View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Physics Letters B 705 (2011) 120–123 Contents lists available at SciVerse ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Perturbative quantum gravity on complex spacetime Mir Faizal Department of Mathematics, Durham University, Durham, DH1 3LE, United Kingdom article i nfo a bstract Article history: In this Letter we will study non-anticommutative perturbative quantum gravity on spacetime with a Received 19 July 2011 complex metric. After analysing the BRST symmetry of this non-anticommutative perturbative quantum Received in revised form 13 September gravity, we will also analyse the effect of shifting all the fields. We will construct a Lagrangian density 2011 which is invariant under the original BRST transformations and the shift transformations in the Batalin– Accepted 15 September 2011 Vilkovisky (BV) formalism. Finally, we will show that the sum of the gauge-fixing term and the ghost Available online 1 October 2011 Editor: M. Cveticˇ term for this shift symmetry invariant Lagrangian density can be elegantly written down in superspace with a single Grassmann parameter. Keywords: © 2011 Elsevier B.V. Open access under CC BY license. BV formalism Non-anticommutative quantum gravity 1. Introduction invariance under the original BRST and shift transformations in the BV formalism. Consequently, we will express our results in super- Noncommutative field theory arises in string theory due to space formalism. presence of the NS antisymmetric tensor background [1–3].How- The BRST symmetry for Yang–Mills theories [22–25] and spon- ever, other backgrounds like the RR background can generate non- taneously broken gauge theories [26] has already been analysed in anticommutative field theory [4,5]. Quantum gravity on noncom- noncommutative spacetime. The invariance of a theory, under the mutative spacetime has been thoroughly studied [6–9].Infact,it original BRST transformations and shift transformations, that oc- is hoped that noncommutativity predicts the existence of the cos- curs naturally in background field method can be analysed in the mological constant which is of the same order as the square of the BV formalism [27–30]. Both the BRST formalism [31,32] and the Hubble’s constant [10]. Perturbative quantum gravity on noncom- BV formalism can be given geometric meaning by the use of su- mutative spacetime has been already analysed [11].Itwasfound perspace [33–36]. that the graviton propagator was the same as that in the commu- BV formalism has been used for quantising W3 gravity [37–39]. tative case. However, the noncommutative nature of spacetime was It has also been used in quantising metric-affine gravity in two experienced at the level of interactions. dimensions [40]. The BRST symmetry for perturbative quantum The idea of noncommutativity of spacetime has been gener- gravity in four-dimensional flat spacetime have been studied by alised to non-anticommutativity. In addition to this, quantum field a number of authors [41–43] and their work has been sum- theory has been studied on non-anticommutative spacetime [12]. marised by N. Nakanishi and I. Ojima [44]. The BRST symmetry Non-anticommutativity of spacetime occurs if the metric is com- in two-dimensional curved spacetime has been studied thoroughly plex. Spacetime with complex metric has been studied as an in- [45–47]. Similarly, the BRST symmetry for topological quantum teresting example of nonsymmetric gravity [13–15].Even though gravity in curved spacetime [48,49] and the BRST symmetry for nonsymmetric quantum gravity was initially studied in an attempt perturbative quantum gravity in both linear and nonlinear gauge’s to unify electromagnetism and gravity [16,17],itisnowmainly has also been analysed [50]. However, so far no work has been studied due to its relevance to string theory [18–20]. Quantum done in analysing the non-anticommutative perturbative quantum gravity on this non-anticommutative complex spacetime has also gravity in superspace BV formalism. This is what we aim to do in been discussed before [21]. this Letter. In this Letter we will discuss the perturbative quantum grav- ity on this non-anticommutative complex spacetime. We will first 2. Perturbative quantum gravity analyse the BRST symmetry of this theory and then study its the We shall analyse perturbative quantum gravity with the follow- E-mail address: [email protected]. ing