*8.2. Conservation Laws*

We have two symmetries, i.e., local Lorentz transformations and diffeomorphisms. They are continuous symmetries, and as such, we expect two conservation laws. Since we are dealing with local symmetries, we shall not find two conserved currents but rather two relations for the variations of the matter lagrangian w.r.t. *e* and *ω*.

These relations directly imply the Bianchi identities of Equations (38) and (56), but we could also do the converse, namely assuming Equations (38) and (56) and then deriving such conservation laws. This means that conservation laws are a consequence of symmetry on the one hand, implemented via the following symmetries (respectively diffeomorphisms and local SO(3, 1))

$$\begin{array}{ll}\delta\_{\xi}e^{a} &= \mathcal{L}\_{\xi}e^{a} = \iota\_{\xi}de^{a} + d\iota\_{\xi}e^{a} \\ \delta\_{\xi}\omega^{ab} &= \mathcal{L}\_{\xi}\omega^{ab} = \iota\_{\xi}d\omega^{ab} + d\iota\_{\xi}\omega^{ab},\end{array} \tag{71}$$

where *ξ* is the generator vector field,

$$\begin{aligned} \delta\_{\Lambda} \varepsilon^{a} &= \Lambda^{ab} \eta\_{bc} \varepsilon^{c} \\ \delta\_{\Lambda} \omega^{ab} &= -d\_{\omega} \Lambda^{ab} \quad \Lambda \in \mathfrak{so} (3,1), \end{aligned} \tag{72}$$

or a direct consequence if we impose field equations and, thus, gravitational dynamics and Bianchi identities on the other hand.

We will follow the shortest derivation, namely to implement the Bianchi identities of Equations (38) and (56) on field Equation (69).

Thanks to Bianchi identities, left hand side of field Equation (69) can be rewritten in the following way:

$$\begin{split} d\_{\omega}(\varepsilon\_{abcd}\varepsilon^{b}\wedge F^{cd}\_{\omega}) &= \iota\_{a}\tilde{\mathbb{Q}}^{b}\wedge(\varepsilon\_{bcde}\varepsilon^{c}\wedge F^{de}\_{\omega}) + \iota\_{a}F^{bc}\_{\omega}\wedge(\varepsilon\_{bcde}\tilde{\mathbb{Q}}^{d}\wedge\varepsilon^{e}) \\ d\_{\omega}(\varepsilon\_{abcd}\tilde{\mathbb{Q}}^{c}\wedge\varepsilon^{d}) &= -\frac{1}{2}(\varepsilon\_{acde}\varepsilon^{c}\wedge F^{de}\_{\omega}\wedge\varepsilon\_{b}-\varepsilon\_{bcde}\varepsilon^{c}\wedge F^{de}\_{\omega}\wedge\varepsilon\_{d}), \end{split} \tag{73}$$

where *ι<sup>a</sup>* = *ιe*¯*<sup>a</sup>* and *e<sup>b</sup>* = *ηbce c* .

However, because of the same field in Equation (69), they reduce to

$$\begin{aligned} d\_{\omega}T\_{a} &= \iota\_{a}\tilde{\mathbb{Q}}^{b} \wedge T\_{b} + \iota\_{a}F^{bc}\_{\omega} \wedge \Sigma\_{bc} \\ d\_{\omega}\Sigma\_{ab} &= \frac{1}{2}T\_{[a} \wedge e\_{b]}\prime \end{aligned} \tag{74}$$

In References [26,27], a more detailed discussion can be found. These are conservation laws for ECSK theory.

In components, as given in Reference [20], they read

$$\begin{aligned} \nabla\_{\mu}T^{\mu\nu} + T\_{\sigma\rho} \mathbf{Q}^{\sigma\rho\nu} - \Sigma\_{\mu\sigma\rho} \mathbf{R}^{\mu\sigma\rho\nu} &= \mathbf{0} \\ \nabla\_{\mu}\Sigma\_{\sigma\omega}{}^{\mu} + \frac{1}{2}T\_{[\sigma\omega]} &= \mathbf{0}. \end{aligned} \tag{75}$$

#### **9. Conclusions**

We have set up all the mathematical background for building ECSK theory, eventually achieving field equations and conservation laws.

In ECSK theory, torsion is only an algebraic constraint and it does not propagate. This is a natural consequence of inserting torsion into the theory without an independent coupling coefficient but simply generalizing the Einstein–Hilbert action (or Palatini action in our formulation) R *R* √ <sup>−</sup>*gd*4*<sup>x</sup>* to a non-torsion-free connection ∇ (or spin connection in our case). In this case, the Ricci scalar contains both curvature and torsion.

It is possible to immediately recover General Relativity by imposing the zero torsion condition, which, in the considered theory, translates to letting the matter field *ϕ* generate a null contribution to the spin tensor Σ*µν σ* . The most natural matter fields which would fit with the theory are spinors; indeed, spinors are the way in which we can have a non-vanishing spin tensor which is also dynamical because of equations of motion for the spinor field.

This review does not want to substitute the well-known literature but to just give a self-contained and mathematically rigorous introduction to ECSK theory, providing also some references for deepening knowledge in the present subjects. Also, we intentionally did not dive too deeply into physical applications to cosmology (like done in References [28–33]), that might be a valid argument for writing another review article.

**Funding:** The APC was funded by Professor Matthias Gaberdiel (Institute for Theoretical Physics, ETH Zürich).

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

### **References**



© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

*Article*
