\( \mathcal{N} \) =2 heterotic string compactifications on orbifolds of K3 × T 2

Journal of High Energy Physics, Jan 2017

We study \( \mathcal{N} \) = 2 compactifications of E 8 × E 8 heterotic string theory on orbifolds of K3 × T 2 by g ′ which acts as an \( {\mathbb{Z}}_N \) automorphism of K3 together with a 1/N shift on a circle of T 2. The orbifold action g ′ corresponds to the 26 conjugacy classes of the Mathieu group M 24. We show that for the standard embedding the new supersymmetric index for these compactifications can always be decomposed into the elliptic genus of K3 twisted by g ′. The difference in one-loop corrections to the gauge couplings are captured by automorphic forms obtained by the theta lifts of the elliptic genus of K3 twisted by g ′. We work out in detail the case for which g ′ belongs to the equivalence class 2B. We then investigate all the non-standard embeddings for K3 realized as a \( {T}^4/{\mathbb{Z}}_{\nu } \) orbifold with ν = 2,4 and g ′ the 2A involution. We show that for non-standard embeddings the new supersymmetric index as well as the difference in one-loop corrections to the gauge couplings are completely characterized by the instanton numbers of the embeddings together with the difference in number of hypermultiplets and vector multiplets in the spectrum.

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\( \mathcal{N} \) =2 heterotic string compactifications on orbifolds of K3 × T 2

Received: November = 2 heterotic string compacti cations on orbifolds Aradhita Chattopadhyaya 0 1 3 Justin R. David 0 1 3 Superstring Vacua 0 1 3 0 C.V. Raman Avenue , Bangalore 560012 , India 1 Centre for High Energy Physics, Indian Institute of Science 2 heterotic string theory on 3 Open Access , c The Authors We study N = 2 compacti cations of E8 orbifolds of K3 T 2 by g0 which acts as an ZN automorphism of K3 together with a 1=N shift on a circle of T 2. The orbifold action g0 corresponds to the 26 conjugacy classes of the Mathieu group M24. We show that for the standard embedding the new supersymmetric index for these compacti cations can always be decomposed into the elliptic genus of K3 twisted by g0. The di erence in one-loop corrections to the gauge couplings are captured by automorphic forms obtained by the theta lifts of the elliptic genus of K3 twisted by g0. We work out in detail the case for which g0 belongs to the equivalence class 2B. We then investigate all the non-standard embeddings for K3 realized as a T 4=Z orbifold with = 2; 4 and g0 the 2A involution. We show that for non-standard embeddings the new supersymmetric index as well as the di erence in one-loop corrections to the gauge couplings are completely characterized by the instanton numbers of the embeddings together with the di erence in number of hypermultiplets and vector multiplets in the spectrum. Superstrings and Heterotic Strings; Conformal Field Models in String Theory - New supersymmetric index and twisted elliptic genus of K3 2.2 Di erence of one loop gauge thresholds Contents 1 Introduction 2 Standard embedding 3 Standard embedding: 2 examples 3.1 Twisted elliptic genus Massless spectrum The new supersymmetric index Twisted