In this section, we will derive the infinitesimal cocycle condition (5.8) within the BV formalism. The crucial insight is that the infinitesimal renormalization group transformation \(\Delta X\), applied to a local functional F, can in fact be identified with the renormalized BV Laplacian \(\triangle _F\) for the interaction F, applied to the vector field \(\partial _X\),

(6.1)

where \(\partial _X\) is the vector field on \(}}(M,}}^n)\) induced by X – see [6]. We rederive this result below in (6.24) and (6.27). The operator \(\triangle _F\) can be expressed by means of a generalization of the AMWI (see (6.23)), so the derivation of the anomaly consistency condition (4.5) given in the current section is essentially equivalent to the previous derivation in Sect. 4. Phrasing it in terms of the BV language, however, is important for showcasing the underlying algebraic structures naturally associated with the space of multivector fields and allows us to make connection with the literature, in particular [5, 9].

The BV formalism gives an interpretation of the space of functionals on the solutions to equations of motion in the language of homological algebra. It also allows one to reformulate the deformation of the pointwise product of functionals into their time-ordered product as a deformation of a certain differential. We will review the main results concerning the application of this formalism in pAQFT, while presenting it in a slightly different way, emphasizing the geometric interpretation.

We start by observing that functions and vector fields on \(}}\) can be considered as elements of a larger algebra, namely the graded commutative algebra \(\textrm(M)\) of multivector fields. This space has an interpretation as a space of functions on a graded manifold, namely the \((-1)\)-shifted cotangent bundle \(T^*[-1]}}(M,}}^n)\) over the configuration space. The latter is just identified with \(}}}}(M)\equiv }}(M,}}^n)\oplus }}_}(M,}}^n)[-1]\), where the number \(-1\) in square brackets indicates that elements of this space are to be seen as odd variables of degree \(-1\).

More concretely, we identify \(\frac\) with degree \(-1\) generators \(\Phi ^\), called antifields, so

$$\begin \Phi _r^(x)[dF[\phi ]]=\frac[\phi ]\,\ F\in }}(M)\ \end$$

(6.2)

and the elements \(}}\in \textrm(M)\) are of the form

$$\begin }}=\sum _\langle f_,\Phi ^\otimes (\Phi ^)^\rangle \end$$

(6.3)

where the compactly supported distributions \(f_\) are symmetric in the first n and antisymmetric in the last m arguments. If \(f_=0\) for \(m\ne 1\), the element \(}}\) can be identified with a vector field on \(}}(M,}}^n)\). The wave front set conditions on f are the same as for the distributions characterizing elements of \(}}(M)\). Analogously as for the functionals on the original configuration space, we introduce the spaces \(\textrm_}(M)\), \(\textrm_}(M)\) and \(\textrm_}(M)\).

The algebra \(\textrm_}(M)\) is equipped with a graded Poisson bracket, the Schouten bracket, also known as antibracket. For a functional \(F\in }}_}(M)\) and a vector field \(}}\in \textrm_}(M)\), it is given by the action of the vector field on the functional as a derivation: \(\}},F\}\doteq }}F\). For two vector fields, we have \(\}},}}\}=[}},}}]\), i.e. , the Lie bracket of vector fields, and for general elements \(}},}}\in \textrm_}(M)\), we invoke the graded Leibniz rule.

In this notation, the antibracket takes the form:

$$\begin \}},}}\}=\left\langle \frac}}},\frac}}}}\right\rangle -\left\langle \frac}}}},\frac}}}\right\rangle \end$$

(6.4)

where \(\delta ^r\) and \(\delta ^l\) signify right and left derivatives, respectively. We will use the convention that if no superscript is present, then the derivative is to be understood as the left derivative (see [6] for more detail).

The physical information about the equations of motion and symmetries of the classical theory with Lagrangian \(}+}\) (both now map into \(\textrm_}(M)\)), is encoded in the classical BV operator. For a compactly supported multivector field \(}}\in \textrm_}(M)\), we define

$$\begin s_}}} }}\doteq \}},}}(f)+}}(f)\}, \end$$

(6.5)

where \(f\equiv 1\) on \( \, }}}\). To simplify the notation, we will often just write \(\}},}}+}}\}\), unless we want to indicate a particular choice of the test function. To avoid signs, in what follows we will assume our Lagrangians and observables to be of the form \(}}=\sum \eta _i}}_i\), \(}}_i\in \textrm_}(M)\) and \(\eta _i\) elements of a multiplier Grassmann algebra such that \(}}\) is even.Footnote 10

