# More Insights into Symmetries in Multisymplectic Field Theories

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## Abstract

**:**

## 1. Introduction

## 2. Symmetries on (Pre)multisymplectic Fiber Bundles

#### 2.1. (Pre)Multisymplectic Bundles

**multisymplectic form**if it is closed and 1-nondegenerate and $(\mathcal{M},\Omega )$ is called a

**multisymplectic manifold**. Alternatively, $\Omega $ is called a

**premultisymplectic form**if it is closed and 1-degenerate and $(\mathcal{M},\Omega )$ is called a

**premultisymplectic manifold**. If $\Omega $ is an exact form, then it is called an

**exact (pre)multisymplectic form**and the couple $(\mathcal{M},\Omega )$ is called a

**(pre)multisymplectic system**.

#### 2.2. Conserved Quantities and Symmetries

**Definition 1.**

**conserved quantity**of the (pre)multisymplectic system $(\mathcal{M},\Omega )$ is a form $\alpha \in {\Omega}^{m-1}\left(\mathcal{M}\right)$ which satisfies $\mathrm{L}\left(\mathbf{X}\right)\alpha ={(-1)}^{m+1}i\left(\mathbf{X}\right)\mathrm{d}\alpha =0$ for every locally decomposable and ϱ-transverse multivector field $\mathbf{X}\in {ker}^{m}\Omega $ (i.e., which satisfies Equations (1) and (2)).

**Remark 1.**

**current**associated with the conserved quantity α. This result associates a

**conservation law**on M to every conserved quantity on $\mathcal{M}$.

**Definition 2.**

- 1.
- A
**symmetry**of the (pre)multisymplectic system $(\mathcal{M},\Omega )$ is a diffeomorphism $\mathrm{\Phi}:\mathcal{M}\to \mathcal{M}$ such that ${\mathrm{\Phi}}_{*}\left({ker}^{m}\Omega \right)\subset {ker}^{m}\Omega $. - 2.
- An
**infinitesimal symmetry**of the (pre)multisymplectic system $(\mathcal{M},\Omega )$ is a vector field $Y\in \mathfrak{X}\left(\mathcal{M}\right)$ whose local flows are local symmetries or, equivalently, $[Y,{ker}^{m}\Omega ]\subset {ker}^{m}\Omega $.

**Remark 2.**

**Theorem 1.**

**Remark 3.**

**spacetime symmetries**. Therefore, in the bundle $\varrho :\mathcal{M}\to M$, the corresponding diffeomorphisms $\mathrm{\Phi}:\mathcal{M}\to \mathcal{M}$ must be fiber preserving and thereby restrict to diffeomorphisms ${\mathrm{\Phi}}_{M}:M\to M$ which satisfy ${\mathrm{\Phi}}_{M}\circ \varrho =\varrho \circ \mathrm{\Phi}$. For infinitesimal symmetries, this means that the vector fields $Y\in \mathfrak{X}\left(\mathcal{M}\right)$ must be ϱ-projectable; hence, there exist ${Y}_{M}\in \mathfrak{X}\left(M\right)$ such that ${\varrho}_{*}Y={Y}_{M}$.

#### 2.3. Noether Symmetries

**Definition 3.**

- 1.
- A
**Noether**or**Cartan symmetry**of the (pre)multisymplectic system $(\mathcal{M},\Omega )$ is a diffeomorphism $\mathrm{\Phi}:\mathcal{M}\to \mathcal{M}$ such that, ${\mathrm{\Phi}}^{*}\Omega =\Omega $. In the particular case where ${\mathrm{\Phi}}^{*}\mathrm{\Theta}=\mathrm{\Theta}$, then Φ is called an**exact Noether**or**exact Cartan symmetry**. - 2.
- An
**infinitesimal Noether**or**Cartan symmetry**of the (pre)multisymplectic system $(\mathcal{M},\Omega )$ is a vector field $Y\in \mathfrak{X}\left(\mathcal{M}\right)$ for which $\mathrm{L}\left(Y\right)\Omega =0$. In the particular case where $\mathrm{L}\left(Y\right)\mathrm{\Theta}=0$, Y is called an**infinitesimal exact Noether**or**infinitesimal exact Cartan symmetry**.

