IITM/PH/TH/2012/1 A non-commuting twist in the partition function Suresh Govindarajan1 Department of Physics, 2 Indian Institute of Technology Madras, 1 Chennai 600036, India. 0 2 n and a J Karthik Inbasekar2 8 Institute of Mathematical Sciences, ] CIT Campus h t Taramani, Chennai 600113, India. - p e h [ Abstract 1 v We compute a twisted index for an orbifold theory when the twist 8 2 generating group does not commute with the orbifold group. The 6 twisted index requires the theory to be defined on moduli spaces that 1 are compatible with the twist. This is carried for CHL models at spe- . 1 cial pointsin themodulispacewherethey admitdihedralsymmetries. 0 The commutator subgroup of the dihedral groups are cyclic groups 2 1 that are used to construct the CHL orbifolds. The residual reflection : v symmetry is chosen to act as a “twist” on the partition function. The i reflection symmetries do not commute with the orbifolding group and X hence we refer to this as a non-commuting twist. We count the de- r a generacy of half-BPS states using the twisted partition function and find that the contribution comes mainly from the untwisted sector. We show that the generating function for these twisted BPS states are related to the Mathieu group M . 24 [email protected] [email protected] Contents 1 Introduction 1 2 Non-abelian orbifolds 3 3 Computing the Twisted Partition Function 7 4 Towards Mathieu representations 16 5 Discussion 17 1 Introduction The microscopic counting of black hole entropy in four-dimensional string theories with N = 4 supersymmetry has turned out to have a surprisingly rich structure [1,2]. This has provided connections to modular forms, Lie algebras [3,4] as well as sporadic groups [5,6]. Due to the large amount of supersymmetry, these theories work as “laboratories” forus to test ideas that presumably should continue to work in situations with fewer super symme- tries. This paper seeks to add another variant to the microscopic counting – we ‘count’ twisted half-BPS states in theories with N = 4 supersymmetry where the twist does not commute with the orbifolding group. This is as a prelude to considering situations where the twist does not commute with supersymmetry. We consider four dimensional CHL Z -orbifolds with N = 4 supersym- n metry [7,8] . These models are asymmetric orbifolds [9,10] constructed by starting with a heterotic string compactified on a T4×S1×S˜1 and then quo- tienting the theory by a Z transformation which involves a 1/n shift along n the S˜1. The Z symmetry has a non-trivial action on the internal conformal n field theory coordinates describing the heterotic compactification on T4. A large class of such models where constructed in [11,12] and were shown to be dual to a type II description compactified on K3×S1×S˜1 via string-string duality [13,14]. By construction CHL models possess maximal supersymmetry and fewer massless vector multiplets at generic points in the moduli space. The re- quirement of maximal supersymmetry restricts one to consider symplectic automorphisms on K3. Symplectic automorphisms leave the holomorphic (2,0) formsinvariant andhence preserve supersymmetry. The actionof these symmetries have fixed points onthe K3 surface andis accompanied by trans- lations on the circle to avoid quotient singularities. So the allowed groups 1 must faithfully represent translations in R2 which implies that the quoti- enting group has to be abelian [15]. The possible abelian groups that act symplectically on K3 where classified and the action of the group on the K3 cohomology was calculated [16]. Once the action on the cohomology is determined one uses string-string duality to map the action to the Heterotic side. The map is allowed provided the supergravity side is free from fixed points, i.e the action on K3 must be accompanied by shifts on the torus. The work of Mukai [17] opened up the possibility that non-abelian groups can act as symplectic automorphisms on the K3 surface. A couple of years ago Garbagnati [18] constructed elliptic K3 surfaces that admit dihedral group as symplectic