On interacting non-Abelian antisymmetric tensor field models

Technical Summary: Interacting Non-Abelian Antisymmetric Tensor Field Models

Problem Statement
Antisymmetric tensor fields (p-forms) are fundamental components of superstring/brane theories and supergravity, characterized by complex, reducible gauge structures. While free p-form models are well-understood as reducible gauge theories (with reducibility stages depending on the form rank), the gauge structure of interacting non-Abelian tensor models remains an open issue. Standard no-go theorems suggest that local, non-Abelian deformations of free p-form gauge transformations are impossible without introducing extra fields. However, models involving non-Abelian tensors interacting with Yang-Mills fields exist in specific contexts (e.g., gauged supergravities, dimensional reductions). The primary challenge addressed in this work is determining the reducibility properties of gauge transformations in these interacting, non-Abelian tensor models, specifically those obtained via dimensional reduction on a group manifold. It is unclear whether the introduction of interactions and mass terms alters the reducibility stages found in their free counterparts.

Methodology
The authors employ a dimensional reduction procedure on a group manifold to generate interacting non-Abelian tensor models from free Abelian higher-rank forms minimally coupled to gravity.

1. Reduction Prescription: Starting from a free Abelian $p$-form field $C_{M_1...M_p}$ in $D$ dimensions coupled to gravity, the coordinates are split into $x^\mu$ (spacetime) and $y^a$ (compact group manifold $G$). The metric and fields are parameterized using Killing vectors $e^a_{\bar{a}}(y)$ of the group $G$, introducing a dimensionful parameter $\kappa$ related to the manifold's size, which acts as the coupling constant.
2. Field Redefinition: The original fields $C$ are redefined into new fields $B$ (and parameters $\Lambda$) using the vielbein of the reduced metric. This transformation ensures that the new fields transform covariantly under the induced Yang-Mills symmetry, removing non-homogeneous terms involving derivatives of the gauge parameters.
3. Algebraic Analysis: The authors explicitly calculate the commutators of the gauge transformations for the reduced fields. They identify the non-Abelian algebra structure and search for "trivial" transformations—sets of parameters for which the gauge variation of all fields vanishes identically.
4. Recursive Reducibility Check: By finding the parameters that generate trivial transformations, the authors determine the stage of reducibility. They verify if these trivial parameters themselves possess gauge redundancies (higher-stage reducibility), comparing the results against the known reducibility of free Abelian theories.

Key Contributions and Results
The paper analyzes models derived from the dimensional reduction of 3-forms, 4-forms, and general $p$-forms.

• 3-Form Model (Section 2):

  • Reduction of a free 3-form yields a system containing a Yang-Mills field, a 2-form, a 1-form, and a scalar, all transforming under a non-Abelian algebra.
  • The gauge transformations are shown to be first-stage reducible for the non-Abelian sector. The trivial transformations are generated by parameters $\omega_{\bar{a}}$ and $\zeta_\mu$, which satisfy specific relations involving the field strength and structure constants.
  • The reducibility stage matches that of the free Abelian 3-form (which is second-stage reducible overall, with the non-Abelian part being first-stage).
  • Stückelberg Fields: The analysis identifies specific fields (e.g., $S_{\mu\bar{a}}$) that act as Stückelberg fields. These fields shift under gauge transformations, effectively restoring the gauge symmetry broken by mass terms ($\sim \kappa^2$) acquired by other tensor fields. This mechanism allows the theory to bypass no-go theorems regarding non-Abelian deformations.

• 4-Form Model (Section 3):

  • Reduction of a free 4-form produces a system with tensors of rank 4, 3, 2, 1, and 0.
  • The gauge transformations are shown to be second-stage reducible for the non-Abelian sector. The trivial transformations are generated by parameters $\omega_{\mu\bar{a}}$ and $\omega_{\bar{a}\bar{b}}$, which themselves are reducible.
  • The reducibility stage again coincides with the free Abelian 4-form (third-stage reducible overall).

• General $p$-Form Model (Section 4):

  • The results are generalized to an arbitrary rank $r+1$ form.
  • The authors demonstrate that the gauge transformations of the resulting non-Abelian system are $p-2$ stage reducible (matching the $p-1$ stage of the free Abelian theory, accounting for the vector/scalar sectors).
  • A general formula for the trivial transformation parameters is derived, showing that the reducibility structure is preserved despite the presence of mass terms and non-Abelian interactions.
  • The paper contrasts this with a specific non-Abelian model from literature [41] which is only first-stage reducible, highlighting that the dimensional reduction method preserves the higher-stage reducibility of the parent free theory.

Significance and Claims
The paper claims that the gauge structure of interacting non-Abelian antisymmetric tensor models obtained via dimensional reduction is robust and retains the reducibility properties of their free Abelian counterparts.

• Preservation of Reducibility: The introduction of interactions and mass terms via dimensional reduction does not alter the stages of reducibility. The interacting models are reducible gauge theories with the same hierarchy of gauge parameters as the free theories.
• Role of Stückelberg Fields: The existence of Stückelberg fields is identified as the mechanism that reconciles the presence of mass terms with gauge invariance. These fields allow the theory to maintain a consistent gauge structure, circumventing the constraints of no-go theorems that assume massless deformations.
• Quantization Implications: The authors state that establishing the reducibility structure is a prerequisite for the consistent quantization of these theories (e.g., via the Batalin-Vilkovisky or BRST-BFV formalisms). The results provide the necessary foundation for constructing path integrals and effective actions for these non-Abelian tensor theories.
• Future Directions: The paper suggests that these methods can be extended to self-dual actions, reductions on manifolds other than groups (e.g., spheres), and the construction of supersymmetric versions. It also notes the potential for studying dualities in the interacting non-Abelian context.

The work does not propose new experimental tests or specific phenomenological applications but focuses on the theoretical consistency and structural properties of these field theories, aiming to fill the gap in understanding the gauge structure of interacting p-forms.