Auxiliary Tensor Fields in Theoretical Physics

• Auxiliary tensor fields are tensor-valued variables used to reformulate complex theories by acting as algebraic or variational tools.
• They enable techniques like the Hubbard–Stratonovich transformation and duality realization, simplifying higher-derivative systems and emergent dynamics.
• These fields are pivotal across quantum gravity, conformal theories, and tensor network states, improving both theoretical clarity and computational efficiency.

An auxiliary tensor field is a tensor-valued variable introduced to reformulate, extend, or render tractable certain classes of physical theories, variational principles, or computational schemes. Auxiliary tensor fields are typically non-dynamical at the outset—meaning their action lacks kinetic terms—but can play indispensable roles in symmetry realization, duality manifestness, field elimination, renormalization, or decomposition of interacting operators. Their specific algebraic or variational properties facilitate either the simplification of higher-derivative systems, enforcement of duality invariances, generation of effective interactions through transformations such as the Hubbard–Stratonovich, or the encoding of symmetries and correlations in tensor network states.

1. Algebraic Auxiliary Tensor Fields in Gravity and Higher-Curvature Theories

Auxiliary tensor fields allow reformulation of higher-derivative gravity theories as systems of second-order equations. For quadratic gravity in $D$ dimensions, one introduces a rank-4 auxiliary field $\Phi_{\mu\nu\rho\sigma}$ carrying the symmetries of the Riemann tensor to rewrite an action with curvature-squared terms as

$$S_\Phi = \frac{1}{16\pi G}\!\int\!d^Dx\,\sqrt{-g} \,\left[ \Phi^{\mu\nu\rho\sigma} R_{\mu\nu\rho\sigma} + b_1\,\Phi^2 + b_2\, (\Phi^{\mu\nu})^2 + b_3\, (\Phi)^2 \right],$$

where $R_{\mu\nu\rho\sigma}$ and its traces are replaced by $\Phi$ via their on-shell algebraic relations. The boundary action is similarly generalized to ensure a well-posed variational principle, and black hole entropy corrections arising from higher curvature terms can be written explicitly in terms of $\Phi$ (Hirochi, 2013). In bigravity, taking the strong-coupling limit removes the kinetic term of the second metric $f_{\mu\nu}$, rendering it auxiliary and non-propagating. The resulting algebraic field equations for $f_{\mu\nu}$ can be solved (possibly with ambiguities) and eliminated, yielding Einstein gravity with a cosmological constant dictated by either the action parameters or integration constants associated with undetermined components of $f_{\mu\nu}$ (Bañados et al., 2013).

2. Duality and Self-Duality: Bispinor and Antisymmetric Tensor Auxiliary Fields

For theories with electromagnetic duality, introducing auxiliary bispinor or antisymmetric tensor fields enables manifest off-shell realization of $O(2)$ or $Sp(2,\mathbb{R})$ dualities. In the Ivanov–Zupnik formalism, U(1)-duality-invariant electrodynamics can be written as

$$L(F,V) = L_2(F,V) + \mathcal{E}(a), \qquad a = v v^*, \quad v = V_{\alpha\beta} V^{\alpha\beta},$$

where the kinetic term is quadratic in $F$ and $V$, and all nonlinearities enter via an $O(2)$-invariant function $\mathcal{E}$. The equations of motion for $V_{\alpha\beta}$ reproduce the nonlinear twisted self-duality constraints, and integrating out $V$ yields the desired duality-invariant nonlinearity (Ivanov et al., 2012, Ivanov et al., 2014). This construction generalizes to Sp(2, R) duality by promoting the auxiliary field to transform appropriately under the duality group and extending $\mathcal{E}$ to Sp(2, R)-invariant functions (Ivanov et al., 2014). The hybrid PST–auxiliary framework integrates the Pasti–Sorokin–Tonin gauge symmetry approach with manifest off-shell duality-covariance by inclusion of an antisymmetric auxiliary tensor, allowing for arbitrary duality-invariant potentials in $V$ without modification of the PST symmetries (Ivanov et al., 2014).

3. Auxiliary Tensor Fields in Quantum Gravity and Composite Emergence

In composite models of gravity, the auxiliary tensor field $H_{\mu\nu}$ is introduced via a Hubbard–Stratonovich transformation to linearize quartic self-interactions of the energy-momentum tensor. Beginning with actions of the form

$$S_{\text{comp}} = \int d^4x \left[ \text{kinetic}_{\text{matter}} - f_2 T_{\mu\nu} T^{\mu\nu} - f_0 (T^\rho{}_\rho)^2 \right],$$

the transformation introduces an auxiliary field coupled bilinearly to matter, plus a mass term for $H_{\mu\nu}$. Renormalization group flow then generates a kinetic term for $H_{\mu\nu}$ in the infrared, converting it into a dynamical spin-2 field with propagating degrees of freedom—a mechanism for emergent gravity wherein the effective kinetic operator matches that of a gauge-fixed graviton fluctuation (Maitiniyazi et al., 9 Dec 2025).

