The ghost fields and the BV extension for finite spectral triples

Iseppi, Roberta A.

doi:10.1063/5.0089795

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 $M_{2} (C)$ ⁠. 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 $M_{2} (C)$ ⁠, 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 $M_{n} (C)$ ⁠.

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 principle⁷ 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.

Definition II.1.

Let $A$ be an involutive unital algebra, $H$ be a Hilbert space, and $D : H \to H$ 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 $a \in A$ ⁠. 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.,

A ≃ ⨁_{i = 1}^{k} M_{n_{i}} (C)

for positive integers n₁, …, 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.

Definition II.2.

Let

(A, H, D)

be a spectral triple, and let γ be a linear map

γ : H \to H

⁠, endowing the Hilbert space

H

with a

Z / 2

-grading. Then,

(A, H, D, γ)

is called an even spectral triple if the following conditions are satisfied:

D γ = - γ D and γ a = a γ

for all

a \in A

⁠.

Definition II.3.

A real structure of KO-dimension n (mod 8) on a spectral triple

(A, H, D)

is an anti-linear isometry

J : H \to H

that satisfies

J^{2} = ε and J D = ε^{'} D J

together with the condition

J γ = ε^{''} γ J

in the even case. The constants ɛ, ɛ′, and ɛ″ depend on the KO-dimension n (mod 8) as follows:

Moreover, we require that for all

a, b \in A

⁠,

the action of $A$ satisfies the commutation rule: $[a, J b^{*} J^{- 1}] = 0$ ⁠, and
the 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.

Definition II.4.

For a real vector space X₀ and a real-valued functional S₀ on X₀, let

F : G \times X_{0} \to X_{0}

be a group action on X₀ for a given group

G

⁠. Then, the pair (X₀, S₀) is called a gauge theory with gauge group

G

if

S_{0} (F (g, φ)) = S_{0} (φ)

for all φ ∈ X₀ and

g \in G

⁠. The space X₀ is referred to as the configuration space, an element φ ∈ X₀ is called a gauge field, and S₀ 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.

Definition II.5.

Given a finite spectral triple

(A, H, D)

and a suitable real-valued function f, the associated spectral action S₀ is defined to be

S_{0} [φ] : = Tr (f (D + φ)),

with, as domain, the set of self-adjoint operators of the form φ = ∑_ja_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 ${Pol}_{R} (x)$ 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.

Definition II.6.

For a finite spectral triple

(A, H, D)

[finite real spectral triple

(A, H, D, J)

], the fermionic action on a subspace

H^{'} \subseteq H

is given by

S_{ferm} [φ] ≔ \frac{1}{2} 〈 φ, D φ 〉 (S_{ferm} [φ] : = \frac{1}{2} 〈 J φ, D φ 〉)

for

φ \in H^{'}

⁠.

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 (X₀, S₀) with gauge group $G$ ⁠.

Proposition II.7.

Given

(A, H, D)

⁠, a finite spectral triple, let X₀ denote the space of inner fluctuations for the operator D,

X_{0} ≔ \{φ = \sum_{j} a_{j} [D, b_{j}] : φ^{*} = φ, a_{j}, b_{j} \in A\},

where we only consider finite sums and ^* denotes the involution structure on

A

. Then, let

G

be the space of unitary elements of

A

⁠,

G : = U (A) = {u \in A : u u^{*} = u^{*} u = 1},

acting on X₀ via the map

F : G \times X_{0} \to X_{0}, (u, φ) \mapsto u φ u^{*} + u [D, u^{*}],

and finally, let S₀ be the spectral action defined on X₀ as

S_{0} [φ] : = Tr (f (D + φ))

for any φ ∈ X₀ and some

f \in {Pol}_{R} (x)

. Then, the pair (X₀, S₀) 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 (X₀, S₀) 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 S₀: indeed, in this context, the action of the gauge group on the configuration space and the gauge invariance of the action functional S₀ determine the presence of orbits of critical points of the action S₀ in the configuration space X₀. 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 Costello¹⁶ 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 (X₀, S₀), the divergences in the path integral can be eliminated by the introduction of auxiliary (non-existing) fields, suggestively called ghost fields.

Definition III.1.

A field/ghost field φ is a graded variable characterized by two integers,

deg (φ) \in Z and ε (φ) \in {0, 1},

with

deg (φ) = ε (φ) (mod Z / 2) .

deg(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,

φ ψ = - ψ φ and φ^{2} = 0 if ε (φ) = ε (ψ) = 1 .

The anti-field/anti-ghost field φ^* corresponding to a field/ghost field φ satisfies

deg (φ^{*}) = - deg (φ) - 1

and

ε (φ^{*}) = ε (φ) + 1 (mod Z / 2) .

Note: In what follows, the term fields are reserved to the initial fields in X₀, 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

After several refinements, among which one should certainly cite the introduction of an anti-bracket structure on the extended configuration space suggested by Zinn-Justin,²¹ the first and key step of the BV construction^22–24 can been seen as an extension process: given an initial gauge theory (X₀, S₀), it determines a new extended theory

(\tilde{X}, \tilde{S})

⁠,

\begin{matrix} (X_{0}, S_{0}) & - - - - - - - - - \to & (\tilde{X}, \tilde{S}), \\ initial gauge theory & + ghost/anti - ghost fields & extended theory \end{matrix}

where the extended configuration space

\tilde{X}

is obtained via the introduction of ghost/anti-ghost fields in the initial configuration space,

\tilde{X} = X_{0} \cup {ghost/anti - ghost fields},

and the extended action

\tilde{S}

is defined by adding extra terms depending on the ghost/anti-ghost fields to the initial action S₀,

\tilde{S} = S_{0} + terms depending on ghost/anti - ghost fields .

However, conditions have to be imposed on 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.

Definition III.2.

Given a gauge theory (X₀, S₀), an extended theory associated with it is a pair

(\tilde{X}, \tilde{S})

where the extended configuration space

\tilde{X} = \oplus_{i \in Z} {[\tilde{X}]}^{i}

is a

Z

-graded super-vector space suitable to be decomposed as

\tilde{X} ≅ F \oplus F^{*} [1] with {[\tilde{X}]}^{0} = X_{0},

(3.1)

where

F = \oplus_{i ⩾ 0} F^{i}

is a graded locally free

O_{X_{0}}

-module with homogeneous components of finite rank for

O_{X_{0}}

⁠, the algebra of regular functions on X₀. Concerning the extended action

\tilde{S} \in {[O_{\tilde{X}}]}^{0}

⁠, it is a real-valued regular function on

\tilde{X}

⁠, with

\tilde{S} |_{X_{0}} = S_{0}

⁠,

\tilde{S} \neq S_{0}

⁠, such that it solves the classical master equation, i.e.,

{\tilde{S}, \tilde{S}} = 0,

(3.2)

where {−, −} denotes the graded Poisson structure on the algebra

O_{\tilde{X}}

⁠.

