Generalized Mathieu Moonshine

Gaberdiel, Matthias R.; Persson, Daniel; Ronellenfitsch, Henrik; Volpato, Roberto

doi:10.4310/cntp.2013.v7.n1.a5

Cited by 58 publications

(146 citation statements)

References 49 publications

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“…This conjecture, originated from an observation of Eguchi, Ooguri and Tachikawa (EOT) in [5], proposes a connection between the elliptic genus of K3 and a finite sporadic simple group, the Mathieu group M 24 . After the original EOT proposal, a considerable amount of evidence in favour of this conjecture has been compiled [6][7][8][9][10] and several different incarnations of the relationship between M 24 and various string compactifications on K3 have been uncovered [11][12][13][14][15][16][17][18][19]. Despite the amount of work on the subject, however, no satisfactory explanation of this phenomenon has been provided so far.…”

Section: Jhep08(2014)094mentioning

confidence: 99%

See 1 more Smart Citation

On symmetries of N $$ \mathcal{N} $$ = (4, 4) sigma models on T 4

Volpato

2014

J. High Energ. Phys.

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Motivated by an analogous result for K3 models, we classify all groups of symmetries of non-linear sigma models on a torus T 4 that preserve the N = (4, 4) superconformal algebra. The resulting symmetry groups are isomorphic to certain subgroups of the Weyl group of E 8 , that plays a role similar to the Conway group for the case of K3 models. Our analysis heavily relies on the triality automorphism of the T-duality group SO(4, 4, Z). As a byproduct of our results, we discover new explicit descriptions of K3 models as asymmetric orbifolds of torus CFTs.

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Section: Jhep08(2014)094mentioning

confidence: 99%

“…In this work, some Siegel modular forms were constructed as the multiplicative lifts of the twisted-twining genera of generalized Mathieu moonshine [11]. Many of these modular forms admit an interpretation as partition functions for 1/4 BPS states in four dimensional CHL models with 16 space-time supersymmetries.…”

Section: Jhep08(2014)094mentioning

confidence: 99%

On symmetries of N $$ \mathcal{N} $$ = (4, 4) sigma models on T 4

Volpato

2014

J. High Energ. Phys.

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“…The reason why M 24 is singled out remains a mystery. From the properties of twining elliptic genera, one may expect a representation of M 24 on a vertex algebra which governs the elliptic genus of K3, as argued in [16,12,17]. However, there are conceptual difficulties in following this lead, particularly in the sector of the elliptic genus corresponding to massless states at leading order.…”

Section: Introductionmentioning

confidence: 99%

A twist in the M 24 moonshine story

Taormina¹,

Wendland²

2016

Confluentes Mathematici

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Prompted by the Mathieu Moonshine observation, we identify a pair of 45-dimensional vector spaces of states that account for the first order term in the massive sector of the elliptic genus of K3 in every Z 2 -orbifold CFT on K3. These generic states are uniquely characterized by the fact that the action of every geometric symmetry group of a Z 2orbifold CFT yields a well-defined faithful representation on them. Moreover, each such representation is obtained by restriction of the 45-dimensional irreducible representation of the Mathieu group M 24 constructed by Margolin. Thus we provide a piece of evidence for Mathieu Moonshine explicitly from SCFTs on K3.The 45-dimensional irreducible representation of M 24 exhibits a twist, which we prove can be undone in the case of Z 2 -orbifold CFTs on K3 for all geometric symmetry groups. This twist however cannot be undone for the combined symmetry group (Z 2 ) 4 A 8 that emerges from surfing the moduli space of Kummer K3s. We conjecture that in general, the untwisted representations are exclusively those of geometric symmetry groups in some geometric interpretation of a CFT on K3. In that light, the twist appears as a representation theoretic manifestation of the maximality constraints in Mukai's classification of geometric symmetry groups of K3.

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“…We represent the elements of this group pictorially as Next we consider the group Z 4 2 of symmetries that is generated by the geometric symmetries s v 2 +v 4 , s v 1 +v 2 , γ 1 and γ 2 as given in (4.39); this is a subgroup of left-right symmetric elements of SU(2) 6 L × SU(2) 6 R . Note that the matrices 0, 1, ω,ω that were introduced in (4.35) act on each tetrad by 8 …”

Section: The Subgroup Gmentioning

confidence: 99%

“…Thus a consistent decomposition of all expansion coefficients in terms of M 24 representations is possible. More recently, evidence was also obtained that the same is true for the twisted twining genera [8,9], i.e. the analogues of Norton's generalised moonshine functions [10].…”

Section: Introductionmentioning

confidence: 99%

A K3 sigma model with $ \mathbb{Z}_2^8 $ : $ {{\mathbb{M}}_{20 }} $ symmetry

Gaberdiel

Taormina

Volpato

et al. 2014

J. High Energ. Phys.

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The K3 sigma model based on the Z 2 -orbifold of the D 4 -torus theory is studied. It is shown that it has an equivalent description in terms of twelve free Majorana fermions, or as a rational conformal field theory based on the affine algebra su(2) 6 . By combining these different viewpoints we show that the N = (4, 4) preserving symmetries of this theory are described by the discrete symmetry group Z 8 2 : M 20 . This model therefore accounts for one of the largest maximal symmetry groups of K3 sigma models. The symmetry group involves also generators that, from the orbifold point of view, map untwisted and twisted sector states into one another.

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Generalized Mathieu Moonshine

Cited by 58 publications

References 49 publications

On symmetries of N $$ \mathcal{N} $$ = (4, 4) sigma models on T 4

On symmetries of N $$ \mathcal{N} $$ = (4, 4) sigma models on T 4

A twist in the M 24 moonshine story

A K3 sigma model with $ \mathbb{Z}_2^8 $ : $ {{\mathbb{M}}_{20 }} $ symmetry

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