generalised moonshine in the elliptic genus of k3 · generalised moonshine in the elliptic genus of...

Generalised Moonshinein the elliptic genus of K3

“Algebra, Geometry and the Physics of BPS-States”Hausdorff Research Institute for Mathematics,

Bonn, November 12, 2012

Daniel Persson

Chalmers University of Technology

Based on work to appear with M. Gaberdiel, H. Ronellenfitsch & R. Volpato

In 2010, Eguchi, Ooguri, Tachikawa conjectured that there is “Moonshine” in the elliptic genus of K3 connected to the finite sporadic group M24 ⇢ S24

EOT observation: Fourier coefficients of K3-elliptic genus are (sums of) dimensions of irreps of M24

representation theory of finite groups

modular forms conformal field theory

The word “moonshine” generally refers to surprising connections between a priori unrelated parts of mathematics and physics, involving:

generalised Kac-Moody algebras

representation theory of finite groups

modular forms conformal field theory

The word “moonshine” generally refers to surprising connections between a priori unrelated parts of mathematics and physics, involving:

The prime example of course being Monstrous Moonshine.

generalised Kac-Moody algebras

Monstrous Moonshine pertains to the “characters” (McKay-Thompson series)

Moonshine conjecture (Conway-Norton): The McKay-Thompson series are modular-invariant under some genus zero

genus zero �g\H ⇠

The conjecture was proven by Borcherds in 1992.

8g 2 M

�g

Tg(⌧) = TrV \(g qL0�1) =1X

n=�1

TrV \n(g)qn

�g ⇢ SL(2, R)

V \ =1M

n=�1

V \n

graded -moduleM

In 1987 Norton proposed a vast generalisation of the monstrous moonshine conjecture, under the name of Generalised Moonshine.

The aim of this talk is to address the analogue of generalised moonshine for the recently discovered connection between the K3 elliptic genus and M24

Motivation?

Of independent mathematical interest to understand new “moonshine phenomena”.

Wall-crossing and (twisted) N=4 dyons in CHL-models.

BPS-algebras.

...and many more

Outline

1. Generalised Monstrous Moonshine

2. Mathieu Moonshine

3. Generalised Mathieu Moonshine

5. Discussion and Outlook

4. Wall-crossing and Automorphic Lifts

Generalised Monstrous Moonshine


Norton proposed a generalisation of the monstrous moonshine conjecture:

To each pair of commuting elements Norton associatesa holomorphic function on the upper half plane , satisfying

g, h 2 MZg,h(⌧) H

(“Maxi-Moonshine”)



Zg,h

(⌧) = Zxgx

�1,xhx

�1(⌧) 8x 2 M



g, h 2 MZg,h(⌧) H



Zg,h

(⌧) = Zxgx

�1,xhx

�1(⌧) 8x 2 M

Zgahc,gbhd(⌧) = �Zg,h

⇣a⌧ + b

c⌧ + d

⌘ ✓a bc d

◆2 SL(2, Z) �24 = 1



g, h 2 MZg,h(⌧) H



Zg,h

(⌧) = Zxgx

�1,xhx

�1(⌧)


⇣a⌧ + b

c⌧ + d

⌘ ✓a bc d

◆2 SL(2, Z) �24 = 1

The coefficients in the q-expansion of are characters of a gradedprojective representation of the centraliser

Zg,h(⌧)CM(g) = {x 2 M |xgx

�1 = g}



g, h 2 MZg,h(⌧) H

8x 2 M



Zg,h

(⌧) = Zxgx

�1,xhx

�1(⌧)


⇣a⌧ + b

c⌧ + d

⌘ ✓a bc d

◆2 SL(2, Z) �24 = 1

Zg,h(⌧) is either constant or is modular invariant under some genus zero congruence subgroup of



g, h 2 MZg,h(⌧) H

8x 2 M

SL(2, R)



�1 = g}



Zg,h

(⌧) = Zxgx

�1,xhx

�1(⌧)


⇣a⌧ + b

c⌧ + d

⌘ ✓a bc d

◆2 SL(2, Z) �24 = 1

Z1,1(⌧) = J(⌧)

Z1,h(⌧) = Th(⌧) } reduces to the original moonshine conjecture



g, h 2 MZg,h(⌧) H

8x 2 M

Zg,h(⌧) is either constant or is modular invariant under some genus zero congruence subgroup of SL(2, R)



