A mass formula for unimodular lattices with no roots

King, Oliver D.

doi:10.1090/s0025-5718-02-01455-2

Cited by 46 publications

(56 citation statements)

References 28 publications

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“…This suggests that we look for c = 32 meromorphic theories with a simple level-1 Kac-Moody algebra, and coset them by a Deligne series CFT. From Table 1 of [25] we see that there are indeed lattices having A 1 , A 2 , D 4 , E 6 as their root systems (but curiously not E 7 ). The CFT on these lattices will have an extra U(1) 32−r symmetry.…”

Section: An Example In Detailmentioning

confidence: 97%

Curiosities above c = 24

Chandra

Mukhi

2019

SciPost Phys.

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Two-dimensional rational CFT are characterised by an integer ℓ, related to the number of zeroes of the Wronskian of the characters. For two-character RCFT's with ℓ < 6 there is a finite number of theories and most of these are classified. Recently it has been shown that for ℓ ≥ 6 there are infinitely many admissible characters that could potentially describe CFT's. In this note we examine the ℓ = 6 case, whose central charges lie between 24 and 32, and propose a classification method based on cosets of meromorphic CFT's. We illustrate the method using theories on Kervaire lattices with complete root systems. In the process we construct the first known two-character RCFT's beyond ℓ = 2.

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Section: An Example In Detailmentioning

confidence: 97%

Curiosities above c = 24

Chandra

Mukhi

2019

SciPost Phys.

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“…4 In the other direction, King [22] classifies all (even) unimodular lattices in dimension 32 with no roots, and finds there to be at least 10 7 such; as the lack of roots implies that the lattices have no vectors of norm 2, it follows that each is extremal. Similarly, Peters [33] shows there are at least 10 51 extremal lattices in dimension 40.…”

Section: Extremal Latticesmentioning

confidence: 99%

On the Extremality of an 80-Dimensional Lattice

Stehlé

Watkins

2010

Lecture Notes in Computer Science

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Abstract. We show that a specific even unimodular lattice of dimension 80, first investigated by Schulze-Pillot and others, is extremal (i.e., the minimal nonzero norm is 8). This is the third known extremal lattice in this dimension. The known part of its automorphism group is isomorphic to SL2(F79), which is smaller (in cardinality) than the two previous examples. The technique to show extremality involves using the positivity of the Θ-series, along with fast vector enumeration techniques including pruning, while also using the automorphisms of the lattice.

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“…However we could show that the Fourier coefficients of 4 (Z, L 32 ) can be made explicit in the sense that if one specified Fourier coefficient of 4 (Z, L 32 ) is obtained then all the Fourier coefficients are in principle computable. One may note that the results in the present paper are free from constructions for the extremal lattices, while the number of non-isometric 32-dimensional even unimodular extremal lattices is at least one billion (King [15,Corollary 17]). …”

Section: Introductionmentioning

confidence: 99%

A Numerical Study of Siegel Theta Series of Various Degrees for the 32-Dimensional Even Unimodular Extremal Lattices

Oura¹,

Ozeki

2016

Kyushu J. Math.

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Abstract. Erokhin showed that the Siegel theta series associated with the even unimodular 32-dimensional extremal lattices of degree up to three is unique. Later, Salvati Manni showed that the difference of the Siegel theta series of degree four associated with the two even unimodular 32-dimensional extremal lattices is a constant multiple of the square J 2 of the Schottky modular form J , which is a Siegel cusp form of degree four and weight eight. In the present paper we show that the Fourier coefficients of the Siegel theta series associated with the even unimodular 32-dimensional extremal lattices of degrees two and three can be computed explicitly, and the Fourier coefficients of the Siegel theta series of degree four for those lattices are computed almost explicitly.

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A mass formula for unimodular lattices with no roots

Cited by 46 publications

References 28 publications

Curiosities above c = 24

Curiosities above c = 24

On the Extremality of an 80-Dimensional Lattice

A Numerical Study of Siegel Theta Series of Various Degrees for the 32-Dimensional Even Unimodular Extremal Lattices

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