**Remark 4.**

**Noether’s theorem**is stated as follows:

**Theorem 2**(Noether).

**Noether current**.

**Remark 5.**

#### 2.4. Gauge Symmetries

**gauge symmetries**. Then,

**infinitesimal gauge symmetries**are vector fields $Y\in \mathfrak{X}\left(\mathcal{M}\right)$ whose local flows are local gauge symmetries and, consequently, are $\varrho $-vertical vector fields. In this respect, it should be noted that, the set of $\varrho $-vertical vector fields are ${\mathfrak{X}}^{V\left(\varrho \right)}\left(\mathcal{M}\right)=ker\phantom{\rule{0.166667em}{0ex}}\omega $.

**geometric (infinitesimal) gauge symmetries**.

**Definition 4.**

**geometric infinitesimal gauge symmetry**, or simply a

**gauge vector field**, of a (pre)multisymplectic system $(\mathcal{M},\Omega )$ is a vector field $Y\in \mathfrak{X}\left(\mathcal{M}\right)$ such that:

- 1.
- $Y\in ker\phantom{\rule{0.166667em}{0ex}}\Omega $.
- 2.
- It is a ϱ-vertical vector field, $Y\in {\mathfrak{X}}^{V\left(\varrho \right)}\left(\mathcal{M}\right)$.
- 3.
- It is tangent to $\mathcal{S}$, $Y\in \underline{\mathfrak{X}\left(\mathcal{S}\right)}$.

**geometric gauge transformations**. Physical states (i.e., sections which are stable solutions to the field equations) related to one another by geometric gauge transformations are called

**geometric gauge equivalent states**, and they are physically equivalent (in the sense that they are physically indistinguishable).

**Remark 6.**

#### 2.5. Multimomentum Map

**Definition 5.**

**multimomentum map**associated with the symmetry group G.

**Noether current**$\mathrm{j}={j}^{\mu}{\mathrm{d}}^{m-1}{x}_{\mu}\in {\Omega}^{m-1}\left(\mathcal{M}\right)$ of a symmetry is obtained as $\mathrm{j}={\mathbf{\psi}}^{*}{\mathrm{J}}_{\xi}$ (see also Remark 1).

## 3. Lifting Transformations from the Base Space of a Jet Bundle

#### 3.1. First-Order Jet Bundles

**holonomic**if $\psi $ is the canonical lift of a section $\varphi ={\pi}^{1}\circ \psi \in \mathrm{\Gamma}\left(\pi \right)$, and hence, $\psi ={j}^{1}({\pi}^{1}\circ \psi )$.

#### 3.2. Lifting Transformations from M to E

**local variation**of the fields. The functions ${\tilde{\xi}}^{i}\left(x\right)$ are not related to the components ${\xi}^{i}(x,y)$ in (6), a priori. The term $-{\xi}^{\mu}\left(x\right)\frac{\partial {y}^{i}}{\partial {x}^{\mu}}$ in (7) is called the

**transport term**, and ${\tilde{\xi}}^{i}\left(x\right)$ is called the

**global variation**of the fields which is given by

**Definition 6.**

**generalized Lie derivative of the section $\varphi $ by Z**is the map $\mathbb{L}\left(Z\right)\varphi :M\to \mathrm{T}E$ defined as

**Lie derivative of the section $\varphi $ by Z**and is given as $\mathrm{L}\left(Z\right)\varphi =\left({x}^{\mu},{\xi}^{\nu}\phantom{\rule{0.166667em}{0ex}}\frac{\partial {\varphi}^{i}}{\partial {x}^{\nu}}-{\xi}^{i}\circ \varphi \right)$ when the local expression for ${Z}_{E}$ is (6).

