Exact renormalization group in Batalin--Vilkovisky theory

In this paper, inspired by the Costello's seminal work, we present a general formulation of exact renormalization group (RG) within the Batalin-Vilkovisky (BV) quantization scheme. In the spirit of effective field theory, the BV bracket and Laplacian structure as well as the BV effective action (EA) depend on an effective energy scale. The BV EA at a certain scale satisfies the BV quantum master equation at that scale. The RG flow of the EA is implemented by BV canonical maps intertwining the BV structures at different scales. Infinitesimally, this generates the BV exact renormalization group equation (RGE). We show that BV RG theory can be extended by augmenting the scale parameter space R to its shifted tangent bundle T[1]R. The extra odd direction in scale space allows for a BV RG supersymmetry that constrains the structure of the BV RGE bringing it to Polchinski's form. We investigate the implications of BV RG supersymmetry in perturbation theory. Finally, we illustrate our findings by constructing free models of BV RG flow and EA exhibiting RG supersymmetry in the degree -1 symplectic framework and studying the perturbation theory thereof. We find in particular that the odd partner of effective action describes perturbatively the deviation of the interacting RG flow from its free counterpart.

a general formulation of exact renormalization group (RG) within the Batalin-Vilkovisky (BV) quantization scheme. In the spirit of effective field theory, the BV bracket and Laplacian structure as well as the BV effective action (EA) depend on an effective energy scale. The BV EA at a certain scale satisfies the BV quantum master equation at that scale. The RG flow of the EA is implemented by BV canonical maps intertwining the BV structures at different scales. Infinitesimally, this generates the BV exact renormalization group equation (RGE). We show that BV RG theory can be extended by augmenting the scale parameter space R to its shifted tangent bundle T r1sR. The extra odd direction in scale space allows for a BV RG supersymmetry that constrains the structure of the BV RGE bringing it to Polchinski's form [6]. We investigate the implications of BV RG supersymmetry in perturbation theory. Finally, we illustrate our findings by constructing free models of BV RG flow and EA exhibiting RG supersymmetry in the degree´1 symplectic framework and studying the perturbation theory thereof. We find in particular that the odd partner of effective action describes perturbatively the deviation of the interacting RG flow from its free counterpart.

Introduction
The Batalin-Vilkovisky (BV) quantization scheme [1,2] is to date the most powerful method for quantizing general classical field theories characterized by possibly non involutive and non freely acting gauge symmetry. It has found applications in ordinary gauge theory as well as supergravity and string theory. Furthermore, since it proceeds in a functional framework, it makes the powerful functional integration methods available. See ref. [3] for a readable introduction to the subject.
In BV theory, ghost fields are added to the classical fields to make Becchi-Rouet-Stora-Tyutin (BRST) symmetry manifest from the onset. Each field is then adjoined by an antifield of opposite statistics. In this way, the total field space is structured as an odd phase space, with fields and antifields playing the role of canonically conjugate coordinates and momenta, respectively. Hamiltonian notions such as those of BV bracket, an odd analog of the Poisson bracket, and canonical map can in this way be formulated and used while keeping manifest covariance and BRST invariance.
The introduction of the antifields doubles the field content. Gauge fixing roughly speaking amounts to the restoration of the original field number and is achieved by restricting to a suitable Lagrangian submanifold of the total field space. Independence of the quantum theory from the choice of the submanifold requires the BV master action (MA) to satisfy the BV quantum master equation (ME). This involves an odd second order differential operator called BV Laplacian in addition to the BV bracket. Both the MA and the Laplacian are affected by UV divergences which need to be suitably regularized and renormalized. Violations of the ME result in gauge anomalies.
The BV quantization procedure, when properly implemented, ensure the existence of well-defined propagators making the usual perturbative Feynman graph methods usable in a manifestly BRST symmetric framework.
The invariant physical content of a quantum field theory is encoded in the partition function with background sources. The renormalization group (RG) [4,5] is a powerful field theoretic construction aimed to probing such content below a certain energy scale, though the content itself is insensitive to the presence of the scale, which serves as a mere analysis device. There are a priori many ways of injecting a scale in a field theory and correspondingly there are many versions of the RG equations (RGE). In this paper, we shall concentrate on the so called exact RG [6,7] (to be called simply RG in the following). See ref. [8] for an updated review.
The theory of RG flow and effective action (EA) has been studied in a BV framework both from a physical [9,10] and a mathematical [11][12][13][14][15] point of view and in a variety of guises. Recent mathematical studies close in spirit to our contribution are [16][17][18].
The study of the BV RG is of considerable interest for its applications to relevant theoretical problems. We mention here a few. In ref. [19], closed string field theory is quantized using the BV algorithm and the BV MA is studied in depth. A family of consistent string vertexes is obtained parametrized by a stub length working as an ultraviolet cutoff. A renormalization group equation for string field theory is yielded in this way. In ref. [20], Kodaira-Spencer theory of gravity [21] is quantized in the BV RG framework of ref. [11] and the genus expansion of the B-model is related to the perturbative expansion of the resulting quantum field theory. In ref. [22], using again the BV RG framework of ref. [11], the holomorphic bosonic string is quantized.

