Batalin-Vilkovisky quantization of fuzzy field theories

We apply the modern Batalin-Vilkovisky quantization techniques of Costello and Gwilliam to noncommutative field theories in the finite-dimensional case of fuzzy spaces. We further develop a generalization of this framework to theories that are equivariant under a triangular Hopf algebra symmetry, which in particular leads to quantizations of finite-dimensional analogs of the field theories proposed recently through the notion of `braided $L_\infty$-algebras'. The techniques are illustrated by computing perturbative correlation functions for scalar and Chern-Simons theories on the fuzzy $2$-sphere, as well as for braided scalar field theories on the fuzzy $2$-torus.


Introduction
Noncommutative quantum field theories are well-known to exhibit many novel features not present in conventional quantum field theory, see e.g. [Sza03] for a review. In particular, despite many years of extensive investigation, the quantization of noncommutative gauge theories is not completely understood, see e.g. [BKSW10] for a review. In this paper we will offer a new perspective on this problem by applying modern incarnations of Batalin-Vilkovisky (BV) quantization to noncommutative field theories. We follow the approach of Costello and Gwilliam [CG16,Gwi12]. We treat only fuzzy field theories, which are by definition finite-dimensional systems, i.e. matrix models, and so can be quantized in a completely rigorous way while avoiding the analytic issues involved when dealing with continuum field theories. These examples will serve to nicely illustrate our formalism while avoiding much technical clutter. We give a general review of these quantization techniques in Section 2. A finite-dimensional BV formalism for certain matrix models is also discussed in [Ise19a,Ise19b], and related to the spectral triple formulation of noncommutative geometry in [IvS17].
Our approach is inspired in part by recent analyses of noncommutative field theories in the framework of L ∞ -algebras. The classical L ∞ -algebra formulation of the standard noncommutative gauge theories was originally presented in [BBKL18]. A new notion of 'braided L ∞ -algebra' was defined more recently in [DCGRS20,DCGRS21], where it was used to construct 'braided field theories' which are equivariant under the action of a triangular Hopf algebra and involve fields with braided noncommutativity. We would like to stress that the term 'braided' in these papers, as well as in our present one, is used to refer to algebraic structures and field theories that are defined in a symmetric braided monoidal category, where however the triangular R-matrix is non-trivial in the sense that it is not the identity. A generalization of [DCGRS20,DCGRS21] and the results of the present paper to the truly (i.e. non-symmetric) braided case is considerably more complicated, see Section 6 for more comments. To handle the theories in [DCGRS20,DCGRS21], we develop a braided version of the BV formalism in Section 4, which fully captures their perturbative quantization and explicitly computes their correlation functions. Our perspective circumvents the issues involved in constructing the classical (Maurer-Cartan) solution space in the equivariant and braided setting that were pointed out in [DCGRS21], as it characterizes this space by its equivariant function dg-algebra. For scalar field theories, our approach agrees with Oeckl's algebraic approach to (symmetric) braided quantum field theory [Oec01,Oec00], which is based on a braided generalization of Wick's theorem and Gaussian integration. Our framework has the advantage of being able to straightforwardly treat theories with gauge symmetries, which are not addressed in Oeckl's approach.
In this paper we treat prototypical fuzzy versions of both types of noncommutative field theories within the BV formalism. These are defined respectively on the fuzzy 2-sphere (Section 3) and on the fuzzy 2-torus with a non-trivial R-matrix (Section 5).
On the fuzzy sphere we illustrate the finite-dimensional BV formalism for both scalar and gauge field theories. We study in detail the example of Φ 4 -theory where we reproduce the known 2-point function at 1-loop order obtained through more traditional techniques [CMS01]. In particular, this nicely illustrates how BV quantization captures the known distinction between planar and non-planar loop corrections in the standard noncommutative field theories, see e.g. [Sza03] for a review. While it is possible to treat Yang-Mills theory on the fuzzy sphere using our techniques, for illustration we consider the simpler example of Chern-Simons gauge theory, which was introduced in [ARS00, GMS01]. The quantization of this theory has so far only been briefly mentioned in [GMS02], without any detailed analysis. In this paper we provide a complete framework in which the perturbative correlation functions of Chern-Simons theory on the fuzzy 2-sphere can be computed. Our results share many similarities with the Chern-Simons model on finite-dimensional commutative dg Frobenius algebras studied in [CM10].
On the fuzzy torus we apply our finite-dimensional braided BV formalism to scalar field theories, which serve to illustrate a host of novelties compared to the standard noncommutative field theories. In particular, our approach produces fuzzy versions of symmetric braided quantum field theories in the continuum [Oec01]. We observe the absence of the notion of non-planar loop corrections as a consequence of the braided symmetry, as pointed out for twist deformed field theories in [Oec00] through algebraic means, and later by [Bal+07] through more heuristic methods. We stress that, in our case, this does not imply that there are no non-trivial braiding effects in the correlation functions; we illustrate this through explicit examples. Nevertheless, we expect the situation to be much different for gauge theories (as also suggested by [Bal+07]), as in this case even the classical braided field theories generally follow a different pattern from the conventional noncommutative gauge theories [DCGRS21]. Unfortunately, unlike the fuzzy sphere, we are not aware of any construction of a differential calculus on the fuzzy torus that could be used to define versions of the standard gauge theories, in contrast to its continuum version, i.e. the noncommutative torus; for a rigorous discussion of this point, see [LLS01].

Batalin-Vilkovisky quantization
In this section we review the necessary background on the Batalin-Vilkovisky (BV) formalism and related tools for the computation of correlation functions of quantum field theories.

Cochain complexes
We briefly recall some preliminary facts about cochain complexes. Let us fix a field K of characteristic 0. We denote by Ch K the category of cochain complexes of K-vector spaces, i.e. we work with cohomological degree conventions in which the differential has degree +1.
Recall that Ch K is a closed symmetric monoidal category. The tensor product V ⊗ W ∈ Ch K of two cochain complexes V, W ∈ Ch K is defined by the graded vector space (2.1a) for all n ∈ Z, and the differential where |v| ∈ Z denotes the degree of a homogeneous element v ∈ V . The monoidal unit is K ∈ Ch K , regarded as a cochain complex in degree 0, and the symmetric braiding is given by the Koszul sign rule The mapping complex (or internal hom) hom(V, W ) ∈ Ch K between two cochain complexes V, W ∈ Ch K is defined by the graded vector space for all n ∈ Z, and the 'adjoint' differential (2.3b) The 0-cocycles of this complex, i.e. f ∈ hom(V, W ) 0 with ∂(f ) = 0, are precisely the cochain maps f : V → W . A cochain homotopy between two cochain maps f, g : V → W is a (−1)-cochain h ∈ hom(V, W ) −1 satisfying f − g = ∂(h).
Recall also that, given any V ∈ Ch K and k ∈ Z, the k-shifted cochain complex V [k] ∈ Ch K is defined by V [k] n := V n+k , for all n ∈ Z, and the differential d V

