Large gauge transformations, gauge invariance, and the QCD θ YM -term
aa r X i v : . [ phy s i c s . h i s t - ph ] J u l Large gauge transformations, gauge invariance, and theQCD θ YM -term Henrique Gomes ∗ and Aldo Riello † July 9, 2020
Abstract
The eliminative view of gauge degrees of freedom—the view that they arisesolely from descriptive redundancy and are therefore eliminable from the theory—is a lively topic of debate in the philosophy of physics. Recent work attempts toleverage properties of the QCD θ YM -term to provide a novel argument against theeliminative view. The argument is based on the claim that the QCD θ YM -termchanges under “large” gauge transformations. Here we review geometrical propo-sitions about fiber bundles that unequivocally falsify these claims: the θ YM -termencodes topological features of the fiber bundle used to represent gauge degrees offreedom, but it is fully gauge-invariant. Nonetheless, within the essentially classi-cal viewpoint pursued here, the physical role of the θ YM -term shows the physicalimportance of bundle topology (or superpositions thereof) and thus weighs against(a naive) eliminativism. Modern philosophers take seriously the ontological status of fields. But what theyusually have in mind are relatively concrete entities, such as the electric and mag-netic fields, and not the elusive gauge fields, such as the electromagnetic potential.How then, to classify “gauge” degrees of freedom? Do these have an ontologicalsignificance similar to electric and magnetic fields, or are they only a notationalconvenience, born of a redundancy in our representations of the world? In thewords of Earman, are gauge degrees of freedom only “redundant descriptive fluff”(Earman, 2004)?The eliminativist view of gauge degrees of freedom advocates not only thatgauge degrees of freedom are redundant, but that they are also eliminable. Oneproponent of eliminativism within the philosophy of physics community is RichardHealey, whose position is laid out in (Healey, 2007). Healey proposes that we shoulduse a different, gauge-invariant basis to describe our physical quantities: non-local,yes, but controllably so; this is called the holonomy-basis. ∗ University of Cambridge, Trinity College, CB2 1TQ, United Kingdom; [email protected] † Perimeter Institute for Theoretical Physics, EC1R 4UP Canada; [email protected] Whether one can really write down a theory—an action functional or a Hamiltonian—in terms of holonomies(or Wilson loops) is challenging, to say the least. But we will not pursue this in this paper. θ YM -term In a recent paper, Dougherty (2019) engages with the details of Healey (2007)’seliminativist program in the context of QCD. Dougherty’s first aim is to convincethe reader that a θ YM -term in the QCD Lagrangian is mandatory.In brief, the argument is as follows: the θ YM -term is necessary to account forcertain experimental facts. To be more specific: the smallness of the masses ofthe up and down quarks gives rise to a chiral symmetry, whose effects (a paritydoubling of the hadron spectrum, cf. (Weinberg, 2005, Sec. 19.10)) are not observedin experiments. This means that this chiral symmetry must be broken somehow.But the spontaneous breaking of this symmetry would generate Goldstone bosons,which are also not observed. Therefore, one must be able to break chiral symmetrywithout creating Goldstone bosons .A solution is to have the breaking be effected through an anomaly. Namely,under chiral transformations (also called a global U(1) A symmetry), it turns outthat the path-integral measure for quark fields fails to be invariant and ratheracquires a phase. Specifically, for a fermion field of flavor f , the chiral symmetryacts by a shift ψ f exp( iγ α f ) ψ f (with γ the fifth gamma-matrix), whereas thefermion path-integral transforms as D ψ D ψ exp i θ YM -term) X f α f ! D ψ D ψ, (1.1)where θ YM -term = 18 π Z tr( F ∧ F ) . (1.2)Therefore, according to this argument, mathematical consistency and experimen-tal evidence—the lack of both the relevant Goldstone bosons and of the paritydoubling of the hadron spectrum—together would provide support for the physicalsignificance of the θ YM -term. It is here important to stress the role fermions play inmaking the θ YM -term inescapable.But this is not the end of the story: such a term would be CP-violating and thusgives rise to other questions of observability. The relation between CP-violation and This solution, however, might not be appropriate in a non-perturbative treatment. See section 1.5. This is the standard argument first put forward by Fujikawa (cf. Bertlmann (1996, Sec. 5.2)). Now, the θ YM -term is a functional of the curvature, F µν , so why does it appear in a change in the measure of purelyfermionic degrees of freedom? In Fujikawa’s implementation of a gauge covariant measure, one writes thefermion field in terms of a basis of eigenfunctions of the Dirac operator, / D, which includes the gauge-covariantderivative D µ = ∂ µ + A µ , inside it (i.e. / D = γ µ D µ , where γ µ are the Dirac matrices). It then turns out that thedeterminant of the Jacobian under a chiral transformation in this orthonormal basis diverges and needs to beregularized. Fujikawa used a gauge-covariant Gaussian cut-off by insertion of the operator lim M →∞ exp ( − / D M ).Ultimately, the curvature appears through the decomposition: / D = D µ D µ + [ γ µ , γ ν ] F µν . One can choose agauge-invariant measure, in which case the anomaly is shifted to counterterms (which necessarily fail to satisfythe same invariances of the Lagrangian, DeWitt (2003, vol 2, ch.28)). θ YM -term is not directly relevant to the central points of this paper, which iswhy we will avoid discussing it. Having set the broader context for the discussion, we now very briefly embedwithin it Dougherty’s criticism of Healey. Before we begin, it should be statedfrom the outset that our intention in this paper is only to set straight a specificmisunderstanding of this criticism: the gauge-invariance properties of the θ YM -term;we will mostly constrain the remit of our discussion accordingly. According to Dougherty (2019) (cf p.1, 7, 8, 16) the underlying reason for Healey’selimination of the θ YM -term is that such a term is only gauge-invariant under gaugetransformations that have a particular behaviour at infinity (or at the relevantboundaries). The idea then is that the non -eliminativist would be comfortablein separating the wheat from the chaff, for they could say: “some ‘gauge trans-formations’ relate distinct physical possibilities while others don’t. Thankfully, I,the non-eliminativist, haven’t eliminated any of them, so I can still tell the twokinds apart!” This strategy, it is claimed, is not available to Healey. The claim isthat, since Healey’s eliminativism does not license a distinction between differenttypes of gauge transformations, no restriction to some type of gauge transformationis allowed. In particular, one cannot keep just those transformations that wouldguarantee invariance of the θ YM -term. Therefore Healey would either have to iden-tify physically distinct states with each other (indeterminacy), or be obliged to set θ YM to zero and thereby fall foul of the fact that allowing for a non-zero θ YM -termis a theoretical requirement. But in fact, no such indeterminacy occurs, since Dougherty’s argument that the θ YM -term is only gauge-invariant under gauge transformations that have a particularbehaviour at the boundaries is incorrect. For the θ YM -term is manifestly gaugeinvariant under the action of all gauge transformations.Nonetheless, there is a subtle and tempting reason to erronously assume thatthe θ YM -term is gauge-variant. For, as Dougherty correctly states, the θ YM -term canalso be expressed as a pure boundary contribution. And it is well-known that thisboundary contribution, which takes the form of a Chern-Simons boundary integral,can acquire different values even on vanishing curvature configurations : the valuesof such terms can differ by an integer