Overview of the BV RG framework of this paper
The effectiveness of the BV RGE as a computational scheme and analysis method depends on the form of the RG flow, which is a basic input datum of the RGE itself. However, there is much that can be learned before committing oneself with a specific choice.
In this paper, we formulate a BV theory of the RG flow and RGE from a very general perspective as a BV effective field theory (EFT) at a varying energy scale t. Our approach is inspired by the seminal work by Costello [11], the indebtedness to which we fully acknowledge, but it differs from it in key points to be discussed later. In an EFT setting, the BV bracket p´,´q t and Laplacian ∆ t depend on t.
The BV MA S t also depends on t and satisfies the BV quantum ME S t can now be viewed as a BV EA. The RG flow of S t must be such to preserve the ME (1.1.1). The essentially only natural way this can come about is through a group ϕ t,s of BV canonical maps, whose pull-back action transforms isomorphically the BV bracket and Laplacian at any scale s into those at another scale t.
The ME remains satisfied identically in t if the EA S t flows as S t " ϕ t,s˚Ss`rϕt,s , where r ϕt,s " 1 2 ln J ϕt,s is the logarithmic Jacobian of ϕ t,s . Infinitesimally, then, S t obeys an RGE of the form in which the first and second term of the right hand side can be identified with the familiar "classical" and "quantum" contributions to the RG flow. We justify this approach to the RG by a biased revisitation of the BV quantization scheme in sect. 2, where the interplay of the RG flow and gauge fixing is also discussed.
In sect. 3, relying only on the general properties shared by every RG flow, we explore the possibility of using as energy scale space more general manifolds than just the real line R. The fact that BV theory belongs to the realm of graded algebra and geometry indicates that the scale space may be promoted to a graded manifold. We have found particularly useful to employ the shifted tangent space T r1sR of R rather R alone for this role. We call this RG set-up "extended" to distinguish it from the customary "basic" set-up. T r1sR is coordinatized by a degree 0 parameter t, to be identified with the usual RG energy scale, and a further degree 1 parameter θ. In the extend set-up, so, the BV bracket p´,´q tθ and Laplacian ∆ tθ depend on both t and θ. The BV MA S tθ also depends on t, θ and satisfies the extended version of the BV ME, S tθ is the extended BV EA. In analogy to basic case, its RG flow is governed by a group ϕ tθ,sζ of canonical maps relating the BV bracket and Laplacian at any extended scale s, ζ to those at another scale t, θ according to the law S tθ " ϕ tθ,sζ˚Ssζ`rϕtθ,sζ . At the infinitesimal level, this entails a more structured RGE than (1.1.2). Expanding in powers of the odd parameter θ, we have p´,´q tθ " p´,´q t˘θ p´,´q ‹ t and ∆ tθ " ∆ t`θ ∆ ‹ t , where p´,´q ‹ t is a degree 0 graded symmetric bracket and ∆ ‹ t is a degree 0 Laplacian. Further, S tθ " S t`θ S ‹ t , where S t is to be identified with the usual BV EA and S ‹ t is a degree´1 partner of it. Then, the extended RGE expresses S ‹ t in terms of S t and and yields an RGE for S t of the form where the last two terms in the right hand side are "seed" terms. In this way, we obtain a version of the BV RGE of the distinctive form of Polchinski's [6] by purely algebraic and geometric means. By adjoining the RG scale with an odd partner, we have further revealed the existence of an intriguing sort of RG supersymmetry with potential implications for perturbation theory.
Since presently we have no proof that the basic RG set-up can be embedded in the extended one in general, it is important to work out non trivial models in which the extended set-up is implemented. In sect. 4, working in the glp1|1q degree´1 symplectic framework originally developed by Costello in ref. [11], we illustrate a free model of BV RG flow and EA in the extended set-up. We obtain explicit expressions for the free EA S 0 t and its odd partner S 0‹ t . We then investigate the associated perturbation theory, where the full EA and partner thereof S t , S ‹ t can be expanded as

1.5)
I t , I ‹ t being interaction terms expressed as formal power series of . We recover the ME and RGE of I t originally obtained by Costello and obtain a further ME involving simultaneously I t , I ‹ t . We also show that, through this latter, the RGE of I t can be cast in a form analogous to Polchinsi's. Finally, we find that the odd interaction action partner I ‹ t describes perturbatively the deviation of the interacting RG flow from its free counterpart.
In the final sect. 5, we speculate on possible ramifications and applications of our results in mathematical physics, geometry and topology.
The present paper aims to present the field theoretic foundations of our BV formulation of RG theory. In a more mathematical oriented paper [23], we shall reformulate the results obtained here in the framework of the abstract theory of BV algebras and manifolds.

Relation to Costello's BV RG theory
In spite of evident formal similarities, there is a basic difference between the RG framework propounded in the present paper and Costello's [11]. Costello's approach is Wilsonian in nature [5]. In the familiar formulation of Wilson's RG, the EFT at the energy scale t is described by an EA S t and functional integration is restricted to the finite dimensional subspace F t of field space F spanned by the field modes of energy up to t. The RG flow is determined by the requirement that the partition function is independent from t. Since flow to a lower energy scale involves a truncation of the functional integration domain F t to a proper subspace, the flow cannot be implemented by invertible field maps. The approach set forth by us is instead Polchinskian [6]. In Polchinski's RG framework, the EFT at the energy scale t is described by an EA S t as in Wilson's case but functional integration is extended to the full field space F . The restriction of integration to the field modes up to the scale t is due to the special dependence of S t on the energy scale t which through the Boltzmann weight e St causes an exponential damping of the contribution of the field modes of energy above t in the functional integral. The RG flow is determined again by the requirement that the partition function is independent from t, but proceeds through invertible field maps [24][25][26]. (Above, we have ignored the complications related to gauge symmetries and gauge fixing for the sake of the argument.) Wilson and Polchinski RG are presumably related but it is difficult formulate this relationship in precise terms. In a Polchinskian perturbative RG framework, the restriction to energy modes below the scale t is due as a rule to a modification of the standard kinetic operator H of the free action S 0 t into H{kpH{tq, where kpxq is some rapidly decreasing cut-off function, e. g. e´H {t . This suppresses the field modes of energy greater than t through the Boltzmann weight e S 0 t . The Wilsonian perturbative RG framework would roughly correspond to replace kpxq with a function taking the value 1 for x ă 1 and 0 for x ą 1. Wilson and Polchinski RG are therefore akin but distinct. So, are Costello's and this paper's BV RG formulations. This also impinges on the way beta functions are defined in the two approaches, though this matter is not dealt with in this paper.

