String-Math 2011, UPenn, Philadelphia
M24, K3 String Theories,
and the
Holographic Moonshines
Miranda Cheng
Harvard University
Friday 10 June 2011
A Short Summary
The moonshine phenomenon, which describes an unexpected relation
between sporadic groups and modular objects, has been one of the most
exciting developments in mathematics in the last century.
Friday 10 June 2011
A Short Summary
The moonshine phenomenon, which describes an unexpected relation
between sporadic groups and modular objects, has been one of the most
exciting developments in mathematics in the last century.
String theory has been proven vital in the understanding of such a
connection, for instance in the case of the famous Monstrous Moonshine.
Friday 10 June 2011
A Short Summary
The moonshine phenomenon, which describes an unexpected relation
between sporadic groups and modular objects, has been one of the most
exciting developments in mathematics in the last century.
String theory has been proven vital in the understanding of such a
connection, for instance in the case of the famous Monstrous Moonshine.
Last year, a new moonshine with many interesting novel features has
been proposed, this time for the largest Mathieu group M24. We will see
how string theory on K3 ties various automorphic objects with
(conjectured) M24 symmetry together into an intricate web.
Friday 10 June 2011
A Short Summary
The moonshine phenomenon, which describes an unexpected relation
between sporadic groups and modular objects, has been one of the most
exciting developments in mathematics in the last century.
String theory has been proven vital in the understanding of such a
connection, for instance in the case of the famous Monstrous Moonshine.
Last year, a new moonshine with many interesting novel features has
been proposed, this time for the largest Mathieu group M24. We will see
how string theory on K3 ties various automorphic objects with
(conjectured) M24 symmetry together into an intricate web.
Finally, we will demonstrate how AdS/CFT considerations
provide natural explanations for some crucial properties of
the modular groups appearing in all known moonshines.
Friday 10 June 2011
Outline
•
•
•
•
Friday 10 June 2011
Automorphic Forms and String Theory
Sporadic Groups and Moonshine Phenomenon
M24 and the K3 Automorphic Forms
Holographic Modularity of the Moonshines
[to appear with John F. Duncan]
Automorphic Forms
in
String Theory
Friday 10 June 2011
String theory is good at producing automorphic forms!
e.g. world-sheet symmetries (mapping class group of Σ)
M
Σ
e.g. space-time symmetries (such as T-, S-dualities)
SL(2,Z)
All symmetries have to be reflected in the
appropriate partition function.
Friday 10 June 2011
Example(I):
Modular Forms From Chiral Bosonic CFT
τ
=
0
Friday 10 June 2011
=
1
→Xd
Example(I):
Modular Forms From Chiral Bosonic CFT
τ
=
0
=
→Xd
1
q-series from L0-grading= modular form of SL(2,Z)
Friday 10 June 2011
Example(I):
Modular Forms From Chiral Bosonic CFT
τ
=
0
=
→Xd
1
q-series from L0-grading= modular form of SL(2,Z)
e.g. 24 chiral bosons
= partition function of chiral half of bosonic strings
= supersymmetric partition function of heterotic
strings
Friday 10 June 2011
Example (II): Weak Jacobi Forms
from Elliptic Genus of Calabi-Yau’s
N=(2,2) 2d sigma model with Calabi-Yau target space X has
and
N=2 SCA, with conserved currents J, G±,T.
Counting
ground states (computing the -cohomology),
graded by quantum numbers of the
SCFA, we get
Friday 10 June 2011
Example (II): Weak Jacobi Forms
from Elliptic Genus of Calabi-Yau’s
N=(2,2) 2d sigma model with Calabi-Yau target space X has
and
N=2 SCA, with conserved currents J, G±,T.
from Kählerity
enhanced to N=4 when X is hyper-Kähler
Counting
ground states (computing the -cohomology),
graded by quantum numbers of the
SCFA, we get
Friday 10 June 2011
Example (II): Weak Jacobi Form
from Elliptic Genus of Calabi-Yau’s
SL(2,Z) + SCA spectral flow symmetry
is a weak Jacobi form of weight 0
They transform nicely under
and have expansions
Friday 10 June 2011
Interlude: Twisting and Orbifolding
(automorphism)
X
g∈G
Friday 10 June 2011
Interlude: Twisting and Orbifolding
G-module
(automorphism)
X
g∈G
g∈G
Twisted P.F.
τ
=
0
Friday 10 June 2011
=
1
→Xd
Interlude: Twisting and Orbifolding
G-module
(automorphism)
orbifold CFT on
X/G
⇐
Twisted sector P.F.
