Correspondence between the twisted N = 2 super-Yang-Mills and conformal Baulieu-Singer theories
aa r X i v : . [ h e p - t h ] J a n Correspondence between the twisted N = 2 superYang-Mills and the conformal Baulieu-Singer theory Octavio C. Junqueira a,b ∗ , Rodrigo F. Sobreiro b † a UFRJ — Universidade Federal do Rio de Janeiro, Instituto de F´ısica,Caixa Postal 68528, Rio de Janeiro, Brasil b UFF — Universidade Federal Fluminense, Instituto de F´ısica,Av. Litoranea s/n, 24210-346, Niter´oi, RJ, Brasil
We characterize the correspondence between the twisted N = 2 super Yang-Mills theoryand the Baulieu-Singer topological theory quantized in the self-dual Landau gauges. Whilethe first is based on an on-shell supersymmetry, the second is based on an off-shell Becchi-Rouet-Stora-Tyutin symmetry. Due to the equivariant cohomology, the twisted N = 2 inthe ultraviolet regime and Baulieu-Singer theories share the same observables, the Donaldsoninvariants for four-manifolds. The triviality of the Gribov copies in the Baulieu-Singer theoryin these gauges shows that working in the instanton moduli space on the twisted N = 2 sideis equivalent to work in the self-dual gauges on the Baulieu-Singer one. After proving thevanishing of the β -function in the Baulieu-Singer theory, we conclude that the twisted N = 2in the ultraviolet regime, in any Riemannian manifold, is correspondent to the Baulieu-Singertheory in the self-dual Landau gauges—a conformal gauge theory defined in Euclidean flatspace. ∗ [email protected] † rodrigo [email protected]ff.br ontents
1. Introduction 32. Topological quantum field theories 4
3. Quantum properties of BS theory in the self-dual Landau gauges 21
4. Perturbative β -functions 26 N = 2 super-Yang-Mills theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2. Baulieu-Singer topological theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2.1. Nonphysical character of the β -function in the off-shell approach . . . . . . . . . . 274.2.2. Conformal structure of the self-dual gauges . . . . . . . . . . . . . . . . . . . . . . 28
5. Characterization of the DW/BS correspondence 31
6. Conclusions 34A. BS Ward identities in the self-dual Landau gauges 35References 36 . Introduction Throughout the 1980s, based on the self-dual Yang-Mills equations introduced by A. Belavin, A. Polyakov,A. Schwartz, and Y. Tyupkin in their study of instantons [1], S. K. Donaldson discovered and describedtopological structures of polynomial invariants for smooth four-manifolds [2–4]. The connection betweenthe Floer theory for three-manifolds [5, 6] and Donaldson invariants for four-manifolds with a non-emptyboundary, i.e. , that assumes values in Floer groups, has led to the Atiyah’s conjecture [7, 8]. In thisconjecture, he proposed that the Floer homology must lead to a relativistic quantum field theory. Thisconjecture was the motivation for the Witten’s topological quantum field theory (TQFT) in four di-mensions, as Witten himself admits [8]. In [7], Atiyah showed that Floer’s results [6] can be seen as aversion of a supersymmetric gauge theory. Answering Atiyah’s conjecture, Witten found a relativisticformulation of [7], capable of reproducing the Donaldson polynomials in the the weak coupling limit ofthe twisted N = 2 SYM theory. This TQFT is commonly referred to as the Donaldson-Witten theory(DW) in the Wess-Zumino gauge [9].In practice, TQFTs have the power to reproduce topological invariants of the basis manifold as ob-servables. The first one to obtain topological invariants from a quantum field theory was A. S. Schwarzin 1978 [10]. He showed that the Ray-Singer analytic torsion [11] can be represented as a partitionfunction of the Abelian Chern-Simons (CS) action, which is invariant by diffeomorphisms. The Schwarztopological theory was the prototype of Witten theories in the 1980s. Indeed the well-known Wittenpaper in which he reproduces the Jones polynomials of knot theory [12] is the non-Abelian generalizationof Schwartz‘s results [10]. In this work Witten was actually able to represent topological invariants ofthree-manifolds as the partition function of the non-Abelian CS theory.After Witten’s result [8], L. Baulieu and I. M. Singer (BS) showed in [13] that the same topologicalobservables can be obtained from a gauge-fixed topological action. In such an approach, the Becchi-Rouet-Stora-Tyutin (BRST) symmetry [14–16] plays a fundamental role. It is not built through a lineartransformation of a supersymmetric gauge theory, like Witten’s TQFT. It is built through a gauge-fixingprocedure of a topological invariant action, in such a way that the BRST operator naturally appears asnilpotent without requiring the use of equations of motion. The geometric interpretation of the BS theoryis that the non-Abelian topological theory lie in an universal space graded as a sum of the ghost numberand the form degree, where the vertical direction of this double complex is determined by the ghostnumber, and the horizontal one, by the form degree. In this space the topological BRST transformationsis written in terms of an universal connection, and its curvature naturally explains the BS approach asa topological Yang-Mills theory with the same global observables of Witten’s TQFT.From the physical point of view, the motivation to study TQFTs comes from the mathematical toolsof such theories, capable of revealing the topological structure of field theories that are independent ofvariations of the metric, and consequently of the background choice. One of the major obstacles to con-struct a quantum theory of gravity is the integration over all metrics. The introduction of a topologicalphase in gravity would have the power to make a theory of gravity to arise from a symmetry brakingmechanism of a background independent topological theory [8, 17]. On the other hand, we can investi-gate conformal properties of field theories via topological models. In three dimensions, for instance, theconnection between the three-dimensional Chern-Simons theory and two-dimensional conformal theoriesis well-known [12]. In four dimensions, TQFTs are intimately connected with the AdS/CFT correspon-dence [18, 19]. More recently, motivated by string dualities, it was proposed a topological gravity phasein the early universe [20]. Such a phase could explain some puzzles concerning early universe cosmology. We must say that the introduction of such a topological phase is one of the intricate problems in topological quantum fieldtheories, since one should develop a mechanism to break the topological symmetry. twisted version of the N = 2 super Yang-Mills theory, the mentioned conformal theory is based on theBaulieu-Singer BRST gauge-fixing approach to a topological action [13]. In recent works [24–26], theexistence of an extra bosonic symmetry was proved in the case of self-dual Landau gauges . This bosonicsymmetry relates the Faddeev-Popov and the topological ghost fields. Together with the known vectorsupersymmetry [27] and the vanishing three-level gauge propagator, one observes that the BS theoryat the self-dual Landau gauges is indeed tree-level exact [26]. Essentially, the proof of this property isdiagrammatic with some help of algebraic renormalization techniques [16]. This remarkable propertyinevitably imply on a vanishing β -function, since it does not receive quantum corrections. Nevertheless,an entire algebraic proof was still lacking until now. It turns out that, for a complete proof of the vanishingof the β -function of the BS theory in the self-dual Landau gauges, one extra property must be considered:the fact that the Gribov copies are inoffensive to the self-dual BS theory [28]. This property establishesthat the self-dual BS theory is conformal, as it allows to recover some discrete symmetries. The use ofthese symmetries make it possible to eliminate the renormalization ambiguities discussed in [25]. Withthis information, we were able to establish the correspondence between self-dual BS theories (a conformalgauge theory defined in Euclidean spaces) for any value of the coupling constant and DW theory at thedeep UV.The paper is organized as follows. Section 2 contains an overview of the main properties of DW andBS theories. We introduce the main aspects of each approach, explaining how each one is constructedfrom different quantization schemes. As the quantum properties of the Witten’s TQFT is well-knownin literature, we dedicate Section 3 to discuss the quantum properties of the BS theory in the self-dualLandau gauges. In Section 4 we analyze and compare the corresponding β -functions of each model, afterproving that the self-dual BS is conformal. Finally, in Section 5, we describe the quantum correspondencebetween Witten and self-dual BS topological theories. Section 6 contains our concluding remarks.
2. Topological quantum field theories
A topological quantum field theory on a smooth manifold is a quantum field theory which is independentof the metric on the basis manifold. Such a theory has no dynamics, no local degrees of freedom, and isonly sensitive to topological invariants which describe the manifold in which the theory is defined. Theobservables of a TQFT are naturally metric independent. The latter statement leads to the main propertyof topological field theories, namely, the metric independence of the observable correlation functions ofthe theory, δδg µν hO α ( φ i ) O α ( φ i ) · · · O α p ( φ i ) i = 0 , (2.1)with hO α ( φ i ) O α ( φ i ) · · · O α p ( φ i ) i = N Z [ Dφ i ] O α ( φ i ) O α ( φ i ) · · · O α p ( φ i ) e − S [ φ ] , (2.2)where g µν is the metric tensor, φ i ( x ) are quantum fields, O α the functional operators of the fields com-posing global observables, S [ φ ] is the classical action, and N the appropriate normalization factor. Atypical operator O α is integrated over the whole space in order to capture the global structures of themanifold. Since there are no particles, the only nontrivial observables are of global nature [29, 30]. For simplicity we will refer to the (anti-)self-dual Landau gauges, defined by instantons and anti-instantons configurations,see gauge condition (3.3), only by the denotation self-dual gauges .
4s a particular result of (2.1), the partition function of a topological theory is itself a topologicalinvariant, δδg µν Z [ J ] = 0 , (2.3)insofar as Z [ J ] represents the expectation value of the vacuum in the presence of a external source, Z [ J ] = h | i J . As discussed in [31], if the action is explicitly independent of the metric, the topologicaltheory is said to be of Schwarz type ; otherwise, if the variation of the action with respect to the metric givesa “BRST-like”-exact term, one says the theory is of
Witten type . More precisely, being δ an infinitesimaltransformation that denotes the symmetry of the action S which characterizes the observables of themodel, then, if the following properties are satisfied, δ O α ( φ i ) = 0 , T µν ( φ i ) = δG µν ( φ i ) , (2.4)where T µν is the energy-momentum tensor of the model, δδg µν S = T µν , (2.5)and G µν some tensor, then the quantum field theory can be regarded as topological. Obviously, in thiscase eq. (2.3) is also satisfied, since the expectation value of the δ -exact term vanishes [8, 13]. In fact,by using (2.4) and (2.5), and assuming that the measure [ Dφ i ] is invariant under δ , δδg µν hO α ( φ i ) O α ( φ i ) · · · O α p ( φ i ) i = − Z [ Dφ i ] O α ( φ i ) O α ( φ i ) · · · O α p ( φ i ) T µν e − S = h δ [ O α ( φ i ) O α ( φ i ) · · · O α ( φ i ) G µν ] i = 0 . (2.6)In the above equation we assumed that all O α are metric independent. Nevertheless this is not a require-ment of the theory. It is also possible to have a more general theory in which δδg µν O α = δ Q µν = 0 , (2.7)that preserves the topological structure of δ g µν hO α · · · O α p i = h δ ( · · · ) i = 0 [31]. Analogously to theBRST operator, eq. (2.6) only makes sense if the δ operator is nilpotent . As mentioned in the introduction, Witten constructed in [8] a four-dimensional generalization of [7],capable of reproducing the Donaldson invariants [2–4] in the weak coupling limit. Such a constructioncan be obtained from the twist transformation of the N = 2 SYM. Let us quickly revise some importantfeatures of such approach. The nilpotent δ -operator works precisely as a BRST operator, and it is well-known that expectation values of BRST-exactterms vanish. For a further analysis concerning renormalization properties, and the definition of physical observables,see [14, 16, 32]. In Donaldson-Witten theory, for instance, such an operator is on-shell nilpotent, i.e. , δ = 0 by using the equations ofmotion. .1.1. The twist transformation The eight supersymmetric charges ( Q iα , ¯ Q j ˙ α ) of N = 2 SYM theories obey the SUSY algebra { Q iα , ¯ Q j ˙ α } = δ ij ( σ µ ) α ˙ α ∂ µ , { Q iα , Q jα } = { ¯ Q i ˙ α , ¯ Q j ˙ α } = 0 , (2.8)where all indices ( i, j, α, ˙ α ) run from one to two. The indices ( i, j ) denote the internal SU (2) symmetryof the N = 2 SYM action, and ( α, ˙ α ) are Weyl spinor