Introduction

Hisham Sati and Urs Schreiber

Abstract. The contributions in this volume are intended to indicate core

aspects of a ﬁrm and workable mathematical foundation for quantum ﬁeld

theory and perturbative string theory. Here we provide some motivational

background, as well as the overall picture in which the various articles ﬁt.

The history of theoretical fundamental physics is the story of a search for the

suitable mathematical notions and structural concepts that naturally model the

physical phenomena in question. It may be worthwhile to recall a few examples:

(1) the identiﬁcation of symplectic geometry as the underlying structure of

classical Hamiltonian mechanics;

(2) the identiﬁcation of (semi-)Riemannian diﬀerential geometry as the un-

derlying structure of gravity;

(3) the identiﬁcation of group and representation theory as the underlying

structure of the zoo of fundamental particles;

(4) the identiﬁcation of Chern-Weil theory and diﬀerential cohomology as the

underlying structure of gauge theories.

All these examples exhibit the identiﬁcation of the precise mathematical language

that naturally captures the physics under investigation. While each of these lan-

guages upon its introduction into theoretical physics originally met with some skep-

ticism or even hostility, we do know in retrospect that the modern insights and

results in the respective areas of theoretical physics would have been literally un-

thinkable without usage of these languages. A famous historical example is the

Wigner-Weyl approach and its hostile dismissal from mainstream physicists of the

time (“Gruppenpest”); we now know that group theory and representation theory

have become indispensible tools for every theoretical and mathematical physicist.

Much time has passed since the last major such formalization success in theo-

retical physics. The rise of quantum ﬁeld theory (QFT) in the middle of the last

century and its stunning successes, despite its notorious lack of formal structural

underpinnings, made theoretical physicists conﬁdent enough to attempt an attack

on the next open structural question – that of the quantum theory of gauge forces

2010 Mathematics Subject Classiﬁcation. Primary 81T40; secondary 81T45, 81T30, 81T60,

81T05, , 57R56, 70S05, 18D05, 55U40, 18D50, 55N34, 19L50, 53C08.

Keywords and phrases. Topological ﬁeld theory, conformal ﬁeld theory, supersymmetric ﬁeld

theory, axiomatic quantum ﬁeld theory, perturbative string theory, conformal nets, monoidal cat-

egories, higher categories, generalized cohomology, diﬀerential cohomology, quantization, operads.

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Proceedings of Symposia in Pure Mathematics

Volume 83, 2011

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