Interacting Gauge Model with Mixed-Antisymmetric Fields

• Interacting Gauge Model with Mixed-Antisymmetric Fields is defined by a non-symmetric rank-2 tensor that splits into symmetric (helicity ±2) and antisymmetric (helicity 0) components.
• The kinetic Lagrangian and generalized Bianchi identities ensure gauge invariance and unitarity by eliminating unphysical modes.
• Non-Abelian interactions introduce cubic and quartic vertices that set this model apart from traditional Yang–Mills, gravitational, and Abelian B-field theories.

An interacting gauge model with mixed-antisymmetric fields generalizes the conventional Yang–Mills framework by extending the gauge sector to include higher-rank tensor fields whose transformation properties and polarizations include both symmetric and antisymmetric components. In the context of non-Abelian gauge theories, the most tractable setting is a second-rank (rank-2) tensor gauge field transforming in the adjoint of a Lie algebra and carrying two space–time indices without symmetry assumptions. The key features of such models include a highly nontrivial mode structure, non-commutative internal charges, generalized gauge and Bianchi identities, and consistent interactions that differ fundamentally from those in gravity or the Abelian Kalb–Ramond theory.

1. Tensor Field Representation and Gauge Structure

The fundamental field is a non-symmetric rank-2 tensor, denoted $A_{\mu\lambda}^a$ ($a$ is a Lie algebra index), in four-dimensional Minkowski spacetime. This field can be decomposed into symmetric and antisymmetric parts in the space–time indices: $$A_{\mu\lambda}^a = A_{(\mu\lambda)}^a + A_{[\mu\lambda]}^a$$ with

$$A_{(\mu\lambda)}^a = \frac{1}{2}\left[A_{\mu\lambda}^a + A_{\lambda\mu}^a\right],\qquad A_{[\mu\lambda]}^a = \frac{1}{2}\left[A_{\mu\lambda}^a - A_{\lambda\mu}^a\right]$$

The non-Abelian field strength is defined as

$$F^a_{\mu\nu,\lambda} = \partial_\mu A^a_{\nu\lambda} - \partial_\nu A^a_{\mu\lambda}$$

This tensor does not possess (anti)symmetry in the first two indices and retains the complexity of the original field.

Upon mode analysis, the 16 components of $A_{\mu\lambda}^a$ reduce, after gauge fixing and using the equations of motion, to three positive-norm propagating degrees of freedom: two associated with a helicity $\pm 2$ charged gauge boson (arising from the symmetric part, akin to the graviton) and one associated with a helicity $0$ charged gauge boson (originating from the antisymmetric part, analogous to the B-field but with internal charges) (0706.0762). The full set of gauge transformations retains the non-Abelian structure, but its realization on the tensor field is highly nontrivial because internal commutators and spacetime index symmetries intertwine nontrivially.

2. Lagrangian and Quadratic Operator

The kinetic (free) Lagrangian is constructed to ensure gauge invariance and positivity of the physical spectrum. Its precise form for the rank-2 non-symmetric tensor is

$$K = -\frac{1}{4}F^a_{\mu\nu,\lambda} F^{a\,\mu\nu,\lambda} + \frac{1}{4}F^a_{\mu\nu,\lambda} F^{a\,\mu\lambda,\nu} + \frac{1}{4}F^a_{\mu\nu,\nu} F^{a\,\mu\lambda,\lambda}$$

The equations of motion in the source-free theory can be compactly expressed in momentum space as $H_{\mu\lambda,\nu\rho}(k)A^{a\,\nu\rho}(k) = 0$, where $H$ is a second-order operator whose algebraic structure allows a counting of propagating vs. gauge/pure-gauge modes. On-shell ($k^2=0$), only three physical polarizations survive: two with helicity $\pm 2$ and one with helicity $0$. All other modes correspond to gauge degrees of freedom or vanish in the free theory (0706.0762).

