Umbral Moonshine, Niemeier Lattices, and String Theory

Umbral Moonshine &
Niemeier Lattices

• A uniform construction of umbral moonshine.
• Illustrates the underlying structure of umbral moonshine.
• The cases previously discussed (1204.2779) correspond to

the “pure A series” with X=24 A1, 12 A2, 8 A3, ....

UM

Wednesday, June 19, 13

Umbral Moonshine &
Niemeier Lattices

• How to construct the umbral module KX?
• What is the role of the Niemeier lattices in the construction?
• What is the physical/geometrical context of umbral

moonshine?

UM

Wednesday, June 19, 13

IV. Umbral Moonshine
and

String Theory

Wednesday, June 19, 13

What is the relation between umbral moonshine and string theory?
At the level of numerical evidence:
First focus on the M24 example

number of massive N=4 multiplets
also dimensions of irreps of M24!!!

M3.CStHatuEsNG UM

Wednesday, June 19, 13

What is the relation between umbral moonshine and string theory?
At the level of numerical evidence:

First focus on the M24 example

number of massive N=4 multiplets
also dimensions of irreps of M24!!!

• A Witten index receiving contribution from the BPS states of the N=(4,4)
theory, computing the Q-cohomology of the spectrum.
• A topological quantity generalising χy, σ,.... Same for all K3 surfaces.
UM
M3.CStHatuEsNG

Wednesday, June 19, 13

Q: Is there a K3 surface whose symmetry is M24?

M3.CStHatuEsNG UM

Wednesday, June 19, 13

Q: Is there a K3 surface whose symmetry is M24?
A: No

All symmetries G of K3 manifold preserving the hyper-
Kähler structure satisfy

G⊂M23⊂M24. [Mukai ’88, Kondo ’98]

M3.CStHatuEsNG UM

Wednesday, June 19, 13

Q: Is there a K3 surface whose symmetry is M24?
A: No

All symmetries G of K3 manifold preserving the hyper-
Kähler structure satisfy

G⊂M23⊂M24. [Mukai ’88, Kondo ’98]

Q: Is there a K3 sigma model whose symmetry is M24?

M3.CStHatuEsNG UM

Wednesday, June 19, 13

Q: Is there a K3 surface whose symmetry is M24?
A: No

All symmetries G of K3 manifold preserving the hyper-
Kähler structure satisfy

G⊂M23⊂M24. [Mukai ’88, Kondo ’98]

Q: Is there a K3 sigma model whose symmetry is M24?
A: No!

•There is no K3 sigma model whose symmetry is M24.
•There exists K3 sigma model whose symmetry is not

contained in M24.

[Gaberdiel–Hohenegger–Volpato ’11]

M3.CStHatuEsNG UM

Wednesday, June 19, 13

Q: Is there a K3 surface whose symmetry is M24?
A: No

All symmetries G of K3 manifold preserving the hyper-
Kähler structure satisfy

G⊂M23⊂M24. [Mukai ’88, Kondo ’98]

Q: Is there a K3 sigma model whose symmetry is M24?
A: No!

•There is no K3 sigma model whose symmetry is M24.
•There exists K3 sigma model whose symmetry is not

contained in M24.

[Gaberdiel–Hohenegger–Volpato ’11]

There’s no contradiction to the idea that M24 acts on susy UM
spectrum of K3 sigma models.

M3.CStHatuEsNGBut... K3 M24 just a coincidence?

Wednesday, June 19, 13

But whenever there is an explicit symmetry G⊂M24 acting
on the sigma model (eg.T4/Zn, Gepner model, ...),

explicit computation of the twisted K3 elliptic genus
EGg(τ,z;K3)
leads to answers compatible with the M24 moonshine.

Moreover, at the non-perturbative level, the 1/2- and 1/4-

BPS spectrum of the N=4, d=4 theory obtained from

compactifying type II strings on K3xT2 is counted by

automorphic forms having a close relation to M24. [MC ’10, MC–Duncan ’12]

Similarly, traces of some other cases of umbral moonshine
can be found in the 1/2- and 1/4-BPS spectrum of the N=4,
d=4 theory obtained from compactifying type II strings on
K3xT2/Zn (the CHL model).

3. Status UM

Wednesday, June 19, 13

But whenever there is an explicit symmetry G⊂M24 acting
on the sigma model (eg.T4/Zn, Gepner model, ...),

explicit computation of the twisted K3 elliptic genus
EGg(τ,z;K3)
leads to answers compatible with the M24 moonshine.

Moreover, at the non-perturbative level, the 1/2- and 1/4-

BPS spectrum of the N=4, d=4 theory obtained from

compactifying type II strings on K3xT2 is counted by

automorphic forms having a close relation to M24. [MC ’10, MC–Duncan ’12]

Similarly, traces of some other cases of umbral moonshine
can be found in the 1/2- and 1/4-BPS spectrum of the N=4,
d=4 theory obtained from compactifying type II strings on
K3xT2/Zn (the CHL model).