**Definition 7.**

- 1.
- Let ${\mathrm{\Phi}}_{M}:M\to M$ be a diffeomorphism. The
**canonical lift of ${\mathrm{\Phi}}_{M}$ to E**is the diffeomorphism ${\mathrm{\Phi}}_{E}:E\to E$ defined as follows: for every $(x,{T}_{x})\in E$ where ${T}_{x}\in {\mathfrak{T}}^{(k,l)}\left({\mathrm{T}}_{x}M\right)$, define ${\mathrm{\Phi}}_{E}(x,{T}_{x}):=({\mathrm{\Phi}}_{M}\left(x\right),\mathcal{T}{\mathrm{\Phi}}_{M}\left({T}_{x}\right))$, where $\mathcal{T}{\mathrm{\Phi}}_{M}$ denotes the canonical transformation of tensors on M induced by ${\mathrm{\Phi}}_{M}$. Thus, $\pi \circ {\mathrm{\Phi}}_{E}={\mathrm{\Phi}}_{M}\circ \pi $. - 2.
- Let $Z\in \mathfrak{X}\left(M\right)$
**be the vector field induced by local one-parameter groups of diffeomorphisms**of M, denoted ${\varphi}_{t}$. The canonical lift of Z to E is the vector field ${Z}_{E}\in \mathfrak{X}\left(E\right)$ induced by local one-parameter groups of diffeomorphisms ${\left({\varphi}_{{}_{E}}\right)}_{t}$ which are the canonical lifts of ${\varphi}_{t}$ to the configuration bundle E.

#### 3.3. Lifting Transformations from E to ${J}^{1}\pi $

**canonical lift**of $\mathrm{\Phi}$ to ${J}^{1}\pi $ is the diffeomorphism ${j}^{1}\mathrm{\Phi}:{J}^{1}\pi \u27f6{J}^{1}\pi $ defined as

**canonical lift**of a $\pi $-projectable vector field $Y\in \mathfrak{X}\left(E\right)$ to ${J}^{1}\pi $ is the vector field ${j}^{1}Y\in \mathfrak{X}\left({J}^{1}\pi \right)$ whose local one-parameter groups of diffeomorphisms are the canonical lifts of the local one-parameter groups of diffeomorphisms of Y. If $Y\in \mathfrak{X}\left(E\right)$ is the canonical lift of $Z\in \mathfrak{X}\left(M\right)$ to E whose expression in local coordinates is given by (6) as $Y={Z}_{E}=-{\xi}^{\mu}\left(x\right)\frac{\partial}{\partial {x}^{\mu}}-{\xi}^{i}(x,y)\frac{\partial}{\partial {y}^{i}}$, then the canonical lift of $Y\in \mathfrak{X}\left(E\right)$ to ${J}^{1}\pi $ is

## 4. Symmetries for Lagrangian and Hamiltonian Field Theories

#### 4.1. First-Order Lagrangian Field Theories

**Poincaré–Cartan**m and

**$(m+1)$-forms**associated with $\mathcal{L}$ are defined as ${\mathrm{\Theta}}_{\mathcal{L}}:=i\left(\mathcal{V}\right)\mathrm{d}\mathcal{L}+\mathcal{L}\in {\Omega}^{m}\left({J}^{1}\pi \right)$ and ${\Omega}_{\mathcal{L}}:=-\mathrm{d}{\mathrm{\Theta}}_{\mathcal{L}}\in {\Omega}^{m+1}\left({J}^{1}\pi \right)$, respectively, and they have the following coordinate expressions:

**Remark 7.**

#### 4.2. De Donder–Weyl Hamiltonian Formalism

**Hamilton–Cartan**m and

**$(m+1)$-forms**; ${\Omega}_{\mathrm{h}}$ is the multisymplectic form on ${J}^{1*}\pi $, and the couple $({J}^{1*}\pi ,{\Omega}_{\mathrm{h}})$ is the Hamiltonian system associated with the (hyper)regular Lagrangian system $({J}^{1}\pi ,{\Omega}_{\mathcal{L}})$. The local expressions for ${\mathrm{\Theta}}_{\mathrm{h}}$ and ${\Omega}_{\mathrm{h}}$ are

**De Donder–Weyl Hamiltonian function**. The field equations can be obtained from the so-called Hamilton–Jacobi variational principle, and their solutions are sections $\psi :M\to {J}^{1*}\pi $ which satisfy

