After arguing why the Batalin–Vilkovisky (BV) formalism is expected to find a natural description within the framework of noncommutative geometry, we explain how this relation takes form for gauge theories induced by finite spectral triples. In particular, we demonstrate how the two extension procedures appearing in the BV formalism, that is, the initial extension via the introduction of ghost/anti-ghost fields and the further extension with auxiliary fields, can be described in the language of noncommutative geometry using the notions of the *BV spectral triple* and *total spectral triple*, respectively. The construction is presented in detail for all *U*(2)-gauge theories induced by spectral triples on the algebra $M2(C)$. Indications are given on how to extend the results to *U*(*n*)-gauge theories for *n* > 2.

## I. THE BV FORMALISM WITHIN NONCOMMUTATIVE GEOMETRY: THE MOTIVATION

In this article, we aim to present a possible approach to the problem of providing the Batalin–Vilkovisky (BV) formalism with a noncommutative geometrical description or, from the symmetrical perspective, to finally encoding this formalism in the language of spectral triples. The connecting point between the BV formalism on one side and noncommutative geometry on the other is the notion of gauge theory: indeed, the BV construction was initially discovered as a method to eliminate the divergences appearing in the path integral when this was applied to quantize theories endowed with a gauge symmetry. The strength of this approach lies in the fact that this compensation of the divergences is performed in a way that still allows us to keep track of all the relevant physical information of the initial theory (cf. Sec. III). On the other hand, the noncommutative geometry also presents a strong connection with gauge theories via the notion of spectral triples: indeed, while spectral triples can be viewed as a kind of noncommutative generalization of the classical and commutative concept of spin manifolds, they also have the property of naturally encoding a gauge theory. In other words, a spectral triple defines not only some kind of geometrical space but also a gauge-symmetrical physical theory (cf. Proposition II.7). As a consequence, because gauge theories are structurally part of the notion of spectral triples, the noncommutative geometry calls for a coherent description of the BV formalism in this context and, vice versa, a novel geometrical approach to the BV formalism could contribute to a better understanding of the mathematical content of this construction, initially developed to solve a problem occurring in a completely physical context.

In the following, we focus on the case of gauge theories induced by a finite spectral triple. As it will be clarified in Sec. II, finite- and infinite-dimensional spectral triples have an analogous structure and encode a gauge theory in the exact same way. However, when looked from the perspective of the BV construction, the methods applied for the introduction of ghost fields and the extension of the initial theory to a new BV-extended theory are pretty different if considered in the context of gauge theories on finite-dimensional affine varieties, as the ones induced by finite spectral triples, or for theories on manifolds, as in the more general context of spectral triples on infinite-dimensional algebras. However, as briefly mentioned in Sec. II, while at a first sight, focusing on the case of finite spectral triples might appear restrictive from a physical point of view, it is actually quite general and pretty interesting in the context of noncommutative geometry: indeed, many results have shown how the gauge content of relevant physical models can be entirely encoded in suitable finite spectral triples (cf. Refs. 1–3).

The construction will be presented in detail for the class of *U*(2)-gauge theories induced by spectral triples on the algebra $M2(C)$, and this will bring to the introduction of the notions of the *BV spectral triple* and *total spectral triple*. These two concepts will permit us to describe not only the extension of the initial gauge theory via the introduction of ghost/anti-ghost fields but also its further extension with auxiliary fields in a coherent way in terms of a suitable spectral triple. Moreover, this approach will reveal a possible noncommutative geometrical interpretation of the key elements entering the BV construction, such as the ghost sector, the ghost degree, and the parity of the ghost fields. Finally, the construction of these two new spectral triples will also be coherent and consistent with the need of also including in the picture the two cohomological theories of BV and BRST complexes, allowing us to relate these two complexes to other cohomological theories naturally appearing on the noncommutative geometrical side. The details regarding these cohomological aspects will soon be presented in a paper devoted to analyze this specific part of the construction.

In detail, the structure of this article is given as follows: in Sec. II, we briefly recall the key notions and constructions from the noncommutative geometry, while in Sec. III, we describe the classical BV construction in the context of finite gauge theories, from the introduction of ghost/anti-ghost fields to the construction of the induced BV/BRST cohomology complexes. After these more introductory parts, we prove how the BV-extension process can be formalized in the language of spectral triples: in Sec. IV, we introduce the notion of *BV spectral triples*, which encodes in the structure of a real spectral triple both the extended configuration space and the extended action, which has to be a solution of the classical master equation. Section V is devoted to the *gauge-fixing process* and explains how the introduction of auxiliary fields, which are needed to define a gauge-fixing fermion, can also be described within the setting of the noncommutative geometry. In particular, the process of adding the auxiliary fields can be seen as a further enlargement of the BV spectral triple. The result is the construction of a so-called *total spectral triple*, which contains all fields/ghost fields and auxiliary fields, such that the induced cohomology is quasi-isomorphic to the BV complex. To conclude, in Sec. VI, we briefly summarize our approach and give indications on how we expect that the construction could extend to the general case of *U*(*n*)-gauge theories induced by spectral triples on the matrix algebras $Mn(C)$.

## II. THE FRAMEWORK: NONCOMMUTATIVE GEOMETRY, FINITE SPECTRAL TRIPLES, AND THE INDUCED GAUGE THEORIES

The notion of a *spectral triple* plays a key role in the contemporary noncommutative geometry, in particular, when the study of this field is approached not only from a purely mathematical perspective but it is actually investigated looking at its relation with mathematical physics and gauge theories, in particular. Indeed, few years after its foundation,^{4,5} the noncommutative geometry revealed to be a novel mathematical setting for an alternative way of thinking the structure of spacetime:^{6} it is in this context that the notion of the spectral triple was first introduced, with the purpose of modifying the usual mathematical description of the spacetime as a Riemannian manifold by making it slightly noncommutative.

If the spectral triple is the notion that encodes the underlying physical spacetime, the *spectral action principle*^{7} is what allows us to enrich the induced physical theory with an action functional and, hence, with a dynamics. This gave origin to a very fruitful line of research that arrived to the key result obtained by Chamseddine, Connes, and Marcolli^{1,8} of deriving the full Standard Model of particles, with neutrino mixing and minimally coupled to gravity, from purely noncommutative geometrical objects, that is, from spectral triples. Since then, many other results were obtained (cf. Refs. 2 and 3), in the perspective of formally going beyond the Standard Model, following a completely mathematical and coherent approach, within the framework of the noncommutative geometry. These achievements contributed to reinforce the idea that the intrinsic relation of the noncommutative geometry with gauge theories makes this mathematical field a very promising setting for the study of gauge theories and any constructions related to them, having the notion of the spectral triple playing a fundamental role. In this paper, we will apply this perspective to the BV formalism. However, before entering into the details, we briefly recall the main notions we will be using for our construction.