multiple of 2 π (times θ ). So, it would benatural to say that these values have some sort of gauge-dependence, i.e. that they Briefly, the field redefinitions above—modifying the definitions of the quarks by a chiral transformation—not only shift the coupling constant θ in front of the θ YM -term in the Yang-Mills Lagrangian by θ θ + P f α f ,but also change the mass terms in the Lagrangian density by m f exp( i α f ) m f . Since physical quantitiescannot be affected by a mere field-redefinition, this means that the only invariant quantity physical systemscan depend on is the product e − iθ Q f m f (cf. (Weinberg, 2005, Sec. 23.6)). This product defines an invariantversion of the θ -coupling, called θ . Thus, if one flavor of quarks had zero mass, the puzzle would be resolved.That doesn’t seem to be the case. Nonetheless, θ is observationally constrained to be close to zero: the currentbound on θ is | θ | < × − [ particle data group, see: this for general citation ]. The question oftheoretical necessity of the θ -term hinges on important issues of naturalness and fine-tuning, and, since there iscurrently experimental reason to believe that it vanishes, one might feel compelled to explain its observationalsmallness. That is, what physicists refer to as the “Strong CP problem”—that Nature conspires to give the CP-violating θ -term a value close to zero—is a real problem that still lacks an agreed explanation. But Doughertydoes not tie his boat to the issue of explanation for the smallness of θ . hange under “large gauge transformations”. This change is the one Doughertywrongly appeals to in his argument. The mistake is subtle, and lies in the construalof the term “large gauge transformation”.Before we briefly sketch our argument in order to clarify this crucial subtlety,we therefore need to define “large gauge transformations”.In practice, the term “large gauge transformation” has been associated with twomeanings:( i ) a smooth Lie-group-valued function on space or spacetime that is not connectedto the group identity, i.e. not infinitesimally generated through exponentiation;( ii ) in the presence of asymptotic boundaries, it is a gauge transformation whichdoes not asymptote to the identity.In this article, we will exclusively use the term “large gauge transformation” inthe sense attached to ( i ), i.e. not being connected to the identity.To make his argument stick, Dougherty must use transformations that satisfyboth ( i ) and ( ii ) (Dougherty, 2019, p. 1), i.e. transformations whose pullbackto the boundary neither vanishes nor is connected to the identity: this is becauseonly such transformations would change the value of the boundary Chern-Simonsintegral which re-expresses the θ YM -term. However, the combination of ( i ) and( ii ), required by Dougherty selects an empty set of functions. This is becausethere is no smooth Lie-group valued function over R that tends at infinity to afunction over ∂ R ∼ = S that is not connected to the identity. This fact is strictlynecessary to ensure the mathematical consistency of the equality between the bulk-integral defining the θ YM -term (which is manifestly gauge invariant under all gaugetransformations) and its expression in terms of Chern-Simons boundary integral(which is not invariant under large-gauge transformations over S ). The goal ofthe following sections is to explain these facts and their basic consequences in gooddetail.Here, we briefly sketch with equations an abstract argument showing that thenecessary transformations cannot be smoothly extended into the bulk (all notationwill be explained later). For now we consider the simplest possible case: that ofa gauge potential A that is pure gauge on a 4-disk D . Thus, A = g − d g for some g : D → G , and its associated curvature vanishes, i.e. F ( A ) = F ( g − d g ) = 0, sothat the θ YM -term, defined as π R D tr( F ∧ F ), manifestly vanishes—in all gauges.Thus,0 = 18 π Z D tr( F ∧ F ) = 124 π I ∂D = S tr( g − d g ∧ g − d g ∧ g − d g ) =: CS S ( h − d h ) , (1.3)where the second equality will be shown in the next section, CS S is by definitionthe Chern-Simons functional (on S ) with h : S → G set to h = g | S .The puzzle arises thus: it is a mathematical fact that certain h ’s yield a non-vanishing CS S ( h − d h ), so how could the above equation avoid mathematical in-consistency? Now, the h ’s that yield these different values are “homotopically”different: they cannot be smoothly deformed into each other, and are thus said todiffer by a “large” transformation. The answer to our question then is that, cru-cially, large transformations that relate different h ’s of this kind cannot be extendedinto the D bulk smoothly and therefore cannot define “gauge transformations” ofthe bulk configuration A = 0; there are no such transformations whose restriction The difference between the two is relevant, as we will show later. Twice-differentiable is sufficient for our purposes. We will deal with the general case in the following sections.
4o the boundary fits in ( i ) above. In other words, the large boundary transforma-tions required to yield a non-zero value of the Chern-Simons functional are not ofthe form h = g | S for a smooth g : D → G . That is: they do not come from bulk gauge transformations of any kind—which, as we know, leave the value of the θ YM -term invariant.Homotopically different h ’s on the right hand side of (1.3) represent physicallydifferent configurations also in the bulk, and indeed must be accompanied by dif-ferent curvatures in the bulk. In due course, we will prove all of these statements,thus avoiding a mathematical contradiction: the gauge-invariance properties of the θ YM -term cannot depend on the way we decide to write it, viz. as a bulk or as aboundary term. This paper will proceed as follows. In section 2, we will give a brief introductionto fibre bundles. We start in section 2.1 by describing fibre bundles as the math-ematical structure underpinning gauge theories. They formalize the notion thatcertain properties that are taken as, in a certain sense, “intrinsic,” such as “be-ing a proton,” are in fact relational. But these relations can have topological, i.e.global features. Some of these features are embodied by Chern classes, which webriefly review in section 2.2. There, we will recall what these classes have to dowith the θ YM term in QCD, and discuss their gauge and topological invariance. Inthe following subsection 2.3, we finally bring in the “large gauge transformations”that underpin Dougherty’s argument and show in particular that they have nothingto do with gauge-transformations: they are quantities that encode the topologicalproperties of the underlying bundle, and are not related to choices of gauge. Suchtopological properties are represented by the particular gluing, or relations, be-tween topologically trivial charts; and the winding numbers encode this ‘gluing’information.These conclusions are valid for manifolds without boundary. In section 3 wedescribe how the previous conclusions can be extended to the context of manifoldswith boundaries. Here it is important to distinguish the Euclidean signature settingfrom the Lorentzian one. In the former case, in section 3.1, we can completeasymptotic boundaries and fall back on the results for the unbounded manifolds.In the latter, of section 3.2, we get two disconnected boundaries, and thus (assumingthe fields behave nicely at space-like infinity) the θ YM topological invariant becomesa difference of two Chern-Simons terms, or of two winding numbers. Nonetheless,the conclusions about their invariance remains, but now it applies to the differenceof winding numbers. In Section 4 we conclude: section 4.1 summarizes the mainpoints made in the paper. Finally, in section 4.2, we briefly smoke a peace-pipe, bygiving a criticism of our own of eliminativism. This criticism does take into accountthe role of the θ YM -term—but not its properties under gauge transformation, which, pace Dougherty, are compatible with eliminativism.
