Type I Green Schwarz Anomaly Cancellation

For the perturbative ten-dimensional $SO(32)$ Type-I $N=1$ branch, adopt Taylor's source-reported anomaly-polynomial and Green–Schwarz field input in its own trace convention; write ${\mathrm{Tr}}_T$ for his adjoint gauge trace and ${\mathrm{tr}}_T$ for his tangent trace. Work locally on a smooth source-free patch $U$ of the ten-dimensional Lorentzian spacetime, and let $P_{SO(32)}\to U$ be the restricted smooth principal gauge bundle and $A$ its connection from the preceding postulate, with curvature $F_A=dA+A\wedge A$. In a chosen local orthonormal frame, let ${\mathrm{\Omega }}_1$ be the Lorentz connection used in this Type-I descent convention, with curvature $R_T=d{\mathrm{\Omega }}_1+{\mathrm{\Omega }}_1\wedge {\mathrm{\Omega }}_1$. Let $C_2\in {\mathrm{\Omega }}^2(U)$ be the RR potential and let ${\omega }_Y,{\omega }_R\in {\mathrm{\Omega }}^3(U)$ be local Chern–Simons forms satisfying $d{\omega }_Y={\mathrm{Tr}}_T(F_A\wedge F_A)$ and $d{\omega }_R={\mathrm{tr}}_T(R_T\wedge R_T)$. In the linearized-connection descent convention, where $F_A=dA$ and $R_T=d{\mathrm{\Omega }}_1$ in the variation terms, take smooth infinitesimal parameters $\mathrm{\Lambda }\in {\mathrm{\Omega }}^0(U,\mathrm{ad}{P}_{SO(32)})$ and $\mathrm{\Theta }\in {\mathrm{\Omega }}^0(U,\mathfrak{s}\mathfrak{o}(1,9))$, with $\delta A=d\mathrm{\Lambda }$, $\delta {\mathrm{\Omega }}_1=d\mathrm{\Theta }$, $\delta {\omega }_Y=d{\mathrm{Tr}}_T(\mathrm{\Lambda }{F}_A)$, and $\delta {\omega }_R=d{\mathrm{tr}}_T(\mathrm{\Theta }{R}_T)$. The modified field and transformations are

$H_3=d{C}_2-{\omega }_Y+30{\omega }_R,\delta C_2={\mathrm{Tr}}_T(\mathrm{\Lambda }{F}_A)-30{\mathrm{tr}}_T(\mathrm{\Theta }{R}_T),\delta A=d\mathrm{\Lambda },\delta {\mathrm{\Omega }}_1=d\mathrm{\Theta }\mathrm{.}$

Adopt

${\widehat{I}}_{12}=-\frac{1}{768}{Y}_4{X}_8,Y_4={\mathrm{tr}}_T{R}_T^2-\frac{1}{30}{\mathrm{Tr}}_T{F}_A^2,$

$X_8={\mathrm{tr}}_T{R}_T^4+\frac{1}{4}({\mathrm{tr}}_T{R}_T^2{)}^2-\frac{1}{30}{\mathrm{Tr}}_T{F}_A^2{\mathrm{tr}}_T{R}_T^2+\frac{1}{3}{\mathrm{Tr}}_T{F}_A^4-\frac{1}{900}({\mathrm{Tr}}_T{F}_A^2{)}^2\mathrm{.}$

At the stated linearized order, these transformations give $\delta H_3=0$ by the displayed descent equations. The source reports a Green–Schwarz coupling proportional to ${\int }_U{C}_2\wedge X_8$; its numerical coefficient is not fixed for later use in this item. This is a quantum-consistency premise, not a local chiral-anomaly computation, a proof that every configuration is globally anomaly-free, or the heterotic NS–NS $B_2$ formulation.

Washington Taylor, §2.4

Mohammad R. Garousi, §1

Taylor's factorization is a source-reported anomaly result. The item does not import its broad conclusion that the two familiar gauge groups exhaust all consistent ten-dimensional $N=1$ theories; only the displayed $SO(32)$ Type-I interface is adopted.

Taylor §2.4, equation (25), sets the Lorentz coefficient in $H_3$ to $k=30$ but prints the adjacent Lorentz term in $\delta B$ without this factor. Under the same $d{\omega }_R={\mathrm{tr}}_T(R_T\wedge R_T)$ and linearized descent convention adopted above, that printed term does not cancel the variation of $+30{\omega }_R$; the coefficient $-30$ above is the one that makes $\delta H_3=0$. No full nonabelian or nonlinear Lorentz transformation is inferred from Taylor's linearized formulas.