automorphisms. These automorphisms are constructed by combining automorphisms which act both on the base and the fiber such that the resulting action is symplectic. In particular [18] determined the ranks of the invariant sublattice and the orthogonal complement and identi- fied the orthogonal complement to the invariant sublattice with the lattices in [19]. However, for compactifications down to four dimensions one cannot quotient by a non-abelian group since these groups do not represent transla- tions faithfully. However, one can consider the theory to be on special points in the moduli space that admit non-abelian symmetries and quotient by the commutator subgroup, which is abelian. In this paper, we consider the CHL Z -orbifold models (3 ≤ n ≤ 6 at n special points in the moduli space where they admit dihedral D = Z ⋊Z n n 2 symmetry3.The Z subgroup is the commutator subgroup of the D group n n and may be quotiented. The special points in moduli space are specified by the elliptic K3 surfaces that admit D ,3 ≤ n ≤ 6 symmetries constructed n in [18]. Since the action of Z group is known on the K3 side, we map it n to the heterotic string using the string-string duality. We then construct the Z Z CHL orbifold in the heterotic picture and let the additional symme- n 2 try act as a twist in the partition function of the orbifolded theory. These twist symmetries are identical to the ones considered in [11,12] but with- out shifts. For N = 4 supersymmetry to be preserved these twists must commute with all the unbroken supersymmetries of the theory. Such twists have been considered in the g ∈ Z twisted partition function [20] for un- n orbifolded theories, which counts the index/degeneracy4 of elementary string states when the theory is restricted to special points in moduli space. The g-twisted helicity index is defined as 1 Bg = Tr[g(−1)2ℓ(2ℓ)2m] , (1) 2m 2m! 3In our notation, D is the dihedral group of order 2n, see section 2. n 4Both are identical for the cases considered in this paper. 2 where g generates a symmetry of finite order, ℓ is the third component of angular momentum of a state in the rest frame, and the trace is taken over all states carrying a given set of charges. States which break less than or equal to 4m g-invariant supersymmetries give non-vanishing contributions to Bg [20]. For the case of 1/2 BPS states that we consider in this paper, 2m the relevant index is Bg. 4 For our case, the choice of the moduli space that has dihedral symmetry is compatible with the g ∈ Z twist. The other requirement that the physical 2 chargeshave tobeg invariant ismet byrequiring thechargesQtotakevalues from lattices invariant under Dihedral symmetry [18,19]. This choice is also compatible with the orbifold action, since these lattices possess invariance Z Z under both and actions. Thus one meets the requirements for the 2 n twist and orbifold action to be well defined. We count the degeneracy of electrically charged 1/2 BPS elementary string states for a fixed charge Q in these theories following the method Z Z described in [21]. The twisted partition function in the orbifold the- 2 n ories receives contribution only from the orbifold untwisted sector for odd n and additionally from the orbifold sector twisted by the element hn/2 for even n. From the point of view of the dihedral group, for even n, the element hn/2 is a nontrivial center of the group and commutes with every element. We derive a generating function for these degeneracies and find that it has the expected asymptotic limit. The generating function for these twisted 1/2 BPS also formsaMathieu representation asit didfor theabeliancases [5,22]. The paper is organised as follows. Following the introductory section, in section 2, we give a pedagogical introduction to non-abelian orbifolds and define the twisted partition function to indicate the contributing orbifold Z twisted sectors. It is followed by the construction of CHL orbifolds in the n heterotic picture and the derivation of the half-BPS degeneracies of g ∈ Z 2 twisted BPS states in section 3. In section 4, we provide a connection with the sporadic Mathieu groupM . We conclude with a summary of our results 24 in section 5. 