hyperbolic complex metric [21], 0370-2693 © 2011 Elsevier B.V. Open access under CC BY license. doi:10.1016/j.physletb.2011.09.062 M. Faizal / Physics Letters B 705 (2011) 120–123 121 ( f ) = ( f ) + ( f ) L = gbc bbc ωabc , (1) g sΨ, (11) where ω is the pure imaginary element of a hyperbolic complex where Clifford algebra with ω2 =+1. Here ω forms a ring of numbers 1 and not a field which the usual system of complex numbers do. ¯a b Ψ = c 3 ∂ hab − k∂ah + ba , (12) The advantage of using ω is that the negative energy states coming 2 from the purely imaginary part of the metric will be avoided. with k = 1. Now the sum of the ghost term, the gauge-fixing term Now we can introduce non-anticommutativity as follows, and the original classical Lagrangian density is invariant under the following BRST transformations xˆa, xˆb = 2xˆaxˆb + iωτ ab, (2) = e 3 a − 3 b a where τ ab is a symmetric tensor. We use Weyl ordering and ex- shab Dab ce, sc cb ∂ c , press the Fourier transformation of this metric as, sc¯a =−ba, sba = 0. (13) ( f ) − ˆ ( f ) gˆ (xˆ) = d4k πe ikx g (k). (3) BV formalism is used to analyse the extended BRST symmetry. ab ab This extended BRST symmetry for perturbative quantum gravity can be obtained by first shifting all the original fields as, Nowwehaveaonetoonemapbetweenafunctionofxˆ to a func- tion of ordinary coordinates y via → − ˜ a → a − ˜a hab hab hab, c c c , ( f ) − ( f ) a a a a a ˜a = 4 ikx c¯ → c¯ − c¯˜ , b → b − b , (14) gab (x) d k πe gab (k). (4) and then requiring the resultant theory to be invariant under both So, the product of ordinary functions is given by the original BRST transformations and these shift transformations. ( f )ab 3 ( f ) This can be achieved by letting the original fields transform as, g (x) gab (x) ω ( f ) a a = ab 2 1 ( f )ab shab = ψab, sc = φ , exp τ ∂a ∂ g (x1)g (x2) = = . (5) 2 b ab x1 x2 x sc¯a = φ¯a, sba = ρa, (15) ( f )a Now R given as, bcd and the shifted fields transform as ( f )a R =−∂ Γ a + ∂ Γ a + Γ a 3 Γ e − Γ a 3 Γ e , (6) bcd d bc c bd ec bd ed bc ˜ = − e 3 − ˜ shab ψab Dab (ce ce), and we also get R = Rd . Thus finally R( f ) is given by bc bcd sc˜a = φa + (c − c˜ ) 3 ∂b(c − c˜ ), b b a a ( f ) a a a ˜a a a ( f ) = ( f )ab 3 sc¯˜ = φ¯ + b − b , sb˜ = . (16) R g Rab . (7) ρ a ¯a The Lagrangian density for pure gravity with cosmological constant Here ψab, φ , φ , and ρa are ghosts associated with the shift sym- λ can now be written as, metry and their BRST transformations vanish, √ L = ( f ) 3 ( f ) − a ¯a a c g R 2λ , (8) sψab = sφ = sφ = sρ = 0. (17) where we have adopted units, such that 16π G = 1. In perturba- We define antifields with opposite parity corresponding to all the ( f ) tive gravity on flat spacetime one splits the full metric gab into original fields. These antifields have the following BRST transfor- ηab which is the metric for the background flat spacetime and hab mations, which is a small perturbation around the background spacetime, ∗ ¯ ∗a a sh = bab, sc = B , ( f ) = + ab g ηab hab. (9) ∗a a ∗a ¯a ab sc¯ = B , sb = b . (18) Here both η and h are complex. The covariant derivatives along ab ab ¯ a a ¯a with the lowering and raising of indices are compatible with the Here bab, B , B , and b are Nakanishi–Lautrup fields and their BRST transformations vanish too, metric for the background spacetime. The small perturbation hab is viewed as the field that is to be quantised. ¯ a a ¯a sbab = sB = sB = sb = 0. (19) 3. BV formalism It is useful to define All the degrees of freedom in hab are not physical as the La- = − ˜ a = a − ˜a hab hab hab, c c c , grangian density for h is invariant under the following gauge ab ¯a ¯a ˜¯a a a ˜a transformations, c = c − c , b = b − b . (20) The physical requirement for the sum of the gauge-fixing term and δ h = De 3 Λ Λ ab ab e the ghost term is that all the fields associated with shift symmetry = e + e + ce 3 δb∂a δa ∂b g (∂chab) vanish. This can be achieved by choosing the following Lagrangian density, + gec 3 h ∂ + ηec 3 h ∂ 3 Λ . (10) ac b cb a e ˜ ¯ ˜ab ∗ab e Lg =−b 3 h − h 3 ψ − D 3 c In order to remove these unphysical degrees of freedom, we need ab ab ab e to fix a gauge by adding a gauge-fixing term along with a ghost a ∗ a b a − B 3 c˜a + c¯ 3 φ + c 3 ∂ c term. In the most general covariant gauge the sum of the gauge- a b + a 3 ¯˜ − ∗ 3 ¯a + a + a 3 ˜ + ∗a 3 fixing term and the ghost term can be expressed as, B ca ca φ b B ba b ρa.