�1 = g}

Generalised Monstrous Moonshine(“Maxi-Moonshine”)

Generalised Moonshine is so far unproven, except in special cases

G. Höhn (Baby Monster)

M. Tuite

S. Carnahan is working towards a complete proof.

Generalised Moonshine and Holomorphic Orbifolds

It was pointed out early on by Dixon-Ginsparg-Harvey and Tuite that Norton’s generalised moonshine conjecture has a natural interpretation in terms of orbifolds of the Frenkel-Lepowsky-Meurman monster CFT . V \


Orbifolding by some element introduces “twisted sectors” V \ g 2 M V \g

and one may consider the twisted CFT characters

g

1

= TrV \g

�qL0�1

�






g

1

= TrV \g

�qL0�1

�

boundary conditions on the torus

0 1

⌧

g

1



g

1

= TrV \g

�qL0�1

�

For any one may further consider the “twisted twined” characterh 2 CM(g)

g

h

= TrV \g

�h qL0�1

�

which then precisely equals the Norton series Zg,h(⌧)



Most of the properties of conjectured by Norton now become natural consequences of known results in orbifold CFT.

Zg,h

Except for the genus zero property which still lacks a satisfactory explanation.


Mathieu Moonshine

Mathieu Moonshine

Non-linear sigma models with target space K3

superconformal algebra with N = (4, 4) c = 6

Large moduli space of such theories:

M = O(4, 20; Z)\O(4, 20; R)/(O(4)⇥O(20))

The physical spectrum varies over moduli space, but there is a graded “partition function”, the elliptic genus, which is constant.

The elliptic genus of only receives contributions from the right-movingRR-ground states.

K3

The elliptic genus is defined by

�K3 = TrHRR

⇣(�1)J0+J̄0qL0�c/24q̄L̄0�c/24yJ0

⌘(q = e2⇡i⌧ , y = e2⇡iz)[Witten][Kawai, Yamada, Yang]

�K3 = TrHRR

⇣(�1)J0+J̄0qL0�c/24q̄L̄0�c/24yJ0

⌘(q = e2⇡i⌧ , y = e2⇡iz)

Cartan generator in the left of SU(2) N = (4, 4)

The elliptic genus is defined by [Witten][Kawai, Yamada, Yang]

�K3 = TrHRR

⇣(�1)J0+J̄0qL0�c/24q̄L̄0�c/24yJ0

⌘(q = e2⇡i⌧ , y = e2⇡iz)

Holomorphic in both and ⌧ z

Modular and elliptic properties:

�K3

⇣a⌧ + b

c⌧ + d,

z

c⌧ + d

⌘= e2⇡i cz2

c⌧+d �K3(⌧, z)

�K3(⌧, z + `⌧ + `0) = e�2⇡i(`2⌧+2`z)�K3(⌧, z) `, `0 2 Z

✓a bc d

◆2 SL(2, Z)


�K3 = TrHRR

⇣(�1)J0+J̄0qL0�c/24q̄L̄0�c/24yJ0

⌘(q = e2⇡i⌧ , y = e2⇡iz)

�K3

⇣a⌧ + b

c⌧ + d,

z

c⌧ + d

⌘= e2⇡i cz2

c⌧+d �K3(⌧, z)

�K3(⌧, z + `⌧ + `0) = e�2⇡i(`2⌧+2`z)�K3(⌧, z) `, `0 2 Z

✓a bc d

◆2 SL(2, Z)

follows from spectral flow




�K3 = TrHRR

⇣(�1)J0+J̄0qL0�c/24q̄L̄0�c/24yJ0

⌘(q = e2⇡i⌧ , y = e2⇡iz)

�K3

⇣a⌧ + b

c⌧ + d,

z

c⌧ + d

⌘= e2⇡i cz2

c⌧+d �K3(⌧, z)

�K3(⌧, z + `⌧ + `0) = e�2⇡i(`2⌧+2`z)�K3(⌧, z) `, `0 2 Z

✓a bc d

◆2 SL(2, Z)

