Generalized Mathieu Moonshine and Siegel Modular Formsmockmodular/pdf/GenMatSieg_IMSc.pdf · g,h 2...

Generalized Mathieu Moonshineand Siegel Modular Forms

IMSc, Chennai, April 18, 2014

Daniel Persson

Chalmers University of Technology

Talk based on:[arXiv:1312.0622] (w/ R. Volpato)[arXiv:1302.5425] (w/ M. Gaberdiel, & R. Volpato)[arXiv:1211.7074] (w/ M. Gaberdiel, H. Ronellenfitsch, R. Volpato)

Mock Modular Forms and Physics

What is Moonshine?

representation theory of finite groups

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

What is Moonshine?


modular forms


What is Moonshine?


modular forms conformal field theory


What is Moonshine?




What is Moonshine?

infinite-dimensionalalgebras





What is Moonshine?




What is Moonshine?

The most famous example is Monstrous Moonshine.


monster group

bosonic string theory on

monster Lie algebra

modular function

M

(T 24/⇤Leech)/Z2)

(holomorphic VOA )V \

m

J(⌧) = q�1 + 196884q + · · ·

Monstrous Moonshine

(Figure stolen from Jeff ’s talk!)

monster group

bosonic string theory on

monster Lie algebra

modular function

M

(T 24/⇤Leech)/Z2)

(holomorphic VOA )V \

m

BRST cohomology

automorphism group

denominator formula

graded dimensionJ(⌧) = q�1 + 196884q + · · ·

Lie algebra automorphisms

Monstrous Moonshine

moonshine

(Figure stolen from Jeff ’s talk!)

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

A completely new moonshine phenomenon to explore!

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

monster group M

Monstrous Moonshine Mathieu Moonshine

Mathieu group M24

bosonic CFT superconformal field theory

Virasoro algebra superconformal algebraN = (4, 4)

J -function elliptic genus of K3

McKay-Thompson series twining genera

monster module V \ ?

mmonster Lie algebra ?[Eguchi, Ooguri, Tachikawa][Cheng][Gaberdiel, Hohenegger, Volpato]

[Eguchi, Hikami][Taormina,Wendland][Gannon]

Despite this amazing progress, we still don’t understand why Mathieu moonshine holds. More precisely, we cannot answer the question:

What does act on?M24



We have considered a “two-step generalization” of Mathieu moonshine that sheds light on this question.

Mathieumoonshine

generalizedMathieu

moonshine

second-quantizedMathieu

moonshine



We have considered a “two-step generalization” of Mathieu moonshine that sheds light on this question.

Mathieumoonshine

generalizedMathieu

moonshine

second-quantizedMathieu

moonshine

Roberto’s talk This talk

Outline

2. Second quantization & black hole counting

5. Summary and outlook

1. Recap: Generalized Mathieu moonshine

4. Connection with umbral moonshine

3. Second quantization of generalized Mathieu moonshine

1. Generalized Mathieu moonshine

Generalized Mathieu Moonshine

Introduce a family of functions, the twisted twining genera:

g, h 2 M24

for each commuting pair�g,h : H⇥ C ! C

such that for we recover the twining generag = e �e,h = �h

[Gaberdiel, D.P., Ronellenfitsch, Volpato]

Generalized Mathieu Moonshine

g, h 2 M24



This is the analogue of Norton’s generalized monstrous moonshine

Zg,h : H ! C g, h 2 M

Ze,h(⌧) = Th(⌧) McKay-Thompson series

[Gaberdiel, D.P., Ronellenfitsch, Volpato]


g, h 2 M24




Zg,h : H ! C g, h 2 M

Partially explained by orbifolds of the FLM monster VOA .V \ [Dixon, Ginsparg, Harvey][Tuite]

Proven in special cases but the full conjecture still open. [Dong, Li, Mason][Höhn][Tuite][Carnahan]

Generalized Mathieu Moonshine[Gaberdiel, D.P., Ronellenfitsch, Volpato]


g, h 2 M24




Zg,h : H ! C g, h 2 M

Can we also interpret generalized Mathieu moonshine in terms of orbifolds?