Finite-dimensional BV formalism
In the following we provide an elementary and self-contained review of the BV quantization techniques developed by Costello and Gwilliam [CG16,Gwi12]. Since the focus of this paper is on matrix models, which are finite-dimensional systems, we can work in a purely algebraic setting and thereby avoid the functional analytic subtleties for continuum field theories addressed in [CG16,Gwi12]. We also refer to [GJF18] for a useful earlier exposition of BV quantization in finite dimensions.
Let us start by recalling the definition of a classical free BV theory from [CG16,Gwi12].
Remark 2.2. The complex E = (E, −Q) should be interpreted as a derived solution space. The elements in degree 0 are the fields of the theory, while the negative degrees encode the ghost fields and the positive degrees encode the antifields. See Section 3 for some explicit examples. The pairing · , · plays the role of a (−1)-shifted symplectic structure.
Since we are interested mainly in matrix models, which are finite-dimensional systems, we shall assume implicitly throughout the whole paper that each homogeneous component E n of the complex E is a finite-dimensional vector space and also that the complex itself is bounded from above and below, i.e. there exists some positive integer N ∈ Z >0 such that E n = 0 for both n > N and n < −N . As a consequence, E is a perfect complex and thus dualizable. △ To every free BV theory (E, −Q, · , · ) one can assign a commutative dg-algebra of polynomial observables. This is defined as the symmetric algebra Sym E * ∈ CAlg(Ch K ), where E * denotes the dual cochain complex of E = (E, −Q). Making use of the non-degenerate pairing · , · , we observe that the dual complex E * ∼ = E[1] is isomorphic to the 1-shifted complex. In fact, the duality pairing between E[1] and E is given by (2.4) Let us also recall that the differential on E[1] acquires an additional minus sign due to the shifting conventions from Section 2.1, hence With the usual abuse of notation, we denote the differential on the dg-algebra of polynomial observables Sym E * ∼ = Sym E[1] by the same symbol Q as the differential on E[1]. As we explain in more detail below, the pairing · , · induces a shifted Poisson bracket on Sym E[1] in the form of a P 0 -algebra structure. We briefly recall this crucial concept and refer to [Saf17] for more details on shifted Poisson structures.  1 We have included a minus sign in the definition of Q in order to avoid unpleasant sign factors in the dual differential on the observables, which we shall use more frequently in the present paper.
where the left vertical isomorphism uses the symmetric braiding τ and is given explicitly by a ⊗ b → (−1) |a| a ⊗ b. When expressed in terms of the bracket { · , · }, the P 0 -algebra axioms take the following form: (i) Compatibility with the differential: For all a, b ∈ A, where d denotes the differential on the unshifted cochain complex A and the minus sign on the left-hand side is due to our shifting conventions from Section 2.1.
(ii) Symmetry: For all a, b ∈ A, (iii) Jacobi identity: For all a, b, c ∈ A, In what follows, we will always describe P 0 -algebras in this more explicit form. △ Let us now explain in more detail how the pairing of a free BV theory (E, −Q, · , · ) induces a P 0 -algebra structure on the symmetric algebra Sym E[1] ∈ CAlg(Ch K ). Analogously to Remark 2.4, the cochain map · , · : E ⊗ E → K[−1] defines a pairing on the shifted complex where the left vertical isomorphism is given by (To understand this sign factor, note that for ϕ ∈ E[1] with E[1]-degree |ϕ|, the E-degree is |ϕ| + 1.) One easily checks that antisymmetry of · , · implies that the cochain map ( · , · ) : trivially because we are considering a constant shifted Poisson structure, and property (i) follows from the fact that ( · , · ) is a cochain map. Altogether, this construction defines a P 0 -algebra (2.13) which is interpreted as the classical observables of the free BV theory.
Interactions and quantization are both described by certain types of deformations of Obs cl , which we shall now briefly review. We start with the interactions. Let λ be a formal parameter, interpreted as a coupling constant, and consider the formal power series extension of Obs cl , which we denote with the usual abuse of notation by the same symbol. Given any 0-cochain I ∈ (Sym E[1]) 0 , interpreted as an interaction term for the classical BV action, we would like to define a deformed differential on Obs cl by the formula (2.14) Using the axioms for P 0 -algebras from Remark 2.4, as well as non-degeneracy of the pairing · , · , one easily checks that the nilpotency condition (Q int ) 2 = 0 of the deformed differential is equivalent to the classical master equation The resulting P 0 -algebra is interpreted as the classical observables of the interacting BV theory corresponding to the interaction term I ∈ (Sym E[1]) 0 , which must satisfy the classical master equation (2.15).
For quantization of the free BV theory, let be another formal parameter, interpreted as Planck's constant, and consider the formal power series extension of Obs cl , denoted again by the same symbol. The deformation of the differential on Obs cl that encodes quantization is given by for all a, b ∈ Sym E[1]. By iterating the latter property and using the P 0 -algebra axioms from Remark 2.4, one finds the explicit expression for the BV Laplacian, for all ϕ 1 , . . . , ϕ n ∈ E[1] with n ≥ 2, where the hat means to omit the corresponding factor. 2 By construction, the BV Laplacian satisfies the two properties the resulting deformed cochain complex, which is interpreted as the quantum observables for the free BV theory. It is important to emphasize that, as a consequence of (2.18b), the deformed differential Q does not respect the multiplication on Sym E[1], i.e. Obs is not a dg-algebra. The algebraic structure of Obs is that of an E 0 -algebra, i.e. a cochain complex with a distinguished 0-cocycle, which in the present case is the unit element ½ ∈ Sym E[1].
The two types of deformations corresponding to interactions and quantization can be combined, which leads to interacting quantum BV theories. Starting from the quantum observables for the free theory (2.21), let us choose again a 0-cochain I ∈ (Sym E[1]) 0 playing the role of an interaction term. We would like to define a deformed differential on Obs by the formula (2.22) The nilpotency condition (Q ,int ) 2 = 0 for this differential is equivalent to the quantum master equation To verify the last statement, it is helpful to note the identity for all a, b ∈ Sym E[1], which may be derived by applying ∆ BV on both sides of (2.18b). The resulting E 0 -algebra is interpreted as the quantum observables of the interacting BV theory corresponding to the interaction term I ∈ (Sym E[1]) 0 , which must satisfy the quantum master equation (2.23).

Cyclic L ∞ -algebras
Let us briefly recall the well-known and powerful construction of interaction terms I ∈ (Sym E[1]) 0 satisfying the classical (and also the quantum) master equation from cyclic L ∞ -algebra structures. See e.g. [JRSW19] for further details.
Definition 2.5. An L ∞ -algebra is a Z-graded vector space L together with a collection {ℓ n : L ⊗n → L} n∈Z ≥1 of graded antisymmetric linear maps of degree |ℓ n | = 2 − n that satisfy the homotopy Jacobi identities for all n ≥ 1, where Sh(n − k; k) ⊂ S n denotes the set of (n − k; k)-shuffled permutations on n letters and τ σ : L ⊗n → L ⊗n denotes the action of the permutation σ via the symmetric braiding on the category of graded vector spaces.