In the absence of interaction, we consider the free classical BV operator:

$$\begin s_0 }}\doteq \}},}}\}, \end$$

(6.6)

We say that the theory satisfies the classical master equation (CME) if

$$\begin \}}(f)+}}(f),}}(f)+}}(f)\}=0. \end$$

In order to ensure that this equation holds exactly (rather than up to terms supported within the support of df), it might be necessary to make some particular choices of the smearing function f (see, e.g., [35]). As a consequence of this equation and the graded Jacobi identity for the Schouten bracket, \(s_}}}\) is a differential,

$$\begin (s_}}})^2(}})=\}},}}+}}\},}}+}}\}=\frac\}},\}}+}},}}+}}\}\}=0\, \end$$

(6.7)

The space of on-shell functionals is encoded in the 0-th cohomology of the differential \(s_}}}\), and the first cohomology gives the space of nontrivial (i.e., not vanishing on shell) symmetries.

Quantizing the theory corresponds to the deformation of the BV differential. For the free theory, we define

$$\begin }_0\doteq T^\circ s_0\circ T \end$$

(6.8)

where the time-ordering operator T is extended to multilocal functionals of fields and antifields, \(}}\in \textrm_}(M)\), by treating antifields as classical sources. (They are not affected by T and the \(\star \) product defined in (2.3) is extended to antifields as the pointwise product.) Since T is linear and \(s_0\) is linear and nilpotent, \(}_0\) is linear and nilpotent.

Operator \(}_0\) has a particularly nice expression on the space of polynomials of linear local functionals. These are called regular functionals \(\textrm_}(M)\subset \textrm_}(M)\) and their functional derivatives at all orders at all points are smooth. We compute

$$\begin }_0(}})=(s_0-i\triangle )(}}) \end$$

(6.9)

with the BV Laplacian

$$\begin \triangle =\int _M\frac(x)}. \end$$

(6.10)

Following [6], we define the renormalized BV Laplacian in the absence of interaction by

$$\begin \triangle _0\doteq i(}_0-s_0). \end$$

(6.11)

In the presence of an interaction \(}}\in \textrm_}(M)\), there are two natural ways to define the quantum BV operator. Assume first that \(}},}}\in \textrm_}(M)\).Footnote 11 On the one hand, we define

$$\begin }_}}}(}})\doteq e^}}}}_0(e^}}}}})\, \end$$

(6.12)

On the other hand, we set:

$$\begin \tilde_}}}(}})\doteq R_}}}^\circ s_0\circ R_}}} (}})\, \end$$

(6.13)

where

$$\begin R_}}}(}})\doteq (Te^}}})^\star T(e^}}}}})\, \end$$

is the retarded Møller operator, which maps functionals to interacting quantum observables. This differential is more natural than \(}_}}}\), since it is the obvious deformation of \(}_0\) when passing from the free quantum theory with the star product \(\star \) to the interacting quantum theory with the star product on \(\textrm_}(M)[[\hbar ]]\), defined by:

$$\begin }}\star _}}}}}\doteq R^_}}}(R_}}}}}\star R_}}}}}). \end$$

Both operators are nilpotent and \(\tilde_}}}\) is in addition a derivation with respect to \(\star _}}}\) (since \(s_0\) is a derivation w.r.t. \(\star \)). A short calculation, relying on the fact that \(s_0\) is a derivation with respect to \(\star \), shows that the relation between the two operators is given by:

$$\begin \tilde_}}}(}})&=R_}}}^\circ s_0((Te^}}})^\star T(e^}}}}}))\\&=R_}}}^\bigl (s_0((Te^}}})^)\star T(e^}}}}})+(Te^}}})^\star s_0(T(e^}}}}}))\bigr )\\&=R_}}}^\bigl (-(Te^}}})^\star s_0(Te^}}})\star (Te^}}})^\star T(e^}}}}})\\&\quad +(Te^}}})^\star T(e^}}}}_}}}(}}))\bigr )\\&=R_}}}^\bigl (- R_}}}\circ }_}}}(1)\star R_}}}(}})+R_}}}(}_}}(}}))\bigr )\\&=}_}}}(}})-}_}}}(1)\star _}}} }}\,, \end$$

so they coincide if \(}_}}}(1)=0\), which can also be expressed as

$$\begin s_0(Te^}}})=0, \end$$

(6.14)

and is the condition that the formal S-matrix is invariant under \(s_0\) [6]. It is equivalent to

$$\begin \}},}}\}+\frac\}},}}\}-i\triangle (}})=0. \end$$

(6.15)