2 Non-abelian orbifolds In this section we describe the standard CFT approach for constructing non- abelian orbifold theories. For a general description of orbifolds in string theory see [23–29]. For some phenomenological model building approaches based on non-abelian orbifold string theories see [30,31]. Orbifold CFT’s are generally constructed by considering a theory T which admits a finite discrete symmetry group G consistent with its allowed interactions and then 3 forming a quotient theory T/G. When the orbifolding group is abelian the entire groupacts as thesymmetry groupwhereas in non-abelianorbifolds the symmetry group is G/[G,G], where [G,G] is the commutator subgroup of G. As an example, for Dihedral groups D of order 2n, the quotienting group n is the cyclic group Z of order n. Before proceeding further, it is useful to n define some notations. Let us denote the worldsheet coordinate as X(τ,σ), with τ and σ being the “space” and “time” directions of the torus. By g ≡ TrHh g qH , (2) h (cid:0) (cid:1) we mean the following closed string boundary conditions are applied simul- taneously. X(τ +2π,σ) = g ·X(τ,σ) X(τ,σ +2π) = h·X(τ,σ). (3) Tr denotes the trace taken in a Hilbert space sector H corresponding to Hh h a spatial twist element h. H denotes the Hamiltonian of the theory. We also denote |G| as the order of the group G. The module g is not well defined h for gh 6= hg as we will explain below. For the CFT to be well defined, the states of the theory must be invariant under the action of the group. Therefore one projects onto G-invariant states by defining a projection operator 1 P = g. (4) |G| X g∈G The projection is implemented by including g in the trace and then by sum- ming over all time twists. The inclusion of g in the trace amounts to twisting the fields by g along the time direction, i.e g · X(τ,σ) = X(τ + 2π,σ) The contribution to the partition function from the spatially untwisted sector of the orbifold CFT is then given by 1 Z = . (5) He |G| X g∈G Modular invariance under SL(2,Z) transformations requires the addition of spatially twisted sectors e , i.e sectors where fields satisfy h · X(τ,σ) = h X(τ,σ + 2π). Each of these spatially h-twisted sectors corresponds to a distinct Hilbert space H and one must project onto the group invariant h 4 states within every Hilbert space. This would mean that the fields would have simultaneous boundary conditions due to the action of g and h. X(τ,σ +2π) = hX(τ,σ) X(τ +2π,σ) = gX(τ,σ) gX(τ,σ+2π) = ghX(τ,σ) hX(τ +2π,σ) = gX(τ,σ) gX(τ,σ+2π) = ghg−1gX(τ,σ) hX(τ +2π,σ) = hgh−1hX(τ,σ) X(τ +2π,σ +2π) = ghX(τ,σ) X(τ +2π,σ+2π) = hgX(τ,σ). (6) From the above equations, one can see that the action of g takes the string in the Hilbert space Hh to the Hilbert space Hghg−1. When g and h do not commute these Hilbert spaces are different. The elements h and h′ = ghg−1 are in the same conjugacy class and hence the projection operator would mix Hilbert spaces corresponding to elements that belong to a given conjugacy class. Thus, one is unable to do a full group invariant projection within the Hilbert spaces in the spatially twisted sectors. In the operator language, the presenceofatimetwistg thatdoesn’tcommutewiththespatialtwistelement h would not allow simultaneous diagonalization of their respective matrix representations. Nevertheless one can choose a basis for g and it acts on the oscillators and eventually on the vacuum. As explained above, the vacuum is not left invariant and the vacuum in Hh taken to the vacuum in Hghg−1. So the trace would be over an off-diagonal matrix with diagonal entries zero and hence would vanish. Or equivalently, the path integral vanishes due to the inconsistent boundary condition (6). Since the spatially twisted sectors are not invariant under the full group. For a given spatially twisted sector H one identifies the little group N consisting of elements that commute h h with h and project onto states invariant under the little group 1 ZHh = |N | g . (7) h X h g∈Nh The various spatially twisted sectors in a given conjugacy class are treated in equal footing and hence the spatially twisted sectors are labelled by their conjugacy class C instead of the group element itself. This follows from i “naive” modular invariance 5. 