Homological perturbation theory
The correlation functions of non-interacting and also interacting quantum BV theories can be computed by employing techniques from homological perturbation theory, see e.g. [CG16,Gwi12].
We will now briefly review the relevant constructions. In the following definition we regard the cohomology H • (V ) of a cochain complex V ∈ Ch K as a cochain complex with trivial differential.
Definition 2.8. A strong deformation retract of a cochain complex V ∈ Ch K onto its cohomology H • (V ) is given by the following data: These data are required to satisfy the following conditions: c) γ 2 = 0, γ ι = 0 and π γ = 0.
A strong deformation retract may be visualized by where we also explicitly display the differentials.
The homological perturbation lemma (see e.g. [Cra04]) states that small perturbations d + δ of the differential d on V lead to perturbations of strong deformation retracts. By a small perturbation one means that, in addition to (d + δ) 2 = 0, the map id V − δ γ is invertible. In particular, the formal deformations of Section 2.2 are always small perturbations in this sense. The precise statement of the homological perturbation lemma is as follows.
Theorem 2.9. Consider any strong deformation retract as in (2.33) and let δ ∈ hom(V, V ) 1 be a small perturbation. Then there exists a strong deformation retract Let us consider now a free BV theory (E, −Q, · , · ) in the sense of Definition 2.1 and choose a strong deformation retract for its dual complex E * ∼ at the level of symmetric algebras. The cochain maps Sym ι and Sym π are given by extending ι and π in the usual way as commutative dg-algebra morphisms, i.e.
, Sym π ϕ 1 · · · ϕ n := π(ϕ 1 ) · · · π(ϕ n ) , (2.37) The cochain homotopy Sym γ is slightly more complicated to define. We first note that ι π : E[1] → E[1] defines a projector, i.e. (ι π) 2 = ι π. Hence one obtains a decomposition and consequently where Sym n denotes the n-th symmetric power. The cochain homotopy Sym γ is then defined by setting The correlation functions of non-interacting and also interacting quantum BV theories can then be determined by applying the homological perturbation lemma from Theorem 2.9 to the strong deformation retract (2.36) and the deformed differentials from Section 2.2. Let us explain this in some more detail. As we have explained in Section 2.2, quantization and including interactions are described by deforming the differential Q of the right complex in the strong deformation retract (2.36). Let us write generically Q + δ for the deformed differential, where δ stands either for an interaction term (2.14), the BV Laplacian (2.17) or the sum of both (2.22). Applying Theorem 2.9 we obtain a deformed strong deformation retract, which we denote by (2.40) The 'smeared' n-point correlation functions are then given by applying the map Sym π on a product of the 'test functions' ϕ 1 , . . . , ϕ n ∈ E[1], i.e. (2.41) This can be computed perturbatively (as formal power series in λ or , or both) by using the explicit formulas from Theorem 2.9. Note that, in general, the correlation functions are not simply numbers, rather they are elements of the symmetric algebra Sym H • (E[1]). The latter should be interpreted as the algebra of polynomial functions on the space of vacua of the theory, which is the cohomology H • (E) of the derived solution complex E, cf. Remark 2.2. Hence the n-point correlation functions in (2.41) are functions on the space of vacua which, when evaluated in a particular vacuum, give the usual numerical correlations of the perturbative field theory around this vacuum. We will illustrate this through concrete examples in Section 3.

Field theories on the fuzzy sphere
We illustrate the formalism of Section 2 by studying scalar field theories and also Chern-Simons theory on the fuzzy 2-sphere. The examples presented in this section are over the field K = C of complex numbers.