Since this article is an answer to Dougherty (2019), we follow here the same, intrin-sically semiclassical, but standard, account of chiral symmetry breaking, cf. e.g.(Weinberg, 2005).However, as we ackowledge in this short intermezzo, a fully non-perturbative5ccount also exists (Strocchi, 2019, Ch. 3). In the field-theoretic path integral,continuous configurations (which are needed to make sense of the “topology ofthe bundle”—see later—as an explanatory device) are of measure zero, and thus anon-perturbative account of the chiral symmetry breaking mechanism which does not rely on topological features of the field configurations is more satisfying (if notnecessary). In this non-perturbative account, it is rather the topology of the gaugegroup that plays a crucial role.Indeed, in the perturbative account of Strocchi (2019, Ch. 3), despite its anoma-lous implementation, chiral symmetry is not “explicitly broken”, but rather givesrise to what one could roughly characterize as a “meta-symmetry” between non-communicating ( θ -)sectors of the theory. These sectors are labeled by their trans-formation properties under central elements of the algebra of observables whichcorrespond to the equivalence classes of gauge transformations which are not con-nected to the identity modulo the ones that are connected to the identity (here werefer to the residual time-independent gauge symmetries not fixed by the choiceof temporal gauge). The technical, but crucial, ingredient entering this account isthe non-weakly-continuous nature of the representation of the symmetries on theHilbert space.Although one immediately sees that this non-perturbative account appears quiteexplicitly incompatible with any naive notion of eliminativism, we will pursue nei-ther a deeper analysis of its philosophical underpinnings nor a clarification of itsrelationship with the (semi)classical approach; both are extremely interesting tasksbut lie well beyond the scope of this article (but see the comments in sections 3.3and 4.2). In this section, we will introduce fiber bundles and their topology, and proceedto assess gauge-invariance of the θ YM -term for closed manifolds in several differentways. In section 2.1 we introduce the basic machinery: the connection-form (and itsrelational interpretation), and the relation between charts, gauge transformationsand transition functions. In section 2.2, we introduce the θ YM -term—also known asthe Chern-number. Seen as a bulk, i.e. spacetime, integral, we show both gaugeand topological invariance of the term. In section 2.3 we relate this invariant tothe appearance of ‘large’ transformations: they appear as Wess-Zumino integralsrelated to transition functions between charts. We also show that gauge transfor-mations on a 4-dimensional disk-region cannot have non-trivial winding number atits boundary. This is entirely compatible and required by our considerations in thispaper. The modern mathematical formalism of gauge theories relies on the theory of prin-cipal (and associated) fibre bundles. We will not give a comprehensive accounthere (cf. (Kobayashi & Nomizu, 1963)), but only introduce the necessary ideas andobjects.Given an n -dimensional manifold M , thought of as representing spacetime (wewill not need any of the metric structure of spacetime however), a standard example Cf. (Strocchi, 2015).
6f a principal fibre bundle with structure group GL ( n ) taking M as the base space,is the space of linear frames over M . The “fibre” over each point of the base space M consists in all of the linear frames of the tangent space there. In this example,there is no “zero” or identity element on each fibre: each point is just a linear framebasis for the tangent space. But there is a one-to-one map between the group GL ( n )and the fibre: we can use the group to go from any frame to any other.The main idea underlying the physical significance of the internal space in a fibrebundle is perhaps best summarized in the original paper by Yang & Mills (1954):The conservation of isotopic spin is identical with the requirement of in-variance of all interactions under isotopic spin rotation. This means thatwhen electromagnetic interactions can be neglected, as we shall hereafterassume to be the case, the orientation of the isotopic spin is of no physicalsignificance. The differentiation between a neutron and a proton is thena purely arbitrary process. As usually conceived, however, this arbitrari-ness is subject to the following limitation: once one chooses what to calla proton, what a neutron, at one space-time point, one is then not freeto make any choices at other space-time points.That is, what is a proton and what is a neutron at a given point is essentially a relational property.The limitations on how to identify “a proton” at two different points of spacetimeare imposed by a connection-form: another structure on the bundle. That is, aconnection-form ω allows us to define which points of neighbouring fibres can betaken as equivalent to an arbitrary starting-off point in an initial fibre. In theexample of linear frames, it gives us a notion of “parallel transport” of the basis aswe go from an initial choice over one point of M , to a neighbouring one. Curvaturethen acquires meaning as non-holonomicity: start-off from the same point andfollow such an identification of bundle points along different paths in the bundle,lifted from a path in M , interpolating between initial and final points. Even if theinitial point on the bundle and the initial and final points on M agree, the finalpoints identified on the bundle may still differ. It is this disagreement that usuallycarries physical consequences.There are two conditions that such a connection-form must satisfy. First, theparallel transport to a neighbouring fibre should commute with the group action;i.e. there is a sense in which it doesn’t really depend on what we choose as thestarting-off basis. Equivalently, there is an equivariance property that ω mustsatisfy. Secondly, there must be exactly one choice of parallel transported frameper direction of M . All the relevant properties of gauge transformations can bederived from these two (cf. footnote 10).We are now going to formalize this intuitive description. Principal fibre bundles
A principal fibre bundle is a smooth manifold P that admitsa smooth action of a (path-connected, semisimple) Lie group, G , i.e. G × P → P with ( g, p ) g · p for some action · and such that for each p ∈ P , the isotropygroup is the identity (i.e. G p := { g ∈ G | g · p = p } = { e } ). Naturally, we constructa projection π : P → M , given p ∼ q ⇔ p = g · q for some g ∈ G . So the base space M is the orbit space of P , M = P/G , with the quotient topology, i.e.: characterizedby an open and continuous π . By definition, G acts transitively on each fibre.Locally over M , it must be possible to choose a smooth embedding of the groupidentity into the fibres. That is, for U ⊂ M , there is a map σ : U → P such that P
7s locally of the form U × G , U ⊂ M , i.e. there is an isomorphism U × G → π − ( U )given by ( x, g ) g · σ ( x ). The maps σ are called local sections of P .On P , we consider an Ehresmann connection ω , which is a 1-form on P valuedin the Lie algebra g that satisfies appropriate compatibility properties with respectto the fibre structure and the group action of G on P . This connection allowsus to locally define “horizontal complements” to the fibres in P (see footnote 10).Through such complements one can horizontally lift paths γ in M to P . Thesehorizontally lifted paths are commonly referred to as “parallel transports” in P along γ with respect to (horizontality as defined by) ω . As you go around a closedcurve in M , parallel transport on P may land you at a different point over theinitial fibre from which you started: e.g. assuming you started from p , you mayend at p ′ = g · p . The relation between p and p ′ (i.e. g ) is the called the holonomyof ω along the closed path γ . Its infinitesimal analogue is the curvature of ω ,Ω = d P ω + ω ∧ P ω, (2.1)where d P is here the exterior derivative on the smooth manifold P , and ∧ P is theexterior product on Λ( P ) (it gives anti-symmetrized tensor products of differentialforms). Gauge transformations v. Transition functions
