Correspondence Between the Twisted N=2 Super-Yang-Mills and Conformal Baulieu-Singer Theories

PHYSICAL REVIEW D 103, 085008 (2021)

Correspondence between the twisted N = 2 super-Yang-Mills and conformal Baulieu-Singer theories

† Octavio C. Junqueira 1,2,* and Rodrigo F. Sobreiro2, 1UFRJ—Universidade Federal do Rio de Janeiro, Instituto de Física, Caixa Postal 68528, Rio de Janeiro, Brasil 2UFF—Universidade Federal Fluminense, Instituto de Física, Avenida Litoranea s/n, 24210-346, Niterói, Rio de Janeiro, Brasil

(Received 29 January 2021; accepted 17 March 2021; published 22 April 2021)

We characterize the correspondence between the twisted N ¼ 2 super-Yang-Mills theory and the Baulieu-Singer topological theory quantized in the self-dual Landau gauges. While the first is based on an on-shell supersymmetry, the second is based on an off-shell Becchi-Rouet-Stora-Tyutin symmetry. Because of the equivariant cohomology, the twisted N ¼ 2 in the ultraviolet regime and Baulieu-Singer theories share the same observables, the Donaldson invariants for 4-manifolds. The triviality of the Gribov copies in the Baulieu-Singer theory in these gauges shows that working in the instanton moduli space on the twisted N ¼ 2 side is equivalent to working in the self-dual gauges on the Baulieu-Singer one. After proving the vanishing of the β function in the Baulieu-Singer theory, we conclude that the twisted N ¼ 2 in the ultraviolet regime, in any Riemannian manifold, is correspondent to the Baulieu-Singer theory in the self- dual Landau gauges—a conformal gauge theory defined in Euclidean flat space.

DOI: 10.1103/PhysRevD.103.085008

I. INTRODUCTION the Donaldson-Witten theory (DW) in the Wess-Zumino gauge [9]. Throughout the 1980s, based on the self-dual Yang-Mills In practice, TQFTs have the power to reproduce topo- equations introduced by A. Belavin et al. in their study of logical invariants of the basis manifold as observables. The instantons [1], S. K. Donaldson discovered and described first one to obtain topological invariants from a quantum topological structures of polynomial invariants for smooth field theory was A. S. Schwarz in 1978 [10]. He showed 4-manifolds [2–4]. The connection between the Floer that the Ray-Singer analytic torsion [11] can be represented theory for 3-manifolds [5,6] and Donaldson invariants as a partition function of the Abelian Chern-Simons (CS) for 4-manifolds with a nonempty boundary, i.e., that action, which is invariant by diffeomorphisms. The assumes values in Floer groups, has led to Atiyah’s Schwarz topological theory was the prototype of Witten conjecture [7,8]. In this conjecture, he proposed that the theories in the 1980s. Indeed, the well-known Witten paper, Floer homology must lead to a relativistic quantum field in which he reproduces the Jones polynomials of knot theory. This conjecture was the motivation for Witten’s theory [12], is the non-Abelian generalization of Schwarz’s topological quantum field theory (TQFT) in four dimen- results [10]. In this work, Witten is actually able to sions, as Witten himself admits [8]. In Ref. [7], Atiyah represent topological invariants of 3-manifolds as the showed that Floer’s results [6] can be seen as a version of a partition function of the non-Abelian CS theory. supersymmetric gauge theory. Answering Atiyah’s con- After Witten’s result [8], L. Baulieu and I. M. Singer jecture, Witten found a relativistic formulation of Ref. [7], (BS) showed in Ref. [13] that the same topological capable of reproducing the Donaldson polynomials in the observables can be obtained from a gauge-fixed topological weak coupling limit of the twisted N ¼ 2 Super-Yang- action. In such an approach, the Becchi-Rouet-Stora-Tyutin Mills (SYM) theory. This TQFT is commonly referred to as (BRST) symmetry [14–16] plays a fundamental role. It is not built through a linear transformation of a supersym- ’ *[email protected] metric gauge theory, like Witten s TQFT. It is built through † [email protected] a gauge-fixing procedure of a topological-invariant action, in such a way that the BRST operator naturally appears as Published by the American Physical Society under the terms of nilpotent without requiring the use of equations of motion. the Creative Commons Attribution 4.0 International license. The geometric interpretation of the BS theory is that the Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, non-Abelian topological theory lies in a universal space and DOI. Funded by SCOAP3. graded as a sum of the ghost number and the form degree,

2470-0010=2021=103(8)=085008(28) 085008-1 Published by the American Physical Society OCTAVIO C. JUNQUEIRA and RODRIGO F. SOBREIRO PHYS. REV. D 103, 085008 (2021) where the vertical direction of this double complex is proof was still lacking until now. It turns out that, for a determined by the ghost number and the horizontal one is complete proof of the vanishing of the β function of the BS determined by the form degree. In this space, the topo- theory in the self-dual Landau gauges, one extra property logical BRST transformations are written in terms of a must be considered: the fact that the Gribov copies are universal connection, and its curvature naturally explains inoffensive to the self-dual BS theory [28]. This property the BS approach as a topological Yang-Mills theory with establishes that the self-dual BS theory is conformal, as it the same global observables of Witten’sTQFT. allows us to recover some discrete symmetries. The use of From the physical point of view, the motivation to study these symmetries makes it possible to eliminate the TQFTs comes from the mathematical tools of such theories, renormalization ambiguities discussed in Ref. [25]. With capable of revealing the topological structure of field this information, we were able to establish the correspon- theories that are independent of variations of the metric, dence between self-dual BS theories (a conformal gauge and consequently of the background choice. One of the theory defined in Euclidean spaces) for any value of the major obstacles to constructing a quantum theory of gravity coupling constant and DW theory at the deep UV. is the integration over all metrics. The introduction of a The paper is organized as follows. Section II contains an topological phase in gravity would have the power to make a overview of the main properties of DW and BS theories. We theory of gravity arise from a symmetry breaking mecha- introduce the main aspects of each approach, explaining nism of a background independent topological theory1 how each one is constructed from different quantization [8,17]. On the other hand, we can investigate conformal schemes. As the quantum properties of the Witten’sTQFT properties of field theories via topological models. In three is well known in literature, we dedicate Sec. III to dimensions, for instance, the connection between the three- discussing the quantum properties of the BS theory in dimensional Chern-Simons theory and two-dimensional the self-dual Landau gauges. In Sec. IV, we analyze and conformal theories is well known [12]. In four dimensions, compare the corresponding β functions of each model, after TQFTs are intimately connected with the AdS=CFT proving that the self-dual BS is conformal. Finally, in correspondence [18,19]. More recently, motivated by string Sec. V, we describe the quantum correspondence between dualities, a topological gravity phase in the early Universe Witten and self-dual BS topological theories. Section VI was proposed [20]. Such a phase could explain some puzzles contains our concluding remarks. concerning early Universe cosmology. The fact that DW theory at the UV regime and BS II. TOPOLOGICAL QUANTUM theories share the same observables is a well-known result FIELD THEORIES in the literature [8,13,21–23]. In this paper, we characterize the correspondence between DW TQFT and a conformal A topological quantum field theory on a smooth mani- BS gauge theory at quantum level. While Witten’s theory is fold is a quantum field theory which is independent of the based on the twisted version of the N ¼ 2 super-Yang-Mills metric on the basis manifold. Such a theory has no theory, the mentioned conformal theory is based on the dynamics nor local degrees of freedom and is only sensitive Baulieu-Singer BRST gauge-fixing approach to a topo- to topological invariants which describe the manifold in logical action [13]. In recent works [24–26], the existence which the theory is defined. The observables of a TQFT are of an extra bosonic symmetry was proved in the case of naturally metric independent. The latter statement leads to self-dual Landau gauges.2 This bosonic symmetry relates the main property of topological field theories, namely, the the Faddeev-Popov and the topological ghost fields. metric independence of the observable correlation func- Together with the known vector supersymmetry [27] and tions of the theory, the vanishing three-level gauge propagator, one observes δ that the BS theory at the self-dual Landau gauges is indeed O φ O φ O φ 0 h α1 ð iÞ α2 ð iÞ α ð iÞi ¼ ; ð2:1Þ tree-level exact [26]. Essentially, the proof of this property δgμν p is diagrammatic with some help of algebraic renormalization techniques [16]. This remarkable property inevitably with implies a vanishing β function, since it does not receive hOα ðφ ÞOα ðφ ÞOα ðφ Þi quantum corrections. Nevertheless, an entire algebraic 1 i Z 2 i p i N φ O φ O φ O φ −S½φ ¼ ½D i α1 ð iÞ α2 ð iÞ α ð iÞe ; ð2:2Þ 1We must say that the introduction of such a topological phase p is one of the intricate problems in topological quantum field theories, since one should develop a mechanism to break the where gμν is the metric tensor, φiðxÞ are quantum fields, Oα topological symmetry. 2 are the functional operators of the fields composing global For simplicity, we will refer to the (anti-)self-dual Landau φ N gauges, defined by instantons and anti-instantons configurations, observables, S½ is the classical action, and is the see gauge condition (3.3.), only by the denotation self-dual appropriate normalization factor. A typical operator Oα is gauges. integrated over the whole space in order to capture the

085008-2 CORRESPONDENCE BETWEEN THE TWISTED N ¼ 2 SUPER- … PHYS. REV. D 103, 085008 (2021) global structures of the manifold. Since there are no which preserves the topological structure of δ O O δ 0 particles, the only nontrivial observables are of global gμν h α1 αp i¼h ð Þi ¼ [31]. Analogously to the nature [29,30]. BRST operator, Eq. (2.6) only makes sense if the δ operator As a particular result of (2.1), the partition function of a is nilpotent.4 topological theory is itself a topological invariant, δ A. Donaldson-Witten theory Z½J¼0; ð2:3Þ δgμν As mentioned in the Introduction, Witten constructed in Ref. [8] a four-dimensional generalization of Ref. [7], – insofar as Z½J represents the expectation value of the capable of reproducing the Donaldson invariants [2 4] in vacuum in the presence of a external source, Z½J¼h0j0i . the weak coupling limit. Such a construction can be J 2 As discussed in Ref. [31], if the action is explicitly obtained from the twist transformation of the N ¼ independent of the metric, the topological theory is said SYM. Let us quickly revise some important features of to be of Schwarz type; otherwise, if the variation of the such approach. action with respect to the metric gives a “BRST-like”-exact term, one says the theory is of Witten type. More precisely, 1. Twist transformation because δ is an infinitesimal transformation that denotes the i ¯ The eight supersymmetric charges (Qα, Qjα_ )ofN ¼ 2 symmetry of the action S which characterizes the observ- SYM theories obey the supersymmetry (SUSY) algebra ables of the model, then, if the following properties are satisfied, i ¯ δi σ ∂ fQα; Qjα_ g¼ jð μÞαα_ μ; i ¯ i ¯ 0 δOαðφiÞ¼0;TμνðφiÞ¼δGμνðφiÞ; ð2:4Þ fQα;Qjαg¼fQα_ ; Qjα_ g¼ ; ð2:8Þ α α_ where Tμν is the energy-momentum tensor of the model, where all indices ði; j; ; Þ run from 1 to 2. The indices ði; jÞ denote the internal SUð2Þ symmetry of the N ¼ 2 δ SYM action, and ðα; α_Þ are Weyl spinor indices: α denotes S ¼ Tμν; ð2:5Þ α_ δgμν right-handed spinors, and denotes left-handed ones. The fact that both indices equally run from 1 to 2 suggests the and Gμν some tensor, the quantum field theory can be identification between spinor and supersymmetry indices, regarded as topological. Obviously, in this case, Eq. (2.3) is ≡ α also satisfied, since the expectation value of the δ-exact i : ð2:9Þ term vanishes3 [8,13]. In fact, by using (2.4) and (2.5), and 2 assuming that the measure ½Dφ is invariant under δ, The N ¼ SYM action theory possesses a gauge group i symmetry given by δ Oα φ Oα φ Oα φ h 1 ð iÞ 2 ð iÞ p ð iÞi SU ð2Þ × SU ð2Þ × SU ð2Þ × U ð1Þ; ð2:10Þ δgμν L R I R Z 2 2 2 − φ O φ O φ O φ −S where SULð Þ × SURð Þ is the rotation group, SUIð Þ is ¼ ½D i α1 ð iÞ α2 ð iÞ αp ð iÞTμνe the internal supersymmetry group labeled by i, and URð1Þ δ O φ O φ O φ is the so-called R-symmetry defined by the supercharges ¼h ½ α1 ð iÞ α2 ð iÞ α2 ð iÞGμνi i ¯ (Qα, Q α_ ), which are assigned eigenvalues (þ1, −1), 0: 2:6 j ¼ ð Þ respectively. The identification performed in Eq. (2.9) amounts to a modification of the rotation group, In the above equation, we assumed that all Oα are metric independent. Nevertheless, this is not a requirement of the SU ð2Þ × SU ð2Þ → SU ð2Þ × SU ð2Þ0; ð2:11Þ theory. It is also possible to have a more general theory in L R L R which 0 where SURð2Þ is the diagonal sum of SURð2Þ and SUIð2Þ. δ The twisted global symmetry of N ¼ 2 SYM takes the O δQ ≠ 0 2 2 0 1 δ α ¼ μν ; ð2:7Þ form SULð Þ × SURð Þ × URð Þ, with the corresponding gμν twisted supercharges

β 3 i ¯ ¯ The nilpotent δ operator works precisely as a BRST operator, Qα → Qα; Qiα¯ → Qαα_ ; ð2:12Þ and it is well known that expectation values of BRST-exact terms vanish. For a further analysis concerning renormalization properties, and the definition of physical observables, see 4In Donaldson-Witten theory, for instance, such an operator is Refs. [14,16,32]. on-shell nilpotent, i.e., δ2 ¼ 0 by using the equations of motion.

