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M24, K3 String Theories,
and the Holographic Moonshines
Miranda Cheng Harvard University
String-Math 2011, UPenn, Philadelphia
Friday 10 June 2011
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.
A Short Summary
Friday 10 June 2011
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.
A Short Summary
Friday 10 June 2011
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.
A Short Summary
Friday 10 June 2011
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.
A Short Summary
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
•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]
Friday 10 June 2011
Automorphic Forms in
String Theory
Friday 10 June 2011
String theory is good at producing automorphic forms!
All symmetries have to be reflected in the appropriate partition function.
e.g. space-time symmetries (such as T-, S-dualities)
SL(2,Z)
Σ
e.g. world-sheet symmetries (mapping class group of Σ)M
Friday 10 June 2011
Example(I): Modular Forms From Chiral Bosonic CFT
0→X d==
1
τ
Friday 10 June 2011
Example(I): Modular Forms From Chiral Bosonic CFT
0→X d==
1
τ
q-series from L0-grading=modular form of SL(2,Z)
Friday 10 June 2011
Example(I): Modular Forms From Chiral Bosonic CFT
e.g. 24 chiral bosons
= partition function of chiral half of bosonic strings = supersymmetric partition function of heterotic strings
0→X d==
1
τ
q-series from L0-grading=modular form of SL(2,Z)
Friday 10 June 2011
Example (II): Weak Jacobi Formsfrom 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 Formsfrom 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 Kählerityenhanced 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
SL(2,Z) + SCA spectral flow symmetry is a weak Jacobi form of weight 0
They transform nicely under and have expansions
Example (II): Weak Jacobi Formfrom Elliptic Genus of Calabi-Yau’s
Friday 10 June 2011
Interlude: Twisting and Orbifolding
(automorphism)X
g∈G
Friday 10 June 2011
Interlude: Twisting and Orbifolding
(automorphism)X
g∈G g∈G
G-module
Twisted P.F.
0
==1
τ
→X d
Friday 10 June 2011
Interlude: Twisting and Orbifolding
(automorphism)X
g∈G g∈G
G-module
Twisted P.F.
0
==1
τ
→X d
⇐X/Gorbifold CFT on
Twisted sector P.F.
0
==1
τ
→X d
Friday 10 June 2011
Interlude: Twisting + Orbifolding
SL(2,Z):
==
== ga
(for gh=hg)
SL(2,Z) → Γg
For Zg(τ) ,
Friday 10 June 2011
M
S 1
Fig. 1: The string configuration corresponding to a twisted sec-tor by a given permutation g ∈ SN . The string disentangles intoseperate strings according to the factorization of g into cyclic per-mutations.
2. The Proof
The Hilbert space ofan orbifold field theory [6]is decomposed into twisted sectorsHg, that are labelled by the conjugacy classes [g]of the orbifold group,in our case thesymmetric group SN . Within each twisted sector,one only keeps the states invariantunder the centralizer subgroup Cof g. We willdenote this Cinvariant subspace by
e.g. (2)(4)(3)
2nd quantised string P.F. on X×S1
[Dijkgraaf-Moore-Verlinde2 ‘97]
Example (III): Automorphic Formsfrom Non-Perturbative String Theory
Friday 10 June 2011
M
S 1
Fig. 1: The string configuration corresponding to a twisted sec-tor by a given permutation g ∈ SN . The string disentangles intoseperate strings according to the factorization of g into cyclic per-mutations.