Given local sections σ α on eachchart U α , i.e. maps σ : U α → P such that π ◦ σ α = id, we define by pullback A α = σ ∗ α ω ∈ Λ ( U α , g ) (here α is a chart index, not a spacetime one). Since thedifferential and the pullback operation “commute”, we also have: F α := σ ∗ α Ω = d A α + A α ∧ A α (2.2)where now d and ∧ are the familiar exterior derivative and products in Λ( M ).Notice that contrary to ω and Ω, the A α ’s and F α are defined over charts of thespacetime M , rather than the bundle P . The price to pay is the introduction of:(a) an (arbitrary) choice of section, and (b)—since global sections might not existin general—of an atlas of charts over M and a corresponding set of A α ’s.In other words, although ω is globally defined on P , the A α ’s are only definedon the respective charts U α of M through the choice of a local section σ α . At fixed ω , and on a given chart U α , different choices of section give A α ’s related by a gaugetransformation. The demand of gauge invariance reflects the arbitrary nature ofthe choice of section. We will come to this in a moment; first we need to worryabout how to patch the charts together.Given an atlas of charts U α ⊂ M , this patching requires us to consider transitionfunctions which relate the A α ’s to each other on the overlaps U αβ = U α ∩ U β :on U αβ : A β = t − αβ A α t αβ + t − αβ d t αβ , (2.3) Given p , the inverse map is a bit more complicated because we must find g ′ such that g ′ · p = σ ( x ), forsome x . It will depend on the form of σ . Given an element of the Lie-algebra g , we define the vertical space V p at a point p ∈ P , as the linear spanof vectors of the form v ξ ( p ) := ddt | t =0 (exp( tξ ) · p ) for ξ ∈ g . And then the conditions on ω are: ω ( v ξ ) = ξ and g ∗ ω = g − ωg, where g ∗ ω p ( v ) = ω g · p ( g ∗ v ) where g ∗ is the push-forward of the tangent space for the map g : P → P . Achoice of connection is equivalent to a choice of covariant ‘horizontal’ complement to the vertical space, i.e. H p ⊕ V p = T p P , with H compatible with the group action. t αβ ≡ t − βα : U α ∩ U β → G. (2.4)These transformation properties translate between choices of local sections acrossoverlapping charts, and must satisfy the cocycle conditions (compatibility overthreefold overlaps U αβγ = U α ∩ U β ∩ U γ ):on U αβγ : t γβ t βα = t γα . (2.5)Transition functions look similar to gauge transformations, and indeed act verysimilarly on the gauge potentials. These similarities reflect the fact that, on theoverlap U αβ , both A α and A β descend from the same ω through different choice ofsections—and, as we will now discuss, the role of gauge transformations is preciselyto translate between different choices of sections.Gauge transformations (i.e. changes of local sections) are encoded in maps g α : U α → G (2.6)that act on the respective A α and t αβ ’s as follows: ( A α g A gα = g − α A α g α + g − α d g α on U α t βα g t gβα = g − β t βα g α on U αβ (2.7)from which one derives using (2.2): F α g F gα = g − α F α g α on U α . (2.8)Notice that both the connection and the transition function transform under theaction of a gauge transformation g α . Thus, under a gauge transformation on U α ,equation (2.3) describing the relation between A β and A α , is left invariant. This isthe basic reason why the transition functions collectively encode the global propertiesof the bundle P while the gauge transformations are simple redundancies. Besides the fact that gauge transformations act on transition functions and notvice versa, another crucial distinction between gauge transformations and transi-tion functions, that underlies their different roles, is that the domain of the gaugetransformations g α ’s is the whole U α , whereas that of t αβ is a subset of U α (viz. itsoverlap with U β ).We reiterate that the introduction of transition functions is generally necessarybecause, global sections do not exist unless the bundle is trivial, i.e. unless P = M × G globally not just locally. In the trivial case, and only in the trivial case, alltransition functions can be trivialized to be the identity, i.e. t βα = g β g − α for somechoices of g α ’s. Only then , equation (2.3) is trivialized and the collection of A α ’syields a global gauge potential 1-form A . Summary
A gauge field configuration can be defined either:(1) “abstractly,” by providing a bundle π : P → M and an Ehresmann connection ω ∈ Λ( P, g ); or(2) “in coordinates,” by providing an atlas of charts U α ⊂ M , a set of sections σ α : U α ∈ P , and compatible transition functions t αβ : U αβ → G (these three The set of all g α ’s on a given U α defines G α := { g α ( x ) } , which inherits from G the structure of an (infinite-dimensional) Lie-group, by pointwise extension of the group multiplication of G over U α . Compatibility is here understood in the sense of equations (2.5). P ), together with a choice of compatible gauge fields A α ∈ Λ ( U α , g ) (this corresponds to the choice of ω ).The coordinate description is redundant because it requires the introduction ofauxiliary choices of sections, σ α ; different choices are related by “gauge transforma-tions” of the A α ’s and of the t αβ ’s. Therefore, gauge invariance requires all physicalobservables to depend on the choice of P and ω only. Crucially, transition functions and gauge transformations play entirely differentroles. Gauge transformations act on the transition functions, but not vice-versa,and a gauge transformation’s domain of definition is the whole chart U α , and notmerely the overlaps U αβ as is the case for the transition functions t αβ ’s. Thesetechnical differences reflect the fact that the g α ’s and t αβ ’s play conceptually dif-ferent roles. From the perspective of P , the gauge transformations g α ’s encode thefreedom of choosing a local section σ α (which is necessarily defined on the whole U α ). Conversely, the t αβ encode—albeit somewhat redundantly—the way in whichthe charts are glued to one another, and thus the global structure of the bundle P . For a closed 4-dimensional manifold M , that is, M is compact and without bound-ary, the quantity (the notation will be explained in a moment, for now it is enoughto notice that the integrand depends on A and is gauge-invariant) Ch [ P ] := Z M ch A is a topological invariant— not of M —but of the fibre bundle P over M . A connection-form ω is defined over P and a collection of local gauge potentials A α is definedover an atlas of M , as above. Since ch A is gauge invariant, the integral can thenbe obtained through an appropriate partition of unity associated to the atlas. Asa topological invariant of P , Ch [ P ] is not only completely gauge invariant, but alsoindependent of the choice of ω over P . We call Ch [ P ] the (second) Chern-number of P .If we write our physics in terms of gauge potentials, and allow them to livein different bundles, e.g. P and P ′ , then the potentials A and A ′ might leadto different values of Ch [ P ]. The question then is: how does A “know about”topological properties’ of P ? And how can Ch [ P ] depend only on the topology of P and not on the detailed choices going into its computation? This is the content ofthe Chern-Weil theorem (Nakahara, 2003, Ch. 11.1), that we briefly review below.From now onwards, we will restrict to G = SU( N ).First, the Chern-number is computed as follows: Ch [ P ] = Z M ch A = 18 π Z M tr( F ∧ F ) (2.9)where ch A := 18 π tr( F ∧ F ) . (2.10)Of course, Ch ( P ) is nothing but the “ θ YM -term,” or, more specifically: the θ YM -termin the QCD Lagrangian can be written using (2.9) as: L θ = θ Ch [ P ] (2.11) Compatibility is here understood in the sense of equations (2.3). Notice that it is possible to change ω (resp A α ) without changing P (resp σ α and t αβ ). θ is just a real-valued coefficient. The integrand ch A defines the secondChern-class of the bundle P . The second Chern-class is manifestly gauge invariant,given the gauge transformation properties of F (2.8) and the cyclicity of the trace. This means that on the overlaps U αβ , ch A α = ch A β , which is why no chart indexappears in the equations above, and why the integral can be performed with nofurther complications.This also immediately tells us that Ch [ P ] can at most depend on the choice of ω , and not of gauge (i.e. of sections). We are now ready to review the Chern-Weiltheorem, which shows that Ch [ P ] is not only gauge invariant but also independentof the choice of ω on P —that is it depends only on the topological properties of P .A first hint of the ‘topological’ nature of Ch [ P ] comes from the observation that itdoes not change under a small arbitrary variation of A (i.e. the equations of motionof the action S [ A ] = R ch A are identically satisfied). This follows immediately from δF = d A δA and the Bianchi identity d A F = 0 where d A := d + [ A, · ] is the exteriorgauge-covariant derivative (for the adjoint representation). But invariance can beproven also for finite, rather than infinitesimal, changes in connection. Consider twoconnections A and A ′ , and now define γ := A ′ − A ∈ Λ ( M ) and a one-parameterfamily of connections A s = A + sγ , s ∈ (0 , A and A ′ (thespace of connections is an affine space). Then, denoting the curvature of A s as F s ,one finds ch A ′ − ch A ≡ π Z dd s tr( F s ∧ F s )d s = 14 π Z tr(d A s γ ∧ F s )d s = 14 π d (cid:16) Z tr( γ ∧ F s )d s (cid:17) , (2.12)Thus the difference ch A ′ − ch A is an exact differential form and thus vanishes whenintegrated over a closed manifold. Since A and A ′ are arbitrary connections, itfollows that R M ch A over a closed manifold P does not depend on the choice ofconnection, i.e. that it is a topological invariant. Summary