Scalar field theories
We consider first the simplest case of scalar field theories, where we show how our formalism reproduces the known 1-loop 2-point function for Φ 4 -theory on the fuzzy sphere, see e.g. [CMS01]. However, in contrast to the traditional approach of [CMS01], our correlation functions are generally disconnected and 1-particle reducible, and involve unamputated external legs.
The fuzzy 2-sphere. To fix our notation and conventions, let us recall the definition of the fuzzy 2-sphere following [CMS01]. Let N ∈ Z >0 be a positive integer and let V denote the irreducible spin N/2 representation of the su(2) Lie algebra. The algebra of functions on the fuzzy sphere S 2 N is defined by where V * denotes the dual representation. Since the underlying vector space of V is (N + 1)dimensional, it follows that A ∼ = Mat N +1 (C) is isomorphic to the algebra of (N + 1) × (N + 1)matrices with complex entries. The action of su(2) on V is encoded by a Lie algebra homomorphism where the Lie bracket on A is the matrix commutator. Let us choose a basis where ǫ ijk is the Levi-Civita symbol and summation over repeated indices is always understood. We introduce the constant Then the elements generate the algebra A and satisfy the fuzzy unit sphere relations where * denotes Hermitian conjugation and δ ij is the Kronecker delta-symbol. Integration on the fuzzy sphere is given by the normalized trace map and the scalar Laplacian reads as (3.6) A vector space basis of A is given by the fuzzy spherical harmonics Y J j ∈ A, for J = 0, 1, . . . , N and −J ≤ j ≤ J. The fuzzy spherical harmonics are eigenfunctions of the scalar Laplacian satisfying the identities There is also an explicit 'fusion formula' for the products Y I i Y J j of fuzzy spherical harmonics in terms of Wigner's 3j and 6j symbols, see e.g. [CMS01], which we however do not need in the present paper.
Free BV theory. We are now ready to describe a non-interacting scalar field theory on the fuzzy sphere as a free BV theory in the sense of Definition 2.1.
Definition 3.1. The free BV theory associated to a scalar field with mass parameter m 2 ≥ 0 on the fuzzy sphere is given by the cochain complex concentrated in degrees 0 and 1, together with the pairing Following the general approach outlined in Section 2.2, we can construct from this input a P 0 -algebra of classical observables of the non-interacting theory, where we note that the complex is concentrated in degrees −1 and 0.
Interactions. In the present case of a scalar field, the dg-algebra Sym E and consequently it automatically satisfies both the classical and quantum master equations (2.15) and (2.23), respectively. This means that every 0-cochain I ∈ (Sym E[1]) 0 provides a well-defined interaction term for a scalar field in both the classical and quantum cases.
Let us nevertheless use the cyclic L ∞ -algebra formalism from Section 2.3 to introduce concrete examples of interaction terms, focusing on the typical m+1-point interactions. The Abelian cyclic L ∞ -algebra corresponding to the scalar field from Definition 3.1 is given by the cochain complex (3.13) and the cyclic structure (3.14) Choosing any m ≥ 2, we can endow this with the compatible m-bracket The interaction term (2.32) corresponding to an m + 1-point interaction then reads concretely as where we stress that the products of the Y J i j i * ∈ E[1] 0 in the second line are not given by matrix multiplication but rather by the product in the symmetric algebra Sym E[1]. The constants can in principle be worked out explicitly in terms of the Wigner 3j and 6j symbols as in [CMS01], but this level of detail is not needed in the present paper. Because of the underlying cyclic L ∞algebra structure, the constants I J 0 J 1 ···Jm j 0 j 1 ···jm are symmetric under the exchange of any neighboring pairs of indices, i.e. (3.18) Strong deformation retract. The cohomology of the complex (3.11) depends on whether one considers a massive or a massless scalar field. Since the spectrum of the scalar Laplacian (3.6) is {J (J + 1) : J = 0, 1, . . . , N }, we obtain In the massive case m 2 > 0, the strong deformation retract is given by where G is the inverse of Q = ∆ + m 2 , i.e. the Green operator, and γ = −G is defined to act as a degree −1 map on E[1] (cf. Definition 2.8 (iii)).
The massless case m 2 = 0 is slightly more complicated because the scalar Laplacian Q = ∆ has a non-trivial kernel, which is given by complex multiples of the unit ½ ∈ A. The linear map N +1 Tr(a) ½, obtained by composing the normalized trace and the unit map η : C → A, defines a projector onto the kernel of ∆, which can be used to decompose By the rank-nullity theorem of linear algebra, the scalar Laplacian restricts to an isomorphism ∆ ⊥ : A ⊥ → A ⊥ and we denote its inverse by G ⊥ : A ⊥ → A ⊥ . Extending G ⊥ by 0 to all of A we obtain the linear map from which the strong deformation retract in the massless case is given by The strong deformation retracts for the massive and massless cases (3.20) and (3.23), respectively, extend to the symmetric algebras via the construction outlined below (2.36).
Correlation functions for m 2 > 0. We shall now explain in some more detail how correlation functions may be computed and provide some explicit examples. We focus here on the massive case m 2 > 0 and comment briefly on the massless case later on.
Recall that the strong deformation retract (3.20) extends to the symmetric algebras. Given any small perturbation δ of the differential Q on Sym E[1], we obtain the deformed strong deformation retract where the tilded quantities are computed through the homological perturbation lemma, cf. Theorem 2.9. To compute the correlation functions (2.41), we have to consider the cochain map where we have simplified the notation by denoting the extensions of maps to symmetric algebras by capital symbols. Recall that the relevant perturbations δ are of the form where ∆ BV is the BV Laplacian (2.19) and λ I ∈ (Sym E[1]) 0 denotes the m + 1-point interaction term (3.17) for some m ≥ 2. We are particularly interested in the correlation functions Π(ϕ 1 · · · ϕ n ) for test functions ϕ 1 , . . . , ϕ n ∈ E[1] 0 of degree zero; these describe the correlators of the physical field, in contrast to correlators involving antifields. To work out the perturbative expansion (3.25) of such correlators, we have to understand how the maps Π and δ Γ act on elements where we recall that G = −γ is the Green operator for Q = ∆ + m 2 . For the second term, we use the axioms of P 0 -algebras (cf. Remark 2.4) and the explicit expression (3.17) for the m + 1-point interaction term (together with its symmetry property (3.18)), resulting in (3.29) The two expressions in (3.28) and (3.29) admit a convenient graphical description. Depicting the element ϕ 1 · · · ϕ n by n vertical lines, the map in (3.28) may be depicted as where the cap indicates a contraction of two elements with respect to ( · , G(·)). The map in (3.29) may be depicted as where the vertex acts on an element as J 0 ,j 0 ,...,Jm,jm I J 0 J 1 ···Jm it turns a single vertical line into m legs.
Example 3.2. Let us set m = 3 and compute the 2-point function of Φ 4 -theory to the lowest non-trivial order in the coupling constant. (Due to our conventions in (3.17), the 4-point interaction vertex has coupling constant λ 2 .) Using our graphical description, we compute (3.33) The 2-fold application of δ Γ is then given by (δ Γ) 2 (ϕ 1 ϕ 2 ) = λ 2 3! 4 + + + + + where the simplification in the second equality uses the symmetry property of the interaction term (3.18). The 3-fold application of δ Γ is given by From this we can compute the 2-point function (3.32) to leading order in the coupling constant as Note that the 2-point function at order λ 2 (and higher) receives both planar and non-planar contributions, analogously to the computation of [CMS01] by traditional perturbative techniques, even though this is not directly apparent in our graphical presentation. The origin of these two kinds of contributions lies in the (graded anti-)symmetrization of the higher L ∞ -algebra bracket (3.15) which enters the definition of the constants I J 0 J 1 J 2 J 3 j 0 j 1 j 2 j 3 in (3.17). ▽ Correlation functions for m 2 = 0. Let us briefly comment on the correlation functions in the massless case m 2 = 0. The relevant cochain map π = 1 N +1 Tr in the massless strong deformation retract (3.23) is not simply the zero map, but the normalized trace. Hence, in contrast to (3.27), the extension of π to symmetric algebras is given in the massless case by Π(½) = 1 ∈ Sym C , Π(ϕ 1 · · · ϕ n ) = π(ϕ 1 ) ⊙ · · · ⊙ π(ϕ n ) ∈ Sym C , (3.37) for all ϕ 1 , . . . , ϕ n ∈ E[1] 0 in degree 0, where we use the symbol ⊙ to denote the product of the symmetric algebra Sym C to distinguish it from the multiplication of complex numbers. Each π(ϕ i ) ∈ Sym C is regarded as a linear function on the space of vacua ker(∆ : where the product (denoted by juxtaposition) in the last step is the usual multiplication of complex numbers. It is convenient to denote (3.37) graphically by attaching vertices on top of the vertical lines which depict empty slots that can be evaluated on classical vacua Φ ∈ ker(∆ : A → A) ∼ = C. These purely classical contributions to the correlation functions are completely analogous to those one would obtain in traditional approaches to quantum field theory by expanding the field operator Φ + Φ around a generic classical solution Φ.
as an element in Sym C. ▽