Strengths and weaknesses of the BV RG framework presented
The BV RG formulation proposed in the present study, relying heavily on functional integral manipulations, is expectedly difficult to put on a firm rigorous ground. E. g., the quantum part of the RG equation is related to the Jacobian of an invertible field map which as a rule requires regularization. Yet, in quantum field theory it is normally possible to cope with this kind of problems relying also on the indications provided by finite dimensional analog models.
Our approach to the BV RG is based on BV geometry. Most studies on this subject available in the literature, including A. Schwarz's foundational work [27], assume a finite dimensional context. Even in this simplified form they retain however a considerable interest for the clues and intuition they provide. This is the point of view taken in this article.
Most of what is stated in the following is therefore strictly true in finite dimensions. A part of analysis is admittedly conjectural, although we feel that the assumptions we made along the way are natural and neatly fit into the whole picture. The main challenge lying ahead is testing the theory in non trivial field theoretic models dealing with the difficult technical problems arising in infinite dimensions. We leave this work for the future.
Costello's BV RG theory [11] is presently on a firm mathematical ground.
With this study, we certainly do not claim having built an alternative BV RG theory of comparable strength and soundness, but only explore an alternative.
The exact RG philosophy we espouse is in fact widely used by quantum field theorists interested in concrete physical applications.

Renormalization group in Batalin-Vilkovisky theory
In this section, we shall first review briefly the foundations of BV theory from a geometric perspective. This will also serve to set our terminology and notation.
We shall then expound our BV formulation of RG theory, its interplay with gauge fixing and its implications for perturbation theory. What is stated below is strictly true only in a finite dimensional setting. In the infinite dimensional context of quantum field theory that interest us, our treatment is merely formal. Most of the geometrical theory was developed in Schwarz's seminal work [27]. See refs. [28,29] for a recent up to date review.

Elements of BV geometry
In this subsection, we review briefly the basic notions of BV geometry highlighting those points which will be relevant in the subsequent applications.
BV geometry is the theory of BV manifolds. A BV manifold M is a Z-graded manifold endowed with a degree´1 symplectic form ω M and a Berezinian µ M , called BV form and measure respectively, obeying a compatibility condition.
Canonically associated with the BV form ω M is a bilinear bracket p´,´q M : FunpMqˆFunpMq Ñ FunpMq, called BV bracket, as follows. For every function f P FunpMq, there is a unique vector field ad M f P VectpMq such that Then, for any function pair f, g P FunpMq, one has pf, gq M " ad M f g.
p´,´q M is a Gerstenhaber bracket, that is it has degree 1, is shifted graded antisymmetric, obeys the shifted graded Jacobi identity and acts as a shifted graded derivation on both its arguments.
The BV measure µ M induces a linear integration map ş M µ M´: FunpMq Ñ C, called BV integral.
for any function f P FunpMq satisfying are the canonical ones, that is those with the property that φ˚ω M " ω M 1 . They preserve the BV bracket, so that for all functions f, g P FunpMq.
The BV integral has the usual covariance properties. If φ : M 1 Ñ M is a canonical map of two BV manifolds M, M 1 P B, then there is a nowhere vanishing degree 0 function J ϕ P FunpM 1 q, called the Jacobian of φ, such that ż for any f P FunpMq. The logarithmic Jacobian is a natural, equivalent substitute for J φ . It obeys the equation The pull-back action of a canonical map φ : well. In this case, (2.1.9) and the identity i φ˚L 1˝φ| L 1 " φ˝i L 1 imply that ż of a Z-graded manifold F with canonical BV form ω can and measure µ can . ω can and µ can are given by the familiar relations in terms of the base and fiber coordinates x a and x˚a, ω can " dx˚adx a and µ can " d dim F xd dim F x˚. The BV bracket and Laplacian are in this case and respectively, where ǫ a " |x a |, ǫ˚a " |x˚a|.
Every BV manifold M with BV form ω and measure µ admits locally special Darboux coordinates x a and x˚a such that ω and µ are expressed as ω can and µ can in terms x a and x˚a. Thus, the local structure of BV manifold M is that of a prototypical BV manifold T˚r´1sF for some F .

BV quantization scheme
In this subsection, we review the BV quantization algorithm or BV theory for short. BV geometry, as formulated in subsect. 2.1, informs BV theory and makes its structure intuitive.
In field theory, one is initially given with a space F cl of degree 0 classical fields ψ cl i and a degree 0 action functional S cl invariant under the action of a distribution G of the tangent space T F cl of F cl , representing gauge symmetries in a broad sense. One is further interested in the computation of quantum correlators of field functionals f of F cl which are invariant under the action of G. In general, G is not freely acting nor it is involutive, though as a rule it is when restricted to the submanifold of F cl of fields obeying the Euler-Lagrange equations, that is the critical locus critpS cl q of S cl in F cl . As well known, a functional integral approach to quantization in a situation of this type is problematic. BV theory offers an elegant solution for this difficulty.
The implementation of the BV quantization algorithm involves the following four steps. First, one enlarges the field space F cl by adding to the classical fields a ghost field of suitable positive degree for each of the independent generators of the distribution G. One obtains in this way a space F of fields ψ a of non negative degree. Second, one adjoins a negatively graded antifield ψ˚a to each field ψ a such that the degrees of ψ˚a and ψ a add up to´1. The resulting total field space is structured as the´1 shifted cotangent bundle T˚r´1sF of F with fields and antifields parametrizing respectively its base and fibers. T˚r´1sF is then endowed with the canonical BV form and measure ω can and µ can , rendering it a BV manifold. Third, the action S cl is extended to an action S, called BV MA, defined on the whole total field space T˚r´1sF and such that S reduces to S cl when all the fields and antifields are set to zero except for the original classical fields ψ cl i .
Fourth, quantization is implemented by restricting the functional integration to a suitable Lagrangian submanifold L of T˚r´1sF with the Boltzmann weight e S inserted. The unnormalized correlator of a functional f on T˚r´1sF is therefore given by an integral of the form L must be carefully chosen in order to avoid the usual diseases associated with the gauge symmetry of S cl . Except for the simplest cases, taking L to be the zero section of T˚r´1sF , corresponding to setting all the antifields to zero, will not work. In general, there are infinitely many choices of L. Consistency requires that the value of quantum correlators is independent from the choice made. This entails a restriction on the form of the BV MA S: the BV quantum ME must be obeyed by S. It also entails a restriction on the allowed functionals f which can be inserted in correlators; they must satisfy the equation where ∆ canS is the degree 1 second order linear differential operator defined by and is therefore characterized by its cohomology.
Inserting the Boltzmann weight e S in the functional integral (2.2.1) amounts to redefine the BV measure from µ can to The BV ME (2.2.2) ensures that the measure µ canS is compatible with ω can as µ can . ∆ canS is just the nilpotent BV Laplacian associated with ω can and µ canS according to (2.1.4).

