Supersymmetry
decomposes
the virtual bundle that underlies the elliptic genus
Geometrical and Enumerative Structures in Supersymmtry,
Special Session 25 of the 2015 International AMS-EMS-SPM Meeting,
Porto, Portugal, June 10-13, 2015
Katrin Wendland
Albert-Ludwigs-Universität Freiburg
Introduction
1. The elliptic genus
2. Decomposing under extended SUSY
3. Interpretation in terms of Mathieu Moonshine
The elliptic genus under extended supersymmetry
Introduction
The elliptic genus of CY manifolds and of SCFTs
2 Decomposing under extended supersymmetry
3 Interpretation in terms of Mathieu Moonshine
1
[Creutzig/W15], work in progress
[W15] K3 en route from geometry to conformal field theory ;
to appear in: Proceedings of the 2013 Summer School “Geometric, Algebraic and
Topological Methods for Quantum Field Theory”, Villa de Leyva, Colombia;
arXiv:1503.08426 [math.DG]
[W14] Snapshots of conformal field theory ;
in “Mathematical Aspects of Quantum Field Theories”, Mathematical Physics
Studies, Springer 2015, pp. 89-129; arXiv:1404.3108 [hep-th]
[Taormina/W13] A twist in the M24 moonshine story, to appear in Confluentes Mathematici;
arXiv:1303.3221 [hep-th]
[Taormina/W12] Symmetry-surfing the moduli space of Kummer K3s, to appear in: Proceedings of
String-Math 2012; arXiv:1303.2931 [hep-th]
[Taormina/W11] The overarching finite symmetry group of Kummer surfaces in the Mathieu group
M24 , JHEP 1308:152 (2013); arXiv:1107.3834 [hep-th]
Katrin Wendland
SUSY decomposes the bundle underlying the elliptic genus
1/7
Introduction
1. The elliptic genus
2. Decomposing under extended SUSY
3. Interpretation in terms of Mathieu Moonshine
1. The elliptic genus of Calabi-Yau manifolds. . .
Let M denote a compact Calabi-Yau D-fold, T := T 1,0 M,
∞
O
∗
− D2
Λ−yqnT ∗ ⊗ Λ−y −1 qnT ⊗ SqnT ∗ ⊗ SqnT ,
Eq,−y := y Λ−yT ⊗
∞
∞
n=1
L
L
where for any bundle E →M, Λx E :=
x m Λm E , S x E :=
xmSmE
m=0
m=0
Definition
With q := e 2πiτ , y := e 2πiz for τ,Z z ∈ C, Im(τ ) > 0,
EM (τ, z) := χ(Eq,−y ) =
Td(M) ch(Eq,−y )
M
is the elliptic genus of M.
Katrin Wendland
SUSY decomposes the bundle underlying the elliptic genus
2/7
Introduction
1. The elliptic genus
2. Decomposing under extended SUSY
3. Interpretation in terms of Mathieu Moonshine
1. The elliptic genus of Calabi-Yau manifolds. . .
Let M denote a compact Calabi-Yau D-fold, T := T 1,0 M,
∞
O
∗
− D2
Λ−yqnT ∗ ⊗ Λ−y −1 qnT ⊗ SqnT ∗ ⊗ SqnT ,
Eq,−y := y Λ−yT ⊗
∞
∞
n=1
L
L
where for any bundle E →M, Λx E :=
x m Λm E , S x E :=
xmSmE
m=0
m=0
Definition
With q := e 2πiτ , y := e 2πiz for τ,Z z ∈ C, Im(τ ) > 0,
EM (τ, z) := χ(Eq,−y ) =
Td(M) ch(Eq,−y )
M
is the elliptic genus of M.
For M: a K3 surface,
that is, a compact Calabi-Yau 2-fold with h1,0 (M) = 0,
2
2
2
ϑ3 (τ,z)
ϑ4 (τ,z)
EK3 (τ, z) = 8 ϑϑ22 (τ,z)
+
8
+
8
.
(τ,0)
ϑ3 (τ,0)
ϑ4 (τ,0)
Katrin Wendland
SUSY decomposes the bundle underlying the elliptic genus
2/7
Introduction
1. The elliptic genus
2. Decomposing under extended SUSY
3. Interpretation in terms of Mathieu Moonshine
1. . . . and for superconformal field theories
CFT elliptic genus
of an N = (2, 2) SCFT at central charges (c, c) with space-time SUSY and integral U(1) charges:
ECFT (τ, z) := sTr HR y J0 q L0 −c/24 q L0 −c/24 ,
HR : Ramond sector,
J0 , L0 , L0 : zero modes of the U(1)-current and Virasoro fields in the SCA
For every N = (2, 2) SCFT at central charges c = c = 6 with
space-time SUSY and integral U(1) charges, the CFT elliptic
genus ECFT (τ, z)
either vanishes, or it agrees with EK3 (τ, z).