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1 which can be rearranged as ψ i → ψ ψ ψ β αβ ¼ 2 ð ðαβÞ þ ½αβÞ; ð2:22Þ 1 ffiffiffi ϵαβ ≡ δ p Qαβ ; ð2:13Þ i αα_ 2 ψ¯ α_ → ψ¯ αα_ → ψ μ ¼ðσμÞ ψ¯ αα_ ; ð2:23Þ

1 ¯ αα_ αα_ together with pffiffiffi Q ðσμÞ ≡ δμ; ð2:14Þ 2 ψ → χ σ αβψ ðαβÞ μν ¼ð μνÞ ðαβÞ; ð2:24Þ 1 αα_ pffiffiffi ðσμνÞ Qαα_ ≡ dμν; ð2:15Þ ψ → η εαβψ 2 ½αβ ¼ ½αβ: ð2:25Þ

αβ μ αα_ αα_ where we adopt the conventions for ϵ , ðσ Þ , and ðσμνÞ The twist consists of a mapping of degrees of freedom. The as the same as in Ref. [33]. The operator dμν is manifestly field ψ¯ αα_ has four independent components as ðα; α_Þ¼ self-dual due to the structure of σμν, f1; 2g and is mapped into the field ψ μ that also has four independent components of the path integral, as the Lorentz 1 λρ index μ ¼f1; 2; 3; 4g in four dimensions. In the other dμν ¼ εμνλρd ; ð2:16Þ 2 mappings (2.22), (2.24) and (2.25), the same occurs, as the ψ ψ symmetric part of αβ, i.e., ðαβÞ has three independent reducing to 3 the number of its independent components. components mapped into the self-dual field χμν, and the The operators δ, δμ, and δμν possess eight independent ψ antisymmetric part, ½αβ, with only one independent components into which the eight original supercharges η ¯ component, into , a scalar field. We must note that Qβα; Qαα_ ð Þ are mapped. These operators obey the following ðψ μ; χμν; ηÞ are anticommuting field variables due to their twisted supersymmetry algebra: spinor origin.

2 Because it is a linear transformation, the twist simply δ ¼ 0; ð2:17Þ corresponds to a change of variables with trivial Jacobian that could be absorbed in the normalization factor; in fδ; δμg¼∂μ; ð2:18Þ other words, both theories (before and after the twist) are perturbatively indistinguishable. Finally, twisting the 2 N¼2 fδμ; δνg¼fdμν; δg¼fdμν;dλρg¼0; ð2:19Þ N ¼ SYM action (SSYM) [8,34], in flat Euclidean space, we obtain the Witten four-dimensional topological Yang- σ Mills action (S ), fδμ;dλρg¼−ðεμλρσ∂ þ gμλ∂ρ − gμρ∂λÞ: ð2:20Þ W

N¼2 ψ i ψ¯ i φ φ¯ → ψ χ φ¯ φ The nilpotent scalar supersymmetry charge δ defines the SSYM½Aμ; α; α_ ; ; SW½Aμ; μ; μν; ; ; ð2:26Þ cohomology of Witten’s TQFT, as its observables appear as cohomology classes of δ, which is invariant under a generic where differential manifold. It is implicit in the anticommutation Z 1 1 relation (2.18) the topological nature of the model, as it 4 þ þμν þ S ¼ Tr d x FμνF − χμνðDμψ ν − Dνψ μÞ allows us to write the common derivative as a δ-exact term. W g2 2 The gauge multiplet of the N ¼ 2 SYM in Wess-Zumino 1 1 η ψ μ − φ¯ μφ φ¯ ψ ψ gauge is given by the fields þ Dμ 2 DμD þ 2 f μ; μg 1 1 1 ψ i ψ¯ i φ φ¯ − φ χ χ − φ η η − φ φ¯ φ φ¯ ðAμ; α; α_ ; ; Þ; ð2:21Þ 2 f μν; μνg 8½ ; 32½ ; ½ ; ; ð2:27Þ

i where ψ α is a Majorana spinor (the supersymmetric þ 6 wherein Fμν is the self-dual field partner of the gauge connection Aμ) and φ is a scalar field, all of them belonging to the adjoint representation þ ˜ ˜ þ þ Fμν ¼ Fμν þ Fμν; ðFμν ¼ FμνÞ; ð2:28Þ of the gauge group. The twist transformation is defined by the identification (2.9) and thus only acts on the fields ˜ 1 i i with Fμν ¼ ϵμναβFαβ, and, analogously, ðψ μ; ψ¯ μÞ, leaving the bosonic fields ðAμ; φ; φ¯ Þ unaltered. 2 Explicitly, the twist transformation is given by the linear 6 transformations5 Following Refs. [8,34], we are considering the positive sign, that corresponds to anti-instantons in the vacuum. A similar construction can be done for instantons, only by changing 5 Φ Φ Φ Φ Φ − Φ Notation: ðαβÞ ¼ αβ þ βα and ½αβ ¼ αβ βα. the sign.

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1 þ transformations, one needs to use the equations of motion ðDμψ ν − Dνψ μÞ ¼ Dμψ ν −Dνψ μ þ εμναβðDαψ β − Dβψ αÞ; 2 to recover the nilpotency of δ [30]. This characterizes the ð2:29Þ DW theory as an on-shell approach. One can easily verify that (see Ref. [8]) with Dμ ≡∂μ − g½Aμ; · being the covariant derivative δ2Φ ¼ 0; for Φ ¼fA; ψ; φ; φ¯ ; ηg; ð2:32Þ in the adjoint representation of the Lie group G, with g 7 being the coupling constant. The Witten action (2.27) but possesses the usual Yang-Mills gauge invariance, denoted 8 by 2 δ χ ¼ equations of motion: ð2:33Þ δYM S ¼ 0: ð2:30Þ gauge W Considering the result of Eq. (2.33), hereafter, we will say The theory, however, does not possess gauge anomalies that the Witten fermionic symmetry is on-shell nilpotent. [36]. The symmetry that defines the cohomology of the This symmetry is associated to an on-shell nilpotent “BRST charge,” Q, according to the definition of the δ theory, also known as equivariant cohomology, is the O fermionic scalar supersymmetry variation of any functional as a transformation on the space of all functionals of field variables, namely, 1 δAμ ¼ −εψ μ; δφ ¼ 0; δφ¯ ¼ 2iεη; δη ¼ ε½φ;φ¯ ; δO − ε Q O Q2 0 2 ¼ i · f ; g; such that jon-shell ¼ : ð2:34Þ þ δψ μ ¼ −εDμφ; δχμν ¼ εF ; ð2:31Þ To verify that Witten theory is valid in curved space- where ε is the supersymmetry fermionic parameter that times, it is worth noting that the commutators of covariant φ carries no spin, ensuring that the propagating modes of derivatives always appear acting in the scalar field , like in δ ψ χ¯ 1 φ χ¯μν commuting and anticommuting fields have the same TrfDμ ν · μνg¼2 Trð½Dμ;Dν · Þ, so the Riemann helicities.9 This symmetry relates bosonic and fermionic tensor does not appear, and the theory could be extended to degrees of freedom, which are identical—an inheritance of any Riemannian manifold. In practice, one can simply take the supersymmetric original action.10 The price of working Z Z pffiffiffi in Wess-Zumino gauge is the fact that, disregarding gauge d4x → d4x g; ð2:35Þ

7 Technically, the Witten action (2.27) is the four-dimensional in order to work in a curved spacetime. Such a theory has generalization of the nonrelativistic topological quantum field theory [7], whose construction is based on the Floer theory for the property of being invariant under infinitesimal changes 3-manifolds M3D, in which the Chern-Simons action is taken as in the metric. This property characterizes the Witten model a Morse function on M3D; see Floer’s original paper [5]. In a few as a topological quantum field theory. Such a feature is words, the critical points of CS action (WCS) yield the curvature verified by the fact that the energy-momentum tensor can δ 1 WCS − εijk jk jk free configurations, as δ a ¼ 2 F , where F is the 2-form Ai be written as the anticommutator curvature in three dimensions, which defines the gradient flows of a Morse function; see Ref. [17]. In the supersymmetric formu- Tμν ¼fQ;Vμνg; ð2:36Þ lation of Ref. [7], the Hamiltonian (H) is obtained via the “supersymmetric charges” dt and dt , from the well-known which means that Tμν is an on-shell BRST-exact term, relation dtdt þ dt dt ¼ 2H, see Ref. [35], whereby dt ¼ −tW tW tW −tW e CS de CS and dt ¼ e CS d e CS , for a real number t, with A δ δ2 0 d being the exterior derivative on the space of all connections , Tμν ¼ Vμν; jon-shell ¼ ; ð2:37Þ δ a ψ a according to the transformation Ai ¼ i , and d being its dual. Before identifying the twist transformation, this formulation (in with four dimensions) was employed by Witten in his original paper [12] to obtain the relativistic topological action (2.27). 8 1 1 The typical Yang-Mills transformations of all fields are σ σ σρ Vμν ¼ Tr Fμσχν þ Fνσχμ − gμνFσρχ implicit in this notation, where the gauge field transforms as 2 2 0 −1 −1 Aμ ¼ S AμS þ S ∂μS with S ∈ SUðNÞ. 9 1 Precisely, the propagating modes of Aμ have helicities þ gμνTrη½φ; φ¯ ð1; −1Þ. For the propagating modes of ðφ; φ¯ Þ, the helicities are 4 η ψ χ (0,0). And the fermionic fields ð ; ; Þ carry helicities 1 ν μ σ ð1; −1; 0; 0Þ. þ Trfψ μD φ¯ þ ψ νD φ¯ − gμνψ σD φ¯ g: ð2:38Þ 10 2 The action SW is also invariant under global scaling with φ φ¯ η ψ χ dimensions (1, 0, 2, 2, 1, 2) for ðA; ; ; ; ; Þ, respectively, and δ 0 preserves an additive U symmetry for the assignments Equation (2.37) together with SW ¼ means that ð0; 2; −2; −1; 1; −1Þ. In the BRST formalism, the latter is Witten theory satisfies (on-shell) the second condition naturally recognized as ghost numbers, as we will see later on. displayed in Eq. (2.4), which allows us to say that SW

085008-5 OCTAVIO C. JUNQUEIRA and RODRIGO F. SOBREIRO PHYS. REV. D 103, 085008 (2021) automatically leads to a four-dimensional topological field giving winding number k and gauge group SUðNÞ.By ˜ model. In other words, perturbing F ¼ F nearby the solution Aμ via a gauge Z transformation Aμ → Aμ þ δAμ, we obtain the self-duality δ δ equation ZW ¼ DΦ − SW expð−SWÞ δgμν δgμν Z α β 1 ffiffiffi DμδAν þ DμδAν þ εμναβD δA ¼ 0: ð2:44Þ − Q 4 p 0 ¼ 2 ; d x gVμν ¼ ; ð2:39Þ g M The solutions of the above equation are called zero modes. Requiring the orthogonal gauge-fixing condition,11 as all expected values of a δ-exact term vanish. It remains to DμAμ ¼ 0, one gets know which kind of topological/differential invariants can be represented by the Feynman path integral of Witten’s DμðδAμÞ¼0: ð2:45Þ TQFT. As we know, it will naturally reproduce the Donaldson invariants for four-manifolds. The Atiyah-Singer index theorem [37,38] counts the number of solutions to Eqs. (2.44) and (2.45). In Euclidean space- 2. Donaldson polynomials in the weak coupling limit times, for instance, the index theorem gives, see Ref. [39], An important feature of the twisted N ¼ 2 SYM is the dimðMÞ¼4kN; ð2:46Þ fact that the theory can be interpreted as quantum fluctua- tions around classical instanton configurations. To find the where the modes due to global gauge transformations of the nontrivial classical minima, one must note that the pure group were included. Looking at fermion zero modes, the χ gauge field terms in S are W equation for S gives Z W 1 gauge 4 ˜ μν ˜ μν α β Dμψ ν þ Dνψ μ þ εμναβD ψ ¼ 0; ð2:47Þ SW ½A¼2 Tr d xðFμν þ FμνÞðF þ F Þ; ð2:40Þ and from the η equation, which is positive semidefinite and only vanishes if the field strength Fμν is anti-self-dual, μ Dμψ ¼ 0: ð2:48Þ ˜ Fμν ¼ −Fμν; ð2:41Þ These are the same equations obtained for the gauge perturbation around an instanton in the orthogonal gauge the same nontrivial vacuum configuration that minimizes fixing, so the number of ψ zero modes is also given by the Yang-Mills action in the case of anti-instantons fields. 12 Mk;N. To get a nonvanishing partition function, Witten Hence, Witten’s action has a nontrivial classical minima for assumed that the moduli space consists of discrete, isolated − ˜ Φ 0 F ¼ F and other fields ¼ . Being precise, the evaluation instantons. Precisely, he assumed that the dimension of the of the twisted N ¼ 2 SYM path integral computes quantum moduli space vanishes.13 corrections to classical anti-instantons solutions. In expanding around an isolated instanton, in the weak Another important property of Witten’s theory is the coupling limit g2 → 0, the action is reduced to quadratic invariance under infinitesimal changes in the coupling terms, 2 constant. The variation of ZW with respect to g yields, for similar reasons as in (2.39), 11This condition is equivalent to the Landau gauge, as ∂ ∂ 1 DμAμ ¼ μAμ. It is important to note that one can promote μ to Dμ in this case, in order to show that Aμ and ψ μ obey the same δ 2 Z ¼ δ 2 − hfQ;Xgi ¼ 0; ð2:42Þ g W g g2 equations. 12As Witten himself admits in his paper [8], “this relation where between the fermion equations and the instanton moduli problem was the motivation for introducing precisely this collection of 1 1 1 fermions.” χμν ψ μφ¯ − η φ φ¯ 13Otherwise, a net violation of the Uð1Þ global symmetry X ¼ 4 TrFμν þ 2 Tr μD 4 Tr ½ ; : ð2:43Þ of SW occurs, and ZW vanishes due to the fermion zero modes; see Refs. [8,40]. The dimension of the instanton moduli The Witten partition function, ZW, is independent of the spaces depends on the bundle, E, and the manifold, M. In the gauge coupling g2; therefore, we can evaluate Z in the SUð2Þ gauge theory, it can be written as dimðMÞ¼ W 8 − 3 χ σ weak coupling limit, i.e., in the regime of very small g2,in kðEÞ 2 ð ðMÞþ ðMÞÞ, where kðEÞ is the first Pontryagin (or winding) number of the bundle E and χðMÞ and σðMÞ are the which ZW is dominated by the classical minima. Euler characteristic and signature of M [38]. [For M ¼ R4, M 4 4 The instanton moduli space k;N is defined to be the χ R σ R 0 E ˜ ð Þ¼ ð Þ¼ .] Thus, one can choose a suitable and space of all solutions to F ¼ F for an instanton with a M in order to get a vanishing dimension, dimðMÞ¼0.