2. The Proof
The Hilbert space ofan orbifold field theory [6]is decomposed into twisted sectorsHg, that are labelled by the conjugacy classes [g]of the orbifold group,in our case thesymmetric group SN . Within each twisted sector,one only keeps the states invariantunder the centralizer subgroup Cof g. We willdenote this Cinvariant subspace by
e.g. (2)(4)(3)
2nd quantised string P.F. on X×S1
Fourier coeff. of
[Dijkgraaf-Moore-Verlinde2 ‘97]
Example (III): Automorphic Formsfrom Non-Perturbative String Theory
Friday 10 June 2011
LIFT: modular forms→automorphic forms
[Gritsenko ’99]almost automorphic for all CY X×(Hodge Anomaly) =
automorphic under O+(3,2;Z)
Example (III): Automorphic Formsfrom Non-Perturbative String Theory
SL(2,Z)~ O+(2,1;Z) →O+(3,2;Z)
[Gritsenko’99]
Friday 10 June 2011
LIFT: modular forms→automorphic forms
[Gritsenko ’99]almost automorphic for all CY X×(Hodge Anomaly) =
automorphic under 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]
Example (III): Automorphic Formsfrom Non-Perturbative String Theory
SL(2,Z)~ O+(2,1;Z) →O+(3,2;Z)
[Gritsenko’99]
Friday 10 June 2011
1/4-BPS States in Type II/K3×T2
The automorphic form counts the 1/4-BPS states of the N=4, d=4 theory: [Dijkgraaf-Verlinde2 ’97]
Friday 10 June 2011
1/4-BPS States in Type II/K3×T2
The automorphic form counts the 1/4-BPS states of the N=4, d=4 theory: [Dijkgraaf-Verlinde2 ’97]
= denominator of a generalised Kac-Moody algebra
Φ
Friday 10 June 2011
1/2-BPS States in Type II/K3×T2
Friday 10 June 2011
1/2-BPS States in Type II/K3×T2
⇐ heterotic/T6
Friday 10 June 2011
1/2-BPS States in Type II/K3×T2
[A. Sen/M.C.-Verlinde ’07, ’08]
1/4-BPS spectrum has to know about the 1/2-BPS spectrum too! 1/2-BPS
1/2-BPS1/4-BPS
Two-Center Bound States↕
Poles in P. F.
⇐ heterotic/T6
Friday 10 June 2011
Weak Jacobi FormZ(τ,z)
lift poles
Automorphic Form Φ(Ω)
Modular Form
Friday 10 June 2011
Sporadic Groupsand
Moonshine Phenomenon
Friday 10 June 2011
Sporadic GroupsThe 26 finite simple groups that don’t come in ∞-families.
Friday 10 June 2011
Sporadic GroupsThe 26 finite simple groups that don’t come in ∞-families.
Friday 10 June 2011
Sporadic GroupsThe 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
modularobjects
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
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
(Partial) Answer: CFT!’88 Frenkel-Lepowsky-Meurmann
(see also Tuite/Dixon-Ginsparg-Harvey)
Example: Monstrous Moonshine
Proven by introducing generalised Kac-Moody algebras and considering the automorphic lifts Φg.
[R. Borcherds ’92]
V♮ = Hilbert space of a chiral CFT (VOA) with -symmetry
Jg(t) = twisted partition function of the CFT( -grading: L0-eigenvalues)
Friday 10 June 2011
Generalised Kac-Moody
Algebra
Sporadic Groups
Automorphic Forms
moonshine
denominator formula
liftModular Objects
automorphism
Friday 10 June 2011
Mathieu 24M24⊂S24 [g]↔”Frame Shape”e.g.
N: one of the 24 Niemeier (24-dim even, self-dual, +-def. ) lattices
NM24
[Mukai ’88, Kondo ’98]
All automorphism G of K3 surfaces preserving the hyper-Kähler structure have G⊂M23⊂M24.
Friday 10 June 2011
Mathieu 24 and
the K3 Automorphic Forms
Friday 10 June 2011
1/2-BPS MoonshineRecall: 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 ’85and 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 [Zwegers ’02/Eguchi-Hikami’10]
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 [Zwegers ’02/Eguchi-Hikami’10]
also dimensions of irreps of M24![Eguchi-Ooguri-Tachikawa ’10]
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 [Zwegers ’02/Eguchi-Hikami’10]
also dimensions of irreps of M24![Eguchi-Ooguri-Tachikawa ’10]
Friday 10 June 2011
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 existThe -cohomology of the N=4 SCFT is an M24 module.
Friday 10 June 2011
Elliptic Genus Moonshine
If this M24-module K = q K1⊕q2 K2⊕q3 K3⊕....
does indeed existThe -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 existThe -cohomology of the N=4 SCFT is an M24 module.
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]
The twisted P.F. transform under some Γg as a wt 1/2 mock modular form.
Friday 10 June 2011
Elliptic Genus MoonshineConjecture 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 coincide with the Zg from M24. [David-Jatkar-Sen ’06]
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 dyons2) It is automorphic under certain subgroups of O+(3,2;R)
[In progress....]Friday 10 June 2011
1/2-BPS
1/2-BPS1/4-BPS
Unifying Moonshine2-Particle Bound States
↕Poles in P. F.
