# Dirac operator zero-modes on a torus

## Abstract

We study Dirac operator zero-modes on a torus for gauge background with uniform field strengths. Under the basic translations of the torus coordinates the wave functions are subject to twisted periodic conditions. In suitable torus coordinates the zero-mode wave functions can be related to holomorphic functions of the complex torus coordinates. Half of the twisted boundary conditions for the holomorphic part of the zero-mode wave function can be made periodic or anti-periodic. The remaining half is until coordinate dependent but diagonal. We completely solve the twisted boundary conditions and construct the zero-mode wave functions. The chirality and the degeneracy of the zero-modes are uniquely determined by the gauge background and are consistent with the index theorem.

- Authors:

- Graduate School of Science and Engineering, Ibaraki University, Mito 310-8512 (Japan)
- Department of Physics, Ibaraki University, Mito 310-8512 (Japan)
- Department of Physics, Ibaraki University, Mito 310-8512 (Japan). E-mail: fujiwara@mx.ibaraki.ac.jp

- Publication Date:

- OSTI Identifier:
- 20976732

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Annals of Physics (New York); Journal Volume: 322; Journal Issue: 2; Other Information: DOI: 10.1016/j.aop.2006.02.013; PII: S0003-4916(06)00063-7; Copyright (c) 2006 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BOUNDARY CONDITIONS; CHIRALITY; COORDINATES; DIRAC OPERATORS; PERIODICITY; WAVE FUNCTIONS

### Citation Formats

```
Tenjinbayashi, Yasushi, Igarashi, Hiroshi, and Fujiwara, Takanori.
```*Dirac operator zero-modes on a torus*. United States: N. p., 2007.
Web. doi:10.1016/j.aop.2006.02.013.

```
Tenjinbayashi, Yasushi, Igarashi, Hiroshi, & Fujiwara, Takanori.
```*Dirac operator zero-modes on a torus*. United States. doi:10.1016/j.aop.2006.02.013.

```
Tenjinbayashi, Yasushi, Igarashi, Hiroshi, and Fujiwara, Takanori. Thu .
"Dirac operator zero-modes on a torus". United States.
doi:10.1016/j.aop.2006.02.013.
```

```
@article{osti_20976732,
```

title = {Dirac operator zero-modes on a torus},

author = {Tenjinbayashi, Yasushi and Igarashi, Hiroshi and Fujiwara, Takanori},

abstractNote = {We study Dirac operator zero-modes on a torus for gauge background with uniform field strengths. Under the basic translations of the torus coordinates the wave functions are subject to twisted periodic conditions. In suitable torus coordinates the zero-mode wave functions can be related to holomorphic functions of the complex torus coordinates. Half of the twisted boundary conditions for the holomorphic part of the zero-mode wave function can be made periodic or anti-periodic. The remaining half is until coordinate dependent but diagonal. We completely solve the twisted boundary conditions and construct the zero-mode wave functions. The chirality and the degeneracy of the zero-modes are uniquely determined by the gauge background and are consistent with the index theorem.},

doi = {10.1016/j.aop.2006.02.013},

journal = {Annals of Physics (New York)},

number = 2,

volume = 322,

place = {United States},

year = {Thu Feb 15 00:00:00 EST 2007},

month = {Thu Feb 15 00:00:00 EST 2007}

}