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Mathieu Moonshine and Heterotic String Compactifications
Based on: M. Cheng, X. Dong, J. Duncan, S. Harrison, S. Kachru, TW to appear N. Paquette, TW to appear
TW 1402.2973 M. Cheng, X. Dong, J. Duncan, J. Harvey, S. Kachru, TW 1306.4981
Texas A&M April 30, 2014
Timm Wrase
Outline
• Introduction to moonshine
• Mathieu Moonshine and
string compactifications
• Physical implications
• A new moonshine phenomena 2
Finite simple groups
There are 18 infinite families, e.g.
• Alternating group of n elements An
e.g. A3: (123) ↔ (231) ↔ (312)
• Cyclic groups of prime order Cp
e.g.: Cp = p = < 2π /p >
3
Finite simple groups
There are also 26 so called sporadic groups that do not come in
infinite families:
4
the largest group
subgroups
Finite simple groups
There are also 26 so called sporadic groups that do not come in
infinite families:
5
|Monster| 8 x 1053
Finite simple groups
There are also 26 so called sporadic groups that do not come in
infinite families:
6
|Monster| 8 x 1053
Largest Mathieu group 2 x 109
Conway group 4 x 1018
|M22| = 443, 520
Modular Forms
Modular function of weight k
),2(,)( ZSL
dc
bafdc
dc
baf k
7
Modular Forms
Modular function of weight k
Jacobi form of weight k and index m
),2(,)( ZSLdc
bafdc
dc
baf k
Z
,,,,
,)(,
)(2
2
2
2
zfezf
zfedcdc
z
dc
baf
zmi
dc
zcmi
k
8
Modular Forms
Modular function of weight k
Jacobi form of weight k and index m
Can Fourier expand
),2(,)( ZSLdc
bafdc
dc
baf k
Z
,,,,
,)(,
)(2
2
2
2
zfezf
zfedcdc
z
dc
baf
zmi
dc
zcmi
k
0 42
),(]2exp[],2exp[,n mnr
rn yqrncziyiqfzf
9
Monstrous Moonshine
• The irreducible representations of the Monster group have dimensions 1, 196 883, 21 296 876, …
• The J-function, that appears in many places in string theory, enjoys the expansion
10
...760493218841961
)( 2 qqq
qJ
Monstrous Moonshine
• The irreducible representations of the Monster group have dimensions 1, 196 883, 21 296 876, …
• The J-function, that appears in many places in string theory, enjoys the expansion
• as observed by John McKay
11
876296218831961 8831961
...760493218841961
)( 2 qqq
qJ
Monstrous Moonshine
1. foolish talk or ideas.
Monstrous Moonshine
1. foolish talk or ideas. 2. NORTH AMERICAN illicitly distilled or smuggled liquor.
Monstrous Moonshine
This surprising connection can be explained by string theory:
• The (left-moving) bosonic string compactified on a 2 orbifold of R24/Λ with Λ the Leech lattice has as its 1-loop partition function the J(q)-function
14
...760493218841961
)(Tr)( 2
H240
qqJqqZ
cL
Monstrous Moonshine
This surprising connection can be explained by string theory:
• The (left-moving) bosonic string compactified on a 2 orbifold of R24/Λ with Λ the Leech lattice has as its 1-loop partition function the J(q)-function
15
tachyon of the bosonic string
no massless q0 states
supermassive string states
...760493218841961
)(Tr)( 2
H240
qqJqqZ
cL
Monstrous Moonshine
This surprising connection can be explained by string theory:
• The (left-moving) bosonic string compactified on a 2 orbifold of R24/Λ with Λ the Leech lattice has as its 1-loop partition function the J(q)-function
• The symmetry group of the compactification space R24/Λ/ 2 is the Monster group.