3. Interactions: Vertices and Non-Abelian Structure

Beyond the free level, the interactions are uniquely controlled by the non-Abelian gauge invariance. The total Lagrangian includes the canonical Yang–Mills term plus interaction terms specific to the rank-2 tensor sector: $$\mathcal{L}_{\text{total}} = \mathcal{L}_{\text{YM}} + g\mathcal{L}_2 + \mathrm{higher}\ \mathrm{order}$$ Both cubic and quartic vertices are nontrivially constrained; their detailed expressions include non-commuting internal indices and tensor contractions not present in gravity or Abelian B-field models. As a result, current exchange amplitudes for the rank-2 field in tree-level processes reveal that the physical spectrum remains unitary, with three independent polarization structures, and the propagator takes a form decomposable into projectors onto these modes.

This interaction profile is distinct: while the free field decomposes as in Abelian or gravitational cases, the presence of non-commuting charge sectors leads to interactions that cannot be mapped onto those of gravity or the Abelian Kalb–Ramond theory.

4. Generalized Bianchi Identity and Gauge Consistency

The consistency of the model is anchored in the generalized Bianchi identity for the field strength: $$\partial_\mu F^a_{\nu\lambda,\rho} + \partial_\nu F^a_{\lambda\mu,\rho} + \partial_\lambda F^a_{\mu\nu,\rho} = 0$$ This identity, connected to the Jacobi identities of the covariant derivative, ensures the invariance of the kinetic term and enforces additional conservation laws on currents coupled to the gauge field: $$\partial_\nu J^a_{\nu\lambda} = 0,\qquad \text{and analogous constraints in the second index}$$ Current conservation ensures unitarity and absence of negative norm (ghost) states at the interacting level. In the Bianchi sector, these identities are essential for eliminating higher-derivative anomalies in the equations of motion and for preserving gauge invariance under both spacetime and internal transformations.

5. Propagation and Mode Analysis

The second-order partial differential equation governing the dynamics is: $$\partial_\mu F^a_{\mu\nu,\lambda} - [\partial_\mu F^a_{\mu\lambda,\nu} + \partial_\lambda F^a_{\mu\nu,\mu} + \partial_\nu F^a_{\lambda\mu,\mu} + \eta_{\nu\lambda}\partial_\mu F^a_{\mu\rho,\rho}] = J^a_{\nu\lambda}$$ Upon Fourier transformation and matrix analysis, the rank and nullity of $H$ reveal three positive-norm physical modes. The symmetric part of the field reduces, on-shell, to the Pauli–Fierz gravity equation (helicity $\pm 2$), while the antisymmetric part satisfies an equation analogous to that of a massless Abelian B-field but with a single physical (helicity 0) state.

The mode content remains robust even when interactions are turned on, as the mixed-antisymmetric structure of the field together with non-Abelian gauge invariance enforces consistent elimination of unphysical degrees of freedom. The spectrum demonstrates that the specific splitting between symmetric and antisymmetric components yields a concrete counting of propagating modes, distinct from generic tensor or higher-spin models.

6. Model Summary and Implications

Sector	Field Content	Propagating Modes
Symmetric (helicity 2)	$A^a_{(\mu\lambda)}$	2 physical polarizations
Antisymmetric (helicity 0)	$A^a_{[\mu\lambda]}$	1 physical polarization
Gauge/pure gauge	Internal/other components	13 non-propagating modes

The interacting gauge model with mixed-antisymmetric fields provides a consistent generalization of Yang–Mills theory with a higher-rank tensor field, producing definite physical spectra and interaction terms that differ both from gravity and from the Abelian B-field despite superficial similarities in mode decomposition. The combination of a generalized Bianchi identity, unitary propagators, and nontrivial cubic and quartic interaction structure sets this framework apart from models built on symmetric or pure antisymmetric tensor fields.

The theoretical formulation is of direct interest for understanding the appearance of higher-rank gauge theories in string theory, the unification of gauge interactions with (higher-spin) charged tensor bosons, and for the development of consistent interacting gauge models that go beyond the familiar vector gauge boson paradigm (0706.0762). Importantly, the delicate balance of constraints ensures that the physically propagating sector is not spoiled by unphysical polarizations, making the model both a robust theoretical construct and a promising candidate for further exploration in extensions of the Standard Model and string-inspired effective field theories.