085008-6 CORRESPONDENCE BETWEEN THE TWISTED N ¼ 2 SUPER- … PHYS. REV. D 103, 085008 (2021) Z ffiffiffi δ ð2Þ 4 p ΦðbÞ ΦðbÞ ΨðfÞ ΨðfÞ which is exact. Using the exterior derivative, d, we can SW ¼ d x gð DB þ i DF Þ; ð2:49Þ M rewrite (2.54) as

b f where Φð Þ ≡ fA; φ; φ¯ g are the bosonic fields and Ψð Þ ≡ dW0 ¼ ifQ;W1g; ð2:55Þ fη; ψ; χg are the fermionic ones. The Gaussian integral over μ all fields gives where W1 is the 1-form Trðφψ μÞdx . A straightforward calculation gives pPfaffffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðDFÞ ZWjg2→0 ¼ ; ð2:50Þ Q Q detðDBÞ dW1 ¼ if ;W2g;dW2 ¼ if ;W3g; ð2:56Þ where PfaffðDFÞ is the Pfaffian of DF, i.e., the square root dW3 ¼ ifQ;W4g;dW4 ¼ 0; ð2:57Þ of the determinant of DF up to a sign. The supersymmetry relates the eigenvalues of the operators DB and DF. The with relation is a standard result in instanton calculus [41], which yields 1 W2 ¼ Tr ψ ∧ ψ þ iφ ∧ F ; ð2:58Þ Y 2 λi Z j 2→0 ¼ pffiffiffiffiffiffiffi ; ð2:51Þ W g λ 2 i j ij W3 ¼ iTrψ ∧ F; ð2:59Þ with i running over all nonzero eigenvalues of DB ðDFÞ. 1 ðkÞ W4 ¼ − TrF ∧ F; ð2:60Þ −1 nk Therefore, for the kth isolated instanton, ZW ¼ð Þ , 2 where nk ¼ 0 or 1 according to the orientation convention of the moduli space (Donaldson proved the orientability of where ∧ is the wedge product; the total charge is U ¼ 4 − k φ ψ the moduli space, i.e., that the definition of the sign of for each Wk; and , , and F are 0-, 1-, and 2-forms on M, Pfaff D is consistent, without global anomalies [4,8]). In respectively. F is the field strength in the p-form formal- ð FÞ 15 the end, summing over all isolated instantons, ism, defined in Eq. (2.73). Considering now the integral X Z 2 −1 nk ZWjg →0 ¼ ð Þ ; ð2:52Þ IðγÞ¼ Wk; ð2:61Þ k γ which is precisely one of topological invariant for with γ being a k-dimensional homology cycle on M, we 4-manifolds described by Donaldson. have The other metric independent observables are con- Z Z structed in the context of Eq. (2.7). These observables fQ;Ig¼ fQ;Wkg¼i dWk−1 ¼ 0: ð2:62Þ can be generated by exploring the descent equations γ γ defined by the equivariant cohomology, i.e., the supersymmetry δ cohomology. For that, with Ui being the global It proves that IðγÞ is δ-invariant and, then, a possible O charge of the operator i (see footnoteQ 10), it must be observable. To be a global observable of the topological O M Punderstood that, for the observable i i, dimð Þ¼ theory, we just have to prove that IðγÞ is BRST exact, which 14 δ i Ui. The simplest -invariant operator, which does can be immediately verified by taking γ as the boundary ∂β not depend explicitly on the metric and cannot be written as and applying the Stokes theorem, δ Q ðXÞ¼f ;Xg (due to the scaling dimensions), is Z Z Z γ Q 1 Ið Þ¼ Wk ¼ dWk ¼ i ; Wkþ1 : ð2:63Þ φ2 4 ∂β β β W0ðxÞ¼2 Tr ðxÞ;UðW0Þ¼ : ð2:53Þ γ δ We conclude, from Eqs. (2.62) and (2.63), that Ið Þ are the Although W0 is not a -exact operator, taking the derivative global observables of the model as their expectation values of W0 with respect of the coordinates, we find produce metric independent quantities, i.e., topological ∂ invariants for 4-manifolds. Finally, the general path integral Q φψ representation of Donaldson invariants in Witten’sTQFT ∂ W0 ¼ if ; Tr μg; ð2:54Þ xμ takes the form

14To construct topological invariants, the net U charge must 15For the definitions and conventions concerning the p-form equal the dimension of the moduli space; see Refs. [8,17]. formalism used here, see Sec. II B 2.

085008-7 OCTAVIO C. JUNQUEIRA and RODRIGO F. SOBREIRO PHYS. REV. D 103, 085008 (2021) Z Z Y (i) the gauge field symmetry, −SW Zðγ1; …; γrÞ¼ DΦ Wk e γ i i i δ a abωb αa Y Z Aμ ¼ Dμ þ μ; ð2:68Þ ¼ Wk ; ð2:64Þ γ i i i (ii) the topological parameter symmetry,

a ab b with moduli space dimension δαμ ¼ Dμ λ ; ð2:69Þ

Xr ab ab abc c where Dμ ≡ δ ∂μ − gf Aμ are the components M 4 − dimð Þ¼ ð krÞ: ð2:65Þ of the covariant derivative in the adjoint representa- i tion of the Lie group G; fabc are the structure a a a G ω αμ λ One of the beautiful results is the appearing of W4 in the constants of ; and , , and are the infini- descent equations. Up to a sign, the observable tesimal G-valued gauge parameters. As a conse- Z Z quence of (2.69), the field strength also transforms 1 as a gauge field,18 W4 ¼ − F ∧ F ð2:66Þ γ 2 γ δ a − abcωb c abαb Fμν ¼ gf Fμν þ D½μ ν: ð2:70Þ is the Pontryagin action written in the formalism of p-forms. The Pontryagin action, a well-known topological The first parameter (ωa) reflects the usual Yang-Mills a invariant of 4-manifolds, naturally appears as one of the symmetry of S½A, whereas the second one (αμ) is the Donaldson polynomials, with a trivial winding number in topological shift associated to the fact that S½A is a this case, since UðW4Þ¼0, and consequently the dimen- topological invariant, i.e., invariant under continuous sion of the moduli space vanishes. deformations. The third gauge parameter (λa) is due to an internal ambiguity present in the gauge transformation B. Baulieu-Singer off-shell approach of the gauge field (2.68). The transformation of the gauge Let us now turn to the main properties Baulieu-Singer field is composed by two independent symmetries. If the αa approach for TQFTs [13], which is based on an off-shell space has a boundary, the parameter μðxÞ must vanish at ωa BRST symmetry, built from the gauge fixing of an original this boundary but not ðxÞ, which explains the internal αa action composed of topological invariants. ambiguity described by (2.69) in which μðxÞ is absorbed into ωaðxÞ, and not the other way around [13]. 1. BRST symmetry in topological gauge theories Following the BRST quantization procedure, the gauge parameters present in the gauge transformations (2.68)– The four-dimensional spacetime is assumed to be ωa → a αa → ψ a 16 (2.70) are promoted to ghost fields: c , μ μ, and Euclidean and flat. The non-Abelian topological action λa → φa; ca is the well-known Faddeev-Popov (FP) ghost; S0½A in four-dimensional spacetime representing the topo- a a ψ μ is a topological fermionic ghost; and φ is a bosonic logical invariants is the Pontryagin action17 ghost. The corresponding BRST transformations are Z 1 4 a ˜ a a − ab b ψ a S0½A¼2 d xFμνFμν; ð2:67Þ sAμ ¼ Dμ c þ μ; g sca ¼ fabccbcc þ φa; which labels topologically inequivalent field configura- 2 2 a abc b c ab b tions, as S0½A¼32π k, in which k is the topological sψ μ ¼ gf c ψ μ þ Dμ φ ; charge known as the winding number. We must note that sφa ¼ gfabccbφc; ð2:71Þ the Pontryagin action has two different gauge symmetries to be fixed, these are: from which one can easily check the nilpotency of the BRST operator, 16Throughout this work, we consider flat Euclidean spacetime. Although the topological action is background independent, the s2 ¼ 0; ð2:72Þ gauge-fixing term entails the introduction of a background. Ultimately, background independence is recovered at the level of correlation functions due to BRST symmetry [13,42,43]. by applying two times the BRST operator s on the fields. 17 It is worth mentioning that the action S0½A could encompass Naturally, S0½A is invariant under the BRST transforma- a wide range of topological gauge theories. The Pontryagin action tions (2.71). The nilpotency property of s defines the is the most common case because it can be defined for all semisimple Lie groups. Nevertheless, other cases can also be considered. For instance, Gauss-Bonnet and Nieh-Yang topo- 18The antisymmetrization index notation here employed means − logical gravities can be formulated for orthogonal groups [44]. that, for a generic tensor, S½μν ¼ Sμν Sνμ.

085008-8 CORRESPONDENCE BETWEEN THE TWISTED N ¼ 2 SUPER- … PHYS. REV. D 103, 085008 (2021) cohomology of the theory, which allows for the gauge such that fixing of the Pontryagin action in a BRST-invariant fashion. Furthermore, such a property is related to the geometric 1 F ¼ d˜ A˜ þ ½A;˜ A˜ ; ð2:78Þ structure of the off-shell BRST transformations in non- 2 Abelian topological gauge theories. with the Bianchi identity

2. Geometric interpretation D˜ F ¼ d˜F þ½A;˜ F¼0: ð2:79Þ To simplify equations in the following sections, we will employ again the formalism of differential forms. In this Here, the space is graded as a sum of form degree and ghost formalism, the fields c and φ are 0-forms, ψ is the 1-form number, in which the BRST operator is the exterior ψ μdxμ, and F is the 2-form differential operator in the moduli space direction A=G, where the gauge fields that differ by a gauge transformation 1 are identified. The whole space is then M × A=G, with M F ¼ dA þ A ∧ A ¼ Fμνdxμ ∧ dxν; ð2:73Þ 2 being a four-dimensional manifold. According to the

μ gauges worked out in this paper, M will be a Euclidean where A is the 1-form Aμdx , ∧ is the wedge product which flat space. indicates that the tensor product is completely antisym- In the definition (2.76) and following equations, we are 19 metric, and d is the exterior derivative. With this, we can adding quantities with different form degrees and ghost then write the BRST transformations in the form numbers as though they were of the same nature. Obviously, this is not being done directly. We must see sA ¼ Dc þ ψ; Eqs. (2.78) and (2.79) as an expansion in form degrees and 1 ghost numbers in which the elements with the same nature φ sc ¼ 2 ½c; cþ ; on both sides have to be compared. The relevant cohomology is defined by the cohomology of M × A=G, d˜ 2 ¼ 0, sψ ¼ Dφ þ½c; ψ; being valid without requiring equations of motion. Such a sφ ¼½c; φ: ð2:74Þ geometric structure reveals the BRST off-shell character of the BS approach.21 We will discuss in Sec. II B 5 how the The geometric meaning of the topological BRST trans- universal curvature F generates the same global observ- formations of (2.74) is revealed from the definition of the ables of Witten theory, i.e., the Donaldson polynomials. extended exterior derivative, d˜, as the sum of the ordinary exterior derivative with the BRST operator, 3. Doublet theorem and gauge fixing: Baulieu-Singer gauges d˜ ¼ d þ s; ð2:75Þ Let us recall the doublet theorem [16], which will be and the generalized connection indispensable for fixing the gauge ambiguities without changing the physical content of the theory. Consider a A˜ ¼ A þ c: ð2:76Þ theory that contains a pair of fields or sources that form a doublet, i.e., By direct inspection, one sees that the BRST transforma- 20 ˆ tions can be written in terms of the generalized curvature δX i ¼ Yi; δˆY ¼ 0; ð2:80Þ F ¼ F þ ψ þ φ; ð2:77Þ i where i is a generic index and δˆ is a fermionic nilpotent 19 X The exterior derivative operation in the space of smooth operator. The field (source) i is assumed to be fermionic. Λ ∶Λ → Λ ω ˆ p-forms, p, d p pþ1, on a generic p-form p, As the operator δ increases the ghost number in one unity, i1 i2 i ω ¼ ω … dx ∧ dx ∧ dx p , is locally defined by X Y p i1;i2; ;ip by definition, and if i is an anticommuting quantity, i is ∂ωi ;i ;…;i ω 1 2 p j ∧ i1 ∧ i2 ∧ ip ω a commuting one. The doublet structure of ðX ; Y Þ in d p ¼ ∂xj dx dx dx dx , with p being i i a p-form and dωp being a (p þ 1)-form. It follows that the Eq. (2.80) implies that such fields (or sources) belong to the exterior derivative is nilpotent, d2 ¼ 0, due to the antisymmetric trivial part of the cohomology of δˆ. The proof is as follows. property of the indices. One assumes that s anticommutes with d, First, we define the operators fs; dg¼0. 20The nature of φ as the “curvature” in the in instanton moduli space direction is implicit in the BRST transformation 21For a detailed study on the geometric interpretation of the of the FP ghost, which can be rewritten in the geometric form universal fiber bundle and its curvature, we suggest, for instance, 1 sc þ 2 ½c; c¼φ. Refs. [21,45].