16
...760493218841961
)(Tr)( 2
H240
qqJqqZ
cL
Monstrous Moonshine
Since we have a Virasoro algebra we can expand the J(q)-function
in terms of Virasoro characters (traces of Verma modules)
17
...)(ch87629621)(ch883196)(ch1
...760493218841961
)(
320
2
qqq
qqq
qJ
)1()(ch,
)1()(ch
1
24/
2
24/
0h n
n
ch
hn
n
c
q
q
18
Complex Analysis Number Theory
(mock) modular forms, Jacobi forms
Group Theory Representation Theory
finite (sporadic) groups String Theory
Monstrous Moonshine
19
Complex Analysis Number Theory
(mock) modular forms, Jacobi forms
Group Theory Representation Theory
finite (sporadic) groups String Theory
Monstrous Moonshine
Very interesting for mathematicians!
Monstrous Moonshine
Compactification of the bosonic string:
we have a tachyon (instability)
spacetime theory has no fermions
Additionally
• Only two spacetime dimensions are non-compact
20
Monstrous Moonshine
Compactification of the bosonic string:
we have a tachyon (instability)
spacetime theory has no fermions
Additionally
• Only two spacetime dimensions are non-compact
21
Not so interesting for physicists!
Mathieu Moonshine
• In 2010 Eguchi, Ooguri and Tachikawa discovered a new moonshine phenomenon that connects K3 to the largest Mathieu group M24
Eguchi, Ooguri, Tachikawa 1004.0956
• They considered a N=(4,4) SCFT with K3 target and calculate an index that is called elliptic genus
22
Mathieu Moonshine
23
2400240 )1()1(Tr),( RRelliptic
cR
cL
LFJLFqyqyqZ
Witten index: No dependence on q
Chemical potential for U(1) in left-moving N=2 theory
Mathieu Moonshine
T. Eguchi, H. Ooguri, A. Taormina, S. -K. Yang Nucl. Phys. B 315, 193 (1989)
We have N=(4,4) world sheet supersymmetry
expand in N=4 Virasoro characters
24
2
4
2
4
2
3
2
3
2
2
2
2K3
)1,(
),(
)1,(
),(
)1,(
),(8),(
elliptic q
yq
q
yq
q
yqyqZ
Mathieu Moonshine
T. Eguchi, H. Ooguri, A. Taormina, S. -K. Yang Nucl. Phys. B 315, 193 (1989)
We have N=(4,4) world sheet supersymmetry
expand in N=4 Virasoro characters
N=4 Virasoro characters are defined by the trace over the
highest weight state and all its descendants
For the case h=c/24 there are short BPS multiplets
25
2
4
2
4
2
3
2
3
2
2
2
2K3
)1,(
),(
)1,(
),(
)1,(
),(8),(
elliptic q
yq
q
yq
q
yqyqZ
0240Tr),(ch lh,
JLyqyq
c
Mathieu Moonshine
T. Eguchi, H. Ooguri, A. Taormina, S. -K. Yang Nucl. Phys. B 315, 193 (1989)
We have N=(4,4) world sheet supersymmetry
expand in N=4 Virasoro characters T. Eguchi, K. Hikami 0904.0911
26
2
4
2
4
2
3
2
3
2
2
2
2K3
)1,(
),(
)1,(
),(
)1,(
),(8),(
elliptic q
yq
q
yq
q
yqyqZ
long
,1
long
,
short
0,
K3
21
41
21
41
41
ellipticchch2ch24
lnhn
nlhlhAZ
...},1440,462,90{nA
Mathieu Moonshine
T. Eguchi, H. Ooguri, A. Taormina, S. -K. Yang Nucl. Phys. B 315, 193 (1989)
We have N=(4,4) world sheet supersymmetry
expand in N=4 Virasoro characters T. Eguchi, K. Hikami 0904.0911
T. Eguchi, H. Ooguri, Y. Tachikawa 1004.0956
27
2
4
2
4
2
3
2
3
2
2
2
2K3
)1,(
),(
)1,(
),(
)1,(
),(8),(
elliptic q
yq
q
yq
q
yqyqZ
...},770770,231231,4545{ nADimensions of Irreps of M24
123
long
,1
long
,
short
0,
K3
21
41
21
41
41
ellipticchch2ch24
lnhn
nlhlhAZ
Mathieu Moonshine
Does this imply a connection between M24 and K3?
• The geometric symmetries of K3 are contained in M23 M24
Mukai, Kondo 1988, 1998
28
Mathieu Moonshine
Does this imply a connection between M24 and K3?