The theory has N = (4, 4) SUSY.
Examples:
toroidal SCFTs, their orbifolds, Gepner models at c = c = 6
Katrin Wendland
SUSY decomposes the bundle underlying the elliptic genus
3/7
Introduction
1. The elliptic genus
2. Decomposing under extended SUSY
3. Interpretation in terms of Mathieu Moonshine
1. . . . and for superconformal field theories
CFT elliptic genus
of an N = (2, 2) SCFT at central charges (c, c) with space-time SUSY and integral U(1) charges:
ECFT (τ, z) := sTr HR y J0 q L0 −c/24 q L0 −c/24 ,
HR : Ramond sector,
J0 , L0 , L0 : zero modes of the U(1)-current and Virasoro fields in the SCA
For every N = (2, 2) SCFT at central charges c = c = 6 with
space-time SUSY and integral U(1) charges, the CFT elliptic
genus ECFT (τ, z)
either vanishes, or it agrees with EK3 (τ, z).
The theory has N = (4, 4) SUSY.
Examples:
toroidal SCFTs, their orbifolds, Gepner models at c = c = 6
Definition
A K3 theory is an N = (4, 4) SCFT at c = c = 6 with spacetime SUSY, integral U(1) charges and CFT elliptic genus EK3 (τ, z).
Katrin Wendland
SUSY decomposes the bundle underlying the elliptic genus
3/7
Introduction
1. The elliptic genus
2. Decomposing under extended SUSY
3. Interpretation in terms of Mathieu Moonshine
2. Decomposing under N = (4, 4) supersymmetry
Assume: D = 2, i.e. c = c = 6 and N = (4, 4) supersymmetry.
3 types of N = 4 irreps H• with χ• (τ, z) = sTr H• y J0 q L0 −1/4 :
vacuum H0 , massless matter Hm.m. , massive matter Hh>0 .
∞
P
An χn (τ, z)
EK3 (τ, z) = −2χ0 (τ, z) + 20χm.m. (τ, z) +
n=1
Katrin Wendland
SUSY decomposes the bundle underlying the elliptic genus
4/7
Introduction
1. The elliptic genus
2. Decomposing under extended SUSY
3. Interpretation in terms of Mathieu Moonshine
2. Decomposing under N = (4, 4) supersymmetry
Assume: D = 2, i.e. c = c = 6 and N = (4, 4) supersymmetry.
3 types of N = 4 irreps H• with χ• (τ, z) = sTr H• y J0 q L0 −1/4 :
vacuum H0 , massless matter Hm.m. , massive matter Hh>0 .
∞
P
An χn (τ, z)
EK3 (τ, z) = −2χ0 (τ, z) + 20χm.m. (τ, z) +
n=1
Z
=
Td(K3)ch(Eq,−y )
K3
Katrin Wendland
SUSY decomposes the bundle underlying the elliptic genus
4/7
Introduction
1. The elliptic genus
2. Decomposing under extended SUSY
3. Interpretation in terms of Mathieu Moonshine
2. Decomposing under N = (4, 4) supersymmetry
Assume: D = 2, i.e. c = c = 6 and N = (4, 4) supersymmetry.
3 types of N = 4 irreps H• with χ• (τ, z) = sTr H• y J0 q L0 −1/4 :
vacuum H0 , massless matter Hm.m. , massive matter Hh>0 .
∞
P
An χn (τ, z)
EK3 (τ, z) = −2χ0 (τ, z) + 20χm.m. (τ, z) +
n=1
Z
=
Td(K3)ch(Eq,−y )
K3
Conjecture [W13]
Let M=K3. There are polynomials pn for every n ∈ N, such that
∞
L
Eq,−y = −OK3 χ0 (τ, z) ⊕ (−T )χm.m. (τ, z) ⊕
pn (T )χn (τ, z),
n=1
R
where An = K3 Td(K3)pn (T )= χ(pn (T )) for all n ∈ N.
Katrin Wendland
SUSY decomposes the bundle underlying the elliptic genus
4/7
Introduction
1. The elliptic genus
2. Decomposing under extended SUSY
3. Interpretation in terms of Mathieu Moonshine
2. Decomposing under N = (4, 4) supersymmetry
Assume: D = 2, i.e. c = c = 6 and N = (4, 4) supersymmetry.