Chern-Simons theory
The fuzzy 2-sphere has a well-known 3-dimensional differential calculus which allows for the definition of a Chern-Simons term on S 2 N , see e.g. [ARS00,GMS01]. Similarly to [GMS01], we shall focus on the Abelian Chern-Simons theory on S 2 N which, due to the noncommutativity of the differential calculus on S 2 N , includes a ternary interaction term; the extension to non-Abelian Chern-Simons theory with matrix gauge algebra such as gl(n) or u(n) is straightforward, as in [ARS00], and presents no essential novelties. This is the simplest example which serves as the prototype for the BV formalism applied to field theories with gauge symmetries. On S 2 N it can be regarded as a fuzzy version of a BF-type theory on the classical 2-sphere S 2 [GMS01].
Differential calculus on the fuzzy 2-sphere. In order to set up Chern-Simons gauge theory within the framework outlined in Section 2, we recall some basic facts about differential forms on the fuzzy 2-sphere. The usual su(2)-equivariant differential calculus on the fuzzy sphere algebra (3.1) is given by the Chevalley-Eilenberg dg-algebra (3.41) The dual of the Lie algebra basis {e i ∈ su(2)} i=1,2,3 defines a basis {θ i ∈ Ω 1 (A)} i=1,2,3 for the A-module of 1-forms, which generates the whole differential calculus Ω • (A). This basis is central, i.e. a θ i = θ i a for all a ∈ A = Ω 0 (A), and θ i ∧ θ j = −θ j ∧ θ i , for all i, j = 1, 2, 3. The de Rham differential is specified by for all a ∈ A = Ω 0 (A) and i = 1, 2, 3, together with the graded Leibniz rule for all ω ∈ Ω p (A) and ζ ∈ Ω • (A). Note that the differential calculus Ω • (A) on the fuzzy 2sphere is 3-dimensional, in contrast to the 2-dimensional calculus on the commutative 2-sphere S 2 . Higher-dimensional (covariant) calculi are a common feature in noncommutative geometry which arise in a broad range of examples, reaching from the fuzzy sphere to quantum groups.
Free BV theory. We can now describe the non-interacting part of Abelian Chern-Simons theory on the fuzzy sphere as a free BV theory in the sense of Definition 2.1.
Definition 3.4. The free BV theory associated to Abelian Chern-Simons theory is given by the cochain complex i.e. Q := d is the de Rham differential, together with the pairing where |α| denotes the cohomological degree of α ∈ E. (Note that the latter differs from the de Rham degree as |α| dR = |α| + 1.) Following the general approach outlined in Section 2.2, we can construct from this input a P 0 -algebra of classical observables of the non-interacting theory, where we note that the complex is concentrated in degrees −2, −1, 0 and 1.
Interactions. Using the cyclic L ∞ -algebra formalism from Section 2.3, we will now introduce an interaction term for the free Chern-Simons theory from Definition 3.4. The Abelian cyclic L ∞ -algebra associated with the free theory is given by the cochain complex and the cyclic structure This can be endowed with the compatible 2-bracket given by the graded commutator in the differential calculus Ω • (A). Note that, in contrast to commutative Chern-Simons theory, the bracket ℓ 2 is not zero because the differential calculus on the fuzzy sphere S 2 N is noncommutative. In order to write down the contracted coordinate functions corresponding to this non-Abelian cyclic dg-Lie algebra, we pick a basis of . The contracted coordinate functions then take the form where the dual basis with respect to the cyclic structure · , · is denoted by c * a ∈ E[1] 1 = Ω 3 (A), The Chern-Simons interaction term thus reads as By the Whitehead lemma, the cohomology of CE • (su(2), (J)) is trivial for all J > 0. This allows us to compute that is concentrated in differential form degrees 0 and 3. From this it follows that the cohomology of the complex (3.52) is given by (3.60) To set up a strong deformation retract as in (2.35), let us first define the cochain maps ι and π for the present example. We define the cochain map via the unit η(1) = ½ ∈ Ω 0 (A) = A and its Hodge dual * η(1) = * (½) ∈ Ω 3 (A). The cochain map is given by integration of top-forms (3.44), where the normalization factor 1 4π is chosen so that π ι = id H • (E[1] In the second line we used the property that the projector ι π commutes with d because it is a cochain map, and also that the Green operator commutes with d because d ∆ ⊥ = ∆ ⊥ d. where λ I is the Chern-Simons interaction term (3.57). The perturbative expansion of the n-point correlation functions can then be computed by using the algebraic properties of Γ = Sym γ (cf. (2.39)), the BV Laplacian ∆ BV (cf. (2.19)) and the graded derivation {λ I, · } (cf. the P 0 -algebra axioms in Remark 2.4). These computations are very similar to those in the case of scalar field theories (cf. Section 3.1), and hence we will not spell out any explicit examples of correlation functions for Chern-Simons theory.

BV quantization of braided field theories
The definitions and constructions in Section 2 can be generalized in a rather straightforward way to the case of theories with a triangular Hopf algebra symmetry. In the following we will spell out the details. The quasi-triangular case is considerably more complicated because it obstructs the formulation of symmetry properties and the Jacobi identity, and we will not treat this more general case in the present paper. Nevertheless, following the terminology of [DCGRS20, DCGRS21], we use the adjective 'braided' (in contrast to the categorically more accurate 'symmetric braided') to emphasize the role of a non-identity triangular R-matrix, in addition to equivariance, in our treatment below.

Triangular Hopf algebras and their representations
We start by recalling some basic concepts and terminology from the theory of Hopf algebras that are required in this paper. More details can be found in e.g. [Maj95,BM20].
Definition 4.1. A Hopf algebra is an associative unital algebra H over K together with two algebra homomorphisms ∆ : H → H ⊗ H (coproduct) and ǫ : H → K (counit), as well as an algebra antihomomorphism S : H → H (antipode) satisfying where µ : H ⊗ H → H denotes the product and η : K → H denotes the unit of the algebra H.
Remark 4.2. We shall often use the Sweedler notation for the coproduct of h ∈ H, and more generally ∆ n (h) = h 1 ⊗ · · · ⊗ h n+1 (summation understood) (4.2b) for the iterated applications of the coproduct. Note that, due to coassociativity (4.1a), it makes sense to write ∆ 2 = (∆ ⊗ id H ) ∆ = (id H ⊗ ∆) ∆ for the two-fold application of the coproduct, and similarly ∆ n for the n-fold applications. In Sweedler notation, the second and third properties in (4.1) read as △ Given any Hopf algebra H, we denote by H Mod the category of left modules over its underlying associative unital algebra. An object in H Mod is a vector space V together with a linear map Using the coproduct and the counit of H, one defines a monoidal structure on the category H Mod. The monoidal product V ⊗ W of two objects V, W ∈ H Mod is given by the tensor product of the underlying vector spaces together with the left tensor product action for all h ∈ H, v ∈ V and w ∈ W . The monoidal unit is given by endowing the one-dimensional vector space K with the trivial left action h ⊲ c = ǫ(h) c, for all h ∈ H and c ∈ K. Using the antipode of H, one observes that this monoidal structure is closed. The internal hom between two objects V, W ∈ H Mod is the vector space hom(V, W ) := Hom K (V, W ) of all (not necessarily H-equivariant) linear maps from V to W together with the left adjoint action To further obtain a closed symmetric monoidal structure on the category H Mod, we require an additional datum on the Hopf algebra H. In the following definition, we denote by ∆ op (h) = h 2 ⊗ h 1 the opposite coproduct and, for an element R = R α ⊗ R α ∈ H ⊗ H (summation over α understood), we write for the flipped element in H ⊗ H and for the associated elements in H ⊗ H ⊗ H.

Definition 4.3. A quasi-triangular structure for a Hopf algebra H is an invertible element
for all h ∈ H. A triangular structure is a quasi-triangular structure R ∈ H ⊗H which additionally satisfies R 21 = R −1 .
Suppose now that R = R α ⊗ R α ∈ H ⊗ H is a quasi-triangular structure for H. Then we can define a braiding for the closed monoidal category H Mod by setting for every pair of objects V, W ∈ H Mod. In the case where R is triangular, this braiding is symmetric, i.e. H Mod is a closed symmetric monoidal category.
Example 4.4. Every finite group G has an associated group Hopf algebra K[G]. This is the free vector space spanned by the group elements g ∈ G, i.e. every element h ∈ K[G] can be written uniquely as h = g∈G h g g for some h g ∈ K. The product on K[G] is given by the bilinear extension of the group operation on G and the unit element is the basis vector associated with the identity element e ∈ G. The coproduct, counit and antipode are defined by ∆(g) = g ⊗ g , ǫ(g) = 1 , S(g) = g −1 , (4.10) for all g ∈ G, and by linear extension.
Depending on the group G, the Hopf algebra K[G] may admit various (quasi-)triangular structures, see below for a concrete example. The trivial choice, which exists for any group G, is the element R triv = e ⊗ e ∈ K[G] ⊗ K[G]. It is easy to check that the corresponding closed symmetric monoidal category K[G] Mod of K[G]-modules is the usual closed symmetric monoidal category Rep K (G) of K-linear representations of G.
The following class of explicit examples of (non-trivial) triangular Hopf algebras features in our study of braided field theories on the fuzzy torus in Section 5. Set the ground field to K = C and let N, n ∈ Z ≥1 be two positive integers. Let us write Z n N := Z N × · · · × Z N n-times (4.11) for the n-fold product of the cyclic group Z N of order N . We denote its elements by where each entry k i will be represented by an integer modulo N , and the group operation is given by addition modulo N . For any N -th root of unity q ∈ C and any n × n-matrix Θ ∈ Mat n (Z) with integer entries, the element where δ s,0 denotes the Kronecker delta-symbol and 0 := (0, . . . , 0) ∈ Z n N is the identity element. In the case where the matrix Θ is antisymmetric, R is further a triangular structure for C[Z n N ]. ▽