• The geometric symmetries of K3 are contained in M23 M24
Mukai, Kondo 1988, 1998
• The symmetry groups of N=(4,4) SCFT with K3 target are never M24 and for some points in moduli space do not even fit into M24
Gaberdiel, Hohenegger, Volpato 1106.4315
29
Mathieu Moonshine
Does this imply a connection between M24 and K3?
• The geometric symmetries of K3 are contained in M23 M24
Mukai, Kondo 1988, 1998
• The symmetry groups of N=(4,4) SCFT with K3 target are never M24 and for some points in moduli space do not even fit into M24
Gaberdiel, Hohenegger, Volpato 1106.4315
• However, all coefficients are positive sums of dimensions of M24
Gannon 1211.5531
30
K3
ellipticZ
Mathieu Moonshine
• K3 has played a central role in string compactifications and string dualities
• What are implications we can derive from Mathieu moonshine for string compactifications?
• Has the elliptic genus of K3 already appeared in the string theory literature?
31
Mathieu Moonshine
• K3 has played a central role in string compactifications and string dualities
• What are implications we can derive from Mathieu moonshine for string compactifications?
• Has the elliptic genus of K3 already appeared in the string theory literature?
32
YES!
Heterotic String Theory
• Consider the heterotic E8 x E8 string theory compactified on K3 x T2
• We need to embed 24 instantons into E8 x E8 (12+n,12-n) n = 0,1,…,12 to satisfy the Bianchi identity for H3
33
Heterotic String Theory
• Consider the heterotic E8 x E8 string theory compactified on K3 x T2
• We need to embed 24 instantons into E8 x E8 (12+n,12-n) n = 0,1,…,12 to satisfy the Bianchi identity for H3
• The resulting four dimensional theories preserves N=2 spacetime supersymmetry
• The 1-loop corrections to the prepotential are related to the new supersymmetric index Znew
Dixon, Kaplunovsky, Louis, de Wit, Lüst, Stieberger, Antoniadis, Narain, Taylor, Gava,
Kiritsis, Kounnas, Harvey, Moore, ….
34 )(),,( 21tree Siloop eOhhUTSh
• The new supersymmetric index is defined as
• The trace is over our internal (22,9) conformal field theory for the heterotic E8xE8 string theory compactified on K3 x T2
2402400)1(Tr 0Rnew
cc LLJqqJZ
Heterotic String Theory
35
• The new supersymmetric index is defined as
• The trace is over our internal (22,9) conformal field theory for the heterotic E8xE8 string theory compactified on K3 x T2
• We have a right moving N=2 SCFT from the T2 and we denote its U(1) generator J(1)
• For the K3 we have an N=4 SCFT with a level one SU(2). We define J(2) = 2 J3 where J3 is the SU(2) Cartan current
• Then J = J(1) + J(2)
2402400)1(Tr 0Rnew
cc LLJqqJZ
Heterotic String Theory
36
• The new supersymmetric index is defined as
240)2(
0240)1(
0
240240)2(
0)1(
0
240240)2(
0)1(
0
)1()1(Tr
)1(Tr
)1()(Tr
)2(
0R
)1(
0R
)2(
0
)1(
0Rnew
cc
cc
cc
LJLJ
LLJJ
LLJJ
qJq
qqJ
qqJJZ
Heterotic String Theory
37
• The new supersymmetric index is defined as
For SU(2) representations eigenvalues of J(2) come in opposite pairs
Heterotic String Theory
38
0)[...])1()1((
)1(Tr 240)2(
0)2(
0K3
n
n
n
LJ
nn
qJc
Z
240)2(
0240)1(
0
240240)2(
0)1(
0
240240)2(
0)1(
0
)1()1(Tr
)1(Tr
)1()(Tr
)2(
0R
)1(
0R
)2(
0
)1(
0Rnew
cc
cc
cc
LJLJ
LLJJ
LLJJ
qJq
qqJ
qqJJZ
• The new supersymmetric index is defined as
Heterotic String Theory