Note: The condition imposed in () reflects the essential requirement of the BV formalism of having a symmetric structure in

\tilde{X}

⁠, 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

\tilde{X}

⁠,

F^{*} [1]

determines the part of anti-fields/anti-ghost fields, with

F^{*} [1]

that denotes the shifted dual module of

F

⁠,

F^{*} [1] = \oplus_{i \in Z} {[F^{*} [1]]}^{i} with {[F^{*} [1]]}^{i} = {[F^{*}]}^{i + 1} .

Moreover, the condition of

\tilde{X}

being a

Z

-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

O_{\tilde{X}}

of real-valued regular functions defined on

\tilde{X}

has the structure of a

Z

-graded algebra. In addition,

O_{\tilde{X}}

can also be endowed with a graded Poisson structure induced by the bracket

{-, -} : {[O_{\tilde{X}}]}^{•} \to {[O_{\tilde{X}}]}^{• + 1}

of degree 1. This structure is completely determined by requiring that, on the generators of

\tilde{X}

⁠, it satisfies the following conditions:

\{β_{i}, β_{j}\} = 0, \{β_{i}^{*}, β_{j}\} = δ_{i j}, and \{β_{i}^{*}, β_{j}^{*}\} = 0

for

β_{i} \in F^{p}

and

β_{i}^{*} \in {[F^{*} [1]]}^{- p - 1}

⁠, while its value on any other possible combination of fields/ghost fields/anti-fields and anti-ghost fields is equal to zero. Then, the definition of the bracket structure on the whole algebra

O_{\tilde{X}}

is obtained by enforcing the bracket being linear and graded Poisson.

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 $\tilde{S}$ ⁠. Based on that result, even though it is not explicitly required in the notion of extended theory, in what follows, we will consider pairs $(\tilde{X}, \tilde{S})$ with a finite level of reducibility.

Definition III.3.

An extended theory $(\tilde{X}, \tilde{S})$ ⁠, with $\tilde{X} = F \oplus F^{*} [1]$ for $F$ being a $Z_{⩾ 0}$ -graded and finitely generated $O_{X_{0}}$ -module, is reducible with a level of reducibility L = k − 1 ⩾ 1 if $F = \oplus_{i = 0}^{k} F_{k}$ ⁠. Otherwise, if L = 0, the theory is called irreducible.

B. The auxiliary fields and the gauge-fixing process

Once the initial theory (X₀, S₀) has been extended via the introduction of ghost/anti-ghost fields, still, the action functional

\tilde{S}

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

(\tilde{X}, \tilde{S})

⁠: given an extended theory, we want to determine a new pair

(\tilde{X}, \tilde{S}) |_{Ψ}

where neither the gauge-fixed configuration space

\tilde{X} |_{Ψ}

nor the gauge-fixed action

\tilde{S} |_{Ψ}

depends on anti-fields/anti-ghost fields. This goal is reached by defining

\tilde{X} |_{Ψ} ≔ [F \oplus F^{*} [1]] |_{φ_{i}^{*} = \frac{\partial Ψ}{\partial φ_{i}}}

and

\tilde{S} (φ_{i}, φ_{i}^{*}) |_{Ψ} ≔ \tilde{S} (φ_{i}, φ_{i}^{*} = \frac{\partial Ψ}{\partial φ_{i}}),

for Ψ being a so-called gauge-fixing fermion.

Definition III.4.

Given an extended configuration space $\tilde{X} = F \oplus F^{*} [1]$ ⁠, a gauge-fixing fermion Ψ on it is a regular function $Ψ \in O_{F}^{- 1}$ ⁠, 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 $(\tilde{X}, \tilde{S}) |_{Ψ}$ is obtained by imposing the gauge-fixing conditions $φ_{i}^{*} = \partial Ψ / \partial φ_{i}$ on both $\tilde{X}$ and $\tilde{S}$ ⁠. In order words, $\tilde{X} |_{Ψ}$ is defined to be the Lagrangian submanifold determined by replacing every anti-field/anti-ghost field $φ_{i}^{*} \in F^{*} [1]$ with the partial derivative of Ψ with respect to the corresponding field/ghost field φ. Similarly, $\tilde{S} |_{Ψ}$ is the restriction of the action $\tilde{S}$ to $\tilde{X} |_{Ψ}$ ⁠.

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

\int_{\tilde{X} |_{ψ}} g d (V o l_{\tilde{X} |_{Ψ}})

(3.3)

for

V o l_{\tilde{X} |_{Ψ}}

being a volume form on

\tilde{X} |_{ψ}

and g being a regular function on

\tilde{X} |_{Ψ}

⁠, do not depend on the choice of Ψ. This condition is satisfied by imposing that the gauge-fixed action is a proper solution of the classical master equation (cf. Refs. 29–31).

However, an additional obstacle occurs in the application of the gauge-fixing procedure if the extended theory

(\tilde{X}, \tilde{S})

we are considering has been obtained by applying the classical BV construction described above: indeed, the absence of ghost fields of negative ghost degree does not allow us to define a gauge-fixing fermion straightforwardly. Hence, an intermediate step needs to be taken: after having performed the first extension by adding ghost/anti-ghost fields, we have to further enlarge the configuration space

\tilde{X}

via the introduction of auxiliary fields and to add extra terms at the extended action

\tilde{S}

⁠,

\begin{matrix} (X_{0}, S_{0}) & - - - - - - - - - \to & (\tilde{X}, \tilde{S}) & - - - - - - - \to & (X_{t}, S_{t}) . \\ initial th. & + ghosts/anti - gh. & extended th. & + auxiliary fld. & total th. \end{matrix}

As it will become clear in Sec. III C, also in this case, the method to introduce these auxiliary fields has to take into account extra constrains, this time related to the need of not modifying the cohomology complex induced by the pair

(\tilde{X}, \tilde{S})

⁠, complex that detects relevant physical information on the initial gauge theory (X₀, S₀) 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

(\tilde{X}, \tilde{S})

⁠, 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).

Definition III.5.

An auxiliary pair is a pair of fields (B, h) such that their ghost degrees and parities satisfy the following relations:

\begin{matrix} deg (h) = deg (B) + 1, & ε (h) = ε (B) + 1 (mod 2) . \end{matrix}

Definition III.6.