indices: α denotes right-handed spinors, and ˙ α ,left-handed ones. The fact that both indices equally run from one to two suggests the identificationbetween spinor and supersymmetry indices, i ≡ α . (2.9)The N = 2 SYM action theory possesses a gauge group symmetry given by SU L (2) × SU R (2) × SU I (2) × U R (1) , (2.10)where SU L (2) × SU R (2) is the rotation group, SU I (2) is the internal supersymmetry group labeledby i , and U R (1), the so-called R -symmetry defined by the supercharges ( Q iα , ¯ Q j ˙ α ) which are assignedeigenvalues (+1, − SU L (2) × SU R (2) → SU L (2) × SU R (2) ′ , (2.11)where SU R (2) ′ is the diagonal sum of SU R (2) and SU I (2). The twisted global symmetry of N = 2 SYMtakes the form SU L (2) × SU R (2) ′ × U R (1), with the corresponding twisted supercharges Q iα → Q βα , ¯ Q i ¯ α → ¯ Q α ˙ α , (2.12)which can be rearranged as 1 √ ǫ αβ Q αβ ≡ δ , (2.13)1 √ Q α ˙ α ( σ µ ) ˙ αα ≡ δ µ , (2.14)1 √ σ µν ) ˙ αα Q ˙ αα ≡ d µν , (2.15)where we adopt the conventions for ǫ αβ , ( σ µ ) α ˙ α and ( σ µν ) ˙ αα as the same of [33]. The operator d µν ismanifestly self-dual due to the structure of σ µν , d µν = 12 ε µνλρ d λρ , (2.16)reducing to three the number of its independent components. The operators δ , δ µ and δ µν possess eightindependent components in which the eight original supercharges ( Q βα , ¯ Q α ˙ α ) are mapped into. Theseoperators obey the following twisted supersymmetry algebra δ = 0 , (2.17) { δ, δ µ } = ∂ µ , (2.18) { δ µ , δ ν } = { d µν , δ } = { d µν , d λρ } = 0 , (2.19) { δ µ , d λρ } = − ( ε µλρσ ∂ σ + g µλ ∂ ρ − g µρ ∂ λ ) . (2.20)6he nilpotent scalar supersymmetry charge δ defines the cohomology of Witten’s TQFT, as its observablesappear as cohomology classes of δ , which is invariant under a generic differential manifold. It is implicitin the anti-commutation relation (2.18) the topological nature of the model, as it allows to write thecommon derivative as a δ -exact term.The gauge multiplet of the N = 2 SYM in Wess-Zumino gauge is given by the fields( A µ , ψ iα , ¯ ψ i ˙ α , φ, ¯ φ ) , (2.21)where ψ iα is a Majorana spinor (the supersymmetric partner of the gauge connection A µ ), and φ , a scalarfield, all of them belonging to the adjoint representation of the gauge group. The twist transformation is defined by the identification eq. (2.9), and thus only acts on the fields ( ψ iµ , ¯ ψ iµ ), leaving the bosonicfields ( A µ , φ, ¯ φ ) unaltered. Explicitly, the twist transformation is given by the linear transformations ψ iβ → ψ αβ = 12 (cid:16) ψ ( αβ ) + ψ [ αβ ] (cid:17) , (2.22)¯ ψ i ˙ α → ¯ ψ α ˙ α → ψ µ = ( σ µ ) α ˙ α ¯ ψ α ˙ α , (2.23)together with ψ ( αβ ) → χ µν = ( σ µν ) αβ ψ ( αβ ) , (2.24) ψ [ αβ ] → η = ε αβ ψ [ αβ ] . (2.25)The twist consists of a mapping of degrees of freedom. The field ¯ ψ α ˙ α has four independent componentsas ( α, ˙ α ) = { , } , and is mapped into the field ψ µ that also has four independent components of the pathintegral, as the Lorentz index µ = { , , , } in four dimensions. In the other mappings the same occurs,as the symmetric part of ψ αβ , i.e. , ψ ( αβ ) has three independent components mapped into the self-dualfield χ µν , and the antisymmetric part, ψ [ αβ ] , with only one independent component, into η , a scalar field.We must note that ( ψ µ , χ µν , η ) are anti-commuting field variables due to their spinor origin.Because it is a linear transformation, the twist simply corresponds to a change of variables withtrivial Jacobian that could be absorbed in the normalization factor, in other words, both theories (beforeand after the twist) are perturbatively indistinguishable. Finally, twisting the N = 2 SYM action( S N =2 SY M ) [8, 34], in flat Euclidean space, we obtain the Witten four-dimensional topological Yang-Millsaction ( S W ), S N =2 SY M [ A µ , ψ iα , ¯ ψ i ˙ α , φ, ¯ φ ] → S W [ A µ , ψ µ , χ µν , ¯ φ, φ ] , (2.26)where S W = 1 g Tr Z d x (cid:18) F + µν F + µν − χ µν ( D µ ψ ν − D ν ψ µ ) + + ηD µ ψ µ −
12 ¯ φD µ D µ φ + 12 ¯ φ { ψ µ , ψ µ } − φ { χ µν , χ µν } −
18 [ φ, η ] η − h φ, ¯ φ i h φ, ¯ φ i(cid:19) , (2.27)wherein F + µν is the self-dual field F + µν = F µν + e F µν , ( e F + µν = F + µν ) , (2.28) Notation: Φ ( αβ ) = Φ αβ + Φ βα and Φ [ αβ ] = Φ αβ − Φ βα . Following [8, 34], we are considering the positive sign, that corresponds to anti-instantons in the vaccum. A similarconstruction can be done for instantons, only by changing the sign. e F µν = ǫ µναβ F αβ , and, analogously,( D µ ψ ν − D ν ψ µ ) + = D µ ψ ν − D ν ψ µ + 12 ε µναβ ( D α ψ β − D β ψ α ) , (2.29)being D µ ≡ ∂ µ − g [ A µ , · ] the covariant derivative in the adjoint representation of the Lie group G , with g the coupling constant. The Witten action (2.27) possesses the usual Yang-Mills gauge invariance,denoted by δ YMgauge S W = 0 . (2.30)The theory, however, does not possess gauge anomalies [36]. The symmetry that defines the cohomologyof the theory, also known as equivariant cohomology , is the fermionic scalar supersymmetry δA µ = − εψ µ , δφ = 0 , δλ = 2 iεη , δη = 12 ε [ φ, ¯ φ ] ,δψ µ = − εD µ φ , δχ µν = εF + , (2.31)where ε is the supersymmetry fermionic parameter that carries no spin, ensuring that the propagatingmodes of commuting and anticommuting fields have the same helicities . This symmetry relates bosonicand fermionic degrees of freedom, which are identical—an inheritance of the supersymmetric originalaction . The price of working in Wess-Zumino gauge is the fact that, disregarding gauge transformations,one needs to use the equations of motion to recover the nilpotency of δ [30]. This characterizes the DWtheory as an on-shell approach . One can easily verify that (see [8]) δ Φ = 0 , for Φ = { A, ψ, φ, ¯ φ, η } , (2.32)but δ χ = equations of motion . (2.33)Considering the result of eq. (2.33), hereafter we will say that the Witten fermionic symmetry is on-shell nilpotent. This symmetry is associated to an on-shell nilpotent “BRST charge”, Q , according to thedefinition of the δ variation of any functional O as a transformation on the space of all functionals offield variables, namely, δ O = − iε · {Q , O} , such that Q | on-shell = 0 . (2.34)In order to verify that Witten theory is valid in curved spacetimes, it is worth noting that the com-mutators of covariant derivatives always appears acting in the scalar field φ , like in δT r { D µ ψ ν · ¯ χ µν } = Technically, the Witten action (2.27) is the four-dimensional generalization of the non-relativistic topological quantum fieldtheory [7], whose construction is based on the Floer theory for three-manifolds M D , in which the Chern-Simons action istaken as a Morse function on M D , see Floer’s original paper [5]. In few words, the critical points of CS action ( W CS ) yieldthe curvature free configurations, as δW CS δA ai = − ε ijk F jk , where F jk is the 2-form curvature in three dimensions, whichdefines the gradient flows of a Morse function, see [17]. In the supersymmetric formulation of [7], the Hamiltonian (H) isobtained via the “supersymmetric charges” d t and d ∗ t , from the well-known relation d t d ∗ t + d ∗ t d t = 2 H , see [35], whereby d t = e − tW CS de tW CS and d ∗ t = e tW CS d ∗ e − tW CS , for a real number t , being d the exterior derivative on the space of allconnections A , according to the transformation δA ai = ψ ai , and d ∗ its dual. Before identifying the twist transformation, thisformulation (in four-dimensions) was employed by Witten in his original paper [12] to obtain the relativistic topologicalaction (2.27). It is implicit in this notation the typical Yang-Mills transformations of all fields, where the gauge field transforms as A ′ µ = S − A µ S + S − ∂ µ S with S ∈ SU ( N ). Precisely, the propagating modes of A µ have helicities (1 , − φ, ¯ φ ), (0 , η, ψ, χ ),helicities (1 , − , , The action S W is also invariant under global scaling with dimensions (1 , , , , ,
2) for (
A, φ, ¯ φ, η, ψ, χ ), respectively; andpreserves an additive U symmetry for the assignments (0 , , − , − , , − T r ([ D µ , D ν ] φ · ¯ χ µν ), so the Riemann tensor does not appear, and the theory could be extended to anyRiemannian manifold. In practice one can simply take Z d x → Z d x √ g , (2.35)in order to work in a curved spacetime. Such a theory has the property of being invariant under infinites-imal changes in the metric. This property characterizes the Witten model as a topological quantum fieldtheory. Such a feature is verified by the fact that the energy-momentum tensor can be written as theanti-commutator T µν = {Q , V µν } , (2.36)which means that T µν is an on-shell BRST-exact term, T µν = δV µν , δ | on-shell = 0 , (2.37)with V µν = 12 Tr { F µσ χ σν + F νσ χ σµ − g µν F σρ χ σρ } + 14 g µν Tr η [ φ, ¯ φ ]+ 12 Tr { ψ µ D ν ¯ φ + ψ ν D µ ¯ φ − g µν ψ σ D σ ¯ φ } . (2.38)Equation (2.37) together with δS W = 0 means that Witten theory satisfies (on-shell) the secondcondition displayed in eq. (2.4), that allows to say that S W automatically leads to a four-dimensionaltopological field model. In other words, δδg µν Z W = Z D Φ( − δδg µν S W )exp( −S W )= − g h{Q , Z M d x √ gV µν }i = 0 , (2.39)as all expected value of a δ -exact term vanish. It remains to know which kind of topological/differentialinvariants can be represented by the Feynman path integral of Witten’s TQFT. As we know, it willnaturally reproduce the Donaldson invariants for four-manifolds. An important feature of the twisted N = 2 SYM is the fact that the theory can be interpreted as quantumfluctuations around classical instanton configurations. To find the nontrivial classical minima one mustnote that the pure gauge field terms in S W are S gaugeW [ A ] = 12 Tr Z d x (cid:16) F µν + e F µν (cid:17) (cid:16) F µν + e F µν (cid:17) , (2.40)which is positive semidefinite, and only vanishes if the field strength F µν is anti-self-dual, F µν = − e F µν , (2.41)the same nontrivial vacuum configuration that minimizes the Yang-Mills action in the case of anti-instantons fields. Hence, Witten’s action has a nontrivial classical minima for F = − e F and Φ other fields = 0.Being precise, the evaluation of the twisted N = 2 SYM path integral computes quantum corrections toclassical anti-instantons solutions. 9nother important property of Witten theory is the invariance under infinitesimal changes in thecoupling constant. The variation of Z W with respect to g yields, for similar reasons to (2.39), δ g Z W = δ g (cid:18) − g (cid:19) h{Q , X }i = 0 , (2.42)where X = 14 Tr F µν χ µν + 12 Tr ψ µ D µ ¯ φ −
14 Tr η [ φ, ¯ φ ] . (2.43)The Witten partition function, Z W , is independent of the gauge coupling g , therefore we can evaluate Z W in the weak coupling limit, i.e. , in the regime of very small g , in which Z W is dominated by theclassical minima.The instanton moduli space M k,N is defined to be the space of all solutions to F = e F for an instantonwith a giving winding number k and gauge group SU ( N ). By perturbing F = e F nearby the solution A µ via a gauge transformation A µ → A µ + δA µ , we obtain the self-duality equation D µ δA ν + D µ δA ν + ε µναβ D α δA β = 0 . (2.44)The solutions of the above equation are called zero modes. Requiring the orthogonal gauge fixing condi-tion , D µ A µ = 0, one gets D µ ( δA µ ) = 0 . (2.45)The Atiyah-Singer index theorem [37, 38] counts the number of solutions to eq. (2.44) and eq. (2.45). InEuclidean spacetimes, for instance, the index theorem gives, see [39]dim( M ) = 4 kN , (2.46)where the modes due to global gauge transformations of the group were included. Looking at fermionzero modes, the χ equation for S W gives D µ ψ ν + D ν ψ µ + ε µναβ D α ψ β = 0 , (2.47)and from the η equation, D µ ψ µ = 0 . (2.48)These are the same equations obtained for the gauge perturbation around an instanton in the orthogonalgauge fixing, so the number of ψ zero modes is also given by M k,N . In order to get a non-vanishing par-tition function, Witten assumed that the moduli space consists of discrete, isolated instantons. Precisely,he assumed that the dimension of the moduli space vanishes . This condition is equivalent to the Landau gauge, as D µ A µ = ∂ µ A µ . It is important to note that one can promote ∂ µ to D µ in this case, in order to show that A µ and ψ µ obey the same equations. As Witten himself admits in his paper [8], “this relation between the fermion equations and the instanton moduli problemwas the motivation for introducing precisely this collection of fermions”. Otherwise, it occurs a net violation of the U (1) global symmetry of S W , and Z W vanishes due to the fermion zero modes,see [8, 40]. The dimension of the instanton moduli spaces depends on the bundle, E , and the manifold, M . In the SU (2)gauge theory, it can be written as dim( M ) = 8 k ( E ) −
32 ( χ ( M ) + σ ( M )) , (2.49)where k ( E ) is the first Pontryagin (or winding) number of the bundle E , and χ ( M ) and σ ( M ) are the Euler characteristicand signature of M [38]. (For M = R , χ ( R ) = σ ( R ) = 0.) Thus one can choose a suitable E and M in order to get avanishing dimension, dim( M ) = 0.
