The Elliptic curves in gauge theory, string theory, and cohomology

TL;DR

This paper explores the role of elliptic curves in gauge and string theories, unifying different physical interpretations through elliptic cohomology and F-theory, and clarifies the mathematical structures involved.

Contribution

It unifies two physical descriptions of elliptic curves in string theory using Sen's orientifold limit, and clarifies the mathematical framework of elliptic cohomology in this context.

Findings

Unified the intersection of M2 and M5 branes with F-theory elliptic fibers.

Clarified the role of the w_4 condition in elliptic cohomology.

Explained the constancy of the elliptic modulus in physical models.

Abstract

Elliptic curves play a natural and important role in elliptic cohomology. In earlier work with I. Kriz, thes 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.