Recall:
[M.C. ’10]
Friday 10 June 2011
1/2-BPS
1/2-BPS1/4-BPS
Unifying Moonshine2-Particle Bound States
↕Poles in P. F.
Recall:
[M.C. ’10]
Friday 10 June 2011
Generalised Kac-Moody
M24Automorphic
Forms Φgmoonshine
denominator formula
liftWeak Jacobi Forms/Mock Modular Forms
Hg
automorphism
Friday 10 June 2011
Generalised Kac-Moody
M24Automorphic
Forms Φgmoonshine
denominator formula
liftWeak Jacobi Forms/Mock Modular Forms
Hg
automorphism
modular formsηg
polesmoonshine
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 prime, Γ0(p)+ is genus zero p| |M|
[Ogg ’73]
Friday 10 June 2011
Genus Zero Property
This has been generalised to• the “generalised moonshine” • groups other than the Monster
see for instanceNorton ’84/Carnahan ’08 Höhn ’03/Duncan ’05, ’06
AN “EXPLANATION”In Monstrous Moonshine, all has genus 0!
Friday 10 June 2011
Genus Zero Property
This has been generalised to• the “generalised moonshine” • groups other than the Monster
see for instanceNorton ’84/Carnahan ’08 Höhn ’03/Duncan ’05, ’06
AN “EXPLANATION”In Monstrous Moonshine, all has genus 0!
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.
[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]
Assuming a CFT has a dual description given by semi-classical AdS gravity
[cf. Heemskerk-Penedones-Polchinski-Sully ’09]
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.
computed from the gravity side by summing over saddle point contributions
The (twisted) partition function Zg(τ) can also be
[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]
Assuming a CFT has a dual description given by semi-classical AdS gravity
[cf. Heemskerk-Penedones-Polchinski-Sully ’09]
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.
computed from the gravity side by summing over saddle point contributions
The (twisted) partition function Zg(τ) can also be
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]
Assuming a CFT has a dual description given by semi-classical AdS gravity
[cf. Heemskerk-Penedones-Polchinski-Sully ’09]
Friday 10 June 2011
Rademacher-Summability
Friday 10 June 2011
Rademacher-Summability
convergent,anomaly-free
Friday 10 June 2011
Rademacher-Summability
[Rademacher 1939]
convergent,anomaly-free
Friday 10 June 2011
Rademacher-Summability
[Rademacher 1939]
[Duncan-Frenkel ’09]
e.g. For the special case that Zg is a modular function (weight 0, weakly holomorphic)
Rademacher-summability
Γg is genus zeroZg
convergent,anomaly-free
Friday 10 June 2011
Rademacher-Summability ofthe M24 Moonshine
Here we have Hg(τ) = weight 1/2 Mock modular form.Rademacher-summability Γg is genus zero
Friday 10 June 2011
Rademacher-Summability ofthe 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 Eguchi-Hikami (’09), we show that all the M24 mock modular forms Hg(τ) can be written as a Rademacher sum.
Friday 10 June 2011
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.
To Summarize
Friday 10 June 2011
All known theories of moonshine have a CFT interpretation.
Zg(τ) = modular functionse.g. Monster moonshine
g=0
Assuming the existence of a dual description
All McKay-Thomson series Zg(τ) have to be Rademacher summable.
To Summarize
Friday 10 June 2011
All known theories of moonshine have a CFT interpretation.
Zg(τ) = modular functionse.g. Monster moonshine
g=0
Zg(τ) = mock modular formsfor the new M24 moonshine
verified
Assuming the existence of a dual description
All McKay-Thomson series Zg(τ) have to be Rademacher summable.
To Summarize
Friday 10 June 2011
Some Whiskey for physicists?
Friday 10 June 2011
AdS/CFT
Friday 10 June 2011
AdS/CFT
AdS/CMT
Friday 10 June 2011
AdS/CFT
AdS/CMT
AdS/QCD(Heavy Ion Physics)
Friday 10 June 2011
AdS/CFT
AdS/CMT
AdS/QCD(Heavy Ion Physics)
AdS/NTAdS/Math
??
Friday 10 June 2011
Thank You!
Friday 10 June 2011
