The umbral moonshine module for the unique unimodular Niemeier root system

Duncan, John F. R.; Harvey, Jeffrey A.

doi:10.2140/ant.2017.11.505

Cited by 24 publications

(35 citation statements)

References 42 publications

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“…Just such an analogue for X = E 3 8 has recently been obtained in [86], where a super vertex operator algebra V X is constructed, together with an action of G X S 3 , such that the components of the vector-valued mock modular forms H X g = H X g,r are recovered from traces of elements of G X on canonically twisted modules for V X . The main ingredient in the construction of [86] is an adaptation of the familiar (to specialists) lattice vertex algebra construction (cf.…”

Section: Theorem 16 (Duncan-griffin-ono) Conjecture 5 Is Truementioning

confidence: 99%

“…The main ingredient in the construction of [86] is an adaptation of the familiar (to specialists) lattice vertex algebra construction (cf. [15,104]), to cones in indefinite lattices.…”

Section: Theorem 16 (Duncan-griffin-ono) Conjecture 5 Is Truementioning

confidence: 99%

“…Just as the vertex operator algebra structure on V gives a strong solution to Thompson's conjecture, Conjecture 1, we can expect concrete constructions of thě K X -whose existence is now guaranteed thanks to Theorem 16 -to be necessary for the elucidation of the physical origins of umbral moonshine. As we have described in Section 9.3, progress on this problem has been obtained recently in [46,86,89], and the related work [40] may also be useful, in the determination of a general, algebraic solution to the module problem for umbral moonshine.…”

Section: Open Problemsmentioning

confidence: 99%

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Moonshine

Duncan

Griffin

Ono

2015

Mathematical Sciences

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illustrate the case of J(τ ) = j(τ ) − 744, whose coefficients turn out to be sums of the dimensions of the 194 irreducible representations of M. Such formulas are dictated by the structure of the graded monstrous moonshine modules. Recent works in moonshine suggest deep relations between number theory and physics. Number theoretic Kloosterman sums have reappeared in quantum gravity, and mock modular forms have emerged as candidates for the computation of black hole degeneracies. This paper is a survey of past and present research on moonshine. We also compute the quantum dimensions of the monster orbifold and obtain exact formulas for the multiplicities of the irreducible components of the moonshine modules. These formulas imply that such multiplicities are asymptotically proportional to dimensions.

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Section: Theorem 16 (Duncan-griffin-ono) Conjecture 5 Is Truementioning

confidence: 99%

“…The main ingredient in the construction of [86] is an adaptation of the familiar (to specialists) lattice vertex algebra construction (cf. [15,104]), to cones in indefinite lattices.…”

Section: Theorem 16 (Duncan-griffin-ono) Conjecture 5 Is Truementioning

confidence: 99%

Section: Open Problemsmentioning

confidence: 99%

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Moonshine

Duncan

Griffin

Ono

2015

Mathematical Sciences

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“…One of the umbral moonshine conjectures then states that there exists a natural way to associate a graded infinite-dimensional module with the finite group G N such that its graded character coincides with the specified mock modular forms. So far, these modules have been shown to exist [38,56], although, with the exception of a special case [39], a "natural" construction, providing an intuitive interpretation of this vector space and of the action of the corresponding group, is still lacking. Although it is not yet clear whether the structure of vertex operator algebra (VOA; or chiral CFT) is as relevant here as in the classical case of monstrous moonshine, the existence of the generalized umbral moonshine [13,55] suggests that certain key features of VOA should be present in the modules underlying umbral moonshine.…”

Section: Introductionmentioning

confidence: 99%

K3 string theory, lattices and moonshine

Cheng

Harrison²,

Volpato

et al. 2018

Res Math Sci

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In this paper, we address the following two closely related questions. First, we complete the classification of finite symmetry groups of type IIA string theory on K 3 × R 6 , where Niemeier lattices play an important role. This extends earlier results by including points in the moduli space with enhanced gauge symmetries in spacetime, or, equivalently, where the world-sheet CFT becomes singular. After classifying the symmetries as abstract groups, we study how they act on the BPS states of the theory. In particular, we classify the conjugacy classes in the T-duality group O + ( 4,20 ) which represent physically distinct symmetries. Subsequently, we make two conjectures regarding the connection between the corresponding twining genera of K 3 CFTs and Conway and umbral moonshine, building upon earlier work on the relation between moonshine and the K 3 elliptic genus.

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“…the recent work[26] which constructs the super vertex operator algebra underlying the X = E3 8 case of umbral moonshine. b The observation that the decomposition into N = 4 characters of (a multiple of ) the function Z(τ , z) returns positive integers that are suggestive of representations of the Mathieu group M 22 was first communicated privately by Jeff Harvey to J.D.…”

mentioning

confidence: 99%

Mock modular Mathieu moonshine modules

et al. 2015

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We construct super vertex operator algebras which lead to modules for moonshine relations connecting the four smaller sporadic simple Mathieu groups with distinguished mock modular forms. Starting with an orbifold of a free fermion theory, any subgroup of Co 0 that fixes a 3dimensional subspace of its unique non-trivial 24-dimensional representation commutes with a certain N = 4 superconformal algebra. Similarly, any subgroup of Co 0 that fixes a 2-dimensional subspace of the 24-dimensional representation commutes with a certain N = 2 superconformal algebra. Through the decomposition of the corresponding twined partition functions into characters of the N = 4 (resp. N = 2) superconformal algebra, we arrive at mock modular forms which coincide with the graded characters of an infinite-dimensional Z-graded module for the corresponding group. The Mathieu groups are singled out amongst various other possibilities by the moonshine property: requiring the corresponding weak Jacobi forms to have certain asymptotic behaviour near cusps. Our constructions constitute the first examples of explicitly realized modules underlying moonshine phenomena relating mock modular forms to sporadic simple groups. Modules for other groups, including the sporadic groups of McLaughlin and Higman-Sims, are also discussed.

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The umbral moonshine module for the unique unimodular Niemeier root system

Cited by 24 publications

References 42 publications

Moonshine

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K3 string theory, lattices and moonshine

Mock modular Mathieu moonshine modules

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