085008-9 OCTAVIO C. JUNQUEIRA and RODRIGO F. SOBREIRO PHYS. REV. D 103, 085008 (2021) Z ˆ ˆ ∂ ∂ cohomology of δ, being thus nonphysical—for a more N ¼ dx X i þ Yi ; ð2:81Þ ∂X i ∂Yi complete analysis, see for instance Refs. [16,46]. Z To fix the three gauge symmetries of the non-Abelian ˆ ∂ topological theory (2.68)–(2.70), we introduce the three A ¼ dxX i ; ð2:82Þ ∂Yi BRST doublets

ˆ ∂ sc¯a ¼ ba;sba ¼ 0; δ ¼ Yi ; ð2:83Þ ∂X i a a a sχ¯μν ¼ Bμν;sBμν ¼ 0; which obey the commutation relations sφ¯ a ¼ η¯a;sη¯a ¼ 0; ð2:92Þ

ˆ ˆ ˆ a a fδ; Ag¼N; ð2:84Þ where χ¯μν and Bμν are (anti-)self-dual fields according to the (negative) positive sign; see (2.95) below. The G-valued ˆ ˆ a a ½δ; N¼0; ð2:85Þ Lagrange multiplier fields b , Bμν, and η¯ have, respectively, ghost numbers 0, 0, and −1, while the antighost fields c¯a, ˆ ˆ2 a a where δ is a nilpotent operator as it is fermionic, δ ¼ 0. χ¯μν, and φ¯ have ghost numbers −1, −1, and −2.(For The operator Nˆ is the counting operator. With △ being a completeness and further use, the quantum numbers of all polynomial in the fields, sources, and parameters, the fields are displayed in Table I.) cohomology of the nilpotent operator δˆ, as we know, is Working in Baulieu-Singer gauges amounts to consi- given by the solutions of dering the constraints [13] ˆ 1 δ△ ¼ 0; ð2:86Þ ∂ a − ρ a μAμ ¼ 2 1b ; ð2:93Þ which is not exact, i.e., which cannot be written in the form ab a Dμ ψ μ ¼ 0; ð2:94Þ △ ¼ δˆΣ: ð2:87Þ 1 a ˜ a a Fμν Fμν ¼ − ρ2Bμν; ð2:95Þ The general expression of △ is then 2 ˜ ρ ρ △ ¼ △ þ δˆΣ; ð2:88Þ where 1 and 2 are real gauge parameters. In a few words, beyond the gauge fixing of the topological ghost (2.94),we ˜ where △ belongs to the nontrivial part of the cohomology; must interpret the requirement of two extra gauge fixings ˜ due to the fact that the gauge field possesses two inde- in other words, it is closed, δˆ △ ¼ 0, but not exact, ˜ ˆ ˜ ˆ pendent gauge symmetries. In this sense, condition (2.93) △ ≠ δ Σ. One can expand △ in eigenvectors of N, a ab b fixes the usual Yang-Mills symmetry δAμ ¼ Dμ ω , and X a a the second one (2.95) fixes the topological shift δAμ ¼ αμ. △ ¼ △ ; ð2:89Þ n The (anti-)self-dual condition for the field strength (in the n≥0 limit ρ2 → 0) is convenient to identify the well-known ˆ observables of topological theories for 4-manifolds (see such that N△n ¼ n△n, where n is the total number of X i and Y in △ . Such an expansion is consistent as each △ is Ref. [17]) given by the Donaldson invariants [2,3], which i n n are described in terms of the instantons. a polynomial in X and Y , and δˆ△ ¼ 0 for ∀ n ≥ 1, i i n The partition functional of the topological action in BS according to (2.80) and the commuting properties of X and i gauges (2.93) takes the form Y . Finally, rewriting (2.89) as i Z X − 1 D D¯Dψ Dχ¯ D DφDφ¯ Dη SBS ˆ ZBS ¼ c c μ μν Bμν e ; ð2:96Þ △ ¼ △0 þ N△n; ð2:90Þ n≥1 n whereby and then using the commuting relation (2.84), we get X ˆ 1 ˆ TABLE I. Quantum numbers of the fields. △ ¼ △0 þ δ A△n ; ð2:91Þ n≥1 n Field A ψ c φ cb¯ φ¯ η¯ χ¯ B Dim 1 1 0 0 2 2 2 2 2 2 which shows that all terms which contain at least one field Ghost no 0112−1 0 −2 −1 −1 0 (source) of the doublet never enter the nontrivial part of the

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BS SBS ¼ S0½AþSgf ; ð2:97Þ

BS with Sgf being the gauge-fixing action which belongs to trivial part of the cohomology, given by Z 1 1 BS 4 ˜ S s d x χ¯μν Fμν Fμν ρ2Bμν φ¯ Dμψ μ c¯ ∂μAμ − ρ1b gf ¼ Tr þ 2 þ þ 2 Z 1 1 4 ˜ ρ χ¯ ψ ε ψ ¼ Tr d x Bμν Fμν Fμν þ 2 2Bμν þ μν D½μ ν 2 μναβD½α β 1 − χ¯ ˜ η ψ φ¯ ψ ψ φ¯ φ − ∂ − ρ − ¯∂ − ¯∂ ψ μν½c; Fμν Fμνþ Dμ μ þ ½ μ; μþ DμDμ b μAμ 2 1b c μDμc c μ μ : ð2:98Þ

1 2 A key observation is that, for ρ1 ¼ ρ2 ¼ 1, one can △ð Þ △ð Þ 0 s 4−form þ d 3−form ¼ : ð2:102Þ eliminate the topological term S0½A, i.e., the Pontryagin action, by integrating out the field Bμν, such that Therefore, radiative corrections which could break the topological property (2.101) at the quantum level are not ˜ 1 Tr BμνðFμν þ FμνÞþ2 BμνBμν expected. The formal proof of the absence of gauge anomalies to all orders in the topological BS theory is ˜ → TrfFμνFμν þ FμνFμνg: ð2:99Þ achieved by employing the isomorphism described in Refs. [22,47]. In this case, we obtain a classical topological action which is equivalent to a Yang-Mills action plus ghost interactions. 4. Absence of gauge anomalies Such an action, however, does not produce local observables as the cohomology of the theory remains the same, as The proof of the absence of gauge anomalies for the we will discuss in more detail later in Sec. II B 5. Slavnov-Taylor identity, Another important property is that the Green’s functions of local observables in (2.96) do not depend on the choice S S 0; : g ð Þ¼ ð2 103Þ of the background metric [13]. Let SBS be an action with gþδg metric choice gμν and S be the same action up to the BS consists in proving that the cohomology of S is empty. In δ change of gμν into gμν þ gμν. As the only terms depending the equation above, S is the classical action for a given on the metric belong to the trivial part of cohomology, we δ gauge choice, and conclude immediately that Sg and Sgþ g only differ by a BS BS Z BRST-exact term, δ S d4x sΦI ; 2:104 Z ¼ ð Þ δΦI ð Þ g − gþδg 4 △ð−1Þ SBS SBS ¼ s d x ; ð2:100Þ where ΦI represents all fields. As S is a Ward identity, in where △ð−1Þ is a polynomial of the fields, with ghost the absence of anomalies, the symmetry (2.103) is also number −1. It means that the expectation values of local valid at the quantum level, i.e., SðΓÞ¼0, with Γ being the operators are the same if computed with a background quantum action in loop expansion. In Eq. (2.104), sΦI ΦI metric gμν or gμν þ δgμν, represents the BRST transformation of each field . The ¯ χ¯ φ¯ η¯ fields c, b, μν, Bμν, , and transform as doublets, δ Y cf. Eq. (2.92). Changing the variables according to the Oα ðφ Þ ¼ 0; ð2:101Þ δ p i redefinitions gμν p

O φ ψ → ψ 0 ¼ ψ − Dc; where αp ð iÞ are functional operators of the quantum fields φ x ; see Eq. (2.6). An anomaly in the topological 1 ið Þ φ → φ0 φ − BRST symmetry would break the above equation. ¼ 2 ½c; c; ð2:105Þ However, there is no 4-form with ghost number 1, △ð1Þ 4−form, defined modulo s- and d-exact terms which obeys the BRST transformations (2.74) are reduced to the doublet (cf. Ref. [13]) transformations

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sA ¼ ψ 0; space M × A=G, where the metric independent observables, known as Chern classes, are constructed in terms of sψ 0 0; ¼ the universal curvature F (2.80). The Donaldson poly- sc ¼ φ0; nomials are naturally recovered, characterized by the so- called equivariant cohomology, which relates the BS sφ0 ¼ 0: ð2:106Þ approach to Witten’s theory at the level of observables. It configures a reduced transformation in which the nonlinear part of the BRST transformations in the Slavnov-Taylor 5. Equivariant cohomology and global observables identity was eliminated. The complete transformation in Witten’s topological theory is constructed without fixing this space is given by the reduced operator its remaining ordinary Yang-Mills gauge symmetry. The Z A G δ theory is developed in the instanton moduli space = .A 4 0I ðWÞ S ¼ d xðsΦ Þ ; ð2:107Þ Oα doublet δΦ0I generic observable of his theory, i , is naturally gauge invariant under Yang-Mills gauge transformations, 0 0 0 where Φ ¼fA; ψ ;c;φ ; c;¯ b; χ¯μν;Bμν; φ¯ ; ηg, which is ðWÞ S s Oα ¼ 0; ð2:111Þ composed of five doublets. It means that doublet has YM i vanishing cohomology (H), where sYM is the nilpotent BRST operator related to the HðS Þ¼⊘; ð2:108Þ ordinary Yang-Mills symmetry, i.e., without including the doublet topological shift, namely, in other words, that any polynomial of the fields Φ0, △ðΦ0Þ, which satisfies sYMAμ ¼ Dμc; Φ Φ sYM adj ¼½c; adj; ð2:112Þ S △ Φ0 0 doubletð ð ÞÞ ¼ ; ð2:109Þ Φ where adj is a generic field in adjoint representation. We S belonging to the trivial part of the cohomology of doublet conclude that we can add an ordinary Yang-Mills gauge (see the doublet theorem in the previous section). The crucial transformation (in the A=G direction) to Witten fermionic S point here is the fact that the cohomology of in the space of symmetry based on the “topological shift” δAμ ∼ ψ μ, local integrated functionals in the fields and sources is H S δ → δ δ isomorphic to a subspace of ð doubletÞ. Consequently, eq ¼ þ sYM; ð2:113Þ S has also vanishing cohomology [22,29,47], in such a way that the descent equations for δ ∼ fQ; ·g will HðSÞ¼⊘: ð2:110Þ remain the same; see (2.34) and (2.56)–(2.60). The operator δ eq is nilpotent when acting on gauge-invariant quantities The result (2.110) shows that there is no room for an under Yang-Mills transformations, defining thus a coho- anomaly in the Salvnov-Taylor identity (2.103). All counter- mology in a space where the fields that differ by a Yang- terms at the quantum level will belong to the trivial part of Mills gauge transformations are identified, known as cohomology, and the condition (2.102) for the existence of equivariant cohomology. Such a property indicates that an anomaly capable of breaking the topological property there is a link between Witten’s theory and the BS approach (2.101) never occurs. Because of the algebraic structure of in which the BRST operator, s, is naturally defined by the theory, Eq. (2.110) proves that all Ward identities are free taking into account the topological shift and the ordinary of gauge anomalies, cf. Ref. [22]. As a consequence of this Yang-Mills transformation in a single formalism. result, the background metric independence is valid to all To prove the link between both approaches, we must orders in perturbation theory. remember that the universal curvature in the space The second point, and not least, is the conclusion that the M × A=G is given by the sum F ¼ F þ ψ þ φ. The BS theory has no local observables. Because of its difference between the on-shell BRST operator, s, and vanishing cohomology (2.110), all BRST-invariant quan- the Witten fermionic symmetry, δ, for X ¼ðF; ψ; φÞ is of tities must belong to the nonphysical (or trivial) part of the the form cohomology of s, and the only possible observables are the global ones, i.e., topological invariants for 4-manifolds. sX ¼ δX þ½X;c; ð2:114Þ Such observables are characterized by the cohomology of s [29,48], in which the observables are globally defined in in other words, in the space of the fields ðF; ψ; φÞ, s and δ agreement with the supersymmetric formulation of J. H. differ by an ordinary Yang-Mills transformation, as Horne [49]. A simple way to identify theses observables is ðF; ψ; φÞ transform in the adjoint representation of the accomplished by studying the cohomology of the extended gauge group. These fields are the only ones we need to