#### 4.3. Symmetries, Conserved Quantities, and Multimomentum Maps

**Definition 8.**

- 1.
- A (Noether) symmetry $\mathrm{\Phi}:{J}^{1}\pi \to {J}^{1}\pi $ of a Lagrangian system $({J}^{1}\pi ,{\Omega}_{\mathcal{L}})$ is said to be
**natural**if Φ is a canonical lift; i.e., $\mathrm{\Phi}={j}^{1}\phi $ for a diffeormorphism $\phi :E\to E$. - 2.
- An infinitesimal (Noether) symmetry $X\in \mathfrak{X}\left({J}^{1}\pi \right)$ of a Lagrangian system $({J}^{1}\pi ,{\Omega}_{\mathcal{L}})$ is said to be
**natural**if X is a canonical lift; i.e., $X={j}^{1}{Z}_{E}$ for some ${Z}_{E}\in \mathfrak{X}\left(E\right)$.

**Proposition 1.**

**Definition 9.**

**natural**if $X={j}^{1}{Z}_{E}$ for some vector field ${Z}_{E}\in \mathfrak{X}\left(E\right)$.

**Definition 10.**

- 1.
- A
**Lagrangian symmetry**of a Lagrangian system $({J}^{1}\pi ,{\Omega}_{\mathcal{L}})$ is a diffeomorphism $\mathrm{\Phi}:{J}^{1}\pi \to {J}^{1}\pi $ that leaves $\mathcal{L}$ invariant: ${\mathrm{\Phi}}^{*}\mathcal{L}=\mathcal{L}$.If $\mathrm{\Phi}={j}^{1}\phi $ for some fiber-preserving diffeomorphism $\phi :E\to E$, then the Lagrangian symmetry is said to be**natural**. - 2.
- An
**infinitesimal Lagrangian symmetry**of a Lagrangian system $({J}^{1}\pi ,{\Omega}_{\mathcal{L}})$ is a vector field $X\in \mathfrak{X}\left({J}^{1}\pi \right)$ that leaves $\mathcal{L}$ invariant.If $X={j}^{1}{Z}_{E}$, for some π-projectable vector field ${Z}_{E}\in \mathfrak{X}\left(E\right)$, then the infinitesimal Lagrangian symmetry is said to be**natural**.

**Definition 11.**

- 1.
- A
**geometric Lagrangian symmetry**of a Lagrangian system $({J}^{1}\pi ,{\Omega}_{\mathcal{L}})$ is a diffeomorphism $\mathrm{\Phi}:{J}^{1}\pi \to {J}^{1}\pi $ such that:- (a)
- ${\mathrm{\Phi}}^{*}\mathcal{L}=\mathcal{L}$.
- (b)
- The canonical geometric structures of ${J}^{1}\pi $ are invariant by Φ.

- 2.
- An
**infinitesimal geometric Lagrangian symmetry**of a Lagrangian system $({J}^{1}\pi ,{\Omega}_{\mathcal{L}})$ is a vector field $X\in \mathfrak{X}\left({J}^{1}\pi \right)$ such that:- (a)
- $\mathrm{L}\left(X\right)\mathcal{L}=0$.
- (b)
- The canonical geometric structures of ${J}^{1}\pi $ are invariant under the action of X.

**gauge equivalent Lagrangians**which give rise to the same Euler–Lagrange equations (see, for instance, [12,42]). Gauge equivalent Lagrangian densities differ by an exact differential form: ${\mathrm{\Phi}}^{*}\mathcal{L}=\mathcal{L}+\mathrm{d}\beta $. Recall that in the physics literature, the analysis of field theories occurs on the jet prolongations ${j}^{1}\varphi :M\to {J}^{1}\pi $ so that $\tilde{\mathcal{L}}=\tilde{L}\omega ={\left({j}^{1}\varphi \right)}^{*}\mathcal{L}\in {C}^{\infty}\left(M\right)$. In this setting, gauge equivalent Lagrangians differ by a total derivative $\delta \tilde{L}={\partial}_{\mu}{\tilde{K}}^{\mu}$ where ${\tilde{K}}^{\mu}={\left({j}^{1}\varphi \right)}^{*}{K}^{\mu}$ for some ${K}^{\mu}$. Furthermore, recall that when the field variations are produced by diffeomorphisms of M generated by some vector field $\xi \in \mathfrak{X}\left(M\right)$, the variation of the Lagrangian function $\tilde{L}$ on ${j}^{1}\varphi $ can be written as a Lie derivative of the local sections ${j}^{1}\varphi $ with respect to $\xi $ so that

