### Abstract

Elliptic curves play a natural and important role in elliptic cohomology. In earlier work with I. Kriz, these elliptic curves were interpreted physically in two ways: as corresponding to the intersection of M2 and M5 in the context of (the reduction of M-theory to) type IIA and as the elliptic fiber leading to F-theory for type IIB. In this paper we elaborate on the physical setting for various generalized cohomology theories, including elliptic cohomology, and we note that the above two seemingly unrelated descriptions can be unified using Sen's picture of the orientifold limit of F-theory compactification on K3, which unifies the Seiberg-Witten curve with the F-theory curve, and through which we naturally explain the constancy of the modulus that emerges from elliptic cohomology. This also clarifies the orbifolding performed in the previous work and justifies the appearance of the w_{4} condition in the elliptic refinement of the mod 2 part of the partition function. We comment on the cohomology theory needed for the case when the modular parameter varies in the base of the elliptic fibration.

Original language | English (US) |
---|---|

Pages (from-to) | 6235-6254 |

Number of pages | 20 |

Journal | Journal of High Energy Physics |

Issue number | 3 |

DOIs | |

State | Published - Mar 1 2006 |

### Fingerprint

### Keywords

- Duality in Gauge Field Theories
- F-Theory
- String Duality

### ASJC Scopus subject areas

- Nuclear and High Energy Physics

### Cite this

**The elliptic curves in gauge theory, string theory, and cohomology.** / Sati, Hisham.

Research output: Contribution to journal › Review article

}

TY - JOUR

T1 - The elliptic curves in gauge theory, string theory, and cohomology

AU - Sati, Hisham

PY - 2006/3/1

Y1 - 2006/3/1

N2 - Elliptic curves play a natural and important role in elliptic cohomology. In earlier work with I. Kriz, these elliptic curves were interpreted physically in two ways: as corresponding to the intersection of M2 and M5 in the context of (the reduction of M-theory to) type IIA and as the elliptic fiber leading to F-theory for type IIB. In this paper we elaborate on the physical setting for various generalized cohomology theories, including elliptic cohomology, and we note that the above two seemingly unrelated descriptions can be unified using Sen's picture of the orientifold limit of F-theory compactification on K3, which unifies the Seiberg-Witten curve with the F-theory curve, and through which we naturally explain the constancy of the modulus that emerges from elliptic cohomology. This also clarifies the orbifolding performed in the previous work and justifies the appearance of the w4 condition in the elliptic refinement of the mod 2 part of the partition function. We comment on the cohomology theory needed for the case when the modular parameter varies in the base of the elliptic fibration.

AB - Elliptic curves play a natural and important role in elliptic cohomology. In earlier work with I. Kriz, these elliptic curves were interpreted physically in two ways: as corresponding to the intersection of M2 and M5 in the context of (the reduction of M-theory to) type IIA and as the elliptic fiber leading to F-theory for type IIB. In this paper we elaborate on the physical setting for various generalized cohomology theories, including elliptic cohomology, and we note that the above two seemingly unrelated descriptions can be unified using Sen's picture of the orientifold limit of F-theory compactification on K3, which unifies the Seiberg-Witten curve with the F-theory curve, and through which we naturally explain the constancy of the modulus that emerges from elliptic cohomology. This also clarifies the orbifolding performed in the previous work and justifies the appearance of the w4 condition in the elliptic refinement of the mod 2 part of the partition function. We comment on the cohomology theory needed for the case when the modular parameter varies in the base of the elliptic fibration.

KW - Duality in Gauge Field Theories

KW - F-Theory

KW - String Duality

UR - http://www.scopus.com/inward/record.url?scp=33645694586&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33645694586&partnerID=8YFLogxK

U2 - 10.1088/1126-6708/2006/03/096

DO - 10.1088/1126-6708/2006/03/096

M3 - Review article

AN - SCOPUS:33645694586

SP - 6235

EP - 6254

JO - Journal of High Energy Physics

JF - Journal of High Energy Physics

SN - 1126-6708

IS - 3

ER -