elliptic genus New supersymmetric index 3.2 The 2B orbifold from K3 based on su(2)6 4 Non-standard embeddings 4.1 4.2 Massless spectrum New supersymmetric index 4.3 Di erence of one loop gauge thresholds 5 Conclusions A Notations, conventions and identities B Threshold integrals C Mathematica les Introduction i cation is the heterotic string on K3 T 2. In the context of string dualities this theory was shown to be independent of the orbifold realization [2, 4, 5]. dex [4{9] which is de ned by Znew(q; q) = TrR F ei F qL0 2c4 qL0 2c4 persymmetric index of K3 T 2 which enumerates BPS states in these compacti cations T 2 by g0 acted elliptic genus of K3. pacti cations of the E8 E8 heterotic string theory on orbifolds of K3 more detail. group to E7 E8. The di erence in one loop corrections of the gauge groups E7 1We use the ATLAS naming for the conjugacy classes of M24 see [13]. by the 2B action. well as Nh equation (4.19). We then examine non-standard embeddings of K3 T 2 compacti cations. This is di erence of the number of hypermultiplets and vector multiplets, Nh Nv of the model. The result can be read o using the table 13 and equation (4.7) In each case we see that of K3 twisted by g0. However there is also a dependence in Nh Nv. We then evaluate results in the paper. Standard embedding heterotic string theory on orbifolds of K3 T 2 by g0 in which the spin connection of K3 is can twist the elliptic genus of K3 [15{17]. E8's be realized in terms of left moving fermions I ; I = 1; 16. The other E8 can be I BaIJ @Xa J + I AiIJ @Xi J + 0I A0iIJ @Xi 0J : Ai; A0i on T 2 are free. Thus we have the 16 standard embedding splits as to have the structure G = (6;6) internal = HD2K3 New supersymmetric index and twisted elliptic genus of K3 Znew = 2 TrR ( 1)F F qL0 c=24qL0 c=24 : T 2 together with the Fermion number on K3 F = F T 2 + F K3: contribution to the index arises from Znew = 2 TrR F T 2 ei (F T 2 +F K3)qL0 c=24qL0 c=24 : the trace reduces to Znew = (2r;2;s)(q; q) 26( ) (r;s) + 36( ) (r;s) The sum over the sectors (r; s) is implied and r; s run from 0 to N 1. The origin and the de nition of each term in the index is as follows. { 4 { 1. The term 2;2 arises from the lattice sum on T 2 together with the left moving bosonic 2 oscillators. The lattice sum is de ned as (2r;2;s)(q; q) = 12 p2R = 12 p2L = mn1;1m=2Z;n+2N2rZ; q p22L q p22R e2 im1s=N ; m1U + m2 + n1T + n2T U j2; and the fact the winding modes are shifted from integers by Nr . in the various sectors are given by ZR(D6; q) = ZNS+ (D6; q) = ZNS (D6; q) = (r;s) = (Nr;Ss)+ = (r;s) = TrR R;gr g ( 1)FR qL0 c=24qL0 c=24i ; h s TrNS R;gr g ( 1)FR qL0 c=24qL0 c=24i ; h s TrNS R;gr g ( 1)FR+FL qL0 c=24qL0 c=24i : h s We will relate them to the twisted elliptic genus of K3 below. elliptic genus of K3 which is de ned as F (r;s)( ; z) = by the following equations (r;s) = F (r;s) (Nr;Ss)+ = q1=4F (r;s) (r;s) = q1=4F (r;s) g0 embedded in M24 following the arguments of [10, 12]. The elliptic genus of K3 twisted by g0 in general can be written as F (0;0)( ; z) = F (0;1)( ; z) = (0;1)A( ; z) + g(00;1) (0;1)( )B( ; z); g0 fg0 where the Jacobi forms A( ; z) and B( ; z) are given by B( ; z) = 12( ; z) a; b; c; d 