Generalized sigma model with dynamical antisymplectic potential and non-Abelian de Rham's differential

For topological sigma models, we propose that their local Lagragian density is allowed to depend non-linearly on the de Rham's"velocities"$D Z^{A}$. Then, by differentiating the Lagrangian density with respect to the latter de Rham's"velocities", we define a"dynamical"anti-symplectic potential, in terms of which a"dynamical"anti-symplectic metric is defined, as well. We define the local and the functional antibracket via the dynamical anti-symplectic metric. Finally, we show that the generalized action of the sigma model satisfies the functional master equation, as required.


Introduction
When formulating a topological sigma model, one proceeds usually with an anti-symplectic configuration space {Z A |ε(Z A ) =: ε A } whose anti-symplectic potential V A (Z), ε(V A ) =: ε A + 1, is given originally [1,2,3,4,5,6,7,8,9,10,11]. At the same time, it is assumed usually that the kinetic part of the original local Lagrangian density L has the form −(DZ A )V A (Z), where D is the de Rham's differential.
In the present paper, we generalize the local original Lagrangian density as to take the form L(Z, DZ) allowed to depend non-linearly on the de Rham's "velocities" DZ A . Then, we define a "dynamical" anti-symplectic potential, V A (Z, DZ) as the derivatives of the new Lagangian density with respect to the mentioned de Rham's "velocities". We define a local anti-symplectic metric in its covariant components E AB (Z, DZ) as the standard vorticity of the "dynamical" anti-symplectic potential V A (Z, DZ), in terms of explicit Z A derivatives.
We define both the local and the functional antibracket by the standard formulae via the "dynamical" anti-symplectic metric in its contravariant components E AB (Z, DZ). Finally, we show that the new action Σ =: dµL(Z, DZ) satisfies the functional master equation, provided the function S(Z, DZ) =: L(Z, DZ) + DZ A V A (Z, DZ) satisfies the local master equation.
2 Non-Abelian de Rham's differential Let Γ be an intrinsic configuration super-manifold, Γ =: {X a , C a |ε(X a ) = 0, ε(C a ) = 1, a = 1, ..., 2m}. (2.1) Let D be a non-Abelian de Rham's differential, as defined by the conditions 2) whose solution is sought for in the form, with Λ b a being invertible, and U d ab being antisymmetric in its subscripts a, b [6]. The conditions The Jacobi relation (2.5) provides for the integrability of the Maurer-Cartan equation (2.4). In terms of the Boson integration measure, the anti-Hermiticity of the differential D implies In turn, it follows from (2.7),

Generalized sigma model
Let Γ ′ be an antisymplectic phase space, with N being an equal number of Boson and Fermion variables among Z A . Let us define the local action of the generalized sigma model in the form, with the measure dµ(Γ) being defined in (2.6). A functional derivative of the action (3.2) has the form, in terms of explicit Z A -derivatives ∂ A . In turn, we have By inserting that into the relation (3.3), we get where . (3.8)

Functional and local master equations
Let the metric (3.6) be invertible, and E AB be its inverse, Let F (Z, DZ), G(Z, DZ) be two arbitrary local functions; their local antibracket is defined as In turn, let F [Z], G[Z] be two arbitrary functionals; their functional antibracket is defined as Due to the definition of the V A , the second in (3.3), together with the definition (3.6) of the E AB , each of the antibrackets, (4.2) and (4.3), satisfies its polarized Jacobi identity, 3 Now, we are in a position to show that the functional master equation is satisfied as for the action Σ[Z] (4.2), provided that the local master equation is satisfied as for the function S(Z, DZ) (3.7). Indeed, by inserting the functional derivative  7). Then, we get the non-polarized form of the functional Jacobi identity, In fact, the second equality in eq.(4.11) holds due to the polarized functional Jacobi identity (4.5) derived in Appendix A. The integrand in eq.(4.11) demonstrates explicitly the structure of the original terms characteristic for the non-polarized functional Jacobi identity. Among other formal properties involved, the simplest one is the nilpotency of the de Rham's differential D, the second in the conditions (2.2), as well as the appearance of the boundary terms of the form "D(anything)", and the local non-polarized Jacobi identity, ((S, S), S) = 0, (4.12) together with its (DZ) -dual identity, Finally, if we introduce the nilpotent functional odd Laplacian, then one can consider formally the functional quantum master equation It looks natural to seek for a solution to the quantum master action functional W in the form of a power series expansion in i , Then, to the i order, it follows from (4.16), Here in (4.18), the second term contains ill-defined functional factor, δ(Γ, Γ). It seems rather difficult to make sense to these expressions in a consistent way. Thus, the status of the functional quantum master equation (4.16) remains an open question. Perhaps, the most realistic idea is that the mentioned ill-defined factor might be canceled from the functional path integral over the field-antifield variables Z A (Γ).

Acknowledgments
The work of I. A. Batalin is supported in part by the RFBR grants 17-01-00429 and 17-02-00317. The work of P. M. Lavrov is supported in part by the RFBR grant 15-02-03594.

Appendix A
In this Appendix, we derive in detail the polarized Jacobi identity (4.5) as for the functional antibracket. First, we rewrite the relation (4.5) in the form 4 Now, the following relation holds The second term in the right-hand side in (A.2) equals to zero due to the second in (3.3). Thus, we get from (A.2), By expressing the right-hand side from (A.3) and then inserting into the left-hand side in (A.1), we observe that each term cancels its counterpart in the cyclic sum, so that the polarized Jacobi identity (4.5) is confirmed.
Notice that the nilpotency of the functional odd Laplacian (4.14) implies the polarized Jacobi identities (4.5) to hold, together with their consequence that the ∆ ′ is a differentiation as for the functional polarized antibracket, and the property [12],

Appendix B
In this Appendix B, we consider in short the local measure, In analogy with (3.2), (3.5), we have from (4.15), Due to the relation (B.2), the second term in (B.1) rewrites in the form which is a natural counterpart to the formula (4.8).

Appendix C
Here, we consider in short the Legendre transformation from the de Rham's "velocities" DZ A to new antifield variables Z * A . Provided that the Hessian matrix Thus, provided that the Hessian matrix (C.1) is invertible, we arrive at the usual situation although formulated within the phase space whose dimension is twice as more than the one of the original antisymplectic phase space (3.1).