39
0
240)2(
0240)1(
0
240240)2(
0)1(
0
240240)2(
0)1(
0
)1()1(Tr
)1(Tr
)1()(Tr
)2(
0R
)1(
0R
)2(
0
)1(
0Rnew
cc
cc
cc
LJLJ
LLJJ
LLJJ
qJq
qqJ
qqJJZ
• The new supersymmetric index is defined as
• Now we calculate it for the standard embedding SU(2) E8 for a compactification on K3 x T2
240)2(
0)1(
0240
240)2(
0240)1(
0
240240)2(
0)1(
0
240240)2(
0)1(
0
)1(Tr
)1()1(Tr
)1(Tr
)1()(Tr
)1(
0R
)2(
0R
)1(
0R
)2(
0
)1(
0Rnew
cc
cc
cc
cc
LJJL
LJLJ
LLJJ
LLJJ
qJq
qJq
qqJ
qqJJZ
Heterotic String Theory
40
0
For K3 x T2 compactifications we have for the standard embedding
that preserves N=(4,4)
Compare to
We will get contributions from at different y-values
240)2(
0)1(
0240 )1(Tr )1(
0Rnew
cc LJJLqJqZ
Heterotic String Theory
41
Witten index
2400240 )1()1(Tr),( RRelliptic
cR
cL
LFJLFqyqyqZ
),(K3
ellipticyqZ
For K3 x T2 compactifications we have for the standard embedding
that preserves N=(4,4)
Harvey, Moore hep-th/9510182
T is the complexified Kähler modulus, U the complex structure
modulus of the T2
),()(
)(),(
)(
)(
)1,()(
)(
(q)
(q)E
)(
),(
2),;(
21
41
21
41
2,2
K3
elliptic
6
4K3
elliptic
6
3
K3
elliptic
6
2
8
4
4new
qqZqq
qqqZq
q
q
qZq
q
q
UTiUTqZ
Heterotic String Theory
42
SO(12) characters
SO(12) characters
Affine E8
For K3 x T2 compactifications we have for the standard embedding
that preserves N=(4,4)
Harvey, Moore hep-th/9510182
So in particular the […] part has an “SO(12)xM24”-expansion:
exactly the same M24 as in Mathieu Moonshine due to N=(4,4)
Heterotic String Theory
43
SO(12) characters
SO(12) characters
Affine E8
),()(
)(),(
)(
)(
)1,()(
)(
(q)
(q)E
)(
),(
2),;(
21
41
21
41
2,2
K3
elliptic
6
4K3
elliptic
6
3
K3
elliptic
6
2
8
4
4new
qqZqq
qqqZq
q
q
qZq
q
q
UTiUTqZ
For K3 x T2 compactifications we have for the standard embedding
that preserves N=(4,4)
n
n
nlhlhlh
lhlhlh
lhlhlh
qAqq
qqqq
q
q
q
qqqq
qqqq
q
q
q
qqqq
qqqq
q
q
q
q
UTiUTqZ
1
long
,
6
4long
,
6
3long
,
6
2
short
,
6
4short
,
6
3short
,
6
2
short
0,
6
4short
0,
6
3short
0,
6
2
8
4
4new
21
41
41
21
21
41
41
21
41
21
21
41
41
21
21
41
41
21
41
21
41
41
21
41
41
41
2,2
ch)(
)(),(ch
)(
)()1,(ch
)(
)(
),(ch)(
)(),(ch
)(
)()1,(ch
)(
)(2
),(ch)(
)(),(ch
)(
)()1,(ch
)(
)(24
(q)
(q)E
)(
),(
2),;(
Heterotic String Theory
44 ...},770770,231231,4545{ nADimensions of Irreps of M24
For K3 x T2 compactifications we have for the standard (24,0)
instanton embedding
Harvey, Moore hep-th/9510182
So in particular the E6(q) has an “SO(12)xM24”-expansion
24
64
12
6
8
4
4new(q)
(q)(q)EE),(
2(q)
(q)E
(q)
(q)E
)(
),(
2),;( 2,22,2
UTi
q
UTiUTqZ
Heterotic String Theory
45
Affine E8
For K3 x T2 compactifications we have for the standard embedding
that preserves N=(4,4)
T is the complexified Kähler modulus, U the complex structure
modulus of the T2
Heterotic String Theory
46
Take away message:
Znew depends on T and U and is connected
to Zelliptic and therefore to M24
Heterotic String Theory
The 1-loop correction to the prepotential is roughly determined
by
and knows about M24 since Znew does
M. Cheng, X. Dong, J. Duncan, J. Harvey, S. Kachru, TW 1306.4981
47
)(),;(),(28
122
new
2
2
QUTeqZ
dUT i
Heterotic String Theory
The 1-loop correction to the prepotential is roughly determined
by
and knows about M24 since Znew does
M. Cheng, X. Dong, J. Duncan, J. Harvey, S. Kachru, TW 1306.4981