1 1 1 ZCi = |C | ZHh = |C | (cid:18)|N | g (cid:19) (8) i X i X h X h h∈Ci h∈Ci g∈Nh The group invariant states in the theory are formed by taking a linear com- bination of states from a sector twisted by a group element g and all other 5modularinvarianceunderPSL(2,Z)transformations,Itisnaivebecausethemodular transformation τ → τ + n can introduce anomalous phases that could spoil modular invariance 5 sectors conjugate to it. The full partition function is then given by summing over all the conjugacy classes Z = Z (9) T/G Ci X Ci Since for any group G, the order of the little group N is the same for every h element h ∈ C 6 , we have |G| = |N ||C | for every conjugacy class C . Thus i h i i the full CFT partition function for a general non-abelian orbifold theory can also be written as 1 ZT/G ≡ g . (10) |G| X h g,h∈G gh=hg We summarize some properties of Dihedral groups which will be useful later. The dihedral group denoted as D is of order 2n. One has the presen- n tation, D ∼= hh,g|hn = e,g2 = e,ghg = h−1i. (11) n where h, g generate of Z and Z symmetries respectively. The group ele- n 2 ments are given by D = {e,h,h2,...,hn−1,g,gh,gh2,...,ghn−1}. The Z n 2 generator acts as an inversion on the axes of reflection, all the elements of the form ghj are of order 2, i.e (ghj)2 = 1. The properties of dihedral group depend on whether n is even or odd. For odd n, D has ⌊n/2⌋ + 2 conju- n gacy classes are given by (the little groups N for each element c in C are ci i i indicated beside) C = {e}; N = D 0 e n C = {g,gh,gh2,...,ghn−1}; N = {e,c } 1 c1 1 C = {h,hn−1},{h2,hn−2},...,{h⌊n/2⌋,h⌊n/2⌋+1}; N = Z (12) k ck n For even n ,D has n/2+3 conjugacy classes which are given by n C = {e}; N = D 0 e n C = {hn/2}; N = D 1 c1 n C = {g,gh2,gh4,...,ghn−2}; N = {e,c ,hn/2,c hn/2} 2 c2 2 2 C = {gh,gh3,gh5,...,ghn−1}; N = {e,c ,hn/2,c hn/2} 3 c3 3 3 C = {h,hn−1},{h2,hn−2},...,{hn/2−1,hn/2+1}; N = Z (13) k ck n 6thisisbecauseeveryelementinaconjugacyclasshasthesameorder,agroupelement h is of order n if hn =1 6 The group invariant projection operator for D has the property n n−1 n−1 1 P = hj + ghj Dn 2n(cid:18) (cid:19) Xj=0 Xj=0 1 n−1 1 1 = gk hj 2 (cid:18)n (cid:19) Xk=0 Xj=0 =PZ .PZ , (14) 2 n which follows from the property of the group elements (11). Even though the element g does not commute with elements h ∈ Z , it commutes with n the projector of Z . Thus if we take g to be a twist, it commutes with the n Z orbifold projection. The partition function is given by n n−1 n−1 1 ZT/Zn = n hj (15) Xj=0 Xk=0 hk Twisting the partition function by g ∈ Z amounts to insertion of g in the 2 trace, Tr g qH (16) Hh (cid:0) (cid:1) By the arguments given in (6) only the following terms contribute to the trace, n−1 n−1 1 ZTg/Zn = n(cid:20) ghj +δn2,[n2] ghj (cid:21) (17) Xj=0 e Xj=0 hn/2 The second sets of terms are there only for even n as can be seen from (13). We refer to this partition function as the “twisted” partition function. Since Z Z the twist generating group does not commute with the orbifold group , 2 n we refer to it as a non-commuting twist. In the following sections, we discuss Z the orbifold action and then evaluate (17) for the CHL -orbifolds. n 3 Computing the Twisted Partition Function We adapt the half-BPS counting method of Sen [21] to compute the twisted partition function. In the notation D = Z ⋊Z = H ⋊G, H is the com- n n 2 mutator subgroup of D which is also the orbifolding group. G represents n an additional symmetry of the theory that appears at special points in the Z moduli spaces. The CHL -orbifoldcan be described as an asymmetric orb- n ifold of the heterotic string compactified on T4×T2. The Z symmetry acts n as a shift on one of the circles in the T2 and as a symmetry transformation 7 on the rest of the CFT involving the T4 coordinates and the 16 