Mathieu moonshine and σ-models on K3 Roberto Volpato Albert-Einstein-Institute Max-Planck-Institut f¨ ur Gravitationsphysik Potsdam, Golm Algebra, Geometry and Physics of BPS States Bonn, 12 November 2012 References M.R.Gaberdiel, R.Volpato, 1206.5143 [hep-th] M.R.Gaberdiel, S.Hohenegger, R.Volpato, 1106.4315, 1008.3778, 1006.0221 [hep-th] Works in progress: M.R.Gaberdiel, D.Persson, H.Ronellenfitsch, R.Volpato M 24 and K3 models Roberto Volpato AEI Potsdam
27
Embed
Mathieu moonshine and -models on K3 - him.uni-bonn.de · A relation between K3 and Mathieu groups A theorem in algebraic geometry gives a connection between K3 and Mathieu groups
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
A theorem in algebraic geometry gives a connection between K3 andMathieu groups
Mukai Theorem
Finite groups of symplectic automorphisms of K3 surf’s are subgroups ofM23 ⊆ M24.[Mukai 1988; Kondo 1998]
A symplectic automorphism of the K3 target space inducesa symmetry of the corresponding σ-model.
But σ-models may have non-geometrical (quantum) symmetries
Can we classify the groups of discrete symmetries of K3 σ-models?(Quantum analogue of Mukai theorem)
M24 and K3 models Roberto Volpato AEI Potsdam
Classification Theorem
Let Co0 be the group of automorphisms of the Leech lattice Λ(even self-dual lattice of dim 24 with no elements of norm 2)
Let G be a group of symmetries of a K3 σ-model that commute withN = (4, 4) and spectral flow.
Theorem
G is a subgroup of Co0 ≡ Aut(Λ) fixing pointwise a sublattice of Λof rank at least 4.
Conversely, any G ⊂ Co0 fixing a sublattice of Λ of rank at least 4 is thesymmetry group of some K3 model.
[Gaberdiel, Hohenegger, R.V. 1106.4315]
Also M24 is a subgroup of Co0, but...
M24 and K3 models Roberto Volpato AEI Potsdam
Subgroups G ⊂ Co0 fixing a sublattice of rank 4:
G ⊂ Z122 oM24 (at least 4 orbits on 24-dim rep)
G = 51+2.Z4
G = Z43 o A6 or G = 31+4.Z2.G
′′ with G ′′ = 1,Z2 or Z22
What can we learn from this description?
There is no G such that M24 ⊆ G
There are some G such that G 6⊂ M24
M24 and K3 models Roberto Volpato AEI Potsdam
Sketch of the proof
Known facts about σ-models on K3:[Aspinwall 9611375, Nahm, Wendland 9912067]
There are 24 RR ground states at h = h = 14 (≡ R4,20)
The ground states are contained in N = (4, 4) supermultiplets:
4 are in one (h, `; h, ¯) = ( 14 ,12 ;
14 ,
12 ) (subspace Π ⊂ R4,20)
20 are in 20 distinct (h, `; h, ¯) = ( 14 , 0;14 , 0)
The D-brane charges form an even self-dual lattice Γ4,20
We can think of Γ4,20 as embedded in (the dual of) R4,20
Moduli space of K3 models:
O(Γ4,20)\O(4, 20,R)/(O(4,R)× O(20,R))
M24 and K3 models Roberto Volpato AEI Potsdam
Sketch of the proof
G : group of symmetries that commute with N = (4, 4) and spectral flow
1 G acts faithfully on the lattice Γ4,20 ⊂ R4,20
2 G fixes pointwise Π ⊂ R4,20, with dimΠ = 4
3 G acts faithfully on the sublattice L = Γ4,20 ∩ Π⊥ with dim L ≤ 20
4 L can be embedded into the Leech Λ and the action of G extends toautomorphisms of Λ
5 The sublattice ΛG ⊂ Λ of vectors fixed by G is the orthogonalcomplement of L ⊂ Λ ↓
G is a subgroup of Aut(Λ) ≡ Co0 that fixes a sublattice of rank≥ 4
M24 and K3 models Roberto Volpato AEI Potsdam
Discussion
Problems:
1 There are no K3 σ-models with symmetry group M24
2 There are groups of symmetries G 6⊂M24 (exceptional cases)
3 All groups are contained in Co0 but no Conway Moonshine
Open questions:
Are there unknown SCFTs with the same elliptic genus?
Is M24 the symmetry of some different ‘structure’ related to K3?
Are exceptional models in some sense ‘special’?
Exceptional models seem related to orbifolds of non-linear σ models on T 4
[Gaberdiel, R.V. 1206.5143]
M24 and K3 models Roberto Volpato AEI Potsdam
Cyclic torus orbifolds
CFT orbifold Given a CFT C with a symmetry g , project on g -invariantstates and introduce new (twisted) sectors
(Cyclic) Torus orbifoldsSpecial families of K3 models are given by orbifolds (in CFT sense)of non-linear σ-models on T 4 by some Zn group of symmetries
Easiest example: non-linear σ-model with target space T 4/ZN ,where ZN is a group of automorphisms of T 4
In general, group ZN might be non-geometric symmetry(e.g. asymmetric orbifolds)