Let $A$ be an involutive unital algebra, $H$ be a Hilbert space, and $D:H\u2192H$ be a self-adjoint operator with a compact resolvent. Then, the triple $(A,H,D)$ is a *spectral triple* if the algebra $A$ can be faithfully represented as operators on $H$ and the commutators [*D*, *a*] are bounded operators for each $a\u2208A$. A spectral triple $(A,H,D)$ is *finite* if the Hilbert space $H$ and, hence, the algebra $A$ are finite dimensional.

*Note:*If we consider a finite spectral triple $(A,H,D)$, a classical result (cf., for example, Ref. 9) allows us to conclude that the condition of being finite dimensional forces the algebra $A$ to be a direct sum of matrix algebras, i.e.,

*n*

_{1}, …,

*n*

_{k}. Moreover, at the level of the operator

*D*, the fact of being in the finite-dimensional setting determines that any additional conditions imposed on the commutators and the resolvent of a self-adjoint operator

*D*would automatically be satisfied.

As anticipated, the notion of the spectral triple can be seen as a noncommutative generalization of the classical commutative concept of geometric space: as proved by Connes in his celebrated reconstruction theorem,^{10} there is a correspondence between, on one side, commutative spectral triples endowed with a *real structure* *J* and a *grading map* *γ* and, on the other hand, compact oriented Riemannian spin manifolds. Clearly, conditions have to be satisfied by these extra structures *J* and *γ*. However, the key point is that if we consider a spectral triple $(A,H,D)$ constructed on a non-commutative algebra $A$, we obtain an entirely new class of purely noncommutative spaces. This is what allows us to see spectral triples as a noncommutative generalization of the classical notion of manifolds.

Because in what follows, the notions of the *real structure* and of *KO-dimension* will play a relevant and physically motivated role; we briefly recall their definitions.

*γ*be a linear map $\gamma :H\u2192H$, endowing the Hilbert space $H$ with a $Z/2$-grading. Then, $(A,H,D,\gamma )$ is called an

*even spectral triple*if the following conditions are satisfied:

*real structure of KO-dimension*

*n*

*(mod*

*8)*on a spectral triple $(A,H,D)$ is an anti-linear isometry $J:H\u2192H$ that satisfies

*ɛ*,

*ɛ*′, and

*ɛ*″ depend on the KO-dimension

*n*(mod 8) as follows:Moreover, we require that for all $a,b\u2208A$,

the action of $A$ satisfies the

*commutation rule*: $a,Jb*J\u22121=0$, andthe operator

*D*fulfills the*first-order condition*: [[*D*,*a*],*Jb*^{*}*J*^{−1}] = 0.

When a spectral triple $(A,H,D)$ is endowed with such a real structure *J*, it is said to be a *real spectral triple* and denoted by $(A,H,D,J)$.

### A. Spectral triples and the induced gauge theories

Classically, a gauge theory is described in terms of connections on a principal bundle, where the structure group of the bundle is the Lie group describing the gauge symmetry of the physical theory. However, next to this approach in the classical and commutative setting of the differential geometry, the notion of *gauge theory* can be reduced to simply encode the idea of having a physical theory that is invariant under the action of a local group of symmetry. This notion, which is precisely stated in Definition II.4, naturally appears in the context of the noncommutative geometry: indeed, each spectral triple, according to this notion, inherently induces a gauge theory. Even though here we restrict to the context of a finite spectral triple, the statement of Proposition II.7 is still valid in a more general context for any kind of spectral triple. This classical property of spectral triples proves their strong and deep relation with gauge theories and the world of quantum field theory.

*X*

_{0}and a real-valued functional

*S*

_{0}on

*X*

_{0}, let $F:G\xd7X0\u2192X0$ be a group action on

*X*

_{0}for a given group $G$. Then, the pair (

*X*

_{0},

*S*

_{0}) is called a

*gauge theory with gauge group*$G$ if

*φ*∈

*X*

_{0}and $g\u2208G$. The space

*X*

_{0}is referred to as the

*configuration space*, an element

*φ*∈

*X*

_{0}is called a

*gauge field*, and

*S*

_{0}is called the

*action functional*.

In order to be able to associate with any given spectral triple a naturally induced gauge theory, we still have to introduce a notion of action in the context of the noncommutative geometry. Actually, there are two definitions of action functionals associated with a spectral triple: while the *spectral action*, introduced in Ref. 7, is the only natural additive spectral invariant of the noncommutative geometry, the *fermionic action*^{1,11} is defined on a subspace of the Hilbert space and can depend on the real structure.

*f*, the associated

*spectral action*

*S*

_{0}is defined to be

*φ*= ∑

_{j}

*a*

_{j}[

*D*,

*b*

_{j}], for

*a*

_{j},

*b*

_{j}being a finite collection of elements in $A$.

*Note:* In the finite-dimensional setting, a family of suitable functions *f* is given by the polynomials in $PolR(x)$ and the trace operator coincides with the standard trace of matrices. Once again, the definition of spectral action has been stated for the case of a finite spectral triple because this is the context we are interested in analyzing. The definition is also meaningful in the more general setting of infinite-dimensional spectral triples, taking into consideration that conditions have to be imposed to ensure that the operator considered is actually a trace-class operator.

*fermionic action*on a subspace $H\u2032\u2286H$ is given by

We conclude by recalling the classical result (cf., for example, Ref. 9) on how to construct, for a finite spectral triple $(A,H,D)$, the induced gauge theory (*X*_{0}, *S*_{0}) with gauge group $G$.

*Given*$(A,H,D)$,

*a finite spectral triple, let*

*X*

_{0}

*denote the space of inner fluctuations for the operator*

*D*,

*where we only consider finite sums and*

^{*}

*denotes the involution structure on*$A$

*. Then, let*$G$

*be the space of unitary elements of*$A$,

*acting on*

*X*

_{0}

*via the map*

*and finally, let*

*S*

_{0}

*be the spectral action defined on*

*X*

_{0}

*as*

*for any*

*φ*∈

*X*

_{0}

*and some*$f\u2208PolR(x)$

*. Then, the pair*(

*X*

_{0},

*S*

_{0})

*is a gauge theory with gauge group*$G$

*.*

## III. THE BATALIN–VILKOVISKY FORMALISM FOR FINITE GAUGE THEORIES

The Batalin–Vilkovisky (BV) construction finds its original motivation in the context of the quantization of a gauge theory (*X*_{0}, *S*_{0}) via the path integral approach.^{12} In particular, this formalism wanted to address the problem of the appearance of divergencies in the path integral due to the presence of local symmetries for the action functional *S*_{0}: indeed, in this context, the action of the gauge group on the configuration space and the gauge invariance of the action functional *S*_{0} determine the presence of orbits of critical points of the action *S*_{0} in the configuration space *X*_{0}. It is precisely the appearance of the critical points in orbits and not as isolated points what prevents to treat these theories using the classical perturbative approach and calls for another perspective on the problem.