left-moving world-sheet scalars associated with the E ×E gauge group. The action of 8 8 a group element h of the orbifold group H is the combination of a shift a h and a rotation R acting on the Narain Lattice Γ(22,6). The action of the h twist g ∈ Z on the K3 side is known [11,16] and has been used to compute 2 twisted indices in [20]. g leaves 14 of the 22 2-cycles of K3 invariant, in other words it exchanges the two E ’s. Furthermore g is not accompanied 8 by shifts. The g ∈ Z insertion in trace requires the physical charges Q to 2 be g-invariant and the orbifolding requires it to be compatible with the Z n orbifold projection. Hence, we let Q takes values in the lattices that are in- variant under D = Z ⋊Z symmetry [19]. For the rest of the computation n n 2 we fix the value of Q. Once this is done the twist g has no further action on the lattice. The set of R ∀ h ∈ H forms a group that describes the rotational part h of H and is represented as R . To preserve N = 4 supersymmetry both H R and g must act trivially on the right movers. In the K3 side this is H enforced by requiring the respective automorphisms to be symplectic. The group H leaves 22 − k of the 22 left moving directions invariant, where k is the number of directions that are not invariant under H. Then, R can H be characterized by k/2 phases φ (h) with j = 1,2,...,k/2. The complex j coordinates Xj represent the planes of rotation and the effect of the rotation R is to multiply the complex oscillators by phases. H The groups also act on the Narain lattice Γ(22,6) and leave a sublattice Λ ⊥ invariant. The orthogonal complement to Λ is denoted as Λ . To preserve ⊥ k N = 4 supersymmetry the right movers take their charge values only from the invariant part of the lattices and the non-invariant part of the lattice is only due to the k left moving directions that are not invariant under the action of the group.. Thus rank(Λ ) = 22 − k , rank(Λ ) = k and ⊥L k rank(Λ ) = 6.7 The total number of U(1) gauge fields in the theory is given ⊥R by rank(Λ ) = 22+ 6− k. For the Z groups, the values of k can be read ⊥ n off from Table 1. We recollect some lattice definitions from [21] for convenience. Let V be the 22 + 6 dimensional vector space in which the Narain lattice Γ(22,6) is embedded. The action of a given group element h ∈ Z on V leaves n a subspace V (h) invariant. The planes of rotation lie along a subspace ⊥ denoted as V (h). It is clear that V (h) and V (h) are mutually orthogonal k k ⊥ to each other. The action of the entire group thus separates the vector space V into an invariant subspace V and its orthogonal complement V which are ⊥ k 7Thiscorrespondstothesixgraviphotonsthatarisefromthetoroidalcompactification. 8 G rank(Λ ) rank(Λ ) k ⊥L Z 8 14 2 Z 12 10 3 Z 14 8 4 Z 16 6 5 Z 16 6 6 Z 18 4 7 Z 18 4 8 D ≃ Z ×Z 12 10 2 2 2 Z ×Z 16 6 2 4 Z ×Z 18 4 2 6 Table 1: For the abelian groups the ranks of the invariant sublattice and the orthogonal complement are given in [32]. defined as8 V = V (h) ; V = V (h) (18) ⊥ ⊥ k k h\∈Zn h[∈Zn The invariant sublattice Λ and its orthogonal complement Λ are defined ⊥ k as Z Λ n := Λ⊥ = Γ V⊥ ; ΛZn := Λk = Γ Vk. (19) \ \ and Λ (h) = Γ V (h) ; Λ (h) = Γ V (h) , (20) ⊥ ⊥ k k \ \ where Λ (h) is the lattice component left invariant by a group element h ⊥ and Λ (h) is the orthogonal complement. The ranks of these lattices are the k dimensions of their respective vector spaces. Inthefollowingwedescribe theheteroticconstruction ofthecounting[21] in the untwisted sector as the non-commuting twist obtains no contribution from the twisted sectors. The projection is unto states invariant under the orbifold group Z . For individual elements, h ∈ Z there will be a non-trivial n n shift vector along with the rotation. In order to obtain expressions for g ∈ Z 2 one has to just put the shift vectors a to zero. For composite elements like g gh one has a rotation due to h followed by a reflection on the axes of rotation 8The sublattice that is invariant under a group G acting on a lattice, Λ, is usually denoted by ΛG and its orthogonalcomplement by Λ . G 9