Too many coincidences for K3 M24 to be a red herring? UM

3. Status

Wednesday, June 19, 13

Obviously, there’s a lot to explore in the relation between
umbral moonshine and strings/K3.

“Niemeier” Umbral Moonshine “Mock”
Finite Modular
Groups
Forms

K3

Wednesday, June 19, 13

K3 is Ubiquitous in String Theories

• The Strominger–Vafa black holes
• Dualities
• One of the first examples of AdS/CFT

F-Theory

type II K3 M24

heterotic Sporadic Quantum
Black Holes

Wednesday, June 19, 13

Heterotic Strings on K3xT2

1306.xxxx
MC, J. Duncan, X. Dong, J. Harvey,

S. Kachru, T. Wrase
See also the talk by Shamit Kachru at Strings 2013

E8xE8 heterotic string on K3xT2

Wednesday, June 19, 13

Heterotic Strings on K3xT2

1306.xxxx
MC, J. Duncan, X. Dong, J. Harvey,

S. Kachru, T. Wrase
See also the talk by Shamit Kachru at Strings 2013

E8xE8 heterotic string on K3xT2
with instanton numbers (12+n,12-n)

Heterotic–IIA Duality

Type IIA string theory on Elliptic-Fibred CY
with base Fn (the n-th Hirzebruch surface)

[Kachru–Vafa ’95]

Wednesday, June 19, 13

Heterotic Strings on K3xT2

We are interested in the “new supersymmetric index” which is closely

related to the threshold corrections. Cecott–Fendley–Intrilligator–Vafa ’92

Antoniadis–Gava–Narain ’92

The CFT factorises:

Dedekind Eta Function

trace over the c=(20,6) internal K3 CFT
with E8xE8 gauge bundle with instanton number (12+n,12-n)

Wednesday, June 19, 13

Heterotic Strings on K3xT2

1. For the standard embedding (of the K3 spin connection into SU(2)⊂E8)
with instanton number (24,0), ZK3 can be related to EG(K3):

where E4, E6 are Eisenstein series that generate all modular forms, M24!
and Zs, Z0 are q-series given by N=4 SCA characters.

Wednesday, June 19, 13

Heterotic Strings on K3xT2

2. Using an argument using properties of modular forms (the only weight
10 modular forms ~ E4 E6), we show that the above result is universal! i.e.
it’s the same for all instanton number (12+n, 12-n) of the gauge bundle
and all embeddings.

Wednesday, June 19, 13

Heterotic Strings on K3xT2

2. Using an argument using properties of modular forms (the only weight
10 modular forms ~ E4 E6), we show that the above result is universal! i.e.
it’s the same for all instanton number (12+n, 12-n) of the gauge bundle
and all embeddings.

3. On the type IIA side, this universality can be viewed as a consequence
of the relation between topological strings amplitudes and (quasi-)
modular forms for elliptic-fibered Calabi-Yaus discussed recently.

[Klemm–Manschot–Woschke/Alim–Scheidegger ’11]

Wednesday, June 19, 13

Heterotic Strings on K3xT2

4. When explicitly computed at the orbifold points (K3~T4/Zn) where
some order 2, 3, 4 symmetries are present, the corresponding “twisted
new susy index”

is consistent with the M24-interpretation.We have found such examples in
both the standard and non-standard embedding models. [in progress]

Wednesday, June 19, 13

Heterotic Strings on K3xT2

4. When explicitly computed at the orbifold points (K3~T4/Zn) where
some order 2, 3, 4 symmetries are present, the corresponding “twisted
new susy index”

is consistent with the M24-interpretation.We have found such examples in
both the standard and non-standard embedding models. [in progress]
5. For the dual Calabi–Yau, this suggests an M24-interpretation of the genus
zero Gromov-Witten invariants.

Wednesday, June 19, 13

Heterotic Strings on K3xT2

4. When explicitly computed at the orbifold points (K3~T4/Zn) where
some order 2, 3, 4 symmetries are present, the corresponding “twisted
new susy index”

is consistent with the M24-interpretation.We have found such examples in
both the standard and non-standard embedding models. [in progress]

5. For the dual Calabi–Yau, this suggests an M24-interpretation of the genus
zero Gromov-Witten invariants.

6. Similar consideration leads to a generalisation of possible M24-
interpretation of genus>0 Gromov-Witten invariants and the cases when
the Wilson lines are turned on.

Wednesday, June 19, 13

The interesting symmetries seem to be even more prevalent
than naively expected!

Obviously, still a lot to be learned here!

“Niemeier” Umbral Moonshine “Mock”
Finite Modular
Groups
Forms

K3

Wednesday, June 19, 13

Thank You &
Please join us in Paris for

String–Math 2016!

Wednesday, June 19, 13