The BV formalism can be described as a cohomological approach to the study of the gauge symmetries. However, recently, other points of views have been added to the purely cohomological one and the BV formalism has, then, been widely investigated approaching it from different mathematical perspectives: using techniques from the differential geometry, in a series of papers,^{13–15} Cattaneo, Mnev, and Reshetikhin considered the case of gauge theories over a spacetime with boundary, while Gwilliam, first with Costello^{16} and then with Haugseng,^{17} followed a more algebraic approach, describing the BV construction in terms of *∞*-categories, and, finally, Fredenhagen and Rejzner^{18,19} addressed the topic using techniques from functional analysis.

Despite all these different mathematical descriptions, at the origin of this sophisticated construction, one would always find a pioneering idea of Faddeev and Popov:^{20} given an initial gauge theory (*X*_{0}, *S*_{0}), the divergences in the path integral can be eliminated by the introduction of auxiliary (non-existing) fields, suggestively called *ghost fields*.

*field/ghost field*

*φ*is a graded variable characterized by two integers,

*φ*) is the

*ghost degree*, while

*ɛ*(

*φ*) is the

*parity*, which distinguishes between the bosonic case, where

*ɛ*(

*φ*) = 0 and

*φ*behaves as a real variable, and the fermionic case, where

*ɛ*(

*φ*) = 1 and

*φ*behaves as a Grassmannian variable,

*anti-field/anti-ghost field*

*φ*

^{*}corresponding to a field/ghost field

*φ*satisfies

*Note:* In what follows, the term *fields* are reserved to the initial fields in *X*_{0}, while *ghost fields* are used to identify the extra fields introduced by the BV construction. Analogously, *anti-fields* are specifically used for the anti-particles corresponding to the initial fields, while the *anti-ghost fields* are the ones corresponding to the ghost fields.

### A. The BV-extended theory

^{21}the first and key step of the BV construction

^{22–24}can been seen as an extension process: given an initial gauge theory (

*X*

_{0},

*S*

_{0}), it determines a new extended theory $(X\u0303,S\u0303)$,

*extended configuration space*$X\u0303$ is obtained via the introduction of

*ghost/anti-ghost fields*in the initial configuration space,

*extended action*$S\u0303$ is defined by adding extra terms depending on the ghost/anti-ghost fields to the initial action

*S*

_{0},

*how*to perform this extension and on how many ghost fields/anti-ghost fields to introduce: indeed, these extra particles reflect the type and complexity of the gauge symmetry of the initial gauge theory.

Although the basic idea is the same, the method to perform this extension varies depending on if we are considering either a finite- or infinite-dimensional gauge theory: in particular, if the latter requires a more involved mathematical structure to describe the corresponding extended configuration space, the finite-dimensional context presents cases where we are forced to introduce new independent variables in any degree, and hence, we have to consider an infinite number of ghost fields. Because in the following we will focus on gauge theories induced by finite spectral triples and, hence, defined on affine configuration spaces, we only present the BV construction for this class of theories, referring to Ref. 25 for a detailed description of the construction in the infinite-dimensional setting.

*X*

_{0},

*S*

_{0}), an

*extended theory*associated with it is a pair $(X\u0303,S\u0303)$ where the extended configuration space $X\u0303=\u2295i\u2208Z[X\u0303]i$ is a $Z$-graded super-vector space suitable to be decomposed as

*X*

_{0}. Concerning the extended action $S\u0303\u2208[OX\u0303]0$, it is a real-valued regular function on $X\u0303$, with $S\u0303|X0=S0$, $S\u0303\u2260S0$, such that it solves the

*classical master equation*, i.e.,

*Note:*The condition imposed in (3.1) reflects the essential requirement of the BV formalism of having a symmetric structure in $X\u0303$, where for each ghost field introduced, we automatically also include the associated anti-ghost field. While $F$ describes the content of fields/ghost-fields of $X\u0303$, $F*[1]$ determines the part of anti-fields/anti-ghost fields, with $F*[1]$ that denotes the shifted dual module of $F$,

*graded super-vector space*simply encodes the fact that ghost/anti-ghost fields of even degree have to be treated as real variables, while they are Grassmannian variables if they have odd ghost degree. Hence, the algebra $OX\u0303$ of real-valued regular functions defined on $X\u0303$ has the structure of a $Z$-graded algebra. In addition, $OX\u0303$ can also be endowed with a graded Poisson structure induced by the bracket

The method used to explicitly determine the collection of ghost fields to introduce in the initial configuration space is based on the Koszul–Tate resolution:^{26} a procedure is described in Ref. 27, which uses the whole Koszul–Tate resolution, ensuring the gauge-invariance of the resulting extended theory, while in Ref. 28, we explained how to select a finite number of ghost fields, which reflects the complexity of the gauge symmetry of the theory and, hence, determines an exact solution of the classical master equation as extended functional $S\u0303$. Based on that result, even though it is not explicitly required in the notion of extended theory, in what follows, we will consider pairs $(X\u0303,S\u0303)$ with a *finite level of reducibility*.

An extended theory $(X\u0303,S\u0303)$, with $X\u0303=F\u2295F*[1]$ for $F$ being a $Z\u2a7e0$-graded and finitely generated $OX0$-module, is *reducible with a level of reducibility* *L* = *k* − 1 ⩾ 1 if $F=\u2295i=0kFk$. Otherwise, if *L* = 0, the theory is called *irreducible*.

### B. The auxiliary fields and the gauge-fixing process

*X*

_{0},

*S*

_{0}) has been extended via the introduction of ghost/anti-ghost fields, still, the action functional $S\u0303$ appears to be in a form that is not suitable for an analysis of the theory through methods coming from perturbation theory. The reason is the presence of anti-fields/anti-ghost fields, which have to be eliminated before being able to compute amplitudes and S-matrix elements. Hence, as a next step in the BV construction, we have to implement a gauge-fixing process on the extended theory $(X\u0303,S\u0303)$: given an extended theory, we want to determine a new pair $(X\u0303,S\u0303)|\Psi $ where neither the

*gauge-fixed configuration space*$X\u0303|\Psi $ nor the

*gauge-fixed action*$S\u0303|\Psi $ depends on anti-fields/anti-ghost fields. This goal is reached by defining

*gauge-fixing fermion*.

Given an extended configuration space $X\u0303=F\u2295F*[1]$, a *gauge-fixing fermion* Ψ on it is a regular function $\Psi \u2208OF\u22121$, that is, a regular function depending only on fields/ghost fields, of total degree −1 and, hence, odd parity.

Thus, the new gauge-fixed theory $(X\u0303,S\u0303)|\Psi $ is obtained by imposing the *gauge-fixing conditions* $\phi i*=\u2202\Psi /\u2202\phi i$ on both $X\u0303$ and $S\u0303$. In order words, $X\u0303|\Psi $ is defined to be the Lagrangian submanifold determined by replacing every anti-field/anti-ghost field $\phi i*\u2208F*[1]$ with the partial derivative of Ψ with respect to the corresponding field/ghost field *φ*. Similarly, $S\u0303|\Psi $ is the restriction of the action $S\u0303$ to $X\u0303|\Psi $.

