Freed–Witten Dp-Brane Anomaly Condition

Let $Y$ be a finite-dimensional Hausdorff second-countable smooth, oriented, time-oriented ten-dimensional Lorentzian spacetime with metric of signature $(-,+,\dots ,+)$. Assume the supplied ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^{+}(1,9)$ lift of the proper-orthochronous orthonormal-frame bundle defined in the preceding postulate: a two-to-one equivariant map of principal bundles induced by the connected double cover ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^{+}(1,9)\to S{O}^{+}(1,9)$. In this item a “spin target” also includes the explicit data $w_1(TY)=w_2(TY)=0$; neither the lift's existence nor a characteristic-class obstruction theorem is being proved. Let $i:Q\hookrightarrow Y$ be a smooth oriented embedded Type II D-brane worldvolume of dimension $p+1$, $0\le p\le 9$. Choose an auxiliary positive-definite bundle metric on $i^{\ast }TY$, and let $\nu$ be the orthogonal normal bundle, equivalently the quotient $i^{\ast }TY\mathrm{/}TQ$, with its induced orientation. Assume the source's oriented Type II worldsheet boundary model on oriented Riemann surfaces $\mathrm{\Sigma }$ whose boundary maps to $Q$.

Let ${\mathcal{G}}_B$ be the NS $B$-field bundle gerbe on $Y$; on a local potential patch write its curvature as $H_3=d{B}_2$, following Supergravity Conventions And Normalizations. Its Dixmier–Douady class $\mathrm{D}\mathrm{D}({\mathcal{G}}_B)={\zeta }_B\in H^3(Y;\mathbb{Z})$ is the integral characteristic class, including torsion. No conversion between the de Rham form $H_3$ and the integral class ${\zeta }_B$ is used here. Let ${\mathcal{G}}_{\nu }$ be the source's anomaly gerbe for the worldsheet Pfaffian, with $\mathrm{D}\mathrm{D}({\mathcal{G}}_{\nu })=W_3(\nu )$. A stable isomorphism between bundle gerbes ${\mathcal{G}}_1,{\mathcal{G}}_2$ means a trivialization of ${\mathcal{G}}_1^{\ast }\otimes {\mathcal{G}}_2$, as in Carey–Johnson–Murray, Definition 4.3. If global anomaly cancellation is implemented by the source's topological Chan–Paton $A$-field, such a stable isomorphism between $i^{\ast }{\mathcal{G}}_B$ and ${\mathcal{G}}_{\nu }$ is required. Then the necessary topological condition is

$$i^{\ast }{\zeta }_B=W_3(\nu )=W_3(TQ),$$

equivalently

$$i^{\ast }{\zeta }_B+W_3(TQ)=0\text{in }{H}^3(Q;\mathbb{Z})\mathrm{.}$$

Here $W_3(E)={\tilde{\beta }}_2(w_2(E))$ is the integral Bockstein class of an oriented real bundle $E$. This is a global anomaly condition, not a Dirac–Born–Infeld equation.

Supergravity Conventions And Normalizations

Integral Third Stiefel–Whitney Class

Third Stiefel–Whitney Classes of Tangent and Normal Bundles

Freed–Witten Chan–Paton Gerbe Input

Dixmier–Douady class and gerbe classification

Freed–Witten's equation (1.12) is sufficient for triviality of the transgressed Pfaffian–$B$ line by itself; its footnote does not identify that line-only statement as necessary. The necessity used here is instead the source's topological Chan–Paton gerbe-isomorphism requirement in §6. The sourced family-index/anomaly calculation is not reconstructed locally, and equality of characteristic classes is not asserted to settle every differential or analytic anomaly condition.

When ${\zeta }_B$ is nonzero, the Chan–Paton data are twisted by the gerbe; the proof does not replace them by an ordinary line bundle. Freed and Witten explicitly exclude orientifolds and Type I strings, so this item makes no claim for either case.

The DD-class conversion in step 1.1 is conditional on the exact current supplier Dixmier–Douady class and gerbe classification. Its Step 1 record is escalated because its proof depends on the supplied Čech-to-singular comparison data [L1]; the comparison input is assumed here and is not proved locally. Accordingly this item remains escalated and does not treat the GTS theorem as proved.