Given an extended theory

(\tilde{X}, \tilde{S})

and a collection of auxiliary pairs

{(B_{i}, h_{i})}_{i}

with 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

\tilde{X}

⁠, (B_i, h_i), and their corresponding anti-fields

(B_{i}^{*}, h_{i}^{*})

for i = 1, …, r,

X_{t} ≔ ⟨ \tilde{X}, B_{i}, h_{i}, B_{i}^{*}, h_{i}^{*} ⟩,

and a total action S_t given by the following sum:

S_{t} ≔ \tilde{S} + S_{aux}, where S_{aux} ≔ \sum_{i = 1}^{r} h_{i} B_{i}^{*} .

Remark III.7.

We note that, by construction, the theories

(\tilde{X}, \tilde{S})

and (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} = E \oplus E^{*} [1],

where

E

is a

Z

-graded finitely generated

O_{X_{0}}

-module. Moreover, the graded Poisson structure on

\tilde{X}

can be extended to the one on X_t by imposing that

{B_{i}^{*}, B_{i}} = {h_{i}^{*}, h_{i}} = 1 and {φ, ξ} = 0,

for φ being any generator in

\tilde{X}

and ξ being any auxiliary fields among B_i, h_i, or their corresponding anti-fields

B_{i}^{*}

and

h_{i}^{*}

⁠, 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

O_{X_{t}}

⁠,

{S_{t}, S_{t}} = 0 .

To conclude, in order to be able to implement a gauge-fixing procedure on a BV-extended theory $(\tilde{X}, \tilde{S})$ ⁠, 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

After having briefly presented the two extension processes in the BV construction, namely, the introduction of ghost/anti-ghost fields first and then the further extension via auxiliary fields to allow the performance of a gauge-fixing procedure, we still have to recall the two cohomology complexes naturally emerging in this framework: the 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 X₀ reflect the complexity of the gauge symmetry of the theory, the conditions imposed on the pair

(\tilde{X}, \tilde{S})

entail that any extended theory naturally induces a cohomology complex: indeed, while the conditions

{[\tilde{X}]}^{0} = X_{0} and \tilde{S} |_{X_{0}} = S_{0}

imply that we are not modifying the initial theory in degree 0 and, hence, we are not interfering with the existing and physically relevant fields, the condition of

\tilde{S}

being a solution of the classical master equation is what ensures that the operator

d_{\tilde{S}} ≔ {\tilde{S}, -}

defines a coboundary operator for the BV complex.

Definition III.8.

Given an extended theory

(\tilde{X}, \tilde{S})

⁠, the induced BV cohomology complex is a cohomology complex whose cochain spaces

C^{i} (\tilde{X}, d_{\tilde{S}})

and coboundary operator

d_{\tilde{S}}

are defined as follows, respectively:

C^{i} (\tilde{X}, d_{\tilde{S}}) ≔ {[S y m_{O_{X_{0}}} (\tilde{X})]}^{i} = {[O_{\tilde{X}}]}^{i}

for

i \in Z

⁠, with

S y m_{O_{X_{0}}} (\tilde{X})

being the

Z

-graded symmetric algebra generated by

\tilde{X}

on the ring

O_{X_{0}}

⁠, and

d_{\tilde{S}} : C^{•} (\tilde{X}, d_{\tilde{S}}) \to C^{• + 1} (\tilde{X}, d_{\tilde{S}}) with d_{\tilde{S}} ≔ {\tilde{S}, -},

for {−, −} denoting the Poisson bracket structure on

O_{\tilde{X}}

⁠.

Note: the fact that the pair $(C^{•} (\tilde{X}, \tilde{S}), d_{\tilde{S}})$ defines a cohomology complex is an immediate consequence of the bracket {, } being of degree 1, with $\tilde{S}$ of degree 0, and satisfying the graded Jacobi identity in addition to $\tilde{S}$ 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 $(\tilde{X}, \tilde{S})$ ⁠, 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 $φ_{i} \in E$ (cf. Ref. 33). However, an explicit computation shows that if we restrict the operator $d_{S_{t}}$ to the algebra $O_{X_{t} |_{Ψ}}$ 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 $\tilde{S}$ (cf. Ref. 32). Hence, it is meaningful and physically relevant to consider what is called the BRST cohomology complex.

Definition III.9.

Given a total theory (X_t, S_t), with

X_{t} = E \oplus E^{*} [1]

and

S_{t} \in {[O_{X_{t}}]}^{0}

⁠, together with a gauge-fixing fermion

Ψ \in {[O_{E}]}^{- 1}

⁠, the induced BRST complex

(C^{j} (X_{t}, d_{S_{t}}) |_{Ψ}, d_{S_{t}} |_{Ψ})

is a cohomology complex with

C^{j} (X_{t}, d_{S_{t}}) |_{Ψ} ≔ {[{Sym}_{O_{X_{0}}} (X_{t} |_{Ψ})]}^{j}

and

d_{S_{t}} |_{Ψ} ≔ {\{S_{t}, -\}|}_{X_{t} |_{Ψ}},

where

j \in Z

and X_t|_Ψ ⊂ X_t is the Lagrangian submanifold defined by the gauge-fixing conditions

{φ_{i}^{*} = \partial Ψ / \partial φ_{i}}

⁠.

First discovered by Becchi, Rouet, and Stora^34–36 and, independently, by Tyutin³⁷ (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,

H^{0} (X_{t}, d_{S_{t}}) |_{Ψ} = {Classical observables for the theory (X_{0}, S_{0})} .

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 (X₀, S₀) via the introduction of ghost and anti-ghost fields. Once obtained the extended theory $(\tilde{X}, \tilde{S})$ ⁠, 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 $H_{B V}$ ⁠: 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

To simplify the exposition, we present our construction on a class of gauge theories induced by the following finite spectral triple:

(A_{0}, H_{0}, D_{0}) : = (M_{n} (C), C^{n}, D_{0}),

where D₀ 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 (X₀, S₀) with gauge group

G

such that

X_{0} = {M \in M_{n} (C) : M^{†} = M},

(4.1)

S_{0} [M] = Tr f (M) and G = U (n),

where † denotes the usual adjoint for a matrix, f is a polynomial in

{Pol}_{R} (x)

⁠, and

G

acts on X₀ 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

(\tilde{X}, \tilde{S})

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

M_{n} (C)

⁠.

For the case n = 2, the model constructed in () can be described explicitly by fixing as a basis for X₀ the one given by Pauli matrices (together with the identity matrix),

σ_{1} = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}), σ_{2} = (\begin{matrix} 0 & - i \\ i & 0 \end{matrix}), σ_{3} = (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}), σ_{4} = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) .