*Note:*in order to properly implement a gauge-fixing procedure, further conditions have to be enforced on the gauge-fixing fermion Ψ. Indeed, to ensure that this process is well defined from a physical point of view, we need that the physically relevant quantities that are computed as

*proper solution*of the classical master equation (cf. Refs. 29–31).

*X*

_{0},

*S*

_{0}) we are interested in analyzing. Hence, to conclude that the BV complex induced by the new pair (

*X*

_{t},

*S*

_{t}) is quasi-isomorphic to the one induced by the extended theory $(X\u0303,S\u0303)$, we require that the auxiliary fields are trivial from a cohomological point of view, that is, they determine

*contractible pairs*in cohomology (cf. Refs. 22 and 23).

*auxiliary pair*is a pair of fields (

*B*,

*h*) such that their ghost degrees and parities satisfy the following relations:

*i*= 1, …,

*r*, the corresponding

*total theory*(

*X*

_{t},

*S*

_{t}) has a

*total configuration space*

*X*

_{t}defined as the $Z$-graded super-vector space generated by $X\u0303$, (

*B*

_{i},

*h*

_{i}), and their corresponding anti-fields $(Bi*,hi*)$ for

*i*= 1, …,

*r*,

*total action*

*S*

_{t}given by the following sum:

*X*

_{t},

*S*

_{t}) satisfy similar properties, the only difference lying in the fact that

*X*

_{t}may also contain negatively graded fields. In particular,

*X*

_{t}also presents the symmetry between fields/ghost fields and anti-fields/anti-ghost fields content, being suitable for the following decomposition:

*X*

_{t}by imposing that

*φ*being any generator in $X\u0303$ and

*ξ*being any auxiliary fields among

*B*

_{i},

*h*

_{i}, or their corresponding anti-fields $Bi*$ and $hi*$, while the value of the bracket on any other possible combination of auxiliary fields and corresponding auxiliary anti-fields is declared to be zero. As a consequence, the total action

*S*

_{t}solves the classical master equation on $OXt$,

To conclude, in order to be able to implement a gauge-fixing procedure on a BV-extended theory $(X\u0303,S\u0303)$, additional auxiliary fields have to be introduced, determining a so-called *total theory* (*X*_{t}, *S*_{t}). These auxiliary fields are introduced in pairs so that they are trivial from a cohomological point view. Finally, the number and type of auxiliary pairs are determined by the level of reducibility of the extended theory so that the corresponding gauge-fixed action *S*_{t}|_{Ψ} determines a proper solution of the classical master equation, property that ensures that all the physically relevant quantities of type (3.3) are invariant with respect to the choice of gauge-fixing fermion Ψ. In addition to the references quoted above, some details on the gauge-fixing procedure for finite-dimensional gauge theory can also be found in Ref. 32.

### C. The BV/BRST cohomology complexes

*BV complex*and the

*BRST complex*. Even though in this paper we will not explicitly face the problem of relating these complexes with other cohomology complexes naturally appearing in the context of spectral triple, still, we think it is worth to mention them and their relevance lying in the fact that they detect important physical information about the initial gauge theory we are examining. Indeed, if the number and type of ghost/anti-ghost fields introduced to extend the initial configuration space

*X*

_{0}reflect the complexity of the gauge symmetry of the theory, the conditions imposed on the pair $(X\u0303,S\u0303)$ entail that any extended theory naturally induces a cohomology complex: indeed, while the conditions

*BV cohomology complex*is a cohomology complex whose cochain spaces $Ci(X\u0303,dS\u0303)$ and coboundary operator $dS\u0303$ are defined as follows, respectively:

*Note:* the fact that the pair $(C\u2022(X\u0303,S\u0303),dS\u0303)$ defines a cohomology complex is an immediate consequence of the bracket {, } being of degree 1, with $S\u0303$ of degree 0, and satisfying the graded Jacobi identity in addition to $S\u0303$ solving the classical master equation. We also remark that this complex is two-sided. Hence, explicitly computing its cohomology groups could require involved computations. This is why we believe that inserting the BV construction in the framework of the noncommutative geometry and, hence, relating the BV complex to other cohomological theories already well-studied in the context of spectral triples could contribute to a better understanding of this interesting complex. While in this article we focus on describing the two extension processes and the gauge-fixing procedure in this new framework via the notions of *BV spectral triple* and *total spectral triples*, the part regarding the cohomological aspects will be presented in detail in another article currently in preparation.

As the extended theory $(X\u0303,S\u0303)$, the total theory (*X*_{t}, *S*_{t}) also naturally induces a cohomology complex, which by construction is quasi-isomorphic to the BV complex. However, once the gauge-fixing process has been implemented, a natural question arises: how does this process reflect at the level of the induced cohomology complex? Is there a residual BRST-symmetry on (*X*_{t}, *S*_{t})|_{Ψ} that induces a new cohomology complex? The answer to this question is always positive if we consider the theory *on-shell*, that is, if we suppose that the equations of motion *∂*(*S*_{t}|_{Ψ})/*∂φ*_{i} = 0 are satisfied for all $\phi i\u2208E$ (cf. Ref. 33). However, an explicit computation shows that if we restrict the operator $dSt$ to the algebra $OXt|\Psi $ of regular functions on the gauge-fixed configuration space, this might still satisfy the coboundary condition also off-shell depending on the explicit form of the action $S\u0303$ (cf. Ref. 32). Hence, it is meaningful and physically relevant to consider what is called the *BRST cohomology complex*.

*X*

_{t},

*S*

_{t}), with $Xt=E\u2295E*[1]$ and $St\u2208[OXt]0$, together with a gauge-fixing fermion $\Psi \u2208[OE]\u22121$, the induced

*BRST complex*$(Cj(Xt,dSt)|\Psi ,dSt|\Psi )$ is a cohomology complex with

*X*

_{t}|

_{Ψ}⊂

*X*

_{t}is the Lagrangian submanifold defined by the gauge-fixing conditions ${\phi i*=\u2202\Psi /\u2202\phi i}$.

^{34–36}and, independently, by Tyutin

^{37}(see also Ref. 38), this cohomology complex clearly shows how these non-existing additional fields introduced in the theory, namely, the ghost fields, are not just a technical tool but they also determine more complicated mathematical structures, which allow us to detect physical properties of the theory we are investigating (see also Refs. 39–41). The physical relevance of this cohomology complex is due to the fact that its zero-degree cohomology group encodes the classical observables of the initial gauge theory,

*Note:* The gauge-fixing procedure determines a reduction of the configuration space and, hence, the generators for cochain spaces. In particular, there are classes of models, among which the one we are interested in, for which the BRST complex turns out to be a one-sided complex. The fact of having only generators of non-negative degree simplifies the computation of the cohomology groups, whose relevance and physical meaning in any strictly positive degree still have to be fully understood.