X
g∈G
g∈G
Twisted P.F.
τ
τ
=
0
Friday 10 June 2011
=
1
→Xd
=
0
=
1
→Xd
=
Interlude: Twisting + Orbifolding
=
g
a
=
SL(2,Z):
=
(for gh=hg)
For Zg(τ) ,
SL(2,Z) → Γg
Friday 10 June 2011
Example (III): Automorphic Forms
from Non-Perturbative String Theory
M
e.g. (2)(4)(3)
2nd quantised string P.F. on X×S1
S1
Fig. 1: The string configuration corresponding to a twisted sector by a given permutation g ∈ SN . The string disentangles into
seperate strings according to the factorization of g into cyclic permutations.
[Dijkgraaf-Moore-Verlinde2 ‘97]
2. The Proof
Friday 10 June 2011
The Hilbert space of an orbifold field theory [6] is decomposed into tw
Hg , that are labelled by the conjugacy classes [g] of the orbifold group, in
symmetric group SN . Within each twisted sector, one only keeps the sta
Example (III): Automorphic Forms
from Non-Perturbative String Theory
M
e.g. (2)(4)(3)
2nd quantised string P.F. on X×S1
S1
Fig. 1: The string configuration corresponding to a twisted secFourier
coeff. of
tor by a given permutation g ∈ S . The string disentangles into
N
seperate strings according to the factorization of g into cyclic permutations.
[Dijkgraaf-Moore-Verlinde2 ‘97]
2. The Proof
Friday 10 June 2011
The Hilbert space of an orbifold field theory [6] is decomposed into tw
Hg , that are labelled by the conjugacy classes [g] of the orbifold group, in
symmetric group SN . Within each twisted sector, one only keeps the sta
Example (III): Automorphic Forms
from Non-Perturbative String Theory
almost automorphic
X[Gritsenko’99]
’99]
×(Hodge
Anomaly) = for all CY[Gritsenko
automorphic under O+(3,2;Z)
LIFT: modular forms→automorphic forms
SL(2,Z)~ O+(2,1;Z) →O+(3,2;Z)
Friday 10 June 2011
Example (III): Automorphic Forms
from Non-Perturbative String Theory
almost automorphic
X[Gritsenko’99]
’99]
×(Hodge
Anomaly) = for all CY[Gritsenko
automorphic under O+(3,2;Z)
LIFT: modular forms→automorphic forms
SL(2,Z)~ O+(2,1;Z) →O+(3,2;Z)
When X=K3,
corresponds to further compactification
type II to 4-dim on K3×T2.
[Shih-Strominger-Yin/ Jatkar-Sen ’05]
Friday 10 June 2011
1/4-BPS States in Type
The automorphic form
N=4, d=4 theory:
Friday 10 June 2011
2
II/K3×T
counts the 1/4-BPS states of the
[Dijkgraaf-Verlinde2 ’97]
1/4-BPS States in Type
The automorphic form
N=4, d=4 theory:
2
II/K3×T
counts the 1/4-BPS states of the
[Dijkgraaf-Verlinde2 ’97]
Φ
= denominator of a generalised Kac-Moody algebra
Friday 10 June 2011
1/2-BPS States in Type
Friday 10 June 2011
2
II/K3×T
1/2-BPS States in Type
⇐
Friday 10 June 2011
2
II/K3×T
heterotic/T6
1/2-BPS States in Type
⇐
2
II/K3×T
heterotic/T6
1/4-BPS spectrum has to know about the 1/2-BPS spectrum too!
Two-Center Bound States
↕
Poles in P. F.
1/2-BPS
1/4-BPS
1/2-BPS
[A. Sen/M.C.-Verlinde ’07, ’08]
Friday 10 June 2011
Automorphic
Form Φ(Ω)
lift
Weak Jacobi Form
Z(τ,z)
Friday 10 June 2011
poles
Modular Form
Sporadic Groups
and
Moonshine Phenomenon
Friday 10 June 2011
Sporadic Groups
The 26 finite simple groups that don’t
come in ∞-families.
Friday 10 June 2011
Sporadic Groups
The 26 finite simple groups that don’t
come in ∞-families.