This equation follows if \(}}\) and \(}}+}}\) satisfy the usual quantum master equation (QME) [36], i.e.,

$$\begin \frac\}}+}},}}+}}\}-i \triangle (}}+}})=0,\quad \frac\}},}}\}-i \triangle (}})=0. \end$$

(6.16)

Typically, we assume that \(}}\) does not depend on antifields, so the latter condition is trivially satisfied. In the following, we will assume that \(}}\) satisfies both the QME and the CME.

Now we generalize the above discussion to the situation where \(}}\) is local, which amounts to renormalization. The effects of renormalization can be understood by the AMWI which was extended to local multivector fields by Hollands and takes the form [5, Prop. 3]:

$$\begin s_0(Te^}}})=i T\Bigl ( \bigl (\tfrac\}}+}},}}+}}\}+A(}})\bigr )e^}}}\Bigr ), \end$$

(6.17)

where A characterizes the anomalies and replaces the ill-defined BV Laplacian \(-i\triangle \) in equation (6.15). (One assumes that \(}}\) satisfies the CME.) It is of the form

$$\begin A(}})=\sum _^\fracA_(}}^n)\, \ }}\in \textrm_}(M) \text , \end$$

(6.18)

where \(A_:\textrm_}(M)\rightarrow \textrm_}(M)\) are linear maps, which reduce the antifield number by 1; hence, \(A(}})\) is odd. In particular, for \(F\in }}_}(M)\) we see that \(A(F)=0\).

The renormalized quantum BV operator \(}_}}}\) is still given by (6.12), so that the generalized AMWI (6.17) can equivalently be written as

$$\begin }_}}(e^}}})=i\,e^}}}\left( \frac\}}+}}+}},}}+}}+}}\}+ A(}}+}})\right) . \end$$

(6.19)

We introduce the interaction-dependent BV Laplacian by

$$\begin \triangle _}}} \doteq i(}_}}}-s_}}}). \end$$

(6.20)

On regular functionals, \(\triangle _}}}=\triangle _0=\triangle \), but due to renormalization, the operators differ in general. Since \(}_0\) is linear, also \(}_}}}\) and \(\triangle _}}}\) are linear. From (6.7) and (6.8), we immediately see that \((}}_0)^2=0\); hence, by (6.12), also \((}}_}}})^2=0\).

The renormalized version of the QME is again (6.14). By using (6.17), it is equivalent to

$$\begin -\triangle _}}}(1)=\frac\}}+}},}}+}}\}+A(}})=0. \end$$

(6.21)

For \(}}\) satisfying QME (6.21), the AMWI (6.19) simplifies to

$$\begin }_}}(e^}}})=i\,e^}}}\left( \}},}}+}}\}+\frac\}},}}\}+ A(}}+}})-A(}})\right) .\qquad \end$$

(6.22)

The relation between A and \(\triangle _}}}\) is obtained from (6.20) and (6.22) by using that \(s_}}}\) is a derivation:

$$\begin -i\triangle _}}}(e^}}})=i\,e^}}}\left( \frac\}},}}\}+ A(}}+}})- A(}})\right) . \end$$

(6.23)

Taking into account that \(\triangle _}}}\) is linear, this formula implies

$$\begin \triangle _}}}(}})=-i\frac\Big \vert _\triangle _}}}(e^}}})=i\langle A'(}}),}}\rangle \, \end$$

(6.24)

for \(}}\) satisfying QME.

Note that for \(}}=\partial _X\eta \) (where \(\eta \) is Grassmann odd in order that \(}}\) is even) the original AMWI (2.27) (or (2.25)) is obtained from the generalized AMWI (6.22) as the coefficient of \(\eta \). Namely, we get

$$\begin T\bigl (e^}_F(\partial _X)\bigr )[\phi ]=s_0\circ T\bigl (e^\partial _X\bigr )[\phi ]=T\bigl (e^\partial _X\langle \Phi ,q\rangle \bigr )[\phi ] \end$$

(6.25)

with \(F\in }}_\textrm\), L used in the definition of the free theory not depending on antifields and \(\phi \) satisfying \(q=\frac[\phi ]\). This is similar to the result of [6], with the difference that here we introduced the external source q. In the last formula on the right-hand side, q can be pulled out from under the time-ordering operator, and after one sets \(q=\frac[\phi ]\), one obtains the same relation between \(}_F\) and \(s_0\) as in [6].