BV RG theory
In a concrete setting in which the fields in T˚r´1sF are sections of bundles on a space-time manifold, the above functional set-up is plagued by ultraviolet (UV) divergences. In fact, the canonical BV form and measure ω can and µ can yield a BV Laplacian ∆ can containing pairs of functional derivatives acting at the same point of space-time and thus ill-defined. In such a situation, the whole theoretical scheme is purely formal and so virtually unusable. To cure this disease, one has to introduce a very high energy scale t 0 to regularize the UV divergences and use the methods of EFT to describe the quantum field theory at a relevant lower energy scale t.
It is natural to expect that, in an EFT description, the formal framework of BV quantization should remain essentially unchanged. BV theory and BV geometry that underlies it constitute a delicate structure unlike to preserve its efficacy and selfconsistency under severe modifications. We assume thus that quantization proceeds much as in the unregularized case but with the unregularized quantities replaced by effective ones at the given scale t. The unregularized BV form and measure ω can and µ can get in this way replaced by an effective BV form ω t and measure µ t and the unregularized BV bracket p´,´q can and Laplacian ∆ can by the associated effective BV bracket p´,´q t and Laplacian ∆ t . There is a price attached to this: ω t , µ t , p´,´q t and ∆ t are mathematically far more complicated than ω can , µ can , p´,´q can and ∆ can are. Further, since the scale t cannot be assigned any a priori special value, we are considering a whole one parameter family of BV manifolds T˚r´1sF t having T˚r´1sF as underlying graded manifold and ω t and µ t as BV form and measure. However, the formal properties of the effective structure are exactly the same as those of the unregularized one, allowing for an in depth theoretical analysis.
From subsect. 2.2, quantization of the EFT involves restriction of integration to a Lagrangian submanifold L t of T˚r´1sF t with insertion of the Boltzmann weight e St of a BV EA S t . Unnormalized correlators thus read as for any functional f on T˚r´1sF . The EA S t must again satisfy the BV ME Further, the functionals f inserted in correlators must obey where ∆ tS is the effective covariant Laplacian defined by Again, by virtue of (2.3.2), ∆ tS is nilpotent and thus characterized by its cohomology.
The variation of the Lagrangian submanifold L t and the BV EA S t under continuous shifts of the energy scale t, the so called RG flow, must be such to preserve the whole BV structure. In our EFT formulation, it is natural to assume that the flow is implemented by scale dependent field redefinitions, that is invertible maps of the total field space T˚r´1sF .
The largest most natural family of maps preserving the BV structure is the set of morphisms of the canonical groupoid of the BV manifold family T˚r´1sF t which we introduced above. In the present modelization, so, the RG flow is given by a two parameter family of canonical maps ϕ t,s : T˚r´1sF t Ñ T˚r´1sF s satisfying the relations for s, t P R. The basic requirement that the RG flow must obey is that the partition function Z S p1q remains constant along it, so that for any two value s, t of the energy scale By the general result (2.1.18), one has ż ϕt,s˚Lt A sufficient condition for the identity (2.3.9) to hold is that the Lagrangian sub- at least up to body homology, and that the RG flow of the BV EA S t is driven by the flow map ϕ t,s , where the logarithmic Jacobian r ϕt,s " r ϕt,s is defined in (2.1.15).
Relation (2.3.12) implies that the RG flow of the BV EA is governed by the first order differential equation, where ϕ ‚ t and r ‚ ϕt are respectively given by any specific assumption about the form of the EA S t .

Derived BV EAs
In the BV theory of the RG, it is often useful to consider parametrized families of BV EA S t whose RG flow is governed by a fixed set of flow maps ϕ t,s . This is most naturally done using the methods of derived geometry along the lines of ref. [13]. Here, we shall present a simplified formulation tailored for a functional integral formulation of BV theory.
In the derived approach, a family of functionals on field space T˚r´1sF is modelled as a functional on T˚r´1sF valued in a differential graded commutative algebra A, which must be thought of as the algebra of functions on a fiducial parameter space. The family might depend on the RG scale t, while A itself does not. The BV Laplacians ∆ t , brackets p´,´q t and RG flow maps ϕ t,s are assumed to be given. The ∆ t , p´,´q t and the pull-backs ϕ t,s˚o f the ϕ t,s act on   A standard application of the above derived geometric set-up occurs in the analysis of BV EA homotopies [13]. In such a case, A " Ω˚pIq is the algebra of differential forms in the interval I " r0, 1s and d A " d dR is the Rham differential.
The derived BV EA is of the form S Ω˚pIqt " S xt`ddR x S˚x t , where S xt , S˚x t have respectively degrees 0,´1. The obstruction form reads as α Ω˚pIq " α x`ddR x α˚x with components α x , α˚x of degrees 1, 0. The deformation S Ω˚pIqt is liftable provided α x " 0. In such a case, one has We can readily absorb α˚x by shifting S xt by the irrelevant constant ş x 0 dx α˚x. Upon doing so, for fixed x, S xt is a BV EA and BS xt {Bx is trivial in the ∆ tSx cohomology.
In many instances, one considers the case where A is the Chevalley-Eilenberg algebra CEpgq of an L 8 algebra g and d A is the associated Chevalley-Eilenberg differential d CEpgq . In such cases, g encodes a generalized symmetry of the EFT and the derived BV EA S CEpgqt describes an action of g on the a BV EFT in the interpretation of ref. [13].
If S CEpgqt can be lifted, then the action of g is called inner. As we have just 0 ÝÑ Cr´1s ÝÑ p g ÝÑ g ÝÑ 0. (2.4.8) The non triviality of the extension is thus tantamount to the non liftability of S CEpgqt . This is how BV anomalies appear.