Friday 10 June 2011
Sporadic Groups
The 26 finite simple groups that don’t
come in ∞-families.
|M|~8×1053
largest Mathieu group
~2×109
Friday 10 June 2011
The Beauty of the Misfits:
The Moonshine Phenomenon
sporadic
group
Friday 10 June 2011
modular
objects
Example: Monstrous Moonshine
Klein invariant
Friday 10 June 2011
Example: Monstrous Moonshine
Klein invariant
Friday 10 June 2011
Example: Monstrous Moonshine
Klein invariant
Friday 10 June 2011
Example: Monstrous Moonshine
If true, can also consider the characters (McKay-Thomson series)
Friday 10 June 2011
Example: Monstrous Moonshine
If true, can also consider the characters (McKay-Thomson series)
Moonshine Conjecture (Conway-Norton ’79):
Jg(τ) is invariant under some genus zero Γg⊂SL(2,R).
Friday 10 June 2011
Example: Monstrous Moonshine
If true, can also consider the characters (McKay-Thomson series)
Moonshine Conjecture (Conway-Norton ’79):
Jg(τ) is invariant under some genus zero Γg⊂SL(2,R).
Q: Why are sporadic groups related to modular forms?
Friday 10 June 2011
Example: Monstrous Moonshine
(Partial) Answer: CFT!
’88 Frenkel-Lepowsky-Meurmann
(see also Tuite/Dixon-Ginsparg-Harvey)
V♮ = Hilbert space of a chiral CFT (VOA) with
( -grading: L0-eigenvalues)
-symmetry
Jg(t) = twisted partition function of the CFT
Proven by introducing generalised Kac-Moody
algebras and considering the automorphic lifts Φg.
[R. Borcherds ’92]
Friday 10 June 2011
Generalised
Kac-Moody
Algebra
automorphism
Automorphic
Forms
Sporadic
Groups
moonshine
lift
Modular
Objects
Friday 10 June 2011
denominator
formula
Mathieu 24
M24⊂S24
[g]↔”Frame Shape”
e.g.
N
M24
N: one of the 24 Niemeier (24-dim
even, self-dual, +-def. ) lattices
All automorphism G of K3 surfaces preserving
the hyper-Kähler structure have G⊂M23⊂M24.
[Mukai ’88, Kondo ’98]
Friday 10 June 2011
Mathieu 24
and
the K3 Automorphic Forms
Friday 10 June 2011
1/2-BPS Moonshine
Recall: 1/2-BPS states counted by
Clearly, the Hilbert space has a M24⊂S24 symmetry and the
corresponding twisted partition functions are given by
[g]↔”Frame Shape” ↔ηg(τ)
e.g.
[See related discussions about Ramanunjan numbers by G. Mason ’85
and a related observation in Govindarajan-Krishna ’09]
Friday 10 June 2011
Elliptic Genus Moonshine
[Eguchi-Ooguri-Taormina-Yang ’89]
Friday 10 June 2011
Elliptic Genus Moonshine
[Eguchi-Ooguri-Taormina-Yang ’89]
number of massive N=4 SCA representations
weight 1/2 Mock Modular Form
Friday 10 June 2011
[Zwegers ’02/
Eguchi-Hikami’10]
Elliptic Genus Moonshine
[Eguchi-Ooguri-Taormina-Yang ’89]
number of massive N=4 SCA representations
also dimensions of irreps of M24!
[Eguchi-Ooguri-Tachikawa ’10]
weight 1/2 Mock Modular Form
Friday 10 June 2011
[Zwegers ’02/
Eguchi-Hikami’10]
Elliptic Genus Moonshine
[Eguchi-Ooguri-Taormina-Yang ’89]
number of massive N=4 SCA representations
also dimensions of irreps of M24!
[Eguchi-Ooguri-Tachikawa ’10]
weight 1/2 Mock Modular Form
Friday 10 June 2011
[Zwegers ’02/
Eguchi-Hikami’10]
Elliptic Genus Moonshine
If this M24-module
K = q K1⊕q2 K2⊕q3 K3⊕....
does indeed exist
Friday 10 June 2011
Elliptic Genus Moonshine
If this M24-module
K = q K1⊕q2 K2⊕q3 K3⊕....
does indeed exist
The
Friday 10 June 2011
-cohomology of the N=4 SCFT is an M24 module.
Elliptic Genus Moonshine
If this M24-module
K = q K1⊕q2 K2⊕q3 K3⊕....
does indeed exist
The
-cohomology of the N=4 SCFT is an M24 module.
The twisted P.F.
transform under some Γg as a wt 1/2 mock modular form.
Friday 10 June 2011
Elliptic Genus Moonshine
If this M24-module
K = q K1⊕q2 K2⊕q3 K3⊕....
does indeed exist
The
-cohomology of the N=4 SCFT is an M24 module.
