Topological $\mathrm {SL} (5,\mathbb {R})$ Gauge-Invariant Action

Abstract

Symmetry plays a predominant role in our perception, although most of the real patterns we observe are far from symmetric. However, in modern particle physics it is much more economical in the mathematical formulation to start from highly symmetric and hence idealized configurations than to try to model the real world directly.

Appendix: Topological Invariants

Let us recall that the $\mathfrak {sl}(5,\mathbb {R})$-valued Chern–Simons term

$$\begin{aligned} \hat{C}:= -{1\over 2}\left( \varGamma _{A}{}^{B}\wedge d\varGamma _{B}{}^{A} - {2\over 3}\varGamma _{A}{}^{B}\wedge \varGamma _{B} {}^{C}\wedge \varGamma _{C}{}^{A}\right) \end{aligned}$$

(13.8.1)

contains a translational CS three-form $C_\mathrm{TT}$ that gives rise to the parity-violating (Mielke 2009) boundary four-form

$$\begin{aligned} dC_\mathrm{TT} = {1\over {2\ell ^2}} \left( T^{\alpha }\wedge T_\alpha +R_{\alpha \beta }\wedge \vartheta ^\alpha \wedge \vartheta ^\beta \right) \end{aligned}$$

(13.8.2)

of Nieh and Yan (NY) (Nieh 2007). On the other hand, the $\mathfrak {gl}(4,\mathbb {R})$-valued Pontryagin four-form

$$\begin{aligned} dC_\mathrm{RR}= -{1\over 2}R_\alpha {}^\beta \wedge R_\beta {}^\alpha \end{aligned}$$

(13.8.3)

is a topological invariant whose variation returns the second Bianchi identity

$$\begin{aligned} D R_\alpha {}^\beta \equiv 0\, , \end{aligned}$$

(13.8.4)

whereas the torsion identity (13.8.2) is based on the first Bianchi identity

$$\begin{aligned} DT^\alpha \equiv R_{\beta }{}^{\alpha }\wedge \vartheta ^\beta \end{aligned}$$

(13.8.5)

in Riemann–Cartan (RC) geometry. In contrast to the metric-free Pontryagin four-form (13.8.3), in the NY term a metric $g_{\alpha \beta }$ is needed to raise and lower the indices, for instance in $T_\alpha =g_{\alpha \beta }T^{\beta }$. Moreover, a fundamental length $\ell$ necessarily enters into (13.8.2) in order to keep all topological invariants dimensionless.

The topological Euler–Poincaré invariant

$$\begin{aligned} \begin{aligned} dC_\mathrm{RR^{(\star )}}&:= \frac{1}{2}\, d \left( \varGamma _{\alpha \beta }\wedge R^{\alpha \beta (\star )} - {1\over 3}\varGamma _\alpha {}^{\beta (\star )}\wedge \varGamma _\beta {}^\gamma \wedge \varGamma _\gamma {}^\alpha \right) \\&\equiv - L_\mathrm{SKY} -2\mathrm{Ric}_{\alpha \beta }\wedge \,^*\mathrm{Ric}^{\alpha \beta } + \frac{1}{2}\mathrm{Ric}_{\alpha }{}^{\alpha }\wedge \,^*\mathrm{Ric}_{\beta }{}^{\beta } \end{aligned} \end{aligned}$$

(13.8.6)

has an equivalent representation in terms of Weyl’s quadratic curvature Lagrangian

$$\begin{aligned} L_\mathrm{SKY}:=-\frac{1}{2}\,R_{\alpha \beta }\wedge \,^* R^{\alpha \beta }, \end{aligned}$$

(13.8.7)

amended by Ricci-squared and curvature scalar-squared terms. The second expression involving the symmetric Ricci tensor, i.e., the zero-form $\mathrm{Ric}_{\alpha \beta }:= (-1)^\mathrm{sig} \, ^*(R_{(\alpha }{}^\delta \wedge \eta _{\delta \vert \beta )})$, is known as the Gauss-Bonnet identity.