4. Auxiliary Tensor and Mixed-Symmetry Fields in Conformal Gravity

In four-dimensional conformal gravity, auxiliary tensor fields correspond to unitary irreducible representations (UIRs) of the de Sitter group that do not contract to the physical Poincaré representation in the flat limit. Specifically, the auxiliary field in dS space transforms according to the UIR $\Pi_{2,1}$ (for rank-2) or $II_{2,1}$ (for mixed-symmetry rank-3), characterized by distinct eigenvalues of the Casimir operator. These fields arise in the group-theoretical formulation via manifestly conformally invariant field equations, e.g.,

$$(Q_2 + 4) K_{AB}(x) = 0,\qquad (Q_3 + 4) F_{ABC}(x) = 0,$$

with appropriate tracelessness, transversality, and divergencelessness constraints. The solutions are expressed in terms of conformally coupled scalar fields and polarization tensors or projectors, and encode degrees of freedom essential for the construction of conformal-invariant gravity. The associated de Sitter-invariant two-point functions can be written as products of polarization tensors/operators and the scalar propagator, and do not possess flat-space analogues—serving as non-physical but group-theoretically crucial fields (Pejhan et al., 2011, Elmizadeh, 2015).

5. Auxiliary Tensor Fields in Quantum Many-Body and Tensor Network States

In the context of quantum many-body systems and quantum field theory, auxiliary tensor fields emerge in the description of continuum tensor network states, such as continuous matrix product states (cMPS) and projected entangled pair states (PEPS). Here, the contraction of a physical state's indices is replaced by the path integration over auxiliary degrees of freedom living in a lower-dimensional "virtual" field system. The physical state's symmetries correspond to Lorentz or Euclidean invariances of the auxiliary field dynamics. This auxiliary structure allows the encoding of area-law entanglement scaling and ensures the completeness of the representable state space as the auxiliary bond dimension increases (Jennings et al., 2012).

6. Auxiliary Tensor Fields in Quantum Monte Carlo Algorithms

In ab initio electronic structure calculations, auxiliary tensor fields are crucial to the auxiliary-field quantum Monte Carlo (AFQMC) method. Here, the electron repulsion integrals are decomposed via tensor factorization (e.g., Cholesky factorization), and the two-body interaction is recast using a sum-of-squares representation: $$H_2 = \frac{1}{2} \sum_{\gamma,\mu} (v_{\gamma\mu})^2,$$ where the $v_{\gamma\mu}$ are one-body operators constructed from low-rank tensor contractions. The Hubbard–Stratonovich transformation introduces auxiliary fields $x_{\gamma\mu}$, sampling over which stochastically simulates quantum propagation. Nested low-rank factorizations reduce computational scaling from quartic to sub-quartic or even cubic in system size, with auxiliary tensor fields at the heart of the scheme's efficiency and flexibility (Motta et al., 2018).

7. Summary and Cross-Disciplinary Significance

Auxiliary tensor fields serve as versatile mathematical and physical tools across a spectrum of contexts: as algebraic devices enabling second-order reformulation of higher-derivative actions and the extraction of boundary dynamics; as off-shell vessels for encoding and manifesting duality symmetries and self-duality constraints; as transformational instruments for generating emergent dynamical fields via renormalization; as non-Poincaré representations facilitating the group-theoretical formulation of conformal or higher-spin field theories; as virtual degrees of freedom organizing quantum entanglement scaling and symmetry properties in the variational classes of many-body states; and as computational intermediates that render otherwise intractable many-electron simulations practical.

Archetypal Roles of Auxiliary Tensor Fields

Context	Auxiliary Field Type	Function
Quadratic gravity, curvature-squared	Rank-4 Riemann-type $\Phi$	Algebraic reduction, well-posed variational principle
Duality-invariant theories	Bispinor/antisymmetric $V$	Manifest duality, self-duality constraint
Composite/emergent gravity	Rank-2 $H_{\mu\nu}$	Dynamical field emergence, RG induced kinetic term
Conformal gravity in dS	Rank-2 or rank-3, UIR fields	Representation-theoretic, conformal invariance
Quantum tensor networks	Virtual auxiliary tensors	State construction, symmetry implementation
AFQMC (many-body computation)	Low-rank auxiliary tensors	Operator decomposition, stochastic sampling

Auxiliary tensor fields thus form a unifying conceptual and computational framework relevant to quantum field theory, gravitation, many-body physics, and computational quantum chemistry, reflecting symmetry, redundancy, and emergence in field-theoretic, algebraic, and algorithmic architectures (Hirochi, 2013, Bañados et al., 2013, Ivanov et al., 2012, Ivanov et al., 2014, Ivanov et al., 2014, Maitiniyazi et al., 9 Dec 2025, Pejhan et al., 2011, Elmizadeh, 2015, Jennings et al., 2012, Motta et al., 2018).