To summarize, we can describe the BV construction as a process made of few steps, where the first is the extension of the initial gauge theory (*X*_{0}, *S*_{0}) via the introduction of ghost and anti-ghost fields. Once obtained the *extended theory* $(X\u0303,S\u0303)$, one can analyze the induced *BV cohomology complex*. The next step in the construction is the *introduction of auxiliary fields*, which are essential to be able to perform a gauge-fixing procedure: after having obtained the so-called *total theory* (*X*_{t}, *S*_{t}), one performs the *gauge-fixing procedure* and finally determines and maybe computes the corresponding *BRST complex*.

## IV. THE BV SPECTRAL TRIPLE FOR *U*(2)-MATRIX MODELS

Starting from this section, we demonstrate how the noncommutative geometry and the language of spectral triples can be used to provide the BV construction with a novel geometric interpretation. In particular, we will focus on the two extension processes present in this construction via ghost/anti-ghost fields first and then with the auxiliary fields. Some preliminary results were already presented in Ref. 42, mostly concerning the introduction of the notion of *BV spectral triples*. However, here, we slightly modify this notion to include the extra information about the ghost degree of the generators in the Hilbert space $HBV$: indeed, keeping track of these degrees is crucial to be able to directly associate with the BV spectral triple a cohomology complex and, more, in general, to extend our preliminary result to include all the steps that enter the BV construction.

### A. The model

*D*

_{0}is a self-adjoint

*n*×

*n*-matrix. By applying the classical construction recalled in Proposition II.7, we deduce that the above spectral triple yields an initial gauge theory (

*X*

_{0},

*S*

_{0}) with gauge group $G$ such that

*f*is a polynomial in $PolR(x)$, and $G$ acts on

*X*

_{0}via the adjoint action. In particular, we focus on the case of

*n*= 2. The reason for restricting to this case does not have to be searched on the side of the spectral triples as their structure and the type of gauge theory induced do not change at the increasing of the dimension of the algebra, but on the side of the BV formalism: indeed, what changes from the

*n*= 2 to

*n*> 2 case is the construction of the ghost sector via the computation of the Koszul–Tate resolution and, then, the research of a solution of the classical master equation on this extended configuration space. Computing the Koszul–Tate resolution in the general case of

*n*> 2 requires more involved computations and extra mathematical tools from the field of the algebraic geometry and invariant theory. As all of this would go beyond the purpose of this article, to the construction of the BV-extended pair $(X\u0303,S\u0303)$ in the general case would be devoted another article, currently in preparation, where all the details will be discussed. However, as more specifically addressed in Sec. VI, we do not expect any further complication appearing on the side of the BV spectral triple, foreseeing the direct generalization of what presented here for the case

*n*= 2 to the general context of

*U*(

*n*)-gauge theories induced by finite spectral triples on the algebra $Mn(C)$.

*n*= 2, the model constructed in (4.1) can be described explicitly by fixing as a basis for

*X*

_{0}the one given by Pauli matrices (together with the identity matrix),

*X*

_{0}is isomorphic to a four-dimensional real vector space generated by four independent initial fields,

*X*

_{0}is the ring of polynomials in the variables

*x*

_{a}, $OX0=PolR(xa)$. Finally, it can be checked that the most general form for a functional

*S*

_{0}on

*X*

_{0}that is invariant under the adjoint action of the gauge group

*U*(2) is given as

*X*

_{0},

*S*

_{0}) as initial gauge theory, one determines the corresponding BV-extended theory $(X\u0303,S\u0303)$ by computing (part of) the Tate resolution of the Jacobian ideal $J(S0)\u2254\u27e8\u2202iS0\u27e9i=1,\u2026,4$ on the ring $OX0$ (cf. Ref. 28 for the detailed construction). The result is that for the case of a generic initial action

*S*

_{0}, that is, if it holds that $GCD(\u2202iS0)i=1,\u2026,4=1$, then the corresponding minimally extended BV theory $(X\u0303,S\u0303)$ has extended configuration space

*ɛ*

_{ijk}being the totally anti-symmetric tensor in three indices

*i*,

*j*,

*k*∈ {1, 2, 3} with

*ɛ*

_{123}= 1.

*BV spectral triple*$(ABV,HBV,DBV,JBV)$ directly from the initial one and, hence, to perform the first step in the BV construction by extrapolating the key properties of our initial gauge theory from the geometric properties of the spectral triple we started with. We discovered that the answer to this question is positive, and in the remaining of this section, we describe in detail how to perform this passage,

### B. BV spectral triple: The aim

The goal is to determine a new spectral triple $(ABV,HBV,DBV,JBV)$ such that

it contains all the ghost/anti-ghost fields appearing in $X\u0303$;

the induced fermionic action coincides with the BV action $SBV\u2254S\u0303\u2212S0$.

The reason why we expect that extending the initial gauge theory via the introduction of ghost/anti-ghost fields enforces, at the level of the corresponding spectral triple, the introduction of a *real structure* *J*_{BV} has to be searched in the different nature of two structures: while the spectral triples are naturally defined in a complex setting, the bosonic fields we consider are actually real. Hence, the need of a real structure in the BV spectral triple. Similarly, we foresee that it would be the *fermionic action*, instead of the spectral action, to play a role because of the appearance of Grassmannian/fermionic fields in the extended action $S\u0303$. Moreover, because the algebra of local gauge symmetries is closed, the extended action is expected to be linear in the anti-fields/anti-ghost fields (cf. Ref. 25), which allows us to search for a *linear operator* *D*_{BV}, inducing the fermionic action, instead of having to consider multi-linear objects. Finally, we expect that the new ghost/anti-ghost fields appear at the level of the Hilbert space $HBV$, which is what plays a role in the fermionic action, while the algebra $ABV$ remains unchanged.

Hence, as a consequence of the above observations, we have to describe the following:

how to extend the initial Hilbert space $H0=C2$ to a new $HBV$, which includes all the ghost/anti-ghost fields, keeping track of their ghost degree. Hence, we are looking for a method to directly obtain the information about the type and number of ghost fields to introduce by looking at the initial spectral triple $(A0,H0,D0)$;

how to define a self-adjoint operator

*D*_{BV}such that the induced fermionic action, added to the initial action*S*_{0}, would give a solution to the classical master equation.

By solving the two points listed above, we would lift the BV-extension mechanism from the algebraic geometric context to the framework of the noncommutative geometry, having as advantage the possibility of skipping the construction of a (partial) Tate resolution for the Jacobian ring *J*(*S*_{0}) and the application of a recursive algorithm to determine the extended action $S\u0303$ as an approximate solution of the classical master equation (cf. Ref. 27)

### C. The Hilbert space $HBV$

How can we determine the ghost sector for the extended theory just by looking at the initial spectral triple? The key idea is that the type and number of these ghost fields would reflect the type and number of symmetries appearing in the initial action *S*_{0}.

*f*of degree

*deg*(

*f*) > 1, then the generic initial action is of the form in (4.3). While the factors

*g*

_{k}(

*x*

_{4}) are completely independent polynomials, the symmetries of the action are enclosed in the term

the presence of three independent symmetries among pairs of variables

*x*_{a}is responsible for the introduction of three independent ghost fields of ghost degree 1, which we denote by*C*_{i}for*i*= 1, 2, 3;the presence of the single symmetry involving all three variables

*x*_{a}determines the introduction of a single ghost field of degree 2, denoted by*E*.