(4.2)

Then, we immediately deduce that the initial configuration space is a four-dimensional real affine space. In order words, if we denote by

{\{x_{a}\}}_{a = 1}^{4}

the dual basis of

{\{σ_{a}\}}_{a = 1}^{4}

⁠, we have that X₀ is isomorphic to a four-dimensional real vector space generated by four independent initial fields,

X_{0} ≃ {⟨ x_{1}, x_{2}, x_{3}, x_{4} ⟩}_{R} .

Hence, the ring of regular functions on X₀ is the ring of polynomials in the variables x_a,

O_{X_{0}} = {Pol}_{R} (x_{a})

⁠. Finally, it can be checked that the most general form for a functional S₀ on X₀ that is invariant under the adjoint action of the gauge group U(2) is given as

S_{0} = \sum_{k = 0}^{r} {(x_{1}^{2} + x_{2}^{2} + x_{3}^{2})}^{k} g_{k} (x_{4})

(4.3)

for

g_{k} (x_{4}) \in {Pol}_{R} (x_{4})

⁠,

r \in N

⁠. Then, taking the pair (X₀, S₀) as initial gauge theory, one determines the corresponding BV-extended theory

(\tilde{X}, \tilde{S})

by computing (part of) the Tate resolution of the Jacobian ideal

J (S_{0}) ≔ {⟨ \partial_{i} S_{0} ⟩}_{i = 1, \dots, 4}

on the ring

O_{X_{0}}

(cf. Ref. 28 for the detailed construction). The result is that for the case of a generic initial action S₀, that is, if it holds that

G C D {(\partial_{i} S_{0})}_{i = 1, \dots, 4} = 1

⁠, then the corresponding minimally extended BV theory

(\tilde{X}, \tilde{S})

has extended configuration space

\tilde{X} = F \oplus F^{*} [1] with F = X_{0} \oplus 〈 C_{1}, C_{2}, C_{3} 〉_{1} \oplus 〈 E 〉_{2},

(4.4)

while the corresponding extended action is

\tilde{S} = S_{0} + \sum_{i, j, k} ε_{ijk} x_{i}^{*} x_{j} C_{k} + \sum_{i, j < k} C_{i}^{*} (x_{i} E + ε_{ijk} C_{j} C_{k}),

(4.5)

for ɛ_ijk being the totally anti-symmetric tensor in three indices i, j, k ∈ {1, 2, 3} with ɛ₁₂₃ = 1.

At this point, a natural question to ask is if it is possible to determine a spectral triple to encode this BV-extended theory. Even more, we want to determine if it is possible to construct this BV spectral triple

(A_{B V}, H_{B V}, D_{B V}, J_{B V})

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,

\begin{matrix} (A_{0}, H_{0}, D_{0}) & - - - - - - - - - - - \to & (A_{B V}, H_{B V}, D_{B V}, J_{B V}) \\ initial sp. tr. & + ghost/anti - ghost fld & BV sp. triple \end{matrix} .

B. BV spectral triple: The aim

The goal is to determine a new spectral triple $(A_{B V}, H_{B V}, D_{B V}, J_{B V})$ such that

it contains all the ghost/anti-ghost fields appearing in $\tilde{X}$ ⁠;
the induced fermionic action coincides with the BV action $S_{B V} ≔ \tilde{S} - S_{0}$ ⁠.

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 $\tilde{S}$ ⁠. 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 $H_{B V}$ ⁠, which is what plays a role in the fermionic action, while the algebra $A_{B V}$ remains unchanged.

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

how to extend the initial Hilbert space $H_{0} = C^{2}$ to a new $H_{B V}$ ⁠, 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 $(A_{0}, H_{0}, D_{0})$ ⁠;
how to define a self-adjoint operator D_BV such that the induced fermionic action, added to the initial action S₀, 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(S₀) and the application of a recursive algorithm to determine the extended action $\tilde{S}$ as an approximate solution of the classical master equation (cf. Ref. 27)

C. The Hilbert space $H_{B V}$

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 S₀.

More precisely, as explained in Ref. 28, if we consider a spectral action defined by a polynomial f of degree deg(f) > 1, then the generic initial action is of the form in (). While the factors g_k(x₄) are completely independent polynomials, the symmetries of the action are enclosed in the term

(x_{1}^{2} + x_{2}^{2} + x_{3}^{2}) .

(4.6)

There, we have the following:

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.

These ghost fields are the ones generating the extended configuration space

\tilde{X}

and entering in the extended action

\tilde{S}

⁠. Hence, they belong to the domain of the fermionic action, that is, the so-called effective subspace

H_{B V, f} \subset H_{B V}

⁠,

H_{B V, f} = Q_{f} \oplus Q_{f}^{*} [1]

for

Q_{f} = {[i s u (2)]}_{0} \oplus {[i s u (2)]}_{1} \oplus {[u (1)]}_{2},

where we recall the required duality between ghost and anti-ghost fields and that the parity, either real or Grassmannian, of the components agrees with the parity of the degree assigned to each term. More explicitly, by fixing basis (), a generic vector

φ \in Q_{f}

has the following form:

φ = {[\begin{matrix} [x_{1} & x_{2} & x_{3} & {0]}_{0} [C_{1} & C_{2} & C_{3} & {0]}_{1} [0 & 0 & 0 & {i E]}_{2} \end{matrix}]}^{T},

where the notation is chosen to agree with the one introduced in Sec. IV A. As it will be discussed in more detail in Remark IV.2,

H_{B V, f}

is taken to be either the subspace preserved by the real structure (up to a product by i) or its complementary depending on the operator D_BV anti-commuting or commuting with J_BV, respectively. Therefore, while

i s u (2)

describes the Hermitian traceless part of

M_{2} (C)

⁠,

u (1)

gets included in

M_{2} (C)

as its trace component, determining that the BV-extending procedure at the level of the Hilbert spaces reads as follows:

\begin{matrix} H_{0} = C^{2} & - - - - - - - - - - - \to & H_{B V} = Q \oplus Q^{*} [1] \\ initial Hilbert sp. & + ghost/anti - ghost fld & BV Hilbert sp \end{matrix}

for

Q = {[M_{2} (C)]}_{0} \oplus {[M_{2} (C)]}_{1} \oplus {[M_{2} (C)]}_{2} .

Concerning the inner product structure, this is given as usual by the Hilbert–Schmidt inner product on each summand

M_{2} (C)

⁠, that is to say,

⟨ φ, φ^{'} ⟩ ≔ Tr (φ {(φ^{'})}^{*})

for

φ, φ^{'} \in M_{2} (C)

⁠.

