**Observations 8:**


$$t\_{\beta} := s\_{\beta}^\* t = g\_{\beta \gamma} t\_{\gamma} g\_{\beta \gamma}^{-1}. \tag{33}$$

The Maurer–Cartan form *θ* reflects the non-horizontality of *ω* to the gauge field, from Equation (30). iii. A difference of two gauge fields like *A* − *A* ′ transforms as Equation (33). In fact, the transformation rule is one of a tensorial form, since the Maurer–Cartan forms simplify.


**Claim.** *This gauge field defines an exterior covariant derivative for bundle-valued forms on M. We denote such a map with*

$$d\_A: \Omega^k(M, P \times\_\rho V) \to \Omega^{k+1}(M, P \times\_\rho V). \tag{34}$$

Anyway, we will further develop this argument in Section 6.1. We can proceed analogously and can define the pull-back of the curvature:

**Definition 18** (Field strength)**.** *Let P* → *M be a G-principal bundle, G be a Lie group with* g *as the respective Lie algebra,* {*Uβ*} *be a cover of M, and s<sup>β</sup>* : *U<sup>β</sup>* → *P be a section.*

*We define the field strength as the pull-back of the curvature form* <sup>Ω</sup> <sup>∈</sup> <sup>Ω</sup><sup>2</sup> *G* (*P*, g) *as*

$$F\_{\mathfrak{F}} = \mathfrak{s}\_{\mathfrak{F}}^{\*} \Omega \in \Omega\_{G}^{2}(\mathcal{U}\_{\mathfrak{F}}, \mathfrak{g}), \tag{35}$$

*which, by definition of* Ω*, is*

$$F\_{\beta} = dA\_{\beta} + \frac{1}{2}[A\_{\beta} \wedge A\_{\beta}].\tag{36}$$

Similarly to what we have done for the gauge field, we can show<sup>15</sup> that the field strength transforms as

$$F\_{\beta} = \mathbf{Ad}\_{\mathcal{S}\beta\gamma} \circ F\_{\gamma} = \mathbf{g}\_{\beta\gamma} F\_{\gamma} \mathbf{g}\_{\beta\gamma'}^{-1} \tag{37}$$

<sup>15</sup> Using the Cartan structure equation for *<sup>θ</sup>*, *<sup>d</sup><sup>θ</sup>* <sup>=</sup> <sup>−</sup> <sup>1</sup> 2 [*θ*, *θ*].

where the last equality holds for matrix Lie groups with *g* and *g* −1 in *G*. This is indeed the transformation of a tensorial form, as in Equation (33).

**Observation 9:** Thanks to our previous observation, i.e., there is a canonical isomorphism between Ω*k G* (*P*, *V*) and Ω*<sup>k</sup>* (*M*, *<sup>P</sup>* <sup>×</sup>*<sup>ρ</sup> <sup>V</sup>*), we can relate <sup>Ω</sup> and *<sup>F</sup><sup>β</sup>* with a form<sup>16</sup> *<sup>F</sup><sup>A</sup>* <sup>∈</sup> <sup>Ω</sup><sup>2</sup> (*M*, ad*P*). Namely there is a canonical isomorphism sending <sup>Ω</sup> <sup>∈</sup> <sup>Ω</sup><sup>2</sup> *G* (*P*, <sup>g</sup>) to *<sup>F</sup><sup>A</sup>* <sup>∈</sup> <sup>Ω</sup><sup>2</sup> (*M*, ad*P*). Indeed, given the transformation law for the field strength in Equation (37), we see that {*Fβ*} is horizontal and equivariant and, thus, forms a global section belonging to Ω<sup>2</sup> (*M*, ad*P*), which is usually denoted as *FA*.

The notation *F<sup>A</sup>* stresses that it is obtained from gauge fields in Ω<sup>1</sup> (*Uβ*, g).

In the case of a trivial bundle, it is also possible to define a global gauge field *<sup>A</sup>* <sup>∈</sup> <sup>Ω</sup><sup>1</sup> (*M*, g).

### *5.2. 2nd Bianchi Identity*

Consider *d<sup>A</sup>* : Ω*<sup>k</sup>* (*M*, *<sup>P</sup>* <sup>×</sup>*<sup>ρ</sup> <sup>V</sup>*) <sup>→</sup> <sup>Ω</sup>*k*+<sup>1</sup> (*M*, *P* ×*<sup>ρ</sup> V*) as the exterior covariant derivative and *F<sup>A</sup>* ∈ Ω2 (*M*, ad *P*) as the field strength.

Then, we have the following:

### **Proposition 3.**

$$d\_A F\_A = 0.\tag{38}$$

*This is the second Bianchi identity.*

**Proof.** Given

$$F\_A = dA + \frac{1}{2}[A \wedge A],\tag{39}$$

$$\begin{aligned} d\_A F\_A &= dF\_A + [A \wedge F\_A] \\ &= d^2 A + \frac{1}{2} d[A \wedge A] + [A \wedge dA] + \frac{1}{2} [A \wedge [A \wedge A]] \\ &= \frac{1}{2} [A \wedge [A \wedge A]] & \qquad \qquad (d^2 A = 0 \text{ and } \frac{1}{2} d[A \wedge A] = -[A \wedge dA]) \\ &= 0. & \qquad \qquad \qquad \qquad \text{(because of Jacobi identity)} \end{aligned} \tag{40}$$

then
