On quantum Galois theory

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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“…§3.1 of [75]). Dong-Mason initiated a vertex algebraic quantum Galois theory in [81] (cf. also [77,82,127]).…”

Section: Corollary 1 In Particular We Have Thatmentioning

confidence: 99%

Moonshine

Duncan

Griffin

Ono

2015

Mathematical Sciences

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“…It follows that the A-modules H λ can be decomposed as 5) where the spaces VĴ are projective S λ -modules and the spacesHλ ,Ĵ areĀ-modules. We make the mild technical assumption that all these modules VĴ andHλ ,Ĵ are irreducible; this holds true in all known examples, and is rigorously proven for the vacuum Ω [7,8] as well as [9] for other A-modules, including twisted sectors. In the case of the vacuum, no multiplicities appear in this decomposition; thus the action of G is genuine and we have U Ω = S Ω = G.…”

Section: (74)mentioning

confidence: 99%

“…8 In fact one should expect that the property of inducing a fusion rule automorphism need not be required independently, but is satisfied automatically as a consequence of the consistency of the relevant orbifold theory. This has been demonstrated in the case of order-two automorphisms in [5].…”

Section: T-dualitymentioning

confidence: 99%

“…Indeed, C(A H ) is just the direct sum of the ideals C (g) (Ā) ⊆ C(A G ) for all g ∈ H. It is also known [8] that the mapping H → A H provides a bijection between the set of subgroups of G and the set of consistent chiral subalgebras of A that contain A G . 11 In our situation the orbifold group G is abelian, but the latter result continues to hold for all nilpotent finite groups [9] and is expected to be true for arbitrary finite orbifold groups.…”

Section: The Universal Classifying Algebramentioning

confidence: 99%

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Symmetry breaking boundaries

Fuchs

Schweigert

2000

Nuclear Physics B

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Various structural properties of the space of symmetry breaking boundary conditions that preserve an orbifold subalgebra are established. To each such boundary condition we associate its automorphism type. We show that correlation functions in the presence of such boundary conditions are expressible in terms of twisted boundary blocks which obey twisted Ward identities. The subset of boundary conditions that share the same automorphism type is controlled by a classifying algebra, whose structure constants are shown to be traces on spaces of chiral blocks. T-duality on boundary conditions is not a one-to-one map in general. These structures are illustrated in a number of examples. Several applications, including the construction of non-BPS boundary conditions in string theory, are exhibited.

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“…C. Dong and G. Mason initiated a systematic search for a vertex operator algebra with a finite automorphism group, which is referred to as the operator content of orbifold models by physicists or as quantum Galois theory for finite groups [6], [9]. There are other Galois correspondences which are relevant to quantum Galois theory for finite groups, especially in the context of subfactors [11], [12].…”

Section: Introductionmentioning

confidence: 99%