Off-shell double copy theories in BV

Technical Summary: Off-shell Double Copy Theories in BV

Problem Statement
The double-copy construction, which generates gravity scattering amplitudes from gauge-theory building blocks via color-kinematics duality, is well-established on-shell. However, extending this duality off-shell to construct local actions for gravity theories remains challenging. While off-shell color-kinematics duality is known to arise from kinematic Lie algebras in specific theories (such as Chern-Simons, 4d BF, and 2d Yang-Mills), a systematic framework for constructing off-shell double-copy actions within the Batalin-Vilkovisky (BV) formalism has been lacking. Furthermore, the geometric interpretation of these resulting double-copy actions, particularly their relation to gravity theories like Kodaira-Spencer gravity, requires formalization.

Methodology
The authors utilize the BV formalism to construct double-copy actions for gauge theories exhibiting off-shell color-kinematics duality. The core methodology relies on identifying a kinematic Lie algebra generated by a second-order operator, $D^\dagger$, acting on a differential graded Lie algebra (dgLa) of fields.

1. Kinematic Algebra Construction: The authors identify a kinematic Lie algebra $K = \text{Im}(D^\dagger)$ where the bracket is defined by the second-order operator $D^\dagger$. This operator satisfies $(D^\dagger)^2 = 0$ and the second-order property (generalized Leibniz rule), ensuring the Jacobi identity holds for the induced bracket.
2. BV Action Formulation: They construct a BV action of the form:
   $$S = \frac{1}{2}\langle A, DA \rangle + \frac{1}{6}\langle A, \{A, A\} \rangle$$
   where $A$ is a superfield, $D$ is a differential operator, and $\{A, A\} = D^\dagger(A^2) - D^\dagger(A)A - (-1)^{|A|}AD^\dagger(A)$ is the bracket restricted to the image of $D^\dagger$. The pairing $\langle \cdot, \cdot \rangle$ is an invariant symplectic form compatible with the operators.
3. Double Copy via Dimension Doubling: To construct the double copy, the authors double the spacetime coordinates (or odd variables), introducing a second set of operators $(\bar{D}, \bar{D}^\dagger)$. The double-copy action is formulated on a space with two sets of odd coordinates, satisfying a generalized Kähler-like identity structure where $[D, \bar{D}^\dagger] = \square$ (the Laplacian).
4. Gauge Fixing and Locality: The construction involves gauge-fixing conditions (e.g., $A \in \text{Im}(D^\dagger)$). The authors demonstrate that while the general formalism often yields non-local actions (due to the inverse Laplacian $\square^{-1}$), specific choices of the kinematic algebra (notably in 4d BF and 2d YM theories) allow for algebraic solutions to the gauge-fixing constraints, resulting in local actions.

Key Contributions and Results

• BV Formulation of CS Double Copy: The authors provide a complete BV formulation for the double copy of Chern-Simons (CS) theory. They show that the previously proposed gauge-fixed action (from Ref. [9]) is the gauge-fixed version of this BV construction. The physical fields of this double copy are identified with deformations of a flat metric, satisfying equations equivalent to the Maurer-Cartan equation for the deformation of a flat metric with a fixed volume form.
• Formal CS and 4d BF Theory: The construction is generalized to formal CS theory and applied to 4d BF theory. A significant finding is that 4d BF theory admits a one-parameter family of kinematic algebras (parameterized by $\lambda$).
  • For $\lambda = 0$, the double copy yields a non-local action similar to the CS double copy.
  • For $\lambda \neq 0$, the gauge-fixing condition can be solved algebraically, yielding a local double-copy action. This local action resembles a deformed BF theory with an infinite-dimensional kinematic algebra acting on spacetime itself.
• 2d Yang-Mills Double Copy: The authors construct the double copy for 2d Yang-Mills theory, resulting in a local action that resembles a deformed BF theory. The physical sector involves fields interpreted as one-forms and zero-forms, with a specific Lie bracket structure derived from the gauge-fixing operator.
• Kodaira-Spencer and Kähler Gravity: The paper demonstrates that Kodaira-Spencer (KS) gravity and Kähler gravity fit naturally into this BV double-copy framework.
  • KS Gravity: The BV action for KS gravity (describing deformations of complex structures) is shown to be a specific instance of the general double-copy construction. The authors extend this to generalized complex structures and to higher dimensions (e.g., 8D), requiring additional odd coordinates to maintain the correct bi-degree of the integration measure.
  • Kähler Gravity: A 6D Kähler gravity theory is constructed using linear combinations of operators ($\partial, \bar{\partial}, \partial^\dagger, \bar{\partial}^\dagger$) satisfying the required algebraic relations.
• Dual BRST Symmetries: A structural feature of these gauge-fixed double-copy actions is the emergence of two anti-commuting BRST symmetries. This arises because the gauge-fixed action can be viewed as the result of applying different gauge-fixing conditions to two distinct BV theories (one defined by $D^\dagger$ and the other by $\bar{D}^\dagger$). The authors suggest this dual BRST structure may be a defining property of off-shell double-copy theories.

Significance and Claims
The paper claims to formalize the construction of off-shell double-copy theories within the BV framework, providing a unified algebraic description that encompasses Chern-Simons, BF, and Yang-Mills theories. By linking these constructions to Kodaira-Spencer gravity and deformations of generalized complex structures, the work offers a potential geometric interpretation for the double copy beyond standard scattering amplitudes.

The authors modestly note that while the framework successfully produces actions for specific theories (particularly those involving 4d BF or 2d YM which yield local actions), the general case often results in non-local actions. They highlight an open question regarding the relationship between the different kinematic algebras found in 4d BF theory (the $\lambda=0$ vs. $\lambda \neq 0$ cases), conjecturing they may be related by a non-local field redefinition. Furthermore, they acknowledge that constructing double copies without off-shell color-kinematics duality (i.e., without a second-order $D^\dagger$) remains an open challenge, as the current formalism relies heavily on the existence of such an operator to define the kinematic Lie bracket.