This identifies with a weak Jacobi form of weight 0 and index 1: �K3

�K3 = �0,1 = 8✓

#2(⌧, z)2

#2(⌧, 0)2+

#3(⌧, z)2

#3(⌧, 0)2+

#4(⌧, z)2

#4(⌧, 0)2

◆

[Eguchi, Ooguri, Taormina, Yang]




The elliptic genus further satisfies

�K3(⌧, z = 0) = �(K3) = 24

Now denote by the subspace of right-moving ground states in HRR

We have a decomposition of into irreps of the left superconformal algebra, which induces a decomposition of the elliptic genus:

where �h,j(⌧, z) = TrHh,j

⇣(�1)J0qL0� c

24 yJ0⌘

HH

�K3(⌧, z) = 20 · �h=

14 ,j=0

� 2 · �h=

14 ,j=

12

+1X

n=1

An · �h=

14+n,j=

12

The elliptic genus further satisfies

�K3(⌧, z = 0) = �(K3) = 24

Now denote by the subspace of right-moving ground states in HRR

We have a decomposition of into irreps of the left superconformal algebra, which induces a decomposition of the elliptic genus:

HH

�K3(⌧, z) = 20 · �h=

14 ,j=0

� 2 · �h=

14 ,j=

12

+1X

n=1

An · �h=

14+n,j=

12

massless representations (BPS)massive representations (non-BPS)

Eguchi, Ooguri, Tachikawa observed

A1 = 90 = 45 + 45

A2 = 462 = 231 + 231

A3 = 1540 = 770 + 770


A1 = 90 = 45 + 45

A2 = 462 = 231 + 231

A3 = 1540 = 770 + 770

Dimensions of irreducible representations of the largest Mathieu group ! M24

M24EOT conjecture: H


A1 = 90 = 45 + 45

A2 = 462 = 231 + 231

A3 = 1540 = 770 + 770

By now, the Mathieu Moonshine conjecture has been established in the sense that all the twining genera have been found.�g

[Cheng][Gaberdiel, Hohenegger, Volpato][Eguchi, Hikami][Gannon]

Generalised Mathieu Moonshine

[Gaberdiel, D.P., Ronellenfitsch, Volpato] (to appear)

Twisted twining elliptic genera

We now want to extend the story and consider the -analogue of Norton’s generalised monstrous moonshine conjecture.

M24

We are thus led to define twisted twining genera as follows:

for commuting pairs g, h 2 M24

�g,h(⌧, z) = TrHg

⇣h (�1)J0+J̄0qL0�c/24q̄L̄0�c/24yJ0

⌘

The trace is taken over the twisted Hilbert space .Hg


We now want to extend the story and consider the -analogue of Norton’s generalised monstrous moonshine conjecture.

M24

We are thus led to define twisted twining genera as follows:

for commuting pairs

The trace is taken over the twisted Hilbert space .

g, h 2 M24

If generalised Mathieu moonshine holds, should be a weight 0, index 1weak Jacobi form for some with generalised multiplier system

�g,h�g,h ⇢ SL(2, Z)

�g,h : �g,h ! U(1)

�g,h

⇣a⌧ + b

c⌧ + d,

z

c⌧ + d

⌘= �g,h

⇣ a bc d

⌘e2⇡i cz2

c⌧+d �g,h(⌧, z)

�g,h(⌧, z) = TrHg

⇣h (�1)J0+J̄0qL0�c/24q̄L̄0�c/24yJ0

⌘

Hg


�g,h : �g,h ! U(1)

�g,h

⇣a⌧ + b

c⌧ + d,

z

c⌧ + d

⌘= �g,h

⇣ a bc d

⌘e2⇡i cz2

c⌧+d �g,h(⌧, z)

In order to make sense of this we recall that the twisted twining genera can be interpreted as characters in an orbifold theory.

To analyze the twisted twining genera and their connection with we must therefore bring in some orbifold machinery.