2 Z; bc = 1: (0;0) = ; (0;1) = p(p + 1) (0;1) = fg(00;1)( ) = Ep( ) = can be obtained by modular transformations using the relation = exp 2 i F (cs+ar;ds+br)( ; z); the following identities q 1=4 q 1=4 in (2.6) and using (2.11) we obtain Znew(q; q) = 1 is understood. Di erence of one loop gauge thresholds Znew(q; q) = lattices sums now are given by (3r;2;s)(q; q) = R = L = m1n; m1=2;Zn+2;Nbr2Z; V T : R + m1n1 + m2n2 + q p22L q p22R e2 im1s=N ; m1U + m2 + n1T + n2(T U The product (T; U; V ) = BG = E~2 = convert the lattice sum with the Wilson line E4;1 ! E~2E4;1 E6;1. This occurs because with the integrand given by BG0 = Here note that E6 ! E~2E6 E42. Using the identities E6;1( ; z)E4) = 144B( ; z); E6;1( ; z)E6 = 576A( ; z); BG0 = G(T; U; V ) G0 (T; U; V ) = the integral well de ned in the 2 ! 1 limit. modular forms. Standard embedding: 2 examples T 2 with by the Z4 which is action given by gs : (x1; x2; y1 + iy2; y3; +iy4) s = 0; 1; 2; 3: orbifold which is a Z2 action given by g0 : (x1; x2; y1; y2; y3; y4) (x1 + ; x2; y1 + ; y2 + ; y3 + ; y4 + ): heterotic string compacti ed on this orbifold K3 T 2 for the standard embedding. Using element 2A. Twisted elliptic genus 8 a;b=0 F (r;s)( ; z) = T rga;g0r ( 1)FL+FR gbg0se2 izFL qL0 qL0 : numbers respectively. We have suppressed the shifts L0 1=4, L0 1=4 in the de nition of the index. Let us further de ne the trace F (a; r; b; s) = T rga;g0r ( 1)FL+FR gbg0se2 izFL qL0 qL0 : Fixed points powers of g; g0. indicates that the xed point moves, while the X indicates the xed point is is summarized in table 1. that the trace reduces to F 0;0( ; z) = ZK3( ; z) = 4A( ; z): where ZK3 is the elliptic genus of K3. does not preserve any of the xed points. Thus we have F (a; 0; b; 1) = 0; for a = 1; 3: untwisted sector we see the contributions are F (0; 0; 0; 1) = 0; F (0; 0; 1; 1) = F (0; 0; 3; 1) = Evaluating the contributions to F (2; 0; b; 1) we obtain F (2; 0; 0; 1) = 0; F (2; 0; 2; 1) = 0; F (2; 0; 1; 1) = F (2; 0; 3; 1) = = 4 2222((z0;; )) A( ; z) 3 E2( )B( ; z): belongs to the class 2A. F (0; 1; a; 0) = 0; for a = 1; 2; 3; arise from the following F (a; 1; b; 0) = for a = 1; 3; b = 0; 2; { 11 { it can be seen that F (a; 1; b; 1) = Again summing up the contributions leads to for a = 1; 3; b = 1; 3; Here P is the E8 E8 lattice vector which is generically of the form identical to the class 2A Massless spectrum together with the shift V = in the E8 The orbifold action g0 (3:2) does not produce any xed points and therefore preserves m2L = NL + (P + nV )2 + En m2R = NR + (r + nv)2 + En 1 = 0; = 0: P = { 12 { class which we denote by A = (n1; n2::::n8) B = D(n) = (n; m) (n; m); with X ni = even integer: En = X ri = odd; v = (0; 0; 1; 1): given by v is a 4 dimensional vector given by (n; m) = exp 2 i (r + nv)mv (P + nV )mV + under the action of gm. is the phase by which the oscillators in the T 4 are rotated by (0; m) = 1; of gm. From table 1 we see that (1; m) = (3; m) = 4; (2; 0) = 16; (2; 1) = 4; (2; 2) = 16; (2; 3) = 4: { 13 { spectrum is that the degeneracy given in (3.23) changes to D(n; g0) = 1 X 1 h (n; m) + (g0)(n; m)i (g0)(0; m) = (0; m) = 1; For the twisted sector, from the tabel 1 we obtain (g0)(1; m) = (g0)(3; m) = 0; (2; 0)(g0) = 0; (2; 1)(g0) = 4; (2; 2)(g0) = 0; (2; 3)(g0) = 4: We are now ready to obtain the spectrum of the model. E8 to E7 E8.2 Thus untwisted sector there are 2 singlet hypers under E7 E8 which we denote as (1; 1) and 2 hypers charged as (56; 1). The twisted sector consists of only hypermultiplets T 2 worked have the conditions r2 = 1; r v = 2We are ignoring the 2 vector multiplets from the one cycles of the T 2. 