Perturbative BV RG theory
In perturbation theory, one assumes that it is possible to turn off the interactions of a given quantum field theory and that a free quantum field theory that is fully under control and simple is yielded in this way.
In BV theory, the effective free field theory is based on a BV structure characterized by a total field space T˚r´1sF 0 and an effective BV form ω 0 t and measure µ 0 t and is governed by a free BV EA S 0 t all depending on the energy scale t. To implement the perturbative program, one considers the algebra of formal power series FunpT˚r´1sF 0 qrr ss over FunpT˚r´1sF 0 q instead of FunpT˚r´1sF 0 q itself as the appropriate algebra of field functionals, the Planck constant working here as formal expansion parameter, and replaces S 0 t by S 0 t { throughout. As required, the free BV EA S 0 t satisfies the ME (2.3.2), Since S 0 t is independent from , the two terms in the left hand side of (2.5.1) must separately vanish and so (2.5.1) effectively breaks up into two independent equations.
The RG flow of the free BV EA S 0 t is governed by a free flow map ϕ 0 t,s . S 0 t obeys accordingly the RGE (2.3.13), Unlike the ME (2.5.1), (2.5.2) does not split into simpler equations, because ϕ 0‚ t and r 0‚ t generally depend on S 0 t { , for reasons explained at the end of subsect. 2.3, making the two terms of the right hand side dependent.
The full BV RG EA S t decomposes perturbatively as

5.3)
where the interaction action I t is an element of FunpT˚r´1sF 0 qrr ss.
The full EA S t also satisfies the ME (2.5.1), which on account of the free ME (2.5.1) reduces effectively to where the BV Laplacian ∆ 0 tS 0 { is defined according to (2.3.4). Since I t is a formal power series in , this equation relates the coefficients of the powers of I t in a non trivial manner.
The RG flow of the full BV EA S t is governed by a full flow map ϕ t,s . S t then obeys the RGE Using the decomposition (2.5.3), this yields a RGE for I t . Since ϕ t,s differs in general from its free counterpart ϕ 0 t,s in a complicated way because of the interaction, ϕ ‚ , r ‚ ϕt are not straightforwardly reducible to ϕ 0‚ t , r 0‚ t . For this reason, there is no simple expression for the RGE of I t , which must be deduced on a case by case basis. We shall came back to this point in a specific setting later in subsect. 4.4.
In ref. [11], a formulation of the perturbative RG in BV theory is provided. Its power and many nice features make its paradigmatic. We hope that the BV RG theory presented in this paper may pave the way to alternative equally fruitful perturbative formulations, satisfying the RGE

Batalin-Vilkovisky renormalization group supersymmetry
In this section, we shall introduce the main novelty of the present work, a BV RG set-up enjoying an RG supersymmetry imposing important constraints on the form of the resulting RGE. This will allow us to establish a connection between the BV RGE we obtain and Polchinski's version of the RGE [6].
To develop our line of thought, we are forced to make a few natural assumptions. So, our results are to an extent conjectural. In particular, we have no general proof that the RG supersymmetry is a general property of BV EFT. In next section, we shall illustrate non trivial models exhibiting it.

BV RG supersymmetry
In BV theory, field space is a graded manifold, field functionals form a graded commutative function algebra and basic structures such as the BV form, measure, bracket and Laplacian are all graded in the appropriate sense. The ubiquitous occurrence of the mathematical structures proper of graded geometry in BV theory suggests that it may be worthy to explore whether extension of the RG energy scale parameter manifold R to a more general graded manifold is capable of providing new useful ways of analyzing BV RG theory.
In what follows, we shall demonstrate that enhancing the energy scale space R to its shifted tangent bundle T r1sR does indeed furnish under certain conditions important structural information on the BV RG flow and RGE. To avoid confusion, we shall call the RG set-ups based on R and T r1sR "basic" and "extended", respectively.
Switching from the R to the T r1sR parameter space involves adding to the usual scale parameter t a degree 1 partner θ. In the extended set-up, so, all the constitutive elements of the BV RG framework get modified by acquiring a θ dependence, while maintaining all their formal properties. Further, on general grounds, because of the nilpotence of θ, they can be expressed as non homogeneous linear polynomials of θ.
In the extended set-up, the BV form ω tθ enjoys the expansion where ω t is the basic set-up form and ω ‹ t is a new partner form. The properties of ω tθ as BV form entail that ω ‹ t has degree´2 and is closed. The extended setup BV measure µ tθ is instead independent form θ because of its multiplicative nature and thus equal to its basic counterpart, µ tθ " µ t .
(3.1.2) By (3.1.1), the extended set-up BV bracket p´,´q tθ has the structure pf, gq tθ " pf, gq t`θ p´1q |f | pf, gq ‹ t (3.1. 3) for f, g P FunpT˚r´1sF q, where p´,´q t is the basic set-up bracket and p´,´q ‹ t is a new partner bilinear bracket. The properties of p´,´q tθ as a BV bracket imply that p´,´q ‹ t has degree 0, is graded symmetric and derivative in both its arguments. The graded Jacobi identity, conversely, fails to hold. By (3.1.1) again, the BV Laplacian ∆ tθ decomposes similarly as

