# A modular-invariant modified Weierstrass sigma-function as a building block for lowest-Landau-level wavefunctions on the torus

## Abstract

A “modified” variant of the Weierstrass sigma, zeta, and elliptic functions is proposed whereby the zeta function is redefined by $$ζ(z) \mapsto \tilde{ζ}(z) ≡ ζ(z) - γ_2z$$, where γ _{2} is a lattice invariant related to the almost-holomorphic modular invariant of the quasi-modular-invariant weight-2 Eisenstein series. If ω _{i} is a primitive half-period, $$\tilde{ζ}(ω_i) = πω^{*}_{i}/A$$, where *A* is the area of the primitive cell of the lattice. The quasiperi-odicity of the modified sigma function is much simpler than that of the original, and it becomes the building-block for the modular-invariant formulation of lowest-Landau-level wavefunctions on the torus. It is suggested that the “modified” sigma function is more natural than the original Weierstrass form, which was formulated before quasi-modular forms were understood. Finally, for the high-symmetry (square and hexagonal) lattices, the modified and original sigma functions coincide.

- Authors:

- Princeton Univ., NJ (United States). Dept. of Physics

- Publication Date:

- Research Org.:
- Princeton Univ., NJ (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1540241

- Alternate Identifier(s):
- OSTI ID: 1459239

- Grant/Contract Number:
- SC0002140

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Mathematical Physics

- Additional Journal Information:
- Journal Volume: 59; Journal Issue: 7; Journal ID: ISSN 0022-2488

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; physics

### Citation Formats

```
Haldane, F. D. M. A modular-invariant modified Weierstrass sigma-function as a building block for lowest-Landau-level wavefunctions on the torus. United States: N. p., 2018.
Web. doi:10.1063/1.5042618.
```

```
Haldane, F. D. M. A modular-invariant modified Weierstrass sigma-function as a building block for lowest-Landau-level wavefunctions on the torus. United States. doi:10.1063/1.5042618.
```

```
Haldane, F. D. M. Fri .
"A modular-invariant modified Weierstrass sigma-function as a building block for lowest-Landau-level wavefunctions on the torus". United States. doi:10.1063/1.5042618. https://www.osti.gov/servlets/purl/1540241.
```

```
@article{osti_1540241,
```

title = {A modular-invariant modified Weierstrass sigma-function as a building block for lowest-Landau-level wavefunctions on the torus},

author = {Haldane, F. D. M.},

abstractNote = {A “modified” variant of the Weierstrass sigma, zeta, and elliptic functions is proposed whereby the zeta function is redefined by $ζ(z) \mapsto \tilde{ζ}(z) ≡ ζ(z) - γ_2z$, where γ2 is a lattice invariant related to the almost-holomorphic modular invariant of the quasi-modular-invariant weight-2 Eisenstein series. If ωi is a primitive half-period, $\tilde{ζ}(ω_i) = πω^{*}_{i}/A$, where A is the area of the primitive cell of the lattice. The quasiperi-odicity of the modified sigma function is much simpler than that of the original, and it becomes the building-block for the modular-invariant formulation of lowest-Landau-level wavefunctions on the torus. It is suggested that the “modified” sigma function is more natural than the original Weierstrass form, which was formulated before quasi-modular forms were understood. Finally, for the high-symmetry (square and hexagonal) lattices, the modified and original sigma functions coincide.},

doi = {10.1063/1.5042618},

journal = {Journal of Mathematical Physics},

number = 7,

volume = 59,

place = {United States},

year = {2018},

month = {7}

}

*Citation information provided by*

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Web of Science

Works referenced in this record:

##
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journal, January 2016

- Charollois, Pierre; Sczech, Robert
- EMS Newsletter, Vol. 2016-9, Issue 101

##
A new construction of Eisenstein’s completion of the Weierstrass zeta function

journal, July 2015

- Rolen, Larry
- Proceedings of the American Mathematical Society, Vol. 144, Issue 4