\( \mathcal{N} \) =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 oneloop 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 nonstandard embeddings for K3 realized as a T 4=Z orbifold with = 2; 4 and g0 the 2A involution. We show that for nonstandard embeddings the new supersymmetric index as well as the di erence in oneloop 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 Nonstandard 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 nonstandard 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 antiparticles. 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 Rsymmetry 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.
Nonstandard 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)/2015EMRI.
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 ;
nonstandard 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.
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