 e B n +1 e B n +2 e A n +2 ! . (6.15)Using the correspondence between the walls of marginal stability and theroots of the BKM Lie superalgebra, one can construct the Cartan matrix for the N = 6 model just as one did for the N = 4 and 5 models. The Cartan matrix forthe BKM Lie superalgebra, e G , corresponding to the N = 6 model is given by A (6) = ( a nm ) where a nm = − (cid:16) Λ − n Λ − m + Λ − m Λ − n (cid:17) + ( − d ( x m )+ d ( x n ) , (6.16)with m, n ∈ Z , Λ , Λ are the roots of the equation r − · r +1 = 0, and d ( x m ) = 0if x m ∈ B L ∪ B R and d ( x m ) = 1 if x m ∈ B C is the grading on the two sets of rootsof the algebra. 23he Γ (6) element γ (6) = (cid:18) − − (cid:19) acts as a translation on the interceptstaking α n α n +1 and β n β n − , while the element δ = (cid:18) − (cid:19) acts as areflection on the intercepts exchanging the walls δ : α n ↔ β n . The two elements( γ (6) , δ ) form a dihedral group for the polygon generated by the roots. General structure of the walls
Now we are ready to put together a general structure for the walls of marginalstability in the case of general N . As one can see from the above analysis, thegeneral structure of the walls has a pattern which is same across N . For all N ,the walls are generated by two sequences, A n , B n , for both the chambers B L ∪ B R and B C . The sequences are generated by linear recurrance relations given by therespective characteristic equations (6.3). The form of this equation is the samefor all the N . The sequences A n , B n in the chambers B L ∪ B R are generated bythe generating functions φ ( x ) = xp ( x ) , and χ ( x ) = 1 + xp ( x ) , (6.17)where p ( x ) is the characteristic equation given in (6.3). The above equationsdepend on N in the denominator p ( x ), but their general form (and even thegeneral form of the characteristic equation (6.3)) remains the same for N = 4 , , B L ∪ B R are given by the general form B L ∪ B R = (cid:8) A n B n , B n N · A n +1 (cid:9) , n ∈ Z , (6.18)for all N . The intercepts in the chamber B C are given by the general form B C = (cid:8) e A n P · e B n , e B n +1 Q · e A n (cid:9) , (6.19)where P, Q are the prime factors of N (5 = 5 · , · α n and β n , denoting the semi-circles in B L and B R withintercepts given by α n = (cid:18) B n − A n N · A n B n (cid:19) and β n = (cid:18) A n +1 B n B n N · A n (cid:19) . (6.20)is the same for all N . Similarly, the sequences, e α n and e β n , denoting the semi-circles in the chamber B C given by e α n = e B − n +1 Q · e A − n e A − n +1 P · e B − n +1 ! and e β n = e A n +1 P · e B n +1 e B n +2 Q · e A n +2 ! (6.21)24s the same for the N = 5 , N . Also, the form of the transformations which translate the intercepts, namelythe Γ ( N ) element γ ( N ) = (cid:18) − N − N (cid:19) acting as a transformation on the inter-cepts taking α n α n +1 and β n β n − (and similarly e α n e α n +1 and e β n e β n − )is the same across N = 4 , ,
6. The transformation that exchanges the intercepts, δ : α n ↔ β n in the chambers B L ∪ B R and δ : e α n ↔ e β n in B C , remains the sametoo.Thus, we see that the general form of the walls of marginal stability for themodels N ≥ N . As mentionedbefore, this is similar to the arithmetic structure found by Cheng and Dabholkar,and extends it to N ≥ Walls of marginal stability for N = 7 Before we conclude this section, let us briefly also analyse the situation for the N = 7 case. The intercepts of the walls in the chamber B L ∪ B R for N = 7 isgiven, as it is for N = 4 , α n and β n , in the B L ∪ B R chamber are given by B L ∪ B R = (cid:8) A n B n , B n · A n +1 (cid:9) , n ∈ Z , (6.22)where the two sequences A n and B n are given by A n = 1 √ (cid:16) Λ n − Λ n (cid:17) , B n = √
21 + 72 √
21 Λ n + √ − √
21 Λ n . (6.23)The limit points for the intercepts in the chamber, B L ∪ B R are given by (1 ± q − ). As before, the element γ ≡ (cid:18) − − (cid:19) , acts as a transformationon the intercepts taking α n α n +1 and β n β n − (and similarly e α n e α n +1 and e β n e β n − ).However, the intercepts of the walls in the central chamber are more compli-cated and are not given by the above analysis. In the previous sections we constructed the BKM Lie superalgebra, e G , and alsostudied the walls of the Weyl chamber of the algebra in relation to the wallsof marginal stability of the -BPS states. While this algebra follows the general25attern of the algebras for N <
5, some peculiar aspects have not been understoodcompletely and this discussion is aimed at underlining them. First, is the factthat the real simple roots from the chamber B C do not have the right sign in theirinner product with respect to the Weyl vector of the algebra. The roots in boththe chambers are exact one-to-one copies of each other, including, having thesame inner product between two corresponding pair of roots in either chamberand only differ in their inner product with ρ . This leads to the possibility thateach of the set of roots forms an algebra by itself. The same also occurs for theroots of e G , which has two chambers in the fundamental domain.Secondly, in Borcherd’s construction of holomorphic infinite products, theBKM Lie superalgebras were related to the modular forms themselves, where asin all the above discussed examples of the CHL strings it is the square root ofthe modular forms which lead to a BKM Lie superalgebra and not the modularforms themselves. The occurrence of the square root is not understood.Finally, the walls of marginal stability of N = 7 seems to differ from the N = 5 and 6 which also have two chambers in the fundamental domain. Whilethe roots in the chamber B L ∪ B R seem to be along the same lines as the analysisfor N = 5 and 6, the intercepts in the central chamber are not given by the samemethod. In this paper we have studied the Z -orbifolded CHL string in some detail, start-ing from the modular forms generating the degeneracy of the half and quarterBPS states in the theory. After explicitly constructing the genus-two Siegel mod-ular form generating the -BPS degeneracies, we see that the modular formsgenerated from the additive and multiplicative lifts do not match for the N = 5case. Taking the product form to be the proper expansion, we constructed theBKM Lie superalgebra corresponding to the Siegel modular form from the Weyl-Kac-Borcherds denominator identity. This BKM Lie superalgebra, like the N = 4case, has an infinite number of real simple roots. We also studied the walls ofmarginal stability for the N = 4 , -BPS states continues to hold for the N = 5 case also. Based on this, we alsopropose a Cartan matrix for the BKM Lie superalgebra e G for the Z -orbifoldedCHL model. 26 cknowledgements I would like to thank Prof. Suresh Govindarajan for numerous helpful discussions.I also thank Prof. Nils Scheithauer and Prof. Dileep Jatkar for clarifications anddiscussions. I am very grateful to Prof. Ashoke Sen, and Prof. Urmie Ray fortheir kind support. Most of this work was done during my stay at HRI, and Iwould like to thank the staff and members of HRI for all the support extendedduring my stay.
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