Generalized Mathieu Moonshine[Gaberdiel, D.P., Ronellenfitsch, Volpato]


Holomorphic Orbifolds and Group Cohomology

Our main assumption is that the twisted twining genera behave similarly as for characters of a holomorphic orbifold



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]

multiplier phases of characters determined by 2-cocycleZg,h(⌧)

cg 2 H2(CG(g), U(1))

obtained from a class via[↵] 2 H3(G,U(1))

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

↵(g1, h, h�1g2h)

In particular, for the S- and T-transformations we have

Zg,h(⌧ + 1) = cg(g, h)Zg,gh(⌧)

Zg,h(�1/⌧) = ch(g, g�1)Zh,g�1(⌧)

[Bantay][Coste, Gannon, Ruelle]

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(⌧)

In particular, for the S- and T-transformations we have

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)


Conjecture (generalized Mathieu moonshine) [GHPV]:For each there exists a graded unitary representation of g 2 M24 Hg N = 4

with central charge carrying a projective representation c = 6

⇢g : CM24(g) ! GL(Hg)

commuting with and determined by a class . N = 4 [↵] 2 H3(M24, U(1))



⇢g : CM24(g) ! GL(Hg)

commuting with and determined by a class . [↵] 2 H3(M24, U(1))N = 4For each commuting pair there exists functions satisfying: g, h 2 M24 �g,h(⌧, z)

�e,h = �h �e,e = �(K3; ⌧, z)and



⇢g : CM24(g) ! GL(Hg)

commuting with and determined by a class . N = 4For each commuting pair there exists functions satisfying: g, h 2 M24 �g,h(⌧, z)

�e,h = �h �e,e = �(K3; ⌧, z)and

�g,h(⌧, z) = ⇠(k)�k�1gk,k�1hk(⌧, z), 8k 2 M24

[↵] 2 H3(M24, U(1))



⇢g : CM24(g) ! GL(Hg)


�e,h = �h �e,e = �(K3; ⌧, z)and

�g,h(⌧, z) = ⇠(k)�k�1gk,k�1hk(⌧, z), 8k 2 M24

�g,h

✓a⌧ + b

c⌧ + d,

z

c⌧ + d

◆= �g,h

�a bc d

�e2⇡i

cz2

c⌧+d�gahc,gbhd(⌧, z),�a bc d

�2 SL(2,Z)

[↵] 2 H3(M24, U(1))

Rg,r representation of a central extension of CM24(g)



⇢g : CM24(g) ! GL(Hg)


�e,h = �h �e,e = �(K3; ⌧, z)and

�g,h(⌧, z) = ⇠(k)�k�1gk,k�1hk(⌧, z), 8k 2 M24

�g,h

✓a⌧ + b

c⌧ + d,

z

c⌧ + d

◆= �g,h

�a bc d

�e2⇡i

cz2


�2 SL(2,Z)

�g,h(⌧, z) =X

r,`

TrRg,r (h)�r+1/4,`(⌧, z), h 2 CM24(g)

[↵] 2 H3(M24, U(1))



⇢g : CM24(g) ! GL(Hg)


�e,h = �h �e,e = �(K3; ⌧, z)and

�g,h(⌧, z) = ⇠(k)�k�1gk,k�1hk(⌧, z), 8k 2 M24

�g,h

✓a⌧ + b

c⌧ + d,

z

c⌧ + d

◆= �g,h

�a bc d

�e2⇡i

cz2


�2 SL(2,Z)

�g,h(⌧, z) =X

r,`

TrRg,r (h)�r+1/4,`(⌧, z), h 2 CM24(g)

The phases , and the central extension of are ⇠g,h �g,h CM24(g)

determined by the same class [↵] 2 H3(M24, U(1))

[↵] 2 H3(M24, U(1))

�8A,2B(⌧, z) =⌘�

⌧2

�6

⌘(⌧)6#1(⌧, z)2

#4(⌧, 0)2

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

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




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(⌧)

�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




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)

Theorem [GHPV]:

For each commuting pair there exists functions satisfying g, h 2 M24 �g,h(⌧, z)

all the expected modular properties with respect to subgroups �g,h ⇢ SL(2,Z)

There is a unique class which determines all the modular phases. [↵] 2 H3(M24, U(1))

Many of the vanish due to cohomological obstructions controlled by �g,h H3(M24, U(1))

(in deriving these results we use the fact that [Ellis, Dutour-Sikiric] )H3(M24, U(1)) ⇠= Z12

Theorem [GHPV]:

For each element there exists projective reps of such that g 2 M24 Rg,r CM24(g)

�g,h(⌧, z) =X

r,`

TrRg,r (h)�r+1/4,`(⌧, z), h 2 CM24(g)

This was verified for the first 500 coefficients.