The gauge invariance of ch A tells us that Ch [ P ] depends at most on ω , and the Chern-Weil theorem tells us that Ch [ P ] does not depend on A (andtherefore on ω ) at all. Therefore, Ch [ P ] can only reflect a (topological) propertyof the bundle P on which the connection is defined. A nontrivial, and extremelydeep, fact is that the second Chern number of P is always an integer Ch [ P ] ∈ Z . (2.13)We conclude this section with a simple remark. The discussion above clearlyshows that the Chern number (2.9) (and thus the θ YM -term) is gauge-invariantunder all possible gauge transformations. And, just to be clear, this even holds atthe level of the integrands: ch A g = ch A ∀ g = g ( x ) (2.14)This fact simply follows from the transformation properties of F (2.8) and the(graded) cyclicity of the trace (for λ, η as p and q-forms, respectively)tr( λ ∧ η ) = ( − pq tr( η ∧ λ ) . (2.15) The proof is simple: tr( g − F g ∧ g − F g ) = tr( g − F ∧ F g ) = tr(
F g ∧ g − F ) = tr( F ∧ F ). For consistency, one should also check that the the 3-form R tr( γ ∧ F s )d s is well defined, i.e. gaugeinvariant. That this is the case follows from the fact that the difference γ between two connections transformsin the adjoint representation under gauge transformations, just like F , and therefore tr( γ ∧ F s ) is point-wisegauge invariant for all values of s . (cf footnote 15). θ YM -term is vetoed by this simple demon-stration. As we have just witnessed, the Chern-number and the so-called θ YM -term, (2.9),is completely gauge-invariant. Thus the inevitable question: whence Dougherty’sclaims? He writes for example that (italic ours) (Dougherty, 2019, p. 7) The Yang- Mills [ θ - ] vacuum term is not preserved by all gauge trans-formations. If the eliminative view of gauge transformations is right,this means that the Yang-Mills vacuum term is physically meaningless.If gauge transformations are redundancies then mathematical differencesbetween gauge equivalent configurations can’t reflect physical differences.So the value of the Yang-Mills vacuum term can’t represent any physicalfact.We will now argue that one way Dougherty might have arrived at this conclusion,ignoring the previous simple argument for the gauge invariance of the θ YM -term, isthrough an incatious invocation of boundaries.Before we get to boundaries of the entire Universe, in section 3, let us revisit thecomputation of the Chern-number under a new guise, by breaking up the manifoldand therefore introducing internal boundaries.First, we recall the Chern density ch A := π tr( F ∧ F ), and the following crucialrelation it has with the Chern-Simons 3-form cs A : ch A = d cs A where cs A := 18 π tr( A ∧ d A + A ∧ A ∧ A ) . (2.16)The subtlety lurking behind this identity is the fact that the Chern-Simons formis, at least naively, not gauge invariant, since: cs A g − cs A = wz g + 116 π d tr(d gg − ∧ A ) (2.17)where the Wess-Zumino term is just a Chern-Simons form of a pure gauge config-uration: wz g := cs g − d g = 124 π tr( g − d g ∧ g − d g ∧ g − d g ) . (2.18)In particle physics lingo, equations (2.14), (2.16), and (2.17) together say that“while the topological charge [ ch A ] is gauge invariant, the topological current [ cs A ]is not.” (Schfer & Shuryak, 1998, p. 31). This is easy to show:d cs A = dtr( A ∧ d A + A ∧ A ∧ A ) = tr(d A ∧ d A + 2 A ∧ A ∧ d A )= tr((d A + A ∧ A )(d A + ∧ A ∧ A )) = ch A where in going from the first to the second line we used (2.15) to infer that tr( A ∧ A ∧ A ∧ A ) ≡ The Chern-Simons functional understood as the action for a 3d boundary theory, defines a classical theoryof connections that is invariant only under gauge transformations that are not large in the sense of ( i ) inSection 1.3. However, quantum mechanically, the situation can be improved, and the Chern-Simons functionalcan define a theory which is invariant under all gauge transformations, provided the coupling constant, i.e.the Chern-Simons “level”, is chosen to be an integer. This is because under large gauge transformations, theChern-Simons action changes at most by a multiple of 2 π —hence allowing the Feynman’s path integral to stillbe invariant. This peculiarity lies at the root of the fascinating phenomenology of Chern-Simons theory and itsquantum-deformed symmetry structure. ch and its relation to cs in (2.16), both sides of (2.17) must be closed 3-forms, andtherefore wz g is necessarily a closed 3-form, i.e. d wz g ≡ . (2.19)Therefore, the gauge invariance of ch A is not affected, even if we write it in termsof the gauge- variant functional cs : ch A g = d cs A g = d( cs A + wz g + d 116 π tr(d gg − ∧ A )) = d cs A = ch A . (2.20)In particular, taking A = 0 and integrating this equation on a manifold with bound-ary, we see that the boundary integral of the Wess-Zumino term associated to agauge transformation in the bulk necessarily vanishes. Equation (2.20) is a firstimportant check, which we will now corroborate with a different calculation.This different computation resolves possible confusion having to do with a par-ticular way of expressing Ch [ P ]. Namely, there is still one manner of computing Ch [ P ] chart by chart, using (2.16), which may confusingly appear gauge-variant. Wewill now set up the puzzle and then dissolve it. Instead of dealing with these issueson a very general basis, we specialize our discussion to a more concrete example.Consider the closed manifold M = S covered by 2 charts, isomorphic to 4-dimensional disks, U , U = D , that overlap on a “transition belt” around theequator, U = S × [ − , A = A t , t ≡ t . Denoting the domainof the charts that lies below/above the equator, respectively, by ˜ U = U \ ( S × [ − , U = U \ ( S × [0 , ∂ ˜ U = − ∂ ˜ U = S ×{ } ≃ S ⊂ U ),we have Ch [ P ] = Z ˜ U ch A + Z ˜ U ch A = I ∂ ˜ U ( cs A − cs A ) = I ∂ ˜ U ( cs A t − cs A ) = I ∂ ˜ U wz t (2.21)where we used (2.17) and (2.18).Thus we see that, setting ∂ ˜ U ≃ S and denoting WZ S ( g ) = R S wz g , Z ∋ Ch [ P ] = WZ S ( t ) . (2.22)This equation is of crucial importance for us. We have not used gauge transfor-mations, and yet, something that “looks like” a gauge-transformation, namely, atransition function (2.3) has appeared in the computation. Now we will verify thatwe cannot get change the Wess-Zumino invariant related to t by applying a gaugetransformation.First of all, as discussed in section 2.1, t encodes a topological property of thebundle. It is therefore not to be interpreted as a gauge transformation, but as partof the definition of P . But things are subtle, because—as we summarized in the lastparagraph of section 2.1— t participates in the definition of P in a way that dependson the choice of gauge, i.e. of sections σ α . As a consequence, under a change in thechoice of sections, the transition functions transform according to (2.7): t g − t g . (2.23) This is a corollary of the fact that tr( A ∧ A ∧ A ∧ A ) ≡ g − d g ) = − g − d g ∧ g − d g . WZ S ( t ) = WZ S ( g − t g ) , (2.24)hold?From a strictly three-dimensional, or boundary, perspective there is no reasonwhy this should be the case. In particular, we could always choose g = e (theidentity of G ) and g such that ( g ) | U = t , thus apparently trivializing the valueof WZ S . However, once we take into account the whole domain of definition ofthe g α ’s , which extends into the four-dimensional bulk of the