10n expanding around an isolated instanton, in the weak coupling limit g →
0, the action is reducedto quadratic terms, S (2) W = Z M d x √ g (cid:16) Φ ( b ) D B Φ ( b ) + i Ψ ( f ) D F Ψ ( f ) (cid:17) , (2.50)where Φ ( b ) ≡ { A, φ, ¯ φ } are the bosonic fields, and Ψ ( f ) ≡ { η, ψ, χ } , the fermionic ones. The Gaussianintegral over all fields gives Z W | g → = Pfaff( D F ) p det( D B ) , (2.51)where Pfaff( D F ) is the Pfaffian of D F , i.e. , the square root of the determinant of D F up to a sign. Thesupersymmetry relates the eigenvalues of the operators D B and D F . The relation is a standard result ininstanton calculus [41], which yields Z W | g → = ± Y i λ i p | λ i | , (2.52)with i running over all non-zero eigenvalues of D B ( D F ). Therefore, for the k th isolated instanton, Z ( k ) W = ( − n k , where n k = 0 or 1 according to the orientation convention of the moduli space (Donaldsonproved the orientability of the moduli space, i.e. , that the definition of the sign of Pfaff( D F ) is consistent,without global anomalies [4, 8]). In the end, summing over all isolated instantons, Z W | g → = X k ( − n k , (2.53)which is precisely one of topological invariant for four-manifolds described by Donaldson.The other metric independent observables are constructed in the context of eq. (2.7). These observablescan be generated by exploring the descent equations defined by the equivariant cohomology, i.e. , thesupersymmetry δ -cohomology. For that, being U i the global charge of the operator O i (see footnote 10on page 8), it must be understood that, for the observable Q i O i , dim( M ) = P i U i . The simplest δ -invariant operator, that does not depend explicitly on the metric, and cannot be written as δ ( X ) = {Q , X } (due to the scaling dimensions) is W ( x ) = 12 Tr φ ( x ) , U ( W ) = 4 . (2.54)Although W is not a δ -exact operator, taking the derivative of W with respect of the coordinates, wefind ∂∂x µ W = i {Q , Tr φψ µ } , (2.55)which is δ -exact. Using the exterior derivative, d , we can rewrite (2.55) as dW = i {Q , W } , (2.56)where W is the 1-form Tr( φψ µ ) dx µ . A straightforward calculation gives dW = i {Q , W } , dW = i {Q , W } , (2.57) dW = i {Q , W } , dW = 0 , (2.58) In order to construct topological invariants, the net U charge must equal the dimension of the moduli space, see [8, 17]. W = Tr( 12 ψ ∧ ψ + iφ ∧ F ) , (2.59) W = i Tr ψ ∧ F , (2.60) W = −
12 Tr F ∧ F , (2.61)where “ ∧ ” is the wedge product, the total charge is U = 4 − k for each W k , and φ, ψ , and F are zero,one, and two forms on M , respectively. F is the field strength in the p -form formalism , defined in eq.(2.74). Considering now the integral I ( γ ) = Z γ W k , (2.62)being γ a k-dimensional homology cycle on M, we have {Q , I } = Z γ {Q , W k } = i Z γ dW k − = 0 . (2.63)It proves that I ( γ ) is δ -invariant and, then, a possible observable. To be a global observable of thetopological theory, we just have to prove that I ( γ ) is BRST exact, which can be immediately verifiedtaking γ as the boundary ∂β , and applying the Stokes theorem, I ( γ ) = Z ∂β W k = Z β dW k = i {Q , Z β W k +1 } . (2.64)We conclude, from equations (2.63) and (2.64), that I ( γ ) are the global observables of the model as theirexpectation values produce metric independent quantities, i.e. , topological invariants for four-manifolds.Finally, the general path integral representation of Donaldson invariants in Witten’s TQFT takes theform Z ( γ , · · · , γ r ) = Z D Φ Y i Z γ i W k i ! e − S W = h Y i Z γ i W k i i , (2.65)with moduli space dimension dim( M ) = r X i (4 − k r ) . (2.66)One of the beautiful results is the appearing of W in the descent equations. Up to a sign, the observable Z γ W = − Z γ F ∧ F (2.67)is the Pontryagin action written in the formalism of p-forms. The Pontryagin action, a well-knowntopological invariant of four-manifolds, naturally appear as one of the Donaldson polynomials—with atrivial winding number in this case, since U ( W ) = 0, and consequently the dimension of the modulispace vanishes. Let us now turn to the main properties Baulieu-Singer approach for TQFTs [13], which is based onan off-shell BRST symmetry, built from the gauge fixing of an original action composed of topologicalinvariants. For the definitions and conventions concerning the p-form formalism used here, see Section 2.2.2. .2.1. BRST symmetry in topological gauge theories The four-dimensional spacetime is assumed to be Euclidean and flat . The non-Abelian topologicalaction S [ A ] in four-dimensional spacetime representing the topological invariants is the Pontryaginaction , S [ A ] = 12 Z d x F aµν e F aµν , (2.68)that labels topologically inequivalent field configurations, as S [ A ] = 32 π k , in which k is the topologicalcharge known as winding number. We must note that the Pontryagin action has two different gaugesymmetries to be fixed, these are:(i) the gauge field symmetry, δA aµ = D abµ ω b + α aµ ; (2.69)(ii) the topological parameter symmetry, δα aµ = D abµ λ b . (2.70)where D abµ ≡ δ ab ∂ µ − gf abc A cµ are the components of the covariant derivative in the adjoint representationof the Lie group G , f abc are the structure constants of G, while ω a , α aµ and λ a are the infinitesimal G -valued gauge parameters. As a consequence of (2.69), the field strength also transforms as a gaugefield , δF aµν = − gf abc ω b F cµν + D ab [ µ α bν ] . (2.71)The first parameter ( ω a ) reflects the usual Yang-Mills symmetry of S [ A ], whereas the second one ( α aµ )is the topological shift associated to the fact that S [ A ] is a topological invariant, i.e. , invariant undercontinuous deformations. The third gauge parameter ( λ a ) is due to an internal ambiguity present in thegauge transformation of the gauge field (2.69). The transformation of the gauge field is composed by twoindependent symmetries. If the space has a boundary, the parameter α aµ ( x ) must vanish at this boundarybut not ω a ( x ), what explains the internal ambiguity described by (2.70) in which α aµ ( x ) is absorbed into ω a ( x ), and not the other way around [13].Following the BRST quantization procedure, the gauge parameters present in the gauge transformations(2.69)-(2.71) are promoted to ghost fields: ω a → c a , α aµ → ψ aµ , and λ a → φ a ; c a is the well-known Faddeev-Popov (FP) ghost; ψ aµ is a topological fermionic ghost; and φ a is a bosonic ghost. The correspondingBRST transformations are sA aµ = − D abµ c b + ψ aµ ,sc a = g f abc c b c c + φ a ,sψ aµ = gf abc c b ψ cµ + D abµ φ b ,sφ a = gf abc c b φ c , (2.72) Throughout this work we consider flat Euclidean spacetime. Although the topological action is background independent,the gauge-fixing term entails the introduction of a background. Ultimately, background independence is recovered at thelevel of correlation functions due to BRST symmetry [13, 42, 43]. It is worth mentioning that the action S [ A ] could encompass a wide range of topological gauge theories. The Pontryaginaction is the most common case because it can be defined for all semi-simple Lie groups. Nevertheless, other cases canalso be considered. For instance, Gauss-Bonnet and Nieh-Yang topological gravities can be formulated for orthogonalgroups [44]. The antisymmetrization index notation here employed means that, for a generic tensor, S [ µν ] = S µν − S νµ . s = 0 , (2.73)by applying two times the BRST operator s on the fields. Naturally, S [ A ] is invariant under the BRSTtransformations (2.72). The nilpotency property of s defines the cohomology of the theory, which allowsfor the gauge fixing of the Pontryagin action in a BRST invariant fashion. Furthermore, such a propertyis related to the geometric structure of the off-shell BRST transformations in non-Abelian topologicalgauge theories. In order to simplify equations in the following sections, we will employ again the formalism of differentialforms. In this formalism, the fields c and φ are 0-forms, ψ is the 1-form ψ µ dx µ , and F , the following2-form: F = dA + A ∧ A = 12 F µν dx µ ∧ dx ν , (2.74)where A is the 1-form A µ dx µ , and “ ∧ ” is the wedge product which indicates that the tensor product iscompletely antisymmetric, and d is the exterior derivative . With this we can then write the BRSTtransformations in the form sA = Dc + ψ ,sc = 12 [ c, c ] + φ ,sψ = Dφ + [ c, ψ ] ,sφ = [ c, φ ] . (2.77)The geometric meaning of the topological BRST transformations of (2.77) is revealed from the definitionof the extended exterior derivative, e d , as the sum of the ordinary exterior derivative with the BRSToperator, e d = d + s , (2.78)and the generalized connection e A = A + c . (2.79)By direct inspection one sees that the BRST transformations can be written in terms of the generalizedcurvature F = F + ψ + φ , (2.80)such that F = e d e A + 12 [ e A, e A ] , (2.81) The exterior derivative operation in the space of smooth p -forms, Λ p , d : Λ p → Λ p +1 , on a generic p -form ω p , ω p = ω i ,i ,...,i p dx i ∧ dx i · · · ∧ dx i p , (2.75)is locally defined by dω p = ∂ω i ,i ,...,i p ∂x j dx j ∧ dx i ∧ dx i · · · ∧ dx i p , (2.76)being ω p a p -form, dω p is a ( p +1)-form. It follows that the exterior derivative is nilpotent, d = 0, due to the antisymmetricproperty of the indices. One assumes that s anticommutes with d , { s, d } = 0. The nature of φ as the “curvature” in the in instanton moduli space direction is implicit in the BRST transformation ofthe FP ghost, that can be rewritten in the geometric form sc + [ c, c ] = φ . e D F = e d F + [ e A, F ] = 0 . (2.82)Here, the space is graded as a sum of form degree and ghost number, in which the BRST operator isthe exterior differential operator in the moduli space direction A / G , where the gauge fields that differ bya gauge transformation are identified. The whole space is then M × A / G , being M a four-dimensionalmanifold. According to the gauges worked out in this paper, M will be an Euclidean flat space.In the definition (2.79) and following equations we are adding quantities with different form degreesand ghost numbers as thought they were of the same nature. Obviously this is not being done directly.We must see equations (2.81) and (2.82) as an expansion in form degrees and ghost numbers in which theelements with the same nature on both sides have to be compared. The relevant cohomology is defined bythe cohomology of M × A / G , e d = 0, being valid without requiring equations of motion. Such a geometricstructure reveals the BRST off-shell character of the BS approach . We will discuss in Section 2.2.5 howthe universal curvature F generates the same global observables of Witten theory, i.e. , the Donaldsonpolynomials. Let us recall the doublet theorem [16] which will be indispensable to fix the gauge ambiguities withoutchanging the physical content of the theory. Suppose a theory that contains a pair of fields or sourcesthat form a doublet, i.e. , ˆ δ X i = Y i , ˆ δ Y i = 0 , (2.83)where i is a generic index, and ˆ δ is a fermionic nilpotent operator. The field (source) X i is assumedto be fermionic. As the operator ˆ δ increases the ghost number in one unity by definition, if X i is ananti-commuting quantity, Y i is a commuting one. The doublet structure of ( X i , Y i ) in eq. (2.83) impliesthat such fields (or sources) belong to the trivial part of the cohomology of ˆ δ . The proof is as follows.First, we define the operators ˆ N = Z dx (cid:18) X i ∂∂ X i + Y i ∂∂ Y i (cid:19) , (2.84)ˆ A = Z dx X i ∂∂ Y i , (2.85)ˆ δ = Y i ∂∂ X i , (2.86)which obey the commutation relations { ˆ δ, ˆ A } = ˆ N , (2.87) h ˆ δ, ˆ N i = 0 , (2.88)where ˆ δ is a nilpotent operator as it is fermionic, ˆ δ = 0. The operator ˆ N is the counting operator. Being △ a polynomial in the fields, sources and parameters, the cohomology of the nilpotent operator ˆ δ , as weknow, is given by the the solutions of ˆ δ △ = 0 , (2.89) For a detailed study on the geometric interpretation of the universal fiber bundle and its curvature, we suggest forinstance [21, 45]. i.e. , that cannot be written in the form △ = ˆ δ Σ . (2.90)The general expression of △ is then △ = e △ + ˆ δ Σ , (2.91)where e △ belongs to the non-trivial part of the cohomology, in other words, it is closed, ˆ δ e △ = 0, but notexact, e △ 6 = ˆ δ e Σ. One can expand △ in eigenvectors of ˆ N , △ = X n ≥ △ n , (2.92)such that ˆ N △ n = n △ n , where n is the total number of X i and Y i in △ n . Such an expansion is consistentas each △ n is a polynomial in X i and Y i , and ˆ δ △ n = 0 for ∀ n ≥