“Almost theorem” [GHPV]:

Theorem [GHPV]:

“Almost theorem” [GHPV]:

For each element there exists projective reps of such that g 2 M24 Rg,r CM24(g)

�g,h(⌧, z) =X

r,`

TrRg,r (h)�r+1/4,`(⌧, z), h 2 CM24(g)

This was verified for the first 500 coefficients.


This is very strong evidence that generalized Mathieu Moonshine holds!

But what is the physical interpretation?





2. Second quantization & black hole counting

Second quantized elliptic genus

Let be a Calabi-Yau manifold and its elliptic genus.X �(X; ⌧, z)

�(X; ⌧, z) is a weak Jacobi form of weight zero and index (dimC X)/2 [Gritsenko]




Dijkgraaf, Moore, Verlinde, Verlinde defined the second quantized elliptic genus as

X(�, ⌧, z) :=1X

n=0

pn�(SnX; ⌧, z) p = e2⇡i�





X(�, ⌧, z) :=1X

n=0

pn�(SnX; ⌧, z) p = e2⇡i�

This is the generating function of elliptic genera of symmetric products of X





X(�, ⌧, z) :=1X

n=0

pn�(SnX; ⌧, z) p = e2⇡i�

DMVV proved the following remarkable formula:

X(�, ⌧, z) = exp

" 1X

L=1

pLTL�(X; ⌧, z)

#=

Y

n>0,m�0`2Z

(1� pnqmy`)�cX(mn,`)





X(�, ⌧, z) :=1X

n=0

pn�(SnX; ⌧, z) p = e2⇡i�

X(�, ⌧, z) = exp

" 1X

L=1

pLTL�(X; ⌧, z)

#=

Y

n>0,m�0`2Z

(1� pnqmy`)�cX(mn,`)

Hecke operator

TL : J0,m ! J0,mL

Fourier coefficients of

�(X; ⌧, z) =X

k�0,`2ZcX(k, `)qky`

DMVV proved the following remarkable formula:


Gritsenko later showed that

�X(�, ⌧, z) :=AX(�, ⌧, z)

X(�, ⌧, z)

is a Siegel modular form of weight cX(0, 0)/2


�X(�, ⌧, z) :=AX(�, ⌧, z)

X(�, ⌧, z)

is a Siegel modular form of weight cX(0, 0)/2

AX is called the “Hodge anomaly”; only depends on the Hodge numbers of X

This is an example of a (multiplicative) Borcherds lift:

� :SL(2,Z)

Jacobi automorphic

SO(3, 2;Z)

Gritsenko later showed that

For a K3-manifold we have that X

�X = �10 = pqyY

m,n,`)>0

(1� pmqny`)c(mn,`)

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


For a K3-manifold we have that X

�X = �10 = pqyY

m,n,`)>0

(1� pmqny`)c(mn,`)

�(K3; ⌧, z) = 2�0,1(⌧, z) =X

n�0,`2Zc(n, `)qny`

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

This is a multiplicative Borcherds lift of the K3 elliptic genus

The inverse is the partition function of 1/4 BPS dyons in or Het/T 6 IIA/(K3⇥ T 2)[Dijkgraaf, Verlinde, Verlinde][Shih, Strominger, Yin]


Counting dyons in string theoryN = 4

(P,Q) 2 �6,22 � �6,22

Large moduli space of such theories:

M = O(6, 22;Z)\O(6, 22;R)/(O(6)⇥O(22))

The discrete duality group preserved the lattice of electric-magnetic charges:

The full non-perturbative duality group is

SL(2,Z)⇥O(6, 22;Z)

(P,Q) transform as a doublet under SL(2,Z)

Hilbert space of states decomposes as

H =O

(P,Q)2�6,22��6,22

HQ,P

These can be realized as charged black holes in the supergravity limit.

Hilbert space of states decomposes as

H =O

(P,Q)2�6,22��6,22

HQ,P

These can be realized as charged black holes in the supergravity limit.