D. The real structure J_BV

As previously observed, the need of going from a structure, which is naturally defined in a complex setting, that is, a finite spectral triple, to a physical model, which deals with real fields, determines the introduction of a real structure. We take it to be given by the following anti-linear isometry:

J_{B V} : H_{B V} \to H_{B V} with J_{B V} (φ_{j}) : = i \cdot (φ_{j}^{†}),

(4.7)

for

φ_{j} \in {[M_{2} (C)]}_{j}

being an element in the direct sum defining

H_{B V}

and † denoting the matrix adjoint.

E. The linear operator D_BV

The expression of the self-adjoint linear operator D_BV acting on the Hilbert space $H_{B V}$ 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 ${[O_{\tilde{X}}]}^{0}$ ⁠, 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 $\tilde{S}$ ⁠, restricted to the initial configuration space X₀, should coincide with the initial action S₀. 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:
$\begin{aligned} A d (p) : M_{2} (C) & \to M_{2} (C), \\ φ & \mapsto [α (p_{i}), φ] \end{aligned}$
and
$\begin{aligned} A b (p) : M_{2} (C) & \to M_{2} (C), \\ φ & \mapsto {[α (p_{i}), φ]}_{+}, \end{aligned}$
where α(p_i) is an Hermitian, traceless 2 × 2-matrix, whose expression in terms of the Pauli matrices is given by
$α = \frac{1}{2} [(- p_{1}) σ_{1} + (- p_{2}) σ_{2} + (- p_{3}) σ_{3}],$
where the notation 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 $M_{2} (C)$ ⁠, while Ab is an odd derivation given in terms of the anti-commutator. Explicitly, in terms of the orthonormal basis (4.2) for $M_{2} (C)$ ⁠, the linear operators Ad and Ab have the following form as 4 × 4-matrices:
$A d (p) ≔ (\begin{matrix} 0 & + i p_{3} & - i p_{2} & 0 \\ - i p_{3} & 0 & + i p_{1} & 0 \\ + i p_{2} & - i p_{1} & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}),$
(4.8)
$A b (p) ≔ (\begin{matrix} 0 & 0 & 0 & p_{1} \\ 0 & 0 & 0 & p_{2} \\ 0 & 0 & 0 & p_{3} \\ p_{1} & p_{2} & p_{3} & 0 \end{matrix}) .$
(4.9)

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 $φ \in H_{B V, f}$ 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 $\tilde{S}$ 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.

Proposition IV.1.

Let D_BV be a self-adjoint operator on

H_{B V}

such that the induced fermionic action

S_{ferm} : H_{B V, f} \to R for φ \mapsto S_{ferm} (φ) : = \frac{1}{2} ⟨ J_{B V} φ, D_{B V} φ ⟩

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:

D_{B V} = (\begin{matrix} 0 & R \\ R^{*} & S \end{matrix})

(4.10)

for

R ≔ (\begin{matrix} 0 & 0 & 0 \\ 0 & \frac{1}{2} A d (C) & A b (x) \\ \frac{1}{2} A d (C) & - \frac{1}{2} A d (x) & 0 \end{matrix}),

S ≔ (\begin{matrix} 0 & A d (x^{*}) & A b (C^{*}) \\ A d (x^{*}) & A d (C^{*}) & 0 \\ A b (C^{*}) & 0 & 0 \end{matrix}) .

Proof.

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 S₀, 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 S₀. 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.

Remark IV.2.

Having introduced both the real structure J_BV and the self-adjoint operator D_BV, we can now explain how, given the Hilbert space

H_{B V}

⁠, one can identify its effective subspace

H_{B V, f}

⁠. The leading idea is to select the effective part of

H_{B V}

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

H_{B V, f} \subset H_{B V}

the one containing self-adjoint or skew-Hermitian matrices, that is, the one satisfying either

J_{B V} (H_{B V, f}) = i H_{B V, f} or J_{B V} (H_{B V, f}) = - i H_{B V, f} .

On the contrary, we have to take a mixed combination of these two conditions,

J_{B V} ({[H_{B V, f}]}^{0} \oplus {[H_{B V, f}]}^{1}) = i ({[H_{B V, f}]}^{0} \oplus {[H_{B V, f}]}^{1})

and

J_{B V} ({[H_{B V, f}]}^{2}) = - i {[H_{B V, f}]}^{2} .

We also remark that the symmetric choice of an effective subspace

H_{B V, f}

such that

J_{B V} ({[H_{B V, f}]}^{0} \oplus {[H_{B V, f}]}^{1}) = - i ({[H_{B V, f}]}^{0} \oplus {[H_{B V, f}]}^{1})

and

J_{B V} ({[H_{B V, f}]}^{2}) = i {[H_{B V, f}]}^{2}

would induce a real fermionic action and would be suitable for the construction.

F. The algebra $A_{B V}$

To complete the definition of the BV spectral triple, we still have to determine the algebra $A_{B V}$ ⁠. However, as previously observed, we anticipate that the BV construction would only act by enlarging the Hilbert space to $H_{B V}$ 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 $A_{B V} = A_{0} = M_{2} (C)$ is optimal: indeed, $M_{2} (C) \subset L (H_{B V})$ is the largest unital subalgebra of the algebra of all linear operators on $H_{B V}$ that satisfies the commutation rule and first-order condition of Definition II.3.

Lemma IV.3.

Let

H_{B V}

, J_BV, and D_BV be as defined above. Then, the maximal unital subalgebra

A_{B V}

\subset L (H_{B V})

of linear operators on

H_{B V}

that satisfies

[a, J_{B V} b^{*} J_{B V}^{- 1}] = 0, [[D_{B V}, a], J_{B V} b^{*} J_{B V}^{- 1}] = 0

for all

a, b \in A_{B V}

is given by

A_{B V} = M_{2} (C)

acting diagonally on

H_{B V}

.

Proof.