085008-12 CORRESPONDENCE BETWEEN THE TWISTED N ¼ 2 SUPER- … PHYS. REV. D 103, 085008 (2021) obtain the Donaldson polynomials as the observables of the theories possess the same observables given by the BS theory, since in the space M × A=G they are constructed Donaldson invariants (2.64). in terms of F. This allows for identifying the equivariant The fact that the observables in the BS approach are δ ≡ operator with the BRST one, eq s, according to the naturally invariant under ordinary Yang-Mills symmetry construction of the observables in Witten’s and BS theories. should not seem surprising, as the nth Chern class is Yang- To understand the above statement, we must invoke the Mills invariant itself (2.115) since F transforms in the ˜ adjoint representation of the gauge group. Equation (2.116) nth Chern class, Wn, defined in terms of the universal curvature by provides a powerful tool to obtain Donaldson polynomials for any ghost number. One must note that we do not have to worry about with the independence of Faddeev-Popov W˜ ¼ TrðF ∧ F ∧ ∧ FÞ; ð2:115Þ n |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ghosts in order to construct the observables in the BS n times approach. Although the gauge-fixed BS action has FP 1 2 3 … ghosts due to the gauge fixing of the Yang-Mills ambiguity, where n ¼f ; ; ; g is the number of wedge products. ¯ W˜ ˜ 22 the ðc; cÞ independence of n is a direct consequence of Wn represents the most general observables of BS theory. ˜ the fact that the universal curvature of the space M × A=G The Weyl theorem [21] ensures that Wn is closed with ˜ does not depend on FP ghosts, but only on F, and the respect to the extended differential operator d ¼ d þ s ghosts ψ and φ. [13,50], i.e., In the weak coupling limit of the twisted N ¼ 2, the observables of both theories are undoubtedly the same: the ˜ ˜ dWn ¼ 0: ð2:116Þ topological Donaldson invariants [21–23]. We might ask if the quantum behavior is also compatible, once BS and If we choose the first Chern class Witten actions does not differ by a BRST-exact term,

˜ W1 ¼ TrðF ∧ FÞ; ð2:117Þ − Σ ≠ SBS SW ¼ G sð Þ: ð2:123Þ the expansion in ghost numbers of Eq. (2.116) yields The relation above does not depend on the gauge choice. sTrðF ∧ FÞ¼dTrð−2ψ ∧ FÞ; ð2:118Þ Consequently, we cannot say, in principle, that BS and Witten partition functions are equivalent at quantum level, 1 since ψ ∧ − ψ ∧ ψ − φ sTrð FÞ¼dTr 2 F ; ð2:119Þ Z Z − − −Σ Z DΦe SBS DΦe SW G ; 2:124 sTrðψ ∧ ψ þ 2φFÞ¼dTrð2ψφÞ; ð2:120Þ BS ¼ ¼ ð Þ 1 ψφ − φφ Σ ≠ sTrð Þ¼dTr ; ð2:121Þ whereinR G is not s exact. At first view, ZBS ZW ¼ 2 −S DΦe W . The fact that ΣG ≠ sð Þ opens the possibility for both theories to have different quantum properties. The φφ 0 sTrð Þ¼ ; ð2:122Þ one-loop exactness of twisted N ¼ 2 SYM β function, for instance, is a well-known result in the literature [34].We – which are the same descent equations obtained in (2.56) will now analyze the Ward identities of the BS theory in δ δ (2.60) following Witten analysis, only replacing (or eq) self-dual Landau gauges, in order to compare the quantum by s, proving that Baulieu-Singer and Witten’s topological properties of the DW and BS theories.

22 It is not possible to construct topological observables III. QUANTUM PROPERTIES OF BS THEORY using the Hodge product, as it is metric dependent. For this reason, we never obtain Yang-Mills terms of the type IN THE SELF-DUAL LANDAU GAUGES μν νσ μ fTrðFμνF Þ; TrðFμνF F σÞ; g, without Levi-Civita tensors In this section, we will summarize the quantum proper- in the internal product, in the place of metric tensors. Moreover, H ties of BS theory in the self-dual Landau (SDL) gauges.23 ðCÞ i Aμdxμ P C the Wilson loop WP ¼ Trf e g is not an observable in Extra details can be found in Refs. [24–27,51]. the non-Abelian topological BS case, as it is not gauge invariant due to the topological shift symmetry. In any case, it does not make sense to discuss confinement in the BS theory, as it is not 23For simplicity, throughout the text, we will refer to the confining to any energy scale. Thence, the only possibilities for Baulieu-Singer theory in the self-dual Landau gauges as self-dual ˜ topological invariants are the wedge products in Wn. BS theory.

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A. Absence of radiative corrections Through the introduction of the three BRST doublets Working in the self-dual Landau gauges amounts to described in Eq. (2.92), the complete gauge-fixed topo- considering the constraints [27] logical action in SDL gauges takes the form

a ∂μAμ ¼ 0; ð3:1Þ S½Φ¼S0½AþSgf½Φ; ð3:4Þ a ∂μψ μ ¼ 0; ð3:2Þ

a ˜ a Fμν Fμν 0: 3:3 ¼ ð Þ with S0½A standing for the Pontryagin action and

Z 1 Φ 4 ¯a∂ a χ¯a a ˜ a φ¯ a∂ ψ a Sgf½ ¼s d z c μAμ þ 2 μνðFμν FμνÞþ μ μ Z 1 4 a∂ a a a ˜ a η¯a − ¯a ∂ ψ a ¯a∂ ab b ¼ d z b μAμ þ 2 BμνðFμν FμνÞþð c Þ μ μ þ c μDμ c þ 1 1 − abcχ¯a b c ˜ c − χ¯a δ δ ϵ abψ b φ¯ a∂ abφb 2 gf μνc ðFμν FμνÞ μν μα νβ 2 μναβ Dα β þ μDμ þ abc a b c þ gf φ¯ ∂μðc ψ μÞ : ð3:5Þ

This action possesses a rich set of symmetries; see Ref. [24] and Appendix. To control the nonlinearity of the Slavnov- Taylor identity [Eq. (A1)] and the bosonic symmetry T [Eq. (A13)], we have to introduce external sources given by the following three BRST doublets [27]:

a a a sτμ ¼ Ωμ;sΩμ ¼ 0; sEa ¼ La;sLa ¼ 0; a a a sΛμν ¼ Kμν;sKμν ¼ 0: ð3:6Þ

The respective external action, written as a BRST-exact contribution preserving the physical content of theory, takes the form Z g 4 τa ab b abc a b c abcΛa bχ¯c Sext ¼ s d z μDμ c þ 2 f E c c þ gf μνc μν Z g 4 Ωa ab b abc a b c abc a bχ¯ c τa abφb abc bψ c ¼ d z μDμ c þ 2 f L c c þ gf Kμνc μν þ μðDμ þ gf c μÞ abc a b c abc a b c abc a b c þ gf E c φ þ gf Λμνc Bμν − gf Λμνφ χ¯μν g2 − abc bdeΛa χ¯c d e 2 f f μν μνc c ; ð3:7Þ with the corresponding quantum number of the external The introduction of the external action does not break the sources displayed in Table II. Therefore, the full classical original symmetries, and the physical limit is obtained by action to be quantized is setting the external sources to zero [16]. One of the symmetries is of particular interest to us: Σ Φ Φ Φ ½ ¼S0½AþSgf½ þSext½ : ð3:8Þ the vector supersymmetry described by Eq. (A12), cf. Refs. [24,27]. By applying BRST-algebraic renormal- TABLE II. Quantum numbers of the external sources. ization techniques [16], and disregarding Gribov ambiguities, it was proved in Ref. [24], with the help of Feynman Source τ Ω ELΛ K diagrams, that all two-point functions are tree-level exact, Dim 3 3 4 4 2 2 as a consequence of the Ward identities of the model. In Ghost no −2 −1 −3 −2 −1 0 particular, as a consequence of the vector supersymmetry

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(A11), the gauge field propagator vanishes to all orders in exact, but all n-point Green functions of the model do not perturbation theory, receive any radiative corrections. This is a direct conse- a b quence of the null gauge propagator (3.9) together with hAμðpÞAν ðqÞi ¼ 0: ð3:9Þ the vertex structure of the full action (3.8). Following the In Ref. [26], this result was generalized: not only are the Feynman rules notation of Ref. [26], we represent the two-point functions of the self-dual BS theory tree-level relevant propagators by

ð3:10Þ

The relevant vertices are represented by

ð3:11Þ

Using these diagrams, one identifies a kind of cascade effect in which the number of internal A legs always increases when trying to construct loop Feynman diagrams, according to the diagram below,

ð3:12Þ

This makes it impossible to close loops without using the hAAi propagator,24 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 which possesses A legs is φ¯ cψ. However, the φ¯ leg could only propagate to the vertex φ¯ Aφ through hφφ¯ i; the c leg could only propagate to cAc¯ through hcc¯ i; and the ψ leg propgates to the vertices χ¯Aψ, χ¯cA,orχ¯cAA through hψχ¯i (hηψ¯ i is not considered because there is no vertex η¯). All possible branches produce at least one remaining internal A leg, and the cascade effect is not avoided, as represented by the diagrams

24The formal proof of this result can be found in Ref. [26].

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ð3:13Þ

c The apparently only nonzero correlation functions are of ΣðΦ0; J 0;g0Þ¼ΣðΦ; J ;gÞþϵΣ ðΦ; J ;gÞ; ð3:17Þ the type with hBBB…bbi¼hsχ¯BB…bbi¼hsðχ¯BB…bbÞi; ð3:14Þ Z c 4 a a a ab b abc a b c Σ ¼ a d xðBμνFμν − 2χ¯μνDμ ψ ν − gf χ¯μνc FμνÞ; a a i.e., with external Bμν or b fields. But (3.14) automatically vanishes as it is BRST exact. ð3:18Þ In a few words, using perturbative techniques, one sees that the tree-level exactness of the BS in the self-dual whereby the resulting Z factors obey the system of equations gauges is a consequence of the vector supersymmetry and BRST symmetry. 1=2 −1=2 −1 ZA ¼ Zb ¼ Zg ; 1=2 1=2 −1=2 −1=2 Z¯ ¼ Zη¯ ¼ Zψ ¼ ZΩ ¼ Z ; B. Renormalization ambiguity c c 1=2 −1=2 −1 −1 Once we have at our disposal all Ward identities, we are Zφ¯ ¼ Zφ ¼ Zτ ¼ ZL ¼ Zg Zc ; able to construct the most general counterterm Σc that can −2 −3=2 ZE ¼ Zg Zc ; absorb the divergences arising in the evaluation of −1 −1=2 −1=2 Feynman graphs. Because of the triviality of the BRST ZK ¼ Zg Zc Zχ¯ ; c cohomology [24,27], Σ belongs to trivial part of the BRST −2 −1 −1=2 ZΛ ¼ Z Z Zχ¯ ; cohomology. The fact that the BS theory is quantum g c 1=2 1=2 1=2 1=2 1 ϵ stable is a well-known result in the literature [24,27,51]. ZB ZA ¼ Zχ¯ Zc ¼ þ a; ð3:19Þ Reference [24] introduced an extra nonlinear bosonic symmetry that relates the topological ghost with the with the independent renormalization parameter denoted by Faddeev-Popov one (among other transformations involv- a. Because of the recursive nature of algebraic renormaliza- ing other fields) through the transformation tion [16], the results (3.19) show that the model is renormalizable to all orders in perturbation theory. a ab b δψ μ ↦ Dμ c ; ð3:15Þ From the algebraic analysis so far, we cannot prove that Zg ¼ 1, as suggested by the tree-level exactness obtained described by the Ward identity T in Eq. (A13). Taking into via the study of the Feynman diagrams. The system of account this extra symmetry, from the multiplicative Z factors (3.19) is undetermined. As we can easily see, the redefinition of the fields, sources, and parameters of the number of equations n and the number of variables z (the model, Z factors) are related by z ¼ n þ 2, indicating that there is a kind of freedom in the choice of two of the Z factors. 1=2 In Ref. [25], the origin of such an ambiguity was Φ0 ¼ Z Φ; Φ explained: it is due to the absence of a kinetic gauge field a a a a a a a a a a Φ0 ¼fAμ; ψ μ;c ; c¯ ; φ ; φ¯ ;b ; η¯ ; χ¯μν;Bμνg; term out from the trivial BRST cohomology and due to the a a a a a a absence of discrete symmetries involving the ghost fields. J 0 ¼ ZJ J ; J ¼fτμ; Ωμ;E ;L ; Λμν;Kμνg; The symmetries of the SDL gauges eliminate the kinetic g0 ¼ Zgg; ð3:16Þ term of the Faddeev-Popov ghost in the counterterm, i.e., one proves the quantum stability of the BS theory in self- ZcZc¯ ¼ 1: ð3:20Þ dual gauges with only one independent renormalization parameter, i.e., that the quantum action Γ ≡ ΣðΦ0; J 0;g0Þ Moreover, from the gauge-ghost vertex (cAc¯ ), which is also at one loop is of the form absent in the counterterm, we achieve