We are interested in BPS-states:

1/2 BPS: Purely electric or magnetic(0, Q) (P, 0)

1/4 BPS (generic): Dyonic (Q,P )

1/2 BPS-states are counted by

1

⌘(⌧)24=

X

n2Zd(n)qn

[Dabholkar, Harvey]

number of 1/2 BPS-states with charge such that d(n) = Q n = Q2/2

In general, 1/4 BPS states are counted by the 6th helicity supertrace

B6(P,Q) :=1

6!TrHP,Q

�(�1)J(2J)6

�J = helicity

[Kiritsis]


1

⌘(⌧)24=

X

n2Zd(n)qn

[Dabholkar, Harvey]


In general, 1/4 BPS states are counted by the 6th helicity supertrace

B6(P,Q) :=1

6!TrHP,Q

�(�1)J(2J)6

�J = helicity

[Kiritsis]


1

⌘(⌧)24=

X

n2Zd(n)qn

[Dabholkar, Harvey]


invariant under SL(2,Z)⇥ SO(6, 22;Z)can only depend on the combinations

P 2, Q2, Q · Plocally constant on M

Generating function:

with the identification

B6(P,Q) = d⇣

Q2

2 , P 2

2 , P ·Q⌘

1

�10(�, ⌧, z)=

X

m,n,`

d(m,n, `)pmqny`

(q := e2⇡i⌧ , y := e2⇡iz, p := e2⇡i�)

Generating function:

1

�10(�, ⌧, z)=

X

m,n,`

d(m,n, `)pmqny`

(q := e2⇡i⌧ , y := e2⇡iz, p := e2⇡i�)

limz!0

�10(�, ⌧, z)

(2⇡iz)2= ⌘(�)24⌘(⌧)24

“wall-crossingformula”

1/4-BPS 1/2-BPS1/2-BPS

�10 has a double pole at . In the limit, we have a factorizationz = 0

with the identification

B6(P,Q) = d⇣

Q2

2 , P 2

2 , P ·Q⌘

3. Second quantization of generalized Mathieu moonshine

Second quantized twisted twining genera

Inspired by the aforementioned results we seek a similar spacetime interpretation for the twisted twining genera of generalized Mathieu moonshine.�g,h(⌧, z)

This generalizes earlier results by Cheng and Govindarajan.




We define the second quantized twisted twining genus as:

g,h(�, ⌧, z) := exp

" 1X

L=1

pLT ↵L �g,h(⌧, z)

#


g,h(�, ⌧, z) := exp

" 1X

L=1


#

where are twisted equivariant Hecke operators, generalizing those used in generalized monstrous moonshine by Ganter & Carnahan.

T ↵L





g,h(�, ⌧, z) := exp

" 1X

L=1


#

where are twisted equivariant Hecke operators, generalizing those used in generalized monstrous moonshine by Ganter & Carnahan.

T ↵L



Note that this depends on the choice of 3-cocycle ↵ 2 H3(M24, U(1))but different representatives in each class simply amounts to a shift of [↵] �


Twisted equivariant Hecke operators

Geometric interpretation following Ganter. Let

moduli space of principal -bundlesM24

on the elliptic curve E⌧

MM24 = P ⇥ (H+ ⇥ C) /M24 ⇥ (SL(2,Z)n Z2)

=





MM24 = P ⇥ (H+ ⇥ C) /M24 ⇥ (SL(2,Z)n Z2)

=

�g,hThe twisted twining genera are sections of a line bundle

L↵g,h ! MM24





MM24 = P ⇥ (H+ ⇥ C) /M24 ⇥ (SL(2,Z)n Z2)

=


L↵g,h ! MM24

T ↵L : L↵

g,h �! (L↵g,h)

⌦L

The twisted equivariant Hecke operators provide a map





MM24 = P ⇥ (H+ ⇥ C) /M24 ⇥ (SL(2,Z)n Z2)

=


L↵g,h ! MM24

T ↵L : L↵

g,h �! (L↵g,h)

⌦L


sections have multiplier phase

sections have multiplier phase �g,h (�g,h)

L

T ↵L : L↵

g,h �! (L↵g,h)

⌦L


T ↵L �g,h(⌧, z) :=

1

L

X

a,d>0ad=L

d�1X

b=0

�g,h

�a b0 d

��gd,g�b,ha

�a⌧+b

d , az�

This is a generalization of similar Hecke operators used in generalized monstrous moonshine by Ganter & Carnahan. (see also [Tuite][Govindarajan])

T ↵L : L↵

g,h �! (L↵g,h)

⌦L


Explicitly one can represent this action by

multiplier phase determined by

[↵] 2 H3(M24, U(1))