1.4)
∆ t being the basic set-up Laplacian and ∆ ‹ t a partner Laplacian, which by the BV Laplacian properties of ∆ tθ has a degree 0 and commutes with ∆ t .
In the extended BV RG framework, the BV RG EA S tθ must likewise be a function of t, θ and therefore expressible as where S t is the basic set-up EA and S ‹ t is a degree´1 partner of it. S tθ must obey the appropriate extended version of the BV quantum ME (2.3.2), ∆ tθ S tθ`1 2 pS tθ , S tθ q tθ " 0. It relates S t and S ‹ t and reads As we shall see in due course, (3.1.7) will play a role in shaping the form of the extended set-up BV RGE.
The RG flow of the extended BV EA S tθ is driven by the flow maps ϕ tθ,sζ , S tθ " ϕ tθ,sζ˚Ssζ`rϕtθ,sζ . obeyed by S t , but it also yields another relation, namely about which we shall have more to say momentarily.
The degree´1 partner S ‹ t of the EA S t allows us to write the infinitesimal generator and logarithmic Jacobian ϕ ‚ t and r ‚ t of the basic set-up BV RG flow ϕ t,s , in terms of which the RGE is written, in the reduced form ϕ ‚ t andr ‚ t being a degree 0 vector field and a degree 0 element of FunpT˚r´1sF q, called reduced infinitesimal generator and Jacobian, respectively. Using these and eq. (3.1.7), the basic set-up BV RGE for the EA S t can be cast as (3.1.14) Recall that, in the RGE (3.1.14), ∆ ‹ t and p´,´q ‹ t are a degree 0 second order differential operator and a degree 0 graded symmetric bracket, respectively. The first two terms in the right hand side, so, have a form analogous to that of the leading terms appearing in Polchinski RGE [6]. Thus, switching to the extended set-up has led us to a more structured RGE than that one would have in the mere basic RG framework. The reduced infinitesimal generator and Jacobianφ ‚ t andr ‚ t turn out under favorable conditions to be simpler than their unreduced counterparts φ ‚ t and r ‚ t . We expect them to correspond to the so-called "seed" terms of the RGE, though at the present level of generality we cannot show this.
To understand the structure of the RGE (3.1.14), we recall that the RG flow of the BV EA S t leaves the partition function invariant. Since the BV structure encoded in the scale dependent BV bracket p´,´q t and measure µ t does itself vary along the flow, the flow rate dS t {dt of S t should comprise in principle two components. The first is such that it would leave the partition function invariant if the variation of the BV structure did not occur. This should therefore be a degree 0 ∆ tS coboundary. The second compensates for the actual variation of the partition function caused by the first due to the variation of the BV structure.
This cannot be a degree 0 ∆ tS coboundary and should not be even a cocycle. The results which we have obtained hinge on the extension of the energy scale space from R to T r1sR by adjoining to the usual scale parameter t an odd partner θ, leading effectively to a sort of RG supersymmetry. Whether there is something more profound behind this seemingly formal procedure we cannot say yet.
Before concluding this subsection, it must be emphasized that in the above analysis we have assumed throughout that the enhancement of the basic set-up to the extended one is possible. However, this cannot be shown in general but only on a case by case basis. In sect. 4, we shall construct a non trivial model admitting an extension.

BV RG flow as a BV MA homotopy
Identifying the degree 1 coordinate θ of T r1sR of the extended RG set-up with the de Rham differential d dR t, the BV EA S tθ given in (3.1.5) can be written as S t`ddR tS t˚a nd so be regarded as a functional on T˚r´1sF valued in the differential graded commutative algebra Ω˚pRq. As such, it represents a homotopy relating the BV MAs at different values of the RG scale t 1 . This homotopy however cannot be described in the derived geometric framework of subsect. 2.4, since in that set-up homotopies relate BV MAs at a fixed value of t. For homotopies of that type, the rate of variation BS xt {Bx is always trivial in the degree 0 ∆ tSx cohomology, while, as discussed at the end of subsect. 3.1, the flow derivative dS t {dt is not and cannot be.
The possibility of viewing the RG flow as a homotopy of BV MAs is implicit in Costello's work [11], but to the best of our knowledge its implications have never been explored to the extent we have done in the present paper.

Perturbative BV RG theory and BV RG supersymmetry
In this subsection, we shall explore the implications of the RG supersymmetry discovered in subsect. 3.1 for perturbative BV RG theory.
When we try to formulate perturbative BV RG theory in the extended set-up, 1 We thank the paper's referee for pointing this out to us.
we have to assume that the free field theory is characterized by an extended BV form ω 0 tθ and measure µ 0 tθ and the associated bracket p´,´q 0 tθ and Laplacian ∆ 0 tθ with the properties described in subsect. 3.1.
Besides the above free BV structure, we have also to assume the existence of an extended free BV RG EA action S 0 tθ so that the basic free EA S 0 t is adjoined by a free degree´1 partner S 0‹ t . Further, not only S 0 t satisfies the BV quantum ME (2.5.1), but S 0 t and S 0‹ t together obey the ME (3.1.7), here reading as As eq. (2.5.1), this too breaks up effectively into two equations due to the independence of S 0 t , S 0‹ t . A free BV RG flow ϕ 0 tθ,sζ governs the t, θ dependence of the extended free EA S 0 tθ . Infinitesimally, this reproduces the basic free BV RGE (2.5.2) for S 0 t and further relates S 0 t and S 0‹ t as in (3.1.11) By virtue of (3.3.1), further, the RGE (2.5.2), can be written in the Polchinski form (3.1.14), In the extended perturbative BV RG framework, the full BV RG EA S tθ decomposes in analogy to (2.5.3) as

3.4)
where the extended interaction action I tθ belongs to the formal power series algebra FunpT˚r´1sF 0 qrr ss. This latter enjoys in turn the expansion