Finite-dimensional braided BV formalism
Let H be a Hopf algebra with triangular structure R. We saw in Section 4. The monoidal product is given by endowing (2.1) with the left tensor product H-action (4.5), the monoidal unit is K endowed with the trivial left H-action and the internal hom is given by endowing (2.3) with the left adjoint H-action (4.6). The symmetric braiding is given by combining (2.2) with (4.9). Explicitly, for all V, W ∈ H Ch, involves both the Koszul signs and the R-matrix R = R α ⊗ R α ∈ H ⊗ H.
The generalization of Definition 2.1 to the present case then reads as follows.
Before we can generalize Definition 2.3, we have to introduce a concept of braided commutative dg-algebra. Because H Ch is a (closed) symmetric monoidal category, there is an associated category CAlg( H Ch) of commutative algebras in H Ch. An object in this category is a triple (A, µ, η) consisting of an object A ∈ H Ch together with two H Ch-morphisms µ : A⊗ A → A and η : K → A satisfying the associativity and unitality axioms. The H-equivariance of the product µ and unit η means explicitly that for all h ∈ H and a, b ∈ A. Commutativity in this case means that µ • τ R = µ, or explicitly for all a, b ∈ A. The main example for us is the braided symmetric algebra Sym R V ∈ CAlg( H Ch) associated with an object V ∈ H Ch. In analogy to the usual case, this H-equivariant dg-algebra is generated by all v ∈ V , modulo the commutation relations involving the R-matrix We are now ready to generalize Definition 2.3, which we shall write in the more explicit format of Remark 2.4.
As in Section 2.2, interactions and quantization are both described by certain types of deformations of the differential Q in (4.25). In order to obtain a deformed cochain complex of left H-modules, i.e. an object in H Ch, one should consider H-invariant deformations. Let us discuss the different types of deformations in detail.
To obtain an interacting braided classical BV theory, we pick a 0-cochain I ∈ (Sym R E[1]) 0 that is H-invariant, i.e. h ⊲ I = ǫ(h) I for all h ∈ H, and satisfies the classical master equation where λ is an H-invariant formal parameter (coupling constant). Then defines a braided P 0 -algebra that is interpreted as the classical observables for the interacting braided BV theory with interaction term I. For quantization, we note that the definition of the BV Laplacian in (2.18) applies to our braided case as well and thereby defines an H Ch-morphism ∆ BV : that squares to 0. However, the explicit formula (2.19) for the ordinary BV Laplacian is modified in the braided case by suitable actions of the R-matrix. Explicitly, one finds that for all ϕ 1 , . . . , ϕ n ∈ E[1] with n ≥ 2. Then defines a braided E 0 -algebra that is interpreted as the quantum observables for the non-interacting braided BV theory. Finally, to obtain an interacting braided quantum BV theory, we consider a 0-cochain I ∈ (Sym R E[1]) 0 that is H-invariant and satisfies the quantum master equation defines a braided E 0 -algebra that is interpreted as the quantum observables for the interacting braided BV theory with interaction term I.

Braided L ∞ -algebras and their cyclic versions
The construction, reviewed in Section 2.3, of interaction terms satisfying the classical (and also the quantum) master equation from cyclic L ∞ -algebra structures generalizes to the case of braided BV theories by using the concept of a braided L ∞ -algebra introduced in [DCGRS20, DCGRS21].
Definition 4.7. A braided L ∞ -algebra is a Z-graded left H-module L together with a collection {ℓ n : L ⊗n → L} n∈Z ≥1 of H-equivariant graded braided antisymmetric linear maps of degree |ℓ n | = 2 − n that satisfy the braided homotopy Jacobi identities for all n ≥ 1, where τ σ R : L ⊗n → L ⊗n denotes the action of the permutation σ via the symmetric braiding τ R on the category of graded left H-modules.
Remark 4.8. Let us spell this out in a bit more detail. The graded braided antisymmetry property of ℓ n : L ⊗n → L means that for all i = 1, . . . , n − 1 and all homogeneous elements v 1 , . . . , v n ∈ L. The permutation action τ σ R : L ⊗n → L ⊗n in (4.33) includes, in addition to the usual Koszul signs, appropriate actions of the R-matrix as in (4.16). Similarly to Remark 2.6, every braided L ∞ -algebra has an underlying cochain complex (L, d L := ℓ 1 ) ∈ H Ch, and ℓ 2 : L ⊗ L → L is an H Ch-morphism. When the only non-vanishing bracket is ℓ 2 , a braided L ∞ -algebra is an example of a braided Lie algebra in the sense of [Maj94]. △ Definition 4.9. A cyclic braided L ∞ -algebra is a braided L ∞ -algebra (L, {ℓ n }) together with a non-degenerate braided symmetric H Ch-morphism · , · : L ⊗ L → K[−3] that satisfies the cyclicity condition for all n ≥ 1 and all homogeneous elements v 0 , v 1 , . . . , v n ∈ L.
Similarly to the ordinary case from Section 2.3, every free braided BV theory (E, −Q, · , · ) as in Definition 4.5 defines an Abelian cyclic braided L ∞ -algebra given by E[−1], ℓ 1 := d E[−1] = Q and cyclic structure (4.36) Introducing an interaction term I ∈ (Sym R E[1]) 0 that satisfies the classical master equation for all homogeneous a, a ′ ∈ Sym R E[1] and v, v ′ ∈ E[−1], and similarly for the brackets ℓ ext n . Remark 4.10. Our claim that (4.37) satisfies the classical master equation (4.27) can be proven by precisely the same calculation as in the ordinary case, see e.g. [JRSW19, Section 4.3]. This is due to the fact that the contracted coordinate functions (4.38) are H-invariant elements of (Sym R E[1]) ⊗ E[−1], which implies that all appearances of R-matrices in the properties of the extended brackets ℓ ext n and the extended pairing · , · ext disappear when they are evaluated on tensor products of the H-invariant element a. By the same argument, one can use the proofs from the ordinary case [JRSW19, Section 4.3] to show that (4.37) is annihilated by the BV Laplacian, i.e. ∆ BV (λ I) = 0, and consequently that it also satisfies the quantum master equation (4.31). △

Braided homological perturbation theory
The usual homological perturbation lemma (cf. Theorem 2.9) extends to our braided setting, provided that we use small perturbations δ ∈ hom(V, V ) 1 that are additionally H-invariant. (Recall that the perturbations corresponding to interactions and quantization from Section 4.2 are H-invariant.) Concretely, the details are spelled out as follows.
Definition 4.11. A braided strong deformation retract of an object V ∈ H Ch onto its cohomology H • (V ) is a strong deformation retract  Proof. By direct inspection, one observes that the explicit formulas in (2.34) satisfy the necessary H-equivariance or H-invariance properties.
Given any free braided BV theory (E, −Q, · , · ) in the sense of Definition 4.5 and any braided strong deformation retract for its dual complex a similar construction as in Section 2.4 defines a braided strong deformation retract (4.44) The correlation functions of braided non-interacting and also interacting quantum BV theories can then be determined by applying Corollary 4.12 to (4.44) and the deformed differentials from Section 4.2. This construction works in a completely analogous way to the ordinary case that we reviewed at the end of Section 2.4.