T ↵L �g,h(⌧, z) :=

1

L

X

a,d>0ad=L

d�1X

b=0

�g,h

�a b0 d

��gd,g�b,ha

�a⌧+b

d , az�

This is a generalization of similar Hecke operators used in generalized monstrous moonshine by Ganter & Carnahan. (see also [Tuite][Govindarajan])

T ↵L : L↵

g,h �! (L↵g,h)

⌦L


Explicitly one can represent this action by

Example: for we haveg, h 2 2B

T ↵2 �g,h(⌧, z) =

1

2

h� �g,e(2⌧, 2z) + �e,h(

⌧2 , z)� �e,gh(

⌧+12 , z)

i

Example: for we haveg, h 2 2B

T ↵2 �g,h(⌧, z) =

1

2

h� �g,e(2⌧, 2z) + �e,h(

⌧2 , z)� �e,gh(

⌧+12 , z)

i

signs come from the multiplier system �g,h

Example: for we have

On the other hand, for we in fact have

by cohomological obstructions from H3(M24, U(1))

g, h 2 2B

T ↵2 �g,h(⌧, z) =

1

2

h� �g,e(2⌧, 2z) + �e,h(

⌧2 , z)� �e,gh(

⌧+12 , z)

i

signs come from the multiplier system �g,h

�g,h(⌧, z) = 0

g, h 2 2B

This implies that for sufficiently large has trivial multiplier phase L T ↵L �g,h

T ↵L : L↵

g,h �! (L↵g,h)

⌦L

Since

Even if vanishes by cohomological obstructions, all the �g,h

second quantized twisted twining genera are unobstructed! g,h

T ↵L : L↵

g,h �! (L↵g,h)

⌦L

Since

g,h(�, ⌧, z) := exp

" 1X

L=1


#

This implies that for sufficiently large has trivial multiplier phase L T ↵L �g,h

The second quantized twisted twining genera satisfy the following properties

Infinite product formula

1

g,h(�, ⌧, z)=

1Y

d=1

1Y

m=0

Y

`2Z

M�1Y

t=0

(1� e2⇡itM q

mN� y`pd)cg,h(d,m,`,t)

Theorem (D.P., Volpato):



M = O(h) N = O(g)

� length of the shortest cycle of in its 24-dim permutation repsg

cg,h(d,m, `, t) :=M�1X

k=0

�N�1X

b=0

e�2⇡itk

M

M

e2⇡ibm�N

�N�g,h

�k b0 d

�cgd,g�bhk(

md

N�, `)

1

g,h(�, ⌧, z)=

1Y

d=1

1Y

m=0

Y

`2Z

M�1Y

t=0

(1� e2⇡itM q





�g,h(�, ⌧, z) :=Ag,h(�, ⌧, z)

g,h(�, ⌧, z)

The ratio

is a Siegel modular form for a subgroup �(2)g,h ⇢ Sp(4;R)

For this was conjectured by Cheng and partially proven by Raum.g = e

1

g,h(�, ⌧, z)=

1Y

d=1

1Y

m=0

Y

`2Z

M�1Y

t=0

(1� e2⇡itM q





�g,h(�, ⌧, z) :=Ag,h(�, ⌧, z)

g,h(�, ⌧, z)

The ratio “Hodge anomaly”

Ag,h = �p#1(⌧, z)2

⌘(⌧)6⌘g,h(⌧)

Mason’s generalized eta-products



1

g,h(�, ⌧, z)=

1Y

d=1

1Y

m=0

Y

`2Z

M�1Y

t=0

(1� e2⇡itM q





�g,h(�, ⌧, z) :=Ag,h(�, ⌧, z)

g,h(�, ⌧, z)

The ratio “Hodge anomaly”

Ag,h = �p#1(⌧, z)2

⌘(⌧)6⌘g,h(⌧)

Mason’s generalized eta-products


limz!0

�g,h(�, ⌧, z)

(2⇡iz)2= ⌘g,h(⌧)⌘g,h(N��)

“Wall-crossing formula”


1

g,h(�, ⌧, z)=

1Y

d=1

1Y

m=0

Y

`2Z

M�1Y

t=0

(1� e2⇡itM q



Automorphy of follow from �g,h

“Electric-magnetic duality”

�g,h(�, ⌧, z) = �g,h0(⌧

N�, N��, z)

where is not necessarily in the same conjugacy class h0 [h]

This generalizes the electric-magnetic duality in �10 [Dijkgraaf, Verlinde, Verlinde]