3We are not keeping track of the U(1) charges in our discussion. (P + 2V )2 = 3=2: { 14 { vectors in both the E8's in the vector conjugacy class. Thus we have + X nj2 = ; 1; n2 = 0 with one of the nj = + X n2k = : Here we can have n1 = of (3.31) and (3.32) form the (56; 1) dimensional representation of E7 E8. Let us condition and satisfy P V = 1=4, and have and V from (3.22) and (3.14) respectively We nd that by 2 to account for the anti-particles. Thus we have 3(56; 1) hypers.4 For these states there is a pair of oscillators each with 1=4. The mL = 0 condition reduces to (P + 2V )2 = 1=2: conjugacy class leading to + X nj2 = : 1; n2 = 1; nj = 0 (2; 1) for 1=4 we obtain (2; 1) = 1. The degeneracy from (3.26) for these states is given by 2 (3 + 1) = 8, here for (3.34) Finally since we have two pairs of oscillators with 1=4 the total number of states is given by have 2 to the E7 4For the model just on T 4=Z4 T 2 we have D(2) = 5 for these states 5For the model without the g0 orbifold the number of such states is 32. { 15 { (T 4=Z4 T 2)=g0 (56; 1) + 2(1; 1) 2(56; 1) + 16(1; 1) 3(56; 1) + 16(1; 1) T 4=Z4 (56; 1) + 2(1; 1) 4(56; 1) + 32(1; 1) 5(56; 1) + 32(1; 1) T 2 with the standard embedding. T 2 with the standard embedding. Nv = Nv = (1; 248) of Nv = the g0 orbifold by evaluating the new supersymmetric index. The new supersymmetric index the actions (3.1), (3.2) with the shift in (3.14) in E8 E8. We adapt the method developed supersymmetric index given in (2.3) splits into the following sectors Znew(q; q) = a;b=0 r;s=0 F (a; r; b; s; q) (2r;2;s)(q; q): tions is given by F (a; r; b; s; q) = Trga g0sR action in (3.1), (3.2) is given by F (a; r; b; s; q) = k(a;r;b;s) 2( )q 1a6 k(a;0;b;0) = 16 B k(a;0;b;1) = 16 B k(a;1;b;0) = 16 B k(a;1;b;1) = 16 B B0 0 0 0C B0 1 0 1C B0 1 0 1C B0 0 4 0C given by B1 1 1 1C B4 1 4 1C B1 0 1 0C B4 1 0 0C { 17 { = ZE(08;3); = ZE(38;0); ZE(08;1) = ZE(18;0) = ZE(18;1) = ZE(18;2) = ZE(18;3) = ZE(28;1) = = ZE(28;3): + 26 h 1 i h 1 i 1=2 1=2 + 46 h 0 i h 0 io 3=2 1=2 6 h 1=2 i h 1=2 i + 26 h 3=2 i h 1=2 i 0 0 6 h 1=2 i h 1=2 i + 26 h 3=2 i h 1=2 i 1=2 1=2 + 46 h 1=2 i h 1=2 i 3=2 1=2 6 h 1=2 i h 1=2 i 6 h 1=2 i h 1=2 i + 26 h 3=2 i h 1=2 i 3=2 3=2 + 46 h 1=2 i h 1=2 i 5=2 1=2 + 26 h 2 i h 0 i 1=2 1=2 Also in the Z2 subgroup sector we have ZE(08;2) = ZE(28;0) = ZE(28;2) = 46 [ 1 ] [ 10 ] + 26 [ 2 ] 2 1 The de nition of the generalized Jacobi theta functions is given by [ ab ] ( ; z) = X q i(k+ a2 )2 e i(k+ a2 )be2 iz(k+ a2 ): (0;0) = 0; (0;1) = obtain the expected results Znew(q; q) = (0;0) = 4; (0;1) = (1;0) = (1;1) = ; (1;0) = (1;1) = ; f2(A0;1)( ) = E2( ); f2(A1;0)( ) = E2 f2(A1;1)( ) = E2 to arrive at the result (3.42). by g0 agrees with that of the 2A orbifold of K3 T 