M24

Holomorphic Orbifolds and Group Cohomology

Consider a holomorphic CFT with .V G = Aut(V)


Zg,h(⌧) = g

h

= TrVg

�h qL0�1

�8h 2 CG(g) = Aut(Vg)


For each there is a unique twisted sector with characterg 2 G Vg


Zg,h(⌧) = g

h

= TrVg

�h qL0�1

�8h 2 CG(g) = Aut(Vg)

The full partition function of the orbifold theory is then V/G

Z(⌧) =X

g

1|CG(g)|

X

h2CG(g)

Zg,h(⌧)

[Dijkgraaf, Vafa, Verlinde, Verlinde]


For each there is a unique twisted sector with characterg 2 G Vg


Fact: Consistent holomorphic orbifolds are classified by . H3(G, U(1))[Dijkgraaf, Witten][Dijkgraaf, Pasquier, Roche][Bantay][Coste, Gannon, Ruelle]


Fact: Consistent holomorphic orbifolds are classified by . H3(G, U(1))[Dijkgraaf, Witten][Dijkgraaf, Pasquier, Roche][Bantay][Coste, Gannon, Ruelle]

Concretely, this works as follows. There exists an element ↵ 2 H3(G, U(1))

↵ : G⇥G⇥G! U(1) (3-cocycle =3-cochain + closedness)

which determines a distinguished element viach 2 H2(CG(h), U(1))

ch(g1, g2) =↵(h, g1, g2)↵(g1, g2, (g1g2)�1h(g1g2))

↵(g1, h, h�1g2h)

Then, under modular transformations the twisted twining characters of a holomorphic orbifold obey the relations

Zg,h(⌧ + 1) = cg(g, h)Zg,gh(⌧)

Zg,h(�1/⌧) = ch(g, g�1)Zh,g�1(⌧)

Moreover, under conjugation of one has the general relationg, h

8k 2 G

[Bantay][Coste, Gannon, Ruelle]

Zg,h(⌧) =cg(h, k)

cg(k, k�1hk)Zk�1gk,k�1hk(⌧)

Cohomological Obstructions from H3(G)

Whenever commutes with both and one findsk g h

Zg,h =cg(h, k)cg(k, h)

Zg,h

Zg,h(⌧) =cg(h, k)


Cohomological Obstructions from H3(G)

Whenever commutes with both and one findsk g h

Zg,h =cg(h, k)cg(k, h)

Zg,h

So unless the 2-cocycle is regular:Zg,h = 0 cg

cg(h, k) = cg(k, h)

When this is not satisfied we have obstructions! [Gannon]

Zg,h(⌧) =cg(h, k)


Twisted Twining Genera

Twisted twining genera of behave similarly to the �g,h M24 Zg,h

�g,h

⇣a⌧ + b

c⌧ + d,

z

c⌧ + d

⌘= �g,h

⇣ a bc d

⌘e2⇡i cz2

c⌧+d �g,h(⌧, z)

⇣ a bc d

⌘2 �g,h ⇢ SL(2, Z)



�g,h

⇣a⌧ + b

c⌧ + d,

z

c⌧ + d

⌘= �g,h

⇣ a bc d

⌘e2⇡i cz2

c⌧+d �g,h(⌧, z)

⇣ a bc d

⌘2 �g,h ⇢ SL(2, Z)

The multiplier phase must be determined by some !�g,h ↵ 2 H3(M24, U(1))



�g,h

⇣a⌧ + b

c⌧ + d,

z

c⌧ + d

⌘= �g,h

⇣ a bc d

⌘e2⇡i cz2

c⌧+d �g,h(⌧, z)

⇣ a bc d

⌘2 �g,h ⇢ SL(2, Z)

The multiplier phase must be determined by some !�g,h ↵ 2 H3(M24, U(1))

The twisted twining genera also form a representation of the full SL(2, Z)

�g,h

⇣a⌧ + b

c⌧ + d,

z

c⌧ + d

⌘= �g,h( a b

c d ) e2⇡i cz2c⌧+d �hcgd,hagb(⌧, z)

✓a bc d

◆2 SL(2, Z)

�g,h

⇣a⌧ + b

c⌧ + d,

z

c⌧ + d

⌘= �g,h( a b

c d ) e2⇡i cz2c⌧+d �hcgd,hagb(⌧, z)

✓a bc d

◆2 SL(2, Z)

�g,h(�1⌧

,z

⌧) = ch(g, g�1)e2⇡i z2

⌧ �h,g�1(⌧, z)

For the - and -transformations we then have:S T

�g,h(⌧ + 1, z) = cg(g, h)�g,gh(⌧, z)

ca(b, c)with is a 2-cocycle descending from .H3(M24, U(1))

Multiplier phases of twisted twining genera determined by the 3rd cohomology of .M24


Luckily, this cohomology group has recently been computed:

H3(M24, U(1)) ⇠= Z12 [Dutour Sikiric, Ellis]



Using GAP,with the HAP homological algebra package, we have been able to construct an explicit representative .↵ 2 H3(M24, U(1))



Using GAP,with the HAP homological algebra package, we have been able to construct an explicit representative .↵ 2 H3(M24, U(1))

The relevant cohomology class of is determined by the condition ↵

�1,h( a bc d ) = e

2⇡icdNk

✓a bc d

◆2 �0(N)

which is the multiplier system of the twining genera . �h = �1,h

Main result:

There exists a unique set of functions , and a uniquecohomology class such that all of Norton’sconjectured properties hold.

�g,h

[↵] 2 H3(M24, U(1))

There are in total 34 genuinely twisted and twined genera.

Their multiplier phases are indeed correctly determinedby the 3-cocycle . ↵ 2 H3(M24, U(1))

Many of the twisted twining genera vanish due to the cohomological obstruction.

There exists a unique set of functions , and a uniquecohomology class such that all of Norton’sconjectured properties hold.

�g,h

[↵] 2 H3(M24, U(1))

Remarks:

Main result:

�8A,2B(⌧, z) =⌘�

⌧2

�6

⌘(⌧)6#1(⌧, z)2

#4(⌧, 0)2

Example: -twist and -twine:8A 2B

�8A,2B(⌧, z) =⌘�

⌧2

�6

⌘(⌧)6#1(⌧, z)2

#4(⌧, 0)2


8A = -conjugacy class of order 8 elements.M24

�8A,2B(⌧, z) =⌘�

⌧2

�6

⌘(⌧)6#1(⌧, z)2

#4(⌧, 0)2



is a Jacobi form of weight 0 index 1 for the group �8A,2B(⌧, z)

�8A,2B :=

n

✓

a bc d

◆

2 SL2(Z) | b ⌘ 0 mod 4

o

= �

0(4)

�8A,2B(⌧, z) =⌘�

⌧2

�6

⌘(⌧)6#1(⌧, z)2

#4(⌧, 0)2



is a Jacobi form of weight 0 index 1 for the group �8A,2B(⌧, z)

Multiplier given by:

�8A,2B(⌧ + 4, z) =Q3

i=0 cg(g, gih)cg4h(g, g�1)cg�1(g4h, g4h)

cg�1(g4h, k)cg�1(k, h)

�8A,2B(⌧) = ��8A,2B(⌧)

using our result for in terms of ↵ 2 H3(M24, U(1))cg1(g2, g3)

�8A,2B :=

n

✓

a bc d

◆

2 SL2(Z) | b ⌘ 0 mod 4

o

= �

0(4)


All twisted twining genera are either weak Jacobi forms of weight 0index 1 for with multiplier system determined by or vanish by the cohomological obstruction.

�g,h

�g,h ⇢ SL(2, Z)↵ 2 H3(M24, U(1))



�g,h

�g,h ⇢ SL(2, Z)↵ 2 H3(M24, U(1))

HgThe twisted sector Hilbert space decompose according to:

Hg =M

r�0

HM24g,r ⌦HN=4

g,r



�g,h

�g,h ⇢ SL(2, Z)↵ 2 H3(M24, U(1))

HgThe twisted sector Hilbert space decompose according to:

Hg =M

r�0

HM24g,r ⌦HN=4

g,r

We have found that further decomposes intoHM24g,r

HM24g,r =

M

i

hi,rRi

projective representation of CM24(g)positive integer multiplicities

determined by the 3-cocycle↵ 2 H3(M24, U(1))


HM24g,r =

M

i

hi,rRi

The twisted twining genera thus admit an expansion

�g,h(⌧, z) = TrHM24g,0

(h)�h=

14 ,`=0

(⌧, z) =1X

r=1

TrHM24g,r

(h)�h=

14+r,`=

12(⌧, z)

TrHM24g,r

(h) =X

i

hi,rTrRi(h)

where each coefficient splits into a positive sum of characters for irreducible projective representations of CM24(g)


HM24g,r =

M

i

hi,rRi

The twisted twining genera thus admit an expansion

�g,h(⌧, z) = TrHM24g,0

(h)�h=

14 ,`=0

(⌧, z) =1X

r=1

TrHM24g,r

(h)�h=

14+r,`=

12(⌧, z)

