Dualities in Field Theories and the Role of K-Theory

It is now known (or in some cases just believed) that many quantum field theories exhibit dualities, equivalences with the same or a different theory in which things appear very different, but the overall physical implications are the same. We will discuss some of these dualities from the point of view of a mathematician, focusing on “charge conservation” and the role played by K-theory and noncommutative geometry. Some of the work described here is joint with Mathai Varghese and Stefan Mendez-Diez; the last section is new.

💡 Research Summary

The paper surveys a variety of dualities that appear in quantum field theory and string theory, emphasizing how these dualities are reflected in the mathematics of charge conservation, K‑theory, and non‑commutative geometry. After a brief review of the basic ingredients of classical field theories—scalars, vector bundles, connections, and actions—the author illustrates how quantization replaces classical observables with operators on a Hilbert space and introduces the path‑integral formulation. Classical examples such as Yang‑Mills theory, general relativity, electric‑magnetic duality, and the harmonic oscillator are used to motivate the notion of a duality as a transformation that exchanges apparently different degrees of freedom while leaving all measurable quantities invariant.

The discussion then turns to Dirac’s quantization condition. By interpreting the electromagnetic potential as a U(1) connection on a complex line bundle, the first Chern class c₁(L)∈H²(M,ℤ) becomes the topological invariant that quantizes electric charge. The magnetic monopole charge is likewise tied to the same cohomology class, leading to a charge group that is essentially ℤ₂ in many simple models. The Montonen‑Olive conjecture is presented as a quantum‑level extension of electric‑magnetic duality, predicting an isomorphism of charge groups between a theory and its dual.

From this physical picture the author abstracts a general framework: a collection C of theories equipped with an action of a discrete duality group G, each theory carrying a charge group C. Whenever an element g∈G relates two theories, it must induce an isomorphism of the corresponding charge groups. In many cases these charge groups are not merely integers but topological invariants such as Pic X=H²(X,ℤ) or, more generally, K‑theory groups.

The paper then focuses on string theory, where D‑branes provide the natural carriers of K‑theoretic charge. In type IIA/IIB theories D‑branes are spinᶜ submanifolds equipped with Chan‑Paton bundles; their charges are given by the Gysin push‑forward ι₊(