2. This result was expected since As a consistency check of our calculations we will evaluate the Nh Nv from the { 18 { where Znew T 2 evaluated in (3.42) we obtain Nv)j2A = elliptic genus of K3 belonging to the class 2A. The 2B orbifold from K3 based on su(2)6 T 2 orbifolded by twisted elliptic genus. Twisted elliptic genus Nv = q1=6 6We have evaluated (Nh Nv) from the new supersymmetric index for all the pA orbifolds of K3 with p = 3; 5; 7; 11. We obtain 134; 256; 317; 376 respectively which indicates that the number of integers in all these situations. 7In [14], g0 was referred to as g, see section 6.1. { 19 { Thus 0 in table 4 represents the su(2) character at level 1 while 1 represent the spinorial su(2) character given by ch1;0 = ch1; 12 = 3(2 ; 2z) 2(2 ; 2z) moving ones. The SU(2)L SU(2)R R-symmetry of K3 is carried by the rst su(2) character with the characters given in the table reduces to that of K3. The g0 orbifold on K3 is implemented by the action g0 = L The SU(2) rotation matrices of g0 on the su(2) characters is given by 3(2 ; 2z) 2(2 ; 2z) 4(2 ; 2z) 1(2 ; 2z) given by F 0;0( ; z) = +5 3(2 ; 2z) 2(2 ) 3(2 )4 5 2(2 ; 2z) 3(2 ) 2(2 )4i components of the twisted elliptic genus to be F (0;1)( ; z) = 2(2 ; 2z) 3(2 ) 4(2 )4 3(2 ; 2z) 2(2 ) 4(2 )4i 2E4( )] B( ; z); F (0;2)( ; z) = [A( ; z) + E2( )B( ; z)] : = 2A( ; z): in (2.12) with the identi cations (0;0) = 2; (0;1) = (0;2) = (0;1) = 0; f2(B0;1) = E2( ) f2(B0;2) = E2( ): (0;2) = New supersymmetric index character in the R+; N S+ and N S sectors. These sectors couple to the corresponding R+; N S+ and N S of the su(2)6 CFT. Comparing tables (5) and (4) we can see how the { 21 { [01 00 00, 01 00 00] [10 11 11, 10 00 00] [00 10 00, 00 10 00] [11 01 11, 00 10 00] [00 01 00, 00 01 00] [11 10 11, 00 01 00] [00 00 10, 00 00 10] [11 11 01, 00 00 10] [00 00 01, 00 00 01] [11 11 10, 00 00 01] [00 00 00, 10 00 00] -[11 11 11, 01 00 00] -[00 00 00, 10 00 00] -[11 11 11, 01 00 00] [11 00 00, 01 00 00] -[00 11 11, 10 00 00] [10 10 00, 00 10 00] -[01 01 11, 00 10 00] [10 01 00, 00 01 00] -[01 10 11, 00 01 00] [10 00 10, 00 00 10] -[01 11 01, 00 00 10] [10 00 01, 00 00 01] -[01 11 10, 00 00 01] [11 00 00, 01 00 00] [00 11 11, 10 00 00] [10 10 00, 00 10 00] [01 01 11, 00 10 00] [10 01 00, 00 01 00] [01 10 11, 00 01 00] [10 00 10, 00 00 10] [01 11 01, 00 00 10] [10 00 01, 00 00 01] [01 11 10, 00 00 01] Let us rst evaluate the component (0;0) in various sectors. Using the character table 5 and the rules in (3.46) and (3.47) we obtain (0;0) = (0;0) = 5 32(2 ) 24(2 ) + 36(2 ) (N0;S0+) = { 22 { (0;1) = 2 2(2 ) 3(2 ) 44(2 ) = [110000; 010000] lead to (0;1) sector the characters which are present are [000000; 100000] and Finally the characters which survive the g0 insertion in [110000; 010000] giving rise to [000000; 100000] and (0;1) = (N0;S1+) = Znewj(0;1) = Here there we have used identities which relate the functions to Eisenstein series which are provided in the appendix. Using the action of g02 which is given by (g0)2 = L Znewj(0;0) = 4 E4E6 : (0;2) = (0;2) = (N0;S2+) = 3 22(2 ) 34(2 ) + 3 24(2 ) 32(2 ) ; Znewj(0;2) = 6 E6 + 3 E2( )E4 : modular transformations. expression (3.43) we obtain Nh Nv = 380 for this model. elliptic genus. Non-standard embeddings T 2 orbifolded by g0 belonging to the conjugacy class 2A. We rst realize K3 as by di erent lattice shifts in the E8 E8. From the spectrum of these embeddings we show Nv which take values 12; 52; 84; 116 for these types. The value 12 as we have seen corresponds only on Nh Nv and the instanton numbers of the embedding. Massless spectrum dings of K3 various embeddings are determined by the lattice shifts in E8 E8. In table 6, we rst { 24 { (1; 1; 06; 08) (12; 06; 2; 07) (56; 2) + 4(1;1) 4(56;1)+16(1;2) (56,2;1)+4(1,1;1) Nv = 12, while the second shift realizes Nh Nv = 116. Gauge group, Shift ( ; ~) (1; 1; 06; 08) (1; 1; 06; 2; 2; 06) (3; 1; 06; 08) 2(56; 1) + 4(1; 1) + 12(1; 1) 6(1; 1; 2) + 2(1; 1; 2) + 2(1; 56; 1) (56; 1) + 2(1; 1) 3(56; 1) + 16(1; 1) (56; 1; 1) + 2(1; 1; 1) 1(56; 1; 1) + 16(1; 1; 1) (12; 2; 1) + (32; 1; 1) + 2(1; 1; 1) 6(1; 2; 1) + 4(12; 1; 1) 2(1; 2; 1) + 2(32; 1; 1) 16(1; 1; 1) + 3(12; 2; 1) + (32; 1; 1) as T 4=Z4 with Nh Nv = T 2 for di erent embeddings belonging to type 0 for K3 half shift given by following orbifold actions (x1; x2; y1; y2; y3; y4) (x1; x2; y1; y2; y3; y4); (x1; x2; y1; y2; y3) (x1 + ; x2; y1 + ; y2; y3; y4): Nv given by 12; 52; 84; 116, the value 12 corresponds to the standard embedding. { 25 { (12; 2; 1; 1) + (32; 1; 1; 1) + 2(1; 1; 1; 1) (56; 1) + 2(1; 1) 3(56; 1) + 16(1; 1) 4(1; 2; 1; 2) + 2(12; 1; 1; 2) 16(1; 1; 1; 1) + (12; 2; 1; 1) + 3(32; 1; 1; 1) (12; 2; 1) + (32; 1; 1) + 2(1; 1; 1) 2(1; 2; 16) 16(1; 1; 1) + 3(12; 2; 10 + (32; 1; 1) (56; 1) + (1; 8) + (1; 56) + 2(1; 1) 6(1; 1) + 2(1; 1) + 2(1; 28) 6(1; 8) + 2(1; 8) (27; 2; 1) + (1; 2; 1) + (1; 1; 64) +2(1; 1; 1) 6(1; 1; 1) + 4(1; 2; 1) +2(27; 1; 1) + 2(1; 1; 14) (1; 2; 14) + 6(1; 2; 1) (1; 1; 06; 4; 07) (3; 1; 06; 2; 2; 06) (3; 1; 06; 4; 07) with Nh Nv = 52. Gauge group, Shift ( ; ~) (1; 1; 06; 17; 1) (2; 1; 1; 05; 2; 07) with Nh Nv = 84. New supersymmetric index Znew for the 2A orbifold of K3 T 2 depends only on the 4 types of the lattice shifts as the instanton number corresponding to the lattice shift. { 26 { (28; 2; 1; 1) + (1; 1; 16; 4) + 2(1; 1; 1; 1) (17; 1; 3; 1; 0) 4(1; 1; 2) + 2(1; 12; 1) + 2(8; 1; 2) (27; 2; 1; 1) + (1; 2; 1; 1) + (1; 1; 16; 4) +2(1; 1; 1; 1) 4(1; 1; 1; 4) + 2(1; 2; 1; 4) +2(1; 1; 16; 1) 3(1; 2; 10; 1) + (1; 2; 1; 6) 2(8; 1; 1; 4) 16(1; 1; 1) + 3(12; 2; 1; 6) + (1; 2; 10; 1) (28; 2; 1) + (1; 1; 64) + 2(1; 1; 1) 4(8; 1; 1) + 2(8; 2; 1) 3(1; 2; 14) + 2(1; 2; 1) (8; 1; 1) + (56; 1; 1) + (1; 12; 1) (1; 32; 1) + 2(1; 1; 1) 6(8; 1; 1) + 2(8; 1; 1) U(1); SO(10) (2; 1; 1; 05; 23; 05) (3; 15; 02; 23; 05) (3; 15; 02; 2; 07) with Nh Nv = 116. (1,1,0,0,0,0,0,0) (0,0,0,0,0,0,0,0) (1,-1,0,0,0,0,0,0) (2,0,0,0,0,0,0,0) Type 0 Type 3 in (2.3) we see that it reduces to Znew(q; q) = a;b=0 r;s=0 F (a; r; b; s; q) (2r;2;s)(q; q); partition function over the shifted E8 lattices are de ned by ZEa;8b(q) = e i a PI8=1 I Y e i a PI8=1 ~I Y { 27 { given by where the k's are read out from the following matrices. (a;1;b;1) = 64 00 e i(2 2)=4 ; 0 (a;0;b;0) = 16BBB14 ee ii8311(62(2 2)2) 4ee ii1814((22 22)) ee ii8113(62(2 2)2)CC; C + (21;2;1) " E6 E2 2 + 1 E4 4E2 2 !