*i*) or its complementary depending on the operator

*D*

_{BV}anti-commuting or commuting with

*J*

_{BV}, respectively. Therefore, while $isu(2)$ describes the Hermitian traceless part of $M2(C)$, $u(1)$ gets included in $M2(C)$ as its trace component, determining that the BV-extending procedure at the level of the Hilbert spaces reads as follows:

### D. The real structure *J*_{BV}

### E. The linear operator *D*_{BV}

The expression of the self-adjoint linear operator *D*_{BV} acting on the Hilbert space $HBV$ is completely determined by implementing the following observations:

The BV action

*S*_{BV}is*linear in the anti-fields/anti-ghost fields*due to the algebra of symmetries being a classical Lie algebra and, hence, being closed already off-shell.*S*_{BV}is a regular function in $[OX\u0303]0$, that is, it has*total degree 0 and bosonic parity*.Because the ghost field

*E*and the corresponding anti-ghost field*E*^{*}are the variables of highest and lowest ghost degrees, respectively, the algorithm stops before and, hence, the operator*D*_{BV}depends neither on*E*not on*E*^{*}.As explicitly requested in the definition of extended theory (cf. Definition III.2), the extended action $S\u0303$, restricted to the initial configuration space

*X*_{0}, should coincide with the initial action*S*_{0}. In other words, the operator*D*_{BV}cannot contain blocks, which both depend and act on the initial fields*x*_{a}for*a*= 1, …, 4, that is, it cannot contribute new terms only depending on the initial fields.- There are two different types of blocks appearing in
*D*_{BV}, given by the following linear operators:and$Ad(p):M2(C)\u2192M2(C),\phi \u21a6[\alpha (pi),\phi ]$where$Ab(p):M2(C)\u2192M2(C),\phi \u21a6[\alpha (pi),\phi ]+,$*α*(*p*_{i}) is an Hermitian, traceless 2 × 2-matrix, whose expression in terms of the Pauli matrices is given bywhere the notation$\alpha =12(\u2212p1)\sigma 1+(\u2212p2)\sigma 2+(\u2212p3)\sigma 3,$*p*_{i}is used to generically denote a ghost/anti-ghost field, to be determined in agreement with the other conditions imposed on the operator*D*_{BV}. We stress that*Ad*is a derivation of $M2(C)$, while*Ab*is an odd derivation given in terms of the anti-commutator. Explicitly, in terms of the orthonormal basis (4.2) for $M2(C)$, the linear operators*Ad*and*Ab*have the following form as 4 × 4-matrices:(4.8)$Ad(p)\u22540+ip3\u2212ip20\u2212ip30+ip10+ip2\u2212ip1000000,$(4.9)$Ab(p)\u2254000p1000p2000p3p1p2p30.$

The criterion to determine which linear operator between *Ad*(*p*) and *Ab*(*p*) is inserted in a block is to look at the parity of the variables involved. Indeed, due to the condition imposed on the action of being bosonic, there are only two possibilities for the parities of the variables in the vectors $\phi \u2208HBV,f$ and in the matrix *D*_{BV}: either they are all bosonic or two of them are fermionic and only one is bosonic. Then, we assign the operators *Ad* and *Ab* to these two cases as follows:

*fermionic-bosonic-fermionic* ⇝ *Ad*, *bosonic-bosonic-bosonic* ⇝ *Ab*.

While all the criteria listed above are somehow related to the conditions imposed to the extended action $S\u0303$ or are consequences of the parity rules for the ghost/anti-ghost fields, if one would try to explicitly construct the operator *D*_{BV} step by step by enforcing these conditions, one would actually note that there would still be the freedom to put a sign in front of the operators *Ad*(*p*_{i}) and *Ab*(*p*_{i}). To fix this remaining freedom, one simply requires that none of the contribution to the induced fermionic action cancels out. Requiring this implies having to add a minus sign in front of the operator *Ad*(*p*_{i}) when it appears in the off-diagonal part of *D*_{BV}, with the *p*_{i} fields being of bosonic parity. Summarizing, implementing the above conditions, one can prove the following proposition.

*Let*

*D*

_{BV}

*be a self-adjoint operator on*$HBV$

*such that the induced fermionic action*

*behaves as a BV action, that is, it satisfies conditions (1)−(5) in the list. Then,*

*D*

_{BV}

*has the following form as*

*the block matrix:*

*for*

The proof is obtained as an immediate consequence of the implementation of conditions (1)–(5): the block structure in (4.10) is due to *D*_{BV} being a self-adjoint operator and to the linearity condition in (1), which determines the appearance of a zero block in the top-left corner. Then, because the BV action does not depend on *E*/*E*^{*} as required in (3), the first line in R and the first element on its second line are 0. The appearance of an anti-diagonal dependence on the variables *C* and *x* is due to the condition imposed on the degree of the action in (2), while the choice between the operators *Ad* and *Ab* in each block is determined by (5). For what concerns the matrix S, its top-left corner is zero due to condition (4), while the diagonal dependence on *x*^{*}/*C*^{*} and the zeros appearing in the bottom-right corner are again due to the degree condition in (2). We only observe that there is some remaining freedom related to the choice of the coefficients in front of the operators *Ad*(*x*) and *Ab*(*x*). While other choices could still give an induced fermionic action, which, added to *S*_{0}, solves the classical master equation, our choice of the factors 1/2 is motivated by the form we previously obtained for the BV action *S*_{BV} for our model of interest.□

*Note:* Explaining why the two operators entering the expression of *D*_{BV} have exactly the form described in (4.8) goes beyond the purpose of this article. However, naturally what plays a role in determining the entries of these matrices is precisely the type of symmetries appearing in the initial action functional *S*_{0}. In other words, these matrices are the ones encoding the *SU*(2)-symmetry appearing in (4.6), and hence, they enter the *D*_{BV} operator for any *U*(2)-gauge theory. However, a more refined structure had to be considered in the case of other types of symmetries appearing in the more general context of *U*(*n*)-gauge theories for *n* > 2.

*J*

_{BV}and the self-adjoint operator

*D*

_{BV}, we can now explain how, given the Hilbert space $HBV$, one can identify its effective subspace $HBV,f$. The leading idea is to select the effective part of $HBV$ in order to obtain an induced fermionic action

*S*

_{ferm}with real coefficients. However, as in

*D*

_{BV}they appear in both real and purely imaginary terms, we cannot globally select as effective subspace $HBV,f\u2282HBV$ the one containing self-adjoint or skew-Hermitian matrices, that is, the one satisfying either

### F. The algebra $ABV$

To complete the definition of the BV spectral triple, we still have to determine the algebra $ABV$. However, as previously observed, we anticipate that the BV construction would only act by enlarging the Hilbert space to $HBV$ and the operator to *D*_{BV}, in addition to requiring the introduction of a real structure, while the algebra was expected to stay unchanged. On the other hand, it can also be proved that the choice of $ABV=A0=M2(C)$ is optimal: indeed, $M2(C)\u2282L(HBV)$ is the largest unital subalgebra of the algebra of all linear operators on $HBV$ that satisfies the *commutation rule* and *first-order condition* of Definition II.3.

