Revisiting the Symmetries of Galilean Electrodynamics

Technical Summary: Revisiting the Symmetries of Galilean Electrodynamics

Problem Statement
Galilean Electrodynamics (GED) and its non-abelian generalization, Galilean Yang-Mills (GYM), serve as fundamental models for non-relativistic field theories, often described as the "hydrogen atom" of this sector. These theories are also central to the non-relativistic AdS$_5$/CFT$_4$ correspondence, specifically as potential duals to the GGK non-relativistic string theory with a String Newton-Cartan target space. However, the literature contains conflicting results regarding the off-shell symmetries of these actions. Previous studies [12, 13] reported finite symmetry algebras for GED in $3+1$ dimensions or claimed the full Galilean Conformal Algebra (GCA) symmetry, while others found infinite-dimensional algebras in $2+1$ dimensions. A primary source of confusion was the strict requirement in earlier works that the Lagrangian transform as a total derivative of a specific form ($\delta L = \partial_\mu (\xi^\mu L)$), rather than the standard Quantum Field Theory (QFT) condition where the Lagrangian transforms as a total derivative of a generic function ($\delta L = \partial_\mu F^\mu$). This paper aims to clarify these discrepancies by systematically revisiting the symmetries of GED and GYM, both in isolation and coupled to scalar fields, in generic spacetime dimensions $d+1$.

Methodology
The authors employ a systematic Noether analysis to determine the off-shell symmetries of the action. The methodology involves:

1. General Ansatz: Proposing the most generic linear transformation ansatz for the spacetime coordinates ($t, x^i$) and the fields (vector potentials $a_i$, scalars $\phi, \tilde{\phi}$, and additional scalars $S_I$). The transformations include spacetime-dependent parameters and linear mixing between fields.
2. Variational Constraint: Requiring that the variation of the Lagrangian density under these transformations results in a total derivative of a generic function $F^\mu$ (fields, coordinates, and parameters), i.e., $\delta L = \partial_\mu F^\mu$. This approach adheres to standard QFT practices where boundary terms are eliminated via suitable fall-off conditions.
3. Parameter Fixing: Solving the resulting differential equations to fix the functional forms of the transformation parameters (e.g., $\xi^t, \xi^i$) and the coefficients of the field transformations.
4. Algebraic Structure: Identifying the generators associated with the free parameters and computing their non-vanishing commutation relations to characterize the symmetry algebra.
5. Comparison with Bulk Theory: Comparing the derived boundary symmetries with the Killing vector symmetries of the GGK non-relativistic string theory in the bulk to assess holographic consistency.

Key Contributions and Results

• GED Symmetries in Generic Dimensions:
  The authors demonstrate that the symmetries of the GED action are always infinite-dimensional in any spacetime dimension $d+1$, correcting previous claims of finite symmetries.

  • In the specific case of $d=3$ ($3+1$ dimensions), the symmetry algebra is identified as the Conformal Milne algebra extended by an additional generator, $D_x$ (spatial dilatation).
  • The algebra includes generators for time translations ($H$), spatial rotations ($J_{ij}$), special conformal transformations ($K$), and an infinite tower of spatial translations ($M^{(n)}_i$).
  • Crucially, the full tower of generators $L^{(n)}$ (which would constitute the full GCA) is not a symmetry; only the finite truncation corresponding to $n = -1, 0, 1$ (generating $H, D, K$) survives, augmented by $D_x$.
  • The paper clarifies that the generator $D_x$ (spatial dilatation) is a symmetry of GED but disappears when a non-abelian structure is introduced.

• GED with 5 Free Static Scalars:
  When GED is coupled to five free static scalar fields (arising from the non-relativistic limit of the DBI action), the spacetime symmetries are preserved.

  • The scalars introduce new internal symmetries: time-dependent rotations ($J^{(n)}_{IJ}$) and time-dependent pseudo-translations ($J^{(n)}_{0I}$).
  • These form an infinite-dimensional algebra alongside the spacetime symmetries.

• GYM Symmetries:
  For the non-abelian Galilean Yang-Mills theory, the authors confirm the result from [13] that the symmetries form the Conformal Milne algebra.

  • Unlike GED, the separate generators $D_t$ (time dilatation) and $D_x$ (spatial dilatation) are not symmetries; only their linear combination $D = D_t + D_x$ is preserved.
  • The algebra is infinite-dimensional, spanned by $H, K, D, J_{ij},$ and $M^{(n)}_i$.

• GYM with 5 Adjoint Scalars:
  When GYM is coupled to five interacting scalar fields in the adjoint representation (with quartic interactions), the Conformal Milne symmetry remains intact.

  • The internal symmetries differ from the free scalar case: the theory admits time-dependent pseudo-translations but only time-independent rotations for the scalars.
  • Shift symmetries for the fields are forbidden due to the presence of Yukawa-type couplings and $S^4$ vertices.

Significance and Holographic Implications
The paper concludes by applying these symmetry results to the non-relativistic AdS$_5$/CFT$_4$ correspondence. The authors compare the derived boundary symmetries with the Killing vector symmetries of the GGK bulk theory:

1. GED with 5 scalars is ruled out as the holographic dual because:
   • It possesses a $D_x$ symmetry not realized in the bulk.
   • It allows time-dependent rotations of the scalars, whereas the bulk only admits time-independent rotations.
   • It admits shift symmetries absent in the GGK theory.
2. GYM with 5 adjoint scalars is identified as the most promising candidate for the dual gauge theory. Its symmetry structure (Conformal Milne algebra + time-dependent pseudo-translations + time-independent rotations) matches the Killing vectors of the bulk theory evaluated at the boundary.

Modest Claims
The authors frame their work as a correction to the literature regarding the off-shell symmetry counting and the specific algebraic structure in $3+1$ dimensions. They do not claim to have solved the full correspondence but rather to have provided the necessary symmetry constraints. They suggest that these symmetries could be used to constrain correlation functions via Ward-Takahashi identities and to match conformal scaling dimensions with the string spectrum, noting that this would be a "highly non-trivial check" given the vanishing of higher-order corrections in the light-cone spectrum. They plan to report on these future developments in subsequent work.