Exact renormalization group in Batalin-Vilkovisky theory

In this paper, inspired by the Costello’s seminal work [11], 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 [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 JHEP03(2018)132 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 (−, −) t and Laplacian ∆ t depend on t. The BV MA S t also depends on t and JHEP03(2018)132 satisfies the BV quantum ME ∆ t S t + 1 2 (S t , S t ) t = 0. (1.1) 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). 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 * S s + r ϕt,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 section 2, where the interplay of the RG flow and gauge fixing is also discussed. In section 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 [1]R 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 [1]R 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 (−, −) 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 * S sζ + r ϕtθ,sζ . At the infinitesimal level, this entails a more structured RGE than (1.2). Expanding in powers of the odd parameter θ, we have (−, −) tθ = (−, −) t ± θ(−, −) t and ∆ tθ = ∆ t + θ∆ t , where (−, −) t is a degree 0 graded symmetric bracket and ∆ t is a degree 0 Laplacian. Further, 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 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 JHEP03(2018)132 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 section 4, working in the gl(1|1) 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 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 section 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 JHEP03(2018)132 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/k(H/t), where k(x) 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 k(x) with a function taking the value 1 for x < 1 and 0 for x > 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 for f ∈ Fun(M). ∆ M is a degree 1 second order differential operator. The compatibility of the BV form ω M and measure µ M consists in ∆ M being nilpotent, This is a non trivial constraint relating ω M and µ M . The BV Laplacian ∆ M enjoys a number of relevant properties. ∆ M encodes the BV bracket (−, −) M as for any function f ∈ Fun(M) satisfying As the BV Laplacian ∆ M is nilpotent by (2.5), the BV cohomology H ∆ M * (M) is defined.
By the BV Stokes' theorem, the L integration map factors through a map L µ M | L 1/2 − : Let B be a fixed collection of BV manifolds which are diffeomorphic as Z-graded manifolds. For instance, B may consist of BV manifolds which are equal as Z-graded manifolds but have different BV forms and measures, the situation mostly considered in this paper.
The natural invertible maps φ : M → M of two BV manifolds M, M ∈ B are the canonical ones, that is those with the property that φ * ω M = ω M . They preserve the BV bracket, so that for all functions f, g ∈ Fun(M).
The BV integral has the usual covariance properties. If φ : M → M is a canonical map of two BV manifolds M, M ∈ B, then there is a nowhere vanishing degree 0 function J ϕ ∈ Fun(M ), called the Jacobian of φ, such that

JHEP03(2018)132
for any f ∈ Fun(M). The logarithmic Jacobian is a natural, equivalent substitute for J φ . It obeys the equation The pull-back action of a canonical map φ : M → M of BV manifolds M, M ∈ B does not intertwine between the BV Laplacians ∆ M , ∆ M . In fact, for f ∈ Fun(M). It does however if φ is special. Relation (2.13) expresses the non covariance of ∆ M for a generic φ.
The BV manifolds of the collection B and their canonical maps constitute a groupoid under composition and inversion, the canonical groupoid of B. The BV manifolds of B together with the special canonical maps constitute a subgroupoid of the canonical groupoid of B, the special canonical groupoid of B. However, we shall not rely on this categorical interpretation in what follows.
If φ : M → M is a canonical map of BV manifolds M, M ∈ B and L is a Lagrangian submanifold of M , then φ * L is a Lagrangian submanifold of M as well. In this case, (2.9) and the identity 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 * a dx a and µ can = d dim F xd dim F x * . The BV bracket and Laplacian are in this case and respectively, where a = |x a |, * 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 * [−1]F 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 subsection 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 crit(S cl ) 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 * [−1]F of F with fields and antifields parametrizing respectively its base and fibers. T * [−1]F 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 * [−1]F 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 * [−1]F with the Boltzmann weight e S inserted. The unnormalized correlator of a functional f on T * [−1]F 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 * [−1]F , 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

JHEP03(2018)132
where ∆ canS is the degree 1 second order linear differential operator defined by ∆ canS is called covariant BV Laplacian. By virtue of (2.22), ∆ canS is nilpotent and is therefore characterized by its cohomology. Inserting the Boltzmann weight e S in the functional integral (2.21) amounts to redefine the BV measure from µ can to The BV ME (2.22) 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 (2.4).