two hemispheres,the above choice might simply be unavailable . That is, if t : S → G is largeaccording to sense ( i ) in Section 1.3—not connected to the identity—there is no smooth extension of it that goes from the belt overlap U = S to the chart domain U = D . An extension would necessarily have to “break” somewhere inside U .Only for t ’s connected to the identity will there be a smooth g such that ( g ) | U = t .We can easily perform a proof by contradiction ( reductio ). For suppose it waspossible to smoothly extend such g α ’s into the interior of their charts, then, followinga radial evolution in the disk U = D , we would find a g ( x, r ) such that g ( x, r =1) = t ( x ) and lim r → g ( r, x ) = g o for all x ∈ S , where g o is some fixed element of G . But exploiting this radial parametrization we can define a 1-parameter familyof gauge transformations { h r ( x ) : S → G | h r ( x ) = g ( r, x ) } r ∈ [0 , , defined at theintersection S , such that WZ ( h r =0 = g o ) = 0 and WZ ( h r =1 = t ) = 0. But thiscannot be right: WZ ( h r ) ∈ Z , and since one cannot continuously jump betweendiscrete values, WZ has to be constant on path-connected components of its domain.Let us prove this explicitly (by adding a differentiability assumption): denoting h r ( x ) = g ( r, x ) and ξ r = d h r d r h − r , we have, for an arbitrary r = r o ,dd r WZ S ( h r ) | r = r o = I S dd r wz h r | r = r o = 124 π I S d tr(d ξ r o ∧ h − r o d h r o ) = 0 (2.25)where the second equality follows from (2.18).In more pictorial terms, WZ S ( h ) computes a “winding number” of the map h : S → G ; this is a topological quantity that cannot be undone by a smoothdeformation of h . However, any smooth map g α ( x, r ) from the 4-disk D into G —agauge transformation according to ( i )— automatically provides through “radialevolution” a homotopy of maps h r ( x ) = g α ( r, x ) : S → G between a constant function h r =0 ( x ) = lim r → g α ( r, x ) = g o (at the central point) and its boundaryvalue h r =1 ( x ) = g α ( r = 1 , x ).It follows that the boundary value of a bulk gauge transformation g α must havetrivial winding number as a map from ∂U α → G , i.e. WZ S ( g α | ∂U α ) ≡
0. That is,the boundary value of any gauge transformation g α ( x, r = 1) on such charts mustbe connected to the identity.From this, it readily follows that t and g − t g are in the same homotopy class asmaps from S into G , and therefore have the same winding number, as per equation(2.24).Therefore, we conclude that in the simple case analyzed here the second Chernnumber of the bundle π : P → S is fully encoded into the winding number of the“equatorial” transition function t : S → G . This winding number is an intrinsicproperty of t that cannot be changed by any gauge transformation. It is clear that transformation which are not smooth to some degree are not allowed. Here we only needthem to be C .
14o far we have discussed bundles on manifolds without boundaries. But tosatisfactorily vanquish all doubts about gauge-invariance, we should also guaran-tee that it emerges when the θ YM -term is expressed not at intersections, but atboundaries. This is only possible when the curvature vanishes at the boundary;e.g. asymptotically. We now turn to this. In the first section, 3.1, we will examine Chern classes within a single bounded,Euclidean manifold and its relation to the Chern-Simons and Wess-Zumino func-tionals. In section 3.2 we briefly examine the Lorentzian case, with two boundaries,one asymptotic past Cauchy surface and one asymptotic future one (as most of theliterature; see e.g. Weinberg (2005, p.454-455)) we neglect spatial boundary termsat infinity (on which A is supposed to vanish). The Chern class then gives a dif-ference of past and future Chern-Simons terms, (naively) representing a transitionbetween different vacua of the theory. In section 3.3, we briefly discern the meaningof non-trivial bundle topology viz. the meaning of individual winding numbers. Setting aside an exhaustive treatment of fibre bundles over manifolds with bound-aries, which goes beyond the scope of this article, we will content ourselves withdiscussing what happens first for M ∼ = D with a boundary S , and then for M ∼ = R complemented with its asymptotic boundary B ∞ ∼ = S .First, we recall that gauge transformations on D induces gauge transformationson ∂D = S that are necessarily connected to the identity (as 3d objects). Armedwith this fact, we can already see why our conclusions of gauge-invariance will holdin the bounded case: even if different enough A ’s give different Chern-numbers(since they may yield different Chern-Simons terms at the boundary, according to(2.16)), such A ’s would not be related by a gauge transformation, as guaranteedby equation (2.20). This proof was easy, but it doesn’t yet get to the bottom ofthe puzzle, which we can only articulate when expressing such integrals in terms ofwinding numbers, i.e. Wess-Zumino functionals. And for that, we need boundaryconditions guaranteeing that the curvature vanishes, which we can treat jointlywith the asymptotic case.Topologically, the space M ∼ = R is just a 4-disk, and we denote it R ∞ ∼ = D to emphasize the addition of a sphere at infinity, ∂ R ∞ = B ∞ ∼ = S . The simpleremark that D constituted one of two hemispheres in the previous discussion willbecome useful later. Note that, for internal boundaries, i.e. for the intersection between charts, we can express the integrals interms of Wess-Zumino integrals, as in (2.16), because it depends on the difference between two Chern-Simonsfunctionals, and smoothness guarantees that this difference can be expressed purely in terms of the transitionfunctions; i.e. Lie-group valued functions. Following Penrose (cf. Hawking & Ellis (1975, Sec. 5)), the physically meaningful way to complement R with a boundary depends on its metric (which so far has played no role whatsoever in our considerations). Thechoice followed here corresponds to the Euclidean 4-dimensional world, rather than a Minkowskian one (whichrequires the introduction of five different typologies of asymptotic boundaries: future and past time-like infinity,future and past null infinity, and spacial inifinity). However, ignoring this complication might be justified sincethe metric one picks on R does not matter for the computation of the θ YM -term. Indeed, the computation in(Weinberg, 2005, Sec. 23.6) also disregards these subtleties. However, we personally find this argument notcompletely satisfactory. For now, we leave this subtle point aside. WZ term?As standard, we start by requiring that the field strength vanishes sufficientlyfast at infinity to render the Yang-Mills action, supplemented by the θ YM term, finite.This implies in particular that the gauge potential must approach a curvature-freeconfiguration at infinity: A x →∞ −−−→ h − d h for some h : B ∞ ∼ = S → G. (3.1)Note that this h need not be seen as a gauge transformation—vanishing curvatureguarantees (3.1)—and thus a characterization as “pure gauge” can be misleading.For such an h may still ‘wind around’ the boundary, in which case A cannot beof the form A = g d g − throughout the region. That is, an A that has non-trivialwinding number at the boundary must have curvature in the bulk. For such an A , from (2.16) and (2.18) one has: Z R ∞ ch A = Z B ∞ wz B ∞ ( h ) . (3.2)(we avoid the Chern-number notation, Ch , because we do not have a closed basemanifold, this preferrence will be maintained in what follows). Again, we know that no gauge transformation—which by definition must be extendible into R ∞ —can belarge at the boundary, nor can it change the local value of ch A , and therefore nonecan change the value of either of the integrals above. This quantity is thereforefully gauge invariant, just as the left-hand side shows manifestly.Intriguingly, even in this, single-boundary case, the Wess-Zumino invariant isstill an integer! Of course, had we computed the quantity R ch A with arbitraryboundary conditions, we can get any (gauge-invariant) quantity, depending on theboundary conditions. WZ B ∞ ( h ) is valued in the integers because of the asymptoticconditions required on the gauge potentials, which are necessary for the integralto converge. As before, this integer counts how many times the boundary map h : S → G winds around the group.A deeper reason why this integral still yields an integer is that, due to theboundary conditions, it can