**Lagrangian multimomentum map**is given as

**canonical energy–momentum tensor**, denoted as usual as ${T}_{\phantom{\rule{4pt}{0ex}}\nu}^{\mu}$, is defined from the terms contracted with ${\xi}^{\nu}$ above:

**Hamiltonian multimomentum map**can be obtained by applying the push-forward of the Legendre map ${\mathcal{FL}}_{*}$ to ${\mathrm{J}}_{\mathcal{L}}\left({X}_{\xi}\right)$ or, equivalently, using the $\mathcal{FL}$-projection of ${X}_{\xi}$ and contracting it with the corresponding (pre)multisymplectic form on the image of the Legendre map. When the field theory under investigation is regular, the calculation is straightforward, and the Hamiltonian multimomentum map is given as

#### 4.4. Symmetries in the Presence of Constraints

- There exists some $X\in \mathfrak{X}\left({J}^{1}\pi \right)$ such that $L\left(X\right){\Omega}_{\mathcal{L}}=0$ and is $\mathcal{FL}$-projectable only from the constraint submanifold ${\mathcal{S}}_{f}\hookrightarrow {J}^{1}\pi $. Then, on the corresponding Hamiltonian constraint submanifold ${P}_{f}\subset {P}_{0}$, there exists the vector field $Y={\mathcal{FL}}_{*}{X|}_{{\mathcal{S}}_{f}}\in \mathfrak{X}\left({P}_{f}\right)$ such that $L\left(Y\right){\Omega}_{h}^{0}{|}_{{P}_{f}}=0$. Furthermore, the vector field $X\in \mathfrak{X}\left({J}^{1}\pi \right)$ may or may not be the local extension of some ${Z|}_{{\mathcal{S}}_{f}}$ to ${J}^{1}\pi $ for some $Z\in \mathfrak{X}\left({J}^{1}\pi \right)$.
- There exists some $X\in \underline{\mathfrak{X}\left({\mathcal{S}}_{f}\right)}$ such that $L\left(X\right){\Omega}_{\mathcal{L}}{|}_{{\mathcal{S}}_{f}}=0$, and X is $\mathcal{FL}$-projectable only from ${\mathcal{S}}_{f}$. Then, on the corresponding Hamiltonian constraint submanifold ${P}_{f}\subset {P}_{0}$, there exists the vector field ${Y|}_{{P}_{f}}={\mathcal{FL}}_{*}X{{|}_{{\mathcal{S}}_{f}}\in \underline{\mathfrak{X}\left({P}_{f}\right)}|}_{{P}_{f}}$ such that $L\left(Y\right){\Omega}_{\mathrm{h}}^{0}{|}_{{P}_{f}}=0$. Furthermore, it is possible to construct a local extension of Y to ${P}_{0}$, denoted as $\tilde{Y}\in \mathfrak{X}\left({P}_{0}\right)$, but $X\in \mathfrak{X}\left({J}^{1}\pi \right)$ may or may not be the local extension of some ${Z|}_{{\mathcal{S}}_{f}}$ to ${J}^{1}\pi $ for some $Z\in \mathfrak{X}\left({J}^{1}\pi \right)$.

## 5. Some Examples

#### 5.1. Bosonic String Theories

#### 5.2. Yang–Mills Theory

#### 5.3. Chern–Simons Theory

#### 5.4. Electric Carrollian Scalar Field Theory

#### 5.5. Magnetic Carrollian Scalar Field Theory

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Multivector Fields on Manifolds and Fiber Bundles

