# Toward a Mechanism for the Emergence of Gravity
## 3 Two Examples: Yang-Mills Theories and Linearized Gravity  
### Yang-Mills Theories  
Let us present here an extended version of Yang-Mills theory that will allow us to illustrate the emergence of gauge symmetries. Our starting point is the most general quadratic and Lorentz-invariant Lagrangian giving rise to second-order differential equations that can be written for a set of \(N\) vector fields \(A^{a}_{\mu}\), with the latin indices \((a,b,c...)\) running from \(1\) to \(N\):  
\[S_{2}=\int d^{4}x\sqrt{-\eta}\left[-\frac{1}{4}F^{a}_{\mu\
u}F^{\mu\
u}_{a}+ \frac{1}{2}\xi_{ab}(\
abla_{\mu}A^{a\mu})(\
abla_{\
u}A^{b\
u})-\frac{1}{2}M_ {ab}A^{a\mu}A^{b}_{\mu}+A_{a\mu}j^{a\mu}\right], \tag{3}\]up to a boundary term which we skip here. We have introduced the notation \(F^{u}_{\mu\
u}=2\
abla_{|\mu}A^{a}_{\
u|}\), where \(\
abla\) is the affine connection compatible with \(\eta_{\mu\
u}\), and we have introduced the non-degenerate matrix \(\xi_{ab}\) and the mass matrix \(M_{ab}\). The case which is especially interesting for us is the massless case \(M_{ab}=0\), since it is a generic feature that field theories emerging around Fermi points in condensed-matter systems give rise to massless excitations [5]. Masses can be acquired in a second step through a Higgs-like mechanism. However, for the moment, we will keep the \(M_{ab}\) term since it is instructive to study the massive and massless cases as representatives of the two possible situations described in the previous section regarding the mechanism for the emergence of gauge symmetries. The equations of motion can be derived straightforwardly from action (3) by taking a simple functional derivative,  
\[\
abla_{\mu}F^{\mu\
u}_{b}-\xi_{ab}\
abla^{\
u}\
abla_{\mu}A^{a\mu}-M_{ab}A^{ a\
u}=j^{\
u}_{b}. \tag{4}\]  
It is possible to identify a set of symmetries for this theory associated with the following transformations:  
\[A^{a}_{\mu}\to A^{a}_{\mu}+\
abla_{\mu}A^{a},\qquad j^{a\mu}\to j^{a\mu}, \tag{5}\]  
where \(\chi^{a}\) are functions that need to obey  
\[(\xi_{ab}\square+M_{ab})\chi^{a}=0. \tag{6}\]  
The currents associated with these symmetries, once evaluated on shell, can be written as follows [15]:  
\[J^{\mu}|_{\text{on\,shell}}=-\
abla_{\
u}(F^{a\mu\
u}\chi_{a})+\xi_{ab}\Big{(} \
abla_{\
u}A^{a\
u}\
abla^{\mu}\chi^{b}-\chi^{b}\
abla^{\mu}\
abla_{\
u}A^{ va}\Big{)} \tag{7}\]  
Notice that the first part of this equation is a superpotential and hence it does not contribute to the charges. Furthermore, the second term vanishes if we restrict ourselves to the subspace \(\mathcal{U}\) of solutions of Equation (4) satisfying \(\
abla_{\mu}A^{a\mu}=0\). Hence, projecting onto the subspace \(\mathcal{U}\) makes the current acquire the form of a gauge current encapsulated in Equation (2). Notice that although Equation (7) is associated with a transformation whose generators \(\chi^{a}\) are spacetime-dependent local functions, it does not constitute a gauge transformation a priori. Gauge transformations are associated with local transformations whose generators are arbitrary functions of spacetime. However, in our case they are constrained to obey Equation (6). Thus, the transformation we are discussing does not constitute a gauge transformation from the beginning although it is generated by local functions. A further discussion of this point with a concrete example can be found in Sections 2.1 and 2.2 from [14]. The projection introduced above is quite natural once we consider the dynamical equations that the scalar fields \(\varphi^{a}=\
abla_{\mu}A^{a\mu}\) obey, which can be obtained by taking the divergence in Equation (4):  
\[\left[\square\xi^{c}_{\ a}+\left(\xi^{-1}\right)^{cb}M_{ab}\right]\varphi^{a }=0. \tag{8}\]  
We have a set of sourceless Klein-Gordon equations for these scalars \(\varphi^{a}\), and hence it is quite natural to restrict ourselves to the subspace \(\mathcal{U}\) defined by \(\varphi^{a}=0\), since there are no Lorentz-invariant sources that might generate excitations of these scalars.