defect

$\mathbf{\text{Classification:}}$
universal fields (focus): considering worldline of defect $M\to X\simeq {\mathbb{R}}^D$, they should be $\sigma$-field (pullback of metric on $X$; still bulk field) and structure of normal bundle.
Suppose metric is fixed, topological,
$M\simeq {\mathbb{R}}^d\cup \{\mathrm{\infty }\}\simeq S^d$, while the change of framing (normal vector field) is given by $M\to SO(D-d)$, such that ${\pi }_d(SO(D-d){)}^{\vee }$ (Pontryagin dual) classifies the change of framing.

background Chern-Simons term? not gauge invariant but detecting the change of framing.
Proposal. $Z[{\mathrm{\Phi }}_b]$ (bulk) is classified by framed cobordism classes of submanifold of $X$ $\stackrel{Pontryagin-Thom}{⟶}{\pi }_D(S^{D-d})$ (whole spacetime$\to$normal bundle).
eg. $d=0$, point defect.
${\pi }_D(S^{D-d})=\mathbb{Z}$, charges.
$J$-homomorphism: ${\pi }_d(SO(D-d))\to {\pi }_D(S^{D-d})$, ${\pi }_D(S^{D-d}{)}^{\vee }\to {\pi }_d(SO(D-d){)}^{\vee }$

Remark: classifying G-structures, ${\pi }_D(MG(d))$. If adding generalized symmetry $F$, ${\pi }_D(MG(d)\wedge B{F}_{+})$. (why similar to a structure? symmetry of the defect, not generated by?)

PS: this works for QFTs, but not yet M-theory and string theory.
spin structure