#) ^b 2 + 1 + 23 ^b E4 : { 28 { (1,-1,0,0,0,0,0,0) (0,0,0,0,0,0,0,0) (1,1,0,0,0,0,0,0) (0,0,0,0,0,0,0,0) Type 0 (1,1,0,0,0,0,0,0) (2,2,0,0,0,0,0,0) (3,1,0,0,0,0,0,0) (0,0,0,0,0,0,0,0) (1,1,0,0,0,0,0,0) (4,0,0,0,0,0,0,0) (3,1,0,0,0,0,0,0) (4,0,0,0,0,0,0,0) Type 1 (3,1,0,0,0,0,0,0) (2,2,0,0,0,0,0,0) (2,1,1,0,0,0,0,0) (2,0,0,0,0,0,0,0) (1,1,0,0,0,0,0,0) (1,1,1,1,1,1,1,-1) Type 2 (2,1,1,0,0,0,0,0) (2,2,2,0,0,0,0,0) (3,1,1,1,1,1,0,0) (2,0,0,0,0,0,0,0) Type 3 (3,1,1,1,1,1,0,0) (2,2,2,0,0,0,0,0) (1,1,1,1,1,1,-1) (3,1,0,0,0,0,0,0) Type 0 Type 1 Type 2 Type 3 can be found by using the equation in (3.43) and is given by Nv = 144^b replaced by for the (0; 1) sector. The lattice sum the other sectors. { 29 { Wilson line is sensitive to the the instanton numbers. For compacti cations on (K3 depends on ^b which is related to Nh Nv of the model by (4.8) and also the instanton number the following compact expression Znew = 12 [n1E4;1E6 + n2E6;1E4] the value of ^b by sum then becomes This modi ed lattice sum ZEa;0b( ; z) is then coupled to the 3;2 lattice using the 8 lattice shifts. The result is given by the expression Znew = sector to 12 E2( 2 ). Similarly in the (1; 1) we have 12 E2( +21 ). We summarize the values orbifolds can be read out. Di erence of one loop gauge thresholds { 30 { Type 0 Type 3 Type 0 Type 1 (1,1,0,0,0,0,0,0) (4,0,0,0,0,0,0,0) (1,-1,0,0,0,0,0,0) (0,0,0,0,0,0,0,0) (1,1,0,0,0,0,0,0) (0,0,0,0,0,0,0,0) (3,1,0,0,0,0,0,0) (0,0,0,0,0,0,0,0) (1,1,0,0,0,0,0,0) (2,2,0,0,0,0,0,0) (3,1,0,0,0,0,0,0) (4,0,0,0,0,0,0,0) (3,1,0,0,0,0,0,0) (2,2,0,0,0,0,0,0) Type 2 (2,1,1,0,0,0,0,0) (2,0,0,0,0,0,0,0) Type 3 (2,1,1,0,0,0,0,0) (2,2,2,0,0,0,0,0) (1,1,0,0,0,0,0,0) (1,1,1,1,1,1,1,-1) (3,1,0,0,0,0,0,0) (1,1,1,1,1,1,1,-1) (3,1,1,1,1,1,0,0) (2,0,0,0,0,0,0,0) (3,1,1,1,1,1,0,0) (2,2,2,0,0,0,0,0) (1,-1,0,0,0,0,0,0) (0,0,0,0,0,0,0,0) (1,-1,0,0,0,0,0,0) (2,0,0,0,0,0,0,0) T 2)=g0 and their a^; ^b; c^ values. T 2)=g0 and their a^; ^b; c^ values. E4;1E~2 E~2E4;1 E6 + n2 E~2E6;1 E4;1E4 E6;1 (E6 + 2E2( )E4) E6;1E~2 E4;1E4 + 2E2( ) E4;1E~2 E4 + 4E4(2 ) E6;1E~2 E4;1E4 + 2E2( )E4;1E~2 we obtain BG = where the terms in the [ ] can be obtained by modular transformation from the correE2 ( ) = (4E4(2 ) + E4) ; { 31 { (n1; n2) BG0 = E6 (E6;1 + 2E2( )E4;1) E6 + 8 E~2(2 )E4(2 ) n1E4;1 E~2E6 E42 + 2E2( )(E4E~2 together with (4.14) and E2( )3 = 4 E4E2( ) + 4 E6 : This results in the following expression for the threshold integral G(T; U; V ) G0(T; U; V ) = E6(2 ) E6;1 + 2E2( )E4;1 n1)A(z) 12B(z)E2( ) + 6B(z)E2 use the following identities E2( ) = 2E~2(2 ) E6(2 ) = E2( ) the di erence Nh G(T; U; V ) G0 (T; U; V ) = 48 log(det(Im( ))6 j 6(U; T; V)j2) log(det Im( ))10 j 10(U; T; V)j2 Here 10 is the unique cusp form of weight 10 under Sp(2; Z), while 6 is the Siegel modular by the 2A orbifold action. 