The twisted P.F.
transform under some Γg as a wt 1/2 mock modular form.
Status: Such mock modular forms Hg(τ) transforming
under Γ0(ord g) have been proposed for all [g]⊂M24.
[M.C. /Gaberdiel-Hohenneger-Volpato/Eguchi-Hikami ’10]
Friday 10 June 2011
Elliptic Genus Moonshine
Conjecture 1:
check 1: Kn have been computed for n≤600.
check 2: when g generates an actual symmetry G of the K3
surface, the full Hilbert space (not just the cohomology) is an G-module and Zg can be
computed explicitly. The ones computed in this way
[David-Jatkar-Sen ’06]
coincide with the Zg from M24.
I bet you that it’s true!
Friday 10 June 2011
1/4-BPS Moonshine
[M.C. ’10]
A Consequence:
The root system of the GKM is an M24-module.
(up to a small subtlety that is not important here)
[In progress....]
Friday 10 June 2011
1/4-BPS Moonshine
[M.C. ’10]
A Consequence:
The root system of the GKM is an M24-module.
(up to a small subtlety that is not important here)
we can compute the twisted denominator Φg from the
twisted elliptic genera Zg.
[In progress....]
Friday 10 June 2011
1/4-BPS Moonshine
[M.C. ’10]
A Consequence:
The root system of the GKM is an M24-module.
(up to a small subtlety that is not important here)
we can compute the twisted denominator Φg from the
twisted elliptic genera Zg.
Conjecture 2:
1) It is the twisted partition function for 1/4-BPS dyons
2) It is automorphic under certain subgroups of O+(3,2;R)
[In progress....]
Friday 10 June 2011
Unifying Moonshine
Recall: 2-Particle Bound States
↕
1/4-BPS
Poles in P. F.
Friday 10 June 2011
[M.C. ’10]
1/2-BPS
1/2-BPS
Unifying Moonshine
Recall: 2-Particle Bound States
↕
1/4-BPS
Poles in P. F.
Friday 10 June 2011
[M.C. ’10]
1/2-BPS
1/2-BPS
Generalised
Kac-Moody
denominator
formula
automorphism
Automorphic
Forms Φg
M24
moonshine
Weak Jacobi Forms/
Mock Modular Forms
Hg
Friday 10 June 2011
lift
Generalised
Kac-Moody
denominator
formula
automorphism
Automorphic
Forms Φg
M24
moonshine
lift
Weak Jacobi Forms/
Mock Modular Forms
Hg
moonshine
poles
modular forms
ηg
Friday 10 June 2011
Holographic Modularity
of the Moonshines
with John Duncan 1106.xxxx [math.RT]
Friday 10 June 2011
Genus Zero Property
Genus zero groups Γ⊂SL(2,R) are rare.
The famous Jack Daniel’s: WHY?
p| |M|
p prime, Γ0(p)+ is genus zero
[Ogg ’73]
Friday 10 June 2011
Genus Zero Property
AN “EXPLANATION”
In Monstrous Moonshine, all
This has been generalised to
• the “generalised moonshine”
• groups other than the Monster
Friday 10 June 2011
has genus 0!
see for instance
Norton ’84/Carnahan ’08
Höhn ’03/Duncan ’05, ’06
Genus Zero Property
AN “EXPLANATION”
In Monstrous Moonshine, all
This has been generalised to
• the “generalised moonshine”
• groups other than the Monster
has genus 0!
see for instance
Norton ’84/Carnahan ’08
Höhn ’03/Duncan ’05, ’06
BUT WHY GENUS ZERO??
Friday 10 June 2011
NO Genus Zero for
the New M24 Moonshine
Heresy!
But true, by inspecting
Friday 10 June 2011
AdS3/CFT2
Recall: Saddle points in Euclidean 3d gravity with aAdS boundary
conditions are labeled by
H3/Γ, Γ⊂SL(2,R)= group of large diffeomorphism transformations.
[see also the “Farey Tail” papers: Dijkgraaf-Maldacena-Moore-Verlinde ’00/
Kraus-Larsen/Dijkgraaf-de Boer-M.C.-Manschot-Verlinde/Denef-Moore/Manschot-Moore ’06]
Friday 10 June 2011
AdS3/CFT2
Recall: Saddle points in Euclidean 3d gravity with aAdS boundary
conditions are labeled by
H3/Γ, Γ⊂SL(2,R)= group of large diffeomorphism transformations.
