Freed–Witten Chan–Paton Gerbe Input

Let $Y$ be a finite-dimensional Hausdorff second-countable smooth, oriented, time-oriented, ten-dimensional Type II spacetime with Lorentzian metric of signature $(-,+,\dots ,+)$. The proper-orthochronous orthonormal-frame bundle $F_{SO}^{+}(Y)\to Y$ has fiber at $y\in Y$ the ordered, oriented orthonormal bases $(e_0,\dots ,e_9)$ of $T_{y}Y$ for which $e_0$ is future-directed; its structure group is $S{O}^{+}(1,9):=\{A\in G{L}_{10}(\mathbb{R}):A^T\eta A=\eta ,\det A=1,A\text{ preserves the future timelike cone}\},\eta =\mathrm{diag}(-1,1,\dots ,1)\mathrm{.}$ Write ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^{+}(1,9)\to S{O}^{+}(1,9)$ for the connected two-fold spin covering. Supply a principal ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^{+}(1,9)$-bundle $\tilde{F}\to Y$ and a two-to-one bundle map $\tilde{F}\to F_{SO}^{+}(Y)$ over the identity that is equivariant with respect to this group covering. This lift is supplied target data, not an existence conclusion. Include $w_1(TY)=w_2(TY)=0$ as explicit target data. Let $i:Q\hookrightarrow Y$ be a smooth embedding of a Hausdorff second-countable oriented D-brane worldvolume $Q$ of dimension $p+1$, with $0\le p\le 9$. Its derivative identifies $TQ$ as a subbundle of $i^{\ast }TY$. Choose an auxiliary positive-definite bundle metric on $i^{\ast }TY$. Define the oriented normal bundle by $\nu :=i^{\ast }TY\mathrm{/}TQ$, identified by this metric with the orthogonal complement of $TQ$ in $i^{\ast }TY$; its orientation is the one for which the orientations of $TQ$ and $\nu$ induce the orientation of $i^{\ast }TY$.

Assume the full Axiom of Choice required by the earlier integral $W_3$ definition and its Bockstein and characteristic-class suppliers.

Here a worldsheet $\mathrm{\Sigma }$ is a smooth oriented real two-manifold with boundary, equipped with the source's bordered Riemann-surface structure: complex charts to half-disks with holomorphic transition maps on interiors, extending smoothly to the boundary and inducing the chosen orientation. The boundary model uses smooth maps $f:\mathrm{\Sigma }\to Y$ with $f(\mathrm{\partial }\mathrm{\Sigma })\subseteq Q$. By a $U(1)$-bundle gerbe with connection and curving, use the earlier bundle-gerbe definition: data $\pi :Z\to M$ given by a surjective smooth submersion with local sections, a Hermitian line bundle $L\to Z^{[2]}$, an associative product $L_{12}\otimes L_{23}\to L_{13}$ over $Z^{[3]}$, a unitary connection preserved by that product, and a curving $C\in {\mathrm{\Omega }}^2(Z)$ satisfying ${\mathrm{pr}}_2^{\ast }C-{\mathrm{pr}}_1^{\ast }C=F_{\mathrm{\nabla }}\mathrm{/}(2\pi i)\mathrm{.}$ Here the stable isomorphism concerns the underlying topological gerbes, with connection and curving forgotten. Let ${\mathcal{G}}_B$ be the NS $B$-field bundle gerbe on $Y$, and let ${\mathcal{G}}_{\nu }$ be the worldsheet Pfaffian anomaly gerbe on $Q$, with integral Dixmier–Douady class $W_3(\nu )={\tilde{\beta }}_2(w_2(\nu ))$. Adopt the source's topological Chan–Paton $A$-field input: global cancellation requires a stable isomorphism

$$i^{\ast }{\mathcal{G}}_B\simeq {\mathcal{G}}_{\nu }\mathrm{.}$$

A stable isomorphism here means a trivialization of $(i^{\ast }{\mathcal{G}}_B{)}^{\ast }\otimes {\mathcal{G}}_{\nu }$, following Carey–Johnson–Murray, Definition 4.3. This records the sourced physical model input; it does not itself assert equality of characteristic classes or reconstruct the family-index anomaly proof.

Integral Third Stiefel–Whitney Class

U(1)-bundle gerbe with connection and curving

The Axiom of Choice

Freed and Witten, §§1 and 5–6

Carey, Johnson, and Murray, §4.4, Definition 4.3