6 was rst constructed as a theta lift in [18]. As expected for Conclusions new supersymmetric index depends only on the di erence Nh Nv of the model and the of string theory of type II on (K3 was the lattice sum 2;2 folded with a holomorphic function which resembled an index. { 33 { Acknowledgments fellowship 09/079(2649)/2015-EMR-I. Notations, conventions and identities modular functions. We also de ne interchangeably in the arguments of the E2(q) = 1 E4(q) = 1 + 240 X E6(q) = 1 n=1 1 n=1 1 n=1 1 12( ; z) = 2(2 ) 3(2 ; 2z) 3(2 ) 2(2 ; 2z); 42( ; z) = 3(2 ) 3(2 ; 2z) 2(2 ) 2(2 ; 2z): 22 = 2 2(2 ) 3(2 ); 32 = 22(2 ) + 32(2 ); 42 = 2 22(2 ) = 32 2 32(2 ) = 32 + 42: Eisenstein series with the U(1) chemical potential are de ned by E4;1 = E4e;v1en even + E4o;d1d(z) odd(z); E6;1 = E6e;v1en even + E6o;d1d(z) odd(z): even(z) = 3(2 ; 2z) odd(z) = 2(2 ; 2z): Then the de nition of fs;1 is iven by E4 = E6 = E4;1(z) = E6;1(z) = 3;2 (even) = 3;2 (odd) = fs;1 = r;s 3;2(even)fse;v1en + r;s 3;2(odd)fso;d1d ; m1;m2;n22Z; n1=Z+ Nr ;b22Z m1;m2;n2;2Z; n1=Z+ Nr ;b22Z+1 R 2 im1s=N R 2 im1s=N : where pL; pR are given in (2.21) and N is the order of the g0 action. E2( )2 = { 35 { using the relation Their modular transformed versions can be simpli ed as: E6(2 ) = E4(2 ) = 8 E2( ) 11E22( ) E23( ) = (E6 + 3E4E2( )): E4( =2) = (5E22( =2) 3 E2 ( =2) = ( 2E6 + 3E4E2( =2)): 38 24 + 28 34 = 28 44 = (E6 + 2E2( )E4) ; We have then the identity E2;1( ; z)2 = 2 36 3(z)2 + 2 46 4(z)2 These are the following identities between E2 and Eisenstein series at 2 . theta functions and E4. This is given by 44(2 ) = index for the 2B model given in (3.60) to Eisenstein series 3 E6 3 E2( )E4 : Threshold integrals Adding and subtracting terms in the integrand we obtain G(T; U; V ) G0(T; U; V ) = I1 = I2 = 3 BE2 [ (0;0) + (0;1) + (1;0) + (1;1)] 4A: n1)A(z) +6B(z)E2 3 BE2( ) + (31;2;0) 3 BE2 2 G(T; U; V ) G0(T; U; V ) = 48 log(det(Im( ))6 j 6(U; T; V)j2) log(det Im( ))10 j 10(U; T; V)j2 from earlier work I1 = I2 = 2 log det(Im( ))10 j 10(U; T; V)j2 ; 2 log det(Im( ))6 j 6(U; T; V)j2 : I~(U; T; V ) = X I~r;s;b ; I~r;s;b = hbr;s( ) = r;s=0 b=0 n2Z b2=4 j22Z+b cbr;s(4n)qn; qp2L=2qp2R=2e2 ism1=N hbr;s; b=0;1 n2Z=N;j22Z+b the result for the integral is given by I~(U; T; V ) = 2 log hdet ~ (U; T; V)i 2 log hdet ~ (U; T; V)i ; non-standard embeddings is the following I3 = 2 h (0;0) + (0;1) + (1;0) + (1;1)i which we will now state. Given the integral of the form with the condition ~ (U; T; V ) = e2 i(~U+ ~T +V ) ~ = ~ = Qr;s = N (cr0;s(0) + 2cr1;s( 1)) ; Q0;0 = (M ) = 24: cbr;s(u) = 2 cb(u): 8A( ; z) = b=0;1 n2Z;j22Z+b Qr;s = 24; ~ = 2; ~ = : = e2 i(2U+T=2+V ) Y 10(2U; T =2; V ): b=0;1 r=0 k02Z; l2Z; j22Z+b k0;l 0; j<0 k0=l=0 of K3. Thus the result of the integral in (B.6) is given by I3 = 2 h (0;0) + (0;1) + (1;0) + (1;1)i Thus we have We can further simplify the expression in (B.11) as follows e2 i(k0T +2lU+jV ) cbr;s(4k0l j2) = e2 i(2U+T=2+V ) Y 66 b=0;1 r=0 k02Z+ r2 ;l2Z; j22Z+b k0;l 0;j<0k0=l=0 k02Z;l2Z; j22Z+b k0;l 0;j<0k0=l=0 b=0;1 64 k02Z;l2Z; j22Z+b Jacobi forms of index 1 and Eisenstein series. E8 lattice Nv as a is also evaluated for the 2B model. Open Access. any medium, provided the original author(s) and source are credited. [INSPIRE]. groups, Oxford University Press (1985). symmetry, JHEP 02 (2014) 022 [arXiv:1309.4127] [INSPIRE]. Theor. 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Aradhita Chattopadhyaya, Justin R. David. \( \mathcal{N} \) =2 heterotic string compactifications on orbifolds of K3 × T 2, Journal of High Energy Physics, 2017, DOI: 10.1007/JHEP01(2017)037