Assuming a CFT has a dual description
given by semi-classical AdS gravity
[cf. Heemskerk-Penedones-Polchinski-Sully ’09]
[see also the “Farey Tail” papers: Dijkgraaf-Maldacena-Moore-Verlinde ’00/
Kraus-Larsen/Dijkgraaf-de Boer-M.C.-Manschot-Verlinde/Denef-Moore/Manschot-Moore ’06]
Friday 10 June 2011
AdS3/CFT2
Recall: Saddle points in Euclidean 3d gravity with aAdS boundary
conditions are labeled by
H3/Γ, Γ⊂SL(2,R)= group of large diffeomorphism transformations.
Assuming a CFT has a dual description
given by semi-classical AdS gravity
[cf. Heemskerk-Penedones-Polchinski-Sully ’09]
The (twisted) partition function Zg(τ) can also be
computed from the gravity side by summing over saddle
point contributions
[see also the “Farey Tail” papers: Dijkgraaf-Maldacena-Moore-Verlinde ’00/
Kraus-Larsen/Dijkgraaf-de Boer-M.C.-Manschot-Verlinde/Denef-Moore/Manschot-Moore ’06]
Friday 10 June 2011
AdS3/CFT2
Recall: Saddle points in Euclidean 3d gravity with aAdS boundary
conditions are labeled by
H3/Γ, Γ⊂SL(2,R)= group of large diffeomorphism transformations.
Assuming a CFT has a dual description
given by semi-classical AdS gravity
[cf. Heemskerk-Penedones-Polchinski-Sully ’09]
The (twisted) partition function Zg(τ) can also be
computed from the gravity side by summing over saddle
point contributions
Zg(τ) has to be Rademacher-summable!
[see also the “Farey Tail” papers: Dijkgraaf-Maldacena-Moore-Verlinde ’00/
Kraus-Larsen/Dijkgraaf-de Boer-M.C.-Manschot-Verlinde/Denef-Moore/Manschot-Moore ’06]
Friday 10 June 2011
Rademacher-Summability
Friday 10 June 2011
Rademacher-Summability
convergent,
anomaly-free
Friday 10 June 2011
Rademacher-Summability
convergent,
anomaly-free
[Rademacher 1939]
Friday 10 June 2011
Rademacher-Summability
convergent,
anomaly-free
[Rademacher 1939]
e.g. For the special case that Zg is a
modular function (weight 0, weakly holomorphic)
Zg
Rademacher-summability
Γg is genus zero
Friday 10 June 2011
[Duncan-Frenkel ’09]
Rademacher-Summability of
the M24 Moonshine
Here we have Hg(τ) = weight 1/2 Mock modular form.
Rademacher-summability
Friday 10 June 2011
Γg is genus zero
Rademacher-Summability of
the M24 Moonshine
Here we have Hg(τ) = weight 1/2 Mock modular form.
Rademacher-summability
Γg is genus zero
Instead, using the results of Bringmann-Ono (’06) and EguchiHikami (’09), we show that all the M24 mock modular
forms Hg(τ) can be written as a Rademacher sum.
Friday 10 June 2011
To Summarize
All known theories of moonshine have a CFT
interpretation.
Assuming the existence of a dual description
All McKay-Thomson series Zg(τ) have to be
Rademacher summable.
Friday 10 June 2011
To Summarize
All known theories of moonshine have a CFT
interpretation.
Assuming the existence of a dual description
All McKay-Thomson series Zg(τ) have to be
Rademacher summable.
Zg(τ) = modular functions
e.g. Monster moonshine
g=0
Friday 10 June 2011
To Summarize
All known theories of moonshine have a CFT
interpretation.
Assuming the existence of a dual description
All McKay-Thomson series Zg(τ) have to be
Rademacher summable.
Zg(τ) = modular functions
e.g. Monster moonshine
g=0
Friday 10 June 2011
Zg(τ) = mock modular forms
for the new M24 moonshine
verified
Some Whiskey for physicists?
Friday 10 June 2011
AdS/CFT
Friday 10 June 2011
AdS/CMT
AdS/CFT
Friday 10 June 2011
AdS/CMT
AdS/CFT
Friday 10 June 2011
AdS/QCD
(Heavy Ion Physics)
AdS/CMT
AdS/CFT
AdS/QCD
(Heavy Ion Physics)
??
Friday 10 June 2011
AdS/NT
AdS/Math
Thank You!
Friday 10 June 2011