1, according to (2.83) and the commutingproperties of X i and Y i . Finally, rewriting (2.92) as △ = △ + X n ≥ n ˆ N △ n , (2.93)and then, using the commuting relation (2.87), we get △ = △ + ˆ δ X n ≥ n ˆ A △ n , (2.94)which shows that all terms which contain at least one field (source) of the doublet never enter thenon-trivial part of the cohomology of ˆ δ , being thus non-physical—for a more complete analysis, see forinstance [16, 46].In order to fix the three gauge symmetries of the non-Abelian topological theory (2.69)-(2.71) weintroduce the following three BRST doublets: s ¯ c a = b a , sb a = 0 ,s ¯ χ aµν = B aµν , sB aµν = 0 ,s ¯ φ a = ¯ η a , s ¯ η a = 0 , (2.95)where ¯ χ aµν and B aµν are (anti-)self-dual fields according to the (negative) positive sign, see (2.98) below.The G -valued Lagrange multiplier fields b a , B aµν and ¯ η have respectively ghost numbers 0, 0, and − c a , ¯ χ aµν and ¯ φ a , ghost numbers − − −
2. (For completeness and furtheruse, the quantum numbers of all fields are displayed in Table 1.)Field
A ψ c φ ¯ c b ¯ φ ¯ η ¯ χ B Dim 1 1 0 0 2 2 2 2 2 2Ghost n o ∂ µ A aµ = − ρ b a , (2.96) D abµ ψ aµ = 0 , (2.97) F aµν ± e F aµν = − ρ B aµν , (2.98)16here ρ and ρ are real gauge parameters. In a few words, beyond the gauge fixing of the topologicalghost (2.97), we must interpret the requirement of two extra gauge fixings due to the fact that thegauge field possesses two independent gauge symmetries. In this sense, condition (2.96) fixes the usualYang-Mills symmetry δA aµ = D abµ ω b , and the second one, (2.98), the topological shift δA aµ = α aµ . The (anti-)self-dual condition for the field strength (in the limit ρ →
0) is convenient to identify the well-knownobservables of topological theories for four-manifolds (see [17]) given by the Donaldson invariants [2, 3],that are described in terms of the instantons.The partition functional of the topological action in BS gauges (2.96) takes the form Z BS = Z D c D ¯ c D ψ µ D ¯ χ µν D B µν D φ D ¯ φ D ηe − S BS , (2.99)whereby S BS = S [ A ] + S BSgf , (2.100)being S BSgf the gauge-fixing action which belongs to trivial part of the cohomology, given by S BSgf = s Tr Z d x (cid:20) ¯ χ µν (cid:18) F µν ± e F µν + 12 ρ B µν (cid:19) + ¯ φD µ ψ µ + ¯ c (cid:18) ∂ µ A µ − ρ b (cid:19)(cid:21) = Tr Z d x (cid:20) B µν (cid:18) F µν ± e F µν + 12 ρ B µν (cid:19) + ¯ χ µν (cid:18) D [ µ ψ ν ] ± ε µναβ D [ α ψ β ] (cid:19) − ¯ χ µν h c, F µν ± e F µν i + ηD µ ψ µ + ¯ φ [ ψ µ , ψ µ ] + ¯ φD µ D µ φ − b (cid:18) ∂ µ A µ − ρ b (cid:19) − ¯ c∂ µ D µ c − ¯ c∂ µ ψ µ ] . (2.101)A key observation is that, for ρ = ρ = 1, one can eliminate the topological term S [ A ], i.e. , thePontryagin action, by integrating out the field B µν , such thatTr { B µν (cid:16) F µν + e F µν (cid:17) + 12 B µν B µν } → Tr { F µν F µν + F µν e F µν } . (2.102)In this case we obtain a classical topological action which is equivalent to a Yang-Mills action plus ghostinteractions. Such an action, however, does not produce local observables as the cohomology of the theoryremain the same, as we will discuss in more detail later in Section 2.2.5.Another important property is that the Green functions of local observables in (2.99) do not dependon the choice of the background metric [13]. Let S gBS be an action with metric choice g µν , and S g + δgBS ,the same action up to the change of g µν into g µν + δg µν . As the only terms depending on the metricbelong to the trivial part of cohomology we conclude immediately that S gBS and S g + δgBS only differ by aBRST-exact term, S gBS − S g + δgBS = s Z d x △ ( − , (2.103)where △ ( − is a polynomial of the fields, with ghost number −
1. It means that the expectation valuesof local operators are the same if computed with a background metric g µν or g µν + δg µν , δδg µν h Y p O α p ( φ i ) i = 0 , (2.104)where O α p ( φ i ) are functional operators of the quantum fields φ i ( x ), see eq. (2.6). An anomaly in thetopological BRST symmetry would break the above equation. However there is no 4-form with ghostnumber 1, △ (1)4 − form , defined modulo s - and d - exact terms which obeys (cf. [13]) s △ (1)4 − form + d △ (2)3 − form = 0 . (2.105)17herefore, radiative corrections which could break the topological property (2.104) at the quantum levelare not expected. The formal proof of the absence of gauge anomalies to all orders in the topological BStheory is achieved by employing the isomorphism described in [22, 47]. The proof of the absence of gauge anomalies for the Slavnov-Taylor identity, S ( S ) = 0 , (2.106)consists in proving that the cohomology of S is empty. In the equation above, S is the classical actionfor a given gauge choice, and S = Z d x ( s Φ I ) δδ Φ I , (2.107)where Φ I represents all fields. As S is a Ward identity, in the absence of anomalies the symmetry (2.106)is also valid at the quantum level, i.e. , S (Γ) = 0, being Γ the quantum action in loop expansion. In eq.(2.107), s Φ I represents the BRST transformation of each field Φ I . The fields ¯ c , b , ¯ χ µν , B µν , ¯ φ and ¯ η transform as doublets, cf. eq. (2.95). Changing the variables according to the redefinitions ψ → ψ ′ = ψ − Dc ,φ → φ ′ = φ −
12 [ c, c ] , (2.108)the BRST transformations (2.77) are reduced to the doublet transformations sA = ψ ′ ,sψ ′ = 0 ,sc = φ ′ ,sφ ′ = 0 . (2.109)It configures a reduced transformation in which the non-linear part of the BRST transformations inthe Slavnov-Taylor identity were eliminated. The complete transformation in this space is given by thereduced operator S doublet = Z d x ( s Φ ′ I ) δδ Φ ′ I , (2.110)where Φ ′ = { A, ψ ′ , c, φ ′ , ¯ c, b, ¯ χ µν , B µν , ¯ φ, η } , which is composed of five doublets. It means that S doublet hasvanishing cohomology ( H ), H ( S doublet ) = ∅ , (2.111)in other words, that any polynomial of the fields Φ ′ , △ (Φ ′ ), that satisfies S doublet ( △ (Φ ′ )) = 0 , (2.112)belonging to the trivial part of the cohomology of S doublet (see the doublet theorem in previous section).The crucial point here is the fact that the cohomology of S in the space of local integrated functionalsin the fields and sources is isomorphic to a subspace of H ( S doublet ). Consequently S has also vanishingcohomology [22, 29, 47], H ( S ) = ∅ . (2.113)18he result (2.113) shows that there is no room for an anomaly in the Salvnov-Taylor identity (2.106).All counterterms at the quantum level will belong to the trivial part of cohomology, and the condition(2.105) for the existence of an anomaly capable of breaking the topological property (2.104) never occurs.Due to the algebraic structure of the theory, eq. (2.113) proves that all Ward identities are free of gaugeanomalies, cf. [22]. As a consequence of this result, the background metric independence is valid to allorders in perturbation theory.The second point, and not least, is the conclusion that the BS theory has no local observables. Due to itsvanishing cohomology (2.113), all BRST-invariant quantities must belong to the non-physical (or trivial)part of the cohomology of s , and the only possible observables are the global ones, i.e. , topologicalinvariants for four-manifolds. Such observables are characterized by the cohomology of s [29, 48], inwhich the observables are globally defined in agreement with the supersymmetric formulation of J. H.Horne [49]. A simple way to identify theses observables is accomplished by studying the cohomologyof the extended space M × A / G , where the metric independent observables, known as Chern classes ,are constructed in terms of the universal curvature F (2.80). The Donaldson polynomials are naturallyrecovered, characterized by the so-called equivariant cohomology , that relates the BS approach to Wittentheory at the level of observables. Witten’s topological theory is constructed without fixing its remaining ordinary Yang-Mills gauge sym-metry. The theory is developed in the instanton moduli space A / G . A generic observable of his theory, O ( W ) α i , is naturally gauge invariant under Yang-Mills gauge transformations, s Y M O ( W ) α i = 0 , (2.114)where s Y M is the nilpotent BRST operator related to the ordinary Yang-Mills symmetry, i.e. , withoutincluding the topological shift, namely, s Y M A µ = D µ c ,s Y M Φ adj = [ c, Φ adj ] , (2.115)where Φ adj is a generic field in adjoint representation. We conclude that we can add an ordinary Yang-Mills gauge transformation (in the A / G direction) to Witten fermionic symmetry based on the “topolog-ical shift” δA µ ∼ ψ µ , δ → δ eq = δ + s Y M , (2.116)in such a way that the descent equations for δ ∼ {Q , · } will remain the same, see (2.34) and (2.57)-(2.61). The operator δ eq is nilpotent when acting on gauge-invariant quantities under YM transformations,defining thus a cohomology in a space where the fields that differ by a Yang-Mills gauge transformationsare identified, known as equivariant cohomology . Such a property indicates that there is a link betweenWitten theory and BS approach in which the BRST operator, s , is naturally defined taking into accountthe topological shift and the ordinary Yang-Mills transformation in a single formalism.To prove the link between both approaches, we must remember that the universal curvature in thespace M × A / G is given by the sum F = F + ψ + φ . The difference between the on-shell BRST operator, s , and the Witten fermionic symmetry, δ , for X = ( F, ψ, φ ) is of the form s X = δ X + [ X , c ] , (2.117)19n other words, in the space of the fields ( F, ψ, φ ), s and δ differ by an ordinary Yang-Mills transformation,as ( F, ψ, φ ) transform in the adjoint representation of the gauge group. These fields are the only ones weneed to obtain the Donaldson polynomials as the observables of the BS theory, since in the space M ×A / G they are constructed in terms of F . This allows for identifying the equivariant operator with the BRSTone, δ eq ≡ s , according to the construction of the observables in Witten and BS theory, respectively.To understand the above statement, we must invoke the n ’th Chern class, f W n , defined in terms of theuniversal curvature by f W n = Tr ( F ∧ F ∧ · · · ∧ F | {z } n times ) (2.118)where n = { , , , · · · } is the number of wedge products. f W n represents the most general observablesof BS theory . Weyl theorem [21] ensures that f W n is closed with respect to the extended differentialoperator e d = d + s [13, 50], i.e. , e d f W n = 0 . (2.120)If we choose the first Chern class f W = Tr ( F ∧ F ) , (2.121)the expansion in ghost numbers of equation (2.120) yields s Tr ( F ∧ F ) = d Tr ( − ψ ∧ F ) , (2.122) s Tr ( ψ ∧ F ) = d Tr ( − ψ ∧ ψ − φF ) , (2.123) s Tr ( ψ ∧ ψ + 2 φF ) = d Tr (2 ψφ ) , (2.124) s Tr ( ψφ ) = d Tr ( − φφ ) , (2.125) s Tr ( φφ ) = 0 , (2.126)which are the same descent equations obtained in (2.57)-(2.61) following Witten analysis, only replacing δ (or δ eq ) by s , proving that Baulieu-Singer and Witten topological theories possess the same observablesgiven by the Donaldson invariants (2.65).It should not seem surprising the fact that the observables in the BS approach are naturally invariantunder ordinary Yang-Mills symmetry, as the n ’th Chern class is Yang-Mills invariant itself (2.118) since F transforms in the adjoint representation of the gauge group. Equation (2.120) provides a powerful toolto obtain Donaldson polynomials for any ghost number. One must note that we do not have to worryabout with the independence of Faddeev-Popov ghosts to construct the observables in the BS approach.Although the gauge-fixed BS action has FP ghosts due to the gauge fixing of the Yang-Mills ambiguity,the ( c, ¯ c ) independence of f W n is a direct consequence of the fact that the universal curvature of the space M × A / G does not depend on FP ghosts, but only on F , and the ghosts ψ and φ .In the weak coupling limit of the twisted N = 2, the observables of both theories are undoubtedlythe same: the topological Donaldson invariants [21–23]. We might ask if the quantum behavior are also It is not possible to construct topological observables using the Hodge product, as it is metric dependent. For this reasonwe never obtain Yang-Mills terms of the type { Tr( F µν F µν ) , Tr( F µν F νσ F µ σ ) , · · · } , without Levi-Civita tensors in theinternal product, in the place of metric tensors. Moreover, the Wilson loop W ( C ) P = Tr {P e i H C A µ dx µ } , (2.119)is not an observable in the non-Abelian topological BS case, as it is not gauge-invariant due to the topological shiftsymmetry. In any case, it does not make sense to discuss confinement in the BS theory, as it is not confining to any energyscale. Thence, the only possibilities for topological invariants are the wedge products in e W n . S BS − S W = Σ G = s ( · · · ) . (2.127)The relation above does not depend on the gauge choice. Consequently, we cannot say, in principle, thatBS and Witten partition functions are equivalent at quantum level, since Z BS = Z D Φ e − S BS = Z D Φ e − S W − Σ G , (2.128)wherein Σ G is not s -exact. At a first view, Z BS = Z W = R D Φ e − S W . The fact that Σ G = s ( · · · ) opensthe possibility for both theories to have different quantum properties. The one-loop exactness of twisted N = 2 SYM β -function for instance is a well-known result in literature [34]. We will now analyze theWard identities of the BS theory in self-dual Landau gauges, in order to compare the quantum propertiesof the DW and BS theories.
