D-branes, T-duality, and Index Theory

TL;DR

This work posits that D-brane transformations under T-duality on a four-torus are governed by the Nahm transform of instantons, naturally exchanging rank and instanton number via ${\rm rank}(\hat{E})={\rm ch}_2(E)$ and ${\rm ch}_2(\hat{E})={\rm rank}(E)$. It generalizes this index-theoretic picture to orthogonal and symplectic gauge groups and to ${\mathbb Z}_2$ orientifolds/orbifolds, via explicit Nahm-type transforms between instantons and orientibundles/orbibundles. The paper then grounds these physical dualities in topology, deriving K-theory isomorphisms that classify D-brane charges under T-duality (e.g., ${\rm KO}$, ${\rm KR}$, ${\rm KSp}$, ${\rm KpR}$) and identifying the precise topological correspondences (e.g., ${\rm K}(T^4)\cong {\rm K}(\hat{T}^4)$ with ${\rm ch}(\hat{E})$ related to ${\rm ch}(E)$ via the Poincaré bundle). Collectively, these results integrate brane dynamics, index theory, derived-category intuition (via Mukai transforms), and K-theory to provide a robust, dimension-spanning framework for D-brane charges and T-duality in orientifold/orbifold contexts.

Abstract

We show that the transformation of D-branes under T-duality on four-torus is represented by Nahm transform of instantons. The argument for this allows us to generalize Nahm transform to the case of orthogonal and symplectic gauge groups as well as to instantons on Z_2 orbifold of four-torus. In addition, we identify the isomorphism of K-theory groups that realizes the transformation of D-brane charges under T-duality on torus of arbitrary dimensions. By the isomorphism we are lead to identify the correct K-theory group that classifies D-brane charges in Type II orientifold.