Elliptic genera and N = 2 superconformal field theory

Kawai, Takeshi; Yamada, Yasuhiko; Yang, Sung–Kil

doi:10.1016/0550-3213(94)90428-6

Cited by 179 publications

(294 citation statements)

References 39 publications

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“…Our discussion here parallels that in [9]; also useful are [12,13]. The focus will be on theories with either (4, 4) or (0, 4) supersymmetry, though in fact replacing any of the 4's by 2's makes almost no difference.…”

Section: Review: Partition Functions In Cftmentioning

confidence: 88%

Partition functions and elliptic genera from supergravity

Kraus¹,

Larsen²

2007

J. High Energy Phys.

125

174

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We develop the spacetime aspects of the computation of partition functions for string/M-theory on AdS 3 × M . Subleading corrections to the semi-classical result are included systematically, laying the groundwork for comparison with CFT partition functions via the AdS 3 /CFT 2 correspondence. This leads to a better understanding of the "Farey tail" expansion of Dijkgraaf et. al. from the point of view of bulk physics. Besides clarifying various issues, we also extend the analysis to the N = 2 setting with higher derivative effects included.

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Section: Review: Partition Functions In Cftmentioning

confidence: 88%

Partition functions and elliptic genera from supergravity

Kraus¹,

Larsen²

2007

J. High Energy Phys.

125

174

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“…The elliptic genus of a CY d-fold, Z CY d (τ, z) with τ, z ∈ C and Im(τ ) > 0, is a weight 0 and index d/2 weak Jacobi form [23]. A Jacobi form φ k,m (τ, z) of weight k and index m satisfies…”

Section: Jacobi Formsmentioning

confidence: 99%

“…The K3 surface is the lowest dimensional non-trivial compact Calabi-Yau manifold. Its geometric symmetries at different points in moduli space have been classified by Mukai and Kondo [11,12] who found that the symplectic automorphisms of any K3 manifold form a subgroup of the Mathieu group M 23 . This means that no particular K3 surface has M 24 as symmetry group.…”

Section: Jhep02(2018)129mentioning

confidence: 99%

“…In particular, one can calculate it for any conformal field theory with Calabi-Yau target space. The elliptic genus for a Calabi-Yau d-fold, where d denotes the complex dimension of the Calabi-Yau manifold, is a Jacobi form with weight 0 and index d 2 [23]. The space of such Jacobi forms, which we denote by J 0, d…”

Section: Jhep02(2018)129mentioning

confidence: 99%

“…we consider the orbifold by Z k+2 of the tensor product of 15 or 10 minimal modes of A k+1 type for k = 1 and k = 2, respectively. It was explained how to calculate the elliptic genus for the orbifolded tensor product of minimal models in [23,50] (this was applied to moonshine in [51]). Each A k+1 minimal model is obtained from a chiral multiplet Φ with superpotential W = Φ k+2 k+2 and gives a multiplicative contribution of…”

Section: Jhep02(2018)129mentioning

confidence: 99%

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Calabi-Yau manifolds and sporadic groups

Banlaki

Chowdhury

Kidambi

et al. 2018

J. High Energ. Phys.

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A few years ago a connection between the elliptic genus of the K3 manifold and the largest Mathieu group M 24 was proposed. We study the elliptic genera for Calabi-Yau manifolds of larger dimensions and discuss potential connections between the expansion coefficients of these elliptic genera and sporadic groups. While the Calabi-Yau 3-fold case is rather uninteresting, the elliptic genera of certain Calabi-Yau d-folds for d > 3 have expansions that could potentially arise from underlying sporadic symmetry groups. We explore such potential connections by calculating twined elliptic genera for a large number of Calabi-Yau 5-folds that are hypersurfaces in weighted projected spaces, for a toroidal orbifold and two Gepner models.

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Quantum Harmonic Analysis and Geometric Invariants

Jaffe

1999

Advances in Mathematics

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We develop two topics in parallel and show their inter-relation. The first centers on the notion of a fractional-differentiable structure on a commutative or a non-commutative space. We call this study quantum harmonic analysis. The second concerns homotopy invariants for these spaces and is an aspect of non-commutative geometry.We study an algebra A, which will be a Banach algebra with unit, represented as an algebra of operators on a Hilbert space H. In order to obtain a geometric interpretation of A, we define a derivative on elements of A. We do this in a Hilbert space context, taking da as a commutator da= [Q, a]. Here Q is a basic self-adjoint operator with discrete spectrum, increasing sufficiently rapidly that exp(&;Q) 2 has a trace whenever ;>0. We can define fractional differentiability of order +, with 0<+ 1, by the boundedness of (Q 2 +I ) +Â2 a(Q 2 +I ) &+Â2 . Alternatively we can require the boundedness of an appropriate smoothing (Bessel potential) of da. We find that it is convenient to assume the boundedness of (, where we choose :, ; 0 such that :+;<1. We show that this also ensures a fractional derivative of order +=1&; in the first sense. We define a family of interpolation spaces J ;, : . Each such space is a Banach algebra of operator, whose elements have a fractional derivative of order +=1&;>0.We concentrate on subalgebras A of J ;, : which have certain additional covariance properties under a group Z 2 _G acting on H by a unitary representation #_U( g). In addition, the derivative Q is assumed to be G-invariant. The geometric interpretation flows from the assumption that elements of A possess an arbitrarily small fractional derivative. We study homotopy invariants of A in terms of equivariant, entire cyclic cohomology. In fact, the existence of a fractional derivative on A allows the construction of the cochain { JLO , which plays the role of the integral of differential forms. We give a simple expression for a homotopy invariant Z Q (a; g), determined by pairing { JLO , with a G-invariant element a # A, such that a is a square root of the identity. This invariant is Z Q (a; g)= (1Â-?) & e &t 2 Tr(#U(g) ae &Q 2 +it da ) dt. This representation of the pairing is reminiscent of the heat-kernel representation for an index. In fact this quantity is an invariant, in the following sense. We isolate a simple condition on a family Q(*) of differentiations that yields a continuouslydifferentiable family { JLO (*) of cochains. Since Z Q (a; g) need not be an integer, continuity of { JLO (*) in * is insufficient to prove the constancy of the pairing. However the existence of the derivative leads to the existence of the homotopy. As Article ID aima.1998.1747, available online at http:ÂÂwww.idealibrary.com on 1 0001-8708Â99 30.00

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Elliptic genera and N = 2 superconformal field theory

Cited by 179 publications

References 39 publications

Partition functions and elliptic genera from supergravity

Partition functions and elliptic genera from supergravity

Calabi-Yau manifolds and sporadic groups

Quantum Harmonic Analysis and Geometric Invariants

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