**m-multivector fields**in $\mathcal{M}$ ($m\le N$) are the sections of the m-multitangent bundle $\stackrel{m}{\bigwedge}\mathrm{T}\mathcal{M}:=\stackrel{m}{\overbrace{\mathrm{T}\mathcal{M}\wedge \dots \wedge \mathrm{T}\mathcal{M}}}$; that is, the skew-symmetric contravariant tensor fields of order m in $\mathcal{M}$; the set of which is denoted ${\mathfrak{X}}^{m}\left(\mathcal{M}\right)$. Then, if $\mathbf{X}\in {\mathfrak{X}}^{m}\left(\mathcal{M}\right)$, for every point $\overline{y}\in \mathcal{M}$, there is an open neighbourhood $U\subset \mathcal{M}$ and ${X}_{1},\dots ,{X}_{r}\in \mathfrak{X}\left(U\right)$ such that, for $m\u2a7dr\u2a7d\mathrm{dim}\phantom{\rule{0.166667em}{0ex}}\mathcal{M}$,

**locally decomposable multivector field**if there exist ${X}_{1},\dots ,{X}_{m}\in \mathfrak{X}\left(U\right)$ such that ${\mathbf{X}|}_{U}={X}_{1}\wedge \dots \wedge {X}_{m}$. The locally decomposable m-multivector fields are locally associated with m-dimensional distributions $D\subset \mathrm{T}\mathcal{M}$, and this splits ${\mathfrak{X}}^{m}\left(\mathcal{M}\right)$ into equivalence classes$\left\{\mathbf{X}\right\}\subset {\mathfrak{X}}^{m}\left(\mathcal{M}\right)$ which are made of the locally decomposable multivector fields associated with the same distribution. If $\mathbf{X},{\mathbf{X}}^{\prime}\in \left\{\mathbf{X}\right\}$ then, for every $U\subset \mathcal{M}$, there exists a non-vanishing function $f\in {\mathrm{C}}^{\infty}\left(U\right)$ such that ${\mathbf{X}}^{\prime}=f\mathbf{X}$ on U.

**contraction**between $\mathbf{X}$ and $\Omega $ is the natural contraction between tensor fields; in particular, it gives zero when $p<m$ and, if $p\ge m$,

**Lie derivative**of $\Omega $ with respect to $\mathbf{X}$ is the graded bracket (of degree $m-1$)

**Schouten-Nijenhuis bracket**of $\mathbf{X},\mathbf{Y}$ is the bilinear map $\mathbf{X},\mathbf{Y}\mapsto [\mathbf{X},\mathbf{Y}]$, where $[\mathbf{X},\mathbf{Y}]$ is a $(i+j-1)$-multivector field obtained as the graded commutator of $\mathrm{L}\left(\mathbf{X}\right)$ and $\mathrm{L}\left(\mathbf{Y}\right)$; that is,

**Lie derivative**of $\mathbf{Y}$ with respect to $\mathbf{X}$, and is denoted $\mathrm{L}\left(\mathbf{X}\right)\mathbf{Y}:=[\mathbf{X},\mathbf{Y}]$. Furthermore, if $Y\in \mathfrak{X}\left(\mathcal{M}\right)$ and $\mathbf{X}\in {\mathfrak{X}}^{m}\left(\mathcal{M}\right)$, then

**integrable multivector fields**, which are those locally decomposable multivector fields whose associated distribution is integrable. Then, if $\mathbf{X}\in {\mathfrak{X}}^{m}\left(\mathcal{M}\right)$ is integrable and $\varrho $-transverse, its integral manifolds are local sections of the projection $\varrho :\mathcal{M}\to M$.

**holonomic**if its integral sections are holonomic sections of ${\overline{\pi}}^{1}$.

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**MDPI and ACS Style**

Guerra, A., IV; Román-Roy, N.
More Insights into Symmetries in Multisymplectic Field Theories. *Symmetry* **2023**, *15*, 390.
https://doi.org/10.3390/sym15020390

**AMA Style**

Guerra A IV, Román-Roy N.
More Insights into Symmetries in Multisymplectic Field Theories. *Symmetry*. 2023; 15(2):390.
https://doi.org/10.3390/sym15020390

**Chicago/Turabian Style**

Guerra, Arnoldo, IV, and Narciso Román-Roy.
2023. "More Insights into Symmetries in Multisymplectic Field Theories" *Symmetry* 15, no. 2: 390.
https://doi.org/10.3390/sym15020390