Higher Structures in M-Theory

The key open problem of string theory remains its non-perturbative completion to M-theory. A decisive hint to its inner workings comes from numerous appearances of higher structures in the limits of M-theory that are already understood, such as higher degree flux fields and their dualities, or the higher algebraic structures governing closed string field theory. These are all controlled by the higher homotopy theory of derived categories, generalised cohomology theories, and $L_\infty$-algebras. This is the introductory chapter to the proceedings of the LMS/EPSRC Durham Symposium on Higher Structures in M-Theory. We first review higher structures as well as their motivation in string theory and beyond. Then we list the contributions in this volume, putting them into context.Comment: 22 pages, Introductory Article to Proceedings of LMS/EPSRC Durham Symposium Higher Structures in M-Theory, August 2018, references update

What do Topologists want from Seiberg--Witten theory? (A review of four-dimensional topology for physicists)

In 1983, Donaldson shocked the topology world by using instantons from physics to prove new theorems about four-dimensional manifolds, and he developed new topological invariants. In 1988, Witten showed how these invariants could be obtained by correlation functions for a twisted N=2 SUSY gauge theory. In 1994, Seiberg and Witten discovered dualities for such theories, and in particular, developed a new way of looking at four-dimensional manifolds that turns out to be easier, and is conjectured to be equivalent to, Donaldson theory. This review describes the development of this mathematical subject, and shows how the physics played a pivotal role in the current understanding of this area of topology.Comment: 51 pages, 10 figures, 8 postscript files. Submitted to International Journal of Modern Physics A, July 2002 Uses Latex 2e with class file ws-ijmpa.cls (included in tar file

Loomis--Sikorski Theorem and Stone Duality for Effect Algebras with Internal State

Recently Flaminio and Montagna, \cite{FlMo}, extended the language of MV-algebras by adding a unary operation, called a state-operator. This notion is introduced here also for effect algebras. Having it, we generalize the Loomis--Sikorski Theorem for monotone $\sigma$-complete effect algebras with internal state. In addition, we show that the category of divisible state-morphism effect algebras satisfying (RDP) and countable interpolation with an order determining system of states is dual to the category of Bauer simplices $\Omega$ such that $\partial_e \Omega$ is an F-space

Duality for convex monoids

Every C*-algebra gives rise to an effect module and a convex space of states, which are connected via Kadison duality. We explore this duality in several examples, where the C*-algebra is equipped with the structure of a finite-dimensional Hopf algebra. When the Hopf algebra is the function algebra or group algebra of a finite group, the resulting state spaces form convex monoids. We will prove that both these convex monoids can be obtained from the other one by taking a coproduct of density matrices on the irreducible representations. We will also show that the same holds for a tensor product of a group and a function algebra.Comment: 13 page