3 types of N = 4 irreps H• with χ• (τ, z) = sTr H• y J0 q L0 −1/4 :
vacuum H0 , massless matter Hm.m. , massive matter Hh>0 .
∞
P
An χn (τ, z)
EK3 (τ, z) = −2χ0 (τ, z) + 20χm.m. (τ, z) +
n=1
Z
=
Td(K3)ch(Eq,−y )
K3
Conjecture [W13]
Let M=K3. There are polynomials pn for every n ∈ N, such that
∞
L
Eq,−y = −OK3 χ0 (τ, z) ⊕ (−T )χm.m. (τ, z) ⊕
pn (T )χn (τ, z),
n=1
R
where An = K3 Td(K3)pn (T )= χ(pn (T )) for all n ∈ N.
proof: [Creutzig/W14]
Katrin Wendland
SUSY decomposes the bundle underlying the elliptic genus
4/7
Introduction
1. The elliptic genus
2. Decomposing under extended SUSY
3. Interpretation in terms of Mathieu Moonshine
Decomposing the virtual bundle: Ideas of proof
Observation:
D
Eq,−y = y − 2 Λ−yT ∗ ⊗
∞ N
Λ−y qnT ∗ ⊗ Λ−y −1 qnT ⊗ SqnT ∗ ⊗ SqnT
n=1
is an SU(2) principal bundle under the holonomy representation.
Katrin Wendland
SUSY decomposes the bundle underlying the elliptic genus
5/7
Introduction
1. The elliptic genus
2. Decomposing under extended SUSY
3. Interpretation in terms of Mathieu Moonshine
Decomposing the virtual bundle: Ideas of proof
Observation:
D
Eq,−y = y − 2 Λ−yT ∗ ⊗
∞ N
Λ−y qnT ∗ ⊗ Λ−y −1 qnT ⊗ SqnT ∗ ⊗ SqnT
n=1
is an SU(2) principal bundle under the holonomy representation.
holonomy distribution
=⇒
∞
L
Eq,−y ∼
Wn ⊗ Mn ,
=
n=1
Wn : SU(2) principal bundle for the n-dimensional irrep of SU(2),
Mn : multiplicity space
Katrin Wendland
SUSY decomposes the bundle underlying the elliptic genus
5/7
Introduction
1. The elliptic genus
2. Decomposing under extended SUSY
3. Interpretation in terms of Mathieu Moonshine
Decomposing the virtual bundle: Ideas of proof
Observation:
D
Eq,−y = y − 2 Λ−yT ∗ ⊗
∞ N
Λ−y qnT ∗ ⊗ Λ−y −1 qnT ⊗ SqnT ∗ ⊗ SqnT
n=1
is an SU(2) principal bundle under the holonomy representation.
holonomy distribution
=⇒
∞
L
Eq,−y ∼
Wn ⊗ Mn ,
=
n=1
Wn : SU(2) principal bundle for the n-dimensional irrep of SU(2),
Mn : multiplicity space (bigraded with respect to a SUSY action)
Katrin Wendland
SUSY decomposes the bundle underlying the elliptic genus
5/7
Introduction
1. The elliptic genus
2. Decomposing under extended SUSY
3. Interpretation in terms of Mathieu Moonshine
Decomposing the virtual bundle: Ideas of proof
Observation:
D
Eq,−y = y − 2 Λ−yT ∗ ⊗
∞ N
Λ−y qnT ∗ ⊗ Λ−y −1 qnT ⊗ SqnT ∗ ⊗ SqnT
n=1
is an SU(2) principal bundle under the holonomy representation.
∞
L
Eq,−y ∼
Wn ⊗ Mn ,
=
holonomy distribution
=⇒
n=1
Wn : SU(2) principal bundle for the n-dimensional irrep of SU(2),
Mn : multiplicity space (bigraded with respect to a SUSY action)
Conclusion (using [Malikov/Schechtman/Vaintrob99]):
Wn = S n−1 (T ) is invariant under the N = 4 SUSY action,
∞
∞
L
L
Eq,−y ∼
Wn ⊗ Mn =
S n−1 (T )κn (τ, z),
=
n=1
n=1
where each κn (τ, z) decomposes into N = 4 characters.
Katrin Wendland
SUSY decomposes the bundle underlying the elliptic genus
5/7
Introduction
1. The elliptic genus
2. Decomposing under extended SUSY
3. Interpretation in terms of Mathieu Moonshine
3. A conjecture of Eguchi, Ooguri and Tachikawa (2010)
EK3 (τ, z) = −2χ0 (τ, z) + 20χm.m. (τ, z) +
∞
P
n=1
Theorem [Gannon12]
´
`
An χn (τ, z), χ• (τ, z) = sTr H• y J0 q L0 −1/4
using results of Cheng, Duncan, Gaberdiel,
Hohenegger, Persson, Ronellenfitsch, Volpato
There exists a representation Rn of M24 for every n ∈ N, s.th.