BV RG theory
In a concrete setting in which the fields in T * [−1]F 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 (−, −) can and Laplacian ∆ can by the associated effective BV bracket (−, −) t and Laplacian ∆ t . There is a price attached to this: ω t , µ t , (−, −) t and ∆ t are mathematically far more complicated than ω can , µ can , (−, −) 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 * [−1]F t having T * [−1]F 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 subsection 2.2, quantization of the EFT involves restriction of integration to a Lagrangian submanifold L t of T * [−1]F t with insertion of the Boltzmann weight e St of a BV EA S t . Unnormalized correlators thus read as

JHEP03(2018)132
for any functional f on T * [−1]F . 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 (2.28), ∆ 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 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 * [−1]F 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 : for s, t ∈ R. The basic requirement that the RG flow must obey is that the partition function Z S (1) remains constant along it, so that for any two value s, t of the energy scale A sufficient condition for the identity (2.35) to hold is that the Lagrangian submanifolds L s , L t are related by 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 (2.15).

JHEP03(2018)132
Relation (2.38) 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 is the BV RGE in its more general form. It has the standard form of an RGE, the first and second term in the right hand side corresponding to the so-called classical and quantum term, respectively. The concrete embodiment of the BV RGE (2.39) obeyed by the BV EA S t depends on the form of the RG flow map ϕ t,s . At first glance, (2.39) appears to be a non homogeneous linear differential equation in S t , but this is a deception of notation. Relation (2.38) on which (2.39) rests is in fact a condition simultaneously constraining and relating ϕ t,s and S t . The flow map ϕ t,s therefore is not a datum independent from the action S t but in general contains built in information on the latter. The realizations of ϕ t,s encountered in concrete applications do indeed depend explicitly on S t . The RGE (2.39), so, normally is nonlinear in S t .
The BV RG framework we have just outlined fits with the Polchinski's RG theory [6] reviewed in subsection 1.1 and is also in line with the RG formulations of refs. [24][25][26], although at this point we have not yet committed ourselves with 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 * [−1]F is modelled as a functional on T * [−1]F 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 (−, −) t and RG flow maps ϕ t,s are assumed to be given. The ∆ t , (−, −) t and the pull-backs ϕ t,s * of the ϕ t,s act on functional families through their A-linear extensions. A derived BV EA is then a t dependent degree 0 A-valued functional S At on T * [−1]F defined modulo A-valued constants and obeying the derived BV quantum ME (cf. eq. (2.28)), d A being the differential of A, and the RGE  Then, eq. (2.43) holds exactly by straightforward differentiation of (2.45) with respect to t. Once this is done, however, eq. (2.42) gets weakened as where α A is a degree 1 element of A. It can be verified that α A is independent from t, that d A α A = 0 and that a different choice of the lift S At alters α A by an amount d A β A for some degree 0 element β A of A. Therefore, there exists a degree 1 cohomology class [α A ] in H 1 (A, d A ) independent from the scale t that obstructs the existence of a lift S At obeying (2.42) exactly. 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 * (I) is the algebra of differential forms in the interval I = [0, 1] and d A = d dR is the Rham differential. The derived BV EA is of the form S Ω * (I)t = S xt + d dR x S * xt , where S xt , S * xt have respectively degrees 0, −1. The obstruction form reads as α Ω * (I) = α x + d dR x α * x with components α x , α * x of degrees 1, 0. The deformation S Ω * (I)t 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 x 0 dx α * x . Upon doing so, for fixed x, S xt is a BV EA and ∂S xt /∂x is trivial in the ∆ tSx cohomology.
In many instances, one considers the case where A is the Chevalley-Eilenberg algebra CE(g) of an L ∞ algebra g and d A is the associated Chevalley-Eilenberg differential d CE(g) . In such cases, g encodes a generalized symmetry of the EFT and the derived BV EA S CE(g)t describes an action of g on the a BV EFT in the interpretation of ref. [13].

JHEP03(2018)132
If S CE(g)t can be lifted, then the action of g is called inner. As we have just shown, the liftability of S CE(g)t is equivalent to the vanishing of the obstruction class [α CE(g) ] in H 1 (CE(g), d CE(g) ). The cohomology H * (CE(g), d CE(g) ) is just the Chevalley-Eilenberg cohomology of g. A class [α CE(g) ] in H 1 (CE(g), d CE(g) ) so describes a −1 shifted central extension of g. 0 The non triviality of the extension is thus tantamount to the non liftability of S CE(g)t . 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 * [−1]F 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 Fun(T 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 (2.28), Since S 0 t is independent from , the two terms in the left hand side of (2.50) must separately vanish and so (2.50) 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 (2.39), Unlike the ME (2.50), (2.51) 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 subsection 2.3, making the two terms of the right hand side dependent.
The full BV RG EA S t decomposes perturbatively as where the interaction action I t is an element of Fun(T The full EA S t also satisfies the ME (2.50),

