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BRST-BV Quantization of Gauge Theories with Global Symmetries

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BRST-BV Quantization of Gauge Theories with Global Symmetries Eur. Phys. J. C (2018) 78:524 https://doi.org/10.1140/epjc/s10052-018-6003-x Regular Article - Theoretical Physics BRST-BV quantization of gauge theories with global symmetries I. L. Buchbinder1,2,3,a,P.M.Lavrov1,2,b 1 Center of Theoretical Physics, Tomsk State Pedagogical University, Kievskaya St. 60, 634061 Tomsk, Russia 2 National Research Tomsk State University, Lenin Av. 36, 634050 Tomsk, Russia 3 Departamento de Física, ICE, Universidade Federal de Juiz de Fora, Campus Universitário, Juiz de Fora, MG 36036-900, Brazil Received: 28 February 2018 / Accepted: 17 June 2018 © The Author(s) 2018 Abstract We consider the general gauge theory with case the important physical properties of the theory can be a closed irreducible gauge algebra possessing the non- violated. Example of such a model is N = 4 supersymmetric anomalous global (super)symmetry in the case when the Yang–Mills theory, where the classical rigid superconformal gauge fixing procedure violates the global invariance of clas- symmetry is conserved at quantum level (see e.g. [18]). As sical action. The theory is quantized in the framework of a result, we arrive at a general problem to study the classical BRST-BV approach in the form of functional integral over all global symmetries in BRST quantization of the gauge theory. fields of the configuration space. It is shown that the global A specific aspect of the above general problem arises symmetry transformations are deformed in the process of in supersymmetric gauge theories. These theories possess a quantization and the full quantum action is invariant under gauge symmetry and rigid or global supersymmetries. Such such deformed global transformations in the configuration theories can be formulated either in the component formalism space. The deformed global transformations are calculated or in the terms of superfields. In the first case the supersym- in an explicit form in the one-loop approximation. metry is non manifest and when we quantize the correspond- ing gauge theory imposing the gauge fixing conditions on vector fields we violate the rigid supersymmetry. Therefore 1 Introduction it is unclear if the quantum effective action should be super- symmetric. At first, this problem arose in N = 1 super Yang– BRST quantization procedure, initiated in the works [1–8], Mills theories and led to the study of the aspects of global which can be realized within the Hamiltonian BFV approach symmetry in quantum gauge theories [19–22] However, this [9,10] or within the Lagrangian BV approach [11,12], is a problem was automatically resolved after formulation of the powerful and universal tool to formulate the quantum gauge N = 1 supergauge theories in terms of N = 1 superfields theories and investigate their structures. This procedure is (see e.g. [23]), where the manifestly N = 1 gauges have applicable to an extremely wide class of gauge theories been used. However, the problem is still remains in extended including the (super) Yang–Mills theories, (super)gravity, super Yang–Mills theories. At present, the best formulation (super)strings and more specific gauge theories like the gauge of 4D, N = 4 rigid super Yang–Mills theory is achieved in antisymmetric field models. All these theories possess many terms of 4D, N = 2 or in terms of 4D, N = 3 harmonic common properties therefore one can talk about a general superfields [24]. In this case only part of supersymmetries are gauge theory and study its quantum aspects, including the manifest, the other remain hidden. The same situation will renormalization issues (see e.g. [13–15]), in general terms. be in 6D, N = (1, 1) rigid super Yang–Mills theory, which Besides the local symmetries, many gauge theories are is similar in many aspects to 4D, N = 4 super Yang–Mills characterized by the rigid symmetries. For example, the role theory. At present, the best formulation of such a theory is which is played by the global chiral symmetry in the Standard achieved in terms of 6D, N = (1, 0) superfields (see e.g. Model (see e.g. [16]) or the role which is played by global [29] and references therein). Again, one part of supersym- conformal group in the String Theory (see e.g. [17]) are well metries is manifest and the other part is hidden. In all such known. In many cases it is essential to preserve the classical theories the gauges preserve the manifest symmetries and global symmetry in quantum theory since in the opposite violate the hidden supersymmetries (see e.g. [25–28,30,31]). Another aspect of the same problem arises at quantization a e-mail: [email protected] of 4D, N = 2 superconformal theories where the gauges b e-mail: [email protected] 123 524 Page 2 of 8 Eur. Phys. J. C (2018) 78:524 preserve the manifest N = 2 supersymmetry but violate the 2 General gauge theory: notations and conventions global superconformal invariance. This aspect was studied in the series of the papers [32–34] where an approach to general Consider the general gauge theory with closed irreducible problem of global symmetries in quantum gauge theories was gauge algebra. It means that the initial action, S0 = S0(ϕ), i i proposed and it was shown that under some assumptions of the fields ϕ ={ϕ }, i = 1, 2,...,n,ε(ϕ) = εi is concerning the structure of the theory, the quantization leads invariant under the gauge transformations to deformation of global symmetries and, in particular, to i i i α S , (ϕ)R (ϕ) = 0,δϕ= R (ϕ)ξ , (2.1) deformation of superconformal transformations. 0 i α α In this paper we study the above problem in general terms where ξ α are the arbitrary functions of space-time coordi- 1 α i and generalize all the previous results. We consider the gen- nates with Grassmann parities ε(ξ ) ≡ εα, and Rα = i i eral gauge theory possessing the non-anomalous global sym- Rα(ϕ), ε(Rα) = εi + εα are generators of gauge transforma- metry and quantize this theory in the framework of the BRST- tions which are assumed to be linear independent in gauge BV procedure. It is assumed that gauges violate the initial indices α. It is convenient to introduce the operators of gauge ˆ ˆ global symmetry of the classical action. Under these condi- transformations, Rα = Rα(ϕ), tions we develop a maximally general approach to the struc- ←− δ ture of the global symmetries at the quantization of gauge ˆ i Rα = Rα, (2.2) theories. We show like in work [32] that the quantization pro- δϕi cedure leads to deformation of classical global transforma- so that the gauge invariance of S0 (2.1) is written in the form tions. However, we use the more general effective action then ˆ in [32] that provides more possibilities for purely algebraic S0 Rα = 0. (2.3) analysis of the global symmetries in quantum gauge theory. ˆ The algebra of the gauge generators Rα has the following We prove that although the vacuum functional is invariant form due to the closure: under classical global transformations, the effective action ˆ ˆ ˆ γ is not invariant and its invariance requires the quantum cor- [Rα, Rβ ]=−Rγ F αβ (2.4) rections to generators of global symmetry transformations. The maximally general form of such deformations is derived. or In particular, we prove that the full quantum action in the i j − (− )εαεβ i j =− i γ , Rα, j Rβ 1 Rβ,j Rα Rγ F αβ (2.5) functional integral is invariant under the deformed transfor- γ γ mations. The one-loop invariance is studied in detail and the where F αβ = F αβ (ϕ) are the structure coefficients depend- corresponding one-loop deformation of the global transfor- ing in general on fields ϕ and obeying the symmetry proper- mations is calculated in explicit form. The results obtained ties are exclusively general and take place for any bosonic or γ ε ε γ F αβ =−(−1) α β F βα. (2.6) fermionic gauge theory with an irreducible closed gauge algebra and with an arbitrary (even open) non-anomalous For Yang–Mills theories they are constants. The Jacobi iden- algebra of global symmetry. tities written in terms of the gauge generators and the struc- The paper is organized as follows. Section 2 is devoted ture coefficients read to description of the general gauge theory with closed irre- σ ρ + σ i (− )εαεγ + (αβγ ) = . ducible gauge algebra and fixing the notations and conven- F αρ F βγ F αβ,i Rγ 1 cycle 0 (2.7) tions. In Sect. 3 we describe the properties of the general For irreducible gauge theories the extended configuration gauge theory with global symmetry. In Sect. 4 we construct space is described by the fields : the quantum action of the theory under consideration and A i α α ¯ α prove that the quantization leads to deformation of the clas- φ = (ϕ , B , C , C ), i α α ¯ α sical global symmetries. In Sect. 5 we consider the above ε(ϕ ) = εi ,ε(B ) = εα,ε(C ) = ε(C ) = εα + 1, α α α deformation in one-loop approximation and construct the gh(ϕi ) = gh(B ) = 0, gh(C ) = 1, ghC¯ ) =−1, deformation in an explicit form. Section 6 summarizes the (2.8) results. In the paper the DeWitt’s condensed notations are used where Bα are Nakanishi-Lautrup auxiliary fields, Cα and C¯ α [35]. We employ the notation ε(A) for the Grassmann parity are the ghost and anti-ghost fields. For gauge theories under of any quantity A. All derivatives with respect to sources consideration which belong to the rank 1 gauge theories in are taken from the left only.
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