The modular invariance of actually tells us that
there is a unique solution. So for all instanton embeddings
(12+n,12-n) the answer is the same. Kiritsis, Kounnas, Petropoulos, Rizos hep-th/9608034
Henningson, Moore hep-th/9608145
48
)(),;(),(28
122
new
2
2
QUTeqZ
dUT i
)(28
12
new2 QZ
Heterotic String Theory
The 1-loop correction to the prepotential is roughly determined
by
and knows about M24 since Znew does
M. Cheng, X. Dong, J. Duncan, J. Harvey, S. Kachru, TW 1306.4981
We have to solve the following second order differential equation Harvey, Moore hep-th/9510182
49
)])()(Re(log[4]2(log[2
)()()(),;(
2
1
)])()(Re(log[1
)1(1
Re
118
8812
2
2
1
11
11
1
28iUiTUT
EbEbQUTqiZ
d
iUJiTJhUTUT
h
Enew
loop
UT
loop
UT
)(),;(),(28
122
new
2
2
QUTeqZ
dUT i
Heterotic String Theory
The solution is given by
where the polylogarithm is given by
50
lk
lUkTiloop eLiklcCUh,
)(2
3
3
311 )()(
1
33 )(n
n
n
xxLi
Heterotic String Theory
The solution is given by
where
51
lk
lUkTiloop eLiklcCUh,
)(2
3
3
311 )()(
Dimensions of M24
1
new )(),(2
),;(2,2
m
mqmcUTi
UTqZ
The solution is given by
where the polylogarithm is given by and the expansion coefficients are the same as in our index (they go along for the ride when integrating)
1
33 )(n
n
n
xxLi
Heterotic String Theory
The solution is given by
where
52
lk
lUkTiloop eLiklcCUh,
)(2
3
3
311 )()(
Dimensions of M24
1
new )(),(2
),;(2,2
m
mqmcUTi
UTqZ
The solution is given by
where the polylogarithm is given by and the expansion coefficients are the same as in our index (they go along for the ride when integrating)
1
33 )(n
n
n
xxLi
Dimensions of M24 (appearing in a spacetime quantity)
Type IIA on CY3
Type IIA string theory
on elliptic fibrations
over Fn (Hirzebruch
surface)
53
Heterotic string
on K3 x T2 with
instanton embedding
(12+n,12-n)
String duality
Type IIA on CY3
Type IIA string theory
on elliptic fibrations
over Fn (Hirzebruch
surface)
Fn
Size of base S2
54
Heterotic string
on K3 x T2 with
instanton embedding
(12+n,12-n)
dilaton S
String duality
Type IIA string theory on elliptic fibrations over Fn:
• Prepotential receives instanton corrections
• These are determined by the Gromow-Witten invariants curve counting (S2, T2, …)
Type IIA on CY3
55
Type IIA string theory on elliptic fibrations over Fn:
• Prepotential receives instanton corrections
• These are determined by the Gromow-Witten invariants curve counting (S2, T2, …)
M. Cheng, X. Dong, J. Duncan, J. Harvey, S. Kachru, TW 1306.4981
M. Alim, E. Scheidegger 1205.1784 A. Klemm, J. Manschot, T. Wotschke 1205.1795
Type IIA on CY3
56
)()()(),,( 2
,
)(2
3
3
31 iS
lk
lUkTi eOeLiklcCUSTUUTSh
Dimensions of M24 (appearing in a spacetime quantity)
Type IIB on CY3
Type IIB string theory
on mirror manifold
57
Type IIA string theory
on elliptic fibrations over
Fn (Hirzebruch surface)
String duality
Type IIB on CY3
Type IIB string theory
on mirror manifold
CY3 manifold Yn
Periods of the holo-morphic 3-form
58
Type IIA string theory
on elliptic fibrations over
Fn (Hirzebruch surface)
CY3 manifold Xn
Gromov-Witten invariants
String duality
Complex Analysis Number Theory
(mock) modular forms, Jacobi forms
(Algebraic) Geometry periods of Calabi-Yau manifolds, Gromov-Witten invariants, elliptic genus, …
Cool new math connections!