The commutation rule

[a, J_{B V} b^{*} J_{B V}^{- 1}] = 0

for all

a, b {\in A}_{B V}

implies that

H_{B V}

carries an

A_{B V}

-bimodule structure. This already restricts

A_{B V}

to be a subalgebra in

A_{B V} \subset {[M_{2} (C)]}^{\oplus 6}

acting diagonally on

H_{B V} = Q \oplus Q^{*} [1]

⁠. Then, by a straightforward computation of the double commutator,

[[D_{B V}, (a_{1}, \dots, a_{6})], J_{B V} (b_{1}, \dots, b_{6}) J_{B V}^{- 1}],

it follows that the first-order condition implies a₁ = ⋯ = a₆ and b₁ = ⋯ = b₆. This selects the subalgebra

M_{2} (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□

Having also determined the algebra

A_{B V}

⁠, we have now defined a quadruple

(A_{B V}, H_{B V}, D_{B V}, J_{B V}) = (M_{2} (C), Q \oplus Q^{*} [1], D_{B V}, J_{B V}),

where

Q = {[M_{2} (C)]}_{0} \oplus {[M_{2} (C)]}_{1} \oplus {[M_{2} (C)]}_{2},

while the operator D_BV is defined in () and the real structure J_BV is defined in (). 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

(\tilde{X}, \tilde{S})

⁠, 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_{B V} = D_{1} + D_{2}

with

D_{1} = (\begin{matrix} 0 & R_{1} \\ R_{1}^{*} & S_{1} \end{matrix}) and D_{2} = (\begin{matrix} 0 & R_{2} \\ R_{2}^{*} & S_{2} \end{matrix})

for

R_{1} = (\begin{matrix} 0 & 0 & 0 \\ 0 & \frac{1}{2} A d (C) & 0 \\ \frac{1}{2} A d (C) & - \frac{1}{2} A d (x) & 0 \end{matrix}), S_{1} = (\begin{matrix} 0 & A d (x^{*}) & 0 \\ A d (x^{*}) & A d (C^{*}) & 0 \\ 0 & 0 & 0 \end{matrix})

and

R_{2} = (\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & A b (x) \\ 0 & 0 & 0 \end{matrix}), S_{2} = (\begin{matrix} 0 & 0 & A b (C^{*}) \\ 0 & 0 & 0 \\ A b (C^{*}) & 0 & 0 \end{matrix}) .

Then, we find that, while D₁ anti-commutes with J_BV, D₂ commutes with it,

J_{B V} D_{1} = - D_{1} J_{B V}, J_{B V} D_{2} = + D_{2} J_{B V} .

Definition IV.4.

Given

(A, H, D)

⁠, a finite spectral triple, and J, an anti-linear isometry on

H

⁠,

(A, H, D, J)

defines a real spectral triple with mixed KO-dimension if J satisfies

J^{2} = \pm I d and [a, J b^{*} J^{- 1}] = 0

for all

a, b \in A

⁠; the operator D is the sum of two self-adjoint operators D₁, D₂, which anti-commutes and commutes, respectively, with J,

D_{1} = - D_{1} J and J D_{2} = D_{2} J,

and the first-order condition holds,

[[D, a], J b^{*} J^{- 1}] = 0, \forall a, b \in A .

We conclude this section by stating our main theorem: the method described above to associate, to a given initial finite spectral triple $(A_{0}, H_{0}, D_{0})$ ⁠, a corresponding real spectral triple $(A_{B V}, H_{B V}, D_{B V}, J_{B V})$ ⁠, 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 $(A_{B V}, H_{B V}, D_{B V}, J_{B V})$ is the BV spectral triple associated with the initial triple $(A_{0}, H_{0}, D_{0})$ ⁠.

Definition IV.5.

Given a finite spectral triple

(A_{0}, H_{0}, D_{0})

with induced gauge theory (X₀, S₀), a real (mixed KO-dimensional) spectral triple

(A_{B V}, H_{B V}, D_{B V}, J_{B V})

is a BV spectral triple associated with the spectral triple

(A_{0}, H_{0}, D_{0})

if the pair

\tilde{X} ≔ (Q_{f} + X_{0}) \oplus (X_{0}^{*} [1] + Q_{f}^{*} [1]), \tilde{S} ≔ S_{0} + \frac{1}{2} S_{ferm}

with

S_{ferm} : H_{B V, f} \to R, φ \mapsto \frac{1}{2} ⟨ J_{B V} (φ), D_{B V} φ ⟩,

for

H_{B V, f} = Q_{f} \oplus Q_{f}^{*} [1] \subset H_{B V}

being the effective part of the Hilbert space

H_{B V} = Q \oplus Q^{*} [1]

⁠, is a BV-extended theory for the gauge theory (X₀, S₀).

Remark IV.6.

In the above definition, in order to be able to perform the sum appearing in the expression for the extended action $\tilde{S}$ ⁠, we are implicitly identifying the functional S_ferm with its representation as a polynomial in the algebra $O_{\tilde{X}}$ ⁠.

Theorem IV.7.

Let

(A_{0}, H_{0}, D_{0}) = (M_{2} (C), C^{2}, D_{0})

be a finite spectral triple with induced gauge theory (X₀, S₀). Then,

(A_{B V}, H_{B V}, D_{B V}, J_{B V}) = (M_{2} (C), Q \oplus Q^{*} [1], D_{B V} J_{B V})

for

Q = {[M_{2} (C)]}_{0} \oplus {[M_{2} (C)]}_{1} \oplus {[M_{2} (C)]}_{2},

and D_BV, J_BV as defined in (4.10) and (4.7), respectively, is a BV spectral triple associated with

(A_{0}, H_{0}, D_{0})

.

Proof.

The proof of the statement consists of two steps: first one has to show that the quadruple

(A_{B V}, H_{B V}, D_{B V}, J_{B V})

defined above determines a real spectral triple of mixed KO-dimension and then one has to show that, by defining the associated pair

(\tilde{X}, \tilde{S})

as prescribed in Definition IV.5, one obtains a BV-extended theory for the gauge theory (X₀, S₀). For what concerns the first point, the fact that

(A_{B V}, H_{B V}, D_{B V}, J_{B V})

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

(A_{B V}, H_{B V}, D_{B V}, J_{B V})

what indicated in Definition IV.5, one would find that

\begin{matrix} \tilde{X} & ≔ (Q_{f} + X_{0}) \oplus (X_{0}^{*} [1] + Q_{f}^{*} [1]) \\ ≅ ({[i u (2)]}_{0} \oplus {[i s u (2)]}_{1} \oplus {[u (1)]}_{2}) \oplus ({[i u (2)]}_{- 1}^{*} \oplus {[i s u (2)]}_{- 2}^{*} \oplus {[u (1)]}_{- 3}^{*}) \\ = F \oplus F^{*} [1], \end{matrix}

for

F

being the non-negatively graded locally free

O_{X_{0}}

-module with homogeneous components of finite rank defined as

F = \oplus_{i ⩾ 0} F^{i} = {[i u (2)]}_{0} \oplus {[i s u (2)]}_{1} \oplus {[u (1)]}_{2} .

