A systematic study of finite BRST-BV transformations within W–X formulation of the standard and the Sp(2)-extended field–antifield formalism

Finite BRST-BV transformations are studied systematically within the W–X formulation of the standard and the Sp(2)-extended field–antifield formalism. The finite BRST-BV transformations are introduced by formulating a new version of the Lie equations. The corresponding finite change of the gauge-fixing master action X and the corresponding Ward identity are derived.

In the present paper we study systematically finite BRST-BV transformations within the W -X formulation both in the standard and Sp(2)-extended field-antifield formalism. We introduce these transformations by formulating the respective Lie equations. Among other things, we derive in this way the effective change in the gauge-fixing master action X , as induced by the finite BRST-BV transformation defined. a e-mail: batalin@lpi.ru b e-mail: bering@physics.muni.cz c e-mail: lavrov@tspu.edu.ru

W -X formulation to the standard field-antifield formalism
Let z A be the complete set of the variables necessary within the standard field-antifield formalism whose Grassmann parities are ε(z A ) = {ε α ; ε α + 1}. (2.2) We denote the respective z A -derivatives as Let Z be the partition function Z = Dz Dλ exp ih W + X , (2.4) where λ α are Lagrange multipliers for gauge-fixing with the Grassmann parity ε(λ α ) = ε α + 1. (2.5) In the partition function (2.4), the dynamical gauge-generating master action W and the gauge-fixing master action X are defined so as to satisfy the respective quantum master equations, In the above quantum master equations (2.6) and (2.7), and ( , ) are the standard nilpotent odd Laplacian 8) and the standard antibracket (2.9) respectively. Equations (2.8) and (2.9) tell us that the anticanonical pairs ( α ; * α ) serve as Darboux coordinates on the flat field-antifield phase space with measure density ρ = 1 and no odd scalar curvature ν ρ = 0. Ath = 0, * α = 0, the W -action coincides with the original action of the theory. As to the X -action, it can be chosen in the form related to the gauge-fixing fermion ( ), is a nilpotent fermionic differential that acts from the right.
In the integrand of the path integral (2.4), consider now the following infinitesimal BRST-BV transformation: where we have defined for later convenience and where μ(z) is an infinitesimal fermionic function with ε(μ) = 1. The Jacobian of the infinitesimal BRST-BV transformation (2.12) has the form (2.14) The complete action in the partition function (2.4) transforms as where σ (X ) is a quantum BRST generator to the X -action in the integrand of the path integral (2.4). We conclude that the partition function (2.4) and the quantum master equation (2.7) for X are both stable under the infinitesimal variation (2.18). Next let t be a bosonic parameter. It is natural to define a one-parameter subgroup t → z A (t) of finite BRST-BV transformations by the Lie equation 1 is the corresponding vector field with components Recall that the antibracket for any fermion F = yμ with itself is zero: (F, F) = 0. This fact yields a conservation law so that the following invariance property holds: The Jacobian of these transformations satisfies the following equation: (2.26) The transformed complete action satisfies the equation (2.27) Due to the transformed master equations (2.6) and (2.7), it follows that where we have defined for later convenience a := 2(σ (X )μ). (2.29) where we have defined the average Here E is the function We have the following Cauchy initial value problem: for arbitrary t. Thereby, we have confirmed that the quantum master equation (2.7) is stable under the finite BRST-BV transformation generated by Eq. (2.19). Of course, the general expression (2.4) itself is stable under the same transformation, as well.
At this point we would like to investigate the quantum master equation where we have denoted the new gauge-fixing master action, (2.37) The exponential exp{A} rewrites in the form where we have defined the first-order operator and we used the formula for Also, there is a potential obstacle that the dynamical master action W actually enters that equation. Thus, the finite parameter μ(z) being restricted in its field-dependence, that circumstance would be a crucial specific feature of the W -X formulation.
One can proceed from a solution A to the quantum master equation (2.37 with ε being a boson parameter, and then expands μ and A in formal power series,

43)
one gets to the first order in ε so that μ 0 remains arbitrary to that order. However, to the second order in ε, one has so that μ 1 remains arbitrary to that order, while (2.49) restricts μ 0 , with H (μ 0 ) being the operator (2.39) as taken at μ = μ 0 .
To the third order in ε, μ 2 remains arbitrary, while μ 1 is restricted to satisfy the condition (2.51) The same situation holds to higher orders in ε: to each subsequent order, the respective coefficient in μ remains arbitrary, while the preceding coefficient in μ becomes restricted. Of course, it looks rather difficult to estimate on being such a strange procedure "convergent" to infinite order in ε.
with O being any operator. Finally we present a simple general argument, based on the integration by parts, that the partition function (2.4) is independent of finite arbitrariness in a solution to the gaugefixing master action X , with X standing for X reduces to the case of the initial X standing for itself. Thus, the partition function is independent of a particular representative of the class (2.54).

