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

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 Ftheory 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

$E_8$ Gauge Theory and Gerbes in String T

“…In particular, we do not know the degrees of freedom of this theory. 18 Nevertheless, the structure of the topological parts of the action seems to indicate that having an index of E 8 implies that we have a curvature F 2 of the bundle, and so the corresponding vector potentials must be present. 19 With this assumption we can look at the Dirac operators coupled to these potentials.…”

The Universal Coefficient Theorem and Quantum Field Theory

“…Plausibility arguments will be given for the usefulness of standard and non-trivial cohomology coefficients in quantum gravity. The main reference is [284] but I will also refer to [285] and [286]. First, one should notice that quantum gravity is currently understood via two rather distinct and in many aspects incompatible [289] theories.…”

“…To make this argument clearer we may start by looking at the generalized elliptic cohomologies. We define an elliptic spectrum [284,285] as consisting of…”

“…When speaking about cohomology theories with coefficient groups defined by means of prime numbers (for example Z/pZ) the prime 2 appears to be important in string theory. This happens because the 2-torsion appears in the fields and can be seen for example in the K-theory calculation of the IIA partition function [284,288]. One can also consider cohomology theories whose formal group laws are elliptic, i.e.…”

“…One method, described also in [284] is to go from an elliptic curve over C to one over Z by means of a curve over the rationals Q. showed that the coefficient groups in (co)homology have important effects at least in the string theoretical interpretation of quantum gravity.…”