Automorphy of follow from �g,h

“Electric-magnetic duality”

�g,h(�, ⌧, z) = �g,h0(⌧

N�, N��, z)

where is not necessarily in the same conjugacy class h0 [h]

This generalizes the electric-magnetic duality in �10 [Dijkgraaf, Verlinde, Verlinde]

Using results of Gritsenko-Nikulin, one also has invariance under (an extension of) the para-modular group

�t(N) = {

0

BB@

⇤ t⇤ ⇤ ⇤⇤ ⇤ ⇤ t�1⇤N⇤ Nt⇤ ⇤ ⇤Nt⇤ Nt⇤ t⇤ ⇤

1

CCA 2 Sp(4,Q), ⇤ 2 Z}

Every is a modular function for some finite index subgroup �g,h �(2)g,h

of a para-modular group for some �t t

Every is a modular function for some finite index subgroup �g,h �(2)g,h

of a para-modular group for some �t t

We can therefore view this our construction as a twisted equivariant generalization of a multiplicative Borcherds lift

MultG[�g,h] := Ag,h(�, ⌧, z)exp

"�

1X

L=1


#

�g,h(⌧, z)

twisted twining genera(weak Jacobi forms)

generalized eta-products(modular forms)

second-quantized twisted twining genera(Siegel modular forms)

twisted equivariant multiplicative lift z ! 0

�g,h(�, ⌧, z)

⌘g,h(⌧)⌘g,h0(N��)

This resolves a puzzle about the connection with Mason’s old version of generalized -moonshine for eta-productsM24

g = e(For this was observed previously by Cheng and Govindarajan. )

(“second quantization”) (“wall-crossing”)

Physical interpretation: CHL-models

Can we interpret the second quantized twisted twining genera as counting spacetime BPS-states?



Suppose are commuting symmetries of the internal superconformal CFT(g, h)

of type or II/(K3⇥ T 2) Het/T 6





Consider the orbifold of this theory by gN = 4new theory

“CHL-model”[Chaudhuri, Hockney, Lykken]





Consider the orbifold of this theory by gN = 4new theory

“CHL-model”[Chaudhuri, Hockney, Lykken]

B6;g,h(P,Q) :=1

6!TrHg

Q,P(h(�1)2J(2J)6) [Sen]

Computed for some pairs of symmetries [Dabholkar, Gaiotto][Dabholkar, Nampuri][Jatkar, Sen][David][Dabholkar, Cheng][Govindarajan][Sen]...

In this orbifold theory we have “twisted” dyon states counted by the twisted BPS-index

B6;g,h(P,Q) = dg,h⇣

Q2

2 , P 2

2 , Q · P⌘

Expanding the second quantized twisted twining genera

1

�g,h(�, ⌧, z)=

X

m,n,`

dg,h(m,n, `)qnpmy`

we find that

B6;g,h(P,Q) = dg,h⇣

Q2

2 , P 2

2 , Q · P⌘

Coincides with Fourier coefficients of

�g,h

for some pairs ! (g, h)

Expanding the second quantized twisted twining genera

1

�g,h(�, ⌧, z)=

X

m,n,`

dg,h(m,n, `)qnpmy`

we find that

Could it be that all of the have interpretations as partition functions for BPS-dyons?

�g,h

4. Connection with umbral moonshine

Umbral moonshine

Cheng, Duncan, Harvey proposed a generalization of Mathieu moonshine involving 23 examples labelled by ADE-type root systems.

(G(`), Z(`)) ` 2 {2, 3, 4, 5, 7, 13}

Here we focus on the 6 cases corresponding to pure A-type root systems.

Umbral moonshine


(G(`), Z(`)) ` 2 {2, 3, 4, 5, 7, 13}

finite group Jacobi form


Umbral moonshine


(G(`), Z(`)) ` 2 {2, 3, 4, 5, 7, 13}

finite group

(G(2), Z(2)) =�M24,�(K3; ⌧, z)

� Mathieu moonshine corresponds to ` = 2

Jacobi form

We shall now see that there appears to be a relation between umbral moonshine and generalized Mathieu moonshine.