3.5)
where I t is the basic set-up interaction action and I ‹ t is a degree´1 partner of it. In this way, in addition to the perturbative splitting (2.5.3), we have also The full EA S tθ obeys further the extended quantum ME (3.1.6) (with p´,´q tθ and ∆ tθ replaced by p´,´q 0 tθ and ∆ 0 tθ ). Insertion of the perturbative expansion (3.3.4) into this latter reproduces upon taking eqs. (2.5.1), (3.3.1) into account eq. (2.5.5) and a further equation where the BV Laplacian ∆ 0 tS 0 { is defined according to (2.3.4) and ∆ 0‹ tS 0 { is the degree 0 Laplacian given by As in the basic perturbative RG framework of subsect. 2.5, the extended BV RG flow ϕ tθ,sζ that governs the t, θ dependence of the extended full EA S tθ differs in principle from it free counterpart ϕ 0 tθ,sζ because of the effect of the interactions. Therefore, the RGE equation for S tθ is not simply that for S 0 tθ with S 0 tθ replaced by S tθ . In particular, the basic full EA S t obeys the RGE (2.5.6), the S ‹ t to S t relation (3.1.11) takes the form and the Polchinski RGE (3.1.14) reads However, as we already mentioned in subsect 3.1, we expect that the reduced full infinitesimal generator and logarithmic Jacobianφ ‚ t andr ‚ ϕt to be somewhat related to the seed terms of the RGE, which, in a perturbative framework, are normally determined by the free EA S 0 t only and are independent from interactions. For this reason, we assume that in the RGE (3.3.10), one has Taking this for granted, we can now subtract the free and full RGE (3.3.3) and (3.3.10) and obtain an RGE equation for the interaction action I t . Upon using relation (3.1.12) for ϕ 0 t,s and S 0 t , we find into account, the following revealing equations are found (3.3.14), (3.3.14) provide an interpretation of interaction action partner I ‹ t : it drives the deviation of the full BV RG flow ϕ t,s from its free counterpart ϕ 0 t,s . Analogously to its non perturbative counterpart, eq. eq. (3.3.9) provides implicitly I ‹ t in terms of I t and so works as an integration formula. We have not been able to recast it in a way that neatly separates the free and interacting contributions. More work is required to elucidate this point.
The conclusions of our analysis rest of course on the assumptions (3.3.11), (3.3.12) and thus fail if these do not obtain. Again, one must check them on a case by case basis. In the model detailed in sec. 4, they are verified to hold. It is also found that the last two seed-like terms in the right hand side of (3.3.13) cancel out leaving a seedless RGE.

Models of Batalin-Vilkovisky renormalization group
An important issue in the BV theory of the RG developed in this paper is the construction of non trivial models which exemplify it. A program of this scope certainly cannot be carried out to its full extent in the limited space of this paper.
Still, a few simple but non trivial models can be built.
In what follows, we work in the degree´1 symplectic framework originally developed by Costello in ref. [11], which has a very rich structure and lends itself particularly well to our task. We first review the framework to set our notation.
We then illustrate a free model of BV RG flow and EA both in the basic and extended set-ups of subsect. 3.1. Finally, we explore the implications of results found for perturbation theory.
All the statements made below hold strictly speaking in finite dimension. Presumably, they can extended also to an infinite dimensional context with limited modifications.

The degree -1 symplectic set-up
In this technical subsection, we review the degree´1 symplectic set-up of ref. [11], which we shall employ to construct a class of models of BV RG theory. A few original results are also presented along the way. A more thorough account of these matter will be provided in ref. [23].
We begin by setting our notation. The basic algebraic structures we shall be concerned with are a graded vector space E, its dual vector space E˚and its internal endomorphism algebra EndpEq. We shall consider further the full degree Rrps. E is a graded vector space, E˚is its dual vector space and EndpEq is its internal endomorphism algebra, as suggested by the notation. For each p P Z, E p , E˚p and End p pEq are just respectively E, Eå nd EndpEq with degree reset to the uniform value p. We shall denote by |´| the degree map of all the graded spaces and algebras considered above. We shall work mostly with E, E˚, EndpEq.
Homogeneous bases of E, E˚come in dual pairs a i , a˚i with the property that |a i |`|a˚i| " 0. Below, we shall set ǫ i "´|a i | " |a˚i| for convenience. Given a dual basis pair a i , a˚i, we can expand homogeneous vectors e P E, covectors l P E˚and endomorphisms A P EndpEq as e " a i e i , l " l i a˚i and A " a i b A i j a˚j, where |e i | " |e|`ǫ i , |l i | " |l|´ǫ i and |A i j | " |A|`ǫ i´ǫj . A homogeneous endomorphism A P EndpEq is characterized by its graded trace, which is defined as It can be shown that grtrpAq is independent from basis choices. Further, one has | grtrpAq| " |A| and grtrpABq " p´1q |A||B| grtrpBAq for A, B P EndpEq. See ref.
In the symplectic set-up, the graded vector space E is equipped with a degreé 1 symplectic pairing, i. e. an antisymmetric bilinear form x´,´y E : EˆE Ñ R with the property that for homogeneous e, f P E, xe, f y E " 0 whenever |e|| f | " 1. x´,´y E induces a bilinear pairing x´,´y E : EˆE Ñ G R such that x´,´y E : E pˆEq Ñ Rr´p´q`1s and, in particular, x´,´y E : E 0ˆE0 Ñ Rr1s.
The pairing x´,´y E can be viewed as degree´1 vector space isomorphism ̟ E : E Ñ E˚. With respect to a given basis a˚i of E˚, ̟ E can be expanded as where ω ij is a non singular antisymmetric real matrix such that ω ij " 0 for ǫ i`ǫj`1 " 0.
Using the matrix ω ij and its inverse ω ij , one can define the symplectic dual bases a i and a˚i of the bases a i and a˚i of an (algebraically) dual pair by a i " a j ω ji and a˚i "´ω ij a˚j. These have degrees |a i | "´ǫ i and |a˚i| " ǫ i , respectively, where ǫ i "´ǫ i´1 . Vectors e P E, covectors l P E˚and endomorphisms A P EndpEq can be expanded with respect the dual bases a i , a˚i e. g. e "´a i e i , with respect the dual bases are related to those with respect to the given bases as expected, e. g. e i "´ω ij e j , l i " l j ω ji , A i j "´ω ik A k l ω lj , etc. Care must be taken when dealing with signs.
The transpose of a homogeneous endomorphism A P EndpEq is the homoge-neous endomorphism A " P EndpEq such that xe, Af y E " xf, A " ey E for e, f P E 0 .
The matrix components of A " are given in terms of those of A by the expression One has |A " | " |A| and pABq " " p´1q 1`|A||B| B " A " for A, B P EndpEq and 1 E " "´1 E . Furthermore, one has grtr A " " grtr A, so that grtr A " 0 whenever Henceforth, we concentrate on the degree 0 subspace E 0 Ă E. E 0 can be endowed with a structure of graded manifold through a set of globally defined graded coordinates x i . Upon picking a basis a˚i of E˚, the x i are the elements of E˚of degree |x i | " ǫ i corresponding to a˚i. The symplectic dual coordinates x i "´ω ij x j of degree |x i | " ǫ i can also be used as coordinates of E 0 .
E 0 is naturally a BV manifold isomorphic to the canonical BV manifold The canonical BV measure is simply for u, v P FunpE 0 q. Further, the canonical BV Laplacian ∆ E associated with ω E and µ E is given by The canonical BV structure of E 0 specified by the BV form and measure ω E , µ E is not sufficient for the formal developments of later subsections. A suitable generalizations of it is required. It is possible to construct a deformation ω A , µ A of ω E , µ E for any endomorphism A P EndpEq that satisfies |A| " 0 and A " "´A and is invertible. The A-deformed BV form reads The A-deformed BV measure equals instead the canonical one The A-deformed BV bracket on FunpE 0 q takes in this way the the form while the A-deformed BV Laplacian reads In the calculations carried out below, we use repeatedly a host of basic identities, which we collect here for convenience in index free form and whose proof will be given in [23]. A part of these involve vector fields of the basic form Another part concern quadratic functions of FunpE 0 q of the form xx, Nxy E (4.1.14) with N P EndpEq an endomorphism of E such that N " "`N. They are characterized by simple deformed BV brackets: if A, B, C P EndpEq are endomorphism such that A " "´A and B " "`B, C " "`C, then pxx, Bxy E , xx, Cxy E q A " 4xx, BACxy E . The action of the deformed Laplacian on them is also simple enough,