Now, applying AMWI (2.25) to the right-hand side, we obtain

$$\begin T\bigl (e^}_F(\partial _X)\bigr )[\phi ]= T\Bigl (e^\bigl (\partial _X(L+F)-\Delta X(F)\bigr )\Bigl )[\phi ]. \end$$

(6.26)

Since \(A(F)=0\), this coincides with the corresponding term for the right-hand side of (6.22), i.e., \(\tfrac\vert _T\bigl (e^}})}\bigl (\lambda \}},L+F\}+\frac\}},}}\}+A(F+\lambda }})\bigr )\bigr )\), with the identification \(\langle A'(F),\partial _X\eta \rangle =-\Delta X(F)\,\eta \), that is,

$$\begin \Delta X(F)=-\frac\Big \vert _A(F+\partial _X\eta )\doteq -\langle A'(F),\partial _X\rangle \, \end$$

(6.27)

hence by (6.24) we indeed obtain the announced relation (6.1) between \(\Delta X\) and \(\triangle _F\).

Within the BV formalism, the anomaly consistency condition is a consequence of the nilpotency of the BV operator, in particular, in the context of pAQFT, this was discussed already in the work of Hollands [5]. It is shown in [5, Prop.5] that the nilpotency of \(}_0\), i.e., \(}_0^2=0\), (or, as observed in [6], the nilpotency of \(}_}}}\)) induces a consistency condition for the anomaly term in (6.19). We recall this result in Proposition 6.2 and provide an alternative (shorter) proof using a result of Fröb [9], which also highlights the \(L_\)-structure underlying the BV quantization.

It was shown by Fröb [9] that there is an \(L_\)-structure on \(\textrm_}(M)\) coming from the AMWI (6.19). The brackets \([\bullet ,\dots ,\bullet ]^}}}_n:\textrm_}(M)^n\rightarrow \textrm_}(M)\) are linear and graded symmetric maps, given in terms of the generating function (for even \(}}\)) by

$$\begin[e^}}}]^}}}\equiv \sum _^\frac[}},\dots ,}}]^}}}_n\doteq e^}}}}_}}}(e^}}}). \end$$

(6.28)

Note that \([e^}}}]^}}}\) is odd. Obviously, with this definition, the AMWI (6.19) can be written as

$$\begin[e^}}}]^}}}=i\bigl (\tfrac\}}+}}+}},}}+}}+}}\}+A(}}+}})\bigr ). \end$$

(6.29)

Crucially, we verify here (streamlining the argument of [9]) that the brackets defined by the formula (6.28) satisfy the generalized Jacobi identity.

Proposition 6.1

The brackets defined by (6.28) satisfy the generalized Jacobi identity:

$$\begin[e^}}},[e^}}}]^}}}]^}}}=0. \end$$

(6.30)

Proof

The result follows directly from the nilpotency of \(}_}}}\) and the fact that \(}_}}}(e^}}})\) is odd. To see this, first note that, for \(}}\in \textrm_}(M)\) even, we obtain

$$\begin[e^}}},}}]^}}}=\frac\Big \vert _[e^}}+\lambda }})}]^}}}\overset -e^}}}}}\,}_}}}(e^}}})+e^}}}}_}}}(e^}}}}}).\nonumber \\ \end$$

(6.31)

Inserting \(}}=\eta \,[e^}}}]^}}}=\eta \,e^}}}}_}}}(e^}}})\) (with \(\eta \) an odd Grassmann variable) and omitting in the resulting formula the factor \(\eta \), we get

$$\begin[e^}}},[e^}}}]^}}}]^}}}= -e^}}}\bigl (}_}}}(e^}}})\bigr )^2+e^}}}}_}}}^2(e^}}})=0. \end$$

\(\square \)

From (6.28), we see that the 0-bracket is

$$\begin[-]_0^}}}=}_}}}(1), \end$$

so vanishes identically if \(}}\) satisfies the QME (6.14) or (6.21), and that the 1-bracket is given by

$$\begin[}}]_1^}}}=}_}}}(}})=s_}}}(}})-i\triangle _}}}(}}). \end$$

(6.32)

From (6.29), we obtain for the 2-bracket

$$\begin[}},}}]_2^}}}=-i\bigl (\}},}}\}+\langle A''(}}),}}\otimes }}\rangle \bigr ) \end$$

(6.33)

and for the n-bracket (with \(n>2\))

$$\begin[}},\dots ,}}]_n^}}}=(-i)^\langle A^(}}),}}^\rangle . \end$$

(6.34)

Hence, we have an \(L_\infty \) structure, provided \(}}+}}\) satisfies the QME. Now we can come back to [5, Prop.5].