3. Quantum properties of BS theory in the self-dual Landau gauges
In this section we will summarize the quantum properties of BS theory in the self-dual Landau (SDL)gauges . Extra details can be found in [24–27, 51]. Working in the self-dual Landau gauges amounts to considering the constraints [27] ∂ µ A aµ = 0 , (3.1) ∂ µ ψ aµ = 0 , (3.2) F aµν ± e F aµν = 0 . (3.3)Through the introduction of the three BRST doublets described in eq. (2.95), the complete gauge-fixedtopological action in SDL gauges takes the form S [Φ] = S [ A ] + S gf [Φ] , (3.4)with S [ A ] standing for the Pontryagin action, and S gf [Φ] = s Z d z (cid:20) ¯ c a ∂ µ A aµ + 12 ¯ χ aµν (cid:16) F aµν ± e F aµν (cid:17) + ¯ φ a ∂ µ ψ aµ (cid:21) = Z d z (cid:20) b a ∂ µ A aµ + 12 B aµν (cid:16) F aµν ± e F aµν (cid:17) + (¯ η a − ¯ c a ) ∂ µ ψ aµ + ¯ c a ∂ µ D abµ c b + − gf abc ¯ χ aµν c b (cid:16) F cµν ± e F cµν (cid:17) − ¯ χ aµν (cid:18) δ µα δ νβ ± ǫ µναβ (cid:19) D abα ψ bβ + ¯ φ a ∂ µ D abµ φ b ++ gf abc ¯ φ a ∂ µ (cid:16) c b ψ cµ (cid:17)i . (3.5) For simplicity, throughout the text we will refer to the Baulieu-Singer theory in the self-dual Landau gauges as self-dualBS theory. T (Equation (A.13)),we have to introduce external sources given by the following three BRST doublets [27] sτ aµ = Ω aµ , s Ω aµ = 0 ,sE a = L a , sL a = 0 ,s Λ aµν = K aµν , sK aµν = 0 . (3.6)The respective external action, written as a BRST-exact contribution preserving the the physical contentof theory, takes the form S ext = s Z d z (cid:18) τ aµ D abµ c b + g f abc E a c b c c + gf abc Λ aµν c b ¯ χ cµν (cid:19) = Z d z (cid:20) Ω aµ D abµ c b + g f abc L a c b c c + gf abc K aµν c b ¯ χ cµν + τ aµ (cid:16) D abµ φ b + gf abc c b ψ cµ (cid:17) + gf abc E a c b φ c + gf abc Λ aµν c b B cµν − gf abc Λ aµν φ b ¯ χ cµν − g f abc f bde Λ aµν ¯ χ cµν c d c e , (3.7)with the corresponding quantum number of the external sources displayed in Table 2 below. Therefore,the full classical action to be quantized isΣ[Φ] = S [ A ] + S gf [Φ] + S ext [Φ] . (3.8)The introduction of the external action does not break the original symmetries, and the physical limit isobtained by setting the external sources to zero [16].Source τ Ω E L Λ K Dim 3 3 4 4 2 2Ghost n o -2 -1 -3 -2 -1 0Table 2: Quantum numbers of the external sources.One of the symmetries are of particular interest to us: the vector supersymmetry described by eq.(A.12), cf. [24,27]. By applying BRST-algebraic renormalization techniques [16], and disregarding Gribovambiguities, it was proved in [24], with the help of Feynman diagrams, that all two-point functions aretree-level exact, as a consequence of the Ward identities of the model. In particular, as a consequenceof the vector supersymmetry (A.11), the gauge field propagator vanishes to all orders in perturbationtheory, h A aµ ( p ) A bν ( q ) i = 0 . (3.9)In [26] this result was generalized: not only the two-point functions of the self-dual BS theory are tree-level exact, but all n -point Green functions of the model do not receive any radiative corrections. Thisis a direct consequence of the null gauge propagator (3.9) together with the vertex structure of the fullaction (3.8). Following the Feynman rules notation of [26], we represent the relevant propagators by h AA i = , h c ¯ c i = , h ¯ χψ i = , h AB i = , h φ ¯ φ i = . (3.10)22he relevant vertexes are represented by: B A A , ¯ χ A c , A ψ ¯ χ , A c ¯ c ,A ¯ φ φ , ¯ φ ψ c , A ¯ χA c . (3.11)Using these diagrams, one identifies a kind of cascade effect in which the number of internal A -legs alwaysincreases when trying to construct loop Feynman diagrams, according to the diagram below,... · · ·· · · ... . (3.12)This makes it impossible to close loops without using the h AA i propagator , which vanishes by means of(3.9). Note that, internally, the A -leg always propagates to the vertex BAA . Looking at the full action(3.8), the only vertex that does possess A -legs is ¯ φcψ , but the ¯ φ -leg could only propagate to the vertex¯ φAφ through h ¯ φφ i ; the c -leg only to ¯ cAc through h ¯ cc i and; the ψ -leg to the vertexes ¯ χAψ , ¯ χcA or ¯ χcAA through h ψ ¯ χ i ( h ¯ ηψ i is not considered because there is no vertex containing ¯ η ). All possible branchesproduce at least one remaining internal A -leg, and the cascade effect is not avoided, as represented bythe diagrams , , . (3.13)The apparently only non-zero correlation functions are of the type h BBB . . . bb i = h s ¯ χBB . . . bb i = h s ( ¯ χBB . . . bb ) i , (3.14) The formal proof of this result can be found in [26]. .e. , with external B aµν or b a fields. But (3.14) automatically vanishes as it is BRST-exact.In a few words, using perturbative techniques, one sees that the tree-level exactness of the BS in theself-dual gauges is a consequence of the vector supersymmetry and BRST symmetry. Once we have at our disposal all Ward identities, we are able to construct the most general countertermΣ c that can absorb the divergences arising in the evaluation of Feynman graphs. Due to the triviality ofthe BRST cohomology [24, 27], Σ c belongs to trivial part of the BRST cohomology. The fact that theBS theory is quantum stable is a well-known result in literature [24, 27, 51]. In [24], it was introducedan extra non-linear bosonic symmetry that relates the topological ghost with the Faddeev-Popov one(among other transformations involving other fields) through the transformation δψ aµ D abµ c b , (3.15)described by the Ward identity T in Equation (A.13). Taking into account this extra symmetry, fromthe multiplicative redefinition of the fields, sources and parameters of the model,Φ = Z / Φ , Φ = { A aµ , ψ aµ , c a , ¯ c a , φ a , ¯ φ a , b a , ¯ η a , ¯ χ aµν , B aµν } , J = Z J J , J = { τ aµ , Ω aµ , E a , L a , Λ aµν , K aµν } ,g = Z g g , (3.16)one proves the quantum stability of the BS theory in self-dual gauges with only one independent renor-malization parameter, i.e. , that the quantum action Γ ≡ Σ(Φ , J , g ) at one-loop is of the formΣ(Φ , J , g ) = Σ(Φ , J , g ) + ǫ Σ c (Φ , J , g ) , (3.17)with Σ c = a Z d x (cid:16) B aµν F aµν − χ aµν D abµ ψ bν − gf abc ¯ χ aµν c b F cµν (cid:17) , (3.18)whereby the resulting Z factors obey the following system of equations: Z / A = Z − / b = Z − g ,Z / c = Z / η = Z − / ψ = Z Ω = Z − / c ,Z / φ = Z − / φ = Z τ = Z L = Z − g Z − c ,Z E = Z − g Z − / c ,Z K = Z − g Z − / c Z − / χ ,Z Λ = Z − g Z − c Z − / χ ,Z / B Z / A = Z / χ Z / c = 1 + ǫa , (3.19)with the independent renormalization parameter denoted by a . Due to the recursive nature of algebraicrenormalization [16], the results (3.19) show that the model is renormalizable to all orders in perturbationtheory.From the algebraic analysis so far, we cannot prove that Z g = 1, as suggested by the tree-level exactnessobtained via the study of the Feynman diagrams. The system of Z factors (3.19) is undetermined. As24e can easily see, the number of equations n and the number of variables z (the Z factors) are relatedby z = n + 2, indicating that there is a kind of freedom in the choice of two of the Z factors.In [25], the origin of such an ambiguity was explained: it is due to the absence of a kinetic gauge fieldterm out from the trivial BRST cohomology, and due to the absence of discrete symmetries involving theghost fields. The symmetries of the SDL gauges eliminate the kinetic term of the Faddeev-Popov ghostin the counterterm, i.e. , Z c Z ¯ c = 1 . (3.20)Moreover, from the gauge-ghost vertex (¯ cAc ), which is also absent in the counterterm, we achieve Z g Z / A = 1 . (3.21)The two relations (3.20) and (3.21) are decoupled, in other words, only by determining Z c or Z ¯ c wedo not get any information about Z g or Z A . As there are no kinetic terms for the gauge field in theclassical action (3.8), the independent determination of Z A becomes impossible. The same analysis canbe performed for the bosonic and topological ghosts, see [25].Extra information is then required in order to determine the system (3.19). In the ordinary Yang-Millstheories (quantized in the Landau gauge), Z c = Z ¯ c which relies on the discrete symmetry c a −→ ¯ c a , ¯ c a −→ − c a . (3.22)This condition, together with the Faddeev-Popov ghost kinetic term, are sufficient to determine Z c and Z ¯ c . It is easy to see that the action (3.8) does not obey such a symmetry. Discrete symmetries betweenthe other ghosts of topological Yang-Mills theories ( φ a and ¯ φ a and; ψ aµ and ¯ χ aµν ) are also not present in(3.8), which explains the second ambiguity. In Witten’s theory, such an ambiguity will not appear bythis reasoning since Witten’s action contains discrete symmetries ensured by the time-reversal symmetry(3.22) in Landau gauge, together with φ → ¯ φ , ¯ φ → φ ,ψ µ → χ µ , χ µ → ψ µ , (3.23)whereby the components of χ µ are defined as follows χ ≡ η , χ i ≡ χ i = 12 ε ijk χ jk , (3.24)implying a “particle-antiparticle” relationship between ¯ c and c , ¯ φ and φ , and ψ µ and χ µ , as demonstratedin [52].This ambiguity is also present in a generalized class of renormalizable gauges [25]. In fact, one couldrelate this ambiguity with the fact that all local degrees of freedom are non-physical ( e.g. the gauge fieldpropagator is totally gauge dependent). In self-dual Landau gauges, where the vector supersymmetryis present, the Feynman diagram structure indicates that imposing Z c = Z ¯ c and Z φ = Z ¯ φ is consistentwith the model. Hence the Z-factor system (3.19) would naturally yield Z g = 1, in accordance with theabsence of radiative corrections in this gauge choice. However, without recovering the discrete symmetriesbetween the ghosts, such an imposition seems to be artificial. As we will see later, the renormalizationambiguity can be solved in the (A)SDL gauges, i.e. , the discrete symmetries can be reconstructed, due tothe triviality of the Gribov copies [28], which allows for a non-local transformation with trivial Jacobian,capable of recovering such symmetries. 25 . Perturbative β -functions Our aim in this section is to compare the DW and BS β -functions to prove that these topological gaugetheories are not completely equivalent at the quantum level, and then identify in which energy regimesthe correspondence could occur. The DW β -function is well known [34, 52], as we will briefly describe. Itremains the task of determining the self-dual BS one to perform the comparison. N = 2 super-Yang-Mills theory In [52] the authors have computed the one-loop β -function of the DW theory. Later, the authors of [34]employed the algebraic renormalization techniques to also study DW theory, and prove that the β -functionof twisted N = 2 SYM ( β N =2 g ) is one-loop exact. The reason is that the composite operator Tr φ ( x )does not renormalize [53]. For that, they considered the fact that the operator d µν , defined in expression(2.15), is redundant [54]. Thence, the definition of an extended BRST operator, namely, S = s Y M + ωδ + ε µ δ µ , (4.1)could be employed. In expression (4.1), ω and ε µ are global ghosts, and δ and δ µ were defined in equations(2.13) and (2.14). The relevant property of the operator S is that it is on-shell nilpotent in the space ofintegrated local functionals, since S = ωε µ ∂ µ + eqs of motion . (4.2)We point out that this extended BRST construction requires the equations of motion to obtain a nilpotentBRST operator—a standard behavior of Witten theory, representing a different quantization scheme ofthe BS theory. Considering the non-renormalization of Tr φ and the on-shell cohomology of the operatordefined in eq. (4.2), the result is that the β -function only receives contributions to one-loop order, andis given by β N =2 g = − Kg , (4.3)with K being a constant. The computation of β N =2 g via Feynman diagrams was performed in [52] byevaluating the one-loop contributions to the gauge field propagator (where the Landau gauge was used tofix the Yang-Mills symmetry of Witten action (2.30)). The behavior of one-loop exactness of the N = 2 β -function had been usually understood in terms of the analogous Adler-Bardeen theorem for the U (1)axial current in the N = 2 SYM [9].Despite the independence of the Witten partition function under changes in the coupling constant, theresult (4.3) should not be surprising. In the twisted version, we can see that the trace of the energy-momentum is not zero, but given by (see [8]) g µν T µν = Tr { D µ φD µ ¯ φ − iD µ ηψ µ + 2 i ¯ φ [ ψ µ , ψ µ ] + 2 iφ [ η, η ] + 12 [ φ, ¯ φ ] ] } , (4.4)meaning that S W is not conformally invariant under the transformation δg µν = h ( x ) g µν , (4.5)for an arbitrary real function h ( x ) on M . Nonetheless, the trace of the energy-momentum tensor can bewritten as a total covariant divergence, g µν T µν = D µ h Tr( ¯ φD µ φ − iηψ µ ) i , (4.6)26hich means that S W is invariant under a global rescaling of the metric, δg µν = wg µν , with w constant [8].The N = 2 β -function only vanishes if we take the weak coupling limit g → β N =2 g ( g →
0) = 0 . (4.7)In this limit, the possibility of loop corrections to the effective action is eliminated, and the Donaldsonpolynomials are reproduced as the observables of the theory. There is no Ward identity, or a particularproperty of the vertices and propagators of S W , capable of eliminating these quantum corrections for anarbitrary energy regime—this situation is distinct from the BS theory in the self-dual Landau gauges. As suggested by the tree-level exactness of the BS theory in the self-dual Landau gauges, according to theanalysis of the Feynman diagrams performed in Section 3, we will formally prove that the self-dual BStheory is conformal. Before proving the vanishing of the BS β -function in this gauge, we will first discussthe non-physical character of the coupling constant in this off-shell approach, since g is introduced in theBS theory as a gauge parameter, in the trivial part of the BRST cohomology. β -function in the off-shell approach In [52], Brooks et al. argued that only one counterterm is required in the on-shell Witten theory, specif-ically for the YM term Tr F µν . In any case, the Donaldson invariants are described by DW theory inthe weak coupling limit g →
0, where the theory is dominated by the classical minima. On the otherhand, it is evident that the BS theory is distinct from Brooks et al. construction because their methodsare based on different BRST quantization schemes, with different cohomological properties. We do notexpect a similar result in the BS theory. According to the cohomology of the BS model, if the β BSg is notzero, we should find that it is Tr ( F µν ± e F µν ) rather than Tr F µν which is renormalized . In this way,the minima of the effective action preserves the instanton configuration at the quantum level, accordingto the global degrees of freedom of the instantons, which defines the observables of the BS theory—theDonaldson invariants.A possible discrepancy between β -functions for the BS approach in different gauge choices cannot beattributed to a gauge anomaly, since it is forbidden in these models due to the trivial BRST cohomology[55], cf. equation (2.105). For instance, if we would had chosen the gauge D abµ ψ bµ = 0 for the topologicalghost, with the covariant derivative instead of the ordinary one, the vector supersymmetry would bebroken, and the gauge propagator would not vanish to all orders anymore. In ordinary Yang-Millstheories, the β -function is an on-shell gauge-invariant physical quantity. Nonetheless, in gauge-fixedBRST topological theories of BS type, the coupling constant is non-physical , introduced in the trivialpart of the cohomology, together with the gauge-fixing action. In these terms, it is not contradictory thatthe β -function is gauge dependent as it computes the running of a non-physical parameter. We mustobserve that the physical observables of the theory, the Donaldson invariants, naturally do not dependon the gauge coupling. So that there is no inconsistency that the observables of this kind of theory,described by topological invariants, i.e. , exact numbers, do not depend on the coupling constant, andconsequently on its running, being g an unobservable quantity.As DW and BS theories possess the same observables, we should then consider the instanton configura-tion not as a gauge fixing condition, but as a physical requirement in order to obtain the correct degrees See [55], where Birmingham et al. had employed the Batalin-Vilkovisky algorithm [56]—a similar quantization to BSapproach, i.e. , with similar cohomological properties.
27f freedom that correspond to the description of all global observables. Furthermore, the Atiyah-Singerindex theorem [37] determines the dimension of the instanton moduli space, in which the Donaldson in-variants are defined—see [57,58] for some exact instanton solutions, whose properties cannot be attributedto gauge artifacts.