Comparing this result with what found in (), we could conclude that the space

\tilde{X}

just defined coincides with the extended configuration space determined for the gauge theory (X₀, S₀) by performing the classical BV-extension procedure. For what concerns the functional

\tilde{S}

⁠, by applying what required in Definition IV.5 to our specific case, we would find that

\begin{matrix} \tilde{S} & ≔ S_{0} + \frac{1}{2} S_{ferm} \\ = S_{0} + \sum_{i, j, k} ε_{ijk} x_{i}^{*} x_{j} C_{k} + \sum_{i, j < k} C_{i}^{*} (x_{i} E + ε_{ijk} C_{j} C_{k}), \end{matrix}

for ɛ_ijk being the totally anti-symmetric tensor in three indices i, j, k ∈ {1, 2, 3} with ɛ₁₂₃ = 1, obtaining a functional

\tilde{S}

coinciding with the extended action functional determined for our model by applying the classical BV construction; see (). Hence, by comparison with what explicitly computed for our model using the classical BV construction, we can conclude that the pair

(\tilde{X}, \tilde{S})

just constructed defines a BV extended theory for the gauge theory (X₀, S₀) induced by the spectral triple

(A_{0}, H_{0}, D_{0})

⁠.□

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 $(\tilde{X}, \tilde{S})$ ⁠, 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

We approach this part of the construction following a similar perspective to what led to the discovery of the BV spectral triple: also, in this case, we aim to construct a new real spectral triple

(A_{t}, H_{t}, D_{t}, J_{t})

such that it encodes all the elements entering the so-called total theory (X_t, S_t). Hence, we expect that the total configuration space X_t, which is obtained as a further extension of

\tilde{X}

via the introduction of pairs of auxiliary fields and anti-fields,

X_{t} = \tilde{X} + \{auxiliary fields and anti - fields\},

would be encoded in the Hilbert space

H_{t}

⁠, while the total action

S_{t} ≔ \tilde{S} + S_{aux}

is expected to be the fermionic action induced by a new operator 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 $\tilde{X}$ 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.

Theorem V.1.

Let

(\tilde{X}, \tilde{S})

be an extended theory with the level of reducibility L. Then, the collection of auxiliary pairs

{(B_{i}^{j}, h_{i}^{j})}

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

deg (B_{i}^{j}) = j - i - 2 if j is odd,

deg (B_{i}^{j}) = i - j + 1 if j is even .

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.

The goal of this section is to present in detail how to perform the following step:

\begin{matrix} (A_{B V}, H_{B V}, D_{B V}, J_{B V}) & - - - - - - \to & (A_{t}, H_{t}, D_{t}, J_{t}) \\ BV spectral triple & + auxiliary fld & total sp. triple \end{matrix}

all within the world of spectral triples. Once again, in order to simplify the presentation of the construction, here, we focus on the 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 $H_{t}$

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 $H_{B V}$ ⁠, 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 ${[i s u (2)]}_{1}$ determine the appearance of the summands
${[s u (2)]}_{- 1} \oplus {[i s u (2)]}_{0} \subseteq {[M_{2} (C)]}_{- 1} \oplus {[M_{2} (C)]}_{0}$
in $H_{t, f} \subseteq H_{t}$ ⁠, which corresponds to three trivial pairs ${(B_{i}, h_{i})}_{i = 1, 2, 3}$ with
$deg (B_{i}) = - 1 and deg (h_{i}) = 0 .$
The ghost field E embedded in the algebra ${[M_{2} (C)]}_{2}$ as the generator of ${[u (1)]}_{2} \subseteq {[M_{2} (C)]}_{2}$ determines the introduction of two trivial pairs (A₁, k₁) and (A₂, k₂) with
$deg (A_{1}) = - 2, deg (k_{1}) = - 1, deg (A_{2}) = 0, deg (k_{2}) = 1 .$

Thus, at the level of the effective total Hilbert space

H_{t, f}

⁠, we also have to add the following terms:

{[u (1)]}_{- 2} \oplus {[u (1)]}_{- 1} \subseteq {[M_{2} (C)]}_{- 2} \oplus {[M_{2} (C)]}_{- 1}

and

{[u (1)]}_{0} \oplus {[u (1)]}_{1} \subseteq {[M_{2} (C)]}_{0} \oplus {[M_{2} (C)]}_{1} .

Finally, as the BV formalism also required to introduce all the anti-fields corresponding to the auxiliary fields just listed, we conclude that the total Hilbert space is given by

H_{t} : = H_{B V} \oplus H_{aux},

where

H_{B V} = {[M_{2} (C)]}^{\oplus 6}, H_{aux} = {[M_{2} (C)]}^{\oplus 12},

and the effective part of

H_{aux}

entering the induced fermionic action is

H_{aux, f} = R_{f} \oplus R_{f}^{*} [1]

for

R_{f} ≔ {[u (1)]}_{- 2} \oplus {[u (1) \oplus s u (2)]}_{- 1} \oplus {[i s u (2) \oplus u (1)]}_{0} \oplus {[u (1)]}_{1} .

Remark V.2.

Even though at this point the fact that the three auxiliary fields B_i should be embedded in $M_{2} (C)$ as the term $[s u (2)]$ while the three fields h_i as $[i s u (2)]$ 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

Considering that we are simply enlarging the BV spectral triple to include additional fields, it is natural to consider as a real structure the one obtained by extending the action of J_BV from

H_{B V}

to

H_{t}

⁠. Explicitly,

J_{t} : H_{t} \to H_{t} with J_{t} (φ_{j}) : = i \cdot (φ_{j}^{†}),

(5.1)

for

φ_{j} \in {[M_{2} (C)]}_{j}

being an element in the direct sum defining

H_{t}

and † denoting the matrix adjoint.

D. The operator D_t

According to the construction described in Sec. III, the auxiliary fields enter the total action 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} = \tilde{S} + S_{aux}, where S_{aux} ≔ \sum_{i = 1}^{3} B_{i}^{*} h_{i} + \sum_{j = 1}^{2} A_{j}^{*} k_{j} .

Because in the action S_t there are no terms involving both the auxiliary fields and the ghost fields/anti-ghost fields already introduced in

\tilde{X}

⁠, the operator D_t has to be representable as a block matrix with

D_{t} = (\begin{matrix} D_{B V} & 0 \\ 0 & D_{aux} \end{matrix}) for D_{aux} ≔ (\begin{matrix} 0 & T \\ T^{*} & 0 \end{matrix}) .

(5.2)

For what concerns the structure of the matrix 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:

T ≔ (\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & i \cdot I d \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & I d & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & i \cdot I d & 0 & 0 & 0 & 0 \end{matrix}) .