Proposition 6.2

The anomaly \(A(}})\) defined by the generalized AMWI (6.19) satisfies the relation

$$\begin 0=\}}+}},A(}})\}+\langle A'(}}),\left( \tfrac\}}+}},}}+}}\}+ A(}})\right) \rangle . \end$$

(6.35)

Proof

We prove this proposition by verifying that the generalized Jacobi identity (6.30) for the particular value \(}}=0\) is precisely the consistency condition (6.35) (which is not surprising since both rely on \(}_}}}^2=0\)). To verify this, we use the fact that

$$\begin[1,}}]^}}}=\frac\Big \vert _[e^}}\lambda }]^}}} \overset\}}+}},}}\}+\langle A'(}}),}}\rangle \ \end$$

(6.36)

for \(}}\in \textrm_}(M)\) odd and \(\lambda \) an odd Grassmann parameter, and applying again the AMWI (6.29), we obtain

$$\begin \begin&0\, =-i\,[1,[1]^}}}]^}}}\overset [1,\bigl (\tfrac\}}+}},}}+}}\}+ A(}})\bigr )]^}}}\\&\overset\}}+}},\,A(}})\} +\langle A'(}}),\bigl (\tfrac\}}+}},}}+}}\}+A(}})\bigr )\rangle , \end \end$$

where we also used the graded Jacobi identity for the antibracket; hence, we arrive at (6.35). \(\square \)

Note that the vector fields \(\partial _X\), are of at most first order in \(\phi \). For these vector fields we have \(\langle A''(F),\partial _X\otimes \partial _Y\rangle =0\) (see [9] for a related result).

To show this, we start with the following lemma.

Lemma 6.3

Let \(}}\in T(\textrm_}(M))\) and \(}}\in \textrm_}(M)\) depend at most linearly on \(\phi \). Then

$$\begin s_0(}}\cdot _T}})=s_0(}})\cdot _T}}+}}\cdot _ s_0(}})+i\}},}}\}. \end$$

(6.37)

with the first-order time-ordered product

$$\begin A\cdot _ B\doteq A\cdot B+\langle A',E^}B'\rangle \,\ A,B\in T(\textrm_}(M)). \end$$

(6.38)

Proof

First note that \(}}\in T(\textrm_}(M))\) implies \(s_0(}})\in T(\textrm_}(M))\), due to the generalized AMWI (6.17), and that \(T}}=}}\), since \(}}\) is at most linear in \(\phi \). We use the off-shell field equation (2.8) and the fact that \(s_0\) is a derivation for the pointwise product. The derivative with respect to \(\phi \) is denoted by \('\). We have

$$\begin \begin&s_0(}}\cdot _T }})=s_0(}}\cdot }}+\langle }}',E^}}}'\rangle )\\&\qquad =s_0(}})\cdot }}+}}\cdot s_0(}})+\langle s_0(}}'),E^}}}'\rangle +\langle }}',E^}s_0(}}')\rangle \\&\quad =s_0(}})\cdot _T}}+}}\cdot _ s_0(}})+\langle s_0(}}')-s_0(}})',E^}}}'\rangle \\ &\qquad +\langle }}',E^}\bigl (s_0(}}')-s_0(}})'\bigr )\rangle \end\nonumber \\ \end$$

(6.39)

But with \(s_0=-\langle \frac,\frac}\rangle \) and

$$\begin \int dy\,\,E^}(z,y)\,\frac=i\delta (z,x)\, \end$$

(6.40)

it holds for any

(6.41)

Hence, the last two terms in (6.39) form the antibracket \(i\}},}}\}\) and we obtain the statement in the lemma. \(\square \)

Next, we show that for elements of first order in \(\phi \), \(\triangle _}}}\) acts as the unrenormalized BV Laplacian, up to an extra term of the form \(\triangle _}}(1)}}}}\).