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1=2 1 transformation with trivial Jacobian, capable of recovering ZgZA ¼ : ð3:21Þ such symmetries. The two relations (3.20) and (3.21) are decoupled; in other β words, only by determining Zc or Zc¯, we do not get any IV. PERTURBATIVE FUNCTIONS Z Z information about g or A. As there are no kinetic terms Our aim in this section is to compare the DW and BS β for the gauge field in the classical action (3.8), the functions to prove that these topological gauge theories are independent determination of ZA becomes impossible. not completely equivalent at the quantum level and then The same analysis can be performed for the bosonic and identify in which energy regimes the correspondence could topological ghosts; see Ref. [25]. occur. The DW β function is well known [34,52], as we will Extra information is then required in order to determine briefly describe. It still lacks the task of determining the the system (3.19). In the ordinary Yang-Mills theories self-dual BS one to perform the comparison. (quantized in the Landau gauge), Zc ¼ Zc¯ , which relies on the discrete symmetry A. Twisted N = 2 super-Yang-Mills theory β ca → c¯a; In Ref. [52], the authors have computed the one-loop function of the DW theory. Later, the authors of Ref. [34] a a c¯ → −c : ð3:22Þ employed the algebraic renormalization techniques to also study DW theory and prove that the β function of twisted This condition, together with the Faddeev-Popov ghost N¼2 N ¼ 2 SYM (βg ) is one-loop exact. The reason is that the kinetic term, is sufficient to determine Z and Z¯ . It is easy 2 c c composite operator Trφ ðxÞ does not renormalize [34].For to see that the action (3.8) does not obey such a symmetry. that, they considered the fact that the operator dμν, defined Discrete symmetries between the other ghosts of topologi- a a a a in expression (2.15), is redundant [53]. Thence, the cal Yang-Mills theories (φ and φ¯ ; ψ μ and χ¯μν) are also not definition of an extended BRST operator, namely, present in (3.8), which explains the second ambiguity. In Witten’s theory, such an ambiguity will not appear by this S ¼ s þ ωδ þ εμδμ; ð4:1Þ reasoning since Witten’s action contains discrete sym- YM metries ensured by the time-reversal symmetry (3.22) in could be employed. In expression (4.1), ω and εμ are global Landau gauge, together with ghosts, and δ and δμ were defined in Eqs. (2.13) and (2.14). The relevant property of the operator S is that it is on-shell φ → φ¯ ; φ¯ → φ; nilpotent in the space of integrated local functionals, since ψ μ → χμ; χμ → ψ μ; ð3:23Þ 2 S ¼ ωεμ∂μ þ equations of motion: ð4:2Þ whereby the components of χμ are defined as We point out that this extended BRST construction 1 requires the equations of motion to obtain a nilpotent χ0 ≡ η; χ ≡ χ0 ε χ ; : i i ¼ 2 ijk jk ð3 24Þ BRST operator—a standard behavior of Witten’s theory, representing a different quantization scheme of the BS implying a “particle-antiparticle” relationship between c¯ theory. Considering the nonrenormalization of Trφ2 and the and c, φ¯ and φ, and ψ μ and χμ, as demonstrated in Ref. [52]. on-shell cohomology of the operator defined in Eq. (4.2), This ambiguity is also present in a generalized class of the result is that the β function only receives contributions renormalizable gauges [25]. In fact, one could relate this to one-loop order and is given by ambiguity with the fact that all local degrees of freedom are N¼2 3 nonphysical (e.g., the gauge field propagator is totally βg ¼ −Kg ; ð4:3Þ gauge dependent). In self-dual Landau gauges, where the N¼2 vector supersymmetry is present, the Feynman diagram with K being a constant. The computation of βg via structure indicates that imposing Zc ¼ Zc¯ and Zφ ¼ Zφ¯ is Feynman diagrams is performed in Ref. [52] by evaluating consistent with the model. Hence, the Z-factor system the one-loop contributions to the gauge field propagator 1 (3.19) would naturally yield Zg ¼ , in accordance with the [where the Landau gauge was used to fix the Yang-Mills absence of radiative corrections in this gauge choice. symmetry of Witten action (2.30)]. The behavior of one- However, without recovering the discrete symmetries loop exactness of the N ¼ 2 β function had been usually between the ghosts, such an imposition seems to be understood in terms of the analogous Adler-Bardeen artificial. As we will see later, the renormalization ambi- theorem for the Uð1Þ axial current in the N ¼ 2 SYM [9]. guity can be solved in the SDL gauges, i.e., the discrete Despite the independence of the Witten partition func- symmetries can be reconstructed, due to the triviality tion under changes in the coupling constant, the result (4.3) of the Gribov copies [28], which allows for a nonlocal should not be surprising. In the twisted version, we can see

085008-17 OCTAVIO C. JUNQUEIRA and RODRIGO F. SOBREIRO PHYS. REV. D 103, 085008 (2021) that the trace of the energy momentum is not zero but given is distinct from Brooks et al. construction because their by (see Ref. [8]) methods are based on different BRST quantization schemes, with different cohomological properties. We do μν φ μφ¯ − 2 ηψ μ 2 φ¯ ψ ψ μ not expect a similar result in the BS theory. According to gμνT ¼ Tr Dμ D iDμ þ i ½ μ; BS the cohomology of the BS model, if the βg is not zero, we ˜ 2 2 1 should find that it is TrðFμν FμνÞ rather than TrFμν 2 φ η η φ φ¯ 2 þ i ½ ; þ2 ½ ; ; ð4:4Þ which is renormalized.25 In this way, the minima of the effective action preserves the instanton configuration at the meaning that SW is not conformally invariant under the quantum level, according to the global degrees of freedom transformation of the instantons, which defines the observables of the BS theory—the Donaldson invariants. β δgμν ¼ hðxÞgμν; ð4:5Þ A possible discrepancy between functions for the BS approach in different gauge choices cannot be attributed to for an arbitrary real function hðxÞ on M. Nonetheless, the a gauge anomaly, since it is forbidden in these models due trace of the energy-momentum tensor can be written as a to the trivial BRST cohomology [54], cf. Eq. (2.102).For ab b total covariant divergence, instance, if we would have chosen the gauge Dμ ψ μ ¼ 0 for the topological ghost, with the covariant derivative instead μν μ μ gμνT ¼ Dμ½Trðφ¯ D φ − 2iηψ Þ; ð4:6Þ of the ordinary one, the vector supersymmetry would be broken, and the gauge propagator would not vanish to all which means that SW is invariant under a global rescaling of orders anymore. In ordinary Yang-Mills theories, the β the metric, δgμν ¼ wgμν, with w constant [8]. The N ¼ 2 β function is an on-shell gauge-invariant physical quantity. function only vanishes if we take the weak coupling limit Nonetheless, in gauge-fixed BRST topological theories of g2 → 0, BS type, the coupling constant is nonphysical, introduced in the trivial part of the cohomology, together with the N¼2 2 βg ðg → 0Þ¼0: ð4:7Þ gauge-fixing action. In these terms, it is not contradictory that the β function is gauge dependent as it computes the In this limit, the possibility of loop corrections to the running of a nonphysical parameter. We must observe that effective action is eliminated, and the Donaldson poly- the physical observables of the theory, the Donaldson nomials are reproduced as the observables of the theory. invariants, naturally do not depend on the gauge coupling. There is no Ward identity, or a particular property of the So that, there is no inconsistency that the observables of vertices and propagators of SW, capable of eliminating this kind of theory, described by topological invariants these quantum corrections for an arbitrary energy regime— (exact numbers) do not depend on the coupling constant, this situation is distinct from the BS theory in the self-dual and consequently on its running. Thence, g is an unob- Landau gauges. servable quantity. As DW and BS theories possess the same observables, B. Baulieu-Singer topological theory we should then consider the instanton configuration not as a As suggested by the tree-level exactness of the BS theory gauge-fixing condition but as a physical requirement in in the self-dual Landau gauges, according to the analysis of order to obtain the correct degrees of freedom that the Feynman diagrams performed in Sec. III, we will correspond to the description of all global observables. formally prove that the self-dual BS theory is conformal. Furthermore, the Atiyah-Singer index theorem [37] deter- Before proving the vanishing of the BS β function in this mines the dimension of the instanton moduli space, in — gauge, we will first discuss the nonphysical character of the which the Donaldson invariants are defined see coupling constant in this off-shell approach, since g is Refs. [56,57] for some exact instanton solutions, whose introduced in the BS theory as a gauge parameter, in the properties cannot be attributed to gauge artifacts. trivial part of the BRST cohomology. 2. Conformal structure of the self-dual gauges β 1. Nonphysical character of the function To prove that the algebraic renormalization is in harmony in the off-shell approach with the Feynman diagram analysis in the self-dual Landau In Ref. [52], Brooks et al. argued that only one counter- gauges, which shows that the BS model does not receive term is required in the on-shell Witten theory, specifically radiative corrections in this gauge, we must invoke a result 2 for the Yang-Mills term TrFμν. In any case, the Donaldson invariants are described by DW theory in the weak coupling 25 2 See Ref. [54], where Birmingham et al. employ the Batalin- limit g → 0, where the theory is dominated by the classical Vilkovisky algorithm [55]—a similar quantization to the BS minima. On the other hand, it is evident that the BS theory approach, i.e., with similar cohomological properties.

085008-18 CORRESPONDENCE BETWEEN THE TWISTED N ¼ 2 SUPER- … PHYS. REV. D 103, 085008 (2021) recently published in Ref. [28]. In this work, it was can always find two configurations A˜ and A obeying the demonstrated that the Gribov ambiguities [58,59] are Landau gauge condition and yet being related by a gauge 26 inoffensive in the self-dual BS theory. The quantization transformation. At the infinitesimal level, the condition for a of this model in a local section of the field space where the configuration A to have a Gribov copy A˜ is that the FP eigenvalues of the Faddeev-Popov determinant are positive operator develops zero modes through is equivalent to its quantization in the whole field space. In other words, the introduction of the Gribov horizon does −∂μDμθ ¼ 0; ð4:12Þ not affect the dynamics of the BS theory in SDL gauges, as its correspondent gap equation forbids the introduction of a with θa taken as an infinitesimal parameter, U ≈ 1 − θaTa. Gribov massive parameter in the gauge field propagator. Equation (4.12) is recognized as the Gribov copies equation This result also suggests that the fiber bundle structure of in the Landau gauge (and also in linear covariant gauges–see the BS theory is trivial [61]. Refs. [64–67]). Equation (4.12) can be seen as an eigenvalue Let us quickly recall the Gribov procedure in the equation for the FP operator where θ is the zero mode of the quantization of non-Abelian gauge theories [58,59].It operator. In Landau gauge, this operator is Hermitian, and essentially consists in eliminating remaining gauge ambi- thus its eigenvalues are real. For values of Aμ sufficiently guities usually present in non-Abelian gauge theories, small, the eigenvalues of the FP operator will be positive, as known as Gribov copies, which are not eliminated in the 2 28 −∂ only has positive eigenvalues. As Aμ increases, it will FP procedure [62,63]. In Yang-Mills theories, the FP attain a first zero mode (4.12). Such a region in which the FP gauge-fixing procedure results in the well-known func- operator has its first vanishing eigenvalue is called Gribov tional generator horizon (wee also Ref. [59]). Gribov’s proposal was to Z restrict the path integral domain to the region Ω (the Gribov − N D −∂ ab δ ∂ SYM ZYM ¼ Aj det½ μDμ j ð μAμÞe region) defined by Z a − S S Ω Aμ; ∂μAμ 0; −∂D>0 : 4:13 ¼ N DADc¯Dce ð YMþ gfÞ; ð4:8Þ ¼f ¼ g ð Þ Such restriction ensures the elimination of all infinitesimal whereby Sgf is the well-known gauge-fixing action given by copies and also guarantees that no independent field is Z eliminated [68]. 1 4 a ab b a 2 The implementation of the restriction to the Gribov Sgf ¼ d x c¯ ∂μDμ c − ð∂μAμÞ : ð4:9Þ 2α region Ω is accomplished by the introduction of a step function Θð−∂DÞ in the Feynman path integral, which α → 0 In (4.9), the limit must be taken in order to reach the leads to the well-known no-pole condition for the FP ghost Landau gauge, propagator hð∂DÞ−1i, which explodes at when a zero mode is attained. The main result of introducing the restriction of ∂ 0 μAμ ¼ : ð4:10Þ the Feynman path integral domain to the Gribov region is a 27 modified gluon propagator, due to the emergence of a Consider a gauge orbit, massive parameter for the gauge field, the so-called Gribov parameter γ. In the presence of the Gribov horizon, the U † i † Aμ ¼ UAμU − ð∂μUÞU ; ð4:11Þ gluon propagator takes the form g 2 a a k −igT θ ðxÞ ∈ θa a b δabδ with U ¼ e jU SUðNÞ, with ðxÞ being the local hAμðkÞAν ðkÞi ¼ ðp þ kÞ 4 γ4 PμνðkÞ; ð4:14Þ gauge parameters of the non-Abelian symmetry and Ta k þ being the generators of the gauge group. The FP hypothesis 2 where PμνðkÞ¼δμν − kμkν=k , and γ is fixed by the gap [62,63] is that there is only one gauge configuration in the orbit (4.11) obeying the Landau gauge condition (4.10).In equation [60,69], his seminal work [58], V. N. Gribov demonstrated that this ∂Γ hypothesis fails at the YM low-energy regime because one 0 ∂γ2 ¼ : ð4:15Þ

26The result was proved to be valid to all orders in perturbation theory by making use of the Zwanziger’s approach [60] to the Gribov problem [58]. 27The gauge orbit is the equivalence class of gauge field 28In Abelian theories, such as QED, −∂2 is the “FP operator,” configurations that only differ by a gauge transformation, and the copy equation only possesses trivial solutions in the representing thus the same physics according to the gauge the thermodynamic limit, meaning that the Gribov copies are gauge principle. inoffensive in this case.