Group Theory Representation Theory
finite (sporadic) groups
N E W
String Theory
59
Moonshine and physics
“I have a sneaking hope, a hope unsupported by any facts or any evidence, that sometime in the twenty-first century physicists will stumble upon the Monster group, built in some unsuspected way into the structure of the Universe.”
– Freeman Dyson (1983)
60
Moonshine and physics
For the K3xT2 compactifications, the 1-loop prepotential
controls the 1-loop corrections to the gauge couplings in
the N=2 spacetime theory
61
Moonshine and physics
For the K3xT2 compactifications, the 1-loop prepotential
controls the 1-loop corrections to the gauge couplings in
the N=2 spacetime theory
For four dimensional N=1 models obtained from orbifold
compactifications of the heterotic E8 x E8 string theory:
62
)(),(),,( 2loop1 SieOUTfSUTSf
Moonshine and physics
For the K3xT2 compactifications, the 1-loop prepotential
controls the 1-loop corrections to the gauge couplings in
the N=2 spacetime theory
For four dimensional N=1 models obtained from orbifold
compactifications of the heterotic E8 x E8 string theory:
The (bulk) moduli dependent 1-loop correction to the gauge
kinetic function arises only from N=2 subsectors! Dixon, Louis, Kaplunovsky Nuclear Physics B 355 (1991)
63
)(),(),,( 2loop1 SieOUTfSUTSf
Moonshine and physics
Example T6/Z6-II = T2 x T2 x T2/Z6-II :
64
),,(),,(:, 32
3/2
1
3/
3216 zzezezzzgg ii
II
Z
Moonshine and physics
Example T6/Z6-II = T2 x T2 x T2/Z6-II :
has two N=2 subsector
65
),,(),,(:, 32
3/2
1
3/
3216 zzezezzzgg ii
II
Z
),,(),,(:},,1{
),,(),,(:},,,1{
321321
33
2
32
3/4
1
3/2
321
242
3
zzzzzzgg
zzezezzzggg ii
Z
Z
Moonshine and physics
Example T6/Z6-II = T2 x T2 x T2/Z6-II :
has two N=2 subsector
For which the internal space is T4/Z3 x T2 or T4/Z2 x T2
respectively and therefore an orbifold limit of T2 x K3.