∞
M
RGan. := (−2)H0 ⊕ 20 Hm.m. ⊕
Rn ⊗ Hn
n=1
(g )
with EK 3 (τ, z)= sTr RGan. g y J0 q L0 −1/4 , g ∈ M24 , “twisted genera”.
Katrin Wendland
SUSY decomposes the bundle underlying the elliptic genus
6/7
Introduction
1. The elliptic genus
2. Decomposing under extended SUSY
3. Interpretation in terms of Mathieu Moonshine
3. A conjecture of Eguchi, Ooguri and Tachikawa (2010)
EK3 (τ, z) = −2χ0 (τ, z) + 20χm.m. (τ, z) +
∞
P
n=1
Theorem [Gannon12]
´
`
An χn (τ, z), χ• (τ, z) = sTr H• y J0 q L0 −1/4
using results of Cheng, Duncan, Gaberdiel,
Hohenegger, Persson, Ronellenfitsch, Volpato
There exists a representation Rn of M24 for every n ∈ N, s.th.
∞
M
RGan. := (−2)H0 ⊕ 20 Hm.m. ⊕
Rn ⊗ Hn
n=1
(g )
with EK 3 (τ, z)= sTr RGan. g y J0 q L0 −1/4 , g ∈ M24 , “twisted genera”.
WHY?
Katrin Wendland
SUSY decomposes the bundle underlying the elliptic genus
6/7
Introduction
1. The elliptic genus
2. Decomposing under extended SUSY
3. Interpretation in terms of Mathieu Moonshine
3. A conjecture of Eguchi, Ooguri and Tachikawa (2010)
EK3 (τ, z) = −2χ0 (τ, z) + 20χm.m. (τ, z) +
∞
P
n=1
Theorem [Gannon12]
´
`
An χn (τ, z), χ• (τ, z) = sTr H• y J0 q L0 −1/4
using results of Cheng, Duncan, Gaberdiel,
Hohenegger, Persson, Ronellenfitsch, Volpato
There exists a representation Rn of M24 for every n ∈ N, s.th.
∞
M
RGan. := (−2)H0 ⊕ 20 Hm.m. ⊕
Rn ⊗ Hn
n=1
(g )
with EK 3 (τ, z)= sTr RGan. g y J0 q L0 −1/4 , g ∈ M24 , “twisted genera”.
HOW?
Katrin Wendland
SUSY decomposes the bundle underlying the elliptic genus
6/7
Introduction
1. The elliptic genus
2. Decomposing under extended SUSY
3. Interpretation in terms of Mathieu Moonshine
3. A conjecture of Eguchi, Ooguri and Tachikawa (2010)
EK3 (τ, z) = −2χ0 (τ, z) + 20χm.m. (τ, z) +
∞
P
n=1
Theorem [Gannon12]
´
`
An χn (τ, z), χ• (τ, z) = sTr H• y J0 q L0 −1/4
using results of Cheng, Duncan, Gaberdiel,
Hohenegger, Persson, Ronellenfitsch, Volpato
There exists a representation Rn of M24 for every n ∈ N, s.th.
∞
M
RGan. := (−2)H0 ⊕ 20 Hm.m. ⊕
Rn ⊗ Hn
n=1
(g )
with EK 3 (τ, z)= sTr RGan. g y J0 q L0 −1/4 , g ∈ M24 , “twisted genera”.
Theorem [Mukai88]
If G is a symmetry group of a K3 surface M,
that is, G fixes the two-forms that define the hyperkähler structure of M,
then G is isomorphic to a subgroup of the Mathieu group M24 ,
and |G | ≤ 960 244.823.040 = |M24 |.
Katrin Wendland
SUSY decomposes the bundle underlying the elliptic genus
6/7
Introduction
1. The elliptic genus
2. Decomposing under extended SUSY
3. Interpretation in terms of Mathieu Moonshine
3. A conjecture of Eguchi, Ooguri and Tachikawa (2010)
EK3 (τ, z) = −2χ0 (τ, z) + 20χm.m. (τ, z) +
∞
P
n=1
Theorem [Gannon12]
´
`
An χn (τ, z), χ• (τ, z) = sTr H• y J0 q L0 −1/4
using results of Cheng, Duncan, Gaberdiel,
Hohenegger, Persson, Ronellenfitsch, Volpato
There exists a representation Rn of M24 for every n ∈ N, s.th.