To prove that the algebraic renormalization is in harmony with the Feynman diagram analysis in theself-dual Landau gauges, which shows that the BS model does not receive radiative corrections in thisgauge, we must invoke a result recently published in [28]. In this work, it was demonstrated that theGribov ambiguities [59, 60] are inoffensive in the self-dual BS theory . The quantization of this modelin a local section of the field space where the eigenvalues of the Faddeev-Popov determinant are positive,is equivalent to its quantization in the whole field space. In other words, the introduction of the Gribovhorizon does not affect the dynamics of the BS theory in SDL gauges, as its correspondent gap equationforbids the introduction of a Gribov massive parameter in the gauge field propagator. This result alsosuggests that the fiber bundle structure of the BS theory is trivial [62].Let us quickly recall the Gribov procedure in the quantization of non-Abelian gauge theories [59, 60].It essentially consists in eliminating remaining gauge ambiguities usually present in non-Abelian gaugetheories, known as Gribov copies, which are not eliminated in the Faddeev-Popov (FP) procedure [63,64].In Yang-Mills theories, the FP gauge-fixing procedure results in the well-known functional generator Z Y M = N Z D A | det[ − ∂ µ D abµ ] | δ ( ∂ µ A µ ) e − S Y M = N Z D A D ¯ c D ce − ( S Y M + S gf ) , (4.8)whereby S gf is the well-known gauge-fixing action given by S gf = Z d x (cid:18) ¯ c a ∂ µ D abµ c b − α ( ∂ µ A aµ ) (cid:19) . (4.9)In (4.9), the limit α −→ ∂ µ A µ = 0 . (4.10)Consider a gauge orbit A Uµ = U A µ U † − ig ( ∂ µ U ) U † , (4.11)with U = e − igT a θ a ( x ) (cid:12)(cid:12)(cid:12) U ∈ SU ( N ) with θ a ( x ) being the local gauge parameters of the non-Abeliansymmetry, and T a the generators of the gauge group. The FP hypothesis [63, 64] is that there is onlyone gauge configuration in the orbit (4.11) obeying the Landau gauge condition (4.10). In his seminalwork [59], V. N. Gribov demonstrated that this hypothesis fail at the YM low energy regime because onecan always find two configurations ˜ A and A obeying the Landau gauge condition and yet being relatedby a gauge transformation. At infinitesimal level, the condition for a configuration A to have a Gribovcopy ˜ A is that the FP operator develops zero-modes through − ∂ µ D µ θ = 0 . (4.12) The result was proved to be valid to all orders in perturbation theory by making use of the Zwanziger’s approach [61] tothe Gribov problem [59]. The gauge orbit is the equivalence class of gauge field configurations that only differ by a gauge transformation, representingthus the same physics according to the gauge the gauge principle. θ a taken as an infinitesimal parameter, U ≈ − θ a T a . Equation (4.12) is recognized as the Gribovcopies equation in the Landau gauge (and also in linear covariant gauges–See [65–68]). Equation (4.12)can be seen as an eigenvalue equation for the FP operator where θ is the zero mode of the operator. InLandau gauge, this operator is Hermitian, and thus, its eigenvalues are real. For values of A µ sufficientlysmall, the eigenvalues of the FP operator will be positive, as − ∂ only has positive eigenvalues . As A µ increases, it will attain a first zero mode (4.12). Such region in which the FP operator has its firstvanishing eigenvalue is called Gribov horizon (See also [60]). Gribov’s proposal was to restrict the pathintegral domain to the region Ω (the Gribov region) defined byΩ = { A aµ ; ∂ µ A µ = 0 , − ∂D > } . (4.13)Such restriction ensures the elimination of all infinitesimal copies and also guarantees that no independentfield is eliminated [69].The implementation of the restriction to the Gribov region Ω is accomplished by the introduction of astep-function Θ( − ∂D ) in the Feynman path integral, that leads to the well-known no-pole condition forthe FP ghost propagator h ( ∂D ) − i , which explodes at when a zero mode is attained. The main resultof introducing the restriction of the Feynman path integral domain to the Gribov region is a modifiedgluon propagator, due to the emergence of a massive parameter for the gauge field, the so called Gribovparameter γ . In the presence of the Gribov horizon, the gluon propagator takes the form h A aµ ( k ) A bν ( k ) i = δ ab δ ( p + k ) k k + γ P µν ( k ) , (4.14)where P µν ( k ) = δ µν − k µ k ν /k , and γ fixed by the gap equation [61, 70], ∂ Γ ∂γ = 0 . (4.15)According to Zwanziger’s generalization [61], the gap equation above is valid to all orders in pertur-bation theory—see [71, 72], where the all-order proof of the equivalence between Gribov and Zwanzigermethods was worked out. An important feature of the Gribov parameter is that it only affects the infrareddynamics. The matching between Gribov-Zwanziger theory and recent lattice data is achieved throughthe introduction of two-dimensional condensates, see [73]. The introduction of the Gribov horizon in theaction explicitly breaks the BRST symmetry. This is usually an unwanted result, as the BRST symmetryis necessary to prove the unitarity, to ensure the renormalizability to all orders, and to define the physicalgauge-invariant observables of the theory [74–76]. This breaking however brought to light the physicalmeaning of the infrared γ parameter, and its intrinsic non-perturbative character. One can prove that theBRST breaking is proportional to γ , in other words, the BRST symmetry is restored in the perturbativeregime. One says that the BRST symmetry is only broken in a soft way, cf. [76–79]. Only more recently,a universal, gauge independent, (non-perturbative) BRST invariant way to introduce the Gribov horizonwas developed [67, 68, 80–82].In the self-dual topological BS theory, it was proved in [28] that all topological gauge copies associatedto the gauge ambiguities (2.69) and (2.70), are eliminated through the introduction of the usual Gribovrestriction, in which the path integral domain is restricted to the region Ω—see eq. (4.13). Moreover, dueto the triviality of the gap equation, it was verified that the Gribov copies does not affect the infrared In Abelian theories, such as QED, − ∂ is the “FP operator”, and the copy equation only possesses trivial solutions in thethermodynamic limit, meaning that the Gribov copies are inoffensive in this case. γ BS = 0 is the only possible solution of the gap equation .Thus, no mass parameter seems to emerge in the BS theory, preserving its conformal character at quantumlevel. Specifically, the tree-level exactness in SDL gauges is preserved. Such a behavior can be inferredfrom the absence of radiative corrections which ensures the semi-positivity of all two-point functions. TheFP ghost propagator, for instance, reads h ¯ c a ( k ) c b ( k ) i = δ ab k , (4.16)which is valid to all orders, demonstrating that the FP operator will remain positive-definite at quantumlevel, consistent with the inverse of the FP propagator being positive, thus proving that we are insidethe Gribov region. Moreover, the gauge two-point function remains trivial, i.e. , h A aµ ( k ) A bν ( k ) i = 0 to allorders.Exploring the positive-definiteness of the FP ghost propagator, we are able to safely perform thefollowing shifts: ¯ η a ¯ η a + ¯ c a ,φ b φ b − gf cde ( ∂ ν D bcν ) − ∂ µ (cid:16) c d ψ eµ (cid:17) , ¯ c a ¯ c a − gf cde ¯ χ dµν ( F ± ) eµν ( ∂ ν D caν ) − . (4.17)It is worth noting that these shifts generate a trivial Jacobian. Calling ρ = α and ρ = β , andimplementing the BS gauge constraints (2.96) and (2.98), together with ∂ µ ψ µ = 0, the final gauge-fixingaction, integrating out the auxiliary fields b and B in the action (2.101), is S gf ( α, β ) = Z d x (cid:20) − α ( ∂A ) − β F ± (cid:21) − Z d x h (¯ η a − ¯ c a ) ∂ µ ψ aµ + ¯ c a ∂ µ D abµ c b − gf abc ¯ χ aµν c b (cid:16) F cµν ± e F cµν (cid:17) − ¯ χ aµν (cid:18) δ µα δ νβ ± ǫ µναβ (cid:19) D abα ψ bβ + ¯ φ a ∂ µ D abµ φ b + gf abc ¯ φ a ∂ µ (cid:16) c b ψ cµ (cid:17)i , (4.18)where F ± = F ± ˜ F and D ± ≡ ( δ µα δ νβ − δ να δ µβ ± ǫ µναβ ) D abα . The self dual Landau gauges is recoveredby setting α, β →
0. Then, applying the shifts (4.17) on the action S gf ( α, β ), one gets S shiftedgf ( α, β ) = Z d x (cid:20) − α ( ∂A ) − β F ± (cid:21) − Z d x h ¯ η a ∂ µ ψ aµ + ¯ c a ∂ µ D abµ c b − ¯ χ aµν (cid:18) δ µα δ νβ ± ǫ µναβ (cid:19) D abα ψ bβ + ¯ φ a ∂ µ D abµ φ b (cid:21) . (4.19)As the Jacobian of the shifts that performs S gf ( α, β ) → S shiftedgf ( α, β ) is trivial, the quantization of bothactions are perturbatively equivalent, cf. [34]. Such a Jacobian only generates a number that can beabsorbed by the normalization factor. This shows that the discrete symmetries (3.22) and (3.23) presentin the Witten theory can be recovered, which naturally impose the relations Z c = Z ¯ c and Z φ = Z ¯ φ (4.20) A similar situation occurs in the N = 4 SYM which possesses a vanishing β -function, indicating the conformal structureof the self-dual BS. The absence of an invariant scale makes it impossible to attach a dynamical meaning to the Gribovparameter [83].
30o be valid in the BS theory. Hence, combining (4.20) with the Z -factor system (3.19), one obtains Z g = 1 , (4.21)which proves that the algebraic analysis is in harmony with the result obtained via the study of theFeynman diagrams in the presence of the vector symmetry, i.e. , that the topological BS theory (followingthe self-dual Landau gauges) is conformal, in accordance with the absence of radiative corrections.
5. Characterization of the DW/BS correspondence
We will characterize in this section the quantum correspondence between the twisted N = 2 SYM in theultraviolet regime and the conformal Baulieu-Singer theory in the SDL gauges. The result obtained in (4.21) in the SDL gauges proves that the self-dual BS β -function vanishes. Thisresult is completely different from the twisted N = 2 SYM which receives one-loop corrections, andpossesses a non-vanishing β -function given by (4.3). The correspondence between the BS and N = 2 β -functions occurs when we take the weak coupling limit ( g →
0) on the N = 2 side. In this limit, β N =2 g →
0. On the BS side, however, the vanishing of the β -function is valid for an arbitrary couplingconstant, and not only for a weak coupling, being the conformal property a consequence of the vectorsupersymmetry which forbids radiative corrections. In DW theory, such a property is obtained by taking g → g = 0), as a consequence of the property that shows that theobservables of DW theory are insensitive under changes of g . That is how Witten computed its partitionfunction that naturally reproduces the Donaldson invariants for four manifolds, i.e. , by eliminating theinfluence of the vertices at g →
0, and taking only the quadratic part of the action. The BS theory isalso invariant under changes of g , as it only appears in the trivial part of the BRST cohomology, but thetree-level exactness is a general property of the BS theory in self-dual Landau gauges, i.e. , it is valid foran arbitrary perturbative regime.The characterization of the correspondence between the twisted N = 2 SYM and a conformal fieldtheory is now complete. The fact that the twisted N = 2 SYM in the weak coupling limit, and theBS theory share the same global observables is a well-known result in literature [22, 23, 84]. In the DWtheory, the Donaldson invariants are defined by the δ -supersymmetry (2.31) according to the bi-descentequations encoded in (2.65). In the BS one, the same bi-descent equations appears, constructed from the n ’th Chern class f W n defined in terms of the universal curvature in the extended space M × A / G . Suchan equivalence is ensured by the equivariant cohomology that allows for the replacement s → δ , as f W n is invariant under ordinary Yang-Mills transformations. We are now defining in which energy regimessuch an equivalence occurs when we employ the self-dual Landau in the BS, and formal proving thecorrespondence between the twisted N = 2 and a conformal gauge theory. The fact that the observablesare the same, as a consequence of the equivariant cohomology, do not characterizes the correspondenceat quantum level (we will provide a counter-example in the next section). The correspondence betweenthe DW and BS observables, given by the equivalence hO DWα ( φ i ) O DWα ( φ i ) · · · O DWα p ( φ i ) i g → = hO BSα ( φ ′ i ) O BSα ( φ ′ i ) · · · O BSα p ( φ ′ i ) i SDL , (5.1)is independent of the gauge choice. The field content that defines the observables are the same in boththeories, φ i ≡ φ ′ i , since the observables are independent of the FP ghosts (which appear in the gauge-fixed BS action). In a few words, O DWα ( φ i ) ≡ O BSα ( φ ′ i ), represented by the product in eq. (2.65). As31emonstrated in Section 2.2.5, the BS observables naturally do not depend on ( c, ¯ c ), due to the invarianceof f W n under s Y M (the Yang-Mills BRST operator). The BS reproduces the Donaldson polynomials onlyas a consequence of the structure of the off-shell BRST transformations (2.72). Witten works exclusivelyin the moduli space A / G , i.e. , without fixing the gauge, being its observables naturally independent ofthe FP ghost.Another crucial point is that the gauge fixing term in which the FP ghosts are introduced in the self-dual BS theory does not allow for the influence of Gribov copies. For this reason, working in the modulispace A / G in the DW theory are completely correspondent to work in the BS theory in SDL gauges foran arbitrary g , since γ = 0. Fixing the remaining YM gauge symmetry of Witten’s action (2.30), insteadof working in A / G , would not break such a correspondence since the Gribov copies could only affect thenon-perturbative regime, being inoffensive at the ultraviolet limit g →
0. The quantum equivalence areillustrated by the agreement between the β - functions, β N =2 g ( g →
0) = β BSg ( g ) = 0.Finally, due to the property of Witten theory (2.35), which ensures that Witten theory can be extendedto any Riemannian manifold, the DW/BS correspondence is characterized as follows: the twsited N = 2SYM at g →
0, in any Riemannian manifold (that can be continuously deformed into each other,including R ) , defined in the instanton moduli space A / G , is correspondent to the topological BStheory in the self-dual Landau gauges in Euclidean spaces, in an arbitrary perturbative regime. Sucha BS theory consists of a conformal field theory, where the gauge copies are inoffensive in the infrared,since the massive infrared parameter originated from the gauge copies vanishes in this gauge—see Table3 bellow. Twisted N = 2 SYM BS in self-dual Landau gauges