Finally, even though we are not going to analyze the cohomological implications of this construction, we can in any case remark that the cohomological triviality of the auxiliary fields clearly reflects in the extremely simple form taken by the operator D_aux.

E. The algebra $A_{t}$

Similarly to what done for the construction of the BV spectral triple, we choose $A_{t}$ to be the maximal algebra that completes the triple $(H_{t}, D_{t}, J_{t})$ to a real spectral triple, satisfying all the commutation/anti-commutation requirements and the first order condition.

Lemma V.3.

Let

H_{t}

, J_t, and D_t be as defined above. Then, the maximal unital subalgebra

A_{t}

\subset L (H_{t})

of linear operators on

H_{t}

that satisfies

[a, J_{t} b^{*} J_{t}^{- 1}] = 0, [[D_{t}, a], J_{t} b^{*} J_{t}^{- 1}] = 0

for all

a, b \in A_{t}

is given by

A_{t} = M_{2} (C)

acting diagonally on

H_{t}

.

Proof.

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□

Definition V.4.

Let

(A_{B V}, H_{B V}, D_{B V}, J_{B V})

be a BV spectral triple for a BV-extended theory

(\tilde{X}, \tilde{S})

⁠. Then, a real spectral triple

(A_{t}, H_{t}, D_{t}, J_{t})

⁠, eventually with mixed KO-dimension, is called to be the total spectral triple for the BV spectral triple

(A_{B V}, H_{B V}, D_{B V}, J_{B V})

if the pair (X_t, S_t) is defined as

X_{t} : = R_{f} \oplus \tilde{X} \oplus R_{f}^{*} [1] and S_{t} ≔ \tilde{S} + \frac{1}{2} S_{ferm} |_{R_{f} \oplus R_{f}^{*} [1]},

for

R_{f}

and

R_{f}^{*} [1]

being the two uniquely defined subspaces such that

H_{t, f} = R_{f} \oplus H_{B V, f} \oplus R_{f}^{*} [1]

⁠, for

H_{t, f}

being the effective part of

H_{t}

⁠, and

S_{ferm} |_{R_{f} \oplus R_{f}^{*} [1]} : R_{f} \oplus R_{f}^{*} [1] \to R, φ \mapsto \frac{1}{2} ⟨ J_{t} (φ), D_{t} φ ⟩

is the total theory associated with the BV-extended theory

(\tilde{X}, \tilde{S})

⁠.

Remark V.5.

In the above definition, we are implicitly identifying the functional S_ferm with its representation as a polynomial in the algebra $O_{X_{t}}$ ⁠. Similarly, the extended action $\tilde{S}$ can also be seen as an element of the same algebra.

Theorem V.6.

The quadruple

(A_{t}, H_{t}, D_{t}, J_{t}) = (M_{2} (C), {[M_{2} (C)]}^{\oplus 18}, D_{t}, J_{t})

with D_t and J_t as defined in (5.1) and (5.2), respectively, is the total spectral triple associated with the BV spectral triple

(M_{2} (C), {[M_{2} (C)]}^{\oplus 6}, D_{B V}, J_{B V})

.

Proof.

Checking that

(A_{t}, H_{t}, D_{t}, J_{t})

defines a spectral triple is pretty straightforward: indeed, from its matrix representation, the operator 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

H_{t}

⁠. 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,

\begin{matrix} D_{t} J_{t} = J_{t} D_{t} & D_{t} J_{t} = - J_{t} D_{t} \\ on the bosons, & on the fermions, \end{matrix}

allowing us to conclude that

(M_{2} (C), {[M_{2} (C)]}^{\oplus 18}, D_{t}, J_{t})

is a real spectral triple with mixed KO-dimension, where the maximality of the choice

A_{t} = M_{2} (C)

was proved in Lemma V.3. On the other hand, the fact that this spectral triple is the total spectral triple corresponding to that BV spectral triple follows immediately from its definition. In particular, the pair (X_t, S_t) obtained by implementing what prescribed in Definition V.4 on the triple

(M_{2} (C), {[M_{2} (C)]}^{\oplus 18}, D_{t}, J_{t})

coincides with the total theory associated with the BV-extended theory

(\tilde{X}, \tilde{S})

according to the classical construction recalled in Theorem V.1.□

Having introduced auxiliary fields with negative degree, we can finally define a gauge-fixing fermion Ψ as

Ψ \in {[O_{X_{0} \oplus Q_{f} \oplus R_{f}}]}^{- 1},

that is, as a regular function of degree −1 defined on the part of the effective total Hilbert space

H_{t, f}

generated by ghost and auxiliary fields.

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 $\tilde{S}$ /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 $(\tilde{X}, \tilde{S})$ ⁠. 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 $M_{n} (C)$ 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 $M_{n} (C)$ ⁠, 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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Published open access through an agreement with Georg-August-Universität Göttingen Mathematisches Institut

The ghost fields and the BV extension for finite spectral triples

I. THE BV FORMALISM WITHIN NONCOMMUTATIVE GEOMETRY: THE MOTIVATION

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

A. Spectral triples and the induced gauge theories

III. THE BATALIN–VILKOVISKY FORMALISM FOR FINITE GAUGE THEORIES

A. The BV-extended theory

B. The auxiliary fields and the gauge-fixing process

C. The BV/BRST cohomology complexes

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

A. The model

B. BV spectral triple: The aim

C. The Hilbert space $H_{B V}$

D. The real structure J_BV

E. The linear operator D_BV

F. The algebra $A_{B V}$

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

A. The total spectral triple

B. The Hilbert space $H_{t}$

C. The real structure J_t

D. The operator D_t

E. The algebra $A_{t}$

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

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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The ghost fields and the BV extension for finite spectral triples

I. THE BV FORMALISM WITHIN NONCOMMUTATIVE GEOMETRY: THE MOTIVATION

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

A. Spectral triples and the induced gauge theories

III. THE BATALIN–VILKOVISKY FORMALISM FOR FINITE GAUGE THEORIES

A. The BV-extended theory

B. The auxiliary fields and the gauge-fixing process

C. The BV/BRST cohomology complexes

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

A. The model

B. BV spectral triple: The aim

C. The Hilbert space HBV

D. The real structure JBV

E. The linear operator DBV

F. The algebra ABV

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

A. The total spectral triple

B. The Hilbert space Ht

C. The real structure Jt

D. The operator Dt

E. The algebra At

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

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Related Content

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

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C. The Hilbert space $H_{B V}$

D. The real structure J_BV

E. The linear operator D_BV

F. The algebra $A_{B V}$

B. The Hilbert space $H_{t}$

C. The real structure J_t

D. The operator D_t

E. The algebra $A_{t}$