Proposition 6.4

Let \(}},}}\in \textrm_}(M)\) be of first order in \(\phi \) and even (we multiply the usual vector fields with Grassman generators – the \(\eta \)-trick), \(}}\in \textrm_}(M)\) even and L independent of antifields. Then

$$\begin \triangle _}}(}}}})=(\triangle _}}}})}}+}}(\triangle _}}}})+\}},}}\}-\triangle _}}(1)}}}}. \end$$

(6.42)

Proof

Since \(\triangle _}}=i(}_}}-s_}})\) and \(s_}}\) is a derivation, the statement is equivalent to

$$\begin }_}}(}}}})=}_}}(}})}}+}}}_}}(}})-i\,\}},}}\}-}_}}(1)}}}}. \end$$

(6.43)

Taking into account that \(}_}}(}})=e^}}}T^s_0(Te^}}}}})\), this is equivalent to

$$\begin s_0(Te^}}}}}}})= & s_0(Te^}}}}})\cdot _T}}+}}\cdot _Ts_0(Te^}}}}}) -i\,T(e^}}}\}},}}\})\nonumber \\ & \quad -s_0(Te^}}})\cdot _T}}\cdot _T}}\, \end$$

(6.44)

by applying (2.5) and that \(T^}}=}}\) (and similarly for \(}}\)). Since also \(T^\}},}}\}=\}},}}\}\), it remains to show that for it holds that

(6.45)

For this purpose, we use Lemma 6.3. We get

(6.46)

where

(6.47)

But Z is just the correction to the derivation property of the antibracket with respect to the time-ordered product, that is, the r.h.s. of (6.46) is indeed equal to . To wit, we have

(6.48)

and, by using (6.40), this coincides with iZ since

$$\begin s_0(}})''(x,y)=-\int dz \Bigl (\frac\frac}}}(z)}+ \frac\frac}}}(z)}\Bigr ).\nonumber \\ \end$$

(6.49)

\(\square \)

Remark 6.5

Note that if, in addition, the quantum master equation (6.14) holds, then \(\triangle _}}(1)=0\) and we obtain the more familiar relation:

$$\begin \triangle _}}(}}}})=(\triangle _}}}})}}+}}(\triangle _}}}})+\}},}}\}. \end$$

(6.50)

The result on \(A''\) follows now directly from Proposition 6.4.

Proposition 6.6

Let A be the anomaly appearing in the AMWI. Let \(}}\in \textrm_}(M)\) even, L independent of antifields and let \(}},}}\in \textrm_}(M)\) be at most linear in \(\phi \). Then

$$\begin \langle A''(}}),}}\otimes }}\rangle =0. \end$$

(6.51)

Proof

Without restriction of generality, we may assume that both \(}}\) and \(}}\) are even (by using the \(\eta \)-trick). Let \(\lambda ,\mu \in }}\). From the generalized AMWI (6.19), we have that:

$$\begin&}_}}(e^}}+\mu }})})\equiv (s_}}-i\triangle _}})(e^}}+\mu }})})\nonumber \\&\overset\ i\,e^}}+\mu }})} \left( \frac\}}+\mu }}+L+}}, \lambda }}+\mu }}+L+}}\}+ A(}}+\lambda }}+\mu }})\right) \end$$

(6.52)

Selecting the terms proportional to \(\lambda \mu \) and using the expression (6.21) for \(\triangle _}}}(1)\), we obtain

$$\begin (s_}}-i\triangle _}})(}}}})&=}}\bigl (\}},L+}}\}-i\triangle _}}(}})\bigr )+}}\bigl (\}},L+}}\}-i\triangle _}}(}})\bigr )\\&\quad -i\}},}}\}+i}}}}\triangle _}}}(1) -i\langle A''(}}),}}\otimes }}\rangle \,, \end$$

by using the analog of (6.24) for \(}}\in \textrm_}(M)\). The statement follows from the derivation property of \(s_}}\) and Proposition 6.4. \(\square \)

For \(F\in }}_}(M)\), one obtains a map

(6.53)

which coincides with the action previously constructed in (5.2). The fact that it is an action was derived from the cocycle relation (5.8) for \(X\mapsto \Delta X\) as a consequence of the cocycle relation for the anomaly map \(\zeta \) in the UAMWI.