On-shell δ -supersymmetry Off-shell BRST + vector supersymmetryDonaldson invariants ( δ ) Donaldson invariants ( s → δ ) g → g Any Riemannian manifold, g µν Euclidean spaces, δ µν A / G gauge-fixed | γ Gribov = 0 β N =2 g → β BSg ( g ) = 0Table 3: Characterization of the DW/BS correspondence.We emphasize that we use perturbative techniques to prove the conformal property of the self-dualBS theory. The fact that the self-dual BS theory in the strong limit g → ∞ is also correspondent toWitten’s TQFT defined at g →
0, can be conjectured by means of the cohomological structure of theoff-shell BRST symmetry. Changing g in the BS theory is equivalent to add a BRST-exact term in theaction, i.e. , it is equivalent to perform a change in the gauge choice. Moreover, the global observablesof BS theory, described by the Chern classes f W n , does not depend on the gauge choice, having onlythe power of reproducing the Donaldson invariants for four-manifolds. Also, the Gribov ambiguities areirrelevant to the BS model (at least in the self-dual Landau gauges), a property that should remain validat the strong regime. This is the only requirement that guarantees that the observables of both sides are correspondent, as the conformal BS isdefined in Euclidean spaces. In the DW theory, spaces that can be continuously deformed, one into the other, representthe same Donaldson invariants for a class of manifolds, since a continuous variation of the metric is equivalent to add a δ -exact term to the action, which does not alter the observables. .2. Considerations about the gauge dependence and possible generalizations Due to the exact nature of the topological Donaldson invariants, which are given by exact numbers, wecan consider the supposition that quantum corrections should not affect the tree-level results, and that thedescription of the Donaldson invariants by the gauge-fixed BS approach should not depend on the gaugechoice. Although intuitive, this argument is not sufficient or complete. As a counterexample, we invokethe β -function obtained by Birmingham et al. in [55], where the Batalin-Vilkovisky (BV) algorithm [56]was employed. Such a model possesses similar cohomological properties to the BS theory. For a particularconfiguration of auxiliary fields used in [55], the BV method recovers the BS gauges with ρ = ρ = 0,together with the constraint D µ ψ µ = 0—see eq. (2.96). This constraint on the topological ghost, with thecovariant derivative instead of the ordinary one, breaks the vector supersymmetry, allowing for quantumcorrections. Consequently, the β -function computed by Birmingham et al. is not zero. As noted by theauthors of [55], it is Tr ( F µν ± e F µν ) rather than Tr F µν which is renormalized, meaning that the vacuumconfigurations are preserved. As expected, the structure of the instanton moduli space, that defines theDonaldson invariants, is not altered.About the gauge dependence of the β -function in off-shell topological gauge models, see Sec. 4.2.1. Thecoupling constant in this model is non-physical, belonging to the trivial part of cohomology. Any changein the unobservable coupling constant only leads to a BRST-exact variation. The only observablesare the global ones, and we must expect that, for different gauge choices, the global observables arenot affected. According to the result of Birmingham et al. in [55], it is possible to obtain non-trivialquantum corrections without destroying the topological properties of the underlying theory, preservingthe observables. Analogously, we can consider the possibility in which the fields could also be consistentlyrenormalized, i.e. , in such a way that the bi-descent equations, that defines the Donaldson invariants,are not altered. This reasoning shows that the renormalization of topological gauge theories, consistentwith the global observables, is not a trivial issue.The vector supersymmetry, that leads to the conformal BS theory, is a particular symmetry of theself-dual Landau gauge. One must note that ∂ µ A µ = D µ A µ , due to antisymmetric property of f abc . Toimpose ∂ µ ψ µ = 0 or D µ ψ µ = 0 automatically preserves the instanton moduli space, where A µ and ψ µ obeythe same equations of motion, according to the Atiyah-Singer theorem that correctly defines its dimension.The preservation of the instanton moduli space is then the only requirement of the topological theory,being the conformal property a particular feature of the self-dual Landau gauges. The dimension of theinstanton moduli space should not depend on the gauge choice, being protected by the Atiyah-Singertheorem.The second generalization that can be worked out is in the direction of developing a model in whichthe BS theory can be constructed for a generic g µν . Again, any change on the Euclidean metric toa generic one corresponds to the addition of a BRST-exact term in the BS theory. This means thatsuch a variation can be interpreted as a change in the gauge choice, and the previous discussion canbe also applied here. The vector supersymmetry is easily defined in flat spaces. In order to reproducethe conformal properties of the SDL gauges in Euclidean space for a generic g µν , we will face the taskof finding a class of metrics whose corresponding action possesses a rich set of Ward identities ( W I ),capable of reproducing the same effect of the self-dual ones, see Appendix A, given by the 11 functionaloperators W BSI ≡ {S , W µ , W a , W a , W a , W a , G aφ , G a , G a , T , G } .Besides that, we will face another inconvenient task of having to study the Gribov copies in curvedspacetimes, which is a highly nontrivial problem. The vanishing of the Gribov parameter in the self-dualBS in Euclidean spaces ensures that the DW/BS correspondence is valid for a generic coupling constanton the BS side. 33 . Conclusions We perform a comparative study between Donaldson-Witten TQFT [8] and the Baulieu-Singer ap-proach [13]. While DW theory is obtained via the twist transformation of the N = 2 SYM, BS theory isbased on the BRST gauge fixing of an action composed of topological invariants—see Sections 2.1 and2.2, respectively. Besides that, Witten theory is defined by an on-shell supersymmetry, according to thefermionic symmetry, see eq. (2.31), whose associated charge is only nilpotent if we use the equations ofmotion. Such a symmetry defines the observables of the theory, given by the Donaldson polynomials.The BS approach, in turn, consists of an off-shell BRST construction, which enjoys the topological BRSTsymmetry (2.72), whose observables are also given by the Donaldson invariants, due to the equivariantcohomology—defined by Witten’s fermionic symmetry—which also characterizes the BS observables thatcan be written as Chern classes for the curvature in the extended space M × A / G , where M is a Rie-mannian manifold, and A / G the instanton moduli space, see Section 2.2.5. Despite sharing the sameobservables, we note that these theories do not necessarily have the same quantum properties, as Wittenand BS actions do not differ by a BRST-exact term, cf. (2.127). In a few words, the BRST quantizationscheme of Witten and BS theories are not equivalent.In harmony with the quantum properties of BS approach in the self-dual Landau gauges, see Section3, we formally prove that the topological self-dual BS theory is conformal. According to the Feynmandiagram analysis performed in [26], it was proved the absence of quantum corrections in the BS theoryin the presence of the vector supersymmetry. In Section 4.2.1, we discussed the non-physical characterof the coupling constant in the off-shell BS approach. Then, to construct an algebraic prove that theself-dual BS is conformal, we first solve the renormalization ambiguity in topological Yang-Mills theoriesdescribed in [25], using a non-local transformation which recovers discrete symmetries between ghost andanti-ghost fields. Such non-local transformations showed to be a freedom of the self-dual BS theory dueto the triviality of the Gribov copies in the SDL gauges [28], see Section 4.2.2. As a consequence of thistriviality, using the Ward identities of the model—together with the symmetry between the topologicaland Faddeed-Popov ghosts introduced in [24]—and employing algebraic renormalization techniques, weconclude that Z g = 1, i.e. , that the self-dual BS β -function indeed vanishes.We observe that these theories do not possess the same quantum structure, by comparing the β -functionof each model, see Section 4. From this analysis we characterize the correspondence between the twisted N = 2 SYM and BS theories at quantum level, defining in which regimes such a correspondence occurs,see Section 5. In spite of having distinct BRST constructions, we conclude that working in the instantonmoduli space A / G on the DW side is completely equivalent to work in the self-dual Landau gauges onthe BS one, since the Gribov copies do not affect its infrared dynamics. In a few words, the twisted N = 2 SYM in any Riemannian manifold (that can be continuously deformed into M = R ), in the weakcoupling limit g →
0, is correspondent to the BS theory in the self-dual Landau gauges in an arbitraryperturbative regime, which consists of a conformal gauge theory defined in Euclidean flat space, see Table3. Such a characterization could shed some light on the connection between supersymmetry, topology,off-shell BRST symmetry, and non-Abelian conformal gauge theories in four dimensions.
Acknowledgments
We would like to thank A. D. Pereira, G. Sadovski, and A. A. Tomaz for enlightening discussions, whichwas indispensable for the development of this work. This study was financed in part by The Coordena¸c˜aode Aperfei¸coamento de Pessoal de N´ıvel Superior - Brasil (CAPES) – Finance Code 001 – and the ConselhoNacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq-Brazil) – Finance Code 159928/2019-2.34 . BS Ward identities in the self-dual Landau gauges
The BS action in the self-dual Landau gauges (3.8) enjoys the following Ward identities:(i) The Slavnov-Taylor identity, which expresses the BRST invariance of the action (3.8): S (Σ) = 0 , (A.1)where S (Σ) = Z d z " ψ aµ − δ Σ δ Ω aµ ! δ Σ δA aµ + δ Σ δτ aµ δ Σ δψ aµ + (cid:18) φ a + δ Σ δL a (cid:19) δ Σ δc a + δ Σ δE a δ Σ δφ a ++ b a δ Σ δ ¯ c a + ¯ η a δ Σ δ ¯ φ a + B aµν δ Σ δ ¯ χ aµν + Ω aµ δ Σ δτ aµ + L a δ Σ δE a + K aµν δ Σ δ Λ aµν . (A.2)(ii) Ordinary Landau gauge fixing and Faddeev-Popov anti-ghost equation: W a Σ = δ Σ δb a = ∂ µ A aµ , W a Σ = δ Σ δ ¯ c a − ∂ µ δ Σ δ Ω aµ = − ∂ µ ψ aµ . (A.3)(iii) Topological Landau gauge fixing and bosonic anti-ghost equation: W a Σ = δ Σ δ ¯ η a = ∂ µ ψ aµ , W a Σ = δ Σ δ ¯ φ a − ∂ µ δ Σ δτ aµ = 0 . (A.4)(iv) Bosonic ghost equation: G aφ Σ = ∆ aφ , (A.5)where G aφ = Z d z (cid:18) δδφ a − gf abc ¯ φ b δδb c (cid:19) , ∆ aφ = gf abc Z d z (cid:16) τ bµ A cµ + E b c c + Λ bµν ¯ χ cµν (cid:17) . (A.6)(v) Ordinary Faddeev-Popov ghost equation: G a Σ = ∆ a , (A.7)where G a = Z d z " δδc a + gf abc ¯ c b δδb c + ¯ φ b δδ ¯ η c + ¯ χ bµν δδB cµν + Λ bµν δδK cµν ! , ∆ a = gf abc Z d z (cid:16) E b φ c − Ω bµ A cµ − τ bµ ψ cµ − L b c c + Λ bµν B cµν − K bµν ¯ χ cµν (cid:17) . (A.8)(vi) Second Faddeev-Popov ghost equation: G a Σ = ∆ a , (A.9)35here G a = Z d z " δδc a − gf abc ¯ φ b δδ ¯ c c + A bµ δδψ cµ + c b δδφ c − ¯ η b δδb c + E b δδL c ! . (A.10)(vii) Vector supersymmetry: W µ Σ = 0 , (A.11)where W µ = Z d z (cid:20) ∂ µ A aν δδψ aν + ∂ µ c a δδφ a + ∂ µ ¯ χ aνα δδB aνα + ∂ µ ¯ φ a (cid:18) δδ ¯ η a + δδ ¯ c a (cid:19) ++ ( ∂ µ ¯ c a − ∂ µ ¯ η a ) δδb a + ∂ µ τ aν δδ Ω aν + ∂ µ E a δδL a + ∂ µ Λ aνα δδK aνα (cid:21) . (A.12)(viii) Bosonic non-linear symmetry: T (Σ) = 0 , (A.13)where T (Σ) = Z d z " δ Σ δ Ω aµ δ Σ δψ aµ − δ Σ δL a δ Σ δφ a − δ Σ δK aµν δ Σ δB aµν + (¯ c a − ¯ η a ) (cid:18) δ Σ δ ¯ c a + δ Σ δ ¯ η a (cid:19) . (ix) Global ghost supersymmetry: G Σ = 0 , (A.14)where G = Z d z " ¯ φ a (cid:18) δδ ¯ η a + δδ ¯ c a (cid:19) − c a δδφ a + τ aµ δδ Ω aµ + 2 E a δδL a + Λ aµν δδK aµν . (A.15)The last two symmetries are the new ones introduced in [24]. The non-linear bosonic symmetry (vii)is precisely the one discussed in Section 3.2, see eq. (3.15) which relates the FP and topological ghosts.We remark that the Faddeev-Popov ghost equations (A.7) and (A.9) can be combined to obtain an exactglobal supersymmetry, ∆ G a Σ = 0 , (A.16)where ∆ G a = G a − G a = Z d z f abc "(cid:16) ¯ c b − ¯ η b (cid:17) δδb c + ¯ φ b (cid:18) δδ ¯ η c + δδ ¯ c c (cid:19) + A bµ δδψ cµ ++ ¯ χ bµν δδB cµν + c b δδφ c + Λ bµν δδK cµν + τ bµ δδ Ω cµ + E b δδL c . (A.17)We observe the similarity of the equation (A.16) with the vector supersymmetry (A.11). It is also worthmentioning that, even though the ghost number of the operator (A.17) is −
1, resembling an anti-BRSTsymmetry, it is not a genuine anti-BRST symmetry—see for instance [85] for the explicit anti-BRSTsymmetry in topological gauge theories.
References [1] A. A. Belavin, A. M. Polyakov, A. S. Schwartz, and Y. S. Tyupkin, “Pseudoparticle solutions ofthe Yang-Mills equations”.
Physics Letters B no. 1, (1975) 85–87.362] S. Donaldson, “Polynomial invariants for smooth four-manifolds”. Topology no. 3, (1990) 257 – 315. .[3] S. K. Donaldson, “An application of gauge theory to four-dimensional topology”. Journal of Differential Geometry no. 2, (1983) 279–315.[4] S. K. Donaldson, “The orientation of yang-mills moduli spaces and 4-manifold topology”. J. Differential Geom. no. 3, (1987) 397–428. https://doi.org/10.4310/jdg/1214441485 .[5] A. Floer, “Morse theory for fixed points of symplectic diffeomorphisms”. Bull. Amer. Math. Soc.(N.S.) no. 2, (04, 1987) 279–281. https://projecteuclid.org:443/euclid.bams/1183553837 .[6] A. Floer, “An instanton-invariant for 3-manifolds”. Comm. Math. Phys. no. 2, (1988) 215–240. https://projecteuclid.org:443/euclid.cmp/1104161987 .[7] M. Atiyah, “NEW INVARIANTS OF THREE-DIMENSIONAL AND FOUR-DIMENSIONALMANIFOLDS”.
Proc. Symp. Pure Math. (1988) 285–299.[8] E. Witten, “Topological quantum field theory”. Communications in Mathematical Physics no. 3, (9, 1988) 353–386.[9] P. West,
Introduction to Supersymmetry and Supergravity . World Scientific, 5, 1990.[10] A. S. Schwarz, “The Partition Function of Degenerate Quadratic Functional and Ray-SingerInvariants”.
Lett. Math. Phys. (1978) 247–252.[11] D. Ray and I. Singer, “Analytic torsion for complex manifolds”. Annals Math. (1973) 154–177.[12] E. Witten, “Quantum field theory and the Jones polynomial”. Communications in Mathematical Physics no. 3, (9, 1989) 351–399.[13] L. Baulieu and I. Singer, “Topological Yang-Mills symmetry”.