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According to Zwanziger’s generalization [60], the Exploring the positive-definite-ness of the FP ghost gap equation above is valid to all orders in perturbation propagator, we are able to safely perform the following theory—see Refs. [70,71], in which the all-order proof of the shifts: equivalence between Gribov and Zwanziger methods was worked out. An important feature of the Gribov parameter is η¯a ↦ η¯a þ c¯a; b b cde bc −1 d e that it only affects the infrared dynamics. The matching φ ↦ φ − gf ð∂νDν Þ ∂μðc ψ μÞ; between Gribov-Zwanziger theory and recent lattice data is 1 achieved through the introduction of two-dimensional con- ¯a ↦ ¯a − cdeχ¯d e ∂ ca −1 c c gf μνðFÞμνð νDν Þ : ð4:17Þ densates; see Ref. [72]. The introduction of the Gribov 2 horizon in the action explicitly breaks the BRST symmetry. It is worth noting that these shifts generate a trivial This is usually an unwanted result, as the BRST symmetry is 1 1 Jacobian. Calling 2 ρ1 ¼ α and 2 ρ2 ¼ β, and implementing necessary to prove the unitarity, to ensure the renormaliz- the BS gauge constraints (2.93) and (2.95), together with ability to all orders, and to define the physical gauge- ∂μψ μ ¼ 0, the final gauge-fixing action, integrating out the – invariant observables of the theory [73 75]. This breaking, auxiliary fields b and B in the action (2.98),is however, brought to light the physical meaning of the Z infrared γ parameter and its intrinsic nonperturbative char- 1 1 S α; β d4x − ∂A 2 − F2 acter. One can prove that the BRST breaking is proportional gfð Þ¼ 2α ð Þ 4β to γ2, in other words, the BRST symmetry is restored in the Z 4 a a a a ab b perturbative regime. One says that the BRST symmetry is − d x ðη¯ − c¯ Þ∂μψ μ þ c¯ ∂μDμ c only broken in a soft way, cf. Refs. [75–78]. Only more 1 recently, a universal, gauge-independent, (nonperturbative) − abcχ¯a b c ˜ c 2 gf μνc ðFμν FμνÞ BRST-invariant way to introduce the Gribov horizon was developed [66,67,79–81]. 1 − χ¯a δ δ ϵ abψ b In the self-dual topological BS theory, it was proved in μν μα νβ 2 μναβ Dα β Ref. [28] that all topological gauge copies associated to the a ab b abc a b c gauge ambiguities (2.68) and (2.69) are eliminated through þ φ¯ ∂μDμ φ þ gf φ¯ ∂μðc ψ μÞ ; ð4:18Þ the introduction of the usual Gribov restriction, in which the path integral domain is restricted to the region Ω—see ˜ ≡ δ δ −δ δ ϵ ab Eq. (4.13). Moreover, because of the triviality of the gap where F ¼ F F and D ð μα νβ να μβ μναβÞDα . equation, it was verified that the Gribov copies do not affect The self-dual Landau gauges is recovered by setting α, the infrared dynamics of the self-dual BS theory because β → 0. Then, applying the shifts (4.17) on the action γ 0 29 S ðα; βÞ, one gets BS ¼ is the only possible solution of the gap equation. gf Z Thus, no mass parameter seems to emerge in the BS theory, 1 1 preserving its conformal character at quantum level. Sshifted α; β d4x − ∂A 2 − F2 gf ð Þ¼ 2α ð Þ 4β Specifically, the tree-level exactness in SDL gauges is Z preserved. Such a behavior can be inferred from the − 4 η¯a∂ ψ a ¯a∂ ab b absence of radiative corrections, which ensures the semi- d x μ μ þ c μDμ c positivity of all two-point functions. The FP ghost propa- 1 gator, for instance, reads − χ¯a δ δ ϵ abψ b μν μα νβ 2 μναβ Dα β 1 a ab b hc¯aðkÞcbðkÞi ¼ δab ; ð4:16Þ þ φ¯ ∂μDμ φ : ð4:19Þ k2 α β → which is valid to all orders, demonstrating that the FP As the Jacobian of the shifts that performs Sgfð ; Þ shifted α β operator will remain positive definite at quantum level, Sgf ð ; Þ is trivial, the quantization of both actions are consistent with the inverse of the FP propagator being perturbatively equivalent, cf. Ref. [34]. Such a Jacobian positive, thus proving that we are inside the Gribov region. only generates a number that can be absorbed by the Moreover, the gauge two-point function remains trivial, normalization factor. This shows that the discrete sym- a b i.e., hAμðkÞAν ðkÞi ¼ 0 to all orders. metries (3.22) and (3.23) (present in the Witten theory) can be recovered, which naturally impose the relations 29A similar situation occurs in the N ¼ 4 SYM, which possesses a vanishing β function, indicating the conformal Zc ¼ Zc¯ and Zφ ¼ Zφ¯ ð4:20Þ structure of the self-dual BS. The absence of an invariant scale makes it impossible to attach a dynamical meaning to the Gribov to be valid in the BS theory. Hence, combining (4.20) with parameter [82]. the Z-factor system (3.19), one obtains

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Zg ¼ 1; ð4:21Þ property being a consequence of the vector supersymmetry which forbids radiative corrections. In DW theory, such a which proves that the algebraic analysis is in harmony with property is obtained by taking g2 → 0 as small as we want the result obtained via the study of the Feynman diagrams (as long as g2 ≠ 0), as a consequence of the property that in the presence of the vector supersymmetry, i.e., that the shows that the observables of DW theory are insensitive topological BS theory (following the self-dual Landau under changes of g. That is how Witten computed its gauges) is conformal, in accordance with the absence of partition function that naturally reproduces the Donaldson radiative corrections. invariants for four manifolds, i.e., by eliminating the The algebraic renormalization and the Feynman diagram influence of the vertices at g2 → 0, and taking only the analysis consist of two independent methods. In the loop quadratic part of the action. The BS theory is also invariant expansion, used to construct the diagrams in Sec. III A,we under changes of g, as it only appears in the trivial part of expand around the trivial A ¼ 0 sector, i.e., around an the BRST cohomology, but the tree-level exactness is a instanton with winding number zero. One may wonder if it general property of the BS theory in self-dual Landau is physically relevant, thinking about the importance of gauges; i.e., it is valid for an arbitrary perturbative regime. instanton configurations in the topological theory, in order The characterization of the correspondence between the to construct the Donaldson invariants. Exactly the topo- twisted N ¼ 2 SYM and a conformal field theory is now logical nature of the off-shell BS theory saves the day here. complete. The fact that the twisted N ¼ 2 SYM in the weak Let us first remark that it is possible to write down a BRST- coupling limit and the BS theory share the same global γ invariant version of the Gribov restriction, that is, if were observables is a well-known result in literature [22,23,83]. to be nonzero, while preserving equivalence with the above 30 In the DW theory, the Donaldson invariants are defined by formalism ; see Refs. [66,79,81] for details. As already the δ supersymmetry (2.31) according to the bidescent reminded before, the topological partition function does not equations encoded in (2.64). In the BS one, the same depend on the coupling g. This means all observables can bidescent equations appears, constructed from the nth be computed in the g → 0 limit. Expanding around a Chern class W˜ defined in terms of the universal curvature nontrivial instanton background rather than around n 2 in the extended space M × A=G. Such an equivalence is 0 −1=g A ¼ would lead to corrections of the type e into ensured by the equivariant cohomology that allows for the the effective action, but the latter vanishes exponentially replacement s → δ,asW˜ is invariant under ordinary fast once g → 0 is considered, which only represents a n Yang-Mills transformations. We are now defining in which liberty of the theory; i.e., it would not affect the global energy regimes such an equivalence occurs when we observables (see [28]). As such, we can a priori work employ the self-dual Landau in the BS and are formally around A ¼ 0, without losing the generality of the result, proving the correspondence between the twisted N ¼ 2 and which will be unaltered for a generic instanton background. a conformal gauge theory. The fact that the observables are the same, as a consequence of the equivariant cohomology, V. CHARACTERIZATION OF THE DW/BS does not characterize the correspondence at quantum level CORRESPONDENCE (we will provide a counterexample in the next section). The We will characterize in this section the quantum corre- correspondence between the DW and BS observables, spondence between the twisted N ¼ 2 SYM in the ultra- given by the equivalence violet regime and the conformal Baulieu-Singer theory in ODW φ ODW φ ODW φ 2 the SDL gauges. h α1 ð iÞ α2 ð iÞ αp ð iÞig →0 BS 0 BS 0 BS 0 ¼hOα ðφ ÞOα ðφ ÞOα ðφ Þi ; ð5:1Þ A. Quantum equivalence between DW 1 i 2 i p i SDL and self-dual BS theories is independent of the gauge choice. The field content that The result obtained in (4.21) in the SDL gauges proves defines the observables is the same in both theories, that the self-dual BS β function vanishes. This result is φ ≡ φ0, since the observables are independent of the FP 2 i i completely different from the twisted N ¼ SYM which ghosts (which appear in the gauge-fixed BS action). In a receives one-loop corrections, and possesses a nonvanish- DW BS 0 few words, Oα ðφ Þ ≡ Oα ðφ Þ, represented by the prod- β i i ing function given by (4.3). The correspondence between uct in Eq. (2.64). As demonstrated in Sec. II B 5, the BS 2 β the BS and N ¼ -= functions occurs when we take the observables naturally do not depend on ðc; c¯Þ, due to the 2 → 0 2 weak coupling limit (g ) on the N ¼ side. In this invariance of W˜ under s (the Yang-Mills BRST βN¼2 → 0 n YM limit, g . On the BS side, however, the vanishing of operator). The BS reproduces the Donaldson polynomials β the function is valid for an arbitrary coupling constant, only as a consequence of the structure of the off-shell and not only for a weak coupling, with the conformal BRST transformations (2.71). Witten works exclusively in the moduli space A=G, i.e., without fixing the gauge, its 30In the sense that all correlation functions will be identical. observables being naturally independent of the FP ghost.

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TABLE III. Characterization of the DW/BS correspondence.