Symplectic gl(1|1) structures
glp1|1q structures, which we review briefly in this subsection, are recurrent in differential geometry, supersymmetric quantum mechanics and topological sigma models. An glp1|1q structure enters also in the symplectic set-up of ref. [11] as one of its constitutive elements.

Free models of BV RG
The basic datum required for the construction of free models of BV RG illustrated below is a degree´1 symplectic vector space E together with a glp1|1q structure Q, Q, H, F on E.
Our aim is working out a free BV EFT on the field space E 0 " T˚r´1sF 0 .
To this end, as explained in subsect. 2.3, we have to replace the unregularized BV form and measure ω E , µ E with effective counterparts ω t , µ t depending on an energy scale parameter t. Inspired by Costello 's formulation [11], we choose ω t , µ t to be the deformations of ω E , µ E associated with the endomorphism e´t H P EndpEq in accordance with (4.1.7), (4.1.8) with the right hand sides defined conforming with (4.1.9), (4.1.10), respectively.
The RG flow ϕ 0 t,s of the free BV EFT is generated infinitesimally by the degree 0 vector field xx, H ad E xy E , ϕ 0 t,s " expˆt´s 2 xx, H ad E xy E˙. The pull-back action ϕ 0 t,s˚o f ϕ 0 t,s on the internal function algebra FunpE 0 q of E 0 is given also by the right hand side of (4.3.5) upon regarding xx, H ad E xy E as a derivation of FunpE 0 q.
The proof that ϕ 0 t,s is a BV RG flow is simple enough. To begin with, we note that relations (2.3.6)-(2.3.8) are trivially satisfied. To show that the maps ϕ 0 t,s are canonical, it is enough to prove that they intertwine the BV brackets p´,´q 0 t as in (2.1.13). This follows from the relation pu, vq 0 t " ϕ 0 t,0˚p ϕ 0 t,0´1˚u , ϕ 0 t,0´1˚v q E , (4.3.7) which can be shown straightforwardly using (4.1.12). The vanishing of r ϕ 0 t,s can be inferred by comparing the relation ∆ 0 t " ϕ 0 t,0˚∆E ϕ 0 t,0´1˚, (4.3.8) which is a simple consequence of (4.1.13), and (2.1.17).
In the extended theory, the appropriate enhancement of the basic theory BV EA is the two-parameter family of free actions S 0 tθ P FunpE 0 q given by S 0 tθ "´1 2 xx, Q e tH xy E´θ 1 4 xx, tQ, Qu e tH xy E , (4.3.21) where tQ, Qu " QQ´QQ. We are now going to show that S 0 tθ has the required properties of a BV RG EA. as can be proven easily using that ϕ 0 0θ,00 " exp`θ 1 2 xx, Q ad E xy E˘b y (4.3.17) and employing (4.1.17). Next, we note that as required.
In the extended set-up, the RGE obeyed by S 0 t " S 0 t0 can be cast as Jacobian of a BV RG flow ϕ t,s . ϕ t,s is the flow generated infinitesimally by the 5 Outlook, beyond renormalization group In this final section, we speculate about possible applications of the BV RG framework we have described in depth in the main body of the paper.
In physical applications of RG theory, much effort is devoted to the derivation of the RGEs of the couplings of the basic fields in the EA and the computation of the appended beta functions. These RGEs are implicit in the RGE of the EA.
It would be interesting to devise systematic methods to obtain them within the BV RG framework of this paper. See ref. [17] for an alternative approach to this problem.
The connection between the weak coupling limit of the RG flow of twodimensional non linear sigma models, originally studied by Friedan in [31][32][33], and the Ricci flow, introduced by Hamilton in [34], is by now established [35,36].
Ricci flow has has played a pivotal role in important developments of geometric analysis, such as Perelman's proof of Thurston's geometrization program [37,38] for three-manifolds and the related Poincaré conjecture [39][40][41].
The Alexandrov-Kontsevich-Schwartz-Zaboronsky (AKSZ) formulation of the BV quantization scheme [42] is a general framework for the construction of the BV MA of a broad class of sigma models, at least in the semiclassical limit. It has had a wide range of field theoretic applications with remarkable mathematical ramifications. See ref. [43] for a recent review of the AKSZ approach.
In the AKSZ approach, the fulfilment of the BV ME by the MA of a sigma model, and so its ultimate quantum consistency, rests on relevant features of the model's target space geometry. The AKSZ approach can therefore be employed to construct for any given target geometry a canonical sigma model capable of probing it. It is conceivable that a carefully designed BV RGE for the model may