A note on Schwartz functions and modular forms

Rolen, Larry; Wagner, Ian

doi:10.1007/s00013-020-01459-y

Cited by 4 publications

(5 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Mazáč [26], which were generalized to a basis by Mazáč and Paulos [54]. Hartman, Mazáč, and Rastelli [9] discovered that these functionals could be adapted to the 2d modular bootstrap with U(1) c or Virasoro symmetry, and special cases were independently constructed by Rolen and Wagner [55] and by Feigenbaum, Grabner, and Hardin [56]. Figure 5 shows the sphere packing bounds obtained from these functionals.…”

Section: Properties Of the Spectrum And Degeneraciesmentioning

confidence: 99%

High-dimensional sphere packing and the modular bootstrap

Afkhami-Jeddi

Cohn

Hartman

et al. 2020

J. High Energ. Phys.

View full text Add to dashboard Cite

We carry out a numerical study of the spinless modular bootstrap for conformal field theories with current algebra U(1)c× U(1)c, or equivalently the linear programming bound for sphere packing in 2c dimensions. We give a more detailed picture of the behavior for finite c than was previously available, and we extrapolate as c → ∞. Our extrapolation indicates an exponential improvement for sphere packing density bounds in high dimen- sions. Furthermore, we study when these bounds can be tight. Besides the known cases c = 1/2, 4, and 12 and the conjectured case c = 1, our calculations numerically rule out sharp bounds for all other c < 90, by combining the modular bootstrap with linear programming bounds for spherical codes.

show abstract

Section: Properties Of the Spectrum And Degeneraciesmentioning

confidence: 99%

High-dimensional sphere packing and the modular bootstrap

Afkhami-Jeddi

Cohn

Hartman

et al. 2020

J. High Energ. Phys.

View full text Add to dashboard Cite

show abstract

“…This spectrum arose in analytic functionals for the 1d conformal bootstrap constructed by Mazáč [25], which were generalized to a basis by Mazáč and Paulos [53]. Hartman, Mazáč, and Rastelli [9] discovered that these functionals could be adapted to the 2d modular bootstrap with U (1) c or Virasoro symmetry, and special cases were independently constructed by Rolen and Wagner [54] and by Feigenbaum, Grabner, and Hardin [55].…”

Section: Properties Of the Spectrum And Degeneraciesmentioning

confidence: 99%

High-dimensional sphere packing and the modular bootstrap

Afkhami-Jeddi,

Cohn,

Hartman

et al. 2020

Preprint

View full text Add to dashboard Cite

We carry out a numerical study of the spinless modular bootstrap for conformal field theories with current algebra U (1) c × U (1) c , or equivalently the linear programming bound for sphere packing in 2c dimensions. We give a more detailed picture of the behavior for finite c than was previously available, and we extrapolate as c → ∞. Our extrapolation indicates an exponential improvement for sphere packing density bounds in high dimensions. Furthermore, we study when these bounds can be tight. Besides the known cases c = 1/2, 4, and 12 and the conjectured case c = 1, our calculations numerically rule out sharp bounds for all other c < 90, by combining the modular bootstrap with linear programming bounds for spherical codes. Contents 1 Introduction 2 The spinless modular bootstrap 3 Numerical results 4 Properties of the spectrum and degeneracies 5 Spherical codes and implied kissing numbers A The limiting case of the spinless modular bootstrap B Convergence of the spinless bootstrap

show abstract

“…The main ingredient of their proof is an interpolation formula for radial Schwartz functions in these dimensions, which allows to interpolate values and first derivatives of f andf in the points √ 2n (n ∈ N). As we were completing this manuscript we became aware of the work by Rolen and Wagner [28], who studied similar questions for dimensions divisible by 8. They were focused on applications for proving packing bounds in these dimensions.…”

Section: B D Where B D Denotes the Volume Of The D-dimensional Unit Ballmentioning

confidence: 99%

“…This shows A − (48) ≤ √ 6. This should be compared to [28]. There it is erroneously stated that for dimension 48 only a +1 eigenfunction with last sign change at distance √ 6 can be obtained by this method.…”

Section: Dimensionmentioning

confidence: 99%

Eigenfunctions of the Fourier transform with specified zeros

Feigenbaum¹,

Grabner²,

Hardin³

2021

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

Eigenfunctions of the Fourier transform with prescribed zeros played a major role in the proof that the E8 and the Leech lattice give the best sphere packings in respective dimensions 8 and 24 by Cohn, Kumar, Miller, Radchenko and Viazovska. The functions used for a linear programming argument were constructed as Laplace transforms of certain modular and quasimodular forms. Similar constructions were used by Cohn and Gonçalves to find a function satisfying an optimal uncertainty principle in dimension 12. This paper gives a unified view on these constructions and develops the machinery to find the underlying forms in all dimensions divisible by 4. Furthermore, the positivity of the Fourier coefficients of the quasimodular forms occurring in this context is discussed.

show abstract

A note on Schwartz functions and modular forms

Cited by 4 publications

References 11 publications

High-dimensional sphere packing and the modular bootstrap

High-dimensional sphere packing and the modular bootstrap

High-dimensional sphere packing and the modular bootstrap

Eigenfunctions of the Fourier transform with specified zeros

Contact Info

Product

Resources

About