**Remark 2.**

*– We observe that d*<sup>2</sup> *<sup>ω</sup><sup>α</sup>* <sup>6</sup><sup>=</sup> <sup>0</sup> *for a general <sup>α</sup>* <sup>∈</sup> <sup>Ω</sup>*<sup>k</sup> G* (*P*, *V*)*, but it is easy to show that it holds*<sup>14</sup>

$$d\_{\omega}^{2}\mathfrak{a} = \Omega \wedge\_{d\rho} \mathfrak{a}\_{\prime} \tag{28}$$

*Thus, for a flat connection such that* Ω = 0*, we have d*<sup>2</sup> *<sup>ω</sup>α* = *d* <sup>2</sup>*α* = 0*.*

$$(\omega \wedge\_{d\rho} a)(v\_1, \dots, v\_{k+1}) = \frac{1}{(1+k)!} \sum\_{\sigma} \text{sign}(\sigma) d\rho \left(\omega(v\_{\sigma(1)})\right) \left(a(v\_{\sigma(2)}, \dots, v\_{\sigma(k+1)})\right).$$

<sup>13</sup> For a general *k*-form:

<sup>14</sup> See the first Bianchi identity in Equation (56) for the proof.


### **5. Gauge Field and Field Strength**