66
),,(),,(:, 32
3/2
1
3/
3216 zzezezzzgg ii
II
Z
),,(),,(:},,1{
),,(),,(:},,,1{
321321
33
2
32
3/4
1
3/2
321
242
3
zzzzzzgg
zzezezzzggg ii
Z
Z
Moonshine and physics
N=2 sectors lead to 1-loop corrections
67
)])()((log[4
)()(log[8
1
),(2
1
||
|'|),(
2
2
,
2
1
3,2,1
loop1
ii
N
i
ii
ii
loop
iUT
i
i
iUiTb
iUJiTJ
UThG
GUTf
ii
N=1 gauge kinetic coupling
N=2 prepotential
Moonshine and physics
N=2 sectors lead to 1-loop corrections
where the prepotential was calculated above
68
)])()((log[4
)()(log[8
1
),(2
1
||
|'|),(
2
2
,
2
1
3,2,1
loop1
ii
N
i
ii
ii
loop
iUT
i
i
iUiTb
iUJiTJ
UThG
GUTf
ii
lk
lUkTiloop eLiklcCUUTh,
)(2
3
3
311 )()(),(
Dimensions of M24
N=1 gauge kinetic coupling
N=2 prepotential
Moonshine and physics
69
No N=2 sectors
Moonshine and physics
70
Moonshine and physics
Four dimensional N=1 models obtained from orbifold
compactifications of the heterotic E8 x E8 string theory
receive universal 1-loop corrections to their gauge
kinetic functions that are related to M24 TW 1402.2973
71
Moonshine and physics
Four dimensional N=1 models obtained from orbifold
compactifications of the heterotic E8 x E8 string theory
receive universal 1-loop corrections to their gauge
kinetic functions that are related to M24 TW 1402.2973
For all T6/ZN, N3,7, and all T6/ZNxZM
72
)(...)()(),,( 2
,
)(2
3
iS
lk
lUkTi
UT eOeLiklcSUTSf
Dimensions of M24
Moonshine and physics
N. Paquette, TW work in progress:
• The holomorphic 3-form plays a role in flux compactifications
Gukov, Vafa, Witten hep-th/9906070 Giddings, Kachru, Polchinski hep-th/0105097
73
... HW
Moonshine and physics
N. Paquette, TW work in progress:
• The holomorphic 3-form plays a role in flux compactifications
Gukov, Vafa, Witten hep-th/9906070 Giddings, Kachru, Polchinski hep-th/0105097
74
... HW
Flux vacua might have large symmetry groups: |M24| 2 x 109
Moonshine and physics
N. Paquette, TW work in progress:
• The holomorphic 3-form plays a role in flux compactifications
Gukov, Vafa, Witten hep-th/9906070 Giddings, Kachru, Polchinski hep-th/0105097
• The Yukawa couplings in heterotic models are given by the third derivative of with respect to the moduli Hosono, Klemm, Theisen, Yau hep-th/9308122
75
... HW
),,( UTShY KJIIJK
New Moonshine phenomena
Consider eight (left-moving) bosons and fermions
compactified on the orbifold T8/ 2 = R8/ΛE8 / 2
The partition function in the NS sector is Frenkel, Lepowsky, Meurman 1985
Duncan math/0502267
76
...1120220482761
)( 23
qqqq
qZ
Sums of dimensions of Conway group
New Moonshine phenomena
Consider eight (left-moving) bosons and fermions
compactified on the orbifold T8/ 2 = R8/ΛE8 / 2
Tensor left- and right movers together in an
asymmetric orbifold where we act with one 2 on
the left and with another one only on the right:
The asymmetric orbifold preserves N=(4,4) symmetry
77
)()(),(partition qZqZqqZ
New Moonshine phenomena
Asymmetric orbifold T8/ 2 x 2 = R8/ΛE8 / 2 x 2
Calculate the elliptic genus and expand in N=4 characters:
Two infinite series, coefficients unrelated to Conway
78
...ch4444ch210
...ch8470ch560
chch21
long
1,
long
1,
long
,
long
,
short
1,
short
0,
25
23
21
25
21
23
21
21
elliptic
lhlh
lhlh
lhlhZ
New Moonshine phenomena
Asymmetric orbifold T8/ 2 x 2 = R8/ΛE8 / 2 x 2
Calculate the elliptic genus and expand in N=4 characters:
All coefficients are dimensions of the Mathieu group M22!
Checks all work and one can see M22 Co preserving N=4 M. Cheng, X. Dong, J. Duncan, S. Harrison, S. Kachru, TW to appear
79
...ch4444ch210
...ch8470ch560
chch21
long
1,
long
1,
long
,
long
,
short
1,
short
0,
25
23
21
25
21
23
21
21
elliptic
lhlh
lhlh
lhlhZ
Conclusion
• Mathieu Moonshine involves K3 that has played a crucial role in superstring compactifications and string dualities
• Certain CY3 manifolds are now also implicated in Mathieu Moonshine
• This leads to a variety of intriguing implications for physically interesting string compactifications
• Much more to come!
80
Conclusion
• Mathieu Moonshine involves K3 that has played a crucial role in superstring compactifications and string dualities
• Certain CY3 manifolds are now also implicated in Mathieu Moonshine
• This leads to a variety of intriguing implications for physically interesting string compactifications
• Much more to come!
THANK YOU! 81
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