Nuclear Physics B - Proceedings Supplements no. 2, (12, 1988) 12–19.[14] C. Becchi, A. Rouet, and R. Stora, “Renormalization of gauge theories”. Annals of Physics no. 2, (6, 1976) 287–321.[15] I. V. Tyutin, “Gauge Invariance in Field Theory and Statistical Physics in Operator Formalism”.[16] O. Piguet and S. P. Sorella, Algebraic Renormalization , vol. 28 of
Lecture Notes in PhysicsMonographs . Springer Berlin Heidelberg, Berlin, Heidelberg, 1995.[17] P. Van Baal, “An introduction to Topological Yang-Mills Theory”.
Acta Physica Polonica
B21 no. 2, (1990) 73.[18] E. Witten, “AdS / CFT correspondence and topological field theory”.
JHEP (1998) 012.[19] P. Benetti Genolini, P. Richmond, and J. Sparks, “Topological AdS/CFT”. JHEP (2017) 039.[20] P. Agrawal, S. Gukov, G. Obied, and C. Vafa, “Topological Gravity as the Early Phase of OurUniverse”.[21] M. Weis, “Topological Aspects of Quantum Gravity”. hep-th/9806179 .[22] F. Delduc, N. Maggiore, O. Piguet, and S. Wolf, “Note on constrained cohomology”. Phys. Lett. B (1996) 132–138.[23] I. S. Boldo, C. P. Constantinidis, O. Piguet, M. Lefranc, J. L. Boldo, C. P. Constantinidis,F. Gieres, M. Lefrancois, and O. Piguet, “Observables in Topological Yang-Mills Theories”.37 nternational Journal of Modern Physics A no. 17n18, (3, 2003) 2971–3004, hep-th/0303053 .[24] O. C. Junqueira, A. D. Pereira, G. Sadovski, R. F. Sobreiro, and A. A. Tomaz, “TopologicalYang-Mills theories in self-dual and anti-self-dual Landau gauges revisited”. Physical Review D no. 8, (10, 2017) 085008, .[25] O. C. Junqueira, A. D. Pereira, G. Sadovski, R. F. Sobreiro, and A. A. Tomaz, “More about therenormalization properties of topological Yang-Mills theories”. Physical Review D no. 10, (11, 2018) 105017, .[26] O. C. Junqueira, A. D. Pereira, G. Sadovski, R. F. Sobreiro, and A. A. Tomaz, “Absence ofradiative corrections in four-dimensional topological Yang-Mills theories”. Physical Review D no. 2, (7, 2018) 21701, .[27] A. Brandhuber, O. Moritsch, M. de Oliveira, O. Piguet, and M. Schweda, “A renormalizedsupersymmetry in the topological Yang-Mills field theory”. Nuclear Physics B no. 1-2, (12, 1994) 173–190, hep-th/9407105 .[28] D. Dudal, C. Felix, O. Junqueira, D. Montes, A. Pereira, G. Sadovski, R. Sobreiro, and A. Tomaz,“Infinitesimal Gribov copies in gauge-fixed topological Yang-Mills theories”.
Phys. Lett. B (2020) 135531.[29] S. Sorella, “Algebraic Characterization of the Topological σ Model”.
Phys. Lett. B (1989) 69–74.[30] A. Blasi and R. Collina, “Basic Cohomology of Topological Quantum Field Theories”.
Phys. Lett. B (1989) 419–424.[31] J. M. F. Labastida and C. Lozano, “Lectures in Topological Quantum Field Theory”. hep-th/9709192 .[32] T. Kugo and I. Ojima, “Local Covariant Operator Formalism of Nonabelian Gauge Theories andQuark Confinement Problem”.
Prog. Theor. Phys. Suppl. (1979) 1–130.[33] J. Wess and J. Bagger, Supersymmetry and supergravity . Princeton University Press, Princeton,NJ, USA, 1992.[34] A. Blasi, V. Lemes, N. Maggiore, S. Sorella, A. Tanzini, O. Ventura, and L. Vilar, “Perturbativebeta function of N=2 superYang-Mills theories”.
JHEP (2000) 039.[35] E. Witten, “Supersymmetry and Morse theory”. J. Diff. Geom. no. 4, (1982) 661–692.[36] N. Maggiore, “Algebraic renormalization of N=2 superYang-Mills theories coupled to matter”. Int. J. Mod. Phys. A (1995) 3781–3802.[37] M. F. Atiyah and I. M. Singer, “The index of elliptic operators on compact manifolds”. Bull. Am. Math. Soc. (1969) 422–433.[38] M. F. Atiyah, N. J. Hitchin, and I. M. Singer, “Self-Duality in Four-Dimensional RiemannianGeometry”. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences no. 1711, (9, 1978) 425–461.[39] D. Tong, “TASI lectures on solitons: Instantons, monopoles, vortices and kinks”. in
TheoreticalAdvanced Study Institute in Elementary Particle Physics: Many Dimensions of String Theory . 6,2005.[40] G. ’t Hooft, “Computation of the quantum effects due to a four-dimensional pseudoparticle”.
Physical Review D no. 12, (12, 1976) 3432–3450.3841] A. D’Adda and P. Di Vecchia, “Supersymmetry and Instantons”. Phys. Lett. B (1978) 162.[42] M. Blau and G. Thompson, “Do metric independent classical actions lead to topological fieldtheories?”. Physics Letters B no. 4, (2, 1991) 535–542.[43] M. Abud and G. Fiore, “Batalin-Vilkovisky approach to the metric independence of TQFT”.
Physics Letters B no. 1-2, (10, 1992) 89–93.[44] A. Mardones and J. Zanelli, “Lovelock-Cartan theory of gravity”.
Classical and Quantum Gravity no. 8, (8, 1991) 1545–1558.[45] M. Daniel and C. M. Viallet, “The geometrical setting of gauge theories of the Yang-Mills type”. Reviews of Modern Physics no. 1, (1, 1980) 175–197.[46] N. Vandersickel, A study of the Gribov-Zwanziger action: from propagators to glueballs . PhD thesis,Ghent University, 4, 2011. .[47] J. A. Dixon, “COHOMOLOGY AND RENORMALIZATION OF GAUGE THEORIES. 2.”.[48] S. Ouvry, R. Stora, and P. Van Baal, “On the algebraic characterization of Witten’s topologicalYang-Mills theory”.
Physics Letters B no. 1-2, (1989) 159–163.[49] J. H. Horne, “Superspace Versions of Topological Theories”.
Nucl. Phys. B (1989) 22–52.[50] H. Kanno, “Weyl Algebra Structure and Geometrical Meaning of BRST Transformation inTopological Quantum Field Theory”.
Z. Phys. C (1989) 477.[51] M. Werneck de Oliveira, “Algebraic renormalization of the topological Yang-Mills field theory”. Phys. Lett. B (1993) 347–352.[52] R. Brooks, D. Montano, and J. Sonnenschein, “Gauge fixing and renormalization in topologicalquantum field theory”.
Physics Letters B no. 1, (11, 1988) 91–97.[53] V. Lemes, N. Maggiore, M. Sarandy, S. Sorella, A. Tanzini, and O. Ventura, “Nonrenormalizationtheorems for N=2 super-Yang-Mills”.[54] F. Fucito, A. Tanzini, L. C. Q. Vilar, O. S. Ventura, C. A. G. Sasaki, and S. P. Sorella, “AlgebraicRenormalization: perturbative twisted considerations on topological Yang-Mills theory and onN=2 supersymmetric gauge theories”. hep-th (7, 1997) 15–19, hep-th/9707209 .[55] D. Birmingham, M. Rakowski, G. Thompson, I. Centre, T. Pto, T. Physics, I. Centre, T. Pttvsk,P. Vi, and P. Jussieu, “Renormalization of topological field theory”.
Nuclear Physics B no. 1, (1, 1990) 83–97.[56] I. Batalin and G. Vilkovisky, “Quantization of Gauge Theories with Linearly DependentGenerators”.
Phys. Rev. D (1983) 2567–2582. [Erratum: Phys.Rev.D 30, 508 (1984)].[57] E. Witten, “Some Exact Multi - Instanton Solutions of Classical Yang-Mills Theory”. Phys. Rev. Lett. (1977) 121–124. [,124(1976)].[58] R. Jackiw, C. Nohl, and C. Rebbi, “Conformal Properties of Pseudoparticle Configurations”. Phys. Rev.
D15 (1977) 1642. [,128(1976)].[59] V. Gribov, “Quantization of non-Abelian gauge theories”.
Nuclear Physics B no. 1-2, (6, 1978) 1–19.[60] R. F. Sobreiro and S. P. Sorella, “Introduction to the Gribov Ambiguities In Euclidean Yang-MillsTheories”. hep-th/0504095 . 3961] D. Zwanziger, “Action From the Gribov Horizon”.
Nucl. Phys. B (1989) 591–604.[62] I. M. Singer, “Some remarks on the Gribov ambiguity”.
Communications in Mathematical Physics no. 1, (2, 1978) 7–12.[63] L. Faddeev and V. Popov, “Feynman diagrams for the Yang-Mills field”. Physics Letters B no. 1, (7, 1967) 29–30.[64] C. Itzykson and J. Zuber, Quantum Field Theory . International Series In Pure and AppliedPhysics. McGraw-Hill, New York, 1980.[65] R. Sobreiro and S. Sorella, “A Study of the Gribov copies in linear covariant gauges in EuclideanYang-Mills theories”.
JHEP (2005) 054.[66] M. Capri, A. Pereira, R. Sobreiro, and S. Sorella, “Non-perturbative treatment of the linearcovariant gauges by taking into account the Gribov copies”. Eur. Phys. J. C no. 10, (2015) 479.[67] M. Capri, D. Dudal, D. Fiorentini, M. Guimaraes, I. Justo, A. Pereira, B. Mintz, L. Palhares,R. Sobreiro, and S. Sorella, “Exact nilpotent nonperturbative BRST symmetry for theGribov-Zwanziger action in the linear covariant gauge”. Phys. Rev. D no. 4, (2015) 045039.[68] M. Capri, D. Fiorentini, M. Guimaraes, B. Mintz, L. Palhares, S. Sorella, D. Dudal, I. Justo,A. Pereira, and R. Sobreiro, “More on the nonperturbative Gribov-Zwanziger quantization of linearcovariant gauges”. Phys. Rev. D no. 6, (2016) 065019.[69] G. Dell’Antonio and D. Zwanziger, “Every gauge orbit passes inside the Gribov horizon”. Communications in Mathematical Physics no. 2, (5, 1991) 291–299.[70] N. Vandersickel, D. Dudal, O. Oliveira, and S. P. Sorella, “From propagators to glueballs in theGribov-Zwanziger framework”.
AIP Conf. Proc. (2011) 155–157.[71] A. J. Gomez, M. S. Guimaraes, R. F. Sobreiro, and S. P. Sorella, “Equivalence betweenZwanziger’s horizon function and Gribov’s no-pole ghost form factor”.
Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics no. 2-3, (10, 2009) 217–221, .[72] M. Capri, D. Dudal, M. Guimaraes, L. Palhares, and S. Sorella, “An all-order proof of theequivalence between Gribov’s no-pole and Zwanziger’s horizon conditions”.
Physics Letters B no. 4-5, (2, 2013) 448–453, .[73] D. Dudal, J. A. Gracey, S. P. Sorella, N. Vandersickel, and H. Verschelde, “A Refinement of theGribov-Zwanziger approach in the Landau gauge: Infrared propagators in harmony with the latticeresults”.
Phys. Rev. D (2008) 065047.[74] A. A. Slavnov, “Physical Unitarity in the { BRST } Approach”.
Phys. Lett. B (1989) 91–94.[75] S. A. Frolov and A. A. Slavnov, “Construction of the Effective Action for General Gauge Theoriesvia Unitarity”.
Nucl. Phys. B (1990) 333–346.[76] D. Dudal, S. P. Sorella, N. Vandersickel, and H. Verschelde, “Gribov no-pole condition, Zwanzigerhorizon function, Kugo-Ojima confinement criterion, boundary conditions, BRST breaking and allthat”.
Phys. Rev. D (2009) 121701.[77] L. Baulieu and S. P. Sorella, “Soft breaking of BRST invariance for introducing non-perturbativeinfrared effects in a local and renormalizable way”. Phys. Lett.
B671 no. 4-5, (8, 2008) 481–485, .[78] S. P. Sorella, “Gribov horizon and BRST symmetry: A Few remarks”.40 hys. Rev. D (2009) 025013.[79] S. P. Sorella, D. Dudal, M. S. Guimaraes, and N. Vandersickel, “Features of the RefinedGribov-Zwanziger theory: Propagators, BRST soft symmetry breaking and glueball masses”. PoS
FACESQCD (2010) 022.[80] M. A. L. Capri, D. Dudal, D. Fiorentini, M. S. Guimaraes, I. F. Justo, A. D. Pereira, B. W. Mintz,L. F. Palhares, R. F. Sobreiro, and S. P. Sorella, “Local and BRST-invariant Yang-Mills theorywithin the Gribov horizon”.
Phys. Rev.
D94 no. 2, (2016) 025035.[81] A. D. Pereira, R. F. Sobreiro, and S. P. Sorella, “Non-perturbative BRST quantization ofEuclidean Yang–Mills theories in Curci–Ferrari gauges”.
Eur. Phys. J. C no. 10, (2016) 528.[82] M. A. L. Capri, D. Dudal, M. S. Guimaraes, A. D. Pereira, B. W. Mintz, L. F. Palhares, and S. P.Sorella, “The universal character of Zwanziger’s horizon function in Euclidean Yang–Mills theories”. Phys. Lett.
B781 (2018) 48–54.[83] M. A. L. Capri, M. S. Guimaraes, I. F. Justo, L. F. Palhares, and S. P. Sorella, “On the irrelevanceof the Gribov issue in N = 4 Super Yang-Mills in the Landau gauge”. Physics Letters B (4, 2014) 277–281, .[84] J. L. Boldo, C. P. Constantinidis, O. Piguet, F. Gieres, and M. Lefran¸cois, “Topological Yang-MillsTheories and their Observables: A Superspace Approach”.
International Journal of Modern Physics A no. 12, (5, 2003) 2119–2125, hep-th/0303084 .[85] N. R. Braga and C. F. Godinho, “Extended BRST invariance in topological Yang-Mills theoryrevisited”. Phys. Rev. D61