JHEP03(2018)132
which on account of the free ME (2.50) reduces effectively to where the BV Laplacian ∆ 0 tS 0 / is defined according to (2.30). 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 (2.52), 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 subsection 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 [1]R 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 [1]R "basic" and "extended", respectively.
Switching from the R to the T [1]R 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 JHEP03(2018)132 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 set-up 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 .
for f, g ∈ Fun(T * [−1]F ), where (−, −) t is the basic set-up bracket and (−, −) t is a new partner bilinear bracket. The properties of (−, −) tθ as a BV bracket imply that (−, −) t has degree 0, is graded symmetric and derivative in both its arguments. The graded Jacobi identity, conversely, fails to hold. By (3.1) again, the BV Laplacian ∆ tθ decomposes similarly as

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 S tθ = S t + θS t , (3.5) 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 (2.28), Insertion of the expansions (3.3), (3.4) and (3.5) into (3.6) yields two equations. The first is the basic BV quantum ME (2.28). The second is novel. It relates S t and S t and reads As we shall see in due course, (3.7) will play a role in shaping the form of the extended set-up BV RGE.

JHEP03(2018)132
where ϕ t,s is the basic set-up RG flow and ϕ t is a degree −1 vector field on T * [−1]F . The associated logarithmic Jacobian r ϕtθ,sζ similarly read as r ϕtθ,sζ = (id Fun(T * [−1]F ) +θϕ t ) * (r ϕt,s + θr ϕt − ζϕ t,s * r ϕs ), (3.9) where r ϕt,s is the basic set-up logarithmic Jacobian and r ϕt is a degree −1 element of Fun(T * [−1]F ). The canonical nature of the flow maps ϕ tθ,sζ entails that ∆ t , ∆ t , ϕ t,s , ϕ t , r ϕt,s and r ϕt obey a host of identities which follow from substituting expressions (3.4), (3.8) and (3.9) into the identity ∆ tθ ϕ tθ,sζ − ϕ tθ,sζ ∆ sζ + ad tθ r ϕtθ,sζ ϕ tθ,sζ = 0 (cf. eq. (2.17)). The RG flow of the extended BV EA S tθ is driven by the flow maps ϕ tθ,sζ , At the infinitesimal level, (3.10) reproduces the basic set-up BV RGE (2.39) 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 Fun(T * [−1]F ), called reduced infinitesimal generator and Jacobian, respectively. Using these and eq. (3.7), the basic set-up BV RGE for the EA S t can be cast as (3.14) Recall that, in the RGE (3.14), ∆ t and (−, −) 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 (3.14), 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 (−, −) 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.

JHEP03(2018)132
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.
Eq. (3.7) shows that the sum of the first two terms in the right hand side of the RGE (3.14) is a degree 0 ∆ tS coboundary. This contribution is so naturally identified with the first component of the flow rate dS t /dt hypothesized in the previous paragraph. The sum of the last two terms in the right hand side of (3.14) should thus constitute the second component. Note that the degree −1 partner S t of S t acts as the ∆ tS integral of the first component. Beyond this, its role needs to be further elucidated. More about this in subsection 3.3.
The results which we have obtained hinge on the extension of the energy scale space from R to T [1]R 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 section 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 [1]R of the extended RG set-up with the de Rham differential d dR t, the BV EA S tθ given in (3.5) can be written as S t + d dR tS t * and so be regarded as a functional on T * [−1]F valued in the differential graded commutative algebra Ω * (R). 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 subsection 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 ∂S xt /∂x is always trivial in the degree 0 ∆ tSx cohomology, while, as discussed at the end of subsection 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 subsection 3.1 for perturbative BV RG theory.
When we try to formulate perturbative BV RG theory in the extended set-up, 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 (−, −) 0 tθ and Laplacian ∆ 0 tθ with the properties described in subsection 3.1.

