Freed–Witten worldvolume anomaly constraint

Let $Y$ be a finite-dimensional Hausdorff second-countable oriented smooth $10$-manifold with a chosen auxiliary Riemannian metric and a supplied spin structure on its oriented tangent bundle. Let $Q$ be a finite-dimensional Hausdorff second-countable oriented smooth manifold without boundary, and let $i:Q\hookrightarrow Y$ be a smooth embedding representing a Type II D-brane worldvolume. Set $\nu =(i^{\ast }TY)\mathrm{/}TQ\cong (TQ{)}^{\perp }$ to be the real normal bundle, oriented by the orientations of $Y$ and $Q$. Let ${\zeta }_B\in H^3(Y;\mathbb{Z})$ be the integral characteristic class of the NS $B$-field, with curvature convention $H_3\mathrm{/}(2\pi )$. Adopt the source's worldvolume anomaly condition $i^{\ast }{\zeta }_B=W_3(\nu )\in H^3(Q;\mathbb{Z})\mathrm{.}$ For an oriented bundle $E$, write $W_3(E)={\tilde{\beta }}_2(w_2(E))$, the integral Bockstein of its second Stiefel–Whitney class. Since $TY{\mathrm{\mid }}_Q\cong TQ\oplus \nu$, the spin condition gives $w_2(TY{\mathrm{\mid }}_Q)=0$, and The integral third Stiefel–Whitney classes of oriented tangent and normal bundles agree when the restricted ambient second class vanishes yields $W_3(TQ)=W_3(\nu )$. Thus the same constraint can be written $i^{\ast }{\zeta }_B=W_3(Q)$. If the $B$-field is topologically trivial, this reduces to $W_3(\nu )=0$; for a positive-rank normal bundle, the Spin-c obstruction criterion of Spin and Spin-c obstruction and torsor classification identifies this with existence of the required Spin-c structure under its full AC and smooth-manifold base hypotheses. For the zero-rank normal bundle, the rank conventions give $w_2(0)=W_3(0)=0$. The negative-square zero-dimensional Clifford algebra is $\mathbb{R}$ and its norm-one even group is $\{\pm 1\}$, so ${\mathrm{Spin}}^c(0)=(\{\pm 1\}\times U(1))\mathrm{/}⟨(-1,-1)⟩\cong U(1)$; its trivial principal bundle supplies a Spin-c lift of the trivial $SO(0)$ frame bundle. Thus the zero-rank case is established directly, without applying a positive-rank theorem.

This is an adopted physical anomaly-cancellation constraint, not a pure mathematical theorem. Freed and Witten explicitly exclude orientifolds and Type I strings; the source's footnote states that its displayed cohomological condition is sufficient. No necessity claim, orientifold extension, or complete torsion/K-theory anomaly classification is made here.

Differential twist and twisted differential K-theory

Spin and Spin-c obstruction and torsor classification

The integral third Stiefel–Whitney classes of oriented tangent and normal bundles agree when the restricted ambient second class vanishes