Ward identities in the standard W -X formulation
Let J A be external sources to the variables z A ; then the integral (2.4) generalizes to the generating functional, Arbitrary variation δz A yields the equations of motion, where . . . J is the source-dependent mean value It follows from Eq. (3.2) that where is the fundamental invertible antibracket. In Eq. (3.1), the BRST-BV variation (2.12) yields Thus we have eliminated the average (2.31) of the gaugefixing master action X from the new Ward identity (3.7). The price is that we have got the non-homogeneity quadratic in the external sources J in the right-hand side in Eq. (3.7). Finally, at the level of finite BRST-BV transformations, Eq. (2.30) yields However, it is impossible to eliminate the average (2.31) of the gauge-fixing master action X from (3.8).

W -X formulation to the Sp(2)-symmetric field-antifield formalism
Let z A be the complete set of the variables necessary to the W -X formulation of the Sp(2)-symmetric field-antifield formalism [15,17,18] z A = { α , π αa ; * αa , * * α } (4.1) whose Grassmann parities are We denote the respective z A derivatives as Let Z be the partition function: where λ α are Lagrange multipliers for gauge-fixing with Grassmann parities, In the partition function (4.4), the dynamical gauge-generating master action W and the gauge-fixing master action X is defined to satisfy the respective quantum master equation In the above quantum master equations (4.6) and (4.7), the a , ( , ) a , V a , and a ± are the Sp(2)-vector-valued odd Laplacian the antibracket (4.9) and the special vector field and a ± := a ± ih V a , (4.11) respectively.
For the W -action, one should require that W is independent of π αa , ( * * α , W ) = 0. (4.12) As to the X -action, it can be chosen in the form related to the gauge-fixing boson F( ), is a Sp(2)-vector-valued fermionic differential that acts from the right.
In the integrand of the path integral (4.4), consider now the following infinitesimal BRST transformation: where we have defined for later convenience 16) and where μ a = μ a (z) is an infinitesimal Sp(2) co-vectorvalued fermionic function. Its Jacobian has the form The complete action in the partition function (4.4) transforms as It follows from Eqs. (4.17) and (4.18) that where is the Sp(2)-vector-valued quantum BRST generator. Equation (4.19) tells us that the BRST transformation (4.15) induces the following variation: . (4.21) to the X -action in the integrand of the path integral (4.4). We conclude that the partition function (4.4) and the quantum master equation (4.7) for X are both stable under the infinitesimal variation (4.21).
Next let t be a bosonic parameter. It is natural to define a one-parameter subgroup t → z A (t) of finite BRST transformations by the Lie equation where is the corresponding vector field with components which cannot be completely integrated explicitly to yield a counterpart to the conservation law (2.25). The Jacobian of the transformation (4.22) satisfies the equation The complete action in Eq. (4.4) satisfies the equation We have the following Cauchy initial value problem: for arbitrary t. Thereby, we have confirmed that the quantum master equation (4.7) is stable under the finite BRST-BV transformation generated by Eq. (4.22). Of course, the general expression (4.4) itself is stable under the same transformations, as well.

Ward identities in the Sp(2)-extended W -X formulation
Let J A be external sources to the variables z A ; then the integral (4.4) generalizes to the generating functional An arbitrary variation δz A yields the equations of motion, where . . . J is the source-dependent mean-value It follows from Eq. (5.2) that is the fundamental Sp(2) antibracket. In Eq. (5.1), the BRST-BV variation (4.15) yields (5.7) Thus we have eliminated the average (4.31) of the gaugefixing master action X from the new Ward identity (5.7). The price is that we have got the non-homogeneity quadratic in the external sources J in the right-hand side in Eq. (5.7).
Finally, at the level of finite BRST-BV transformations, the relation (4.30) yields However, it is impossible to eliminate the average (4.31) of the gauge-fixing master action X from Eq. (5.8).

Conclusions
Notice that, on one hand (and in contrast to the original Sp(2)-formulation [15,17,18]), in the Sp(2)-symmetric W -X formulation, the anticanonical dynamical activity of the variables {π αa , * * α } [22], as represented by the second term in Eq. (4.8) and in the square bracket of Eq. (4.9), is of crucial importance to satisfy the quantum master equation (4.7) with the anzatz (4.13) for X .
On the other hand, π αa and * * α are kept as dynamically passive (antibracket-commuting) variables in the W -action. Thus, one may realize what the price is of the coexistence of the Sp(2)-symmetry and the complementary W -X duality of the quantum master equations (4.6) and (4.7).
In this appendix we present the general formal algebra of the σ -operators, both in the standard and the Sp(2) case.
In the standard case we introduce the σ -operator and where the quantum master equations for W and X are used.
In the Sp(2) case the set of operators σ a (F), σ a ± (F) for any bosonic functional F is introduced (A.6) The Sp(2) nilpotency reads where