*Let*$HBV$

*,*

*J*

_{BV},

*and*

*D*

_{BV}

*be as defined above. Then, the maximal unital subalgebra*$ABV$ $\u2282L(HBV)$

*of linear operators on*$HBV$

*that satisfies*

*for all*$a,b\u2208ABV$

*is given by*$ABV=M2(C)$

*acting diagonally on*$HBV$

*.*

*a*

_{1}= ⋯ =

*a*

_{6}and

*b*

_{1}= ⋯ =

*b*

_{6}. This selects the subalgebra $M2(C)$ as the maximal one for which both of the above conditions are satisfied. Alternatively, one can prove the statement using a graphical method based on the notion of

*Krajewski diagram*.

^{9,43}□

*D*

_{BV}is defined in (4.10) and the real structure

*J*

_{BV}is defined in (4.7). However, before proving that this quadruple is actually the BV spectral triple we have been looking for, namely, that it really encodes the extended theory $(X\u0303,S\u0303)$, we first remark that in this spectral triple, it appears as the phenomenon of a

*mixed KO-dimension*, related to the fact that the operator

*D*

_{BV}partially commutes and partially anti-commutes with the real structure

*J*

_{BV}. Indeed, we may decompose

*D*

_{BV}as

*D*

_{1}

*anti-commutes*with

*J*

_{BV},

*D*

_{2}

*commutes*with it,

*J*, an anti-linear isometry on $H$, $(A,H,D,J)$ defines a

*real spectral triple with mixed KO-dimension*if

*J*satisfies

*D*is the sum of two self-adjoint operators

*D*

_{1},

*D*

_{2}, which

*anti*-commutes and commutes, respectively, with

*J*,

We conclude this section by stating our main theorem: the method described above to associate, to a given initial finite spectral triple $(A0,H0,D0)$, a corresponding real spectral triple $(ABV,HBV,DBV,JBV)$, with mixed KO-dimensions 1 and 7 (mod 8) accomplishes our goal of encoding the construction of a BV extension for our *U*(2)-gauge theory all within the framework of the noncommutative geometry. In other words, the spectral triple $(ABV,HBV,DBV,JBV)$ is the *BV spectral triple* associated with the initial triple $(A0,H0,D0)$.

*X*

_{0},

*S*

_{0}), a real (mixed KO-dimensional) spectral triple $(ABV,HBV,DBV,JBV)$ is a

*BV spectral triple*associated with the spectral triple $(A0,H0,D0)$ if the pair

*X*

_{0},

*S*

_{0}).

In the above definition, in order to be able to perform the sum appearing in the expression for the extended action $S\u0303$, we are implicitly identifying the functional *S*_{ferm} with its representation as a polynomial in the algebra $OX\u0303$.

*Let*

*be a finite spectral triple with induced gauge theory*(

*X*

_{0},

*S*

_{0})

*. Then,*

*for*

*and*

*D*

_{BV}

*,*

*J*

_{BV}

*as defined in*

*(4.10)*

*and*

*(4.7)*,

*respectively, is a BV spectral triple associated with*$(A0,H0,D0)$

*.*

*X*

_{0},

*S*

_{0}). For what concerns the first point, the fact that $(ABV,HBV,DBV,JBV)$ defines a real spectral triple is a straightforward consequence of what observed and proved in the first part of this section and, in particular, in Lemma IV.3. On the other hand, by applying to the spectral triple $(ABV,HBV,DBV,JBV)$ what indicated in Definition IV.5, one would find that

*X*

_{0},

*S*

_{0}) by performing the classical BV-extension procedure. For what concerns the functional $S\u0303$, by applying what required in Definition IV.5 to our specific case, we would find that

*ɛ*

_{ijk}being the totally anti-symmetric tensor in three indices

*i*,

*j*,

*k*∈ {1, 2, 3} with

*ɛ*

_{123}= 1, obtaining a functional $S\u0303$ coinciding with the extended action functional determined for our model by applying the classical BV construction; see (4.5). Hence, by comparison with what explicitly computed for our model using the classical BV construction, we can conclude that the pair $(X\u0303,S\u0303)$ just constructed defines a BV extended theory for the gauge theory (

*X*

_{0},

*S*

_{0}) induced by the spectral triple $(A0,H0,D0)$.□

## V. THE GAUGE-FIXING PROCEDURE IN TERMS OF NONCOMMUTATIVE GEOMETRY

In this section, we also prove how the second extension process entering the BV construction, namely, the introduction of auxiliary fields in the BV-extended theory $(X\u0303,S\u0303)$, can be inserted in the framework of the noncommutative geometry and described using the language of spectral triples. This goal will be achieved by the introduction of the notion of *total spectral triples*. The interesting aspect to underline is that both these spectral triples, that is, the BV and the total spectral triple, present the same structure. This coherence confirms that the approach we suggest to insert the BV formalism in the framework of the noncommutative geometry has an intrinsic consistency.

### A. The total spectral triple

*total theory*(

*X*

_{t},

*S*

_{t}). Hence, we expect that the total configuration space

*X*

_{t}, which is obtained as a further extension of $X\u0303$ via the introduction of pairs of auxiliary fields and anti-fields,

*D*

_{t}, which will be an extension of the operator

*D*

_{BV}entering the BV spectral triple.

The introduction of the auxiliary pairs is just a technical step, which only depends on the type and number of ghost fields introduced in $X\u0303$ and is not affected by other characteristics of the model. More precisely, what plays a key role is the so-called *level of reducibility of theory*, whose precise definition has been stated in Definition III.3. Knowing the level of reducibility of the theory, one can determine the minimal list of auxiliary pairs by applying the following theorem.

*Let*$(X\u0303,S\u0303)$

*be an extended theory with the level of reducibility L. Then, the collection of auxiliary pairs*${(Bij,hij)}$

*with*

*i*= 0, …,

*L*

*,*

*j*= 1, …,

*i*+ 1

*that has to be introduced so that the gauge-fixed action*

*S*

_{t}|

_{Ψ}

*of the corresponding total theory*(

*X*

_{t},

*S*

_{t})

*is a proper solution of the classical master equation is completely determined by imposing that*

This theorem was first proved inductively by Batalin and Vilkovisky (cf. Refs. 22 and 23). In addition, we refer to Ref. 30 for an extended explanation of the reason for enforcing the condition of being a proper solution of the classical master equation imposed on the gauge-fixed action.

*U*(2)-model already taken into analysis. However, because the only aspect that plays a role is the level of reducibility of the theory, we expect that this procedure could straightforwardly be applied to a generic BV spectral triple, going beyond the specificity of this class of model.