TrHM24g,r

(h) =X

i

hi,rTrRi(h)

where each coefficient splits into a positive sum of characters for irreducible projective representations of CM24(g)

This establishes the -analogue of Generalised MoonshineM24

Wall-Crossing and Automorphic Lifts

Mason’s -Moonshine for Eta-Products M24

In 1990, Mason proposed a generalised moonshine for involving eta-products.M24



M24 admits a permutation representation of degree 24 (“defining representation”).

each conjugacy class can be characterised by its cycle shape:[g] 2 M24

[g] = 1Aidentity cycle shape 124 (1)(2)(3) · · · (22)(23)(24)

[g] = 2Aorder 2 cycle shape 1828 (1) · · · (8)(9 10)(11 12) · · · (23 24)



M24 admits a permutation representation of degree 24 (“defining representation”).

each conjugacy class can be characterised by its cycle shape:[g] 2 M24

[g] = 1Aidentity cycle shape 124 (1)(2)(3) · · · (22)(23)(24)

[g] = 2Aorder 2 cycle shape 1828 (1) · · · (8)(9 10)(11 12) · · · (23 24)

For each cycle shape one associates an eta-product :

⌘1A(⌧) = ⌘(⌧)24

⌘g(⌧)

⌘2A(⌧) = ⌘(⌧)8⌘(2⌧)8


Mason proposed eta-products corresponding to Norton series ⌘g,h(⌧) Zg,h(⌧)

⌘2A,2A(⌧) = ⌘(2⌧)12

⌘2A,2B(⌧) = ⌘(2⌧)4⌘(4⌧)4

for all pairwise commuting elements g, h 2 M24

Examples:


Mason proposed eta-products corresponding to Norton series ⌘g,h(⌧) Zg,h(⌧)

⌘2A,2A(⌧) = ⌘(2⌧)12

⌘2A,2B(⌧) = ⌘(2⌧)4⌘(4⌧)4

for all pairwise commuting elements g, h 2 M24

Examples:

He showed that all these eta products satisfy the requirements of Norton.⌘g,h(⌧)

In particular, they decompose into projective characters of :

⌘g,h(⌧) =1X

n=1

an(g, h)qn/D

CM24(g)

So, is Mason’s generalised -moonshine also related to to ? M24 K3

So, is Mason’s generalised -moonshine also related to to ? M24 K3

Yes! They are linked via the process of automorphic lift and “wall-crossing”.

Multiplicative automorphic lift

Lift : Jacobi forms Siegel modular forms[Borcherds]


Lift : Jacobi forms Siegel modular forms

For the elliptic genus of this yields: K3

[Borcherds]

�K3(⌧, z) =X

n�0`2Z

c(4n� `2)qny` pqyY

n>0,m�0,`2Z

�1� pnqmy`

�c(4mn�`2)

Igusa cusp form of weight 10 for Sp(4, Z)�10

[Gritsenko, Nikulin]




[Borcherds]

�K3(⌧, z) =X

n�0`2Z


n>0,m�0,`2Z

�1� pnqmy`

�c(4mn�`2)


is the partition function of 1/4 BPS dyons in string theory. N = 4��110

[Dijkgraaf, Verlinde, Verlinde][Shih, Strominger, Yin]




[Borcherds]

�K3(⌧, z) =X

n�0`2Z


n>0,m�0,`2Z

�1� pnqmy`

�c(4mn�`2)

Alternative presentation:


[Dijkgraaf, Moore, Verlinde, Verlinde]

Hecke operatorautomorphic correction


�10(⌧, z, ⇢) = A(⌧, z, ⇢) exp

"�1X

N=0

pN�TN�K3(⌧, z)

�#

The automorphic lift provides a link from Jacobi forms to eta-products.