Let us consider the case when in g, h 2 2A M24

�g,h = 0 but T ↵2 �g,h 2 Jweak

0,2



0,2

T ↵2 �g,h = Z(3)(⌧, z)

In fact, this is nothing but the umbral Jacobi form for ` = 3



0,2

In fact, this is nothing but the umbral Jacobi form for

T ↵2 �g,h = Z(3)(⌧, z)

The same holds for a few other conjugacy classes in that we checked

T ↵3 �g,h = Z(4)(⌧, z)(3A, 3A)

(4B, 4B) T ↵4 �g,h = Z(5)(⌧, z)

` = 3

M24

Starting from the umbral Jacobi forms Cheng-Duncan-Harvey constructed a class of Siegel modular forms using a standard Borcherds lift:

�(`) = Mult[Z(`)] = pA(`)qB(`)yC(`)Y

(m,n,r)>0

(1� pmqnyr)c(`)(mn,r)


For one has ` 2 {2, 3, 4, 5} �(`) = (�k)2 k = 7�`

`�1

�k = weight Siegel modular forms constructed by Gritsenko-Nikulink

�(`) = Mult[Z(`)] = pA(`)qB(`)yC(`)Y

(m,n,r)>0



For one has ` 2 {2, 3, 4, 5} �(`) = (�k)2 k = 7�`

`�1

�(`) = Mult[Z(`)] = pA(`)qB(`)yC(`)Y

(m,n,r)>0


(2A, 2A) : �g,h = (�2)2 = �(3)

(3A, 3A) : �g,h = (�1)2 = �(4)

(4B, 4B) : �g,h = (�1/2)2 = �(5)


We observe that these Siegel modular forms coincide with some of the second quantized twisted twining genera in generalized Mathieu moonshine:


For one has ` 2 {2, 3, 4, 5} �(`) = (�k)2 k = 7�`

`�1

We observe that these Siegel modular forms coincide with some of the second quantized twisted twining genera in generalized Mathieu moonshine:

(2A, 2A) : �g,h = (�2)2 = �(3)

(3A, 3A) : �g,h = (�1)2 = �(4)

(4B, 4B) : �g,h = (�1/2)2 = �(5)

conjugacy classes in

M24

Overlap between umbral moonshine and generalized Mathieu moonshine!

�(`) = Mult[Z(`)] = pA(`)qB(`)yC(`)Y

(m,n,r)>0



(2A, 2A) : �g,h = (�2)2 = �(3)

(3A, 3A) : �g,h = (�1)2 = �(4)

(4B, 4B) : �g,h = (�1/2)2 = �(5)


M24


(2A, 2A) : �g,h = (�2)2 = �(3)

(3A, 3A) : �g,h = (�1)2 = �(4)

(4B, 4B) : �g,h = (�1/2)2 = �(5)


M24


Note that this is non-trivial since the LHS is constructed using an equivariant lift while the RHS is constructed using a standard Borcherds lift:

MultG[�g,h] = Mult[Z(`)]

These Siegel modular forms also appear in CHL-models. [Sen][Govindarajan]

In fact, following an observation by Govindarajan, for these cases one can also show that the same functions can be obtained using an additive lift from the “Hodge anomaly’‘ Ag,h(�, ⌧, z)

(2A, 2A) : �g,h = (�2)2 = �(3)

(3A, 3A) : �g,h = (�1)2 = �(4)

(4B, 4B) : �g,h = (�1/2)2 = �(5)


M24


MultG[�g,h] = Mult[Z(`)]

These Siegel modular forms also appear in CHL-models. [Sen][Govindarajan]

In fact, following an observation by Govindarajan, for these cases one can also show that the same functions can be obtained using an additive lift from the “Hodge anomaly’‘ Ag,h(�, ⌧, z)

A modular coincidence or an indication of some deeper relation?

Note that this is non-trivial since the LHS is constructed using an equivariant lift while the RHS is constructed using a standard Borcherds lift:

5. 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)

All the second quantized twisted twining genera found and verified to be Siegel modular forms

Some of these correspond to partition functions of twisted dyons in CHL-models

Intriguing connection with umbral moonshine

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])

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

Recent interesting results indicate that there is are N=2 and N=1 versions of Mathieu Moonshine in heterotic string theory.

[Cheng, Dong, Duncan, Harvey, Kachru, Wrase][Harrison, Kachru, Paquette][Wrase]

Can one construct an action of on the (cohomology) of the chiral de Rham complex of K3?

M24

Twisted equivariant additive lifts: ?AddG[Ag,h]

Does play a role in mirror symmetry?M24

(see also [Eguchi, Hikami])

See Katrin’s talk!


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

...but which one remains a mystery...

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