Braided field theories on the fuzzy torus
We illustrate the formalism of Section 4 in the example of scalar field theories on the fuzzy 2torus, reproducing from our perspective many facets of Oeckl's braided quantum field theory for symmetric braidings [Oec00]. In this section we work over the field K = C of complex numbers.
The fuzzy 2-torus. To fix our notation and conventions, let us recall the definition of the fuzzy 2-torus and its triangular Hopf algebra symmetry, see e.g. [BG19,BSS17] for further details. Let us fix a positive integer N ∈ Z >0 and set q := e 2π i /N ∈ C . (5.1) The algebra of functions on the fuzzy torus T 2 N is defined as the noncommutative * -algebra generated by two elements U and V , modulo the * -ideal generated by the displayed relations. One should think of the generators U and V as the two exponential functions corresponding to the two 1-cycles of T 2 N . Every element of A may be written uniquely as a = i,j∈Z N a ij U i V j , where a ij ∈ C play the role of Fourier coefficients. 4 The fuzzy 2-torus has a (discrete) translation symmetry that is given by a left action ⊲ : H ⊗ A → A of the group Hopf algebra H := C[Z 2 N ] introduced in Example 4.4. Explicitly, the basis vectors k = (k 1 , k 2 ) ∈ H, with k 1 , k 2 ∈ Z N integers modulo N , act on the generators of A as This action is extended to all of A by demanding that A is a left H-module algebra, i.e. k ⊲(a b) which is motivated from the fact that with this choice A becomes a braided commutative left H-module algebra, i.e. a b = (R α ⊲ b) (R α ⊲ a) for all a, b ∈ A. The latter statement can be easily checked by considering, without loss of generality, two basis elements a = U i V j and b = U k V l , for some i, j, k, l ∈ Z N . Using the commutation relation in (5.2), one computes Using now the definition of the R-matrix (5.4) and of the left action (5.3), one computes where the last step follows from (4.14). The two expressions coincide, showing that A is braided commutative.
Integration on the fuzzy torus is defined through the linear map where we have chosen the square root q 1/2 := e π i /N ∈ C of q. The scalar Laplacian is also H-equivariant under the action (5.3) because the powers of q resulting from the action on U and on U * compensate each other, and similarly for those from V and V * . A basis of eigenfunctions of the scalar Laplacian is given by for all k = (k 1 , k 2 ) ∈ Z 2 N , and the corresponding eigenvalues are given by where the q-numbers are defined as [n] q := q n/2 − q −n/2 q 1/2 − q −1/2 . (5.9b) For later use, let us record the properties e * k = q −k 1 k 2 e −k , e k e l = q −l 1 k 2 e k+l , e * k e l = δ k,l , (5.10a) and τ R (e k ⊗ e l ) = q −kΘl e l ⊗ e k = q lΘk e l ⊗ e k , (5.10b) for all k, l ∈ Z 2 N , which imply in particular that the dual of the basis {e k } under the integration pairing is given by {e * k }.
Free braided BV theory. We are now ready to describe a non-interacting scalar field theory on the fuzzy torus as a free braided BV theory in the sense of Definition 4.5.
Definition 5.1. The free braided BV theory associated to a scalar field with mass parameter m 2 ≥ 0 on the fuzzy torus is given by the H Ch-object (5.14) In the second step we used H-equivariance of the pairing to write · , · = (−t) ⊲ · , · = (−t) ⊲ · , (−t) ⊲ · and in the fourth step we used (4.14). Together with (5.13), it then follows that the pairing (5.12) is both strictly antisymmetric and also braided antisymmetric, the latter property being as required by Definition 4.5. △ Following the general approach outlined in Section 4.2, we can construct from this input a braided P 0 -algebra of classical observables of the non-interacting theory. Let us recall that Sym R denotes the braided symmetric algebra (defined via (4.20)) and note that the complex is concentrated in degrees −1 and 0.
Interactions. Analogously to the scalar field on the fuzzy sphere discussed in Section 3.1, the braided symmetric algebra Sym As a concrete example, let us introduce the m+1-point interaction by using the cyclic braided L ∞ -algebra formalism from Section 4.3. The scalar field from Definition 5.1 defines an Abelian cyclic braided L ∞ -algebra given by the complex A ∈ H Ch (5.18) and the cyclic structure Choosing any m ≥ 2, we can endow this with the compatible m-bracket given by the multiplication in the left H-module algebra A. Note that, for degree reasons, this is only non-vanishing if each ϕ i ∈ E[−1] is of degree 1 in E[−1] and that the braided graded antisymmetry property of ℓ m is a consequence of the fact that A is braided commutative. Indeed, we find where the factors of q arise from the braiding identity in (5.10). We stress again that the products of the elements e * k i in the second line are taken in the braided symmetric algebra Sym R E[1] and not in A.
The constants can be worked out explicitly by using (5.19), (5.20), (5.6) and (5.10), from which one finds where the double subscript notation indicates the components k i = (k i1 , k i2 ) ∈ Z 2 N for i = 0, 1, . . . , m. The constants I k 0 k 1 ···k m satisfy the q-deformed symmetry property for any exchange of neighboring indices, which in particular implies the strict cyclicity property I k 0 k 1 ···k m = I k 1 ···k m k 0 (5.24b) by further using momentum conservation imposed by the Kronecker delta-symbol δ k 0 +k 1 +···+k m ,0 .
Braided strong deformation retract. In the remainder of this section we consider only the massive case m 2 > 0. (The massless case is slightly more involved, but it can be treated analogously to the fuzzy sphere example in Section 3. where we have adopted the same notation for the basis vectors as in (5.22). The strong deformation retract (5.25) clearly satisfies the requisite H-equivariance and H-invariance properties to be a braided strong deformation retract in the sense of Definition 4.11.
Correlation functions. We shall now explain in more detail how correlation functions may be computed and provide some explicit examples. The braided strong deformation retract (5.25) extends to the braided symmetric algebras. Given any small H-invariant perturbation δ of the differential Q on Sym R E[1], we obtain via Corollary 4.12 the deformed braided strong deformation retract where the tilded quantities are computed through the homological perturbation lemma, cf. Theorem 2.9. The correlation functions may be computed by the H Ch-morphism where we use again the abbreviations Π := Sym R π and Γ := Sym R γ. The relevant perturbations δ are of the form where ∆ BV is the BV Laplacian (4.29) and λ I ∈ (Sym R E[1]) 0 denotes the m+1-point interaction term (5.23) for some m ≥ 2.
In order to evaluate the correlation functions Π(ϕ 1 · · · ϕ n ) for test functions of degree zero, we have to understand how the maps Π and δ Γ act on elements ϕ 1 · · · ϕ n ∈ Sym R E[1] with all ϕ i ∈ E[1] 0 of degree zero. For Π this is very easy and we find Π(½) = 1 , Π(ϕ 1 · · · ϕ n ) = 0 , (5.30) for all n ≥ 1. To describe δ Γ = ∆ BV Γ + {λ I, · } Γ, it is convenient to consider the two summands individually. Using the explicit formula (4.29) for the BV Laplacian, one finds for the first term Using now the explicit expression (5.23) for the m + 1-point interaction term, together with its properties (5.24), one finds for the second term Similarly to Section 3.1, the two expressions (5.31) and (5.32) may be visualized graphically. Depicting again the element ϕ 1 · · · ϕ n by n vertical lines, the map in (5.31) may be depicted as ∆ BV Γ · · · = − 2 n · · · + · · · + · · · + · · · , (5.33) where the cap indicates a contraction of two elements with respect to ( · , G(·)). In such pictures it is understood that the right leg of the contraction is permuted via the symmetric braiding τ R across the intermediate vertical lines, which leads precisely to the R-matrix insertions in (5.31). The map in (5.32) may be depicted as λ I , Γ · · · = − λ m−1 m! n m legs · · · + · · · + · · · m legs , (5.34) where the vertex acts on an element as k 0 ,...,k m ∈Z 2 Example 5.3. To illustrate the pattern of R-matrix insertions, as a first simple example let us compute the 4-point function Let us explain in more detail the simplification in the last step: Using H-equivariance of the pairing ( · , · ) and the standard identity (S ⊗ id H )R = R −1 = R 21 for a triangular R-matrix, it follows that hence the second and fifth term in the first line of (5.37) coincide, yielding the second term in the second line. Furthermore, using again H-equivariance of the pairing ( · , · ), the third R-matrix property in (4.8) and the normalization condition ǫ(R α ) R α = 1, it follows that hence the third and fourth term in the first line of (5.37) coincide, yielding the third term in the second line.
Altogether we find that the 2-point function of Φ 4 -theory on the fuzzy torus reads to leading order in the coupling constant as Π ϕ 1 ϕ 2 = − − λ 2 2 2 + O(λ 4 ) = − ϕ 1 , G(ϕ 2 ) − λ 2 2 2 k,l∈Z 2 N e * k , G(ϕ 1 ) e k , G(ϕ 2 ) [l 1 ] 2 q + [l 2 ] 2 q + m 2 where we also used the explicit expression (5.26) for the Green operator. We emphasize that in our interacting 2-point function (5.45) there is no distinction between planar and non-planar loop corrections, in contrast to the traditional (unbraided) approaches to noncommutative quantum field theory [IIKK00,MVRS00]. This feature is intimately tied to the braided commutativity property of the fuzzy torus algebra (5.2), which in particular implies that the higher L ∞ -algebra brackets (5.20) are automatically braided (graded anti-)symmetric without the need for graded antisymmetrization as in the fuzzy sphere case (3.15). The absence of non-planar features in loop corrections has similarly been observed by Oeckl in his framework of (symmetric) braided quantum field theory [Oec00]. ▽ Example 5.5. We stress that the disappearance of the q-factors from the interaction term (5.23) in the tadpole diagram (5.45) is only due to the shape of this diagram and not a general feature of our formalism. As an illustrative example, let us consider again Φ 4 -theory and compute the connected part of the 4-point function to the first non-trivial order in the coupling constant λ 2 . Using the same arguments as in the previous examples, in particular the q-deformed symmetry property (5.24) of the interaction term, one finds Π(ϕ 1 ϕ 2 ϕ 3 ϕ 4 ) connected = λ 2 3 + O(λ 4 ) (5.46) = λ 2 3 k 0 ,k 1 ,k 2 ,k 3 ∈Z 2 N I k 0 k 1 k 2 k 3 e * k 0 , G(ϕ 1 ) e * k 1 , G(ϕ 4 ) e * k 2 , G(ϕ 3 ) e * k 3 , G(ϕ 2 ) + O(λ 4 ) .
Recalling now the explicit expression (5.23c) for the constants I k 0 k 1 k 2 k 3 , we see that this correlation function includes the expected q-factors from the interaction term. ▽