∞
M
RGan. := (−2)H0 ⊕ 20 Hm.m. ⊕
Rn ⊗ Hn
n=1
(g )
with EK 3 (τ, z)= sTr RGan. g y J0 q L0 −1/4 , g ∈ M24 , “twisted genera”.
Theorem [Mukai88]
If G is a symmetry group of a K3 surface M,
then G is isomorphic to a subgroup of the Mathieu group M24 ,
and |G | ≤ 960 244.823.040 = |M24 |.
[Gaberdiel/Hohenegger/Volpato11]
M24 cannot act as symmetry group of a K3 theory.
Katrin Wendland
SUSY decomposes the bundle underlying the elliptic genus
6/7
Introduction
1. The elliptic genus
2. Decomposing under extended SUSY
3. Interpretation in terms of Mathieu Moonshine
3. A conjecture of Eguchi, Ooguri and Tachikawa (2010)
EK3 (τ, z) = −2χ0 (τ, z) + 20χm.m. (τ, z) +
∞
P
n=1
Theorem [Gannon12]
´
`
An χn (τ, z), χ• (τ, z) = sTr H• y J0 q L0 −1/4
using results of Cheng, Duncan, Gaberdiel,
Hohenegger, Persson, Ronellenfitsch, Volpato
There exists a representation Rn of M24 for every n ∈ N, s.th.
∞
M
RGan. := (−2)H0 ⊕ 20 Hm.m. ⊕
Rn ⊗ Hn
n=1
(g )
with EK 3 (τ, z)= sTr RGan. g y J0 q L0 −1/4 , g ∈ M24 , “twisted genera”.
Theorem [Mukai88]
If G is a symmetry group of a K3 surface M,
then G is isomorphic to a subgroup of the Mathieu group M24 ,
and |G | ≤ 960 244.823.040 = |M24 |.
[Gaberdiel/Hohenegger/Volpato11]
M24 cannot act as symmetry group of a K3 theory.
[Creutzig/W14]
For every n ∈ N, An = χ(pn (T )).
Katrin Wendland
SUSY decomposes the bundle underlying the elliptic genus
6/7
Introduction
1. The elliptic genus
2. Decomposing under extended SUSY
3. Interpretation in terms of Mathieu Moonshine
Solving Mathieu Moonshine by Symmetry Surfing?
Conjecture [Taormina/W10-13]
In every geometric interpretation, we have HR HR,gen , where
∞
L
Rn ⊗ Hn = RGan.
HR,gen ∼
= (−2)H0 ⊕ Rm.m. ⊗ Hm.m. ⊕
n=1
as a representation of the geometric symmetry group G ⊂ M24 ;
RGan. collects symmetries from distinct pts. of the moduli space.
We call this procedure symmetry surfing.
Katrin Wendland
SUSY decomposes the bundle underlying the elliptic genus
7/7
Introduction
1. The elliptic genus
2. Decomposing under extended SUSY
3. Interpretation in terms of Mathieu Moonshine
Solving Mathieu Moonshine by Symmetry Surfing?
Conjecture [Taormina/W10-13]
In every geometric interpretation, we have HR HR,gen , where
∞
L
Rn ⊗ Hn = RGan.
HR,gen ∼
= (−2)H0 ⊕ Rm.m. ⊗ Hm.m. ⊕
n=1
as a representation of the geometric symmetry group G ⊂ M24 ;
RGan. collects symmetries from distinct pts. of the moduli space.
We call this procedure symmetry surfing.
Results [Taormina/W11&12&13]
Restricting to the geometric Z2 -orbifold CFTs on K3:
• The joint action of all geometric symmetry groups yields
the maximal subgroup (Z2 )4 o A8 ⊂ M24 .
Note: (Z2 )4 o A8 is not a subgroup of M23 .
• R1 arises as a space of common states with an action of
(Z2 )4 o A8 induced from the 45 ⊕ 45 of M24 .
Note: There is a twist in this action.
Katrin Wendland
SUSY decomposes the bundle underlying the elliptic genus
7/7
Introduction
1. The elliptic genus
2. Decomposing under extended SUSY
3. Interpretation in terms of Mathieu Moonshine
Supersymmetry decomposes the virtual bundle that underlies the elliptic genus
Katrin Wendland
SUSY decomposes the bundle underlying the elliptic genus
7/7
Introduction
1. The elliptic genus
2. Decomposing under extended SUSY
3. Interpretation in terms of Mathieu Moonshine
Supersymmetry decomposes the virtual bundle that underlies the elliptic genus
Thank you
for your attention!
Katrin Wendland
SUSY decomposes the bundle underlying the elliptic genus
7/7