JHEP03(2018)132
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 (2.50), but S 0 t and S 0 t together obey the ME (3.7), here reading as As eq. (2.50), 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 (2.51) for S 0 t and further relates S 0 t and S 0 t as in (3.11) By virtue of (3.15), further, the RGE (2.51), can be written in the Polchinski form (3.14), In the extended perturbative BV RG framework, the full BV RG EA S tθ decomposes in analogy to (2.52) as S tθ = S 0 tθ + I tθ , (3.18) where the extended interaction action I tθ belongs to the formal power series algebra . This latter enjoys in turn the expansion 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 (2.52), we have also (3.20) The full EA S tθ obeys further the extended quantum ME (3.6) (with (−, −) tθ and ∆ tθ replaced by (−, −) 0 tθ and ∆ 0 tθ ). Insertion of the perturbative expansion (3.18) into this latter reproduces upon taking eqs. (2.50), (3.15) into account eq. (2.54) and a further equa- where the BV Laplacian ∆ 0 tS 0 / is defined according to (2.30) and ∆ 0 tS 0 / is the degree 0 Laplacian given by 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 (2.55), the S t to S t relation (3.11) takes the form S t = ϕ t S t + r ϕt (3.23) and the Polchinski RGE (3.14) 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 (3.24), one hasφ Taking this for granted, we can now subtract the free and full RGE (3.17) and (3.24) and obtain an RGE equation for the interaction action I t . Upon using relation (3.12) for ϕ 0 t,s and S 0 t , we find (3.29) (3.28), (3.28) 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. (3.7), discussed in subsection 3.1, eq. (3.21) expresses roughly the triviality in the degree 0 ∆ tS / cohomology of the non seed terms of the RGE (3.27), the degree −1 partner I t of the interaction action I t being their integral. On account of (2.52) and (3.18), eq. (3.23) 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 (3.25), (3.26) and thus fail if these do not obtain. Again, one must check them on a case by case basis. In the model detailed in section 4, they are verified to hold. It is also found that the last two seed-like terms in the right hand side of (3.27) cancel out leaving a seedless RGE.

JHEP03(2018)132 4 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 subsection 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 End(E). We shall consider further the full degree prolongations . E is a graded vector space, E * is its dual vector space and End(E) is its internal endomorphism algebra, as suggested by the notation. For each p ∈ Z, E p , E * p and End p (E) are just respectively E, E * and End(E) 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 * , End(E).
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 i = −|a i | = |a * i | for convenience. Given a dual basis pair a i , a * i , we can expand homogeneous vectors e ∈ E, covectors l ∈ E * and endomorphisms A ∈ End(E) as e = a i e i , l = l i a * i and A = a i ⊗ A i j a * j , where |e i | = |e| + i , |l i | = |l| − i and |A i j | = |A| + i − j . A homogeneous endomorphism A ∈ End(E) is characterized by its graded trace, which is defined as It can be shown that grtr(A) is independent from basis choices. Further, one has | grtr(A)| = |A| and grtr(AB) = (−1) |A||B| grtr(BA) for A, B ∈ End(E). See ref. [30] for an analogous notion.
In the symplectic set-up, the graded vector space E is equipped with a degree −1 symplectic pairing, i.e. an antisymmetric bilinear form −, − E : E × E → R with the property that for homogeneous e, f ∈ E, e, f E = 0 whenever |e| + |f | = 1. −, − E induces JHEP03(2018)132 The pairing −, − E can be viewed as degree −1 vector space isomorphism E : E → E * . With respect to a given basis a * i of E * , E can be expanded as E = a * i ⊗ ω ij a * i , where ω ij is a non singular antisymmetric real matrix such that ω ij = 0 for i + j + 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 | = − i and |a * i | = i , respectively, where i = − i −1. Vectors e ∈ E, covectors l ∈ E * and endomorphisms A ∈ End(E) can be expanded with respect the dual bases a i , a * i e. g.
The components 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 ∈ End(E) is the homogeneous endomorphism A ∼ ∈ End(E) such that e, Af E = f, A ∼ e E for e, f ∈ E 0 . The matrix components of A ∼ are given in terms of those of A by the expression One has |A ∼ | = |A| and (AB) Furthermore, one has grtr A ∼ = grtr A, so that grtr A = 0 whenever A ∼ = −A.
Henceforth, we concentrate on the degree 0 subspace E 0 ⊂ 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 | = i corresponding to a * i . The symplectic dual coordinates x i = −ω ij x j of degree |x i | = i can also be used as coordinates of E 0 .
E 0 is naturally a BV manifold isomorphic to the canonical BV manifold T * [−1]F 0 of some graded vector space F 0 regarded as a graded manifold. The Darboux coordinates of E 0 are the coordinate functions x i associated to a chosen Darboux basis a i for the symplectic pairing −, − E . The canonical BV form of E 0 is the degree −1 symplectic form The canonical BV measure is simply Let Fun(E 0 ) be the internal graded commutative algebra of smooth functions on E 0 with the usual grading |−|. The canonical BV bracket (−, −) E on Fun(E 0 ) associated with for u, v ∈ Fun(E 0 ). Further, the canonical BV Laplacian ∆ E associated with ω E and µ E is given by for u ∈ Fun(E 0 ).