Concluding remarks and outlook
In this paper we have initiated the study of noncommutative quantum field theories using modern tools from BV quantization [CG16,Gwi12]. We focused on the case of fuzzy field theories, which are finite-dimensional models and therefore do not require regularization and renormalization. We discussed two different flavors of models, namely ordinary noncommutative field theories (see Sections 2 and 3) and so-called 'braided' noncommutative field theories (see Sections 4 and 5). Our ordinary noncommutative field theories are described at the classical level by ordinary L ∞algebras as in [BBKL18] and, as illustrated in the present paper, their BV quantization can be carried out using precisely the same methods as in commutative field theory [CG16,Gwi12]. We would like to emphasize that this does not mean that such theories are insensitive to noncommutative geometry, which enters through the explicit form of the propagators and the interaction vertices. In particular, we recover from our formalism well-known noncommutative features such as non-planar contributions to loop diagrams in noncommutative scalar field theories, see Example 3.2.
The second flavor of models we have studied are the so-called 'braided' noncommutative theories, which are described at the classical level by the 'braided L ∞ -algebras' proposed in [DCGRS20,DCGRS21]. These are field theories that are defined in the representation category of a Hopf algebra that is endowed with a non-identity triangular R-matrix. The BV quantization techniques of [CG16,Gwi12] generalize in a rather straightforward way to these models (because the representation category of a triangular Hopf algebra is a symmetric braided monoidal category) and we have spelled out the details in Section 4. When applied to a scalar field, our approach coincides with the (symmetric) braided quantum field theory of Oeckl [Oec00]. In particular, we observed in Example 5.4 that non-planar contributions to loop diagrams are absent in the braided framework.
The main lesson of our paper is that modern BV quantization as in [CG16,Gwi12] provides a collection of systematic and powerful tools to study both ordinary and 'braided' noncommutative quantum field theories. In particular, well-known noncommutative features, such as the appearance of non-planar loop contributions or the absence of those in 'braided' field theories, are recovered from this approach. Compared to more traditional approaches, the main advantage of these more abstract BV quantization techniques is that they readily apply to noncommutative gauge theories, as we have illustrated with an explicit example in Section 3.2.
We believe that there are two particularly interesting avenues for future research. Firstly, it would be interesting to study the regularization and renormalization of noncommutative quantum field theories on infinite-dimensional algebras, e.g. the Moyal plane, from our point of view of BV quantization. This requires an adaption of the analytical aspects of Costello and Gwilliam's work [CG16,Gwi12] and an understanding of their interplay with noncommutative phenomena such as UV/IR-mixing. Secondly, it would be interesting to generalize the 'braided BV formalism' in Section 4 to the truly (i.e. non-symmetric) braided case governed by quasi-triangular Hopf algebras. We expect that this will be considerably more difficult than the symmetric braided case discussed in this paper, because it is not at all straightforward to define truly braided analogs of the relevant algebraic structures, such as P 0 -algebras and the BV Laplacian. For instance, already the definition of truly braided analogs of Lie algebras [Maj94] is rather non-intuitive and involved, and we expect that this will also be the case for truly braided P 0 -algebras. As a minimalistic approach, one could skip the P 0 -algebras and try to generalize the explicit form of the BV Laplacian (4.29) to the case of a quasi-triangular R-matrix. The resulting formula agrees with Oeckl's braided Wick Theorem [Oec01], but unfortunately in the truly braided case there are obstructions (due to quasi-triangularity) to the important square-zero condition ∆ 2 BV = 0 of the BV Laplacian. We currently do not know how to resolve these issues and why they seem to play no role in Oeckl's truly braided approach.