### B. The Hilbert space $Ht$

As just recalled, we first have to determine the level of reducibility of our theory. This can be done immediately by looking at the graded structure of $HBV$, finding in our case *L* = 1. Hence, according to Theorem V.1, we have the following:

- The three ghost fields
*C*_{i}generating the term $[isu(2)]1$ determine the appearance of the summandsin $Ht,f\u2286Ht$, which corresponds to three trivial pairs ${(Bi,hi)}i=1,2,3$ with$[su(2)]\u22121\u2295[isu(2)]0\u2286[M2(C)]\u22121\u2295[M2(C)]0$$deg(Bi)=\u22121anddeg(hi)=0.$ - The ghost field
*E*embedded in the algebra $[M2(C)]2$ as the generator of $[u(1)]2\u2286[M2(C)]2$ determines the introduction of two trivial pairs (*A*_{1},*k*_{1}) and (*A*_{2},*k*_{2}) with$deg(A1)=\u22122,deg(k1)=\u22121,deg(A2)=0,deg(k2)=1.$

Even though at this point the fact that the three auxiliary fields *B*_{i} should be embedded in $M2(C)$ as the term $[su(2)]$ while the three fields *h*_{i} as $[isu(2)]$ might appear arbitrary, one can check that this choice is the only one that would allow us to comply with the requirement of having an induced auxiliary action *S*_{aux} with *real* coefficients.

### C. The real structure *J*_{t}

*J*

_{BV}from $HBV$ to $Ht$. Explicitly,

### D. The operator *D*_{t}

*S*

_{t}with trivial terms so that they would determine contractible pairs in the induced BV cohomology complex. Hence, the operator

*D*

_{t}should be defined so that the corresponding fermionic action would be given by

*S*

_{t}there are no terms involving both the auxiliary fields and the ghost fields/anti-ghost fields already introduced in $X\u0303$, the operator

*D*

_{t}has to be representable as a block matrix with

*T*, it is determined by recalling the pairing of the auxiliary fields in trivial pairs. Explicitly, the matrix

*T*has to have the following block-structure:

*D*

_{aux}.

### E. The algebra $At$

Similarly to what done for the construction of the BV spectral triple, we choose $At$ to be the maximal algebra that completes the triple $(Ht,Dt,Jt)$ to a real spectral triple, satisfying all the commutation/anti-commutation requirements and the first order condition.

*Let*$Ht$

*,*

*J*

_{t},

*and*

*D*

_{t}

*be as defined above. Then, the maximal unital subalgebra*$At$ $\u2282L(Ht)$

*of linear operators on*$Ht$

*that satisfies*

*for all*$a,b\u2208At$

*is given by*$At=M2(C)$

*acting diagonally on*$Ht$

*.*

The proof of this lemma is analogous to the one given in Lemma IV.3: also, in this case, the statement can be verified either with a direct computation or using the tool of the Krajewski diagrams.^{43}□

*total spectral triple*for the BV spectral triple $(ABV,HBV,DBV,JBV)$ if the pair (

*X*

_{t},

*S*

_{t}) is defined as

In the above definition, we are implicitly identifying the functional *S*_{ferm} with its representation as a polynomial in the algebra $OXt$. Similarly, the extended action $S\u0303$ can also be seen as an element of the same algebra.

*D*

_{t}is clearly self-adjoint. Moreover, because the map

*J*

_{t}has been simply obtained as an extension of the real structure

*J*

_{BV}, it clearly defines a real structure on the whole Hilbert space $Ht$. In addition, by a direct computation, one can verify that once again the operator

*D*

_{t}partially commutes and partially anti-commutes with the real structure,

*X*

_{t},

*S*

_{t}) obtained by implementing what prescribed in Definition V.4 on the triple $(M2(C),[M2(C)]\u229518,Dt,Jt)$ coincides with the total theory associated with the BV-extended theory $(X\u0303,S\u0303)$ according to the classical construction recalled in Theorem V.1.□

*gauge-fixing fermion*Ψ as

All this allows us to conclude that not only the process of including in an extended theory the required pairs of auxiliary fields but also the concept of gauge-fixing fermions perfectly fit in the framework provided by the noncommutative geometry: in particular, using the notion of spectral triples, we obtain a (noncommutative) geometric interpretation of the BV construction, where the appearance of the new phenomenon of a mixed KO-dimension is a sign of the two natures of the ghost fields introduced in the model, part of which are bosonic and part fermionic.

## VI. OUTLOOKS: THE WHOLE BV CONSTRUCTION IN THE SETTING OF NONCOMMUTATIVE GEOMETRY

In this article, we reported about our approach to the problem of trying to include the BV construction in the world of noncommutative geometry. In particular, focusing on gauge theories induced by finite spectral triples, here, we proved how both the extension procedures appearing in this construction can be described as further enlargements of the spectral triples, with ghost/anti-ghost/auxiliary fields entering the Hilbert space and the self-adjoint operators *D*_{BV}/*D*_{t} determining the extended action $S\u0303$/the total action *S*_{t}, respectively, as their induced fermionic actions. At this point, the natural question to ask regards the BV/BRST cohomology complexes and how they enter this framework. These questions have already found an answer, which will be presented in detail in an paper currently in preparation and completely devoted to explain all the details. However, briefly, the idea is that this BV spectral triple has been constructed in such a way that it naturally induces a cohomology complex within the language of noncommutative geometry. The key result is that this complex coincides with the BV complex induced by the extended theory $(X\u0303,S\u0303)$. Moreover, similarly, the total spectral triple also induces such a complex, which is quasi-isomorphic to the BV complex. Finally, after having performed the gauge-fixing procedure, the BRST complex also finds its description in terms of a cohomology complex naturally appearing in the context of NCG. All this allows us to conclude that the approach we presented here provides a coherent and consistent description of the whole BV construction, from the introduction of ghost fields to the computation of the BRST complex, in terms of the key objects in noncommutative geometry, namely, in terms of spectral triples and the cohomology theory they determine. A last remark to do concerns the extension of the construction presented here to the general case of *U*(*n*)-gauge theories induced by spectral triples on the algebra $Mn(C)$ for *n* > 2: the reasons why we focused on the *n* = 2 in this article have to be searched on the increasing complexity of computing the BV extension of a generic *U*(*n*)-gauge theory for *n* > 2. This construction required the use of more advanced tools from algebraic geometry and the invariant theory and will be described in detail in an article currently in preparation. However, there are no particular phenomena appearing when we pass to consider an higher dimensional gauge theory. Similarly, on the side of finite spectral triples, we also have that their structure does not change at the increase of the dimension of the algebra. Hence, the approach presented here is expected to straightforwardly extend to the entire class of *U*(*n*)-gauge theories induced by spectral triples on the algebra $Mn(C)$, concluding the quest for a noncommutative geometric description of the BV construction, at least for the case of gauge theories induced by finite spectral triples.

## AUTHOR DECLARATIONS

### Conflict of Interest

The author has no conflicts to disclose.

### Author Contributions

**Roberta Anna Iseppi**: Conceptualization (lead), Formal analysis (lead), Investigation (lead), Writing – original draft (lead), Writing – review & editing (lead).

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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