In the limit one has:z ! 0


limz!0

�10(⌧, ⇢, z)(2⇡iz)2

= ⌘(⌧)24 ⌘(⇢)24


1/4-BPS 1/2-BPS 1/2-BPS Wall-crossing:


limz!0

�10(⌧, ⇢, z)(2⇡iz)2

= ⌘(⌧)24 ⌘(⇢)24


Note: ⌘(⌧)24⌘(⇢)24 = ⌘1A(⌧)⌘1A(⇢)



limz!0

�10(⌧, ⇢, z)(2⇡iz)2

= ⌘(⌧)24 ⌘(⇢)24


Note: ⌘(⌧)24⌘(⇢)24 = ⌘1A(⌧)⌘1A(⇢)

This fact generalises to all twining genera:

�g(⌧, z) �g(⌧, z, ⇢) ⌘g(⌧)⌘g(⇢)

[Cheng][Govindarajan][Eguchi, Hikami]

lift z ! 0



limz!0

�10(⌧, ⇢, z)(2⇡iz)2

= ⌘(⌧)24 ⌘(⇢)24

Multiplicative Lift for Generalised Mathieu Moonshine

Following Dijkgraaf, Moore, Verlinde, Verlinde we define the second quantized twisted twining genus via the lift

�g,h(⌧, z) g,h = exp

⇥ 1X

N=0

pN(T�

N�g,h)(⌧, z)

⇤

Here is the twisted equivariant Hecke operator:TN

[Ganter][Carnahan]

�g,h(⌧, z)


g,h = exp

⇥ 1X

N=0

pN(T�

N�g,h)(⌧, z)

⇤

(T�N�g,h)(⌧, z) =

1N

X

a,d>0,ad=N

d�1X

b=0

�g,h( a b0 d ) �gd,g�bha

�a⌧ + b

d, az

�


TN

�g,h(⌧, z)


g,h = exp

⇥ 1X

N=0

pN(T�

N�g,h)(⌧, z)

⇤

(T�N�g,h)(⌧, z) =

1N

X

a,d>0,ad=N

d�1X

b=0


�a⌧ + b

d, az

�

T�N : Jweak

0,1 ! Jweak0,N

Here is the twisted equivariant Hecke operator:


[Ganter][Carnahan]

TN

�g,h(⌧, z)


g,h = exp

⇥ 1X

N=0

pN(T�

N�g,h)(⌧, z)

⇤

(T�N�g,h)(⌧, z) =

1N

X

a,d>0,ad=N

d�1X

b=0


�a⌧ + b

d, az

�

Here is the twisted equivariant Hecke operator:


[Ganter][Carnahan]

We expect that has modular properties w.r.t. �g,h ⌘ A g,h

�(2)g,h ⇢ Sp(4;Z)

�g,h(⌧, z)


g,h = exp

⇥ 1X

N=0

pN(T�

N�g,h)(⌧, z)

⇤

By taking the limit of we reproducez ! 0 g,h(⌧, ⇢, z)⌘g,h(⌧)all of Mason’s generalised eta-products !

lim

z!0

p

g,h(⇢, ⌧, z)= exp[�

1X

N=0

pN+1TN�g,h(⌧, 0)] = ⌘g,h(⇢)


We have thus given strong evidence for the following “triality” of generalised -moonshines:

�g,h(⌧, z, ⇢)Siegel modular forms

⌘g,h(⌧)

Mason’s generalised eta-products

�g,h(⌧, z)

twisted twining genera(Jacobi forms)

M24

M24

multiplicative lift(“second quantization”)

wall-crossing

Summary and Outlook

Summary

We have established that generalised Mathieu moonshine holds by computing all twisted twining genera .�g,h

A key role is played by the third cohomology group . H3(M24, U(1))

Twisted twining genera can be expanded in projective characters of .CM24(g)

The automorphic lift provides a link to Mason’s generalised eta-products.

Outlook

Can one construct a generalised Kac-Moody algebra for each conjugacy class ?[g] 2 M24

Relation with BPS-algebras à la Harvey Moore...?

(c.f. [Borcherds][Carnahan])

It would be interesting to revisit the Generalised Monstrous Moonshine conjecture in light of our results for . M24

Some of the Norton series vanish; could this be explained via obstructions arising from ?

Zg,h

H3(M, U(1))

not known but believed to contain .H3(M, U(1)) Z48 [Mason]

Relation to twisted dyons in CHL models, wall-crossing...?

Generalised Umbral Moonshine...? [Cheng, Duncan, Harvey]

What does act on?M24

Our results strongly suggests that there is a holomorphic vertex operator algebra underlying Mathieu Moonshine...

generalised moonshine in the elliptic genus of k3 · generalised moonshine in the elliptic genus of...

Documents