Actually, the cocycle relation for \(\Delta \) in the form of the equivalent consistency relation (4.5) derives directly from the BV consistency condition (6.35).

Proposition 6.7

The BV consistency relation (6.35) implies the extended Wess–Zumino consistency relation (4.5).

Proof

Let L be independent of antifields. We insert \(}}=F+\partial _\eta _1+\partial _\eta _2\) into the BV consistency relation (6.35), where \(F\in }}_}(M)\) and \(\eta _1,\eta _2\) are Grassmann generators. We use the fact that \(\langle A''(F),\partial _\otimes \partial _\rangle =0\). Since \(A(F)=0\), we obtain the following finite Taylor expansion in \(\eta _1,\eta _2\):

$$\begin A(}})=-\Delta X_1(F)\,\eta _1-\Delta X_2(F)\,\eta _2, \end$$

(6.54)

where we also used (6.27). In particular, note that \(\frac}})}=0\). With that, we obtain

$$\begin \}},A(}})\}&=-\\eta _1+\partial _\eta _2),\bigl (\Delta X_1(F)\,\eta _1+\Delta X_2(F)\,\eta _2\bigr )\}\nonumber \\&=\bigl (-\partial _\Delta X_2(F)+\partial _\Delta X_1(F)\bigr )\,\eta _1\eta _2\ . \end$$

(6.55)

Note that

$$\begin&\langle A'(}}),(G+\partial _Z\eta )\rangle =\frac\Big \vert _A\bigl (}}+\tau (G+\partial _Z\eta )\bigr )\end$$

(6.56)

$$\begin&\quad =-\frac\Big \vert _ \Bigl (\Delta X_1(F+\tau G)\,\eta _1+\Delta X_2(F+\tau G)\,\eta _2+\tau \Delta Z(F+\tau G)\,\eta \Bigr )\nonumber \\&\quad =-\langle (\Delta X_1)'(F),G\rangle \eta _1-\langle (\Delta X_2)'(F),G\rangle \eta _2 -\Delta Z(F)\,\eta \ , \end$$

(6.57)

where \(G\in }}_}(M)\), and \(\eta \) is another Grassmann generator. Hence, using (2.23), we obtain

$$\begin&\langle A'(}}),\left( \tfrac\}},L+}}\}\right) \rangle = \langle A'(}}),\left( (\partial _\eta _1+\partial _\eta _2)(L+F)-\partial _\,\eta _1\eta _2\right) \rangle \nonumber \\&\quad =\Bigl (-\langle (\Delta X_1)'(F), \partial _(L+F)\rangle +\langle (\Delta X_2)'(F),\partial _(L+F)\rangle +\Delta [X_1,X_2](F)\Bigr )\eta _1\eta _2\ \end$$

(6.58)

and

$$\begin \langle A'(}}),\, A(}})\rangle&=-\langle A'(}}),\left( \Delta X_1(F)\,\eta _1+\Delta X_2(F)\,\eta _2\right) \rangle \nonumber \\&=\Bigl (\langle (\Delta X_1)'(F),\Delta X_2(F)\rangle -\langle (\Delta X_2)'(F),\Delta X_1(F)\rangle \Bigr )\eta _1\eta _2\,. \end$$

(6.59)

Composing (6.55), (6.58) and (6.59), we obtain the consistency equation (4.5). \(\square \)

Remark 6.8

Note that by tracing back the arguments given in this section, we can see that Proposition 6.7 essentially states that the generalized Wess–Zumino consistency condition is the consequence of \(}^2_}}}=0\). We can compare this with a simple fact that the nilpotency of the nonrenormalized BV Lapalacian \(\triangle \) (see (6.10)) implies an analogous statement for vector fields. Without the loss of generality, we assume \(}}\) and \(}}\) to be even (we multiply the usual vector fields with Grassman parameters) \(},}\in \textrm_}(M)\) (regular multivector fields):

$$\begin 0 & =\triangle ^2(}}}})=\triangle \bigl ((\triangle }}) }}+}}(\triangle }})+\}},}}\}\bigr ) \\ & =\partial _}}}(\triangle }})+\partial _}}}(\triangle }})+\triangle (\}},}}\})\, \end$$

by using that \(\Delta \) satisfies a relation analogous to (6.42), where \(\partial _}}}\), \(\partial _}}}\) denotes the natural action of vector fields on functionals as derivations.