Heterotic Green Schwarz Bianchi And Anomaly Cancellation

In the perturbative heterotic field basis through the displayed first ${\alpha }^{\mathrm{\prime }}$ correction, work on a nonempty open set $U$ of the smooth oriented Lorentzian ten-manifold $(M,g)$ from the preceding postulate, with all fields and connections smooth on $U$ and with no localized source current in this local relation. Restrict the gauge bundle and geometric connection of the preceding postulate to $U$, write their curvature as $F_A\in {\mathrm{\Omega }}^2(U,\mathrm{ad}{P}_G)$, and let $B_2\in {\mathrm{\Omega }}^2(U)$ be a local two-form potential with field strength $H_3\in {\mathrm{\Omega }}^3(U)$. Define the tangent connections by $({\mathrm{\nabla }}_M^{\pm }V{)}^N=({\mathrm{\nabla }}_M^{\mathrm{L}\mathrm{C}}V{)}^N\pm \frac{1}{2}(H_3{)}_M{}^N{}_P{V}^P$ for a smooth vector field $V$, and write $R_H:=R_{+}$ for the curvature of ${\mathrm{\nabla }}^{+}$. In the McOrist–Picard Bianchi convention the gauge curvature is represented by antihermitian matrices. Write ${\mathrm{Tr}}_H$ and ${\mathrm{tr}}_H$ for the gauge and tangent instances of that source's ordinary matrix trace, which McOrist–Picard denote by $\mathrm{tr}$. In the chosen gauge matrix representation its generators $T_a$ are normalized by ${\mathrm{Tr}}_H(T_a{T}_b)=-{\delta }_{ab}$, and $F_A=dA+A\wedge A$. For the tangent term, ${\mathrm{tr}}_H(R_H\wedge R_H)$ is the ordinary endomorphism trace of the curvature acting on $TU$; this does not assert that the invariant bilinear form on the noncompact real algebra $\mathfrak{s}\mathfrak{o}(1,9)$ is negative definite. McOrist–Picard's string-frame conversion fixes the displayed coefficient in the convention whose action normalization has ${\kappa }_{\mathrm{M}\mathrm{P}}^2\mathrm{/}{g}_{\mathrm{Y}\mathrm{M},\mathrm{M}\mathrm{P}}^2={\alpha }^{\mathrm{\prime }}\mathrm{/}4$, where ${\kappa }_{\mathrm{M}\mathrm{P}}$ and $g_{\mathrm{Y}\mathrm{M},\mathrm{M}\mathrm{P}}$ are that source's gravitational and gauge couplings, not the spacetime metric $g$. Choose local Chern–Simons 3-forms ${\omega }_{3,\mathrm{Y}\mathrm{M}}$ and ${\omega }_{3,\mathrm{L}}$ satisfying $d{\omega }_{3,\mathrm{Y}\mathrm{M}}={\mathrm{Tr}}_H(F_A\wedge F_A)$ and $d{\omega }_{3,\mathrm{L}}={\mathrm{tr}}_H(R_H\wedge R_H)$. Adopt the perturbative local relation, understood order-by-order in ${\alpha }^{\mathrm{\prime }}$,

$H_3=d{B}_2+\frac{{\alpha }^{\mathrm{\prime }}}{4}({\omega }_{3,\mathrm{Y}\mathrm{M}}-{\omega }_{3,\mathrm{L}}),d{H}_3=\frac{{\alpha }^{\mathrm{\prime }}}{4}({\mathrm{Tr}}_H(F_A\wedge F_A)-{\mathrm{tr}}_H(R_H\wedge R_H))\mathrm{.}$

Separately, in Taylor's anomaly-polynomial convention, choose a local orthonormal frame and let ${\mathrm{\Omega }}_T$ be the local tangent/Lorentz connection used there; its curvature is $R_T=d{\mathrm{\Omega }}_T+{\mathrm{\Omega }}_T\wedge {\mathrm{\Omega }}_T$. This is the tangent curvature denoted $R$ in Taylor's anomaly polynomial; it is not the Hull curvature $R_H$ in the Bianchi identity. Let ${\mathrm{Tr}}_T$ be Taylor's adjoint gauge trace and ${\mathrm{tr}}_T$ his tangent trace. For the source's admissible $so(32)$ or $e_8\oplus e_8$ gauge data, adopt

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

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

The Bianchi identity and the anomaly-polynomial factorization are separate adopted physical inputs in their stated conventions. They do not assert a conversion between ${\mathrm{Tr}}_H,{\mathrm{tr}}_H$ and ${\mathrm{Tr}}_T,{\mathrm{tr}}_T$, a local reconstruction of the chiral determinant, or global anomaly cancellation.

Jock McOrist and Sebastien Picard, §1.1

Jock McOrist and Sebastien Picard, Appendix B.1

Washington Taylor, §2.4

The canonical plan's Chern–Simons derivative labels the gauge trace as $\mathrm{Tr}$ but writes $\mathrm{tr}$ in the Bianchi formula. The carrier uses one consistently labeled McOrist–Picard gauge characteristic form in both places; this is a notation correction, not a conversion to Taylor's anomaly-polynomial trace convention. Taylor's $R_T$ and the heterotic Hull curvature $R_H$ are separate connection choices and are never identified here. The separate Type-I RR Bianchi input is recorded by its own postulate.