Degree-six cubic trace polynomial for a chiral representation

Let $H=SU(3)\times U(1)$, where $SU(3)=\{g\in M_3(\mathbb{C}):g^{†}g=I_3,\det g=1\}$ and $U(1)=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$. For a finite-dimensional unitary representation $\rho :H\to U(V_{\rho })$, let $A_{\rho }$ be a smooth anti-Hermitian matrix-valued one-form on a smooth manifold $X$, representing a connection in a local trivialization, and set $F_{\rho }=d{A}_{\rho }+A_{\rho }\wedge A_{\rho }$. Define the degree-six invariant polynomial

${\mathcal{I}}_6(\rho ,A_{\rho }):=2\pi {[{\mathrm{tr}}_{V_{\rho }}\exp \text{\hspace{0.17em}⁣}(-\frac{i{F}_{\rho }}{2\pi })]}_6=\frac{i}{24{\pi }^2}{\mathrm{tr}}_{V_{\rho }}(F_{\rho }\wedge F_{\rho }\wedge F_{\rho }),$

where the exponential is its formal power series and $[\cdot {]}_6$ selects the differential-form component of degree six. This is a homogeneous polynomial in the formal curvature; on a four-dimensional physical spacetime it is interpreted on the formal six-dimensional extension used for descent, not integrated as a six-form over spacetime. The normalization fixes anti-Hermitian generators, the fundamental trace, and the overall coefficient once and for all.

For the defining representation $E={\mathbb{C}}^3$ with ${\rho }_E(g,z)=zg$, a background connection is $A_E=A+iB{I}_3$, where $A$ is an $\mathfrak{s}\mathfrak{u}(3)$-valued one-form and $B$ is a real $U(1)$ connection one-form. The dual representation is $E^{\ast }={\mathrm{Hom}}_{\mathbb{C}}(E,\mathbb{C})$ with ${\rho }_{E^{\ast }}(g,z)={\rho }_E(g,z{)}^{-T}$; its induced connection and curvature are $A_{E^{\ast }}=-A_E^T$ and $F_{E^{\ast }}=-F_E^T$.

Bilal, §10.4, equations (10.49)–(10.54)

Alvarez, §§2.1–2.2