JHEP03(2018)132
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 ∈ End(E) 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 Fun(E 0 ) takes in this way the form while the A-deformed BV Laplacian reads where again u, v ∈ Fun(E 0 ). When A = 1 E , the A-deformed BV form and measure reduce to the canonical ones as do the deformed BV bracket and Laplacian. It is convenient however to proceed in another direction and relax some of the restriction on the deforming endomorphism A ∈ End(E) imposing only that A ∼ = −A but requiring that neither |A| = 0 nor A is invertible. The A deformed BV bracket (−, −) A and Laplacian ∆ A can still be defined through (4.9) and (4.10). (−, −) A exhibits properties generalizing those of the canonical BV bracket (−, −) E except for the graded Jacobi identity. Similarly, ∆ A enjoys properties extending those of the canonical BV Laplacian ∆ E except for nilpotence. For (−, −) A and ∆ A to have all the properties of a BV bracket and Laplacian, respectively, it is required in addition that |A| = 0 mod 2.
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 where K ∈ End(E) is an endomorphism of E such that K ∼ = −K. These act naturally on deformed brackets: for A, B ∈ End(E) such that A ∼ = −A, |A| = 0 mod 2 and B ∼ = −B, one has

JHEP03(2018)132
Another part concern quadratic functions of Fun(E 0 ) of the form x, N x E (4.14) with N ∈ End(E) an endomorphism of E such that N ∼ = +N . They are characterized by simple deformed BV brackets: if A, B, C ∈ End(E) are endomorphism such that A ∼ = −A and B ∼ = +B, C ∼ = +C, then The action of the deformed Laplacian on them is also simple enough, Finally, we have

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

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 gl(1|1) structure Q, Q, H, F on E.
Our aim is working out a free BV EFT on the field space E 0 = T * [−1]F 0 . To this end, as explained in subsection 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 −tH ∈ End(E) in accordance with (4.7), (4.8) Since | e −tH | = 0 and e −tH∼ = − e −tH and e −tH is invertible, this can be consistently done. The effective BV bracket and Laplacian (−, −) 0 t , ∆ 0 t are then given by the concomitant deformations of (−, −) E , ∆ E , viz ∆ 0 t = ∆ e −tH (4.39) with the right hand sides defined conforming with (4.9), (4.10), respectively.

JHEP03(2018)132
In perturbation theory, one promotes the function algebra Fun(E 0 ) to the formal power series algebra Fun(E 0 )[[ ]] of the parameter over Fun(E 0 ). Further, for fixed t, the full action S t ∈ Fun(E 0 )[[ ]] is rescaled by −1 . It is further assumed that S t satisfies the same BV quantum ME as the free action S 0 t , ∆ e −tH S t + 1 2 (S t , S t ) e −tH = 0. (4.62) Aiming to a perturbative analysis of the BV RG, one splits the action S t as where S 0 t is the free action (4.44) and I t ∈ Fun(E 0 )[[ ]] is an interaction term at least cubic in x mod .
A simple calculation shows that I t obeys the BV quantum ME ∆ e −tH I t − QI t + 1 2 (I t , I t ) e −tH = 0, (4.64) where Q is the degree 1 first order differential operator defined by  64) is precisely of the same form as the ME for I t obtained by Costello in ref. [11]. The next problem we have to address is determining the BV RG flow of the full action S t . We assume as a working hypothesis that this is governed by the Polchinski's RGE with seed action S 0 t reducing to the free RGE (4.61) in the limit of vanishing I t is The last term may be absorbed by adding a constant term t 2 grtr(QQ) to S t and for this reason is usually neglected, but we shall keep it for the time being. Next, we are going to argue that (4.66) is the correct full RGE.
By the perturbative decomposition (4.63), the RGE (4.66) can be cast as introduced by Hamilton in [34], is by now established [35,36]. Ricci flow 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 yield a flow equation analogous to Ricci flow potentially useful for the study of important geometrical and topological issues of the target manifold. This remains at the moment an unexplored possibility.