be recast as an integral over a closed manifold, asbefore. That is, in the Euclidean case being studied here, we can connect the abovecomputations with the previous ones performed for the closed manifold case, at theend of Section 2.3. It turns out that given the asymptotic boundary conditions(3.1), there is a “minimal” way to extend the bundle over M = R ∞ ∼ = D to abundle P over a closed manifold M ∼ = S (where we denote the closure by anoverbar). Then, with this extension, Ch [ P ] = Z R ∞ ch A . (3.3)To understand P , it is enough to observe that the asymptotic boundary condi-tions (3.1) are just the minimal requirements to be able to compactify R to S .If the field strength vanishes at infinity rapidly enough, we can compactify R to The proof follows the one showing a gauge transformation can only have a trivial winding number, in theprevious section. Here we are ignoring subtleties related to rapidity of the fall-offs at infinity and smoothness in the com-pactified manifold. by simply adding one point at infinity—the North Pole in the stereographicprojection of S —and declaring that at this point F = 0—the only value it canassume by continuity. This compactification will take us back to our previouslycovered example. But there is still one remaining piece of the puzzle. Much of what we have doneis based on an Euclidean-signature intuition for the manifold R ∞ : the θ YM -termmeasures the topology of a canonically defined bundle on P → S and WZ S ∞ ( h )measures the winding number of the asymptotic field configuration around the 3-sphere at infinity. Thinking about the Lorentzian case opens new perspectives.To think about the manifold with Lorentzian signature, we can imagine squish-ing the boundary at infinity B ∞ ∼ S from opposite sides, making it look moreand more like a ‘thin lens’. This effectively separates the boundary into three com-ponents: a past and a future Cauchy surface, Σ ± , and a “celestial sphere” S ∞ atspatial infinity. Each Cauchy surface supports some (asymptotic) gauge-potentialconfiguration that encodes a classical state of the theory. In our case, these stateshave half of their support on the northern (southern) hemisphere of S ∞ correspond-ing to the asymptotic past (future, respectively) Cauchy surfaces.It is easy to find configurations that are curvature-free at asymptotic past andfuture infinities, Σ ±∞ . For the same reason as in the previous case, asymptoticconditions guarantee that the Chern-Simons terms are numbers, n ± . And due tothe fixed orientation of these surfaces, the Chern class gives a difference betweenthese numbers, i.e. R ch A = n + − n − .Therefore, in a similar fashion to what we did throughout the paper, we can rec-oncile the fact that curvature-free boundary states h (3.1) can encode the physical,i.e. gauge-invariant, value of the θ YM -term— which only depends on the curvature .To summarize some of these results from different contexts: while it is true thatonly the curvatures figure in the argument of R ch A , this term is only related toChern-Simons terms on the boundaries of the manifold (cf. (2.16)), and these latterterms do not depend on the curvature. For closed unbounded manifolds, windingnumbers appear as differences of Chern-Simons terms at transition patches; forEuclidean bounded manifolds, the boundary is connected and we obtain a singlewinding-number (that cannot be changed by gauge transformations that properlyextend into the bulk); but here, since the configurations are “pure gauge” at dis-connected boundaries, we extract winding numbers from each connected boundaryChern-Simons term. The θ YM -term, R ch A , will thus be related to a difference ofwinding numbers due to the inward/outward orientation of the two Cauchy sliceswith respect to the 4-dimensional bulk.But, as emphasized after equation (3.1), curvature-free vacuum states with dif-ferent nontrivial winding numbers, although perfectly admissible, must includecurvature in the bulk. This means that, although the individual boundary windingnumbers associated to each boundary are not distinguishable by curvature invari-ants, transitions between them are. And this is because, crucially, the transition As opposed to a three-sphere. See however footnote 22. Together with assumptions about the field behaviour at spatial infinity, see e.g. Weinberg (2005, Ch. 23.5p. 454-455). Extra conditions at S ∞ may be needed to have well defined winding numbers on the past and futureCauchy surfaces independently. We will ignore this issue, since we can resolve the puzzle without it. states with non-trivial winding numberscan never proceed through curvature-free histories . Within the bulk of spacetime,one has to go through non-vanishing values of F that contribute to ch A , and valueswhich are uncontroversially encoded in the holonomies. θ -vacuum The quantity R ch A itself is computable even from an eliminativist perspective, sinceit is fully based on curvature observables encoded e.g. in infinitesimal holonomies.Therefore, even if the eliminativist view is incapable of describing the different,spatial and curvature-free A ’s—the different winding numbers,—the integral R ch A could still have physical significance.A suggestive comparison can be carried out with the observability of the totalenergy of a subsystem in classical mechanics. The total energy [ ∼ winding number ofa vacuum state] due to one boundary is not a well defined concept, nor a physicallymeaningful one. Nonetheless differences in energies [ ∼ non-vanishing values of the θ YM -term] are meaningful and physically measurable. In classical mechanics thereis no absolute concept of energy, but differences in energy are perfectly kosherphysical quantities, and one can get by just fine referring solely to such differences.Similarly, one could express all physical quantities solely with the θ YM -term withoutappeal to the individual winding numbers. Indeed, Healey (2007, p. 179) makes avery similar analogy:Models related by a “large” gauge transformation are characterized bydifferent Chern-Simons numbers, and one might take these to exhibita difference in the intrinsic properties of the situations they represent.But it is questionalble whether the Chern-Simons number of a gauge-configuration represents an intrinsic property of that configuration, evenif a difference in Chern-Simons numbers represents an intrinsic difference between gauge-configuration. Perhaps Chern-Simons numbers are likevelocities in models of special relativity.This observations then underpin the second role of the θ YM -term. That is,gauge theory allows the existence of distinct boundary states (e.g. initial andfinal states) that are all curvature-free but labelled by different winding numbers.These boundary states then represent different choices of initial and final vacua forthe theory and the θ YM -term can represent, in a semiclassical (“instanton”) approx-imation, a transition from one such curvature-free boundary state to a differentone A.A. Belavin (1975); ’t Hooft (1976). That is, as we saw, for asymptoticallyflat configurations, the Chern number gives a difference between winding numbers, R ch A = n + − n − =: ν . If one wants to include configurations with different wind-ing numbers in the path integral, with weight factors f ( ν ) for each sector, clusterdecomposition of expectation values requires that f ( ν ) = exp( iθν ), where θ is afree-parameter (cf Weinberg (2005, p. 456)). Thus the inclusion of the θ YM -termin the Lagrangian corresponds to allowing a superposition of all winding numbers,and the same parameter in the path integral will be included in the superpositionof vacuum states.Indeed, the impossibility of distinguishing vacuum states with different windingnumber ( | n i ) from each other via local observables , jointly with the physical signif- This follows from the same arguments exposed below equation (2.24). A global observable that is capable of this is CS ( A ). difference between winding numbers, allows the (formal) introductionof the θ -vacuum state: | θ i = X n e iθn | n i (3.4)which transforms by a phase under shifts of the winding number. Then, each θ -vacuum defines an independent sector of the quantum theory.One important point to observe from this argument, vis `a vis eliminativism, isthat it is at least a logical possibility to have a representation of L θ in the physicsand yet have no way of discerning the individual winding numbers