Twisted N ¼ 2 SYM BS in self-dual Landau gauges On-shell δ supersymmetry Off-shell BRST þ vector supersymmetry Donaldson invariants (δ) Donaldson invariants (s → δ) g2 → 0 Arbitrary g Any Riemannian manifold, gμν Euclidean spaces, δμν A G γ4 0 = Gauge fixedj Gribov ¼ N¼2 BS βg → 0 βg ðgÞ¼0

Another crucial point is that the gauge-fixing term in the Donaldson invariants for 4-manifolds. Also, the Gribov which the FP ghosts are introduced in the self-dual BS ambiguities are irrelevant to the BS model (at least in the theory does not allow for the influence of Gribov copies. self-dual Landau gauges), a property that should remain For this reason, working in the moduli space, A=G in the valid at the strong regime. DW theory are completely correspondent to work in the BS theory in SDL gauges for an arbitrary g, since γ4 ¼ 0. B. Considerations about the gauge dependence Fixing the remaining YM gauge symmetry of Witten’s and possible generalizations A G action (2.30), instead of working in = , would not break Because of the exact nature of the topological Donaldson such a correspondence since the Gribov copies could invariants, which are given by exact numbers, we can only affect the nonperturbative regime, being inoffensive consider the supposition that quantum corrections should 2 → 0 at the ultraviolet limit g . The quantum equivalence is not affect the tree-level results and that the description β illustrated by the agreement between the functions, of the Donaldson invariants by the gauge-fixed BS βN¼2 2 → 0 βBS 0 g ðg Þ¼ g ðgÞ¼ . approach should not depend on the gauge choice. Finally, because of the property of Witten’s theory Although intuitive, this argument is not sufficient or (2.35), which ensures that Witten’s theory can be extended complete. As a counterexample, we invoke the β function to any Riemannian manifold, the DW/BS correspondence obtained by Birmingham et al. in Ref. [54], in which the is characterized as follows: the twisted N ¼ 2 SYM at Batalin-Vilkovisky (BV) algorithm [55] was employed. 2 g → 0, in any Riemannian manifold (that can be contin- Such a model possesses cohomological properties similar 4 31 uously deformed into each other, including R ), defined to those in the BS theory. For a particular configuration of in the instanton moduli space A=G, is correspondent to the auxiliary fields used in Ref. [54], the BV method recovers topological BS theory in the self-dual Landau gauges in the BS gauges with ρ1 ¼ ρ2 ¼ 0, together with the con- Euclidean spaces, in an arbitrary perturbative regime. Such straint Dμψ μ ¼ 0—see Eq. (2.93). This constraint on the a BS theory consists of a conformal field theory, where the topological ghost, with the covariant derivative instead of gauge copies are inoffensive in the infrared, since the the ordinary one, breaks the vector supersymmetry, massive infrared parameter originated from the gauge allowing for quantum corrections. Consequently, the β copies vanishes in this gauge—see Table III. function computed by Birmingham et al. is not zero. As We emphasize that we use perturbative techniques to ˜ 2 noted by the authors of Ref. [54],itisTrðFμν FμνÞ prove the conformal property of the self-dual BS theory. 2 rather than TrFμν which is renormalized, meaning that The fact that the self-dual BS theory in the strong limit the vacuum configurations are preserved. As expected, the g2 → ∞ is also correspondent to Witten’s TQFT defined at 2 → 0 structure of the instanton moduli space, that defines the g can be conjectured by means of the cohomological Donaldson invariants, is not altered. structure of the off-shell BRST symmetry. Changing g in About the gauge dependence of the β function in off- the BS theory is equivalent to adding a BRST-exact term in shell topological gauge models, see Sec. IV B 1. The the action; i.e., it is equivalent to performing a change in the coupling constant in this model is nonphysical, belonging gauge choice. Moreover, the global observables of BS to the trivial part of cohomology. Any change in the W˜ theory, described by the Chern classes n, do not depend unobservable coupling constant only leads to a BRST- on the gauge choice, having only the power of reproducing exact variation. The only observables are the global ones, and we must expect that, for different gauge choices, the 31This is the only requirement that guarantees that the observ- global observables are not affected. According to the result ables of both sides are correspondent, as the conformal BS is of Birmingham et al. in Ref. [54], it is possible to obtain defined in Euclidean spaces. In the DW theory, spaces that can be nontrivial quantum corrections without destroying the continuously deformed, one into the other, represent the same Donaldson invariants for a class of manifolds, since a continuous topological properties of the underlying theory, preserving variation of the metric is equivalent to adding a δ-exact term to the the observables. Analogously, we can consider the pos- action, which does not alter the observables. sibility in which the fields could also be consistently

085008-22 CORRESPONDENCE BETWEEN THE TWISTED N ¼ 2 SUPER- … PHYS. REV. D 103, 085008 (2021) renormalized, i.e., in such a way that the bidescent are also given by the Donaldson invariants, due to the equations, that define the Donaldson invariants, are not equivariant cohomology—defined by Witten’s fermionic altered. This reasoning shows that the renormalization of symmetry—which also characterizes the BS observables topological gauge theories, consistent with the global that can be written as Chern classes for the curvature in the observables, is not a trivial issue. extended space M × A=G, where M is a Riemannian The vector supersymmetry, that leads to the conformal manifold and A=G is the instanton moduli space; see BS theory, is a particular symmetry of the self-dual Landau Sec. II B 5. Despite sharing the same observables, we note gauge. One must note that ∂μAμ ¼ DμAμ, due to antisym- that these theories do not necessarily have the same abc metric property of f . To impose ∂μψ μ ¼ 0 or Dμψ μ ¼ 0 quantum properties, as Witten and BS actions do not differ automatically preserves the instanton moduli space, where by a BRST-exact term, cf. Eq. (2.123). In a few words, the Aμ and ψ μ obey the same equations of motion, according to BRST quantization schemes of Witten’s and BS theories the Atiyah-Singer theorem that correctly defines its dimen- are not equivalent. sion. The preservation of the instanton moduli space is then In harmony with the quantum properties of BS approach the only requirement of the topological theory, being the in the self-dual Landau gauges, see Sec. III, we formally conformal property a particular feature of the self-dual prove that the topological self-dual BS theory is conformal. Landau gauges. The dimension of the instanton moduli According to the Feynman diagram analysis performed in space should not depend on the gauge choice, being Ref. [26], we proved the absence of quantum corrections in protected by the Atiyah-Singer theorem. the BS theory in the presence of the vector supersymmetry. The second generalization that can be worked out is in the In Sec. IV B 1, we discussed the nonphysical character of direction of developing a model in which the BS theory can the coupling constant in the off-shell BS approach. Then, to be constructed for a generic gμν. Again, any change on the construct an algebraic proof that the self-dual BS is Euclidean metric to a generic one corresponds to the conformal, we first solved the renormalization ambiguity addition of a BRST-exact term in the BS theory. This means in topological Yang-Mills theories described in Ref. [25], that such a variation can be interpreted as a change in the using a nonlocal transformation which recovers discrete gauge choice, and the previous discussion can be also symmetries between ghost and antighost fields. Such applied here. The vector supersymmetry is easily defined nonlocal transformations showed to be a freedom of the in flat spaces. To reproduce the conformal properties of the self-dual BS theory due to the triviality of the Gribov copies SDL gauges in Euclidean space for a generic gμν, we will in the SDL gauges [28]; see Sec. IV B 2. As a consequence face the task of finding a class of metrics whose correspond- of this triviality, using the Ward identities of the model— together with the symmetry between the topological and ing action possesses a rich set of Ward identities (WI), capable of reproducing the same effect of the self-dual Faddeed-Popov ghosts introduced in Ref. [24]—and ones, see Appendix, given by the 11 functional opera- employing algebraic renormalization techniques, we con- WBS ≡ S W Wa Wa Wa Wa Ga Ga Ga T G clude that Zg ¼ 1, i.e., that the self-dual BS β function tors I f ; μ; 1; 2; 3; 4; φ; 1; 2; ; 3g. Besides that, we will face another inconvenient task of indeed vanishes. having to study the Gribov copies in curved spacetimes, We observed that these theories do not possess the same β which is a highly nontrivial problem. The vanishing of the quantum structure, by comparing the function of each Gribov parameter in the self-dual BS in Euclidean spaces model; see Sec. IV. From this analysis, we characterized the 2 ensures that the DW/BS correspondence is valid for a correspondence between the twisted N ¼ SYM and BS generic coupling constant on the BS side. theories at quantum level, defining in which regimes such a correspondence occurs; see Sec. V. In spite of having distinct BRST constructions, we conclude that working in VI. CONCLUSIONS the instanton moduli space A=G on the DW side is We perform a comparative study between Donaldson- completely equivalent to working in the self-dual Witten TQFT [8] and the Baulieu-Singer approach [13]. Landau gauges on the BS one, since the Gribov copies While DW theory is obtained via the twist transformation do not affect its infrared dynamics. In a few words, the of the N ¼ 2 SYM, BS theory is based on the BRST gauge twisted N ¼ 2 SYM in any Riemannian manifold (that can 4 fixing of an action composed of topological invariants—see be continuously deformed into M ¼ R ), in the weak Secs. II A and II B. Besides that, Witten’s theory is defined coupling limit g2 → 0, is correspondent to the BS theory by an on-shell supersymmetry, according to the fermionic in the self-dual Landau gauges in an arbitrary perturbative symmetry, see Eq. (2.31), whose associated charge is only regime, which consists of a conformal gauge theory nilpotent if we use the equations of motion. Such a defined in Euclidean flat space; see Table III. Such a symmetry defines the observables of the theory, given characterization could shed some light on the connection by the Donaldson polynomials. The BS approach, in turn, between supersymmetry, topology, off-shell BRST sym- consists of an off-shell BRST construction, which enjoys metry, and non-Abelian conformal gauge theories in four the topological BRST symmetry (2.71), whose observables dimensions.

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ACKNOWLEDGMENTS APPENDIX: BS WARD IDENTITIES IN THE SELF-DUAL LANDAU GAUGES We would like to thank A. D. Pereira, G. Sadovski, and A. A. Tomaz for enlightening discussions, which were The BS action in the self-dual Landau gauges (3.8) indispensable for the development of this work. This study enjoys the following Ward identities: was financed in part by The Coordenação de (i) The Slavnov-Taylor identity, which expresses the Aperfeiçoamento de Pessoal de Nível Superior—Brasil BRST invariance of the action (3.8), (CAPES), Finance Code No. 001, and the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq-Brazil), Finance Code No. 159928/2019-2. SðΣÞ¼0; ðA1Þ

where

Z δΣ δΣ δΣ δΣ δΣ δΣ δΣ δΣ S Σ 4 ψ a − φa ð Þ¼ d z μ a a þ a a þ þ a a þ a a δΩμ δAμ δτμ δψ μ δL δc δE δφ δΣ δΣ δΣ δΣ δΣ δΣ a η¯a a Ωa a a þ b a þ a þ Bμν a þ μ a þ L a þ Kμν a : ðA2Þ δc¯ δφ¯ δχ¯μν δτμ δE δΛμν

(ii) Ordinary Landau gauge fixing and the Faddeev-Popov antighost equation:

δΣ a a W Σ ¼ ¼ ∂μAμ; 1 δba δΣ δΣ WaΣ − ∂ −∂ ψ a 2 ¼ a μ a ¼ μ μ: ðA3Þ δc¯ δΩμ

(iii) Topological Landau gauge fixing and the bosonic antighost equation:

δΣ WaΣ ∂ ψ a; 3 ¼ δη¯a ¼ μ μ δΣ δΣ WaΣ − ∂ 0 4 ¼ a μ a ¼ : ðA4Þ δφ¯ δτμ

(iv) The bosonic ghost equation,

a a GφΣ ¼ Δφ; ðA5Þ

where

Z δ δ a 4 abc b Gφ ¼ d z − gf φ¯ ; δφa δbc Z a abc 4 b c b c b c Δφ ¼ gf d zðτμAμ þ E c þ Λμνχ¯μνÞ: ðA6Þ

(v) The ordinary Faddeev-Popov ghost equation,

a a G1Σ ¼ Δ ; ðA7Þ

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where Z δ δ δ δ δ Ga 4 abc ¯b φ¯ b χ¯b Λb 1 ¼ d z a þ gf c c þ c þ μν c þ μν c ; δc δb δη¯ δBμν δKμν Z a abc 4 b c b c b c b c b c b c Δ ¼ gf d zðE φ − ΩμAμ − τμψ μ − L c þ ΛμνBμν − Kμνχ¯ μνÞ: ðA8Þ

(vi) The second Faddeev-Popov ghost equation,

a a G2Σ ¼ Δ ; ðA9Þ

where Z δ δ δ δ δ δ Ga 4 − abc φ¯ b b b − η¯b b 2 ¼ d z a gf c þ Aμ c þ c c c þ E c : ðA10Þ δc δc¯ δψ μ δφ δb δL

(vii) Vector supersymmetry,

WμΣ ¼ 0; ðA11Þ

where Z δ δ δ δ δ W 4 ∂ a ∂ a ∂ χ¯a ∂ φ¯ a μ ¼ d z μAν a þ μc a þ μ να a þ μ a þ a δψ ν δφ δBνα δη¯ δc¯ δ δ δ δ ∂ ¯a − ∂ η¯a ∂ τa ∂ a ∂ Λa þð μc μ Þ a þ μ ν a þ μE a þ μ να a : ðA12Þ δb δΩν δL δKνα

(viii) Bosonic nonlinear symmetry,

T ðΣÞ¼0; ðA13Þ

where Z δΣ δΣ δΣ δΣ δΣ δΣ δΣ δΣ T Σ 4 − − ¯a − η¯a ð Þ¼ d z a a a a a a þðc Þ a þ a : δΩμ δψμ δL δφ δKμν δBμν δc¯ δη¯

(ix) Global ghost supersymmetry,

G3Σ ¼ 0; ðA14Þ

where Z δ δ δ δ δ δ G 4 φ¯ a − a τa 2 a Λa 3 ¼ d z a þ a c a þ μ a þ E a þ μν a : ðA15Þ δη¯ δc¯ δφ δΩμ δL δKμν

The last two symmetries are the new ones introduced in Ref. [24]. The nonlinear bosonic symmetry (vii) is precisely the one discussed in Sec. III. B, see Eq. (3.15), which relates the FP and topological ghosts. We remark that the Faddeev-Popov ghost equations (A7) and (A9) can be combined to obtain an exact global supersymmetry,

ΔGaΣ ¼ 0; ðA16Þ where

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ΔGa ¼ Ga − Ga Z1 2 δ δ δ δ δ δ δ δ δ 4 abc ¯b − η¯b φ¯ b b χ¯b b Λb τb b ¼ d zf ðc Þ c þ c þ c þ Aμ c þ μν c þ c c þ μν c þ μ c þ E c : ðA17Þ δb δη¯ δc¯ δψ μ δBμν δφ δKμν δΩμ δL

We observe the similarity of Eq. (A16) with the vector supersymmetry (A11). It is also worth mentioning that, even though the ghost number of the operator (A17) is −1, resembling an anti-BRST symmetry, it is not a genuine anti-BRST symmetry—see, for instance, Ref. [84] for the explicit anti-BRST symmetry in topological gauge theories.

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