entering the θ -vacuum.But there are other possibilities. Accounting for certain non-perturbative prop-erties of the quantization of a gauge system (Strocchi, 2019, Ch. 3), there maybe no place for a physically significant topological θ YM -term, and yet chiral in-variance should still be broken without introducing Goldstone bosons. Indeed,non-perturbative resolutions of the U(1) A -puzzle could not resort to the topologicalproperties of the bundle (since the path integral supposedly has a zero-measure sup-port on continuous fields); rather, they resort to topological properties of the groupof local (time-independent) gauge transformations G that survive the impositionof temporal gauge:The topological invariants [of the group of local gauge transformations G ] defines elements of the center of the local algebra of observables; forYang-Mills theories such elements [...] are labeled by the winding num-ber [...] their spectrum labels the factorial representations of the localalgebra of observables, the corresponding ground states being the θ YM -vacua. They are unstable under the chiral transformations [..] andtherefore chiral transformations are inevitably broken [within each facto-rial representation (sector) defined by a choice of θ YM -vacuum ...] Thus,the topology [of G ] provides an explanation of chiral symmetry breakingin QCD, without recourse to the instanton semiclassical approximation.Strocchi (2015, p. 12)In sum, depending on the level of mathematical rigor or the validity of the semi-classical approximation, different accounts of the resolutions of the U(1) A -puzzle canbe found in the literature. And even if holonomies are incapable of having a rep-resentation of the different connected components of G , it could still be true thatchiral symmetry is broken without the addition of Goldstone modes, as claimed byFort & Gambini (2000)—a claim we will not assess.Here, we should again emphasize: in this paper, our intent was not to examinethe full, non-perturbative quantum picture, nor (Fort & Gambini, 2000)’s claims,nor their relation to (Healey, 2007)’s, and thus we have refrained from assessing thesignificance of the θ YM -term in these respective domains. Our intent was rather tocorrect a mistreatment of gauge in the semiclassical picture—irrespective of whetherthis picture provides a completely satisfactory account of chiral symmetry breakingor not. The instability mentioned in this quote is due to the fact that the chiral symmetry acts as what we couldcall a “meta-symmetry” between different θ YM -vacua, θ θ + λ . Key to the consistency of this formulationis the fact that the limit λ → Conclusions
On the eliminative view and the gauge-invariant properties of the θ YM -term, Dougherty(2019, p. 16) concludes:[I] showed that if the eliminative view were true then the vacuum Yang-Mills θ YM -term [(2.9)] [...] would lead to inconsistency when integratedover any region [...] By Stokes’ theorem it is a matter of mathematicalfact that this integral coincides with the integral of cs A . But this inte-gral varies under large gauge transformations. So if I were to eliminategauge from the theory then each configuration would be assigned con-tradictory values for the vacuum Yang-Mills term of the action: one foreach class of representative gauge potentials that differ by a large gaugetransformation.Our discussion has explained, qualified, and rectified Dougherty’s statement.The θ YM -term is manifestly gauge invariant under all gauge transformations, asshown in section 2. This is just a consequence of the cyclic trace identity and thetransformation properties of the curvature—and Stokes’ theorem cannot changethis fact.Nonetheless, we felt it was important to explain some sources of confusion sur-rounding the θ YM -term. For instance, it may be expressed as Wess-Zumino in-tegrals on gluing surfaces, and the arguments of these integrals look like gaugetransformations; doesn’t that indicate their gauge-variance, contrary to the brutefact mentioned above?This puzzle is straighened once we take into account that the arguments ofthese integrals on the gluing surfaces are transition functions, and not gauge trans-formations, and that in fact, non-trivial transition functions cannot be trivializedby gauge transformations. Gauge transformations are smooth, and they are as-sociated to charts of the manifold. These two simple requirements mean gaugetransformations cannot affect the value of the integral of cs A on the boundary ofthe manifold, in accord with the invariance of the Chern number.For configurations that are asymptotically curvature-free, the only way to obtaina non-trivial winding number at the asymptotic boundary requires a non-vanishingcurvature for A in the bulk— A is not a “pure-gauge” configuration. That is howthe winding number can be represented by the θ YM -term—which depends only onthe curvature. In Lorentzian signature (with appropriate boundary conditions atspacelike infinity) this means that transitions over time between winding numbersmust be associated with curvature at some point in time.Regarding Dougherty (2019)’s criticism: it invokes a “size distinction” by as-suming there is a choice to be made on whether to accept gauge transformationsas acting solely on the boundary of the manifold or not. Moreover, he chains theeliminativist to the more permissive choice, where a restricted gauge transforma-tion, e.g. acting solely at the boundary, is bona-fide. But no such choice exists:a size-distinction would lead to two different and incompatible notions of gauge.A boundary transformation that changes the (total) winding number cannot besystematically extended to a bulk transformation that sends one solution of theequations of motion to another—as a gauge transformation would—and thereforethis transformation cannot be called a symmetry. Indeed, the bulk configuration—including its curvature—has to be changed alongside the change at the boundary20ecessary for a different winding number.While it is true that on a manifold with asymptotic boundaries one can nonethe-less use Stokes’ theorem to extract interesting and nontrivial features of the vacuumstructure of Yang-Mills theory, none of these features provide a smoking gun againstthe eliminative view of gauge, at least in the forms discussed by Healey (2007, Sec.6.6). Having arrived at the end of this paper, we can smoke a peace-pipe with Dougherty.As tobacco acceptable to both parties, we notice that the most developed under-standing of the solution to the U(1) A -puzzle (i.e. the breaking of chiral symmetrywithout the introduction of Goldstone bosons), requires the physical significance ofstructures associated to the existence of the gauge symmetry: be it the role of thefibre bundle topology in the standard semi-classical account, or the role of differ-ent connected components of G in the non-perturbative one. In both cases, thearguments bode against any naive implementation of eliminativism.More broadly, eliminativism of gauge fields is unwarranted for many reasons,some of which we now briefly summarize. Gauge degrees of freedom simplify mathe-matical treatments of physical theories by allowing us to write our theories in termsof Lorentz invariant action functionals (and path integrals): there is no availablelocal Hamiltonian or Lagrangian, even in the Abelian case (i.e. electromagnetism)that traffics only in electric and magnetic fields. Gauge fields are also necessary tomaintain certain composition properties: e.g. regionally reduced theories cannotbe composed (cf. (Gomes, 2019a,b, 2020; Gomes & Riello, 2019; Rovelli, 2014)).Moreover, gauge degrees of freedom are introduced to mandate the local Gauss law:action functionals that employ them automatically ensure both the local Gauss lawand charge conservation. In this sense, gauge degrees of freedom fill an explanatorygap: e.g. they guarantee conservation laws along much else.Fibre bundles provide a yet deeper explanation of these degrees of freedomthrough a sort of ‘internal relationism’, in accord with Yang and Mills’ originalinterpretation (cf. Section 2.1). That is, fibre bundles formalize the notion thatcertain properties that are taken as, in a certain sense, “intrinsic”, such as “being aproton”, are in fact relational. Empirical consequences of these relations—Gaussand conservation laws—follow from this realist-friendly explanation. References
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